From ae51d8cf4a1b814211c2fb0df054171e54d38423 Mon Sep 17 00:00:00 2001 From: Omid Bodaghi <42227752+omibo@users.noreply.github.com> Date: Fri, 27 Feb 2026 17:18:05 -0800 Subject: [PATCH 01/34] Implement Hachi PCS protocol with required primitives (#1) MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit * chore: add toolchain and formatting config Pin Rust 1.88 with minimal profile (cargo, rustc, clippy, rustfmt). Co-authored-by: Cursor * chore(ci): switch to actions-rust-lang/setup-rust-toolchain Respects rust-toolchain.toml automatically. Also normalize clippy flags to use --all --all-targets consistently. Co-authored-by: Cursor * feat(primitives): add u128/i128 serialization support Required by the Fp128 field backend. Co-authored-by: Cursor * feat(algebra): add prime fields, extensions, and modules Introduces the algebra module with: - Fp32/Fp64/Fp128 prime field backends with branchless constant-time add/sub/neg and rejection-sampled random - U256 helper for Fp128 wide multiplication - Fp2/Fp4 tower extensions with Karatsuba-ready structure - VectorModule fixed-length vector module over any field - Poly fixed-size polynomial container Co-authored-by: Cursor * feat(algebra): add NTT small-prime arithmetic and CRT helpers Adds the ntt submodule with: - NttPrime: per-prime Montgomery-like fpmul, Barrett-like fpred, branchless csubq/caddq/center - LimbQ/QData: radix-2^14 limb arithmetic for big-q coefficients - logq=32 parameter preset (six NTT-friendly primes, CRT constants) Co-authored-by: Cursor * test(algebra): add comprehensive algebra test suite 24 tests covering: - Field arithmetic, identities, and distributivity (Fp32/Fp64/Fp128) - Zero inversion returns None - Serialization round-trips (all field types, extensions, VectorModule) - Fp2 conjugate, norm, and distributivity - U256 wide multiply and bit access - LimbQ round-trip and add/sub inverse - QData consistency with preset constants - NTT normalize range and fpmul commutativity - Poly add/sub/neg Co-authored-by: Cursor * docs: add and update progress tracking document Records Phase 0 status: all field types, extensions, NTT scaffolding, constant-time arithmetic, and 24-test suite. Reflects the fields/ntt/module/poly directory layout. Co-authored-by: Cursor * refactor(ntt): Rust-ify NTT/CRT port from C Overhaul the NTT small-prime arithmetic and CRT modules: - Add MontCoeff newtype (#[repr(transparent)] i16 wrapper) to enforce Montgomery-domain vs canonical-domain separation at the type level - NttPrime methods now take/return MontCoeff instead of bare i16: fpmul→mul, fpred→reduce, csubq→csubp, caddq→caddp - Add domain conversion: from_canonical (i16→Mont), to_canonical (Mont→i16) - Delete free functions (pointwise_mul etc), replaced by methods on NttPrime - LimbQ: replace add_limbs/sub_limbs/less_than with std Add/Sub/Ord impls - LimbQ: replace from_u128/to_u128 with From/TryFrom for u128 - LimbQ: add Display impl, branchless csub_mod - Rename all LABRADOR* constants to project-native Q32_* names - Add #[cfg(test)] verification that re-derives pinv/v/mont/montsq from p - Add MontCoeff round-trip and LimbQ ordering tests (28 total) Co-authored-by: Cursor * chore: remove section banners, update progress doc Remove // ---- Section ---- banner comments from prime.rs and crt.rs. Add non-negotiable rules to HACHI_PROGRESS.md: - No section-banner comments - No commit/push without explicit user approval Co-authored-by: Cursor * feat(ring): add CyclotomicRing, CyclotomicNtt, and NTT butterfly Milestone 1 - CyclotomicRing (coefficient form): - Schoolbook negacyclic convolution Mul (X^D = -1) - Add/Sub/Neg/AddAssign/SubAssign/MulAssign, scale, zero/one/x - HachiSerialize/HachiDeserialize Milestone 2 - NTT butterfly + CyclotomicNtt: - Merged negacyclic Cooley-Tukey forward NTT (twist folded into twiddles) - Gentleman-Sande inverse NTT with D^{-1} scaling - Runtime primitive-root finder and twiddle table computation (TODO: migrate to compile-time const tables) - CyclotomicNtt with per-prime pointwise Add/Sub/Neg/Mul - Ring<->Ntt transforms with CRT reconstruction Co-authored-by: Cursor * test(algebra): add ring and NTT tests, wrap in mod tests Add 12 new tests: - CyclotomicRing: negacyclic X^D=-1, mul identity/zero, commutativity, distributivity, associativity, additive inverse, serde, degree-64 - NTT: forward/inverse round-trip (single prime + all primes), NTT mul matches schoolbook cross-check Wrap all integration tests in a single mod tests block and remove section-banner comments. Co-authored-by: Cursor * fix(algebra): harden ring-NTT conversion and field decoding Constrain ring/NTT conversions to explicit field backends and replace fragile CRT reconstruction with deterministic modular lifting. Enforce canonical deserialization checks in validated field decoding paths to reject malformed encodings. Co-authored-by: Cursor * test(algebra): add CRT round-trip and serialization guard coverage Add end-to-end ring->NTT->ring CRT round-trip tests plus reduced-ops stability checks. Expand serialization coverage for Fp4/Poly and verify checked deserialization rejects non-canonical field encodings. Co-authored-by: Cursor * chore(bench): add ring_ntt benchmark target and CT tracking docs Add a dedicated ring/NTT benchmark harness and register it in Cargo metadata. Record current constant-time review status and sync the implementation progress board with new milestones and test coverage. Co-authored-by: Cursor * refactor(field): split core, canonical, and sampling capabilities Break the monolithic Field trait into FieldCore, CanonicalField, and FieldSampling, and update algebra primitives to depend on explicit capabilities for cleaner semantics and future backend integration. Co-authored-by: Cursor * feat(fields): add pow2-offset pseudo-mersenne registry and checks Introduce the curated 2^k-offset prime registry and typed field aliases, then add dedicated Miller-Rabin regression tests to enforce probable primality for all enabled profiles. Co-authored-by: Cursor * refactor(ring): introduce crt-ntt backend/domain layering Rename the ring NTT representation to explicit CRT+NTT semantics and route conversions through backend traits, adding scalar backend and domain aliases for a cleaner representation-vs-execution boundary. Co-authored-by: Cursor * test(algebra): cover backend parity and pow2-offset invariants Expand algebra tests to validate default-vs-backend CRT+NTT equivalence, sampling bounds, and pow2-offset registry consistency under the new field and ring abstractions. Co-authored-by: Cursor * docs(algebra): update progress notes and add prime analysis references Refresh progress and constant-time notes to match the new CRT+NTT naming and field scope, and add the NTT prime analysis document plus local NIST standards artifacts used for parameter rationale. Co-authored-by: Cursor * fix(algebra): harden fp128 reduction and CRT reconstruction arithmetic Make Fp128 reduction and CRT inner accumulation paths more timing-stable with branchless modular operations, and refresh ring/docs/tests status after the hardening cleanup pass. Co-authored-by: Cursor * feat(protocol): add transcript and commitment scaffold Introduce Hachi protocol-layer interfaces and placeholder types with Blake2b/Keccak transcript backends plus phase-aligned labels, while making transcript absorption label-directed at call sites. Co-authored-by: Cursor * test(protocol): add transcript and commitment contract coverage Add deterministic transcript schedule checks (including keccak) and protocol commitment contract tests so transcript ordering and challenge derivation behavior are locked down. Co-authored-by: Cursor * docs(protocol): align transcript spec and progress status Document the protocol scaffold as in-progress, capture the commitment-focused transcript label vocabulary, and clarify deferred Jolt adapter expectations. Co-authored-by: Cursor * feat(protocol): add ring commitment core and seeded matrix derivation Implement the ring-native commitment setup/commit core with config validation, utility modules, and seeded domain-separated public matrix derivation, while wiring prover/verifier stub modules for the next open-check phase. Co-authored-by: Cursor * test(protocol): consolidate ring commitment and stub contract coverage Unify ring commitment core and config validation checks in one test file and add explicit prover/verifier stub contract tests to lock current placeholder behavior before open-check implementation. Co-authored-by: Cursor * docs(progress): update phase 2 status after commitment core landing Record that ring-native §4.1 commitment setup/commit and protocol wiring are in place, and clarify that open-check prove/verify remains the next unfinished protocol milestone. Co-authored-by: Cursor * fix(algebra): harden CT inversion path and CRT final projection Add a constant-time inversion helper for prime fields and replace scalar CRT's final `% q` projection with a division-free fixed-iteration reducer, so secret-bearing arithmetic paths avoid variable-latency behavior. Co-authored-by: Cursor * refactor(algebra): rename inversion helper API without ct suffix Rename the secret-path inversion helper to `Invertible::inv_or_zero` while preserving constant-time semantics via doc contracts, and update CT tracking docs to match the new API names. Co-authored-by: Cursor * test(algebra): clean inversion test naming and normalize formatting Rename the inversion helper test to match the new API naming and keep the ring commitment test formatting consistent after linting. Co-authored-by: Cursor * feat(protocol): add sumcheck core module and tests Introduce core sumcheck building blocks (univariate messages, compression, and transcript-driving prover/verifier driver) and add unit/integration tests. Update progress doc to reflect sumcheck core landing. Co-authored-by: Cursor * Add reference PDF papers * Add local agent instruction files * Add Hachi and SuperNOVA digest docs * Add general field, ring, and multilinear utilities * Add sparse Fiat-Shamir challenge sampling * Implement Polynomail Evaluation as Quadradic Equation * Rename stub to prover and verifier * Refactor code organization * Replace decopose with balanced decompose * Transform polynomial over Fq to ring * Refactor function names * Impl commitment_scheme API * Add SolinasFp128 backend for sparse 128-bit primes Introduce `SolinasFp128` with two-fold Solinas reduction for `p = 2^128 - c` (sparse `c`), plus `U256::sqr_u128`. Export descriptive prime aliases, add BigUint-backed correctness tests, and include a Criterion bench for mul/inv. Co-authored-by: Cursor * Tighten docs and minor clippy cleanups Add missing rustdoc Errors/Panics sections and apply small simplifications suggested by clippy. Co-authored-by: Cursor * Add reduction steps to iteration prover * Optimize Solinas mul/add/sub: fused u64-limb schoolbook + csel canonicalize Rewrite mul_raw as a fused 2×2 schoolbook multiply with two-fold Solinas reduction using explicit u64 limbs and mac helper, bypassing U256. Replace mask-based canonicalize with carry-flag-based pattern that compiles to adds+adcs+csel+csel (4 insns) instead of 10 on AArch64. Add pure-mul, sqr, and throughput microbenchmarks. Made-with: Cursor * Switch SolinasFp128 repr from u128 to [u64; 2] for 8-byte alignment Storage is now [u64; 2] (lo, hi) which halves alignment from 16 to 8 bytes, improving struct packing. Arithmetic hot paths convert to u128 for LLVM-optimal codegen (adds/adcs pairs), so no perf regression. Made-with: Cursor * Fuse overflow correction with canonicalize in fold2_canonicalize When fold-2 overflows, the wrapped value s < C², so s + C < C(C+1) < P — meaning s + C is already canonical. This lets us replace the separate overflow-correction + canonicalize (3 + 4 insns) with a single fused `if (overflow | carry) { s + C } else { s }` select, saving 2 instructions on the critical path. Add compile-time assertion enforcing C(C+1) < P. Made-with: Cursor * Unify Fp128 with Solinas-optimized arithmetic, delete SolinasFp128 Replace the generic Fp128 (binary-long-division via U256) with the Solinas-optimized implementation. Fp128 now uses [u64; 2] storage, fused schoolbook 2x2 + two-fold Solinas reduction (~23 cycles/mul on AArch64/x86-64), and compile-time validation that P = 2^128 - C with C < 2^64. Delete SolinasFp128, SolinasParams, solinas128.rs, and u256.rs. All call sites updated; prime type aliases (Prime128M13M4P0 etc.) are now simple Fp128<...> aliases in fp128.rs. Blanket PseudoMersenneField impl for all Fp128

. Made-with: Cursor * Use git deps for ark-bn254/ark-ff instead of local paths Switch from local path dependencies to the a16z/arkworks-algebra git repo (branch dev/twist-shout) so collaborators can compile without needing a local checkout of arkworks-algebra-jolt. Made-with: Cursor * Add template for sumchecks * Optimize Fp128 mul path and expand Rust field benchmarks. Refine Fp128 multiply/fold carry handling for better generated code and add isolated, passthrough, independent, and long-chain Rust microbenches to separate latency and throughput effects when comparing against BN254. Made-with: Cursor * Add 2^a±1 Fp128 reduction specialization and benches. Detect C = 2^a ± 1 at compile time and route fold multiplications through a specialized shift-based path with generic fallback, plus add benchmark coverage for sparse 128-bit primes using this shape. Made-with: Cursor * Add packed Fp128 field backend scaffolding and focused benchmarks. This introduces AArch64-first packed field abstractions with a scalar fallback and adds dedicated field-only validation/benchmark coverage before any ring or protocol integration. Made-with: Cursor * Refactor packed Fp128 backend to true SoA layout and stabilize benchmarking. This switches packed lane storage to SoA with NEON add/sub kernels and a SoA mul path, and updates packed-field APIs and benches so scalar-vs-packed latency/throughput comparisons are measured consistently. Made-with: Cursor * Optimize packed Fp128 mul throughput with array-backed SoA lanes. This keeps mul in true SoA form while removing repeated vector transmute overhead and inlining the limb-level Solinas lane kernel, improving packed mul throughput and latency against scalar baselines. Made-with: Cursor * Add Fp128 widening multiply API and specialized Solinas reduction Expose mul_wide_u64, mul_wide, mul_wide_u128, solinas_reduce, and to_limbs for deferred-reduction patterns needed by jolt-hachi. Hand-optimized reduce paths for 3/4/5 limbs avoid generic loop overhead. Refactor mul_raw to reuse mul_wide + reduce_4 (zero overhead). Add 9 unit tests and widening/accumulator benchmarks. Made-with: Cursor * Clean up fp128: remove section banners, hoist std::ops imports, rename mul_wide free fn Rename free function mul_wide → mul64_wide to avoid shadowing Fp128::mul_wide. Move reduce_4 next to fold2_canonicalize. Replace fully qualified std::ops::{Add,Sub,Mul,Neg} with use imports. Made-with: Cursor * Constrain Fp32/Fp64 to pseudo-Mersenne primes with Solinas reduction Rework fp32.rs and fp64.rs to require p = 2^k - c (small c), matching fp128's design. Compile-time constants BITS/C/MASK derived from P with static assertions. Replace bit-serial reduction with two-fold Solinas reduction (reduce_product for hot path, loop-based reduce_u64/u128 for arbitrary inputs). Add widening ops (mul_wide, square, solinas_reduce). Fix FieldSampling to use direct modular reduction instead of rejection sampling. Blanket-impl PseudoMersenneField, remove manual impls. Rename const generic MODULUS -> P at all call sites. Add latency + throughput benchmarks. Hoist mid-function imports in tests/algebra.rs. Made-with: Cursor * Specialize Fp64 sub-word primes to u64-only arithmetic For BITS < 64 (e.g. 2^40-195), avoid u128 intermediates in reduce_product, add_raw, and sub_raw. Use mul_c_narrow which splits C*high into u32x32->u64 widening multiplies (umaddl on AArch64), preventing LLVM from promoting to u128. Brings 40-bit mul throughput within 4% of 64-bit (690 vs 716 Melem/s), up from ~20% gap. Made-with: Cursor * Add 2^30 and 2^31 pseudo-Mersenne primes and expand benchmarks Add Pow2Offset30Field (2^30-35) and Pow2Offset31Field (2^31-19) prime definitions and type aliases. Refactor fp32/fp64 latency benchmarks with chain_bench! macro, add throughput benchmarks for all new primes. Made-with: Cursor * Add NEON packed backends for Fp32 (4-wide) and Fp64 (2-wide) PackedFp32Neon: 4 lanes in uint32x4_t with full NEON Solinas reduction for mul (vmull_u32 + 2-fold reduce), umin trick for add/sub (BITS<=31), overflow-aware paths for BITS==32. C_SHIFT_KIND optimization for C=2^a+/-1. PackedFp64Neon: 2 lanes in uint64x2_t with NEON add/sub (conditional P for BITS<=62, carry-aware for BITS>=63), scalar-per-lane mul (no native 64x64->128 on NEON). Fp32 packed achieves 2.4-3.5x mul throughput and 3.5-5.0x add/sub throughput over scalar. Includes HasPacking impls, type aliases, NoPacking fallbacks, 7 correctness tests, and throughput benchmarks. Made-with: Cursor * Optimize packed Fp32/Fp64 Solinas multiply hot paths on NEON For packed Fp32, remove the shift/add C-special-case in the Solinas fold and always use vmull_u32 with a hoisted C broadcast, which improves stability and removes the 24-bit mul regression. For packed Fp64, replace per-lane Fp64 wrapper multiplication with packed-local per-lane 64x64->128 products plus specialized Solinas reduction (including the sub-word u64 fold path), reducing mul overhead for both 40-bit and 64-bit packed variants. Made-with: Cursor * Tune packed Fp64 mul folding and add reducer/codegen probes Switch packed Fp64 sub-word fold multiplication to direct `C*x`, which improves packed mul throughput in repeated A/B runs. Add dedicated reducer and codegen probe benches so we can compare 40-bit and 64-bit fold paths with instruction-level visibility. Made-with: Cursor * Optimize x86 BMI2 multiply paths for fp64/fp128 fields Use BMI2 widening multiplies in scalar field hot paths and specialize x86 sub-word fold multiplication to a single 64-bit multiply, improving 40-bit fp64 throughput while keeping 64-bit and 128-bit paths stable. Made-with: Cursor * Optimize fp128 wide-limb multiply path for Jolt integration Raise Hachi MSRV to 1.88, add specialized Fp128 mul_wide_limbs kernels for M={3,4} and OUT={4,5,6}, and add field_arith benches that track mul_wide_limbs-only and roundtrip costs to catch regressions. Made-with: Cursor * Specialize Fp128 CanonicalField small-int constructors Make from_u64 use a direct canonical limb construction (no reduction path), fix from_i64 to use unsigned_abs to avoid i64::MIN overflow, and add a regression test for the min-value case. Made-with: Cursor * Impl sumchecks for hachi * Add optimized one-hot commitment path for regular sparse witnesses Exploits the structure of one-hot vectors (T chunks of K field elements, each chunk with exactly one 1) to eliminate all inner ring multiplications. Gadget decomposition of {0,1} coefficients is trivial (only level-0 digit is nonzero), and the inner Ajtai t = A*s reduces to summing selected columns of A with O(D) negacyclic rotations instead of O(D^2) ring muls. Handles both K >= D and D >= K as long as one divides the other: - K >= D: each nonzero ring element is a monomial X^j (single rotation) - D >= K: each ring element is a sum of D/K monomials (multiple rotations) Total inner cost: N_A * T * D coefficient additions (zero multiplications), vs N_A * 2^M * delta * D^2 coefficient multiplications in the dense path. Made-with: Cursor * Apply rustfmt formatting to fp128 and field_arith bench Made-with: Cursor * Inject sumchecks to Hachi prover * Add commitment to w to transcript * Add AVX2 and AVX-512 packed field backends for Fp32, Fp64, Fp128 Implement vectorized SIMD arithmetic for x86_64: - AVX2: 8-wide Fp32, 4-wide Fp64, 2-wide Fp128 (scalar delegation) - AVX-512: 16-wide Fp32, 8-wide Fp64, 4-wide Fp128 (scalar delegation) Fp32 uses even/odd lane split with 2-fold Solinas reduction. Fp64 uses vectorized 64×64→128 schoolbook multiply (adapted from plonky3 Goldilocks) with custom Solinas reduction for pseudo-Mersenne primes p = 2^k - c. Also: extract NEON backend into packed_neon.rs, add cfg-gated module selection (AVX-512 > AVX2 > NEON > NoPacking), enable nightly stdarch_x86_avx512 feature, add sumcheck-mix benchmark, and fix minor clippy lints in fp64/fp128. Made-with: Cursor * Vectorize Fp128 packed add/sub on AVX-512 (8-wide) and AVX2 (4-wide) Convert Fp128 packed backends from scalar delegation (AoS) to SoA layout with vectorized add/sub via __m512i / __m256i. Mul remains scalar per-lane. Add FIELD_OPS_PERF.md with Zen 5 benchmark results. Fp128 packed add: +114% (1.08 → 2.31 Gelem/s on Zen 5 AVX-512) Fp128 packed sub: +137% (1.34 → 3.18 Gelem/s) Made-with: Cursor * Add M4 Pro NEON benchmarks, remove mul_add experiment Populate FIELD_OPS_PERF.md with Apple M4 Pro (NEON) results for all primes across scalar, packed, and sumcheck MACC workloads. Remove the experimental mul_add trait method (vectorized add already optimal after inlining; scalar fused approach was 16% slower). Made-with: Cursor * Change sumcheck API * Separate ring switch logic * Rename sumchecks to NormSumcheck and RelationSumcheck * Remove iteration prover * Eliminate O(D^2) schoolbook ring multiplication from protocol hot paths At production parameters (D=256/1024), schoolbook CyclotomicRing multiplication is catastrophically expensive. Every protocol hot path has exploitable operand structure that avoids the full D^2 cost: - Add CyclotomicRing::mul_by_sparse for O(omega*D) sparse challenge multiplication (90-140x speedup in compute_z_hat) - Change RingOpeningPoint to store Vec scalars; use scale() instead of ring mul in compute_w_hat (256-1024x speedup) - Add kron_scalars, kron_row_scale, kron_sparse_scale; refactor generate_m to use scalar-aware Kronecker products - Add zero-skip and scalar-detect in compute_r_via_poly_division - Add sample_sparse_challenges, store Vec in QuadraticEquation throughout prover and verifier paths Made-with: Cursor * lint: section banner removal, naming hoist, cfg(test) for test-only paths - Remove section banner comments (----, =====) repo-wide in src, tests, benches - commitment_scheme: hoist RingCommitment, RingOpeningPoint, transcript labels to top-level use; add #[cfg(test)] use for rederive_alpha_and_m_a body (Blake2bTranscript, eval_ring_matrix_at, expand_m_a, labels) so that function uses short names without polluting lib build - Leave mod tests imports in place (no hoisting of test-module use blocks) Made-with: Cursor * Fix CI issues --------- Co-authored-by: Quang Dao Co-authored-by: Cursor --- .github/workflows/ci.yml | 14 +- AGENTS.md | 35 + CLAUDE.md | 1 + CONSTANT_TIME_NOTES.md | 42 + Cargo.lock | 401 ++++- Cargo.toml | 21 +- HACHI_PROGRESS.md | 200 +++ NTT_PRIME_ANALYSIS.md | 146 ++ README.md | 4 + benches/field_arith.rs | 1448 ++++++++++++++++++ benches/fp64_reduce_probe.rs | 118 ++ benches/ring_ntt.rs | 67 + docs/FIELD_EMBEDDINGS_SUPERNEO_VS_HACHI.md | 698 +++++++++ docs/FIELD_OPS_PERF.md | 237 +++ docs/HACHI_DIGEST.md | 1362 ++++++++++++++++ docs/SUPERNEO_DIGEST.md | 271 ++++ docs/TRANSCRIPT_COMMITMENT_COMPAT_SPEC.md | 106 ++ examples/codegen_probe_special.rs | 143 ++ paper/fri-binius.pdf | Bin 0 -> 819617 bytes paper/greyhound.pdf | Bin 0 -> 557273 bytes paper/labrador.pdf | Bin 0 -> 569118 bytes paper/standards/NIST.FIPS.203.pdf | Bin 0 -> 1252341 bytes paper/standards/NIST.FIPS.204.pdf | Bin 0 -> 3291746 bytes paper/superneo.pdf | Bin 0 -> 740185 bytes rust-toolchain.toml | 4 + rustfmt.toml | 5 + src/algebra/backend/mod.rs | 7 + src/algebra/backend/scalar.rs | 191 +++ src/algebra/backend/traits.rs | 56 + src/algebra/domains/coeff.rs | 6 + src/algebra/domains/crt_ntt.rs | 6 + src/algebra/domains/mod.rs | 7 + src/algebra/fields/ext.rs | 446 ++++++ src/algebra/fields/fp128.rs | 1032 +++++++++++++ src/algebra/fields/fp32.rs | 455 ++++++ src/algebra/fields/fp64.rs | 604 ++++++++ src/algebra/fields/lift.rs | 43 + src/algebra/fields/mod.rs | 40 + src/algebra/fields/packed.rs | 434 ++++++ src/algebra/fields/packed_avx2.rs | 704 +++++++++ src/algebra/fields/packed_avx512.rs | 666 ++++++++ src/algebra/fields/packed_neon.rs | 731 +++++++++ src/algebra/fields/pseudo_mersenne.rs | 177 +++ src/algebra/mod.rs | 34 + src/algebra/module.rs | 202 +++ src/algebra/ntt/butterfly.rs | 155 ++ src/algebra/ntt/crt.rs | 189 +++ src/algebra/ntt/mod.rs | 10 + src/algebra/ntt/prime.rs | 180 +++ src/algebra/ntt/tables.rs | 149 ++ src/algebra/poly.rs | 95 ++ src/algebra/ring/crt_ntt_repr.rs | 246 +++ src/algebra/ring/cyclotomic.rs | 439 ++++++ src/algebra/ring/mod.rs | 9 + src/algebra/ring/sparse_challenge.rs | 164 ++ src/lib.rs | 20 +- src/primitives/arithmetic.rs | 56 +- src/primitives/mod.rs | 1 + src/primitives/multilinear_evals.rs | 155 ++ src/primitives/poly.rs | 15 +- src/primitives/serialization.rs | 2 + src/primitives/transcript.rs | 4 +- src/protocol/challenges/mod.rs | 6 + src/protocol/challenges/sparse.rs | 121 ++ src/protocol/commitment/commit.rs | 146 ++ src/protocol/commitment/config.rs | 186 +++ src/protocol/commitment/mod.rs | 17 + src/protocol/commitment/onehot.rs | 268 ++++ src/protocol/commitment/scheme.rs | 277 ++++ src/protocol/commitment/transcript_append.rs | 13 + src/protocol/commitment/types.rs | 157 ++ src/protocol/commitment/utils/linear.rs | 44 + src/protocol/commitment/utils/math.rs | 14 + src/protocol/commitment/utils/matrix.rs | 131 ++ src/protocol/commitment/utils/mod.rs | 6 + src/protocol/commitment/utils/norm.rs | 48 + src/protocol/commitment_scheme.rs | 425 +++++ src/protocol/mod.rs | 30 + src/protocol/opening_point.rs | 20 + src/protocol/proof.rs | 59 + src/protocol/quadratic_equation.rs | 744 +++++++++ src/protocol/ring_switch.rs | 453 ++++++ src/protocol/sumcheck/mod.rs | 641 ++++++++ src/protocol/sumcheck/norm_sumcheck.rs | 255 +++ src/protocol/sumcheck/relation_sumcheck.rs | 345 +++++ src/protocol/transcript/blake2b.rs | 82 + src/protocol/transcript/keccak.rs | 82 + src/protocol/transcript/labels.rs | 69 + src/protocol/transcript/mod.rs | 37 + src/test_utils.rs | 123 ++ tests/algebra.rs | 1058 +++++++++++++ tests/commitment_contract.rs | 236 +++ tests/hachi_sumcheck.rs | 225 +++ tests/label_schedule.rs | 63 + tests/onehot_commitment.rs | 145 ++ tests/primality.rs | 124 ++ tests/ring_commitment_core.rs | 184 +++ tests/sparse_challenge.rs | 98 ++ tests/sumcheck_core.rs | 304 ++++ tests/sumcheck_prover_driver.rs | 98 ++ tests/transcript.rs | 136 ++ 101 files changed, 20489 insertions(+), 34 deletions(-) create mode 100644 AGENTS.md create mode 120000 CLAUDE.md create mode 100644 CONSTANT_TIME_NOTES.md create mode 100644 HACHI_PROGRESS.md create mode 100644 NTT_PRIME_ANALYSIS.md create mode 100644 benches/field_arith.rs create mode 100644 benches/fp64_reduce_probe.rs create mode 100644 benches/ring_ntt.rs create mode 100644 docs/FIELD_EMBEDDINGS_SUPERNEO_VS_HACHI.md create mode 100644 docs/FIELD_OPS_PERF.md create mode 100644 docs/HACHI_DIGEST.md create mode 100644 docs/SUPERNEO_DIGEST.md create mode 100644 docs/TRANSCRIPT_COMMITMENT_COMPAT_SPEC.md create mode 100644 examples/codegen_probe_special.rs create mode 100644 paper/fri-binius.pdf create mode 100644 paper/greyhound.pdf create mode 100644 paper/labrador.pdf create mode 100644 paper/standards/NIST.FIPS.203.pdf create mode 100644 paper/standards/NIST.FIPS.204.pdf create mode 100644 paper/superneo.pdf create mode 100644 rust-toolchain.toml create mode 100644 rustfmt.toml create mode 100644 src/algebra/backend/mod.rs create mode 100644 src/algebra/backend/scalar.rs create mode 100644 src/algebra/backend/traits.rs create mode 100644 src/algebra/domains/coeff.rs create mode 100644 src/algebra/domains/crt_ntt.rs create mode 100644 src/algebra/domains/mod.rs create mode 100644 src/algebra/fields/ext.rs create mode 100644 src/algebra/fields/fp128.rs create mode 100644 src/algebra/fields/fp32.rs create mode 100644 src/algebra/fields/fp64.rs create mode 100644 src/algebra/fields/lift.rs create mode 100644 src/algebra/fields/mod.rs create mode 100644 src/algebra/fields/packed.rs create mode 100644 src/algebra/fields/packed_avx2.rs create mode 100644 src/algebra/fields/packed_avx512.rs create mode 100644 src/algebra/fields/packed_neon.rs create mode 100644 src/algebra/fields/pseudo_mersenne.rs create mode 100644 src/algebra/mod.rs create mode 100644 src/algebra/module.rs create mode 100644 src/algebra/ntt/butterfly.rs create mode 100644 src/algebra/ntt/crt.rs create mode 100644 src/algebra/ntt/mod.rs create mode 100644 src/algebra/ntt/prime.rs create mode 100644 src/algebra/ntt/tables.rs create mode 100644 src/algebra/poly.rs create mode 100644 src/algebra/ring/crt_ntt_repr.rs create mode 100644 src/algebra/ring/cyclotomic.rs create mode 100644 src/algebra/ring/mod.rs create mode 100644 src/algebra/ring/sparse_challenge.rs create mode 100644 src/primitives/multilinear_evals.rs create mode 100644 src/protocol/challenges/mod.rs create mode 100644 src/protocol/challenges/sparse.rs create mode 100644 src/protocol/commitment/commit.rs create mode 100644 src/protocol/commitment/config.rs create mode 100644 src/protocol/commitment/mod.rs create mode 100644 src/protocol/commitment/onehot.rs create mode 100644 src/protocol/commitment/scheme.rs create mode 100644 src/protocol/commitment/transcript_append.rs create mode 100644 src/protocol/commitment/types.rs create mode 100644 src/protocol/commitment/utils/linear.rs create mode 100644 src/protocol/commitment/utils/math.rs create mode 100644 src/protocol/commitment/utils/matrix.rs create mode 100644 src/protocol/commitment/utils/mod.rs create mode 100644 src/protocol/commitment/utils/norm.rs create mode 100644 src/protocol/commitment_scheme.rs create mode 100644 src/protocol/mod.rs create mode 100644 src/protocol/opening_point.rs create mode 100644 src/protocol/proof.rs create mode 100644 src/protocol/quadratic_equation.rs create mode 100644 src/protocol/ring_switch.rs create mode 100644 src/protocol/sumcheck/mod.rs create mode 100644 src/protocol/sumcheck/norm_sumcheck.rs create mode 100644 src/protocol/sumcheck/relation_sumcheck.rs create mode 100644 src/protocol/transcript/blake2b.rs create mode 100644 src/protocol/transcript/keccak.rs create mode 100644 src/protocol/transcript/labels.rs create mode 100644 src/protocol/transcript/mod.rs create mode 100644 src/test_utils.rs create mode 100644 tests/algebra.rs create mode 100644 tests/commitment_contract.rs create mode 100644 tests/hachi_sumcheck.rs create mode 100644 tests/label_schedule.rs create mode 100644 tests/onehot_commitment.rs create mode 100644 tests/primality.rs create mode 100644 tests/ring_commitment_core.rs create mode 100644 tests/sparse_challenge.rs create mode 100644 tests/sumcheck_core.rs create mode 100644 tests/sumcheck_prover_driver.rs create mode 100644 tests/transcript.rs diff --git a/.github/workflows/ci.yml b/.github/workflows/ci.yml index 7aa166de..5715a7f5 100644 --- a/.github/workflows/ci.yml +++ b/.github/workflows/ci.yml @@ -20,7 +20,7 @@ jobs: runs-on: ubuntu-latest steps: - uses: actions/checkout@v4 - - uses: dtolnay/rust-toolchain@stable + - uses: actions-rust-lang/setup-rust-toolchain@v1 with: components: rustfmt - name: Check formatting @@ -31,20 +31,20 @@ jobs: runs-on: ubuntu-latest steps: - uses: actions/checkout@v4 - - uses: dtolnay/rust-toolchain@stable + - uses: actions-rust-lang/setup-rust-toolchain@v1 with: components: clippy - name: Clippy (all features) - run: cargo clippy -q --message-format=short --all-features --all-targets -- -D warnings + run: cargo clippy --all --all-targets --all-features -- -D warnings - name: Clippy (no default features) - run: cargo clippy -q --message-format=short --no-default-features --lib -- -D warnings + run: cargo clippy --all --all-targets --no-default-features -- -D warnings doc: name: Documentation runs-on: ubuntu-latest steps: - uses: actions/checkout@v4 - - uses: dtolnay/rust-toolchain@stable + - uses: actions-rust-lang/setup-rust-toolchain@v1 - name: Build documentation run: cargo doc -q --no-deps --all-features env: @@ -55,8 +55,8 @@ jobs: runs-on: ubuntu-latest steps: - uses: actions/checkout@v4 - - uses: dtolnay/rust-toolchain@stable + - uses: actions-rust-lang/setup-rust-toolchain@v1 - name: Install cargo-nextest uses: taiki-e/install-action@nextest - name: Run tests - run: cargo nextest run -q --all-features + run: cargo nextest run --all-features diff --git a/AGENTS.md b/AGENTS.md new file mode 100644 index 00000000..68f667fa --- /dev/null +++ b/AGENTS.md @@ -0,0 +1,35 @@ +# AGENTS.md + +**Compatibility notice (explicit): This repo makes NO backward-compatibility guarantees. Breaking changes are allowed and expected.** + +## Project Overview + +Hachi is a lattice-based polynomial commitment scheme (PCS) with transparent setup and post-quantum security. Built in Rust. Intended to replace Dory in Jolt. + +## Essential Commands + +```bash +cargo clippy --all --message-format=short -q -- -D warnings +cargo fmt -q +cargo test # no nextest yet +``` + +## Crate Structure + +Two workspace members: `hachi-pcs` (root) and `derive` (proc macros). + +- `src/primitives/` — Core traits: `FieldCore`, `Module`, `MultilinearLagrange`, `Transcript`, serialization +- `src/algebra/` — Concrete backends: prime fields, extension fields, cyclotomic rings, NTT, domains +- `src/protocol/` — Protocol layer: commitment, prover, verifier, opening (ring-switch), challenges, transcript +- `src/error.rs` — Error types + +## Key Abstractions + +- `CommitmentScheme` / `StreamingCommitmentScheme` — top-level PCS traits +- `FieldCore` + `PseudoMersenneField` + `Module` — arithmetic over lattice-friendly fields and rings +- `MultilinearLagrange` — multilinear polynomial in Lagrange basis +- `Transcript` — Fiat-Shamir + +## Feature Flags + +- `parallel` — Rayon parallelization diff --git a/CLAUDE.md b/CLAUDE.md new file mode 120000 index 00000000..47dc3e3d --- /dev/null +++ b/CLAUDE.md @@ -0,0 +1 @@ +AGENTS.md \ No newline at end of file diff --git a/CONSTANT_TIME_NOTES.md b/CONSTANT_TIME_NOTES.md new file mode 100644 index 00000000..dac24600 --- /dev/null +++ b/CONSTANT_TIME_NOTES.md @@ -0,0 +1,42 @@ +# Constant-Time Review Notes (Phase 0/1 Algebra) + +This note tracks timing-sensitive implementation decisions for the current +algebra and ring stack. + +## Reviewed Components + +- `src/algebra/fields/fp32.rs` +- `src/algebra/fields/fp64.rs` +- `src/algebra/fields/fp128.rs` +- `src/algebra/ntt/prime.rs` +- `src/algebra/ntt/butterfly.rs` +- `src/algebra/ring/cyclotomic.rs` +- `src/algebra/ring/crt_ntt_repr.rs` + +## Current State + +- Branchless primitives are in place for: + - `Fp32/Fp64/Fp128` add/sub/neg raw helpers. + - `Fp128` multiplication reduction (`reduce_u256`) with branchless conditional subtract. + - `Fp32/Fp64` multiplication reduction (division-free fixed-iteration paths). + - NTT helper operations `csubp`, `caddp`, and `center`. +- NTT butterfly arithmetic runs in fixed loop structure independent of data. +- Ring multiplication (`CyclotomicRing`) is fixed-structure schoolbook over `D`. +- CRT reconstruction inner accumulation now uses fixed-trip, branchless + modular add/mul-by-small-factor helpers. +- Prime fields now expose `Invertible::inv_or_zero()` for secret-bearing + inversion use-cases without input-dependent branching on zero. +- CRT reconstruction final projection now uses a division-free fixed-iteration + reducer (`reduce_u128_divfree`) instead of `% q`. + +## Known Timing Risks / Follow-ups + +- `FieldCore::inv()` still returns `Option` and therefore branches on zero; + treat that API as public-value oriented. Use `Invertible::inv_or_zero()` + in secret-dependent paths. + +## Action Items Before Production-Critical Use + +1. Wire secret-bearing call sites to `Invertible::inv_or_zero()` as + protocol code matures. +2. Add dedicated CT review tests/checklists for any arithmetic subsystem changes. diff --git a/Cargo.lock b/Cargo.lock index a505ce5c..e184f5cc 100644 --- a/Cargo.lock +++ b/Cargo.lock @@ -2,6 +2,18 @@ # It is not intended for manual editing. version = 3 +[[package]] +name = "ahash" +version = "0.8.12" +source = "registry+https://github.com/rust-lang/crates.io-index" +checksum = "5a15f179cd60c4584b8a8c596927aadc462e27f2ca70c04e0071964a73ba7a75" +dependencies = [ + "cfg-if", + "once_cell", + "version_check", + "zerocopy", +] + [[package]] name = "aho-corasick" version = "1.1.4" @@ -11,6 +23,33 @@ dependencies = [ "memchr", ] +[[package]] +name = "allocative" +version = "0.3.4" +source = "registry+https://github.com/rust-lang/crates.io-index" +checksum = "8fac2ce611db8b8cee9b2aa886ca03c924e9da5e5295d0dbd0526e5d0b0710f7" +dependencies = [ + "allocative_derive", + "ctor", +] + +[[package]] +name = "allocative_derive" +version = "0.3.3" +source = 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[ + "proc-macro2", + "quote", + "syn 2.0.110", ] diff --git a/Cargo.toml b/Cargo.toml index 47abb4d8..d6e45663 100644 --- a/Cargo.toml +++ b/Cargo.toml @@ -6,7 +6,7 @@ resolver = "2" name = "hachi-pcs" version = "0.1.0" edition = "2021" -rust-version = "1.75" +rust-version = "1.88" authors = [ "Markos Georghiades ", ] @@ -37,14 +37,31 @@ parallel = ["dep:rayon"] [dependencies] thiserror = "2.0" -rand_core = "0.6" +rand_core = { version = "0.6", features = ["getrandom"] } hachi-derive = { version = "0.1.0", path = "derive" } tracing = "0.1" rayon = { version = "1.10", optional = true } +blake2 = "0.10.6" +sha3 = "0.10.8" [dev-dependencies] rand = "0.8" criterion = { version = "0.5", features = ["html_reports"] } +num-bigint = "0.4.6" +ark-bn254 = { git = "https://github.com/a16z/arkworks-algebra", branch = "dev/twist-shout", features = ["scalar_field"] } +ark-ff = { git = "https://github.com/a16z/arkworks-algebra", branch = "dev/twist-shout" } + +[[bench]] +name = "ring_ntt" +harness = false + +[[bench]] +name = "field_arith" +harness = false + +[[bench]] +name = "fp64_reduce_probe" +harness = false [lints.rust] missing_docs = "warn" diff --git a/HACHI_PROGRESS.md b/HACHI_PROGRESS.md new file mode 100644 index 00000000..afca63e6 --- /dev/null +++ b/HACHI_PROGRESS.md @@ -0,0 +1,200 @@ +## Hachi PCS implementation progress + +This file is the **single source of truth** for implementation status and near-term priorities. + +### Goals (project-level) + +- **Production-ready implementation**: correctness, security, maintainability, and performance are first-class goals. +- **Standalone codebase**: implementation and comments should stand on their own; external acknowledgements live in `README.md`. +- **Constant-time cryptographic core**: arithmetic and protocol-critical paths must be constant-time with respect to secret data. +- **No shortcuts / no fallback design**: avoid temporary or degraded code paths in the core implementation. + +### Non-negotiable requirements + +- **Constant-time discipline** + - No secret-dependent branches or memory access patterns in cryptographic hot paths. + - No secret-indexed table lookups; table access patterns must be independent of secret data. + - Keep data representations and reductions explicit and auditable for timing behavior. + - Add targeted tests/reviews for constant-time-sensitive code as features land. +- **Code quality bar** + - Clear naming, explicit invariants, small cohesive modules, and API docs for public interfaces. + - No placeholder crypto logic in mainline code (no "temporary" arithmetic shortcuts). + - Tests are required for correctness-critical arithmetic before dependent protocol code is built. + - No section-banner comments (e.g., `// ---- Section ----`, `// === ... ===`). Let the code and doc-comments speak for themselves. +- **Standalone implementation policy** + - Do not mention external inspirations/ports in core code comments. + - Keep terminology and structure internally coherent and project-native. + - Keep external attribution limited to dedicated docs (for now: `README.md` acknowledgements). +- **Git discipline** + - Do not commit or push without explicit user approval. + +### Implementation workflow (cautious + approval-driven) + +- Before each major subsystem, present implementation options with trade-offs. +- Seek explicit approval before proceeding with a selected option. +- Pause at milestone boundaries for review and feedback before continuing. +- Prefer slow, verifiable progress over rapid, high-risk changes. +- Ask for user input frequently when requirements are ambiguous or involve design trade-offs. + +### Definition of Done (all crypto-critical work) + +- **Security / constant-time** + - Secret-independent control flow and memory access in cryptographic paths. + - Constant-time review notes included for non-trivial arithmetic/ring changes. +- **Correctness** + - Unit tests for edge cases and algebraic identities. + - Cross-check vectors/reference checks added where practical. +- **Code quality** + - Clear naming, explicit invariants, and no placeholder logic in core paths. + - Public interfaces documented sufficiently for safe usage. +- **Performance** + - Hot-path performance impact evaluated (benchmark or measured rationale). +- **Tooling + CI** + - `cargo fmt --all --check` passes. + - `cargo clippy --all --all-targets --all-features` passes. + - `cargo test` (or targeted suite for touched modules) passes. +- **Process** + - Implementation options reviewed with user before major subsystem changes. + - Milestone update recorded in this file. + +### Scope (current) + +- **Implemented so far (Phase 0 + Phase 1 functional core)**: prime fields (32/64/128-bit representations), extension fields, cyclotomic `R_q = Z_q[X]/(X^d + 1)`, CRT+NTT representation, backend/domain layering, ring automorphisms, and functional gadget decomposition. +- **Phase 2+ protocol status**: interface scaffold plus ring-native §4.1 commitment core are present (`Transcript`, Blake2b/Keccak backends, phase-grounded labels, `RingCommitmentScheme`, config layer, and setup/commit implementation). Sumcheck core building blocks (univariate messages + transcript-driving prover/verifier driver) are now implemented, with tests. Open-check prover/verifier paths remain stubbed. +- **Deferred future phase**: integration into Jolt (replacement of Dory with Hachi) is intentionally out of current execution scope; cross-repo analysis is design input only. + +### Critical review snapshot (2026-02-13) + +- **Phase 1 functional milestone appears complete** + - Ring/gadget components listed in Phase 1 are implemented and currently checked off. + - Conversion and arithmetic paths in coefficient and CRT+NTT domains are exercised by passing tests. +- **Not yet "production-ready" despite functional completion** + - Constant-time hardening follow-ups narrowed: secret-bearing call-sites still need to migrate from `FieldCore::inv()` to `Invertible::inv_or_zero()` as protocol code lands (see `CONSTANT_TIME_NOTES.md`). + - Current ring multiplication in coefficient form remains `O(D^2)` schoolbook (`src/algebra/ring/cyclotomic.rs`), with CRT+NTT available as the faster domain path. +- **Tooling/quality gate status (current branch snapshot)** + - `cargo test` passes, including protocol transcript/label/commitment contract tests and new ring-commitment core/config/stub tests. + - `cargo fmt --all --check` passes. + - `cargo clippy --all --all-targets --all-features` passes. +- **Phase 2 scaffold + commitment core landed; proof-system work still pending** + - `src/protocol/*` now provides transcript + commitment abstraction boundaries with `Transcript` naming. + - Two transcript backends are wired (`Blake2bTranscript`, `KeccakTranscript`) with deterministic replay/order/reset tests. + - Hachi-native labels are now calibrated to paper-stage phases (§4.1, §4.2, §4.3, §4.5). + - Commitment absorption is label-directed at call sites (`AppendToTranscript` no longer hardcodes commitment labels). + - Ring-native commitment setup/commit flow for §4.1 is implemented in `src/protocol/commitment/commit.rs` behind `RingCommitmentScheme`. + - Sumcheck core module landed (`src/protocol/sumcheck.rs`) with unit/integration tests (`tests/sumcheck_core.rs`, `tests/sumcheck_prover_driver.rs`). + - Prover/verifier split folders are wired with explicit stubs (`src/protocol/prover/stub.rs`, `src/protocol/verifier/stub.rs`) for future open-check implementation. +- **Conclusion** + - Treat **Phase 1 as functionally complete**. + - Treat **Phase 2 as active/in-progress** (commitment core implemented; prove/verify and later reductions still open). + - Remaining strict CT follow-ups stay tracked in `CONSTANT_TIME_NOTES.md`. + +### Status board + +#### Phase 0 — Algebra + +- [x] Prime field `Fp32` (u32 storage; u64 mul) implementing `FieldCore + CanonicalField` (`src/algebra/fields/fp32.rs`) +- [x] Prime field `Fp64` (u64 storage; u128 mul) implementing `FieldCore + CanonicalField` (`src/algebra/fields/fp64.rs`) +- [x] Prime field `Fp128` (u128 storage; 256-bit intermediate) implementing `FieldCore + CanonicalField` (`src/algebra/fields/fp128.rs`, `src/algebra/fields/u256.rs`) +- [x] Branchless constant-time `add_raw`, `sub_raw`, `neg` for all field types +- [x] Constant-time inversion helper for prime fields: `Invertible::inv_or_zero()` (`src/primitives/arithmetic.rs`, `src/algebra/fields/fp*.rs`) +- [x] Division-free fixed-iteration reduction for `Fp32/Fp64` multiplication paths +- [x] Division-free fixed-iteration CRT final projection (replaced `% q` in scalar reconstruction path) +- [x] Rejection-sampled `FieldSampling::sample()` for all field types (no modular bias) +- [x] Pow2Offset pseudo-Mersenne registry + aliases (`q = 2^k - offset`, bounded `k <= 128`, `q % 8 == 5`) (`src/algebra/fields/pseudo_mersenne.rs`) +- [x] Constant-time review notes for current algebra/ring paths (`CONSTANT_TIME_NOTES.md`) +- [x] Deterministic parameter presets + - [x] `q = 2^32 - 99` constants scaffold (`src/algebra/ntt/tables.rs`) + - [x] `Pow2Offset` presets selected for 64/128-bit path: + - `q = 2^64 - 59` (`POW2_OFFSET_MODULUS_64`) + - `q = 2^128 - 275` (`POW2_OFFSET_MODULUS_128`) + - source: `src/algebra/fields/pseudo_mersenne.rs` +- [x] `Module` implementations: + - [x] `VectorModule` (fixed-length vectors; `Module` via scalar*vector mul) (`src/algebra/module.rs`) + - [x] `PolyModule` removed from current scope (not needed for near-term Hachi milestones) +- [ ] Extension fields: + - [x] `Fp2` quadratic extension (`src/algebra/fields/ext.rs`) + - [x] `Fp4` tower extension (`src/algebra/fields/ext.rs`) +- [x] Serialization for algebra types (`HachiSerialize` / `HachiDeserialize`) (+ `u128/i128` primitives in `src/primitives/serialization.rs`) +- [x] NTT small-prime arithmetic: Montgomery-like `fpmul`, Barrett-like `fpred`, branchless `csubq`/`caddq`/`center` (`src/algebra/ntt/prime.rs`) +- [x] CRT limb arithmetic: `LimbQ`, `QData` (`src/algebra/ntt/crt.rs`) +- [x] Tests (49 total in `tests/algebra.rs`): + - [x] field arithmetic, identities, distributivity (Fp32/Fp64/Fp128) + - [x] zero inversion returns None + - [x] serialization round-trips (all field types, extensions, Poly, VectorModule) + - [x] Fp2 conjugate, norm, distributivity + - [x] U256 wide multiply and bit access + - [x] LimbQ round-trip, add/sub inverse, QData consistency + - [x] NTT normalize range, fpmul commutativity + - [x] Poly add/sub/neg + - [x] Cyclotomic ring identities and serialization (D=4, D=64) + - [x] NTT forward/inverse round-trips (single prime and all Q32 primes) + - [x] Cyclotomic CRT+NTT full round-trip (`from_ring` -> `to_ring`) + - [x] Scalar backend path equivalence (`*_with_backend` vs default path) + - [x] Pow2Offset profile invariants (`q = 2^k - offset`, `q % 8 == 5`) + - [x] `FieldSampling::sample()` output bound checks + - [x] Checked deserialization rejects non-canonical field encodings + - [x] Galois automorphism checks (`sigma` composition + multiplicativity) + - [x] Functional gadget decompose/recompose round-trip checks + - [x] Sparse `+/-1` challenge support checks (`hamming_weight = omega`) +- [x] Dedicated Pow2Offset primality regression tests (`tests/primality.rs`) + - [x] Miller-Rabin probable-prime checks for all registered Pow2Offset moduli + - [x] Composite sanity rejection checks + +#### Phase 1 — Ring + gadgets (functional core) + +- [x] Cyclotomic ring `Rq` with `X^D = -1` (`src/algebra/ring/cyclotomic.rs`) +- [x] CRT+NTT-domain ring representation + CRT conversion (`src/algebra/ring/crt_ntt_repr.rs`) +- [x] Backend/domain layering for ring execution (`src/algebra/backend/*`, `src/algebra/domains/*`) +- [x] Galois automorphisms `sigma_i: X ↦ X^i` (odd `i`) +- [x] Functional gadget decomposition/recomposition (`G^{-1}` / `G` behavior) for base-`2^d` digits, without materializing dense gadget matrices +- [x] sparse short challenges (paper: `||c||_1 ≤ ω`, sparse ±1) + +#### Phase 2+ — Protocol (later) + +- [x] Protocol module scaffold (`src/protocol/*`) and top-level re-exports +- [x] Transcript interface (`Transcript`) plus Blake2b/Keccak implementations +- [x] Hachi-native transcript label schedule aligned to paper phases (§4.1/§4.2/§4.3/§4.5) +- [x] Commitment trait surface + streaming trait surface + contract tests +- [x] Label-directed transcript absorption for commitments (`AppendToTranscript` takes label at call site) +- [x] ring-native commitment core (`RingCommitmentScheme`, `commit.rs`, config wiring) for §4.1 setup/commit +- [x] protocol prover/verifier folder split with explicit stubs (`prover/stub.rs`, `verifier/stub.rs`) +- [x] ring-commitment tests (`ring_commitment_core`, `ring_commitment_config`, `prover_verifier_stub_contract`) +- [x] sumcheck core building blocks (univariate messages + transcript-driving prover/verifier driver) (`src/protocol/sumcheck.rs`) +- [x] sumcheck core tests (`tests/sumcheck_core.rs`, `tests/sumcheck_prover_driver.rs`) +- [ ] commitment open-check prove/verify implementation (currently stubs) +- [ ] evaluation → linear relation (paper §4.2) +- [ ] ring-switching constraints as sumcheck instances (paper §4.3, Fig. 4–7) +- [ ] recursion / “stop condition” + optional Greyhound composition (§4.5) + +#### Phase 3 — Integration into Jolt (deferred; not active now) + +- [ ] Define compatibility boundary document (what must match Jolt/Dory behavior vs what can remain Hachi-native) +- [ ] Provide Jolt-facing transcript adapter design (`Jolt` transcript pattern ↔ Hachi transcript object) +- [ ] Provide Jolt-facing PCS shim design (`CommitmentScheme`/`StreamingCommitmentScheme` mapping) +- [ ] Add transcript/commitment compatibility tests for integration-readiness (without wiring into Jolt yet) + +### Conventions + +- **Correctness first**: lock arithmetic with tests before touching protocol code. +- **Security first**: enforce constant-time behavior for secret-dependent operations. +- **Lean deps**: avoid heavyweight crypto crates until there is a clear need. +- **Explicit parameter sets**: each field/ring preset lives in code with a clear name and rationale. + +### Module layout + +``` +src/algebra/ +├── backend/ Backend execution traits + scalar backend +├── domains/ Domain-level aliases (coefficient / CRT+NTT) +├── fields/ Prime fields, pseudo-mersenne registry, u256, and extensions +├── ntt/ NTT kernels (butterfly), prime kernels (prime), CRT helpers (crt), presets (tables) +├── module.rs VectorModule +├── poly.rs Poly container +└── ring/ Cyclotomic ring and CRT+NTT representation +``` + +### References + +- Hachi paper: `paper/hachi.pdf` +- Core traits: `src/primitives/arithmetic.rs`, `src/primitives/serialization.rs` + diff --git a/NTT_PRIME_ANALYSIS.md b/NTT_PRIME_ANALYSIS.md new file mode 100644 index 00000000..e6b9c85c --- /dev/null +++ b/NTT_PRIME_ANALYSIS.md @@ -0,0 +1,146 @@ +# NTT Prime Analysis (Pow2Offset / Solinas Context) + +This note records the current analysis for small NTT primes and CRT coverage targets. + +## References + +- NIST ML-KEM: `paper/standards/NIST.FIPS.203.pdf` +- NIST ML-DSA: `paper/standards/NIST.FIPS.204.pdf` +- Current small-prime table: `src/algebra/ntt/tables.rs` +- Labrador generator heuristic: `../labrador/data.py` + +## Why does `2D` divide `p - 1`? + +For negacyclic NTT on `Z_p[X]/(X^D + 1)`, we need a primitive `2D`-th root `psi` such that: + +- `psi^D = -1 (mod p)` +- `psi^(2D) = 1 (mod p)` + +Over prime fields, `F_p^*` is cyclic of size `p - 1`, so an element of order `2D` exists iff: + +- `2D | (p - 1)` + +So yes, the `128 | (p - 1)` condition is directly tied to `D = 64`. + +## What if `D = 1024`? + +Then requirement becomes: + +- `2D = 2048`, so `2048 | (p - 1)`. + +Under the current "small prime" cap (`p < 2^14`), this is extremely restrictive. + +## Why `< 2^14` in current code? + +This is a backend implementation constraint, not a hard NTT math requirement: + +- Current small-prime NTT backend stores modulus/coefficients in signed 16-bit lanes (`i16`). +- It relies on centered signed arithmetic and butterfly add/sub before full normalization. +- Keeping `p < 2^14` leaves practical headroom in those 16-bit operations. +- Current CRT limb code is also radix-`2^14`, matching this design style. + +So the `2^14` cap is about the present `i16` scalar kernel design. If we introduce an `i32` backend, this cap can be raised substantially. + +## Exhaustive counts (for `p < 2^14`) + +We classify exact Solinas as: + +- `p = 2^x - 2^y + 1`. + +Results: + +- `D = 64` (`128 | p-1`) + - all small NTT primes: **31** + - exact Solinas NTT primes: **6** + - all-prime set: + - `257, 641, 769, 1153, 1409, 2689, 3329, 3457, 4481, 4993, 6529, 7297, 7681, 7937, 9473, 9601, 9857, 10369, 10753, 11393, 11777, 12161, 12289, 13313, 13441, 13697, 14081, 14593, 15233, 15361, 16001` + - Solinas set: `257, 769, 7681, 7937, 12289, 15361` +- `D = 256` (`512 | p-1`) + - all small NTT primes: **6** + - exact Solinas NTT primes: **3** + - Solinas set: `7681, 12289, 15361` +- `D = 1024` (`2048 | p-1`) + - all small NTT primes: **1** + - exact Solinas NTT primes: **1** + - Solinas set: `12289` + +Conclusion: for higher `D`, the small-prime pool shrinks rapidly. + +## 30-bit exploration (`p < 2^30`) with NTT constraints + +To assess a larger-prime backend direction, we scanned for primes under `2^30` with: + +- `p ≡ 1 (mod 2D)`. + +Below are the **full outputs of the bounded search run** (top 30 largest primes found by descending scan): + +### `D = 64` (`2D = 128`) + +- Top-30 list: + - `1073741441, 1073739649, 1073738753, 1073736449, 1073735297, 1073734913, 1073732993, 1073732609, 1073731201, 1073731073, 1073730817, 1073728897, 1073727617, 1073726977, 1073722753, 1073719681, 1073717377, 1073716993, 1073713409, 1073712769, 1073712257, 1073710721, 1073708929, 1073707009, 1073703809, 1073702657, 1073702401, 1073698817, 1073696257, 1073693441` +- Coverage for `q = 2^128 - 275`: + - `P > q`: **5** limbs + - `P > 128*q^2`: **9** limbs + +### `D = 1024` (`2D = 2048`) + +- Top-30 list: + - `1073707009, 1073698817, 1073692673, 1073682433, 1073668097, 1073655809, 1073651713, 1073643521, 1073620993, 1073600513, 1073569793, 1073563649, 1073551361, 1073539073, 1073522689, 1073510401, 1073508353, 1073479681, 1073453057, 1073442817, 1073440769, 1073430529, 1073412097, 1073391617, 1073385473, 1073354753, 1073350657, 1073330177, 1073299457, 1073268737` +- Coverage for `q = 2^128 - 275`: + - `P > q`: **5** limbs + - `P > 128*q^2`: **9** limbs + +### Bit-estimate sanity check + +- `ceil(128 / 30) = 5` +- `ceil(263 / 30) = 9` (for `128*q^2 ~ 2^263`) + +This matches the concrete product counts above. + +## CRT size targets for `q = 2^128 - 275` + +Two common thresholds: + +1. Minimal uniqueness target: + - `P = prod(p_i) > q` +2. Labrador conservative heuristic: + - `P > 128 * q^2` (from `data.py`, with `FIXME` comment) + +### Limb counts at `D = 64` with current small-prime pool + +- Using all small NTT primes (`31` available): + - `P > q` achievable with **10** limbs + - `P > 128*q^2` achievable with **20** limbs +- Using only exact Solinas NTT primes (`6` available): + - `P > q`: **not achievable** + - `P > 128*q^2`: **not achievable** + - total product is only about `2^70` + +### Limb counts at `D = 1024` with current small-prime pool + +- Only one qualifying prime (`12289`) under `p < 2^14`, so neither threshold is achievable. + +## What is Labrador's safety margin doing? + +In Labrador code, prime selection stops at: + +- `P > 128 * q^2` + +Interpretation: + +- `q^2` tracks product-scale growth, +- extra factor `128` gives additional headroom (for `N=64`, this is `2N`), +- but their own `FIXME` comment indicates this is a conservative engineering bound, not a tight proof. + +So treat this as a robust heuristic rather than a formal minimum. + +## Practical implication for Hachi + +- If we stay with `D=64` and small i16-ish primes, we need non-Solinas primes in the CRT set. +- If we push to `D=1024`, we must either: + - lift prime size beyond `<2^14`, or + - change CRT strategy (fewer larger limbs / different backend), or + - avoid strict small-prime CRT-NTT at that degree. +- A mixed backend model is sensible: + - keep the current `i16` backend for small-prime kernels, + - add an `i32`/wider backend for larger-prime kernels (e.g., up to ~30-bit). diff --git a/README.md b/README.md index 492b4d12..8036344f 100644 --- a/README.md +++ b/README.md @@ -3,3 +3,7 @@ A high performance and modular implementation of the Hachi polynomial commitment scheme. Hachi is a lattice-based polynomial commitment scheme with transparent setup and post-quantum security. + +## Acknowledgements + +The CRT/NTT and small-prime arithmetic design in this repository is informed by the Labrador/Greyhound C implementation family. In particular, the current pseudo-Mersenne profile uses moduli of the form `q = 2^k - offset` (smallest prime below `2^k` with `q % 8 == 5`). Hachi provides a Rust-native architecture and APIs, while drawing algorithmic inspiration from those implementations. diff --git a/benches/field_arith.rs b/benches/field_arith.rs new file mode 100644 index 00000000..9caf70d8 --- /dev/null +++ b/benches/field_arith.rs @@ -0,0 +1,1448 @@ +#![allow(missing_docs)] + +use ark_bn254::Fr as BN254Fr; +use ark_ff::{AdditiveGroup, Field}; +use criterion::{black_box, criterion_group, criterion_main, Criterion, Throughput}; +use hachi_pcs::algebra::fields::fp128::{Prime128M18M0, Prime128M54P0}; +use hachi_pcs::algebra::fields::fp32::Fp32; +use hachi_pcs::algebra::{HasPacking, PackedField, PackedValue, Prime128M13M4P0, Prime128M8M4M1M0}; +use hachi_pcs::algebra::{ + Pow2Offset24Field, Pow2Offset30Field, Pow2Offset31Field, Pow2Offset32Field, Pow2Offset40Field, + Pow2Offset48Field, Pow2Offset56Field, Pow2Offset64Field, +}; +use hachi_pcs::{CanonicalField, FieldCore, FieldSampling, Invertible}; +use rand::{rngs::StdRng, RngCore, SeedableRng}; +use std::env; +#[cfg(feature = "parallel")] +use std::thread; + +#[cfg(feature = "parallel")] +use rayon::prelude::*; +#[cfg(feature = "parallel")] +use rayon::ThreadPoolBuilder; + +fn rand_u128(rng: &mut R) -> u128 { + let lo = rng.next_u64() as u128; + let hi = rng.next_u64() as u128; + lo | (hi << 64) +} + +fn env_usize(name: &str, default: usize) -> usize { + env::var(name) + .ok() + .and_then(|v| v.parse::().ok()) + .unwrap_or(default) +} + +fn bench_mul(c: &mut Criterion) { + type F13 = Prime128M13M4P0; + type F275 = Prime128M8M4M1M0; + type F2p18p1 = Prime128M18M0; + type F2p54m1 = Prime128M54P0; + + let mut rng = StdRng::seed_from_u64(0x5eed); + let inputs_u128: Vec = (0..2048).map(|_| rand_u128(&mut rng)).collect(); + + let inputs_f13: Vec = inputs_u128 + .iter() + .copied() + .map(F13::from_canonical_u128_reduced) + .collect(); + + let inputs_f275: Vec = inputs_u128 + .iter() + .copied() + .map(F275::from_canonical_u128_reduced) + .collect(); + let inputs_f2p18p1: Vec = inputs_u128 + .iter() + .copied() + .map(F2p18p1::from_canonical_u128_reduced) + .collect(); + let inputs_f2p54m1: Vec = inputs_u128 + .iter() + .copied() + .map(F2p54m1::from_canonical_u128_reduced) + .collect(); + + let mut group = c.benchmark_group("field_mul"); + + group.bench_function("fp128_prime128m13m4p0", |b| { + b.iter(|| { + let mut acc = F13::one(); + for x in inputs_f13.iter() { + acc = acc * *x + acc; + } + black_box(acc) + }) + }); + + group.bench_function("fp128_prime128m8m4m1m0", |b| { + b.iter(|| { + let mut acc = F275::one(); + for x in inputs_f275.iter() { + acc = acc * *x + acc; + } + black_box(acc) + }) + }); + + group.bench_function("fp128_prime128m18m0_shift_special", |b| { + b.iter(|| { + let mut acc = F2p18p1::one(); + for x in inputs_f2p18p1.iter() { + acc = acc * *x + acc; + } + black_box(acc) + }) + }); + + group.bench_function("fp128_prime128m54p0_shift_special", |b| { + b.iter(|| { + let mut acc = F2p54m1::one(); + for x in inputs_f2p54m1.iter() { + acc = acc * *x + acc; + } + black_box(acc) + }) + }); + + group.finish(); +} + +fn bench_mul_only(c: &mut Criterion) { + type F13 = Prime128M13M4P0; + type F2p18p1 = Prime128M18M0; + type F2p54m1 = Prime128M54P0; + + let mut rng = StdRng::seed_from_u64(0x5eed); + let inputs_f13: Vec = (0..2048) + .map(|_| F13::from_canonical_u128_reduced(rand_u128(&mut rng))) + .collect(); + let inputs_f2p18p1: Vec = (0..2048) + .map(|_| F2p18p1::from_canonical_u128_reduced(rand_u128(&mut rng))) + .collect(); + let inputs_f2p54m1: Vec = (0..2048) + .map(|_| F2p54m1::from_canonical_u128_reduced(rand_u128(&mut rng))) + .collect(); + + let mut group = c.benchmark_group("field_mul_only"); + + group.bench_function("mul_chain_2048", |b| { + b.iter(|| { + let mut acc = F13::one(); + for x in inputs_f13.iter() { + acc = acc * *x; + } + black_box(acc) + }) + }); + + group.bench_function("mul_chain_16384", |b| { + b.iter(|| { + let mut acc = F13::one(); + for _ in 0..8 { + for x in inputs_f13.iter() { + acc = acc * *x; + } + } + black_box(acc) + }) + }); + + group.bench_function("mul_parallel_1024", |b| { + b.iter(|| { + let mut sum = F13::zero(); + for pair in inputs_f13.chunks_exact(2) { + sum = sum + pair[0] * pair[1]; + } + black_box(sum) + }) + }); + + group.bench_function("mul_chain_2048_special_m18m0", |b| { + b.iter(|| { + let mut acc = F2p18p1::one(); + for x in inputs_f2p18p1.iter() { + acc = acc * *x; + } + black_box(acc) + }) + }); + + group.bench_function("mul_chain_2048_special_m54p0", |b| { + b.iter(|| { + let mut acc = F2p54m1::one(); + for x in inputs_f2p54m1.iter() { + acc = acc * *x; + } + black_box(acc) + }) + }); + + group.finish(); +} + +fn bench_mul_isolated(c: &mut Criterion) { + use ark_ff::UniformRand; + + type F13 = Prime128M13M4P0; + + let mut rng = StdRng::seed_from_u64(0x5eed); + let a_fp128 = F13::from_canonical_u128_reduced(rand_u128(&mut rng)); + let b_fp128 = F13::from_canonical_u128_reduced(rand_u128(&mut rng)); + let a_bn254 = BN254Fr::rand(&mut rng); + let b_bn254 = BN254Fr::rand(&mut rng); + + let mut group = c.benchmark_group("field_mul_isolated"); + + group.bench_function("fp128_black_box_only", |b| b.iter(|| black_box(a_fp128))); + + group.bench_function("bn254_black_box_only", |b| b.iter(|| black_box(a_bn254))); + + group.bench_function("fp128_pair_passthrough", |b| { + b.iter(|| { + let x = black_box(a_fp128); + let y = black_box(b_fp128); + black_box((x, y)) + }) + }); + + group.bench_function("bn254_pair_passthrough", |b| { + b.iter(|| { + let x = black_box(a_bn254); + let y = black_box(b_bn254); + black_box((x, y)) + }) + }); + + group.bench_function("fp128_mul_single", |b| { + b.iter(|| { + let x = black_box(a_fp128); + let y = black_box(b_fp128); + black_box(x * y) + }) + }); + + group.bench_function("bn254_mul_single", |b| { + b.iter(|| { + let x = black_box(a_bn254); + let y = black_box(b_bn254); + black_box(x * y) + }) + }); + + let lanes_fp128: [(F13, F13); 8] = std::array::from_fn(|_| { + ( + F13::from_canonical_u128_reduced(rand_u128(&mut rng)), + F13::from_canonical_u128_reduced(rand_u128(&mut rng)), + ) + }); + let lanes_bn254: [(BN254Fr, BN254Fr); 8] = + std::array::from_fn(|_| (BN254Fr::rand(&mut rng), BN254Fr::rand(&mut rng))); + + group.bench_function("fp128_mul_8way_independent", |b| { + b.iter(|| { + let lanes = black_box(&lanes_fp128); + let p0 = lanes[0].0 * lanes[0].1; + let p1 = lanes[1].0 * lanes[1].1; + let p2 = lanes[2].0 * lanes[2].1; + let p3 = lanes[3].0 * lanes[3].1; + let p4 = lanes[4].0 * lanes[4].1; + let p5 = lanes[5].0 * lanes[5].1; + let p6 = lanes[6].0 * lanes[6].1; + let p7 = lanes[7].0 * lanes[7].1; + black_box([p0, p1, p2, p3, p4, p5, p6, p7]) + }) + }); + + group.bench_function("fp128_8way_passthrough", |b| { + b.iter(|| { + let lanes = black_box(&lanes_fp128); + let p0 = lanes[0].0; + let p1 = lanes[1].0; + let p2 = lanes[2].0; + let p3 = lanes[3].0; + let p4 = lanes[4].0; + let p5 = lanes[5].0; + let p6 = lanes[6].0; + let p7 = lanes[7].0; + black_box([p0, p1, p2, p3, p4, p5, p6, p7]) + }) + }); + + group.bench_function("bn254_mul_8way_independent", |b| { + b.iter(|| { + let lanes = black_box(&lanes_bn254); + let p0 = lanes[0].0 * lanes[0].1; + let p1 = lanes[1].0 * lanes[1].1; + let p2 = lanes[2].0 * lanes[2].1; + let p3 = lanes[3].0 * lanes[3].1; + let p4 = lanes[4].0 * lanes[4].1; + let p5 = lanes[5].0 * lanes[5].1; + let p6 = lanes[6].0 * lanes[6].1; + let p7 = lanes[7].0 * lanes[7].1; + black_box([p0, p1, p2, p3, p4, p5, p6, p7]) + }) + }); + + group.bench_function("bn254_8way_passthrough", |b| { + b.iter(|| { + let lanes = black_box(&lanes_bn254); + let p0 = lanes[0].0; + let p1 = lanes[1].0; + let p2 = lanes[2].0; + let p3 = lanes[3].0; + let p4 = lanes[4].0; + let p5 = lanes[5].0; + let p6 = lanes[6].0; + let p7 = lanes[7].0; + black_box([p0, p1, p2, p3, p4, p5, p6, p7]) + }) + }); + + group.finish(); +} + +fn bench_sqr(c: &mut Criterion) { + type F13 = Prime128M13M4P0; + + let mut rng = StdRng::seed_from_u64(0x5eed); + let start = F13::from_canonical_u128_reduced(rand_u128(&mut rng)); + + let mut group = c.benchmark_group("field_sqr"); + + group.bench_function("sqr_chain_2048", |b| { + b.iter(|| { + let mut acc = start; + for _ in 0..2048 { + acc = acc.square(); + } + black_box(acc) + }) + }); + + group.bench_function("mul_self_chain_2048", |b| { + b.iter(|| { + let mut acc = start; + for _ in 0..2048 { + acc = acc * acc; + } + black_box(acc) + }) + }); + + group.finish(); +} + +fn bench_inv(c: &mut Criterion) { + type F13 = Prime128M13M4P0; + + let mut rng = StdRng::seed_from_u64(0x1a2b_3c4d_5e6f_7788); + let inputs: Vec = (0..256) + .map(|_| F13::from_canonical_u128_reduced(rand_u128(&mut rng))) + .collect(); + + c.bench_function("fp128_inv_or_zero_prime128m13m4p0", |b| { + b.iter(|| { + let mut acc = F13::one(); + for x in inputs.iter() { + acc = acc * x.inv_or_zero(); + } + black_box(acc) + }) + }); +} + +fn bench_bn254(c: &mut Criterion) { + use ark_ff::UniformRand; + + let mut rng = StdRng::seed_from_u64(0x5eed); + let inputs: Vec = (0..2048).map(|_| BN254Fr::rand(&mut rng)).collect(); + + let mut group = c.benchmark_group("bn254_fr"); + + group.bench_function("mul_add_chain_2048", |b| { + b.iter(|| { + let mut acc = BN254Fr::ONE; + for x in inputs.iter() { + acc = acc * x + acc; + } + black_box(acc) + }) + }); + + group.bench_function("mul_chain_2048", |b| { + b.iter(|| { + let mut acc = BN254Fr::ONE; + for x in inputs.iter() { + acc *= x; + } + black_box(acc) + }) + }); + + group.bench_function("mul_chain_16384", |b| { + b.iter(|| { + let mut acc = BN254Fr::ONE; + for _ in 0..8 { + for x in inputs.iter() { + acc *= x; + } + } + black_box(acc) + }) + }); + + group.bench_function("mul_parallel_1024", |b| { + b.iter(|| { + let mut sum = BN254Fr::ZERO; + for pair in inputs.chunks_exact(2) { + sum += pair[0] * pair[1]; + } + black_box(sum) + }) + }); + + group.bench_function("sqr_chain_2048", |b| { + b.iter(|| { + let mut acc = inputs[0]; + for _ in 0..2048 { + acc.square_in_place(); + } + black_box(acc) + }) + }); + + group.bench_function("inv_256", |b| { + b.iter(|| { + let mut acc = BN254Fr::ONE; + for x in inputs[..256].iter() { + acc *= x.inverse().unwrap_or(BN254Fr::ZERO); + } + black_box(acc) + }) + }); + + group.finish(); +} + +fn bench_packed_fp128_backend(c: &mut Criterion) { + type F = Prime128M13M4P0; + type PF = ::Packing; + let packed_streams = env_usize("HACHI_BENCH_PACKED_STREAMS", 8); + let latency_iters = env_usize("HACHI_BENCH_LATENCY_ITERS", 4096); + let throughput_iters = env_usize("HACHI_BENCH_THROUGHPUT_ITERS", 256); + let stream_iters = env_usize("HACHI_BENCH_STREAM_ITERS", 2048); + let mix_iters = env_usize("HACHI_BENCH_MIX_ITERS", 256); + let mix_muls = env_usize("HACHI_BENCH_MIX_MULS", 3); + let mix_adds = env_usize("HACHI_BENCH_MIX_ADDS", 1); + let mix_subs = env_usize("HACHI_BENCH_MIX_SUBS", 1); + + assert!(packed_streams > 0, "HACHI_BENCH_PACKED_STREAMS must be > 0"); + assert!(latency_iters > 0, "HACHI_BENCH_LATENCY_ITERS must be > 0"); + assert!( + throughput_iters > 0, + "HACHI_BENCH_THROUGHPUT_ITERS must be > 0" + ); + assert!(stream_iters > 0, "HACHI_BENCH_STREAM_ITERS must be > 0"); + assert!(mix_iters > 0, "HACHI_BENCH_MIX_ITERS must be > 0"); + + let muls_per_stream = throughput_iters + 1; + let mix_ops = mix_muls + mix_adds + mix_subs; + assert!(mix_ops > 0, "at least one mix operation must be enabled"); + + let backend = if cfg!(all(target_arch = "aarch64", target_feature = "neon")) { + "aarch64_neon" + } else { + "scalar_fallback" + }; + let mut group = c.benchmark_group(format!("field_packed_backend/{backend}/w{}", PF::WIDTH)); + + let mut rng = StdRng::seed_from_u64(0xd00d_f00d_1122_3344); + let scalar_stream_len = PF::WIDTH * stream_iters; + let lhs: Vec = (0..scalar_stream_len) + .map(|_| F::from_canonical_u128_reduced(rand_u128(&mut rng))) + .collect(); + let rhs: Vec = (0..scalar_stream_len) + .map(|_| F::from_canonical_u128_reduced(rand_u128(&mut rng))) + .collect(); + + let packed_lhs: Vec = PF::pack_slice(&lhs); + let packed_rhs: Vec = PF::pack_slice(&rhs); + let scalar_latency_inputs: Vec = (0..latency_iters) + .map(|_| F::from_canonical_u128_reduced(rand_u128(&mut rng))) + .collect(); + let packed_latency_inputs: Vec = (0..latency_iters) + .map(|_| PF::from_fn(|_| F::from_canonical_u128_reduced(rand_u128(&mut rng)))) + .collect(); + + let scalar_streams = packed_streams * PF::WIDTH; + let scalar_lanes: Vec<(F, F)> = (0..scalar_streams) + .map(|_| { + ( + F::from_canonical_u128_reduced(rand_u128(&mut rng)), + F::from_canonical_u128_reduced(rand_u128(&mut rng)), + ) + }) + .collect(); + let packed_lanes: Vec<(PF, PF)> = (0..packed_streams) + .map(|_| { + ( + PF::from_fn(|_| F::from_canonical_u128_reduced(rand_u128(&mut rng))), + PF::from_fn(|_| F::from_canonical_u128_reduced(rand_u128(&mut rng))), + ) + }) + .collect(); + + group.throughput(Throughput::Elements(scalar_stream_len as u64)); + group.bench_function("scalar_add_stream", |b| { + let mut out = lhs.clone(); + b.iter(|| { + for (dst, src) in out.iter_mut().zip(rhs.iter()) { + *dst = *dst + *src; + } + black_box(out[0]) + }) + }); + + group.throughput(Throughput::Elements(scalar_stream_len as u64)); + group.bench_function("packed_add_stream", |b| { + let mut out = packed_lhs.clone(); + b.iter(|| { + for (dst, src) in out.iter_mut().zip(packed_rhs.iter()) { + *dst = *dst + *src; + } + black_box(out[0].extract(0)) + }) + }); + + group.throughput(Throughput::Elements(latency_iters as u64)); + group.bench_function("scalar_mul_latency_chain", |b| { + b.iter(|| { + let mut acc = F::one(); + for x in scalar_latency_inputs.iter() { + acc = acc * *x; + } + black_box(acc) + }) + }); + + group.throughput(Throughput::Elements((latency_iters * PF::WIDTH) as u64)); + group.bench_function("packed_mul_latency_chain", |b| { + b.iter(|| { + let mut acc = PF::broadcast(F::one()); + for x in packed_latency_inputs.iter() { + acc = acc * *x; + } + black_box(acc.extract(0)) + }) + }); + + group.throughput(Throughput::Elements( + (scalar_streams * muls_per_stream) as u64, + )); + group.bench_function("scalar_mul_throughput_8way", |b| { + b.iter(|| { + let lanes = black_box(&scalar_lanes); + let mut acc: Vec = lanes.iter().map(|(a, b)| *a * *b).collect(); + for _ in 0..throughput_iters { + for (acc_i, lane) in acc.iter_mut().zip(lanes.iter()) { + *acc_i = *acc_i * lane.0; + } + } + black_box(acc[0]) + }) + }); + + group.throughput(Throughput::Elements( + (packed_streams * muls_per_stream * PF::WIDTH) as u64, + )); + group.bench_function("packed_mul_throughput_8way", |b| { + b.iter(|| { + let lanes = black_box(&packed_lanes); + let mut acc: Vec = lanes.iter().map(|(a, b)| *a * *b).collect(); + for _ in 0..throughput_iters { + for (acc_i, lane) in acc.iter_mut().zip(lanes.iter()) { + *acc_i = *acc_i * lane.0; + } + } + black_box(acc[0].extract(0)) + }) + }); + + group.throughput(Throughput::Elements( + (scalar_streams * mix_iters * mix_ops) as u64, + )); + group.bench_function("scalar_mix_sumcheck_like", |b| { + b.iter(|| { + let lanes = black_box(&scalar_lanes); + let mut acc: Vec = lanes.iter().map(|(a, b)| *a + *b).collect(); + for _ in 0..mix_iters { + for (acc_i, lane) in acc.iter_mut().zip(lanes.iter()) { + let (x, y) = *lane; + for _ in 0..mix_muls { + *acc_i = *acc_i * x; + } + for _ in 0..mix_adds { + *acc_i = *acc_i + y; + } + for _ in 0..mix_subs { + *acc_i = *acc_i - x; + } + } + } + black_box(acc[0]) + }) + }); + + group.throughput(Throughput::Elements( + (packed_streams * PF::WIDTH * mix_iters * mix_ops) as u64, + )); + group.bench_function("packed_mix_sumcheck_like", |b| { + b.iter(|| { + let lanes = black_box(&packed_lanes); + let mut acc: Vec = lanes.iter().map(|(a, b)| *a + *b).collect(); + for _ in 0..mix_iters { + for (acc_i, lane) in acc.iter_mut().zip(lanes.iter()) { + let (x, y) = *lane; + for _ in 0..mix_muls { + *acc_i = *acc_i * x; + } + for _ in 0..mix_adds { + *acc_i = *acc_i + y; + } + for _ in 0..mix_subs { + *acc_i = *acc_i - x; + } + } + } + black_box(acc[0].extract(0)) + }) + }); + + group.finish(); +} + +fn bench_fp32_fp64_mul(c: &mut Criterion) { + let mut rng = StdRng::seed_from_u64(0x3264_3264); + let n = 2048; + + let inputs_24: Vec = + (0..n).map(|_| FieldSampling::sample(&mut rng)).collect(); + let inputs_30: Vec = + (0..n).map(|_| FieldSampling::sample(&mut rng)).collect(); + let inputs_31: Vec = + (0..n).map(|_| FieldSampling::sample(&mut rng)).collect(); + let inputs_32: Vec = + (0..n).map(|_| FieldSampling::sample(&mut rng)).collect(); + let inputs_40: Vec = + (0..n).map(|_| FieldSampling::sample(&mut rng)).collect(); + let inputs_64: Vec = + (0..n).map(|_| FieldSampling::sample(&mut rng)).collect(); + + let mut group = c.benchmark_group("fp32_fp64_mul"); + + macro_rules! chain_bench { + ($name:expr, $ty:ty, $inputs:expr) => { + group.bench_function(concat!($name, "_mul_chain_2048"), |b| { + b.iter(|| { + let mut acc = <$ty>::one(); + for x in $inputs.iter() { + acc = acc * *x; + } + black_box(acc) + }) + }); + group.bench_function(concat!($name, "_mul_add_chain_2048"), |b| { + b.iter(|| { + let mut acc = <$ty>::one(); + for x in $inputs.iter() { + acc = acc * *x + acc; + } + black_box(acc) + }) + }); + }; + } + + chain_bench!("fp32_2pow24m3", Pow2Offset24Field, inputs_24); + chain_bench!("fp32_2pow30m35", Pow2Offset30Field, inputs_30); + chain_bench!("fp32_2pow31m19", Pow2Offset31Field, inputs_31); + chain_bench!("fp32_2pow32m99", Pow2Offset32Field, inputs_32); + chain_bench!("fp64_2pow40m195", Pow2Offset40Field, inputs_40); + chain_bench!("fp64_2pow64m59", Pow2Offset64Field, inputs_64); + + group.finish(); +} + +fn bench_widening_ops(c: &mut Criterion) { + type F = Prime128M8M4M1M0; + + let mut rng = StdRng::seed_from_u64(0x01de_be0c_0001); + let a = F::from_canonical_u128_reduced(rand_u128(&mut rng)); + let b = F::from_canonical_u128_reduced(rand_u128(&mut rng)); + let b_u64 = rng.next_u64(); + + let mut group = c.benchmark_group("widening_ops"); + + group.bench_function("mul_wide_u64_only", |bench| { + bench.iter(|| black_box(black_box(a).mul_wide_u64(black_box(b_u64)))) + }); + + group.bench_function("mul_wide_only", |bench| { + bench.iter(|| black_box(black_box(a).mul_wide(black_box(b)))) + }); + + let limbs3 = [rng.next_u64(), rng.next_u64(), rng.next_u64()]; + let limbs4 = [ + rng.next_u64(), + rng.next_u64(), + rng.next_u64(), + rng.next_u64(), + ]; + + group.bench_function("mul_wide_limbs_3_to_5_only", |bench| { + bench.iter(|| black_box(black_box(a).mul_wide_limbs::<3, 5>(black_box(limbs3)))) + }); + group.bench_function("mul_wide_limbs_3_to_4_only", |bench| { + bench.iter(|| black_box(black_box(a).mul_wide_limbs::<3, 4>(black_box(limbs3)))) + }); + group.bench_function("mul_wide_limbs_4_to_5_only", |bench| { + bench.iter(|| black_box(black_box(a).mul_wide_limbs::<4, 5>(black_box(limbs4)))) + }); + group.bench_function("mul_wide_limbs_4_to_4_only", |bench| { + bench.iter(|| black_box(black_box(a).mul_wide_limbs::<4, 4>(black_box(limbs4)))) + }); + + group.bench_function("full_mul_u64_reduce", |bench| { + bench.iter(|| black_box(black_box(a) * F::from_u64(black_box(b_u64)))) + }); + + group.bench_function("full_mul_reduce", |bench| { + bench.iter(|| black_box(black_box(a) * black_box(b))) + }); + + let wide3 = a.mul_wide_u64(b_u64); + let wide4 = a.mul_wide(b); + let wide5 = { + let mut l = [0u64; 5]; + l[..3].copy_from_slice(&wide3); + l[4] = rng.next_u64() & 0xFF; + l + }; + + group.bench_function("solinas_reduce_3_limbs", |bench| { + bench.iter(|| black_box(F::solinas_reduce(black_box(&wide3)))) + }); + + group.bench_function("solinas_reduce_4_limbs", |bench| { + bench.iter(|| black_box(F::solinas_reduce(black_box(&wide4)))) + }); + + group.bench_function("solinas_reduce_5_limbs", |bench| { + bench.iter(|| black_box(F::solinas_reduce(black_box(&wide5)))) + }); + + group.bench_function("mul_wide_u64_roundtrip", |bench| { + bench.iter(|| { + let x = black_box(a); + let y = black_box(b_u64); + black_box(F::solinas_reduce(&x.mul_wide_u64(y))) + }) + }); + + group.bench_function("mul_wide_roundtrip", |bench| { + bench.iter(|| { + let x = black_box(a); + let y = black_box(b); + black_box(F::solinas_reduce(&x.mul_wide(y))) + }) + }); + + group.bench_function("mul_wide_limbs_3_to_5_roundtrip", |bench| { + bench.iter(|| { + let x = black_box(a); + let m = black_box(limbs3); + black_box(F::solinas_reduce(&x.mul_wide_limbs::<3, 5>(m))) + }) + }); + group.bench_function("mul_wide_limbs_3_to_4_roundtrip", |bench| { + bench.iter(|| { + let x = black_box(a); + let m = black_box(limbs3); + black_box(F::solinas_reduce(&x.mul_wide_limbs::<3, 4>(m))) + }) + }); + group.bench_function("mul_wide_limbs_4_to_5_roundtrip", |bench| { + bench.iter(|| { + let x = black_box(a); + let m = black_box(limbs4); + black_box(F::solinas_reduce(&x.mul_wide_limbs::<4, 5>(m))) + }) + }); + group.bench_function("mul_wide_limbs_4_to_4_roundtrip", |bench| { + bench.iter(|| { + let x = black_box(a); + let m = black_box(limbs4); + black_box(F::solinas_reduce(&x.mul_wide_limbs::<4, 4>(m))) + }) + }); + + group.finish(); +} + +fn bench_accumulator_pattern(c: &mut Criterion) { + type F = Prime128M8M4M1M0; + + let mut rng = StdRng::seed_from_u64(0xacc0_1a70_0002); + let inputs_a: Vec = (0..256) + .map(|_| F::from_canonical_u128_reduced(rand_u128(&mut rng))) + .collect(); + let inputs_b_u64: Vec = (0..256).map(|_| rng.next_u64()).collect(); + let inputs_b_f: Vec = (0..256) + .map(|_| F::from_canonical_u128_reduced(rand_u128(&mut rng))) + .collect(); + + let mut group = c.benchmark_group("accumulator_pattern"); + + for &n in &[16, 64, 256] { + group.bench_function(format!("eager_mul_u64_{n}"), |bench| { + bench.iter(|| { + let a_s = black_box(&inputs_a[..n]); + let b_s = black_box(&inputs_b_u64[..n]); + let mut acc = F::zero(); + for i in 0..n { + acc = acc + a_s[i] * F::from_u64(b_s[i]); + } + black_box(acc) + }) + }); + + group.bench_function(format!("widening_accum_u64_{n}"), |bench| { + bench.iter(|| { + let a_s = black_box(&inputs_a[..n]); + let b_s = black_box(&inputs_b_u64[..n]); + let mut acc = [0u64; 5]; + for i in 0..n { + let wide = a_s[i].mul_wide_u64(b_s[i]); + let mut carry: u64 = 0; + for j in 0..3 { + let sum = acc[j] as u128 + wide[j] as u128 + carry as u128; + acc[j] = sum as u64; + carry = (sum >> 64) as u64; + } + for item in &mut acc[3..5] { + let sum = *item as u128 + carry as u128; + *item = sum as u64; + carry = (sum >> 64) as u64; + } + } + black_box(F::solinas_reduce(&acc)) + }) + }); + + group.bench_function(format!("eager_mul_full_{n}"), |bench| { + bench.iter(|| { + let a_s = black_box(&inputs_a[..n]); + let b_s = black_box(&inputs_b_f[..n]); + let mut acc = F::zero(); + for i in 0..n { + acc = acc + a_s[i] * b_s[i]; + } + black_box(acc) + }) + }); + + group.bench_function(format!("widening_accum_full_{n}"), |bench| { + bench.iter(|| { + let a_s = black_box(&inputs_a[..n]); + let b_s = black_box(&inputs_b_f[..n]); + let mut acc = [0u64; 6]; + for i in 0..n { + let wide = a_s[i].mul_wide(b_s[i]); + let mut carry: u64 = 0; + for j in 0..4 { + let sum = acc[j] as u128 + wide[j] as u128 + carry as u128; + acc[j] = sum as u64; + carry = (sum >> 64) as u64; + } + for item in &mut acc[4..6] { + let sum = *item as u128 + carry as u128; + *item = sum as u64; + carry = (sum >> 64) as u64; + } + } + black_box(F::solinas_reduce(&acc)) + }) + }); + } + + group.finish(); +} + +fn bench_throughput(c: &mut Criterion) { + let n = 4096u64; + let mut rng = StdRng::seed_from_u64(0xdead_cafe); + + type M31 = Fp32<{ (1u32 << 31) - 1 }>; + + let a24: Vec = (0..n).map(|_| FieldSampling::sample(&mut rng)).collect(); + let b24: Vec = (0..n).map(|_| FieldSampling::sample(&mut rng)).collect(); + let a30: Vec = (0..n).map(|_| FieldSampling::sample(&mut rng)).collect(); + let b30: Vec = (0..n).map(|_| FieldSampling::sample(&mut rng)).collect(); + let a31: Vec = (0..n).map(|_| FieldSampling::sample(&mut rng)).collect(); + let b31: Vec = (0..n).map(|_| FieldSampling::sample(&mut rng)).collect(); + let am31: Vec = (0..n).map(|_| FieldSampling::sample(&mut rng)).collect(); + let bm31: Vec = (0..n).map(|_| FieldSampling::sample(&mut rng)).collect(); + let a32: Vec = (0..n).map(|_| FieldSampling::sample(&mut rng)).collect(); + let b32: Vec = (0..n).map(|_| FieldSampling::sample(&mut rng)).collect(); + let a40: Vec = (0..n).map(|_| FieldSampling::sample(&mut rng)).collect(); + let b40: Vec = (0..n).map(|_| FieldSampling::sample(&mut rng)).collect(); + let a48: Vec = (0..n).map(|_| FieldSampling::sample(&mut rng)).collect(); + let b48: Vec = (0..n).map(|_| FieldSampling::sample(&mut rng)).collect(); + let a56: Vec = (0..n).map(|_| FieldSampling::sample(&mut rng)).collect(); + let b56: Vec = (0..n).map(|_| FieldSampling::sample(&mut rng)).collect(); + let a64: Vec = (0..n).map(|_| FieldSampling::sample(&mut rng)).collect(); + let b64: Vec = (0..n).map(|_| FieldSampling::sample(&mut rng)).collect(); + let a128: Vec = (0..n) + .map(|_| Prime128M8M4M1M0::from_canonical_u128_reduced(rand_u128(&mut rng))) + .collect(); + let b128: Vec = (0..n) + .map(|_| Prime128M8M4M1M0::from_canonical_u128_reduced(rand_u128(&mut rng))) + .collect(); + + let mut out24 = vec![Pow2Offset24Field::zero(); n as usize]; + let mut out30 = vec![Pow2Offset30Field::zero(); n as usize]; + let mut out31 = vec![Pow2Offset31Field::zero(); n as usize]; + let mut outm31 = vec![M31::zero(); n as usize]; + let mut out32 = vec![Pow2Offset32Field::zero(); n as usize]; + let mut out40 = vec![Pow2Offset40Field::zero(); n as usize]; + let mut out48 = vec![Pow2Offset48Field::zero(); n as usize]; + let mut out56 = vec![Pow2Offset56Field::zero(); n as usize]; + let mut out64 = vec![Pow2Offset64Field::zero(); n as usize]; + let mut out128 = vec![Prime128M8M4M1M0::zero(); n as usize]; + + let mut group = c.benchmark_group("throughput"); + group.throughput(Throughput::Elements(n)); + + macro_rules! bench_op { + ($name:expr, $a:expr, $b:expr, $out:expr, $op:tt) => { + group.bench_function($name, |bench| { + bench.iter(|| { + let a = black_box(&$a); + let b = black_box(&$b); + let out = &mut $out; + for i in 0..n as usize { + out[i] = a[i] $op b[i]; + } + }) + }); + }; + } + + bench_op!("fp32_24b_mul", a24, b24, out24, *); + bench_op!("fp32_24b_add", a24, b24, out24, +); + bench_op!("fp32_30b_mul", a30, b30, out30, *); + bench_op!("fp32_30b_add", a30, b30, out30, +); + bench_op!("fp32_31b_mul", a31, b31, out31, *); + bench_op!("fp32_31b_add", a31, b31, out31, +); + bench_op!("fp32_m31_mul", am31, bm31, outm31, *); + bench_op!("fp32_m31_add", am31, bm31, outm31, +); + bench_op!("fp32_32b_mul", a32, b32, out32, *); + bench_op!("fp32_32b_add", a32, b32, out32, +); + bench_op!("fp64_40b_mul", a40, b40, out40, *); + bench_op!("fp64_40b_add", a40, b40, out40, +); + bench_op!("fp64_48b_mul", a48, b48, out48, *); + bench_op!("fp64_48b_add", a48, b48, out48, +); + bench_op!("fp64_56b_mul", a56, b56, out56, *); + bench_op!("fp64_56b_add", a56, b56, out56, +); + bench_op!("fp64_64b_mul", a64, b64, out64, *); + bench_op!("fp64_64b_add", a64, b64, out64, +); + bench_op!("fp128_mul", a128, b128, out128, *); + bench_op!("fp128_add", a128, b128, out128, +); + + group.finish(); +} + +fn bench_packed_throughput(c: &mut Criterion) { + use hachi_pcs::algebra::{Fp128Packing, Fp32Packing, Fp64Packing}; + + let n = 4096u64; + let mut rng = StdRng::seed_from_u64(0xbeef_cafe); + + macro_rules! packed_bench { + ($group:expr, $label:expr, $field:ty, $packing:ty, $rng:expr, $n:expr) => {{ + let lhs: Vec<$field> = (0..$n).map(|_| FieldSampling::sample($rng)).collect(); + let rhs: Vec<$field> = (0..$n).map(|_| FieldSampling::sample($rng)).collect(); + let lhs_p = <$packing>::pack_slice(&lhs); + let rhs_p = <$packing>::pack_slice(&rhs); + let mut out_p = vec![<$packing>::broadcast(<$field>::zero()); lhs_p.len()]; + + $group.bench_function(concat!($label, "_packed_mul"), |b| { + b.iter(|| { + let a = black_box(&lhs_p); + let b_v = black_box(&rhs_p); + let out = &mut out_p; + for i in 0..out.len() { + out[i] = a[i] * b_v[i]; + } + }) + }); + $group.bench_function(concat!($label, "_packed_add"), |b| { + b.iter(|| { + let a = black_box(&lhs_p); + let b_v = black_box(&rhs_p); + let out = &mut out_p; + for i in 0..out.len() { + out[i] = a[i] + b_v[i]; + } + }) + }); + $group.bench_function(concat!($label, "_packed_sub"), |b| { + b.iter(|| { + let a = black_box(&lhs_p); + let b_v = black_box(&rhs_p); + let out = &mut out_p; + for i in 0..out.len() { + out[i] = a[i] - b_v[i]; + } + }) + }); + }}; + } + + let mut group = c.benchmark_group("packed_throughput"); + group.throughput(Throughput::Elements(n)); + + use hachi_pcs::algebra::fields::pseudo_mersenne::*; + type M31 = Fp32<{ (1u32 << 31) - 1 }>; + + type P24 = Fp32Packing<{ POW2_OFFSET_MODULUS_24 }>; + type P30 = Fp32Packing<{ POW2_OFFSET_MODULUS_30 }>; + type P31 = Fp32Packing<{ POW2_OFFSET_MODULUS_31 }>; + type PM31 = Fp32Packing<{ (1u32 << 31) - 1 }>; + type P32 = Fp32Packing<{ POW2_OFFSET_MODULUS_32 }>; + type P40 = Fp64Packing<{ POW2_OFFSET_MODULUS_40 }>; + type P48 = Fp64Packing<{ POW2_OFFSET_MODULUS_48 }>; + type P56 = Fp64Packing<{ POW2_OFFSET_MODULUS_56 }>; + type P64 = Fp64Packing<{ POW2_OFFSET_MODULUS_64 }>; + type P128 = Fp128Packing<{ POW2_OFFSET_MODULUS_128 }>; + + packed_bench!(group, "fp32_24b", Pow2Offset24Field, P24, &mut rng, n); + packed_bench!(group, "fp32_30b", Pow2Offset30Field, P30, &mut rng, n); + packed_bench!(group, "fp32_31b", Pow2Offset31Field, P31, &mut rng, n); + packed_bench!(group, "fp32_m31", M31, PM31, &mut rng, n); + packed_bench!(group, "fp32_32b", Pow2Offset32Field, P32, &mut rng, n); + packed_bench!(group, "fp64_40b", Pow2Offset40Field, P40, &mut rng, n); + packed_bench!(group, "fp64_48b", Pow2Offset48Field, P48, &mut rng, n); + packed_bench!(group, "fp64_56b", Pow2Offset56Field, P56, &mut rng, n); + packed_bench!(group, "fp64_64b", Pow2Offset64Field, P64, &mut rng, n); + packed_bench!(group, "fp128", Prime128M8M4M1M0, P128, &mut rng, n); + + group.finish(); +} + +#[cfg(feature = "parallel")] +fn bench_parallel_throughput(c: &mut Criterion) { + use hachi_pcs::algebra::{Fp32Packing, Fp64Packing}; + + let profile = env::var("HACHI_BENCH_PAR_PROFILE").unwrap_or_else(|_| "dev".to_string()); + let default_n = match profile.as_str() { + "scale" | "large" => 1 << 20, + "xlarge" => 1 << 22, + _ => 1 << 15, + }; + let n = env_usize("HACHI_BENCH_PAR_N", default_n); + let default_chunk = match profile.as_str() { + "scale" | "large" => 1 << 14, + "xlarge" => 1 << 15, + _ => 1 << 12, + }; + let chunk = env_usize("HACHI_BENCH_PAR_CHUNK", default_chunk); + let threads = env_usize( + "HACHI_BENCH_PAR_THREADS", + thread::available_parallelism() + .map(|v| v.get()) + .unwrap_or(1), + ); + + assert!(threads > 0, "HACHI_BENCH_PAR_THREADS must be > 0"); + assert!(n > 0, "HACHI_BENCH_PAR_N must be > 0"); + assert!(chunk > 0, "HACHI_BENCH_PAR_CHUNK must be > 0"); + assert!(n % 4 == 0, "HACHI_BENCH_PAR_N must be divisible by 4"); + + let pool = ThreadPoolBuilder::new() + .num_threads(threads) + .build() + .expect("failed to build rayon pool"); + + let mut rng = StdRng::seed_from_u64(0xfeed_face); + + let lhs31: Vec = (0..n).map(|_| FieldSampling::sample(&mut rng)).collect(); + let rhs31: Vec = (0..n).map(|_| FieldSampling::sample(&mut rng)).collect(); + let lhs64: Vec = (0..n).map(|_| FieldSampling::sample(&mut rng)).collect(); + let rhs64: Vec = (0..n).map(|_| FieldSampling::sample(&mut rng)).collect(); + let lhs128: Vec = (0..n) + .map(|_| Prime128M13M4P0::from_canonical_u128_reduced(rand_u128(&mut rng))) + .collect(); + let rhs128: Vec = (0..n) + .map(|_| Prime128M13M4P0::from_canonical_u128_reduced(rand_u128(&mut rng))) + .collect(); + + type P31 = Fp32Packing<{ hachi_pcs::algebra::fields::pseudo_mersenne::POW2_OFFSET_MODULUS_31 }>; + type P64 = Fp64Packing<{ hachi_pcs::algebra::fields::pseudo_mersenne::POW2_OFFSET_MODULUS_64 }>; + type F128 = Prime128M13M4P0; + type P128 = ::Packing; + let chunk31_p = (chunk / P31::WIDTH).max(1); + let chunk64_p = (chunk / P64::WIDTH).max(1); + let chunk128_p = (chunk / P128::WIDTH).max(1); + + let lhs31_p = P31::pack_slice(&lhs31); + let rhs31_p = P31::pack_slice(&rhs31); + let lhs64_p = P64::pack_slice(&lhs64); + let rhs64_p = P64::pack_slice(&rhs64); + let lhs128_p = P128::pack_slice(&lhs128); + let rhs128_p = P128::pack_slice(&rhs128); + + let mut out31 = vec![Pow2Offset31Field::zero(); n]; + let mut out64 = vec![Pow2Offset64Field::zero(); n]; + let mut out128 = vec![F128::zero(); n]; + let mut out31_p = vec![P31::broadcast(Pow2Offset31Field::zero()); lhs31_p.len()]; + let mut out64_p = vec![P64::broadcast(Pow2Offset64Field::zero()); lhs64_p.len()]; + let mut out128_p = vec![P128::broadcast(F128::zero()); lhs128_p.len()]; + + let mut group = c.benchmark_group(format!( + "parallel_throughput/{profile}/t{threads}/n{n}/c{chunk}" + )); + group.throughput(Throughput::Elements(n as u64)); + + group.bench_function("fp32_31b_mul_seq", |b| { + b.iter(|| { + let a = black_box(&lhs31); + let b_v = black_box(&rhs31); + let out = &mut out31; + for i in 0..out.len() { + out[i] = a[i] * b_v[i]; + } + black_box(out[0]) + }) + }); + + group.bench_function("fp32_31b_mul_par_zip", |b| { + b.iter(|| { + let a = black_box(&lhs31); + let b_v = black_box(&rhs31); + let out = &mut out31; + pool.install(|| { + out.par_iter_mut() + .zip(a.par_iter()) + .zip(b_v.par_iter()) + .for_each(|((dst, lhs), rhs)| *dst = *lhs * *rhs); + }); + black_box(out[0]) + }) + }); + + group.bench_function("fp32_31b_mul_par_chunked", |b| { + b.iter(|| { + let a = black_box(&lhs31); + let b_v = black_box(&rhs31); + let out = &mut out31; + pool.install(|| { + out.par_chunks_mut(chunk) + .zip(a.par_chunks(chunk)) + .zip(b_v.par_chunks(chunk)) + .for_each(|((dst, lhs), rhs)| { + for i in 0..dst.len() { + dst[i] = lhs[i] * rhs[i]; + } + }); + }); + black_box(out[0]) + }) + }); + + group.bench_function("fp32_31b_packed_mul_seq", |b| { + b.iter(|| { + let a = black_box(&lhs31_p); + let b_v = black_box(&rhs31_p); + let out = &mut out31_p; + for i in 0..out.len() { + out[i] = a[i] * b_v[i]; + } + black_box(out[0].extract(0)) + }) + }); + + group.bench_function("fp32_31b_packed_mul_par_zip", |b| { + b.iter(|| { + let a = black_box(&lhs31_p); + let b_v = black_box(&rhs31_p); + let out = &mut out31_p; + pool.install(|| { + out.par_iter_mut() + .zip(a.par_iter()) + .zip(b_v.par_iter()) + .for_each(|((dst, lhs), rhs)| *dst = *lhs * *rhs); + }); + black_box(out[0].extract(0)) + }) + }); + + group.bench_function("fp32_31b_packed_mul_par_chunked", |b| { + b.iter(|| { + let a = black_box(&lhs31_p); + let b_v = black_box(&rhs31_p); + let out = &mut out31_p; + pool.install(|| { + out.par_chunks_mut(chunk31_p) + .zip(a.par_chunks(chunk31_p)) + .zip(b_v.par_chunks(chunk31_p)) + .for_each(|((dst, lhs), rhs)| { + for i in 0..dst.len() { + dst[i] = lhs[i] * rhs[i]; + } + }); + }); + black_box(out[0].extract(0)) + }) + }); + + group.bench_function("fp64_64b_mul_seq", |b| { + b.iter(|| { + let a = black_box(&lhs64); + let b_v = black_box(&rhs64); + let out = &mut out64; + for i in 0..out.len() { + out[i] = a[i] * b_v[i]; + } + black_box(out[0]) + }) + }); + + group.bench_function("fp64_64b_mul_par_zip", |b| { + b.iter(|| { + let a = black_box(&lhs64); + let b_v = black_box(&rhs64); + let out = &mut out64; + pool.install(|| { + out.par_iter_mut() + .zip(a.par_iter()) + .zip(b_v.par_iter()) + .for_each(|((dst, lhs), rhs)| *dst = *lhs * *rhs); + }); + black_box(out[0]) + }) + }); + + group.bench_function("fp64_64b_mul_par_chunked", |b| { + b.iter(|| { + let a = black_box(&lhs64); + let b_v = black_box(&rhs64); + let out = &mut out64; + pool.install(|| { + out.par_chunks_mut(chunk) + .zip(a.par_chunks(chunk)) + .zip(b_v.par_chunks(chunk)) + .for_each(|((dst, lhs), rhs)| { + for i in 0..dst.len() { + dst[i] = lhs[i] * rhs[i]; + } + }); + }); + black_box(out[0]) + }) + }); + + group.bench_function("fp64_64b_packed_mul_seq", |b| { + b.iter(|| { + let a = black_box(&lhs64_p); + let b_v = black_box(&rhs64_p); + let out = &mut out64_p; + for i in 0..out.len() { + out[i] = a[i] * b_v[i]; + } + black_box(out[0].extract(0)) + }) + }); + + group.bench_function("fp64_64b_packed_mul_par_zip", |b| { + b.iter(|| { + let a = black_box(&lhs64_p); + let b_v = black_box(&rhs64_p); + let out = &mut out64_p; + pool.install(|| { + out.par_iter_mut() + .zip(a.par_iter()) + .zip(b_v.par_iter()) + .for_each(|((dst, lhs), rhs)| *dst = *lhs * *rhs); + }); + black_box(out[0].extract(0)) + }) + }); + + group.bench_function("fp64_64b_packed_mul_par_chunked", |b| { + b.iter(|| { + let a = black_box(&lhs64_p); + let b_v = black_box(&rhs64_p); + let out = &mut out64_p; + pool.install(|| { + out.par_chunks_mut(chunk64_p) + .zip(a.par_chunks(chunk64_p)) + .zip(b_v.par_chunks(chunk64_p)) + .for_each(|((dst, lhs), rhs)| { + for i in 0..dst.len() { + dst[i] = lhs[i] * rhs[i]; + } + }); + }); + black_box(out[0].extract(0)) + }) + }); + + group.bench_function("fp128_mul_seq", |b| { + b.iter(|| { + let a = black_box(&lhs128); + let b_v = black_box(&rhs128); + let out = &mut out128; + for i in 0..out.len() { + out[i] = a[i] * b_v[i]; + } + black_box(out[0]) + }) + }); + + group.bench_function("fp128_mul_par_chunked", |b| { + b.iter(|| { + let a = black_box(&lhs128); + let b_v = black_box(&rhs128); + let out = &mut out128; + pool.install(|| { + out.par_chunks_mut(chunk) + .zip(a.par_chunks(chunk)) + .zip(b_v.par_chunks(chunk)) + .for_each(|((dst, lhs), rhs)| { + for i in 0..dst.len() { + dst[i] = lhs[i] * rhs[i]; + } + }); + }); + black_box(out[0]) + }) + }); + + group.bench_function("fp128_packed_mul_seq", |b| { + b.iter(|| { + let a = black_box(&lhs128_p); + let b_v = black_box(&rhs128_p); + let out = &mut out128_p; + for i in 0..out.len() { + out[i] = a[i] * b_v[i]; + } + black_box(out[0].extract(0)) + }) + }); + + group.bench_function("fp128_packed_mul_par_chunked", |b| { + b.iter(|| { + let a = black_box(&lhs128_p); + let b_v = black_box(&rhs128_p); + let out = &mut out128_p; + pool.install(|| { + out.par_chunks_mut(chunk128_p) + .zip(a.par_chunks(chunk128_p)) + .zip(b_v.par_chunks(chunk128_p)) + .for_each(|((dst, lhs), rhs)| { + for i in 0..dst.len() { + dst[i] = lhs[i] * rhs[i]; + } + }); + }); + black_box(out[0].extract(0)) + }) + }); + + group.finish(); +} + +#[cfg(not(feature = "parallel"))] +fn bench_parallel_throughput(_: &mut Criterion) {} + +fn bench_packed_sumcheck_mix(c: &mut Criterion) { + use hachi_pcs::algebra::{Fp128Packing, Fp32Packing, Fp64Packing}; + + let n = 4096u64; + let mut rng = StdRng::seed_from_u64(0xface_bead); + + macro_rules! sumcheck_bench { + ($group:expr, $label:expr, $field:ty, $packing:ty, $rng:expr, $n:expr) => {{ + let eq: Vec<$field> = (0..$n).map(|_| FieldSampling::sample($rng)).collect(); + let poly: Vec<$field> = (0..$n).map(|_| FieldSampling::sample($rng)).collect(); + let eq_p = <$packing>::pack_slice(&eq); + let poly_p = <$packing>::pack_slice(&poly); + let mut acc = <$packing>::broadcast(<$field>::zero()); + + $group.bench_function(concat!($label, "_packed_macc"), |b| { + b.iter(|| { + let e = black_box(&eq_p); + let p_v = black_box(&poly_p); + acc = <$packing>::broadcast(<$field>::zero()); + for i in 0..e.len() { + acc = acc + e[i] * p_v[i]; + } + black_box(acc) + }) + }); + }}; + } + + let mut group = c.benchmark_group("packed_sumcheck_mix"); + group.throughput(Throughput::Elements(n)); + + use hachi_pcs::algebra::fields::pseudo_mersenne::*; + type M31 = Fp32<{ (1u32 << 31) - 1 }>; + + type P24 = Fp32Packing<{ POW2_OFFSET_MODULUS_24 }>; + type P30 = Fp32Packing<{ POW2_OFFSET_MODULUS_30 }>; + type P31 = Fp32Packing<{ POW2_OFFSET_MODULUS_31 }>; + type PM31 = Fp32Packing<{ (1u32 << 31) - 1 }>; + type P32 = Fp32Packing<{ POW2_OFFSET_MODULUS_32 }>; + type P40 = Fp64Packing<{ POW2_OFFSET_MODULUS_40 }>; + type P48 = Fp64Packing<{ POW2_OFFSET_MODULUS_48 }>; + type P56 = Fp64Packing<{ POW2_OFFSET_MODULUS_56 }>; + type P64 = Fp64Packing<{ POW2_OFFSET_MODULUS_64 }>; + type P128 = Fp128Packing<{ POW2_OFFSET_MODULUS_128 }>; + + sumcheck_bench!(group, "fp32_24b", Pow2Offset24Field, P24, &mut rng, n); + sumcheck_bench!(group, "fp32_30b", Pow2Offset30Field, P30, &mut rng, n); + sumcheck_bench!(group, "fp32_31b", Pow2Offset31Field, P31, &mut rng, n); + sumcheck_bench!(group, "fp32_m31", M31, PM31, &mut rng, n); + sumcheck_bench!(group, "fp32_32b", Pow2Offset32Field, P32, &mut rng, n); + sumcheck_bench!(group, "fp64_40b", Pow2Offset40Field, P40, &mut rng, n); + sumcheck_bench!(group, "fp64_48b", Pow2Offset48Field, P48, &mut rng, n); + sumcheck_bench!(group, "fp64_56b", Pow2Offset56Field, P56, &mut rng, n); + sumcheck_bench!(group, "fp64_64b", Pow2Offset64Field, P64, &mut rng, n); + sumcheck_bench!(group, "fp128", Prime128M8M4M1M0, P128, &mut rng, n); + + group.finish(); +} + +criterion_group!( + field_arith, + bench_mul, + bench_mul_only, + bench_mul_isolated, + bench_sqr, + bench_inv, + bench_packed_fp128_backend, + bench_bn254, + bench_fp32_fp64_mul, + bench_widening_ops, + bench_accumulator_pattern, + bench_throughput, + bench_packed_throughput, + bench_packed_sumcheck_mix, + bench_parallel_throughput +); +criterion_main!(field_arith); diff --git a/benches/fp64_reduce_probe.rs b/benches/fp64_reduce_probe.rs new file mode 100644 index 00000000..073544a9 --- /dev/null +++ b/benches/fp64_reduce_probe.rs @@ -0,0 +1,118 @@ +#![allow(missing_docs)] + +use criterion::{black_box, criterion_group, criterion_main, Criterion, Throughput}; + +const P40: u64 = hachi_pcs::algebra::fields::pseudo_mersenne::POW2_OFFSET_MODULUS_40; +const P64: u64 = hachi_pcs::algebra::fields::pseudo_mersenne::POW2_OFFSET_MODULUS_64; +const C40: u64 = (1u64 << 40) - P40; // 195 +const C64: u64 = 0u64.wrapping_sub(P64); // 59 +const MASK40: u64 = (1u64 << 40) - 1; +const MASK64_U128: u128 = u64::MAX as u128; + +#[inline(always)] +fn mul_c40_split(x: u64) -> u64 { + let c = C40 as u32; + let x_lo = x as u32; + let x_hi = (x >> 32) as u32; + (c as u64 * x_lo as u64).wrapping_add((c as u64 * x_hi as u64) << 32) +} + +#[inline(always)] +fn reduce40_split(lo: u64, hi: u64) -> u64 { + let high = (lo >> 40) | (hi << 24); + let f1 = (lo & MASK40).wrapping_add(mul_c40_split(high)); + let f2 = (f1 & MASK40).wrapping_add(mul_c40_split(f1 >> 40)); + let reduced = f2.wrapping_sub(P40); + let borrow = reduced >> 63; + reduced.wrapping_add(borrow.wrapping_neg() & P40) +} + +#[inline(always)] +fn reduce40_direct(lo: u64, hi: u64) -> u64 { + let high = (lo >> 40) | (hi << 24); + let f1 = (lo & MASK40).wrapping_add(C40.wrapping_mul(high)); + let f2 = (f1 & MASK40).wrapping_add(C40.wrapping_mul(f1 >> 40)); + let reduced = f2.wrapping_sub(P40); + let borrow = reduced >> 63; + reduced.wrapping_add(borrow.wrapping_neg() & P40) +} + +#[inline(always)] +fn reduce64(lo: u64, hi: u64) -> u64 { + let f1 = (lo as u128) + (C64 as u128) * (hi as u128); + let f2 = (f1 & MASK64_U128) + (C64 as u128) * ((f1 >> 64) as u64 as u128); + let reduced = f2.wrapping_sub(P64 as u128); + let borrow = reduced >> 127; + reduced.wrapping_add(borrow.wrapping_neg() & (P64 as u128)) as u64 +} + +#[inline(always)] +fn next_u64(state: &mut u64) -> u64 { + let mut x = *state; + x ^= x << 13; + x ^= x >> 7; + x ^= x << 17; + *state = x; + x +} + +fn bench_fp64_reduce_probe(c: &mut Criterion) { + let n = 1 << 13; + let mut seed = 0x9e37_79b9_7f4a_7c15u64; + + let mut pairs40 = Vec::with_capacity(n); + let mut pairs64 = Vec::with_capacity(n); + for _ in 0..n { + let a40 = next_u64(&mut seed) % P40; + let b40 = next_u64(&mut seed) % P40; + let x40 = (a40 as u128) * (b40 as u128); + pairs40.push((x40 as u64, (x40 >> 64) as u64)); + + let a64 = next_u64(&mut seed); + let b64 = next_u64(&mut seed); + let x64 = (a64 as u128) * (b64 as u128); + pairs64.push((x64 as u64, (x64 >> 64) as u64)); + } + + for &(lo, hi) in &pairs40 { + assert_eq!(reduce40_split(lo, hi), reduce40_direct(lo, hi)); + } + + let mut group = c.benchmark_group("fp64_reduce_probe"); + group.throughput(Throughput::Elements(n as u64)); + + group.bench_function("reduce40_split", |b| { + b.iter(|| { + let mut acc = 0u64; + for &(lo, hi) in black_box(&pairs40) { + acc ^= reduce40_split(lo, hi); + } + black_box(acc) + }) + }); + + group.bench_function("reduce40_direct", |b| { + b.iter(|| { + let mut acc = 0u64; + for &(lo, hi) in black_box(&pairs40) { + acc ^= reduce40_direct(lo, hi); + } + black_box(acc) + }) + }); + + group.bench_function("reduce64", |b| { + b.iter(|| { + let mut acc = 0u64; + for &(lo, hi) in black_box(&pairs64) { + acc ^= reduce64(lo, hi); + } + black_box(acc) + }) + }); + + group.finish(); +} + +criterion_group!(fp64_reduce_probe, bench_fp64_reduce_probe); +criterion_main!(fp64_reduce_probe); diff --git a/benches/ring_ntt.rs b/benches/ring_ntt.rs new file mode 100644 index 00000000..546f9b9d --- /dev/null +++ b/benches/ring_ntt.rs @@ -0,0 +1,67 @@ +#![allow(missing_docs)] + +use criterion::{black_box, criterion_group, criterion_main, Criterion}; +use hachi_pcs::algebra::ntt::butterfly::{forward_ntt, inverse_ntt, NttTwiddles}; +use hachi_pcs::algebra::tables::{Q32_DATA, Q32_MODULUS, Q32_NUM_PRIMES, Q32_PRIMES}; +use hachi_pcs::algebra::{CyclotomicCrtNtt, CyclotomicRing, Fp64, MontCoeff}; +use hachi_pcs::CanonicalField; + +type F = Fp64<{ Q32_MODULUS }>; +type R = CyclotomicRing; +type N = CyclotomicCrtNtt; + +fn sample_ring(seed: u64) -> R { + let coeffs = std::array::from_fn(|i| { + let x = seed + .wrapping_mul(31) + .wrapping_add((i as u64).wrapping_mul(17)); + F::from_u64(x % Q32_MODULUS) + }); + R::from_coefficients(coeffs) +} + +fn bench_ring_schoolbook_mul(c: &mut Criterion) { + let lhs = sample_ring(3); + let rhs = sample_ring(11); + c.bench_function("ring_schoolbook_mul_d64", |b| { + b.iter(|| black_box(lhs) * black_box(rhs)) + }); +} + +fn bench_ntt_single_prime_round_trip(c: &mut Criterion) { + let prime = Q32_PRIMES[0]; + let tw = NttTwiddles::<64>::compute(prime); + let base: [MontCoeff; 64] = + std::array::from_fn(|i| prime.from_canonical(((i * 5 + 7) as i16) % prime.p)); + + c.bench_function("ntt_single_prime_forward_inverse_d64", |b| { + b.iter(|| { + let mut a = base; + forward_ntt(&mut a, prime, &tw); + inverse_ntt(&mut a, prime, &tw); + black_box(a) + }) + }); +} + +fn bench_crt_round_trip(c: &mut Criterion) { + let ring = sample_ring(19); + let twiddles: [NttTwiddles<64>; Q32_NUM_PRIMES] = + std::array::from_fn(|k| NttTwiddles::<64>::compute(Q32_PRIMES[k])); + + c.bench_function("ring_ntt_crt_round_trip_d64_k6", |b| { + b.iter(|| { + let ntt = N::from_ring(black_box(&ring), &Q32_PRIMES, &twiddles); + let back: R = ntt.to_ring(&Q32_PRIMES, &twiddles, &Q32_DATA); + black_box(back) + }) + }); +} + +criterion_group!( + ring_ntt, + bench_ring_schoolbook_mul, + bench_ntt_single_prime_round_trip, + bench_crt_round_trip +); +criterion_main!(ring_ntt); diff --git a/docs/FIELD_EMBEDDINGS_SUPERNEO_VS_HACHI.md b/docs/FIELD_EMBEDDINGS_SUPERNEO_VS_HACHI.md new file mode 100644 index 00000000..90b631c2 --- /dev/null +++ b/docs/FIELD_EMBEDDINGS_SUPERNEO_VS_HACHI.md @@ -0,0 +1,698 @@ +# Field embeddings in SuperNeo vs Hachi (math-first notes) + +This note focuses **only** on the math in the two papers: + +- `docs/superneo.pdf` (“Neo and SuperNeo: Post-quantum folding with pay-per-bit costs over small fields”) +- `paper/hachi.pdf` (“Hachi: Efficient Lattice-Based Multilinear Polynomial Commitments over Extension Fields”) + +The shared theme is: lattice commitments naturally live over **cyclotomic rings**, but we want the *interactive proof logic* (sum-check, norm checks, etc.) to live over a **(small) field or a small extension field**. Both works build *embeddings/reductions* that let you: + +- commit in a ring/module world (Ajtai/Module-SIS commitments), while +- proving the needed algebraic statements using field arithmetic, and +- keeping norms under control (for binding) and enabling linear-combination/folding operations. + +--- + +## 1) Common background and notation (as used in SuperNeo) + +SuperNeo sets up a base field \(F = \mathbb{F}_q\), an extension field \(K/F\) of minimal degree such that \(1/|K| = \mathrm{negl}(\lambda)\), and a cyclotomic ring + +\[ +R_F := F[X]/(\Phi(X)), \quad R_K := K[X]/(\Phi(X)), +\] + +where \(\Phi(X)\) is an \(\eta\)-th cyclotomic polynomial of degree \(d\). It explicitly treats + +\[ +F \subseteq R_F \subseteq R_K, \qquad F \subseteq K +\] + +as nested substructures (SuperNeo, Def. 1; `docs/superneo.pdf`, p. 19–20: “-- 19 of 60 --”, “-- 20 of 60 --”). + +Two coefficient maps show up everywhere: + +- coefficient vector: \(\mathrm{cf}(a)\in F^d\) for \(a\in R_F\), +- constant term: \(\mathrm{ct}(a)\in F\) for \(a\in R_F\), + +and similarly over \(R_K\) (SuperNeo, Def. 2; `docs/superneo.pdf`, p. 20: “-- 20 of 60 --”). + +### 1.1 What I mean by the “Gram operator” in this note + +Fix: + +- an \(F\)-basis \(\{e_0,\dots,e_{d-1}\}\) of the \(F\)-vector space \(R_F\) (most often \(e_i = X^i\), the coefficient basis), and +- a bilinear form \(B: R_F\times R_F \to F\). + +Typical examples in these papers are: + +- \(B(u,v)=\mathrm{ct}(u\cdot v)\) (SuperNeo’s constant-term functional), possibly with an automorphism inserted, or +- \(B(u,v)=\mathrm{Tr}_H(u\cdot \sigma_{-1}(v))\) (Hachi’s trace-to-subfield functional). + +Then the **Gram matrix** of \(B\) in the basis \(\{e_i\}\) is the \(d\times d\) matrix +\[ +G_{ij} := B(e_i,e_j). +\] +This matrix encodes the pairing: +if \(u=\sum_i a_i e_i\) and \(v=\sum_j b_j e_j\), then +\[ +B(u,v) = a^\top\, G\, b. +\] + +The corresponding **Gram operator** is just the linear map \(g: F^d\to F^d\) given by +\[ +g(b) := G\,b, +\] +so that \(B(u,v)=a^\top g(b)\). + +When SuperNeo says “there exists a linear transform \(T\) such that \(\mathrm{ct}(T(a)\cdot b)=\langle a,b\rangle\)” (Thm. 3), +one way to interpret it is: choose a bilinear form \(B_0(u,v)=\mathrm{ct}(u\cdot v)\) (or a close variant), write down its Gram matrix \(G\) in the coefficient basis, and take \(T = G^{-{\top}}\). This makes the pairing become the standard dot product in coordinates. + +--- + +## 2) What “field embedding” means in SuperNeo (the core problem) + +### 2.1 The Ajtai-commitment mismatch + +Ajtai/Module-SIS style commitments are *ring-module* commitments: + +- Commit to \(z \in R_F^n\) via a linear map \(c = A z\) over \(R_F\). + +But CCS witnesses (and CCS arithmetic) are naturally vectors over \(F\). So you need a map: + +\[ +\iota: F^{N} \longrightarrow R_F^n +\] + +that is compatible with: + +- **norm constraints** (binding only holds for “small-norm” openings), +- **field constraint checking** via sum-check over \(F\) or \(K\), +- **folding** (random linear combinations of commitments and claims). + +SuperNeo frames this as “embed field vectors (CCS witnesses) into the ring vectors that Ajtai commitments operate over” and calls out that the embedding must preserve **norm bounds** and an **evaluation homomorphism** needed for sum-check-based folding (SuperNeo, §1.2; `docs/superneo.pdf`, p. 5–6: “-- 5 of 60 --”, “-- 6 of 60 --”). + +### 2.2 What went wrong before (NTT embedding) + +Prior lattice folding used an NTT/SIMD isomorphism that maps ring elements into a product of extension fields; this makes field-constraint checking look “ring-native”, but: + +- the NTT map is **not norm-preserving**, so small bit-width witnesses become arbitrary-norm ring elements, +- the commitment must then decompose regardless of bit-width ⇒ **no pay-per-bit**, +- packing efficiency is limited by the factor \(t\) in \(F_{q^t}\) (SuperNeo, §1.2.1; `docs/superneo.pdf`, p. 6–7: “-- 6 of 60 --”, “-- 7 of 60 --”). + +--- + +## 3) SuperNeo’s key innovation: *norm-preserving embeddings + evaluation homomorphism* + +SuperNeo’s abstract summarizes the core as: + +> “two new norm-preserving embeddings of field vectors into ring vectors that respect an evaluation homomorphism required for folding” +(`docs/superneo.pdf`, p. 1: “-- 1 of 60 --”). + +The paper describes both a “Neo embedding” (SIMD-friendly) and a “SuperNeo embedding” (general, non-SIMD), and then focuses the rest of the paper on SuperNeo (see `docs/superneo.pdf`, p. 10–11: “-- 10 of 60 --”, “-- 11 of 60 --”). + +### 3.1 Neo embedding (high level: “coefficients-as-SIMD lanes”) + +Neo’s embedding idea: pack **\(d\) field vectors** \(z^{(1)},\dots,z^{(d)}\in F^n\) into the **coefficient slots** of a ring vector \(z \in R_F^n\), so the coefficient matrix \(\mathrm{cf}(z)\in F^{d\times n}\) literally equals those \(d\) vectors (SuperNeo, §1.2.2 “Contribution 1”; `docs/superneo.pdf`, p. 8–10: “-- 8 of 60 --”, “-- 9 of 60 --”, “-- 10 of 60 --”). + +Key consequences: + +- **norm-preserving**: small field entries ⇒ small ring coefficients ⇒ binding is aligned with bit-width. +- **optimal SIMD packing**: achieves \(d\cdot n\) field elements per length-\(n\) ring vector (under a SIMD constraint system). +- **evaluation homomorphism for folding**: if you fold commitments with a *short ring* challenge \(\delta\in R_F\), the embedded evaluation claims can be folded consistently by embedding the \(d\) field evaluations as a ring element \(y=\sum_i y^{(i)}X^{i-1}\) (SuperNeo, p. 9: “-- 9 of 60 --”). + +But Neo still needs SIMD constraints: “the same constraint system must be applied to all \(d\) underlying field vectors” (SuperNeo, p. 10: “-- 10 of 60 --”). + +### 3.2 SuperNeo embedding (formal: coefficient embedding of **one** length-\(d n_R\) vector) + +SuperNeo removes SIMD by embedding a **single** long field vector \(z\in F^{n_F}\) where \(n_F = d\cdot n_R\), by chunking it into \(n_R\) blocks of length \(d\), and mapping each block to one ring element’s coefficient vector. + +This is defined formally as “Coefficient Embedding” (SuperNeo, §5, Def. 7; `docs/superneo.pdf`, p. 23: “-- 23 of 60 --”): + +- element: \(v\in F^d \mapsto \mathbf{v}\in R_F\) with \(\mathrm{cf}(\mathbf{v})=v\) +- vector: \(z\in F^{d n_R}\mapsto \mathbf{z}\in R_F^{n_R}\) by splitting into \(d\)-sized blocks +- matrix: \(M\in F^{m\times d n_R}\mapsto \mathbf{M}\in R_F^{m\times n_R}\) row-wise. + +**Why this matters:** + +- **optimal packing without SIMD**: you pack \(d\cdot n_R\) field elements into \(n_R\) ring elements. +- **norm-preserving**: the committed object’s coefficients are exactly the witness entries. +- **field-native checks become possible**: you can write constraints directly over the underlying field vector \(z\) and use sum-check over \(F\) or \(K\). + +### 3.3 The nontrivial part: lifting *field* products to *ring* products while keeping folding linear + +SuperNeo’s obstacle is: commitments/folding live over the ring, but sum-check outputs **field** multilinear evaluation claims like + +\[ +M z \;\widetilde{}\; (r) \in K +\] + +for some random \(r\), and folding wants to take ring-linear combinations \(z'' = z + \delta z'\) with \(\delta \in R_F\) and have the **claims** fold “the same way”. + +SuperNeo’s main embedding tool is Section 5 “Embedding products with evaluation homomorphism”: + +1. Use a cyclotomic **inner-product automorphism trick** to turn coefficient inner products into ring constant terms. + +SuperNeo states an “Inner Product Transform” (Thm. 3; `docs/superneo.pdf`, p. 23: “-- 23 of 60 --”): + +> there exists a linear transform \(\bar{\cdot}: F^d \to F^d\) such that for all \(a,b\in F^d\), +> \[ +> \mathrm{ct}(\overline{a}\cdot \mathbf{b}) = \langle a,b\rangle. +> \] + +Conceptually: cyclotomic rings have many Galois automorphisms (e.g. “conjugation” \(X\mapsto X^{-1}\)), and by applying an appropriate automorphism/linear transform to one operand, the constant coefficient of a ring product recovers a dot product of coefficient vectors. (SuperNeo explicitly attributes this to “(Galois, conjugation, or inner product) automorphism trick” in §1.2.2; `docs/superneo.pdf`, p. 10–11: “-- 10 of 60 --”, “-- 11 of 60 --”.) + +2. Extend this transform blockwise to vectors/matrices (Def. 8; `docs/superneo.pdf`, p. 23: “-- 23 of 60 --”). + +3. Obtain a **matrix-vector product transform** (Thm. 4; `docs/superneo.pdf`, p. 23–24: “-- 23 of 60 --”, “-- 24 of 60 --”): + +> For \(M\in F^{m\times n_F}\), \(z\in F^{n_F}\), +> \[ +> M z = \mathrm{ct}(\overline{M}\,\mathbf{z}), +> \] +> i.e. the field product equals the vector of constant terms of a ring product. + +4. Lift this to evaluation claims and prove the **evaluation homomorphism** (Thm. 5; `docs/superneo.pdf`, p. 24: “-- 24 of 60 --”): + +Roughly: if you linearly combine committed ring vectors with ring scalars \(\rho_i\in R_F\), then the lifted ring-evaluation objects combine linearly as well, and constant terms track the underlying field evaluations. + +This is the formal engine that makes “field-native sum-check + ring-linear folding” composable. + +### 3.3.1 Explicit inner-product transforms for two cyclotomics you care about + +SuperNeo’s Theorem 3 is an *existence* statement: there is a linear map \(T: F^d \to F^d\) (write \(T(a)=\bar a\)) such that for all \(a,b\in F^d\), + +\[ +\mathrm{ct}(\mathbf{\bar a}\cdot \mathbf{b}) = \langle a,b\rangle +\] + +where \(\mathbf{v}\in R_F\) denotes the coefficient embedding of \(v\in F^d\). + +Below are **concrete closed forms** for \(T\) in two important special cases. + +#### (A) Power-of-two cyclotomic: \(\Phi(X)=X^d+1\) (negacyclic ring) + +Let \(R_F = F[X]/(X^d+1)\), and write +\(a(X)=\sum_{i=0}^{d-1} a_i X^i\), \(b(X)=\sum_{i=0}^{d-1} b_i X^i\). + +Define \(\bar a(X)\) by the coefficient rule: + +- \(\bar a_0 := a_0\) +- for \(i=1,\dots,d-1\): \(\bar a_i := -a_{d-i}\) + +Equivalently, +\[ +\bar a(X)=a_0 - \sum_{i=1}^{d-1} a_i X^{d-i}. +\] + +Then in \(R_F\), +\[ +\mathrm{ct}(\bar a(X)\,b(X)) = \sum_{i=0}^{d-1} a_i b_i. +\] + +Reason (one-line): the term \((-a_i X^{d-i})(b_i X^i)=-a_i b_i X^d\) contributes \(+a_i b_i\) to the constant term since \(X^d=-1\); other cross-terms cannot reduce to constants without leaving a nonzero power of \(X\). + +This is exactly the classical “conjugation/inversion automorphism” trick specialized to \(X^d+1\). + +#### (B) Trinomial cyclotomic: \(\Phi_{81}(X)=X^{54}+X^{27}+1\) + +Let \(R_F = F[X]/(X^{54}+X^{27}+1)\) (so \(d=54\)). Write +\(a(X)=\sum_{i=0}^{53} a_i X^i\), \(b(X)=\sum_{i=0}^{53} b_i X^i\). + +One valid “inner product transform” \(T(a)=\bar a\) with +\(\mathrm{ct}(\bar a(X)\,b(X))=\sum_i a_i b_i\) +is: + +- \(\bar a_0 := a_0\) +- for \(i=1,\dots,26\): \(\bar a_i := -(a_{27-i} + a_{54-i})\) +- for \(i=27,\dots,53\): \(\bar a_i := -a_{54-i}\) + +(indices are in \(\{0,\dots,53\}\)). + +Notable features: + +- **extremely sparse**: each output coefficient depends on at most 2 input coefficients, +- **\(O(d)\)** time with only adds + sign flips, matching SuperNeo’s efficiency remark for power-of-two / trinomial cyclotomics. + +--- + +### 3.4 Why this is the “field-embedding innovation” (in one sentence) + +SuperNeo’s innovation on the embedding side is: + +- **embed a single field witness vector into ring coefficients in a norm-preserving way**, and +- **systematically lift field matrix products/evaluations to ring expressions whose constant terms recover the field values**, so that +- **ring-linear folding preserves the field evaluation claims** (evaluation homomorphism), + +thereby enabling a HyperNova-like folding architecture where *sum-check and norm checks run over \(K\)* rather than over the ring, while commitments still live over \(R_F\). + +--- + +## 4) What “field embedding” means in Hachi (the PCS perspective) + +Hachi is not a folding scheme; it is a **multilinear PCS**. But the verification bottleneck is similar: classic lattice PCS machinery lives in cyclotomic rings \(R_q = \mathbb{Z}_q[X]/(X^d+1)\), whereas sum-check is naturally over a field. + +Hachi’s abstract states its two embedding/reduction ideas (Hachi, Abstract; `paper/hachi.pdf`, p. 1: “-- 1 of 33 --”): + +1. **Ring-switching + sum-check**: integrate Greyhound with ring-switching so the verifier avoids ring multiplication. +2. **Generic reduction (extension field → ring)**: convert evaluation proofs over \( \mathbb{F}_{q^k}\) into statements over cyclotomic rings \(R_q\). + +### 4.1 Embedding extension fields inside cyclotomic rings (fixed rings under automorphisms) + +In Hachi’s technical overview (Hachi, §1.3; `paper/hachi.pdf`, p. 4–5: “-- 4 of 33 --”, “-- 5 of 33 --”), it identifies finite fields \( \mathbb{F}_{q^k}\) *inside* \(R_q\) using fixed subrings under a subgroup of the Galois group: + +- Let \(R = \mathbb{Z}[X]/(X^d+1)\) and \(R_q = R/(q)\) with \(d=2^\alpha\). +- For automorphisms \(\sigma_i: X\mapsto X^i\), define the fixed ring + \[ + R_q^H := \{x\in R_q : \forall \sigma\in H,\;\sigma(x)=x\}. + \] + +Then (Lemma 1, informal; `paper/hachi.pdf`, p. 5: “-- 5 of 33 --”): + +> for suitable \(q\) (notably \(q\equiv 5\pmod 8\)) and \(k\mid d/2\), there exists \(H\) such that \(R_q^H\) is a **subfield** of \(R_q\) isomorphic to \(\mathbb{F}_{q^k}\). + +This is a literal *field embedding into the ring*: \(\mathbb{F}_{q^k}\hookrightarrow R_q\) realized as a fixed subring. + +### 4.2 Inner products via trace + automorphisms (a close cousin of SuperNeo’s transform) + +Hachi then uses a trace map \(\mathrm{Tr}_H:R_q\to R_q^H\) and an automorphism (notably \(\sigma_{-1}\)) to turn ring products into **field inner products** (Theorem 1, informal; `paper/hachi.pdf`, p. 5: “-- 5 of 33 --”): + +> there exists a bijection \(\psi:(R_q^H)^{d/k}\to R_q\) such that +> \[ +> \mathrm{Tr}_H\big(\psi(a)\cdot \sigma_{-1}(\psi(b))\big) = (d/k)\cdot \langle a,b\rangle. +> \] + +This is structurally very similar to SuperNeo’s “inner product transform → constant term” idea, except Hachi uses: + +- a **trace to a subfield** \(R_q^H \cong \mathbb{F}_{q^k}\), +- whereas SuperNeo phrases it as a **linear transform** on coefficient vectors whose ring-product constant term recovers the dot product. + +### 4.3 Ring-switching: from ring equations to extension-field equations (so the verifier stays field-native) + +Hachi’s “ring switching and sum-check over extension fields” overview (Hachi, §1.3; `paper/hachi.pdf`, p. 6–7: “-- 6 of 33 --”, “-- 7 of 33 --”) sketches: + +- lift a relation over \(R_q\) to an identity over \(\mathbb{Z}_q[X]\) with an explicit multiple of \((X^d+1)\), +- sample \(\alpha \leftarrow \mathbb{F}_{q^k}\) and substitute \(X=\alpha\), +- reducing the ring relation to a **field inner product / sum-check-type claim** over \(\mathbb{F}_{q^k}\), +- then run sum-check over the field and recurse. + +This is the PCS analogue of “avoid ring operations during sum-check”: Hachi’s verifier can avoid cyclotomic ring multiplications even though the underlying assumption/commitments are lattice/ring-based. + +--- + +## 5) Same vs different (embedding viewpoint) + +### 5.1 The same (high-level mathematical pattern) + +- **Cyclotomic rings + automorphisms are the bridge.** + - SuperNeo: an automorphism/linear transform makes \(\mathrm{ct}(\bar{a}\cdot b)\) become \(\langle a,b\rangle\) (SuperNeo Thm. 3; `docs/superneo.pdf`, p. 23: “-- 23 of 60 --”). + - Hachi: trace + automorphisms make \(\mathrm{Tr}_H(\psi(a)\cdot \sigma_{-1}(\psi(b)))\) become \(\langle a,b\rangle\) (Hachi Thm. 1; `paper/hachi.pdf`, p. 5: “-- 5 of 33 --”). + +- **Field-native sum-check is the goal.** + - SuperNeo explicitly targets “field-native arithmetic” where “sum-check and norm checks run purely over a small field” (SuperNeo Abstract; `docs/superneo.pdf`, p. 1: “-- 1 of 60 --”). + - Hachi’s verifier similarly reduces to sum-check over \(\mathbb{F}_{q^k}\) after ring switching (Hachi §1.3; `paper/hachi.pdf`, p. 6–7: “-- 6 of 33 --”, “-- 7 of 33 --”). + +- **Linearity matters.** + Both constructions rely on the fact that the commitment operation is linear in the ring/module, and they build embeddings/reductions so the *claimed evaluations* transform linearly under the same combinations (SuperNeo Thm. 5; `docs/superneo.pdf`, p. 24: “-- 24 of 60 --”). + +### 5.2 The different (what each paper is optimizing for) + +- **Direction of “embedding”:** + - **SuperNeo**: embeds *field witnesses* into *ring vectors* to make Ajtai commitments “pay-per-bit” and folding-friendly. + - **Hachi**: embeds *extension-field evaluation statements* into *ring statements* (and back), to make PCS verification fast via sum-check. + +- **Primary object being preserved:** + - **SuperNeo**: preserves **norm** (for binding/pay-per-bit) *and* preserves **evaluation homomorphism** (for folding). + - **Hachi**: preserves the **truth of evaluation claims** (over \(\mathbb{F}_{q^k}\)) when translated into ring relations; norm constraints are handled via sum-check after ring switching. + +- **Ring family emphasis:** + - **Hachi**’s core ring is explicitly power-of-two cyclotomic \(X^d+1\) (Hachi Abstract; `paper/hachi.pdf`, p. 1: “-- 1 of 33 --”). + - **SuperNeo** broadens to more general cyclotomics to support fields like Goldilocks without “full splitting” issues, and explicitly mentions supporting trinomials in parameter sets (SuperNeo intro + parameters discussion; `docs/superneo.pdf`, p. 1–4 and later “concrete parameters” sections). + +- **Protocol role:** + - **SuperNeo** needs an embedding compatible with **folding** (random linear combination of instances/commitments). + - **Hachi** needs a reduction compatible with **PCS recursion and verifier-time reduction** (ring switching + sum-check). + +### 5.3 Are the embeddings “the same”, mathematically? + +It depends what you mean by “same”. There are two distinct layers: + +- **Layer 1: the raw embedding map(s) \(F^{dn}\to R_F^n\) vs \(F_{q^k}\to R_q\)** + These are **not the same functions**. + - SuperNeo’s core embedding is literally the **coefficient embedding** (Def. 7 in their §5), i.e. *place field coordinates into ring coefficients* (`docs/superneo.pdf`, “Definition 7”; see the quote range in this repo at lines 1222–1238 of the extracted text). + - Hachi’s key “field embedding” is to realize \(F_{q^k}\) as a **fixed subfield** \(R_q^H \subseteq R_q\), then use a **basis-dependent bijection** \(\psi:(R_q^H)^{d/k}\to R_q\) (`paper/hachi.pdf`, lines 290–300 in the extracted text). + +- **Layer 2: the algebraic *mechanism* (automorphisms/trace giving inner products, and linearity of evaluations)** + At this layer, they are **the same underlying idea**: both are exploiting a canonical bilinear pairing on cyclotomic rings derived from Galois automorphisms (e.g. \(\sigma_{-1}\)) plus a linear functional (constant term or trace). + - SuperNeo packages it as “there exists a linear transform \(T\) so that \(\mathrm{ct}(T(a)\cdot b)=\langle a,b\rangle\)” (Thm. 3). + - Hachi packages it as “there exists a bijection \(\psi\) so that \(\mathrm{Tr}_H(\psi(a)\cdot\sigma_{-1}(\psi(b)))=(d/k)\langle a,b\rangle\)” (Thm. 1). + +Concretely: SuperNeo’s \(T\) is (mathematically) the **inverse Gram operator** of the ring’s trace/constant-term pairing *expressed in the coefficient basis*. Hachi’s \(\psi\) is a **choice of basis** that identifies \(R_q\) as a free module over a subfield \(R_q^H\cong F_{q^k}\), so that the same pairing looks like a scaled dot product over that subfield. + +So the correct crisp statement is: + +- **Not identical as maps** (different domains/codomains and different basis choices), but +- **equivalent in the sense that both instantiate the same cyclotomic “automorphism + linear functional = inner product” backbone**, just presented in different coordinates. + +--- + +## 6) Practical mental model (how to read both papers through one lens) + +Both papers can be read as building “interfaces” between three layers: + +1. **Ring/module commitment layer** (Ajtai / Module-SIS commitments) +2. **Field arithmetic layer** (sum-check, equality tests, low-degree checks) +3. **Embedding/reduction layer** (the math glue) + +SuperNeo’s embedding layer is primarily: + +- coefficient embedding \(F^{d n_R} \leftrightarrow R_F^{n_R}\), +- inner-product transform + constant-term extraction, +- evaluation homomorphism for ring-linear combinations. + +Hachi’s embedding layer is primarily: + +- fixed-field embedding \( \mathbb{F}_{q^k} \cong R_q^H \subseteq R_q\), +- trace + automorphisms for inner products, +- ring-switching by evaluation \(X=\alpha\) to move verifier work into \( \mathbb{F}_{q^k}\). + +If you want a single phrase: + +> **SuperNeo**: “make folding happen over small fields even though commitments live over rings.” +> **Hachi**: “make PCS verification happen over extension fields even though commitments live over rings.” + + +--- + +## 7) SuperNeo's bilinear form, explicitly identified + +SuperNeo's Theorem 3 states existence of a linear transform \(T\) but never identifies it as a Galois automorphism. Here we close that gap. + +### 7.1 Claim: \(T = \sigma_{-1}\) for \(\Phi(X) = X^d + 1\) + +In \(R_q = F_q[X]/(X^d+1)\), we have \(X^d = -1\), so \(X^{-1} = -X^{d-1}\). The automorphism \(\sigma_{-1}: X \mapsto X^{-1}\) acts on monomials as: + +\[ +\sigma_{-1}(X^0) = 1, \qquad \sigma_{-1}(X^i) = X^{-i} = -X^{d-i} \;\text{ for } 1 \le i \le d-1. +\] + +So on a polynomial \(a(X) = \sum_i a_i X^i\): + +\[ +\sigma_{-1}(a) = a_0 - \sum_{i=1}^{d-1} a_i\, X^{d-i}. +\] + +This is *exactly* the transform \(\bar{a}(X)\) from §3.3.1(A). + +### 7.2 Why the Gram matrix is the identity + +Consider the bilinear form \(B(a,b) = \mathrm{ct}(\sigma_{-1}(a)\cdot b)\) in the monomial basis \(\{1, X, \dots, X^{d-1}\}\). The Gram matrix is: + +\[ +G_{ij} = \mathrm{ct}\big(X^{j-i} \bmod (X^d+1)\big). +\] + +For \(j = i\): \(\mathrm{ct}(1) = 1\). + +For \(j \ne i\): the exponent \(j-i\) satisfies \(-(d-1) \le j-i \le d-1\) and \(j-i \ne 0\). +- If \(j > i\): \(\mathrm{ct}(X^{j-i}) = 0\) since \(1 \le j-i \le d-1\). +- If \(j < i\): \(X^{j-i} = X^{j-i+d}\cdot X^{-d} = -X^{d+j-i}\) where \(1 \le d+j-i \le d-1\), so \(\mathrm{ct}(-X^{d+j-i}) = 0\). + +Therefore \(G = I\) (the identity matrix). This means: + +\[ +\mathrm{ct}(\sigma_{-1}(a)\cdot b) = \mathrm{cf}(a)^\top\, I\, \mathrm{cf}(b) = \langle \mathrm{cf}(a),\,\mathrm{cf}(b)\rangle. +\] + +No Gram correction is needed. The pairing \((\sigma_{-1}, \mathrm{ct})\) gives the standard dot product directly. + +### 7.3 For the trinomial \(\Phi_{81}(X) = X^{54}+X^{27}+1\), the Gram matrix is NOT the identity + +Here \(X^{81} = 1\) but \(X^{27} \ne 1\) in \(R_F\). So \(X^{-27} = X^{54} = -X^{27}-1\), which gives: + +\[ +\mathrm{ct}(X^{-27}) = \mathrm{ct}(-X^{27}-1) = -1. +\] + +The Gram matrix \(G_{ij} = \mathrm{ct}(X^{j-i} \bmod \Phi_{81})\) therefore has off-diagonal entries \(G_{i,\,i+27} = -1\) (and their transposes). The inner-product transform for the trinomial (§3.3.1(B)) is \(T = G^{-1}\circ \sigma_{-1}\), which is why its formula involves sums like \(-(a_{27-i}+a_{54-i})\) rather than a simple sign flip. + +### 7.4 Bottom line + +SuperNeo's bilinear form is: + +\[ +B_{\mathrm{SuperNeo}}(a,b) = \mathrm{ct}\big(\sigma_{-1}(a)\cdot b\big) \in F_q. +\] + +Hachi's bilinear form is: + +\[ +B_{\mathrm{Hachi}}(a,b) = \mathrm{Tr}_H\big(a\cdot \sigma_{-1}(b)\big) \in R_q^H \cong \mathbb{F}_{q^k}. +\] + +Same involution \(\sigma_{-1}\). Different linear functional (\(\mathrm{ct}\) vs \(\mathrm{Tr}_H\)). Different target (\(F_q\) vs \(\mathbb{F}_{q^k}\)). + +--- + +## 8) Classification of non-degenerate bilinear forms on cyclotomic rings + +### 8.1 The general template + +For a cyclotomic ring \(R_q = F_q[X]/(\Phi(X))\) of degree \(d\), with Galois group \(\mathrm{Gal} = (\mathbb{Z}/\eta\mathbb{Z})^\times\) acting by \(\sigma_i: X\mapsto X^i\), the general bilinear form from an automorphism + linear functional is: + +\[ +B_{\sigma,\lambda}(a,b) = \lambda\big(a\cdot \sigma(b)\big) +\] + +where \(\sigma \in \mathrm{Gal}\) and \(\lambda: R_q \to T\) is a linear functional into some target \(T\). + +### 8.2 When is this non-degenerate? (Frobenius algebra theory) + +\(R_q = F_q[X]/(\Phi(X))\) is a **Frobenius algebra** over \(F_q\) (since cyclotomic polynomials are squarefree). This gives a clean classification: + +**Theorem (Frobenius classification)**: A linear functional \(\lambda: R_q \to F_q\) makes the bilinear form \(B(a,b) = \lambda(a\cdot b)\) non-degenerate if and only if \(\lambda\) is a **generating functional**, meaning the induced map \(R_q \to R_q^\vee\) (sending \(a\) to the functional \(b\mapsto \lambda(ab)\)) is an isomorphism. + +Moreover: **the set of all generating functionals is \(\{\lambda(g\,\cdot\,{-}) : g \in R_q^\times\}\)**. Once you have one non-degenerate form, all others come from pre-multiplying by a unit. + +For the form \(B_{\sigma,\lambda}(a,b)=\lambda(a\cdot\sigma(b))\), non-degeneracy reduces to: \(\lambda\circ(\text{mult by }\sigma(\cdot))\) is generating. Since \(\sigma\) is an automorphism (hence maps units to units), \(B_{\sigma,\lambda}\) is non-degenerate iff \(\lambda\) is generating. + +### 8.3 Complete menu of linear functionals + +The dual space \(R_q^\vee = \mathrm{Hom}_{F_q}(R_q, F_q)\) is \(d\)-dimensional. Here are the "natural" families: + +#### (A) Coefficient extraction: \(\mathrm{ct}_j(a) = [X^j](a)\) + +Target: \(F_q\). There are \(d\) such functionals (one per coefficient index \(j\)). + +\(\mathrm{ct} = \mathrm{ct}_0\) is the constant term. For \(\Phi = X^d+1\) with \(\sigma_{-1}\), the Gram matrix of \(\mathrm{ct}_0\) is the identity (§7.2). Other \(\mathrm{ct}_j\) give non-degenerate forms too, but with permuted/signed Gram matrices. + +*Generating?* Yes for many cyclotomics including \(X^d+1\). Can fail for pathological factorizations. + +#### (B) Absolute trace: \(\mathrm{Tr}_{R_q/F_q}(a) = \sum_{\sigma\in\mathrm{Gal}} \sigma(a)\) + +Target: \(F_q\). This is the canonical choice from algebraic number theory. + +*Generating?* Always yes for separable algebras (which cyclotomic rings are). + +Relation to \(\mathrm{ct}\): since both are generating, \(\mathrm{Tr}(a) = \mathrm{ct}(g\cdot a)\) for some explicit unit \(g\in R_q^\times\) (related to the "different ideal" of the cyclotomic extension). + +#### (C) Relative/partial trace: \(\mathrm{Tr}_H(a) = \sum_{\sigma\in H}\sigma(a) \in R_q^H\) + +Target: \(R_q^H \cong \mathbb{F}_{q^k}\) (the fixed subfield under a subgroup \(H \subseteq \mathrm{Gal}\)). + +This is what Hachi uses. The key upgrade: the target is an extension field \(\mathbb{F}_{q^k}\), not just \(F_q\). You get an \(\mathbb{F}_{q^k}\)-valued bilinear form, which means inner products live natively in the sumcheck domain. + +*Generating?* As a map \(R_q \to R_q^H\), yes: the induced \(R_q^H\)-bilinear form on \(R_q\) (viewed as a free \(R_q^H\)-module of rank \(d/k\)) is non-degenerate. + +Trace tower: \(\mathrm{Tr}_{R_q/F_q} = \mathrm{Tr}_{R_q^H/F_q}\circ \mathrm{Tr}_H\). So the relative trace "refines" the absolute trace. + +#### (D) Evaluation at a root: \(\mathrm{ev}_\zeta(a) = a(\zeta)\) for \(\zeta\) a root of \(\Phi\) in some \(\mathbb{F}_{q^t}\) + +Target: \(\mathbb{F}_{q^t}\). + +*Generating?* If \(\Phi\) is irreducible over \(F_q\) (so \(R_q \cong \mathbb{F}_{q^d}\)), every nonzero functional is generating. If \(\Phi\) factors, \(\mathrm{ev}_\zeta\) projects onto one CRT slot and annihilates the others, so it is **not** generating. + +**Fatal for norms**: the evaluation map is equivalent to NTT, which is not norm-preserving. This is exactly the problem SuperNeo identifies with prior approaches (§2.2). + +#### (E) CRT-slot projection: \(\pi_s: R_q \twoheadrightarrow \mathbb{F}_{q^t}\) + +When \(\Phi = \prod_i \phi_i\) factors over \(F_q\) (each \(\phi_i\) irreducible of degree \(t\)), \(R_q \cong \prod_i \mathbb{F}_{q^t}\). The projection \(\pi_s\) onto one factor is: + +Target: \(\mathbb{F}_{q^t}\). + +*Generating?* **No** --- it annihilates the other factors. + +**Same norm problem as (D)**: the CRT isomorphism is an NTT-like map. + +#### (F) Arbitrary \(F_q\)-linear combinations of the above + +Any \(\lambda = \sum_j c_j \,\mathrm{ct}_j\) for scalars \(c_j \in F_q\) gives a functional. It is generating iff the associated element \(g = \sum_j c_j X^j \in R_q\) is a unit. + +### 8.4 Practical takeaway + +If you want **norm preservation**, you must stay in "coefficient-land" (functionals A, B, C, F). Evaluation-based functionals (D, E) destroy norms. + +If you want **extension-field-valued inner products** (for native sumcheck over \(\mathbb{F}_{q^k}\)), the only natural coefficient-land option is **(C) the relative trace** \(\mathrm{Tr}_H\). + +--- + +## 9) The evaluation homomorphism (SuperNeo Theorem 5), spelled out + +This is the key property that makes SuperNeo's folding work. The paper states it abstractly; here is the fully explicit version. + +### 9.1 Setup + +Fix: +- Base field \(F = \mathbb{F}_q\), extension \(K = \mathbb{F}_{q^2}\) (or any extension with \(1/|K| = \mathrm{negl}(\lambda)\)). +- Cyclotomic ring \(R_F = F[X]/(\Phi(X))\), \(R_K = K[X]/(\Phi(X))\), degree \(d\). +- A CCS matrix \(M \in F^{m\times n_F}\) with \(n_F = d\cdot n_R\). +- Witness vectors \(z_1, \dots, z_\ell \in F^{n_F}\), coefficient-embedded as \(\mathbf{z}_i \in R_F^{n_R}\). +- Folding challenges \(\rho_1, \dots, \rho_\ell \in C \subset R_F\) (short ring elements). +- An evaluation point \(r \in K^{\log m}\) (from a prior sumcheck round). +- Any \(R_F\)-module homomorphisms \(L: R_F^{n_R}\to C\) (commitment) and \(\mathrm{Lin}: R_F^{n_R}\to R_F^{n_{R,\mathrm{in}}}\) (input extraction). + +### 9.2 Per-instance "lifted" claims + +For each instance \(i\in[\ell]\), define three objects: + +- **Commitment**: \(c_i := L(\mathbf{z}_i) \in C\) (Ajtai commitment of the ring vector). +- **Input**: \(x_i := \mathrm{Lin}(\mathbf{z}_i) \in R_F^{n_{R,\mathrm{in}}}\) (public input part). +- **Lifted evaluation**: \(y_i := \overline{M}\,\widetilde{\mathbf{z}_i}(r) \in R_K\) (apply the transformed matrix \(\overline{M}\) to the multilinear extension of the ring vector \(\mathbf{z}_i\), evaluated at \(r\)). + +The lifted evaluation \(y_i\) is a **ring element in \(R_K\)** (not a field element in \(K\)). Its constant term recovers the field-level evaluation (by Theorem 4 / Remark 2): + +\[ +\mathrm{ct}(y_i) = M\,\widetilde{z_i}(r) \in K. +\] + +In other words: \(y_i\) has \(d\) coefficients in \(K\); the constant coefficient is the "real" field evaluation; the other \(d-1\) coefficients carry extra information that is needed for folding consistency. + +### 9.3 The evaluation homomorphism (Theorem 5) + +**Statement**: Define the folded objects: + +\[ +\mathbf{z} := \sum_{i\in[\ell]} \rho_i\,\mathbf{z}_i \in R_F^{n_R}, \quad +c := \sum_{i\in[\ell]} \rho_i\, c_i \in C, \quad +x := \sum_{i\in[\ell]} \rho_i\, x_i \in R_F^{n_{R,\mathrm{in}}}, +\quad +y := \sum_{i\in[\ell]} \rho_i\, y_i \in R_K. +\] + +Then **all three of the following hold simultaneously**: + +1. **Commitment homomorphism**: \(c = L(\mathbf{z})\). +2. **Lifted-evaluation homomorphism**: \(y = \overline{M}\,\widetilde{\mathbf{z}}(r)\). +3. **Field-evaluation consistency**: \(\mathrm{ct}(y) = M\,\widetilde{z}(r) \in K\) where \(z = \sum_i \rho_i z_i \in F^{n_F}\). + +(1) holds because \(L\) is \(R_F\)-linear. (2) holds because multilinear extension is \(K\)-linear and matrix multiplication is \(R_K\)-linear. (3) follows from applying \(\mathrm{ct}\) to (2) and invoking Theorem 4. + +### 9.4 Why ring-level claims are essential (the subtle point) + +The folding challenge \(\rho_i \in R_F\) is a **ring** scalar, not a field scalar. At the field level, multiplying by \(\rho_i\) acts on the coefficient vector \(z_i \in F^{n_F}\) by **convolution** (polynomial multiplication scrambles coefficients), not by coordinate-wise scaling. + +This means: +- **At the field level**: after folding, the individual CCS constraints \(Mz = 0\) do NOT hold in any simple sense. The "meaning" of the folded witness as a CCS satisfier is lost. +- **At the ring level**: the commitment, lifted evaluation claims, and input extraction all fold cleanly. The accumulator stores \((c, x, r, y)\) as ring-level objects. + +This is why SuperNeo's accumulator relation is **CE (CCS Evaluation)**, not CCS itself: it stores ring-level evaluation claims \(y_j \in R_K\), and the decider checks them by opening the commitment and verifying ring-level equalities. The \(\mathrm{ct}\) extraction (field-level check) happens only once, at decision time. + +### 9.5 Contrast with Hachi + +Hachi does not need an "evaluation homomorphism" in SuperNeo's sense because Hachi does not fold multiple instances. Instead: + +- Hachi's protocol reduces a *single* opening claim by splitting the coefficient table into blocks, folding with a sparse ring challenge \(c\), and then using ring-switching + sumcheck. +- The analogue of "linearity under ring multiplication" in Hachi is the fold relations (Eq. (18)--(19)): \(a^\top G_{2^m} z = (c^\top\otimes G_1)\hat{w}\) and \(Az = (c^\top\otimes G_{n_A})\hat{t}\), which are linear in the sparse challenge \(c\). + +Both papers exploit the same algebraic fact (ring multiplication is \(R_F\)-linear), but for different protocol purposes: folding (SuperNeo) vs opening reduction (Hachi). + +--- + +## 10) Unified embedding framework (best of both worlds) + +### 10.1 The core observation + +The two bilinear forms use the **same involution** \(\sigma_{-1}\) and differ only in the "readout functional": + +| | SuperNeo | Hachi | +|---|---|---| +| Form | \(\mathrm{ct}(\sigma_{-1}(a)\cdot b)\) | \(\mathrm{Tr}_H(\sigma_{-1}(a)\cdot b)\) | +| Target | \(F_q\) | \(\mathbb{F}_{q^k}\) | +| Gram (for \(X^d+1\)) | \(I\) (identity) | Non-trivial (absorbed into \(\psi\)) | + +These are **compatible**: they use the same product \(\sigma_{-1}(a)\cdot b\) in \(R_q\) and differ only in what you project onto. The trace tower connects them: + +\[ +\mathrm{Tr}_{R_q/F_q} = \mathrm{Tr}_{\mathbb{F}_{q^k}/F_q} \circ \mathrm{Tr}_H. +\] + +### 10.2 The unified pairing abstraction + +**Definition (Cyclotomic Pairing).** Given: +- cyclotomic ring \(R_q\) of degree \(d\), +- a subfield \(S \cong \mathbb{F}_{q^k}\) realized as \(R_q^H \subseteq R_q\) (or \(S = F_q\) as the trivial case \(k=1\)), +- the involution \(\sigma_{-1} \in \mathrm{Aut}(R_q)\), +- the relative trace \(\mathrm{Tr}_H: R_q \to S\), + +define the **level-\(k\) pairing**: + +\[ +B_k(a,b) := \mathrm{Tr}_H\big(\sigma_{-1}(a)\cdot b\big) \in S \cong \mathbb{F}_{q^k}. +\] + +Special cases: +- **\(k = 1\)** (\(H = \mathrm{Gal}\), \(S = F_q\)): \(B_1\) is (up to unit scaling) the absolute-trace pairing. Closely related to SuperNeo's \(\mathrm{ct}\)-based form. +- **\(k = d\)** (\(H = \{1\}\), \(S = R_q\)): \(B_d(a,b) = \sigma_{-1}(a)\cdot b\) (the ring product itself, no projection). +- **\(1 < k < d\)** (\(H\) a proper subgroup): Hachi's setting. \(B_k\) is an \(\mathbb{F}_{q^k}\)-valued non-degenerate bilinear form on the free \(\mathbb{F}_{q^k}\)-module \(R_q \cong \mathbb{F}_{q^k}^{d/k}\). + +### 10.3 Combined embedding pipeline + +``` +Field witness z ∈ F_q^{d·n_R} + │ coefficient embedding (SuperNeo Def. 7, norm-preserving) + ▼ +Ring vector z ∈ R_q^{n_R} ← ‖z‖_∞ preserved + │ Ajtai commit: c = Az + ▼ +Commitment c ∈ R_q^κ + │ + ├── MODE A (SuperNeo-style folding): + │ Use B_1: ct(σ_{-1}(a)·b) = ⟨cf(a), cf(b)⟩ ∈ F_q + │ → sumcheck over K = F_{q²}, fold with ρ ∈ R_F, track ring-level claims y ∈ R_K + │ + └── MODE B (Hachi-style PCS opening): + Use B_k: Tr_H(σ_{-1}(a)·b) ∈ F_{q^k} + → ring-switch at α ∈ F_{q^k}, sumcheck over F_{q^k}, recurse on smaller instance +``` + +The committed object \(\mathbf{z}\in R_q^{n_R}\) is the same in both modes. The choice of functional only affects how you read out inner products and run the interactive proof. + +### 10.4 What you gain + +| Property | SuperNeo alone | Hachi alone | Unified | +|---|---|---|---| +| Norm-preserving embedding | Yes | N/A (PCS, not CCS) | Yes | +| Pay-per-bit commitments | Yes | No | Yes | +| Extension-field inner products | No (\(\mathrm{ct}\to F_q\)) | Yes (\(\mathrm{Tr}_H\to \mathbb{F}_{q^k}\)) | Yes | +| Sumcheck natively over \(\mathbb{F}_{q^k}\) | No (needs \(K\) externally) | Yes | Yes | +| Power-of-two cyclotomics | Supported | Required | Supported | +| General cyclotomics (trinomials) | Supported | Not discussed | Supported (Gram \(\ne I\)) | + +### 10.5 The Gram cost of switching from \(\mathrm{ct}\) to \(\mathrm{Tr}_H\) + +For \(\Phi = X^d+1\) with \(\sigma_{-1}\): +- \(\mathrm{ct}\): Gram = \(I\). No correction needed. Beautifully simple. +- \(\mathrm{Tr}_H\): Gram \(\ne I\). Hachi absorbs this into the packing map \(\psi:(R_q^H)^{d/k}\to R_q\) (Theorem 2, Eq. (8)). + +If you use \(\mathrm{Tr}_H\) instead of \(\mathrm{ct}\), you need: +- the packing map \(\psi\) (one \(O(d)\) linear map per block), +- its inverse \(\psi^{-1}\) (for verification). + +Both are explicit, \(O(d)\)-time, and defined once per parameter set. The Galois subgroup \(H\) and the map \(\psi\) are static public data. + +### 10.6 When you would want each mode + +- **Folding** (SuperNeo-style IVC) with pay-per-bit: use \(\mathrm{ct}\) mode. The evaluation homomorphism (§9) works cleanly because the Gram is trivial. +- **PCS opening** (Hachi-style recursive reduction): switch to \(\mathrm{Tr}_H\) mode when you need ring-switching at \(\alpha \in \mathbb{F}_{q^k}\). +- **Hybrid fold-then-open**: fold in \(\mathrm{ct}\) mode (accumulating ring-level claims), then at the end, open the accumulated commitment using \(\mathrm{Tr}_H\) mode for efficient PCS verification. + +The handoff between modes is seamless because the committed object \(\mathbf{z}\in R_q^{n_R}\) is the same either way. diff --git a/docs/FIELD_OPS_PERF.md b/docs/FIELD_OPS_PERF.md new file mode 100644 index 00000000..13a90846 --- /dev/null +++ b/docs/FIELD_OPS_PERF.md @@ -0,0 +1,237 @@ +# Field Operations Performance + +Benchmark results for scalar and packed (SIMD) field arithmetic across +platforms. All numbers are **element throughput** (median) reported by +`criterion` over 4096-element arrays (`cargo bench --bench field_arith`). + +## Prime selection + +All primes (except M31) are pseudo-Mersenne: `q = 2^k − c` where `c` is +the **smallest positive offset** such that `q` is prime and `q ≡ 5 +(mod 8)`. The congruence `q ≡ 5 (mod 8)` is required so that the +cyclotomic ring `Z_q[X]/(X^d + 1)` splits fully via NTT when `d` is a +power of two (equivalently, `−1` is a quadratic residue but not a quartic +residue mod `q`). + +The constraint `q ≡ 5 (mod 8)` forces `c ≡ 2^k − 5 (mod 8)`: + +| k mod 8 | required c mod 8 | examples | +|---------|------------------|----------| +| 0 | 3 | k=24,32,40,48,56,64,128 | +| 6 | 1 | k=30 | +| 7 | 2 | k=31 | + +For each `k`, smaller candidates were checked and found composite. For +instance at `k = 31` (`c ≡ 3 mod 8`): `2^31 − 3 = 5 × 429496729`, +`2^31 − 11 = 3 × 715827879`, so the first prime is `2^31 − 19`. + +**M31** (`2^31 − 1`, Mersenne prime, `q ≡ 7 mod 8`) is included for +comparison with plonky3 even though it does not satisfy `q ≡ 5 (mod 8)`. + +## Primes benchmarked + +| Label | Modulus | Offset | Rust type | SIMD width | +|-------|---------|--------|-----------|------------| +| fp32_24b | `2^24 − 3` | 3 | `Fp32` | AVX-512: 16, AVX2: 8, NEON: 4 | +| fp32_30b | `2^30 − 35` | 35 | `Fp32` | AVX-512: 16, AVX2: 8, NEON: 4 | +| fp32_31b | `2^31 − 19` | 19 | `Fp32` | AVX-512: 16, AVX2: 8, NEON: 4 | +| fp32_m31 | `2^31 − 1` | 1 | `Fp32` | AVX-512: 16, AVX2: 8, NEON: 4 | +| fp32_32b | `2^32 − 99` | 99 | `Fp32` | AVX-512: 16, AVX2: 8, NEON: 4 | +| fp64_40b | `2^40 − 195` | 195 | `Fp64` | AVX-512: 8, AVX2: 4, NEON: 2 | +| fp64_48b | `2^48 − 59` | 59 | `Fp64` | AVX-512: 8, AVX2: 4, NEON: 2 | +| fp64_56b | `2^56 − 27` | 27 | `Fp64` | AVX-512: 8, AVX2: 4, NEON: 2 | +| fp64_64b | `2^64 − 59` | 59 | `Fp64` | AVX-512: 8, AVX2: 4, NEON: 2 | +| fp128 | `2^128 − 275` | 275 | `Fp128` | AVX-512: 8 (SoA), AVX2: 4 (SoA), NEON: 2 (SoA) | + +--- + +## AMD Zen 5 (Ryzen 9950X / leopard) + +Backend: **AVX-512** (16-wide Fp32, 8-wide Fp64, 8-wide Fp128 SoA with +vectorized add/sub and scalar-per-lane mul). +`RUSTFLAGS='-C target-cpu=native'`, nightly toolchain. + +### Scalar (`throughput/`) + +| Field | mul | add | +|-------|-----|-----| +| fp32_24b | 1.224 Gelem/s | 2.050 Gelem/s | +| fp32_30b | 1.220 Gelem/s | 2.026 Gelem/s | +| fp32_31b | 1.212 Gelem/s | 1.866 Gelem/s | +| fp32_m31 | 1.355 Gelem/s | 1.993 Gelem/s | +| fp32_32b | 1.219 Gelem/s | 1.955 Gelem/s | +| fp64_40b | 1.018 Gelem/s | 2.074 Gelem/s | +| fp64_48b | 1.021 Gelem/s | 2.073 Gelem/s | +| fp64_56b | 0.945 Gelem/s | 2.060 Gelem/s | +| fp64_64b | 0.927 Gelem/s | 1.840 Gelem/s | +| fp128 | 0.452 Gelem/s | 1.127 Gelem/s | + +### Packed (`packed_throughput/`) + +| Field | mul | add | sub | +|-------|-----|-----|-----| +| fp32_24b | 5.362 Gelem/s | 12.76 Gelem/s | 12.74 Gelem/s | +| fp32_30b | 6.145 Gelem/s | 13.53 Gelem/s | 13.55 Gelem/s | +| fp32_31b | 6.187 Gelem/s | 13.53 Gelem/s | 13.54 Gelem/s | +| fp32_m31 | 6.943 Gelem/s | 13.56 Gelem/s | 13.50 Gelem/s | +| fp32_32b | 6.785 Gelem/s | 13.02 Gelem/s | 12.66 Gelem/s | +| fp64_40b | 1.961 Gelem/s | 5.847 Gelem/s | 5.861 Gelem/s | +| fp64_48b | 1.942 Gelem/s | 5.852 Gelem/s | 5.853 Gelem/s | +| fp64_56b | 1.937 Gelem/s | 5.847 Gelem/s | 5.796 Gelem/s | +| fp64_64b | 1.742 Gelem/s | 5.278 Gelem/s | 5.760 Gelem/s | +| fp128 | 0.284 Gelem/s | 2.314 Gelem/s | 3.175 Gelem/s | + +### Packed speedup over scalar + +| Field | mul | add | +|-------|-----|-----| +| fp32_24b | **4.4x** | **6.2x** | +| fp32_30b | **5.0x** | **6.7x** | +| fp32_31b | **5.1x** | **7.3x** | +| fp32_m31 | **5.1x** | **6.8x** | +| fp32_32b | **5.6x** | **6.7x** | +| fp64_40b | **1.9x** | **2.8x** | +| fp64_48b | **1.9x** | **2.8x** | +| fp64_56b | **2.0x** | **2.8x** | +| fp64_64b | **1.9x** | **2.9x** | +| fp128 | **0.6x** | **2.1x** | + +### Sumcheck MACC (`packed_sumcheck_mix/`) + +`acc += eq[i] * poly[i]` loop (dominant inner loop in sumcheck provers). + +| Field | MACC | % of pure mul | +|-------|------|---------------| +| fp32_24b | 4.764 Gelem/s | 89% | +| fp32_30b | 5.352 Gelem/s | 87% | +| fp32_31b | 5.351 Gelem/s | 86% | +| fp32_m31 | 6.097 Gelem/s | 88% | +| fp32_32b | 3.409 Gelem/s | 50% | +| fp64_40b | 1.488 Gelem/s | 76% | +| fp64_48b | 1.492 Gelem/s | 77% | +| fp64_56b | 1.491 Gelem/s | 77% | +| fp64_64b | 1.141 Gelem/s | 65% | +| fp128 | 0.323 Gelem/s | 114% | + +--- + +## Apple M4 Pro (macOS / aarch64) + +Backend: **NEON** (4-wide Fp32, 2-wide Fp64, 2-wide Fp128 SoA). +`RUSTFLAGS='-C target-cpu=native'`, nightly toolchain. + +### Scalar (`throughput/`) + +| Field | mul | add | +|-------|-----|-----| +| fp32_24b | 1.129 Gelem/s | 1.426 Gelem/s | +| fp32_30b | 1.133 Gelem/s | 1.425 Gelem/s | +| fp32_31b | 1.043 Gelem/s | 1.433 Gelem/s | +| fp32_m31 | 1.319 Gelem/s | 1.435 Gelem/s | +| fp32_32b | 1.135 Gelem/s | 1.423 Gelem/s | +| fp64_40b | 0.871 Gelem/s | 1.446 Gelem/s | +| fp64_48b | 0.886 Gelem/s | 1.385 Gelem/s | +| fp64_56b | 0.891 Gelem/s | 1.442 Gelem/s | +| fp64_64b | 0.923 Gelem/s | 1.443 Gelem/s | +| fp128 | 0.444 Gelem/s | 0.938 Gelem/s | + +### Packed (`packed_throughput/`) + +| Field | mul | add | sub | +|-------|-----|-----|-----| +| fp32_24b | 3.717 Gelem/s | 5.272 Gelem/s | 5.278 Gelem/s | +| fp32_30b | 3.719 Gelem/s | 5.281 Gelem/s | 5.275 Gelem/s | +| fp32_31b | 3.719 Gelem/s | 5.283 Gelem/s | 5.268 Gelem/s | +| fp32_m31 | 3.720 Gelem/s | 5.263 Gelem/s | 5.263 Gelem/s | +| fp32_32b | 2.524 Gelem/s | 5.296 Gelem/s | 5.253 Gelem/s | +| fp64_40b | 1.253 Gelem/s | 2.648 Gelem/s | 2.645 Gelem/s | +| fp64_48b | 1.254 Gelem/s | 2.650 Gelem/s | 2.643 Gelem/s | +| fp64_56b | 1.255 Gelem/s | 2.632 Gelem/s | 2.650 Gelem/s | +| fp64_64b | 1.399 Gelem/s | 2.639 Gelem/s | 2.602 Gelem/s | +| fp128 | 0.480 Gelem/s | 1.724 Gelem/s | 2.107 Gelem/s | + +### Packed speedup over scalar + +| Field | mul | add | +|-------|-----|-----| +| fp32_24b | **3.3x** | **3.7x** | +| fp32_30b | **3.3x** | **3.7x** | +| fp32_31b | **3.6x** | **3.7x** | +| fp32_m31 | **2.8x** | **3.7x** | +| fp32_32b | **2.2x** | **3.7x** | +| fp64_40b | **1.4x** | **1.8x** | +| fp64_48b | **1.4x** | **1.9x** | +| fp64_56b | **1.4x** | **1.8x** | +| fp64_64b | **1.5x** | **1.8x** | +| fp128 | **1.1x** | **1.8x** | + +### Sumcheck MACC (`packed_sumcheck_mix/`) + +| Field | MACC | % of pure mul | +|-------|------|---------------| +| fp32_24b | 2.652 Gelem/s | 71% | +| fp32_30b | 2.660 Gelem/s | 72% | +| fp32_31b | 2.662 Gelem/s | 72% | +| fp32_m31 | 2.661 Gelem/s | 72% | +| fp32_32b | 1.991 Gelem/s | 79% | +| fp64_40b | 0.990 Gelem/s | 79% | +| fp64_48b | 0.991 Gelem/s | 79% | +| fp64_56b | 0.993 Gelem/s | 79% | +| fp64_64b | 0.795 Gelem/s | 57% | +| fp128 | 0.450 Gelem/s | 94% | + +--- + +## Notes + +### Zen 5 AVX-512 observations + +- **Fp32 add/sub** saturate at ~13–13.5 Gelem/s, close to 1 cycle per + 16-wide vector at 5 GHz. +- **M31 is the fastest Fp32 prime** for packed mul (6.94 Gelem/s) and + sumcheck MACC (6.10 Gelem/s), because `C = 1` minimizes the reduction + chain. +- **Fp32 mul** is latency-bound by the 2-fold Solinas reduction chain + (~18 cycles per vector). +- **Fp64 sub-word** primes (40b, 48b, 56b) show nearly identical packed + mul throughput (~1.94 Gelem/s), since the vectorized schoolbook + multiply + Solinas reduction dominates regardless of bit-width. +- **Fp64 64b** is slower than sub-word variants due to multi-stage + overflow tracking in the Solinas reduction. +- **Fp32 32b sumcheck MACC** drops to 50% of pure mul, the worst ratio, + because the carry-based add correction creates additional dependencies + in the `acc += eq * poly` loop. +- **Fp128** packed backend now uses SoA layout (8-wide) with vectorized + add/sub via `__m512i`. Add improved **2.1x** (1.13 → 2.31 Gelem/s), + sub improved **2.4x** (1.34 → 3.18 Gelem/s). Mul remains scalar + per-lane and regressed 0.6x (0.45 → 0.28 Gelem/s) due to SoA + pack/unpack overhead. Sumcheck MACC is -7% (0.35 → 0.32 Gelem/s); + MACC exceeds pure-mul throughput (114%) because the accumulation loop + avoids the SoA store overhead that the throughput benchmark incurs. + +### M4 Pro NEON observations + +- **NEON is 4-wide for Fp32, 2-wide for Fp64/Fp128**, so maximum + speedup is 4x and 2x respectively (vs 16x/8x for AVX-512). +- **Fp32 packed mul** is uniform at ~3.72 Gelem/s for all sub-word + primes (24b–31b, including M31), unlike Zen 5 where M31 is notably + faster. The 4-wide NEON `vmull_u32` + reduction is the bottleneck. +- **Fp32 32b packed mul** drops to 2.52 Gelem/s (carry-based path). +- **Fp64 packed mul** ~1.25 Gelem/s for all sub-word primes, ~1.40 + for 64b — the 64b prime is *faster* on NEON, opposite to Zen 5. +- **Fp128 packed add** 1.72 Gelem/s = **1.8x** scalar speedup; sub + 2.11 Gelem/s = **2.2x**, both close to the theoretical 2x from + 2-wide `uint64x2_t`. Mul 0.48 Gelem/s ≈ **1.1x** (scalar per-lane). +- **Sumcheck MACC** is 71–72% of pure mul for sub-word Fp32, 79% for + Fp64 sub-word, and 94% for Fp128 — higher than Zen 5 ratios. + +### Reduction strategy by field width + +| Field | Reduction method | +|-------|-----------------| +| Fp32, BITS ≤ 31 | `min(t, t−P)` — single unsigned compare + blend | +| Fp32, BITS = 32 | carry-based: detect overflow, conditionally add `C`, then subtract `P` if `≥ P` | +| Fp64, BITS ≤ 62 | 2-fold Solinas in u64: `(lo & mask) + c * (lo >> k)`, repeat | +| Fp64, BITS = 64 | vectorized schoolbook 64×64→128 + multi-stage Solinas with overflow tracking | +| Fp128 add/sub | vectorized 128-bit add/sub with carry/borrow propagation via `__m512i` | +| Fp128 mul | scalar per-lane: 9-limb Solinas via u64 decomposition | diff --git a/docs/HACHI_DIGEST.md b/docs/HACHI_DIGEST.md new file mode 100644 index 00000000..c0d0328b --- /dev/null +++ b/docs/HACHI_DIGEST.md @@ -0,0 +1,1362 @@ +# Hachi Digest (for side-by-side comparison with SuperNeo) + +This file captures the parts of `paper/hachi.pdf` that are most actionable for understanding Hachi’s **parameterization** and **protocol shape**, in a compact AI-readable format. + +Primary source: `paper/hachi.pdf` (“Hachi: Efficient Lattice-Based Multilinear Polynomial Commitments over Extension Fields”). + +## Canonical parameter tuples (from paper) + +The paper gives a concrete benchmark parameter set (ℓ = 30) in Figure 9 and uses it to estimate proof size (~55KB) in §5.2. + +```yaml +hachi_parameter_tuples: + - id: l30_benchmark_first_round + witness_num_vars_ell: 30 + base_field: + q: 4294967197 # ~2^32, prime (paper Fig. 9) + ring_coeffs: "Z_q (aka F_q)" + extension_field_for_sumcheck: + k: 4 + field: "F_{q^4}" # paper §5.1, §5.4 + cyclotomic_ring: + phi: "X^d + 1" + alpha: 10 + d: 1024 + ring: "R_q = Z_q[X]/(X^d + 1)" + split_and_fold_params: + m: 10 + r: 10 + commitment_matrices_heights: + nA: 1 + nB: 1 + nD: 1 + decomposition_and_challenges: + decomposition_base_b: 16 + delta: 8 # decomposition length (Fig. 9) + tau: 4 # expansion factor for decomposing z (Fig. 9) + omega_L1: 16 + c_nonzero_coeffs_in_challenge: 16 + norm_bounds: + z_Linf_bound: 30583 + next_witness_Linf_bound: 8 + next_round_witness: + size: 226 + proof_size_estimate: + first_round_sumcheck: "~7.3KB" + adaptation_overhead: "~4.8KB" + greyhound_subproof: "~43KB" + total: "~55.1KB" + timings_reported: + verify_ms_server: 227 # paper §5.2 narrative (first round), plus Fig. 8 context + verify_ms_server_greyhound: 130 # cited as Greyhound estimate in §5.2 narrative +``` + +Notes: + +- The paper’s 55.1KB estimate is explicitly derived in §5.2 (“To conclude, the total evaluation proof can be estimated to be 7.3 + 4.8 + 43 KB = 55.1KB.”). +- This tuple is the “Hachi + compose with Greyhound” estimate for ℓ = 30, not “Hachi alone forever”; Hachi’s design explicitly allows switching to Greyhound/LaBRADOR at small witness sizes. + +## Exact counts (from the paper’s concrete section) + +- Total explicit concrete parameter tables in the paper: **1** (Figure 9; ℓ = 30). +- Concrete benchmark variable counts shown: **3** (ℓ ∈ {26, 28, 30} in Figure 8 timings), but only ℓ = 30 is fully parameterized in Figure 9. +- Unique cyclotomic family used: **1** (`X^d + 1` with `d` power of two). + +## What Hachi suggests (for its purpose) + +High-level message (from the abstract + technical overview): + +- Use **sum-check** to get fast verification, but avoid doing sum-check “over the ring”. +- Use **ring switching** (evaluate at a random α in an extension field) so that the verifier’s checks are field-native and do not require expensive \(R_q\) multiplication. +- Use a **generic reduction** to convert evaluation proofs over extension fields \(F_{q^k}\) into equivalent ring statements over \(R_q\), enabling extension-field evaluation support for lattice PCS. + +Concrete implication (paper §5.3–§5.4): + +- Hachi can pick **larger ring dimensions** (e.g. \(d=1024\)) than Greyhound’s typical \(d=64\) and still keep verification efficient; larger \(d\) helps commitment time (fewer ring mults, NTT-friendly structure) and enables very sparse challenges. + +## Fit into this repo design (easy vs harder) + +- Easy fit (already aligned): + - Hachi’s core ring family is power-of-two cyclotomic \(R_q = Z_q[X]/(X^d+1)\), which matches this repo’s existing algebra/ring direction. +- Harder pieces (protocol-level, not yet fully implemented here): + - Ring switching pipeline (lifting ring equations to \(Z_q[X]\), evaluate at random α in \(F_{q^k}\), sum-check over \(F_{q^k}\)). + - The “compose with Greyhound” handoff (treating the reduced witness as a short ring instance for an existing PCS/proof system). + +## Notation glossary (from `paper/hachi.pdf`) + +Hachi’s paper reuses common lattice-proof-system symbols; here are the ones that tend to confuse on first read. + +### Base objects + +- **\(q\)**: prime modulus; base ring/field is \(Z_q\) (paper treats \(Z_q\) and \(F_q\) interchangeably). +- **\(d = 2^\alpha\)**: cyclotomic ring dimension (power of two). +- **\(\alpha\)**: shorthand for \(\log_2 d\) (used heavily in §3’s “variable count after transformation” formulas). +- **\(R\)**: integer cyclotomic ring \(Z[X]/(X^d+1)\). +- **\(R_q\)**: cyclotomic ring mod \(q\): \(Z_q[X]/(X^d+1)\). +- **\(F_{q^k}\)**: extension field used to run sum-check with negligible soundness error. +- **\(\kappa\)**: shorthand for \(\log_2 k\) when \(k\) is a power of two (this is the \(\kappa\) used in §3.2 and in the “\(\ell-\alpha+\kappa\)” variable-count formulas). +- **\(\sigma_i\)**: Galois automorphism \(X \mapsto X^i\) on \(R\) / \(R_q\) (with \(i\in (\mathbb{Z}/2d\mathbb{Z})^\times\)). +- **\(R_q^H\)**: fixed ring under a subgroup \(H\) of automorphisms; becomes a subfield isomorphic to \(F_{q^k}\) under conditions (Lemma 1 informal in §1.3). +- **\(\mathrm{Tr}_H\)**: trace map \(R_q \to R_q^H\). +- **\(\psi\)**: an efficiently computable bijection \((R_q^H)^{d/k} \to R_q\) used to turn trace-of-product into inner products (Theorem 1 informal in §1.3). + +### Multilinear polynomials / sizes + +- **\(\ell\)**: number of variables of the multilinear polynomial (so witness length is \(2^\ell\)). +- **\(L := 2^\ell\)**: number of coefficients / evaluation-table length. + +### Split-and-fold / decomposition parameters (Figure 9) + +The paper uses **\(m, r\)** as “folding parameters” (they are *not* the same \(m\) as “#constraints” in SuperNeo’s CCS section). + +- **\(m, r\)**: split-and-fold parameters controlling the shape of the quadratic relation after one reduction step. +- **\(b\)**: decomposition base (e.g. 16 in Fig. 9). +- **\(\delta\)**: decomposition length for base witness (e.g. 8 in Fig. 9; essentially \(\lceil \log_b q\rceil\) in spirit). +- **\(\tau\)**: expansion factor / extra length parameter for decomposing intermediate vectors (e.g. 4 in Fig. 9). +- **\(\omega\)**: \(\ell_1\)-norm bound on a sparse challenge (Fig. 9). +- **\(c\)**: number of nonzero coefficients in a sparse challenge polynomial (Fig. 9). + +### Commitment matrix “heights” (Figure 9) + +Hachi uses commitment matrices \(A,B,D\) (not the same “Ajtai A” notation as in SuperNeo). + +- **\(n_A, n_B, n_D\)**: heights (row counts) of the corresponding commitment matrices in the composed relation (Fig. 9 uses all 1). + +## Protocol overview (what is proved, and what is sent) + +This is an end-to-end walkthrough of the **full Hachi protocol**, in the order it’s built in `paper/hachi.pdf`. + +### What Hachi is (PCS statement + design goal, **field-first**) + +The digest previously jumped straight to the §4 “ring PCS statement”. That is **not the full picture**: the paper’s *natural* PCS interface is the usual one where the **witness coefficients are in the base field** \(Z_q \cong F_q\) but the **evaluation point is in an extension field** \(F_{q^k}\) (because sumcheck / batching wants negligible soundness error). + +Concretely, the “true” opening statement the paper starts from (Intro + §3.2) is: + +- **Witness polynomial (true witness)**: \(f \in Z_q^{\le 1}[X_1,\dots,X_\ell]\) (equivalently \(F_q^{\le1}[\cdot]\)), with coefficient table \((f_i)_{i\in\{0,1\}^\ell}\subset Z_q\). +- **Claim (extension-field point/value)**: for a public point \(x=(x_1,\dots,x_\ell)\in F_{q^k}^\ell\), prove \(f(x)=y\in F_{q^k}\). + +Hachi is engineered so that the verifier’s heavy checking runs as **sumcheck over \(F_{q^k}\)** (fast), while commitments / Module-SIS structure remain over the cyclotomic ring \(R_q = Z_q[X]/(X^d+1)\). + +### Step 0 (paper §3.2 → §3.1): the missing “embedding” step (from **\(F_q\)-witness @ \(F_{q^k}\)-point** to a ring statement) + +The paper’s §3 is exactly the bridge from the “true” statement above to the §4 ring PCS statement. + +#### 0.A (paper §3.2): reduce **\(f\in Z_q[\cdot]\)** at **\(x\in F_{q^k}^\ell\)** to one evaluation over \(F_{q^k}\) + +Assume \(k\) is a power of two and write \(k = 2^\kappa\) (this \(\kappa\) is the one used in §3). Split variables into the first \(\kappa\) and the remaining \(\ell-\kappa\). The evaluation can be rewritten (paper Eq. (11)) as: + +\[ +y \;=\;\sum_{i\in\{0,1\}^\kappa}\Big(\prod_{t=1}^{\kappa} x_t^{i_t}\Big)\cdot y_i, +\quad\text{where}\quad +y_i \;:=\; f_{i}(\,x_{\kappa+1},\dots,x_\ell\,)\in F_{q^k}. +\] + +So the prover can send the \(k=2^\kappa\) *partial evaluations* \((y_i)_i\) (and the verifier checks the recombination); what remains is to prove each \(y_i\) is well-formed. + +Paper detail (§3.2, right after Eq. (11)): the verifier can compute \(y_{0\ldots 0}\) from the claimed \(y\) and the other \(y_i\), so in principle only **\(k-1\)** of the partial evaluations need to be transmitted. + +To make that “prove all \(y_i\)” look like **one** extension-field evaluation, §3.2 defines \(F_{q^k} := F_q[Z]/\varphi(Z)\) and builds an \((\ell-\kappa)\)-variate multilinear polynomial \(f' \in F_{q^k}[X_{\kappa+1},\dots,X_\ell]\) by *embedding* the \(k\) slices \((f_i)_i\) into the \(F_q\)-basis \(1,Z,Z^2,\dots,Z^{k-1}\) (paper §3.2): + +\[ +f'(X_{\kappa+1},\dots,X_\ell) +:=\sum_{i\in\{0,1\}^\kappa} +f_i(X_{\kappa+1},\dots,X_\ell)\cdot Z^{\sum_{t=1}^{\kappa} i_t 2^{t-1}}. +\] + +Then \(f'(x_{\kappa+1},\dots,x_\ell)=\sum_i y_i\cdot Z^{(\cdot)}\) holds as an algebraic identity. + +**Critical caveat (security / binding):** it is **not** generally sound to claim that “proving the single packed value \(f'(x_{\kappa+1},\dots,x_\ell)\)” suffices to prove that *each* \(y_i\) is correct, because the coefficients \(y_i\) live in \(F_{q^k}\) (not in the ground field \(F_q\)). + +Concretely, the linear map + +\[ +(y_i)_{i\in\{0,1\}^\kappa}\in (F_{q^k})^{2^\kappa} +\;\longmapsto\; +\sum_i y_i \cdot Z^{(\cdot)} \in F_{q^k} +\] + +is **\(F_{q^k}\)-linear** and therefore has a large kernel whenever \(2^\kappa>1\). So many distinct tuples \((y_i)_i\) produce the *same* packed sum. A toy example for \(k=2\) (basis \(\{1,Z\}\)): the two different pairs \((y_0,y_1)\) and \((y_0+Z,\; y_1-1)\) satisfy + +\[ +(y_0+Z) + (y_1-1)\cdot Z \;=\; y_0 + y_1\cdot Z, +\] + +so the packed value alone does not pin down \(y_0,y_1\). + +This is exactly the “basis is only independent over the ground field” pitfall called out in `paper/fri-binius.pdf`, where they explain that a basis \((\beta_v)\) is linearly independent over \(K\) but **not** over its extension \(L\); hence basis-combining \(L\)-valued claims is insecure. See the strawman discussion around Figure 1 in: + +- `paper/fri-binius.pdf` §1.3 “Ring-Switching”, “A strawman approach”, Figure 1, and the paragraph beginning “While this protocol is complete, it’s not secure.” (pages 4–6 in this PDF copy). + +**How to fix (high-level, sumcheck-style):** you must reduce the “\(y_i\) are well-formed” constraints into **ground-field (\(F_q\)) constraints** before basis-combining / packing them. + +The standard way (as in Fri-Binius ring-switching) is: + +1. **Basis-decompose each \(y_i\) over \(F_q\)**: write \(y_i = \sum_{u=0}^{k-1} y_{u,i}\,Z^u\) with \(y_{u,i}\in F_q\). +2. Also basis-decompose the extension-field weights (the equality-polynomial weights / monomial weights) into \(F_q\) coefficients. +3. Check the resulting family of \(F_q\)-valued equalities “slice-wise” (over the \(u\) index), and then **batch them with an additional sumcheck** so the verifier only pays polylog overhead. + +After this extra sumcheck layer pins down the \(F_q\)-slices, packing becomes injective again (because it is now combining **\(F_q\)-vectors** against an \(F_q\)-basis), and the reduction from “many partial claims” to “one packed claim” becomes sound. + +##### 0.A.1 Concrete “extra sumcheck” shape (Fri-Binius Eq. (12) style), and how many rounds it costs + +This subsection spells out the *exact* reason the extra sumcheck has \(\ell\) rounds (and how it fits the objects already in §3.2). + +Let \(K:=F_q\) and \(L:=F_{q^k}\), and assume \(k=2^\kappa\) for some \(\kappa\) (as in §3.2). Fix a \(K\)-basis \((\beta_u)_{u\in\{0,1\}^\kappa}\) of \(L\) (in §3.2 the paper chooses \(1,Z,\dots,Z^{k-1}\), which is just one such basis). + +The goal is to prove the family of claims (paper after Eq. (11)): + +\[ +\forall v\in\{0,1\}^\kappa:\quad +y_v \stackrel{?}{=} f_v(x_{\kappa+1},\dots,x_\ell)\in L, +\] + +where \(f_v\in K^{\le1}[X_{\kappa+1},\dots,X_\ell]\) is the “slice” of \(f\) with the first \(\kappa\) variables fixed to \(v\). + +The unsafe step is to basis-combine these \(L\)-valued equalities directly. The safe replacement is to basis-decompose everything so the equalities become \(K\)-valued first. + +1) **Basis-decompose the prover’s \(L\)-claims**: for each \(v\in\{0,1\}^\kappa\), write + +\[ +y_v = \sum_{u\in\{0,1\}^\kappa} y_{u,v}\,\beta_u, +\quad\text{with } y_{u,v}\in K. +\] + +2) **Expand \(f_v(x_{\kappa+1},\dots,x_\ell)\) as a \(K\)-weighted sum.** Let \(\ell':=\ell-\kappa\) and index \(w\in\{0,1\}^{\ell'}\). Then + +\[ +f_v(x_{\kappa+1},\dots,x_\ell) += +\sum_{w\in\{0,1\}^{\ell'}} +\mathrm{eq}(x_{\kappa+1},\dots,x_\ell;\,w)\cdot f(v,w), +\] + +where \(f(v,w)\in K\) is the \((v,w)\) Lagrange coefficient of \(f\), and \(\mathrm{eq}(\cdot;\,w)\in L\) is the multilinear equality indicator value. + +3) **Basis-decompose the (public) weights**: for each \(w\), decompose the \(L\)-element \(\mathrm{eq}(x_{\kappa+1},\dots,x_\ell;\,w)\) in the same basis: + +\[ +\mathrm{eq}(x_{\kappa+1},\dots,x_\ell;\,w) += +\sum_{u\in\{0,1\}^\kappa} A_{w,u}\,\beta_u, +\quad\text{with } A_{w,u}\in K, +\] + +where the \(A_{w,u}\) are deterministically computable from the public point \((x_{\kappa+1},\dots,x_\ell)\) and the chosen basis. + +4) **Now each coordinate is a \(K\)-statement**: equating coefficients of \(\beta_u\) yields the family of \(K\)-equalities + +\[ +\forall u,v\in\{0,1\}^\kappa:\quad +y_{u,v} \stackrel{?}{=} \sum_{w\in\{0,1\}^{\ell'}} A_{w,u}\cdot f(v,w). +\] + +5) **Pack the \(f(v,w)\) table into one \(L\)-multilinear** (this is the same “packing” idea as §3.2, but applied to a \(K\)-table so it is information-preserving): + +\[ +f'(w) := \sum_{v\in\{0,1\}^\kappa} f(v,w)\,\beta_v \in L, +\] + +so \(f'\) has \(\ell'=\ell-\kappa\) variables over \(L\). + +6) **Batch and sumcheck.** Define the combined \(L\)-valued claims (these are the secure analog of the strawman’s linear combination step): + +\[ +\hat y_u := \sum_{v\in\{0,1\}^\kappa} y_{u,v}\,\beta_v \in L. +\] + +Then the \(K\)-equalities above imply the \(L\)-equalities + +\[ +\forall u\in\{0,1\}^\kappa:\quad +\hat y_u \stackrel{?}{=} \sum_{w\in\{0,1\}^{\ell'}} A_{w,u}\cdot f'(w). +\] + +Finally, batch over \(u\) with a random point \(r''\in L^\kappa\) and apply sumcheck to the identity + +\[ +\sum_{u\in\{0,1\}^\kappa} \mathrm{eq}(u;\,r'')\cdot \hat y_u +\stackrel{?}{=} +\sum_{w\in\{0,1\}^{\ell'}} +\Big(\sum_{u\in\{0,1\}^\kappa}\mathrm{eq}(u;\,r'')\cdot A_{w,u}\Big)\cdot f'(w). +\] + +This is exactly the structural form of Fri-Binius Eq. (12). The sum ranges over \((u,w)\in\{0,1\}^\kappa\times\{0,1\}^{\ell'}\), so the sumcheck has **\(\kappa+\ell'=\ell\) rounds**, i.e. it costs **\(+\kappa=\log_2 k\)** more rounds than a hypothetical scheme that only needed to open \(f'\) (which has \(\ell'\) variables). + +Asymptotically, prover time for this added sumcheck is linear in the domain size \(2^{\kappa+\ell'}=2^\ell\) (times poly\((\ell,k)\) factors), i.e. \(\Theta(2^\ell)\) field operations in \(L\) for the sumcheck layer, plus one opening of the \(L\)-PCS on \(f'\) at the final point. + +#### 0.B (paper §3.1): embed \(F_{q^k}\) inside \(R_q\) and turn extension-field inner products into trace statements over \(R_q\) + +Now treat the remaining claim as an evaluation over \(F_{q^k}\) (equivalently over the subfield \(R_q^H \cong F_{q^k}\) from Lemma 5). §3.1 then provides: + +- **A subfield of \(R_q\)**: for \(q \equiv 5 \pmod 8\) and \(k\mid d/2\), define \(H=\langle\sigma^{-1},\sigma^{4k+1}\rangle\subset\mathrm{Aut}(R)\). Then the fixed ring \(R_q^H\) is a field and \(R_q^H \cong F_{q^k}\). (Lemma 5.) +- **A packing bijection** \(\psi:(R_q^H)^{d/k}\to R_q\). (Eq. (8), Theorem 2.) +- **A trace/inner-product identity** (Theorem 2): + + \[ + \mathrm{Tr}_H\big(\psi(a)\cdot \sigma^{-1}(\psi(b))\big) \;=\; \frac{d}{k}\cdot \langle a,b\rangle, + \quad a,b \in (R_q^H)^{d/k}. + \] + +Concretely (paper around Eq. (10)), pick \(\alpha:=\log_2 d\) and split the \(\ell\) variables into an “outer” prefix of length \(\ell-\alpha+\kappa\) and an “inner” suffix of length \(\alpha-\kappa\). Write indices as \(i\in\{0,1\}^{\ell-\alpha+\kappa}\) and \(j\in\{0,1\}^{\alpha-\kappa}\). Define: + +- **Packed coefficient blocks**: \(F_i := \psi\big((f_{i\parallel j})_{j\in\{0,1\}^{\alpha-\kappa}}\big)\in R_q\). +- **Packed monomial block at the suffix**: + + \[ + v := \psi\Big(\big(\prod_{t=1}^{\alpha-\kappa} x_{\ell-\alpha+\kappa+t}^{\,j_t}\big)_{j\in\{0,1\}^{\alpha-\kappa}}\Big)\in R_q. + \] + +Then the extension-field evaluation is reduced to checking a **single trace equation** in \(R_q\) involving: + +\[ +Y \;:=\; \sum_{i\in\{0,1\}^{\ell-\alpha+\kappa}} +\Big(\prod_{t=1}^{\ell-\alpha+\kappa} x_t^{i_t}\Big)\cdot F_i +\;\in\; R_q, +\] + +namely \(\mathrm{Tr}_H\big(Y\cdot \sigma^{-1}(v)\big)=\tfrac{d}{k}\,y\). The prover sends this single ring element \(Y\). + +What remains is to prove that \(Y\) is well-formed, i.e. that it is *exactly* the evaluation of the \((\ell-\alpha+\kappa)\)-variate ring polynomial \(F := (F_i)_i \in R_q^{\le 1}[X_1,\dots,X_{\ell-\alpha+\kappa}]\) at the point \((x_1,\dots,x_{\ell-\alpha+\kappa})\) (viewing those \(x_t\) as elements of the subfield \(R_q^H\subset R_q\)). This is the “smaller multilinear evaluation claim over \(R_q\)” that becomes the input to the §4 core PCS. + +### Step 1 (paper §4): the internal ring PCS statement Hachi actually proves + +After §3’s transformation, Hachi reduces extension-field evaluation claims to the §4 “core” ring PCS statement: + +- **Witness polynomial (ring core)**: \(f \in R_q^{\le 1}[X_1,\dots,X_{\ell'}]\) with coefficients in \(R_q\) (this \(f\) is the reduced ring polynomial constructed in Step 0.B; i.e., it is the \(F := (F_i)_i\) from above, just returning to the paper’s §4 notation), +- **Claim**: for a public point \(x\in R_q^{\ell'}\) (in the reduction flow, typically \(x\in (R_q^H)^{\ell'}\subset R_q^{\ell'}\)), prove \(f(x)=u\in R_q\), + +where (per §3.2) \(\ell' = \ell - \alpha\) in the important special case “coefficients in \(Z_q\), point in \(F_{q^k}\)”, with \(d=2^\alpha\). + +From here on, we are *inside* §4. The paper continues to write the ring instance as \(\ell\)-variate; in digest notation, you can read the \(\ell\) used in Steps A/B/C below as \(\ell'\) (the post-§3 reduced variable count). + +### Step A (paper §4.1): Commit to the coefficient table (inner + outer commitments) + +Hachi’s commitment structure is Greyhound-style: it commits to the coefficient table in blocks using two Ajtai commitments. + +Split \(\ell = m + r\) with \(m \approx r\). Define the block slices \(f_i^\top := (f_{i\|j})_{j\in\{0,1\}^m}\in R_q^{2^m}\) for each outer index \(i\in\{0,1\}^r\). (This is the “\(f_i\)” notation right below Equation (12).) + +Commitment construction (Equations (13)–(14)): + +- **Decompose each slice**: \(s_i := G^{-1}_{2^m}(f_i)\in R_q^{2^m\delta}\), where \(\delta=\lceil \log_b q\rceil\). (Eq. (13).) +- **Inner Ajtai commit**: \(t_i := A s_i \in R_q^{n_A}\) for all \(i\in[2^r]\). +- **Decompose the inner commit**: \(\hat t_i := G^{-1}_{n_A}(t_i)\). +- **Outer Ajtai commit**: stack all \(\hat t_i\) and commit: + - \(u := B[\hat t_1;\dots;\hat t_{2^r}] \in R_q^{n_B}\). (Eq. (14).) + +So: + +- **Commitment** is \(u \in R_q^{n_B}\). +- **Opening witness** (conceptually) is \((s_i,\hat t_i)_i\). +- Knowledge/binding is phrased in terms of “weak openings” (definition right after Eq. (14), Lemma 7). + +### Step B (paper §4.2): Reduce “open \(f(x)=u\)” to “prove knowledge of short vectors satisfying a public linear system” + +This is the core reduction pipeline of Hachi’s opening proof. + +#### B.0. Descriptive names for the “hat” variables (implementation-minded) + +The paper uses “hats” (\(\hat{\cdot}\)) for **digit-decomposed** objects (small coefficients), and uses a few single-letter vectors that are easy to lose track of. Here is a naming map that matches the paper’s definitions in §4.1–§4.2 and is meant to be used as docstring text during implementation. + +- **\(s_i\)** = **block_digits** (digits of the \(i\)-th coefficient block): + - \(s_i = G^{-1}_{2^m}(f_i)\). (Eq. (13).) + - Shape: vector of ring elements, length \(2^m\cdot\delta\). + +- **\(t_i\)** = **inner_commitment_block_i** (Ajtai commit of the \(i\)-th block digits): + - \(t_i := A s_i \in R_q^{n_A}\). + +- **\(\hat t_i\)** = **inner_commitment_digits_block_i** (digits of \(t_i\)): + - \(\hat t_i := G^{-1}_{n_A}(t_i)\). + +- **\(u\)** = **outer_commitment** (the actual PCS commitment output): + - \(u := B[\hat t_1;\dots;\hat t_{2^r}] \in R_q^{n_B}\). (Eq. (14).) + +- **\(w_i\)** = **block_partial_eval_i** (block \(i\)’s contribution after plugging the “\(a\)” half of the opening point): + - \(w_i := a^\top G_{2^m} s_i \in R_q\). (Defined right before Eq. (16).) + +- **\(\hat w_i\)** = **block_partial_eval_digits_i** (digits of \(w_i\)): + - \(\hat w_i := G^{-1}_1(w_i) \in R_q^\delta\). + +- **\(v\)** = **aux_commitment_to_block_partials** (commitment to all \(\hat w_i\)): + - \(v := D\hat w \in R_q^{n_D}\). (Eq. (16).) + +- **\(c\)** = **fold_challenge_vector** (short/sparse ring challenge used to fold blocks): + - \(c=(c_1,\dots,c_{2^r})\in C^{2^r}\), with \(\|c_i\|_1\le \omega\). + +- **\(z\)** = **folded_block_digits** (folded witness over block digits): + - \(z := \sum_{i=1}^{2^r} c_i s_i\). (Eq. (18)/(19) discussion.) + - This is the key “compress all blocks into one witness” object. + +- **\(\hat z\)** = **folded_block_digits_redecomposed** (extra decomposition of \(z\) after coefficient growth): + - \(\hat z := J^{-1}_{2^m}(z)\), where \(J\) is the gadget matrix sized for \(\tau\) digits (so \(\hat z\) has length \(2^m\cdot\delta\cdot\tau\)). (Right after Eq. (20).) + +- **\(r\)** = **modulus_quotient_witness** / **slack_witness** (ring-switching quotient): + - \(Mz = y + (X^d+1)\cdot r\) over \(Z_q[X]\). (§4.3.) + - In the “real” protocol, \(r\) is digit-decomposed as \(r=\sum_u b^u r_u\) and the prover commits to the digit vectors \(r_u\). (See §4.3 and our C.2.1.) + +#### B.1. Rewrite evaluation as a bilinear form (Eq. (12)) + +Define the monomial vectors from the evaluation point \(x\): + +- \(b^\top := (x_1^{i_1}\cdots x_r^{i_r})_{i\in\{0,1\}^r}\in R_q^{2^r}\) +- \(a^\top := (x_{r+1}^{j_1}\cdots x_\ell^{j_m})_{j\in\{0,1\}^m}\in R_q^{2^m}\) + +Then the evaluation can be written as Equation (12), which motivates everything that follows. + +#### B.2. Introduce intermediate values \(w_i\) and commit to them (Eq. (16)) + +Using the decomposed slices \(s_i\), Equation (15) rewrites the opening equation using \(s_i\). + +Define the intermediate ring elements: + +- \(w_i := a^\top G_{2^m} s_i \in R_q\). (Defined right before Eq. (16).) + +Then: + +- \(u = b^\top w\). (Eq. (17).) + +The prover also commits to \(w\) using another Ajtai-style commitment (Eq. (16)): + +- decompose \(w_i\) to \(\hat w_i := G^{-1}_1(w_i)\in R_q^\delta\), +- stack \(\hat w := (\hat w_1,\dots,\hat w_{2^r})\), +- compute **\(v := D \hat w \in R_q^{n_D}\)**. (Eq. (16).) + +This is the first prover message in Figure 3: + +- **P → V**: send \(v\). + +#### B.3. Fold all the \(s_i\) using a short challenge vector \(c\) (Eqs. (18)–(19)) + +The verifier samples a short/sparse challenge vector: + +- **V → P**: \(c=(c_1,\dots,c_{2^r}) \leftarrow C^{2^r}\), where \(C \subset \{c\in R_q : \|c\|_1 \le \omega\}\). (See the paragraph introducing \(c\).) + +The prover folds the witness: + +- \(z := \sum_{i=1}^{2^r} c_i s_i \in R_q^{2^m\delta}\). (Immediately after defining \(c\).) + +Two crucial linear identities then hold: + +- \(a^\top G_{2^m} z = (c^\top \otimes G_1)\hat w\). (Eq. (18).) +- \(A z = (c^\top \otimes G_{n_A})\hat t\). (Eq. (19).) + +These relate the folded witness \(z\) to the already-committed objects \(\hat w,\hat t\). + +#### B.4. Decompose \(z\) further and form one big “unstructured linear relation” (Eq. (20)) + +Because coefficients of \(z\) are larger, the prover further decomposes \(z\) using another gadget matrix \(J\): + +- pick a bound \(\beta\) on \(\|z\|_\infty\), define \(\tau:=\lceil \log_b \beta\rceil\), +- compute \(\hat z := J^{-1}_{2^m}(z)\in R_q^{2^m\delta\tau}\). + +At this point, the prover’s remaining task becomes: + +> prove knowledge of a short vector \((\hat w,\hat t,\hat z)\) satisfying a public linear system over \(R_q\). (Eq. (20).) + +Equation (20) is exactly that public linear system: + +- it includes the matrices \(D,B,A\), +- it includes the evaluation-derived vectors \(a,b\), +- it includes the verifier challenge \(c\), +- and it enforces: (i) consistency with the commitments \(u,v\), (ii) the opening equation, and (iii) the fold relations (18)–(19). + +##### B.4.1 Eq. (20) rewritten as a list of constraints (no hats, descriptive meaning) + +Mentally, Eq. (20) is just a *bundling* of several checks into one linear system \(M_{\text{big}}\cdot \text{witness} = \text{statement}\). Written as explicit constraints, the prover is proving existence of **small** objects: + +- **`block_partial_eval_digits`** = \(\hat w\) +- **`inner_commitment_digits`** = \(\hat t\) +- **`folded_block_digits_redecomposed`** = \(\hat z\) (and implicitly \(z = J\hat z\)) + +such that all of the following hold: + +1. **Aux-commitment consistency (ties \(\hat w\) to the sent commitment \(v\))**: + - \(D\hat w = v\). (Eq. (16).) + +2. **Main commitment consistency (ties \(\hat t\) to the sent commitment \(u\))**: + - \(B\hat t = u\). (Eq. (14).) + +3. **Evaluation equation (ties the claimed opening value \(u\) to the partials \(w\))**: + - If \(w := G_{2^r}\hat w\) then \(b^\top w = u\). (Eq. (17).) + +4. **Fold-consistency: partial-eval folding matches the folded witness \(z\)**: + - Let \(z := J\hat z\) (so \(z\) is the “recomposed” folded block-digits witness). + - Then \(a^\top G_{2^m} z = (c^\top\otimes G_1)\hat w\). (Eq. (18).) + +5. **Fold-consistency: inner-commitment folding matches the same folded witness \(z\)**: + - \(A z = (c^\top\otimes G_{n_A})\hat t\). (Eq. (19).) + +6. **Smallness/range constraints (coefficient bounds)**: + - Coefficients of \(\hat w,\hat t,\hat z\) lie in the intended digit ranges (bounded by the gadget decomposition), and the protocol’s range-check machinery in §4.3 ultimately enforces these via the \(H_0\) constraint. + +Figure 3 shows this “conceptual protocol”, ending with the prover *sending* \((\hat w,\hat t,\hat z)\), but the paper immediately notes: + +- in the **final scheme**, the prover does not send these in the clear; instead it proves knowledge of them. + +### Step C (paper §4.3): Prove the unstructured linear relation + coefficient smallness via ring switching + sumcheck over \(F_{q^k}\) + +This is the main verifier-efficiency trick: transform ring relations into field relations (via evaluation at a random \(\alpha\)), then apply sumcheck over \(F_{q^k}\). + +#### C.1. The generic linear relation they want to prove + +They define: + +- \(R^{\mathrm{lin}}_{q,d,n,\mu,b}\): given public \(M\in R_q^{n\times \mu}\), \(y\in R_q^n\), prove knowledge of \(z\in R_q^\mu\) such that \(Mz=y\) and \(\|z\|_\infty \le b-1\). + +Eq. (20) is an instance of this relation (with \(z=(\hat w,\hat t,\hat z)\)). + +#### C.2. Ring switching (Figure 4): lift equality in \(R_q\) to an identity in \(Z_q[X]\), then evaluate at random \(\alpha\) + +Because \(R_q = Z_q[X]/(X^d+1)\), the ring equation \(Mz=y\) holds iff there exists a “slack” polynomial vector \(r\) such that (paper §4.3, “Ring switching.”): + +- \(Mz = y + (X^d+1)\cdot r\) over \(Z_q[X]\). + +##### C.2.1 Critical “hidden detail”: \(r\) is gadget-decomposed, and the protocol commits to the digits + +The paper explicitly notes that **both \(z\) and \(r\)** are prover witnesses, and while the verifier ultimately needs a check involving \(r\), the prover *does not* want to commit to a large-\(q\) object and then range-check it “as-is”. + +So, for “notation and implementation simplicity”, Hachi performs a base-\(b\) gadget decomposition of the quotient witness \(r\) (paper §4.3): + +- \(r = \sum_u b^u \cdot r_u\), +- the prover commits to \((z, r_1, \dots, r_{\log_b(q)})\) instead of \((z,r)\), +- and the prover proves \(\|r_u\|_\infty \le b-1\) for every digit-vector \(r_u\). + +The paper then says it **omits the subscript \(u\)** from this point on, and that to incorporate this in the sumcheck view, “we can modify the multilinear extension correspondingly.” + +Concretely, the “real” linear constraint is: + +\[ +Mz = y + (X^d+1)\cdot \sum_u b^u r_u \quad\text{over } Z_q[X], +\] + +and the “real” smallness constraints are: + +\[ +\|z\|_\infty \le b-1 +\quad\text{and}\quad +\|r_u\|_\infty \le b-1 \ \forall u. +\] + +Protocol idea (Figure 4): + +- **P → V**: commit to \((z,r)\): \(t := \mathrm{Com}(z,r)\). (In the “real” protocol: commit to \((z,(r_u)_u)\) as above; the paper keeps writing \((z,r)\) after omitting the digit index.) +- **V → P**: sample \(\alpha \leftarrow F_{q^k}\). +- Reduce to checking the *field* equations: + - \(M(\alpha)z(\alpha) = y(\alpha) + (\alpha^d+1)r(\alpha)\) over \(F_{q^k}\). (Figure 4.) + +Soundness of this “evaluate at random \(\alpha\)” step is formalized as \(2d\)-special soundness (Lemma 9), reflecting degree \(\le 2d-1\). + +They also need to enforce that the witness coefficients are genuinely in \(Z_q\) and small (not arbitrary in \(F_{q^k}\)), and they fold those checks into the next “sumcheck view”. + +#### C.3. Represent the constraints as multilinear polynomials and batch them (Eqs. (21)–(23), Figure 5) + +This is where the **full constraint system** (including the range check) is spelled out in the paper. + +##### C.3.1 The witness polynomial \(\tilde w\) (Eq. (21)) + +The prover’s committed witness for sumcheck is a multilinear polynomial \(\tilde w\) that encodes coefficient tables. + +**Important:** because of §4.3’s hidden gadget decomposition \(r=\sum_u b^u r_u\), the *real* committed witness should be thought of as encoding coefficient tables of: + +- the vector of polynomials \(z\), and +- all digit-vectors \(r_u\), + +not just a single undigitized \(r\). The paper immediately omits the digit index \(u\) and keeps writing \((z,r)\); this is why Eq. (21) below only has one \(r\). + +The paper defines a multilinear polynomial \(e_w\) (same role as \(\tilde w\) in our prose) as: + +- \(e_w(u,\ell) = z_{u,\ell}\) if \(u \le \mu\), +- \(e_w(u,\ell) = r_{u-\mu,\ell}\) if \(\mu < u \le \mu+n\). (Eq. (21).) + +Here \(u\) and \(\ell\) are treated as binary strings indexing \([\,\mu+n\,]\) and \([\,d\,]\). + +To “incorporate the gadget decomposition” in the exact same shape, one natural flattened encoding is: + +- let \(\delta := \lceil \log_b(q)\rceil\) be the number of digits, +- treat the committed table as indexed by \([\,\mu + n\cdot \delta\,]\times[\,d\,]\), +- keep \(e_w(u,\ell)=z_{u,\ell}\) for \(u\in[\mu]\), +- and set \(e_w(\mu + u'\cdot n + i,\ell) := (r_{u'})_{i,\ell}\) for digit index \(u'\in[\delta]\) and row index \(i\in[n]\). + +With this flattening, the “range check” constraints apply to **every** coordinate of \(z\) and **every** coordinate of every digit vector \(r_{u'}\). + +##### C.3.1.1 Digit-aware \(e_{M_\alpha}\): the fully expanded constraint coefficient function + +The paper’s simplified definition (after Eq. (21)) encodes the linear constraint at \(\alpha\) by defining a public function \(e_{M_\alpha}(i,u)\) that (informally) gives the coefficient multiplying the polynomial \(w_u(\alpha)\) inside row \(i\). + +Once you flatten the digit vectors \((r_{u'})_{u'\in[\delta]}\) into the witness index \(u\in[\,\mu+n\delta\,]\), the *literal* digit-aware version is: + +- Let \(u\in[\,\mu+n\delta\,]\). +- Define the “decoded” witness coordinate \(W_u(\alpha)\) by: + - for \(1 \le u \le \mu\): \(W_u(\alpha) := z_u(\alpha)\), + - for \(\mu + (u'-1)n + i\) with \(u'\in[\delta]\) and \(i\in[n]\): \(W_{\mu + (u'-1)n + i}(\alpha) := (r_{u'})_i(\alpha)\). + +Then define: + +\[ +e^{\text{dig}}_{M_\alpha}(i,u) := +\begin{cases} +M_{i,u}(\alpha) & \text{if } 1 \le u \le \mu, \\ +-\,b^{u'-1}\cdot(\alpha^d+1) & \text{if } u = \mu + (u'-1)n + i \text{ for some } u'\in[\delta], \\ +0 & \text{otherwise.} +\end{cases} +\] + +With this explicit \(e^{\text{dig}}_{M_\alpha}\), the digitized ring-switch check for each row \(i\in[n]\) is exactly: + +\[ +\sum_{u=1}^{\mu+n\delta} e^{\text{dig}}_{M_\alpha}(i,u)\cdot W_u(\alpha) \;=\; y_i(\alpha). +\] + +This is just the statement: + +\[ +\sum_{j=1}^{\mu} M_{i,j}(\alpha)\,z_j(\alpha)\;-\;(\alpha^d+1)\sum_{u'=1}^{\delta} b^{u'-1}\,(r_{u'})_i(\alpha) \;=\; y_i(\alpha), +\] + +which is equivalent to the “real” lifted identity \(Mz = y + (X^d+1)\sum_{u'}b^{u'-1}r_{u'}\) after evaluation at \(X=\alpha\). + +##### C.3.2 The two *literal* constraints before batching (paper right before Eq. (22)/(23)) + +For a fixed ring-switch challenge \(\alpha\in F_{q^k}\), the verifier wants to enforce: + +1. **Linear constraints (ring switching at \(\alpha\))**: for each row \(i\in[n]\), + + \[ + \sum_{u=1}^{\mu+n} e_{M_\alpha}(i,u)\cdot \sum_{\ell} e_w(u,\ell)\cdot e_\alpha(\ell)\;=\;y_i(\alpha), + \] + + where \(e_\alpha(\ell)=\alpha^\ell\) and \(e_{M_\alpha}(i,u)\) is the public multilinear encoding of \(M(\alpha)\) plus the extra \(-(\alpha^d+1)\) term on the \(r\)-coordinates. (This is the first bullet list item under Figure 5 in the paper.) + + With the hidden gadget decomposition \(r=\sum_u b^u r_u\), the same constraint is enforced except the right-hand side becomes \(y_i(\alpha) + (\alpha^d+1)\sum_u b^u r_{i,u}(\alpha)\). In the “\(e_{M_\alpha}\cdot e_w\)” encoding, this is handled by: + + - expanding \(e_w\) to include all digit-vectors \((r_u)_u\) (as described in C.3.1), and + - modifying the \(-(\alpha^d+1)\) part of \(e_{M_\alpha}\) to include the appropriate digit weights \(-b^u(\alpha^d+1)\) on the digit blocks. + +2. **Smallness / range constraints (this is the “range check”)**: for *every* coordinate \((u,\ell)\), + + \[ + P_b\big(e_w(u,\ell)\big) = 0 + \quad\text{where}\quad + P_b(T) := \prod_{t=-(b-1)}^{b-1}(T-t). + \] + + The paper writes this explicitly as the vanishing product: + \(e_w(u,\ell)\cdot(e_w(u,\ell)-1)\cdot(e_w(u,\ell)+1)\cdots(e_w(u,\ell)-b+1)\cdot(e_w(u,\ell)+b-1)=0\). + +This is **not optional**: it is exactly how Hachi enforces that the coefficients the prover is claiming for \(z\) (and the \(r\)-side witness they commit) are small integers (embedded into \(F_{q^k}\)), i.e. a range/membership proof via a root-check polynomial. + +##### C.3.3 Batching with equality polynomials: the exact \(H_\alpha\) and \(H_0\) (Eqs. (22)–(23)) + +Let the multilinear equality polynomial be: + +- \(e_{eq}(t,i) = \prod_j (t_j i_j + (1-t_j)(1-i_j))\). + +Then the paper defines: + +- **Linear constraint batch** \(H_\alpha(t)\) (Eq. (22)): + + \[ + H_\alpha(t) := + \sum_{i\in[n]} e_{eq}(t,i)\cdot + \Big(\sum_{u,\ell} e_{M_\alpha}(i,u)\cdot e_w(u,\ell)\cdot e_\alpha(\ell) - y_i(\alpha)\Big). + \] + +- **Smallness/range batch** \(H_0(t)\) (Eq. (23)): + + \[ + H_0(t) := + \sum_{u,\ell} e_{eq}\big(t,(u,\ell)\big)\cdot P_b\big(e_w(u,\ell)\big). + \] + +The goal is to prove both are **identically zero polynomials**, which they reduce (Figure 5) to random-point checks: + +- **V → P**: send random points \(\tau_0,\tau_1\), +- prove \(H_0(\tau_0)=0\) and \(H_\alpha(\tau_1)=0\). + +#### C.4. Use sumcheck to prove the remaining batched sums (Figure 6) + +After fixing \(\tau_0,\tau_1\), they rewrite \(H_0(\tau_0)\) and \(H_\alpha(\tau_1)\) as sums over \((u,\ell)\) of polynomials \(F_{0,\tau_0}\) and \(F_{\alpha,\tau_1}\), and then apply **sumcheck** over \(F_{q^k}\) (discussion right after Eq. (23)). + +Figure 6 gives the “single-round view” of sumcheck: prover sends univariate \(g_i\), verifier sends random challenge scalars \(a_i\), and in the end the verifier reduces everything to checking an evaluation of the witness polynomial \(\tilde w\) at one final random point. + +Crucially, the sumcheck ends in: + +- a **final evaluation claim** of the committed \(\tilde w\) at a random point \(r^\*\), +- plus the requirement to prove this evaluation is consistent with the commitment \(t\). + +That “prove the evaluation claim for \(\tilde w\)” is exactly where recursion happens: you invoke the PCS again on the smaller committed object. + +##### C.4.1 Full message flow (Figure 7), spelled out + +Figure 7 in the paper is the full composition of Figures 4, 5, and 6. Written as an explicit transcript skeleton: + +- **P → V**: \(t := \mathrm{Com}(z,r)\). (In the “real” protocol: \(t := \mathrm{Com}(z,(r_u)_u)\) with \(r=\sum_u b^u r_u\).) +- **V → P**: sample challenges: + - \(\alpha \leftarrow F_{q^k}\), + - \(\tau_0 \leftarrow F_{q^k}^{\log(\mu)+\log(d)}\), + - \(\tau_1 \leftarrow F_{q^k}^{\log(n)}\). +- **P ↔ V (sumcheck)**: run sumcheck to prove \(H_0(\tau_0)=0\) and \(H_\alpha(\tau_1)=0\): + - in each sumcheck round, **P → V** sends a univariate polynomial \(g_i(X_i)\), + - and **V → P** responds with a random scalar challenge \(a_i \leftarrow F_{q^k}\). (Figure 6.) +- **P → V (final opening)**: open the commitment \(t\) at the final point determined by \((a_1,\dots,a_\ell)\) to provide the needed value(s) of \(\tilde w\). +- **V (final checks)**: + - evaluate the public multilinear extensions (notably \(e_{M_\alpha}\) / \(e_\alpha\)) at the same final point, + - and check the final sumcheck identities (Figure 6’s last-round checks). + +### Step D (paper §3 + §5): recursion shape and concrete proof-size accounting + +Recursion shape (high-level): + +- One invocation reduces the big opening proof into opening proofs for **smaller** committed objects (smaller \(\ell\), smaller coefficient domains after decomposition, and evaluation points living in \(F_{q^k}\)). +- Eventually, Hachi suggests switching to different sub-proofs when the witness is small (paper discusses both switching to LaBRADOR/JL and composing with Greyhound). + +Concrete proof-size estimate for \(\ell=30\) (paper §5.2): + +- Sumcheck for the first round: ~7.3KB. +- “Adaptation + Greyhound subproof” gives total ~55.1KB for an evaluation proof. + +This is the explicit estimate in §5.2: + +- first-round sumcheck: \(7.3\)KB, +- plus preparation/adaptation \(4.8\)KB, +- plus Greyhound evaluation subproof \(43\)KB, +- total \(7.3 + 4.8 + 43 = 55.1\)KB. + +#### D.1 What the “adaptation to Greyhound” actually means (paper §4.5, §5.2) + +When the witness becomes small enough, Hachi can stop running its own §4.3 ring-switch + sumcheck recursion and instead reduce the remaining claim into a **Greyhound-native** opening proof. + +The key shape (paper §4.5 and the concrete instantiation in §5.2) is: + +- you end up with an evaluation claim where the verifier has reduced everything to checking that a committed multilinear object \(\tilde w\) evaluates correctly at a random point; +- the prover groups / “packs” the needed coefficients into extension-field elements, and then uses the §3 embedding machinery (the \(\psi\) map and trace identity) to turn the remaining check into a ring statement supported by Greyhound. + +In the concrete accounting, the additional prover communication for this handoff is bounded as (Equation (28) in the paper excerpted in §4.5): + +- \((k-1)\cdot k \cdot \log q\) bits for the sent partial evaluations \((y_i)\), plus +- \(d'\cdot \log q\) bits for sending one ring element \(p\in R'_{q}\) (where \(d'\) is the Greyhound ring dimension, e.g. \(d'=64\) in §5.2). + +This is why §5.2 reports the “adaptation overhead” as small (~0.3KB for the non-commitment part), and then adds the cost to **commit** to the new Greyhound witness element(s). + +### Message flow summary (first / dominant round) + +If you want a “wire-format mental model” aligned to Figures 3–6, the dominant first round contains: + +- **Ring commitments**: the main commitment \(u\in R_q^{n_B}\) (Eq. (14)) and the auxiliary commitment \(v\in R_q^{n_D}\) (Eq. (16)). +- **Ring challenge**: short/sparse \(c\in C^{2^r}\) used to fold the opening witness (Eq. (18)–(19)). +- **Field challenge**: \(\alpha\in F_{q^k}\) for ring switching (Figure 4), and random points \(\tau_0,\tau_1\) (Figure 5). +- **Sumcheck transcript**: univariate polynomials \(g_i\) and challenge scalars \(a_i\) (Figure 6), ending in a point \(r^\*\) and an evaluation value \(\tilde w(r^\*)\). +- **Opening proof(s)**: recursive PCS openings that prove the claimed evaluations of the committed \(\tilde w\) values match the commitments. + +## Implementation-oriented PCS flow spec (what we need before coding) + +This section re-states the end-to-end PCS as a **software spec**: what the prover/verifier compute, what objects exist (and their “witness” roles), and what must be pinned down *before* we implement §4.3 “witness table embedding” or the ring-switch sumcheck instances. + +### E.0 The key mental model: “stacked linear relation” → “ring switch” → “sumcheck” → “new opening claim” + +The paper explains §4.3 in the simplified setting \(Mz=y\) for \(z\in R_q^\mu\). In the actual PCS opening proof, §4.2 produces exactly such a relation, but with: + +- a **stacked witness vector** \(z\) that bundles multiple unknowns (notably \(\hat w,\hat t,\hat z\) from Eq. (20)), and +- a stacked statement \(y\) and matrix \(M\) derived from public matrices \(A,B,D\), the opening point \(x\) (via \(a,b\)), the claimed opening value, and the verifier challenge \(c\). + +Once the prover and verifier agree on that concrete \((M,y)\), §4.3 proceeds as: + +1. **Ring switching introduces a quotient witness** \(r\) such that: + + \[ + Mz = y + (X^d+1)\cdot r \quad \text{over } Z_q[X]. + \] + +2. **Digitize the quotient witness** \(r = \sum_{u'=0}^{\delta-1} b^{u'} r_{u'}\) and range-check each digit block. +3. Encode the (digitized) coefficient tables of \((z,(r_{u'})_{u'})\) into one multilinear object \(\tilde w\) (paper Eq. (21), with the digit index omitted in the paper’s notation). +4. Prove the linear and range constraints by sumcheck, which ends in an evaluation claim \(\tilde w(r^\*)\) at a random point \(r^\*\). +5. **Recursion boundary**: the next PCS subproblem is “open the commitment to \(\tilde w\) at point \(r^\*\)”. + +### E.1 What the prover’s “witness” is after the first interaction rounds (ring switching + sumcheck) + +If by “after the first rounds” you mean “after the verifier samples \(\alpha,\tau_0,\tau_1\) and sumcheck begins”, then the prover’s relevant witnesses are: + +- **The stacked linear-relation witness** \(z\) from §4.2 (Eq. (20)’s unknown vector; in the paper’s naming, this is \((\hat w,\hat t,\hat z)\) with the implicit recomposition \(z := J\hat z\)). +- **The ring-switch quotient witness digits** \((r_{u'})_{u'\in[\delta]}\) satisfying: + + \[ + Mz - y = (X^d+1)\sum_{u'} b^{u'} r_{u'}. + \] + +These are not sent directly. Instead they are **committed** and then accessed *only* through: + +- evaluations of the committed multilinear object \(\tilde w\) during sumcheck, and finally +- one PCS opening of \(\tilde w\) at the final sumcheck point \(r^\*\). + +So the short answer is: + +> After ring switching starts, the “PCS witness” (for the next recursion layer) is the committed multilinear polynomial \(\tilde w\) that encodes the coefficient table of the stacked witness \(z\) and the digitized quotient witness \((r_{u'})_{u'}\). Sumcheck reduces everything to opening \(\tilde w\) at one random point \(r^\*\). + +### E.2 What “witness table embedding” (Eq. (21)) really must encode in the full PCS (not the simplified \(Mz=y\) story) + +Eq. (21) defines \(e_w(u,\ell)\) in the simplified \((z,r)\) notation. For implementation, the important points are: + +- The “\(z\)” rows are not just “some vector”; in the PCS they are the **stacked unknowns** of Eq. (20), i.e. the prover’s hidden objects that tie together: + - the main commitment \(u\), + - the aux commitment \(v\), + - the opening equation, + - the fold identities (18)–(19), + - and the redecomposition via \(J\). +- The “\(r\)” rows are not optional: they are the **quotient witness** for the lifted equation over \(Z_q[X]\), and in the real protocol are digitized as \((r_{u'})_{u'}\) and range-checked. + +In other words, the witness-table encoder we build should take as input: + +- the stacked witness vector (call it `linear_relation_witness` instead of paper’s `z`), and +- the digitized quotient witness blocks `quotient_digits` (instead of paper’s `r`), + +and output a padded, evaluation-form multilinear object \(\tilde w\). + +### E.3 Why the PoC’s “next witness” contains “extra quotient terms” (and why they don’t contradict the paper) + +The PoC does not implement §4.3 for a single abstract relation \(Mz=y\). It already constructs (a variant of) Eq. (20)’s **stacked linear system**, which bundles multiple constraints. + +Each lifted constraint row contributes its own quotient polynomial(s) when you rewrite it as: + +\[ +\text{(row LHS)} - \text{(row RHS)} = (X^d+1)\cdot r_i(X), +\] + +so the PoC ends up with several quotient vectors (for several different stacked sub-constraints), and it concatenates them into the “next witness table” before padding to a power of two. + +This is an implementation manifestation of the same principle the paper uses: + +- §4.2 stacks many checks into one linear system (Eq. (20)), +- §4.3 introduces quotient witnesses for that system when lifted to \(Z_q[X]\), +- §4.3 then commits to a single coefficient-table object \(\tilde w\) representing *all* witness coordinates and *all* quotient-digit coordinates. + +So: seeing “more quotient chunks” in a prototype is expected whenever you expand the simplified \(Mz=y\) exposition into the concrete PCS relation. + +### E.4 Concrete “before we code” checklist (MVP up through §4.3) + +To avoid implementing §4.3 machinery “in the dark”, we should pin down the following concrete spec items first. + +#### E.4.1 Fix the exact stacked linear relation produced by §4.2 + +We need a concrete `LinearRelationInstance` spec with: + +- **Public statement**: + - the stacked matrix \(M\in R_q^{n\times \mu}\), + - the stacked right-hand side \(y\in R_q^n\), + - and the coefficient smallness parameters (base \(b\), digit length \(\delta\), and any redecomposition \(\tau\) where relevant). +- **Witness semantics**: + - what each coordinate of the stacked witness vector means (e.g., slices corresponding to \(\hat w,\hat t,\hat z\)), + - and which coordinates are subject to the range-check polynomial \(P_b\). + +Without this, we cannot correctly define how many “\(z\) rows” Eq. (21) has (the paper’s \(\mu\)). + +#### E.4.2 Fix the quotient witness structure introduced by ring switching + +For the chosen \((M,y)\), ring switching requires: + +- the quotient witness vector \(r\in (Z_q[X]_{ idx\) to become “evaluations on the hypercube”. + +This is the “witness table embedding” deliverable: it is the object whose commitment is opened at the end of sumcheck (Figure 6 / Figure 7). + +#### E.4.4 Fix the public multilinear encodings needed by §4.3 constraints + +To define \(H_\alpha\) and \(H_0\) (Eqs. (22)–(23)), we need: + +- `AlphaPowers`: the table \(e_\alpha(\ell)=\alpha^\ell\), +- `LinearCoeffEncoding`: the digit-aware coefficient function \(e^{dig}_{M_\alpha}(i,u)\) (i.e., how \(M(\alpha)\) and \(-(\alpha^d+1)\cdot b^{u'}\) are encoded), +- and the equality polynomials \(e_{eq}\) used for batching. + +#### E.4.5 Fix how sumcheck’s final oracle check becomes a PCS opening claim + +Our sumcheck core deliberately stops at “here is the final point \(r^\*\)”; the ring-switch module must: + +- compute the expected final value using the public parts, and +- reduce to a **single opening claim** of the committed \(\tilde w\) at \(r^\*\). + +This is exactly where the PCS prover/verifier “open-check” logic plugs in (currently stubbed in this repo). + +## Modulus switching / cross-prime sumcheck (Jolt-motivated extension; not in the Hachi paper) + +This section sketches how to adapt the Hachi “ring switch \(\to\) sumcheck” pipeline to the setting where: + +- **Commitments** are over a *small* prime field / ring modulus \(q\) (e.g. \(\approx 2^{32}\)), because that makes commitment-time arithmetic and NTT/CRT layouts fast. +- **Sumcheck / arithmetization** must run over a *large* prime field \(F_{q'}\) (e.g. 128-bit prime), because the application (e.g. Jolt) requires characteristic large enough to avoid wrap-around in \(u64\cdot u64\) accumulation. + +This is *similar in spirit* to §3’s extension-field story, but **strictly harder**: there is no field embedding \(F_q \hookrightarrow F_{q'}\) that preserves addition/multiplication mod the prime, so we must explicitly control an **integer lift** via range checks. + +### F.0 Target statement (“foreign-field opening”) + +Let \(q\) be a small prime, \(q'\) a large prime, and let \(f\) be an \(\ell\)-variate multilinear with **small coefficients**, ideally bits: + +\[ +f \in F_q^{\le 1}[X_1,\dots,X_\ell],\quad f_i \in \{0,1\}\subset F_q. +\] + +We commit to the coefficient table \((f_i)_{i\in\{0,1\}^\ell}\) using the Hachi/Greyhound-style commitment core over \(R_q\) (or over \(F_q\) as the \(d=1\) special case). + +The opening claim we want to support is over the **large prime field**: + +\[ +\text{given } x\in F_{q'}^\ell,\ y\in F_{q'},\ \text{prove } f(x)=y\ \text{(interpreting the coefficients as small integers in }F_{q'}\text{).} +\] + +Because the coefficients are in \(\{0,1\}\), there is a canonical injection \(\iota:\{0,1\}\to F_{q'}\) (map to the same integers). The only remaining job is to enforce that the committed coefficients are indeed in \(\{0,1\}\) (bitness) and that every algebraic check is performed with respect to this integer lift. + +#### F.0.1 Important clarification: this is **not** “digit-decompose \(x,y\)” + +It is tempting to think “we have an evaluation claim \(f(x)=y\) in the large field \(F_{q'}\), so we should decompose \(x\) and \(y\) into base-\(b\) digits and prove digit-wise subclaims.” That is **not** what the modulus-switching reduction does, and it generally does not work (polynomial evaluation does not decompose into independent digit evaluations). + +Instead: + +- The evaluation point \(x\in F_{q'}^\ell\) and claimed value \(y\in F_{q'}\) are already *native* to the field where Jolt runs sumcheck. There is typically no reason to decompose them. +- The lift issue arises because the **committed objects** live over the *small* modulus \(q\) (i.e. values are only defined modulo \(q\)), while the verifier wants to check equations **inside \(F_{q'}\)**. + +So the purpose of \(\mathrm{lift}_q\) is to assign a **canonical integer meaning** to committed mod-\(q\) values (made unambiguous by range constraints like “bitness”), so those values can be interpreted inside \(F_{q'}\). + +Warm-up (scalar version). For integers \(\tilde A,\tilde B\in Z\), the statement “\(A=B\) in \(Z_q\)” is exactly: + +\[ +\tilde A \equiv \tilde B \pmod q +\quad\Longleftrightarrow\quad +\exists s\in Z:\ \tilde A - \tilde B = q\cdot s. +\] + +The \(q\cdot s\) term is the “modulus switching slack”. The ring/polynomial version in F.1 is the same idea applied coefficient-wise (and with an additional cyclotomic slack \((X^d+1)\cdot r\) for the ring quotient). + +#### F.0.2 How this plugs into Hachi’s six core constraints (B.4.1) in the Jolt Stage-8 setting + +Jolt Stage 8 asks the PCS to prove openings of the form “\(P(\mathbf r)=v\)” where both the point \(\mathbf r\) and value \(v\) live in the **Jolt field** \(F\) (in your target case, \(F=F_{q'}\)). See: + +- `../jolt/jolt-core/src/poly/opening_proof.rs`: `pub type Opening = (OpeningPoint<..., F>, F);` +- `../jolt/jolt-core/src/zkvm/prover.rs`: Stage 8 calls `PCS::prove(..., &opening_point.r, ...)` where `opening_point.r` is a vector of field challenges. + +Hachi’s Step B constraints (the six items in B.4.1) are written as equalities over the **ring** \(R_q\). To use a Hachi-style PCS under Jolt’s interface, we keep the *same witness objects* \((\hat w,\hat t,\hat z,\dots)\) and the *same logical constraints*, but we do **not** expect the verifier to check them natively in \(R_q\) (and we cannot even form the “\(a,b\in R_q\) from the opening point” parts when \(\mathbf r\in F_{q'}^\ell\)). + +Instead, the PCS opening proof checks these constraints **after ring switching**, i.e. after applying an evaluation map + +\[ +\mathrm{ev}_\alpha: R_q \to F_{q'} +\] + +at a random \(\alpha\in F_{q'}\) (and including the modulus-switch slack \(q\cdot s\) to make “mod \(q\)” equalities meaningful in \(F_{q'}\)). + +Concretely: + +- Constraints **(1), (2), and the purely ring-linear parts** of (4),(5) are still “ring equations”, but they are verified in \(F_{q'}\) by checking \(\mathrm{ev}_\alpha(\text{LHS}-\text{RHS})=0\) (with quotient witnesses for \((X^d+1)\) and \(q\) as in F.1). +- Constraints **(3)–(5)** are the ones that *mention the evaluation point* (via the monomial vectors \(a,b\)) and therefore must be interpreted in the **field**: + - compute \(a,b\) directly from the Stage-8 opening point \(\mathbf r\in F_{q'}^\ell\), + - treat unknown ring elements like \(w_i\in R_q\) only through their field images \(w_i(\alpha):=\mathrm{ev}_\alpha(w_i)\in F_{q'}\), + - and enforce the same algebraic equalities (Eq. (17)–(19)) in \(F_{q'}\). + +So the “generalization to account for \(q'\)” is not “change a few constraints while keeping the rest in \(q\)”; it is: + +> keep the constraint *structure*, but **move their verification domain** from \(R_q\) to \(F_{q'}\) via ring switching (plus a \(q\cdot s\) slack), because Stage 8’s statement lives in \(F_{q'}\). + +#### F.0.3 Why you cannot keep the *entire* opening proof “purely in \(F_q/R_q\)” under Jolt’s interface + +Under Jolt, the opening point \(\mathbf r\) is sampled as transcript challenges in the **same field as the sumchecks**, i.e. \(\mathbf r\in F_{q'}^m\) (see `OpeningPoint<..., F>` in `../jolt/jolt-core/src/poly/opening_proof.rs`). + +Any PCS that plugs into Stage 8 must therefore convince the verifier of a statement that is *parameterized by* these \(F_{q'}\) elements: + +\[ +P(\mathbf r)=v\quad\text{for }\mathbf r,v\in F_{q'}. +\] + +If a verifier refuses to do any \(F_{q'}\) arithmetic, it cannot even *evaluate the public weights* (equality polynomials) at \(\mathbf r\), nor can it check the final identity that defines “evaluation at \(\mathbf r\)”. The only way around this would be to represent every \(F_{q'}\) element (including \(\mathbf r\), \(v\), and all derived weights) as **non-native data** over \(F_q\) (e.g. base-\(b\) limbs) and then prove that these limbs satisfy the mod-\(q'\) arithmetic via additional quotient/carry constraints. + +That “all-\(F_q\) verification” route is a different, much heavier design: it is essentially a SNARK for \(F_{q'}\) arithmetic implemented over \(F_q\), and it introduces digit/carry constraints for every multiplication/addition in the opening protocol. This is *not* what Hachi’s ring switching is optimizing for. + +So the realistic design space is: + +- commitments / witness representation over \(F_q\) / \(R_q\) for performance, +- but verifier-side checking (sumcheck challenges, equality weights, and final identities) in \(F_{q'}\). + +#### F.0.4 Recursion / “next folding steps” in the cross-prime setting + +In Hachi, the “recursion boundary” is: sumcheck reduces a large set of constraints to **one new opening claim** for a committed multilinear object (the \(\tilde w\) table). The next layer repeats the same pattern on a smaller witness. + +In the cross-prime Stage-8 adaptation: + +- the **commitment domain stays** \(F_q/R_q\) at every layer (you keep committing to digitized tables over the small modulus), +- the **opening points stay** in \(F_{q'}\) at every layer, because they are derived from sumcheck challenges / transcript in \(F_{q'}\), +- and the verifier continues to check constraints in \(F_{q'}\) via ring switching (evaluation at \(\alpha\in F_{q'}\) plus the \((X^d+1)\cdot r\) and \(q\cdot s\) quotient witnesses). + +So “continuing in \(F_{q'}\)” does not mean “switch commitments to \(q'\)”; it means the *interactive checking algebra* (sumcheck, batching, evaluation weights) remains in the Jolt field where the statement is expressed. + +#### F.0.5 One full “folding/recursion step” of the adapted PCS (clean, end-to-end) + +This subsection gives a clean one-layer view: how we go from **one opening claim at point \(\mathbf r\in F_{q'}^m\)** to **a smaller opening claim at a new point \(\mathbf r^\*\in F_{q'}^{m'}\)**, and why that output is amenable to repeating the same procedure. + +We describe this as a PCS protocol (what Stage 8 wants), independent of Jolt’s earlier stages. + +##### Inputs (statement) and commitment domain + +- **Commitment domain** (small): commitments are to coefficient tables over \(F_q\) (or ring elements over \(R_q\)) using Hachi’s §4.1 Ajtai-style structure. +- **Opening statement domain** (large): the opening point \(\mathbf r\) and claimed value \(v\) are in \(F_{q'}\) (Jolt’s `JoltField`). + +So the opening statement is: + +\[ +\text{Given commitment } C \text{ to a multilinear table } P,\ \text{and public } \mathbf r\in F_{q'}^m,\ v\in F_{q'},\ \text{prove } P(\mathbf r)=v. +\] + +Here \(P(\mathbf r)\) is defined by interpreting the committed coefficients as integers (via \(\mathrm{lift}_q\)) and reducing them into \(F_{q'}\); for bit/one-hot tables, this interpretation is canonical. + +##### Step 1 (Hachi Step B “split-and-fold”): reduce evaluation to a structured linear relation witness + +This step is unchanged in *spirit* from Hachi: we rewrite “\(P(\mathbf r)=v\)” into a small set of algebraic constraints involving intermediate witnesses (partial evaluations, decomposed digits, and folded combinations). + +The key point in the two-field setting is typing: + +- any time the original §4.2 constraints multiply “point-derived scalars” into witness quantities, those scalars are in \(F_{q'}\) and we interpret the witness quantities through their images in \(F_{q'}\) (via ring switching in Step 2), rather than trying to multiply \(F_{q'}\) scalars by \(R_q\) elements directly. + +Operationally, the prover still constructs the same witness objects (\(\hat w,\hat t,\hat z\), folded \(z\), etc.) and commits to the same ring elements (\(u,v\), etc.). The verifier will not check the ring equations directly; it will check their *ring-switched* images in Step 2. + +##### Step 2 (Hachi Step C generalized): ring switching + modulus switching to get field-native constraints + +The verifier samples a random \(\alpha\leftarrow F_{q'}\) and defines the evaluation map: + +\[ +\mathrm{ev}_\alpha: R_q \to F_{q'} +\] + +as “lift coefficients to integers (per chosen \(\mathrm{lift}_q\)), then evaluate the polynomial at \(X=\alpha\) in \(F_{q'}\)”. + +Now take each ring equation from the Step B constraint set (the six constraints in B.4.1) and convert it into a field equation at \(\alpha\) by: + +1. lifting it from \(R_q\) to \(Z[X]\) with a cyclotomic quotient witness \(r\) (as in the paper), and +2. adding a modulus quotient witness \(s\) so that “mod \(q\) equality” becomes an *integer* equality plus a \(q\cdot s\) slack (Section F.1). + +After evaluating at \(X=\alpha\), every check becomes an identity in \(F_{q'}\) involving: + +- public scalars derived from \(\mathbf r\) (equality weights / monomial vectors), +- the prover’s unknowns only through \(\mathrm{ev}_\alpha(\cdot)\), +- and the quotient witnesses \(r(\alpha), s(\alpha)\). + +##### Step 3 (Hachi §4.3 sumcheck): batch all constraints and reduce to one oracle evaluation + +Exactly as in Hachi, we encode the relevant coefficient tables into one committed multilinear object \(\tilde w\) (“witness table embedding”), except that in the cross-prime setting \(\tilde w\) must encode the extra modulus quotient witness \(s\) as well as \(r\). + +Then we define batched constraint polynomials (the analogs of \(H_\alpha\) and \(H_0\)), and run sumcheck over **\(F_{q'}\)** (not over \(R_q\)). The sumcheck transcript produces a final random point: + +\[ +\mathbf r^\* \in F_{q'}^{m'} +\] + +and reduces verification to a **single evaluation claim** of the committed witness-table multilinear: + +\[ +\tilde w(\mathbf r^\*) = v^\*\in F_{q'}. +\] + +##### Step 4 (recursion boundary): output a smaller opening claim of the same type + +This is the crucial “amenable to further folding” point: + +- The new claim “\(\tilde w(\mathbf r^\*)=v^\*\)” has the **same shape** as the original opening claim, just on a different committed multilinear and a different point. +- The commitment to \(\tilde w\) is again over the **small modulus domain** (it is built from digitized tables over \(F_q/R_q\)). +- The opening point \(\mathbf r^\*\) is again in the **large field** \(F_{q'}\), because it is derived from the sumcheck challenges (and thus from the Jolt transcript). + +Therefore, we can repeat the same 4-step pipeline on \(\tilde w\): + +\[ +P(\mathbf r)=v +\ \leadsto\ +\tilde w(\mathbf r^\*)=v^\* +\ \leadsto\ +\tilde w^{(2)}(\mathbf r^{\*\*})=v^{\*\*} +\ \leadsto\ \cdots +\] + +and eventually hand off to a base PCS / small-instance prover once the witness table is small enough (exactly the same “stop recursion when small” design choice as Hachi’s §5 composition discussion). + +#### F.0.6 Option B (for comparison): decompose the \(F_{q'}\) opening point/value and prove everything over \(F_q/R_q\) + +This subsection works out the alternative you asked for: **avoid doing any verifier arithmetic in \(F_{q'}\)** by representing all \(F_{q'}\) elements (the opening point \(\mathbf r\) and claimed value \(v\), plus all derived weights) as *digits/limbs* over the small field. + +This is not “ring switching”; it is **non-native (foreign-field) arithmetic**: we simulate mod-\(q'\) arithmetic inside a proof system whose native arithmetic is mod-\(q\). + +##### Why this is a different interface than Jolt Stage 8 + +In Jolt, Stage 8’s opening statement is parameterized by an actual \(\mathbf r\in F_{q'}^m\) sampled from the transcript (see `OpeningPoint<..., F>`). + +Option B instead treats \(\mathbf r\) and \(v\) as **public limb vectors over \(F_q\)**. To plug this into Jolt unchanged, you would need to either: + +- change Jolt’s transcript/challenges to live in the small field (not compatible with the stated “char \(> u64\cdot u64\)” requirement), or +- keep \(\mathbf r\in F_{q'}\) as usual but additionally provide a limb decomposition of \(\mathbf r\) and then **prove inside the PCS** that those limbs reconstruct the same \(\mathbf r\) (which reintroduces \(F_{q'}\) operations unless you also non-natively model the transcript). + +So Option B is best viewed as a “theoretical comparison point”, not a drop-in Stage-8 replacement. + +##### Representation: limbs for \(F_{q'}\) elements + +Fix a radix \(B=2^t\) (e.g. \(B=2^{16}\) so limbs fit comfortably under a 32-bit-ish \(q\)), and let + +\[ +L := \left\lceil \log_B(q') \right\rceil +\] + +be the limb count (for a 128-bit prime and \(B=2^{16}\), \(L\approx 8\)). + +We represent a field element \(a\in F_{q'}\) by an integer representative \(\tilde a\in[0,q')\) (canonical lift mod \(q'\)) and a limb decomposition: + +\[ +\tilde a = \sum_{j=0}^{L-1} a_j B^j,\quad a_j\in\{0,1,\dots,B-1\}. +\] + +All limbs \(a_j\) are then encoded as small elements of \(F_q\) (and range-checked to lie in \([0,B)\)). + +##### Non-native arithmetic constraints (core gadget) + +To simulate \(F_{q'}\) arithmetic, every operation becomes an integer identity plus a \(q'\)-multiple slack: + +- **Addition**: to enforce \(c \equiv a+b \pmod{q'}\), prove + + \[ + \tilde a + \tilde b - \tilde c = q'\cdot k + \] + + for some integer \(k\) (with \(k\in\{0,1\}\) if \(\tilde a,\tilde b,\tilde c\in[0,q')\)). + +- **Multiplication**: to enforce \(c \equiv a\cdot b \pmod{q'}\), prove + + \[ + \tilde a\cdot \tilde b - \tilde c = q'\cdot k + \] + + for some integer \(k\) with \(0 \le k < q'\). + +In practice, you do not materialize \(\tilde a\) as a single huge integer; you enforce these identities **in base \(B\) with carries**: + +- introduce carry variables \(u_t\) so that the schoolbook convolution of limbs matches the target limbs, +- and range-check carries so that “equality mod \(q\)” implies “equality over integers” (no wrap-around) at the limb level. + +Cost model: one non-native multiplication of two \(L\)-limb numbers costs + +\[ +\Theta(L^2)\ \text{small-field multiplications} \quad + \quad \Theta(L)\ \text{carry/range constraints}. +\] + +##### How to prove an opening claim \(P(\mathbf r)=v\) using Option B + +Let \(P\) be a multilinear polynomial committed over \(F_q\) (or \(R_q\)) whose coefficients are small integers (bits/one-hot is the cleanest case). + +Public input for Option B: + +- limb decompositions of each coordinate \(r_i\in F_{q'}\) (so \(\tilde r_i=\sum_j r_{i,j}B^j\)), +- limb decomposition of \(v\in F_{q'}\), +- and the modulus \(q'\) itself (public constant). + +Prover witness: + +- limb decompositions for all intermediate \(F_{q'}\) values needed to evaluate \(P\) at \(\mathbf r\), +- plus all carry/quotient witnesses for the modular reduction constraints above. + +Then enforce, inside the proof system over \(F_q/R_q\): + +1. **Range**: every limb is in \([0,B)\) (bitness is the special case \(B=2\)). +2. **Recomposition**: the limb vectors correspond to integers in \([0,q')\) (often implemented by providing a quotient \(t\) such that \(\sum_j a_jB^j = \tilde a + q' t\) and constraining \(\tilde a CE(b,L)^(K+k) -- + | + Π_RLC + | + CE(B,L) + | + Π_DEC + | + next accumulator: CE(b,L)^k +``` + +### What the prover sends in each sub-protocol + +Below is the communication “payload” you should have in mind (Fiat–Shamir makes verifier randomness transcript-derived, but sizes are the same). + +#### Π_CCS (paper §7.3) + +Prover payload: + +- **Sum-check transcript over \(K\)** for \(\log m\) rounds (one univariate polynomial per round; degree is small in typical CCS, e.g. R1CS-like \(u=2\)). +- **Oracle answers after sum-check**: for every claim \(i\in[K+k]\) and every matrix \(j\in[t]\), send + \[ + y'_{i,j} = \bar M_j \tilde z_i(r') \in R_K. + \] + +Rule of thumb: the \(y'_{i,j}\) payload dominates. + +#### Π_RLC (paper §7.4) + +This is just random linear combination using \(\rho_i\in C\): + +\[ +c := \sum_i \rho_i c_i,\quad y_j := \sum_i \rho_i y_{i,j},\quad z := \sum_i \rho_i z_i. +\] + +Prover payload: essentially **nothing new** (the reduction is “algebraic bookkeeping”; in FS the \(\rho_i\) come from the transcript). + +#### Π_DEC (paper §7.5) + +Prover payload: + +- Decompose \(z\) into digits \((z_1,\dots,z_k)=\mathrm{split}_b(z)\). +- Send **\(k\) new commitments** \(c_i = L(z_i)\). +- Send **\(k\cdot t\)** new lifted evaluations \(y_{i,j} = \bar M_j \tilde z_i(r)\in R_K\). + +### Proof size accounting (symbolic, per fold step) + +Let: + +- commitment output be \(c\in R_F^\kappa\) (Ajtai: \(\kappa\) ring elements over \(R_F\)), +- each lifted evaluation be one ring element in \(R_K\) (degree \(d\) over \(K\)), +- \(\ell := \log m\) be the sum-check arity. + +Then per fold step, prover sends approximately: + +- **Π\_CCS**: + - sum-check: \(O(\ell)\) field elements in \(K\), + - evaluations: \((K+k)\cdot t\) ring elements in \(R_K\). +- **Π\_DEC**: + - commitments: \(k\cdot \kappa\) ring elements in \(R_F\), + - evaluations: \(k\cdot t\) ring elements in \(R_K\). + +Total (dominant terms): + +\[ +\text{#}(R_K\text{-elements}) \approx (K+k)t + kt = (K+2k)t, +\] +\[ +\text{#}(R_F\text{-elements}) \approx k\kappa, +\] +plus a small \(\tilde O(\log m)\) number of \(K\)-field elements from sum-check. + +To convert to bytes for appendix parameter sets where \(K=\mathbb{F}_{q^2}\) and \(q\) is 61–64 bits: + +- 1 \(F\)-element ≈ 8 bytes +- 1 \(K\)-element ≈ 16 bytes +- 1 \(R_F\)-element ≈ \(d\cdot 8\) bytes +- 1 \(R_K\)-element ≈ \(d\cdot 16\) bytes +- 1 commitment \(c\in R_F^\kappa\) ≈ \(\kappa\cdot d\cdot 8\) bytes + +So a back-of-the-envelope fold proof byte size is: + +\[ +\approx (K+2k)t\cdot(d\cdot 16)\;+\;k\kappa\cdot(d\cdot 8)\;+\;\tilde O(\log m)\cdot 16. +\] + +## Exact counts + +- Total concrete field/cyclotomic tuples in SuperNeo appendix: **3** +- Unique base fields in those tuples: **3** +- Unique cyclotomic polynomials in those tuples: **2** + - `X^64 + 1` + - `X^54 + X^27 + 1` +- Strict Solinas-only tuples (`q = 2^x - 2^y + 1`): **2 / 3** + - Goldilocks (`2^64 - 2^32 + 1`) + - Mersenne61 (`2^61 - 1 = 2^61 - 2^1 + 1`) +- Non-Solinas tuple: **1 / 3** + - Almost-Goldilocks (`2^64 - 2^32 - 31`) + +## What SuperNeo suggests (for their purpose) + +- Keep sum-check and norm-check over a small field (or small extension), not over ring arithmetic. +- Do not restrict to only power-of-two cyclotomics (`X^d + 1`); using broader cyclotomics can unlock better field compatibility. +- For Goldilocks/M61, the paper uses a trinomial cyclotomic (`X^54 + X^27 + 1`) instead of `X^d + 1`. +- They explicitly discuss why power-of-two cyclotomics are problematic for some small fields (full splitting / security issues in prior designs). + +## Fit into Hachi design (easy vs harder) + +- Easy fit now: + - Almost-Goldilocks-like setup with `d = 64` and `X^64 + 1` is structurally close to Hachi's current ring form. +- Requires refactor: + - Goldilocks and M61 with `X^54 + X^27 + 1` do not match the current hardcoded negacyclic `X^D + 1` shape. + +## Minimal integration plan into current architecture + +1. Add a cyclotomic profile abstraction (`CyclotomicProfile`) that defines modulus polynomial behavior. +2. Keep current profile as `Negacyclic` (`X^D + 1`) for existing NTT/CRT path. +3. Add a `Trinomial54` profile (`X^54 + X^27 + 1`) for SuperNeo-style field/ring experiments. +4. Keep backend split: + - current CRT+NTT backend for `Negacyclic` only, + - coefficient-domain backend first for `Trinomial54` (no forced NTT dependency). +5. Add explicit domain aliases for profile-bound rings so APIs remain clear (`CoeffDomain`). + +## Practical implication if we insist on Solinas-only fields + +- Within SuperNeo's concrete options, we still have **2 viable Solinas choices**: + - Goldilocks + `X^54 + X^27 + 1` + - M61 + `X^54 + X^27 + 1` +- If we also insist on power-of-two cyclotomics only (`X^d + 1`), SuperNeo's concrete Solinas count drops to **0**. diff --git a/docs/TRANSCRIPT_COMMITMENT_COMPAT_SPEC.md b/docs/TRANSCRIPT_COMMITMENT_COMPAT_SPEC.md new file mode 100644 index 00000000..03fceb7a --- /dev/null +++ b/docs/TRANSCRIPT_COMMITMENT_COMPAT_SPEC.md @@ -0,0 +1,106 @@ +# Transcript and Commitment Compatibility Spec (Hachi Core) + +This document specifies Hachi's protocol-layer transcript and commitment interfaces. + +## Scope + +- Applies to Hachi core (`src/protocol/*`). +- Uses **Hachi-native** transcript labels and ordering. +- Does **not** wire Hachi into Jolt in this phase. +- Any future cross-system interop (for example Jolt-facing adaptation) must be handled by an adapter layer outside core label definitions. + +## Transcript Contract + +Hachi protocol transcripts implement: + +- `new(domain_label)` +- `append_bytes(label, bytes)` +- `append_field(label, x)` +- `append_serde(label, s)` +- `challenge_scalar(label)` +- `reset(domain_label)` + +Current core implementations: + +- `Blake2bTranscript` (Blake2b-512) +- `KeccakTranscript` (Keccak-256, matching Jolt's `sha3` crate usage) + +### Byte Framing + +All absorbed bytes use deterministic framing: + +- `label || len_le64 || bytes` + +This framing is applied uniformly for raw bytes, fields, and serializable protocol objects. + +### Field Encoding + +- Field elements are encoded through canonical representatives (little-endian `u128` bytes). +- Challenge derivation maps transcript digest bytes into field elements via canonical reduction. + +### Label Namespace + +All labels are defined in `src/protocol/transcript/labels.rs` and are Hachi-native. + +Reserved core labels include: + +- Domain label: `hachi/protocol` +- Commitment phase (§4.1): commitment +- Reduction phase (§4.2): evaluation-claims + linear-relation challenge +- Ring-switch phase (§4.3): ring-switch-message + ring-switch challenge +- Sumcheck phase (§4.3): sumcheck-round + sumcheck-round challenge +- Recursion stop phase (§4.5): stop-condition + stop-condition challenge + +Forbidden in Hachi core transcript constants: + +- Dory label literals (for example `vmv_c`, `beta`, `alpha`, `gamma`, `final_e1`, `final_e2`, `d`) + +## Commitment Contract + +Hachi protocol commitment interfaces include: + +- `CommitmentScheme` +- `StreamingCommitmentScheme` +- `AppendToTranscript` + +The commitment layer defines: + +- setup split (`setup_prover`, `setup_verifier`) +- commitment/opening APIs (`commit`, `prove`, `verify`) +- homomorphic combination APIs (`combine_commitments`, `combine_hints`) +- optional streaming/chunked path for large inputs +- label-directed transcript absorption (`AppendToTranscript` call sites choose event labels) + +## Determinism Requirements + +- Prover and verifier must absorb the same labeled byte sequence in the same order. +- Transcript challenges must be reproducible for identical input schedules. +- Commitment/proof objects absorbed via `append_serde` must use deterministic `HachiSerialize` encoding. + +## Test Requirements + +Tests should enforce: + +- Transcript replay determinism (same schedule => same challenges). +- Label/order sensitivity (different labels/order => diverging challenges). +- Framing stability. +- No Dory-label leakage in Hachi label constants/schedules. +- Commitment/hint combination algebraic sanity. + +## Deferred Integration Note + +Integration into Jolt is a separate, deferred phase tracked in `HACHI_PROGRESS.md`. +When started, an adapter should translate between external transcript conventions and Hachi core interfaces without changing Hachi-native core labels. + +## Deferred Adapter Contract (Design Only) + +`JoltToHachiTranscript` is deferred, but its expected behavior is fixed now: + +- Owns a mutable reference to a Jolt transcript object. +- Implements Hachi `Transcript` by forwarding absorption/challenge calls. +- Performs label translation at the boundary (Jolt-side naming to Hachi-side API events). +- Never mutates or extends Hachi core label constants. +- Maintains deterministic call ordering: prover and verifier adapter paths must replay identical absorb/challenge sequences. +- Supports domain initialization and explicit reset semantics. + +This adapter lives outside Hachi core protocol modules and is not part of this phase's implementation. diff --git a/examples/codegen_probe_special.rs b/examples/codegen_probe_special.rs new file mode 100644 index 00000000..bc6d5b50 --- /dev/null +++ b/examples/codegen_probe_special.rs @@ -0,0 +1,143 @@ +#![allow(missing_docs)] + +//! Codegen probe for packed/scalar Fp64 multiply kernels. +//! +//! Build with: +//! `cargo rustc --example codegen_probe_special --release -- --emit=asm` + +use hachi_pcs::algebra::fields::pseudo_mersenne::{POW2_OFFSET_MODULUS_40, POW2_OFFSET_MODULUS_64}; +use hachi_pcs::algebra::{Fp64, Fp64Packing, PackedValue}; +use hachi_pcs::CanonicalField; + +const MASK40: u64 = (1u64 << 40) - 1; +const P40: u64 = POW2_OFFSET_MODULUS_40; +const C40: u64 = (1u64 << 40) - P40; // 195 +const P64: u64 = POW2_OFFSET_MODULUS_64; +const C64: u64 = 0u64.wrapping_sub(P64); // 59 + +#[inline(always)] +fn mul_c40_split(x: u64) -> u64 { + let c = C40 as u32; + let x_lo = x as u32; + let x_hi = (x >> 32) as u32; + (c as u64 * x_lo as u64).wrapping_add((c as u64 * x_hi as u64) << 32) +} + +#[inline(always)] +fn mul_c40_shiftadd(x: u64) -> u64 { + // 195x = (128 + 64 + 2 + 1) * x + (x << 7) + .wrapping_add(x << 6) + .wrapping_add(x << 1) + .wrapping_add(x) +} + +#[inline(always)] +fn reduce40_with_mulc(lo: u64, hi: u64, mulc: fn(u64) -> u64) -> u64 { + let high = (lo >> 40) | (hi << 24); + let f1 = (lo & MASK40).wrapping_add(mulc(high)); + let f2 = (f1 & MASK40).wrapping_add(mulc(f1 >> 40)); + let reduced = f2.wrapping_sub(P40); + let borrow = reduced >> 63; + reduced.wrapping_add(borrow.wrapping_neg() & P40) +} + +#[inline(always)] +fn reduce64(lo: u64, hi: u64) -> u64 { + let f1 = (lo as u128) + (C64 as u128) * (hi as u128); + let f2 = (f1 as u64 as u128) + (C64 as u128) * ((f1 >> 64) as u64 as u128); + let reduced = f2.wrapping_sub(P64 as u128); + let borrow = reduced >> 127; + reduced.wrapping_add(borrow.wrapping_neg() & (P64 as u128)) as u64 +} + +#[inline(never)] +#[no_mangle] +pub extern "C" fn probe_reduce40_split(lo: u64, hi: u64) -> u64 { + reduce40_with_mulc(lo, hi, mul_c40_split) +} + +#[inline(never)] +#[no_mangle] +pub extern "C" fn probe_reduce40_shiftadd(lo: u64, hi: u64) -> u64 { + reduce40_with_mulc(lo, hi, mul_c40_shiftadd) +} + +#[inline(never)] +#[no_mangle] +pub extern "C" fn probe_reduce64(lo: u64, hi: u64) -> u64 { + reduce64(lo, hi) +} + +#[inline(never)] +#[no_mangle] +pub extern "C" fn probe_packed_fp64_40_mul(a0: u64, a1: u64, b0: u64, b1: u64) -> u64 { + type F = Fp64<{ POW2_OFFSET_MODULUS_40 }>; + type PF = Fp64Packing<{ POW2_OFFSET_MODULUS_40 }>; + + let a = PF::from_fn(|i| { + if i == 0 { + F::from_canonical_u64(a0) + } else { + F::from_canonical_u64(a1) + } + }); + let b = PF::from_fn(|i| { + if i == 0 { + F::from_canonical_u64(b0) + } else { + F::from_canonical_u64(b1) + } + }); + let c = a * b; + (c.extract(0).to_canonical_u128() as u64) ^ (c.extract(1).to_canonical_u128() as u64) +} + +#[inline(never)] +#[no_mangle] +pub extern "C" fn probe_packed_fp64_64_mul(a0: u64, a1: u64, b0: u64, b1: u64) -> u64 { + type F = Fp64<{ POW2_OFFSET_MODULUS_64 }>; + type PF = Fp64Packing<{ POW2_OFFSET_MODULUS_64 }>; + + let a = PF::from_fn(|i| { + if i == 0 { + F::from_canonical_u64(a0) + } else { + F::from_canonical_u64(a1) + } + }); + let b = PF::from_fn(|i| { + if i == 0 { + F::from_canonical_u64(b0) + } else { + F::from_canonical_u64(b1) + } + }); + let c = a * b; + (c.extract(0).to_canonical_u128() as u64) ^ (c.extract(1).to_canonical_u128() as u64) +} + +#[inline(never)] +#[no_mangle] +pub extern "C" fn probe_scalar_fp64_40_mul(a: u64, b: u64) -> u64 { + type F = Fp64<{ POW2_OFFSET_MODULUS_40 }>; + (F::from_canonical_u64(a) * F::from_canonical_u64(b)).to_canonical_u128() as u64 +} + +#[inline(never)] +#[no_mangle] +pub extern "C" fn probe_scalar_fp64_64_mul(a: u64, b: u64) -> u64 { + type F = Fp64<{ POW2_OFFSET_MODULUS_64 }>; + (F::from_canonical_u64(a) * F::from_canonical_u64(b)).to_canonical_u128() as u64 +} + +fn main() { + let x = probe_packed_fp64_40_mul(1, 2, 3, 4) + ^ probe_packed_fp64_64_mul(5, 6, 7, 8) + ^ probe_scalar_fp64_40_mul(9, 10) + ^ probe_scalar_fp64_64_mul(11, 12) + ^ probe_reduce40_split(13, 14) + ^ probe_reduce40_shiftadd(15, 16) + ^ probe_reduce64(17, 18); + std::hint::black_box(x); +} diff --git a/paper/fri-binius.pdf b/paper/fri-binius.pdf new file mode 100644 index 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zUQHVK!{j?bxkDqr8pa$uCxx1E#+*X;3HVPtiAFWOD^8MUMsFL2a%jBb)&yMl{{^fh zNt|Ve3{4BmU6b0leHl;SwW4(WGTwj^S}y<@Dtjgk-6f)xJ8Pf(E=Dm=c58Tlq%R{> zwRZ0y(^l!?!2JCh0<5sI&S(<#xahHlS*XxXZVTr2#K`q%L< zB?qxQUG=)t%?;cl)d!Ug_SH;+R%|<7OqsQ*ZCzyt?G3;8Ol1fw>4H8NC{&<#1tJhu zaXw}jFjXkONKDsjLnz+44>Moa#9v!h_-`h3GgEYybPsN5Y$k26US!mF?^hr=#^c}5 zl%V?m9EQ7NW4*YIgNs*#Us!qWrS>%|?&c_f>+;nM-kpF`>uBocqF}jria`CnMS+l8 zgLglw9!$9LZnFppwf(nIu~*MxOf7+W$iq-=I++6<6{jBd&I}QA`ZRB6<1j0*Tcu=_RDOXQH&7(9_ViZBq)S$ zY-BeC9bp`xHd%^oUG{ea{rES3KC(nEpT@neMeSh+?PdD>d83y1qnFJ-KDfTQ?x%G` zIH4N!--{%K)j0-jAnp7$3INEj6N#_sXdRK1WkG12>G4UR_ozGbP=^R_j7=%+cFf9u zi~|_Za~%-YMfzn7ao698_gioPg%gs0Qp^*~Q^}=jcUrSIA9zp NttPrimeOps for ScalarBackend { + #[inline] + fn from_canonical(prime: NttPrime, value: i16) -> MontCoeff { + prime.from_canonical(value) + } + + #[inline] + fn to_canonical(prime: NttPrime, value: MontCoeff) -> i16 { + prime.to_canonical(value) + } + + #[inline] + fn reduce(prime: NttPrime, value: MontCoeff) -> MontCoeff { + prime.reduce(value) + } + + #[inline] + fn pointwise_mul( + prime: NttPrime, + out: &mut [MontCoeff; D], + lhs: &[MontCoeff; D], + rhs: &[MontCoeff; D], + ) { + prime.pointwise_mul(out, lhs, rhs); + } +} + +impl NttTransform for ScalarBackend { + #[inline] + fn forward_ntt(limb: &mut [MontCoeff; D], prime: NttPrime, twiddles: &NttTwiddles) { + forward_ntt(limb, prime, twiddles); + } + + #[inline] + fn inverse_ntt(limb: &mut [MontCoeff; D], prime: NttPrime, twiddles: &NttTwiddles) { + inverse_ntt(limb, prime, twiddles); + } +} + +impl CrtReconstruct for ScalarBackend { + fn reconstruct( + primes: &[NttPrime; K], + canonical_limbs: &[[i16; D]; K], + qdata: &QData, + ) -> [F; D] { + let q = qdata.q_u128().expect("q must fit in u128"); + let prime_moduli: [u128; K] = std::array::from_fn(|k| { + u128::try_from(primes[k].p).expect("CRT prime modulus must be positive") + }); + let big_p = prime_moduli.iter().fold(1u128, |acc, p| { + acc.checked_mul(*p) + .expect("product of CRT primes must fit u128") + }); + let crt_m: [u128; K] = std::array::from_fn(|k| big_p / prime_moduli[k]); + let crt_inv: [u16; K] = std::array::from_fn(|k| { + let inv = mod_inverse( + (crt_m[k] % prime_moduli[k]) as i128, + prime_moduli[k] as i128, + ); + u16::try_from(inv).expect("CRT inverse must fit u16 for small-prime backend") + }); + + let mut coeffs = [F::zero(); D]; + for (d, coeff) in coeffs.iter_mut().enumerate() { + let mut acc: u128 = 0; + for k in 0..K { + let ck_i16 = canonical_limbs[k][d]; + debug_assert!(ck_i16 >= 0 && ck_i16 < primes[k].p); + let ck = u16::try_from(ck_i16).expect("canonical residue must fit u16"); + + // Multiply by tiny residues/inverses (<= 15 bits) in fixed-time + // loops, then accumulate modulo P. + let term = mul_mod_by_small_u16(crt_m[k], ck, big_p); + let term = mul_mod_by_small_u16(term, crt_inv[k], big_p); + acc = add_mod_u128(acc, term, big_p); + } + + // Final projection into [0, q) without variable-latency division. + let reduced = reduce_u128_divfree(acc, q); + *coeff = F::from_q_residue_u128(reduced); + } + + coeffs + } +} + +fn mod_inverse(a: i128, modulus: i128) -> i128 { + let (mut t, mut new_t) = (0i128, 1i128); + let (mut r, mut new_r) = (modulus, ((a % modulus) + modulus) % modulus); + + while new_r != 0 { + let q = r / new_r; + (t, new_t) = (new_t, t - q * new_t); + (r, new_r) = (new_r, r - q * new_r); + } + + assert_eq!(r, 1, "CRT inverse does not exist"); + (t % modulus + modulus) % modulus +} + +#[inline] +fn add_mod_u128(a: u128, b: u128, modulus: u128) -> u128 { + debug_assert!(a < modulus); + debug_assert!(b < modulus); + + let (sum_lo, carry) = a.overflowing_add(b); + let hi = carry as u128; + let (sub_lo, borrow) = sum_lo.overflowing_sub(modulus); + let sum_ge_modulus = (!borrow) as u128; + let should_sub = hi | sum_ge_modulus; + let mask = should_sub.wrapping_neg(); + (sum_lo & !mask) | (sub_lo & mask) +} + +#[inline] +fn mul_mod_by_small_u16(a: u128, b: u16, modulus: u128) -> u128 { + debug_assert!(a < modulus); + let mut acc = 0u128; + let mut cur = a; + for i in 0..16 { + let candidate = add_mod_u128(acc, cur, modulus); + let bit = ((b >> i) & 1) as u128; + let mask = bit.wrapping_neg(); + acc = (acc & !mask) | (candidate & mask); + cur = add_mod_u128(cur, cur, modulus); + } + acc +} + +#[inline] +fn reduce_u128_divfree(x: u128, modulus: u128) -> u128 { + debug_assert!(modulus > 0); + let mut rem = 0u128; + for i in (0..128).rev() { + rem = (rem << 1) | ((x >> i) & 1); + let (candidate, borrow) = rem.overflowing_sub(modulus); + let mask = 0u128.wrapping_sub((!borrow) as u128); + rem = (candidate & mask) | (rem & !mask); + } + rem +} + +#[cfg(test)] +mod tests { + use super::{add_mod_u128, mul_mod_by_small_u16, reduce_u128_divfree}; + + #[test] + fn add_mod_matches_native_when_sum_fits_u128() { + let modulus = (1u128 << 100) - 159; + for i in 0..4096u128 { + let a = (i * 104_729 + 17) % modulus; + let b = (i * 130_363 + 31) % modulus; + let expected = (a + b) % modulus; + assert_eq!(add_mod_u128(a, b, modulus), expected); + } + } + + #[test] + fn mul_mod_small_matches_native_when_product_fits_u128() { + let modulus = (1u128 << 100) - 159; + for i in 0..4096u128 { + let a = (i * 786_433 + 19) % modulus; + let b = ((i * 97 + 7) & 0xFFFF) as u16; + let expected = (a * (b as u128)) % modulus; + assert_eq!(mul_mod_by_small_u16(a, b, modulus), expected); + } + } + + #[test] + fn reduce_u128_divfree_matches_native_mod() { + let modulus = (1u128 << 100) - 159; + for i in 0..4096u128 { + let x = (i.wrapping_mul(1_146_989_321_207u128)) + ^ (i.rotate_left(37)) + ^ 0xA5A5_A5A5_A5A5_A5A5_A5A5_A5A5_A5A5_A5A5u128; + assert_eq!(reduce_u128_divfree(x, modulus), x % modulus); + } + } +} diff --git a/src/algebra/backend/traits.rs b/src/algebra/backend/traits.rs new file mode 100644 index 00000000..d6c8a5a0 --- /dev/null +++ b/src/algebra/backend/traits.rs @@ -0,0 +1,56 @@ +//! Backend traits for CRT+NTT execution semantics. + +use crate::algebra::ntt::butterfly::NttTwiddles; +use crate::algebra::ntt::crt::QData; +use crate::algebra::ntt::prime::{MontCoeff, NttPrime}; +use crate::algebra::ring::CrtNttConvertibleField; + +/// Per-prime arithmetic primitives used by CRT+NTT domains. +pub trait NttPrimeOps { + /// Convert canonical coefficient to backend prime representation. + fn from_canonical(prime: NttPrime, value: i16) -> MontCoeff; + + /// Convert backend prime representation back to canonical coefficient. + fn to_canonical(prime: NttPrime, value: MontCoeff) -> i16; + + /// Range-reduce one backend prime coefficient. + fn reduce(prime: NttPrime, value: MontCoeff) -> MontCoeff; + + /// Pointwise multiplication in backend prime representation. + fn pointwise_mul( + prime: NttPrime, + out: &mut [MontCoeff; D], + lhs: &[MontCoeff; D], + rhs: &[MontCoeff; D], + ); +} + +/// Forward/inverse transform kernels for one NTT limb. +pub trait NttTransform { + /// Forward transform from coefficient limb to NTT limb. + fn forward_ntt(limb: &mut [MontCoeff; D], prime: NttPrime, twiddles: &NttTwiddles); + + /// Inverse transform from NTT limb to coefficient limb. + fn inverse_ntt(limb: &mut [MontCoeff; D], prime: NttPrime, twiddles: &NttTwiddles); +} + +/// CRT reconstruction contract from per-prime canonical coefficients. +pub trait CrtReconstruct { + /// Reconstruct coefficient-domain values from canonical CRT residues. + fn reconstruct( + primes: &[NttPrime; K], + canonical_limbs: &[[i16; D]; K], + qdata: &QData, + ) -> [F; D]; +} + +/// Convenience composition trait for full ring backend capability. +pub trait RingBackend: + NttPrimeOps + NttTransform + CrtReconstruct +{ +} + +impl RingBackend for T where + T: NttPrimeOps + NttTransform + CrtReconstruct +{ +} diff --git a/src/algebra/domains/coeff.rs b/src/algebra/domains/coeff.rs new file mode 100644 index 00000000..461a2fd4 --- /dev/null +++ b/src/algebra/domains/coeff.rs @@ -0,0 +1,6 @@ +//! Coefficient-domain representation boundary. + +use crate::algebra::ring::CyclotomicRing; + +/// Coefficient-domain ring representation. +pub type CoeffDomain = CyclotomicRing; diff --git a/src/algebra/domains/crt_ntt.rs b/src/algebra/domains/crt_ntt.rs new file mode 100644 index 00000000..48fe2263 --- /dev/null +++ b/src/algebra/domains/crt_ntt.rs @@ -0,0 +1,6 @@ +//! CRT+NTT-domain representation boundary. + +use crate::algebra::ring::CyclotomicCrtNtt; + +/// CRT+NTT-domain ring representation. +pub type CrtNttDomain = CyclotomicCrtNtt; diff --git a/src/algebra/domains/mod.rs b/src/algebra/domains/mod.rs new file mode 100644 index 00000000..fa140308 --- /dev/null +++ b/src/algebra/domains/mod.rs @@ -0,0 +1,7 @@ +//! Explicit domain-layer aliases for ring representations. + +pub mod coeff; +pub mod crt_ntt; + +pub use coeff::CoeffDomain; +pub use crt_ntt::CrtNttDomain; diff --git a/src/algebra/fields/ext.rs b/src/algebra/fields/ext.rs new file mode 100644 index 00000000..29061e9b --- /dev/null +++ b/src/algebra/fields/ext.rs @@ -0,0 +1,446 @@ +//! Quadratic and quartic extension fields. + +use crate::algebra::module::VectorModule; +use crate::primitives::serialization::{ + Compress, HachiDeserialize, HachiSerialize, SerializationError, Valid, Validate, +}; +use crate::{FieldCore, FieldSampling}; +use rand_core::RngCore; +use std::io::{Read, Write}; +use std::marker::PhantomData; +use std::ops::{Add, Mul, Neg, Sub}; + +/// Parameters for an `Fp2` quadratic extension over base field `F`. +pub trait Fp2Config { + /// Non-residue `NR` such that `u^2 = NR`. + fn non_residue() -> F; +} + +/// Quadratic extension element `c0 + c1 * u` with `u^2 = NR`. +pub struct Fp2> { + /// Constant term. + pub c0: F, + /// Coefficient of `u`. + pub c1: F, + _cfg: PhantomData C>, +} + +impl> Fp2 { + /// Construct `c0 + c1 * u`. + #[inline] + pub fn new(c0: F, c1: F) -> Self { + Self { + c0, + c1, + _cfg: PhantomData, + } + } + + /// Return the conjugate `c0 - c1 * u`. + #[inline] + pub fn conjugate(self) -> Self { + Self::new(self.c0, -self.c1) + } + + /// Return the norm in the base field: `c0^2 - NR * c1^2`. + #[inline] + pub fn norm(self) -> F { + let nr = C::non_residue(); + (self.c0 * self.c0) - (nr * (self.c1 * self.c1)) + } +} + +impl> Clone for Fp2 { + fn clone(&self) -> Self { + *self + } +} + +impl> Copy for Fp2 {} + +impl> Default for Fp2 { + fn default() -> Self { + Self::new(F::zero(), F::zero()) + } +} + +impl> PartialEq for Fp2 { + fn eq(&self, other: &Self) -> bool { + self.c0 == other.c0 && self.c1 == other.c1 + } +} + +impl> Add for Fp2 { + type Output = Self; + fn add(self, rhs: Self) -> Self::Output { + Self::new(self.c0 + rhs.c0, self.c1 + rhs.c1) + } +} +impl> Sub for Fp2 { + type Output = Self; + fn sub(self, rhs: Self) -> Self::Output { + Self::new(self.c0 - rhs.c0, self.c1 - rhs.c1) + } +} +impl> Neg for Fp2 { + type Output = Self; + fn neg(self) -> Self::Output { + Self::new(-self.c0, -self.c1) + } +} +impl> Mul for Fp2 { + type Output = Self; + fn mul(self, rhs: Self) -> Self::Output { + let nr = C::non_residue(); + let a0 = self.c0; + let a1 = self.c1; + let b0 = rhs.c0; + let b1 = rhs.c1; + Self::new((a0 * b0) + (nr * (a1 * b1)), (a0 * b1) + (a1 * b0)) + } +} + +impl<'a, F: FieldCore, C: Fp2Config> Add<&'a Self> for Fp2 { + type Output = Self; + fn add(self, rhs: &'a Self) -> Self::Output { + self + *rhs + } +} +impl<'a, F: FieldCore, C: Fp2Config> Sub<&'a Self> for Fp2 { + type Output = Self; + fn sub(self, rhs: &'a Self) -> Self::Output { + self - *rhs + } +} +impl<'a, F: FieldCore, C: Fp2Config> Mul<&'a Self> for Fp2 { + type Output = Self; + fn mul(self, rhs: &'a Self) -> Self::Output { + self * *rhs + } +} + +impl> Valid for Fp2 { + fn check(&self) -> Result<(), SerializationError> { + self.c0.check()?; + self.c1.check()?; + Ok(()) + } +} + +impl> HachiSerialize for Fp2 { + fn serialize_with_mode( + &self, + mut writer: W, + compress: Compress, + ) -> Result<(), SerializationError> { + self.c0.serialize_with_mode(&mut writer, compress)?; + self.c1.serialize_with_mode(&mut writer, compress)?; + Ok(()) + } + + fn serialized_size(&self, compress: Compress) -> usize { + self.c0.serialized_size(compress) + self.c1.serialized_size(compress) + } +} + +impl> HachiDeserialize for Fp2 { + fn deserialize_with_mode( + mut reader: R, + compress: Compress, + validate: Validate, + ) -> Result { + let c0 = F::deserialize_with_mode(&mut reader, compress, validate)?; + let c1 = F::deserialize_with_mode(&mut reader, compress, validate)?; + let out = Self::new(c0, c1); + if matches!(validate, Validate::Yes) { + out.check()?; + } + Ok(out) + } +} + +impl> FieldCore for Fp2 { + fn zero() -> Self { + Self::new(F::zero(), F::zero()) + } + + fn one() -> Self { + Self::new(F::one(), F::zero()) + } + + fn is_zero(&self) -> bool { + self.c0.is_zero() && self.c1.is_zero() + } + + fn add(&self, rhs: &Self) -> Self { + *self + *rhs + } + + fn sub(&self, rhs: &Self) -> Self { + *self - *rhs + } + + fn mul(&self, rhs: &Self) -> Self { + *self * *rhs + } + + fn inv(self) -> Option { + if self.is_zero() { + return None; + } + let inv_n = self.norm().inv()?; + Some(Self::new(self.c0 * inv_n, (-self.c1) * inv_n)) + } +} + +impl> FieldSampling for Fp2 { + fn sample(rng: &mut R) -> Self { + Self::new(F::sample(rng), F::sample(rng)) + } +} + +/// Parameters for an `Fp4` quadratic extension over `Fp2`. +pub trait Fp4Config> { + /// Non-residue `NR2` in `Fp2` such that `v^2 = NR2`. + fn non_residue() -> Fp2; +} + +/// Quartic extension element `c0 + c1 * v` over `Fp2`, where `v^2 = NR2`. +pub struct Fp4, C4: Fp4Config> { + /// Constant term. + pub c0: Fp2, + /// Coefficient of `v`. + pub c1: Fp2, + _cfg: PhantomData C4>, +} + +impl, C4: Fp4Config> Fp4 { + /// Construct `c0 + c1 * v`. + #[inline] + pub fn new(c0: Fp2, c1: Fp2) -> Self { + Self { + c0, + c1, + _cfg: PhantomData, + } + } + + /// Return the norm in `Fp2`: `c0^2 - NR2 * c1^2`. + #[inline] + pub fn norm(self) -> Fp2 { + let nr2 = C4::non_residue(); + (self.c0 * self.c0) - (nr2 * (self.c1 * self.c1)) + } +} + +impl, C4: Fp4Config> Clone for Fp4 { + fn clone(&self) -> Self { + *self + } +} + +impl, C4: Fp4Config> Copy for Fp4 {} + +impl, C4: Fp4Config> Default for Fp4 { + fn default() -> Self { + Self::new( + Fp2::new(F::zero(), F::zero()), + Fp2::new(F::zero(), F::zero()), + ) + } +} + +impl, C4: Fp4Config> PartialEq for Fp4 { + fn eq(&self, other: &Self) -> bool { + self.c0 == other.c0 && self.c1 == other.c1 + } +} + +impl, C4: Fp4Config> Add for Fp4 { + type Output = Self; + fn add(self, rhs: Self) -> Self::Output { + Self::new(self.c0 + rhs.c0, self.c1 + rhs.c1) + } +} +impl, C4: Fp4Config> Sub for Fp4 { + type Output = Self; + fn sub(self, rhs: Self) -> Self::Output { + Self::new(self.c0 - rhs.c0, self.c1 - rhs.c1) + } +} +impl, C4: Fp4Config> Neg for Fp4 { + type Output = Self; + fn neg(self) -> Self::Output { + Self::new(-self.c0, -self.c1) + } +} +impl, C4: Fp4Config> Mul for Fp4 { + type Output = Self; + fn mul(self, rhs: Self) -> Self::Output { + let nr2 = C4::non_residue(); + let a0 = self.c0; + let a1 = self.c1; + let b0 = rhs.c0; + let b1 = rhs.c1; + Self::new((a0 * b0) + (nr2 * (a1 * b1)), (a0 * b1) + (a1 * b0)) + } +} + +impl<'a, F: FieldCore, C2: Fp2Config, C4: Fp4Config> Add<&'a Self> for Fp4 { + type Output = Self; + fn add(self, rhs: &'a Self) -> Self::Output { + self + *rhs + } +} +impl<'a, F: FieldCore, C2: Fp2Config, C4: Fp4Config> Sub<&'a Self> for Fp4 { + type Output = Self; + fn sub(self, rhs: &'a Self) -> Self::Output { + self - *rhs + } +} +impl<'a, F: FieldCore, C2: Fp2Config, C4: Fp4Config> Mul<&'a Self> for Fp4 { + type Output = Self; + fn mul(self, rhs: &'a Self) -> Self::Output { + self * *rhs + } +} + +impl, C4: Fp4Config> Valid for Fp4 { + fn check(&self) -> Result<(), SerializationError> { + self.c0.check()?; + self.c1.check()?; + Ok(()) + } +} + +impl, C4: Fp4Config> HachiSerialize for Fp4 { + fn serialize_with_mode( + &self, + mut writer: W, + compress: Compress, + ) -> Result<(), SerializationError> { + self.c0.serialize_with_mode(&mut writer, compress)?; + self.c1.serialize_with_mode(&mut writer, compress)?; + Ok(()) + } + + fn serialized_size(&self, compress: Compress) -> usize { + self.c0.serialized_size(compress) + self.c1.serialized_size(compress) + } +} + +impl, C4: Fp4Config> HachiDeserialize + for Fp4 +{ + fn deserialize_with_mode( + mut reader: R, + compress: Compress, + validate: Validate, + ) -> Result { + let c0 = Fp2::::deserialize_with_mode(&mut reader, compress, validate)?; + let c1 = Fp2::::deserialize_with_mode(&mut reader, compress, validate)?; + let out = Self::new(c0, c1); + if matches!(validate, Validate::Yes) { + out.check()?; + } + Ok(out) + } +} + +impl, C4: Fp4Config> FieldCore for Fp4 { + fn zero() -> Self { + Self::new(Fp2::zero(), Fp2::zero()) + } + + fn one() -> Self { + Self::new(Fp2::one(), Fp2::zero()) + } + + fn is_zero(&self) -> bool { + self.c0.is_zero() && self.c1.is_zero() + } + + fn add(&self, rhs: &Self) -> Self { + *self + *rhs + } + + fn sub(&self, rhs: &Self) -> Self { + *self - *rhs + } + + fn mul(&self, rhs: &Self) -> Self { + *self * *rhs + } + + fn inv(self) -> Option { + if self.is_zero() { + return None; + } + let inv_n = self.norm().inv()?; + Some(Self::new(self.c0 * inv_n, (-self.c1) * inv_n)) + } +} + +impl, C4: Fp4Config> FieldSampling + for Fp4 +{ + fn sample(rng: &mut R) -> Self { + Self::new(Fp2::sample(rng), Fp2::sample(rng)) + } +} + +// Scalar * VectorModule impls for extension scalars. + +impl Mul, N>> for Fp2 +where + F: FieldCore + Valid, + C: Fp2Config, +{ + type Output = VectorModule, N>; + fn mul(self, rhs: VectorModule, N>) -> Self::Output { + let mut out = rhs.0; + for coeff in &mut out { + *coeff = self * *coeff; + } + VectorModule(out) + } +} + +impl<'a, F, C, const N: usize> Mul<&'a VectorModule, N>> for Fp2 +where + F: FieldCore + Valid, + C: Fp2Config, +{ + type Output = VectorModule, N>; + fn mul(self, rhs: &'a VectorModule, N>) -> Self::Output { + self * *rhs + } +} + +impl Mul, N>> for Fp4 +where + F: FieldCore + Valid, + C2: Fp2Config, + C4: Fp4Config, +{ + type Output = VectorModule, N>; + fn mul(self, rhs: VectorModule, N>) -> Self::Output { + let mut out = rhs.0; + for coeff in &mut out { + *coeff = self * *coeff; + } + VectorModule(out) + } +} + +impl<'a, F, C2, C4, const N: usize> Mul<&'a VectorModule, N>> for Fp4 +where + F: FieldCore + Valid, + C2: Fp2Config, + C4: Fp4Config, +{ + type Output = VectorModule, N>; + fn mul(self, rhs: &'a VectorModule, N>) -> Self::Output { + self * *rhs + } +} diff --git a/src/algebra/fields/fp128.rs b/src/algebra/fields/fp128.rs new file mode 100644 index 00000000..09248062 --- /dev/null +++ b/src/algebra/fields/fp128.rs @@ -0,0 +1,1032 @@ +//! 128-bit prime field for primes of the form `p = 2^128 − c` with `c < 2^64`. +//! +//! Uses Solinas-style two-fold reduction: no Montgomery form, ~23 cycles/mul +//! on both AArch64 and x86-64. The offset `c` is computed at compile time +//! from the const-generic modulus `P`. +//! +//! ## Naming convention for built-in primes +//! +//! The built-in type names encode the **signed terms as they appear in the +//! modulus `p`** (excluding the leading `+2^128` term). For example, +//! `Prime128M13M4P0` denotes `p = 2^128 − 2^13 − 2^4 + 2^0`. + +use std::io::{Read, Write}; +use std::ops::{Add, Mul, Neg, Sub}; + +use rand_core::RngCore; + +use crate::primitives::serialization::{ + Compress, HachiDeserialize, HachiSerialize, SerializationError, Valid, Validate, +}; +use crate::{CanonicalField, FieldCore, FieldSampling, Invertible, PseudoMersenneField}; + +/// Pack two u64 limbs into `[lo, hi]`. +#[inline(always)] +const fn pack(lo: u64, hi: u64) -> [u64; 2] { + [lo, hi] +} + +/// Convert `u128` → `[u64; 2]`. +#[inline(always)] +const fn from_u128(x: u128) -> [u64; 2] { + [x as u64, (x >> 64) as u64] +} + +/// Convert `[u64; 2]` → `u128`. +#[inline(always)] +const fn to_u128(x: [u64; 2]) -> u128 { + x[0] as u128 | (x[1] as u128) << 64 +} + +/// `a * b` widening to 128 bits; returns `(lo64, hi64)`. +#[inline(always)] +fn mul64_wide(a: u64, b: u64) -> (u64, u64) { + #[cfg(all(target_arch = "x86_64", target_feature = "bmi2"))] + { + // Safety: this block is compiled only when `bmi2` is enabled. + unsafe { mul64_wide_bmi2(a, b) } + } + #[cfg(not(all(target_arch = "x86_64", target_feature = "bmi2")))] + { + let prod = (a as u128) * (b as u128); + (prod as u64, (prod >> 64) as u64) + } +} + +#[cfg(all(target_arch = "x86_64", target_feature = "bmi2"))] +#[inline(always)] +unsafe fn mul64_wide_bmi2(a: u64, b: u64) -> (u64, u64) { + let mut hi = 0; + // Safety: caller guarantees CPU support via cfg-gated compilation. + let lo = unsafe { std::arch::x86_64::_mulx_u64(a, b, &mut hi) }; + (lo, hi) +} + +#[inline(always)] +const fn is_pow2_u64(x: u64) -> bool { + x != 0 && (x & (x - 1)) == 0 +} + +#[inline(always)] +const fn log2_pow2_u64(mut x: u64) -> u32 { + let mut k = 0u32; + while x > 1 { + x >>= 1; + k += 1; + } + k +} + +/// 128-bit prime field element for primes `p = 2^128 − c` with `c < 2^64`. +/// +/// Stored as `[u64; 2]` (lo, hi) for 8-byte alignment and direct limb access. +/// +/// The offset `c = 2^128 − p` and all derived constants are computed at +/// compile time from the const-generic `P`. Instantiating `Fp128` with a +/// modulus that is not of this form is a compile-time error. +#[derive(Debug, Clone, Copy, Default)] +pub struct Fp128(pub(crate) [u64; 2]); + +impl PartialEq for Fp128

{ + fn eq(&self, other: &Self) -> bool { + self.0 == other.0 + } +} + +impl Eq for Fp128

{} + +impl Fp128

{ + /// Offset `c = 2^128 − p`. Validated at compile time. + const C: u128 = { + let c = 0u128.wrapping_sub(P); + assert!(P != 0, "modulus must be nonzero"); + assert!(P & 1 == 1, "modulus must be odd"); + assert!(c < (1u128 << 64), "P must be 2^128 - c with c < 2^64"); + // Fused overflow+canonicalize requires C(C+1) < P. + assert!( + c * (c + 1) < P, + "C(C+1) < P required for fused canonicalize" + ); + c + }; + const C_LO: u64 = Self::C as u64; + /// +1 means `C = 2^a + 1`, -1 means `C = 2^a - 1`, 0 means generic. + const C_SHIFT_KIND: i8 = { + let c = Self::C_LO; + if c > 1 && is_pow2_u64(c - 1) { + 1 + } else if c == u64::MAX || is_pow2_u64(c + 1) { + -1 + } else { + 0 + } + }; + const C_SHIFT: u32 = { + let c = Self::C_LO; + if Self::C_SHIFT_KIND == 1 { + log2_pow2_u64(c - 1) + } else if Self::C_SHIFT_KIND == -1 { + if c == u64::MAX { + 64 + } else { + log2_pow2_u64(c + 1) + } + } else { + 0 + } + }; + + /// Multiply by `C = 2^128 - P`. For `C = 2^a ± 1`, this is shift/add or + /// shift/sub only; otherwise it falls back to generic widening multiply. + #[inline(always)] + fn mul_c_wide(x: u64) -> (u64, u64) { + if Self::C_SHIFT_KIND == 1 { + let v = ((x as u128) << Self::C_SHIFT) + x as u128; + (v as u64, (v >> 64) as u64) + } else if Self::C_SHIFT_KIND == -1 { + let v = ((x as u128) << Self::C_SHIFT) - x as u128; + (v as u64, (v >> 64) as u64) + } else { + mul64_wide(Self::C_LO, x) + } + } + + /// Create from a canonical representative in `[0, p)`. + #[inline] + pub fn from_canonical_u128(x: u128) -> Self { + debug_assert!(x < P); + Self(from_u128(x)) + } + + /// Return the canonical representative in `[0, p)`. + #[inline] + pub fn to_canonical_u128(self) -> u128 { + to_u128(self.0) + } + + #[inline(always)] + fn add_raw(a: [u64; 2], b: [u64; 2]) -> [u64; 2] { + let (s, carry) = to_u128(a).overflowing_add(to_u128(b)); + let (reduced, borrow) = s.overflowing_sub(P); + from_u128(if carry | !borrow { + reduced + } else { + reduced.wrapping_add(P) + }) + } + + #[inline(always)] + fn sub_raw(a: [u64; 2], b: [u64; 2]) -> [u64; 2] { + let (diff, borrow) = to_u128(a).overflowing_sub(to_u128(b)); + from_u128(if borrow { diff.wrapping_add(P) } else { diff }) + } + + /// Fold 2 + canonicalize: reduce `[t0, t1] + t2·2^128` into `[0, p)`. + /// + /// Correctness argument for the fused overflow+canonicalize: + /// + /// Let `v = base + C·t2` (mathematical, not mod 2^128). + /// From the fold-1 mac chain, `t2 ≤ C`, so `C·t2 ≤ C²`. + /// + /// - **No overflow** (`v < 2^128`): `s = v`, and the standard + /// canonicalize applies — `s + C` carries iff `s ≥ P`. + /// - **Overflow** (`v ≥ 2^128`): `s = v − 2^128`, so `s < C·t2 ≤ C²`. + /// The correct reduced value is `s + C` (since `2^128 ≡ C mod P`). + /// Because `s + C < C² + C = C(C+1)` and `C(C+1) < P` for all + /// `C < 2^64`, the value `s + C` is already in `[0, P)` — no + /// further canonicalization is needed, and `s + C < 2^128` so the + /// add does NOT carry. + /// + /// Therefore `if (overflow | carry) { s + C } else { s }` is correct + /// in both cases, fusing the overflow correction with canonicalization. + #[inline(always)] + fn fold2_canonicalize(t0: u64, t1: u64, t2: u64) -> [u64; 2] { + let (ct2_lo, ct2_hi) = Self::mul_c_wide(t2); + + let (s0, carry0) = t0.overflowing_add(ct2_lo); + let (s1a, carry1a) = t1.overflowing_add(ct2_hi); + let (s1, carry1b) = s1a.overflowing_add(carry0 as u64); + let overflow = carry1a | carry1b; + + let (r0, carry2) = s0.overflowing_add(Self::C_LO); + let (r1, carry3) = s1.overflowing_add(carry2 as u64); + + pack( + if overflow | carry3 { r0 } else { s0 }, + if overflow | carry3 { r1 } else { s1 }, + ) + } + + /// Solinas fold for exactly 4 limbs: `[r0,r1] + C·[r2,r3]` → 3 limbs, + /// then `fold2_canonicalize`. + #[inline(always)] + fn reduce_4(r0: u64, r1: u64, r2: u64, r3: u64) -> [u64; 2] { + let (cr2_lo, cr2_hi) = Self::mul_c_wide(r2); + let (cr3_lo, cr3_hi) = Self::mul_c_wide(r3); + + let t0_sum = r0 as u128 + cr2_lo as u128; + let t0 = t0_sum as u64; + let carryf = (t0_sum >> 64) as u64; + + let t1_sum = r1 as u128 + cr2_hi as u128 + cr3_lo as u128 + carryf as u128; + let t1 = t1_sum as u64; + + let t2_sum = cr3_hi as u128 + (t1_sum >> 64); + let t2 = t2_sum as u64; + debug_assert_eq!(t2_sum >> 64, 0); + + Self::fold2_canonicalize(t0, t1, t2) + } + + #[inline(always)] + fn mul_raw(a: [u64; 2], b: [u64; 2]) -> [u64; 2] { + let [r0, r1, r2, r3] = Self(a).mul_wide(Self(b)); + Self::reduce_4(r0, r1, r2, r3) + } + + #[inline(always)] + fn sqr_raw(a: [u64; 2]) -> [u64; 2] { + Self::mul_raw(a, a) + } + + /// Squaring, equivalent to `self * self`. + #[inline(always)] + pub fn square(self) -> Self { + Self(Self::sqr_raw(self.0)) + } + + fn pow_u128(self, mut exp: u128) -> Self { + let mut base = self; + let mut acc = Self::one(); + while exp > 0 { + if (exp & 1) == 1 { + acc = acc * base; + } + base = Self(Self::sqr_raw(base.0)); + exp >>= 1; + } + acc + } + + /// Extract the canonical `[lo, hi]` limb representation. + #[inline(always)] + pub fn to_limbs(self) -> [u64; 2] { + self.0 + } + + /// 128×64 → 192-bit widening multiply, **no reduction**. + /// + /// Returns `[lo, mid, hi]` representing `self · other` as a 192-bit + /// integer. Cost: 2 widening `mul64`. + #[inline(always)] + pub fn mul_wide_u64(self, other: u64) -> [u64; 3] { + let (a0, a1) = (self.0[0], self.0[1]); + let (p0_lo, p0_hi) = mul64_wide(a0, other); + let (p1_lo, p1_hi) = mul64_wide(a1, other); + let mid = p0_hi as u128 + p1_lo as u128; + let hi = p1_hi + (mid >> 64) as u64; + [p0_lo, mid as u64, hi] + } + + /// 128×128 → 256-bit widening multiply, **no reduction**. + /// + /// Returns `[r0, r1, r2, r3]` representing `self · other` as a 256-bit + /// integer. This is the schoolbook 2×2 portion of the Solinas multiply, + /// without the reduction fold. Cost: 4 widening `mul64`. + #[inline(always)] + pub fn mul_wide(self, other: Self) -> [u64; 4] { + let (a0, a1) = (self.0[0], self.0[1]); + let (b0, b1) = (other.0[0], other.0[1]); + let (p00_lo, p00_hi) = mul64_wide(a0, b0); + let (p01_lo, p01_hi) = mul64_wide(a0, b1); + let (p10_lo, p10_hi) = mul64_wide(a1, b0); + let (p11_lo, p11_hi) = mul64_wide(a1, b1); + + let row1 = p00_hi as u128 + p01_lo as u128 + p10_lo as u128; + let r0 = p00_lo; + let r1 = row1 as u64; + let carry1 = (row1 >> 64) as u64; + + let row2 = p01_hi as u128 + p10_hi as u128 + p11_lo as u128 + carry1 as u128; + let r2 = row2 as u64; + let carry2 = (row2 >> 64) as u64; + + let row3 = p11_hi as u128 + carry2 as u128; + let r3 = row3 as u64; + debug_assert_eq!(row3 >> 64, 0); + + [r0, r1, r2, r3] + } + + /// 128×128 → 256-bit widening multiply with a raw `u128` operand, + /// **no reduction**. + #[inline(always)] + pub fn mul_wide_u128(self, other: u128) -> [u64; 4] { + self.mul_wide(Self(from_u128(other))) + } + + /// 128×(64*M) → (64*OUT) widening multiply, **no reduction**. + /// + /// Multiplies a canonical Fp128 value (`[u64; 2]`) by an arbitrary + /// little-endian limb array and returns the little-endian product + /// truncated/extended to `OUT` limbs. + #[inline(always)] + pub fn mul_wide_limbs(self, other: [u64; M]) -> [u64; OUT] { + let (a0, a1) = (self.0[0], self.0[1]); + + // Hot-path specializations used by Jolt (M in {3,4}, OUT in {4,5}). + // These avoid loop/control-flow overhead in tight sumcheck FMAs. + if M == 3 && OUT == 5 { + let b0 = other[0]; + let b1 = other[1]; + let b2 = other[2]; + + let (p00_lo, p00_hi) = mul64_wide(a0, b0); + let (p01_lo, p01_hi) = mul64_wide(a0, b1); + let (p02_lo, p02_hi) = mul64_wide(a0, b2); + let (p10_lo, p10_hi) = mul64_wide(a1, b0); + let (p11_lo, p11_hi) = mul64_wide(a1, b1); + let (p12_lo, p12_hi) = mul64_wide(a1, b2); + + let r0 = p00_lo; + + let row1 = p00_hi as u128 + p01_lo as u128 + p10_lo as u128; + let r1 = row1 as u64; + let carry1 = row1 >> 64; + + let row2 = p01_hi as u128 + p02_lo as u128 + p10_hi as u128 + p11_lo as u128 + carry1; + let r2 = row2 as u64; + let carry2 = row2 >> 64; + + let row3 = p02_hi as u128 + p11_hi as u128 + p12_lo as u128 + carry2; + let r3 = row3 as u64; + let carry3 = row3 >> 64; + + let row4 = p12_hi as u128 + carry3; + let r4 = row4 as u64; + debug_assert_eq!(row4 >> 64, 0); + + let mut out = [0u64; OUT]; + out[0] = r0; + out[1] = r1; + out[2] = r2; + out[3] = r3; + out[4] = r4; + return out; + } + if M == 3 && OUT == 4 { + let b0 = other[0]; + let b1 = other[1]; + let b2 = other[2]; + + let (p00_lo, p00_hi) = mul64_wide(a0, b0); + let (p01_lo, p01_hi) = mul64_wide(a0, b1); + let (p02_lo, p02_hi) = mul64_wide(a0, b2); + let (p10_lo, p10_hi) = mul64_wide(a1, b0); + let (p11_lo, p11_hi) = mul64_wide(a1, b1); + let p12_lo = a1.wrapping_mul(b2); + + let r0 = p00_lo; + + let row1 = p00_hi as u128 + p01_lo as u128 + p10_lo as u128; + let r1 = row1 as u64; + let carry1 = row1 >> 64; + + let row2 = p01_hi as u128 + p02_lo as u128 + p10_hi as u128 + p11_lo as u128 + carry1; + let r2 = row2 as u64; + let carry2 = row2 >> 64; + + let row3 = p02_hi as u128 + p11_hi as u128 + p12_lo as u128 + carry2; + let r3 = row3 as u64; + + let mut out = [0u64; OUT]; + out[0] = r0; + out[1] = r1; + out[2] = r2; + out[3] = r3; + return out; + } + if M == 4 && OUT == 6 { + let b0 = other[0]; + let b1 = other[1]; + let b2 = other[2]; + let b3 = other[3]; + + let (p00_lo, p00_hi) = mul64_wide(a0, b0); + let (p01_lo, p01_hi) = mul64_wide(a0, b1); + let (p02_lo, p02_hi) = mul64_wide(a0, b2); + let (p03_lo, p03_hi) = mul64_wide(a0, b3); + let (p10_lo, p10_hi) = mul64_wide(a1, b0); + let (p11_lo, p11_hi) = mul64_wide(a1, b1); + let (p12_lo, p12_hi) = mul64_wide(a1, b2); + let (p13_lo, p13_hi) = mul64_wide(a1, b3); + + let r0 = p00_lo; + + let row1 = p00_hi as u128 + p01_lo as u128 + p10_lo as u128; + let r1 = row1 as u64; + let carry1 = row1 >> 64; + + let row2 = p01_hi as u128 + p02_lo as u128 + p10_hi as u128 + p11_lo as u128 + carry1; + let r2 = row2 as u64; + let carry2 = row2 >> 64; + + let row3 = p02_hi as u128 + p03_lo as u128 + p11_hi as u128 + p12_lo as u128 + carry2; + let r3 = row3 as u64; + let carry3 = row3 >> 64; + + let row4 = p03_hi as u128 + p12_hi as u128 + p13_lo as u128 + carry3; + let r4 = row4 as u64; + let carry4 = row4 >> 64; + + let row5 = p13_hi as u128 + carry4; + let r5 = row5 as u64; + debug_assert_eq!(row5 >> 64, 0); + + let mut out = [0u64; OUT]; + out[0] = r0; + out[1] = r1; + out[2] = r2; + out[3] = r3; + out[4] = r4; + out[5] = r5; + return out; + } + if M == 4 && OUT == 5 { + let b0 = other[0]; + let b1 = other[1]; + let b2 = other[2]; + let b3 = other[3]; + + let (p00_lo, p00_hi) = mul64_wide(a0, b0); + let (p01_lo, p01_hi) = mul64_wide(a0, b1); + let (p02_lo, p02_hi) = mul64_wide(a0, b2); + let (p03_lo, p03_hi) = mul64_wide(a0, b3); + let (p10_lo, p10_hi) = mul64_wide(a1, b0); + let (p11_lo, p11_hi) = mul64_wide(a1, b1); + let (p12_lo, p12_hi) = mul64_wide(a1, b2); + let p13_lo = a1.wrapping_mul(b3); + + let r0 = p00_lo; + + let row1 = p00_hi as u128 + p01_lo as u128 + p10_lo as u128; + let r1 = row1 as u64; + let carry1 = row1 >> 64; + + let row2 = p01_hi as u128 + p02_lo as u128 + p10_hi as u128 + p11_lo as u128 + carry1; + let r2 = row2 as u64; + let carry2 = row2 >> 64; + + let row3 = p02_hi as u128 + p03_lo as u128 + p11_hi as u128 + p12_lo as u128 + carry2; + let r3 = row3 as u64; + let carry3 = row3 >> 64; + + let row4 = p03_hi as u128 + p12_hi as u128 + p13_lo as u128 + carry3; + let r4 = row4 as u64; + + let mut out = [0u64; OUT]; + out[0] = r0; + out[1] = r1; + out[2] = r2; + out[3] = r3; + out[4] = r4; + return out; + } + if M == 4 && OUT == 4 { + let b0 = other[0]; + let b1 = other[1]; + let b2 = other[2]; + let b3 = other[3]; + + let (p00_lo, p00_hi) = mul64_wide(a0, b0); + let (p01_lo, p01_hi) = mul64_wide(a0, b1); + let (p02_lo, p02_hi) = mul64_wide(a0, b2); + let p03_lo = a0.wrapping_mul(b3); + let (p10_lo, p10_hi) = mul64_wide(a1, b0); + let (p11_lo, p11_hi) = mul64_wide(a1, b1); + let p12_lo = a1.wrapping_mul(b2); + + let r0 = p00_lo; + + let row1 = p00_hi as u128 + p01_lo as u128 + p10_lo as u128; + let r1 = row1 as u64; + let carry1 = row1 >> 64; + + let row2 = p01_hi as u128 + p02_lo as u128 + p10_hi as u128 + p11_lo as u128 + carry1; + let r2 = row2 as u64; + let carry2 = row2 >> 64; + + let row3 = p02_hi as u128 + p03_lo as u128 + p11_hi as u128 + p12_lo as u128 + carry2; + let r3 = row3 as u64; + + let mut out = [0u64; OUT]; + out[0] = r0; + out[1] = r1; + out[2] = r2; + out[3] = r3; + return out; + } + + let mut out = [0u64; OUT]; + + for (i, &b) in other.iter().enumerate() { + if i >= OUT { + break; + } + + let (p0_lo, p0_hi) = mul64_wide(a0, b); + let (p1_lo, p1_hi) = mul64_wide(a1, b); + + let s0 = out[i] as u128 + p0_lo as u128; + out[i] = s0 as u64; + let mut carry = s0 >> 64; + + if i + 1 >= OUT { + continue; + } + let s1 = out[i + 1] as u128 + p0_hi as u128 + p1_lo as u128 + carry; + out[i + 1] = s1 as u64; + carry = s1 >> 64; + + if i + 2 >= OUT { + continue; + } + let s2 = out[i + 2] as u128 + p1_hi as u128 + carry; + out[i + 2] = s2 as u64; + + let mut carry_hi = s2 >> 64; + let mut j = i + 3; + while carry_hi != 0 && j < OUT { + let sj = out[j] as u128 + carry_hi; + out[j] = sj as u64; + carry_hi = sj >> 64; + j += 1; + } + } + + out + } + + /// Reduce an arbitrary-width little-endian limb array to a canonical + /// field element via iterated Solinas folding. + /// + /// Each fold splits at the 128-bit boundary and replaces + /// `hi · 2^128` with `hi · C`, reducing width by one limb per + /// iteration. Supports 0–10 input limbs (up to 640 bits). + /// + /// # Panics + /// + /// Panics if `limbs.len() > 10`. + #[inline(always)] + pub fn solinas_reduce(limbs: &[u64]) -> Self { + match limbs.len() { + 0 => Self::zero(), + 1 => Self(pack(limbs[0], 0)), + 2 => Self::from_canonical_u128_reduced(to_u128([limbs[0], limbs[1]])), + 3 => Self(Self::fold2_canonicalize(limbs[0], limbs[1], limbs[2])), + 4 => Self(Self::reduce_4(limbs[0], limbs[1], limbs[2], limbs[3])), + 5 => { + let (l0, l1, l2, l3, l4) = (limbs[0], limbs[1], limbs[2], limbs[3], limbs[4]); + let (c2_lo, c2_hi) = Self::mul_c_wide(l2); + let (c3_lo, c3_hi) = Self::mul_c_wide(l3); + let (c4_lo, c4_hi) = Self::mul_c_wide(l4); + + let s0 = l0 as u128 + c2_lo as u128; + let s1 = l1 as u128 + c2_hi as u128 + c3_lo as u128 + (s0 >> 64); + let s2 = c3_hi as u128 + c4_lo as u128 + (s1 >> 64); + let s3 = c4_hi as u128 + (s2 >> 64); + debug_assert_eq!(s3 >> 64, 0); + + Self(Self::reduce_4(s0 as u64, s1 as u64, s2 as u64, s3 as u64)) + } + n => { + assert!(n <= 10, "solinas_reduce supports at most 10 limbs"); + let mut buf = [0u64; 11]; + buf[..n].copy_from_slice(limbs); + let mut len = n; + let c = Self::C_LO; + + while len > 5 { + let high_len = len - 2; + let mut next = [0u64; 11]; + + let mut carry: u64 = 0; + for i in 0..high_len { + let wide = c as u128 * buf[i + 2] as u128 + carry as u128; + next[i] = wide as u64; + carry = (wide >> 64) as u64; + } + next[high_len] = carry; + + let s0 = next[0] as u128 + buf[0] as u128; + next[0] = s0 as u64; + let s1 = next[1] as u128 + buf[1] as u128 + (s0 >> 64); + next[1] = s1 as u64; + let mut c_out = (s1 >> 64) as u64; + for limb in &mut next[2..=high_len] { + if c_out == 0 { + break; + } + let s = *limb as u128 + c_out as u128; + *limb = s as u64; + c_out = (s >> 64) as u64; + } + debug_assert_eq!(c_out, 0); + + buf = next; + len -= 1; + while len > 5 && buf[len - 1] == 0 { + len -= 1; + } + } + + Self::solinas_reduce(&buf[..len]) + } + } + } +} + +impl Add for Fp128

{ + type Output = Self; + #[inline] + fn add(self, rhs: Self) -> Self::Output { + Self(Self::add_raw(self.0, rhs.0)) + } +} + +impl Sub for Fp128

{ + type Output = Self; + #[inline] + fn sub(self, rhs: Self) -> Self::Output { + Self(Self::sub_raw(self.0, rhs.0)) + } +} + +impl Mul for Fp128

{ + type Output = Self; + #[inline] + fn mul(self, rhs: Self) -> Self::Output { + Self(Self::mul_raw(self.0, rhs.0)) + } +} + +impl Neg for Fp128

{ + type Output = Self; + #[inline] + fn neg(self) -> Self::Output { + Self(Self::sub_raw(pack(0, 0), self.0)) + } +} + +impl<'a, const P: u128> Add<&'a Self> for Fp128

{ + type Output = Self; + #[inline] + fn add(self, rhs: &'a Self) -> Self::Output { + self + *rhs + } +} + +impl<'a, const P: u128> Sub<&'a Self> for Fp128

{ + type Output = Self; + #[inline] + fn sub(self, rhs: &'a Self) -> Self::Output { + self - *rhs + } +} + +impl<'a, const P: u128> Mul<&'a Self> for Fp128

{ + type Output = Self; + #[inline] + fn mul(self, rhs: &'a Self) -> Self::Output { + self * *rhs + } +} + +impl Valid for Fp128

{ + fn check(&self) -> Result<(), SerializationError> { + if to_u128(self.0) < P { + Ok(()) + } else { + Err(SerializationError::InvalidData("Fp128 out of range".into())) + } + } +} + +impl HachiSerialize for Fp128

{ + fn serialize_with_mode( + &self, + mut writer: W, + _compress: Compress, + ) -> Result<(), SerializationError> { + to_u128(self.0).serialize_with_mode(&mut writer, Compress::No)?; + Ok(()) + } + + fn serialized_size(&self, _compress: Compress) -> usize { + 16 + } +} + +impl HachiDeserialize for Fp128

{ + fn deserialize_with_mode( + mut reader: R, + _compress: Compress, + validate: Validate, + ) -> Result { + let x = u128::deserialize_with_mode(&mut reader, Compress::No, validate)?; + if matches!(validate, Validate::Yes) && x >= P { + return Err(SerializationError::InvalidData( + "Fp128 out of range".to_string(), + )); + } + + // Without validation, reduce without division. + // For `p = 2^128 − c` with `c < 2^64` we have `p > 2^127`, + // so any `u128` is in `[0, 2p)` and one conditional subtract suffices. + let out = if matches!(validate, Validate::Yes) { + x + } else { + let (sub, borrow) = x.overflowing_sub(P); + if borrow { + x + } else { + sub + } + }; + Ok(Self(from_u128(out))) + } +} + +impl FieldCore for Fp128

{ + fn zero() -> Self { + Self(pack(0, 0)) + } + + fn one() -> Self { + Self(pack(1, 0)) + } + + fn is_zero(&self) -> bool { + self.0 == [0, 0] + } + + fn add(&self, rhs: &Self) -> Self { + *self + *rhs + } + + fn sub(&self, rhs: &Self) -> Self { + *self - *rhs + } + + fn mul(&self, rhs: &Self) -> Self { + *self * *rhs + } + + fn inv(self) -> Option { + let inv = self.inv_or_zero(); + if self.is_zero() { + None + } else { + Some(inv) + } + } +} + +impl Invertible for Fp128

{ + fn inv_or_zero(self) -> Self { + let candidate = self.pow_u128(P.wrapping_sub(2)); + let v = to_u128(self.0); + let nz = ((v | v.wrapping_neg()) >> 127) & 1; + let mask = 0u128.wrapping_sub(nz); + let masked = to_u128(candidate.0) & mask; + Self(from_u128(masked)) + } +} + +impl FieldSampling for Fp128

{ + fn sample(rng: &mut R) -> Self { + loop { + let lo = rng.next_u64(); + let hi = rng.next_u64(); + let x = lo as u128 | (hi as u128) << 64; + if x < P { + return Self(pack(lo, hi)); + } + } + } +} + +impl CanonicalField for Fp128

{ + fn from_u64(val: u64) -> Self { + // For Fp128 pseudo-Mersenne primes, p = 2^128 - c with c < 2^64. + // Therefore any u64 is always canonical (< p), so this can be a + // direct limb construction with no reduction path. + Self(from_u128(val as u128)) + } + + fn from_i64(val: i64) -> Self { + if val >= 0 { + Self::from_u64(val as u64) + } else { + // unsigned_abs avoids overflow for i64::MIN. + -Self::from_u64(val.unsigned_abs()) + } + } + + fn to_canonical_u128(self) -> u128 { + to_u128(self.0) + } + + fn from_canonical_u128_checked(val: u128) -> Option { + if val < P { + Some(Self(from_u128(val))) + } else { + None + } + } + + fn from_canonical_u128_reduced(val: u128) -> Self { + let (sub, borrow) = val.overflowing_sub(P); + Self(from_u128(if borrow { val } else { sub })) + } +} + +impl PseudoMersenneField for Fp128

{ + const MODULUS_BITS: u32 = 128; + const MODULUS_OFFSET: u128 = Self::C; +} + +/// `p = 2^128 − 2^13 − 2^4 + 2^0` (C = 8207). +pub type Prime128M13M4P0 = Fp128<0xffffffffffffffffffffffffffffdff1>; +/// `p = 2^128 − 2^37 + 2^3 + 2^0` (C = 137438953463). +pub type Prime128M37P3P0 = Fp128<0xffffffffffffffffffffffe000000009>; +/// `p = 2^128 − 2^52 − 2^3 + 2^0` (C = 4503599627370487). +pub type Prime128M52M3P0 = Fp128<0xffffffffffffffffffeffffffffffff9>; +/// `p = 2^128 − 2^54 + 2^4 + 2^0` (C = 18014398509481967). +pub type Prime128M54P4P0 = Fp128<0xffffffffffffffffffc0000000000011>; +/// `p = 2^128 − 2^8 − 2^4 − 2^1 − 2^0` (C = 275). +pub type Prime128M8M4M1M0 = Fp128<0xfffffffffffffffffffffffffffffeed>; +/// `p = 2^128 − 2^18 − 2^0` (C = 2^18 + 1). +pub type Prime128M18M0 = Fp128<0xfffffffffffffffffffffffffffbffff>; +/// `p = 2^128 − 2^54 + 2^0` (C = 2^54 − 1). +pub type Prime128M54P0 = Fp128<0xffffffffffffffffffc0000000000001>; + +#[cfg(test)] +mod tests { + use super::*; + use crate::{FieldSampling, PseudoMersenneField}; + use rand::rngs::StdRng; + use rand::SeedableRng; + use rand_core::RngCore; + + type F = Prime128M8M4M1M0; + + #[test] + fn to_limbs_roundtrip() { + let mut rng = StdRng::seed_from_u64(0xdead_beef_cafe_1234); + for _ in 0..1000 { + let a: F = FieldSampling::sample(&mut rng); + assert_eq!(Fp128(a.to_limbs()), a); + } + } + + #[test] + fn mul_wide_u64_matches_full_mul() { + let mut rng = StdRng::seed_from_u64(0x1122_3344_5566_7788); + for _ in 0..1000 { + let a: F = FieldSampling::sample(&mut rng); + let b = rng.next_u64(); + let expected = a * F::from_u64(b); + let reduced = F::solinas_reduce(&a.mul_wide_u64(b)); + assert_eq!(reduced, expected); + } + } + + #[test] + fn mul_wide_matches_full_mul() { + let mut rng = StdRng::seed_from_u64(0xaabb_ccdd_eeff_0011); + for _ in 0..1000 { + let a: F = FieldSampling::sample(&mut rng); + let b: F = FieldSampling::sample(&mut rng); + let expected = a * b; + let reduced = F::solinas_reduce(&a.mul_wide(b)); + assert_eq!(reduced, expected); + } + } + + #[test] + fn mul_wide_u128_matches_full_mul() { + let mut rng = StdRng::seed_from_u64(0x9988_7766_5544_3322); + for _ in 0..1000 { + let a: F = FieldSampling::sample(&mut rng); + let b = rng.next_u64() as u128 | ((rng.next_u64() as u128) << 64); + let expected = a * F::from_canonical_u128_reduced(b); + let reduced = F::solinas_reduce(&a.mul_wide_u128(b)); + assert_eq!(reduced, expected); + } + } + + #[test] + fn mul_wide_limbs_roundtrips_through_reduction() { + let mut rng = StdRng::seed_from_u64(0x1bad_f00d_0ddc_afe1); + for _ in 0..1000 { + let a: F = FieldSampling::sample(&mut rng); + let b3 = [rng.next_u64(), rng.next_u64(), rng.next_u64()]; + let b4 = [ + rng.next_u64(), + rng.next_u64(), + rng.next_u64(), + rng.next_u64(), + ]; + + let got3_full = a.mul_wide_limbs::<3, 5>(b3); + let got3_trunc = a.mul_wide_limbs::<3, 4>(b3); + assert_eq!( + got3_trunc, + [got3_full[0], got3_full[1], got3_full[2], got3_full[3]] + ); + let exp3 = a * F::solinas_reduce(&b3); + assert_eq!(F::solinas_reduce(&got3_full), exp3); + + let got4_full = a.mul_wide_limbs::<4, 6>(b4); + let got4_trunc = a.mul_wide_limbs::<4, 4>(b4); + assert_eq!( + got4_trunc, + [got4_full[0], got4_full[1], got4_full[2], got4_full[3]] + ); + let exp4 = a * F::solinas_reduce(&b4); + assert_eq!(F::solinas_reduce(&got4_full), exp4); + } + } + + #[test] + fn solinas_reduce_small_inputs() { + assert_eq!(F::solinas_reduce(&[]), F::zero()); + assert_eq!(F::solinas_reduce(&[42]), F::from_u64(42)); + let one_shifted = F::from_canonical_u128_reduced(1u128 << 64); + assert_eq!(F::solinas_reduce(&[0, 1]), one_shifted); + } + + #[test] + fn solinas_reduce_4_limbs_max() { + // 2^256 - 1 ≡ C² - 1 (mod P), since 2^128 ≡ C + let c = F::from_canonical_u128_reduced(::MODULUS_OFFSET); + let expected = c * c - F::one(); + assert_eq!(F::solinas_reduce(&[u64::MAX; 4]), expected); + } + + #[test] + fn solinas_reduce_9_limbs() { + // 1 + 2^512 = 1 + (2^128)^4 ≡ 1 + C^4 + let c = F::from_canonical_u128_reduced(::MODULUS_OFFSET); + let expected = F::one() + c * c * c * c; + assert_eq!(F::solinas_reduce(&[1, 0, 0, 0, 0, 0, 0, 0, 1]), expected); + } + + #[test] + fn solinas_reduce_accumulated_products() { + let mut rng = StdRng::seed_from_u64(0xfeed_face_0bad_c0de); + let mut acc = [0u64; 5]; + let mut expected = F::zero(); + + for _ in 0..200 { + let a: F = FieldSampling::sample(&mut rng); + let b = rng.next_u64(); + let wide = a.mul_wide_u64(b); + + let mut carry: u64 = 0; + for j in 0..5 { + let addend = if j < 3 { wide[j] } else { 0 }; + let sum = acc[j] as u128 + addend as u128 + carry as u128; + acc[j] = sum as u64; + carry = (sum >> 64) as u64; + } + assert_eq!(carry, 0); + expected = expected + a * F::from_u64(b); + } + + assert_eq!(F::solinas_reduce(&acc), expected); + } + + #[test] + fn solinas_reduce_cross_prime() { + // Verify with Prime128M18M0 (C = 2^18 + 1, shift+add path) + type G = Prime128M18M0; + let c = G::from_canonical_u128_reduced(::MODULUS_OFFSET); + let expected = c * c - G::one(); + assert_eq!(G::solinas_reduce(&[u64::MAX; 4]), expected); + + // Verify with Prime128M54P0 (C = 2^54 - 1, shift-sub path) + type H = Prime128M54P0; + let c = H::from_canonical_u128_reduced(::MODULUS_OFFSET); + let expected = c * c - H::one(); + assert_eq!(H::solinas_reduce(&[u64::MAX; 4]), expected); + } + + #[test] + fn from_i64_handles_min_without_overflow() { + let x = F::from_i64(i64::MIN); + let y = F::from_u64(i64::MIN.unsigned_abs()); + assert_eq!(x + y, F::zero()); + } +} diff --git a/src/algebra/fields/fp32.rs b/src/algebra/fields/fp32.rs new file mode 100644 index 00000000..0a834698 --- /dev/null +++ b/src/algebra/fields/fp32.rs @@ -0,0 +1,455 @@ +//! Prime field for primes of the form `p = 2^k − c` with `c` small, backed +//! by `u32` storage. +//! +//! Uses Solinas-style two-fold reduction: the offset `c` and fold point `k` +//! are computed at compile time from the const-generic modulus `P`. + +use std::ops::{Add, Mul, Neg, Sub}; + +use rand_core::RngCore; + +use crate::primitives::serialization::{ + Compress, HachiDeserialize, HachiSerialize, SerializationError, Valid, Validate, +}; +use crate::{CanonicalField, FieldCore, FieldSampling, Invertible, PseudoMersenneField}; +use std::io::{Read, Write}; + +/// Prime field element for primes `p = 2^k − c` stored as `u32`. +/// +/// The fold point `k` and offset `c = 2^k − p` are computed at compile time +/// from the const-generic `P`. Instantiating with a modulus that does not +/// satisfy the Solinas conditions is a compile-time error. +#[derive(Debug, Clone, Copy, PartialEq, Eq, Default)] +pub struct Fp32(pub(crate) u32); + +impl Fp32

{ + /// Fold point: smallest `k` such that `P ≤ 2^k`. + const BITS: u32 = 32 - P.leading_zeros(); + + /// Offset `c = 2^k − P`. + const C: u32 = { + let c = if Self::BITS == 32 { + 0u32.wrapping_sub(P) + } else { + (1u32 << Self::BITS) - P + }; + assert!(P != 0, "modulus must be nonzero"); + assert!(P & 1 == 1, "modulus must be odd"); + assert!( + (c as u64) * (c as u64 + 1) < P as u64, + "C(C+1) < P required for fused canonicalize" + ); + c + }; + + /// Mask for extracting the low `BITS` bits from a u64. + const MASK: u64 = if Self::BITS == 32 { + u32::MAX as u64 + } else { + (1u64 << Self::BITS) - 1 + }; + + /// Create from a canonical representative in `[0, P)`. + #[inline] + pub fn from_canonical_u32(x: u32) -> Self { + debug_assert!(x < P); + Self(x) + } + + /// Return the canonical representative in `[0, P)`. + #[inline] + pub fn to_canonical_u32(self) -> u32 { + self.0 + } + + /// Solinas reduction: fold a u64 at bit `BITS` until the value fits, + /// then conditionally subtract `P`. + /// + /// For multiplication products (< 2^{2·BITS}) exactly 2 folds suffice; + /// for arbitrary u64 inputs (e.g. `from_u64`) the loop runs at most + /// `ceil(64 / BITS)` iterations. + #[inline(always)] + fn reduce_u64(x: u64) -> u32 { + let c = Self::C as u64; + let mut v = x; + while v >> Self::BITS != 0 { + v = (v & Self::MASK) + c * (v >> Self::BITS); + } + let reduced = v.wrapping_sub(P as u64); + let borrow = reduced >> 63; + reduced.wrapping_add(borrow.wrapping_neg() & (P as u64)) as u32 + } + + /// Reduce a `u128` to canonical form (for `from_canonical_u128_reduced`). + #[inline(always)] + fn reduce_u128(x: u128) -> u32 { + let c = Self::C as u128; + let bits = Self::BITS; + let mask = if bits == 32 { + u32::MAX as u128 + } else { + (1u128 << bits) - 1 + }; + let mut v = x; + while v >> bits != 0 { + v = (v & mask) + c * (v >> bits); + } + let f = v as u64; + let reduced = f.wrapping_sub(P as u64); + let borrow = reduced >> 63; + reduced.wrapping_add(borrow.wrapping_neg() & (P as u64)) as u32 + } + + /// Two-fold Solinas reduction for multiplication products. + /// + /// Input must be < 2^{2·BITS} (guaranteed for `a*b` where `a,b < P`). + /// Exactly 2 folds + conditional subtract, no loop. + #[inline(always)] + fn reduce_product(x: u64) -> u32 { + let c = Self::C as u64; + let f1 = (x & Self::MASK) + c * (x >> Self::BITS); + let f2 = (f1 & Self::MASK) + c * (f1 >> Self::BITS); + let reduced = f2.wrapping_sub(P as u64); + let borrow = reduced >> 63; + reduced.wrapping_add(borrow.wrapping_neg() & (P as u64)) as u32 + } + + #[inline(always)] + fn add_raw(a: u32, b: u32) -> u32 { + let s = (a as u64) + (b as u64); + let reduced = s.wrapping_sub(P as u64); + let borrow = reduced >> 63; + reduced.wrapping_add(borrow.wrapping_neg() & (P as u64)) as u32 + } + + #[inline(always)] + fn sub_raw(a: u32, b: u32) -> u32 { + let diff = (a as u64).wrapping_sub(b as u64); + let borrow = diff >> 63; + diff.wrapping_add(borrow.wrapping_neg() & (P as u64)) as u32 + } + + #[inline(always)] + fn mul_raw(a: u32, b: u32) -> u32 { + Self::reduce_product((a as u64) * (b as u64)) + } + + #[inline(always)] + fn sqr_raw(a: u32) -> u32 { + Self::mul_raw(a, a) + } + + /// Squaring, equivalent to `self * self`. + #[inline(always)] + pub fn square(self) -> Self { + Self(Self::sqr_raw(self.0)) + } + + fn pow(self, mut exp: u64) -> Self { + let mut base = self; + let mut acc = Self::one(); + while exp > 0 { + if (exp & 1) == 1 { + acc = acc * base; + } + base = base.square(); + exp >>= 1; + } + acc + } + + /// Extract the canonical value. + #[inline(always)] + pub fn to_limbs(self) -> u32 { + self.0 + } + + /// 32×32 → 64-bit widening multiply, **no reduction**. + #[inline(always)] + pub fn mul_wide(self, other: Self) -> u64 { + (self.0 as u64) * (other.0 as u64) + } + + /// 32×32 → 64-bit widening multiply with a raw `u32` operand, + /// **no reduction**. + #[inline(always)] + pub fn mul_wide_u32(self, other: u32) -> u64 { + (self.0 as u64) * (other as u64) + } + + /// Reduce a u64 value via Solinas folding to a canonical field element. + #[inline(always)] + pub fn solinas_reduce(x: u64) -> Self { + Self(Self::reduce_u64(x)) + } +} + +impl Add for Fp32

{ + type Output = Self; + #[inline] + fn add(self, rhs: Self) -> Self::Output { + Self(Self::add_raw(self.0, rhs.0)) + } +} + +impl Sub for Fp32

{ + type Output = Self; + #[inline] + fn sub(self, rhs: Self) -> Self::Output { + Self(Self::sub_raw(self.0, rhs.0)) + } +} + +impl Mul for Fp32

{ + type Output = Self; + #[inline] + fn mul(self, rhs: Self) -> Self::Output { + Self(Self::mul_raw(self.0, rhs.0)) + } +} + +impl Neg for Fp32

{ + type Output = Self; + #[inline] + fn neg(self) -> Self::Output { + Self(Self::sub_raw(0, self.0)) + } +} + +impl<'a, const P: u32> Add<&'a Self> for Fp32

{ + type Output = Self; + #[inline] + fn add(self, rhs: &'a Self) -> Self::Output { + self + *rhs + } +} + +impl<'a, const P: u32> Sub<&'a Self> for Fp32

{ + type Output = Self; + #[inline] + fn sub(self, rhs: &'a Self) -> Self::Output { + self - *rhs + } +} + +impl<'a, const P: u32> Mul<&'a Self> for Fp32

{ + type Output = Self; + #[inline] + fn mul(self, rhs: &'a Self) -> Self::Output { + self * *rhs + } +} + +impl Valid for Fp32

{ + fn check(&self) -> Result<(), SerializationError> { + if self.0 < P { + Ok(()) + } else { + Err(SerializationError::InvalidData("Fp32 out of range".into())) + } + } +} + +impl HachiSerialize for Fp32

{ + fn serialize_with_mode( + &self, + mut writer: W, + _compress: Compress, + ) -> Result<(), SerializationError> { + self.0.serialize_with_mode(&mut writer, Compress::No)?; + Ok(()) + } + + fn serialized_size(&self, _compress: Compress) -> usize { + 4 + } +} + +impl HachiDeserialize for Fp32

{ + fn deserialize_with_mode( + mut reader: R, + _compress: Compress, + validate: Validate, + ) -> Result { + let x = u32::deserialize_with_mode(&mut reader, Compress::No, validate)?; + if matches!(validate, Validate::Yes) && x >= P { + return Err(SerializationError::InvalidData( + "Fp32 out of range".to_string(), + )); + } + let out = if matches!(validate, Validate::Yes) { + Self(x) + } else { + Self(Self::reduce_u64(x as u64)) + }; + Ok(out) + } +} + +impl FieldCore for Fp32

{ + fn zero() -> Self { + Self(0) + } + + fn one() -> Self { + Self(if P > 1 { 1 } else { 0 }) + } + + fn is_zero(&self) -> bool { + self.0 == 0 + } + + fn add(&self, rhs: &Self) -> Self { + *self + *rhs + } + + fn sub(&self, rhs: &Self) -> Self { + *self - *rhs + } + + fn mul(&self, rhs: &Self) -> Self { + *self * *rhs + } + + fn inv(self) -> Option { + let inv = self.inv_or_zero(); + if self.is_zero() { + None + } else { + Some(inv) + } + } +} + +impl Invertible for Fp32

{ + fn inv_or_zero(self) -> Self { + let candidate = self.pow((P as u64).wrapping_sub(2)); + let nz = ((self.0 | self.0.wrapping_neg()) >> 31) & 1; + let mask = 0u32.wrapping_sub(nz); + Self(candidate.0 & mask) + } +} + +impl FieldSampling for Fp32

{ + fn sample(rng: &mut R) -> Self { + Self(Self::reduce_u64(rng.next_u64())) + } +} + +impl CanonicalField for Fp32

{ + fn from_u64(val: u64) -> Self { + Self(Self::reduce_u64(val)) + } + + fn from_i64(val: i64) -> Self { + if val >= 0 { + Self::from_u64(val as u64) + } else { + -Self::from_u64((-val) as u64) + } + } + + fn to_canonical_u128(self) -> u128 { + self.0 as u128 + } + + fn from_canonical_u128_checked(val: u128) -> Option { + if val < P as u128 { + Some(Self(val as u32)) + } else { + None + } + } + + fn from_canonical_u128_reduced(val: u128) -> Self { + Self(Self::reduce_u128(val)) + } +} + +impl PseudoMersenneField for Fp32

{ + const MODULUS_BITS: u32 = Self::BITS; + const MODULUS_OFFSET: u128 = Self::C as u128; +} + +#[cfg(test)] +mod tests { + use super::*; + use rand::rngs::StdRng; + use rand::SeedableRng; + + type F = Fp32<251>; // 2^8 - 5 + + #[test] + fn solinas_constants() { + assert_eq!(F::BITS, 8); + assert_eq!(F::C, 5); + assert_eq!(F::MASK, 255); + + type G = Fp32<{ (1u32 << 24) - 3 }>; // 2^24 - 3 + assert_eq!(G::BITS, 24); + assert_eq!(G::C, 3); + } + + #[test] + fn basic_arithmetic() { + let a = F::from_u64(100); + let b = F::from_u64(200); + assert_eq!((a + b).to_canonical_u32(), (100 + 200) % 251); + assert_eq!((a * b).to_canonical_u32(), (100 * 200) % 251); + assert_eq!((b - a).to_canonical_u32(), 100); + assert_eq!((-a).to_canonical_u32(), 251 - 100); + } + + #[test] + fn mul_wide_matches_full_mul() { + let mut rng = StdRng::seed_from_u64(0x1234_5678); + for _ in 0..1000 { + let a: F = FieldSampling::sample(&mut rng); + let b: F = FieldSampling::sample(&mut rng); + let expected = a * b; + let reduced = F::solinas_reduce(a.mul_wide(b)); + assert_eq!(reduced, expected); + } + } + + #[test] + fn mul_wide_u32_matches() { + let mut rng = StdRng::seed_from_u64(0xabcd_ef01); + for _ in 0..1000 { + let a: F = FieldSampling::sample(&mut rng); + let b = rng.next_u32() % 251; + let expected = a * F::from_canonical_u32(b); + let reduced = F::solinas_reduce(a.mul_wide_u32(b)); + assert_eq!(reduced, expected); + } + } + + #[test] + fn reduce_large_values() { + assert_eq!( + F::from_u64(u64::MAX).to_canonical_u32(), + (u64::MAX % 251) as u32 + ); + assert_eq!(F::from_u64(0).to_canonical_u32(), 0); + assert_eq!(F::from_u64(251).to_canonical_u32(), 0); + assert_eq!(F::from_u64(252).to_canonical_u32(), 1); + } + + #[test] + fn pseudo_mersenne_trait() { + assert_eq!(::MODULUS_BITS, 8); + assert_eq!(::MODULUS_OFFSET, 5); + } + + #[test] + fn cross_prime_32bit() { + type G = Fp32<{ u32::MAX - 98 }>; // 2^32 - 99 + assert_eq!(G::BITS, 32); + assert_eq!(G::C, 99); + + let a = G::from_u64(1_000_000); + let b = G::from_u64(2_000_000); + let product = (1_000_000u64 * 2_000_000u64) % ((1u64 << 32) - 99); + assert_eq!((a * b).to_canonical_u32(), product as u32); + } +} diff --git a/src/algebra/fields/fp64.rs b/src/algebra/fields/fp64.rs new file mode 100644 index 00000000..a56e3c1c --- /dev/null +++ b/src/algebra/fields/fp64.rs @@ -0,0 +1,604 @@ +//! Prime field for primes of the form `p = 2^k − c` with `c` small, backed +//! by `u64` storage. +//! +//! Uses Solinas-style two-fold reduction. For `c = 2^a ± 1` the fold +//! multiply is replaced by shift+add/sub, saving a u128 widening multiply. + +use std::ops::{Add, Mul, Neg, Sub}; + +use rand_core::RngCore; + +use crate::primitives::serialization::{ + Compress, HachiDeserialize, HachiSerialize, SerializationError, Valid, Validate, +}; +use crate::{CanonicalField, FieldCore, FieldSampling, Invertible, PseudoMersenneField}; +use std::io::{Read, Write}; + +#[inline(always)] +const fn is_pow2_u64(x: u64) -> bool { + x != 0 && (x & (x - 1)) == 0 +} + +#[inline(always)] +const fn log2_pow2_u64(mut x: u64) -> u32 { + let mut k = 0u32; + while x > 1 { + x >>= 1; + k += 1; + } + k +} + +#[inline(always)] +fn mul64_wide(a: u64, b: u64) -> (u64, u64) { + #[cfg(all(target_arch = "x86_64", target_feature = "bmi2"))] + { + unsafe { mul64_wide_bmi2(a, b) } + } + #[cfg(not(all(target_arch = "x86_64", target_feature = "bmi2")))] + { + let prod = (a as u128) * (b as u128); + (prod as u64, (prod >> 64) as u64) + } +} + +#[cfg(all(target_arch = "x86_64", target_feature = "bmi2"))] +#[inline(always)] +unsafe fn mul64_wide_bmi2(a: u64, b: u64) -> (u64, u64) { + let mut hi = 0; + // Safety: caller guarantees CPU support via cfg-gated compilation. + let lo = unsafe { std::arch::x86_64::_mulx_u64(a, b, &mut hi) }; + (lo, hi) +} + +/// Prime field element for primes `p = 2^k − c` stored as `u64`. +/// +/// The fold point `k` and offset `c = 2^k − p` are computed at compile time +/// from the const-generic `P`. For `c = 2^a ± 1`, the fold multiply is +/// replaced by shift+add/sub. +#[derive(Debug, Clone, Copy, PartialEq, Eq, Default)] +pub struct Fp64(pub(crate) u64); + +impl Fp64

{ + /// Fold point: smallest `k` such that `P ≤ 2^k`. + const BITS: u32 = 64 - P.leading_zeros(); + + /// Offset `c = 2^k − P`. + const C: u64 = { + let c = if Self::BITS == 64 { + 0u64.wrapping_sub(P) + } else { + (1u64 << Self::BITS) - P + }; + assert!(P != 0, "modulus must be nonzero"); + assert!(P & 1 == 1, "modulus must be odd"); + assert!( + (c as u128) * (c as u128 + 1) < P as u128, + "C(C+1) < P required for fused canonicalize" + ); + c + }; + + /// +1 means `C = 2^a + 1`, -1 means `C = 2^a - 1`, 0 means generic. + const C_SHIFT_KIND: i8 = { + let c = Self::C; + if c > 1 && is_pow2_u64(c - 1) { + 1 + } else if c == u64::MAX || is_pow2_u64(c + 1) { + -1 + } else { + 0 + } + }; + + const C_SHIFT: u32 = { + let c = Self::C; + if Self::C_SHIFT_KIND == 1 { + log2_pow2_u64(c - 1) + } else if Self::C_SHIFT_KIND == -1 { + if c == u64::MAX { + 64 + } else { + log2_pow2_u64(c + 1) + } + } else { + 0 + } + }; + + /// Mask for extracting the low `BITS` bits from a u128. + const MASK: u128 = if Self::BITS == 64 { + u64::MAX as u128 + } else { + (1u128 << Self::BITS) - 1 + }; + + /// u64-width mask (only valid when BITS < 64). + const MASK64: u64 = if Self::BITS < 64 { + (1u64 << Self::BITS) - 1 + } else { + u64::MAX + }; + + /// Whether Solinas folding of a multiplication product can stay + /// entirely in u64. True when BITS < 64 and C·2^BITS < 2^64. + const FOLD_IN_U64: bool = Self::BITS < 64 && (Self::C as u128) < (1u128 << (64 - Self::BITS)); + + /// u64 multiply by C, split into u32-wide halves so LLVM emits + /// `umull` (32×32→64) instead of promoting to u128. + /// Only valid when C fits in u32 (always true: C < sqrt(P) < 2^32). + #[inline(always)] + fn mul_c_narrow(x: u64) -> u64 { + #[cfg(target_arch = "x86_64")] + { + // x86_64 has fast scalar 64-bit multiply; use one multiply instead + // of two widened 32-bit multiplies in the fold hot path. + Self::C.wrapping_mul(x) + } + #[cfg(not(target_arch = "x86_64"))] + { + let c = Self::C as u32; + let x_lo = x as u32; + let x_hi = (x >> 32) as u32; + (c as u64 * x_lo as u64).wrapping_add((c as u64 * x_hi as u64) << 32) + } + } + + /// Multiply `x` by `C`. For `C = 2^a ± 1` uses shift+add/sub. + #[inline(always)] + fn mul_c(x: u64) -> u128 { + if Self::C_SHIFT_KIND == 1 { + ((x as u128) << Self::C_SHIFT) + x as u128 + } else if Self::C_SHIFT_KIND == -1 { + ((x as u128) << Self::C_SHIFT) - x as u128 + } else { + (Self::C as u128) * (x as u128) + } + } + + /// Create from a canonical representative in `[0, P)`. + #[inline] + pub fn from_canonical_u64(x: u64) -> Self { + debug_assert!(x < P); + Self(x) + } + + /// Return the canonical representative in `[0, P)`. + #[inline] + pub fn to_canonical_u64(self) -> u64 { + self.0 + } + + /// Solinas reduction: fold a u128 at bit `BITS` until the value fits, + /// then conditionally subtract `P`. + /// + /// For multiplication products (< 2^{2·BITS}) exactly 2 folds suffice; + /// for arbitrary u128 inputs the loop runs at most `ceil(128 / BITS)` + /// iterations. + #[inline(always)] + fn reduce_u128(x: u128) -> u64 { + let mut v = x; + while v >> Self::BITS != 0 { + v = (v & Self::MASK) + Self::mul_c((v >> Self::BITS) as u64); + } + let reduced = v.wrapping_sub(P as u128); + let borrow = reduced >> 127; + reduced.wrapping_add(borrow.wrapping_neg() & (P as u128)) as u64 + } + + /// Two-fold Solinas reduction for multiplication products. + /// + /// Input must be < 2^{2·BITS} (guaranteed for `a*b` where `a,b < P`). + /// Exactly 2 folds + conditional subtract, no loop. + /// + /// When `FOLD_IN_U64` is true the entire reduction stays in u64, + /// avoiding expensive u128 mask/shift on sub-word primes. + #[inline(always)] + fn reduce_product(x: u128) -> u64 { + if Self::FOLD_IN_U64 { + let lo = x as u64; + let hi = (x >> 64) as u64; + let high = (lo >> Self::BITS) | (hi << (64 - Self::BITS)); + let f1 = (lo & Self::MASK64) + Self::mul_c_narrow(high); + let f2 = (f1 & Self::MASK64) + Self::mul_c_narrow(f1 >> Self::BITS); + let reduced = f2.wrapping_sub(P); + let borrow = reduced >> 63; + reduced.wrapping_add(borrow.wrapping_neg() & P) + } else { + let f1 = (x & Self::MASK) + Self::mul_c((x >> Self::BITS) as u64); + let f2 = (f1 & Self::MASK) + Self::mul_c((f1 >> Self::BITS) as u64); + let reduced = f2.wrapping_sub(P as u128); + let borrow = reduced >> 127; + reduced.wrapping_add(borrow.wrapping_neg() & (P as u128)) as u64 + } + } + + /// BMI2 fast path: avoid re-materializing `u128` product in the common + /// sub-word configuration where reduction stays in `u64`. + #[cfg(all(target_arch = "x86_64", target_feature = "bmi2"))] + #[inline(always)] + fn reduce_product_wide(lo: u64, hi: u64) -> u64 { + if Self::FOLD_IN_U64 { + let high = (lo >> Self::BITS) | (hi << (64 - Self::BITS)); + let f1 = (lo & Self::MASK64) + Self::mul_c_narrow(high); + let f2 = (f1 & Self::MASK64) + Self::mul_c_narrow(f1 >> Self::BITS); + let reduced = f2.wrapping_sub(P); + let borrow = reduced >> 63; + reduced.wrapping_add(borrow.wrapping_neg() & P) + } else { + Self::reduce_product(lo as u128 | ((hi as u128) << 64)) + } + } + + #[inline(always)] + fn add_raw(a: u64, b: u64) -> u64 { + if Self::BITS <= 62 { + let s = a + b; + let reduced = s.wrapping_sub(P); + let borrow = reduced >> 63; + reduced.wrapping_add(borrow.wrapping_neg() & P) + } else { + let s = (a as u128) + (b as u128); + let reduced = s.wrapping_sub(P as u128); + let borrow = reduced >> 127; + reduced.wrapping_add(borrow.wrapping_neg() & (P as u128)) as u64 + } + } + + #[inline(always)] + fn sub_raw(a: u64, b: u64) -> u64 { + if Self::BITS <= 62 { + let diff = a.wrapping_sub(b); + let borrow = diff >> 63; + diff.wrapping_add(borrow.wrapping_neg() & P) + } else { + let diff = (a as u128).wrapping_sub(b as u128); + let borrow = diff >> 127; + diff.wrapping_add(borrow.wrapping_neg() & (P as u128)) as u64 + } + } + + #[inline(always)] + fn mul_raw(a: u64, b: u64) -> u64 { + #[cfg(all(target_arch = "x86_64", target_feature = "bmi2"))] + { + let (lo, hi) = mul64_wide(a, b); + Self::reduce_product_wide(lo, hi) + } + #[cfg(not(all(target_arch = "x86_64", target_feature = "bmi2")))] + { + Self::reduce_product((a as u128) * (b as u128)) + } + } + + #[inline(always)] + fn sqr_raw(a: u64) -> u64 { + Self::mul_raw(a, a) + } + + /// Squaring, equivalent to `self * self`. + #[inline(always)] + pub fn square(self) -> Self { + Self(Self::sqr_raw(self.0)) + } + + fn pow(self, mut exp: u64) -> Self { + let mut base = self; + let mut acc = Self::one(); + while exp > 0 { + if (exp & 1) == 1 { + acc = acc * base; + } + base = base.square(); + exp >>= 1; + } + acc + } + + /// Extract the canonical value. + #[inline(always)] + pub fn to_limbs(self) -> u64 { + self.0 + } + + /// 64×64 → 128-bit widening multiply, **no reduction**. + #[inline(always)] + pub fn mul_wide(self, other: Self) -> u128 { + let (lo, hi) = mul64_wide(self.0, other.0); + lo as u128 | ((hi as u128) << 64) + } + + /// 64×64 → 128-bit widening multiply with a raw `u64` operand, + /// **no reduction**. + #[inline(always)] + pub fn mul_wide_u64(self, other: u64) -> u128 { + let (lo, hi) = mul64_wide(self.0, other); + lo as u128 | ((hi as u128) << 64) + } + + /// Reduce a u128 value via Solinas folding to a canonical field element. + #[inline(always)] + pub fn solinas_reduce(x: u128) -> Self { + Self(Self::reduce_u128(x)) + } +} + +impl Add for Fp64

{ + type Output = Self; + #[inline] + fn add(self, rhs: Self) -> Self::Output { + Self(Self::add_raw(self.0, rhs.0)) + } +} + +impl Sub for Fp64

{ + type Output = Self; + #[inline] + fn sub(self, rhs: Self) -> Self::Output { + Self(Self::sub_raw(self.0, rhs.0)) + } +} + +impl Mul for Fp64

{ + type Output = Self; + #[inline] + fn mul(self, rhs: Self) -> Self::Output { + Self(Self::mul_raw(self.0, rhs.0)) + } +} + +impl Neg for Fp64

{ + type Output = Self; + #[inline] + fn neg(self) -> Self::Output { + Self(Self::sub_raw(0, self.0)) + } +} + +impl<'a, const P: u64> Add<&'a Self> for Fp64

{ + type Output = Self; + #[inline] + fn add(self, rhs: &'a Self) -> Self::Output { + self + *rhs + } +} + +impl<'a, const P: u64> Sub<&'a Self> for Fp64

{ + type Output = Self; + #[inline] + fn sub(self, rhs: &'a Self) -> Self::Output { + self - *rhs + } +} + +impl<'a, const P: u64> Mul<&'a Self> for Fp64

{ + type Output = Self; + #[inline] + fn mul(self, rhs: &'a Self) -> Self::Output { + self * *rhs + } +} + +impl Valid for Fp64

{ + fn check(&self) -> Result<(), SerializationError> { + if self.0 < P { + Ok(()) + } else { + Err(SerializationError::InvalidData("Fp64 out of range".into())) + } + } +} + +impl HachiSerialize for Fp64

{ + fn serialize_with_mode( + &self, + mut writer: W, + _compress: Compress, + ) -> Result<(), SerializationError> { + self.0.serialize_with_mode(&mut writer, Compress::No)?; + Ok(()) + } + + fn serialized_size(&self, _compress: Compress) -> usize { + 8 + } +} + +impl HachiDeserialize for Fp64

{ + fn deserialize_with_mode( + mut reader: R, + _compress: Compress, + validate: Validate, + ) -> Result { + let x = u64::deserialize_with_mode(&mut reader, Compress::No, validate)?; + if matches!(validate, Validate::Yes) && x >= P { + return Err(SerializationError::InvalidData( + "Fp64 out of range".to_string(), + )); + } + let out = if matches!(validate, Validate::Yes) { + Self(x) + } else { + Self(Self::reduce_u128(x as u128)) + }; + Ok(out) + } +} + +impl FieldCore for Fp64

{ + fn zero() -> Self { + Self(0) + } + + fn one() -> Self { + Self(if P > 1 { 1 } else { 0 }) + } + + fn is_zero(&self) -> bool { + self.0 == 0 + } + + fn add(&self, rhs: &Self) -> Self { + *self + *rhs + } + + fn sub(&self, rhs: &Self) -> Self { + *self - *rhs + } + + fn mul(&self, rhs: &Self) -> Self { + *self * *rhs + } + + fn inv(self) -> Option { + let inv = self.inv_or_zero(); + if self.is_zero() { + None + } else { + Some(inv) + } + } +} + +impl Invertible for Fp64

{ + fn inv_or_zero(self) -> Self { + let candidate = self.pow(P.wrapping_sub(2)); + let nz = ((self.0 | self.0.wrapping_neg()) >> 63) & 1; + let mask = 0u64.wrapping_sub(nz); + Self(candidate.0 & mask) + } +} + +impl FieldSampling for Fp64

{ + fn sample(rng: &mut R) -> Self { + let lo = rng.next_u64() as u128; + let hi = rng.next_u64() as u128; + Self(Self::reduce_u128(lo | (hi << 64))) + } +} + +impl CanonicalField for Fp64

{ + fn from_u64(val: u64) -> Self { + Self(Self::reduce_u128(val as u128)) + } + + fn from_i64(val: i64) -> Self { + if val >= 0 { + Self::from_u64(val as u64) + } else { + -Self::from_u64((-val) as u64) + } + } + + fn to_canonical_u128(self) -> u128 { + self.0 as u128 + } + + fn from_canonical_u128_checked(val: u128) -> Option { + if val < P as u128 { + Some(Self(val as u64)) + } else { + None + } + } + + fn from_canonical_u128_reduced(val: u128) -> Self { + Self(Self::reduce_u128(val)) + } +} + +impl PseudoMersenneField for Fp64

{ + const MODULUS_BITS: u32 = Self::BITS; + const MODULUS_OFFSET: u128 = Self::C as u128; +} + +#[cfg(test)] +mod tests { + use super::*; + use rand::rngs::StdRng; + use rand::SeedableRng; + + type F40 = Fp64<{ (1u64 << 40) - 195 }>; // 2^40 - 195 + type F64 = Fp64<{ u64::MAX - 58 }>; // 2^64 - 59 + + #[test] + fn solinas_constants() { + assert_eq!(F40::BITS, 40); + assert_eq!(F40::C, 195); + + assert_eq!(F64::BITS, 64); + assert_eq!(F64::C, 59); + } + + #[test] + fn basic_arithmetic_sub_word() { + let a = F40::from_u64(1_000_000); + let b = F40::from_u64(2_000_000); + let p = (1u64 << 40) - 195; + assert_eq!((a + b).to_canonical_u64(), 3_000_000); + assert_eq!( + (a * b).to_canonical_u64(), + (1_000_000u128 * 2_000_000u128 % p as u128) as u64 + ); + } + + #[test] + fn basic_arithmetic_full_word() { + let a = F64::from_u64(1_000_000_000); + let b = F64::from_u64(2_000_000_000); + let p = u64::MAX - 58; + assert_eq!( + (a * b).to_canonical_u64(), + (1_000_000_000u128 * 2_000_000_000u128 % p as u128) as u64 + ); + } + + #[test] + fn mul_wide_matches_full_mul() { + let mut rng = StdRng::seed_from_u64(0xdead_beef); + for _ in 0..1000 { + let a: F40 = FieldSampling::sample(&mut rng); + let b: F40 = FieldSampling::sample(&mut rng); + let expected = a * b; + let reduced = F40::solinas_reduce(a.mul_wide(b)); + assert_eq!(reduced, expected); + } + } + + #[test] + fn mul_wide_u64_matches() { + let mut rng = StdRng::seed_from_u64(0xcafe_d00d); + for _ in 0..1000 { + let a: F40 = FieldSampling::sample(&mut rng); + let b = rng.next_u64() % ((1u64 << 40) - 195); + let expected = a * F40::from_canonical_u64(b); + let reduced = F40::solinas_reduce(a.mul_wide_u64(b)); + assert_eq!(reduced, expected); + } + } + + #[test] + fn pseudo_mersenne_trait() { + assert_eq!(::MODULUS_BITS, 40); + assert_eq!(::MODULUS_OFFSET, 195); + assert_eq!(::MODULUS_BITS, 64); + assert_eq!(::MODULUS_OFFSET, 59); + } + + #[test] + fn shift_optimization_detected() { + type G = Fp64<{ (1u64 << 56) - 27 }>; // C = 27, not 2^a±1 + assert_eq!(G::C_SHIFT_KIND, 0); + + type H = Fp64<{ u64::MAX - 58 }>; // C = 59, not 2^a±1 + assert_eq!(H::C_SHIFT_KIND, 0); + } + + #[test] + fn reduce_u128_large() { + assert_eq!(F64::from_canonical_u128_reduced(u128::MAX), { + let p = u64::MAX as u128 - 58; + F64::from_canonical_u64((u128::MAX % p) as u64) + }); + } +} diff --git a/src/algebra/fields/lift.rs b/src/algebra/fields/lift.rs new file mode 100644 index 00000000..5e243c9c --- /dev/null +++ b/src/algebra/fields/lift.rs @@ -0,0 +1,43 @@ +//! Helpers for embedding base fields into extension fields. + +use crate::algebra::fields::ext::{Fp2, Fp2Config, Fp4, Fp4Config}; +use crate::primitives::serialization::Valid; +use crate::FieldCore; + +/// Lift a base-field element into an extension field. +/// +/// This is intentionally small: for extension towers we embed into the constant term. +pub trait LiftBase: FieldCore { + /// Embed `x ∈ F` as a constant in `Self`. + fn lift_base(x: F) -> Self; +} + +impl LiftBase for F { + #[inline] + fn lift_base(x: F) -> Self { + x + } +} + +impl LiftBase for Fp2 +where + F: FieldCore + Valid, + C: Fp2Config, +{ + #[inline] + fn lift_base(x: F) -> Self { + Self::new(x, F::zero()) + } +} + +impl LiftBase for Fp4 +where + F: FieldCore + Valid, + C2: Fp2Config, + C4: Fp4Config, +{ + #[inline] + fn lift_base(x: F) -> Self { + Self::new(Fp2::new(x, F::zero()), Fp2::new(F::zero(), F::zero())) + } +} diff --git a/src/algebra/fields/mod.rs b/src/algebra/fields/mod.rs new file mode 100644 index 00000000..d255852a --- /dev/null +++ b/src/algebra/fields/mod.rs @@ -0,0 +1,40 @@ +//! Prime fields and extension field towers. + +pub mod ext; +pub mod fp128; +pub mod fp32; +pub mod fp64; +pub mod lift; +pub mod packed; +#[cfg(all( + target_arch = "x86_64", + target_feature = "avx2", + not(all(target_feature = "avx512f", target_feature = "avx512dq")) +))] +pub mod packed_avx2; +#[cfg(all( + target_arch = "x86_64", + target_feature = "avx512f", + target_feature = "avx512dq" +))] +pub mod packed_avx512; +#[cfg(all(target_arch = "aarch64", target_feature = "neon"))] +pub mod packed_neon; +pub mod pseudo_mersenne; + +pub use ext::{Fp2, Fp2Config, Fp4, Fp4Config}; +pub use fp128::{ + Fp128, Prime128M13M4P0, Prime128M37P3P0, Prime128M52M3P0, Prime128M54P4P0, Prime128M8M4M1M0, +}; +pub use fp32::Fp32; +pub use fp64::Fp64; +pub use lift::LiftBase; +pub use packed::{ + Fp128Packing, Fp32Packing, Fp64Packing, HasPacking, NoPacking, PackedField, PackedValue, +}; +pub use pseudo_mersenne::{ + is_pow2_offset, pow2_offset, pseudo_mersenne_modulus, Pow2Offset128Field, Pow2Offset24Field, + Pow2Offset30Field, Pow2Offset31Field, Pow2Offset32Field, Pow2Offset40Field, Pow2Offset48Field, + Pow2Offset56Field, Pow2Offset64Field, Pow2OffsetPrimeSpec, POW2_OFFSET_IMPLEMENTED_MAX_BITS, + POW2_OFFSET_MAX, POW2_OFFSET_PRIMES, POW2_OFFSET_TABLE, +}; diff --git a/src/algebra/fields/packed.rs b/src/algebra/fields/packed.rs new file mode 100644 index 00000000..e66063a9 --- /dev/null +++ b/src/algebra/fields/packed.rs @@ -0,0 +1,434 @@ +//! Packed field abstractions and architecture-specific SIMD backends. + +use crate::algebra::fields::{Fp128, Fp32, Fp64}; +use crate::FieldCore; +use core::ops::{Add, Mul, Sub}; + +/// Array-like packed values over a scalar type. +pub trait PackedValue: 'static + Copy + Send + Sync { + /// Scalar value type carried by each lane. + type Value: 'static + Copy + Send + Sync; + + /// Number of scalar lanes. + const WIDTH: usize; + + /// Build from a lane generator. + fn from_fn(f: F) -> Self + where + F: FnMut(usize) -> Self::Value; + + /// Extract one lane. + fn extract(&self, lane: usize) -> Self::Value; + + /// Pack a scalar slice into packed values. + /// + /// # Panics + /// + /// Panics if the length is not divisible by `WIDTH`. + #[inline] + fn pack_slice(buf: &[Self::Value]) -> Vec { + assert!( + buf.len() % Self::WIDTH == 0, + "slice length {} must be divisible by WIDTH {}", + buf.len(), + Self::WIDTH + ); + buf.chunks_exact(Self::WIDTH) + .map(|chunk| Self::from_fn(|i| chunk[i])) + .collect() + } + + /// Packed prefix + scalar suffix split. + #[inline] + fn pack_slice_with_suffix(buf: &[Self::Value]) -> (Vec, &[Self::Value]) { + let split = buf.len() - (buf.len() % Self::WIDTH); + let (packed, suffix) = buf.split_at(split); + (Self::pack_slice(packed), suffix) + } + + /// Unpack packed values into a flat scalar vector. + #[inline] + fn unpack_slice(buf: &[Self]) -> Vec { + let mut out = Vec::with_capacity(buf.len() * Self::WIDTH); + for packed in buf { + for lane in 0..Self::WIDTH { + out.push(packed.extract(lane)); + } + } + out + } +} + +/// Packed arithmetic over a scalar field. +pub trait PackedField: + PackedValue + Add + Sub + Mul +{ + /// Scalar field type. + type Scalar: FieldCore; + + /// Broadcast one scalar across all lanes. + fn broadcast(value: Self::Scalar) -> Self; +} + +/// Scalar fallback packed type with one lane. +#[derive(Clone, Copy, Debug, Default, PartialEq, Eq)] +#[repr(transparent)] +pub struct NoPacking(pub [T; 1]); + +impl PackedValue for NoPacking +where + T: 'static + Copy + Send + Sync, +{ + type Value = T; + const WIDTH: usize = 1; + + #[inline] + fn from_fn(mut f: F) -> Self + where + F: FnMut(usize) -> Self::Value, + { + Self([f(0)]) + } + + #[inline] + fn extract(&self, lane: usize) -> Self::Value { + debug_assert_eq!(lane, 0); + self.0[0] + } +} + +impl Add for NoPacking { + type Output = Self; + #[inline] + fn add(self, rhs: Self) -> Self { + Self([self.0[0] + rhs.0[0]]) + } +} + +impl Sub for NoPacking { + type Output = Self; + #[inline] + fn sub(self, rhs: Self) -> Self { + Self([self.0[0] - rhs.0[0]]) + } +} + +impl Mul for NoPacking { + type Output = Self; + #[inline] + fn mul(self, rhs: Self) -> Self { + Self([self.0[0] * rhs.0[0]]) + } +} + +impl PackedField for NoPacking { + type Scalar = T; + + #[inline] + fn broadcast(value: Self::Scalar) -> Self { + Self([value]) + } +} + +/// Scalar field -> packed field association. +pub trait HasPacking: FieldCore { + /// Packed representation for this scalar field. + type Packing: PackedField; +} + +/// Selected packed backend for `Fp128`. +#[cfg(all(target_arch = "aarch64", target_feature = "neon"))] +pub type Fp128Packing = super::packed_neon::PackedFp128Neon

; + +/// Selected packed backend for `Fp128`. +#[cfg(all( + target_arch = "x86_64", + target_feature = "avx512f", + target_feature = "avx512dq" +))] +pub type Fp128Packing = super::packed_avx512::PackedFp128Avx512

; + +/// Selected packed backend for `Fp128`. +#[cfg(all( + target_arch = "x86_64", + target_feature = "avx2", + not(all(target_feature = "avx512f", target_feature = "avx512dq")) +))] +pub type Fp128Packing = super::packed_avx2::PackedFp128Avx2

; + +/// Selected packed backend for `Fp128`. +#[cfg(not(any( + all(target_arch = "aarch64", target_feature = "neon"), + all(target_arch = "x86_64", target_feature = "avx2") +)))] +pub type Fp128Packing = NoPacking>; + +impl HasPacking for Fp128

{ + type Packing = Fp128Packing

; +} + +/// Selected packed backend for `Fp32`. +#[cfg(all(target_arch = "aarch64", target_feature = "neon"))] +pub type Fp32Packing = super::packed_neon::PackedFp32Neon

; + +/// Selected packed backend for `Fp32`. +#[cfg(all( + target_arch = "x86_64", + target_feature = "avx512f", + target_feature = "avx512dq" +))] +pub type Fp32Packing = super::packed_avx512::PackedFp32Avx512

; + +/// Selected packed backend for `Fp32`. +#[cfg(all( + target_arch = "x86_64", + target_feature = "avx2", + not(all(target_feature = "avx512f", target_feature = "avx512dq")) +))] +pub type Fp32Packing = super::packed_avx2::PackedFp32Avx2

; + +/// Selected packed backend for `Fp32`. +#[cfg(not(any( + all(target_arch = "aarch64", target_feature = "neon"), + all(target_arch = "x86_64", target_feature = "avx2") +)))] +pub type Fp32Packing = NoPacking>; + +impl HasPacking for Fp32

{ + type Packing = Fp32Packing

; +} + +/// Selected packed backend for `Fp64`. +#[cfg(all(target_arch = "aarch64", target_feature = "neon"))] +pub type Fp64Packing = super::packed_neon::PackedFp64Neon

; + +/// Selected packed backend for `Fp64`. +#[cfg(all( + target_arch = "x86_64", + target_feature = "avx512f", + target_feature = "avx512dq" +))] +pub type Fp64Packing = super::packed_avx512::PackedFp64Avx512

; + +/// Selected packed backend for `Fp64`. +#[cfg(all( + target_arch = "x86_64", + target_feature = "avx2", + not(all(target_feature = "avx512f", target_feature = "avx512dq")) +))] +pub type Fp64Packing = super::packed_avx2::PackedFp64Avx2

; + +/// Selected packed backend for `Fp64`. +#[cfg(not(any( + all(target_arch = "aarch64", target_feature = "neon"), + all(target_arch = "x86_64", target_feature = "avx2") +)))] +pub type Fp64Packing = NoPacking>; + +impl HasPacking for Fp64

{ + type Packing = Fp64Packing

; +} + +#[cfg(test)] +mod tests { + use super::{HasPacking, PackedField, PackedValue}; + use crate::algebra::fields::{ + Pow2Offset24Field, Pow2Offset31Field, Pow2Offset32Field, Pow2Offset40Field, + Pow2Offset64Field, Prime128M13M4P0, + }; + use crate::{CanonicalField, FieldCore, FieldSampling}; + use rand::{rngs::StdRng, RngCore, SeedableRng}; + + fn rand_u128(rng: &mut R) -> u128 { + let lo = rng.next_u64() as u128; + let hi = rng.next_u64() as u128; + lo | (hi << 64) + } + + fn check_packed_add_sub_mul(seed: u64) + where + F: FieldCore + FieldSampling + PartialEq + std::fmt::Debug, + PF: PackedField + PackedValue, + { + let mut rng = StdRng::seed_from_u64(seed); + let len = PF::WIDTH * 17 + 3; + let lhs: Vec = (0..len).map(|_| FieldSampling::sample(&mut rng)).collect(); + let rhs: Vec = (0..len).map(|_| FieldSampling::sample(&mut rng)).collect(); + + let (lhs_p, lhs_s) = PF::pack_slice_with_suffix(&lhs); + let (rhs_p, rhs_s) = PF::pack_slice_with_suffix(&rhs); + + let add_p: Vec = lhs_p + .iter() + .zip(rhs_p.iter()) + .map(|(&a, &b)| a + b) + .collect(); + let sub_p: Vec = lhs_p + .iter() + .zip(rhs_p.iter()) + .map(|(&a, &b)| a - b) + .collect(); + let mul_p: Vec = lhs_p + .iter() + .zip(rhs_p.iter()) + .map(|(&a, &b)| a * b) + .collect(); + + let mut add_out = PF::unpack_slice(&add_p); + let mut sub_out = PF::unpack_slice(&sub_p); + let mut mul_out = PF::unpack_slice(&mul_p); + + for (&a, &b) in lhs_s.iter().zip(rhs_s.iter()) { + add_out.push(a + b); + sub_out.push(a - b); + mul_out.push(a * b); + } + + for i in 0..len { + assert_eq!( + add_out[i], + lhs[i] + rhs[i], + "packed add mismatch at lane {i}" + ); + assert_eq!( + sub_out[i], + lhs[i] - rhs[i], + "packed sub mismatch at lane {i}" + ); + assert_eq!( + mul_out[i], + lhs[i] * rhs[i], + "packed mul mismatch at lane {i}" + ); + } + } + + fn check_broadcast_roundtrip(val: F) + where + F: FieldCore + PartialEq + std::fmt::Debug, + PF: PackedField + PackedValue, + { + let p = PF::broadcast(val); + for lane in 0..PF::WIDTH { + assert_eq!(p.extract(lane), val); + } + } + + #[test] + fn packed_fp128_add_sub_mul_match_scalar() { + type F = Prime128M13M4P0; + type PF = ::Packing; + + let mut rng = StdRng::seed_from_u64(0x55aa_4422_1177_0033); + let len = PF::WIDTH * 17 + 3; + let lhs: Vec = (0..len) + .map(|_| F::from_canonical_u128_reduced(rand_u128(&mut rng))) + .collect(); + let rhs: Vec = (0..len) + .map(|_| F::from_canonical_u128_reduced(rand_u128(&mut rng))) + .collect(); + + let (lhs_p, lhs_s) = PF::pack_slice_with_suffix(&lhs); + let (rhs_p, rhs_s) = PF::pack_slice_with_suffix(&rhs); + + let add_p: Vec = lhs_p + .iter() + .zip(rhs_p.iter()) + .map(|(&a, &b)| a + b) + .collect(); + let sub_p: Vec = lhs_p + .iter() + .zip(rhs_p.iter()) + .map(|(&a, &b)| a - b) + .collect(); + let mul_p: Vec = lhs_p + .iter() + .zip(rhs_p.iter()) + .map(|(&a, &b)| a * b) + .collect(); + + let mut add_out = PF::unpack_slice(&add_p); + let mut sub_out = PF::unpack_slice(&sub_p); + let mut mul_out = PF::unpack_slice(&mul_p); + + for (&a, &b) in lhs_s.iter().zip(rhs_s.iter()) { + add_out.push(a + b); + sub_out.push(a - b); + mul_out.push(a * b); + } + + for i in 0..len { + assert_eq!( + add_out[i], + lhs[i] + rhs[i], + "packed add mismatch at lane {i}" + ); + assert_eq!( + sub_out[i], + lhs[i] - rhs[i], + "packed sub mismatch at lane {i}" + ); + assert_eq!( + mul_out[i], + lhs[i] * rhs[i], + "packed mul mismatch at lane {i}" + ); + } + } + + #[test] + fn fp128_broadcast_and_extract_roundtrip() { + type F = Prime128M13M4P0; + type PF = ::Packing; + check_broadcast_roundtrip::(F::from_u64(42)); + } + + #[test] + fn packed_fp32_24b_add_sub_mul() { + type F = Pow2Offset24Field; + type PF = ::Packing; + check_packed_add_sub_mul::(0xaa24_bb24_cc24_dd24); + } + + #[test] + fn packed_fp32_31b_add_sub_mul() { + type F = Pow2Offset31Field; + type PF = ::Packing; + check_packed_add_sub_mul::(0xaa31_bb31_cc31_dd31); + } + + #[test] + fn packed_fp32_32b_add_sub_mul() { + type F = Pow2Offset32Field; + type PF = ::Packing; + check_packed_add_sub_mul::(0xaa32_bb32_cc32_dd32); + } + + #[test] + fn fp32_broadcast_and_extract_roundtrip() { + type F = Pow2Offset24Field; + type PF = ::Packing; + check_broadcast_roundtrip::(F::from_u64(42)); + } + + #[test] + fn packed_fp64_40b_add_sub_mul() { + type F = Pow2Offset40Field; + type PF = ::Packing; + check_packed_add_sub_mul::(0xaa40_bb40_cc40_dd40); + } + + #[test] + fn packed_fp64_64b_add_sub_mul() { + type F = Pow2Offset64Field; + type PF = ::Packing; + check_packed_add_sub_mul::(0xaa64_bb64_cc64_dd64); + } + + #[test] + fn fp64_broadcast_and_extract_roundtrip() { + type F = Pow2Offset40Field; + type PF = ::Packing; + check_broadcast_roundtrip::(F::from_u64(42)); + } +} diff --git a/src/algebra/fields/packed_avx2.rs b/src/algebra/fields/packed_avx2.rs new file mode 100644 index 00000000..95468959 --- /dev/null +++ b/src/algebra/fields/packed_avx2.rs @@ -0,0 +1,704 @@ +//! AVX2 packed backends for Fp32, Fp64, Fp128. +//! +//! Techniques adapted from plonky2 (Goldilocks) and plonky3 (Mersenne-31). + +use super::packed::{PackedField, PackedValue}; +use crate::algebra::fields::{Fp128, Fp32, Fp64}; +use core::arch::x86_64::*; +use core::fmt; +use core::mem::transmute; +use core::ops::{Add, Mul, Sub}; + +/// Duplicate high 32 bits of each 64-bit lane into the low 32 bits. +/// Uses the float `movehdup` instruction which runs on port 5 (doesn't compete +/// with multiply on ports 0/1). +#[inline(always)] +unsafe fn movehdup_epi32(x: __m256i) -> __m256i { + _mm256_castps_si256(_mm256_movehdup_ps(_mm256_castsi256_ps(x))) +} + +#[inline(always)] +unsafe fn moveldup_epi32(x: __m256i) -> __m256i { + _mm256_castps_si256(_mm256_moveldup_ps(_mm256_castsi256_ps(x))) +} + +/// 64×64→128 schoolbook multiply using 32×32→64 partial products. +/// Returns (hi, lo) representing the 128-bit product. +#[inline] +unsafe fn mul64_64_256(x: __m256i, y: __m256i) -> (__m256i, __m256i) { + let x_hi = movehdup_epi32(x); + let y_hi = movehdup_epi32(y); + + let mul_ll = _mm256_mul_epu32(x, y); + let mul_lh = _mm256_mul_epu32(x, y_hi); + let mul_hl = _mm256_mul_epu32(x_hi, y); + let mul_hh = _mm256_mul_epu32(x_hi, y_hi); + + let mul_ll_hi = _mm256_srli_epi64::<32>(mul_ll); + let t0 = _mm256_add_epi64(mul_hl, mul_ll_hi); + let mask32 = _mm256_set1_epi64x(0xFFFF_FFFF_i64); + let t0_lo = _mm256_and_si256(t0, mask32); + let t0_hi = _mm256_srli_epi64::<32>(t0); + let t1 = _mm256_add_epi64(mul_lh, t0_lo); + let t2 = _mm256_add_epi64(mul_hh, t0_hi); + let t1_hi = _mm256_srli_epi64::<32>(t1); + let res_hi = _mm256_add_epi64(t2, t1_hi); + + let t1_lo = moveldup_epi32(t1); + let res_lo = _mm256_blend_epi32::<0b10101010>(mul_ll, t1_lo); + + (res_hi, res_lo) +} + +/// Number of `Fp32` lanes in an AVX2 packed vector. +pub const FP32_WIDTH: usize = 8; + +/// AVX2 packed arithmetic for `Fp32

`, processing 8 lanes. +#[derive(Clone, Copy)] +#[repr(transparent)] +pub struct PackedFp32Avx2(pub [Fp32

; FP32_WIDTH]); + +impl PackedFp32Avx2

{ + const BITS: u32 = 32 - P.leading_zeros(); + + const C: u32 = { + let c = if Self::BITS == 32 { + 0u32.wrapping_sub(P) + } else { + (1u32 << Self::BITS) - P + }; + assert!(P != 0, "modulus must be nonzero"); + assert!(P & 1 == 1, "modulus must be odd"); + assert!( + (c as u64) * (c as u64 + 1) < P as u64, + "C(C+1) < P required for fused canonicalize" + ); + c + }; + + const MASK_U64: u64 = if Self::BITS == 32 { + u32::MAX as u64 + } else { + (1u64 << Self::BITS) - 1 + }; + + #[inline(always)] + fn to_vec(self) -> __m256i { + unsafe { transmute(self) } + } + + #[inline(always)] + unsafe fn from_vec(v: __m256i) -> Self { + unsafe { transmute(v) } + } +} + +impl Default for PackedFp32Avx2

{ + #[inline] + fn default() -> Self { + Self([Fp32(0); FP32_WIDTH]) + } +} + +impl fmt::Debug for PackedFp32Avx2

{ + fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result { + f.debug_tuple("PackedFp32Avx2").field(&self.0).finish() + } +} + +impl PartialEq for PackedFp32Avx2

{ + #[inline] + fn eq(&self, other: &Self) -> bool { + self.0 == other.0 + } +} + +impl Eq for PackedFp32Avx2

{} + +impl Add for PackedFp32Avx2

{ + type Output = Self; + #[inline] + fn add(self, rhs: Self) -> Self { + unsafe { + let a = self.to_vec(); + let b = rhs.to_vec(); + let p = _mm256_set1_epi32(P as i32); + + let result = if Self::BITS <= 31 { + let t = _mm256_add_epi32(a, b); + let u = _mm256_sub_epi32(t, p); + _mm256_min_epu32(t, u) + } else { + let c = _mm256_set1_epi32(Self::C as i32); + let t = _mm256_add_epi32(a, b); + // Emulate unsigned compare: XOR with 0x80000000 converts u32 compare to i32 + let sign32 = _mm256_set1_epi32(i32::MIN); + let overflow = + _mm256_cmpgt_epi32(_mm256_xor_si256(a, sign32), _mm256_xor_si256(t, sign32)); + // Step 1: correct overflow (2^32 ≡ C mod P) + let t2 = _mm256_add_epi32(t, _mm256_and_si256(overflow, c)); + // Step 2: subtract P if t2 >= P + let r = _mm256_sub_epi32(t2, p); + _mm256_min_epu32(t2, r) + }; + + Self::from_vec(result) + } + } +} + +impl Sub for PackedFp32Avx2

{ + type Output = Self; + #[inline] + fn sub(self, rhs: Self) -> Self { + unsafe { + let a = self.to_vec(); + let b = rhs.to_vec(); + let p = _mm256_set1_epi32(P as i32); + + let result = if Self::BITS <= 31 { + let t = _mm256_sub_epi32(a, b); + let u = _mm256_add_epi32(t, p); + _mm256_min_epu32(t, u) + } else { + let t = _mm256_sub_epi32(a, b); + let sign32 = _mm256_set1_epi32(i32::MIN); + let underflow = + _mm256_cmpgt_epi32(_mm256_xor_si256(b, sign32), _mm256_xor_si256(a, sign32)); + let corrected = _mm256_add_epi32(t, p); + _mm256_blendv_epi8(t, corrected, underflow) + }; + + Self::from_vec(result) + } + } +} + +impl Mul for PackedFp32Avx2

{ + type Output = Self; + #[inline] + fn mul(self, rhs: Self) -> Self { + unsafe { + let a = self.to_vec(); + let b = rhs.to_vec(); + + let prod_evn = _mm256_mul_epu32(a, b); + let a_odd = movehdup_epi32(a); + let b_odd = movehdup_epi32(b); + let prod_odd = _mm256_mul_epu32(a_odd, b_odd); + + let mask = _mm256_set1_epi64x(Self::MASK_U64 as i64); + let c_vec = _mm256_set1_epi64x(Self::C as i64); + let shift = _mm_set_epi64x(0, Self::BITS as i64); + + // Fold 1 + let evn_lo = _mm256_and_si256(prod_evn, mask); + let evn_hi = _mm256_srl_epi64(prod_evn, shift); + let evn_f1 = _mm256_add_epi64(evn_lo, _mm256_mul_epu32(evn_hi, c_vec)); + + let odd_lo = _mm256_and_si256(prod_odd, mask); + let odd_hi = _mm256_srl_epi64(prod_odd, shift); + let odd_f1 = _mm256_add_epi64(odd_lo, _mm256_mul_epu32(odd_hi, c_vec)); + + // Fold 2 + let evn_f1_lo = _mm256_and_si256(evn_f1, mask); + let evn_f1_hi = _mm256_srl_epi64(evn_f1, shift); + let evn_f2 = _mm256_add_epi64(evn_f1_lo, _mm256_mul_epu32(evn_f1_hi, c_vec)); + + let odd_f1_lo = _mm256_and_si256(odd_f1, mask); + let odd_f1_hi = _mm256_srl_epi64(odd_f1, shift); + let odd_f2 = _mm256_add_epi64(odd_f1_lo, _mm256_mul_epu32(odd_f1_hi, c_vec)); + + // Recombine even/odd: shift odd results into high 32-bit positions, blend. + let odd_shifted = _mm256_slli_epi64::<32>(odd_f2); + let combined = _mm256_blend_epi32::<0b10101010>(evn_f2, odd_shifted); + + // Conditional subtract P + let p_vec = _mm256_set1_epi32(P as i32); + let reduced = _mm256_sub_epi32(combined, p_vec); + Self::from_vec(_mm256_min_epu32(combined, reduced)) + } + } +} + +impl PackedValue for PackedFp32Avx2

{ + type Value = Fp32

; + const WIDTH: usize = FP32_WIDTH; + + #[inline] + fn from_fn(mut f: F) -> Self + where + F: FnMut(usize) -> Self::Value, + { + Self([f(0), f(1), f(2), f(3), f(4), f(5), f(6), f(7)]) + } + + #[inline] + fn extract(&self, lane: usize) -> Self::Value { + debug_assert!(lane < FP32_WIDTH); + self.0[lane] + } +} + +impl PackedField for PackedFp32Avx2

{ + type Scalar = Fp32

; + + #[inline] + fn broadcast(value: Self::Scalar) -> Self { + Self([value; FP32_WIDTH]) + } +} + +/// Number of `Fp64` lanes in an AVX2 packed vector. +pub const FP64_WIDTH: usize = 4; + +/// AVX2 packed arithmetic for `Fp64

`, processing 4 lanes. +#[derive(Clone, Copy)] +#[repr(transparent)] +pub struct PackedFp64Avx2(pub [Fp64

; FP64_WIDTH]); + +impl PackedFp64Avx2

{ + const BITS: u32 = 64 - P.leading_zeros(); + + const C_LO: u64 = { + let c = if Self::BITS == 64 { + 0u64.wrapping_sub(P) + } else { + (1u64 << Self::BITS) - P + }; + assert!(P != 0, "modulus must be nonzero"); + assert!(P & 1 == 1, "modulus must be odd"); + c + }; + + const MASK64: u64 = if Self::BITS < 64 { + (1u64 << Self::BITS) - 1 + } else { + u64::MAX + }; + + #[inline(always)] + fn to_vec(self) -> __m256i { + unsafe { transmute(self) } + } + + #[inline(always)] + unsafe fn from_vec(v: __m256i) -> Self { + unsafe { transmute(v) } + } + + #[inline] + unsafe fn reduce128_vec(hi: __m256i, lo: __m256i) -> __m256i { + if Self::BITS < 64 { + Self::reduce128_small_k(hi, lo) + } else { + Self::reduce128_full_k(hi, lo) + } + } + + /// Reduction for BITS < 64. All intermediates fit in u64 — no overflow. + #[inline] + unsafe fn reduce128_small_k(hi: __m256i, lo: __m256i) -> __m256i { + let mask_k = _mm256_set1_epi64x(Self::MASK64 as i64); + let c_vec = _mm256_set1_epi64x(Self::C_LO as i64); + let p_vec = _mm256_set1_epi64x(P as i64); + let shift_k = _mm_set_epi64x(0, Self::BITS as i64); + let shift_64mk = _mm_set_epi64x(0, (64 - Self::BITS) as i64); + + let lo_k = _mm256_and_si256(lo, mask_k); + let lo_upper = _mm256_srl_epi64(lo, shift_k); + let hi_shifted = _mm256_sll_epi64(hi, shift_64mk); + let hi_k = _mm256_or_si256(lo_upper, hi_shifted); + + let c_hi_lo = _mm256_mul_epu32(c_vec, hi_k); + let hi_k_top = _mm256_srli_epi64::<32>(hi_k); + let c_hi_top = _mm256_mul_epu32(c_vec, hi_k_top); + let c_hi_top_shifted = _mm256_slli_epi64::<32>(c_hi_top); + let c_hi_full = _mm256_add_epi64(c_hi_lo, c_hi_top_shifted); + + let fold1 = _mm256_add_epi64(lo_k, c_hi_full); + + let fold1_lo_k = _mm256_and_si256(fold1, mask_k); + let fold1_hi = _mm256_srl_epi64(fold1, shift_k); + let c_fold1_hi = _mm256_mul_epu32(c_vec, fold1_hi); + let fold2 = _mm256_add_epi64(fold1_lo_k, c_fold1_hi); + + let reduced = _mm256_sub_epi64(fold2, p_vec); + let sign = _mm256_set1_epi64x(i64::MIN); + let fold2_s = _mm256_xor_si256(fold2, sign); + let reduced_s = _mm256_xor_si256(reduced, sign); + let fold2_lt = _mm256_cmpgt_epi64(reduced_s, fold2_s); + _mm256_blendv_epi8(reduced, fold2, fold2_lt) + } + + /// Reduction for BITS == 64. Uses XOR-with-SIGN_BIT trick for unsigned + /// overflow detection. + #[inline] + unsafe fn reduce128_full_k(hi: __m256i, lo: __m256i) -> __m256i { + let c_vec = _mm256_set1_epi64x(Self::C_LO as i64); + let p_vec = _mm256_set1_epi64x(P as i64); + let sign = _mm256_set1_epi64x(i64::MIN); + let c_hi_lo = _mm256_mul_epu32(c_vec, hi); + let hi_hi = _mm256_srli_epi64::<32>(hi); + let c_hi_hi = _mm256_mul_epu32(c_vec, hi_hi); + + let c_hi_hi_lo32 = _mm256_slli_epi64::<32>(c_hi_hi); + let c_hi_carry = _mm256_srli_epi64::<32>(c_hi_hi); + + let sum_lo = _mm256_add_epi64(c_hi_lo, c_hi_hi_lo32); + let c_hi_lo_s = _mm256_xor_si256(c_hi_lo, sign); + let sum_lo_s = _mm256_xor_si256(sum_lo, sign); + let carry0 = _mm256_cmpgt_epi64(c_hi_lo_s, sum_lo_s); + let overflow = _mm256_sub_epi64(c_hi_carry, carry0); + + let s = _mm256_add_epi64(lo, sum_lo); + let lo_s = _mm256_xor_si256(lo, sign); + let s_s = _mm256_xor_si256(s, sign); + let carry1 = _mm256_cmpgt_epi64(lo_s, s_s); + let total_overflow = _mm256_sub_epi64(overflow, carry1); + + let final_corr = _mm256_mul_epu32(c_vec, total_overflow); + let result = _mm256_add_epi64(s, final_corr); + let s2_s = _mm256_xor_si256(s, sign); + let result_s = _mm256_xor_si256(result, sign); + let carry_f = _mm256_cmpgt_epi64(s2_s, result_s); + let corr_f = _mm256_and_si256(carry_f, c_vec); + let result = _mm256_add_epi64(result, corr_f); + + let result_s2 = _mm256_xor_si256(result, sign); + let p_s = _mm256_xor_si256(p_vec, sign); + let lt_p = _mm256_cmpgt_epi64(p_s, result_s2); + let sub_amt = _mm256_andnot_si256(lt_p, p_vec); + _mm256_sub_epi64(result, sub_amt) + } +} + +impl Default for PackedFp64Avx2

{ + #[inline] + fn default() -> Self { + Self([Fp64(0); FP64_WIDTH]) + } +} + +impl fmt::Debug for PackedFp64Avx2

{ + fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result { + f.debug_tuple("PackedFp64Avx2").field(&self.0).finish() + } +} + +impl PartialEq for PackedFp64Avx2

{ + #[inline] + fn eq(&self, other: &Self) -> bool { + self.0 == other.0 + } +} + +impl Eq for PackedFp64Avx2

{} + +impl Add for PackedFp64Avx2

{ + type Output = Self; + #[inline] + fn add(self, rhs: Self) -> Self { + unsafe { + let a = self.to_vec(); + let b = rhs.to_vec(); + let p = _mm256_set1_epi64x(P as i64); + + let result = if Self::BITS <= 62 { + // a + b < 2P < 2^63: no overflow. + let s = _mm256_add_epi64(a, b); + let r = _mm256_sub_epi64(s, p); + // s < P? Use signed compare after shift trick. + let sign = _mm256_set1_epi64x(i64::MIN); + let s_s = _mm256_xor_si256(s, sign); + let p_s = _mm256_xor_si256(p, sign); + let borrow = _mm256_cmpgt_epi64(p_s, s_s); + _mm256_blendv_epi8(r, s, borrow) + } else { + // a + b can overflow u64. + let s = _mm256_add_epi64(a, b); + let sign = _mm256_set1_epi64x(i64::MIN); + let a_s = _mm256_xor_si256(a, sign); + let s_s = _mm256_xor_si256(s, sign); + let overflow = _mm256_cmpgt_epi64(a_s, s_s); + let c = _mm256_set1_epi64x(Self::C_LO as i64); + let s_plus_c = _mm256_add_epi64(s, c); + let s_minus_p = _mm256_sub_epi64(s, p); + let p_s = _mm256_xor_si256(p, sign); + let lt_p = _mm256_cmpgt_epi64(p_s, s_s); + let no_of = _mm256_blendv_epi8(s_minus_p, s, lt_p); + _mm256_blendv_epi8(no_of, s_plus_c, overflow) + }; + + Self::from_vec(result) + } + } +} + +impl Sub for PackedFp64Avx2

{ + type Output = Self; + #[inline] + fn sub(self, rhs: Self) -> Self { + unsafe { + let a = self.to_vec(); + let b = rhs.to_vec(); + let p = _mm256_set1_epi64x(P as i64); + let d = _mm256_sub_epi64(a, b); + + let sign = _mm256_set1_epi64x(i64::MIN); + let a_s = _mm256_xor_si256(a, sign); + let b_s = _mm256_xor_si256(b, sign); + let underflow = _mm256_cmpgt_epi64(b_s, a_s); + let corrected = _mm256_add_epi64(d, p); + Self::from_vec(_mm256_blendv_epi8(d, corrected, underflow)) + } + } +} + +impl Mul for PackedFp64Avx2

{ + type Output = Self; + #[inline] + fn mul(self, rhs: Self) -> Self { + unsafe { + let (hi, lo) = mul64_64_256(self.to_vec(), rhs.to_vec()); + Self::from_vec(Self::reduce128_vec(hi, lo)) + } + } +} + +impl PackedValue for PackedFp64Avx2

{ + type Value = Fp64

; + const WIDTH: usize = FP64_WIDTH; + + #[inline] + fn from_fn(mut f: F) -> Self + where + F: FnMut(usize) -> Self::Value, + { + Self([f(0), f(1), f(2), f(3)]) + } + + #[inline] + fn extract(&self, lane: usize) -> Self::Value { + debug_assert!(lane < FP64_WIDTH); + self.0[lane] + } +} + +impl PackedField for PackedFp64Avx2

{ + type Scalar = Fp64

; + + #[inline] + fn broadcast(value: Self::Scalar) -> Self { + Self([value; FP64_WIDTH]) + } +} + +/// Number of `Fp128` lanes in an AVX2 packed vector. +pub const FP128_WIDTH: usize = 4; + +/// AVX2 packed arithmetic for `Fp128

`, 4 lanes in SoA layout. +/// +/// Stores 4 elements as separate `lo` and `hi` `u64` arrays, enabling +/// vectorized add/sub via `__m256i`. Mul remains scalar per-lane. +#[derive(Clone, Copy)] +pub struct PackedFp128Avx2 { + lo: [u64; FP128_WIDTH], + hi: [u64; FP128_WIDTH], +} + +impl PackedFp128Avx2

{ + const P_LO: u64 = P as u64; + const P_HI: u64 = (P >> 64) as u64; +} + +impl Default for PackedFp128Avx2

{ + #[inline] + fn default() -> Self { + Self { + lo: [0; FP128_WIDTH], + hi: [0; FP128_WIDTH], + } + } +} + +impl fmt::Debug for PackedFp128Avx2

{ + fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result { + let elems: Vec<_> = (0..FP128_WIDTH).map(|i| self.extract(i)).collect(); + f.debug_tuple("PackedFp128Avx2").field(&elems).finish() + } +} + +impl PartialEq for PackedFp128Avx2

{ + #[inline] + fn eq(&self, other: &Self) -> bool { + self.lo == other.lo && self.hi == other.hi + } +} + +impl Eq for PackedFp128Avx2

{} + +impl Add for PackedFp128Avx2

{ + type Output = Self; + #[inline] + fn add(self, rhs: Self) -> Self { + unsafe { + let a_lo = _mm256_loadu_si256(self.lo.as_ptr().cast()); + let a_hi = _mm256_loadu_si256(self.hi.as_ptr().cast()); + let b_lo = _mm256_loadu_si256(rhs.lo.as_ptr().cast()); + let b_hi = _mm256_loadu_si256(rhs.hi.as_ptr().cast()); + let p_lo = _mm256_set1_epi64x(Self::P_LO as i64); + let p_hi = _mm256_set1_epi64x(Self::P_HI as i64); + let sign = _mm256_set1_epi64x(i64::MIN); + let one = _mm256_set1_epi64x(1); + + // 128-bit add with unsigned compare emulation (XOR sign bit) + let sum_lo = _mm256_add_epi64(a_lo, b_lo); + let carry_lo = + _mm256_cmpgt_epi64(_mm256_xor_si256(a_lo, sign), _mm256_xor_si256(sum_lo, sign)); + let carry_lo_bit = _mm256_and_si256(carry_lo, one); + + let hi_tmp = _mm256_add_epi64(a_hi, b_hi); + let ov1 = + _mm256_cmpgt_epi64(_mm256_xor_si256(a_hi, sign), _mm256_xor_si256(hi_tmp, sign)); + let sum_hi = _mm256_add_epi64(hi_tmp, carry_lo_bit); + let ov2 = _mm256_cmpgt_epi64( + _mm256_xor_si256(hi_tmp, sign), + _mm256_xor_si256(sum_hi, sign), + ); + let carry_128 = _mm256_or_si256(ov1, ov2); + + // 128-bit subtract P + let red_lo = _mm256_sub_epi64(sum_lo, p_lo); + let borrow_lo = + _mm256_cmpgt_epi64(_mm256_xor_si256(p_lo, sign), _mm256_xor_si256(sum_lo, sign)); + let borrow_lo_bit = _mm256_and_si256(borrow_lo, one); + + let red_hi_tmp = _mm256_sub_epi64(sum_hi, p_hi); + let bw1 = + _mm256_cmpgt_epi64(_mm256_xor_si256(p_hi, sign), _mm256_xor_si256(sum_hi, sign)); + let red_hi = _mm256_sub_epi64(red_hi_tmp, borrow_lo_bit); + let bw2 = _mm256_cmpgt_epi64( + _mm256_xor_si256(borrow_lo_bit, sign), + _mm256_xor_si256(red_hi_tmp, sign), + ); + let borrow = _mm256_or_si256(bw1, bw2); + + // use_reduced = carry_128 | !borrow + let not_borrow = _mm256_xor_si256(borrow, _mm256_set1_epi64x(-1)); + let use_reduced = _mm256_or_si256(carry_128, not_borrow); + let out_lo = _mm256_blendv_epi8(sum_lo, red_lo, use_reduced); + let out_hi = _mm256_blendv_epi8(sum_hi, red_hi, use_reduced); + + let mut result = Self::default(); + _mm256_storeu_si256(result.lo.as_mut_ptr().cast(), out_lo); + _mm256_storeu_si256(result.hi.as_mut_ptr().cast(), out_hi); + result + } + } +} + +impl Sub for PackedFp128Avx2

{ + type Output = Self; + #[inline] + fn sub(self, rhs: Self) -> Self { + unsafe { + let a_lo = _mm256_loadu_si256(self.lo.as_ptr().cast()); + let a_hi = _mm256_loadu_si256(self.hi.as_ptr().cast()); + let b_lo = _mm256_loadu_si256(rhs.lo.as_ptr().cast()); + let b_hi = _mm256_loadu_si256(rhs.hi.as_ptr().cast()); + let p_lo = _mm256_set1_epi64x(Self::P_LO as i64); + let p_hi = _mm256_set1_epi64x(Self::P_HI as i64); + let sign = _mm256_set1_epi64x(i64::MIN); + let one = _mm256_set1_epi64x(1); + + // 128-bit sub + let diff_lo = _mm256_sub_epi64(a_lo, b_lo); + let borrow_lo = + _mm256_cmpgt_epi64(_mm256_xor_si256(b_lo, sign), _mm256_xor_si256(a_lo, sign)); + let borrow_lo_bit = _mm256_and_si256(borrow_lo, one); + + let hi_tmp = _mm256_sub_epi64(a_hi, b_hi); + let bw1 = + _mm256_cmpgt_epi64(_mm256_xor_si256(b_hi, sign), _mm256_xor_si256(a_hi, sign)); + let diff_hi = _mm256_sub_epi64(hi_tmp, borrow_lo_bit); + let bw2 = _mm256_cmpgt_epi64( + _mm256_xor_si256(borrow_lo_bit, sign), + _mm256_xor_si256(hi_tmp, sign), + ); + let borrow_128 = _mm256_or_si256(bw1, bw2); + + // Correction: add P back where underflow occurred + let corr_lo = _mm256_add_epi64(diff_lo, p_lo); + let carry_lo = _mm256_cmpgt_epi64( + _mm256_xor_si256(diff_lo, sign), + _mm256_xor_si256(corr_lo, sign), + ); + let carry_lo_bit = _mm256_and_si256(carry_lo, one); + let corr_hi = _mm256_add_epi64(diff_hi, p_hi); + let corr_hi = _mm256_add_epi64(corr_hi, carry_lo_bit); + + let out_lo = _mm256_blendv_epi8(diff_lo, corr_lo, borrow_128); + let out_hi = _mm256_blendv_epi8(diff_hi, corr_hi, borrow_128); + + let mut result = Self::default(); + _mm256_storeu_si256(result.lo.as_mut_ptr().cast(), out_lo); + _mm256_storeu_si256(result.hi.as_mut_ptr().cast(), out_hi); + result + } + } +} + +impl Mul for PackedFp128Avx2

{ + type Output = Self; + #[inline] + fn mul(self, rhs: Self) -> Self { + let mut out = Self::default(); + for i in 0..FP128_WIDTH { + let a = Fp128::

([self.lo[i], self.hi[i]]); + let b = Fp128::

([rhs.lo[i], rhs.hi[i]]); + let r = a * b; + out.lo[i] = r.0[0]; + out.hi[i] = r.0[1]; + } + out + } +} + +impl PackedValue for PackedFp128Avx2

{ + type Value = Fp128

; + const WIDTH: usize = FP128_WIDTH; + + #[inline] + fn from_fn(mut f: F) -> Self + where + F: FnMut(usize) -> Self::Value, + { + let mut lo = [0u64; FP128_WIDTH]; + let mut hi = [0u64; FP128_WIDTH]; + for i in 0..FP128_WIDTH { + let v = f(i); + lo[i] = v.0[0]; + hi[i] = v.0[1]; + } + Self { lo, hi } + } + + #[inline] + fn extract(&self, lane: usize) -> Self::Value { + debug_assert!(lane < FP128_WIDTH); + Fp128([self.lo[lane], self.hi[lane]]) + } +} + +impl PackedField for PackedFp128Avx2

{ + type Scalar = Fp128

; + + #[inline] + fn broadcast(value: Self::Scalar) -> Self { + Self { + lo: [value.0[0]; FP128_WIDTH], + hi: [value.0[1]; FP128_WIDTH], + } + } +} diff --git a/src/algebra/fields/packed_avx512.rs b/src/algebra/fields/packed_avx512.rs new file mode 100644 index 00000000..1102ac95 --- /dev/null +++ b/src/algebra/fields/packed_avx512.rs @@ -0,0 +1,666 @@ +//! AVX-512 packed backends for Fp32, Fp64, Fp128. +//! +//! Requires AVX-512F + AVX-512DQ. Uses native unsigned comparisons and mask +//! registers for branchless conditionals. + +use super::packed::{PackedField, PackedValue}; +use crate::algebra::fields::{Fp128, Fp32, Fp64}; +use crate::FieldCore; +use core::arch::x86_64::*; +use core::fmt; +use core::mem::transmute; +use core::ops::{Add, Mul, Sub}; + +#[inline(always)] +unsafe fn movehdup_epi32_512(x: __m512i) -> __m512i { + _mm512_castps_si512(_mm512_movehdup_ps(_mm512_castsi512_ps(x))) +} + +#[inline(always)] +unsafe fn moveldup_epi32_512(x: __m512i) -> __m512i { + _mm512_castps_si512(_mm512_moveldup_ps(_mm512_castsi512_ps(x))) +} + +/// 64×64→128 schoolbook multiply using 32×32→64 partial products. +/// Returns (hi, lo) representing the 128-bit product. +/// Adapted from plonky3's Goldilocks AVX-512 backend. +#[inline] +unsafe fn mul64_64_512(x: __m512i, y: __m512i) -> (__m512i, __m512i) { + let x_hi = movehdup_epi32_512(x); + let y_hi = movehdup_epi32_512(y); + + let mul_ll = _mm512_mul_epu32(x, y); + let mul_lh = _mm512_mul_epu32(x, y_hi); + let mul_hl = _mm512_mul_epu32(x_hi, y); + let mul_hh = _mm512_mul_epu32(x_hi, y_hi); + + let mul_ll_hi = _mm512_srli_epi64::<32>(mul_ll); + let t0 = _mm512_add_epi64(mul_hl, mul_ll_hi); + let mask32 = _mm512_set1_epi64(0xFFFF_FFFF_i64); + let t0_lo = _mm512_and_si512(t0, mask32); + let t0_hi = _mm512_srli_epi64::<32>(t0); + let t1 = _mm512_add_epi64(mul_lh, t0_lo); + let t2 = _mm512_add_epi64(mul_hh, t0_hi); + let t1_hi = _mm512_srli_epi64::<32>(t1); + let res_hi = _mm512_add_epi64(t2, t1_hi); + + let t1_lo = moveldup_epi32_512(t1); + let res_lo = _mm512_mask_blend_epi32(0b0101_0101_0101_0101, t1_lo, mul_ll); + + (res_hi, res_lo) +} + +/// Number of `Fp32` lanes in an AVX-512 packed vector. +pub const FP32_WIDTH: usize = 16; + +/// AVX-512 packed arithmetic for `Fp32

`, processing 16 lanes. +#[derive(Clone, Copy)] +#[repr(transparent)] +pub struct PackedFp32Avx512(pub [Fp32

; FP32_WIDTH]); + +impl PackedFp32Avx512

{ + const BITS: u32 = 32 - P.leading_zeros(); + + const C: u32 = { + let c = if Self::BITS == 32 { + 0u32.wrapping_sub(P) + } else { + (1u32 << Self::BITS) - P + }; + assert!(P != 0, "modulus must be nonzero"); + assert!(P & 1 == 1, "modulus must be odd"); + assert!( + (c as u64) * (c as u64 + 1) < P as u64, + "C(C+1) < P required for fused canonicalize" + ); + c + }; + + const MASK_U64: u64 = if Self::BITS == 32 { + u32::MAX as u64 + } else { + (1u64 << Self::BITS) - 1 + }; + + #[inline(always)] + fn to_vec(self) -> __m512i { + unsafe { transmute(self) } + } + + #[inline(always)] + unsafe fn from_vec(v: __m512i) -> Self { + unsafe { transmute(v) } + } +} + +impl Default for PackedFp32Avx512

{ + #[inline] + fn default() -> Self { + Self([Fp32(0); FP32_WIDTH]) + } +} + +impl fmt::Debug for PackedFp32Avx512

{ + fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result { + f.debug_tuple("PackedFp32Avx512").field(&self.0).finish() + } +} + +impl PartialEq for PackedFp32Avx512

{ + #[inline] + fn eq(&self, other: &Self) -> bool { + self.0 == other.0 + } +} + +impl Eq for PackedFp32Avx512

{} + +impl Add for PackedFp32Avx512

{ + type Output = Self; + #[inline] + fn add(self, rhs: Self) -> Self { + unsafe { + let a = self.to_vec(); + let b = rhs.to_vec(); + let p = _mm512_set1_epi32(P as i32); + + let result = if Self::BITS <= 31 { + let t = _mm512_add_epi32(a, b); + let u = _mm512_sub_epi32(t, p); + _mm512_min_epu32(t, u) + } else { + let c = _mm512_set1_epi32(Self::C as i32); + let t = _mm512_add_epi32(a, b); + // Step 1: correct overflow (2^32 ≡ C mod P) + let overflow = _mm512_cmplt_epu32_mask(t, a); + let t2 = _mm512_mask_add_epi32(t, overflow, t, c); + // Step 2: subtract P if t2 >= P + let geq_p = _mm512_cmpge_epu32_mask(t2, p); + _mm512_mask_sub_epi32(t2, geq_p, t2, p) + }; + + Self::from_vec(result) + } + } +} + +impl Sub for PackedFp32Avx512

{ + type Output = Self; + #[inline] + fn sub(self, rhs: Self) -> Self { + unsafe { + let a = self.to_vec(); + let b = rhs.to_vec(); + let p = _mm512_set1_epi32(P as i32); + + let result = if Self::BITS <= 31 { + let t = _mm512_sub_epi32(a, b); + let u = _mm512_add_epi32(t, p); + _mm512_min_epu32(t, u) + } else { + let t = _mm512_sub_epi32(a, b); + let underflow = _mm512_cmplt_epu32_mask(a, b); + _mm512_mask_add_epi32(t, underflow, t, p) + }; + + Self::from_vec(result) + } + } +} + +impl Mul for PackedFp32Avx512

{ + type Output = Self; + #[inline] + fn mul(self, rhs: Self) -> Self { + unsafe { + let a = self.to_vec(); + let b = rhs.to_vec(); + + let prod_evn = _mm512_mul_epu32(a, b); + let a_odd = movehdup_epi32_512(a); + let b_odd = movehdup_epi32_512(b); + let prod_odd = _mm512_mul_epu32(a_odd, b_odd); + + let mask = _mm512_set1_epi64(Self::MASK_U64 as i64); + let c_vec = _mm512_set1_epi64(Self::C as i64); + let shift = _mm_set_epi64x(0, Self::BITS as i64); + + // Fold 1 + let evn_lo = _mm512_and_si512(prod_evn, mask); + let evn_hi = _mm512_srl_epi64(prod_evn, shift); + let evn_f1 = _mm512_add_epi64(evn_lo, _mm512_mul_epu32(evn_hi, c_vec)); + + let odd_lo = _mm512_and_si512(prod_odd, mask); + let odd_hi = _mm512_srl_epi64(prod_odd, shift); + let odd_f1 = _mm512_add_epi64(odd_lo, _mm512_mul_epu32(odd_hi, c_vec)); + + // Fold 2 + let evn_f1_lo = _mm512_and_si512(evn_f1, mask); + let evn_f1_hi = _mm512_srl_epi64(evn_f1, shift); + let evn_f2 = _mm512_add_epi64(evn_f1_lo, _mm512_mul_epu32(evn_f1_hi, c_vec)); + + let odd_f1_lo = _mm512_and_si512(odd_f1, mask); + let odd_f1_hi = _mm512_srl_epi64(odd_f1, shift); + let odd_f2 = _mm512_add_epi64(odd_f1_lo, _mm512_mul_epu32(odd_f1_hi, c_vec)); + + // Recombine even/odd + let odd_shifted = _mm512_slli_epi64::<32>(odd_f2); + let combined = _mm512_mask_blend_epi32(0b1010101010101010, evn_f2, odd_shifted); + + // Conditional subtract P + let p_vec = _mm512_set1_epi32(P as i32); + let reduced = _mm512_sub_epi32(combined, p_vec); + Self::from_vec(_mm512_min_epu32(combined, reduced)) + } + } +} + +impl PackedValue for PackedFp32Avx512

{ + type Value = Fp32

; + const WIDTH: usize = FP32_WIDTH; + + #[inline] + fn from_fn(mut f: F) -> Self + where + F: FnMut(usize) -> Self::Value, + { + Self([ + f(0), + f(1), + f(2), + f(3), + f(4), + f(5), + f(6), + f(7), + f(8), + f(9), + f(10), + f(11), + f(12), + f(13), + f(14), + f(15), + ]) + } + + #[inline] + fn extract(&self, lane: usize) -> Self::Value { + debug_assert!(lane < FP32_WIDTH); + self.0[lane] + } +} + +impl PackedField for PackedFp32Avx512

{ + type Scalar = Fp32

; + + #[inline] + fn broadcast(value: Self::Scalar) -> Self { + Self([value; FP32_WIDTH]) + } +} + +/// Number of `Fp64` lanes in an AVX-512 packed vector. +pub const FP64_WIDTH: usize = 8; + +/// AVX-512 packed arithmetic for `Fp64

`, processing 8 lanes. +#[derive(Clone, Copy)] +#[repr(transparent)] +pub struct PackedFp64Avx512(pub [Fp64

; FP64_WIDTH]); + +impl PackedFp64Avx512

{ + const BITS: u32 = 64 - P.leading_zeros(); + + const C_LO: u64 = { + let c = if Self::BITS == 64 { + 0u64.wrapping_sub(P) + } else { + (1u64 << Self::BITS) - P + }; + assert!(P != 0, "modulus must be nonzero"); + assert!(P & 1 == 1, "modulus must be odd"); + c + }; + + const MASK64: u64 = if Self::BITS < 64 { + (1u64 << Self::BITS) - 1 + } else { + u64::MAX + }; + + #[inline(always)] + fn to_vec(self) -> __m512i { + unsafe { transmute(self) } + } + + #[inline(always)] + unsafe fn from_vec(v: __m512i) -> Self { + unsafe { transmute(v) } + } + + /// Vectorized 128-bit Solinas reduction for p = 2^BITS - C. + /// Given (hi, lo) = 128-bit product, computes result ≡ (hi*2^64 + lo) mod p. + #[inline] + unsafe fn reduce128_vec(hi: __m512i, lo: __m512i) -> __m512i { + if Self::BITS < 64 { + Self::reduce128_small_k(hi, lo) + } else { + Self::reduce128_full_k(hi, lo) + } + } + + /// Reduction for BITS < 64 (e.g. 40-bit prime). No overflow issues: all + /// intermediates fit in u64. + #[inline] + unsafe fn reduce128_small_k(hi: __m512i, lo: __m512i) -> __m512i { + let mask_k = _mm512_set1_epi64(Self::MASK64 as i64); + let c_vec = _mm512_set1_epi64(Self::C_LO as i64); + let p_vec = _mm512_set1_epi64(P as i64); + let shift_k = _mm_set_epi64x(0, Self::BITS as i64); + let shift_64mk = _mm_set_epi64x(0, (64 - Self::BITS) as i64); + + let lo_k = _mm512_and_si512(lo, mask_k); + let lo_upper = _mm512_srl_epi64(lo, shift_k); + let hi_shifted = _mm512_sll_epi64(hi, shift_64mk); + let hi_k = _mm512_or_si512(lo_upper, hi_shifted); + + // c * hi_k: hi_k may exceed 32 bits, split into lo32 and top + let c_hi_lo = _mm512_mul_epu32(c_vec, hi_k); + let hi_k_top = _mm512_srli_epi64::<32>(hi_k); + let c_hi_top = _mm512_mul_epu32(c_vec, hi_k_top); + let c_hi_top_shifted = _mm512_slli_epi64::<32>(c_hi_top); + let c_hi_full = _mm512_add_epi64(c_hi_lo, c_hi_top_shifted); + + let fold1 = _mm512_add_epi64(lo_k, c_hi_full); + + let fold1_lo_k = _mm512_and_si512(fold1, mask_k); + let fold1_hi = _mm512_srl_epi64(fold1, shift_k); + let c_fold1_hi = _mm512_mul_epu32(c_vec, fold1_hi); + let fold2 = _mm512_add_epi64(fold1_lo_k, c_fold1_hi); + + let reduced = _mm512_sub_epi64(fold2, p_vec); + _mm512_min_epu64(fold2, reduced) + } + + /// Reduction for BITS == 64 (e.g. p = 2^64 - 87). Tracks overflow from + /// c*hi exceeding 64 bits, using native unsigned comparisons. + #[inline] + unsafe fn reduce128_full_k(hi: __m512i, lo: __m512i) -> __m512i { + let c_vec = _mm512_set1_epi64(Self::C_LO as i64); + let p_vec = _mm512_set1_epi64(P as i64); + let one = _mm512_set1_epi64(1); + + // c * hi_lo32 + let c_hi_lo = _mm512_mul_epu32(c_vec, hi); + // c * hi_hi32 + let hi_hi = _mm512_srli_epi64::<32>(hi); + let c_hi_hi = _mm512_mul_epu32(c_vec, hi_hi); + + let c_hi_hi_lo32 = _mm512_slli_epi64::<32>(c_hi_hi); + let c_hi_carry = _mm512_srli_epi64::<32>(c_hi_hi); + + // Lower 64 bits of c * hi + let sum_lo = _mm512_add_epi64(c_hi_lo, c_hi_hi_lo32); + let carry0 = _mm512_cmplt_epu64_mask(sum_lo, c_hi_lo); + let overflow = _mm512_mask_add_epi64(c_hi_carry, carry0, c_hi_carry, one); + + // lo + sum_lo + let s = _mm512_add_epi64(lo, sum_lo); + let carry1 = _mm512_cmplt_epu64_mask(s, lo); + let total_overflow = _mm512_mask_add_epi64(overflow, carry1, overflow, one); + + // Fold overflow: total_overflow * c (at most ~2^15) + let final_corr = _mm512_mul_epu32(c_vec, total_overflow); + let result = _mm512_add_epi64(s, final_corr); + let carry_f = _mm512_cmplt_epu64_mask(result, s); + let result = _mm512_mask_add_epi64(result, carry_f, result, c_vec); + + let ge_mask = _mm512_cmpge_epu64_mask(result, p_vec); + _mm512_mask_sub_epi64(result, ge_mask, result, p_vec) + } +} + +impl Default for PackedFp64Avx512

{ + #[inline] + fn default() -> Self { + Self([Fp64(0); FP64_WIDTH]) + } +} + +impl fmt::Debug for PackedFp64Avx512

{ + fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result { + f.debug_tuple("PackedFp64Avx512").field(&self.0).finish() + } +} + +impl PartialEq for PackedFp64Avx512

{ + #[inline] + fn eq(&self, other: &Self) -> bool { + self.0 == other.0 + } +} + +impl Eq for PackedFp64Avx512

{} + +impl Add for PackedFp64Avx512

{ + type Output = Self; + #[inline] + fn add(self, rhs: Self) -> Self { + unsafe { + let a = self.to_vec(); + let b = rhs.to_vec(); + let p = _mm512_set1_epi64(P as i64); + + let result = if Self::BITS <= 62 { + let s = _mm512_add_epi64(a, b); + let geq_p = _mm512_cmpge_epu64_mask(s, p); + _mm512_mask_sub_epi64(s, geq_p, s, p) + } else { + let s = _mm512_add_epi64(a, b); + let overflow = _mm512_cmplt_epu64_mask(s, a); + let c = _mm512_set1_epi64(Self::C_LO as i64); + let geq_p = _mm512_cmpge_epu64_mask(s, p); + let no_of = _mm512_mask_sub_epi64(s, geq_p, s, p); + let s_plus_c = _mm512_add_epi64(s, c); + _mm512_mask_blend_epi64(overflow, no_of, s_plus_c) + }; + + Self::from_vec(result) + } + } +} + +impl Sub for PackedFp64Avx512

{ + type Output = Self; + #[inline] + fn sub(self, rhs: Self) -> Self { + unsafe { + let a = self.to_vec(); + let b = rhs.to_vec(); + let p = _mm512_set1_epi64(P as i64); + let d = _mm512_sub_epi64(a, b); + let underflow = _mm512_cmplt_epu64_mask(a, b); + Self::from_vec(_mm512_mask_add_epi64(d, underflow, d, p)) + } + } +} + +impl Mul for PackedFp64Avx512

{ + type Output = Self; + #[inline] + fn mul(self, rhs: Self) -> Self { + unsafe { + let (hi, lo) = mul64_64_512(self.to_vec(), rhs.to_vec()); + Self::from_vec(Self::reduce128_vec(hi, lo)) + } + } +} + +impl PackedValue for PackedFp64Avx512

{ + type Value = Fp64

; + const WIDTH: usize = FP64_WIDTH; + + #[inline] + fn from_fn(mut f: F) -> Self + where + F: FnMut(usize) -> Self::Value, + { + Self([f(0), f(1), f(2), f(3), f(4), f(5), f(6), f(7)]) + } + + #[inline] + fn extract(&self, lane: usize) -> Self::Value { + debug_assert!(lane < FP64_WIDTH); + self.0[lane] + } +} + +impl PackedField for PackedFp64Avx512

{ + type Scalar = Fp64

; + + #[inline] + fn broadcast(value: Self::Scalar) -> Self { + Self([value; FP64_WIDTH]) + } +} + +/// Number of `Fp128` lanes in an AVX-512 packed vector. +pub const FP128_WIDTH: usize = 8; + +/// AVX-512 packed arithmetic for `Fp128

`, 8 lanes in SoA layout. +/// +/// Stores 8 elements as separate `lo` and `hi` `u64` arrays, enabling +/// vectorized add/sub via `__m512i`. Mul remains scalar per-lane. +#[derive(Clone, Copy)] +pub struct PackedFp128Avx512 { + lo: [u64; FP128_WIDTH], + hi: [u64; FP128_WIDTH], +} + +impl PackedFp128Avx512

{ + const P_LO: u64 = P as u64; + const P_HI: u64 = (P >> 64) as u64; +} + +impl Default for PackedFp128Avx512

{ + #[inline] + fn default() -> Self { + Self { + lo: [0; FP128_WIDTH], + hi: [0; FP128_WIDTH], + } + } +} + +impl fmt::Debug for PackedFp128Avx512

{ + fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result { + let elems: Vec<_> = (0..FP128_WIDTH).map(|i| self.extract(i)).collect(); + f.debug_tuple("PackedFp128Avx512").field(&elems).finish() + } +} + +impl PartialEq for PackedFp128Avx512

{ + #[inline] + fn eq(&self, other: &Self) -> bool { + self.lo == other.lo && self.hi == other.hi + } +} + +impl Eq for PackedFp128Avx512

{} + +impl Add for PackedFp128Avx512

{ + type Output = Self; + #[inline] + fn add(self, rhs: Self) -> Self { + unsafe { + let a_lo = _mm512_loadu_si512(self.lo.as_ptr().cast()); + let a_hi = _mm512_loadu_si512(self.hi.as_ptr().cast()); + let b_lo = _mm512_loadu_si512(rhs.lo.as_ptr().cast()); + let b_hi = _mm512_loadu_si512(rhs.hi.as_ptr().cast()); + let p_lo = _mm512_set1_epi64(Self::P_LO as i64); + let p_hi = _mm512_set1_epi64(Self::P_HI as i64); + let one = _mm512_set1_epi64(1); + + // 128-bit add: (sum_hi, sum_lo) = (a_hi, a_lo) + (b_hi, b_lo) + let sum_lo = _mm512_add_epi64(a_lo, b_lo); + let carry_lo = _mm512_cmplt_epu64_mask(sum_lo, a_lo); + let hi_tmp = _mm512_add_epi64(a_hi, b_hi); + let ov1 = _mm512_cmplt_epu64_mask(hi_tmp, a_hi); + let sum_hi = _mm512_mask_add_epi64(hi_tmp, carry_lo, hi_tmp, one); + let ov2 = _mm512_cmplt_epu64_mask(sum_hi, hi_tmp); + let carry_128 = ov1 | ov2; + + // 128-bit subtract P: (red_hi, red_lo) = (sum_hi, sum_lo) - P + let red_lo = _mm512_sub_epi64(sum_lo, p_lo); + let borrow_lo = _mm512_cmplt_epu64_mask(sum_lo, p_lo); + let red_hi_tmp = _mm512_sub_epi64(sum_hi, p_hi); + let bw1 = _mm512_cmplt_epu64_mask(sum_hi, p_hi); + let red_hi = _mm512_mask_sub_epi64(red_hi_tmp, borrow_lo, red_hi_tmp, one); + let bw2 = _mm512_cmplt_epu64_mask(red_hi_tmp, _mm512_maskz_mov_epi64(borrow_lo, one)); + let borrow = bw1 | bw2; + + // Use reduced if: overflow happened OR subtraction didn't borrow + let use_reduced = carry_128 | !borrow; + let out_lo = _mm512_mask_blend_epi64(use_reduced, sum_lo, red_lo); + let out_hi = _mm512_mask_blend_epi64(use_reduced, sum_hi, red_hi); + + let mut result = Self::default(); + _mm512_storeu_si512(result.lo.as_mut_ptr().cast(), out_lo); + _mm512_storeu_si512(result.hi.as_mut_ptr().cast(), out_hi); + result + } + } +} + +impl Sub for PackedFp128Avx512

{ + type Output = Self; + #[inline] + fn sub(self, rhs: Self) -> Self { + unsafe { + let a_lo = _mm512_loadu_si512(self.lo.as_ptr().cast()); + let a_hi = _mm512_loadu_si512(self.hi.as_ptr().cast()); + let b_lo = _mm512_loadu_si512(rhs.lo.as_ptr().cast()); + let b_hi = _mm512_loadu_si512(rhs.hi.as_ptr().cast()); + let p_lo = _mm512_set1_epi64(Self::P_LO as i64); + let p_hi = _mm512_set1_epi64(Self::P_HI as i64); + let one = _mm512_set1_epi64(1); + + // 128-bit sub: (diff_hi, diff_lo) = (a_hi, a_lo) - (b_hi, b_lo) + let diff_lo = _mm512_sub_epi64(a_lo, b_lo); + let borrow_lo = _mm512_cmplt_epu64_mask(a_lo, b_lo); + let hi_tmp = _mm512_sub_epi64(a_hi, b_hi); + let bw1 = _mm512_cmplt_epu64_mask(a_hi, b_hi); + let diff_hi = _mm512_mask_sub_epi64(hi_tmp, borrow_lo, hi_tmp, one); + let bw2 = _mm512_cmplt_epu64_mask(hi_tmp, _mm512_maskz_mov_epi64(borrow_lo, one)); + let borrow_128 = bw1 | bw2; + + // Correction: add P back where underflow occurred + let corr_lo = _mm512_add_epi64(diff_lo, p_lo); + let carry_lo = _mm512_cmplt_epu64_mask(corr_lo, diff_lo); + let corr_hi = _mm512_add_epi64(diff_hi, p_hi); + let corr_hi = _mm512_mask_add_epi64(corr_hi, carry_lo, corr_hi, one); + + let out_lo = _mm512_mask_blend_epi64(borrow_128, diff_lo, corr_lo); + let out_hi = _mm512_mask_blend_epi64(borrow_128, diff_hi, corr_hi); + + let mut result = Self::default(); + _mm512_storeu_si512(result.lo.as_mut_ptr().cast(), out_lo); + _mm512_storeu_si512(result.hi.as_mut_ptr().cast(), out_hi); + result + } + } +} + +impl Mul for PackedFp128Avx512

{ + type Output = Self; + #[inline] + fn mul(self, rhs: Self) -> Self { + let mut out = Self::default(); + for i in 0..FP128_WIDTH { + let a = Fp128::

([self.lo[i], self.hi[i]]); + let b = Fp128::

([rhs.lo[i], rhs.hi[i]]); + let r = a * b; + out.lo[i] = r.0[0]; + out.hi[i] = r.0[1]; + } + out + } +} + +impl PackedValue for PackedFp128Avx512

{ + type Value = Fp128

; + const WIDTH: usize = FP128_WIDTH; + + #[inline] + fn from_fn(mut f: F) -> Self + where + F: FnMut(usize) -> Self::Value, + { + let mut lo = [0u64; FP128_WIDTH]; + let mut hi = [0u64; FP128_WIDTH]; + for i in 0..FP128_WIDTH { + let v = f(i); + lo[i] = v.0[0]; + hi[i] = v.0[1]; + } + Self { lo, hi } + } + + #[inline] + fn extract(&self, lane: usize) -> Self::Value { + debug_assert!(lane < FP128_WIDTH); + Fp128([self.lo[lane], self.hi[lane]]) + } +} + +impl PackedField for PackedFp128Avx512

{ + type Scalar = Fp128

; + + #[inline] + fn broadcast(value: Self::Scalar) -> Self { + Self { + lo: [value.0[0]; FP128_WIDTH], + hi: [value.0[1]; FP128_WIDTH], + } + } +} diff --git a/src/algebra/fields/packed_neon.rs b/src/algebra/fields/packed_neon.rs new file mode 100644 index 00000000..aeeea0ef --- /dev/null +++ b/src/algebra/fields/packed_neon.rs @@ -0,0 +1,731 @@ +//! AArch64 NEON packed backends for Fp32, Fp64, Fp128. + +use super::packed::{PackedField, PackedValue}; +use crate::algebra::fields::{Fp128, Fp32, Fp64}; +use crate::FieldCore; +use core::arch::aarch64::{ + uint32x2_t, uint32x4_t, uint64x2_t, vaddq_u32, vaddq_u64, vandq_u64, vbslq_u32, vbslq_u64, + vcgtq_u64, vcltq_u32, vcltq_u64, vcombine_u32, vdup_n_u32, vdupq_n_s64, vdupq_n_u32, + vdupq_n_u64, veorq_u64, vget_low_u32, vminq_u32, vmovn_u64, vmull_high_u32, vmull_u32, + vorrq_u64, vshlq_u64, vsubq_u32, vsubq_u64, +}; +use core::fmt; +use core::mem::transmute; +use core::ops::{Add, Mul, Sub}; + +/// Number of packed `Fp128` lanes in this backend. +pub const WIDTH: usize = 2; + +/// True SoA layout for two packed `Fp128` lanes. +/// +/// `lo = [lane0.lo, lane1.lo]` +/// `hi = [lane0.hi, lane1.hi]` +#[derive(Clone, Copy)] +pub struct PackedFp128Neon { + lo: [u64; 2], + hi: [u64; 2], +} + +#[inline(always)] +fn to_vec(x: [u64; 2]) -> uint64x2_t { + unsafe { transmute::<[u64; 2], uint64x2_t>(x) } +} + +#[inline(always)] +fn from_vec(v: uint64x2_t) -> [u64; 2] { + unsafe { transmute::(v) } +} + +#[inline(always)] +fn mask_to_bit(mask: uint64x2_t) -> uint64x2_t { + // SAFETY: NEON intrinsics are available under this cfg. + unsafe { vandq_u64(mask, vdupq_n_u64(1)) } +} + +#[inline(always)] +const fn modulus_lo() -> u64 { + P as u64 +} + +#[inline(always)] +const fn modulus_hi() -> u64 { + (P >> 64) as u64 +} + +#[inline(always)] +const fn is_pow2_u64(x: u64) -> bool { + x != 0 && (x & (x - 1)) == 0 +} + +#[inline(always)] +const fn log2_pow2_u64(mut x: u64) -> u32 { + let mut k = 0u32; + while x > 1 { + x >>= 1; + k += 1; + } + k +} + +impl PackedFp128Neon

{ + const C: u128 = { + let c = 0u128.wrapping_sub(P); + assert!(P != 0, "modulus must be nonzero"); + assert!(P & 1 == 1, "modulus must be odd"); + assert!(c < (1u128 << 64), "P must be 2^128 - c with c < 2^64"); + assert!( + c * (c + 1) < P, + "C(C+1) < P required for fused canonicalize" + ); + c + }; + const C_LO: u64 = Self::C as u64; + const C_SHIFT_KIND: i8 = { + let c = Self::C_LO; + if c > 1 && is_pow2_u64(c - 1) { + 1 + } else if c == u64::MAX || is_pow2_u64(c + 1) { + -1 + } else { + 0 + } + }; + const C_SHIFT: u32 = { + let c = Self::C_LO; + if Self::C_SHIFT_KIND == 1 { + log2_pow2_u64(c - 1) + } else if Self::C_SHIFT_KIND == -1 { + if c == u64::MAX { + 64 + } else { + log2_pow2_u64(c + 1) + } + } else { + 0 + } + }; + + #[inline(always)] + fn mul_wide_u64(a: u64, b: u64) -> (u64, u64) { + let prod = (a as u128) * (b as u128); + (prod as u64, (prod >> 64) as u64) + } + + #[inline(always)] + fn mul_c_wide(x: u64) -> (u64, u64) { + if Self::C_SHIFT_KIND == 1 { + let v = ((x as u128) << Self::C_SHIFT) + x as u128; + (v as u64, (v >> 64) as u64) + } else if Self::C_SHIFT_KIND == -1 { + let v = ((x as u128) << Self::C_SHIFT) - x as u128; + (v as u64, (v >> 64) as u64) + } else { + Self::mul_wide_u64(Self::C_LO, x) + } + } + + #[inline(always)] + fn fold2_canonicalize(t0: u64, t1: u64, t2: u64) -> (u64, u64) { + let (ct2_lo, ct2_hi) = Self::mul_c_wide(t2); + + let (s0, carry0) = t0.overflowing_add(ct2_lo); + let (s1a, carry1a) = t1.overflowing_add(ct2_hi); + let (s1, carry1b) = s1a.overflowing_add(carry0 as u64); + let overflow = carry1a | carry1b; + + let (r0, carry2) = s0.overflowing_add(Self::C_LO); + let (r1, carry3) = s1.overflowing_add(carry2 as u64); + + if overflow | carry3 { + (r0, r1) + } else { + (s0, s1) + } + } + + #[inline(always)] + fn mul_raw_lane(a0: u64, a1: u64, b0: u64, b1: u64) -> (u64, u64) { + let (p00_lo, p00_hi) = Self::mul_wide_u64(a0, b0); + let (p01_lo, p01_hi) = Self::mul_wide_u64(a0, b1); + let (p10_lo, p10_hi) = Self::mul_wide_u64(a1, b0); + let (p11_lo, p11_hi) = Self::mul_wide_u64(a1, b1); + + let row1 = p00_hi as u128 + p01_lo as u128 + p10_lo as u128; + let r0 = p00_lo; + let r1 = row1 as u64; + let carry1 = (row1 >> 64) as u64; + + let row2 = p01_hi as u128 + p10_hi as u128 + p11_lo as u128 + carry1 as u128; + let r2 = row2 as u64; + let carry2 = (row2 >> 64) as u64; + + let row3 = p11_hi as u128 + carry2 as u128; + let r3 = row3 as u64; + debug_assert_eq!(row3 >> 64, 0); + + let (cr2_lo, cr2_hi) = Self::mul_c_wide(r2); + let (cr3_lo, cr3_hi) = Self::mul_c_wide(r3); + + let t0_sum = r0 as u128 + cr2_lo as u128; + let t0 = t0_sum as u64; + let carryf = (t0_sum >> 64) as u64; + + let t1_sum = r1 as u128 + cr2_hi as u128 + cr3_lo as u128 + carryf as u128; + let t1 = t1_sum as u64; + + let t2_sum = cr3_hi as u128 + (t1_sum >> 64); + let t2 = t2_sum as u64; + debug_assert_eq!(t2_sum >> 64, 0); + + Self::fold2_canonicalize(t0, t1, t2) + } +} + +impl Default for PackedFp128Neon

{ + #[inline] + fn default() -> Self { + Self::broadcast(Fp128::zero()) + } +} + +impl fmt::Debug for PackedFp128Neon

{ + fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result { + f.debug_tuple("PackedFp128Neon") + .field(&[self.extract(0), self.extract(1)]) + .finish() + } +} + +impl PartialEq for PackedFp128Neon

{ + #[inline] + fn eq(&self, other: &Self) -> bool { + self.extract(0) == other.extract(0) && self.extract(1) == other.extract(1) + } +} + +impl Eq for PackedFp128Neon

{} + +impl PackedValue for PackedFp128Neon

{ + type Value = Fp128

; + const WIDTH: usize = WIDTH; + + #[inline] + fn from_fn(mut f: F) -> Self + where + F: FnMut(usize) -> Self::Value, + { + let x0 = f(0); + let x1 = f(1); + Self { + lo: [x0.0[0], x1.0[0]], + hi: [x0.0[1], x1.0[1]], + } + } + + #[inline] + fn extract(&self, lane: usize) -> Self::Value { + debug_assert!(lane < WIDTH); + Fp128([self.lo[lane], self.hi[lane]]) + } +} + +impl Add for PackedFp128Neon

{ + type Output = Self; + #[inline] + fn add(self, rhs: Self) -> Self { + let lo_a = to_vec(self.lo); + let hi_a = to_vec(self.hi); + let lo_b = to_vec(rhs.lo); + let hi_b = to_vec(rhs.hi); + + let (out_lo, out_hi) = unsafe { + let p_lo = vdupq_n_u64(modulus_lo::

()); + let p_hi = vdupq_n_u64(modulus_hi::

()); + + let sum_lo = vaddq_u64(lo_a, lo_b); + let carry_lo = mask_to_bit(vcltq_u64(sum_lo, lo_a)); + + let hi_tmp = vaddq_u64(hi_a, hi_b); + let carry_hi1 = vcltq_u64(hi_tmp, hi_a); + let sum_hi = vaddq_u64(hi_tmp, carry_lo); + let carry_hi2 = vcltq_u64(sum_hi, hi_tmp); + let carry_128 = vorrq_u64(carry_hi1, carry_hi2); + + let red_lo = vsubq_u64(sum_lo, p_lo); + let borrow_lo = mask_to_bit(vcgtq_u64(p_lo, sum_lo)); + + let red_hi_tmp = vsubq_u64(sum_hi, p_hi); + let borrow_hi1 = vcgtq_u64(p_hi, sum_hi); + let red_hi = vsubq_u64(red_hi_tmp, borrow_lo); + let borrow_hi2 = vcltq_u64(red_hi_tmp, borrow_lo); + let borrow = vorrq_u64(borrow_hi1, borrow_hi2); + + let not_borrow = veorq_u64(borrow, vdupq_n_u64(u64::MAX)); + let use_reduced = vorrq_u64(carry_128, not_borrow); + let out_lo = vbslq_u64(use_reduced, red_lo, sum_lo); + let out_hi = vbslq_u64(use_reduced, red_hi, sum_hi); + (out_lo, out_hi) + }; + + Self { + lo: from_vec(out_lo), + hi: from_vec(out_hi), + } + } +} + +impl Sub for PackedFp128Neon

{ + type Output = Self; + #[inline] + fn sub(self, rhs: Self) -> Self { + let lo_a = to_vec(self.lo); + let hi_a = to_vec(self.hi); + let lo_b = to_vec(rhs.lo); + let hi_b = to_vec(rhs.hi); + + let (out_lo, out_hi) = unsafe { + let p_lo = vdupq_n_u64(modulus_lo::

()); + let p_hi = vdupq_n_u64(modulus_hi::

()); + + let diff_lo = vsubq_u64(lo_a, lo_b); + let borrow_lo = mask_to_bit(vcltq_u64(lo_a, lo_b)); + + let diff_hi_tmp = vsubq_u64(hi_a, hi_b); + let borrow_hi1 = vcltq_u64(hi_a, hi_b); + let diff_hi = vsubq_u64(diff_hi_tmp, borrow_lo); + let borrow_hi2 = vcltq_u64(diff_hi_tmp, borrow_lo); + let borrow_128 = vorrq_u64(borrow_hi1, borrow_hi2); + + let corr_lo = vaddq_u64(diff_lo, p_lo); + let carry_lo = mask_to_bit(vcltq_u64(corr_lo, diff_lo)); + + let corr_hi_tmp = vaddq_u64(diff_hi, p_hi); + let corr_hi = vaddq_u64(corr_hi_tmp, carry_lo); + + let out_lo = vbslq_u64(borrow_128, corr_lo, diff_lo); + let out_hi = vbslq_u64(borrow_128, corr_hi, diff_hi); + (out_lo, out_hi) + }; + + Self { + lo: from_vec(out_lo), + hi: from_vec(out_hi), + } + } +} + +impl Mul for PackedFp128Neon

{ + type Output = Self; + #[inline] + fn mul(self, rhs: Self) -> Self { + let (o0_lo, o0_hi) = Self::mul_raw_lane(self.lo[0], self.hi[0], rhs.lo[0], rhs.hi[0]); + let (o1_lo, o1_hi) = Self::mul_raw_lane(self.lo[1], self.hi[1], rhs.lo[1], rhs.hi[1]); + + Self { + lo: [o0_lo, o1_lo], + hi: [o0_hi, o1_hi], + } + } +} + +impl PackedField for PackedFp128Neon

{ + type Scalar = Fp128

; + + #[inline] + fn broadcast(value: Self::Scalar) -> Self { + Self::from_fn(|_| value) + } +} + +/// Number of packed `Fp32` lanes. +pub const FP32_WIDTH: usize = 4; + +/// NEON packed `Fp32` backend: 4 lanes in `uint32x4_t`. +#[derive(Clone, Copy)] +pub struct PackedFp32Neon { + vals: [u32; 4], +} + +#[inline(always)] +fn to_vec32(x: [u32; 4]) -> uint32x4_t { + unsafe { transmute::<[u32; 4], uint32x4_t>(x) } +} + +#[inline(always)] +fn from_vec32(v: uint32x4_t) -> [u32; 4] { + unsafe { transmute::(v) } +} + +impl PackedFp32Neon

{ + const BITS: u32 = 32 - P.leading_zeros(); + + const C: u32 = { + let c = if Self::BITS == 32 { + 0u32.wrapping_sub(P) + } else { + (1u32 << Self::BITS) - P + }; + assert!(P != 0, "modulus must be nonzero"); + assert!(P & 1 == 1, "modulus must be odd"); + assert!( + (c as u64) * (c as u64 + 1) < P as u64, + "C(C+1) < P required for fused canonicalize" + ); + c + }; + + const MASK_U64: u64 = if Self::BITS == 32 { + u32::MAX as u64 + } else { + (1u64 << Self::BITS) - 1 + }; + + #[inline(always)] + fn mul_c_u64(hi: uint64x2_t, c: uint32x2_t) -> uint64x2_t { + unsafe { + let hi_narrow = vmovn_u64(hi); + vmull_u32(hi_narrow, c) + } + } + + #[inline(always)] + fn solinas_reduce(prod_lo: uint64x2_t, prod_hi: uint64x2_t) -> uint32x4_t { + unsafe { + let mask = vdupq_n_u64(Self::MASK_U64); + let neg_bits = vdupq_n_s64(-(Self::BITS as i64)); + let c = vdup_n_u32(Self::C); + + let f1_lo = vaddq_u64( + vandq_u64(prod_lo, mask), + Self::mul_c_u64(vshlq_u64(prod_lo, neg_bits), c), + ); + let f1_hi = vaddq_u64( + vandq_u64(prod_hi, mask), + Self::mul_c_u64(vshlq_u64(prod_hi, neg_bits), c), + ); + + let f2_lo = vaddq_u64( + vandq_u64(f1_lo, mask), + Self::mul_c_u64(vshlq_u64(f1_lo, neg_bits), c), + ); + let f2_hi = vaddq_u64( + vandq_u64(f1_hi, mask), + Self::mul_c_u64(vshlq_u64(f1_hi, neg_bits), c), + ); + + if Self::BITS < 32 { + let result = vcombine_u32(vmovn_u64(f2_lo), vmovn_u64(f2_hi)); + let p = vdupq_n_u32(P); + vminq_u32(result, vsubq_u32(result, p)) + } else { + let p_u64 = vdupq_n_u64(P as u64); + + let red_lo = vsubq_u64(f2_lo, p_u64); + let keep_lo = vcltq_u64(f2_lo, p_u64); + let out_lo = vbslq_u64(keep_lo, f2_lo, red_lo); + + let red_hi = vsubq_u64(f2_hi, p_u64); + let keep_hi = vcltq_u64(f2_hi, p_u64); + let out_hi = vbslq_u64(keep_hi, f2_hi, red_hi); + + vcombine_u32(vmovn_u64(out_lo), vmovn_u64(out_hi)) + } + } + } +} + +impl Default for PackedFp32Neon

{ + #[inline] + fn default() -> Self { + Self { vals: [0; 4] } + } +} + +impl fmt::Debug for PackedFp32Neon

{ + fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result { + f.debug_tuple("PackedFp32Neon").field(&self.vals).finish() + } +} + +impl PartialEq for PackedFp32Neon

{ + #[inline] + fn eq(&self, other: &Self) -> bool { + self.vals == other.vals + } +} + +impl Eq for PackedFp32Neon

{} + +impl Add for PackedFp32Neon

{ + type Output = Self; + #[inline] + fn add(self, rhs: Self) -> Self { + let a = to_vec32(self.vals); + let b = to_vec32(rhs.vals); + let result = unsafe { + let p = vdupq_n_u32(P); + if Self::BITS <= 31 { + let t = vaddq_u32(a, b); + vminq_u32(t, vsubq_u32(t, p)) + } else { + let c = vdupq_n_u32(Self::C); + let t = vaddq_u32(a, b); + let overflow = vcltq_u32(t, a); + let t_plus_c = vaddq_u32(t, c); + let no_of = vminq_u32(t, vsubq_u32(t, p)); + vbslq_u32(overflow, t_plus_c, no_of) + } + }; + Self { + vals: from_vec32(result), + } + } +} + +impl Sub for PackedFp32Neon

{ + type Output = Self; + #[inline] + fn sub(self, rhs: Self) -> Self { + let a = to_vec32(self.vals); + let b = to_vec32(rhs.vals); + let result = unsafe { + let p = vdupq_n_u32(P); + if Self::BITS <= 31 { + let t = vsubq_u32(a, b); + vminq_u32(t, vaddq_u32(t, p)) + } else { + let t = vsubq_u32(a, b); + let underflow = vcltq_u32(a, b); + vbslq_u32(underflow, vaddq_u32(t, p), t) + } + }; + Self { + vals: from_vec32(result), + } + } +} + +impl Mul for PackedFp32Neon

{ + type Output = Self; + #[inline] + fn mul(self, rhs: Self) -> Self { + let a = to_vec32(self.vals); + let b = to_vec32(rhs.vals); + let result = unsafe { + let prod_lo = vmull_u32(vget_low_u32(a), vget_low_u32(b)); + let prod_hi = vmull_high_u32(a, b); + Self::solinas_reduce(prod_lo, prod_hi) + }; + Self { + vals: from_vec32(result), + } + } +} + +impl PackedValue for PackedFp32Neon

{ + type Value = Fp32

; + const WIDTH: usize = FP32_WIDTH; + + #[inline] + fn from_fn(mut f: F) -> Self + where + F: FnMut(usize) -> Self::Value, + { + Self { + vals: [f(0).0, f(1).0, f(2).0, f(3).0], + } + } + + #[inline] + fn extract(&self, lane: usize) -> Self::Value { + debug_assert!(lane < FP32_WIDTH); + Fp32(self.vals[lane]) + } +} + +impl PackedField for PackedFp32Neon

{ + type Scalar = Fp32

; + + #[inline] + fn broadcast(value: Self::Scalar) -> Self { + Self { vals: [value.0; 4] } + } +} + +/// Number of packed `Fp64` lanes. +pub const FP64_WIDTH: usize = 2; + +/// NEON packed `Fp64` backend: 2 lanes in `uint64x2_t`. +#[derive(Clone, Copy)] +pub struct PackedFp64Neon { + vals: [u64; 2], +} + +impl PackedFp64Neon

{ + const BITS: u32 = 64 - P.leading_zeros(); + + const C_LO: u64 = { + let c = if Self::BITS == 64 { + 0u64.wrapping_sub(P) + } else { + (1u64 << Self::BITS) - P + }; + assert!(P != 0, "modulus must be nonzero"); + assert!(P & 1 == 1, "modulus must be odd"); + c + }; + + const MASK64: u64 = if Self::BITS < 64 { + (1u64 << Self::BITS) - 1 + } else { + u64::MAX + }; + + const MASK_U128: u128 = if Self::BITS == 64 { + u64::MAX as u128 + } else { + (1u128 << Self::BITS) - 1 + }; + + const FOLD_IN_U64: bool = + Self::BITS < 64 && (Self::C_LO as u128) < (1u128 << (64 - Self::BITS)); + + #[inline(always)] + fn mul_c_narrow(x: u64) -> u64 { + Self::C_LO.wrapping_mul(x) + } + + #[inline(always)] + fn reduce_product(x: u128) -> u64 { + if Self::FOLD_IN_U64 { + let lo = x as u64; + let hi = (x >> 64) as u64; + let high = (lo >> Self::BITS) | (hi << (64 - Self::BITS)); + let f1 = (lo & Self::MASK64).wrapping_add(Self::mul_c_narrow(high)); + let f2 = (f1 & Self::MASK64).wrapping_add(Self::mul_c_narrow(f1 >> Self::BITS)); + let reduced = f2.wrapping_sub(P); + let borrow = reduced >> 63; + reduced.wrapping_add(borrow.wrapping_neg() & P) + } else { + let f1 = + (x & Self::MASK_U128) + (Self::C_LO as u128) * ((x >> Self::BITS) as u64 as u128); + let f2 = + (f1 & Self::MASK_U128) + (Self::C_LO as u128) * ((f1 >> Self::BITS) as u64 as u128); + let reduced = f2.wrapping_sub(P as u128); + let borrow = reduced >> 127; + reduced.wrapping_add(borrow.wrapping_neg() & (P as u128)) as u64 + } + } +} + +impl Default for PackedFp64Neon

{ + #[inline] + fn default() -> Self { + Self { vals: [0; 2] } + } +} + +impl fmt::Debug for PackedFp64Neon

{ + fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result { + f.debug_tuple("PackedFp64Neon").field(&self.vals).finish() + } +} + +impl PartialEq for PackedFp64Neon

{ + #[inline] + fn eq(&self, other: &Self) -> bool { + self.vals == other.vals + } +} + +impl Eq for PackedFp64Neon

{} + +impl Add for PackedFp64Neon

{ + type Output = Self; + #[inline] + fn add(self, rhs: Self) -> Self { + let a = to_vec(self.vals); + let b = to_vec(rhs.vals); + let result = unsafe { + let p = vdupq_n_u64(P); + if Self::BITS <= 62 { + let s = vaddq_u64(a, b); + let r = vsubq_u64(s, p); + let borrow = vcltq_u64(s, p); + vbslq_u64(borrow, s, r) + } else { + let s = vaddq_u64(a, b); + let overflow = vcltq_u64(s, a); + let c = vdupq_n_u64(Self::C_LO); + let s_plus_c = vaddq_u64(s, c); + let s_minus_p = vsubq_u64(s, p); + let borrow = vcltq_u64(s, p); + let no_of = vbslq_u64(borrow, s, s_minus_p); + vbslq_u64(overflow, s_plus_c, no_of) + } + }; + Self { + vals: from_vec(result), + } + } +} + +impl Sub for PackedFp64Neon

{ + type Output = Self; + #[inline] + fn sub(self, rhs: Self) -> Self { + let a = to_vec(self.vals); + let b = to_vec(rhs.vals); + let result = unsafe { + let p = vdupq_n_u64(P); + let d = vsubq_u64(a, b); + let underflow = vcltq_u64(a, b); + vbslq_u64(underflow, vaddq_u64(d, p), d) + }; + Self { + vals: from_vec(result), + } + } +} + +impl Mul for PackedFp64Neon

{ + type Output = Self; + #[inline] + fn mul(self, rhs: Self) -> Self { + let x0 = (self.vals[0] as u128) * (rhs.vals[0] as u128); + let x1 = (self.vals[1] as u128) * (rhs.vals[1] as u128); + let r0 = Self::reduce_product(x0); + let r1 = Self::reduce_product(x1); + Self { vals: [r0, r1] } + } +} + +impl PackedValue for PackedFp64Neon

{ + type Value = Fp64

; + const WIDTH: usize = FP64_WIDTH; + + #[inline] + fn from_fn(mut f: F) -> Self + where + F: FnMut(usize) -> Self::Value, + { + Self { + vals: [f(0).0, f(1).0], + } + } + + #[inline] + fn extract(&self, lane: usize) -> Self::Value { + debug_assert!(lane < FP64_WIDTH); + Fp64(self.vals[lane]) + } +} + +impl PackedField for PackedFp64Neon

{ + type Scalar = Fp64

; + + #[inline] + fn broadcast(value: Self::Scalar) -> Self { + Self { vals: [value.0; 2] } + } +} diff --git a/src/algebra/fields/pseudo_mersenne.rs b/src/algebra/fields/pseudo_mersenne.rs new file mode 100644 index 00000000..17920899 --- /dev/null +++ b/src/algebra/fields/pseudo_mersenne.rs @@ -0,0 +1,177 @@ +//! `2^k - offset` pseudo-Mersenne registry and field aliases. +//! +//! This module models the specific flavor where each modulus is the smallest +//! prime below `2^k` with `q % 8 == 5`, written as `q = 2^k - offset`. + +use super::{Fp128, Fp32, Fp64}; + +/// Offset table (`q = 2^k - offset[k]`) imported from `labrador/data.py`. +pub const POW2_OFFSET_TABLE: [i16; 256] = [ + -1, -1, -1, 3, 3, 3, 3, 19, 27, 3, 3, 19, 3, 75, 3, 19, 99, 91, 11, 19, 3, 19, 3, 27, 3, 91, + 27, 115, 299, 3, 35, 19, 99, 355, 131, 451, 243, 123, 107, 19, 195, 75, 11, 67, 539, 139, 635, + 115, 59, 123, 27, 139, 395, 315, 131, 67, 27, 195, 27, 99, 107, 259, 171, 259, 59, 115, 203, + 19, 83, 19, 35, 411, 107, 475, 35, 427, 123, 43, 11, 67, 1307, 51, 315, 139, 35, 19, 35, 67, + 299, 99, 75, 315, 83, 51, 3, 211, 147, 595, 51, 115, 99, 99, 483, 339, 395, 139, 1187, 171, 59, + 91, 195, 835, 75, 211, 11, 67, 3, 451, 563, 867, 395, 531, 3, 67, 59, 579, 203, 507, 275, 315, + 27, 315, 347, 99, 603, 795, 243, 339, 203, 187, 27, 171, 1491, 355, 83, 355, 1371, 387, 347, + 99, 3, 195, 539, 171, 243, 499, 195, 19, 155, 91, 75, 1011, 627, 867, 155, 115, 1811, 771, + 1467, 643, 195, 19, 155, 531, 3, 267, 563, 339, 563, 507, 107, 283, 267, 147, 59, 339, 371, + 1411, 363, 819, 11, 19, 915, 123, 75, 915, 459, 75, 627, 459, 75, 1035, 195, 187, 1515, 1219, + 1443, 91, 299, 451, 171, 1099, 99, 3, 395, 1147, 683, 675, 243, 355, 395, 3, 875, 235, 363, + 1131, 155, 835, 723, 91, 27, 235, 875, 3, 83, 259, 875, 1515, 731, 531, 467, 819, 267, 475, + 1923, 163, 107, 411, 387, 75, 2331, 355, 1515, 1723, 1427, 19, +]; + +/// Maximum supported offset in this `2^k - offset` specialization. +pub const POW2_OFFSET_MAX: u128 = 1u128 << 16; + +/// Current active bit-size bound for concrete field aliases in this phase. +pub const POW2_OFFSET_IMPLEMENTED_MAX_BITS: u32 = 128; + +/// Metadata describing a `2^k - offset` pseudo-Mersenne modulus. +#[derive(Debug, Clone, Copy, PartialEq, Eq)] +pub struct Pow2OffsetPrimeSpec { + /// `k` in `2^k - offset`. + pub bits: u32, + /// `offset` in `2^k - offset`. + pub offset: u16, + /// Modulus value. + pub modulus: u128, +} + +/// Return table offset for `q = 2^k - offset` when available and positive. +pub const fn pow2_offset(bits: u32) -> Option { + if bits as usize >= POW2_OFFSET_TABLE.len() { + return None; + } + let offset = POW2_OFFSET_TABLE[bits as usize]; + if offset <= 0 { + None + } else { + Some(offset as u16) + } +} + +/// Compute `2^k - offset` for `k <= 128`. +pub const fn pseudo_mersenne_modulus(bits: u32, offset: u128) -> Option { + if bits == 0 || bits > 128 || offset == 0 { + return None; + } + if bits == 128 { + Some(u128::MAX - (offset - 1)) + } else { + Some((1u128 << bits) - offset) + } +} + +/// Check whether `(k, offset)` is accepted by the `2^k - offset` policy. +pub const fn is_pow2_offset(bits: u32, offset: u128) -> bool { + if bits > POW2_OFFSET_IMPLEMENTED_MAX_BITS || offset > POW2_OFFSET_MAX { + return false; + } + match pow2_offset(bits) { + Some(qoff) => (qoff as u128) == offset, + None => false, + } +} + +/// `offset` for `k = 24`. +pub const POW2_OFFSET_24: u16 = 3; +/// `offset` for `k = 30`. +pub const POW2_OFFSET_30: u16 = 35; +/// `offset` for `k = 31`. +pub const POW2_OFFSET_31: u16 = 19; +/// `offset` for `k = 32`. +pub const POW2_OFFSET_32: u16 = 99; +/// `offset` for `k = 40`. +pub const POW2_OFFSET_40: u16 = 195; +/// `offset` for `k = 48`. +pub const POW2_OFFSET_48: u16 = 59; +/// `offset` for `k = 56`. +pub const POW2_OFFSET_56: u16 = 27; +/// `offset` for `k = 64`. +pub const POW2_OFFSET_64: u16 = 59; +/// `offset` for `k = 128`. +pub const POW2_OFFSET_128: u16 = 275; + +/// `2^24 - 3`. +pub const POW2_OFFSET_MODULUS_24: u32 = ((1u128 << 24) - (POW2_OFFSET_24 as u128)) as u32; +/// `2^30 - 35`. +pub const POW2_OFFSET_MODULUS_30: u32 = ((1u128 << 30) - (POW2_OFFSET_30 as u128)) as u32; +/// `2^31 - 19`. +pub const POW2_OFFSET_MODULUS_31: u32 = ((1u128 << 31) - (POW2_OFFSET_31 as u128)) as u32; +/// `2^32 - 99`. +pub const POW2_OFFSET_MODULUS_32: u32 = ((1u128 << 32) - (POW2_OFFSET_32 as u128)) as u32; +/// `2^40 - 195`. +pub const POW2_OFFSET_MODULUS_40: u64 = ((1u128 << 40) - (POW2_OFFSET_40 as u128)) as u64; +/// `2^48 - 59`. +pub const POW2_OFFSET_MODULUS_48: u64 = ((1u128 << 48) - (POW2_OFFSET_48 as u128)) as u64; +/// `2^56 - 27`. +pub const POW2_OFFSET_MODULUS_56: u64 = ((1u128 << 56) - (POW2_OFFSET_56 as u128)) as u64; +/// `2^64 - 59`. +pub const POW2_OFFSET_MODULUS_64: u64 = u64::MAX - ((POW2_OFFSET_64 as u64) - 1); +/// `2^128 - 275`. +pub const POW2_OFFSET_MODULUS_128: u128 = u128::MAX - (POW2_OFFSET_128 as u128 - 1); + +/// Alias for `2^24 - offset`. +pub type Pow2Offset24Field = Fp32; +/// Alias for `2^30 - offset`. +pub type Pow2Offset30Field = Fp32; +/// Alias for `2^31 - offset`. +pub type Pow2Offset31Field = Fp32; +/// Alias for `2^32 - offset`. +pub type Pow2Offset32Field = Fp32; +/// Alias for `2^40 - offset`. +pub type Pow2Offset40Field = Fp64; +/// Alias for `2^48 - offset`. +pub type Pow2Offset48Field = Fp64; +/// Alias for `2^56 - offset`. +pub type Pow2Offset56Field = Fp64; +/// Alias for `2^64 - offset`. +pub type Pow2Offset64Field = Fp64; +/// Alias for `2^128 - offset`. +pub type Pow2Offset128Field = Fp128; + +/// `2^k - offset` profiles currently enabled in-code. +/// +/// Each listed modulus satisfies `q % 8 == 5`. +pub const POW2_OFFSET_PRIMES: [Pow2OffsetPrimeSpec; 7] = [ + Pow2OffsetPrimeSpec { + bits: 24, + offset: POW2_OFFSET_24, + modulus: POW2_OFFSET_MODULUS_24 as u128, + }, + Pow2OffsetPrimeSpec { + bits: 32, + offset: POW2_OFFSET_32, + modulus: POW2_OFFSET_MODULUS_32 as u128, + }, + Pow2OffsetPrimeSpec { + bits: 40, + offset: POW2_OFFSET_40, + modulus: POW2_OFFSET_MODULUS_40 as u128, + }, + Pow2OffsetPrimeSpec { + bits: 48, + offset: POW2_OFFSET_48, + modulus: POW2_OFFSET_MODULUS_48 as u128, + }, + Pow2OffsetPrimeSpec { + bits: 56, + offset: POW2_OFFSET_56, + modulus: POW2_OFFSET_MODULUS_56 as u128, + }, + Pow2OffsetPrimeSpec { + bits: 64, + offset: POW2_OFFSET_64, + modulus: POW2_OFFSET_MODULUS_64 as u128, + }, + Pow2OffsetPrimeSpec { + bits: 128, + offset: POW2_OFFSET_128, + modulus: POW2_OFFSET_MODULUS_128, + }, +]; + +// All PseudoMersenneField impls for Fp32/Fp64/Fp128 are blanket impls in +// their respective modules (fp32.rs, fp64.rs, fp128.rs). diff --git a/src/algebra/mod.rs b/src/algebra/mod.rs new file mode 100644 index 00000000..8a5cc194 --- /dev/null +++ b/src/algebra/mod.rs @@ -0,0 +1,34 @@ +//! Concrete algebra backends and arithmetic building blocks. +//! +//! This module includes: +//! - Generic prime fields and extensions (`fields`) +//! - Module and polynomial containers (`module`, `poly`) +//! - Low-level NTT and CRT+NTT arithmetic scaffolding (`ntt`) + +pub mod backend; +pub mod domains; +pub mod fields; +pub mod module; +pub mod ntt; +pub mod poly; +pub mod ring; + +// Flat re-exports for convenience. +pub use backend::{CrtReconstruct, NttPrimeOps, NttTransform, RingBackend, ScalarBackend}; +pub use domains::{CoeffDomain, CrtNttDomain}; +pub use fields::{ + is_pow2_offset, pow2_offset, pseudo_mersenne_modulus, Fp128, Fp128Packing, Fp2, Fp2Config, + Fp32, Fp32Packing, Fp4, Fp4Config, Fp64, Fp64Packing, HasPacking, NoPacking, PackedField, + PackedValue, Pow2Offset128Field, Pow2Offset24Field, Pow2Offset30Field, Pow2Offset31Field, + Pow2Offset32Field, Pow2Offset40Field, Pow2Offset48Field, Pow2Offset56Field, Pow2Offset64Field, + Pow2OffsetPrimeSpec, Prime128M13M4P0, Prime128M37P3P0, Prime128M52M3P0, Prime128M54P4P0, + Prime128M8M4M1M0, POW2_OFFSET_IMPLEMENTED_MAX_BITS, POW2_OFFSET_MAX, POW2_OFFSET_PRIMES, + POW2_OFFSET_TABLE, +}; +pub use module::VectorModule; +pub use ntt::tables; +pub use ntt::{LimbQ, MontCoeff, NttPrime, QData, RADIX_BITS}; +pub use ring::{ + CrtNttConvertibleField, CyclotomicCrtNtt, CyclotomicRing, SparseChallenge, + SparseChallengeConfig, +}; diff --git a/src/algebra/module.rs b/src/algebra/module.rs new file mode 100644 index 00000000..6cfdb554 --- /dev/null +++ b/src/algebra/module.rs @@ -0,0 +1,202 @@ +//! Simple module implementations. + +use super::fields::{Fp128, Fp32, Fp64}; +use crate::primitives::serialization::{ + Compress, HachiDeserialize, HachiSerialize, SerializationError, Valid, Validate, +}; +use crate::{CanonicalField, FieldCore, FieldSampling, Module}; +use rand_core::RngCore; +use std::io::{Read, Write}; +use std::ops::{Add, Mul, Neg, Sub}; + +/// Fixed-length vector module over a scalar type `F`. +#[derive(Debug, Clone, Copy, PartialEq, Eq)] +pub struct VectorModule(pub [F; N]); + +impl VectorModule { + /// Construct the zero vector. + #[inline] + pub fn zero_vec() -> Self { + Self([F::zero(); N]) + } +} + +impl Add for VectorModule { + type Output = Self; + fn add(self, rhs: Self) -> Self::Output { + let mut out = self.0; + for (dst, src) in out.iter_mut().zip(rhs.0.iter()) { + *dst = *dst + *src; + } + Self(out) + } +} + +impl Sub for VectorModule { + type Output = Self; + fn sub(self, rhs: Self) -> Self::Output { + let mut out = self.0; + for (dst, src) in out.iter_mut().zip(rhs.0.iter()) { + *dst = *dst - *src; + } + Self(out) + } +} + +impl Neg for VectorModule { + type Output = Self; + fn neg(self) -> Self::Output { + let mut out = self.0; + for coeff in &mut out { + *coeff = -*coeff; + } + Self(out) + } +} + +impl<'a, F: FieldCore, const N: usize> Add<&'a Self> for VectorModule { + type Output = Self; + fn add(self, rhs: &'a Self) -> Self::Output { + self + *rhs + } +} + +impl<'a, F: FieldCore, const N: usize> Sub<&'a Self> for VectorModule { + type Output = Self; + fn sub(self, rhs: &'a Self) -> Self::Output { + self - *rhs + } +} + +impl Valid for VectorModule { + fn check(&self) -> Result<(), SerializationError> { + for x in self.0.iter() { + x.check()?; + } + Ok(()) + } +} + +impl HachiSerialize for VectorModule { + fn serialize_with_mode( + &self, + mut writer: W, + compress: Compress, + ) -> Result<(), SerializationError> { + for x in self.0.iter() { + x.serialize_with_mode(&mut writer, compress)?; + } + Ok(()) + } + + fn serialized_size(&self, compress: Compress) -> usize { + self.0.iter().map(|x| x.serialized_size(compress)).sum() + } +} + +impl HachiDeserialize for VectorModule { + fn deserialize_with_mode( + mut reader: R, + compress: Compress, + validate: Validate, + ) -> Result { + let mut arr = [F::zero(); N]; + for coeff in &mut arr { + *coeff = F::deserialize_with_mode(&mut reader, compress, validate)?; + } + let out = Self(arr); + if matches!(validate, Validate::Yes) { + out.check()?; + } + Ok(out) + } +} + +impl Module for VectorModule +where + F: FieldCore + + CanonicalField + + FieldSampling + + Valid + + Mul, Output = VectorModule> + + for<'a> Mul<&'a VectorModule, Output = VectorModule>, +{ + type Scalar = F; + + fn zero() -> Self { + Self::zero_vec() + } + + fn add(&self, rhs: &Self) -> Self { + *self + *rhs + } + + fn neg(&self) -> Self { + -*self + } + + fn scale(&self, k: &Self::Scalar) -> Self { + // Delegate to Scalar * VectorModule to satisfy the Module trait’s scalar bounds. + *k * *self + } + + fn random(rng: &mut R) -> Self { + Self(std::array::from_fn(|_| F::sample(rng))) + } +} + +// Scalar * VectorModule impls for our local prime field types. + +impl Mul, N>> for Fp32

{ + type Output = VectorModule, N>; + fn mul(self, rhs: VectorModule, N>) -> Self::Output { + let mut out = rhs.0; + for coeff in &mut out { + *coeff = self * *coeff; + } + VectorModule(out) + } +} + +impl<'a, const P: u32, const N: usize> Mul<&'a VectorModule, N>> for Fp32

{ + type Output = VectorModule, N>; + fn mul(self, rhs: &'a VectorModule, N>) -> Self::Output { + self * *rhs + } +} + +impl Mul, N>> for Fp64

{ + type Output = VectorModule, N>; + fn mul(self, rhs: VectorModule, N>) -> Self::Output { + let mut out = rhs.0; + for coeff in &mut out { + *coeff = self * *coeff; + } + VectorModule(out) + } +} + +impl<'a, const P: u64, const N: usize> Mul<&'a VectorModule, N>> for Fp64

{ + type Output = VectorModule, N>; + fn mul(self, rhs: &'a VectorModule, N>) -> Self::Output { + self * *rhs + } +} + +impl Mul, N>> for Fp128

{ + type Output = VectorModule, N>; + fn mul(self, rhs: VectorModule, N>) -> Self::Output { + let mut out = rhs.0; + for coeff in &mut out { + *coeff = self * *coeff; + } + VectorModule(out) + } +} + +impl<'a, const P: u128, const N: usize> Mul<&'a VectorModule, N>> for Fp128

{ + type Output = VectorModule, N>; + fn mul(self, rhs: &'a VectorModule, N>) -> Self::Output { + self * *rhs + } +} diff --git a/src/algebra/ntt/butterfly.rs b/src/algebra/ntt/butterfly.rs new file mode 100644 index 00000000..9f58da94 --- /dev/null +++ b/src/algebra/ntt/butterfly.rs @@ -0,0 +1,155 @@ +//! NTT butterfly transforms for negacyclic rings `Z_p[X]/(X^D + 1)`. +//! +//! Uses a merged negacyclic Cooley-Tukey / Gentleman-Sande butterfly where +//! the twist factors for `X^D + 1` are folded directly into the twiddles. +//! Twiddle factors are powers of `psi`, a primitive `2D`-th root of unity +//! (`psi^D = -1 mod p`), rather than a `D`-th root. +//! +//! TODO(perf): migrate twiddle tables to compile-time `const` arrays once +//! parameter sets are finalized. + +use super::prime::{MontCoeff, NttPrime}; + +/// Precomputed twiddle factors for a specific prime and degree `D`. +/// +/// `D` must be a power of two. +pub struct NttTwiddles { + /// Twiddle factors in Montgomery form, indexed by bit-reversed position. + /// `zetas[i] = from_canonical(psi^{brv(i)})` for `i = 0..D`. + /// Index 0 is unused by the forward NTT but set to the Montgomery form of 1. + pub(crate) zetas: [MontCoeff; D], + /// `D^{-1} mod p` in Montgomery form, used for inverse NTT final scaling. + pub(crate) d_inv: MontCoeff, +} + +impl NttTwiddles { + /// Compute twiddle factors for the given prime. + /// + /// Finds a primitive `2D`-th root of unity mod `p`, then fills the + /// twiddle table in bit-reversed order. All values are stored in + /// Montgomery form. + /// + /// # Panics + /// + /// Panics if `D` is not a power of two, or if `2D` does not divide `p - 1`. + pub fn compute(prime: NttPrime) -> Self { + assert!(D.is_power_of_two(), "D must be a power of two"); + let p = prime.p as i64; + assert!( + (p - 1) % (2 * D as i64) == 0, + "2D must divide p - 1 for NTT roots to exist" + ); + + let n = D.trailing_zeros(); + let psi = find_primitive_root_2d(p, D); + + let mut zetas = [MontCoeff::from_raw(0); D]; + for (i, z) in zetas.iter_mut().enumerate() { + let brv_i = bit_reverse(i, n); + let power = pow_mod(psi, brv_i as i64, p) as i16; + *z = prime.from_canonical(power); + } + + let d_inv_canonical = pow_mod(D as i64, p - 2, p) as i16; + let d_inv = prime.from_canonical(d_inv_canonical); + + Self { zetas, d_inv } + } +} + +/// Forward negacyclic NTT (Cooley-Tukey, decimation-in-time). +/// +/// Transforms `D` coefficients in-place from coefficient form to NTT +/// evaluation form. Both outputs of each butterfly are Barrett-reduced +/// to prevent overflow. +/// +/// After this call, the coefficients represent evaluations at +/// `psi^{2*brv(i)+1}` for `i = 0..D-1`. +pub fn forward_ntt(a: &mut [MontCoeff; D], prime: NttPrime, tw: &NttTwiddles) { + let mut k = 1usize; + let mut len = D / 2; + while len >= 1 { + let mut start = 0; + while start < D { + let zeta = tw.zetas[k]; + k += 1; + for j in start..(start + len) { + let t = prime.mul(a[j + len], zeta); + let diff = a[j].raw().wrapping_sub(t.raw()); + let sum = a[j].raw().wrapping_add(t.raw()); + a[j + len] = prime.reduce(MontCoeff::from_raw(diff)); + a[j] = prime.reduce(MontCoeff::from_raw(sum)); + } + start += 2 * len; + } + len /= 2; + } +} + +/// Inverse negacyclic NTT (Gentleman-Sande, decimation-in-frequency). +/// +/// Transforms `D` evaluations in-place back to coefficient form. +/// Includes the final `D^{-1}` scaling. +pub fn inverse_ntt(a: &mut [MontCoeff; D], prime: NttPrime, tw: &NttTwiddles) { + let mut k = D - 1; + let mut len = 1; + while len <= D / 2 { + let mut start = 0; + while start < D { + let zeta = tw.zetas[k]; + k = k.wrapping_sub(1); + for j in start..(start + len) { + let t = a[j]; + let sum = t.raw().wrapping_add(a[j + len].raw()); + let diff = t.raw().wrapping_sub(a[j + len].raw()); + a[j] = prime.reduce(MontCoeff::from_raw(sum)); + // Multiply difference by negative twiddle: negate zeta. + let neg_zeta = MontCoeff::from_raw(zeta.raw().wrapping_neg()); + a[j + len] = prime.mul(MontCoeff::from_raw(diff), neg_zeta); + } + start += 2 * len; + } + len *= 2; + } + + // Scale by D^{-1}. + for c in a.iter_mut() { + *c = prime.mul(*c, tw.d_inv); + } +} + +/// Bit-reverse an `n`-bit integer. +fn bit_reverse(x: usize, n: u32) -> usize { + x.reverse_bits() >> (usize::BITS - n) +} + +/// Find a primitive `2D`-th root of unity mod `p`. +/// +/// Returns `psi` in `[0, p)` such that `psi^D = -1 mod p`. +fn find_primitive_root_2d(p: i64, d: usize) -> i64 { + // Find the smallest quadratic non-residue a (i.e., a^{(p-1)/2} = -1 mod p). + let half = (p - 1) / 2; + let exp = (p - 1) / (2 * d as i64); + for a in 2..p { + if pow_mod(a, half, p) == p - 1 { + let psi = pow_mod(a, exp, p); + debug_assert_eq!(pow_mod(psi, d as i64, p), p - 1, "psi^D != -1"); + return psi; + } + } + panic!("no primitive root found for p={p}"); +} + +/// Modular exponentiation: `base^exp mod modulus`. +fn pow_mod(mut base: i64, mut exp: i64, modulus: i64) -> i64 { + let mut result = 1i64; + base %= modulus; + while exp > 0 { + if exp & 1 == 1 { + result = result * base % modulus; + } + base = base * base % modulus; + exp >>= 1; + } + result +} diff --git a/src/algebra/ntt/crt.rs b/src/algebra/ntt/crt.rs new file mode 100644 index 00000000..7df39710 --- /dev/null +++ b/src/algebra/ntt/crt.rs @@ -0,0 +1,189 @@ +//! Limb-based helpers for `q = 2^k - c` style moduli. +//! +//! Big-`q` coefficients are stored in radix `2^14` limbs. +//! This module provides a portable scalar representation with the same limb layout. + +use std::cmp::Ordering; +use std::fmt; +use std::ops::{Add, Sub}; + +/// Limb radix bit-width (`2^14`). +pub const RADIX_BITS: u32 = 14; +const RADIX: i32 = 1 << RADIX_BITS; +const RADIX_MASK: i32 = RADIX - 1; + +/// Fixed-width radix-`2^14` integer. +/// +/// Limbs are little-endian: `limbs[0]` is least significant. +#[derive(Debug, Clone, Copy, PartialEq, Eq)] +pub struct LimbQ { + /// Little-endian limbs. + pub limbs: [u16; L], +} + +impl Default for LimbQ { + #[inline] + fn default() -> Self { + Self::zero() + } +} + +impl LimbQ { + /// Zero value. + #[inline] + pub const fn zero() -> Self { + Self { limbs: [0; L] } + } + + /// Construct directly from limbs. + #[inline] + pub const fn from_limbs(limbs: [u16; L]) -> Self { + Self { limbs } + } + + /// Conditional subtraction: if `self >= modulus`, return `self - modulus` (branchless). + #[inline] + pub fn csub_mod(self, modulus: Self) -> Self { + // Compute self - modulus, tracking the final borrow. + let mut diff = [0u16; L]; + let mut borrow = 0i32; + for (i, df) in diff.iter_mut().enumerate() { + let d = self.limbs[i] as i32 - modulus.limbs[i] as i32 + borrow; + borrow = d >> 31; + if i + 1 < L { + *df = (d - borrow * RADIX) as u16; + } else { + *df = d as u16; + } + } + // borrow = -1 if self < modulus (underflowed), 0 otherwise. + // Branchless select: mask = 0xFFFF if underflow (keep self), 0 if not (keep diff). + let mask = borrow as u16; + let mut result = [0u16; L]; + for (i, r) in result.iter_mut().enumerate() { + *r = (self.limbs[i] & mask) | (diff[i] & !mask); + } + Self { limbs: result } + } +} + +impl From for LimbQ { + fn from(mut x: u128) -> Self { + let mut out = [0u16; L]; + for (i, limb) in out.iter_mut().enumerate() { + if i + 1 < L { + *limb = (x & (RADIX_MASK as u128)) as u16; + x >>= RADIX_BITS; + } else { + *limb = x as u16; + } + } + Self { limbs: out } + } +} + +impl TryFrom> for u128 { + type Error = &'static str; + + fn try_from(limb: LimbQ) -> Result { + if (L as u32) * RADIX_BITS > 128 { + return Err("LimbQ too wide for u128"); + } + let mut acc = 0u128; + for i in (0..L).rev() { + acc <<= RADIX_BITS; + acc |= limb.limbs[i] as u128; + } + Ok(acc) + } +} + +impl PartialOrd for LimbQ { + #[inline] + fn partial_cmp(&self, other: &Self) -> Option { + Some(self.cmp(other)) + } +} + +impl Ord for LimbQ { + fn cmp(&self, other: &Self) -> Ordering { + for i in (0..L).rev() { + match self.limbs[i].cmp(&other.limbs[i]) { + Ordering::Equal => continue, + ord => return ord, + } + } + Ordering::Equal + } +} + +impl Add for LimbQ { + type Output = Self; + + fn add(self, rhs: Self) -> Self { + let mut out = [0u16; L]; + let mut carry = 0i32; + for (i, out_limb) in out.iter_mut().enumerate() { + let s = self.limbs[i] as i32 + rhs.limbs[i] as i32 + carry; + if i + 1 < L { + carry = s >> RADIX_BITS; + *out_limb = (s & RADIX_MASK) as u16; + } else { + *out_limb = s as u16; + } + } + Self { limbs: out } + } +} + +impl Sub for LimbQ { + type Output = Self; + + fn sub(self, rhs: Self) -> Self { + let mut out = [0u16; L]; + let mut borrow = 0i32; + for (i, out_limb) in out.iter_mut().enumerate() { + let d = self.limbs[i] as i32 - rhs.limbs[i] as i32 + borrow; + if i + 1 < L { + borrow = d >> 31; + *out_limb = (d - borrow * RADIX) as u16; + } else { + *out_limb = d as u16; + } + } + Self { limbs: out } + } +} + +impl fmt::Display for LimbQ { + fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result { + if let Ok(val) = u128::try_from(*self) { + write!(f, "{val}") + } else { + write!(f, "LimbQ{:?}", self.limbs) + } + } +} + +/// CRT/q constants for a given parameter set. +#[derive(Debug, Clone, Copy, PartialEq, Eq)] +pub struct QData { + /// The modulus `q` in radix-`2^14` limbs. + pub q: LimbQ, + /// `-P mod q` in limbs (where `P = prod p_i`). + pub pmq: LimbQ, + /// `P/p_i mod q` in limbs. + pub xvec: [LimbQ; K], + /// `k` in `q = 2^k - c`. + pub logq: u32, + /// `c` in `q = 2^k - c`. + pub qoff: u16, +} + +impl QData { + /// `q` as `u128` when representable. + #[inline] + pub fn q_u128(self) -> Option { + u128::try_from(self.q).ok() + } +} diff --git a/src/algebra/ntt/mod.rs b/src/algebra/ntt/mod.rs new file mode 100644 index 00000000..fffbf284 --- /dev/null +++ b/src/algebra/ntt/mod.rs @@ -0,0 +1,10 @@ +//! NTT-friendly small-prime arithmetic and CRT helpers. + +pub mod butterfly; +pub mod crt; +pub mod prime; +pub mod tables; + +pub use butterfly::NttTwiddles; +pub use crt::{LimbQ, QData, RADIX_BITS}; +pub use prime::{MontCoeff, NttPrime}; diff --git a/src/algebra/ntt/prime.rs b/src/algebra/ntt/prime.rs new file mode 100644 index 00000000..9b19cf22 --- /dev/null +++ b/src/algebra/ntt/prime.rs @@ -0,0 +1,180 @@ +//! Small-prime arithmetic kernels for NTT-friendly parameter sets. +//! +//! Per-prime scalar operations: +//! - Montgomery multiplication ([`NttPrime::mul`]) +//! - Barrett-style reduction ([`NttPrime::reduce`]) +//! - Branchless conditional add/sub +//! +//! Coefficients in Montgomery domain are wrapped in [`MontCoeff`] to prevent +//! accidental mixing with canonical `i16` values. + +use std::fmt; + +/// A coefficient in Montgomery domain for an NTT prime. +/// +/// Wraps an `i16` representing `a * R mod p` (where `R = 2^16`). +/// Use [`NttPrime::from_canonical`] to enter and [`NttPrime::to_canonical`] +/// to leave Montgomery domain. +#[derive(Clone, Copy, PartialEq, Eq, Default)] +#[repr(transparent)] +pub struct MontCoeff(i16); + +impl MontCoeff { + /// Wrap a raw Montgomery-domain value. + /// + /// The caller must ensure `val` is a valid Montgomery-domain representative + /// for the intended prime. + #[inline] + pub const fn from_raw(val: i16) -> Self { + Self(val) + } + + /// Extract the raw `i16` (still in Montgomery domain). + #[inline] + pub const fn raw(self) -> i16 { + self.0 + } +} + +impl fmt::Debug for MontCoeff { + fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result { + write!(f, "Mont({})", self.0) + } +} + +/// Per-prime constants for NTT arithmetic over a small prime `p < 2^14`. +/// +/// All scalar operations use Montgomery representation internally. Constants +/// are stored as raw `i16` in centered signed form. +#[derive(Debug, Clone, Copy, PartialEq, Eq)] +pub struct NttPrime { + /// Prime modulus. + pub p: i16, + /// `p^{-1} mod 2^{16}` (centered signed). Used in Montgomery reduction. + pub pinv: i16, + /// `round(2^{27} / p)` for Barrett-style range reduction. + pub v: i16, + /// `R mod p` where `R = 2^{16}` (centered signed). Montgomery form of 1. + pub mont: i16, + /// `R^2 mod p` (centered signed). Used for canonical → Montgomery conversion. + pub montsq: i16, + /// NTT injection scale factor (centered signed). + pub s: i16, + /// Inverse-NTT post-scale factor (centered signed). + pub f: i16, + /// CRT reconstruction factor (centered signed). + pub t: i16, +} + +impl NttPrime { + /// Montgomery product: `a * b * R^{-1} mod p` where `R = 2^{16}`. + #[inline] + pub fn mul(self, a: MontCoeff, b: MontCoeff) -> MontCoeff { + MontCoeff(self.mont_mul_raw(a.0, b.0)) + } + + /// Raw Montgomery multiply on bare `i16` values. + /// + /// Computes `(a*b - ((a*b mod R) * p_inv mod R) * p) / R`. + #[inline] + pub(crate) fn mont_mul_raw(self, a: i16, b: i16) -> i16 { + let c = (a as i32) * (b as i32); + let t = (c as i16).wrapping_mul(self.pinv) as i32; + ((c - t * (self.p as i32)) >> 16) as i16 + } + + /// Barrett-style reduction: tighten a Montgomery coefficient's range + /// toward `[0, p)` without full normalization. + #[inline] + pub fn reduce(self, a: MontCoeff) -> MontCoeff { + let val = a.0; + let mut t = (((self.v as i32) * (val as i32) + (1 << 26)) >> 27) as i16; + t = t.wrapping_mul(self.p); + MontCoeff(val.wrapping_sub(t)) + } + + /// Conditionally subtract `p` if `a >= p` (branchless). + #[inline] + pub fn csubp(self, a: MontCoeff) -> MontCoeff { + let diff = a.0.wrapping_sub(self.p); + let mask = diff >> 15; + MontCoeff(diff.wrapping_add(mask & self.p)) + } + + /// Conditionally add `p` if `a < 0` (branchless). + #[inline] + pub fn caddp(self, a: MontCoeff) -> MontCoeff { + let mask = a.0 >> 15; + MontCoeff(a.0.wrapping_add(mask & self.p)) + } + + /// Fully normalize a Montgomery coefficient to `[0, p)`. + #[inline] + pub fn normalize(self, a: MontCoeff) -> MontCoeff { + self.csubp(self.caddp(self.reduce(a))) + } + + /// Convert a canonical `i16` into Montgomery domain: `a ↦ a * R mod p`. + #[inline] + pub fn from_canonical(self, a: i16) -> MontCoeff { + MontCoeff(self.mont_mul_raw(a, self.montsq)) + } + + /// Convert from Montgomery domain to canonical `[0, p)`. + #[inline] + pub fn to_canonical(self, a: MontCoeff) -> i16 { + let raw = MontCoeff(self.mont_mul_raw(a.0, 1)); + self.normalize(raw).0 + } + + /// Center a canonical value from approximately `(-p, p)` into `[-p/2, p/2)`. + /// + /// First conditionally adds `p` (if negative), then subtracts `p` (if > p/2). + /// Both steps are branchless. + #[inline] + pub fn center(self, a: i16) -> i16 { + let mask_neg = a >> 15; + let canonical = a.wrapping_add(mask_neg & self.p); + let half = self.p / 2; + let needs_sub = half.wrapping_sub(canonical) >> 15; + canonical.wrapping_add(needs_sub & self.p.wrapping_neg()) + } + + /// Pointwise Montgomery multiplication of two coefficient slices. + /// + /// # Panics + /// + /// Panics if slices have different lengths. + #[inline] + pub fn pointwise_mul(self, out: &mut [MontCoeff], lhs: &[MontCoeff], rhs: &[MontCoeff]) { + assert_eq!(out.len(), lhs.len()); + assert_eq!(lhs.len(), rhs.len()); + for ((o, a), b) in out.iter_mut().zip(lhs.iter()).zip(rhs.iter()) { + *o = self.mul(*a, *b); + } + } + + /// In-place Montgomery scaling by a constant. + #[inline] + pub fn scale_in_place(self, coeffs: &mut [MontCoeff], scalar: MontCoeff) { + for c in coeffs { + *c = self.mul(*c, scalar); + } + } + + /// In-place Barrett reduction on a coefficient slice. + #[inline] + pub fn reduce_in_place(self, coeffs: &mut [MontCoeff]) { + for c in coeffs { + *c = self.reduce(*c); + } + } + + /// In-place centering of canonical values to `[-p/2, p/2)`. + #[inline] + pub fn center_slice(self, coeffs: &mut [i16]) { + for c in coeffs { + *c = self.center(*c); + } + } +} diff --git a/src/algebra/ntt/tables.rs b/src/algebra/ntt/tables.rs new file mode 100644 index 00000000..613ccc95 --- /dev/null +++ b/src/algebra/ntt/tables.rs @@ -0,0 +1,149 @@ +//! Deterministic parameter presets for small-prime CRT arithmetic. +//! +//! Initial scope: `logq = 32` preset with six NTT-friendly primes. + +use super::crt::{LimbQ, QData}; +use super::prime::NttPrime; + +/// Polynomial degree for the base ring `Z_q[X]/(X^d + 1)`. +pub const RING_DEGREE: usize = 64; + +/// Number of CRT primes for the `logq = 32` parameter set. +pub const Q32_NUM_PRIMES: usize = 6; + +/// Number of radix-`2^14` limbs for `logq = 32`. +pub const Q32_NUM_LIMBS: usize = 3; + +/// Offset `c` in `q = 2^32 - c`. +pub const Q32_OFFSET: u16 = 99; + +/// `log2(q)` for the `logq = 32` parameter set. +pub const Q32_LOG_MODULUS: u32 = 32; + +/// The modulus `q = 2^32 - 99`. +pub const Q32_MODULUS: u64 = (1u64 << 32) - (Q32_OFFSET as u64); + +/// CRT primes and per-prime Montgomery constants for `logq = 32`. +pub const Q32_PRIMES: [NttPrime; Q32_NUM_PRIMES] = [ + NttPrime { + p: 13697, + pinv: 2689, + v: 9799, + mont: -2949, + montsq: -994, + s: 4705, + f: 3540, + t: -5758, + }, + NttPrime { + p: 13441, + pinv: 2945, + v: 9986, + mont: -1669, + montsq: 3274, + s: 3777, + f: -5468, + t: -1680, + }, + NttPrime { + p: 13313, + pinv: -13311, + v: 10082, + mont: -1029, + montsq: -6199, + s: 3325, + f: 4553, + t: -948, + }, + NttPrime { + p: 12289, + pinv: -12287, + v: 10922, + mont: 4091, + montsq: -1337, + s: -3, + f: 354, + t: 4472, + }, + NttPrime { + p: 12161, + pinv: 4225, + v: 11037, + mont: 4731, + montsq: -6040, + s: -383, + f: 5993, + t: 1653, + }, + NttPrime { + p: 11777, + pinv: -11775, + v: 11397, + mont: -5126, + montsq: 1389, + s: -1475, + f: -3812, + t: 1191, + }, +]; + +/// CRT limb constants for the `logq = 32` parameter set. +pub const Q32_DATA: QData = QData { + q: LimbQ::from_limbs([16285, 16383, 15]), + pmq: LimbQ::from_limbs([8747, 999, 12]), + xvec: [ + LimbQ::from_limbs([9406, 14930, 2]), + LimbQ::from_limbs([9295, 15936, 2]), + LimbQ::from_limbs([16336, 1061, 5]), + LimbQ::from_limbs([1006, 14584, 9]), + LimbQ::from_limbs([1315, 5273, 10]), + LimbQ::from_limbs([7927, 10169, 6]), + ], + logq: Q32_LOG_MODULUS, + qoff: Q32_OFFSET, +}; + +#[cfg(test)] +mod tests { + use super::*; + + /// Verify that derived constants (`pinv`, `v`, `mont`, `montsq`) are + /// consistent with the prime `p`. Turns the magic numbers into auditable + /// values. + #[test] + fn verify_prime_derived_constants() { + for prime in &Q32_PRIMES { + let p = prime.p; + + // pinv: p * pinv ≡ 1 (mod 2^16) + assert_eq!(p.wrapping_mul(prime.pinv), 1, "pinv failed for p={p}"); + + // v: round(2^27 / p) + let expected_v = ((1i64 << 27) + (p as i64 / 2)) / (p as i64); + assert_eq!(prime.v as i64, expected_v, "v failed for p={p}"); + + // mont: 2^16 mod p, centered to [-p/2, p/2) + let raw_mont = ((1i32 << 16) % (p as i32)) as i16; + let centered_mont = if raw_mont > p / 2 { + raw_mont - p + } else { + raw_mont + }; + assert_eq!(prime.mont, centered_mont, "mont failed for p={p}"); + + // montsq: 2^32 mod p, centered to [-p/2, p/2) + let raw_montsq = ((1i64 << 32) % (p as i64)) as i16; + let centered_montsq = if raw_montsq > p / 2 { + raw_montsq - p + } else { + raw_montsq + }; + assert_eq!(prime.montsq, centered_montsq, "montsq failed for p={p}"); + } + } + + #[test] + fn verify_q_value() { + assert_eq!(Q32_DATA.q_u128().unwrap(), Q32_MODULUS as u128); + } +} diff --git a/src/algebra/poly.rs b/src/algebra/poly.rs new file mode 100644 index 00000000..684aa7d8 --- /dev/null +++ b/src/algebra/poly.rs @@ -0,0 +1,95 @@ +//! Fixed-size polynomial container. + +use crate::primitives::serialization::{ + Compress, HachiDeserialize, HachiSerialize, SerializationError, Valid, Validate, +}; +use crate::FieldCore; +use std::io::{Read, Write}; + +/// A degree-(pub [F; D]); + +impl Poly { + /// Construct the zero polynomial. + pub fn zero() -> Self { + Self([F::zero(); D]) + } +} + +impl std::ops::Add for Poly { + type Output = Self; + fn add(self, rhs: Self) -> Self::Output { + let mut out = self.0; + for (dst, src) in out.iter_mut().zip(rhs.0.iter()) { + *dst = *dst + *src; + } + Self(out) + } +} + +impl std::ops::Sub for Poly { + type Output = Self; + fn sub(self, rhs: Self) -> Self::Output { + let mut out = self.0; + for (dst, src) in out.iter_mut().zip(rhs.0.iter()) { + *dst = *dst - *src; + } + Self(out) + } +} + +impl std::ops::Neg for Poly { + type Output = Self; + fn neg(self) -> Self::Output { + let mut out = self.0; + for coeff in &mut out { + *coeff = -*coeff; + } + Self(out) + } +} + +impl Valid for Poly { + fn check(&self) -> Result<(), SerializationError> { + for x in self.0.iter() { + x.check()?; + } + Ok(()) + } +} + +impl HachiSerialize for Poly { + fn serialize_with_mode( + &self, + mut writer: W, + compress: Compress, + ) -> Result<(), SerializationError> { + for x in self.0.iter() { + x.serialize_with_mode(&mut writer, compress)?; + } + Ok(()) + } + + fn serialized_size(&self, compress: Compress) -> usize { + self.0.iter().map(|x| x.serialized_size(compress)).sum() + } +} + +impl HachiDeserialize for Poly { + fn deserialize_with_mode( + mut reader: R, + compress: Compress, + validate: Validate, + ) -> Result { + let mut arr = [F::zero(); D]; + for coeff in &mut arr { + *coeff = F::deserialize_with_mode(&mut reader, compress, validate)?; + } + let out = Self(arr); + if matches!(validate, Validate::Yes) { + out.check()?; + } + Ok(out) + } +} diff --git a/src/algebra/ring/crt_ntt_repr.rs b/src/algebra/ring/crt_ntt_repr.rs new file mode 100644 index 00000000..18e4772a --- /dev/null +++ b/src/algebra/ring/crt_ntt_repr.rs @@ -0,0 +1,246 @@ +//! CRT+NTT-domain representation of cyclotomic ring elements. + +use crate::algebra::backend::{CrtReconstruct, NttPrimeOps, NttTransform, ScalarBackend}; +use crate::algebra::fields::{Fp128, Fp32, Fp64}; +use crate::algebra::ntt::butterfly::NttTwiddles; +use crate::algebra::ntt::crt::QData; +use crate::algebra::ntt::prime::{MontCoeff, NttPrime}; +use crate::{CanonicalField, FieldCore}; + +use super::cyclotomic::CyclotomicRing; + +/// CRT+NTT-domain representation of a cyclotomic ring element. +/// +/// Stores `K` arrays of `D` [`MontCoeff`] values, one per CRT prime. +/// Multiplication is pointwise per prime - O(K*D) vs O(D^2) for coefficient form. +/// +/// Use [`CyclotomicCrtNtt::from_ring`] and [`CyclotomicCrtNtt::to_ring`] to convert +/// between coefficient and CRT+NTT domains. +#[derive(Debug, Clone, PartialEq, Eq)] +pub struct CyclotomicCrtNtt { + pub(crate) limbs: [[MontCoeff; D]; K], +} + +/// Field types that can safely convert to and from the CRT+NTT representation. +pub trait CrtNttConvertibleField: FieldCore + CanonicalField { + /// Reduce this field element modulo a small prime `p`. + fn mod_small_prime(self, p: i16) -> i16; + + /// Reconstruct from a residue in `[0, q)` after CRT combination. + fn from_q_residue_u128(x: u128) -> Self; +} + +impl CrtNttConvertibleField for Fp32

{ + #[inline] + fn mod_small_prime(self, p: i16) -> i16 { + (self.to_canonical_u32() % (p as u32)) as i16 + } + + #[inline] + fn from_q_residue_u128(x: u128) -> Self { + let narrowed = u32::try_from(x).expect("CRT residue does not fit in u32"); + Self::from_u64(narrowed as u64) + } +} + +impl CrtNttConvertibleField for Fp64

{ + #[inline] + fn mod_small_prime(self, p: i16) -> i16 { + (self.to_canonical_u64() % (p as u64)) as i16 + } + + #[inline] + fn from_q_residue_u128(x: u128) -> Self { + let narrowed = u64::try_from(x).expect("CRT residue does not fit in u64"); + Self::from_u64(narrowed) + } +} + +impl CrtNttConvertibleField for Fp128

{ + #[inline] + fn mod_small_prime(self, p: i16) -> i16 { + (self.to_canonical_u128() % (p as u128)) as i16 + } + + #[inline] + fn from_q_residue_u128(x: u128) -> Self { + Self::from_canonical_u128_reduced(x) + } +} + +impl CyclotomicCrtNtt { + /// The additive identity (all zeros in every CRT limb). + pub fn zero() -> Self { + Self { + limbs: [[MontCoeff::from_raw(0); D]; K], + } + } + + /// Convert a coefficient-form ring element into CRT+NTT domain + /// using the default scalar backend. + pub fn from_ring( + ring: &CyclotomicRing, + primes: &[NttPrime; K], + twiddles: &[NttTwiddles; K], + ) -> Self { + Self::from_ring_with_backend::(ring, primes, twiddles) + } + + /// Convert a coefficient-form ring element into CRT+NTT domain + /// through an explicit backend implementation. + pub fn from_ring_with_backend< + F: CrtNttConvertibleField, + B: NttPrimeOps + NttTransform, + >( + ring: &CyclotomicRing, + primes: &[NttPrime; K], + twiddles: &[NttTwiddles; K], + ) -> Self { + let mut limbs = [[MontCoeff::from_raw(0); D]; K]; + for ((limb, prime), tw) in limbs.iter_mut().zip(primes.iter()).zip(twiddles.iter()) { + // Reduce each coefficient mod p and convert to Montgomery form. + for (dst, src) in limb.iter_mut().zip(ring.coeffs.iter()) { + let val = src.mod_small_prime(prime.p); + *dst = B::from_canonical(*prime, val); + } + B::forward_ntt(limb, *prime, tw); + } + Self { limbs } + } + + /// Convert from CRT+NTT domain back to coefficient form + /// using the default scalar backend. + pub fn to_ring( + &self, + primes: &[NttPrime; K], + twiddles: &[NttTwiddles; K], + qdata: &QData, + ) -> CyclotomicRing { + self.to_ring_with_backend::(primes, twiddles, qdata) + } + + /// Convert from CRT+NTT domain back to coefficient form + /// through an explicit backend implementation. + pub fn to_ring_with_backend< + F: CrtNttConvertibleField, + B: NttPrimeOps + NttTransform + CrtReconstruct, + const L: usize, + >( + &self, + primes: &[NttPrime; K], + twiddles: &[NttTwiddles; K], + qdata: &QData, + ) -> CyclotomicRing { + let mut canonical = [[0i16; D]; K]; + for (k, ((can, prime), tw)) in canonical + .iter_mut() + .zip(primes.iter()) + .zip(twiddles.iter()) + .enumerate() + { + let mut limb = self.limbs[k]; + B::inverse_ntt(&mut limb, *prime, tw); + for (dst, src) in can.iter_mut().zip(limb.iter()) { + *dst = B::to_canonical(*prime, *src); + } + } + + let coeffs = B::reconstruct::(primes, &canonical, qdata); + + CyclotomicRing::from_coefficients(coeffs) + } + + /// Add another CRT+NTT element and reduce each coefficient with the matching + /// prime to maintain valid Montgomery ranges using the scalar backend. + pub fn add_reduced(&self, rhs: &Self, primes: &[NttPrime; K]) -> Self { + self.add_reduced_with_backend::(rhs, primes) + } + + /// Add another CRT+NTT element and reduce each coefficient with the matching + /// prime through an explicit backend implementation. + pub fn add_reduced_with_backend>( + &self, + rhs: &Self, + primes: &[NttPrime; K], + ) -> Self { + let mut out = self.clone(); + for (k, (limb, rhs_limb)) in out.limbs.iter_mut().zip(rhs.limbs.iter()).enumerate() { + let prime = primes[k]; + for (a, b) in limb.iter_mut().zip(rhs_limb.iter()) { + let sum = MontCoeff::from_raw(a.raw().wrapping_add(b.raw())); + *a = B::reduce(prime, sum); + } + } + out + } + + /// Subtract another CRT+NTT element and reduce each coefficient with the + /// matching prime to maintain valid Montgomery ranges using the scalar backend. + pub fn sub_reduced(&self, rhs: &Self, primes: &[NttPrime; K]) -> Self { + self.sub_reduced_with_backend::(rhs, primes) + } + + /// Subtract another CRT+NTT element and reduce each coefficient with the + /// matching prime through an explicit backend implementation. + pub fn sub_reduced_with_backend>( + &self, + rhs: &Self, + primes: &[NttPrime; K], + ) -> Self { + let mut out = self.clone(); + for (k, (limb, rhs_limb)) in out.limbs.iter_mut().zip(rhs.limbs.iter()).enumerate() { + let prime = primes[k]; + for (a, b) in limb.iter_mut().zip(rhs_limb.iter()) { + let diff = MontCoeff::from_raw(a.raw().wrapping_sub(b.raw())); + *a = B::reduce(prime, diff); + } + } + out + } + + /// Negate each CRT+NTT coefficient and reduce with the matching prime + /// using the scalar backend. + pub fn neg_reduced(&self, primes: &[NttPrime; K]) -> Self { + self.neg_reduced_with_backend::(primes) + } + + /// Negate each CRT+NTT coefficient and reduce with the matching prime + /// through an explicit backend implementation. + pub fn neg_reduced_with_backend>(&self, primes: &[NttPrime; K]) -> Self { + let mut out = self.clone(); + for (k, limb) in out.limbs.iter_mut().enumerate() { + let prime = primes[k]; + for a in limb.iter_mut() { + let neg = MontCoeff::from_raw(a.raw().wrapping_neg()); + *a = B::reduce(prime, neg); + } + } + out + } + + /// Pointwise multiplication in CRT+NTT domain. + /// + /// Each limb is multiplied pointwise using the corresponding prime's + /// Montgomery multiplication. + pub fn pointwise_mul(&self, rhs: &Self, primes: &[NttPrime; K]) -> Self { + self.pointwise_mul_with_backend::(rhs, primes) + } + + /// Pointwise multiplication in CRT+NTT domain through an explicit backend implementation. + pub fn pointwise_mul_with_backend>( + &self, + rhs: &Self, + primes: &[NttPrime; K], + ) -> Self { + let mut out = [[MontCoeff::from_raw(0); D]; K]; + for (k, ((o, a), b)) in out + .iter_mut() + .zip(self.limbs.iter()) + .zip(rhs.limbs.iter()) + .enumerate() + { + B::pointwise_mul(primes[k], o, a, b); + } + Self { limbs: out } + } +} diff --git a/src/algebra/ring/cyclotomic.rs b/src/algebra/ring/cyclotomic.rs new file mode 100644 index 00000000..5dd2f0fc --- /dev/null +++ b/src/algebra/ring/cyclotomic.rs @@ -0,0 +1,439 @@ +//! Cyclotomic ring `Z_q[X]/(X^D + 1)` in coefficient form. + +use super::sparse_challenge::SparseChallenge; +use crate::primitives::serialization::{ + Compress, HachiDeserialize, HachiSerialize, SerializationError, Valid, Validate, +}; +use crate::{CanonicalField, FieldCore, FieldSampling}; +use rand_core::RngCore; +use std::io::{Read, Write}; +use std::ops::{Add, AddAssign, Mul, MulAssign, Neg, Sub, SubAssign}; + +/// Element of the cyclotomic ring `Z_q[X]/(X^D + 1)`. +/// +/// Stored as `D` coefficients in the base field `F`, representing +/// `a_0 + a_1*X + ... + a_{D-1}*X^{D-1}`. +/// +/// Multiplication is negacyclic convolution: `X^D = -1`, so a product +/// term at index `i + j >= D` wraps to index `(i + j) - D` with a sign flip. +#[derive(Debug, Clone, Copy, PartialEq, Eq)] +pub struct CyclotomicRing { + pub(crate) coeffs: [F; D], +} + +impl CyclotomicRing { + /// Construct from a coefficient array. + #[inline] + pub fn from_coefficients(coeffs: [F; D]) -> Self { + Self { coeffs } + } + + /// Borrow the coefficient array. + #[inline] + pub fn coefficients(&self) -> &[F; D] { + &self.coeffs + } + + /// The additive identity (all-zero polynomial). + #[inline] + pub fn zero() -> Self { + Self { + coeffs: [F::zero(); D], + } + } + + /// The multiplicative identity (`1 + 0*X + ... + 0*X^{D-1}`). + #[inline] + pub fn one() -> Self { + let mut coeffs = [F::zero(); D]; + coeffs[0] = F::one(); + Self { coeffs } + } + + /// The monomial `X` (i.e., `[0, 1, 0, ..., 0]`). + /// + /// # Panics + /// + /// Panics if `D < 2`. + #[inline] + pub fn x() -> Self { + assert!(D >= 2, "ring degree must be at least 2"); + let mut coeffs = [F::zero(); D]; + coeffs[1] = F::one(); + Self { coeffs } + } + + /// Scalar multiplication: multiply every coefficient by `k`. + #[inline] + pub fn scale(&self, k: &F) -> Self { + let mut out = self.coeffs; + for c in &mut out { + *c = *c * *k; + } + Self { coeffs: out } + } + + /// Apply the cyclotomic automorphism `sigma_k: X -> X^k` for odd `k`. + /// + /// In `Z_q[X]/(X^D + 1)`, this permutes/sign-flips coefficients using + /// exponent reduction modulo `2D`. + /// + /// # Panics + /// + /// Panics if `D == 0` or `k` is not odd modulo `2D`. + pub fn sigma(&self, k: usize) -> Self { + assert!(D > 0, "ring degree must be non-zero"); + let two_d = 2 * D; + let k_mod = k % two_d; + assert!(k_mod % 2 == 1, "sigma_k requires odd k in Z_q[X]/(X^D + 1)"); + + let mut out = [F::zero(); D]; + for (j, coeff) in self.coeffs.iter().copied().enumerate() { + let idx = (j * k_mod) % two_d; + if idx < D { + out[idx] = out[idx] + coeff; + } else { + out[idx - D] = out[idx - D] - coeff; + } + } + Self { coeffs: out } + } + + /// Apply `sigma_{-1}` (`X -> X^{-1} = X^{2D-1}` in this ring). + /// + /// # Panics + /// + /// Panics if `D == 0`. + pub fn sigma_m1(&self) -> Self { + assert!(D > 0, "ring degree must be non-zero"); + self.sigma(2 * D - 1) + } + + /// Multiply by `X^k` in `Z_q[X]/(X^D + 1)` via O(D) coefficient rotation. + /// + /// Since `X^D = -1`, coefficients that wrap past index `D` get negated. + #[inline] + pub fn negacyclic_shift(&self, k: usize) -> Self { + let k = k % D; + if k == 0 { + return *self; + } + let mut out = [F::zero(); D]; + for i in 0..D { + let target = i + k; + if target < D { + out[target] = self.coeffs[i]; + } else { + out[target - D] = -self.coeffs[i]; + } + } + Self { coeffs: out } + } + + /// Multiply `self` by a sum of monomials `X^{k_1} + X^{k_2} + ...` + /// + /// Each term is a negacyclic shift, so the total cost is + /// `O(positions.len() * D)` field additions with zero multiplications. + pub fn mul_by_monomial_sum(&self, nonzero_positions: &[usize]) -> Self { + let mut result = Self::zero(); + for &k in nonzero_positions { + result += self.negacyclic_shift(k); + } + result + } + + /// Multiply `self` by a sparse challenge element. + /// + /// Cost: `O(omega * D)` field additions instead of `O(D^2)` multiplications. + /// For `omega=23, D=256` this is 5,888 adds vs 65,536 muls. + pub fn mul_by_sparse(&self, challenge: &SparseChallenge) -> Self + where + F: CanonicalField, + { + let mut result = Self::zero(); + for (&pos, &coeff) in challenge.positions.iter().zip(challenge.coeffs.iter()) { + let shifted = self.negacyclic_shift(pos as usize); + match coeff { + 1 => result += shifted, + -1 => result -= shifted, + c => result += shifted.scale(&F::from_i64(c as i64)), + } + } + result + } + + /// Check whether all coefficients are zero. + #[inline] + pub fn is_zero(&self) -> bool { + self.coeffs.iter().all(|c| c.is_zero()) + } + + /// Count non-zero coefficients. + #[inline] + pub fn hamming_weight(&self) -> usize { + self.coeffs.iter().filter(|c| !c.is_zero()).count() + } + + /// Sample a sparse challenge with exactly `omega` non-zeros in `{+1, -1}`. + /// + /// # Panics + /// + /// Panics if `omega > D` or `D == 0` with non-zero `omega`. + pub fn sample_sparse_pm1(rng: &mut R, omega: usize) -> Self { + assert!(omega <= D, "omega must be <= ring degree"); + assert!(D > 0 || omega == 0, "ring degree must be non-zero"); + + let mut coeffs = [F::zero(); D]; + let mut placed = 0usize; + while placed < omega { + let idx = (rng.next_u64() % (D as u64)) as usize; + if coeffs[idx].is_zero() { + coeffs[idx] = if (rng.next_u32() & 1) == 0 { + F::one() + } else { + -F::one() + }; + placed += 1; + } + } + Self { coeffs } + } +} + +impl CyclotomicRing { + /// Functional gadget recomposition (`G * digits`) for base `2^log_basis`. + /// + /// Coefficients from each part are interpreted as one digit plane and + /// recombined back into canonical integers (then reduced into the field). + /// + /// # Panics + /// + /// Panics if `log_basis == 0`, `log_basis >= 128`, or `parts.len() * log_basis > 128`. + pub fn gadget_recompose_pow2(parts: &[Self], log_basis: u32) -> Self { + if parts.is_empty() { + return Self::zero(); + } + + assert!(log_basis > 0 && log_basis < 128, "invalid log_basis"); + + let b = F::from_canonical_u128_reduced(1u128 << log_basis); + let coeffs = std::array::from_fn(|i| { + let mut acc = F::zero(); + let mut power = F::one(); + for part in parts.iter() { + acc = acc + part.coeffs[i] * power; + power = power * b; + } + acc + }); + Self { coeffs } + } + + /// Balanced (centered) base-`2^log_basis` gadget decomposition: `G^{-1}`. + /// + /// Each coefficient `c` (centered into `(-q/2, q/2]`) is decomposed into + /// `levels` balanced digits `d_k ∈ [-b/2, b/2)` satisfying + /// `c ≡ Σ_k d_k · b^k (mod q)`. + /// + /// Negative digits are stored as their field representation (`q + d`). + /// + /// # Panics + /// + /// Panics if `log_basis == 0`, `log_basis >= 128`, or `levels * log_basis > 128`. + pub fn balanced_decompose_pow2(&self, levels: usize, log_basis: u32) -> Vec { + assert!(log_basis > 0 && log_basis < 128, "invalid log_basis"); + assert!( + (levels as u32).saturating_mul(log_basis) <= 128, + "levels * log_basis must be <= 128" + ); + + let b = 1i128 << log_basis; + let half_b = b / 2; + let q = (-F::one()).to_canonical_u128() as i128 + 1; + let half_q = q / 2; + + let mut digit_planes: Vec<[F; D]> = (0..levels).map(|_| [F::zero(); D]).collect(); + + for i in 0..D { + let canonical = self.coeffs[i].to_canonical_u128() as i128; + let mut c = if canonical > half_q { + canonical - q + } else { + canonical + }; + + for plane in digit_planes.iter_mut() { + let d = c.rem_euclid(b); + let balanced = if d >= half_b { d - b } else { d }; + c = (c - balanced) / b; + + plane[i] = if balanced >= 0 { + F::from_canonical_u128_reduced(balanced as u128) + } else { + F::from_canonical_u128_reduced((q + balanced) as u128) + }; + } + } + + digit_planes + .into_iter() + .map(Self::from_coefficients) + .collect() + } +} + +impl CyclotomicRing { + /// Generate a random ring element. + pub fn random(rng: &mut R) -> Self { + Self { + coeffs: std::array::from_fn(|_| F::sample(rng)), + } + } +} + +impl AddAssign for CyclotomicRing { + fn add_assign(&mut self, rhs: Self) { + for (dst, src) in self.coeffs.iter_mut().zip(rhs.coeffs.iter()) { + *dst = *dst + *src; + } + } +} + +impl SubAssign for CyclotomicRing { + fn sub_assign(&mut self, rhs: Self) { + for (dst, src) in self.coeffs.iter_mut().zip(rhs.coeffs.iter()) { + *dst = *dst - *src; + } + } +} + +impl Add for CyclotomicRing { + type Output = Self; + fn add(mut self, rhs: Self) -> Self { + self += rhs; + self + } +} + +impl Sub for CyclotomicRing { + type Output = Self; + fn sub(mut self, rhs: Self) -> Self { + self -= rhs; + self + } +} + +impl Neg for CyclotomicRing { + type Output = Self; + fn neg(self) -> Self { + let mut out = self.coeffs; + for c in &mut out { + *c = -*c; + } + Self { coeffs: out } + } +} + +impl MulAssign for CyclotomicRing { + fn mul_assign(&mut self, rhs: Self) { + *self = *self * rhs; + } +} + +impl<'a, F: FieldCore, const D: usize> Add<&'a Self> for CyclotomicRing { + type Output = Self; + fn add(self, rhs: &'a Self) -> Self { + self + *rhs + } +} + +impl<'a, F: FieldCore, const D: usize> Sub<&'a Self> for CyclotomicRing { + type Output = Self; + fn sub(self, rhs: &'a Self) -> Self { + self - *rhs + } +} + +impl<'a, F: FieldCore, const D: usize> Mul<&'a Self> for CyclotomicRing { + type Output = Self; + fn mul(self, rhs: &'a Self) -> Self { + self * *rhs + } +} + +/// Schoolbook negacyclic convolution: O(D^2). +/// +/// For each pair `(i, j)`: +/// - If `i + j < D`: accumulate `a_i * b_j` at index `i + j`. +/// - If `i + j >= D`: accumulate `-(a_i * b_j)` at index `(i + j) - D`. +impl Mul for CyclotomicRing { + type Output = Self; + fn mul(self, rhs: Self) -> Self { + let mut out = [F::zero(); D]; + for i in 0..D { + for j in 0..D { + let product = self.coeffs[i] * rhs.coeffs[j]; + let idx = i + j; + if idx < D { + out[idx] = out[idx] + product; + } else { + out[idx - D] = out[idx - D] - product; + } + } + } + Self { coeffs: out } + } +} + +impl Valid for CyclotomicRing { + fn check(&self) -> Result<(), SerializationError> { + for x in self.coeffs.iter() { + x.check()?; + } + Ok(()) + } +} + +impl HachiSerialize for CyclotomicRing { + fn serialize_with_mode( + &self, + mut writer: W, + compress: Compress, + ) -> Result<(), SerializationError> { + for x in self.coeffs.iter() { + x.serialize_with_mode(&mut writer, compress)?; + } + Ok(()) + } + + fn serialized_size(&self, compress: Compress) -> usize { + self.coeffs + .iter() + .map(|x| x.serialized_size(compress)) + .sum() + } +} + +impl HachiDeserialize for CyclotomicRing { + fn deserialize_with_mode( + mut reader: R, + compress: Compress, + validate: Validate, + ) -> Result { + let mut coeffs = [F::zero(); D]; + for c in &mut coeffs { + *c = F::deserialize_with_mode(&mut reader, compress, validate)?; + } + let out = Self { coeffs }; + if matches!(validate, Validate::Yes) { + out.check()?; + } + Ok(out) + } +} + +impl Default for CyclotomicRing { + fn default() -> Self { + Self::zero() + } +} diff --git a/src/algebra/ring/mod.rs b/src/algebra/ring/mod.rs new file mode 100644 index 00000000..1b64555e --- /dev/null +++ b/src/algebra/ring/mod.rs @@ -0,0 +1,9 @@ +//! Cyclotomic ring types and NTT representations. + +pub mod crt_ntt_repr; +pub mod cyclotomic; +pub mod sparse_challenge; + +pub use crt_ntt_repr::{CrtNttConvertibleField, CyclotomicCrtNtt}; +pub use cyclotomic::CyclotomicRing; +pub use sparse_challenge::{SparseChallenge, SparseChallengeConfig}; diff --git a/src/algebra/ring/sparse_challenge.rs b/src/algebra/ring/sparse_challenge.rs new file mode 100644 index 00000000..4f55b06a --- /dev/null +++ b/src/algebra/ring/sparse_challenge.rs @@ -0,0 +1,164 @@ +//! Sparse ring challenges for cyclotomic protocols. +//! +//! Many lattice protocols sample "short/sparse" ring challenges whose coefficients +//! are mostly zero and whose non-zero coefficients come from a tiny integer alphabet +//! (e.g. `{±1}` or `{±1,±2}`), with a fixed Hamming weight `ω`. +//! +//! This module provides a minimal representation that is: +//! - independent of any specific protocol (Hachi/Greyhound/SuperNeo, etc.), +//! - easy to sample deterministically from Fiat–Shamir at the protocol layer, +//! - and efficient to evaluate at a point `α` using precomputed powers. + +use super::CyclotomicRing; +use crate::{CanonicalField, FieldCore}; + +/// Configuration for sampling a sparse challenge. +/// +/// This intentionally avoids redundant knobs: the distribution is determined by: +/// - exact `weight` (Hamming weight), +/// - and a list of allowed **non-zero** integer coefficients. +#[derive(Debug, Clone, PartialEq, Eq)] +pub struct SparseChallengeConfig { + /// Exact Hamming weight ω. + pub weight: usize, + /// Allowed non-zero coefficients (small signed integers). + /// + /// Examples: + /// - `{±1}`: `vec![-1, 1]` + /// - `{±1,±2}`: `vec![-2, -1, 1, 2]` + pub nonzero_coeffs: Vec, +} + +impl SparseChallengeConfig { + /// Validate basic invariants for a given ring degree `D`. + /// + /// # Errors + /// + /// Returns an error if `weight > D`, if `nonzero_coeffs` is empty, or if it + /// contains `0`. + pub fn validate(&self) -> Result<(), &'static str> { + if self.weight > D { + return Err("weight must be <= ring degree D"); + } + if self.nonzero_coeffs.is_empty() { + return Err("nonzero_coeffs must be non-empty"); + } + if self.nonzero_coeffs.contains(&0) { + return Err("nonzero_coeffs must not contain 0"); + } + Ok(()) + } +} + +/// Sparse polynomial in `F[X]/(X^D+1)` represented by its non-zero terms. +/// +/// Invariants: +/// - `positions.len() == coeffs.len()` +/// - all positions are `< D` +/// - positions are unique +/// - all coeffs are non-zero +#[derive(Debug, Clone, PartialEq, Eq)] +pub struct SparseChallenge { + /// Coefficient indices (powers of `X`) where the polynomial is non-zero. + pub positions: Vec, + /// Small integer coefficients at the corresponding positions. + pub coeffs: Vec, +} + +impl SparseChallenge { + /// Construct an empty (all-zero) challenge. + #[inline] + pub fn zero() -> Self { + Self { + positions: Vec::new(), + coeffs: Vec::new(), + } + } + + /// Number of non-zero coefficients (Hamming weight). + #[inline] + pub fn hamming_weight(&self) -> usize { + debug_assert_eq!(self.positions.len(), self.coeffs.len()); + self.positions.len() + } + + /// ℓ₁ norm over integers: `Σ |coeff_i|`. + #[inline] + pub fn l1_norm(&self) -> u64 { + self.coeffs + .iter() + .map(|&c| (c as i32).unsigned_abs() as u64) + .sum() + } + + /// Validate structural invariants for a ring degree `D`. + /// + /// # Errors + /// + /// Returns an error if lengths mismatch, if any coefficient is zero, if any + /// position is out of range, or if positions contain duplicates. + pub fn validate(&self) -> Result<(), &'static str> { + if self.positions.len() != self.coeffs.len() { + return Err("positions and coeffs must have same length"); + } + // Check coeffs are non-zero and positions are in range + unique. + let mut seen = vec![false; D]; + for (&pos, &c) in self.positions.iter().zip(self.coeffs.iter()) { + if c == 0 { + return Err("coeffs must not contain 0"); + } + let p = pos as usize; + if p >= D { + return Err("position out of range"); + } + if seen[p] { + return Err("positions must be unique"); + } + seen[p] = true; + } + Ok(()) + } + + /// Convert to a dense ring element by placing coefficients in the canonical + /// coefficient basis. + /// + /// # Errors + /// + /// Returns an error if the sparse representation violates structural invariants. + pub fn to_dense( + &self, + ) -> Result, &'static str> { + self.validate::()?; + let mut out = [F::zero(); D]; + for (&pos, &c) in self.positions.iter().zip(self.coeffs.iter()) { + out[pos as usize] = out[pos as usize] + F::from_i64(c as i64); + } + Ok(CyclotomicRing::from_coefficients(out)) + } + + /// Evaluate this sparse polynomial at `α` in `E`, given precomputed powers + /// `[α^0, α^1, ..., α^{D-1}]`. + /// + /// This is `O(weight)` and is intended to be used for verifier-side oracles + /// where `D` may be large but `weight` is small. + /// + /// # Errors + /// + /// Returns an error if structural invariants fail or if `alpha_pows.len() != D`. + pub fn eval_at_alpha(&self, alpha_pows: &[E]) -> Result + where + F: FieldCore + CanonicalField, + E: FieldCore + crate::algebra::fields::LiftBase, + { + self.validate::()?; + if alpha_pows.len() != D { + return Err("alpha_pows length mismatch"); + } + let mut acc = E::zero(); + for (&pos, &c) in self.positions.iter().zip(self.coeffs.iter()) { + let coeff_f = F::from_i64(c as i64); + acc = acc + (E::lift_base(coeff_f) * alpha_pows[pos as usize]); + } + Ok(acc) + } +} diff --git a/src/lib.rs b/src/lib.rs index e3859c54..eb5ac81c 100644 --- a/src/lib.rs +++ b/src/lib.rs @@ -1,3 +1,7 @@ +#![cfg_attr( + all(target_arch = "x86_64", target_feature = "avx512f"), + feature(stdarch_x86_avx512) +)] //! # hachi //! //! A high performance and modular implementation of the Hachi polynomial commitment scheme. @@ -35,7 +39,21 @@ pub mod error; /// Primitive traits and operations pub mod primitives; +/// Concrete algebra backends (prime fields, extensions, rings) +pub mod algebra; + +/// Protocol-layer transcript and commitment abstractions +pub mod protocol; + +#[doc(hidden)] +#[allow(missing_docs)] +pub mod test_utils; + pub use error::HachiError; -pub use primitives::arithmetic::{Field, HachiRoutines, Module}; +pub use primitives::arithmetic::{ + CanonicalField, FieldCore, FieldSampling, HachiRoutines, Invertible, Module, + PseudoMersenneField, +}; pub use primitives::poly::{MultilinearLagrange, Polynomial}; pub use primitives::serialization::{HachiDeserialize, HachiSerialize}; +pub use protocol::{CommitmentScheme, StreamingCommitmentScheme, Transcript}; diff --git a/src/primitives/arithmetic.rs b/src/primitives/arithmetic.rs index ccaaa4ae..903b3a17 100644 --- a/src/primitives/arithmetic.rs +++ b/src/primitives/arithmetic.rs @@ -3,8 +3,8 @@ use super::{HachiDeserialize, HachiSerialize}; use rand_core::RngCore; -/// Field trait for lattice-based arithmetic -pub trait Field: +/// Core field operations required across algebra backends. +pub trait FieldCore: Sized + Clone + Copy @@ -39,17 +39,55 @@ pub trait Field: /// Field multiplication fn mul(&self, rhs: &Self) -> Self; - /// Field inversion + /// Field inversion. + /// + /// This API may branch on zero-check and is intended for public/non-secret + /// values. For secret-bearing paths, use [`Invertible::inv_or_zero`]. fn inv(self) -> Option; +} - /// Generate random field element - fn random(rng: &mut R) -> Self; +/// Constant-time inversion helper for secret-bearing code paths. +/// +/// Implementations return `0` when the input is `0`, and `x^{-1}` otherwise, +/// without branching on the input value. +pub trait Invertible: FieldCore { + /// Constant-time inversion with zero-mapping behavior. + fn inv_or_zero(self) -> Self; +} - /// Convert from u64 +/// Canonical conversion operations for field elements. +pub trait CanonicalField: FieldCore { + /// Convert from `u64`. fn from_u64(val: u64) -> Self; - /// Convert from i64 + /// Convert from `i64`. fn from_i64(val: i64) -> Self; + + /// Return canonical integer representation as `u128`. + fn to_canonical_u128(self) -> u128; + + /// Construct from canonical value if it is in range. + fn from_canonical_u128_checked(val: u128) -> Option; + + /// Construct from canonical value reduced modulo the field modulus. + fn from_canonical_u128_reduced(val: u128) -> Self; +} + +/// Optional sampling support for field elements. +/// +/// This is intentionally separate from core field algebra and may evolve. +pub trait FieldSampling: FieldCore { + /// Generate a sampled field element. + fn sample(rng: &mut R) -> Self; +} + +/// Metadata for pseudo-Mersenne style moduli (`2^k - c`). +pub trait PseudoMersenneField: CanonicalField { + /// Exponent `k` in `2^k - c`. + const MODULUS_BITS: u32; + + /// Offset `c` in `2^k - c`. + const MODULUS_OFFSET: u128; } /// Module trait for lattice-based algebraic structures @@ -73,7 +111,9 @@ pub trait Module: + for<'a> std::ops::Sub<&'a Self, Output = Self> { /// Scalar type (field/ring elements) - type Scalar: Field + type Scalar: FieldCore + + CanonicalField + + FieldSampling + std::ops::Mul + for<'a> std::ops::Mul<&'a Self, Output = Self>; diff --git a/src/primitives/mod.rs b/src/primitives/mod.rs index 9f53c2f1..21376422 100644 --- a/src/primitives/mod.rs +++ b/src/primitives/mod.rs @@ -2,6 +2,7 @@ //! This submodule defines the basic lattice arithmetic and cryptographic tools that Hachi is built upon pub mod arithmetic; +pub mod multilinear_evals; pub mod poly; pub mod serialization; pub mod transcript; diff --git a/src/primitives/multilinear_evals.rs b/src/primitives/multilinear_evals.rs new file mode 100644 index 00000000..2ba3b95a --- /dev/null +++ b/src/primitives/multilinear_evals.rs @@ -0,0 +1,155 @@ +//! Dense multilinear polynomials in evaluation form. +//! +//! This module intentionally follows the same high-level representation style as +//! Jolt's `DensePolynomial` for multilinear extensions (MLEs): store the values +//! of the multilinear polynomial on the Boolean hypercube `{0,1}^n` and provide +//! binding/evaluation by iterative folding. +//! +//! The key convention for this repo (used by the ring-switch witness table) is: +//! +//! - An evaluation index `idx` is interpreted in binary. +//! - The **lowest** index bit is the **first** variable bound under +//! [`BindingOrder::LowToHigh`]. +//! +//! This matches the row-major flattening `idx = row * d + col` when `d` is a +//! power of two: the low `log2(d)` bits correspond to the `col` coordinate. + +use crate::primitives::arithmetic::FieldCore; +use crate::primitives::poly::Polynomial; + +/// The order in which variables are bound/evaluated. +#[derive(Debug, Clone, Copy, PartialEq, Eq, Default)] +pub enum BindingOrder { + /// Bind the lowest index bit first (LSB → MSB). + #[default] + LowToHigh, + /// Bind the highest index bit first (MSB → LSB). + HighToLow, +} + +/// Dense multilinear polynomial in evaluation form over `{0,1}^num_vars`. +#[derive(Debug, Clone, PartialEq, Eq)] +pub struct DenseMultilinearEvals { + /// Number of variables in the multilinear extension. + pub num_vars: usize, + /// Active length (decreases as variables are bound). + pub len: usize, + /// Evaluations on the hypercube, length is a power of two. + pub evals: Vec, +} + +impl Default for DenseMultilinearEvals { + fn default() -> Self { + Self { + num_vars: 0, + len: 1, + evals: vec![F::zero()], + } + } +} + +impl DenseMultilinearEvals { + /// Construct from evaluations, padding with zeros to a power of two. + /// + /// The variable count is derived from the padded length. + pub fn new_padded(mut evals: Vec) -> Self { + if evals.is_empty() { + evals.push(F::zero()); + } + while !evals.len().is_power_of_two() { + evals.push(F::zero()); + } + let num_vars = evals.len().trailing_zeros() as usize; + let len = evals.len(); + Self { + num_vars, + len, + evals, + } + } + + /// Return the original (backing) evaluation length. + pub fn original_len(&self) -> usize { + self.evals.len() + } + + /// Bind one variable in-place, reducing `len` by a factor of 2. + /// + /// After binding, the polynomial has one fewer variable. + /// + /// # Panics + /// + /// Panics if the current backing length is not a power of two, or if attempting + /// to bind a constant (length-1) polynomial. + pub fn bind_in_place(&mut self, r: F, order: BindingOrder) { + assert!(self.len.is_power_of_two()); + assert!(self.len >= 2, "cannot bind variable of constant polynomial"); + match order { + BindingOrder::LowToHigh => self.bind_lsb_in_place(r), + BindingOrder::HighToLow => self.bind_msb_in_place(r), + } + } + + #[inline] + fn bind_lsb_in_place(&mut self, r: F) { + let next_len = self.len / 2; + for i in 0..next_len { + let v0 = self.evals[i << 1]; + let v1 = self.evals[(i << 1) | 1]; + // (1-r)*v0 + r*v1 = v0 + r*(v1-v0) + self.evals[i] = v0 + r * (v1 - v0); + } + self.len = next_len; + self.num_vars = self.num_vars.saturating_sub(1); + } + + #[inline] + fn bind_msb_in_place(&mut self, r: F) { + let next_len = self.len / 2; + let (left, right) = self.evals.split_at_mut(next_len); + for i in 0..next_len { + let v0 = left[i]; + let v1 = right[i]; + left[i] = v0 + r * (v1 - v0); + } + self.len = next_len; + self.num_vars = self.num_vars.saturating_sub(1); + } + + /// Evaluate without mutating `self`. + /// + /// # Panics + /// + /// Panics if `point.len() != self.num_vars`. + pub fn evaluate_with_order(&self, point: &[F], order: BindingOrder) -> F { + if point.is_empty() { + return self.evals[0]; + } + assert_eq!( + point.len(), + self.num_vars, + "point dimension mismatch: expected {}, got {}", + self.num_vars, + point.len() + ); + let mut tmp = self.clone(); + for r in point.iter().copied() { + tmp.bind_in_place(r, order); + } + tmp.evals[0] + } +} + +impl Polynomial for DenseMultilinearEvals { + fn num_vars(&self) -> usize { + self.num_vars + } + + fn evaluate(&self, point: &[F]) -> F { + self.evaluate_with_order(point, BindingOrder::LowToHigh) + } + + fn coeffs(&self) -> Vec { + self.evals[..self.len].to_vec() + } +} diff --git a/src/primitives/poly.rs b/src/primitives/poly.rs index 1e9a8b63..16f034cd 100644 --- a/src/primitives/poly.rs +++ b/src/primitives/poly.rs @@ -1,9 +1,9 @@ -//! Polynomial trait for multilinear polynomials +//! Polynomial trait for multilinear polynomials. -use super::arithmetic::Field; +use super::arithmetic::FieldCore; /// Trait for multilinear Lagrange polynomial operations -pub trait MultilinearLagrange: Polynomial { +pub trait MultilinearLagrange: Polynomial { /// Compute multilinear Lagrange basis evaluations at a point /// /// For variables (r₀, r₁, ..., r_{n-1}), computes all 2^n basis polynomial evaluations. @@ -30,7 +30,7 @@ pub trait MultilinearLagrange: Polynomial { /// Trait for multilinear polynomials /// /// Represents a polynomial in evaluation form (coefficients at hypercube points). -pub trait Polynomial { +pub trait Polynomial { /// Number of variables fn num_vars(&self) -> usize; @@ -52,6 +52,9 @@ pub trait Polynomial { /// # Returns /// Polynomial evaluation result fn evaluate(&self, point: &[F]) -> F; + + /// Return the coefficient/evaluation table on `{0,1}^n` in LSB-first order. + fn coeffs(&self) -> Vec; } /// Compute multilinear Lagrange basis evaluations at a point @@ -62,7 +65,7 @@ pub trait Polynomial { /// Uses an iterative doubling approach: /// - Start with [1-r₀, r₀] /// - For each variable rᵢ, split each value v into [v*(1-rᵢ), v*rᵢ] -pub(crate) fn multilinear_lagrange_basis(output: &mut [F], point: &[F]) { +pub(crate) fn multilinear_lagrange_basis(output: &mut [F], point: &[F]) { assert!( output.len() <= (1 << point.len()), "Output length must be at most 2^point.len()" @@ -115,7 +118,7 @@ pub(crate) fn multilinear_lagrange_basis(output: &mut [F], point: &[F] /// polynomial_evaluation(point) = L^T × M × R /// /// Splits variables between rows and columns based on sigma and nu. -pub fn compute_left_right_vectors( +pub fn compute_left_right_vectors( point: &[F], nu: usize, sigma: usize, diff --git a/src/primitives/serialization.rs b/src/primitives/serialization.rs index f4a0bf1e..eaca5481 100644 --- a/src/primitives/serialization.rs +++ b/src/primitives/serialization.rs @@ -182,11 +182,13 @@ mod primitive_impls { impl_primitive_serialization!(u16, 2); impl_primitive_serialization!(u32, 4); impl_primitive_serialization!(u64, 8); + impl_primitive_serialization!(u128, 16); impl_primitive_serialization!(usize, std::mem::size_of::()); impl_primitive_serialization!(i8, 1); impl_primitive_serialization!(i16, 2); impl_primitive_serialization!(i32, 4); impl_primitive_serialization!(i64, 8); + impl_primitive_serialization!(i128, 16); impl Valid for bool { fn check(&self) -> Result<(), SerializationError> { diff --git a/src/primitives/transcript.rs b/src/primitives/transcript.rs index 89ebc29a..ce2e8f53 100644 --- a/src/primitives/transcript.rs +++ b/src/primitives/transcript.rs @@ -2,13 +2,13 @@ #![allow(missing_docs)] -use crate::primitives::arithmetic::{Field, Module}; +use crate::primitives::arithmetic::{CanonicalField, FieldCore, Module}; use crate::primitives::HachiSerialize; /// Transcript for Fiat-Shamir transformations pub trait Transcript { /// Field type for challenges - type Field: Field; + type Field: FieldCore + CanonicalField; /// Append raw bytes to the transcript fn append_bytes(&mut self, label: &[u8], bytes: &[u8]); diff --git a/src/protocol/challenges/mod.rs b/src/protocol/challenges/mod.rs new file mode 100644 index 00000000..524941ed --- /dev/null +++ b/src/protocol/challenges/mod.rs @@ -0,0 +1,6 @@ +//! Protocol-level Fiat–Shamir challenge samplers. +//! +//! These utilities derive structured challenges (e.g. sparse ring elements) from +//! the transcript while keeping the low-level representations in the algebra layer. + +pub mod sparse; diff --git a/src/protocol/challenges/sparse.rs b/src/protocol/challenges/sparse.rs new file mode 100644 index 00000000..aca54aaf --- /dev/null +++ b/src/protocol/challenges/sparse.rs @@ -0,0 +1,121 @@ +//! Sparse challenge sampling via Fiat–Shamir. + +use crate::algebra::ring::{CyclotomicRing, SparseChallenge, SparseChallengeConfig}; +use crate::error::HachiError; +use crate::protocol::transcript::labels::{ABSORB_SPARSE_CHALLENGE, CHALLENGE_SPARSE_CHALLENGE}; +use crate::protocol::transcript::Transcript; +use crate::{CanonicalField, FieldCore}; + +/// Sample a sparse ring challenge (exact weight ω) from a transcript. +/// +/// This is intentionally deterministic and label-aware: +/// - first we absorb the sampling context under `ABSORB_SPARSE_CHALLENGE`, +/// - then we derive as many `CHALLENGE_SPARSE_CHALLENGE` scalars as needed. +/// +/// Notes: +/// - Indices are sampled with a simple `mod D` reduction. For the intended +/// regimes (small `D`, cryptographic transcript), any bias is negligible. +/// - Duplicate indices are rejected to enforce exact Hamming weight. +/// +/// # Errors +/// +/// Returns an error if the provided config is invalid for degree `D`. +pub fn sparse_challenge_from_transcript( + transcript: &mut T, + label: &[u8], + instance_idx: u64, + cfg: &SparseChallengeConfig, +) -> Result +where + F: FieldCore + CanonicalField, + T: Transcript, +{ + cfg.validate::() + .map_err(|e| HachiError::InvalidInput(format!("invalid sparse challenge config: {e}")))?; + + // Absorb domain-separating context so different call sites can't collide. + transcript.append_bytes(ABSORB_SPARSE_CHALLENGE, label); + transcript.append_bytes(ABSORB_SPARSE_CHALLENGE, &instance_idx.to_le_bytes()); + transcript.append_bytes(ABSORB_SPARSE_CHALLENGE, &(D as u64).to_le_bytes()); + transcript.append_bytes(ABSORB_SPARSE_CHALLENGE, &(cfg.weight as u64).to_le_bytes()); + // Include the coefficient alphabet (as little-endian i16 stream). + let mut coeff_bytes = Vec::with_capacity(cfg.nonzero_coeffs.len() * 2); + for &c in cfg.nonzero_coeffs.iter() { + coeff_bytes.extend_from_slice(&c.to_le_bytes()); + } + transcript.append_bytes(ABSORB_SPARSE_CHALLENGE, &coeff_bytes); + + let mut seen = vec![false; D]; + let mut positions = Vec::with_capacity(cfg.weight); + let mut coeffs = Vec::with_capacity(cfg.weight); + + while positions.len() < cfg.weight { + let r = transcript + .challenge_scalar(CHALLENGE_SPARSE_CHALLENGE) + .to_canonical_u128(); + let lo = r as u64; + let hi = (r >> 64) as u64; + + let pos = (lo % (D as u64)) as usize; + if seen[pos] { + continue; + } + seen[pos] = true; + positions.push(pos as u32); + + let coeff_idx = (hi % (cfg.nonzero_coeffs.len() as u64)) as usize; + let c = cfg.nonzero_coeffs[coeff_idx]; + debug_assert_ne!(c, 0); + coeffs.push(c); + } + + Ok(SparseChallenge { positions, coeffs }) +} + +/// Sample `n` sparse challenges from a transcript, returning the sparse +/// representation directly. +/// +/// # Errors +/// +/// Returns an error if challenge sampling fails. +pub fn sample_sparse_challenges( + transcript: &mut T, + label: &[u8], + n: usize, + cfg: &SparseChallengeConfig, +) -> Result, HachiError> +where + F: FieldCore + CanonicalField, + T: Transcript, +{ + (0..n) + .map(|i| sparse_challenge_from_transcript::(transcript, label, i as u64, cfg)) + .collect() +} + +/// Sample `n` sparse challenges from a transcript and convert them to dense +/// `CyclotomicRing` elements. +/// +/// # Errors +/// +/// Returns an error if challenge sampling or dense conversion fails. +pub fn sample_dense_challenges( + transcript: &mut T, + label: &[u8], + n: usize, + cfg: &SparseChallengeConfig, +) -> Result>, HachiError> +where + F: FieldCore + CanonicalField, + T: Transcript, +{ + (0..n) + .map(|i| { + let sparse = + sparse_challenge_from_transcript::(transcript, label, i as u64, cfg)?; + sparse + .to_dense::() + .map_err(|e| HachiError::InvalidInput(e.to_string())) + }) + .collect() +} diff --git a/src/protocol/commitment/commit.rs b/src/protocol/commitment/commit.rs new file mode 100644 index 00000000..485a5570 --- /dev/null +++ b/src/protocol/commitment/commit.rs @@ -0,0 +1,146 @@ +//! Ring-native §4.1 commitment core implementation. + +use super::config::{ + ensure_block_layout, ensure_matrix_shape, ensure_supported_num_vars, validate_and_derive_layout, +}; +use super::onehot::{inner_ajtai_onehot, map_onehot_to_sparse_blocks}; +use super::scheme::RingCommitmentScheme; +use super::types::RingCommitment; +use super::utils::linear::{decompose_block, decompose_rows, mat_vec_mul_unchecked}; +use super::utils::matrix::{derive_public_matrix, sample_public_matrix_seed, PublicMatrixSeed}; +use super::CommitmentConfig; +use crate::algebra::ring::CyclotomicRing; +use crate::error::HachiError; +use crate::{CanonicalField, FieldCore, FieldSampling}; + +/// Unified setup for the ring-native commitment (§4.1) and prover (§4.2). +#[allow(non_snake_case)] +#[derive(Debug, Clone, PartialEq, Eq)] +pub struct RingCommitmentSetup { + /// Maximum supported variable count. + pub max_num_vars: usize, + /// Public seed used to derive commitment matrices. + pub public_matrix_seed: PublicMatrixSeed, + /// Inner matrix `A`. + pub A: Vec>>, + /// Outer matrix `B`. + pub B: Vec>>, + /// Prover matrix `D ∈ R_q^{n_D × δ·2^R}` (§4.2). + pub D: Vec>>, +} + +/// Concrete §4.1 commitment core. +#[derive(Clone, Copy, Default)] +pub struct HachiCommitmentCore; + +impl RingCommitmentScheme for HachiCommitmentCore +where + F: FieldCore + CanonicalField + FieldSampling, + Cfg: CommitmentConfig, +{ + type ProverSetup = RingCommitmentSetup; + type VerifierSetup = RingCommitmentSetup; + type Commitment = RingCommitment; + + fn setup(max_num_vars: usize) -> Result<(Self::ProverSetup, Self::VerifierSetup), HachiError> { + let layout = validate_and_derive_layout::()?; + ensure_supported_num_vars(max_num_vars, layout.required_vars)?; + let public_matrix_seed = sample_public_matrix_seed(); + let a_matrix = + derive_public_matrix::(Cfg::N_A, layout.inner_width, &public_matrix_seed, b"A"); + let b_matrix = + derive_public_matrix::(Cfg::N_B, layout.outer_width, &public_matrix_seed, b"B"); + let d_matrix = derive_public_matrix::( + Cfg::N_D, + layout.d_matrix_width, + &public_matrix_seed, + b"D", + ); + + let setup = RingCommitmentSetup { + max_num_vars, + public_matrix_seed, + A: a_matrix, + B: b_matrix, + D: d_matrix, + }; + ensure_matrix_shape(&setup.A, Cfg::N_A, layout.inner_width, "A")?; + ensure_matrix_shape(&setup.B, Cfg::N_B, layout.outer_width, "B")?; + ensure_matrix_shape(&setup.D, Cfg::N_D, layout.d_matrix_width, "D")?; + Ok((setup.clone(), setup)) + } + + #[allow(clippy::type_complexity)] + fn commit_ring_blocks( + f_blocks: &[Vec>], + setup: &Self::ProverSetup, + ) -> Result< + ( + Self::Commitment, + Vec>>, + Vec>>, + ), + HachiError, + > { + let layout = validate_and_derive_layout::()?; + ensure_supported_num_vars(setup.max_num_vars, layout.required_vars)?; + ensure_block_layout(f_blocks, layout)?; + ensure_matrix_shape(&setup.A, Cfg::N_A, layout.inner_width, "A")?; + ensure_matrix_shape(&setup.B, Cfg::N_B, layout.outer_width, "B")?; + + let mut s_all: Vec>> = Vec::with_capacity(layout.num_blocks); + let mut t_hat_all: Vec>> = Vec::with_capacity(layout.num_blocks); + let mut t_hat_flat: Vec> = Vec::with_capacity(layout.outer_width); + for block in f_blocks { + let s_i = decompose_block(block, Cfg::DELTA, Cfg::LOG_BASIS); + + let t_i = mat_vec_mul_unchecked(&setup.A, &s_i); + let t_hat_i = decompose_rows(&t_i, Cfg::DELTA, Cfg::LOG_BASIS); + t_hat_flat.extend(t_hat_i.iter().copied()); + + s_all.push(s_i); + t_hat_all.push(t_hat_i); + } + + let u = mat_vec_mul_unchecked(&setup.B, &t_hat_flat); + Ok((RingCommitment { u }, s_all, t_hat_all)) + } + + #[allow(clippy::type_complexity)] + fn commit_onehot( + onehot_k: usize, + indices: &[usize], + setup: &Self::ProverSetup, + ) -> Result< + ( + Self::Commitment, + Vec>>, + Vec>>, + ), + HachiError, + > { + let layout = validate_and_derive_layout::()?; + ensure_supported_num_vars(setup.max_num_vars, layout.required_vars)?; + ensure_matrix_shape(&setup.A, Cfg::N_A, layout.inner_width, "A")?; + ensure_matrix_shape(&setup.B, Cfg::N_B, layout.outer_width, "B")?; + + let sparse_blocks = map_onehot_to_sparse_blocks(onehot_k, indices, Cfg::R, Cfg::M, D)?; + + let mut s_all: Vec>> = Vec::with_capacity(layout.num_blocks); + let mut t_hat_all: Vec>> = Vec::with_capacity(layout.num_blocks); + let mut t_hat_flat: Vec> = Vec::with_capacity(layout.outer_width); + + for block_entries in &sparse_blocks { + let (t_i, s_i) = + inner_ajtai_onehot(&setup.A, block_entries, layout.block_len, Cfg::DELTA); + let t_hat_i = decompose_rows(&t_i, Cfg::DELTA, Cfg::LOG_BASIS); + t_hat_flat.extend(t_hat_i.iter().copied()); + + s_all.push(s_i); + t_hat_all.push(t_hat_i); + } + + let u = mat_vec_mul_unchecked(&setup.B, &t_hat_flat); + Ok((RingCommitment { u }, s_all, t_hat_all)) + } +} diff --git a/src/protocol/commitment/config.rs b/src/protocol/commitment/config.rs new file mode 100644 index 00000000..3150554e --- /dev/null +++ b/src/protocol/commitment/config.rs @@ -0,0 +1,186 @@ +//! Configuration presets for ring-native commitment construction. + +use super::utils::math::checked_pow2; +use crate::algebra::ring::CyclotomicRing; +use crate::error::HachiError; +use crate::FieldCore; + +/// Parameter bundle for the ring-native commitment core (§4.1–§4.2). +pub trait CommitmentConfig: Clone + Send + Sync + 'static { + /// Ring degree used by `CyclotomicRing`. + const D: usize; + /// Number of variables inside each committed block (`2^M` entries). + const M: usize; + /// Number of block-select variables (`2^R` blocks). + const R: usize; + /// Inner Ajtai matrix row count. + const N_A: usize; + /// Outer commitment matrix row count. + const N_B: usize; + /// Prover commitment matrix `D` row count (§4.2). + const N_D: usize; + /// Base-2 logarithm of gadget decomposition base. + const LOG_BASIS: u32; + /// Decomposition levels `delta`. + const DELTA: usize; + /// Decomposition levels for the folded witness `z` (`τ` in the paper). + const TAU: usize; + /// L∞ norm bound for `z` (`β` in the paper). Prover aborts if exceeded. + const BETA: u128; + /// Hamming weight of sparse challenges (`ω` in the paper). + const CHALLENGE_WEIGHT: usize; +} + +/// Runtime-derived dimensions from a `CommitmentConfig`. +#[derive(Debug, Clone, Copy)] +pub(super) struct CommitmentLayout { + /// Number of committed blocks (`2^R`). + pub(super) num_blocks: usize, + /// Number of ring elements per block (`2^M`). + pub(super) block_len: usize, + /// Width of inner matrix `A`. + pub(super) inner_width: usize, + /// Width of outer matrix `B`. + pub(super) outer_width: usize, + /// Width of prover matrix `D` (`δ · 2^R`). + pub(super) d_matrix_width: usize, + /// Minimum variable count supported by this config. + pub(super) required_vars: usize, +} + +/// Validate static config invariants and derive runtime dimensions. +/// +/// # Errors +/// +/// Returns an error when config constants are inconsistent or overflow. +pub(super) fn validate_and_derive_layout( +) -> Result { + if D != Cfg::D { + return Err(HachiError::InvalidSetup(format!( + "const D={D} mismatches config D={}", + Cfg::D + ))); + } + if Cfg::LOG_BASIS == 0 || Cfg::LOG_BASIS >= 128 { + return Err(HachiError::InvalidSetup("invalid LOG_BASIS".to_string())); + } + if (Cfg::DELTA as u32).saturating_mul(Cfg::LOG_BASIS) > 128 { + return Err(HachiError::InvalidSetup( + "DELTA * LOG_BASIS must be <= 128".to_string(), + )); + } + + let num_blocks = checked_pow2(Cfg::R)?; + let block_len = checked_pow2(Cfg::M)?; + let inner_width = block_len + .checked_mul(Cfg::DELTA) + .ok_or_else(|| HachiError::InvalidSetup("inner width overflow".to_string()))?; + let outer_width = Cfg::N_A + .checked_mul(Cfg::DELTA) + .and_then(|x| x.checked_mul(num_blocks)) + .ok_or_else(|| HachiError::InvalidSetup("outer width overflow".to_string()))?; + let d_matrix_width = Cfg::DELTA + .checked_mul(num_blocks) + .ok_or_else(|| HachiError::InvalidSetup("D-matrix width overflow".to_string()))?; + let required_vars = Cfg::M + .checked_add(Cfg::R) + .ok_or_else(|| HachiError::InvalidSetup("variable count overflow".to_string()))?; + + Ok(CommitmentLayout { + num_blocks, + block_len, + inner_width, + outer_width, + d_matrix_width, + required_vars, + }) +} + +/// Ensure `max_num_vars` is sufficient for config dimensions. +/// +/// # Errors +/// +/// Returns an error when `max_num_vars < required_vars`. +pub(super) fn ensure_supported_num_vars( + max_num_vars: usize, + required_vars: usize, +) -> Result<(), HachiError> { + if max_num_vars < required_vars { + return Err(HachiError::InvalidSetup(format!( + "max_num_vars {max_num_vars} is smaller than required {required_vars}" + ))); + } + Ok(()) +} + +/// Ensure input blocks match the expected config-derived layout. +/// +/// # Errors +/// +/// Returns an error when block count or per-block size mismatch. +pub(super) fn ensure_block_layout( + f_blocks: &[Vec>], + layout: CommitmentLayout, +) -> Result<(), HachiError> { + if f_blocks.len() != layout.num_blocks { + return Err(HachiError::InvalidSize { + expected: layout.num_blocks, + actual: f_blocks.len(), + }); + } + for block in f_blocks { + if block.len() != layout.block_len { + return Err(HachiError::InvalidSize { + expected: layout.block_len, + actual: block.len(), + }); + } + } + Ok(()) +} + +/// Ensure matrix shape matches expected dimensions. +/// +/// # Errors +/// +/// Returns an error if row count or row width mismatch. +pub(super) fn ensure_matrix_shape( + mat: &[Vec], + expected_rows: usize, + expected_cols: usize, + name: &str, +) -> Result<(), HachiError> { + if mat.len() != expected_rows { + return Err(HachiError::InvalidSize { + expected: expected_rows, + actual: mat.len(), + }); + } + for (row_idx, row) in mat.iter().enumerate() { + if row.len() != expected_cols { + return Err(HachiError::InvalidSetup(format!( + "{name} row {row_idx} has width {}, expected {expected_cols}", + row.len() + ))); + } + } + Ok(()) +} + +/// Default correctness-first config for early protocol integration. +#[derive(Clone, Copy, Debug, Default)] +pub struct DefaultCommitmentConfig; + +impl CommitmentConfig for DefaultCommitmentConfig { + const D: usize = 16; + const M: usize = 4; + const R: usize = 2; + const N_A: usize = 8; + const N_B: usize = 4; + const N_D: usize = 4; + const LOG_BASIS: u32 = 4; + const DELTA: usize = 9; + const TAU: usize = 4; + const BETA: u128 = 1_000_000; + const CHALLENGE_WEIGHT: usize = 3; +} diff --git a/src/protocol/commitment/mod.rs b/src/protocol/commitment/mod.rs new file mode 100644 index 00000000..3a3b7170 --- /dev/null +++ b/src/protocol/commitment/mod.rs @@ -0,0 +1,17 @@ +//! Protocol commitment abstraction layer. + +mod commit; +mod config; +pub(crate) mod onehot; +mod scheme; +mod transcript_append; +mod types; +pub(crate) mod utils; + +pub use commit::{HachiCommitmentCore, RingCommitmentSetup}; +pub use config::{CommitmentConfig, DefaultCommitmentConfig}; +pub use scheme::{CommitmentScheme, RingCommitmentScheme, StreamingCommitmentScheme}; +pub use transcript_append::AppendToTranscript; +pub use types::{ + DummyProof, HachiCommitment, HachiOpeningClaim, HachiOpeningPoint, RingCommitment, +}; diff --git a/src/protocol/commitment/onehot.rs b/src/protocol/commitment/onehot.rs new file mode 100644 index 00000000..ccf5998d --- /dev/null +++ b/src/protocol/commitment/onehot.rs @@ -0,0 +1,268 @@ +//! One-hot commitment path for regular one-hot ring elements. +//! +//! Exploits the sparsity of one-hot witnesses (coefficients in {0,1}) to +//! eliminate all inner ring multiplications. The inner Ajtai `t = A * s` +//! reduces to summing selected columns of `A` with negacyclic rotations. + +use std::collections::BTreeMap; + +use crate::algebra::ring::CyclotomicRing; +use crate::error::HachiError; +use crate::{CanonicalField, FieldCore}; + +/// Describes a nonzero ring element within one block of the commitment layout. +#[derive(Debug, Clone)] +pub(crate) struct SparseBlockEntry { + /// Position within the block (0..2^M). + pub pos_in_block: usize, + /// Coefficient indices that are 1 within this ring element. + pub nonzero_coeffs: Vec, +} + +/// Map a regular one-hot witness to sparse ring block entries. +/// +/// - `onehot_k`: chunk size K. The witness has T chunks of K field elements, +/// each chunk containing exactly one 1. +/// - `indices`: length-T slice where `indices[c]` is the hot position in +/// chunk `c` (must be in `[0, K)`). +/// - `r`, `m`: commitment config parameters (2^R blocks of 2^M ring elements). +/// - `D`: ring degree (const generic on caller side, passed as runtime here). +/// +/// Returns one `Vec` per block (outer len = 2^R). +/// +/// # Errors +/// +/// Returns an error if K and D are not "nicely matched" (one must divide +/// the other), if any index is out of range, or if the dimensions don't +/// fill the commitment layout. +pub(crate) fn map_onehot_to_sparse_blocks( + onehot_k: usize, + indices: &[usize], + r: usize, + m: usize, + d: usize, +) -> Result>, HachiError> { + if onehot_k == 0 || d == 0 { + return Err(HachiError::InvalidInput( + "onehot_k and D must be nonzero".into(), + )); + } + if !(onehot_k % d == 0 || d % onehot_k == 0) { + return Err(HachiError::InvalidInput(format!( + "K={onehot_k} and D={d} must be nicely matched (one divides the other)" + ))); + } + + let num_chunks = indices.len(); + let total_field_elems = num_chunks + .checked_mul(onehot_k) + .ok_or_else(|| HachiError::InvalidInput("T*K overflow".into()))?; + if total_field_elems % d != 0 { + return Err(HachiError::InvalidInput(format!( + "T*K={total_field_elems} is not divisible by D={d}" + ))); + } + let total_ring_elems = total_field_elems / d; + let num_blocks = 1usize << r; + let block_len = 1usize << m; + if total_ring_elems != num_blocks * block_len { + return Err(HachiError::InvalidSize { + expected: num_blocks * block_len, + actual: total_ring_elems, + }); + } + + // Accumulate nonzero coefficients per ring element index. + let mut ring_elem_map: BTreeMap> = BTreeMap::new(); + for (c, &idx) in indices.iter().enumerate() { + if idx >= onehot_k { + return Err(HachiError::InvalidInput(format!( + "index {idx} out of range for chunk size K={onehot_k} at position {c}" + ))); + } + let field_pos = c * onehot_k + idx; + let ring_elem_idx = field_pos / d; + let coeff_idx = field_pos % d; + ring_elem_map + .entry(ring_elem_idx) + .or_default() + .push(coeff_idx); + } + + // Distribute into blocks using the same interleaved layout as commit_coeffs: + // block_idx = ring_elem_idx % num_blocks + // pos_in_block = ring_elem_idx / num_blocks + // (equivalently: flat_idx = (pos_in_block << R) + block_idx) + let mut blocks: Vec> = vec![Vec::new(); num_blocks]; + for (ring_elem_idx, nonzero_coeffs) in ring_elem_map { + let block_idx = ring_elem_idx & (num_blocks - 1); + let pos_in_block = ring_elem_idx >> r; + blocks[block_idx].push(SparseBlockEntry { + pos_in_block, + nonzero_coeffs, + }); + } + + Ok(blocks) +} + +/// Sparse inner Ajtai: compute `t = A * s` for a one-hot block. +/// +/// Instead of materializing the full decomposed vector `s` and doing a dense +/// matvec, we accumulate only the nonzero contributions: +/// +/// ```text +/// t[a] = sum_{entry} A[a][entry.pos * delta].mul_by_monomial_sum(entry.nonzero_coeffs) +/// ``` +/// +/// Also returns `s` (densely materialized) for the opening proof hint. +#[allow(non_snake_case)] +pub(crate) fn inner_ajtai_onehot( + A: &[Vec>], + sparse_entries: &[SparseBlockEntry], + block_len: usize, + delta: usize, +) -> (Vec>, Vec>) { + let n_a = A.len(); + let inner_width = block_len * delta; + + // Build s: mostly zeros, with level-0 entries for nonzero ring elements. + let mut s = vec![CyclotomicRing::::zero(); inner_width]; + for entry in sparse_entries { + let mut coeffs = [F::zero(); D]; + for &ci in &entry.nonzero_coeffs { + coeffs[ci] = F::one(); + } + s[entry.pos_in_block * delta] = CyclotomicRing::from_coefficients(coeffs); + } + + // Compute t[a] = sum over nonzero entries of A[a][pos*delta] * f_j, + // where f_j is the monomial sum at that position. + let mut t = vec![CyclotomicRing::::zero(); n_a]; + for entry in sparse_entries { + let col = entry.pos_in_block * delta; + for a in 0..n_a { + t[a] += A[a][col].mul_by_monomial_sum(&entry.nonzero_coeffs); + } + } + + (t, s) +} + +#[cfg(test)] +mod tests { + use super::*; + use crate::test_utils::F; + + #[test] + fn map_onehot_k_gt_d() { + // K=16, D=4, T=2 chunks => 32 field elements => 8 ring elements + // R=1 (2 blocks), M=2 (4 per block) => 8 ring elements total + let k = 16; + let d = 4; + let indices = vec![3, 10]; + let blocks = map_onehot_to_sparse_blocks(k, &indices, 1, 2, d).unwrap(); + + assert_eq!(blocks.len(), 2); + let total_entries: usize = blocks.iter().map(|b| b.len()).sum(); + assert_eq!(total_entries, 2, "T=2 nonzero ring elements"); + + for block in &blocks { + for entry in block { + assert_eq!(entry.nonzero_coeffs.len(), 1, "K>D => single monomial"); + } + } + } + + #[test] + fn map_onehot_k_eq_d() { + // K=4, D=4, T=4 chunks => 16 field elements => 4 ring elements + // R=1 (2 blocks), M=1 (2 per block) + let k = 4; + let d = 4; + let indices = vec![0, 2, 3, 1]; + let blocks = map_onehot_to_sparse_blocks(k, &indices, 1, 1, d).unwrap(); + + assert_eq!(blocks.len(), 2); + let total_entries: usize = blocks.iter().map(|b| b.len()).sum(); + assert_eq!(total_entries, 4, "K=D => every ring element is nonzero"); + + for block in &blocks { + for entry in block { + assert_eq!(entry.nonzero_coeffs.len(), 1, "K=D => single monomial"); + } + } + } + + #[test] + fn map_onehot_k_lt_d() { + // K=4, D=8, T=8 chunks => 32 field elements => 4 ring elements + // R=1 (2 blocks), M=1 (2 per block) + let k = 4; + let d = 8; + let indices = vec![0, 2, 3, 1, 0, 0, 3, 3]; + let blocks = map_onehot_to_sparse_blocks(k, &indices, 1, 1, d).unwrap(); + + assert_eq!(blocks.len(), 2); + let total_entries: usize = blocks.iter().map(|b| b.len()).sum(); + assert_eq!(total_entries, 4, "D>K => all ring elements nonzero"); + + for block in &blocks { + for entry in block { + assert_eq!( + entry.nonzero_coeffs.len(), + 2, + "D=2K => 2 nonzero coeffs per ring element" + ); + } + } + } + + #[test] + fn map_onehot_rejects_non_divisible() { + let result = map_onehot_to_sparse_blocks(3, &[0, 1], 0, 1, 4); + assert!(result.is_err()); + } + + #[test] + fn inner_ajtai_onehot_single_monomial() { + const D: usize = 4; + type R = CyclotomicRing; + + // A is 2x4 (N_A=2, inner_width = block_len * delta = 2 * 2 = 4) + let a: Vec> = vec![ + vec![ + R::from_coefficients(std::array::from_fn(|i| F::from_u64((i + 1) as u64))), + R::from_coefficients(std::array::from_fn(|i| F::from_u64((i + 10) as u64))), + R::from_coefficients(std::array::from_fn(|i| F::from_u64((i + 20) as u64))), + R::from_coefficients(std::array::from_fn(|i| F::from_u64((i + 30) as u64))), + ], + vec![ + R::from_coefficients(std::array::from_fn(|i| F::from_u64((i + 5) as u64))), + R::from_coefficients(std::array::from_fn(|i| F::from_u64((i + 15) as u64))), + R::from_coefficients(std::array::from_fn(|i| F::from_u64((i + 25) as u64))), + R::from_coefficients(std::array::from_fn(|i| F::from_u64((i + 35) as u64))), + ], + ]; + + // One nonzero entry at pos=1, coefficient index 2 => monomial X^2 + let entries = vec![SparseBlockEntry { + pos_in_block: 1, + nonzero_coeffs: vec![2], + }]; + + let (t, s) = inner_ajtai_onehot(&a, &entries, 2, 2); + + // t[row] should equal A[row][1*2] * X^2 = A[row][2].negacyclic_shift(2) + for row in 0..2 { + let expected = a[row][2].negacyclic_shift(2); + assert_eq!(t[row], expected); + } + + // s should have a nonzero entry at position 1*2 = 2 + assert_eq!(s[2].coefficients()[2], F::one()); + assert!(s[0] == R::zero()); + assert!(s[1] == R::zero()); + assert!(s[3] == R::zero()); + } +} diff --git a/src/protocol/commitment/scheme.rs b/src/protocol/commitment/scheme.rs new file mode 100644 index 00000000..df0c4301 --- /dev/null +++ b/src/protocol/commitment/scheme.rs @@ -0,0 +1,277 @@ +//! Commitment-scheme trait surface for Hachi protocol code. + +use super::config::CommitmentConfig; +use super::transcript_append::AppendToTranscript; +use super::utils::math::checked_pow2; +use crate::algebra::ring::CyclotomicRing; +use crate::error::HachiError; +use crate::protocol::transcript::Transcript; +use crate::{CanonicalField, FieldCore, Polynomial}; + +/// Output type for batched commitments. +pub(crate) type BatchCommitOutput = Result, HachiError>; + +/// Generic commitment-scheme interface used by Hachi protocol code. +pub trait CommitmentScheme: Clone + Send + Sync + 'static +where + F: FieldCore + CanonicalField, +{ + /// Prover setup parameters. + type ProverSetup: Clone + Send + Sync; + /// Verifier setup parameters. + type VerifierSetup: Clone + Send + Sync; + /// Commitment object. + type Commitment: Clone + PartialEq + Send + Sync + AppendToTranscript; + /// Evaluation/opening proof object. + type Proof: Clone + Send + Sync; + /// Optional prover-side hint produced at commitment time. + type OpeningProofHint: Clone + Send + Sync; + + /// Build prover setup for maximum polynomial dimension. + fn setup_prover(max_num_vars: usize) -> Self::ProverSetup; + + /// Derive verifier setup from prover setup. + fn setup_verifier(setup: &Self::ProverSetup) -> Self::VerifierSetup; + + /// Commit to one polynomial. + /// + /// # Errors + /// + /// Returns an error when setup/parameter constraints are not satisfied. + fn commit>( + poly: &P, + setup: &Self::ProverSetup, + ) -> Result<(Self::Commitment, Self::OpeningProofHint), HachiError>; + + /// Commit to many polynomials. + /// + /// # Errors + /// + /// Returns an error if any per-polynomial commitment fails. + fn batch_commit>( + polys: &[P], + setup: &Self::ProverSetup, + ) -> BatchCommitOutput { + polys.iter().map(|p| Self::commit(p, setup)).collect() + } + + /// Produce an opening proof at `opening_point`. + /// + /// # Errors + /// + /// Returns an error if the opening point is invalid or proof generation fails. + fn prove, P: Polynomial>( + setup: &Self::ProverSetup, + poly: &P, + opening_point: &[F], + hint: Option, + transcript: &mut T, + commitment: &Self::Commitment, + ) -> Result; + + /// Verify an opening proof. + /// + /// # Errors + /// + /// Returns an error when verification fails. + fn verify>( + proof: &Self::Proof, + setup: &Self::VerifierSetup, + transcript: &mut T, + opening_point: &[F], + opening: &F, + commitment: &Self::Commitment, + ) -> Result<(), HachiError>; + + /// Homomorphic commitment combination. + fn combine_commitments(commitments: &[Self::Commitment], coeffs: &[F]) -> Self::Commitment; + + /// Homomorphic hint combination. + fn combine_hints(hints: Vec, coeffs: &[F]) -> Self::OpeningProofHint; + + /// Protocol identifier. + fn protocol_name() -> &'static [u8]; +} + +/// Streaming extension for chunked commitment workflows. +pub trait StreamingCommitmentScheme: CommitmentScheme +where + F: FieldCore + CanonicalField, +{ + /// Intermediate chunk state. + type ChunkState: Clone + Send + Sync + PartialEq + std::fmt::Debug; + + /// Process one chunk of field elements. + fn process_chunk(setup: &Self::ProverSetup, chunk: &[F]) -> Self::ChunkState; + + /// Process one chunk of one-hot values. + fn process_chunk_onehot( + setup: &Self::ProverSetup, + onehot_k: usize, + chunk: &[Option], + ) -> Self::ChunkState; + + /// Aggregate chunk states into one commitment + hint. + fn aggregate_chunks( + setup: &Self::ProverSetup, + onehot_k: Option, + chunks: &[Self::ChunkState], + ) -> (Self::Commitment, Self::OpeningProofHint); +} + +/// Ring-native commitment interface for §4.1 implementation work. +pub trait RingCommitmentScheme: Clone + Send + Sync + 'static +where + F: FieldCore + CanonicalField, + Cfg: CommitmentConfig, +{ + /// Prover setup parameters. + type ProverSetup: Clone + Send + Sync; + /// Verifier setup parameters. + type VerifierSetup: Clone + Send + Sync; + /// Ring-native commitment type. + type Commitment: Clone + PartialEq + Send + Sync; + + /// Construct commitment setup for at most `max_num_vars` variables. + /// + /// # Errors + /// + /// Returns an error if dimensions are inconsistent with `Cfg`. + fn setup(max_num_vars: usize) -> Result<(Self::ProverSetup, Self::VerifierSetup), HachiError>; + + /// Commit to ring blocks arranged as `2^R` vectors of length `2^M`. + /// + /// Returns `(commitment, s, t_hat)` where `s` and `t_hat` are the + /// decomposed witness vectors from §4.1. + /// + /// # Errors + /// + /// Returns an error if block layout mismatches config or commitment fails. + #[allow(clippy::type_complexity)] + fn commit_ring_blocks( + f_blocks: &[Vec>], + setup: &Self::ProverSetup, + ) -> Result< + ( + Self::Commitment, + Vec>>, + Vec>>, + ), + HachiError, + >; + + /// Commit to a flat coefficient table `(f_i)_{i∈{0,1}^ℓ}` in ring form. + /// + /// The input is indexed in LSB-first order (same convention as + /// `DenseMultilinearEvals`). We split the variables as: + /// + /// - outer index `i` for the first `R` variables, + /// - inner index `j` for the last `M` variables, + /// + /// We then form blocks `f_i = (f_{i||j})_{j}` (Eq. (12) in the paper). + /// + /// This prepares `f_blocks` and delegates to `commit_ring_blocks`. + /// + /// # Errors + /// + /// Returns an error if `f_coeffs.len()` does not match the configured block + /// layout, if internal index computations overflow, or if the underlying + /// commitment routine fails. + #[allow(clippy::type_complexity)] + fn commit_coeffs( + f_coeffs: &[CyclotomicRing], + setup: &Self::ProverSetup, + ) -> Result< + ( + Self::Commitment, + Vec>>, + Vec>>, + ), + HachiError, + > { + let num_blocks = checked_pow2(Cfg::R)?; + let block_len = checked_pow2(Cfg::M)?; + let expected_len = num_blocks + .checked_mul(block_len) + .ok_or_else(|| HachiError::InvalidSetup("coefficient length overflow".to_string()))?; + if f_coeffs.len() != expected_len { + return Err(HachiError::InvalidSize { + expected: expected_len, + actual: f_coeffs.len(), + }); + } + + let mut blocks = Vec::with_capacity(num_blocks); + for i in 0..num_blocks { + let mut block = Vec::with_capacity(block_len); + for j in 0..block_len { + let idx = (j << Cfg::R) + .checked_add(i) + .ok_or_else(|| HachiError::InvalidSetup("index overflow".to_string()))?; + block.push(f_coeffs[idx]); + } + blocks.push(block); + } + + Self::commit_ring_blocks(&blocks, setup) + } + + /// Commit to a regular one-hot witness. + /// + /// The witness represents `T` chunks of `onehot_k` field elements, each + /// chunk containing exactly one 1 and all other entries 0. `indices[c]` + /// gives the hot position in chunk `c` (must be in `[0, onehot_k)`). + /// + /// Requires `D` and `onehot_k` to be "nicely matched": one must divide + /// the other. + /// + /// The default implementation materializes the full one-hot field vector, + /// packs it into ring elements via coefficient embedding, and delegates + /// to `commit_coeffs`. Implementations may override this with a + /// sparse-aware path that avoids all inner ring multiplications. + /// + /// # Errors + /// + /// Returns an error if dimensions are inconsistent or any index is out + /// of range. + #[allow(clippy::type_complexity)] + fn commit_onehot( + onehot_k: usize, + indices: &[usize], + setup: &Self::ProverSetup, + ) -> Result< + ( + Self::Commitment, + Vec>>, + Vec>>, + ), + HachiError, + > { + let num_chunks = indices.len(); + let total_field_elems = num_chunks + .checked_mul(onehot_k) + .ok_or_else(|| HachiError::InvalidInput("T*K overflow".into()))?; + if total_field_elems % D != 0 { + return Err(HachiError::InvalidInput(format!( + "T*K={total_field_elems} is not divisible by D={D}" + ))); + } + + // Materialize the full one-hot vector as ring elements. + let total_ring_elems = total_field_elems / D; + let mut ring_coeffs = vec![CyclotomicRing::::zero(); total_ring_elems]; + for (c, &idx) in indices.iter().enumerate() { + if idx >= onehot_k { + return Err(HachiError::InvalidInput(format!( + "index {idx} out of range for chunk size K={onehot_k} at position {c}" + ))); + } + let field_pos = c * onehot_k + idx; + let ring_idx = field_pos / D; + let coeff_idx = field_pos % D; + ring_coeffs[ring_idx].coeffs[coeff_idx] = F::one(); + } + + Self::commit_coeffs(&ring_coeffs, setup) + } +} diff --git a/src/protocol/commitment/transcript_append.rs b/src/protocol/commitment/transcript_append.rs new file mode 100644 index 00000000..6510ccb5 --- /dev/null +++ b/src/protocol/commitment/transcript_append.rs @@ -0,0 +1,13 @@ +//! Traits for appending commitment objects to protocol transcripts. + +use crate::protocol::transcript::Transcript; +use crate::{CanonicalField, FieldCore}; + +/// Protocol object that can be absorbed into a transcript. +pub trait AppendToTranscript +where + F: FieldCore + CanonicalField, +{ + /// Append this object to a transcript using the provided event label. + fn append_to_transcript>(&self, label: &[u8], transcript: &mut T); +} diff --git a/src/protocol/commitment/types.rs b/src/protocol/commitment/types.rs new file mode 100644 index 00000000..339a1a03 --- /dev/null +++ b/src/protocol/commitment/types.rs @@ -0,0 +1,157 @@ +//! Protocol commitment/opening wrapper types. + +use super::transcript_append::AppendToTranscript; +use crate::algebra::ring::CyclotomicRing; +use crate::primitives::serialization::{ + Compress, HachiDeserialize, HachiSerialize, SerializationError, Valid, Validate, +}; +use crate::protocol::transcript::Transcript; +use crate::{CanonicalField, FieldCore}; +use std::io::{Read, Write}; + +/// A Hachi opening point represented as field coordinates. +#[derive(Debug, Clone, PartialEq, Eq)] +pub struct HachiOpeningPoint { + /// Point coordinates used for multilinear opening. + pub r: Vec, +} + +/// A Hachi opening claim `(point, value)`. +#[derive(Debug, Clone, PartialEq, Eq)] +pub struct HachiOpeningClaim { + /// Opening point. + pub point: HachiOpeningPoint, + /// Claimed value at `point`. + pub value: F, +} + +/// Minimal commitment wrapper used by protocol traits/tests. +#[derive(Debug, Clone, Copy, PartialEq, Eq, Default)] +pub struct HachiCommitment(pub u128); + +/// Minimal proof wrapper used by protocol trait stubs and tests. +#[derive(Debug, Clone, Copy, PartialEq, Eq, Default)] +pub struct DummyProof(pub u128); + +impl Valid for HachiCommitment { + fn check(&self) -> Result<(), SerializationError> { + Ok(()) + } +} + +impl HachiSerialize for HachiCommitment { + fn serialize_with_mode( + &self, + mut writer: W, + _compress: Compress, + ) -> Result<(), SerializationError> { + self.0.serialize_with_mode(&mut writer, Compress::No) + } + + fn serialized_size(&self, _compress: Compress) -> usize { + 16 + } +} + +impl HachiDeserialize for HachiCommitment { + fn deserialize_with_mode( + mut reader: R, + _compress: Compress, + validate: Validate, + ) -> Result { + let value = u128::deserialize_with_mode(&mut reader, Compress::No, validate)?; + Ok(Self(value)) + } +} + +impl Valid for DummyProof { + fn check(&self) -> Result<(), SerializationError> { + Ok(()) + } +} + +impl HachiSerialize for DummyProof { + fn serialize_with_mode( + &self, + mut writer: W, + _compress: Compress, + ) -> Result<(), SerializationError> { + self.0.serialize_with_mode(&mut writer, Compress::No) + } + + fn serialized_size(&self, _compress: Compress) -> usize { + 16 + } +} + +impl HachiDeserialize for DummyProof { + fn deserialize_with_mode( + mut reader: R, + _compress: Compress, + validate: Validate, + ) -> Result { + let value = u128::deserialize_with_mode(&mut reader, Compress::No, validate)?; + Ok(Self(value)) + } +} + +impl AppendToTranscript for HachiCommitment +where + F: FieldCore + CanonicalField, +{ + fn append_to_transcript>(&self, label: &[u8], transcript: &mut T) { + transcript.append_serde(label, self); + } +} + +/// Ring-native commitment object `u in R_q^{n_B}` used by §4.1. +#[derive(Debug, Clone, PartialEq, Eq, Default)] +pub struct RingCommitment { + /// Outer commitment vector. + pub u: Vec>, +} + +impl Valid for RingCommitment { + fn check(&self) -> Result<(), SerializationError> { + self.u.check() + } +} + +impl HachiSerialize for RingCommitment { + fn serialize_with_mode( + &self, + mut writer: W, + compress: Compress, + ) -> Result<(), SerializationError> { + self.u.serialize_with_mode(&mut writer, compress) + } + + fn serialized_size(&self, compress: Compress) -> usize { + self.u.serialized_size(compress) + } +} + +impl HachiDeserialize for RingCommitment { + fn deserialize_with_mode( + mut reader: R, + compress: Compress, + validate: Validate, + ) -> Result { + let u = + Vec::>::deserialize_with_mode(&mut reader, compress, validate)?; + let out = Self { u }; + if matches!(validate, Validate::Yes) { + out.check()?; + } + Ok(out) + } +} + +impl AppendToTranscript for RingCommitment +where + F: FieldCore + CanonicalField, +{ + fn append_to_transcript>(&self, label: &[u8], transcript: &mut T) { + transcript.append_serde(label, self); + } +} diff --git a/src/protocol/commitment/utils/linear.rs b/src/protocol/commitment/utils/linear.rs new file mode 100644 index 00000000..daca6d24 --- /dev/null +++ b/src/protocol/commitment/utils/linear.rs @@ -0,0 +1,44 @@ +//! Linear algebra helpers for ring commitment. + +use crate::algebra::ring::CyclotomicRing; +use crate::{CanonicalField, FieldCore}; + +pub(crate) fn mat_vec_mul_unchecked( + mat: &[Vec>], + vec: &[CyclotomicRing], +) -> Vec> { + let mut out = Vec::with_capacity(mat.len()); + for row in mat { + debug_assert_eq!(row.len(), vec.len()); + let mut acc = CyclotomicRing::::zero(); + for (a, x) in row.iter().zip(vec.iter()) { + acc += *a * *x; + } + out.push(acc); + } + out +} + +pub(crate) fn decompose_block( + block: &[CyclotomicRing], + delta: usize, + log_basis: u32, +) -> Vec> { + let mut out = Vec::with_capacity(block.len() * delta); + for coeff_vec in block { + out.extend(coeff_vec.balanced_decompose_pow2(delta, log_basis)); + } + out +} + +pub(crate) fn decompose_rows( + rows: &[CyclotomicRing], + delta: usize, + log_basis: u32, +) -> Vec> { + let mut out = Vec::with_capacity(rows.len() * delta); + for row in rows { + out.extend(row.balanced_decompose_pow2(delta, log_basis)); + } + out +} diff --git a/src/protocol/commitment/utils/math.rs b/src/protocol/commitment/utils/math.rs new file mode 100644 index 00000000..16cf8618 --- /dev/null +++ b/src/protocol/commitment/utils/math.rs @@ -0,0 +1,14 @@ +//! Small math helpers for commitment internals. + +use crate::error::HachiError; + +/// Compute `2^exp` with overflow checks. +/// +/// # Errors +/// +/// Returns `InvalidSetup` if `2^exp` does not fit in `usize`. +pub(in crate::protocol::commitment) fn checked_pow2(exp: usize) -> Result { + 1usize + .checked_shl(exp as u32) + .ok_or_else(|| HachiError::InvalidSetup(format!("2^{exp} does not fit usize"))) +} diff --git a/src/protocol/commitment/utils/matrix.rs b/src/protocol/commitment/utils/matrix.rs new file mode 100644 index 00000000..fa0b80ce --- /dev/null +++ b/src/protocol/commitment/utils/matrix.rs @@ -0,0 +1,131 @@ +//! Matrix sampling helpers for setup. + +use crate::algebra::ring::CyclotomicRing; +use crate::{FieldCore, FieldSampling}; +use rand_core::{CryptoRng, RngCore}; +use sha3::digest::{ExtendableOutput, Update, XofReader}; +use sha3::Shake256; + +/// Public seed used to derive commitment matrices. +pub(crate) type PublicMatrixSeed = [u8; 32]; + +const PUBLIC_MATRIX_DOMAIN: &[u8] = b"hachi/commitment/public-matrix"; + +/// Fixed public seed for deterministic, reproducible setup. +pub(crate) fn sample_public_matrix_seed() -> PublicMatrixSeed { + let mut seed = [0u8; 32]; + seed[..8].copy_from_slice(&0xDEAD_BEEF_CAFE_BABEu64.to_le_bytes()); + seed +} + +/// Derive a public matrix from a seed using domain-separated SHAKE expansion. +/// +/// This follows the same high-level pattern used in NIST lattice specs: +/// derive deterministic public structure from a seed + indices, then sample +/// coefficients via rejection-sampling at the field layer. +/// +/// NOTE: Potential future hardening: +/// move toward stricter ML-KEM/ML-DSA-style byte layout and parsing rules +/// (fixed-format seed/index encoding and scheme-specific expansion details) +/// if we decide to maximize standards-shape interoperability. +pub(crate) fn derive_public_matrix( + rows: usize, + cols: usize, + seed: &PublicMatrixSeed, + matrix_label: &[u8], +) -> Vec>> { + (0..rows) + .map(|r| { + (0..cols) + .map(|c| { + let mut entry_rng = ShakeXofRng::new(seed, matrix_label, rows, cols, r, c); + CyclotomicRing::random(&mut entry_rng) + }) + .collect() + }) + .collect() +} + +struct ShakeXofRng { + reader: Box, +} + +impl ShakeXofRng { + fn new( + seed: &PublicMatrixSeed, + matrix_label: &[u8], + rows: usize, + cols: usize, + row: usize, + col: usize, + ) -> Self { + let mut xof = Shake256::default(); + absorb_len_prefixed(&mut xof, b"domain", PUBLIC_MATRIX_DOMAIN); + absorb_len_prefixed(&mut xof, b"seed", seed); + absorb_len_prefixed(&mut xof, b"matrix", matrix_label); + absorb_len_prefixed(&mut xof, b"rows", &(rows as u64).to_le_bytes()); + absorb_len_prefixed(&mut xof, b"cols", &(cols as u64).to_le_bytes()); + absorb_len_prefixed(&mut xof, b"row", &(row as u64).to_le_bytes()); + absorb_len_prefixed(&mut xof, b"col", &(col as u64).to_le_bytes()); + Self { + reader: Box::new(xof.finalize_xof()), + } + } +} + +impl RngCore for ShakeXofRng { + fn next_u32(&mut self) -> u32 { + let mut buf = [0u8; 4]; + self.fill_bytes(&mut buf); + u32::from_le_bytes(buf) + } + + fn next_u64(&mut self) -> u64 { + let mut buf = [0u8; 8]; + self.fill_bytes(&mut buf); + u64::from_le_bytes(buf) + } + + fn fill_bytes(&mut self, dest: &mut [u8]) { + self.reader.read(dest); + } + + fn try_fill_bytes(&mut self, dest: &mut [u8]) -> Result<(), rand_core::Error> { + self.fill_bytes(dest); + Ok(()) + } +} + +impl CryptoRng for ShakeXofRng {} + +fn absorb_len_prefixed(xof: &mut Shake256, label: &[u8], data: &[u8]) { + xof.update(&(label.len() as u64).to_le_bytes()); + xof.update(label); + xof.update(&(data.len() as u64).to_le_bytes()); + xof.update(data); +} + +#[cfg(test)] +mod tests { + use super::*; + use crate::algebra::Fp64; + + type F = Fp64<4294967197>; + const D: usize = 64; + + #[test] + fn matrix_derivation_is_deterministic_for_same_seed() { + let seed = [42u8; 32]; + let m1 = derive_public_matrix::(3, 5, &seed, b"A"); + let m2 = derive_public_matrix::(3, 5, &seed, b"A"); + assert_eq!(m1, m2); + } + + #[test] + fn matrix_derivation_domain_separates_labels() { + let seed = [7u8; 32]; + let a = derive_public_matrix::(2, 3, &seed, b"A"); + let b = derive_public_matrix::(2, 3, &seed, b"B"); + assert_ne!(a, b); + } +} diff --git a/src/protocol/commitment/utils/mod.rs b/src/protocol/commitment/utils/mod.rs new file mode 100644 index 00000000..39f38327 --- /dev/null +++ b/src/protocol/commitment/utils/mod.rs @@ -0,0 +1,6 @@ +//! Utility helpers for commitment internals. + +pub(crate) mod linear; +pub(crate) mod math; +pub(crate) mod matrix; +pub(crate) mod norm; diff --git a/src/protocol/commitment/utils/norm.rs b/src/protocol/commitment/utils/norm.rs new file mode 100644 index 00000000..cc01f9a7 --- /dev/null +++ b/src/protocol/commitment/utils/norm.rs @@ -0,0 +1,48 @@ +//! Infinity norm utilities for ring elements over Z_q. + +use crate::algebra::ring::CyclotomicRing; +use crate::CanonicalField; + +/// Detect the field modulus from the canonical representation. +/// +/// Uses the identity: the canonical form of `−1` in `Z_q` is `q − 1`. +pub(crate) fn detect_field_modulus() -> u128 { + (-F::one()).to_canonical_u128() + 1 +} + +/// Centered absolute value of a field element. +/// +/// Maps canonical representation `v ∈ [0, q)` to `min(v, q − v)`. +#[inline] +pub(crate) fn centered_abs(x: F, modulus: u128) -> u128 { + let v = x.to_canonical_u128(); + let half = modulus / 2; + if v <= half { + v + } else { + modulus - v + } +} + +/// L∞ norm of a single ring element (maximum centered coefficient magnitude). +pub(crate) fn ring_inf_norm( + r: &CyclotomicRing, + modulus: u128, +) -> u128 { + r.coefficients() + .iter() + .map(|c| centered_abs(*c, modulus)) + .max() + .unwrap_or(0) +} + +/// L∞ norm of a vector of ring elements. +pub(crate) fn vec_inf_norm( + v: &[CyclotomicRing], + modulus: u128, +) -> u128 { + v.iter() + .map(|r| ring_inf_norm(r, modulus)) + .max() + .unwrap_or(0) +} diff --git a/src/protocol/commitment_scheme.rs b/src/protocol/commitment_scheme.rs new file mode 100644 index 00000000..6da64c79 --- /dev/null +++ b/src/protocol/commitment_scheme.rs @@ -0,0 +1,425 @@ +//! Commitment scheme trait implementation. + +use crate::algebra::ring::CyclotomicRing; +use crate::error::HachiError; +use crate::primitives::poly::multilinear_lagrange_basis; +use crate::protocol::commitment::{ + CommitmentConfig, CommitmentScheme, DefaultCommitmentConfig, HachiCommitmentCore, + RingCommitment, RingCommitmentScheme, RingCommitmentSetup, +}; +use crate::protocol::opening_point::RingOpeningPoint; +use crate::protocol::proof::{HachiCommitmentHint, HachiProof, SumcheckAux}; +use crate::protocol::quadratic_equation::QuadraticEquation; +use crate::protocol::ring_switch::{ring_switch_prover, ring_switch_verifier}; +use crate::protocol::sumcheck::norm_sumcheck::{NormSumcheckProver, NormSumcheckVerifier}; +use crate::protocol::sumcheck::relation_sumcheck::{ + RelationSumcheckProver, RelationSumcheckVerifier, +}; +use crate::protocol::sumcheck::{prove_sumcheck, verify_sumcheck}; +use crate::protocol::transcript::labels::CHALLENGE_SUMCHECK_ROUND; +use crate::protocol::transcript::Transcript; +use crate::{CanonicalField, FieldCore, FieldSampling, Polynomial}; + +#[cfg(test)] +use crate::protocol::ring_switch::{eval_ring_matrix_at, expand_m_a}; +#[cfg(test)] +use crate::protocol::transcript::labels::{ + ABSORB_SUMCHECK_W, CHALLENGE_RING_SWITCH, DOMAIN_HACHI_PROTOCOL, +}; +#[cfg(test)] +use crate::protocol::transcript::Blake2bTranscript; + +type Cfg = DefaultCommitmentConfig; +const D: usize = Cfg::D; + +/// Placeholder for the end-to-end PCS wrapper. +#[derive(Clone, Copy, Debug, Default)] +pub struct HachiCommitmentScheme; + +impl CommitmentScheme for HachiCommitmentScheme +where + F: FieldCore + CanonicalField + FieldSampling, +{ + type ProverSetup = RingCommitmentSetup; + type VerifierSetup = RingCommitmentSetup; + type Commitment = RingCommitment; + type Proof = HachiProof; + type OpeningProofHint = HachiCommitmentHint; + + fn setup_prover(max_num_vars: usize) -> Self::ProverSetup { + let (setup, _) = + >::setup(max_num_vars) + .expect("commitment setup failed"); + setup + } + + fn setup_verifier(setup: &Self::ProverSetup) -> Self::VerifierSetup { + setup.clone() + } + + fn commit>( + poly: &P, + setup: &Self::ProverSetup, + ) -> Result<(Self::Commitment, Self::OpeningProofHint), HachiError> { + let ring_coeffs = + reduce_coeffs_to_ring_elements::(poly.num_vars(), &poly.coeffs())?; + let (commitment, s, t_hat) = >::commit_coeffs(&ring_coeffs, setup)?; + let hint = HachiCommitmentHint { + s, + t_hat, + ring_coeffs, + }; + Ok((commitment, hint)) + } + + fn prove, P: Polynomial>( + setup: &Self::ProverSetup, + poly: &P, + opening_point: &[F], + hint: Option, + transcript: &mut T, + commitment: &Self::Commitment, + ) -> Result { + let hint = hint.ok_or_else(|| { + HachiError::InvalidInput("missing commitment hint for proving".to_string()) + })?; + let num_vars = poly.num_vars(); + let alpha = Cfg::D.trailing_zeros() as usize; + let reduced_len = num_vars + .checked_sub(alpha) + .ok_or_else(|| HachiError::InvalidSetup("reduction length underflow".to_string()))?; + if opening_point.len() < reduced_len { + return Err(HachiError::InvalidPointDimension { + expected: reduced_len, + actual: opening_point.len(), + }); + } + + let ring_opening_point = + ring_opening_point_from_field::(&opening_point[..reduced_len], Cfg::R, Cfg::M)?; + + let y_ring = + evaluate_packed_ring_poly::(&hint.ring_coeffs, &opening_point[..reduced_len]); + + // §4.2 Quadratic equation + let quad_eq = QuadraticEquation::::new_prover( + setup, + &ring_opening_point, + &hint, + transcript, + commitment, + &y_ring, + )?; + + // §4.3 Ring switch + let rs = ring_switch_prover::(&quad_eq, transcript)?; + + // Norm sumcheck (range check on w) + let mut norm_prover = NormSumcheckProver::new(&rs.tau0, rs.w_evals.clone(), rs.b); + let (norm_proof, ..) = + prove_sumcheck::(&mut norm_prover, transcript, |tr| { + tr.challenge_scalar(CHALLENGE_SUMCHECK_ROUND) + })?; + + // Relation sumcheck (evaluation relation) + let mut relation_prover = RelationSumcheckProver::new( + rs.w_evals, + &rs.alpha_evals_y, + &rs.m_evals_x, + rs.num_u, + rs.num_l, + ); + let (relation_proof, ..) = + prove_sumcheck::(&mut relation_prover, transcript, |tr| { + tr.challenge_scalar(CHALLENGE_SUMCHECK_ROUND) + })?; + + Ok(HachiProof { + v: quad_eq.v, + y_ring, + norm_proof, + relation_proof, + sumcheck_aux: SumcheckAux { w: rs.w }, + w_commitment: rs.w_commitment, + }) + } + + fn verify>( + proof: &Self::Proof, + setup: &Self::VerifierSetup, + transcript: &mut T, + opening_point: &[F], + opening: &F, + commitment: &Self::Commitment, + ) -> Result<(), HachiError> { + let alpha_bits = Cfg::D.trailing_zeros() as usize; + let reduced_len = opening_point.len().checked_sub(alpha_bits).ok_or_else(|| { + HachiError::InvalidSetup("opening point length underflow".to_string()) + })?; + let reduced_opening_point = &opening_point[..reduced_len]; + let inner_point = &opening_point[reduced_len..]; + + // §3.1 trace check + let v = reduce_inner_openings_to_ring_elements::(inner_point)?; + let d = F::from_u64(Cfg::D as u64); + let trace_lhs = trace::(&(proof.y_ring * v.sigma_m1())); + let trace_rhs = d * *opening; + if trace_lhs != trace_rhs { + return Err(HachiError::InvalidProof); + } + + // §4.2 Quadratic equation + let ring_opening_point = + ring_opening_point_from_field::(reduced_opening_point, Cfg::R, Cfg::M)?; + let quad_eq = QuadraticEquation::::new_verifier( + setup, + &ring_opening_point, + &proof.v, + transcript, + commitment, + &proof.y_ring, + )?; + + // §4.3 Ring switch (verifier side) + let rs = ring_switch_verifier::( + &quad_eq, + &proof.sumcheck_aux.w, + &proof.w_commitment, + transcript, + )?; + + // Norm sumcheck verification (range check) + let norm_verifier = NormSumcheckVerifier::new(rs.tau0, rs.w_evals.clone(), rs.b); + verify_sumcheck::(&proof.norm_proof, &norm_verifier, transcript, |tr| { + tr.challenge_scalar(CHALLENGE_SUMCHECK_ROUND) + })?; + + // Relation sumcheck verification (evaluation relation) + let relation_verifier = RelationSumcheckVerifier::new( + rs.w_evals, + rs.alpha_evals_y, + rs.m_evals_x, + rs.tau1, + proof.v.clone(), + commitment.u.clone(), + proof.y_ring, + rs.alpha, + rs.num_u, + rs.num_l, + ); + verify_sumcheck::( + &proof.relation_proof, + &relation_verifier, + transcript, + |tr| tr.challenge_scalar(CHALLENGE_SUMCHECK_ROUND), + )?; + + Ok(()) + } + + fn combine_commitments(_commitments: &[Self::Commitment], _coeffs: &[F]) -> Self::Commitment { + unimplemented!() + } + + fn combine_hints(_hints: Vec, _coeffs: &[F]) -> Self::OpeningProofHint { + unimplemented!() + } + + fn protocol_name() -> &'static [u8] { + unimplemented!() + } +} + +/// Re-derive the ring-switch challenge `alpha` and the expanded `M_a` vector +/// by replaying the transcript from the proof data and setup, exactly as the +/// verifier does. +#[cfg(test)] +pub(crate) fn rederive_alpha_and_m_a( + proof: &HachiProof, + setup: &>::ProverSetup, + opening_point: &[F], + commitment: &>::Commitment, +) -> Result<(F, Vec), HachiError> +where + F: FieldCore + CanonicalField + FieldSampling + 'static, +{ + let alpha_bits = Cfg::D.trailing_zeros() as usize; + let reduced_len = opening_point + .len() + .checked_sub(alpha_bits) + .ok_or_else(|| HachiError::InvalidSetup("opening point length underflow".to_string()))?; + let ring_opening_point = + ring_opening_point_from_field::(&opening_point[..reduced_len], Cfg::R, Cfg::M)?; + let mut transcript = Blake2bTranscript::::new(DOMAIN_HACHI_PROTOCOL); + let quad_eq = QuadraticEquation::::new_verifier( + setup, + &ring_opening_point, + &proof.v, + &mut transcript, + commitment, + &proof.y_ring, + )?; + transcript.append_serde(ABSORB_SUMCHECK_W, &proof.w_commitment); + let alpha: F = transcript.challenge_scalar(CHALLENGE_RING_SWITCH); + let m_a = eval_ring_matrix_at::(quad_eq.m(), &alpha); + let m_a_vec = expand_m_a::(&m_a, alpha)?; + Ok((alpha, m_a_vec)) +} + +fn lagrange_weights(point: &[F]) -> Vec { + let len = 1usize << point.len(); + let mut weights = vec![F::zero(); len]; + multilinear_lagrange_basis(&mut weights, point); + weights +} + +fn ring_opening_point_from_field( + opening_point: &[F], + r_vars: usize, + m_vars: usize, +) -> Result, HachiError> { + let expected_len = r_vars + .checked_add(m_vars) + .ok_or_else(|| HachiError::InvalidSetup("opening point length overflow".to_string()))?; + if opening_point.len() != expected_len { + return Err(HachiError::InvalidPointDimension { + expected: expected_len, + actual: opening_point.len(), + }); + } + + let b = lagrange_weights(&opening_point[..r_vars]); + let a = lagrange_weights(&opening_point[r_vars..]); + Ok(RingOpeningPoint { a, b }) +} + +fn reduce_coeffs_to_ring_elements( + num_vars: usize, + coeffs: &[F], +) -> Result>, HachiError> { + if D == 0 || !D.is_power_of_two() { + return Err(HachiError::InvalidInput(format!( + "ring degree D={D} is not a power of two" + ))); + } + let alpha = D.trailing_zeros() as usize; + if num_vars < alpha { + return Err(HachiError::InvalidInput(format!( + "num_vars {num_vars} is smaller than alpha {alpha}" + ))); + } + + let expected_len = 1usize + .checked_shl(num_vars as u32) + .ok_or_else(|| HachiError::InvalidInput(format!("2^{num_vars} does not fit usize")))?; + if coeffs.len() != expected_len { + return Err(HachiError::InvalidSize { + expected: expected_len, + actual: coeffs.len(), + }); + } + + let outer_vars = num_vars - alpha; + let outer_len = 1usize + .checked_shl(outer_vars as u32) + .ok_or_else(|| HachiError::InvalidInput(format!("2^{outer_vars} does not fit usize")))?; + + let mut out = Vec::with_capacity(outer_len); + for i in 0..outer_len { + let coeffs = std::array::from_fn(|j| { + let idx = i + (j << outer_vars); + coeffs[idx] + }); + out.push(CyclotomicRing::from_coefficients(coeffs)); + } + Ok(out) +} + +fn reduce_inner_openings_to_ring_elements( + inner_point: &[F], +) -> Result, HachiError> { + let weights = lagrange_weights(inner_point); + if weights.len() != D { + return Err(HachiError::InvalidInput(format!( + "inner basis length {} does not match D={D}", + weights.len() + ))); + } + let coeffs = std::array::from_fn(|i| weights[i]); + Ok(CyclotomicRing::from_coefficients(coeffs)) +} + +fn evaluate_packed_ring_poly( + packed_coeffs: &[CyclotomicRing], + outer_point: &[F], +) -> CyclotomicRing { + let weights = lagrange_weights(outer_point); + debug_assert_eq!(weights.len(), packed_coeffs.len()); + packed_coeffs + .iter() + .zip(weights.iter()) + .fold(CyclotomicRing::::zero(), |acc, (f_i, w_i)| { + acc + f_i.scale(w_i) + }) +} + +fn trace(u: &CyclotomicRing) -> F { + let d = F::from_u64(D as u64); + u.coefficients()[0] * d +} + +#[cfg(test)] +mod tests { + use super::*; + use crate::primitives::multilinear_evals::DenseMultilinearEvals; + use crate::protocol::commitment::CommitmentConfig; + use crate::protocol::transcript::Blake2bTranscript; + use crate::test_utils::F; + use crate::{CommitmentScheme, Polynomial}; + + #[test] + fn verify_passes_for_consistent_opening() { + let alpha = Cfg::D.trailing_zeros() as usize; + let num_vars = Cfg::R + Cfg::M + alpha; + let len = 1usize << num_vars; + + let evals: Vec = (0..len).map(|i| F::from_u64(i as u64)).collect(); + let poly = DenseMultilinearEvals::new_padded(evals); + + let setup = >::setup_prover(num_vars); + let verifier_setup = >::setup_verifier(&setup); + + let (commitment, hint) = + >::commit(&poly, &setup).unwrap(); + + let opening_point: Vec = (0..num_vars).map(|i| F::from_u64((i + 2) as u64)).collect(); + let opening = poly.evaluate(&opening_point); + + let mut prover_transcript = Blake2bTranscript::::new(b"test/prove"); + let proof = >::prove( + &setup, + &poly, + &opening_point, + Some(hint), + &mut prover_transcript, + &commitment, + ) + .unwrap(); + + let mut verifier_transcript = Blake2bTranscript::::new(b"test/prove"); + let result = >::verify( + &proof, + &verifier_setup, + &mut verifier_transcript, + &opening_point, + &opening, + &commitment, + ); + + assert!(result.is_ok()); + } +} diff --git a/src/protocol/mod.rs b/src/protocol/mod.rs new file mode 100644 index 00000000..49997845 --- /dev/null +++ b/src/protocol/mod.rs @@ -0,0 +1,30 @@ +//! Protocol-layer transcript and commitment abstractions. +//! +//! This module defines the Hachi-native protocol interfaces used by higher-level +//! proof logic. It intentionally stays independent from external integration +//! details (for example, Jolt wiring). + +pub mod challenges; +pub mod commitment; +pub mod commitment_scheme; +pub mod opening_point; +pub mod proof; +pub mod quadratic_equation; +pub mod ring_switch; +pub mod sumcheck; +pub mod transcript; + +pub use commitment::{ + AppendToTranscript, CommitmentConfig, CommitmentScheme, DefaultCommitmentConfig, DummyProof, + HachiCommitment, HachiCommitmentCore, HachiOpeningClaim, HachiOpeningPoint, RingCommitment, + RingCommitmentScheme, RingCommitmentSetup, StreamingCommitmentScheme, +}; +pub use commitment_scheme::HachiCommitmentScheme; +pub use opening_point::RingOpeningPoint; +pub use proof::{HachiProof, SumcheckAux}; +pub use quadratic_equation::QuadraticEquation; +pub use sumcheck::{ + prove_sumcheck, verify_sumcheck, CompressedUniPoly, SumcheckInstanceProver, + SumcheckInstanceVerifier, SumcheckProof, UniPoly, +}; +pub use transcript::{Blake2bTranscript, KeccakTranscript, Transcript}; diff --git a/src/protocol/opening_point.rs b/src/protocol/opening_point.rs new file mode 100644 index 00000000..154402a9 --- /dev/null +++ b/src/protocol/opening_point.rs @@ -0,0 +1,20 @@ +//! Ring-native opening point for the Hachi protocol. + +use crate::FieldCore; + +/// Ring-native opening point storing field scalars (Lagrange weights). +/// +/// Contains the two vectors used by the §4.2 prover: +/// - `a`: evaluation vector of length `2^m` (inner-block coordinates). +/// - `b`: block-select vector of length `2^r` (outer coordinates). +/// +/// These are raw field scalars, not ring elements — they originate from +/// multilinear Lagrange basis evaluations and are always constant (scalar) +/// ring elements when embedded into the ring. +#[derive(Debug, Clone, PartialEq, Eq)] +pub struct RingOpeningPoint { + /// Evaluation vector of length `2^m` (field scalars). + pub a: Vec, + /// Block-select vector of length `2^r` (field scalars). + pub b: Vec, +} diff --git a/src/protocol/proof.rs b/src/protocol/proof.rs new file mode 100644 index 00000000..9a8c07e3 --- /dev/null +++ b/src/protocol/proof.rs @@ -0,0 +1,59 @@ +//! Proof structures for the Hachi protocol. + +use crate::algebra::ring::CyclotomicRing; +use crate::primitives::serialization::Compress; +use crate::protocol::commitment::RingCommitment; +use crate::protocol::sumcheck::SumcheckProof; +use crate::{FieldCore, HachiSerialize}; + +/// Prover-side hint produced at commitment time. +#[derive(Debug, Clone, PartialEq, Eq)] +pub struct HachiCommitmentHint { + /// Decomposed `s_i` blocks from the commitment phase. + pub s: Vec>>, + /// Decomposed `t̂_i` blocks from the commitment phase. + pub t_hat: Vec>>, + /// Ring coefficients from the §3.1 reduction (evaluation table). + pub ring_coeffs: Vec>, +} + +/// Temporary auxiliary data the verifier needs for sumcheck output verification. +/// +/// Will be removed once recursive PCS evaluation proofs replace the direct +/// oracle check at the end of each sumcheck instance. +#[derive(Debug, Clone, PartialEq, Eq)] +pub struct SumcheckAux { + /// `w` coefficients (z and r coefficients, concatenated). The verifier + /// reshapes this into sumcheck evaluation form to compute the expected + /// output claims for F_0 and F_alpha. + pub w: Vec, +} + +/// Hachi Proof for One Iteration. +#[derive(Debug, Clone, PartialEq, Eq)] +pub struct HachiProof { + /// `y_ring` from the §3.1 reduction. + pub y_ring: CyclotomicRing, + /// `v = D · ŵ`. + pub v: Vec>, + /// Range-check (norm) sumcheck proof (§4.3, F_0). + pub norm_proof: SumcheckProof, + /// Evaluation-relation sumcheck proof (§4.3, F_α). + pub relation_proof: SumcheckProof, + /// Temporary verifier auxiliary (will be removed with recursive PCS). + pub sumcheck_aux: SumcheckAux, + /// Commitment to the sumcheck witness `w`. + pub w_commitment: RingCommitment, +} + +impl HachiProof { + /// Returns the proof size in bytes (uncompressed). + pub fn size(&self) -> usize { + self.v.serialized_size(Compress::No) + + self.y_ring.serialized_size(Compress::No) + + self.sumcheck_aux.w.serialized_size(Compress::No) + + self.norm_proof.serialized_size(Compress::No) + + self.relation_proof.serialized_size(Compress::No) + + self.w_commitment.serialized_size(Compress::No) + } +} diff --git a/src/protocol/quadratic_equation.rs b/src/protocol/quadratic_equation.rs new file mode 100644 index 00000000..16114ffa --- /dev/null +++ b/src/protocol/quadratic_equation.rs @@ -0,0 +1,744 @@ +//! Quadratic equation builder for the Hachi PCS (§4.2). +//! +//! This module encapsulates the stage-1 prover logic and the generation of +//! the quadratic equation components M, y, z, and v. + +use crate::algebra::ring::{CyclotomicRing, SparseChallenge, SparseChallengeConfig}; +use crate::error::HachiError; +use crate::protocol::challenges::sparse::sample_sparse_challenges; +use crate::protocol::commitment::utils::linear::mat_vec_mul_unchecked; +use crate::protocol::commitment::utils::norm::{detect_field_modulus, vec_inf_norm}; +use crate::protocol::commitment::{CommitmentConfig, RingCommitment, RingCommitmentSetup}; +use crate::protocol::opening_point::RingOpeningPoint; +use crate::protocol::proof::HachiCommitmentHint; +use crate::protocol::transcript::labels::{ABSORB_PROVER_V, CHALLENGE_STAGE1_FOLD}; +use crate::protocol::transcript::Transcript; +use crate::{CanonicalField, FieldCore}; + +/// **Steps 1–3.** Compute `w_i = a^T G_{2^m} s_i` and decompose: `ŵ_i = G_1^{-1}(w_i)`. +fn compute_w_hat( + opening_point: &RingOpeningPoint, + s: &[Vec>], +) -> Vec>> +where + F: FieldCore + CanonicalField, + Cfg: CommitmentConfig, +{ + let a = &opening_point.a; + let block_len = 1usize << Cfg::M; + let delta = Cfg::DELTA; + let log_basis = Cfg::LOG_BASIS; + + debug_assert_eq!(a.len(), block_len); + + s.iter() + .map(|s_i| { + let mut w_i = CyclotomicRing::::zero(); + for (j, a_j) in a.iter().enumerate().take(block_len) { + let start = j * delta; + let end = start + delta; + let recomp_j = CyclotomicRing::gadget_recompose_pow2(&s_i[start..end], log_basis); + w_i += recomp_j.scale(a_j); + } + w_i.balanced_decompose_pow2(delta, log_basis) + }) + .collect() +} + +/// **Step 4.** Compute `v = D · ŵ` (first prover message). +#[allow(non_snake_case)] +fn compute_v( + d: &[Vec>], + w_hat: &[Vec>], +) -> Vec> { + let w_hat_flat: Vec> = + w_hat.iter().flat_map(|v| v.iter().copied()).collect(); + mat_vec_mul_unchecked(d, &w_hat_flat) +} + +/// **Steps 7–9.** Fold `z = Σ c_i · s_i`, check `‖z‖_∞ ≤ β`, and decompose `ẑ = J^{-1}(z)`. +fn compute_z_hat( + s: &[Vec>], + challenges: &[SparseChallenge], +) -> Result>, HachiError> +where + F: FieldCore + CanonicalField, + Cfg: CommitmentConfig, +{ + debug_assert_eq!(challenges.len(), s.len()); + let len = s[0].len(); + let mut z = vec![CyclotomicRing::::zero(); len]; + for (c_i, s_i) in challenges.iter().zip(s.iter()) { + for (z_j, s_ij) in z.iter_mut().zip(s_i.iter()) { + *z_j += s_ij.mul_by_sparse(c_i); + } + } + + let modulus = detect_field_modulus::(); + let norm = vec_inf_norm(&z, modulus); + if norm > Cfg::BETA { + return Err(HachiError::InvalidInput(format!( + "prover abort: ||z||_inf = {norm} > beta = {}", + Cfg::BETA + ))); + } + + Ok(z.iter() + .flat_map(|z_j| z_j.balanced_decompose_pow2(Cfg::TAU, Cfg::LOG_BASIS)) + .collect()) +} + +/// Stage-1 quadratic equation state for the Hachi protocol. +/// +/// Encapsulates the relation $M(x) \cdot z = y(x) + (X^D + 1) \cdot r(x)$ +/// along with intermediate prover witness data (`w_hat`, `z_hat`, `hint`). +pub struct QuadraticEquation { + /// Stage-1 proof vector `v = D · ŵ`. + pub v: Vec>, + /// Stage-1 folding challenges (sparse representation). + pub challenges: Vec, + /// Matrix `M`. + m: Vec>>, + /// Vector `y`. + y: Vec>, + /// Vector `z` (prover only). + z: Option>>, + /// Decomposed `ŵ_i = G_1^{-1}(w_i)` (prover only). + w_hat: Option>>>, + /// Decomposed `ẑ = J^{-1}(z)` (prover only). + z_hat: Option>>, + /// Commitment hint (prover only). + hint: Option>, + + _marker: std::marker::PhantomData, +} + +impl QuadraticEquation +where + F: FieldCore + CanonicalField, + Cfg: CommitmentConfig, +{ + /// Prover constructor: runs §4.2 stage 1 and builds all equation components. + /// + /// # Errors + /// + /// Returns an error if the norm check, challenge sampling, or matrix + /// generation fails. + pub fn new_prover>( + setup: &RingCommitmentSetup, + ring_opening_point: &RingOpeningPoint, + hint: &HachiCommitmentHint, + transcript: &mut T, + commitment: &RingCommitment, + y_ring: &CyclotomicRing, + ) -> Result { + // Steps 1–3: w_i = a^T G_{2^m} s_i, then ŵ_i = G_1^{-1}(w_i) + let w_hat = compute_w_hat::(ring_opening_point, &hint.s); + + // Step 4: v = D · ŵ + let v = compute_v(&setup.D, &w_hat); + + // Step 5: append v to transcript + transcript.append_serde(ABSORB_PROVER_V, &v); + + // Step 6: sample sparse folding challenges + let challenge_cfg = SparseChallengeConfig { + weight: Cfg::CHALLENGE_WEIGHT, + nonzero_coeffs: vec![-1, 1], + }; + let challenges = sample_sparse_challenges::( + transcript, + CHALLENGE_STAGE1_FOLD, + 1usize << Cfg::R, + &challenge_cfg, + )?; + + // Steps 7–9: z = Σ c_i · s_i, check ‖z‖_∞ ≤ β, then ẑ = J^{-1}(z) + let z_hat = compute_z_hat::(&hint.s, &challenges)?; + + let m = generate_m::(setup, ring_opening_point, &challenges)?; + let y = generate_y::(&v, &commitment.u, y_ring)?; + let z = generate_z(&w_hat, &hint.t_hat, &z_hat); + + Ok(Self { + v, + challenges, + m, + y, + z: Some(z), + w_hat: Some(w_hat), + z_hat: Some(z_hat), + hint: Some(hint.clone()), + _marker: std::marker::PhantomData, + }) + } + + /// Verifier constructor: Derives challenges and computes M and y. + /// + /// # Errors + /// + /// Returns an error if challenge derivation fails. + pub fn new_verifier>( + setup: &RingCommitmentSetup, + ring_opening_point: &RingOpeningPoint, + v: &Vec>, + transcript: &mut T, + commitment: &RingCommitment, + y_ring: &CyclotomicRing, + ) -> Result { + let challenges = derive_stage1_challenges::(transcript, v)?; + let m = generate_m::(setup, ring_opening_point, &challenges)?; + let y = generate_y::(v, &commitment.u, y_ring)?; + + Ok(Self { + v: v.to_vec(), + challenges, + m, + y, + z: None, + w_hat: None, + z_hat: None, + hint: None, + _marker: std::marker::PhantomData, + }) + } + + /// Get the matrix M. + pub fn m(&self) -> &[Vec>] { + &self.m + } + + /// Get the vector y. + pub fn y(&self) -> &[CyclotomicRing] { + &self.y + } + + /// Get the vector z (returns None if constructed by verifier). + pub fn z(&self) -> Option<&[CyclotomicRing]> { + self.z.as_deref() + } + + /// Get the vector v. + pub fn v(&self) -> &[CyclotomicRing] { + &self.v + } + + /// Get the decomposed witness `ŵ` (prover only). + pub fn w_hat(&self) -> Option<&[Vec>]> { + self.w_hat.as_deref() + } + + /// Get the decomposed folded witness `ẑ` (prover only). + pub fn z_hat(&self) -> Option<&[CyclotomicRing]> { + self.z_hat.as_deref() + } + + /// Get the commitment hint (prover only). + pub fn hint(&self) -> Option<&HachiCommitmentHint> { + self.hint.as_ref() + } +} + +pub(crate) fn derive_stage1_challenges( + transcript: &mut T, + v: &Vec>, +) -> Result, HachiError> +where + F: FieldCore + CanonicalField, + T: Transcript, +{ + let challenge_cfg = SparseChallengeConfig { + weight: Cfg::CHALLENGE_WEIGHT, + nonzero_coeffs: vec![-1, 1], + }; + let num_blocks = 1usize + .checked_shl(Cfg::R as u32) + .ok_or_else(|| HachiError::InvalidSetup("2^R does not fit usize".to_string()))?; + transcript.append_serde(ABSORB_PROVER_V, v); + sample_sparse_challenges::( + transcript, + CHALLENGE_STAGE1_FOLD, + num_blocks, + &challenge_cfg, + ) +} + +fn constant_ring(value: F) -> CyclotomicRing { + let mut coeffs = [F::zero(); D]; + coeffs[0] = value; + CyclotomicRing::from_coefficients(coeffs) +} + +fn gadget_row_scalars(levels: usize, log_basis: u32) -> Vec { + let base = F::from_canonical_u128_reduced(1u128 << log_basis); + let mut out = Vec::with_capacity(levels); + let mut power = F::one(); + for _ in 0..levels { + out.push(power); + power = power * base; + } + out +} + +/// Kronecker product of two scalar vectors, producing constant ring elements. +/// Cost: O(1) per pair (just a field multiplication + ring construction). +fn kron_scalars( + left: &[F], + right: &[F], +) -> Vec> { + let mut out = Vec::with_capacity(left.len().saturating_mul(right.len())); + for &l in left { + for &r in right { + out.push(constant_ring(l * r)); + } + } + out +} + +/// Kronecker product where the right operand is field scalars. +/// Cost: O(D) per pair instead of O(D^2). +fn kron_row_scale( + left: &[CyclotomicRing], + right: &[F], +) -> Vec> { + let mut out = Vec::with_capacity(left.len().saturating_mul(right.len())); + for l in left { + for &r in right { + out.push(l.scale(&r)); + } + } + out +} + +/// Kronecker product of sparse challenges with scalar gadget entries. +/// Cost: O(omega + D) per pair instead of O(D^2). +fn kron_sparse_scale( + left: &[SparseChallenge], + right: &[F], +) -> Vec> { + let mut out = Vec::with_capacity(left.len().saturating_mul(right.len())); + for l in left { + let dense: CyclotomicRing = l.to_dense().expect("valid sparse challenge"); + for &r in right { + out.push(dense.scale(&r)); + } + } + out +} + +fn gadget_block_diag_scalars(blocks: usize, row: &[F]) -> Vec> { + let row_len = row.len(); + let mut rows = Vec::with_capacity(blocks); + for i in 0..blocks { + let mut out = vec![F::zero(); blocks * row_len]; + let start = i * row_len; + out[start..start + row_len].copy_from_slice(row); + rows.push(out); + } + rows +} + +pub(crate) fn generate_m( + setup: &RingCommitmentSetup, + opening_point: &RingOpeningPoint, + challenges: &[SparseChallenge], +) -> Result>>, HachiError> +where + F: FieldCore + CanonicalField, +{ + let num_blocks = 1usize + .checked_shl(Cfg::R as u32) + .ok_or_else(|| HachiError::InvalidSetup("2^R does not fit usize".to_string()))?; + let block_len = 1usize + .checked_shl(Cfg::M as u32) + .ok_or_else(|| HachiError::InvalidSetup("2^M does not fit usize".to_string()))?; + let w_len = Cfg::DELTA + .checked_mul(num_blocks) + .ok_or_else(|| HachiError::InvalidSetup("w length overflow".to_string()))?; + let t_len = Cfg::DELTA + .checked_mul(Cfg::N_A) + .and_then(|v| v.checked_mul(num_blocks)) + .ok_or_else(|| HachiError::InvalidSetup("t length overflow".to_string()))?; + let z_len = Cfg::TAU + .checked_mul(Cfg::DELTA) + .and_then(|v| v.checked_mul(block_len)) + .ok_or_else(|| HachiError::InvalidSetup("z length overflow".to_string()))?; + let total_cols = w_len + .checked_add(t_len) + .and_then(|v| v.checked_add(z_len)) + .ok_or_else(|| HachiError::InvalidSetup("matrix width overflow".to_string()))?; + + if opening_point.b.len() != num_blocks { + return Err(HachiError::InvalidPointDimension { + expected: num_blocks, + actual: opening_point.b.len(), + }); + } + if opening_point.a.len() != block_len { + return Err(HachiError::InvalidPointDimension { + expected: block_len, + actual: opening_point.a.len(), + }); + } + if challenges.len() != num_blocks { + return Err(HachiError::InvalidSize { + expected: num_blocks, + actual: challenges.len(), + }); + } + if setup.D.len() != Cfg::N_D { + return Err(HachiError::InvalidSize { + expected: Cfg::N_D, + actual: setup.D.len(), + }); + } + if setup.B.len() != Cfg::N_B { + return Err(HachiError::InvalidSize { + expected: Cfg::N_B, + actual: setup.B.len(), + }); + } + if setup.A.len() != Cfg::N_A { + return Err(HachiError::InvalidSize { + expected: Cfg::N_A, + actual: setup.A.len(), + }); + } + if setup.A.first().map(|row| row.len()) != Some(block_len * Cfg::DELTA) { + return Err(HachiError::InvalidSetup("A row width mismatch".to_string())); + } + + let g1 = gadget_row_scalars::(Cfg::DELTA, Cfg::LOG_BASIS); + let j1 = gadget_row_scalars::(Cfg::TAU, Cfg::LOG_BASIS); + + let row3_w = kron_scalars::(&opening_point.b, &g1); + let row4_w = kron_sparse_scale::(challenges, &g1); + + let ag = kron_scalars::(&opening_point.a, &g1); + let ag_scalars: Vec = ag.iter().map(|r| r.coefficients()[0]).collect(); + let row4_z = kron_scalars::(&ag_scalars, &j1) + .into_iter() + .map(|x| -x) + .collect::>(); + + let g_na = gadget_block_diag_scalars::(Cfg::N_A, &g1); + let row5_mid = g_na + .iter() + .map(|row| kron_sparse_scale::(challenges, row)) + .collect::>(); + let row5_right = setup + .A + .iter() + .map(|row| kron_row_scale(row, &j1).into_iter().map(|x| -x).collect()) + .collect::>>(); + + let zero = CyclotomicRing::::zero(); + let mut rows = Vec::with_capacity(Cfg::N_D + Cfg::N_B + 1usize + 1usize + Cfg::N_A); + + for row in setup.D.iter() { + if row.len() != w_len { + return Err(HachiError::InvalidSetup("D row width mismatch".to_string())); + } + let mut full = vec![zero; total_cols]; + full[..w_len].copy_from_slice(row); + rows.push(full); + } + + for row in setup.B.iter() { + if row.len() != t_len { + return Err(HachiError::InvalidSetup("B row width mismatch".to_string())); + } + let mut full = vec![zero; total_cols]; + full[w_len..w_len + t_len].copy_from_slice(row); + rows.push(full); + } + + let mut row3 = vec![zero; total_cols]; + row3[..w_len].copy_from_slice(&row3_w); + rows.push(row3); + + let mut row4 = vec![zero; total_cols]; + row4[..w_len].copy_from_slice(&row4_w); + row4[w_len + t_len..].copy_from_slice(&row4_z); + rows.push(row4); + + for (mid, right) in row5_mid.into_iter().zip(row5_right.into_iter()) { + let mut row = vec![zero; total_cols]; + row[w_len..w_len + t_len].copy_from_slice(&mid); + row[w_len + t_len..].copy_from_slice(&right); + rows.push(row); + } + + Ok(rows) +} + +pub(crate) fn generate_z( + w_hat: &[Vec>], + t_hat: &[Vec>], + z_hat: &[CyclotomicRing], +) -> Vec> { + let mut out = Vec::with_capacity( + w_hat.len() + + t_hat.len() + + z_hat.len() + + w_hat.iter().map(|v| v.len()).sum::() + + t_hat.iter().map(|v| v.len()).sum::(), + ); + for w in w_hat { + out.extend(w.iter().copied()); + } + for t in t_hat { + out.extend(t.iter().copied()); + } + out.extend_from_slice(z_hat); + out +} + +pub(crate) fn generate_y( + v: &[CyclotomicRing], + u: &[CyclotomicRing], + u_eval: &CyclotomicRing, +) -> Result>, HachiError> +where + F: FieldCore, +{ + if v.len() != Cfg::N_D { + return Err(HachiError::InvalidSize { + expected: Cfg::N_D, + actual: v.len(), + }); + } + if u.len() != Cfg::N_B { + return Err(HachiError::InvalidSize { + expected: Cfg::N_B, + actual: u.len(), + }); + } + let mut out = Vec::with_capacity(Cfg::N_D + Cfg::N_B + 1 + 1 + Cfg::N_A); + out.extend_from_slice(v); + out.extend_from_slice(u); + out.push(*u_eval); + out.push(CyclotomicRing::::zero()); + out.extend(std::iter::repeat_n( + CyclotomicRing::::zero(), + Cfg::N_A, + )); + Ok(out) +} + +#[cfg(test)] +mod tests { + use super::*; + use crate::algebra::ring::SparseChallengeConfig; + use crate::algebra::CyclotomicRing; + use crate::protocol::challenges::sparse::sample_sparse_challenges; + use crate::protocol::commitment::{HachiCommitmentCore, RingCommitmentScheme}; + use crate::protocol::proof::HachiCommitmentHint; + use crate::protocol::transcript::Blake2bTranscript; + use crate::test_utils::*; + use crate::Transcript; + + const TRANSCRIPT_SEED: &[u8] = b"test/prover-relation"; + + fn replay_challenges(v: &Vec>) -> Vec> { + let mut transcript = Blake2bTranscript::::new(TRANSCRIPT_SEED); + transcript.append_serde(ABSORB_PROVER_V, v); + + let challenge_cfg = SparseChallengeConfig { + weight: TinyConfig::CHALLENGE_WEIGHT, + nonzero_coeffs: vec![-1, 1], + }; + let sparse = sample_sparse_challenges::, D>( + &mut transcript, + CHALLENGE_STAGE1_FOLD, + NUM_BLOCKS, + &challenge_cfg, + ) + .unwrap(); + sparse + .iter() + .map(|c| c.to_dense::().unwrap()) + .collect() + } + + struct Fixture { + setup: RingCommitmentSetup, + commitment_u: Vec>, + point: RingOpeningPoint, + blocks: Vec>>, + quad_eq: QuadraticEquation, + /// Challenges re-derived via transcript replay (cross-check). + challenges: Vec>, + } + + fn build_fixture() -> Fixture { + let (setup, _) = + >::setup(16).unwrap(); + + let blocks = sample_blocks(); + let (commitment, s, t_hat) = + >::commit_ring_blocks( + &blocks, &setup, + ) + .unwrap(); + + let point = RingOpeningPoint { + a: sample_a(), + b: sample_b(), + }; + + let hint = HachiCommitmentHint { + s, + t_hat, + ring_coeffs: Vec::new(), + }; + let mut transcript = Blake2bTranscript::::new(TRANSCRIPT_SEED); + let y_ring = CyclotomicRing::::zero(); + let quad_eq = QuadraticEquation::::new_prover( + &setup, + &point, + &hint, + &mut transcript, + &commitment, + &y_ring, + ) + .unwrap(); + + let challenges = replay_challenges(&quad_eq.v); + + Fixture { + setup, + commitment_u: commitment.u.clone(), + point, + blocks, + quad_eq, + challenges, + } + } + + /// Row 1: D · ŵ = v + #[test] + fn row1_d_times_w_hat_equals_v() { + let f = build_fixture(); + + let w_hat = f.quad_eq.w_hat().unwrap(); + let w_hat_flat: Vec> = + w_hat.iter().flat_map(|v| v.iter().copied()).collect(); + let lhs = mat_vec_mul(&f.setup.D, &w_hat_flat); + + assert_eq!(lhs, f.quad_eq.v(), "Row 1 failed: D · ŵ ≠ v"); + } + + /// Row 2: B · t̂ = u (commitment vector) + #[test] + fn row2_b_times_t_hat_equals_u_commitment() { + let f = build_fixture(); + + let hint = f.quad_eq.hint().unwrap(); + let t_hat_flat: Vec> = + hint.t_hat.iter().flat_map(|v| v.iter().copied()).collect(); + let lhs = mat_vec_mul(&f.setup.B, &t_hat_flat); + + assert_eq!(lhs, f.commitment_u, "Row 2 failed: B · t̂ ≠ u"); + } + + /// Row 3: b^T · G_{2^r} · ŵ = u_eval + #[test] + fn row3_bt_gadget_w_hat_equals_u_eval() { + let f = build_fixture(); + + let w_hat = f.quad_eq.w_hat().unwrap(); + let w_recomposed: Vec> = w_hat + .iter() + .map(|w_hat_i| CyclotomicRing::gadget_recompose_pow2(w_hat_i, LOG_BASIS)) + .collect(); + + let u_eval = w_recomposed + .iter() + .zip(f.point.b.iter()) + .fold(CyclotomicRing::::zero(), |acc, (w_i, b_i)| { + acc + w_i.scale(b_i) + }); + + let u_eval_direct = f.blocks.iter().zip(f.point.b.iter()).fold( + CyclotomicRing::::zero(), + |acc, (block_i, b_i)| { + let inner: CyclotomicRing = block_i + .iter() + .zip(f.point.a.iter()) + .fold(CyclotomicRing::::zero(), |acc2, (f_ij, a_j)| { + acc2 + f_ij.scale(a_j) + }); + acc + inner.scale(b_i) + }, + ); + + assert_eq!( + u_eval, u_eval_direct, + "Row 3 failed: b^T G ŵ ≠ Σ b_i (a^T f_i)" + ); + } + + /// Row 4: (c^T ⊗ G_1) · ŵ = a^T · G_{2^m} · J · ẑ + #[test] + fn row4_challenge_fold_w_equals_a_gadget_j_z_hat() { + let f = build_fixture(); + + let w_hat = f.quad_eq.w_hat().unwrap(); + let w: Vec> = w_hat + .iter() + .map(|w_hat_i| CyclotomicRing::gadget_recompose_pow2(w_hat_i, LOG_BASIS)) + .collect(); + + let lhs = f + .challenges + .iter() + .zip(w.iter()) + .fold(CyclotomicRing::::zero(), |acc, (c_i, w_i)| { + acc + (*c_i * *w_i) + }); + + let z_recovered = recompose_z_hat(f.quad_eq.z_hat().unwrap()); + let rhs = a_transpose_gadget_times_vec(&f.point.a, &z_recovered); + + assert_eq!(lhs, rhs, "Row 4 failed: (c^T ⊗ G_1)ŵ ≠ a^T G J ẑ"); + } + + /// Row 5: (c^T ⊗ G_{n_A}) · t̂ = A · J · ẑ + #[test] + fn row5_challenge_fold_t_equals_a_j_z_hat() { + let f = build_fixture(); + + let hint = f.quad_eq.hint().unwrap(); + let mut lhs = vec![CyclotomicRing::::zero(); N_A]; + for (c_i, t_hat_i) in f.challenges.iter().zip(hint.t_hat.iter()) { + let t_i = gadget_recompose_vec(t_hat_i); + assert_eq!(t_i.len(), N_A); + for (lhs_j, t_ij) in lhs.iter_mut().zip(t_i.iter()) { + *lhs_j += *c_i * *t_ij; + } + } + + let z_recovered = recompose_z_hat(f.quad_eq.z_hat().unwrap()); + let rhs = mat_vec_mul(&f.setup.A, &z_recovered); + + assert_eq!(lhs, rhs, "Row 5 failed: (c^T ⊗ G_nA)t̂ ≠ A · J · ẑ"); + } + + #[test] + fn prove_output_shapes_are_correct() { + let f = build_fixture(); + + assert_eq!(f.quad_eq.v().len(), TinyConfig::N_D); + + let w_hat = f.quad_eq.w_hat().unwrap(); + assert_eq!(w_hat.len(), NUM_BLOCKS); + assert!(w_hat.iter().all(|v| v.len() == DELTA)); + + let hint = f.quad_eq.hint().unwrap(); + assert_eq!(hint.t_hat.len(), NUM_BLOCKS); + assert!(hint.t_hat.iter().all(|v| v.len() == N_A * DELTA)); + + assert_eq!(f.quad_eq.z_hat().unwrap().len(), BLOCK_LEN * DELTA * TAU); + } +} diff --git a/src/protocol/ring_switch.rs b/src/protocol/ring_switch.rs new file mode 100644 index 00000000..9a9d19a7 --- /dev/null +++ b/src/protocol/ring_switch.rs @@ -0,0 +1,453 @@ +//! Ring switching logic for the Hachi PCS (Section 4.3). +//! +//! Handles the transition from the ring-based quadratic equation to field-based +//! sumcheck instances by expanding the ring elements into their coefficient +//! vectors and setting up the evaluation tables. + +use crate::algebra::ring::CyclotomicRing; +use crate::error::HachiError; +use crate::protocol::commitment::utils::norm::detect_field_modulus; +use crate::protocol::commitment::{ + CommitmentConfig, DefaultCommitmentConfig, HachiCommitmentCore, RingCommitment, + RingCommitmentScheme, +}; +use crate::protocol::quadratic_equation::QuadraticEquation; +use crate::protocol::sumcheck::eq_evals; +use crate::protocol::transcript::labels::{ + ABSORB_SUMCHECK_W, CHALLENGE_RING_SWITCH, CHALLENGE_TAU0, CHALLENGE_TAU1, +}; +use crate::protocol::transcript::Transcript; +use crate::{CanonicalField, FieldCore, FieldSampling}; + +/// Output of the ring switch prover, containing everything needed for sumchecks. +pub struct RingSwitchOutput { + /// The witness vector w (concatenation of z and r coefficients). + pub w: Vec, + /// Commitment to w. + pub w_commitment: RingCommitment, + /// Evaluation table of w (reordered for sumcheck). + pub w_evals: Vec, + /// Evaluation table of M_alpha(x) (tau1-weighted). + pub m_evals_x: Vec, + /// Evaluation table of alpha powers (y dimension). + pub alpha_evals_y: Vec, + /// Number of upper variable bits in w_evals. + pub num_u: usize, + /// Number of lower variable bits in w_evals. + pub num_l: usize, + /// Challenge tau0 for F_0 sumcheck. + pub tau0: Vec, + /// Challenge tau1 for F_alpha sumcheck. + pub tau1: Vec, + /// Basis size b = 2^LOG_BASIS. + pub b: usize, + /// Ring-switch challenge alpha. + pub alpha: F, +} + +/// Execute the prover side of the ring switching protocol (Section 4.3). +/// +/// # Errors +/// +/// Returns an error if z is missing, commitment fails, or matrix expansion fails. +pub fn ring_switch_prover( + quad_eq: &QuadraticEquation, + transcript: &mut T, +) -> Result, HachiError> +where + F: FieldCore + CanonicalField + FieldSampling, + T: Transcript, + Cfg: CommitmentConfig, +{ + let z = quad_eq + .z() + .ok_or_else(|| HachiError::InvalidInput("missing z in prover".to_string()))?; + let r = compute_r_via_poly_division::(quad_eq.m(), z, quad_eq.y()); + let w = build_w_coeffs::(z, &r); + + let w_commitment = commit_w::(&w)?; + transcript.append_serde(ABSORB_SUMCHECK_W, &w_commitment); + + let alpha: F = transcript.challenge_scalar(CHALLENGE_RING_SWITCH); + + let m_a = eval_ring_matrix_at::(quad_eq.m(), &alpha); + let m_a_vec = expand_m_a::(&m_a, alpha)?; + let m_rows = m_row_count::(); + let m_cols = if m_a.is_empty() { + 0 + } else { + m_a_vec.len() / m_a.len() + }; + + let (w_evals, num_u, num_l) = build_w_evals(&w, D); + let alpha_evals_y = build_alpha_evals_y(alpha, D); + + let num_sc_vars = num_u + num_l; + let tau0 = sample_tau::(transcript, CHALLENGE_TAU0, num_sc_vars); + + let num_i = m_rows.next_power_of_two().trailing_zeros() as usize; + let tau1 = sample_tau::(transcript, CHALLENGE_TAU1, num_i); + + let m_evals_x = build_m_evals_x::(&m_a_vec, m_rows, m_cols, &tau1); + + Ok(RingSwitchOutput { + w, + w_commitment, + w_evals, + m_evals_x, + alpha_evals_y, + num_u, + num_l, + tau0, + tau1, + b: 1usize << Cfg::LOG_BASIS, + alpha, + }) +} + +/// Replay the verifier side of ring switching to reconstruct evaluation tables. +/// +/// # Errors +/// +/// Returns an error if matrix expansion fails. +pub fn ring_switch_verifier( + quad_eq: &QuadraticEquation, + w: &[F], + w_commitment: &RingCommitment, + transcript: &mut T, +) -> Result, HachiError> +where + F: FieldCore + CanonicalField + FieldSampling, + T: Transcript, + Cfg: CommitmentConfig, +{ + transcript.append_serde(ABSORB_SUMCHECK_W, w_commitment); + + let alpha: F = transcript.challenge_scalar(CHALLENGE_RING_SWITCH); + + let m_a = eval_ring_matrix_at::(quad_eq.m(), &alpha); + let m_a_vec = expand_m_a::(&m_a, alpha)?; + let m_rows = m_row_count::(); + let m_cols = if m_a.is_empty() { + 0 + } else { + m_a_vec.len() / m_a.len() + }; + + let (w_evals, num_u, num_l) = build_w_evals(w, D); + let alpha_evals_y = build_alpha_evals_y(alpha, D); + + let num_sc_vars = num_u + num_l; + let tau0 = sample_tau::(transcript, CHALLENGE_TAU0, num_sc_vars); + + let num_i = m_rows.next_power_of_two().trailing_zeros() as usize; + let tau1 = sample_tau::(transcript, CHALLENGE_TAU1, num_i); + + let m_evals_x = build_m_evals_x::(&m_a_vec, m_rows, m_cols, &tau1); + + Ok(RingSwitchOutput { + w: w.to_vec(), + w_commitment: w_commitment.clone(), + w_evals, + m_evals_x, + alpha_evals_y, + num_u, + num_l, + tau0, + tau1, + b: 1usize << Cfg::LOG_BASIS, + alpha, + }) +} + +pub(crate) fn compute_r_via_poly_division( + m: &[Vec>], + z: &[CyclotomicRing], + y: &[CyclotomicRing], +) -> Vec> { + let poly_len = 2 * D - 1; + m.iter() + .zip(y.iter()) + .map(|(row, y_i)| { + let mut poly = vec![F::zero(); poly_len]; + for (m_ij, z_j) in row.iter().zip(z.iter()) { + if m_ij.is_zero() { + continue; + } + let a = m_ij.coefficients(); + let b = z_j.coefficients(); + + // Detect constant (scalar) ring elements: only a[0] is non-zero. + let is_scalar = a[1..].iter().all(|c| c.is_zero()); + if is_scalar { + let scalar = a[0]; + for s in 0..D { + poly[s] = poly[s] + scalar * b[s]; + } + } else { + for t in 0..D { + for s in 0..D { + poly[t + s] = poly[t + s] + a[t] * b[s]; + } + } + } + } + let y_coeffs = y_i.coefficients(); + for k in 0..D { + poly[k] = poly[k] - y_coeffs[k]; + } + let mut quotient = vec![F::zero(); D]; + for k in (D..poly_len).rev() { + let q = poly[k]; + quotient[k - D] = q; + poly[k - D] = poly[k - D] - q; + } + let coeffs: [F; D] = std::array::from_fn(|k| quotient[k]); + CyclotomicRing::from_coefficients(coeffs) + }) + .collect() +} + +#[derive(Clone, Copy, Debug)] +struct WCommitmentConfig; + +impl CommitmentConfig for WCommitmentConfig { + const D: usize = DefaultCommitmentConfig::D; + const M: usize = 11; + const R: usize = 0; + const N_A: usize = DefaultCommitmentConfig::N_A; + const N_B: usize = DefaultCommitmentConfig::N_B; + const N_D: usize = DefaultCommitmentConfig::N_D; + const LOG_BASIS: u32 = DefaultCommitmentConfig::LOG_BASIS; + const DELTA: usize = DefaultCommitmentConfig::DELTA; + const TAU: usize = DefaultCommitmentConfig::TAU; + const BETA: u128 = DefaultCommitmentConfig::BETA; + const CHALLENGE_WEIGHT: usize = DefaultCommitmentConfig::CHALLENGE_WEIGHT; +} + +fn commit_w(w: &[F]) -> Result, HachiError> +where + F: FieldCore + CanonicalField + FieldSampling, +{ + let d = DefaultCommitmentConfig::D; + let block_len = 1usize << WCommitmentConfig::M; + + let ring_elems: Vec> = w + .chunks(d) + .map(|chunk| { + let coeffs: [F; DefaultCommitmentConfig::D] = + std::array::from_fn(|i| if i < chunk.len() { chunk[i] } else { F::zero() }); + CyclotomicRing::from_coefficients(coeffs) + }) + .collect(); + + let mut padded = ring_elems; + padded.resize( + block_len, + CyclotomicRing::::zero(), + ); + + let blocks = vec![padded]; + + let (w_setup, _) = >::setup(WCommitmentConfig::M)?; + + let (commitment, _, _) = >::commit_ring_blocks(&blocks, &w_setup)?; + + Ok(commitment) +} + +pub(crate) fn eval_ring_at(r: &CyclotomicRing, alpha: &F) -> F { + let mut acc = F::zero(); + let mut power = F::one(); + for coeff in r.coefficients() { + acc = acc + (*coeff * power); + power = power * *alpha; + } + acc +} + +pub(crate) fn eval_ring_vec_at( + v: &[CyclotomicRing], + alpha: &F, +) -> Vec { + v.iter().map(|r| eval_ring_at(r, alpha)).collect() +} + +pub(crate) fn eval_ring_matrix_at( + m: &[Vec>], + alpha: &F, +) -> Vec> { + m.iter().map(|row| eval_ring_vec_at(row, alpha)).collect() +} + +pub(crate) fn r_decomp_levels() -> usize { + let modulus = detect_field_modulus::(); + let bits = 128 - (modulus.saturating_sub(1)).leading_zeros() as usize; + let log_basis = Cfg::LOG_BASIS as usize; + let mut levels = (bits + log_basis.saturating_sub(1)) / log_basis.max(1); + if levels == 0 { + levels = 1; + } + + let b = 1u128 << Cfg::LOG_BASIS; + let half_q = modulus / 2; + let half_b_minus_1 = b / 2 - 1; + let b_minus_1 = b - 1; + let mut b_pow = 1u128; + for _ in 0..levels { + b_pow = b_pow.saturating_mul(b); + } + let max_positive = half_b_minus_1.saturating_mul((b_pow - 1) / b_minus_1); + if max_positive < half_q { + levels += 1; + } + + levels +} + +pub(crate) fn expand_m_a( + m_a: &[Vec], + alpha: F, +) -> Result, HachiError> { + if m_a.is_empty() { + return Ok(Vec::new()); + } + let rows = m_a.len(); + let cols = m_a[0].len(); + if cols == 0 { + return Ok(vec![F::zero(); rows]); + } + for row in m_a.iter() { + if row.len() != cols { + return Err(HachiError::InvalidSize { + expected: cols, + actual: row.len(), + }); + } + } + + let levels = r_decomp_levels::(); + let total_cols = cols + .checked_add( + rows.checked_mul(levels) + .ok_or_else(|| HachiError::InvalidSetup("expanded M width overflow".to_string()))?, + ) + .ok_or_else(|| HachiError::InvalidSetup("expanded M width overflow".to_string()))?; + + let base = F::from_canonical_u128_reduced(1u128 << Cfg::LOG_BASIS); + let mut gadget_row = Vec::with_capacity(levels); + let mut power = F::one(); + for _ in 0..levels { + gadget_row.push(power); + power = power * base; + } + + let mut alpha_pow = F::one(); + for _ in 0..D { + alpha_pow = alpha_pow * alpha; + } + let denom = alpha_pow + F::one(); + + let mut out = vec![F::zero(); rows * total_cols]; + for (i, m_a_row) in m_a.iter().enumerate() { + let row_start = i * total_cols; + out[row_start..row_start + cols].copy_from_slice(m_a_row); + let r_start = row_start + cols + i * levels; + for (j, g) in gadget_row.iter().enumerate() { + out[r_start + j] = -denom * *g; + } + } + Ok(out) +} + +pub(crate) fn build_w_evals(w: &[F], d: usize) -> (Vec, usize, usize) { + let num_l = d.trailing_zeros() as usize; + let num_ring_elems = w.len() / d.max(1); + let num_u = num_ring_elems.next_power_of_two().trailing_zeros() as usize; + let x_len = 1usize << num_u; + let y_len = 1usize << num_l; + let n = x_len * y_len; + + let mut evals = vec![F::zero(); n]; + for x in 0..x_len { + for y in 0..y_len { + let src = y + (x << num_l); + if src < w.len() { + let dst = x + (y << num_u); + evals[dst] = w[src]; + } + } + } + (evals, num_u, num_l) +} + +pub(crate) fn m_row_count() -> usize { + Cfg::N_D + Cfg::N_B + 1 + 1 + Cfg::N_A +} + +pub(crate) fn build_m_evals_x( + m_a_flat: &[F], + rows: usize, + cols: usize, + tau1: &[F], +) -> Vec { + let eq_tau1 = eq_evals(tau1); + let x_len = cols.next_power_of_two(); + let mut m_evals = vec![F::zero(); x_len]; + for x in 0..x_len { + let mut acc = F::zero(); + for i in 0..eq_tau1.len() { + let row_val = if i < rows && x < cols { + m_a_flat[i * cols + x] + } else { + F::zero() + }; + acc = acc + eq_tau1[i] * row_val; + } + m_evals[x] = acc; + } + m_evals +} + +pub(crate) fn build_alpha_evals_y(alpha: F, d: usize) -> Vec { + let mut out = vec![F::zero(); d]; + let mut power = F::one(); + for val in out.iter_mut() { + *val = power; + power = power * alpha; + } + out +} + +pub(crate) fn sample_tau>( + transcript: &mut T, + label: &[u8], + n: usize, +) -> Vec { + (0..n).map(|_| transcript.challenge_scalar(label)).collect() +} + +pub(crate) fn build_w_coeffs( + z: &[CyclotomicRing], + r: &[CyclotomicRing], +) -> Vec { + let levels = r_decomp_levels::(); + let r_hat: Vec> = r + .iter() + .flat_map(|ri| ri.balanced_decompose_pow2(levels, Cfg::LOG_BASIS)) + .collect(); + + let mut out = Vec::with_capacity((z.len() + r_hat.len()) * D); + for elem in z.iter().chain(r_hat.iter()) { + out.extend_from_slice(elem.coefficients()); + } + out +} diff --git a/src/protocol/sumcheck/mod.rs b/src/protocol/sumcheck/mod.rs new file mode 100644 index 00000000..2bba5ca0 --- /dev/null +++ b/src/protocol/sumcheck/mod.rs @@ -0,0 +1,641 @@ +//! Sumcheck core building blocks (univariate messages + proof transcript driver). +//! +//! This module provides: +//! - Univariate polynomials over a field `E` (`UniPoly`). +//! - A compressed univariate representation (`CompressedUniPoly`) that omits the +//! linear term and reconstructs it from the per-round sumcheck hint `g(0)+g(1)`. +//! - A minimal sumcheck proof container (`SumcheckProof`) with a verifier-side +//! transcript driver that returns the derived point and final claimed value. +//! +//! It intentionally does **not** implement a concrete sumcheck prover for any +//! particular oracle/table representation. Higher-level protocols (e.g. Hachi §4.3) +//! should implement the prover logic that constructs each round's univariate `g_i(X)`. +//! +//! ## Temporary duplication notice (Jolt integration) +//! +//! Jolt already has a mature, streaming-aware sumcheck implementation. Long-term, we +//! expect to extract the common sumcheck machinery into a dedicated crate and depend +//! on it from both Hachi and Jolt. Until that exists, this module intentionally +//! duplicates the essential sumcheck data types and transcript-driving logic as a +//! pragmatic workaround. + +pub mod norm_sumcheck; +pub mod relation_sumcheck; + +use crate::error::HachiError; +use crate::primitives::serialization::{ + Compress, HachiDeserialize, HachiSerialize, SerializationError, Valid, Validate, +}; +use crate::protocol::transcript::labels; +use crate::protocol::transcript::Transcript; +use crate::CanonicalField; +use crate::FieldCore; +use std::io::{Read, Write}; + +/// Univariate polynomial in coefficient form: `p(X) = Σ_{i=0}^d coeffs[i] * X^i`. +#[derive(Debug, Clone, PartialEq, Eq)] +pub struct UniPoly { + /// Coefficients from low degree to high degree. + pub coeffs: Vec, +} + +impl UniPoly { + /// Construct from coefficients in increasing-degree order. + pub fn from_coeffs(coeffs: Vec) -> Self { + Self { coeffs } + } + + /// Degree of the polynomial. + /// + /// # Panics + /// + /// Panics if the polynomial has no coefficients. + pub fn degree(&self) -> usize { + self.coeffs + .len() + .checked_sub(1) + .expect("UniPoly must have at least one coefficient") + } + + /// Evaluate at `x` via Horner's method. + pub fn evaluate(&self, x: &E) -> E { + // Horner from highest degree. + let mut acc = E::zero(); + for c in self.coeffs.iter().rev() { + acc = acc * *x + *c; + } + acc + } + + /// Compress this polynomial by omitting the linear coefficient. + /// + /// The verifier can reconstruct/evaluate the missing linear coefficient using + /// the per-round hint `g(0)+g(1)` from the sumcheck protocol. + /// + /// This matches the technique used by Jolt's sumcheck (`CompressedUniPoly`). + pub fn compress(&self) -> CompressedUniPoly { + let coeffs = &self.coeffs; + if coeffs.is_empty() { + return CompressedUniPoly { + coeffs_except_linear_term: Vec::new(), + }; + } + if coeffs.len() == 1 { + return CompressedUniPoly { + coeffs_except_linear_term: vec![coeffs[0]], + }; + } + // Keep coeff[0], drop coeff[1], keep coeff[2..]. + let mut out = Vec::with_capacity(coeffs.len().saturating_sub(1)); + out.push(coeffs[0]); + out.extend_from_slice(&coeffs[2..]); + CompressedUniPoly { + coeffs_except_linear_term: out, + } + } +} + +impl UniPoly { + /// Interpolate from evaluations at equispaced integer points `x = 0, 1, ..., d`. + /// + /// Uses Newton forward-difference interpolation: compute divided differences, + /// then expand via Horner on the nested Newton form. + /// + /// # Panics + /// + /// Panics if any required factorial inverse does not exist (field characteristic + /// must exceed the number of evaluation points). + pub fn from_evals(evals: &[E]) -> Self { + let n = evals.len(); + if n == 0 { + return Self::from_coeffs(vec![]); + } + if n == 1 { + return Self::from_coeffs(vec![evals[0]]); + } + + // Forward differences: delta^k[0] + let mut table = evals.to_vec(); + let mut deltas = vec![table[0]]; + for _ in 1..n { + for j in 0..table.len() - 1 { + table[j] = table[j + 1] - table[j]; + } + table.pop(); + deltas.push(table[0]); + } + + // Divided differences: dd[k] = delta^k / k! + let mut factorial = E::one(); + let mut divided_diffs = vec![deltas[0]]; + for (k, delta_k) in deltas.iter().enumerate().skip(1) { + factorial = factorial * E::from_u64(k as u64); + divided_diffs.push( + *delta_k + * factorial + .inv() + .expect("field characteristic too small for interpolation"), + ); + } + + // Horner expansion of the Newton nested form: + // p(x) = dd[0] + (x-0)(dd[1] + (x-1)(dd[2] + ...)) + let mut coeffs = vec![divided_diffs[n - 1]]; + + for k in (0..n - 1).rev() { + let shift = E::from_u64(k as u64); + let old_len = coeffs.len(); + let mut new_coeffs = vec![E::zero(); old_len + 1]; + + new_coeffs[0] = divided_diffs[k]; + for i in 0..old_len { + new_coeffs[i + 1] = new_coeffs[i + 1] + coeffs[i]; + new_coeffs[i] = new_coeffs[i] - shift * coeffs[i]; + } + + coeffs = new_coeffs; + } + + while coeffs.len() > 1 && coeffs.last().is_some_and(|c| c.is_zero()) { + coeffs.pop(); + } + + Self::from_coeffs(coeffs) + } +} + +impl Valid for UniPoly { + fn check(&self) -> Result<(), SerializationError> { + self.coeffs.check() + } +} + +impl HachiSerialize for UniPoly { + fn serialize_with_mode( + &self, + mut writer: W, + compress: Compress, + ) -> Result<(), SerializationError> { + self.coeffs.serialize_with_mode(&mut writer, compress) + } + + fn serialized_size(&self, compress: Compress) -> usize { + self.coeffs.serialized_size(compress) + } +} + +impl HachiDeserialize for UniPoly { + fn deserialize_with_mode( + mut reader: R, + compress: Compress, + validate: Validate, + ) -> Result { + let coeffs = Vec::::deserialize_with_mode(&mut reader, compress, validate)?; + let out = Self { coeffs }; + if matches!(validate, Validate::Yes) { + out.check()?; + } + Ok(out) + } +} + +/// Compressed univariate polynomial representation omitting the linear term. +/// +/// We store `[c0, c2, c3, ..., cd]`. Given the sumcheck hint `hint = g(0)+g(1)`, +/// the missing linear coefficient is: +/// +/// `c1 = hint - 2*c0 - Σ_{i=2..d} ci`. +#[derive(Debug, Clone, PartialEq, Eq)] +pub struct CompressedUniPoly { + /// Coefficients excluding the linear term: `[c0, c2, c3, ..., cd]`. + pub coeffs_except_linear_term: Vec, +} + +impl CompressedUniPoly { + /// Degree of the underlying uncompressed polynomial. + /// + /// For degree `d`, this stores `d` coefficients (all except the linear term). + pub fn degree(&self) -> usize { + self.coeffs_except_linear_term.len().saturating_sub(1) + } + + fn recover_linear_term(&self, hint: &E) -> E { + if self.coeffs_except_linear_term.is_empty() { + // Treat empty as the zero polynomial. + return E::zero(); + } + + // linear = hint - 2*c0 - Σ_{i>=2} ci + let c0 = self.coeffs_except_linear_term[0]; + let mut linear = *hint - c0 - c0; + for c in self.coeffs_except_linear_term.iter().skip(1) { + linear = linear - *c; + } + linear + } + + /// Decompress using `hint = g(0)+g(1)`. + pub fn decompress(&self, hint: &E) -> UniPoly { + if self.coeffs_except_linear_term.is_empty() { + return UniPoly::from_coeffs(Vec::new()); + } + let linear = self.recover_linear_term(hint); + // Always materialize the missing linear coefficient. + // For truly-constant polynomials, the hint forces `linear = 0`, so this is harmless. + let mut coeffs = Vec::with_capacity(self.coeffs_except_linear_term.len() + 1); + coeffs.push(self.coeffs_except_linear_term[0]); + coeffs.push(linear); + coeffs.extend_from_slice(&self.coeffs_except_linear_term[1..]); + UniPoly::from_coeffs(coeffs) + } + + /// Evaluate the uncompressed polynomial at `x`, using `hint = g(0)+g(1)`. + /// + /// This avoids materializing the full coefficient list. + pub fn eval_from_hint(&self, hint: &E, x: &E) -> E { + if self.coeffs_except_linear_term.is_empty() { + return E::zero(); + } + + let linear = self.recover_linear_term(hint); + let mut acc = self.coeffs_except_linear_term[0] + (*x * linear); + + // Add Σ_{i=2..d} c_i * x^i, where stored slice is [c2, c3, ..., cd] + let mut pow = *x * *x; // x^2 + for c in self.coeffs_except_linear_term.iter().skip(1) { + acc = acc + (*c * pow); + pow = pow * *x; + } + acc + } +} + +impl Valid for CompressedUniPoly { + fn check(&self) -> Result<(), SerializationError> { + self.coeffs_except_linear_term.check() + } +} + +impl HachiSerialize for CompressedUniPoly { + fn serialize_with_mode( + &self, + mut writer: W, + compress: Compress, + ) -> Result<(), SerializationError> { + self.coeffs_except_linear_term + .serialize_with_mode(&mut writer, compress) + } + + fn serialized_size(&self, compress: Compress) -> usize { + self.coeffs_except_linear_term.serialized_size(compress) + } +} + +impl HachiDeserialize for CompressedUniPoly { + fn deserialize_with_mode( + mut reader: R, + compress: Compress, + validate: Validate, + ) -> Result { + let coeffs_except_linear_term = + Vec::::deserialize_with_mode(&mut reader, compress, validate)?; + let out = Self { + coeffs_except_linear_term, + }; + if matches!(validate, Validate::Yes) { + out.check()?; + } + Ok(out) + } +} + +/// Sumcheck proof containing one compressed univariate polynomial per round. +#[derive(Debug, Clone, PartialEq, Eq)] +pub struct SumcheckProof { + /// One compressed univariate polynomial per sumcheck round. + pub round_polys: Vec>, +} + +impl Valid for SumcheckProof { + fn check(&self) -> Result<(), SerializationError> { + self.round_polys.check() + } +} + +impl HachiSerialize for SumcheckProof { + fn serialize_with_mode( + &self, + mut writer: W, + compress: Compress, + ) -> Result<(), SerializationError> { + self.round_polys.serialize_with_mode(&mut writer, compress) + } + + fn serialized_size(&self, compress: Compress) -> usize { + self.round_polys.serialized_size(compress) + } +} + +impl HachiDeserialize for SumcheckProof { + fn deserialize_with_mode( + mut reader: R, + compress: Compress, + validate: Validate, + ) -> Result { + let round_polys = + Vec::>::deserialize_with_mode(&mut reader, compress, validate)?; + let out = Self { round_polys }; + if matches!(validate, Validate::Yes) { + out.check()?; + } + Ok(out) + } +} + +impl SumcheckProof { + /// Verifier-side sumcheck transcript driver. + /// + /// This method: + /// - absorbs the per-round prover message (compressed univariate), + /// - samples one challenge per round via `sample_challenge`, + /// - updates the running claim using `eval_from_hint`. + /// + /// It does **not** perform the final oracle check `final_claim == f(r*)`. + /// Callers (e.g. ring-switching) must compute `f(r*)` themselves and compare. + /// + /// # Errors + /// + /// Returns an error if the proof length does not match `num_rounds` or if any + /// per-round polynomial exceeds `degree_bound`. + pub fn verify( + &self, + mut claim: E, + num_rounds: usize, + degree_bound: usize, + transcript: &mut T, + mut sample_challenge: S, + ) -> Result<(E, Vec), HachiError> + where + F: crate::FieldCore + crate::CanonicalField, + T: Transcript, + S: FnMut(&mut T) -> E, + { + if self.round_polys.len() != num_rounds { + return Err(HachiError::InvalidSize { + expected: num_rounds, + actual: self.round_polys.len(), + }); + } + + let mut r = Vec::with_capacity(num_rounds); + for poly in &self.round_polys { + if poly.degree() > degree_bound { + return Err(HachiError::InvalidInput(format!( + "sumcheck round poly degree {} exceeds bound {}", + poly.degree(), + degree_bound + ))); + } + + transcript.append_serde(labels::ABSORB_SUMCHECK_ROUND, poly); + let r_i = sample_challenge(transcript); + r.push(r_i); + + claim = poly.eval_from_hint(&claim, &r_i); + } + + Ok((claim, r)) + } +} + +/// Prover-side sumcheck instance interface. +/// +/// This trait encapsulates the protocol-specific logic required to compute each +/// per-round univariate polynomial `g_j(X)` and to update (fold) internal state +/// after receiving the verifier challenge `r_j`. +/// +/// Hachi §4.3 will implement concrete instances for `H_0` and `H_α`. +pub trait SumcheckInstanceProver: Send + Sync { + /// Number of rounds (i.e. number of variables bound by sumcheck). + fn num_rounds(&self) -> usize; + + /// Maximum allowed degree for any round univariate polynomial. + fn degree_bound(&self) -> usize; + + /// The initial claimed sum that this sumcheck instance is proving. + fn input_claim(&self) -> E; + + /// Compute the prover message `g_round(X)` given the previous running claim. + /// + /// In standard sumcheck, `previous_claim` is the expected value of the + /// remaining sum after binding previous challenges, and must satisfy: + /// + /// `g_round(0) + g_round(1) == previous_claim`. + fn compute_round_univariate(&mut self, round: usize, previous_claim: E) -> UniPoly; + + /// Ingest the verifier challenge `r_round` to fold/bind the current variable. + fn ingest_challenge(&mut self, round: usize, r_round: E); + + /// Optional end-of-protocol hook after the last challenge has been ingested. + fn finalize(&mut self) {} +} + +/// Verifier-side sumcheck instance interface. +/// +/// Implementations provide the initial claim and the oracle evaluation at the +/// challenge point, enabling the verifier to perform the final consistency check. +pub trait SumcheckInstanceVerifier: Send + Sync { + /// Number of rounds (i.e. number of variables bound by sumcheck). + fn num_rounds(&self) -> usize; + + /// Maximum allowed degree for any round univariate polynomial. + fn degree_bound(&self) -> usize; + + /// The initial claimed sum that this sumcheck instance is proving. + fn input_claim(&self) -> E; + + /// Compute the expected final evaluation `f(r_0, ..., r_{n-1})` at the + /// challenge point derived during the protocol. + fn expected_output_claim(&self, challenges: &[E]) -> E; +} + +/// Produce a sumcheck proof for a single instance, driving the Fiat–Shamir transcript. +/// +/// This method: +/// - does **not** absorb the initial claim into the transcript (callers should do so), +/// - appends each round message under `labels::ABSORB_SUMCHECK_ROUND`, +/// - samples one challenge per round via `sample_challenge`, +/// - updates the running claim using the per-round hint (`g(0)+g(1)`). +/// +/// It returns the proof, the derived point `r`, and the final claimed value at `r`. +/// +/// # Errors +/// +/// Returns an error if any per-round polynomial exceeds the instance's degree bound. +pub fn prove_sumcheck( + instance: &mut Inst, + transcript: &mut T, + mut sample_challenge: S, +) -> Result<(SumcheckProof, Vec, E), HachiError> +where + F: crate::FieldCore + crate::CanonicalField, + T: Transcript, + E: FieldCore, + S: FnMut(&mut T) -> E, + Inst: SumcheckInstanceProver, +{ + let mut claim = instance.input_claim(); + transcript.append_serde(labels::ABSORB_SUMCHECK_CLAIM, &claim); + + let num_rounds = instance.num_rounds(); + let degree_bound = instance.degree_bound(); + + let mut round_polys = Vec::with_capacity(num_rounds); + let mut r = Vec::with_capacity(num_rounds); + + for round in 0..num_rounds { + let g = instance.compute_round_univariate(round, claim); + // Optional sanity: enforce hint relation in debug builds. + debug_assert!( + g.evaluate(&E::zero()) + g.evaluate(&E::one()) == claim, + "sumcheck round univariate does not match previous claim hint" + ); + + let compressed = g.compress(); + if compressed.degree() > degree_bound { + return Err(HachiError::InvalidInput(format!( + "sumcheck round poly degree {} exceeds bound {}", + compressed.degree(), + degree_bound + ))); + } + + transcript.append_serde(labels::ABSORB_SUMCHECK_ROUND, &compressed); + let r_i = sample_challenge(transcript); + r.push(r_i); + + // Update running claim (this is g(r_i)). + claim = compressed.eval_from_hint(&claim, &r_i); + + instance.ingest_challenge(round, r_i); + round_polys.push(compressed); + } + + instance.finalize(); + + let proof = SumcheckProof { round_polys }; + Ok((proof, r, claim)) +} + +/// Verify a single-instance sumcheck proof. +/// +/// This function: +/// - absorbs the initial claim into the transcript, +/// - delegates round-by-round verification to [`SumcheckProof::verify`], +/// - performs the final oracle check: `final_claim == verifier.expected_output_claim(r)`. +/// +/// Returns the challenge point `r` on success. +/// +/// # Errors +/// +/// Returns [`HachiError::InvalidProof`] if the final sumcheck claim does not +/// match the oracle evaluation, or propagates any error from the per-round +/// verification (e.g. degree-bound violation, round-count mismatch). +pub fn verify_sumcheck( + proof: &SumcheckProof, + verifier: &V, + transcript: &mut T, + sample_challenge: S, +) -> Result, HachiError> +where + F: crate::FieldCore + crate::CanonicalField, + T: Transcript, + E: FieldCore, + S: FnMut(&mut T) -> E, + V: SumcheckInstanceVerifier, +{ + let claim = verifier.input_claim(); + transcript.append_serde(labels::ABSORB_SUMCHECK_CLAIM, &claim); + let (final_claim, challenges) = proof.verify::( + claim, + verifier.num_rounds(), + verifier.degree_bound(), + transcript, + sample_challenge, + )?; + + let expected = verifier.expected_output_claim(&challenges); + if final_claim != expected { + return Err(HachiError::InvalidProof); + } + + Ok(challenges) +} + +/// Build the full eq polynomial evaluation table. +/// +/// Returns a vector of size `2^{tau.len()}` where entry `b` (interpreted as a +/// little-endian bit string, i.e. bit `k` of `b` corresponds to `τ[k]`) equals +/// `ẽq(τ, b) = Π_i (τ_i·b_i + (1−τ_i)(1−b_i))`. +pub fn eq_evals(tau: &[E]) -> Vec { + let size = 1usize << tau.len(); + let mut evals = vec![E::zero(); size]; + evals[0] = E::one(); + let mut len = 1usize; + for &t in tau.iter().rev() { + let one_minus_t = E::one() - t; + for j in (0..len).rev() { + evals[2 * j + 1] = evals[j] * t; + evals[2 * j] = evals[j] * one_minus_t; + } + len *= 2; + } + evals +} + +/// Evaluate ẽq(τ, r) at a single point. +pub fn eq_eval(tau: &[E], point: &[E]) -> E { + debug_assert_eq!(tau.len(), point.len()); + tau.iter().zip(point).fold(E::one(), |acc, (&t, &r)| { + acc * (t * r + (E::one() - t) * (E::one() - r)) + }) +} + +/// Evaluate the range-check polynomial `w · Π_{k=1}^{b−1} (w − k)(w + k)`. +/// +/// This polynomial vanishes exactly when `w ∈ {−(b−1), …, b−1}`. +/// Total degree in `w` is `2b − 1`. +pub fn range_check_eval(w: E, b: usize) -> E { + let mut acc = w; + for k in 1..b { + let k_e = E::from_u64(k as u64); + acc = acc * (w - k_e) * (w + k_e); + } + acc +} + +/// Evaluate a multilinear polynomial (given by boolean-hypercube evaluations in +/// little-endian bit order) at an arbitrary point via iterated folding. +pub fn multilinear_eval(evals: &[E], point: &[E]) -> E { + let mut current = evals.to_vec(); + for &r in point { + let half = current.len() / 2; + let mut next = Vec::with_capacity(half); + for i in 0..half { + next.push(current[2 * i] + r * (current[2 * i + 1] - current[2 * i])); + } + current = next; + } + current[0] +} + +/// Fold an evaluation table by binding its first variable to `r`, halving the +/// table size. +pub(crate) fn fold_evals(evals: &[E], r: E) -> Vec { + let half = evals.len() / 2; + let mut out = Vec::with_capacity(half); + for i in 0..half { + out.push(evals[2 * i] + r * (evals[2 * i + 1] - evals[2 * i])); + } + out +} diff --git a/src/protocol/sumcheck/norm_sumcheck.rs b/src/protocol/sumcheck/norm_sumcheck.rs new file mode 100644 index 00000000..e1b72490 --- /dev/null +++ b/src/protocol/sumcheck/norm_sumcheck.rs @@ -0,0 +1,255 @@ +//! Norm (range-check) sumcheck instance (F_0). +//! +//! **F_{0,τ₀}(x, y)** = ẽq(τ₀,(x,y)) · w̃(x,y) · (w̃−1)(w̃+1)···(w̃−b+1)(w̃+b−1) +//! +//! Proves that all entries of w̃ lie in {−(b−1), …, b−1}; the sum over the +//! boolean hypercube should equal zero when the range constraint holds. + +use super::{eq_eval, eq_evals, fold_evals, multilinear_eval, range_check_eval}; +use super::{SumcheckInstanceProver, SumcheckInstanceVerifier, UniPoly}; +use crate::{CanonicalField, FieldCore}; + +/// Prover for `F_{0,τ₀}(x,y) = ẽq(τ₀,(x,y)) · w̃(x,y) · range_check(w̃(x,y), b)`. +/// +/// Stores `eq` and `w` evaluation tables separately so the composite can be +/// evaluated at the `2b + 1` points needed per round (degree `2b`). +pub struct NormSumcheckProver { + eq_table: Vec, + w_table: Vec, + num_vars: usize, + b: usize, +} + +impl NormSumcheckProver { + /// Create a new norm (range-check) sumcheck prover. + /// + /// # Panics + /// + /// Panics if `w_evals.len() != 2^tau.len()`. + pub fn new(tau: &[E], w_evals: Vec, b: usize) -> Self { + let num_vars = tau.len(); + assert_eq!(w_evals.len(), 1 << num_vars); + Self { + eq_table: eq_evals(tau), + w_table: w_evals, + num_vars, + b, + } + } +} + +impl SumcheckInstanceProver for NormSumcheckProver { + fn num_rounds(&self) -> usize { + self.num_vars + } + + fn degree_bound(&self) -> usize { + 2 * self.b + } + + fn input_claim(&self) -> E { + E::zero() + } + + fn compute_round_univariate(&mut self, _round: usize, _previous_claim: E) -> UniPoly { + let half = self.eq_table.len() / 2; + let degree = 2 * self.b; + let num_points = degree + 1; + let mut round_evals = vec![E::zero(); num_points]; + + for j in 0..half { + let eq_0 = self.eq_table[2 * j]; + let eq_1 = self.eq_table[2 * j + 1]; + let w_0 = self.w_table[2 * j]; + let w_1 = self.w_table[2 * j + 1]; + + for (t, eval) in round_evals.iter_mut().enumerate() { + let t_e = E::from_u64(t as u64); + let eq_t = eq_0 + t_e * (eq_1 - eq_0); + let w_t = w_0 + t_e * (w_1 - w_0); + *eval = *eval + eq_t * range_check_eval(w_t, self.b); + } + } + + UniPoly::from_evals(&round_evals) + } + + fn ingest_challenge(&mut self, _round: usize, r: E) { + self.eq_table = fold_evals(&self.eq_table, r); + self.w_table = fold_evals(&self.w_table, r); + } +} + +/// Verifier for the norm (range-check) sumcheck `F_{0,τ₀}`. +pub struct NormSumcheckVerifier { + tau: Vec, + w_evals: Vec, + num_vars: usize, + b: usize, +} + +impl NormSumcheckVerifier { + /// Create a new norm (range-check) sumcheck verifier. + /// + /// # Panics + /// + /// Panics if `w_evals.len() != 2^tau.len()`. + pub fn new(tau: Vec, w_evals: Vec, b: usize) -> Self { + let num_vars = tau.len(); + assert_eq!(w_evals.len(), 1 << num_vars); + Self { + tau, + w_evals, + num_vars, + b, + } + } +} + +impl SumcheckInstanceVerifier for NormSumcheckVerifier { + fn num_rounds(&self) -> usize { + self.num_vars + } + + fn degree_bound(&self) -> usize { + 2 * self.b + } + + fn input_claim(&self) -> E { + E::zero() + } + + fn expected_output_claim(&self, challenges: &[E]) -> E { + let eq_val = eq_eval(&self.tau, challenges); + let w_val = multilinear_eval(&self.w_evals, challenges); + eq_val * range_check_eval(w_val, self.b) + } +} + +#[cfg(test)] +mod tests { + use super::*; + use crate::algebra::ring::CyclotomicRing; + use crate::algebra::Fp64; + use crate::primitives::multilinear_evals::DenseMultilinearEvals; + use crate::protocol::ring_switch::build_w_coeffs; + use crate::protocol::sumcheck::eq_evals; + use crate::protocol::transcript::labels; + use crate::protocol::{ + prove_sumcheck, verify_sumcheck, Blake2bTranscript, CommitmentConfig, CommitmentScheme, + DefaultCommitmentConfig, HachiCommitmentScheme, Transcript, + }; + use crate::{CanonicalField, FieldCore}; + + type F = Fp64<4294967197>; + const D: usize = 8; + + fn ring_with_small_coeff(value: u64) -> CyclotomicRing { + let coeffs = std::array::from_fn(|_| F::from_u64(value)); + CyclotomicRing::from_coefficients(coeffs) + } + + #[test] + fn norm_sumcheck_uses_commitment_w_evals() { + let z = vec![ + ring_with_small_coeff(1), + ring_with_small_coeff(2), + ring_with_small_coeff(3), + ]; + let r = vec![ring_with_small_coeff(0), ring_with_small_coeff(1)]; + let mut w_evals = build_w_coeffs::(&z, &r); + + let target_len = w_evals.len().next_power_of_two(); + w_evals.resize(target_len, F::zero()); + let num_vars = target_len.trailing_zeros() as usize; + let tau: Vec = (0..num_vars).map(|i| F::from_u64((i + 2) as u64)).collect(); + let b = 1usize << DefaultCommitmentConfig::LOG_BASIS; + + let eq_table = eq_evals(&tau); + let _claim: F = (0..w_evals.len()) + .map(|i| eq_table[i] * range_check_eval(w_evals[i], b)) + .fold(F::zero(), |a, v| a + v); + + let mut prover = NormSumcheckProver::new(&tau, w_evals.clone(), b); + let mut pt = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + let (proof, prover_challenges, final_claim) = + prove_sumcheck::(&mut prover, &mut pt, |tr| { + tr.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND) + }) + .unwrap(); + + let oracle = eq_eval(&tau, &prover_challenges) + * range_check_eval(multilinear_eval(&w_evals, &prover_challenges), b); + assert_eq!(final_claim, oracle, "prover final claim != oracle eval"); + + let verifier = NormSumcheckVerifier::new(tau, w_evals, b); + let mut vt = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + let verifier_challenges = + verify_sumcheck::(&proof, &verifier, &mut vt, |tr| { + tr.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND) + }) + .unwrap(); + + assert_eq!(prover_challenges, verifier_challenges); + } + + #[test] + fn norm_sumcheck_uses_prove_w_evals() { + let alpha = DefaultCommitmentConfig::D.trailing_zeros() as usize; + let num_vars = DefaultCommitmentConfig::R + DefaultCommitmentConfig::M + alpha; + let len = 1usize << num_vars; + let evals: Vec = (0..len).map(|i| F::from_u64(i as u64)).collect(); + let poly = DenseMultilinearEvals::new_padded(evals); + + let setup = HachiCommitmentScheme::setup_prover(num_vars); + let (commitment, hint) = HachiCommitmentScheme::commit(&poly, &setup).unwrap(); + + let opening_point: Vec = (0..num_vars).map(|i| F::from_u64((i + 2) as u64)).collect(); + let mut prover_transcript = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + let proof = HachiCommitmentScheme::prove( + &setup, + &poly, + &opening_point, + Some(hint), + &mut prover_transcript, + &commitment, + ) + .unwrap(); + + let mut w_evals = proof.sumcheck_aux.w.clone(); + let target_len = w_evals.len().next_power_of_two(); + w_evals.resize(target_len, F::zero()); + let num_sumcheck_vars = target_len.trailing_zeros() as usize; + let tau: Vec = (0..num_sumcheck_vars) + .map(|i| F::from_u64((i + 3) as u64)) + .collect(); + let b = 1usize << DefaultCommitmentConfig::LOG_BASIS; + + let eq_table = eq_evals(&tau); + let _claim: F = (0..w_evals.len()) + .map(|i| eq_table[i] * range_check_eval(w_evals[i], b)) + .fold(F::zero(), |a, v| a + v); + + let mut prover = NormSumcheckProver::new(&tau, w_evals.clone(), b); + let mut pt = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + let (proof_sc, prover_challenges, final_claim) = + prove_sumcheck::(&mut prover, &mut pt, |tr| { + tr.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND) + }) + .unwrap(); + + let oracle = eq_eval(&tau, &prover_challenges) + * range_check_eval(multilinear_eval(&w_evals, &prover_challenges), b); + assert_eq!(final_claim, oracle, "prover final claim != oracle eval"); + + let verifier = NormSumcheckVerifier::new(tau, w_evals, b); + let mut vt = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + let verifier_challenges = + verify_sumcheck::(&proof_sc, &verifier, &mut vt, |tr| { + tr.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND) + }) + .unwrap(); + + assert_eq!(prover_challenges, verifier_challenges); + } +} diff --git a/src/protocol/sumcheck/relation_sumcheck.rs b/src/protocol/sumcheck/relation_sumcheck.rs new file mode 100644 index 00000000..12de5a7f --- /dev/null +++ b/src/protocol/sumcheck/relation_sumcheck.rs @@ -0,0 +1,345 @@ +//! Evaluation-relation sumcheck instance (F_α). +//! +//! **F_{α,τ₁}(x, y)** = w̃(x,y) · α̃(y) · m(x) +//! where m(x) = Σ_i ẽq(τ₁,i) · M̃_α(i,x). +//! +//! Proves the evaluation relation; sum equals `a = Σ_i ẽq(τ₁,i) · y_i(α)`. + +use super::{eq_evals, fold_evals, multilinear_eval}; +use super::{SumcheckInstanceProver, SumcheckInstanceVerifier, UniPoly}; +use crate::algebra::ring::CyclotomicRing; +use crate::protocol::ring_switch::eval_ring_at; +use crate::{CanonicalField, FieldCore}; + +/// Prover for `F_{α,τ₁}(x,y) = w̃(x,y) · α̃(y) · m(x)`. +/// +/// All three constituent evaluation tables are stored at full domain size +/// (`2^{num_u + num_l}`). `α̃` is replicated along x dimensions and `m` along +/// y dimensions so that a uniform fold-by-pairs works in every round. +/// +/// Round polynomial degree is 2 (product of at most two multilinear factors +/// depending on any single variable). +pub struct RelationSumcheckProver { + w_table: Vec, + alpha_table: Vec, + m_table: Vec, + num_vars: usize, +} + +impl RelationSumcheckProver { + /// Construct from the three constituent evaluation tables. + /// + /// - `w_evals`: evaluations of `w̃` over `{0,1}^{num_u + num_l}` (full domain). + /// - `alpha_evals_y`: evaluations of `α̃` over `{0,1}^{num_l}` (compact). + /// - `m_evals_x`: evaluations of `m` over `{0,1}^{num_u}` (compact). + /// + /// The constructor extends the compact tables to the full domain by replication. + /// + /// # Panics + /// + /// Panics if table sizes don't match `2^num_u`, `2^num_l`, or `2^(num_u+num_l)`. + pub fn new( + w_evals: Vec, + alpha_evals_y: &[E], + m_evals_x: &[E], + num_u: usize, + num_l: usize, + ) -> Self { + let num_vars = num_u + num_l; + let n = 1usize << num_vars; + assert_eq!(w_evals.len(), n); + assert_eq!(alpha_evals_y.len(), 1 << num_l); + assert_eq!(m_evals_x.len(), 1 << num_u); + + let x_mask = (1usize << num_u) - 1; + let alpha_table: Vec = (0..n).map(|idx| alpha_evals_y[idx >> num_u]).collect(); + let m_table: Vec = (0..n).map(|idx| m_evals_x[idx & x_mask]).collect(); + + Self { + w_table: w_evals, + alpha_table, + m_table, + num_vars, + } + } +} + +impl SumcheckInstanceProver for RelationSumcheckProver { + fn num_rounds(&self) -> usize { + self.num_vars + } + + fn degree_bound(&self) -> usize { + 2 + } + + fn input_claim(&self) -> E { + self.w_table + .iter() + .zip(self.alpha_table.iter()) + .zip(self.m_table.iter()) + .fold(E::zero(), |acc, ((&w, &a), &m)| acc + w * a * m) + } + + fn compute_round_univariate(&mut self, _round: usize, _previous_claim: E) -> UniPoly { + let half = self.w_table.len() / 2; + let num_points = 3; // degree 2 → 3 evaluation points + let mut round_evals = vec![E::zero(); num_points]; + + for j in 0..half { + let w_0 = self.w_table[2 * j]; + let w_1 = self.w_table[2 * j + 1]; + let a_0 = self.alpha_table[2 * j]; + let a_1 = self.alpha_table[2 * j + 1]; + let m_0 = self.m_table[2 * j]; + let m_1 = self.m_table[2 * j + 1]; + + for (t, eval) in round_evals.iter_mut().enumerate() { + let t_e = E::from_u64(t as u64); + let w_t = w_0 + t_e * (w_1 - w_0); + let a_t = a_0 + t_e * (a_1 - a_0); + let m_t = m_0 + t_e * (m_1 - m_0); + *eval = *eval + w_t * a_t * m_t; + } + } + + UniPoly::from_evals(&round_evals) + } + + fn ingest_challenge(&mut self, _round: usize, r: E) { + self.w_table = fold_evals(&self.w_table, r); + self.alpha_table = fold_evals(&self.alpha_table, r); + self.m_table = fold_evals(&self.m_table, r); + } +} + +/// Verifier for the evaluation-relation sumcheck `F_{α,τ₁}`. +pub struct RelationSumcheckVerifier { + w_evals: Vec, + alpha_evals_y: Vec, + m_evals_x: Vec, + tau: Vec, + v: Vec>, + u: Vec>, + y_ring: CyclotomicRing, + alpha: F, + num_u: usize, + num_l: usize, +} + +impl RelationSumcheckVerifier { + /// Create a new evaluation-relation sumcheck verifier. + /// + /// # Panics + /// + /// Panics if table sizes don't match `2^num_u`, `2^num_l`, or `2^(num_u+num_l)`. + #[allow(clippy::too_many_arguments)] + pub fn new( + w_evals: Vec, + alpha_evals_y: Vec, + m_evals_x: Vec, + tau: Vec, + v: Vec>, + u: Vec>, + y_ring: CyclotomicRing, + alpha: F, + num_u: usize, + num_l: usize, + ) -> Self { + assert_eq!(w_evals.len(), 1 << (num_u + num_l)); + assert_eq!(alpha_evals_y.len(), 1 << num_l); + assert_eq!(m_evals_x.len(), 1 << num_u); + Self { + w_evals, + alpha_evals_y, + m_evals_x, + tau, + v, + u, + y_ring, + alpha, + num_u, + num_l, + } + } +} + +impl SumcheckInstanceVerifier + for RelationSumcheckVerifier +{ + fn num_rounds(&self) -> usize { + self.num_u + self.num_l + } + + fn degree_bound(&self) -> usize { + 2 + } + + fn input_claim(&self) -> F { + let y_a: Vec = self + .v + .iter() + .chain(self.u.iter()) + .chain(std::iter::once(&self.y_ring)) + .map(|r| eval_ring_at(r, &self.alpha)) + .collect(); + + let eq_tau = eq_evals(&self.tau); + let mut acc = F::zero(); + for (i, eq_i) in eq_tau.iter().enumerate() { + let y_i = if i < y_a.len() { y_a[i] } else { F::zero() }; + acc = acc + *eq_i * y_i; + } + acc + } + + fn expected_output_claim(&self, challenges: &[F]) -> F { + let (x_challenges, y_challenges) = challenges.split_at(self.num_u); + let w_val = multilinear_eval(&self.w_evals, challenges); + let alpha_val = multilinear_eval(&self.alpha_evals_y, y_challenges); + let m_val = multilinear_eval(&self.m_evals_x, x_challenges); + w_val * alpha_val * m_val + } +} + +#[cfg(test)] +mod tests { + use super::*; + use crate::algebra::Fp64; + use crate::primitives::multilinear_evals::DenseMultilinearEvals; + use crate::protocol::commitment_scheme::rederive_alpha_and_m_a; + use crate::protocol::sumcheck::eq_evals; + use crate::protocol::transcript::labels; + use crate::protocol::{ + prove_sumcheck, verify_sumcheck, Blake2bTranscript, CommitmentConfig, CommitmentScheme, + DefaultCommitmentConfig, HachiCommitmentScheme, Transcript, + }; + use crate::{CanonicalField, FieldCore}; + + type F = Fp64<4294967197>; + + #[test] + fn relation_sumcheck_uses_prove_w_evals() { + let alpha_bits = DefaultCommitmentConfig::D.trailing_zeros() as usize; + let num_vars = DefaultCommitmentConfig::R + DefaultCommitmentConfig::M + alpha_bits; + let len = 1usize << num_vars; + let evals: Vec = (0..len).map(|i| F::from_u64(i as u64)).collect(); + let poly = DenseMultilinearEvals::new_padded(evals); + + let setup = HachiCommitmentScheme::setup_prover(num_vars); + let (commitment, hint) = HachiCommitmentScheme::commit(&poly, &setup).unwrap(); + + let opening_point: Vec = (0..num_vars).map(|i| F::from_u64((i + 2) as u64)).collect(); + let mut prover_transcript = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + let proof = HachiCommitmentScheme::prove( + &setup, + &poly, + &opening_point, + Some(hint), + &mut prover_transcript, + &commitment, + ) + .unwrap(); + + let (alpha, m_a_vec) = + rederive_alpha_and_m_a(&proof, &setup, &opening_point, &commitment).unwrap(); + + let d = DefaultCommitmentConfig::D; + assert_eq!(proof.sumcheck_aux.w.len() % d, 0); + let w_u = proof.sumcheck_aux.w.len() / d; + let rows = DefaultCommitmentConfig::N_D + + DefaultCommitmentConfig::N_B + + 1 + + 1 + + DefaultCommitmentConfig::N_A; + assert!(rows > 0); + assert_eq!(m_a_vec.len() % rows, 0); + let cols = m_a_vec.len() / rows; + assert_eq!(w_u, cols); + + let num_u = cols.next_power_of_two().trailing_zeros() as usize; + let num_l = alpha_bits; + let n = 1usize << (num_u + num_l); + + let mut w_evals = vec![F::zero(); n]; + let y_len = 1usize << num_l; + let x_len = 1usize << num_u; + for x in 0..x_len { + for y in 0..y_len { + let src = y + (x << num_l); + if src < proof.sumcheck_aux.w.len() { + let dst = x + (y << num_u); + w_evals[dst] = proof.sumcheck_aux.w[src]; + } + } + } + + let num_i = rows.next_power_of_two().trailing_zeros() as usize; + let tau1: Vec = (0..num_i).map(|i| F::from_u64((i + 5) as u64)).collect(); + let eq_tau1 = eq_evals(&tau1); + + let mut m_evals_x = vec![F::zero(); x_len]; + for x in 0..x_len { + let mut acc = F::zero(); + for i in 0..(1usize << num_i) { + let row_val = if i < rows && x < cols { + m_a_vec[i * cols + x] + } else { + F::zero() + }; + acc = acc + eq_tau1[i] * row_val; + } + m_evals_x[x] = acc; + } + + let mut alpha_evals_y = vec![F::zero(); y_len]; + let mut power = F::one(); + for val in alpha_evals_y.iter_mut() { + *val = power; + power = power * alpha; + } + + let x_mask = x_len - 1; + let alpha_full: Vec = (0..n).map(|idx| alpha_evals_y[idx >> num_u]).collect(); + let m_full: Vec = (0..n).map(|idx| m_evals_x[idx & x_mask]).collect(); + let _claim: F = (0..n) + .map(|i| w_evals[i] * alpha_full[i] * m_full[i]) + .fold(F::zero(), |a, v| a + v); + + let mut prover = + RelationSumcheckProver::new(w_evals.clone(), &alpha_evals_y, &m_evals_x, num_u, num_l); + let mut pt = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + let (proof_sc, prover_challenges, final_claim) = + prove_sumcheck::(&mut prover, &mut pt, |tr| { + tr.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND) + }) + .unwrap(); + + let (x_ch, y_ch) = prover_challenges.split_at(num_u); + let oracle = multilinear_eval(&w_evals, &prover_challenges) + * multilinear_eval(&alpha_evals_y, y_ch) + * multilinear_eval(&m_evals_x, x_ch); + assert_eq!(final_claim, oracle, "prover final claim != oracle eval"); + + let verifier = RelationSumcheckVerifier::new( + w_evals, + alpha_evals_y, + m_evals_x, + tau1, + proof.v.clone(), + commitment.u.clone(), + proof.y_ring, + alpha, + num_u, + num_l, + ); + let mut vt = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + let verifier_challenges = + verify_sumcheck::(&proof_sc, &verifier, &mut vt, |tr| { + tr.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND) + }) + .unwrap(); + + assert_eq!(prover_challenges, verifier_challenges); + } +} diff --git a/src/protocol/transcript/blake2b.rs b/src/protocol/transcript/blake2b.rs new file mode 100644 index 00000000..c72fba39 --- /dev/null +++ b/src/protocol/transcript/blake2b.rs @@ -0,0 +1,82 @@ +//! Blake2b transcript implementation for protocol-layer Fiat-Shamir. + +use super::Transcript; +use crate::primitives::serialization::HachiSerialize; +use crate::{CanonicalField, FieldCore}; +use blake2::{Blake2b512, Digest}; +use std::marker::PhantomData; + +/// Blake2b512 transcript with labeled framing. +#[derive(Clone)] +pub struct Blake2bTranscript +where + F: FieldCore + CanonicalField + 'static, +{ + hasher: Blake2b512, + _field: PhantomData, +} + +impl Blake2bTranscript +where + F: FieldCore + CanonicalField + 'static, +{ + #[inline] + fn append_bytes_impl(&mut self, label: &[u8], bytes: &[u8]) { + self.hasher.update(label); + self.hasher.update((bytes.len() as u64).to_le_bytes()); + self.hasher.update(bytes); + } + + #[inline] + fn challenge_bytes(&mut self, label: &[u8]) -> [u8; 64] { + self.hasher.update(label); + let digest = self.hasher.clone().finalize(); + let mut out = [0u8; 64]; + out.copy_from_slice(digest.as_slice()); + self.hasher.update(out); + out + } +} + +impl Transcript for Blake2bTranscript +where + F: FieldCore + CanonicalField + 'static, +{ + fn new(domain_label: &[u8]) -> Self { + let mut hasher = Blake2b512::new(); + hasher.update(domain_label); + Self { + hasher, + _field: PhantomData, + } + } + + fn append_bytes(&mut self, label: &[u8], bytes: &[u8]) { + self.append_bytes_impl(label, bytes); + } + + fn append_field(&mut self, label: &[u8], x: &F) { + self.append_bytes_impl(label, &x.to_canonical_u128().to_le_bytes()); + } + + fn append_serde(&mut self, label: &[u8], s: &S) { + let mut bytes = Vec::new(); + s.serialize_compressed(&mut bytes) + .expect("HachiSerialize should not fail"); + self.append_bytes_impl(label, &bytes); + } + + fn challenge_scalar(&mut self, label: &[u8]) -> F { + let bytes = self.challenge_bytes(label); + let mut lo = [0u8; 16]; + lo.copy_from_slice(&bytes[..16]); + let sampled = u128::from_le_bytes(lo); + F::from_canonical_u128_reduced(sampled) + } + + fn reset(&mut self, domain_label: &[u8]) { + let mut hasher = Blake2b512::new(); + hasher.update(domain_label); + self.hasher = hasher; + } +} diff --git a/src/protocol/transcript/keccak.rs b/src/protocol/transcript/keccak.rs new file mode 100644 index 00000000..8fa3855c --- /dev/null +++ b/src/protocol/transcript/keccak.rs @@ -0,0 +1,82 @@ +//! Keccak transcript implementation for protocol-layer Fiat-Shamir. + +use super::Transcript; +use crate::primitives::serialization::HachiSerialize; +use crate::{CanonicalField, FieldCore}; +use sha3::{Digest, Keccak256}; +use std::marker::PhantomData; + +/// Keccak256 transcript with labeled framing. +#[derive(Clone)] +pub struct KeccakTranscript +where + F: FieldCore + CanonicalField + 'static, +{ + hasher: Keccak256, + _field: PhantomData, +} + +impl KeccakTranscript +where + F: FieldCore + CanonicalField + 'static, +{ + #[inline] + fn append_bytes_impl(&mut self, label: &[u8], bytes: &[u8]) { + self.hasher.update(label); + self.hasher.update((bytes.len() as u64).to_le_bytes()); + self.hasher.update(bytes); + } + + #[inline] + fn challenge_bytes(&mut self, label: &[u8]) -> [u8; 32] { + self.hasher.update(label); + let digest = self.hasher.clone().finalize(); + let mut out = [0u8; 32]; + out.copy_from_slice(digest.as_slice()); + self.hasher.update(out); + out + } +} + +impl Transcript for KeccakTranscript +where + F: FieldCore + CanonicalField + 'static, +{ + fn new(domain_label: &[u8]) -> Self { + let mut hasher = Keccak256::new(); + hasher.update(domain_label); + Self { + hasher, + _field: PhantomData, + } + } + + fn append_bytes(&mut self, label: &[u8], bytes: &[u8]) { + self.append_bytes_impl(label, bytes); + } + + fn append_field(&mut self, label: &[u8], x: &F) { + self.append_bytes_impl(label, &x.to_canonical_u128().to_le_bytes()); + } + + fn append_serde(&mut self, label: &[u8], s: &S) { + let mut bytes = Vec::new(); + s.serialize_compressed(&mut bytes) + .expect("HachiSerialize should not fail"); + self.append_bytes_impl(label, &bytes); + } + + fn challenge_scalar(&mut self, label: &[u8]) -> F { + let bytes = self.challenge_bytes(label); + let mut lo = [0u8; 16]; + lo.copy_from_slice(&bytes[..16]); + let sampled = u128::from_le_bytes(lo); + F::from_canonical_u128_reduced(sampled) + } + + fn reset(&mut self, domain_label: &[u8]) { + let mut hasher = Keccak256::new(); + hasher.update(domain_label); + self.hasher = hasher; + } +} diff --git a/src/protocol/transcript/labels.rs b/src/protocol/transcript/labels.rs new file mode 100644 index 00000000..f5d93952 --- /dev/null +++ b/src/protocol/transcript/labels.rs @@ -0,0 +1,69 @@ +//! Hachi-native transcript labels. +//! +//! These constants are the single source of truth for protocol transcript +//! labels in Hachi core. External integrations should translate at adapter +//! boundaries instead of introducing foreign label names here. + +/// Top-level protocol domain label. +pub const DOMAIN_HACHI_PROTOCOL: &[u8] = b"hachi/protocol"; + +/// Absorb commitment object(s) (paper §4.1). +pub const ABSORB_COMMITMENT: &[u8] = b"hachi/absorb/commitment"; +/// Absorb claimed openings/evaluations before relation reduction (paper §4.2). +pub const ABSORB_EVALUATION_CLAIMS: &[u8] = b"hachi/absorb/evaluation-claims"; +/// Challenge for the evaluation-to-linear-relation reduction (paper §4.2). +pub const CHALLENGE_LINEAR_RELATION: &[u8] = b"hachi/challenge/linear-relation"; +/// Absorb ring-switch relation messages (paper §4.3). +pub const ABSORB_RING_SWITCH_MESSAGE: &[u8] = b"hachi/absorb/ring-switch-message"; +/// Challenge used by ring-switching relation checks (paper §4.3). +pub const CHALLENGE_RING_SWITCH: &[u8] = b"hachi/challenge/ring-switch"; +/// Absorb sparse-challenge sampling context (e.g. for short/sparse ring `c`). +pub const ABSORB_SPARSE_CHALLENGE: &[u8] = b"hachi/absorb/sparse-challenge"; +/// Challenge bytes used to sample sparse challenges (e.g. ring `c` with weight ω). +pub const CHALLENGE_SPARSE_CHALLENGE: &[u8] = b"hachi/challenge/sparse-challenge"; +/// Absorb the initial sumcheck claim before round messages begin. +pub const ABSORB_SUMCHECK_CLAIM: &[u8] = b"hachi/absorb/sumcheck-claim"; +/// Absorb per-round sumcheck messages (paper §4.3). +pub const ABSORB_SUMCHECK_ROUND: &[u8] = b"hachi/absorb/sumcheck-round"; +/// Challenge sampled per sumcheck round (paper §4.3). +pub const CHALLENGE_SUMCHECK_ROUND: &[u8] = b"hachi/challenge/sumcheck-round"; +/// Absorb recursion/stop-condition message payloads (paper §4.5). +pub const ABSORB_STOP_CONDITION: &[u8] = b"hachi/absorb/stop-condition"; +/// Challenge sampled for recursion stop-condition checks (paper §4.5). +pub const CHALLENGE_STOP_CONDITION: &[u8] = b"hachi/challenge/stop-condition"; + +/// Absorb the prover's stage-1 message `v = D · ŵ` (paper §4.2, Figure 3). +pub const ABSORB_PROVER_V: &[u8] = b"hachi/absorb/prover-stage1-v"; +/// Challenge label for stage-1 fold (sampling sparse `c_i`). +pub const CHALLENGE_STAGE1_FOLD: &[u8] = b"hachi/challenge/stage1-fold"; + +/// Absorb the `w` coefficient vector before sumcheck (paper §4.3). +pub const ABSORB_SUMCHECK_W: &[u8] = b"hachi/absorb/sumcheck-w"; +/// Challenge for sampling `τ₀` (F_0 range-check batching point, paper §4.3). +pub const CHALLENGE_TAU0: &[u8] = b"hachi/challenge/tau0"; +/// Challenge for sampling `τ₁` (F_α evaluation-relation batching point, paper §4.3). +pub const CHALLENGE_TAU1: &[u8] = b"hachi/challenge/tau1"; + +/// Return all Hachi-core transcript labels. +pub fn all_labels() -> &'static [&'static [u8]] { + &[ + DOMAIN_HACHI_PROTOCOL, + ABSORB_COMMITMENT, + ABSORB_EVALUATION_CLAIMS, + CHALLENGE_LINEAR_RELATION, + ABSORB_RING_SWITCH_MESSAGE, + CHALLENGE_RING_SWITCH, + ABSORB_SPARSE_CHALLENGE, + CHALLENGE_SPARSE_CHALLENGE, + ABSORB_SUMCHECK_CLAIM, + ABSORB_SUMCHECK_ROUND, + CHALLENGE_SUMCHECK_ROUND, + ABSORB_STOP_CONDITION, + CHALLENGE_STOP_CONDITION, + ABSORB_PROVER_V, + CHALLENGE_STAGE1_FOLD, + ABSORB_SUMCHECK_W, + CHALLENGE_TAU0, + CHALLENGE_TAU1, + ] +} diff --git a/src/protocol/transcript/mod.rs b/src/protocol/transcript/mod.rs new file mode 100644 index 00000000..7ccd5be9 --- /dev/null +++ b/src/protocol/transcript/mod.rs @@ -0,0 +1,37 @@ +//! Protocol transcript contracts and implementations. + +mod blake2b; +mod keccak; +pub mod labels; + +use crate::{CanonicalField, FieldCore, HachiSerialize}; + +pub use blake2b::Blake2bTranscript; +pub use keccak::KeccakTranscript; + +/// Transcript interface for protocol Fiat-Shamir transforms. +/// +/// The protocol layer is label-aware and uses deterministic byte encoding for +/// all absorbed values. +pub trait Transcript: Clone + Send + Sync + 'static +where + F: FieldCore + CanonicalField, +{ + /// Construct a new transcript under a domain label. + fn new(domain_label: &[u8]) -> Self; + + /// Append labeled raw bytes. + fn append_bytes(&mut self, label: &[u8], bytes: &[u8]); + + /// Append a field element with deterministic encoding. + fn append_field(&mut self, label: &[u8], x: &F); + + /// Append a serializable protocol value. + fn append_serde(&mut self, label: &[u8], s: &S); + + /// Derive a challenge scalar under the provided label. + fn challenge_scalar(&mut self, label: &[u8]) -> F; + + /// Reset transcript state under a new domain label. + fn reset(&mut self, domain_label: &[u8]); +} diff --git a/src/test_utils.rs b/src/test_utils.rs new file mode 100644 index 00000000..2bf775f0 --- /dev/null +++ b/src/test_utils.rs @@ -0,0 +1,123 @@ +//! Shared test configuration and helpers. +//! +//! This module is only compiled under `#[cfg(test)]` and provides common +//! building blocks for both unit tests (inside `src/`) and integration +//! tests (inside `tests/`). + +use crate::algebra::{CyclotomicRing, Fp64}; +use crate::protocol::commitment::CommitmentConfig; +use crate::{CanonicalField, FieldCore}; + +pub type F = Fp64<4294967197>; +pub const D: usize = 64; + +#[derive(Clone)] +pub struct TinyConfig; + +impl CommitmentConfig for TinyConfig { + const D: usize = 64; + const M: usize = 1; + const R: usize = 1; + const N_A: usize = 2; + const N_B: usize = 2; + const N_D: usize = 2; + const LOG_BASIS: u32 = 4; + const DELTA: usize = 9; + const TAU: usize = 4; + const BETA: u128 = 1_000_000; + const CHALLENGE_WEIGHT: usize = 3; +} + +pub const BLOCK_LEN: usize = 1 << TinyConfig::M; +pub const NUM_BLOCKS: usize = 1 << TinyConfig::R; +pub const DELTA: usize = TinyConfig::DELTA; +pub const LOG_BASIS: u32 = TinyConfig::LOG_BASIS; +pub const N_A: usize = TinyConfig::N_A; +pub const TAU: usize = TinyConfig::TAU; + +pub fn mat_vec_mul( + mat: &[Vec>], + vec: &[CyclotomicRing], +) -> Vec> { + mat.iter() + .map(|row| { + assert_eq!(row.len(), vec.len()); + row.iter() + .zip(vec.iter()) + .fold(CyclotomicRing::::zero(), |acc, (a, x)| { + acc + (*a * *x) + }) + }) + .collect() +} + +pub fn sample_blocks() -> Vec>> { + (0..NUM_BLOCKS) + .map(|bi| { + (0..BLOCK_LEN) + .map(|bj| { + let coeffs = + std::array::from_fn(|k| F::from_u64((bi * 1_000 + bj * 100 + k) as u64)); + CyclotomicRing::from_coefficients(coeffs) + }) + .collect() + }) + .collect() +} + +pub fn sample_a() -> Vec { + (0..BLOCK_LEN) + .map(|j| F::from_u64((j * 10 + 1) as u64)) + .collect() +} + +pub fn sample_b() -> Vec { + (0..NUM_BLOCKS) + .map(|i| F::from_u64((i * 7 + 3) as u64)) + .collect() +} + +pub fn field_gadget_recompose( + parts: &[CyclotomicRing], + log_basis: u32, +) -> CyclotomicRing { + let b = F::from_u64(1u64 << log_basis); + let mut result = CyclotomicRing::::zero(); + let mut b_power = F::one(); + for part in parts { + result += part.scale(&b_power); + b_power = b_power * b; + } + result +} + +pub fn recompose_z_hat(z_hat: &[CyclotomicRing]) -> Vec> { + z_hat + .chunks(TAU) + .map(|chunk| field_gadget_recompose(chunk, LOG_BASIS)) + .collect() +} + +pub fn gadget_recompose_vec(x_hat: &[CyclotomicRing]) -> Vec> { + x_hat + .chunks(DELTA) + .map(|chunk| field_gadget_recompose(chunk, LOG_BASIS)) + .collect() +} + +pub fn field_gadget_recompose_vec(v: &[CyclotomicRing]) -> Vec> { + v.chunks(DELTA) + .map(|chunk| field_gadget_recompose(chunk, LOG_BASIS)) + .collect() +} + +pub fn a_transpose_gadget_times_vec(a: &[F], z: &[CyclotomicRing]) -> CyclotomicRing { + let recomposed = field_gadget_recompose_vec(z); + assert_eq!(recomposed.len(), a.len()); + recomposed + .iter() + .zip(a.iter()) + .fold(CyclotomicRing::::zero(), |acc, (z_j, a_j)| { + acc + z_j.scale(a_j) + }) +} diff --git a/tests/algebra.rs b/tests/algebra.rs new file mode 100644 index 00000000..cbb51aeb --- /dev/null +++ b/tests/algebra.rs @@ -0,0 +1,1058 @@ +#![allow(missing_docs)] + +#[cfg(test)] +mod tests { + use num_bigint::BigUint; + use rand::{rngs::StdRng, SeedableRng}; + + use hachi_pcs::algebra::ntt::butterfly::{forward_ntt, inverse_ntt, NttTwiddles}; + use hachi_pcs::algebra::poly::Poly; + use hachi_pcs::algebra::tables::{Q32_DATA, Q32_MODULUS, Q32_NUM_PRIMES, Q32_PRIMES}; + use hachi_pcs::algebra::{ + pseudo_mersenne_modulus, Pow2Offset128Field, Pow2OffsetPrimeSpec, POW2_OFFSET_MAX, + POW2_OFFSET_PRIMES, POW2_OFFSET_TABLE, + }; + use hachi_pcs::algebra::{ + CyclotomicCrtNtt, CyclotomicRing, Fp128, Fp2, Fp2Config, Fp32, Fp4, Fp4Config, Fp64, LimbQ, + MontCoeff, Prime128M13M4P0, Prime128M37P3P0, Prime128M52M3P0, Prime128M54P4P0, + Prime128M8M4M1M0, ScalarBackend, VectorModule, + }; + use hachi_pcs::primitives::serialization::SerializationError; + use hachi_pcs::{ + CanonicalField, FieldCore, FieldSampling, HachiDeserialize, HachiSerialize, Invertible, + Module, PseudoMersenneField, + }; + + const P_159: u128 = 340282366920938463463374607431768211297u128; + + #[test] + fn fp32_basic_arith() { + type F = Fp32<251>; + let a = F::from_u64(17); + let b = F::from_u64(99); + assert_eq!((a + b).to_canonical_u32(), 116); + assert_eq!((a * b).to_canonical_u32(), (17u32 * 99) % 251); + + let inv = a.inv().unwrap(); + assert_eq!(a * inv, F::one()); + } + + #[test] + fn fp64_hachi_q_inv() { + type F = Fp64<4294967197>; + let two = F::from_u64(2); + let inv2 = two.inv().unwrap(); + assert_eq!(two * inv2, F::one()); + } + + #[test] + fn fp128_basic_arith() { + type F = Fp128; + + let a = F::from_u64(123); + let b = F::from_u64(456); + let c = a * b + a - b; + let inv = c.inv().unwrap(); + assert_eq!(c * inv, F::one()); + } + + fn rand_u128(rng: &mut R) -> u128 { + let lo = rng.next_u64() as u128; + let hi = rng.next_u64() as u128; + lo | (hi << 64) + } + + fn biguint_to_u128(x: &num_bigint::BigUint) -> u128 { + let mut bytes = x.to_bytes_le(); + bytes.resize(16, 0); + let mut arr = [0u8; 16]; + arr.copy_from_slice(&bytes[..16]); + u128::from_le_bytes(arr) + } + + fn big_mul_mod_u128(a: u128, b: u128, p: u128) -> u128 { + let n = BigUint::from(a) * BigUint::from(b); + let r = n % BigUint::from(p); + biguint_to_u128(&r) + } + + fn check_solinas_prime< + S: CanonicalField + FieldCore + Invertible + PseudoMersenneField + std::fmt::Debug, + >( + p: u128, + iters: usize, + seed: u64, + ) { + assert_eq!(::MODULUS_BITS, 128); + assert_eq!( + ::MODULUS_OFFSET, + 0u128.wrapping_sub(p) + ); + assert_eq!(std::mem::size_of::(), 16); + + let mut rng = StdRng::seed_from_u64(seed); + + for _ in 0..iters { + let a_raw = rand_u128(&mut rng); + let b_raw = rand_u128(&mut rng); + + let a = S::from_canonical_u128_reduced(a_raw); + let b = S::from_canonical_u128_reduced(b_raw); + + // Canonical range invariant. + assert!(a.to_canonical_u128() < p); + assert!(b.to_canonical_u128() < p); + + // Add/sub/neg identities. + assert_eq!(a + S::zero(), a); + assert_eq!(a - S::zero(), a); + assert_eq!(a + (-a), S::zero()); + + // Multiplicative identity. + assert_eq!(a * S::one(), a); + + // BigUint oracle for mul and sqr (exercises reduction). + let aa = a.to_canonical_u128(); + let bb = b.to_canonical_u128(); + let got_mul = (a * b).to_canonical_u128(); + let exp_mul = big_mul_mod_u128(aa, bb, p); + assert_eq!(got_mul, exp_mul); + + let got_sqr = (a * a).to_canonical_u128(); + let exp_sqr = big_mul_mod_u128(aa, aa, p); + assert_eq!(got_sqr, exp_sqr); + + // Inversion checks (skip explicit inv on zero). + let inv = a.inv_or_zero(); + if a.is_zero() { + assert_eq!(inv, S::zero()); + } else { + assert_eq!(a * inv, S::one()); + assert_eq!(a.inv().unwrap(), inv); + } + } + } + + #[test] + fn fp128_sparse_primes_match_biguint_oracle() { + // These are the five sparse `2^128 - c` primes we care about. + const P13: u128 = 0xffffffffffffffffffffffffffffdff1u128; + const P37: u128 = 0xffffffffffffffffffffffe000000009u128; + const P52: u128 = 0xffffffffffffffffffeffffffffffff9u128; + const P54: u128 = 0xffffffffffffffffffc0000000000011u128; + const P275: u128 = 0xfffffffffffffffffffffffffffffeedu128; + + check_solinas_prime::(P13, 2_000, 13); + check_solinas_prime::(P37, 2_000, 37); + check_solinas_prime::(P52, 2_000, 52); + check_solinas_prime::(P54, 2_000, 54); + check_solinas_prime::(P275, 2_000, 275); + } + + struct NR; + impl Fp2Config> for NR { + fn non_residue() -> Fp32<251> { + -Fp32::<251>::one() + } + } + + struct NR4; + impl Fp4Config, NR> for NR4 { + fn non_residue() -> Fp2, NR> { + Fp2::new(Fp32::<251>::zero(), Fp32::<251>::one()) + } + } + + #[test] + fn fp2_fp4_inversion_smoke() { + type F = Fp32<251>; + type F2 = Fp2; + type F4 = Fp4; + + let x = F2::new(F::from_u64(3), F::from_u64(7)); + let inv = x.inv().unwrap(); + assert!((x * inv) == F2::one()); + + let y = F4::new( + F2::new(F::from_u64(5), F::from_u64(1)), + F2::new(F::from_u64(2), F::from_u64(9)), + ); + let invy = y.inv().unwrap(); + assert!((y * invy) == F4::one()); + } + + #[test] + fn vector_module_ops() { + type F = Fp32<251>; + + let a = VectorModule::([F::from_u64(1), F::from_u64(2), F::from_u64(3)]); + let b = VectorModule::([F::from_u64(3), F::from_u64(4), F::from_u64(5)]); + + let c = a + b; + assert_eq!(c.0[0], F::from_u64(4)); + + let d = a.scale(&F::from_u64(7)); + assert_eq!(d.0[1], F::from_u64(14)); + } + + #[test] + fn inv_zero_returns_none() { + assert!(Fp32::<251>::zero().inv().is_none()); + assert!(Fp64::<4294967197>::zero().inv().is_none()); + assert!(Fp128::::zero().inv().is_none()); + } + + #[test] + fn inv_or_zero_behavior_for_prime_fields() { + type F32 = Fp32<251>; + assert_eq!(F32::zero().inv_or_zero(), F32::zero()); + let x32 = F32::from_u64(17); + let inv32 = x32.inv_or_zero(); + assert_eq!(x32 * inv32, F32::one()); + + type F64 = Fp64<4294967197>; + assert_eq!(F64::zero().inv_or_zero(), F64::zero()); + let x64 = F64::from_u64(2); + let inv64 = x64.inv_or_zero(); + assert_eq!(x64 * inv64, F64::one()); + + type F128 = Fp128; + assert_eq!(F128::zero().inv_or_zero(), F128::zero()); + let x128 = F128::from_u64(12345); + let inv128 = x128.inv_or_zero(); + assert_eq!(x128 * inv128, F128::one()); + } + + #[test] + fn field_identities_fp32() { + type F = Fp32<251>; + let a = F::from_u64(42); + let b = F::from_u64(73); + let c = F::from_u64(11); + + // Additive identity + assert_eq!(a + F::zero(), a); + // Multiplicative identity + assert_eq!(a * F::one(), a); + // Additive inverse + assert_eq!(a + (-a), F::zero()); + // Distributivity + assert_eq!(a * (b + c), a * b + a * c); + // Commutativity + assert_eq!(a * b, b * a); + assert_eq!(a + b, b + a); + } + + #[test] + fn field_identities_fp64() { + type F = Fp64<4294967197>; + let a = F::from_u64(123456); + let b = F::from_u64(789012); + let c = F::from_u64(345678); + + assert_eq!(a + F::zero(), a); + assert_eq!(a * F::one(), a); + assert_eq!(a + (-a), F::zero()); + assert_eq!(a * (b + c), a * b + a * c); + } + + #[test] + fn field_identities_fp128() { + type F = Fp128; + let a = F::from_u64(999999); + let b = F::from_u64(888888); + let c = F::from_u64(777777); + + assert_eq!(a + F::zero(), a); + assert_eq!(a * F::one(), a); + assert_eq!(a + (-a), F::zero()); + assert_eq!(a * (b + c), a * b + a * c); + } + + #[test] + fn serialization_round_trip_fp32() { + type F = Fp32<251>; + let val = F::from_u64(42); + let mut buf = Vec::new(); + val.serialize_compressed(&mut buf).unwrap(); + let restored = F::deserialize_compressed(&buf[..]).unwrap(); + assert_eq!(val, restored); + } + + #[test] + fn serialization_round_trip_fp64() { + type F = Fp64<4294967197>; + let val = F::from_u64(123456789); + let mut buf = Vec::new(); + val.serialize_compressed(&mut buf).unwrap(); + let restored = F::deserialize_compressed(&buf[..]).unwrap(); + assert_eq!(val, restored); + } + + #[test] + fn serialization_round_trip_fp128() { + type F = Fp128; + let val = F::from_u64(999999999); + let mut buf = Vec::new(); + val.serialize_compressed(&mut buf).unwrap(); + let restored = F::deserialize_compressed(&buf[..]).unwrap(); + assert_eq!(val, restored); + } + + #[test] + fn serialization_round_trip_ext() { + type F = Fp32<251>; + type F2 = Fp2; + let val = F2::new(F::from_u64(3), F::from_u64(7)); + let mut buf = Vec::new(); + val.serialize_compressed(&mut buf).unwrap(); + let restored = F2::deserialize_compressed(&buf[..]).unwrap(); + assert!(val == restored); + } + + #[test] + fn serialization_round_trip_fp4() { + type F = Fp32<251>; + type F2 = Fp2; + type F4 = Fp4; + + let val = F4::new( + F2::new(F::from_u64(5), F::from_u64(1)), + F2::new(F::from_u64(2), F::from_u64(9)), + ); + let mut buf = Vec::new(); + val.serialize_compressed(&mut buf).unwrap(); + let restored = F4::deserialize_compressed(&buf[..]).unwrap(); + assert!(val == restored); + } + + #[test] + fn serialization_round_trip_vector_module() { + type F = Fp32<251>; + let val = VectorModule::([F::from_u64(1), F::from_u64(2), F::from_u64(3)]); + let mut buf = Vec::new(); + val.serialize_compressed(&mut buf).unwrap(); + let restored = VectorModule::::deserialize_compressed(&buf[..]).unwrap(); + assert_eq!(val, restored); + } + + #[test] + fn serialization_round_trip_poly() { + type F = Fp32<251>; + + let val = Poly::([ + F::from_u64(7), + F::from_u64(11), + F::from_u64(13), + F::from_u64(29), + ]); + let mut buf = Vec::new(); + val.serialize_compressed(&mut buf).unwrap(); + let restored = Poly::::deserialize_compressed(&buf[..]).unwrap(); + assert_eq!(val, restored); + } + + #[test] + fn deserialize_checked_rejects_non_canonical_field_elements() { + type F32 = Fp32<251>; + let bad32 = 251u32.to_le_bytes(); + let err32 = F32::deserialize_compressed(&bad32[..]).unwrap_err(); + assert!(matches!(err32, SerializationError::InvalidData(_))); + let unchecked32 = F32::deserialize_compressed_unchecked(&bad32[..]).unwrap(); + assert_eq!(unchecked32, F32::zero()); + + type F64 = Fp64<4294967197>; + let bad64 = 4294967197u64.to_le_bytes(); + let err64 = F64::deserialize_compressed(&bad64[..]).unwrap_err(); + assert!(matches!(err64, SerializationError::InvalidData(_))); + let unchecked64 = F64::deserialize_compressed_unchecked(&bad64[..]).unwrap(); + assert_eq!(unchecked64, F64::zero()); + + type F128 = Fp128; + let bad128 = P_159.to_le_bytes(); + let err128 = F128::deserialize_compressed(&bad128[..]).unwrap_err(); + assert!(matches!(err128, SerializationError::InvalidData(_))); + let unchecked128 = F128::deserialize_compressed_unchecked(&bad128[..]).unwrap(); + assert_eq!(unchecked128, F128::zero()); + + // Sparse 128-bit prime: same checked/unchecked behavior. + type S13 = Prime128M13M4P0; + const P13: u128 = 0xffffffffffffffffffffffffffffdff1u128; + let bad13 = P13.to_le_bytes(); + let err13 = S13::deserialize_compressed(&bad13[..]).unwrap_err(); + assert!(matches!(err13, SerializationError::InvalidData(_))); + let unchecked13 = S13::deserialize_compressed_unchecked(&bad13[..]).unwrap(); + assert_eq!(unchecked13, S13::zero()); + } + + #[test] + fn fp2_conjugate_and_norm() { + type F = Fp32<251>; + type F2 = Fp2; + let x = F2::new(F::from_u64(3), F::from_u64(7)); + let conj = x.conjugate(); + assert!(conj == F2::new(F::from_u64(3), -F::from_u64(7))); + // For Fp2 with u^2 = -1: norm = c0^2 + c1^2 = 9 + 49 = 58 + assert_eq!(x.norm(), F::from_u64(58)); + // x * conjugate(x) should embed the norm into Fp2 + let prod = x * conj; + assert!(prod == F2::new(F::from_u64(58), F::zero())); + } + + #[test] + fn fp2_distributivity() { + type F = Fp32<251>; + type F2 = Fp2; + let a = F2::new(F::from_u64(3), F::from_u64(7)); + let b = F2::new(F::from_u64(11), F::from_u64(5)); + let c = F2::new(F::from_u64(2), F::from_u64(9)); + assert!(a * (b + c) == a * b + a * c); + } + + #[test] + fn limbq_from_to_u128_round_trip() { + for &val in &[0u128, 1, 12345, 123456789, (1u128 << 28) - 1] { + let limb: LimbQ<3> = LimbQ::from(val); + assert_eq!( + u128::try_from(limb).unwrap(), + val, + "round-trip failed for {val}" + ); + } + } + + #[test] + fn limbq_add_sub_inverse() { + let a: LimbQ<3> = LimbQ::from(12345u128); + let b: LimbQ<3> = LimbQ::from(6789u128); + let sum = a + b; + let diff = sum - b; + assert_eq!(diff, a); + } + + #[test] + fn limbq_ordering() { + let a: LimbQ<3> = LimbQ::from(100u128); + let b: LimbQ<3> = LimbQ::from(200u128); + assert!(a < b); + assert!(b > a); + assert_eq!(a, a); + } + + #[test] + fn qdata_q_matches_const() { + let q_from_data = Q32_DATA.q_u128().unwrap(); + assert_eq!(q_from_data, Q32_MODULUS as u128); + } + + #[test] + fn ntt_normalize_in_range() { + for prime in &Q32_PRIMES { + for &a in &[0i16, 1, -1, 100, -100, prime.p - 1, -(prime.p - 1)] { + let n = prime.normalize(MontCoeff::from_raw(a)); + assert!( + n.raw() >= 0 && n.raw() < prime.p, + "normalize({a}) = {} for p={}", + n.raw(), + prime.p + ); + } + } + } + + #[test] + fn ntt_mul_commutative() { + let prime = Q32_PRIMES[0]; + let a = MontCoeff::from_raw(1234); + let b = MontCoeff::from_raw(5678); + assert_eq!(prime.mul(a, b), prime.mul(b, a)); + } + + #[test] + fn mont_coeff_round_trip() { + for prime in &Q32_PRIMES { + for &val in &[0i16, 1, 2, 100, prime.p - 1] { + let mont = prime.from_canonical(val); + let back = prime.to_canonical(mont); + assert_eq!(back, val, "round-trip failed for val={val}, p={}", prime.p); + } + } + } + + #[test] + fn poly_add_sub_neg() { + type F = Fp32<251>; + let a = Poly::([F::from_u64(1), F::from_u64(2), F::from_u64(3)]); + let b = Poly::([F::from_u64(10), F::from_u64(20), F::from_u64(30)]); + + let sum = a + b; + assert_eq!(sum.0[0], F::from_u64(11)); + assert_eq!(sum.0[1], F::from_u64(22)); + assert_eq!(sum.0[2], F::from_u64(33)); + + let diff = b - a; + assert_eq!(diff.0[0], F::from_u64(9)); + + let neg_a = -a; + assert_eq!(a + neg_a, Poly::zero()); + } + + #[test] + fn cyclotomic_ring_negacyclic_property() { + type F = Fp32<251>; + type R = CyclotomicRing; + + // X in the ring: [0, 1, 0, 0] + let x = R::x(); + + // X^2 + let x2 = x * x; + let expected_x2 = R::from_coefficients([F::zero(), F::zero(), F::one(), F::zero()]); + assert_eq!(x2, expected_x2); + + // X^4 should equal -1 (because X^4 + 1 = 0 in the ring) + let x4 = x2 * x2; + assert_eq!(x4, -R::one(), "X^D should equal -1 in Z_q[X]/(X^D + 1)"); + } + + #[test] + fn cyclotomic_ring_mul_identity() { + type F = Fp32<251>; + type R = CyclotomicRing; + + let a = R::from_coefficients([ + F::from_u64(3), + F::from_u64(7), + F::from_u64(11), + F::from_u64(42), + ]); + assert_eq!(a * R::one(), a); + assert_eq!(R::one() * a, a); + } + + #[test] + fn cyclotomic_ring_mul_zero() { + type F = Fp32<251>; + type R = CyclotomicRing; + + let a = R::from_coefficients([ + F::from_u64(3), + F::from_u64(7), + F::from_u64(11), + F::from_u64(42), + ]); + assert_eq!(a * R::zero(), R::zero()); + } + + #[test] + fn cyclotomic_ring_commutativity() { + type F = Fp32<251>; + type R = CyclotomicRing; + + let a = R::from_coefficients([ + F::from_u64(3), + F::from_u64(7), + F::from_u64(11), + F::from_u64(42), + ]); + let b = R::from_coefficients([ + F::from_u64(5), + F::from_u64(13), + F::from_u64(99), + F::from_u64(1), + ]); + assert_eq!(a * b, b * a); + } + + #[test] + fn cyclotomic_ring_distributivity() { + type F = Fp32<251>; + type R = CyclotomicRing; + + let a = R::from_coefficients([ + F::from_u64(3), + F::from_u64(7), + F::from_u64(11), + F::from_u64(42), + ]); + let b = R::from_coefficients([ + F::from_u64(5), + F::from_u64(13), + F::from_u64(99), + F::from_u64(1), + ]); + let c = R::from_coefficients([ + F::from_u64(2), + F::from_u64(9), + F::from_u64(50), + F::from_u64(77), + ]); + assert_eq!(a * (b + c), a * b + a * c); + } + + #[test] + fn cyclotomic_ring_associativity() { + type F = Fp32<251>; + type R = CyclotomicRing; + + let a = R::from_coefficients([ + F::from_u64(3), + F::from_u64(7), + F::from_u64(11), + F::from_u64(42), + ]); + let b = R::from_coefficients([ + F::from_u64(5), + F::from_u64(13), + F::from_u64(99), + F::from_u64(1), + ]); + let c = R::from_coefficients([ + F::from_u64(2), + F::from_u64(9), + F::from_u64(50), + F::from_u64(77), + ]); + assert_eq!((a * b) * c, a * (b * c)); + } + + #[test] + fn cyclotomic_ring_additive_inverse() { + type F = Fp32<251>; + type R = CyclotomicRing; + + let a = R::from_coefficients([ + F::from_u64(3), + F::from_u64(7), + F::from_u64(11), + F::from_u64(42), + ]); + assert_eq!(a + (-a), R::zero()); + } + + #[test] + fn cyclotomic_ring_serialization_round_trip() { + type F = Fp32<251>; + type R = CyclotomicRing; + + let a = R::from_coefficients([ + F::from_u64(3), + F::from_u64(7), + F::from_u64(11), + F::from_u64(42), + ]); + let mut buf = Vec::new(); + a.serialize_compressed(&mut buf).unwrap(); + let restored = R::deserialize_compressed(&buf[..]).unwrap(); + assert_eq!(a, restored); + } + + #[test] + fn cyclotomic_ring_degree_64() { + type F = Fp64<4294967197>; + type R = CyclotomicRing; + + // X^64 = -1 in Z_q[X]/(X^64 + 1) + let x = R::x(); + let mut power = R::one(); + for _ in 0..64 { + power *= x; + } + assert_eq!(power, -R::one(), "X^64 should equal -1"); + } + + #[test] + fn ntt_forward_inverse_round_trip() { + let prime = Q32_PRIMES[0]; + let tw = NttTwiddles::<64>::compute(prime); + + // Create a test polynomial in Montgomery form. + let original: [MontCoeff; 64] = + std::array::from_fn(|i| prime.from_canonical((i as i16) % prime.p)); + + // Forward then inverse should give back the original. + let mut a = original; + forward_ntt(&mut a, prime, &tw); + inverse_ntt(&mut a, prime, &tw); + + // Normalize and compare. + for (i, (got, expected)) in a.iter().zip(original.iter()).enumerate() { + let got_canon = prime.to_canonical(prime.normalize(*got)); + let exp_canon = prime.to_canonical(prime.normalize(*expected)); + assert_eq!( + got_canon, exp_canon, + "NTT round-trip mismatch at index {i}: got {got_canon}, expected {exp_canon}" + ); + } + } + + #[test] + fn ntt_forward_inverse_all_primes() { + for (pi, prime) in Q32_PRIMES.iter().enumerate() { + let tw = NttTwiddles::<64>::compute(*prime); + + let original: [_; 64] = + std::array::from_fn(|i| prime.from_canonical(((i * (pi + 1)) as i16) % prime.p)); + + let mut a = original; + forward_ntt(&mut a, *prime, &tw); + inverse_ntt(&mut a, *prime, &tw); + + for (i, (got, expected)) in a.iter().zip(original.iter()).enumerate() { + let g = prime.to_canonical(prime.normalize(*got)); + let e = prime.to_canonical(prime.normalize(*expected)); + assert_eq!( + g, e, + "prime[{pi}] p={}: round-trip mismatch at index {i}", + prime.p + ); + } + } + } + + #[test] + fn cyclotomic_ntt_crt_round_trip_q32() { + type F = Fp64<{ Q32_MODULUS }>; + type R = CyclotomicRing; + type N = CyclotomicCrtNtt; + + let twiddles: [NttTwiddles<64>; Q32_NUM_PRIMES] = + std::array::from_fn(|k| NttTwiddles::<64>::compute(Q32_PRIMES[k])); + + let coeffs: [F; 64] = + std::array::from_fn(|i| F::from_u64(((i as u64 * 17) + 5) % Q32_MODULUS)); + let ring = R::from_coefficients(coeffs); + let ntt = N::from_ring(&ring, &Q32_PRIMES, &twiddles); + let round_trip = ntt.to_ring(&Q32_PRIMES, &twiddles, &Q32_DATA); + + assert_eq!(ring, round_trip); + } + + #[test] + fn cyclotomic_ntt_reduced_ops_are_stable() { + type F = Fp64<{ Q32_MODULUS }>; + type R = CyclotomicRing; + type N = CyclotomicCrtNtt; + + let twiddles: [NttTwiddles<64>; Q32_NUM_PRIMES] = + std::array::from_fn(|k| NttTwiddles::<64>::compute(Q32_PRIMES[k])); + + let a = R::from_coefficients(std::array::from_fn(|i| { + F::from_u64(((i as u64 * 3) + 1) % Q32_MODULUS) + })); + let b = R::from_coefficients(std::array::from_fn(|i| { + F::from_u64(((i as u64 * 11) + 7) % Q32_MODULUS) + })); + + let ntt_a = N::from_ring(&a, &Q32_PRIMES, &twiddles); + let ntt_b = N::from_ring(&b, &Q32_PRIMES, &twiddles); + + let sum = ntt_a.add_reduced(&ntt_b, &Q32_PRIMES); + let back = sum.sub_reduced(&ntt_b, &Q32_PRIMES); + assert_eq!(back, ntt_a); + + let zero_ntt = ntt_a.add_reduced(&ntt_a.neg_reduced(&Q32_PRIMES), &Q32_PRIMES); + let zero_ring = zero_ntt.to_ring(&Q32_PRIMES, &twiddles, &Q32_DATA); + assert_eq!(zero_ring, R::zero()); + } + + #[test] + fn backend_path_matches_default_scalar_path() { + type F = Fp64<{ Q32_MODULUS }>; + type R = CyclotomicRing; + type N = CyclotomicCrtNtt; + + let twiddles: [NttTwiddles<64>; Q32_NUM_PRIMES] = + std::array::from_fn(|k| NttTwiddles::<64>::compute(Q32_PRIMES[k])); + let ring = R::from_coefficients(std::array::from_fn(|i| { + F::from_u64(((i as u64 * 13) + 9) % Q32_MODULUS) + })); + + let default_ntt = N::from_ring(&ring, &Q32_PRIMES, &twiddles); + let backend_ntt = + N::from_ring_with_backend::(&ring, &Q32_PRIMES, &twiddles); + assert_eq!(default_ntt, backend_ntt); + + let default_back = default_ntt.to_ring(&Q32_PRIMES, &twiddles, &Q32_DATA); + let backend_back = backend_ntt.to_ring_with_backend::( + &Q32_PRIMES, + &twiddles, + &Q32_DATA, + ); + assert_eq!(default_back, backend_back); + } + + #[test] + fn field_sampling_respects_modulus() { + type F = Fp32<251>; + let mut rng = StdRng::seed_from_u64(42); + for _ in 0..1024 { + let x = F::sample(&mut rng); + assert!(x.to_canonical_u32() < 251); + } + } + + #[test] + fn pow2_offset_registry_is_consistent() { + fn assert_is_pseudo_mersenne() {} + assert_is_pseudo_mersenne::(); + + for Pow2OffsetPrimeSpec { + bits, + offset, + modulus, + .. + } in POW2_OFFSET_PRIMES + { + assert!((offset as u128) <= POW2_OFFSET_MAX); + assert_eq!(POW2_OFFSET_TABLE[bits as usize], offset as i16); + assert_eq!( + Some(modulus), + pseudo_mersenne_modulus(bits, offset as u128), + "2^k-offset modulus mismatch for k={bits}, offset={offset}" + ); + assert_eq!(modulus % 8, 5); + } + + let x = Pow2Offset128Field::from_u64(1234567); + let inv = x.inv().unwrap(); + assert_eq!(x * inv, Pow2Offset128Field::one()); + } + + #[test] + fn cyclotomic_sigma_is_ring_automorphism() { + type F = Fp32<251>; + type R = CyclotomicRing; + let a = R::from_coefficients(std::array::from_fn(|i| F::from_u64((3 * i + 1) as u64))); + let b = R::from_coefficients(std::array::from_fn(|i| F::from_u64((5 * i + 2) as u64))); + + let k1 = 3usize; + let k2 = 5usize; + let two_d = 16usize; + + assert_eq!(a.sigma(1), a); + assert_eq!(a.sigma_m1().sigma_m1(), a); + assert_eq!(a.sigma(k1).sigma(k2), a.sigma((k1 * k2) % two_d)); + assert_eq!((a * b).sigma(k1), a.sigma(k1) * b.sigma(k1)); + } + + #[test] + fn cyclotomic_balanced_pow2_decompose_recompose_round_trip() { + type F = Fp64<{ Q32_MODULUS }>; + type R = CyclotomicRing; + + let ring = R::from_coefficients(std::array::from_fn(|i| { + F::from_u64(((i as u64 * 73) + 17) % Q32_MODULUS) + })); + + // Q32 balanced base-16: 9 levels absorb the carry-out near q/2. + let digits = ring.balanced_decompose_pow2(9, 4); + let round_trip = R::gadget_recompose_pow2(&digits, 4); + assert_eq!(round_trip, ring); + } + + #[test] + fn sparse_pm1_challenge_has_expected_weight() { + type F = Fp32<251>; + type R = CyclotomicRing; + + let mut rng = StdRng::seed_from_u64(123); + let challenge = R::sample_sparse_pm1(&mut rng, 11); + assert_eq!(challenge.hamming_weight(), 11); + + for c in challenge.coefficients() { + let x = c.to_canonical_u32(); + if x != 0 { + assert!(x == 1 || x == 250, "nonzero coefficient must be +/-1"); + } + } + } + + #[test] + fn negacyclic_shift_equals_mul_by_monomial() { + type F = Fp32<251>; + type R = CyclotomicRing; + + let a = R::from_coefficients(std::array::from_fn(|i| F::from_u64((3 * i + 1) as u64))); + + for k in 0..8 { + let mut monomial_coeffs = [F::zero(); 8]; + monomial_coeffs[k] = F::one(); + let monomial = R::from_coefficients(monomial_coeffs); + assert_eq!( + a.negacyclic_shift(k), + a * monomial, + "negacyclic_shift({k}) != mul by X^{k}" + ); + } + + assert_eq!(a.negacyclic_shift(0), a); + assert_eq!( + a.negacyclic_shift(8), + a, + "shift by D should be identity mod D" + ); + } + + #[test] + fn negacyclic_shift_degree_64() { + type F = Fp64<4294967197>; + type R = CyclotomicRing; + + let a = R::from_coefficients(std::array::from_fn(|i| F::from_u64((7 * i + 3) as u64))); + let x = R::x(); + let mut x_pow = R::one(); + for k in 0..64 { + assert_eq!( + a.negacyclic_shift(k), + a * x_pow, + "negacyclic_shift({k}) mismatch at D=64" + ); + x_pow *= x; + } + } + + #[test] + fn mul_by_monomial_sum_matches_ring_mul() { + type F = Fp32<251>; + type R = CyclotomicRing; + + let a = R::from_coefficients(std::array::from_fn(|i| F::from_u64((5 * i + 2) as u64))); + + // Sum of X^1 + X^3 + X^5 + let positions = [1, 3, 5]; + let mut sparse = [F::zero(); 8]; + for &p in &positions { + sparse[p] = F::one(); + } + let sparse_ring = R::from_coefficients(sparse); + + assert_eq!( + a.mul_by_monomial_sum(&positions), + a * sparse_ring, + "mul_by_monomial_sum should equal ring mul by sparse element" + ); + } + + #[test] + fn mul_by_monomial_sum_single_position_equals_shift() { + type F = Fp32<251>; + type R = CyclotomicRing; + + let a = R::from_coefficients(std::array::from_fn(|i| F::from_u64((i + 1) as u64))); + for k in 0..8 { + assert_eq!( + a.mul_by_monomial_sum(&[k]), + a.negacyclic_shift(k), + "single-position monomial_sum should equal negacyclic_shift" + ); + } + } + + #[test] + fn mul_by_monomial_sum_empty_is_zero() { + type F = Fp32<251>; + type R = CyclotomicRing; + + let a = R::from_coefficients(std::array::from_fn(|i| F::from_u64((i + 1) as u64))); + assert_eq!(a.mul_by_monomial_sum(&[]), R::zero()); + } + + #[test] + fn mul_by_sparse_matches_schoolbook() { + use hachi_pcs::algebra::SparseChallenge; + + type F = Fp64<4294967197>; + type R = CyclotomicRing; + + let a = R::from_coefficients(std::array::from_fn(|i| F::from_u64((3 * i + 7) as u64))); + + let challenge = SparseChallenge { + positions: vec![2, 17, 41], + coeffs: vec![1, -1, 1], + }; + let dense: R = challenge.to_dense().unwrap(); + + let via_sparse = a.mul_by_sparse(&challenge); + let via_schoolbook = a * dense; + + assert_eq!( + via_sparse, via_schoolbook, + "mul_by_sparse must equal schoolbook multiplication" + ); + } + + #[test] + fn mul_by_sparse_with_all_negative_coeffs() { + use hachi_pcs::algebra::SparseChallenge; + + type F = Fp64<4294967197>; + type R = CyclotomicRing; + + let a = R::from_coefficients(std::array::from_fn(|i| F::from_u64((i + 1) as u64))); + + let challenge = SparseChallenge { + positions: vec![0, 5, 63], + coeffs: vec![-1, -1, -1], + }; + let dense: R = challenge.to_dense().unwrap(); + + assert_eq!(a.mul_by_sparse(&challenge), a * dense); + } + + #[test] + fn is_zero_detects_zero_and_nonzero() { + type F = Fp32<251>; + type R = CyclotomicRing; + + assert!(R::zero().is_zero()); + assert!(!R::one().is_zero()); + + let a = R::from_coefficients(std::array::from_fn(|i| F::from_u64(i as u64))); + assert!(!a.is_zero()); + } + + #[test] + fn kron_scalars_matches_kron_row_constant_rings() { + type F = Fp64<4294967197>; + type R = CyclotomicRing; + + let scalars_a: Vec = (0..4).map(|i| F::from_u64(i * 3 + 1)).collect(); + let scalars_b: Vec = (0..3).map(|i| F::from_u64(i * 7 + 2)).collect(); + + let rings_a: Vec = scalars_a + .iter() + .map(|&s| { + let mut c = [F::zero(); 16]; + c[0] = s; + R::from_coefficients(c) + }) + .collect(); + let rings_b: Vec = scalars_b + .iter() + .map(|&s| { + let mut c = [F::zero(); 16]; + c[0] = s; + R::from_coefficients(c) + }) + .collect(); + + let via_ring: Vec = rings_a + .iter() + .flat_map(|l| rings_b.iter().map(move |r| *l * *r)) + .collect(); + + let via_scalar: Vec = scalars_a + .iter() + .flat_map(|&l| { + scalars_b.iter().map(move |&r| { + let mut c = [F::zero(); 16]; + c[0] = l * r; + R::from_coefficients(c) + }) + }) + .collect(); + + assert_eq!(via_ring, via_scalar); + } +} diff --git a/tests/commitment_contract.rs b/tests/commitment_contract.rs new file mode 100644 index 00000000..dc300613 --- /dev/null +++ b/tests/commitment_contract.rs @@ -0,0 +1,236 @@ +#![allow(missing_docs)] + +use hachi_pcs::algebra::Fp64; +use hachi_pcs::protocol::commitment::{DummyProof, HachiCommitment}; +use hachi_pcs::protocol::transcript::labels; +use hachi_pcs::protocol::{ + AppendToTranscript, Blake2bTranscript, CommitmentScheme, StreamingCommitmentScheme, Transcript, +}; +use hachi_pcs::{CanonicalField, FieldCore, HachiError, Polynomial}; + +type F = Fp64<4294967197>; + +#[derive(Clone)] +struct SimplePoly { + coeffs: Vec, +} + +impl Polynomial for SimplePoly { + fn num_vars(&self) -> usize { + self.coeffs.len().saturating_sub(1) + } + + fn evaluate(&self, point: &[F]) -> F { + assert_eq!(point.len(), self.num_vars()); + let mut acc = self.coeffs[0]; + for (i, r_i) in point.iter().enumerate() { + acc = acc + self.coeffs[i + 1] * *r_i; + } + acc + } + + fn coeffs(&self) -> Vec { + self.coeffs.clone() + } +} + +#[derive(Clone)] +struct DummySetup { + _max_num_vars: usize, +} + +#[derive(Clone)] +struct DummyScheme; + +impl CommitmentScheme for DummyScheme { + type ProverSetup = DummySetup; + type VerifierSetup = DummySetup; + type Commitment = HachiCommitment; + type Proof = DummyProof; + type OpeningProofHint = HachiCommitment; + + fn setup_prover(max_num_vars: usize) -> Self::ProverSetup { + DummySetup { + _max_num_vars: max_num_vars, + } + } + + fn setup_verifier(setup: &Self::ProverSetup) -> Self::VerifierSetup { + setup.clone() + } + + fn commit>( + poly: &P, + _setup: &Self::ProverSetup, + ) -> Result<(Self::Commitment, Self::OpeningProofHint), HachiError> { + let zero = vec![F::zero(); poly.num_vars()]; + let c = HachiCommitment(poly.evaluate(&zero).to_canonical_u128()); + Ok((c, c)) + } + + fn prove, P: Polynomial>( + setup: &Self::ProverSetup, + poly: &P, + opening_point: &[F], + hint: Option, + transcript: &mut T, + _commitment: &Self::Commitment, + ) -> Result { + if opening_point.len() != poly.num_vars() { + return Err(HachiError::InvalidPointDimension { + expected: poly.num_vars(), + actual: opening_point.len(), + }); + } + + let absorb_commitment = if let Some(h) = hint { + h + } else { + Self::commit(poly, setup)?.0 + }; + absorb_commitment.append_to_transcript(labels::ABSORB_COMMITMENT, transcript); + + let q = transcript.challenge_scalar(labels::CHALLENGE_LINEAR_RELATION); + let opening = poly.evaluate(opening_point); + Ok(DummyProof( + opening.to_canonical_u128() ^ q.to_canonical_u128(), + )) + } + + fn verify>( + proof: &Self::Proof, + _setup: &Self::VerifierSetup, + transcript: &mut T, + _opening_point: &[F], + opening: &F, + commitment: &Self::Commitment, + ) -> Result<(), HachiError> { + commitment.append_to_transcript(labels::ABSORB_COMMITMENT, transcript); + let q = transcript.challenge_scalar(labels::CHALLENGE_LINEAR_RELATION); + let expected = opening.to_canonical_u128() ^ q.to_canonical_u128(); + if proof.0 == expected { + Ok(()) + } else { + Err(HachiError::InvalidProof) + } + } + + fn combine_commitments(commitments: &[Self::Commitment], coeffs: &[F]) -> Self::Commitment { + let acc = commitments + .iter() + .zip(coeffs.iter()) + .fold(0u128, |sum, (c, coeff)| { + sum.wrapping_add(c.0.wrapping_mul(coeff.to_canonical_u128())) + }); + HachiCommitment(acc) + } + + fn combine_hints(hints: Vec, coeffs: &[F]) -> Self::OpeningProofHint { + let acc = hints + .iter() + .zip(coeffs.iter()) + .fold(0u128, |sum, (h, coeff)| { + sum.wrapping_add(h.0.wrapping_mul(coeff.to_canonical_u128())) + }); + HachiCommitment(acc) + } + + fn protocol_name() -> &'static [u8] { + b"HachiDummy" + } +} + +impl StreamingCommitmentScheme for DummyScheme { + type ChunkState = HachiCommitment; + + fn process_chunk(_setup: &Self::ProverSetup, chunk: &[F]) -> Self::ChunkState { + let sum = chunk + .iter() + .fold(0u128, |acc, x| acc.wrapping_add(x.to_canonical_u128())); + HachiCommitment(sum) + } + + fn process_chunk_onehot( + _setup: &Self::ProverSetup, + onehot_k: usize, + chunk: &[Option], + ) -> Self::ChunkState { + let sum = chunk.iter().fold(0u128, |acc, x| { + let v = x.unwrap_or(0) as u128; + acc.wrapping_add(v) + }); + HachiCommitment(sum.wrapping_add(onehot_k as u128)) + } + + fn aggregate_chunks( + _setup: &Self::ProverSetup, + _onehot_k: Option, + chunks: &[Self::ChunkState], + ) -> (Self::Commitment, Self::OpeningProofHint) { + let sum = chunks.iter().fold(0u128, |acc, c| acc.wrapping_add(c.0)); + let c = HachiCommitment(sum); + (c, c) + } +} + +#[test] +fn commitment_scheme_round_trip() { + let poly = SimplePoly { + coeffs: vec![F::from_u64(3), F::from_u64(5), F::from_u64(7)], + }; + let opening_point = [F::from_u64(11), F::from_u64(13)]; + let opening = poly.evaluate(&opening_point); + + let psetup = DummyScheme::setup_prover(poly.num_vars()); + let vsetup = DummyScheme::setup_verifier(&psetup); + + let (commitment, hint) = DummyScheme::commit(&poly, &psetup).unwrap(); + + let mut prover_t = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + let proof = DummyScheme::prove( + &psetup, + &poly, + &opening_point, + Some(hint), + &mut prover_t, + &commitment, + ) + .unwrap(); + + let mut verifier_t = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + DummyScheme::verify( + &proof, + &vsetup, + &mut verifier_t, + &opening_point, + &opening, + &commitment, + ) + .unwrap(); +} + +#[test] +fn combine_commitments_and_hints_are_consistent() { + let c1 = HachiCommitment(10); + let c2 = HachiCommitment(20); + let coeffs = [F::from_u64(3), F::from_u64(7)]; + + let combined_c = DummyScheme::combine_commitments(&[c1, c2], &coeffs); + let combined_h = DummyScheme::combine_hints(vec![c1, c2], &coeffs); + + let expected = 10u128 + .wrapping_mul(coeffs[0].to_canonical_u128()) + .wrapping_add(20u128.wrapping_mul(coeffs[1].to_canonical_u128())); + assert_eq!(combined_c.0, expected); + assert_eq!(combined_h.0, expected); +} + +#[test] +fn streaming_chunk_path_aggregates() { + let setup = DummyScheme::setup_prover(4); + let c1 = DummyScheme::process_chunk(&setup, &[F::from_u64(1), F::from_u64(2)]); + let c2 = DummyScheme::process_chunk_onehot(&setup, 8, &[Some(3), None, Some(5)]); + + let (commitment, hint) = DummyScheme::aggregate_chunks(&setup, Some(8), &[c1, c2]); + assert_eq!(commitment, hint); +} diff --git a/tests/hachi_sumcheck.rs b/tests/hachi_sumcheck.rs new file mode 100644 index 00000000..69f16bf4 --- /dev/null +++ b/tests/hachi_sumcheck.rs @@ -0,0 +1,225 @@ +#![allow(missing_docs)] + +use hachi_pcs::algebra::ring::CyclotomicRing; +use hachi_pcs::algebra::Fp64; +use hachi_pcs::protocol::sumcheck::norm_sumcheck::{NormSumcheckProver, NormSumcheckVerifier}; +use hachi_pcs::protocol::sumcheck::relation_sumcheck::{ + RelationSumcheckProver, RelationSumcheckVerifier, +}; +use hachi_pcs::protocol::sumcheck::{eq_eval, eq_evals, multilinear_eval, range_check_eval}; +use hachi_pcs::protocol::transcript::labels; +use hachi_pcs::protocol::{prove_sumcheck, verify_sumcheck, Blake2bTranscript, Transcript}; +use hachi_pcs::{CanonicalField, FieldCore, FieldSampling}; +use rand::rngs::StdRng; +use rand::SeedableRng; +use std::time::Instant; + +type F = Fp64<4294967197>; + +fn run_f0_e2e(num_u: usize, num_l: usize, b: usize) { + let num_vars = num_u + num_l; + let n = 1usize << num_vars; + let mut rng = StdRng::seed_from_u64(0xF0); + + let w_evals: Vec = (0..n).map(|i| F::from_u64((i % b) as u64)).collect(); + let tau0: Vec = (0..num_vars).map(|_| F::sample(&mut rng)).collect(); + + let t0 = Instant::now(); + let mut prover = NormSumcheckProver::new(&tau0, w_evals.clone(), b); + let mut pt = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + + let (proof, prover_challenges, final_claim) = + prove_sumcheck::(&mut prover, &mut pt, |tr| { + tr.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND) + }) + .unwrap(); + let prove_time = t0.elapsed(); + + // Sanity: prover's final claim matches oracle evaluation. + let oracle = eq_eval(&tau0, &prover_challenges) + * range_check_eval(multilinear_eval(&w_evals, &prover_challenges), b); + assert_eq!(final_claim, oracle, "prover final claim != oracle eval"); + + // ---- Verifier ---- + let t1 = Instant::now(); + let verifier = NormSumcheckVerifier::new(tau0, w_evals, b); + let mut vt = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + + let verifier_challenges = verify_sumcheck::(&proof, &verifier, &mut vt, |tr| { + tr.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND) + }) + .unwrap(); + let verify_time = t1.elapsed(); + + assert_eq!(prover_challenges, verifier_challenges); + + eprintln!( + "[F0 e2e] num_u={num_u} num_l={num_l} b={b} n=2^{num_vars}={n} \ + prove={prove_time:.2?} verify={verify_time:.2?} \ + rounds={} degree={}", + proof.round_polys.len(), + 2 * b, + ); +} + +#[test] +fn f0_sumcheck_e2e_small() { + run_f0_e2e(3, 2, 2); +} + +#[test] +fn f0_sumcheck_e2e() { + run_f0_e2e(4, 3, 2); +} + +#[test] +fn f0_sumcheck_e2e_larger_b() { + run_f0_e2e(3, 3, 3); +} + +fn run_f_alpha_e2e(num_u: usize, num_i: usize) { + let num_l = D.trailing_zeros() as usize; + let num_vars = num_u + num_l; + let n = 1usize << num_vars; + let mut rng = StdRng::seed_from_u64(0xFA); + + let w_evals: Vec = (0..n).map(|_| F::sample(&mut rng)).collect(); + let alpha_evals_y: Vec = (0..D).map(|_| F::sample(&mut rng)).collect(); + let m_alpha_evals: Vec = (0..(1usize << (num_i + num_u))) + .map(|_| F::sample(&mut rng)) + .collect(); + let tau1: Vec = (0..num_i).map(|_| F::sample(&mut rng)).collect(); + + // Compute m(x) = Σ_i ẽq(τ₁, i) · M̃_α(i, x) + let eq_tau1 = eq_evals(&tau1); + let num_x = 1usize << num_u; + let m_evals_x: Vec = (0..num_x) + .map(|x_idx| { + (0..(1usize << num_i)) + .map(|i_idx| eq_tau1[i_idx] * m_alpha_evals[i_idx * num_x + x_idx]) + .fold(F::zero(), |a, v| a + v) + }) + .collect(); + + // Compute y_a[i] = Σ_x M̃_α(i,x) · w_α(x), where w_α(x) = Σ_y w(x,y) · α̃(y) + let num_y = D; + let num_rows = 1usize << num_i; + let w_alpha: Vec = (0..num_x) + .map(|x| { + (0..num_y) + .map(|y| w_evals[x + y * num_x] * alpha_evals_y[y]) + .fold(F::zero(), |a, v| a + v) + }) + .collect(); + let y_a: Vec = (0..num_rows) + .map(|i| { + (0..num_x) + .map(|x| m_alpha_evals[i * num_x + x] * w_alpha[x]) + .fold(F::zero(), |a, v| a + v) + }) + .collect(); + + // Embed y_a values as constant ring elements for the verifier. + let v_rings: Vec> = y_a + .iter() + .map(|&val| { + let mut coeffs = [F::zero(); D]; + coeffs[0] = val; + CyclotomicRing::from_coefficients(coeffs) + }) + .collect(); + let u_rings: Vec> = vec![]; + let u_eval_ring = CyclotomicRing::::zero(); + let ring_alpha = F::one(); + + let t0 = Instant::now(); + let mut prover = + RelationSumcheckProver::new(w_evals.clone(), &alpha_evals_y, &m_evals_x, num_u, num_l); + let mut pt = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + + let (proof, prover_challenges, final_claim) = + prove_sumcheck::(&mut prover, &mut pt, |tr| { + tr.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND) + }) + .unwrap(); + let prove_time = t0.elapsed(); + + // Sanity: prover's final claim matches oracle evaluation. + let (x_ch, y_ch) = prover_challenges.split_at(num_u); + let oracle = multilinear_eval(&w_evals, &prover_challenges) + * multilinear_eval(&alpha_evals_y, y_ch) + * multilinear_eval(&m_evals_x, x_ch); + assert_eq!(final_claim, oracle, "prover final claim != oracle eval"); + + let t1 = Instant::now(); + let verifier = RelationSumcheckVerifier::::new( + w_evals, + alpha_evals_y, + m_evals_x, + tau1, + v_rings, + u_rings, + u_eval_ring, + ring_alpha, + num_u, + num_l, + ); + let mut vt = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + + let verifier_challenges = verify_sumcheck::(&proof, &verifier, &mut vt, |tr| { + tr.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND) + }) + .unwrap(); + let verify_time = t1.elapsed(); + + assert_eq!(prover_challenges, verifier_challenges); + + eprintln!( + "[Fα e2e] num_u={num_u} num_l={num_l} num_i={num_i} n=2^{num_vars}={n} \ + prove={prove_time:.2?} verify={verify_time:.2?} \ + rounds={} degree=2", + proof.round_polys.len(), + ); +} + +#[test] +fn f_alpha_sumcheck_e2e_small() { + run_f_alpha_e2e::<4>(3, 2); +} + +#[test] +fn f_alpha_sumcheck_e2e() { + run_f_alpha_e2e::<8>(4, 3); +} + +#[test] +fn f_alpha_sumcheck_e2e_asymmetric() { + run_f_alpha_e2e::<4>(5, 4); +} + +#[test] +fn from_evals_matches_direct_polynomial() { + use hachi_pcs::protocol::UniPoly; + + // Verify that interpolation at integer points reproduces the polynomial. + let mut rng = StdRng::seed_from_u64(0xEE); + + for degree in 0..6usize { + let coeffs: Vec = (0..=degree).map(|_| F::sample(&mut rng)).collect(); + let poly = UniPoly::from_coeffs(coeffs); + + let evals: Vec = (0..=degree) + .map(|t| poly.evaluate(&F::from_u64(t as u64))) + .collect(); + let reconstructed = UniPoly::from_evals(&evals); + + for x_u64 in [0u64, 1, 2, 3, 7, 13] { + let x = F::from_u64(x_u64); + assert_eq!( + poly.evaluate(&x), + reconstructed.evaluate(&x), + "degree {degree}, x={x_u64}" + ); + } + } +} diff --git a/tests/label_schedule.rs b/tests/label_schedule.rs new file mode 100644 index 00000000..c943b83b --- /dev/null +++ b/tests/label_schedule.rs @@ -0,0 +1,63 @@ +#![allow(missing_docs)] + +use hachi_pcs::algebra::Fp64; +use hachi_pcs::protocol::transcript::labels; +use hachi_pcs::protocol::{Blake2bTranscript, Transcript}; + +type F = Fp64<4294967197>; + +#[test] +fn label_namespace_does_not_include_dory_literals() { + let banned = ["vmv_", "beta", "alpha", "gamma", "final_e", "dory"]; + for label in labels::all_labels() { + let text = std::str::from_utf8(label).expect("labels must be valid utf8 literals"); + for needle in &banned { + assert!( + !text.contains(needle), + "label `{text}` must not contain banned token `{needle}`" + ); + } + } +} + +fn run_hachi_schedule>(transcript: &mut T) -> (F, F, F) { + transcript.append_bytes(labels::ABSORB_COMMITMENT, b"C"); + transcript.append_bytes(labels::ABSORB_EVALUATION_CLAIMS, b"O"); + let c_linear_relation = transcript.challenge_scalar(labels::CHALLENGE_LINEAR_RELATION); + + transcript.append_bytes(labels::ABSORB_RING_SWITCH_MESSAGE, b"RS"); + let c_ring_switch = transcript.challenge_scalar(labels::CHALLENGE_RING_SWITCH); + + transcript.append_bytes(labels::ABSORB_SUMCHECK_ROUND, b"SC1"); + let c_sumcheck = transcript.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND); + transcript.append_bytes(labels::ABSORB_STOP_CONDITION, b"STOP"); + let _ = transcript.challenge_scalar(labels::CHALLENGE_STOP_CONDITION); + + (c_linear_relation, c_ring_switch, c_sumcheck) +} + +#[test] +fn schedule_is_replayable_with_hachi_labels() { + let mut prover = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + let mut verifier = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + assert_eq!( + run_hachi_schedule(&mut prover), + run_hachi_schedule(&mut verifier) + ); +} + +#[test] +fn schedule_detects_reordered_round_messages() { + let mut t1 = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + let mut t2 = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + + t1.append_bytes(labels::ABSORB_COMMITMENT, b"C"); + t1.append_bytes(labels::ABSORB_EVALUATION_CLAIMS, b"O"); + let a = t1.challenge_scalar(labels::CHALLENGE_LINEAR_RELATION); + + t2.append_bytes(labels::ABSORB_EVALUATION_CLAIMS, b"O"); + t2.append_bytes(labels::ABSORB_COMMITMENT, b"C"); + let b = t2.challenge_scalar(labels::CHALLENGE_LINEAR_RELATION); + + assert_ne!(a, b); +} diff --git a/tests/onehot_commitment.rs b/tests/onehot_commitment.rs new file mode 100644 index 00000000..f338f266 --- /dev/null +++ b/tests/onehot_commitment.rs @@ -0,0 +1,145 @@ +#![allow(missing_docs)] + +use hachi_pcs::protocol::commitment::{HachiCommitmentCore, RingCommitmentScheme}; +use hachi_pcs::test_utils::*; +use hachi_pcs::{CanonicalField, FieldCore}; + +type Core = HachiCommitmentCore; + +fn psetup() -> >::ProverSetup { + >::setup(16) + .unwrap() + .0 +} + +/// Compare the optimized one-hot path against the default dense path. +/// +/// The default implementation materializes the full vector and calls +/// `commit_coeffs`. The optimized impl uses sparse inner Ajtai. +/// Both must produce identical (commitment, s_all, t_hat_all). +fn assert_onehot_matches_dense(onehot_k: usize, indices: &[usize]) { + let setup = psetup(); + + // Optimized sparse path. + let (c_sparse, s_sparse, t_sparse) = + >::commit_onehot(onehot_k, indices, &setup) + .unwrap(); + + // Reference: materialize the full one-hot vector, pack into ring elements, + // and commit via the dense path. + let total_field = indices.len() * onehot_k; + let total_ring = total_field / D; + let mut field_elems = vec![F::zero(); total_field]; + for (c, &idx) in indices.iter().enumerate() { + field_elems[c * onehot_k + idx] = F::from_u64(1); + } + let ring_coeffs: Vec> = (0..total_ring) + .map(|r| { + let coeffs: [F; D] = std::array::from_fn(|i| field_elems[r * D + i]); + hachi_pcs::algebra::CyclotomicRing::from_coefficients(coeffs) + }) + .collect(); + let (c_dense, s_dense, t_dense) = + >::commit_coeffs(&ring_coeffs, &setup) + .unwrap(); + + assert_eq!(c_sparse, c_dense, "commitments must match"); + assert_eq!(s_sparse, s_dense, "s_all (decomposed witness) must match"); + assert_eq!( + t_sparse, t_dense, + "t_hat_all (decomposed inner output) must match" + ); +} + +#[test] +fn onehot_k_gt_d_basic() { + // K=128, D=64 => K/D=2, T=2 => T*K=256 => 4 ring elements + assert_onehot_matches_dense(128, &[0, 64]); +} + +#[test] +fn onehot_k_gt_d_various_positions() { + assert_onehot_matches_dense(128, &[127, 0]); + assert_onehot_matches_dense(128, &[63, 65]); + assert_onehot_matches_dense(128, &[32, 96]); +} + +#[test] +fn onehot_k_much_gt_d() { + // K=256, D=64 => K/D=4, T=1 => T*K=256 => 4 ring elements + assert_onehot_matches_dense(256, &[0]); + assert_onehot_matches_dense(256, &[63]); + assert_onehot_matches_dense(256, &[64]); + assert_onehot_matches_dense(256, &[255]); + assert_onehot_matches_dense(256, &[100]); +} + +#[test] +fn onehot_k_eq_d_basic() { + // K=64=D, T=4 => 4 ring elements, each is a monomial X^{idx}. + assert_onehot_matches_dense(64, &[0, 0, 0, 0]); +} + +#[test] +fn onehot_k_eq_d_varied() { + assert_onehot_matches_dense(64, &[0, 31, 32, 63]); + assert_onehot_matches_dense(64, &[1, 2, 3, 4]); + assert_onehot_matches_dense(64, &[63, 63, 63, 63]); +} + +#[test] +fn onehot_k_lt_d_basic() { + // K=16, D=64 => D/K=4, T=16 => T*K=256 => 4 ring elements. + // Each ring element spans 4 chunks, so has 4 nonzero coefficients. + let indices: Vec = (0..16).map(|i| i % 16).collect(); + assert_onehot_matches_dense(16, &indices); +} + +#[test] +fn onehot_k_lt_d_all_zeros() { + let indices = vec![0; 16]; + assert_onehot_matches_dense(16, &indices); +} + +#[test] +fn onehot_k_lt_d_all_max() { + let indices = vec![15; 16]; + assert_onehot_matches_dense(16, &indices); +} + +#[test] +fn onehot_k_lt_d_mixed() { + let indices = vec![0, 15, 7, 3, 12, 1, 8, 14, 5, 10, 2, 9, 6, 11, 4, 13]; + assert_onehot_matches_dense(16, &indices); +} + +#[test] +fn onehot_k_lt_d_ratio_2() { + // K=32, D=64 => D/K=2, T=8 => T*K=256 => 4 ring elements. + let indices = vec![0, 31, 16, 8, 24, 4, 12, 20]; + assert_onehot_matches_dense(32, &indices); +} + +#[test] +fn onehot_rejects_non_divisible_k_and_d() { + let setup = psetup(); + let result = + >::commit_onehot(17, &[0; 4], &setup); + assert!(result.is_err()); +} + +#[test] +fn onehot_rejects_out_of_range_index() { + let setup = psetup(); + let result = + >::commit_onehot(64, &[0, 64, 0, 0], &setup); + assert!(result.is_err()); +} + +#[test] +fn onehot_rejects_wrong_total_size() { + let setup = psetup(); + let result = + >::commit_onehot(64, &[0, 0, 0], &setup); + assert!(result.is_err()); +} diff --git a/tests/primality.rs b/tests/primality.rs new file mode 100644 index 00000000..0d7a3a8f --- /dev/null +++ b/tests/primality.rs @@ -0,0 +1,124 @@ +#![allow(missing_docs)] + +use hachi_pcs::algebra::{pseudo_mersenne_modulus, Pow2OffsetPrimeSpec, POW2_OFFSET_PRIMES}; + +// Strong probable-prime test using multiple fixed bases. +// This is not a formal primality certificate, but is sufficient as a +// practical regression guard for the current Pow2Offset profiles. +fn is_probable_prime_miller_rabin(n: u128) -> bool { + if n < 2 { + return false; + } + if n % 2 == 0 { + return n == 2; + } + + const SMALL_PRIMES: [u128; 11] = [3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37]; + for p in SMALL_PRIMES { + if n == p { + return true; + } + if n % p == 0 { + return false; + } + } + + let (d, s) = decompose_pow2(n - 1); + const BASES: [u128; 24] = [ + 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, + ]; + + 'outer: for a in BASES { + if a >= n { + continue; + } + let mut x = pow_mod(a, d, n); + if x == 1 || x == n - 1 { + continue; + } + for _ in 1..s { + x = mul_mod(x, x, n); + if x == n - 1 { + continue 'outer; + } + } + return false; + } + + true +} + +fn decompose_pow2(mut d: u128) -> (u128, u32) { + let mut s = 0u32; + while d % 2 == 0 { + d >>= 1; + s += 1; + } + (d, s) +} + +fn pow_mod(mut base: u128, mut exp: u128, modulus: u128) -> u128 { + let mut result = 1u128; + base %= modulus; + while exp > 0 { + if (exp & 1) == 1 { + result = mul_mod(result, base, modulus); + } + base = mul_mod(base, base, modulus); + exp >>= 1; + } + result +} + +fn mul_mod(mut a: u128, mut b: u128, modulus: u128) -> u128 { + let mut result = 0u128; + a %= modulus; + b %= modulus; + while b > 0 { + if (b & 1) == 1 { + result = add_mod(result, a, modulus); + } + a = add_mod(a, a, modulus); + b >>= 1; + } + result +} + +fn add_mod(a: u128, b: u128, modulus: u128) -> u128 { + if a >= modulus - b { + a - (modulus - b) + } else { + a + b + } +} + +#[test] +fn pow2_offset_profiles_are_probable_primes() { + for Pow2OffsetPrimeSpec { + bits, + offset, + modulus, + } in POW2_OFFSET_PRIMES + { + assert_eq!( + Some(modulus), + pseudo_mersenne_modulus(bits, offset as u128), + "profile formula mismatch for bits={bits}, offset={offset}" + ); + assert!( + is_probable_prime_miller_rabin(modulus), + "Miller-Rabin rejected bits={bits}, offset={offset}, q={modulus}" + ); + } +} + +#[test] +fn miller_rabin_rejects_known_composites() { + let composites: [u128; 9] = [4, 9, 15, 21, 341, 561, 645, 1105, 1729]; + for n in composites { + assert!( + !is_probable_prime_miller_rabin(n), + "composite unexpectedly accepted: {n}" + ); + } +} diff --git a/tests/ring_commitment_core.rs b/tests/ring_commitment_core.rs new file mode 100644 index 00000000..2fa20bee --- /dev/null +++ b/tests/ring_commitment_core.rs @@ -0,0 +1,184 @@ +#![allow(missing_docs)] + +use hachi_pcs::algebra::CyclotomicRing; +use hachi_pcs::error::HachiError; +use hachi_pcs::protocol::commitment::{ + CommitmentConfig, DefaultCommitmentConfig, HachiCommitmentCore, RingCommitmentScheme, +}; +use hachi_pcs::test_utils::*; + +#[derive(Clone)] +struct BadDegreeConfig; + +impl CommitmentConfig for BadDegreeConfig { + const D: usize = 32; + const M: usize = 4; + const R: usize = 2; + const N_A: usize = 8; + const N_B: usize = 4; + const N_D: usize = 4; + const LOG_BASIS: u32 = 4; + const DELTA: usize = 8; + const TAU: usize = 4; + const BETA: u128 = 1_000_000; + const CHALLENGE_WEIGHT: usize = 3; +} + +#[derive(Clone)] +struct BadDigitBudgetConfig; + +impl CommitmentConfig for BadDigitBudgetConfig { + const D: usize = 64; + const M: usize = 4; + const R: usize = 2; + const N_A: usize = 8; + const N_B: usize = 4; + const N_D: usize = 4; + const LOG_BASIS: u32 = 32; + const DELTA: usize = 5; // 160 > 128 + const TAU: usize = 4; + const BETA: u128 = 1_000_000; + const CHALLENGE_WEIGHT: usize = 3; +} + +#[test] +fn setup_shape_is_consistent() { + let (p1, v1) = + >::setup(16).unwrap(); + let (p2, v2) = + >::setup(16).unwrap(); + + assert_eq!(p1.max_num_vars, 16); + assert_eq!(v1.max_num_vars, 16); + assert_eq!(p2.max_num_vars, 16); + assert_eq!(v2.max_num_vars, 16); + assert_eq!(p1.A.len(), TinyConfig::N_A); + assert_eq!(p1.A[0].len(), (1usize << TinyConfig::M) * TinyConfig::DELTA); + assert_eq!(p1.B.len(), TinyConfig::N_B); + assert_eq!( + p1.B[0].len(), + TinyConfig::N_A * TinyConfig::DELTA * (1usize << TinyConfig::R) + ); +} + +#[test] +fn commit_is_deterministic_and_shape_consistent() { + let (psetup, _) = + >::setup(16).unwrap(); + let blocks = sample_blocks(); + + let (c1, s1, t1) = + >::commit_ring_blocks( + &blocks, &psetup, + ) + .unwrap(); + let (c2, s2, t2) = + >::commit_ring_blocks( + &blocks, &psetup, + ) + .unwrap(); + + assert_eq!(c1, c2); + assert_eq!(s1, s2); + assert_eq!(t1, t2); + + let num_blocks = 1usize << TinyConfig::R; + let block_len = 1usize << TinyConfig::M; + assert_eq!(c1.u.len(), TinyConfig::N_B); + assert_eq!(s1.len(), num_blocks); + assert_eq!(t1.len(), num_blocks); + assert!(s1.iter().all(|s| s.len() == block_len * TinyConfig::DELTA)); + assert!(t1 + .iter() + .all(|t| t.len() == TinyConfig::N_A * TinyConfig::DELTA)); +} + +#[test] +fn commit_ring_coeffs_matches_block_commitment() { + let (psetup, _) = + >::setup(16).unwrap(); + let blocks = sample_blocks(); + + let (c_blocks, s_blocks, t_blocks) = + >::commit_ring_blocks( + &blocks, &psetup, + ) + .unwrap(); + + let num_blocks = 1usize << TinyConfig::R; + let block_len = 1usize << TinyConfig::M; + let mut f_coeffs = Vec::with_capacity(num_blocks * block_len); + for j in 0..block_len { + for block in blocks.iter().take(num_blocks) { + f_coeffs.push(block[j]); + } + } + + let (c_coeffs, s_coeffs, t_coeffs) = + >::commit_coeffs( + &f_coeffs, &psetup, + ) + .unwrap(); + + assert_eq!(c_blocks, c_coeffs); + assert_eq!(s_blocks, s_coeffs); + assert_eq!(t_blocks, t_coeffs); +} + +#[test] +fn opening_satisfies_inner_and_outer_equations() { + let (psetup, _) = + >::setup(16).unwrap(); + let blocks = sample_blocks(); + let (commitment, s, t_hat) = + >::commit_ring_blocks( + &blocks, &psetup, + ) + .unwrap(); + + for i in 0..s.len() { + let lhs = mat_vec_mul(&psetup.A, &s[i]); + let rhs: Vec> = (0..TinyConfig::N_A) + .map(|j| { + let start = j * TinyConfig::DELTA; + let end = start + TinyConfig::DELTA; + CyclotomicRing::gadget_recompose_pow2(&t_hat[i][start..end], TinyConfig::LOG_BASIS) + }) + .collect(); + assert_eq!(lhs, rhs); + } + + let t_hat_flat: Vec> = + t_hat.iter().flat_map(|x| x.iter().copied()).collect(); + let outer = mat_vec_mul(&psetup.B, &t_hat_flat); + assert_eq!(outer, commitment.u); +} + +#[test] +fn default_config_has_expected_shape() { + assert_eq!(DefaultCommitmentConfig::D, 16); + assert_eq!(1usize << DefaultCommitmentConfig::M, 16); + assert_eq!(1usize << DefaultCommitmentConfig::R, 4); + let delta = DefaultCommitmentConfig::DELTA; + assert!(delta > 0); +} + +#[test] +fn setup_rejects_mismatched_degree() { + let err = >::setup(16) + .unwrap_err(); + match err { + HachiError::InvalidSetup(msg) => assert!(msg.contains("mismatches")), + other => panic!("unexpected error: {other:?}"), + } +} + +#[test] +fn setup_rejects_invalid_digit_budget() { + let err = >::setup(16) + .unwrap_err(); + match err { + HachiError::InvalidSetup(msg) => assert!(msg.contains("DELTA * LOG_BASIS")), + other => panic!("unexpected error: {other:?}"), + } +} diff --git a/tests/sparse_challenge.rs b/tests/sparse_challenge.rs new file mode 100644 index 00000000..16eb8ee4 --- /dev/null +++ b/tests/sparse_challenge.rs @@ -0,0 +1,98 @@ +#![allow(missing_docs)] + +use hachi_pcs::algebra::fields::LiftBase; +use hachi_pcs::algebra::ring::{CyclotomicRing, SparseChallenge, SparseChallengeConfig}; +use hachi_pcs::algebra::Fp64; +use hachi_pcs::protocol::challenges::sparse::sparse_challenge_from_transcript; +use hachi_pcs::protocol::transcript::labels::DOMAIN_HACHI_PROTOCOL; +use hachi_pcs::protocol::transcript::{Blake2bTranscript, Transcript}; +use hachi_pcs::{CanonicalField, FieldCore}; + +type F = Fp64<4294967197>; + +const D: usize = 16; + +fn dense_eval>(alpha: E, x: &CyclotomicRing) -> E { + let mut acc = E::zero(); + let mut pow = E::one(); + for c in x.coefficients().iter().copied() { + acc = acc + E::lift_base(c) * pow; + pow = pow * alpha; + } + acc +} + +#[test] +fn sparse_challenge_validate_and_to_dense() { + let cfg = SparseChallengeConfig { + weight: 3, + nonzero_coeffs: vec![-1, 1], + }; + cfg.validate::().unwrap(); + + let s = SparseChallenge { + positions: vec![0, 7, 12], + coeffs: vec![1, -1, 1], + }; + s.validate::().unwrap(); + assert_eq!(s.hamming_weight(), 3); + assert_eq!(s.l1_norm(), 3); + + let dense = s.to_dense::().unwrap(); + assert_eq!(dense.hamming_weight(), 3); + assert_eq!(dense.coefficients()[0], F::one()); + assert_eq!(dense.coefficients()[7], -F::one()); + assert_eq!(dense.coefficients()[12], F::one()); +} + +#[test] +fn sparse_eval_at_alpha_matches_dense_eval() { + let alpha = F::from_u64(5); + let alpha_pows = { + let mut out = Vec::with_capacity(D); + let mut acc = F::one(); + for _ in 0..D { + out.push(acc); + acc = acc * alpha; + } + out + }; + + let s = SparseChallenge { + positions: vec![1, 3, 9], + coeffs: vec![2, -1, 1], + }; + let dense = s.to_dense::().unwrap(); + + let sparse_eval = s.eval_at_alpha::(&alpha_pows).unwrap(); + let dense_eval = dense_eval::(alpha, &dense); + assert_eq!(sparse_eval, dense_eval); +} + +#[test] +fn sparse_challenge_sampling_is_deterministic_and_exact_weight() { + let cfg = SparseChallengeConfig { + weight: 8, + nonzero_coeffs: vec![-1, 1], + }; + + let mut t1 = Blake2bTranscript::::new(DOMAIN_HACHI_PROTOCOL); + let mut t2 = Blake2bTranscript::::new(DOMAIN_HACHI_PROTOCOL); + + // Make transcript state non-empty to avoid degenerate behavior. + t1.append_field(b"seed", &F::from_u64(123)); + t2.append_field(b"seed", &F::from_u64(123)); + + let c1 = sparse_challenge_from_transcript::(&mut t1, b"c", 0, &cfg).unwrap(); + let c2 = sparse_challenge_from_transcript::(&mut t2, b"c", 0, &cfg).unwrap(); + assert_eq!(c1, c2); + c1.validate::().unwrap(); + assert_eq!(c1.hamming_weight(), cfg.weight); + assert_eq!(c1.l1_norm(), cfg.weight as u64); + + // Different instance_idx should change the sample. + let mut t3 = Blake2bTranscript::::new(DOMAIN_HACHI_PROTOCOL); + t3.append_field(b"seed", &F::from_u64(123)); + let c3 = sparse_challenge_from_transcript::(&mut t3, b"c", 1, &cfg).unwrap(); + assert_ne!(c1, c3); +} diff --git a/tests/sumcheck_core.rs b/tests/sumcheck_core.rs new file mode 100644 index 00000000..a9c58f28 --- /dev/null +++ b/tests/sumcheck_core.rs @@ -0,0 +1,304 @@ +#![allow(missing_docs)] + +use std::time::Instant; + +use hachi_pcs::algebra::Fp64; +use hachi_pcs::protocol::transcript::labels; +use hachi_pcs::protocol::{ + prove_sumcheck, verify_sumcheck, Blake2bTranscript, CompressedUniPoly, SumcheckInstanceProver, + SumcheckInstanceVerifier, SumcheckProof, Transcript, UniPoly, +}; +use hachi_pcs::{CanonicalField, FieldCore, FieldSampling}; +use rand::rngs::StdRng; +use rand::RngCore; +use rand::SeedableRng; + +type F = Fp64<4294967197>; + +#[test] +fn compressed_unipoly_round_trip_and_eval() { + let mut rng = StdRng::seed_from_u64(123); + + for degree in 0..8usize { + let coeffs: Vec = (0..=degree).map(|_| F::sample(&mut rng)).collect(); + let poly = UniPoly::from_coeffs(coeffs); + + // Hint is g(0) + g(1). + let hint = poly.evaluate(&F::zero()) + poly.evaluate(&F::one()); + + let compressed = poly.compress(); + let decompressed = compressed.decompress(&hint); + + // Decompression should be functionally equivalent (it may materialize + // a trailing zero linear term for constant polynomials). + for x_u64 in [0u64, 1, 2, 3, 17] { + let x = F::from_u64(x_u64); + let direct = poly.evaluate(&x); + let decompressed_direct = decompressed.evaluate(&x); + let via_hint = compressed.eval_from_hint(&hint, &x); + assert_eq!(direct, decompressed_direct); + assert_eq!(direct, via_hint); + } + } +} + +#[test] +fn sumcheck_proof_verifier_driver_is_transcript_deterministic() { + // This test checks that the verifier driver absorbs messages and samples challenges + // consistently, and that the returned (final_claim, r_vec) matches a manual replay. + let mut rng = StdRng::seed_from_u64(999); + + let num_rounds = 5usize; + let degree_bound = 7usize; + + // Build random per-round univariates (degree <= degree_bound), compress them. + let round_polys: Vec> = (0..num_rounds) + .map(|_| { + let deg = (rng.next_u32() as usize) % (degree_bound + 1); + let coeffs: Vec = (0..=deg).map(|_| F::sample(&mut rng)).collect(); + UniPoly::from_coeffs(coeffs).compress() + }) + .collect(); + + let proof = SumcheckProof { round_polys }; + let claim0 = F::sample(&mut rng); + + // Verifier run. + let mut t1 = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + let (final_claim_1, r_1) = proof + .verify::(claim0, num_rounds, degree_bound, &mut t1, |tr| { + tr.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND) + }) + .unwrap(); + + // Manual replay with a fresh transcript (must match). + let mut t2 = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + let mut claim = claim0; + let mut r_manual = Vec::with_capacity(num_rounds); + for poly in &proof.round_polys { + t2.append_serde(labels::ABSORB_SUMCHECK_ROUND, poly); + let r_i = t2.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND); + r_manual.push(r_i); + claim = poly.eval_from_hint(&claim, &r_i); + } + + assert_eq!(r_1, r_manual); + assert_eq!(final_claim_1, claim); +} + +/// Evaluate a multilinear polynomial (given by its boolean-hypercube evaluations +/// in little-endian bit order) at an arbitrary point via iterated folding. +fn multilinear_eval(evals: &[E], point: &[E]) -> E { + let mut current = evals.to_vec(); + for r in point { + let half = current.len() / 2; + let mut next = Vec::with_capacity(half); + for i in 0..half { + next.push(current[2 * i] + *r * (current[2 * i + 1] - current[2 * i])); + } + current = next; + } + current[0] +} + +struct DenseSumcheckProver { + evals: Vec, + num_vars: usize, +} + +impl SumcheckInstanceProver for DenseSumcheckProver { + fn num_rounds(&self) -> usize { + self.num_vars + } + + fn degree_bound(&self) -> usize { + 1 + } + + fn input_claim(&self) -> E { + self.evals.iter().copied().fold(E::zero(), |a, b| a + b) + } + + fn compute_round_univariate(&mut self, _round: usize, _previous_claim: E) -> UniPoly { + let half = self.evals.len() / 2; + let mut eval_0 = E::zero(); + let mut eval_1 = E::zero(); + for i in 0..half { + eval_0 = eval_0 + self.evals[2 * i]; + eval_1 = eval_1 + self.evals[2 * i + 1]; + } + // g(X) = eval_0 + (eval_1 - eval_0) * X (degree 1) + UniPoly::from_coeffs(vec![eval_0, eval_1 - eval_0]) + } + + fn ingest_challenge(&mut self, _round: usize, r: E) { + let half = self.evals.len() / 2; + let mut new_evals = Vec::with_capacity(half); + for i in 0..half { + new_evals.push(self.evals[2 * i] + r * (self.evals[2 * i + 1] - self.evals[2 * i])); + } + self.evals = new_evals; + } +} + +struct DenseSumcheckVerifier { + evals: Vec, + num_vars: usize, + claim: E, +} + +impl SumcheckInstanceVerifier for DenseSumcheckVerifier { + fn num_rounds(&self) -> usize { + self.num_vars + } + + fn degree_bound(&self) -> usize { + 1 + } + + fn input_claim(&self) -> E { + self.claim + } + + fn expected_output_claim(&self, challenges: &[E]) -> E { + multilinear_eval(&self.evals, challenges) + } +} + +#[test] +fn prove_and_verify_single_sumcheck() { + let num_vars = 4; + let n = 1 << num_vars; + + let evals: Vec = (1..=n).map(|i| F::from_u64(i as u64)).collect(); + let claim: F = evals.iter().copied().fold(F::zero(), |a, b| a + b); + + let mut prover = DenseSumcheckProver { + evals: evals.clone(), + num_vars, + }; + + let mut prover_transcript = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + + let (proof, prover_challenges, _final_claim) = + prove_sumcheck::(&mut prover, &mut prover_transcript, |tr| { + tr.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND) + }) + .unwrap(); + + let verifier = DenseSumcheckVerifier { + evals, + num_vars, + claim, + }; + + let mut verifier_transcript = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + + let verifier_challenges = + verify_sumcheck::(&proof, &verifier, &mut verifier_transcript, |tr| { + tr.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND) + }) + .unwrap(); + + assert_eq!(prover_challenges, verifier_challenges); +} + +#[test] +fn verify_rejects_wrong_claim() { + let num_vars = 3; + let n = 1 << num_vars; + + let evals: Vec = (1..=n).map(|i| F::from_u64(i as u64)).collect(); + let correct_claim: F = evals.iter().copied().fold(F::zero(), |a, b| a + b); + let wrong_claim = correct_claim + F::one(); + + // Prove with correct claim. + let mut prover = DenseSumcheckProver { + evals: evals.clone(), + num_vars, + }; + let mut pt = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + + let (proof, _, _) = prove_sumcheck::(&mut prover, &mut pt, |tr| { + tr.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND) + }) + .unwrap(); + + // Verify with *wrong* claim — should fail. + let verifier = DenseSumcheckVerifier { + evals, + num_vars, + claim: wrong_claim, + }; + let mut vt = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + + let result = verify_sumcheck::(&proof, &verifier, &mut vt, |tr| { + tr.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND) + }); + + assert!(result.is_err()); +} + +/// End-to-end sumcheck over 2^20 random field elements. +/// +/// The prover holds a multilinear polynomial f with 2^20 evaluations and +/// proves that Σ_{b ∈ {0,1}^20} f(b) = claimed_sum. The verifier checks the +/// proof using only the proof transcript and the oracle evaluation f(r). +#[test] +fn e2e_sumcheck_2_pow_20() { + let num_vars = 20; + let n: usize = 1 << num_vars; // 1,048,576 + + let mut rng = StdRng::seed_from_u64(42); + let evals: Vec = (0..n).map(|_| F::sample(&mut rng)).collect(); + let claim: F = evals.iter().copied().fold(F::zero(), |a, b| a + b); + + let t0 = Instant::now(); + + let mut prover = DenseSumcheckProver { + evals: evals.clone(), + num_vars, + }; + let mut prover_transcript = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + + let (proof, prover_challenges, final_claim) = + prove_sumcheck::(&mut prover, &mut prover_transcript, |tr| { + tr.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND) + }) + .unwrap(); + + let prove_time = t0.elapsed(); + + // Proof is just 20 compressed univariate polynomials (degree 1 each). + assert_eq!(proof.round_polys.len(), num_vars); + + // Sanity: final claim must equal f evaluated at the challenge point. + let oracle_eval = multilinear_eval(&evals, &prover_challenges); + assert_eq!(final_claim, oracle_eval); + + let t1 = Instant::now(); + + let verifier = DenseSumcheckVerifier { + evals, + num_vars, + claim, + }; + let mut verifier_transcript = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + + let verifier_challenges = + verify_sumcheck::(&proof, &verifier, &mut verifier_transcript, |tr| { + tr.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND) + }) + .unwrap(); + + let verify_time = t1.elapsed(); + + assert_eq!(prover_challenges, verifier_challenges); + + eprintln!( + "[e2e_sumcheck_2_pow_20] n=2^{num_vars}={n} \ + prove={prove_time:.2?} verify={verify_time:.2?} \ + rounds={} degree=1", + proof.round_polys.len() + ); +} diff --git a/tests/sumcheck_prover_driver.rs b/tests/sumcheck_prover_driver.rs new file mode 100644 index 00000000..d309a36d --- /dev/null +++ b/tests/sumcheck_prover_driver.rs @@ -0,0 +1,98 @@ +#![allow(missing_docs)] + +use hachi_pcs::algebra::Fp64; +use hachi_pcs::protocol::transcript::labels; +use hachi_pcs::protocol::{ + prove_sumcheck, Blake2bTranscript, SumcheckInstanceProver, Transcript, UniPoly, +}; +use hachi_pcs::{FieldCore, FieldSampling}; +use rand::rngs::StdRng; +use rand::SeedableRng; + +type F = Fp64<4294967197>; + +/// A tiny prover-side sumcheck instance for a multilinear function in evaluation-table form. +/// +/// Variable order convention: the current round binds the least-significant index bit first, +/// i.e. pairs are `(i<<1)|0` and `(i<<1)|1` (matches the common LSB-first sumcheck table fold). +struct DenseTableSumcheck { + table: Vec, +} + +impl DenseTableSumcheck { + fn new(table: Vec) -> Self { + assert!(table.len().is_power_of_two()); + Self { table } + } +} + +impl SumcheckInstanceProver for DenseTableSumcheck { + fn num_rounds(&self) -> usize { + self.table.len().trailing_zeros() as usize + } + + fn degree_bound(&self) -> usize { + 1 + } + + fn input_claim(&self) -> F { + self.table.iter().copied().fold(F::zero(), |a, b| a + b) + } + + fn compute_round_univariate(&mut self, _round: usize, _previous_claim: F) -> UniPoly { + let half = self.table.len() / 2; + let mut s0 = F::zero(); + let mut s1 = F::zero(); + for i in 0..half { + s0 = s0 + self.table[i << 1]; + s1 = s1 + self.table[(i << 1) | 1]; + } + // g(X) = s0 + (s1 - s0) X + UniPoly::from_coeffs(vec![s0, s1 - s0]) + } + + fn ingest_challenge(&mut self, _round: usize, r_round: F) { + let half = self.table.len() / 2; + let mut next = Vec::with_capacity(half); + let one_minus = F::one() - r_round; + for i in 0..half { + let v0 = self.table[i << 1]; + let v1 = self.table[(i << 1) | 1]; + next.push(one_minus * v0 + r_round * v1); + } + self.table = next; + } +} + +#[test] +fn prover_driver_produces_proof_that_verifier_replays() { + let mut rng = StdRng::seed_from_u64(2026); + let num_rounds = 8usize; + let n = 1usize << num_rounds; + + let table: Vec = (0..n).map(|_| F::sample(&mut rng)).collect(); + let mut prover_inst = DenseTableSumcheck::new(table.clone()); + let mut prover_t = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + let (proof, r_vec, final_claim) = + prove_sumcheck::(&mut prover_inst, &mut prover_t, |tr| { + tr.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND) + }) + .unwrap(); + + // After folding all variables, the table should be a single value equal to f(r*). + assert_eq!(prover_inst.table.len(), 1); + assert_eq!(final_claim, prover_inst.table[0]); + + // Verifier replay must derive the same (final_claim, r_vec). + let initial_claim = table.iter().copied().fold(F::zero(), |acc, x| acc + x); + let mut verifier_t = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + verifier_t.append_serde(labels::ABSORB_SUMCHECK_CLAIM, &initial_claim); + let (final_claim_v, r_vec_v) = proof + .verify::(initial_claim, num_rounds, 1, &mut verifier_t, |tr| { + tr.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND) + }) + .unwrap(); + + assert_eq!(r_vec_v, r_vec); + assert_eq!(final_claim_v, final_claim); +} diff --git a/tests/transcript.rs b/tests/transcript.rs new file mode 100644 index 00000000..50335bec --- /dev/null +++ b/tests/transcript.rs @@ -0,0 +1,136 @@ +#![allow(missing_docs)] + +use hachi_pcs::algebra::Fp64; +use hachi_pcs::protocol::transcript::labels; +use hachi_pcs::protocol::{Blake2bTranscript, KeccakTranscript, Transcript}; + +type F = Fp64<4294967197>; + +fn sample_schedule>(transcript: &mut T) -> F { + transcript.append_bytes(labels::ABSORB_COMMITMENT, b"commitment-a"); + transcript.append_bytes(labels::ABSORB_COMMITMENT, b"commitment-b"); + transcript.append_serde(labels::ABSORB_EVALUATION_CLAIMS, &42u64); + let rho = transcript.challenge_scalar(labels::CHALLENGE_LINEAR_RELATION); + + transcript.append_bytes(labels::ABSORB_RING_SWITCH_MESSAGE, b"ring-switch"); + let zeta = transcript.challenge_scalar(labels::CHALLENGE_RING_SWITCH); + + transcript.append_field(labels::ABSORB_SUMCHECK_ROUND, &(rho + zeta)); + let r = transcript.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND); + + transcript.append_field(labels::ABSORB_STOP_CONDITION, &r); + transcript.challenge_scalar(labels::CHALLENGE_STOP_CONDITION) +} + +#[test] +fn transcript_is_deterministic_for_identical_schedule() { + let mut t1 = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + let mut t2 = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + + let c1 = sample_schedule(&mut t1); + let c2 = sample_schedule(&mut t2); + assert_eq!(c1, c2); +} + +#[test] +fn transcript_differs_when_label_changes() { + let mut t1 = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + let mut t2 = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + + t1.append_bytes(labels::ABSORB_COMMITMENT, b"same-bytes"); + t2.append_bytes(labels::ABSORB_EVALUATION_CLAIMS, b"same-bytes"); + let c1 = t1.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND); + let c2 = t2.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND); + assert_ne!(c1, c2); +} + +#[test] +fn transcript_differs_when_absorb_order_changes() { + let mut t1 = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + let mut t2 = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + + t1.append_bytes(labels::ABSORB_COMMITMENT, b"A"); + t1.append_bytes(labels::ABSORB_EVALUATION_CLAIMS, b"B"); + + t2.append_bytes(labels::ABSORB_EVALUATION_CLAIMS, b"B"); + t2.append_bytes(labels::ABSORB_COMMITMENT, b"A"); + + let c1 = t1.challenge_scalar(labels::CHALLENGE_LINEAR_RELATION); + let c2 = t2.challenge_scalar(labels::CHALLENGE_LINEAR_RELATION); + assert_ne!(c1, c2); +} + +#[test] +fn transcript_reset_restores_domain_state() { + let mut t = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + t.append_bytes(labels::ABSORB_COMMITMENT, b"before-reset"); + let _ = t.challenge_scalar(labels::CHALLENGE_STOP_CONDITION); + + t.reset(labels::DOMAIN_HACHI_PROTOCOL); + let after_reset = sample_schedule(&mut t); + + let mut fresh = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + let fresh_challenge = sample_schedule(&mut fresh); + assert_eq!(after_reset, fresh_challenge); +} + +#[test] +fn keccak_transcript_is_deterministic_for_identical_schedule() { + let mut t1 = KeccakTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + let mut t2 = KeccakTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + + let c1 = sample_schedule(&mut t1); + let c2 = sample_schedule(&mut t2); + assert_eq!(c1, c2); +} + +#[test] +fn keccak_transcript_differs_when_label_changes() { + let mut t1 = KeccakTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + let mut t2 = KeccakTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + + t1.append_bytes(labels::ABSORB_COMMITMENT, b"same-bytes"); + t2.append_bytes(labels::ABSORB_EVALUATION_CLAIMS, b"same-bytes"); + let c1 = t1.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND); + let c2 = t2.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND); + assert_ne!(c1, c2); +} + +#[test] +fn keccak_transcript_differs_when_absorb_order_changes() { + let mut t1 = KeccakTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + let mut t2 = KeccakTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + + t1.append_bytes(labels::ABSORB_COMMITMENT, b"A"); + t1.append_bytes(labels::ABSORB_EVALUATION_CLAIMS, b"B"); + + t2.append_bytes(labels::ABSORB_EVALUATION_CLAIMS, b"B"); + t2.append_bytes(labels::ABSORB_COMMITMENT, b"A"); + + let c1 = t1.challenge_scalar(labels::CHALLENGE_LINEAR_RELATION); + let c2 = t2.challenge_scalar(labels::CHALLENGE_LINEAR_RELATION); + assert_ne!(c1, c2); +} + +#[test] +fn keccak_transcript_reset_restores_domain_state() { + let mut t = KeccakTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + t.append_bytes(labels::ABSORB_COMMITMENT, b"before-reset"); + let _ = t.challenge_scalar(labels::CHALLENGE_STOP_CONDITION); + + t.reset(labels::DOMAIN_HACHI_PROTOCOL); + let after_reset = sample_schedule(&mut t); + + let mut fresh = KeccakTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + let fresh_challenge = sample_schedule(&mut fresh); + assert_eq!(after_reset, fresh_challenge); +} + +#[test] +fn blake2b_and_keccak_diverge_on_same_schedule() { + let mut blake = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + let mut keccak = KeccakTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + let b = sample_schedule(&mut blake); + let k = sample_schedule(&mut keccak); + assert_ne!(b, k); +} From 4980e1ab92686bb7e89fe223ea5e4c4cf8e7a4ab Mon Sep 17 00:00:00 2001 From: Omid Bodaghi <42227752+omibo@users.noreply.github.com> Date: Fri, 27 Feb 2026 21:39:13 -0800 Subject: [PATCH 02/34] Implement Batched Sumcheck and Gruen EQ (#2) --- src/protocol/commitment_scheme.rs | 39 ++-- src/protocol/mod.rs | 1 + src/protocol/proof.rs | 9 +- src/protocol/ring_switch.rs | 4 +- src/protocol/sumcheck/batched_sumcheck.rs | 258 +++++++++++++++++++++ src/protocol/sumcheck/eq_poly.rs | 229 ++++++++++++++++++ src/protocol/sumcheck/mod.rs | 32 +-- src/protocol/sumcheck/norm_sumcheck.rs | 57 +++-- src/protocol/sumcheck/relation_sumcheck.rs | 9 +- src/protocol/sumcheck/split_eq.rs | 213 +++++++++++++++++ src/protocol/transcript/labels.rs | 3 + tests/hachi_sumcheck.rs | 7 +- 12 files changed, 773 insertions(+), 88 deletions(-) create mode 100644 src/protocol/sumcheck/batched_sumcheck.rs create mode 100644 src/protocol/sumcheck/eq_poly.rs create mode 100644 src/protocol/sumcheck/split_eq.rs diff --git a/src/protocol/commitment_scheme.rs b/src/protocol/commitment_scheme.rs index 6da64c79..23ecc795 100644 --- a/src/protocol/commitment_scheme.rs +++ b/src/protocol/commitment_scheme.rs @@ -11,11 +11,14 @@ use crate::protocol::opening_point::RingOpeningPoint; use crate::protocol::proof::{HachiCommitmentHint, HachiProof, SumcheckAux}; use crate::protocol::quadratic_equation::QuadraticEquation; use crate::protocol::ring_switch::{ring_switch_prover, ring_switch_verifier}; +use crate::protocol::sumcheck::batched_sumcheck::{ + prove_batched_sumcheck, verify_batched_sumcheck, +}; use crate::protocol::sumcheck::norm_sumcheck::{NormSumcheckProver, NormSumcheckVerifier}; use crate::protocol::sumcheck::relation_sumcheck::{ RelationSumcheckProver, RelationSumcheckVerifier, }; -use crate::protocol::sumcheck::{prove_sumcheck, verify_sumcheck}; +use crate::protocol::sumcheck::SumcheckInstanceProver; use crate::protocol::transcript::labels::CHALLENGE_SUMCHECK_ROUND; use crate::protocol::transcript::Transcript; use crate::{CanonicalField, FieldCore, FieldSampling, Polynomial}; @@ -118,14 +121,8 @@ where // §4.3 Ring switch let rs = ring_switch_prover::(&quad_eq, transcript)?; - // Norm sumcheck (range check on w) + // Batched sumcheck: norm + relation let mut norm_prover = NormSumcheckProver::new(&rs.tau0, rs.w_evals.clone(), rs.b); - let (norm_proof, ..) = - prove_sumcheck::(&mut norm_prover, transcript, |tr| { - tr.challenge_scalar(CHALLENGE_SUMCHECK_ROUND) - })?; - - // Relation sumcheck (evaluation relation) let mut relation_prover = RelationSumcheckProver::new( rs.w_evals, &rs.alpha_evals_y, @@ -133,16 +130,18 @@ where rs.num_u, rs.num_l, ); - let (relation_proof, ..) = - prove_sumcheck::(&mut relation_prover, transcript, |tr| { + + let instances: Vec<&mut dyn SumcheckInstanceProver> = + vec![&mut norm_prover, &mut relation_prover]; + let (sumcheck_proof, ..) = + prove_batched_sumcheck::(instances, transcript, |tr| { tr.challenge_scalar(CHALLENGE_SUMCHECK_ROUND) })?; Ok(HachiProof { v: quad_eq.v, y_ring, - norm_proof, - relation_proof, + sumcheck_proof, sumcheck_aux: SumcheckAux { w: rs.w }, w_commitment: rs.w_commitment, }) @@ -192,13 +191,8 @@ where transcript, )?; - // Norm sumcheck verification (range check) + // Batched sumcheck verification: norm (F_0) + relation (F_α) let norm_verifier = NormSumcheckVerifier::new(rs.tau0, rs.w_evals.clone(), rs.b); - verify_sumcheck::(&proof.norm_proof, &norm_verifier, transcript, |tr| { - tr.challenge_scalar(CHALLENGE_SUMCHECK_ROUND) - })?; - - // Relation sumcheck verification (evaluation relation) let relation_verifier = RelationSumcheckVerifier::new( rs.w_evals, rs.alpha_evals_y, @@ -211,9 +205,12 @@ where rs.num_u, rs.num_l, ); - verify_sumcheck::( - &proof.relation_proof, - &relation_verifier, + + let verifiers: Vec<&dyn crate::protocol::sumcheck::SumcheckInstanceVerifier> = + vec![&norm_verifier, &relation_verifier]; + verify_batched_sumcheck::( + &proof.sumcheck_proof, + verifiers, transcript, |tr| tr.challenge_scalar(CHALLENGE_SUMCHECK_ROUND), )?; diff --git a/src/protocol/mod.rs b/src/protocol/mod.rs index 49997845..6f4d5581 100644 --- a/src/protocol/mod.rs +++ b/src/protocol/mod.rs @@ -23,6 +23,7 @@ pub use commitment_scheme::HachiCommitmentScheme; pub use opening_point::RingOpeningPoint; pub use proof::{HachiProof, SumcheckAux}; pub use quadratic_equation::QuadraticEquation; +pub use sumcheck::batched_sumcheck::{prove_batched_sumcheck, verify_batched_sumcheck}; pub use sumcheck::{ prove_sumcheck, verify_sumcheck, CompressedUniPoly, SumcheckInstanceProver, SumcheckInstanceVerifier, SumcheckProof, UniPoly, diff --git a/src/protocol/proof.rs b/src/protocol/proof.rs index 9a8c07e3..4a119329 100644 --- a/src/protocol/proof.rs +++ b/src/protocol/proof.rs @@ -36,10 +36,8 @@ pub struct HachiProof { pub y_ring: CyclotomicRing, /// `v = D · ŵ`. pub v: Vec>, - /// Range-check (norm) sumcheck proof (§4.3, F_0). - pub norm_proof: SumcheckProof, - /// Evaluation-relation sumcheck proof (§4.3, F_α). - pub relation_proof: SumcheckProof, + /// Batched sumcheck proof (F_0 norm + F_α relation, §4.3). + pub sumcheck_proof: SumcheckProof, /// Temporary verifier auxiliary (will be removed with recursive PCS). pub sumcheck_aux: SumcheckAux, /// Commitment to the sumcheck witness `w`. @@ -52,8 +50,7 @@ impl HachiProof { self.v.serialized_size(Compress::No) + self.y_ring.serialized_size(Compress::No) + self.sumcheck_aux.w.serialized_size(Compress::No) - + self.norm_proof.serialized_size(Compress::No) - + self.relation_proof.serialized_size(Compress::No) + + self.sumcheck_proof.serialized_size(Compress::No) + self.w_commitment.serialized_size(Compress::No) } } diff --git a/src/protocol/ring_switch.rs b/src/protocol/ring_switch.rs index 9a9d19a7..cb4bc0f9 100644 --- a/src/protocol/ring_switch.rs +++ b/src/protocol/ring_switch.rs @@ -12,7 +12,7 @@ use crate::protocol::commitment::{ RingCommitmentScheme, }; use crate::protocol::quadratic_equation::QuadraticEquation; -use crate::protocol::sumcheck::eq_evals; +use crate::protocol::sumcheck::eq_poly::EqPolynomial; use crate::protocol::transcript::labels::{ ABSORB_SUMCHECK_W, CHALLENGE_RING_SWITCH, CHALLENGE_TAU0, CHALLENGE_TAU1, }; @@ -399,7 +399,7 @@ pub(crate) fn build_m_evals_x( cols: usize, tau1: &[F], ) -> Vec { - let eq_tau1 = eq_evals(tau1); + let eq_tau1 = EqPolynomial::evals(tau1); let x_len = cols.next_power_of_two(); let mut m_evals = vec![F::zero(); x_len]; for x in 0..x_len { diff --git a/src/protocol/sumcheck/batched_sumcheck.rs b/src/protocol/sumcheck/batched_sumcheck.rs new file mode 100644 index 00000000..15673987 --- /dev/null +++ b/src/protocol/sumcheck/batched_sumcheck.rs @@ -0,0 +1,258 @@ +//! Batched sumcheck protocol. +//! +//! Implements the standard technique for batching parallel sumchecks to reduce +//! verifier cost and proof size. +//! +//! For details, refer to Jim Posen's ["Perspectives on Sumcheck Batching"](https://hackmd.io/s/HyxaupAAA). +//! We do what they describe as "front-loaded" batch sumcheck. +//! +//! Adapted from Jolt's `BatchedSumcheck` implementation. + +use super::{SumcheckInstanceProver, SumcheckInstanceVerifier, SumcheckProof, UniPoly}; +use crate::error::HachiError; +use crate::protocol::transcript::labels; +use crate::protocol::transcript::Transcript; +use crate::{CanonicalField, FieldCore}; + +fn mul_pow_2(x: E, k: usize) -> E { + let mut result = x; + for _ in 0..k { + result = result + result; + } + result +} + +fn linear_combination(polys: &[UniPoly], coeffs: &[E]) -> UniPoly { + let max_len = polys.iter().map(|p| p.coeffs.len()).max().unwrap_or(0); + let mut result = vec![E::zero(); max_len]; + for (poly, coeff) in polys.iter().zip(coeffs.iter()) { + for (i, c) in poly.coeffs.iter().enumerate() { + result[i] = result[i] + *c * *coeff; + } + } + UniPoly::from_coeffs(result) +} + +/// Produce a batched sumcheck proof for multiple instances sharing the same +/// variable space, driving the Fiat–Shamir transcript. +/// +/// This function: +/// - absorbs each instance's initial claim, +/// - samples batching coefficients (one per instance), +/// - computes a single batched round polynomial per round as a linear +/// combination of the individual round polynomials, +/// - returns a single [`SumcheckProof`] and the derived challenge vector. +/// +/// Instances with fewer rounds than the maximum are padded with constant +/// "dummy" round polynomials (the Jolt "front-loaded" approach). +/// +/// # Panics +/// +/// Panics if `instances` is empty or if 2 is not invertible in the field. +/// +/// # Errors +/// +/// Returns an error if the field inverse of 2 does not exist. +pub fn prove_batched_sumcheck( + mut instances: Vec<&mut dyn SumcheckInstanceProver>, + transcript: &mut T, + mut sample_challenge: S, +) -> Result<(SumcheckProof, Vec), HachiError> +where + F: FieldCore + CanonicalField, + T: Transcript, + E: FieldCore + CanonicalField, + S: FnMut(&mut T) -> E, +{ + assert!(!instances.is_empty()); + + let max_num_rounds = instances + .iter() + .map(|inst| inst.num_rounds()) + .max() + .unwrap(); + + // Absorb individual input claims. + for inst in instances.iter() { + let claim = inst.input_claim(); + transcript.append_serde(labels::ABSORB_SUMCHECK_CLAIM, &claim); + } + + // Sample one batching coefficient per instance. + let batching_coeffs: Vec = (0..instances.len()) + .map(|_| sample_challenge(transcript)) + .collect(); + + // To see why we may need to scale by a power of two, consider a batch of + // two sumchecks: + // claim_a = \sum_x P(x) where x \in {0, 1}^M + // claim_b = \sum_{x, y} Q(x, y) where x \in {0, 1}^M, y \in {0, 1}^N + // Then the batched sumcheck is: + // \sum_{x, y} A * P(x) + B * Q(x, y) where A and B are batching coefficients + // = A * \sum_y \sum_x P(x) + B * \sum_{x, y} Q(x, y) + // = A * \sum_y claim_a + B * claim_b + // = A * 2^N * claim_a + B * claim_b + let mut individual_claims: Vec = instances + .iter() + .map(|inst| { + let n = inst.num_rounds(); + let claim = inst.input_claim(); + mul_pow_2(claim, max_num_rounds - n) + }) + .collect(); + + let two_inv = E::from_u64(2) + .inv() + .expect("2 must be invertible in the field"); + + let mut round_polys = Vec::with_capacity(max_num_rounds); + let mut challenges = Vec::with_capacity(max_num_rounds); + + for round in 0..max_num_rounds { + let univariate_polys: Vec> = instances + .iter_mut() + .zip(individual_claims.iter()) + .map(|(inst, previous_claim)| { + let n = inst.num_rounds(); + let offset = max_num_rounds - n; + let active = round >= offset && round < offset + n; + if active { + inst.compute_round_univariate(round - offset, *previous_claim) + } else { + // Variable is "dummy" for this instance: polynomial is independent of it, + // so the round univariate is constant with H(0)=H(1)=previous_claim/2. + UniPoly::from_coeffs(vec![*previous_claim * two_inv]) + } + }) + .collect(); + + let batched_poly = linear_combination(&univariate_polys, &batching_coeffs); + + #[cfg(debug_assertions)] + { + let g0 = batched_poly.evaluate(&E::zero()); + let g1 = batched_poly.evaluate(&E::one()); + let batched_claim: E = individual_claims + .iter() + .zip(batching_coeffs.iter()) + .map(|(c, b)| *c * *b) + .fold(E::zero(), |a, v| a + v); + debug_assert!( + g0 + g1 == batched_claim, + "round {round}: H(0) + H(1) != batched claim" + ); + } + + let compressed = batched_poly.compress(); + transcript.append_serde(labels::ABSORB_SUMCHECK_ROUND, &compressed); + let r_j = sample_challenge(transcript); + challenges.push(r_j); + + // Update individual claims from each instance's own univariate. + for (claim, poly) in individual_claims.iter_mut().zip(univariate_polys.iter()) { + *claim = poly.evaluate(&r_j); + } + + // Ingest challenge into each active instance. + for inst in instances.iter_mut() { + let n = inst.num_rounds(); + let offset = max_num_rounds - n; + let active = round >= offset && round < offset + n; + if active { + inst.ingest_challenge(round - offset, r_j); + } + } + + round_polys.push(compressed); + } + + for inst in instances.iter_mut() { + inst.finalize(); + } + + Ok((SumcheckProof { round_polys }, challenges)) +} + +/// Verify a batched sumcheck proof. +/// +/// This function: +/// - absorbs each verifier instance's initial claim, +/// - re-derives the batching coefficients, +/// - computes the batched initial claim, +/// - verifies the proof against the batched claim, +/// - checks that the final output matches the batched expected output claims. +/// +/// Returns the challenge vector on success. +/// +/// # Panics +/// +/// Panics if `verifiers` is empty. +/// +/// # Errors +/// +/// Returns [`HachiError::InvalidProof`] if the batched output claim does not +/// match the expected value, or propagates per-round verification errors. +pub fn verify_batched_sumcheck( + proof: &SumcheckProof, + verifiers: Vec<&dyn SumcheckInstanceVerifier>, + transcript: &mut T, + mut sample_challenge: S, +) -> Result, HachiError> +where + F: FieldCore + CanonicalField, + T: Transcript, + E: FieldCore + CanonicalField, + S: FnMut(&mut T) -> E, +{ + assert!(!verifiers.is_empty()); + + let max_degree = verifiers.iter().map(|v| v.degree_bound()).max().unwrap(); + let max_num_rounds = verifiers.iter().map(|v| v.num_rounds()).max().unwrap(); + + // Absorb individual input claims. + for v in verifiers.iter() { + let claim = v.input_claim(); + transcript.append_serde(labels::ABSORB_SUMCHECK_CLAIM, &claim); + } + + // Re-derive batching coefficients. + let batching_coeffs: Vec = (0..verifiers.len()) + .map(|_| sample_challenge(transcript)) + .collect(); + + // Compute the combined initial claim with power-of-two scaling. + let batched_claim: E = verifiers + .iter() + .zip(batching_coeffs.iter()) + .map(|(v, coeff)| { + let n = v.num_rounds(); + let claim = v.input_claim(); + mul_pow_2(claim, max_num_rounds - n) * *coeff + }) + .fold(E::zero(), |a, v| a + v); + + let (output_claim, r_sumcheck) = proof.verify::( + batched_claim, + max_num_rounds, + max_degree, + transcript, + &mut sample_challenge, + )?; + + // Compute the expected batched output claim from each verifier instance. + let expected_output_claim: E = verifiers + .iter() + .zip(batching_coeffs.iter()) + .map(|(v, coeff)| { + let offset = max_num_rounds - v.num_rounds(); + let r_slice = &r_sumcheck[offset..offset + v.num_rounds()]; + v.expected_output_claim(r_slice) * *coeff + }) + .fold(E::zero(), |a, v| a + v); + + if output_claim != expected_output_claim { + return Err(HachiError::InvalidProof); + } + + Ok(r_sumcheck) +} diff --git a/src/protocol/sumcheck/eq_poly.rs b/src/protocol/sumcheck/eq_poly.rs new file mode 100644 index 00000000..1b562cd0 --- /dev/null +++ b/src/protocol/sumcheck/eq_poly.rs @@ -0,0 +1,229 @@ +//! Utilities for the equality polynomial `eq(x, y) = Πᵢ (xᵢ yᵢ + (1 − xᵢ)(1 − yᵢ))`. +//! +//! The equality polynomial evaluates to 1 when `x = y` (over the boolean hypercube) +//! and 0 otherwise. Its multilinear extension (MLE) is used throughout sumcheck +//! protocols. +//! +//! Adapted from Jolt's `EqPolynomial` implementation. +//! +//! ## Bit / index order: Little-endian +//! +//! The evaluation tables produced by this module use **little-endian** bit order: +//! entry `b` (as an integer index) corresponds to the boolean vector where +//! bit `k` of `b` equals `x[k]`. In other words, `r[0]` corresponds to the +//! **least-significant bit** (bit 0) and `r[n-1]` to the MSB. + +use crate::FieldCore; +use std::marker::PhantomData; + +/// Utilities for the equality polynomial `eq(x, y) = Πᵢ (xᵢ yᵢ + (1 − xᵢ)(1 − yᵢ))`. +pub struct EqPolynomial(PhantomData); + +impl EqPolynomial { + /// Compute the MLE of the equality polynomial at two points: + /// `eq(x, y) = Πᵢ (xᵢ yᵢ + (1 − xᵢ)(1 − yᵢ))`. + /// + /// # Panics + /// + /// Panics if `x.len() != y.len()`. + pub fn mle(x: &[E], y: &[E]) -> E { + assert_eq!(x.len(), y.len()); + x.iter() + .zip(y.iter()) + .map(|(&x_i, &y_i)| x_i * y_i + (E::one() - x_i) * (E::one() - y_i)) + .fold(E::one(), |acc, v| acc * v) + } + + /// Compute the zero selector: `eq(r, 0) = Πᵢ (1 − rᵢ)`. + pub fn zero_selector(r: &[E]) -> E { + r.iter().fold(E::one(), |acc, &r_i| acc * (E::one() - r_i)) + } + + /// Compute the full evaluation table `{ eq(r, x) : x ∈ {0,1}^n }`. + /// + /// Uses **little-endian** bit order: entry `b` has bit `k` of `b` + /// corresponding to `r[k]`. + /// + /// For a scaled table, use [`Self::evals_with_scaling`]. + pub fn evals(r: &[E]) -> Vec { + Self::evals_with_scaling(r, None) + } + + /// Compute the full evaluation table with optional scaling: + /// `scaling_factor · eq(r, x)` for all `x ∈ {0,1}^n`. + /// + /// Uses the same **little-endian** index order as [`Self::evals`]. + /// If `scaling_factor` is `None`, defaults to 1 (no scaling). + pub fn evals_with_scaling(r: &[E], scaling_factor: Option) -> Vec { + #[cfg(feature = "parallel")] + { + const PARALLEL_THRESHOLD: usize = 16; + if r.len() > PARALLEL_THRESHOLD { + return Self::evals_parallel(r, scaling_factor); + } + } + Self::evals_serial(r, scaling_factor) + } + + /// Serial (single-threaded) version of [`Self::evals_with_scaling`]. + /// + /// Uses **little-endian** index order. + pub fn evals_serial(r: &[E], scaling_factor: Option) -> Vec { + let size = 1usize << r.len(); + let mut evals = vec![E::zero(); size]; + evals[0] = scaling_factor.unwrap_or(E::one()); + let mut len = 1usize; + for &t in r.iter().rev() { + let one_minus_t = E::one() - t; + for j in (0..len).rev() { + evals[2 * j + 1] = evals[j] * t; + evals[2 * j] = evals[j] * one_minus_t; + } + len *= 2; + } + evals + } + + /// Compute eq evaluations and cache intermediate tables. + /// + /// Returns `result` where `result[j]` contains evaluations for the prefix + /// `r[..j]`: `result[j][x] = eq(r[..j], x)` for `x ∈ {0,1}^j`. + /// + /// So `result[0] = [1]`, `result[1]` has 2 entries, ..., and `result[n]` + /// equals [`Self::evals(r)`]. + pub fn evals_cached(r: &[E]) -> Vec> { + Self::evals_cached_with_scaling(r, None) + } + + /// Like [`Self::evals_cached`], but with optional scaling. + pub fn evals_cached_with_scaling(r: &[E], scaling_factor: Option) -> Vec> { + let mut result: Vec> = (0..r.len() + 1).map(|i| vec![E::zero(); 1 << i]).collect(); + result[0][0] = scaling_factor.unwrap_or(E::one()); + for j in 0..r.len() { + let idx = r.len() - 1 - j; + let t = r[idx]; + let one_minus_t = E::one() - t; + let prev_len = 1 << j; + for i in (0..prev_len).rev() { + result[j + 1][2 * i + 1] = result[j][i] * t; + result[j + 1][2 * i] = result[j][i] * one_minus_t; + } + } + result + } + + /// Parallel version of [`Self::evals_with_scaling`]. + /// + /// Uses rayon to compute the largest layers of the DP tree in parallel. + /// Uses the same **little-endian** index order as [`Self::evals`]. + #[cfg(feature = "parallel")] + pub fn evals_parallel(r: &[E], scaling_factor: Option) -> Vec { + use rayon::prelude::*; + + let final_size = 1usize << r.len(); + let mut evals = vec![E::zero(); final_size]; + evals[0] = scaling_factor.unwrap_or(E::one()); + let mut size = 1; + + // Forward iteration (r[0] first) produces little-endian ordering. + for &r_i in r.iter() { + let (evals_left, evals_right) = evals.split_at_mut(size); + let (evals_right, _) = evals_right.split_at_mut(size); + + evals_left + .par_iter_mut() + .zip(evals_right.par_iter_mut()) + .for_each(|(x, y)| { + *y = *x * r_i; + *x = *x - *y; + }); + + size *= 2; + } + + evals + } +} + +#[cfg(test)] +mod tests { + use super::*; + use crate::algebra::Fp64; + use crate::{CanonicalField, FieldSampling}; + use rand::rngs::StdRng; + use rand::SeedableRng; + + type F = Fp64<4294967197>; + + #[test] + fn evals_matches_mle_pointwise() { + let mut rng = StdRng::seed_from_u64(0xEE); + for n in 1..8 { + let r: Vec = (0..n).map(|_| F::sample(&mut rng)).collect(); + let table = EqPolynomial::evals(&r); + assert_eq!(table.len(), 1 << n); + for (idx, &val) in table.iter().enumerate() { + let bits: Vec = (0..n) + .map(|k| { + if (idx >> k) & 1 == 1 { + F::one() + } else { + F::zero() + } + }) + .collect(); + let expected = EqPolynomial::mle(&r, &bits); + assert_eq!(val, expected, "n={n} idx={idx}"); + } + } + } + + #[test] + fn evals_with_scaling_scales_uniformly() { + let mut rng = StdRng::seed_from_u64(0xAB); + let r: Vec = (0..5).map(|_| F::sample(&mut rng)).collect(); + let scale = F::from_u64(7); + let unscaled = EqPolynomial::evals(&r); + let scaled = EqPolynomial::evals_with_scaling(&r, Some(scale)); + for (u, s) in unscaled.iter().zip(scaled.iter()) { + assert_eq!(*s, *u * scale); + } + } + + #[test] + fn evals_cached_last_matches_evals() { + let mut rng = StdRng::seed_from_u64(0xCD); + for n in 1..8 { + let r: Vec = (0..n).map(|_| F::sample(&mut rng)).collect(); + let table = EqPolynomial::evals(&r); + let cached = EqPolynomial::evals_cached(&r); + assert_eq!(cached.len(), n + 1); + assert_eq!(cached[0], vec![F::one()]); + assert_eq!(*cached.last().unwrap(), table); + } + } + + #[test] + fn zero_selector_matches_mle_at_origin() { + let mut rng = StdRng::seed_from_u64(0x00); + for n in 1..8 { + let r: Vec = (0..n).map(|_| F::sample(&mut rng)).collect(); + let zeros = vec![F::zero(); n]; + let expected = EqPolynomial::mle(&r, &zeros); + let actual = EqPolynomial::zero_selector(&r); + assert_eq!(actual, expected, "n={n}"); + } + } + + #[cfg(feature = "parallel")] + #[test] + fn evals_parallel_matches_serial() { + let mut rng = StdRng::seed_from_u64(0xFF); + for n in 1..20 { + let r: Vec = (0..n).map(|_| F::sample(&mut rng)).collect(); + let serial = EqPolynomial::evals_serial(&r, None); + let parallel = EqPolynomial::evals_parallel(&r, None); + assert_eq!(serial, parallel, "n={n}"); + } + } +} diff --git a/src/protocol/sumcheck/mod.rs b/src/protocol/sumcheck/mod.rs index 2bba5ca0..9788b60e 100644 --- a/src/protocol/sumcheck/mod.rs +++ b/src/protocol/sumcheck/mod.rs @@ -19,8 +19,11 @@ //! duplicates the essential sumcheck data types and transcript-driving logic as a //! pragmatic workaround. +pub mod batched_sumcheck; +pub mod eq_poly; pub mod norm_sumcheck; pub mod relation_sumcheck; +pub mod split_eq; use crate::error::HachiError; use crate::primitives::serialization::{ @@ -572,35 +575,6 @@ where Ok(challenges) } -/// Build the full eq polynomial evaluation table. -/// -/// Returns a vector of size `2^{tau.len()}` where entry `b` (interpreted as a -/// little-endian bit string, i.e. bit `k` of `b` corresponds to `τ[k]`) equals -/// `ẽq(τ, b) = Π_i (τ_i·b_i + (1−τ_i)(1−b_i))`. -pub fn eq_evals(tau: &[E]) -> Vec { - let size = 1usize << tau.len(); - let mut evals = vec![E::zero(); size]; - evals[0] = E::one(); - let mut len = 1usize; - for &t in tau.iter().rev() { - let one_minus_t = E::one() - t; - for j in (0..len).rev() { - evals[2 * j + 1] = evals[j] * t; - evals[2 * j] = evals[j] * one_minus_t; - } - len *= 2; - } - evals -} - -/// Evaluate ẽq(τ, r) at a single point. -pub fn eq_eval(tau: &[E], point: &[E]) -> E { - debug_assert_eq!(tau.len(), point.len()); - tau.iter().zip(point).fold(E::one(), |acc, (&t, &r)| { - acc * (t * r + (E::one() - t) * (E::one() - r)) - }) -} - /// Evaluate the range-check polynomial `w · Π_{k=1}^{b−1} (w − k)(w + k)`. /// /// This polynomial vanishes exactly when `w ∈ {−(b−1), …, b−1}`. diff --git a/src/protocol/sumcheck/norm_sumcheck.rs b/src/protocol/sumcheck/norm_sumcheck.rs index e1b72490..3fd5aa83 100644 --- a/src/protocol/sumcheck/norm_sumcheck.rs +++ b/src/protocol/sumcheck/norm_sumcheck.rs @@ -5,16 +5,21 @@ //! Proves that all entries of w̃ lie in {−(b−1), …, b−1}; the sum over the //! boolean hypercube should equal zero when the range constraint holds. -use super::{eq_eval, eq_evals, fold_evals, multilinear_eval, range_check_eval}; +use super::eq_poly::EqPolynomial; +use super::split_eq::GruenSplitEq; +use super::{fold_evals, multilinear_eval, range_check_eval}; use super::{SumcheckInstanceProver, SumcheckInstanceVerifier, UniPoly}; use crate::{CanonicalField, FieldCore}; /// Prover for `F_{0,τ₀}(x,y) = ẽq(τ₀,(x,y)) · w̃(x,y) · range_check(w̃(x,y), b)`. /// -/// Stores `eq` and `w` evaluation tables separately so the composite can be -/// evaluated at the `2b + 1` points needed per round (degree `2b`). -pub struct NormSumcheckProver { - eq_table: Vec, +/// Uses the Gruen/Dao-Thaler optimization: the eq polynomial is factored into +/// a running scalar and split tables instead of being stored as a full table +/// and folded each round. The round polynomial is computed as `l(X) · q(X)` +/// where `l(X)` is the linear eq factor and `q(X)` is the inner sum without +/// the current-variable eq contribution. +pub struct NormSumcheckProver { + split_eq: GruenSplitEq, w_table: Vec, num_vars: usize, b: usize, @@ -30,7 +35,7 @@ impl NormSumcheckProver { let num_vars = tau.len(); assert_eq!(w_evals.len(), 1 << num_vars); Self { - eq_table: eq_evals(tau), + split_eq: GruenSplitEq::new(tau), w_table: w_evals, num_vars, b, @@ -52,30 +57,36 @@ impl SumcheckInstanceProver for NormSumcheckPr } fn compute_round_univariate(&mut self, _round: usize, _previous_claim: E) -> UniPoly { - let half = self.eq_table.len() / 2; - let degree = 2 * self.b; - let num_points = degree + 1; - let mut round_evals = vec![E::zero(); num_points]; + let half = self.w_table.len() / 2; + let degree_q = 2 * self.b - 1; + let num_points_q = degree_q + 1; + let mut q_evals = vec![E::zero(); num_points_q]; + + let (e_first, e_second) = self.split_eq.remaining_eq_tables(); + let num_first = e_first.len(); + let first_bits = num_first.trailing_zeros(); for j in 0..half { - let eq_0 = self.eq_table[2 * j]; - let eq_1 = self.eq_table[2 * j + 1]; + let j_low = j & (num_first - 1); + let j_high = j >> first_bits; + let eq_rem = e_first[j_low] * e_second[j_high]; + let w_0 = self.w_table[2 * j]; let w_1 = self.w_table[2 * j + 1]; - for (t, eval) in round_evals.iter_mut().enumerate() { + for (t, eval) in q_evals.iter_mut().enumerate() { let t_e = E::from_u64(t as u64); - let eq_t = eq_0 + t_e * (eq_1 - eq_0); let w_t = w_0 + t_e * (w_1 - w_0); - *eval = *eval + eq_t * range_check_eval(w_t, self.b); + *eval = *eval + eq_rem * range_check_eval(w_t, self.b); } } - UniPoly::from_evals(&round_evals) + let q_poly = UniPoly::from_evals(&q_evals); + self.split_eq.gruen_mul(&q_poly) } fn ingest_challenge(&mut self, _round: usize, r: E) { - self.eq_table = fold_evals(&self.eq_table, r); + self.split_eq.bind(r); self.w_table = fold_evals(&self.w_table, r); } } @@ -120,7 +131,7 @@ impl SumcheckInstanceVerifier for NormSumcheck } fn expected_output_claim(&self, challenges: &[E]) -> E { - let eq_val = eq_eval(&self.tau, challenges); + let eq_val = EqPolynomial::mle(&self.tau, challenges); let w_val = multilinear_eval(&self.w_evals, challenges); eq_val * range_check_eval(w_val, self.b) } @@ -133,7 +144,7 @@ mod tests { use crate::algebra::Fp64; use crate::primitives::multilinear_evals::DenseMultilinearEvals; use crate::protocol::ring_switch::build_w_coeffs; - use crate::protocol::sumcheck::eq_evals; + use crate::protocol::sumcheck::eq_poly::EqPolynomial; use crate::protocol::transcript::labels; use crate::protocol::{ prove_sumcheck, verify_sumcheck, Blake2bTranscript, CommitmentConfig, CommitmentScheme, @@ -165,7 +176,7 @@ mod tests { let tau: Vec = (0..num_vars).map(|i| F::from_u64((i + 2) as u64)).collect(); let b = 1usize << DefaultCommitmentConfig::LOG_BASIS; - let eq_table = eq_evals(&tau); + let eq_table = EqPolynomial::evals(&tau); let _claim: F = (0..w_evals.len()) .map(|i| eq_table[i] * range_check_eval(w_evals[i], b)) .fold(F::zero(), |a, v| a + v); @@ -178,7 +189,7 @@ mod tests { }) .unwrap(); - let oracle = eq_eval(&tau, &prover_challenges) + let oracle = EqPolynomial::mle(&tau, &prover_challenges) * range_check_eval(multilinear_eval(&w_evals, &prover_challenges), b); assert_eq!(final_claim, oracle, "prover final claim != oracle eval"); @@ -225,7 +236,7 @@ mod tests { .collect(); let b = 1usize << DefaultCommitmentConfig::LOG_BASIS; - let eq_table = eq_evals(&tau); + let eq_table = EqPolynomial::evals(&tau); let _claim: F = (0..w_evals.len()) .map(|i| eq_table[i] * range_check_eval(w_evals[i], b)) .fold(F::zero(), |a, v| a + v); @@ -238,7 +249,7 @@ mod tests { }) .unwrap(); - let oracle = eq_eval(&tau, &prover_challenges) + let oracle = EqPolynomial::mle(&tau, &prover_challenges) * range_check_eval(multilinear_eval(&w_evals, &prover_challenges), b); assert_eq!(final_claim, oracle, "prover final claim != oracle eval"); diff --git a/src/protocol/sumcheck/relation_sumcheck.rs b/src/protocol/sumcheck/relation_sumcheck.rs index 12de5a7f..44e3c3c2 100644 --- a/src/protocol/sumcheck/relation_sumcheck.rs +++ b/src/protocol/sumcheck/relation_sumcheck.rs @@ -5,7 +5,8 @@ //! //! Proves the evaluation relation; sum equals `a = Σ_i ẽq(τ₁,i) · y_i(α)`. -use super::{eq_evals, fold_evals, multilinear_eval}; +use super::eq_poly::EqPolynomial; +use super::{fold_evals, multilinear_eval}; use super::{SumcheckInstanceProver, SumcheckInstanceVerifier, UniPoly}; use crate::algebra::ring::CyclotomicRing; use crate::protocol::ring_switch::eval_ring_at; @@ -184,7 +185,7 @@ impl SumcheckInstanceVerifier .map(|r| eval_ring_at(r, &self.alpha)) .collect(); - let eq_tau = eq_evals(&self.tau); + let eq_tau = EqPolynomial::evals(&self.tau); let mut acc = F::zero(); for (i, eq_i) in eq_tau.iter().enumerate() { let y_i = if i < y_a.len() { y_a[i] } else { F::zero() }; @@ -208,7 +209,7 @@ mod tests { use crate::algebra::Fp64; use crate::primitives::multilinear_evals::DenseMultilinearEvals; use crate::protocol::commitment_scheme::rederive_alpha_and_m_a; - use crate::protocol::sumcheck::eq_evals; + use crate::protocol::sumcheck::eq_poly::EqPolynomial; use crate::protocol::transcript::labels; use crate::protocol::{ prove_sumcheck, verify_sumcheck, Blake2bTranscript, CommitmentConfig, CommitmentScheme, @@ -276,7 +277,7 @@ mod tests { let num_i = rows.next_power_of_two().trailing_zeros() as usize; let tau1: Vec = (0..num_i).map(|i| F::from_u64((i + 5) as u64)).collect(); - let eq_tau1 = eq_evals(&tau1); + let eq_tau1 = EqPolynomial::evals(&tau1); let mut m_evals_x = vec![F::zero(); x_len]; for x in 0..x_len { diff --git a/src/protocol/sumcheck/split_eq.rs b/src/protocol/sumcheck/split_eq.rs new file mode 100644 index 00000000..1f10dbfe --- /dev/null +++ b/src/protocol/sumcheck/split_eq.rs @@ -0,0 +1,213 @@ +//! Gruen/Dao-Thaler split equality polynomial for efficient sumcheck. +//! +//! Factors `eq(τ, x)` into a running scalar, a linear factor for the current +//! variable, and precomputed split tables for the remaining variables. This +//! avoids maintaining and folding a full-size eq table during sumcheck. +//! +//! For details, see . +//! +//! Adapted from Jolt's `GruenSplitEqPolynomial`. +//! +//! ## Variable Layout (forward binding, little-endian) +//! +//! ```text +//! τ = [τ_current, τ_first_half, τ_second_half] +//! 1 var m vars (n-1-m) vars +//! ``` +//! +//! where `m = (n-1) / 2` and `n = τ.len()`. +//! +//! After binding `τ_current`, the next variable comes from `τ_first_half`, +//! then from `τ_second_half`. Suffix-cached eq tables for each half enable +//! O(1) pops per round instead of an O(2^n) fold. + +use super::eq_poly::EqPolynomial; +use super::UniPoly; +use crate::{CanonicalField, FieldCore}; + +/// Split equality polynomial with Gruen scalar accumulation. +/// +/// Instead of storing and folding a full eq table each round, this struct +/// maintains: +/// - `current_scalar`: accumulated `eq(τ_bound, r_bound)` from already-bound +/// variables +/// - `E_first` / `E_second`: suffix-cached eq tables for two halves of the +/// remaining (unbound, non-current) variables +/// +/// The eq contribution for a pair index `j` in the inner sum is: +/// ```text +/// eq_remaining(j) = E_first[j_low] · E_second[j_high] +/// ``` +/// and the full round polynomial is `l(X) · q(X)` where `l(X)` is the linear +/// eq factor for the current variable. +#[allow(non_snake_case)] +pub struct GruenSplitEq { + tau: Vec, + current_round: usize, + current_scalar: E, + /// Suffix-cached eq tables for the first half of remaining variables. + /// `E_first[k]` = `eq(τ[split-k..split], ·)` with `2^k` entries. + /// Invariant: never empty; `E_first[0] = [1]`. + E_first: Vec>, + /// Suffix-cached eq tables for the second half of remaining variables. + /// `E_second[k]` = `eq(τ[n-k..n], ·)` with `2^k` entries. + /// Invariant: never empty; `E_second[0] = [1]`. + E_second: Vec>, +} + +#[allow(non_snake_case)] +impl GruenSplitEq { + /// Create a new split-eq from the full challenge vector `τ`. + /// + /// Precomputes suffix-cached eq tables for two halves of `τ[1..n]`. + /// + /// # Panics + /// + /// Panics if `tau` is empty. + pub fn new(tau: &[E]) -> Self { + let n = tau.len(); + assert!(n >= 1); + let m = (n - 1) / 2; + let split = 1 + m; + let first_half = &tau[1..split]; + let second_half = &tau[split..n]; + let E_first = EqPolynomial::evals_cached(first_half); + let E_second = EqPolynomial::evals_cached(second_half); + Self { + tau: tau.to_vec(), + current_round: 0, + current_scalar: E::one(), + E_first, + E_second, + } + } + + /// The accumulated scalar `Π_{k < current_round} eq(τ[k], r[k])`. + pub fn current_scalar(&self) -> E { + self.current_scalar + } + + /// The τ value for the variable about to be bound. + pub fn current_tau(&self) -> E { + self.tau[self.current_round] + } + + /// Return the current top-level split-eq tables `(E_first, E_second)`. + /// + /// For a pair index `j` in the inner sum, the eq factor for the + /// remaining (non-current) variables is: + /// ```text + /// eq_remaining(j) = E_first[j & (E_first.len()-1)] + /// · E_second[j >> E_first.len().trailing_zeros()] + /// ``` + /// + /// # Panics + /// + /// Panics if either `E_first` or `E_second` is empty (invariant violation). + pub fn remaining_eq_tables(&self) -> (&[E], &[E]) { + ( + self.E_first.last().expect("E_first is never empty"), + self.E_second.last().expect("E_second is never empty"), + ) + } + + /// Bind the current variable to challenge `r`, advancing to the next round. + /// + /// Updates `current_scalar` with `eq(τ[current_round], r)` and pops the + /// appropriate split table level. + pub fn bind(&mut self, r: E) { + let tau_k = self.tau[self.current_round]; + self.current_scalar = + self.current_scalar * (tau_k * r + (E::one() - tau_k) * (E::one() - r)); + self.current_round += 1; + if self.E_first.len() > 1 { + self.E_first.pop(); + } else if self.E_second.len() > 1 { + self.E_second.pop(); + } + } + + /// Compute the round polynomial `s(X) = l(X) · q(X)` from the inner + /// polynomial `q` (given as evaluations at integer points `0, 1, ..., d`). + /// + /// `l(X) = current_scalar · eq(τ_current, X)` is the linear eq factor + /// for the current variable. The result has degree `d + 1`. + pub fn gruen_mul(&self, q_poly: &UniPoly) -> UniPoly { + let tau_k = self.current_tau(); + let scalar = self.current_scalar(); + let l_0 = scalar * (E::one() - tau_k); + let l_1 = scalar * tau_k; + let slope = l_1 - l_0; + let mut coeffs = vec![E::zero(); q_poly.coeffs.len() + 1]; + for (i, &c) in q_poly.coeffs.iter().enumerate() { + coeffs[i] = coeffs[i] + c * l_0; + coeffs[i + 1] = coeffs[i + 1] + c * slope; + } + UniPoly::from_coeffs(coeffs) + } +} + +#[cfg(test)] +mod tests { + use super::*; + use crate::algebra::Fp64; + use crate::protocol::sumcheck::fold_evals; + use crate::FieldSampling; + use rand::rngs::StdRng; + use rand::SeedableRng; + + type F = Fp64<4294967197>; + + #[test] + fn gruen_eq_matches_full_eq_table() { + let mut rng = StdRng::seed_from_u64(0xBB); + for n in 1..10 { + let tau: Vec = (0..n).map(|_| F::sample(&mut rng)).collect(); + let mut full_eq = EqPolynomial::evals(&tau); + let mut split_eq = GruenSplitEq::new(&tau); + + for _round in 0..n { + let half = full_eq.len() / 2; + let (e_first, e_second) = split_eq.remaining_eq_tables(); + let num_first = e_first.len(); + + for j in 0..half { + let j_low = j & (num_first - 1); + let j_high = j >> num_first.trailing_zeros(); + let eq_rem = e_first[j_low] * e_second[j_high]; + + let tau_k = split_eq.current_tau(); + let scalar = split_eq.current_scalar(); + let eq_0 = scalar * (F::one() - tau_k) * eq_rem; + let eq_1 = scalar * tau_k * eq_rem; + + assert_eq!(eq_0, full_eq[2 * j], "n={n} round={_round} j={j} eq_0"); + assert_eq!(eq_1, full_eq[2 * j + 1], "n={n} round={_round} j={j} eq_1"); + } + + let r = F::sample(&mut rng); + full_eq = fold_evals(&full_eq, r); + split_eq.bind(r); + } + } + } + + #[test] + fn gruen_mul_matches_direct_product() { + let mut rng = StdRng::seed_from_u64(0xCC); + let tau: Vec = (0..5).map(|_| F::sample(&mut rng)).collect(); + let split_eq = GruenSplitEq::new(&tau); + + let q = UniPoly::from_coeffs(vec![F::from_u64(3), F::from_u64(7), F::from_u64(2)]); + let s = split_eq.gruen_mul(&q); + + let tau_k = split_eq.current_tau(); + let scalar = split_eq.current_scalar(); + for t in 0..10u64 { + let x = F::from_u64(t); + let l_x = scalar * (tau_k * x + (F::one() - tau_k) * (F::one() - x)); + let q_x = q.evaluate(&x); + assert_eq!(s.evaluate(&x), l_x * q_x, "t={t}"); + } + } +} diff --git a/src/protocol/transcript/labels.rs b/src/protocol/transcript/labels.rs index f5d93952..0cfa6d4a 100644 --- a/src/protocol/transcript/labels.rs +++ b/src/protocol/transcript/labels.rs @@ -27,6 +27,8 @@ pub const ABSORB_SUMCHECK_CLAIM: &[u8] = b"hachi/absorb/sumcheck-claim"; pub const ABSORB_SUMCHECK_ROUND: &[u8] = b"hachi/absorb/sumcheck-round"; /// Challenge sampled per sumcheck round (paper §4.3). pub const CHALLENGE_SUMCHECK_ROUND: &[u8] = b"hachi/challenge/sumcheck-round"; +/// Challenge for batched sumcheck coefficient sampling. +pub const CHALLENGE_SUMCHECK_BATCH: &[u8] = b"hachi/challenge/sumcheck-batch"; /// Absorb recursion/stop-condition message payloads (paper §4.5). pub const ABSORB_STOP_CONDITION: &[u8] = b"hachi/absorb/stop-condition"; /// Challenge sampled for recursion stop-condition checks (paper §4.5). @@ -58,6 +60,7 @@ pub fn all_labels() -> &'static [&'static [u8]] { ABSORB_SUMCHECK_CLAIM, ABSORB_SUMCHECK_ROUND, CHALLENGE_SUMCHECK_ROUND, + CHALLENGE_SUMCHECK_BATCH, ABSORB_STOP_CONDITION, CHALLENGE_STOP_CONDITION, ABSORB_PROVER_V, diff --git a/tests/hachi_sumcheck.rs b/tests/hachi_sumcheck.rs index 69f16bf4..86b843f0 100644 --- a/tests/hachi_sumcheck.rs +++ b/tests/hachi_sumcheck.rs @@ -2,11 +2,12 @@ use hachi_pcs::algebra::ring::CyclotomicRing; use hachi_pcs::algebra::Fp64; +use hachi_pcs::protocol::sumcheck::eq_poly::EqPolynomial; use hachi_pcs::protocol::sumcheck::norm_sumcheck::{NormSumcheckProver, NormSumcheckVerifier}; use hachi_pcs::protocol::sumcheck::relation_sumcheck::{ RelationSumcheckProver, RelationSumcheckVerifier, }; -use hachi_pcs::protocol::sumcheck::{eq_eval, eq_evals, multilinear_eval, range_check_eval}; +use hachi_pcs::protocol::sumcheck::{multilinear_eval, range_check_eval}; use hachi_pcs::protocol::transcript::labels; use hachi_pcs::protocol::{prove_sumcheck, verify_sumcheck, Blake2bTranscript, Transcript}; use hachi_pcs::{CanonicalField, FieldCore, FieldSampling}; @@ -36,7 +37,7 @@ fn run_f0_e2e(num_u: usize, num_l: usize, b: usize) { let prove_time = t0.elapsed(); // Sanity: prover's final claim matches oracle evaluation. - let oracle = eq_eval(&tau0, &prover_challenges) + let oracle = EqPolynomial::mle(&tau0, &prover_challenges) * range_check_eval(multilinear_eval(&w_evals, &prover_challenges), b); assert_eq!(final_claim, oracle, "prover final claim != oracle eval"); @@ -91,7 +92,7 @@ fn run_f_alpha_e2e(num_u: usize, num_i: usize) { let tau1: Vec = (0..num_i).map(|_| F::sample(&mut rng)).collect(); // Compute m(x) = Σ_i ẽq(τ₁, i) · M̃_α(i, x) - let eq_tau1 = eq_evals(&tau1); + let eq_tau1 = EqPolynomial::evals(&tau1); let num_x = 1usize << num_u; let m_evals_x: Vec = (0..num_x) .map(|x_idx| { From 10e53dd19877143917def886dfa229e47e325612 Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Fri, 27 Feb 2026 22:02:49 -0800 Subject: [PATCH 03/34] Add rayon parallelism behind `parallel` feature flag (enabled by default) - New src/parallel.rs with cfg_iter!/cfg_into_iter!/cfg_chunks! macros that dispatch to rayon parallel iterators when `parallel` is enabled - Parallelize protocol hot paths: ring polynomial division, w_evals construction, M_alpha evaluation, ring vector evaluation, packed ring poly evaluation, coefficients-to-ring reduction, quadratic equation folding, and sumcheck round polynomial computation - All 174 tests pass with and without the parallel feature Made-with: Cursor --- Cargo.toml | 2 +- src/lib.rs | 5 + src/parallel.rs | 56 ++++++++++ src/protocol/commitment_scheme.rs | 46 ++++++--- src/protocol/quadratic_equation.rs | 22 ++-- src/protocol/ring_switch.rs | 113 +++++++++++++-------- src/protocol/sumcheck/norm_sumcheck.rs | 65 +++++++++--- src/protocol/sumcheck/relation_sumcheck.rs | 106 ++++++++++++++----- 8 files changed, 312 insertions(+), 103 deletions(-) create mode 100644 src/parallel.rs diff --git a/Cargo.toml b/Cargo.toml index d6e45663..7de0b328 100644 --- a/Cargo.toml +++ b/Cargo.toml @@ -32,7 +32,7 @@ include = [ all-features = true [features] -default = [] +default = ["parallel"] parallel = ["dep:rayon"] [dependencies] diff --git a/src/lib.rs b/src/lib.rs index eb5ac81c..eb936ec1 100644 --- a/src/lib.rs +++ b/src/lib.rs @@ -45,6 +45,11 @@ pub mod algebra; /// Protocol-layer transcript and commitment abstractions pub mod protocol; +/// Conditional parallelism utilities (`cfg_iter!`, `cfg_into_iter!`, etc.) +#[doc(hidden)] +#[allow(missing_docs)] +pub mod parallel; + #[doc(hidden)] #[allow(missing_docs)] pub mod test_utils; diff --git a/src/parallel.rs b/src/parallel.rs new file mode 100644 index 00000000..d73c9381 --- /dev/null +++ b/src/parallel.rs @@ -0,0 +1,56 @@ +//! Conditional parallelism utilities. +//! +//! When the `parallel` feature is enabled, the `cfg_iter!` family of macros +//! expand to rayon's parallel iterators. Otherwise they fall back to standard +//! sequential iterators. + +#[cfg(feature = "parallel")] +pub use rayon::prelude::*; + +/// Returns `.par_iter()` when `parallel` is enabled, `.iter()` otherwise. +#[macro_export] +macro_rules! cfg_iter { + ($e:expr) => {{ + #[cfg(feature = "parallel")] + let it = $e.par_iter(); + #[cfg(not(feature = "parallel"))] + let it = $e.iter(); + it + }}; +} + +/// Returns `.par_iter_mut()` when `parallel` is enabled, `.iter_mut()` otherwise. +#[macro_export] +macro_rules! cfg_iter_mut { + ($e:expr) => {{ + #[cfg(feature = "parallel")] + let it = $e.par_iter_mut(); + #[cfg(not(feature = "parallel"))] + let it = $e.iter_mut(); + it + }}; +} + +/// Returns `.into_par_iter()` when `parallel` is enabled, `.into_iter()` otherwise. +#[macro_export] +macro_rules! cfg_into_iter { + ($e:expr) => {{ + #[cfg(feature = "parallel")] + let it = $e.into_par_iter(); + #[cfg(not(feature = "parallel"))] + let it = $e.into_iter(); + it + }}; +} + +/// Returns `.par_chunks(n)` when `parallel` is enabled, `.chunks(n)` otherwise. +#[macro_export] +macro_rules! cfg_chunks { + ($e:expr, $n:expr) => {{ + #[cfg(feature = "parallel")] + let it = $e.par_chunks($n); + #[cfg(not(feature = "parallel"))] + let it = $e.chunks($n); + it + }}; +} diff --git a/src/protocol/commitment_scheme.rs b/src/protocol/commitment_scheme.rs index 23ecc795..4655fc9b 100644 --- a/src/protocol/commitment_scheme.rs +++ b/src/protocol/commitment_scheme.rs @@ -1,7 +1,10 @@ //! Commitment scheme trait implementation. use crate::algebra::ring::CyclotomicRing; +use crate::cfg_into_iter; use crate::error::HachiError; +#[cfg(feature = "parallel")] +use crate::parallel::*; use crate::primitives::poly::multilinear_lagrange_basis; use crate::protocol::commitment::{ CommitmentConfig, CommitmentScheme, DefaultCommitmentConfig, HachiCommitmentCore, @@ -325,14 +328,15 @@ fn reduce_coeffs_to_ring_elements( .checked_shl(outer_vars as u32) .ok_or_else(|| HachiError::InvalidInput(format!("2^{outer_vars} does not fit usize")))?; - let mut out = Vec::with_capacity(outer_len); - for i in 0..outer_len { - let coeffs = std::array::from_fn(|j| { - let idx = i + (j << outer_vars); - coeffs[idx] - }); - out.push(CyclotomicRing::from_coefficients(coeffs)); - } + let out: Vec> = cfg_into_iter!(0..outer_len) + .map(|i| { + let ring_coeffs = std::array::from_fn(|j| { + let idx = i + (j << outer_vars); + coeffs[idx] + }); + CyclotomicRing::from_coefficients(ring_coeffs) + }) + .collect(); Ok(out) } @@ -356,12 +360,26 @@ fn evaluate_packed_ring_poly( ) -> CyclotomicRing { let weights = lagrange_weights(outer_point); debug_assert_eq!(weights.len(), packed_coeffs.len()); - packed_coeffs - .iter() - .zip(weights.iter()) - .fold(CyclotomicRing::::zero(), |acc, (f_i, w_i)| { - acc + f_i.scale(w_i) - }) + #[cfg(feature = "parallel")] + { + packed_coeffs + .par_iter() + .zip(weights.par_iter()) + .fold( + || CyclotomicRing::::zero(), + |acc, (f_i, w_i)| acc + f_i.scale(w_i), + ) + .reduce(|| CyclotomicRing::::zero(), |a, b| a + b) + } + #[cfg(not(feature = "parallel"))] + { + packed_coeffs + .iter() + .zip(weights.iter()) + .fold(CyclotomicRing::::zero(), |acc, (f_i, w_i)| { + acc + f_i.scale(w_i) + }) + } } fn trace(u: &CyclotomicRing) -> F { diff --git a/src/protocol/quadratic_equation.rs b/src/protocol/quadratic_equation.rs index 16114ffa..71f415da 100644 --- a/src/protocol/quadratic_equation.rs +++ b/src/protocol/quadratic_equation.rs @@ -5,6 +5,8 @@ use crate::algebra::ring::{CyclotomicRing, SparseChallenge, SparseChallengeConfig}; use crate::error::HachiError; +#[cfg(feature = "parallel")] +use crate::parallel::*; use crate::protocol::challenges::sparse::sample_sparse_challenges; use crate::protocol::commitment::utils::linear::mat_vec_mul_unchecked; use crate::protocol::commitment::utils::norm::{detect_field_modulus, vec_inf_norm}; @@ -13,6 +15,7 @@ use crate::protocol::opening_point::RingOpeningPoint; use crate::protocol::proof::HachiCommitmentHint; use crate::protocol::transcript::labels::{ABSORB_PROVER_V, CHALLENGE_STAGE1_FOLD}; use crate::protocol::transcript::Transcript; +use crate::{cfg_into_iter, cfg_iter}; use crate::{CanonicalField, FieldCore}; /// **Steps 1–3.** Compute `w_i = a^T G_{2^m} s_i` and decompose: `ŵ_i = G_1^{-1}(w_i)`. @@ -31,7 +34,7 @@ where debug_assert_eq!(a.len(), block_len); - s.iter() + cfg_iter!(s) .map(|s_i| { let mut w_i = CyclotomicRing::::zero(); for (j, a_j) in a.iter().enumerate().take(block_len) { @@ -67,12 +70,17 @@ where { debug_assert_eq!(challenges.len(), s.len()); let len = s[0].len(); - let mut z = vec![CyclotomicRing::::zero(); len]; - for (c_i, s_i) in challenges.iter().zip(s.iter()) { - for (z_j, s_ij) in z.iter_mut().zip(s_i.iter()) { - *z_j += s_ij.mul_by_sparse(c_i); - } - } + + let z: Vec> = cfg_into_iter!(0..len) + .map(|j| { + challenges + .iter() + .zip(s.iter()) + .fold(CyclotomicRing::::zero(), |acc, (c_i, s_i)| { + acc + s_i[j].mul_by_sparse(c_i) + }) + }) + .collect(); let modulus = detect_field_modulus::(); let norm = vec_inf_norm(&z, modulus); diff --git a/src/protocol/ring_switch.rs b/src/protocol/ring_switch.rs index cb4bc0f9..1356bc22 100644 --- a/src/protocol/ring_switch.rs +++ b/src/protocol/ring_switch.rs @@ -6,6 +6,8 @@ use crate::algebra::ring::CyclotomicRing; use crate::error::HachiError; +#[cfg(feature = "parallel")] +use crate::parallel::*; use crate::protocol::commitment::utils::norm::detect_field_modulus; use crate::protocol::commitment::{ CommitmentConfig, DefaultCommitmentConfig, HachiCommitmentCore, RingCommitment, @@ -17,6 +19,7 @@ use crate::protocol::transcript::labels::{ ABSORB_SUMCHECK_W, CHALLENGE_RING_SWITCH, CHALLENGE_TAU0, CHALLENGE_TAU1, }; use crate::protocol::transcript::Transcript; +use crate::{cfg_into_iter, cfg_iter}; use crate::{CanonicalField, FieldCore, FieldSampling}; /// Output of the ring switch prover, containing everything needed for sumchecks. @@ -169,29 +172,54 @@ pub(crate) fn compute_r_via_poly_division( m.iter() .zip(y.iter()) .map(|(row, y_i)| { - let mut poly = vec![F::zero(); poly_len]; - for (m_ij, z_j) in row.iter().zip(z.iter()) { - if m_ij.is_zero() { - continue; - } - let a = m_ij.coefficients(); - let b = z_j.coefficients(); - - // Detect constant (scalar) ring elements: only a[0] is non-zero. - let is_scalar = a[1..].iter().all(|c| c.is_zero()); - if is_scalar { - let scalar = a[0]; - for s in 0..D { - poly[s] = poly[s] + scalar * b[s]; + let column_contribution = + |m_ij: &CyclotomicRing, z_j: &CyclotomicRing| -> Vec { + let mut local = vec![F::zero(); poly_len]; + if m_ij.is_zero() { + return local; } - } else { - for t in 0..D { + let a = m_ij.coefficients(); + let b = z_j.coefficients(); + let is_scalar = a[1..].iter().all(|c| c.is_zero()); + if is_scalar { + let scalar = a[0]; for s in 0..D { - poly[t + s] = poly[t + s] + a[t] * b[s]; + local[s] = scalar * b[s]; + } + } else { + for t in 0..D { + for s in 0..D { + local[t + s] = local[t + s] + a[t] * b[s]; + } } } + local + }; + + let pointwise_add = |mut a: Vec, b: Vec| -> Vec { + for (ai, bi) in a.iter_mut().zip(b.iter()) { + *ai = *ai + *bi; } - } + a + }; + + #[cfg(feature = "parallel")] + let mut poly = row + .par_iter() + .zip(z.par_iter()) + .fold( + || vec![F::zero(); poly_len], + |acc, (m_ij, z_j)| pointwise_add(acc, column_contribution(m_ij, z_j)), + ) + .reduce(|| vec![F::zero(); poly_len], pointwise_add); + + #[cfg(not(feature = "parallel"))] + let mut poly = row + .iter() + .zip(z.iter()) + .fold(vec![F::zero(); poly_len], |acc, (m_ij, z_j)| { + pointwise_add(acc, column_contribution(m_ij, z_j)) + }); let y_coeffs = y_i.coefficients(); for k in 0..D { poly[k] = poly[k] - y_coeffs[k]; @@ -278,7 +306,7 @@ pub(crate) fn eval_ring_vec_at( v: &[CyclotomicRing], alpha: &F, ) -> Vec { - v.iter().map(|r| eval_ring_at(r, alpha)).collect() + cfg_iter!(v).map(|r| eval_ring_at(r, alpha)).collect() } pub(crate) fn eval_ring_matrix_at( @@ -373,19 +401,20 @@ pub(crate) fn build_w_evals(w: &[F], d: usize) -> (Vec, usize, let num_ring_elems = w.len() / d.max(1); let num_u = num_ring_elems.next_power_of_two().trailing_zeros() as usize; let x_len = 1usize << num_u; - let y_len = 1usize << num_l; - let n = x_len * y_len; + let n = x_len << num_l; - let mut evals = vec![F::zero(); n]; - for x in 0..x_len { - for y in 0..y_len { + let evals: Vec = cfg_into_iter!(0..n) + .map(|dst| { + let x = dst & (x_len - 1); + let y = dst >> num_u; let src = y + (x << num_l); if src < w.len() { - let dst = x + (y << num_u); - evals[dst] = w[src]; + w[src] + } else { + F::zero() } - } - } + }) + .collect(); (evals, num_u, num_l) } @@ -401,20 +430,20 @@ pub(crate) fn build_m_evals_x( ) -> Vec { let eq_tau1 = EqPolynomial::evals(tau1); let x_len = cols.next_power_of_two(); - let mut m_evals = vec![F::zero(); x_len]; - for x in 0..x_len { - let mut acc = F::zero(); - for i in 0..eq_tau1.len() { - let row_val = if i < rows && x < cols { - m_a_flat[i * cols + x] - } else { - F::zero() - }; - acc = acc + eq_tau1[i] * row_val; - } - m_evals[x] = acc; - } - m_evals + cfg_into_iter!(0..x_len) + .map(|x| { + let mut acc = F::zero(); + for i in 0..eq_tau1.len() { + let row_val = if i < rows && x < cols { + m_a_flat[i * cols + x] + } else { + F::zero() + }; + acc = acc + eq_tau1[i] * row_val; + } + acc + }) + .collect() } pub(crate) fn build_alpha_evals_y(alpha: F, d: usize) -> Vec { diff --git a/src/protocol/sumcheck/norm_sumcheck.rs b/src/protocol/sumcheck/norm_sumcheck.rs index 3fd5aa83..00bb5bc0 100644 --- a/src/protocol/sumcheck/norm_sumcheck.rs +++ b/src/protocol/sumcheck/norm_sumcheck.rs @@ -9,6 +9,8 @@ use super::eq_poly::EqPolynomial; use super::split_eq::GruenSplitEq; use super::{fold_evals, multilinear_eval, range_check_eval}; use super::{SumcheckInstanceProver, SumcheckInstanceVerifier, UniPoly}; +#[cfg(feature = "parallel")] +use crate::parallel::*; use crate::{CanonicalField, FieldCore}; /// Prover for `F_{0,τ₀}(x,y) = ẽq(τ₀,(x,y)) · w̃(x,y) · range_check(w̃(x,y), b)`. @@ -60,26 +62,59 @@ impl SumcheckInstanceProver for NormSumcheckPr let half = self.w_table.len() / 2; let degree_q = 2 * self.b - 1; let num_points_q = degree_q + 1; - let mut q_evals = vec![E::zero(); num_points_q]; let (e_first, e_second) = self.split_eq.remaining_eq_tables(); let num_first = e_first.len(); let first_bits = num_first.trailing_zeros(); - - for j in 0..half { - let j_low = j & (num_first - 1); - let j_high = j >> first_bits; - let eq_rem = e_first[j_low] * e_second[j_high]; - - let w_0 = self.w_table[2 * j]; - let w_1 = self.w_table[2 * j + 1]; - - for (t, eval) in q_evals.iter_mut().enumerate() { - let t_e = E::from_u64(t as u64); - let w_t = w_0 + t_e * (w_1 - w_0); - *eval = *eval + eq_rem * range_check_eval(w_t, self.b); + let b = self.b; + + #[cfg(feature = "parallel")] + let q_evals = { + (0..half) + .into_par_iter() + .fold( + || vec![E::zero(); num_points_q], + |mut evals, j| { + let j_low = j & (num_first - 1); + let j_high = j >> first_bits; + let eq_rem = e_first[j_low] * e_second[j_high]; + let w_0 = self.w_table[2 * j]; + let w_1 = self.w_table[2 * j + 1]; + for (t, eval) in evals.iter_mut().enumerate() { + let t_e = E::from_u64(t as u64); + let w_t = w_0 + t_e * (w_1 - w_0); + *eval = *eval + eq_rem * range_check_eval(w_t, b); + } + evals + }, + ) + .reduce( + || vec![E::zero(); num_points_q], + |mut a, b_vec| { + for (ai, bi) in a.iter_mut().zip(b_vec.iter()) { + *ai = *ai + *bi; + } + a + }, + ) + }; + #[cfg(not(feature = "parallel"))] + let q_evals = { + let mut evals = vec![E::zero(); num_points_q]; + for j in 0..half { + let j_low = j & (num_first - 1); + let j_high = j >> first_bits; + let eq_rem = e_first[j_low] * e_second[j_high]; + let w_0 = self.w_table[2 * j]; + let w_1 = self.w_table[2 * j + 1]; + for (t, eval) in evals.iter_mut().enumerate() { + let t_e = E::from_u64(t as u64); + let w_t = w_0 + t_e * (w_1 - w_0); + *eval = *eval + eq_rem * range_check_eval(w_t, b); + } } - } + evals + }; let q_poly = UniPoly::from_evals(&q_evals); self.split_eq.gruen_mul(&q_poly) diff --git a/src/protocol/sumcheck/relation_sumcheck.rs b/src/protocol/sumcheck/relation_sumcheck.rs index 44e3c3c2..8ab342aa 100644 --- a/src/protocol/sumcheck/relation_sumcheck.rs +++ b/src/protocol/sumcheck/relation_sumcheck.rs @@ -9,6 +9,9 @@ use super::eq_poly::EqPolynomial; use super::{fold_evals, multilinear_eval}; use super::{SumcheckInstanceProver, SumcheckInstanceVerifier, UniPoly}; use crate::algebra::ring::CyclotomicRing; +use crate::cfg_into_iter; +#[cfg(feature = "parallel")] +use crate::parallel::*; use crate::protocol::ring_switch::eval_ring_at; use crate::{CanonicalField, FieldCore}; @@ -53,8 +56,12 @@ impl RelationSumcheckProver { assert_eq!(m_evals_x.len(), 1 << num_u); let x_mask = (1usize << num_u) - 1; - let alpha_table: Vec = (0..n).map(|idx| alpha_evals_y[idx >> num_u]).collect(); - let m_table: Vec = (0..n).map(|idx| m_evals_x[idx & x_mask]).collect(); + let alpha_table: Vec = cfg_into_iter!(0..n) + .map(|idx| alpha_evals_y[idx >> num_u]) + .collect(); + let m_table: Vec = cfg_into_iter!(0..n) + .map(|idx| m_evals_x[idx & x_mask]) + .collect(); Self { w_table: w_evals, @@ -75,34 +82,85 @@ impl SumcheckInstanceProver for RelationSumche } fn input_claim(&self) -> E { - self.w_table - .iter() - .zip(self.alpha_table.iter()) - .zip(self.m_table.iter()) - .fold(E::zero(), |acc, ((&w, &a), &m)| acc + w * a * m) + #[cfg(feature = "parallel")] + { + self.w_table + .par_iter() + .zip(self.alpha_table.par_iter()) + .zip(self.m_table.par_iter()) + .fold( + || E::zero(), + |acc, ((&w, &a), &m)| acc + w * a * m, + ) + .reduce(|| E::zero(), |a, b| a + b) + } + #[cfg(not(feature = "parallel"))] + { + self.w_table + .iter() + .zip(self.alpha_table.iter()) + .zip(self.m_table.iter()) + .fold(E::zero(), |acc, ((&w, &a), &m)| acc + w * a * m) + } } fn compute_round_univariate(&mut self, _round: usize, _previous_claim: E) -> UniPoly { let half = self.w_table.len() / 2; let num_points = 3; // degree 2 → 3 evaluation points - let mut round_evals = vec![E::zero(); num_points]; - - for j in 0..half { - let w_0 = self.w_table[2 * j]; - let w_1 = self.w_table[2 * j + 1]; - let a_0 = self.alpha_table[2 * j]; - let a_1 = self.alpha_table[2 * j + 1]; - let m_0 = self.m_table[2 * j]; - let m_1 = self.m_table[2 * j + 1]; - - for (t, eval) in round_evals.iter_mut().enumerate() { - let t_e = E::from_u64(t as u64); - let w_t = w_0 + t_e * (w_1 - w_0); - let a_t = a_0 + t_e * (a_1 - a_0); - let m_t = m_0 + t_e * (m_1 - m_0); - *eval = *eval + w_t * a_t * m_t; + + #[cfg(feature = "parallel")] + let round_evals = { + (0..half) + .into_par_iter() + .fold( + || vec![E::zero(); num_points], + |mut evals, j| { + let w_0 = self.w_table[2 * j]; + let w_1 = self.w_table[2 * j + 1]; + let a_0 = self.alpha_table[2 * j]; + let a_1 = self.alpha_table[2 * j + 1]; + let m_0 = self.m_table[2 * j]; + let m_1 = self.m_table[2 * j + 1]; + for (t, eval) in evals.iter_mut().enumerate() { + let t_e = E::from_u64(t as u64); + let w_t = w_0 + t_e * (w_1 - w_0); + let a_t = a_0 + t_e * (a_1 - a_0); + let m_t = m_0 + t_e * (m_1 - m_0); + *eval = *eval + w_t * a_t * m_t; + } + evals + }, + ) + .reduce( + || vec![E::zero(); num_points], + |mut a, b| { + for (ai, bi) in a.iter_mut().zip(b.iter()) { + *ai = *ai + *bi; + } + a + }, + ) + }; + #[cfg(not(feature = "parallel"))] + let round_evals = { + let mut evals = vec![E::zero(); num_points]; + for j in 0..half { + let w_0 = self.w_table[2 * j]; + let w_1 = self.w_table[2 * j + 1]; + let a_0 = self.alpha_table[2 * j]; + let a_1 = self.alpha_table[2 * j + 1]; + let m_0 = self.m_table[2 * j]; + let m_1 = self.m_table[2 * j + 1]; + for (t, eval) in evals.iter_mut().enumerate() { + let t_e = E::from_u64(t as u64); + let w_t = w_0 + t_e * (w_1 - w_0); + let a_t = a_0 + t_e * (a_1 - a_0); + let m_t = m_0 + t_e * (m_1 - m_0); + *eval = *eval + w_t * a_t * m_t; + } } - } + evals + }; UniPoly::from_evals(&round_evals) } From 0ed273689a8306d709469e216085ffe44dce57d7 Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Fri, 27 Feb 2026 22:27:26 -0800 Subject: [PATCH 04/34] Add e2e benchmark and make HachiCommitmentScheme generic over config - Make HachiCommitmentScheme generic over so different configs (and thus num_vars) can be used without code duplication. - Remove hardcoded DefaultCommitmentConfig::D from ring_switch.rs; WCommitmentConfig and commit_w now flow D generically. - Add benches/hachi_e2e.rs with configs sweeping nv=10,14,18,20. Made-with: Cursor --- Cargo.toml | 4 + benches/hachi_e2e.rs | 167 +++++++++++++++++++++ src/protocol/commitment_scheme.rs | 50 +++--- src/protocol/ring_switch.rs | 73 +++++---- src/protocol/sumcheck/norm_sumcheck.rs | 8 +- src/protocol/sumcheck/relation_sumcheck.rs | 10 +- 6 files changed, 244 insertions(+), 68 deletions(-) create mode 100644 benches/hachi_e2e.rs diff --git a/Cargo.toml b/Cargo.toml index 7de0b328..6b88584f 100644 --- a/Cargo.toml +++ b/Cargo.toml @@ -63,6 +63,10 @@ harness = false name = "fp64_reduce_probe" harness = false +[[bench]] +name = "hachi_e2e" +harness = false + [lints.rust] missing_docs = "warn" unreachable_pub = "warn" diff --git a/benches/hachi_e2e.rs b/benches/hachi_e2e.rs new file mode 100644 index 00000000..06028d53 --- /dev/null +++ b/benches/hachi_e2e.rs @@ -0,0 +1,167 @@ +#![allow(missing_docs)] + +use criterion::{black_box, criterion_group, criterion_main, BenchmarkId, Criterion}; +use hachi_pcs::algebra::Fp64; +use hachi_pcs::primitives::multilinear_evals::DenseMultilinearEvals; +use hachi_pcs::protocol::commitment::DefaultCommitmentConfig; +use hachi_pcs::protocol::commitment_scheme::HachiCommitmentScheme; +use hachi_pcs::protocol::transcript::Blake2bTranscript; +use hachi_pcs::protocol::CommitmentConfig; +use hachi_pcs::{CanonicalField, CommitmentScheme, Polynomial, Transcript}; +use std::time::Duration; + +type F = Fp64<4294967197>; + +const D: usize = DefaultCommitmentConfig::D; + +macro_rules! bench_config { + ($name:ident, M = $m:expr, R = $r:expr) => { + #[derive(Clone, Copy, Debug)] + struct $name; + impl CommitmentConfig for $name { + const D: usize = D; + const M: usize = $m; + const R: usize = $r; + const N_A: usize = DefaultCommitmentConfig::N_A; + const N_B: usize = DefaultCommitmentConfig::N_B; + const N_D: usize = DefaultCommitmentConfig::N_D; + const LOG_BASIS: u32 = DefaultCommitmentConfig::LOG_BASIS; + const DELTA: usize = DefaultCommitmentConfig::DELTA; + const TAU: usize = DefaultCommitmentConfig::TAU; + const BETA: u128 = DefaultCommitmentConfig::BETA; + const CHALLENGE_WEIGHT: usize = DefaultCommitmentConfig::CHALLENGE_WEIGHT; + } + }; +} + +bench_config!(CfgNv10, M = 4, R = 2); +bench_config!(CfgNv14, M = 6, R = 4); +bench_config!(CfgNv18, M = 8, R = 6); +bench_config!(CfgNv20, M = 8, R = 8); + +fn num_vars() -> usize { + let alpha = Cfg::D.trailing_zeros() as usize; + Cfg::R + Cfg::M + alpha +} + +fn make_poly(nv: usize) -> DenseMultilinearEvals { + let len = 1usize << nv; + let evals: Vec = (0..len).map(|i| F::from_u64(i as u64)).collect(); + DenseMultilinearEvals::new_padded(evals) +} + +fn opening_point(nv: usize) -> Vec { + (0..nv).map(|i| F::from_u64((i + 2) as u64)).collect() +} + +fn bench_phases(c: &mut Criterion, label: &str) +where + HachiCommitmentScheme: CommitmentScheme, +{ + type S = HachiCommitmentScheme; + let nv = num_vars::(); + let poly = make_poly(nv); + let pt = opening_point(nv); + + let mut group = c.benchmark_group(format!("hachi/{label}/nv{nv}")); + if nv >= 18 { + group.sample_size(10); + group.measurement_time(Duration::from_secs(30)); + } + + group.bench_function("setup", |b| { + b.iter(|| black_box( as CommitmentScheme>::setup_prover(black_box(nv)))) + }); + + let setup = as CommitmentScheme>::setup_prover(nv); + + group.bench_function("commit", |b| { + b.iter(|| { + black_box( + as CommitmentScheme>::commit(black_box(&poly), black_box(&setup)) + .unwrap(), + ) + }) + }); + + let (commitment, hint) = as CommitmentScheme>::commit(&poly, &setup).unwrap(); + + group.bench_function("prove", |b| { + b.iter(|| { + let mut transcript = Blake2bTranscript::::new(b"bench"); + black_box( + as CommitmentScheme>::prove( + black_box(&setup), + black_box(&poly), + black_box(&pt), + Some(hint.clone()), + &mut transcript, + black_box(&commitment), + ) + .unwrap(), + ) + }) + }); + + let verifier_setup = as CommitmentScheme>::setup_verifier(&setup); + let opening = poly.evaluate(&pt); + let mut prover_transcript = Blake2bTranscript::::new(b"bench"); + let proof = as CommitmentScheme>::prove( + &setup, + &poly, + &pt, + Some(hint), + &mut prover_transcript, + &commitment, + ) + .unwrap(); + + group.bench_function("verify", |b| { + b.iter(|| { + let mut transcript = Blake2bTranscript::::new(b"bench"); + black_box( + as CommitmentScheme>::verify( + black_box(&proof), + black_box(&verifier_setup), + &mut transcript, + black_box(&pt), + black_box(&opening), + black_box(&commitment), + ) + .unwrap(), + ) + }) + }); + + group.bench_function(BenchmarkId::new("e2e", nv), |b| { + b.iter(|| { + let (cm, h) = as CommitmentScheme>::commit(&poly, &setup).unwrap(); + let mut pt_tr = Blake2bTranscript::::new(b"bench"); + let pf = as CommitmentScheme>::prove( + &setup, &poly, &pt, Some(h), &mut pt_tr, &cm, + ) + .unwrap(); + let mut vt_tr = Blake2bTranscript::::new(b"bench"); + as CommitmentScheme>::verify( + &pf, + &verifier_setup, + &mut vt_tr, + &pt, + &opening, + &cm, + ) + .unwrap(); + black_box(()) + }) + }); + + group.finish(); +} + +fn bench_nv10(c: &mut Criterion) { bench_phases::(c, "fp64"); } +fn bench_nv14(c: &mut Criterion) { bench_phases::(c, "fp64"); } +fn bench_nv18(c: &mut Criterion) { bench_phases::(c, "fp64"); } +fn bench_nv20(c: &mut Criterion) { bench_phases::(c, "fp64"); } + +criterion_group!(hachi_benches, bench_nv10, bench_nv14, bench_nv18, bench_nv20); +criterion_main!(hachi_benches); diff --git a/src/protocol/commitment_scheme.rs b/src/protocol/commitment_scheme.rs index 4655fc9b..f5bcc13d 100644 --- a/src/protocol/commitment_scheme.rs +++ b/src/protocol/commitment_scheme.rs @@ -35,22 +35,22 @@ use crate::protocol::transcript::labels::{ #[cfg(test)] use crate::protocol::transcript::Blake2bTranscript; -type Cfg = DefaultCommitmentConfig; -const D: usize = Cfg::D; - -/// Placeholder for the end-to-end PCS wrapper. +/// End-to-end PCS wrapper, generic over ring degree `D` and config `Cfg`. #[derive(Clone, Copy, Debug, Default)] -pub struct HachiCommitmentScheme; +pub struct HachiCommitmentScheme { + _cfg: std::marker::PhantomData, +} -impl CommitmentScheme for HachiCommitmentScheme +impl CommitmentScheme for HachiCommitmentScheme where F: FieldCore + CanonicalField + FieldSampling, + Cfg: CommitmentConfig, { - type ProverSetup = RingCommitmentSetup; - type VerifierSetup = RingCommitmentSetup; - type Commitment = RingCommitment; - type Proof = HachiProof; - type OpeningProofHint = HachiCommitmentHint; + type ProverSetup = RingCommitmentSetup; + type VerifierSetup = RingCommitmentSetup; + type Commitment = RingCommitment; + type Proof = HachiProof; + type OpeningProofHint = HachiCommitmentHint; fn setup_prover(max_num_vars: usize) -> Self::ProverSetup { let (setup, _) = @@ -238,14 +238,15 @@ where /// by replaying the transcript from the proof data and setup, exactly as the /// verifier does. #[cfg(test)] -pub(crate) fn rederive_alpha_and_m_a( - proof: &HachiProof, - setup: &>::ProverSetup, +pub(crate) fn rederive_alpha_and_m_a( + proof: &HachiProof, + setup: &RingCommitmentSetup, opening_point: &[F], - commitment: &>::Commitment, + commitment: &RingCommitment, ) -> Result<(F, Vec), HachiError> where F: FieldCore + CanonicalField + FieldSampling + 'static, + Cfg: CommitmentConfig, { let alpha_bits = Cfg::D.trailing_zeros() as usize; let reduced_len = opening_point @@ -255,7 +256,7 @@ where let ring_opening_point = ring_opening_point_from_field::(&opening_point[..reduced_len], Cfg::R, Cfg::M)?; let mut transcript = Blake2bTranscript::::new(DOMAIN_HACHI_PROTOCOL); - let quad_eq = QuadraticEquation::::new_verifier( + let quad_eq = QuadraticEquation::::new_verifier( setup, &ring_opening_point, &proof.v, @@ -265,8 +266,8 @@ where )?; transcript.append_serde(ABSORB_SUMCHECK_W, &proof.w_commitment); let alpha: F = transcript.challenge_scalar(CHALLENGE_RING_SWITCH); - let m_a = eval_ring_matrix_at::(quad_eq.m(), &alpha); - let m_a_vec = expand_m_a::(&m_a, alpha)?; + let m_a = eval_ring_matrix_at::(quad_eq.m(), &alpha); + let m_a_vec = expand_m_a::(&m_a, alpha)?; Ok((alpha, m_a_vec)) } @@ -396,6 +397,9 @@ mod tests { use crate::test_utils::F; use crate::{CommitmentScheme, Polynomial}; + type Cfg = DefaultCommitmentConfig; + type Scheme = HachiCommitmentScheme<{ Cfg::D }, Cfg>; + #[test] fn verify_passes_for_consistent_opening() { let alpha = Cfg::D.trailing_zeros() as usize; @@ -405,17 +409,17 @@ mod tests { let evals: Vec = (0..len).map(|i| F::from_u64(i as u64)).collect(); let poly = DenseMultilinearEvals::new_padded(evals); - let setup = >::setup_prover(num_vars); - let verifier_setup = >::setup_verifier(&setup); + let setup = >::setup_prover(num_vars); + let verifier_setup = >::setup_verifier(&setup); let (commitment, hint) = - >::commit(&poly, &setup).unwrap(); + >::commit(&poly, &setup).unwrap(); let opening_point: Vec = (0..num_vars).map(|i| F::from_u64((i + 2) as u64)).collect(); let opening = poly.evaluate(&opening_point); let mut prover_transcript = Blake2bTranscript::::new(b"test/prove"); - let proof = >::prove( + let proof = >::prove( &setup, &poly, &opening_point, @@ -426,7 +430,7 @@ mod tests { .unwrap(); let mut verifier_transcript = Blake2bTranscript::::new(b"test/prove"); - let result = >::verify( + let result = >::verify( &proof, &verifier_setup, &mut verifier_transcript, diff --git a/src/protocol/ring_switch.rs b/src/protocol/ring_switch.rs index 1356bc22..b4ac4688 100644 --- a/src/protocol/ring_switch.rs +++ b/src/protocol/ring_switch.rs @@ -10,8 +10,7 @@ use crate::error::HachiError; use crate::parallel::*; use crate::protocol::commitment::utils::norm::detect_field_modulus; use crate::protocol::commitment::{ - CommitmentConfig, DefaultCommitmentConfig, HachiCommitmentCore, RingCommitment, - RingCommitmentScheme, + CommitmentConfig, HachiCommitmentCore, RingCommitment, RingCommitmentScheme, }; use crate::protocol::quadratic_equation::QuadraticEquation; use crate::protocol::sumcheck::eq_poly::EqPolynomial; @@ -56,7 +55,7 @@ pub struct RingSwitchOutput { pub fn ring_switch_prover( quad_eq: &QuadraticEquation, transcript: &mut T, -) -> Result, HachiError> +) -> Result, HachiError> where F: FieldCore + CanonicalField + FieldSampling, T: Transcript, @@ -68,7 +67,7 @@ where let r = compute_r_via_poly_division::(quad_eq.m(), z, quad_eq.y()); let w = build_w_coeffs::(z, &r); - let w_commitment = commit_w::(&w)?; + let w_commitment = commit_w::(&w)?; transcript.append_serde(ABSORB_SUMCHECK_W, &w_commitment); let alpha: F = transcript.challenge_scalar(CHALLENGE_RING_SWITCH); @@ -116,9 +115,9 @@ where pub fn ring_switch_verifier( quad_eq: &QuadraticEquation, w: &[F], - w_commitment: &RingCommitment, + w_commitment: &RingCommitment, transcript: &mut T, -) -> Result, HachiError> +) -> Result, HachiError> where F: FieldCore + CanonicalField + FieldSampling, T: Transcript, @@ -237,57 +236,55 @@ pub(crate) fn compute_r_via_poly_division( } #[derive(Clone, Copy, Debug)] -struct WCommitmentConfig; +struct WCommitmentConfig { + _cfg: std::marker::PhantomData, +} -impl CommitmentConfig for WCommitmentConfig { - const D: usize = DefaultCommitmentConfig::D; +impl CommitmentConfig for WCommitmentConfig { + const D: usize = D; const M: usize = 11; const R: usize = 0; - const N_A: usize = DefaultCommitmentConfig::N_A; - const N_B: usize = DefaultCommitmentConfig::N_B; - const N_D: usize = DefaultCommitmentConfig::N_D; - const LOG_BASIS: u32 = DefaultCommitmentConfig::LOG_BASIS; - const DELTA: usize = DefaultCommitmentConfig::DELTA; - const TAU: usize = DefaultCommitmentConfig::TAU; - const BETA: u128 = DefaultCommitmentConfig::BETA; - const CHALLENGE_WEIGHT: usize = DefaultCommitmentConfig::CHALLENGE_WEIGHT; + const N_A: usize = Cfg::N_A; + const N_B: usize = Cfg::N_B; + const N_D: usize = Cfg::N_D; + const LOG_BASIS: u32 = Cfg::LOG_BASIS; + const DELTA: usize = Cfg::DELTA; + const TAU: usize = Cfg::TAU; + const BETA: u128 = Cfg::BETA; + const CHALLENGE_WEIGHT: usize = Cfg::CHALLENGE_WEIGHT; } -fn commit_w(w: &[F]) -> Result, HachiError> +fn commit_w(w: &[F]) -> Result, HachiError> where F: FieldCore + CanonicalField + FieldSampling, + Cfg: CommitmentConfig, { - let d = DefaultCommitmentConfig::D; - let block_len = 1usize << WCommitmentConfig::M; + type WCfg = WCommitmentConfig; + let block_len = 1usize << WCfg::::M; - let ring_elems: Vec> = w - .chunks(d) + let ring_elems: Vec> = w + .chunks(D) .map(|chunk| { - let coeffs: [F; DefaultCommitmentConfig::D] = + let coeffs: [F; D] = std::array::from_fn(|i| if i < chunk.len() { chunk[i] } else { F::zero() }); CyclotomicRing::from_coefficients(coeffs) }) .collect(); let mut padded = ring_elems; - padded.resize( - block_len, - CyclotomicRing::::zero(), - ); + padded.resize(block_len, CyclotomicRing::::zero()); let blocks = vec![padded]; - let (w_setup, _) = >::setup(WCommitmentConfig::M)?; - - let (commitment, _, _) = >::commit_ring_blocks(&blocks, &w_setup)?; + let (w_setup, _) = + >>::setup( + WCfg::::M, + )?; + + let (commitment, _, _) = + >>::commit_ring_blocks( + &blocks, &w_setup, + )?; Ok(commitment) } diff --git a/src/protocol/sumcheck/norm_sumcheck.rs b/src/protocol/sumcheck/norm_sumcheck.rs index 00bb5bc0..a5c7cf5d 100644 --- a/src/protocol/sumcheck/norm_sumcheck.rs +++ b/src/protocol/sumcheck/norm_sumcheck.rs @@ -189,6 +189,8 @@ mod tests { type F = Fp64<4294967197>; const D: usize = 8; + type Cfg = DefaultCommitmentConfig; + type Scheme = HachiCommitmentScheme<{ Cfg::D }, Cfg>; fn ring_with_small_coeff(value: u64) -> CyclotomicRing { let coeffs = std::array::from_fn(|_| F::from_u64(value)); @@ -247,12 +249,12 @@ mod tests { let evals: Vec = (0..len).map(|i| F::from_u64(i as u64)).collect(); let poly = DenseMultilinearEvals::new_padded(evals); - let setup = HachiCommitmentScheme::setup_prover(num_vars); - let (commitment, hint) = HachiCommitmentScheme::commit(&poly, &setup).unwrap(); + let setup = Scheme::setup_prover(num_vars); + let (commitment, hint) = Scheme::commit(&poly, &setup).unwrap(); let opening_point: Vec = (0..num_vars).map(|i| F::from_u64((i + 2) as u64)).collect(); let mut prover_transcript = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); - let proof = HachiCommitmentScheme::prove( + let proof = Scheme::prove( &setup, &poly, &opening_point, diff --git a/src/protocol/sumcheck/relation_sumcheck.rs b/src/protocol/sumcheck/relation_sumcheck.rs index 8ab342aa..0fec983c 100644 --- a/src/protocol/sumcheck/relation_sumcheck.rs +++ b/src/protocol/sumcheck/relation_sumcheck.rs @@ -276,6 +276,8 @@ mod tests { use crate::{CanonicalField, FieldCore}; type F = Fp64<4294967197>; + type Cfg = DefaultCommitmentConfig; + type Scheme = HachiCommitmentScheme<{ Cfg::D }, Cfg>; #[test] fn relation_sumcheck_uses_prove_w_evals() { @@ -285,12 +287,12 @@ mod tests { let evals: Vec = (0..len).map(|i| F::from_u64(i as u64)).collect(); let poly = DenseMultilinearEvals::new_padded(evals); - let setup = HachiCommitmentScheme::setup_prover(num_vars); - let (commitment, hint) = HachiCommitmentScheme::commit(&poly, &setup).unwrap(); + let setup = Scheme::setup_prover(num_vars); + let (commitment, hint) = Scheme::commit(&poly, &setup).unwrap(); let opening_point: Vec = (0..num_vars).map(|i| F::from_u64((i + 2) as u64)).collect(); let mut prover_transcript = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); - let proof = HachiCommitmentScheme::prove( + let proof = Scheme::prove( &setup, &poly, &opening_point, @@ -301,7 +303,7 @@ mod tests { .unwrap(); let (alpha, m_a_vec) = - rederive_alpha_and_m_a(&proof, &setup, &opening_point, &commitment).unwrap(); + rederive_alpha_and_m_a::(&proof, &setup, &opening_point, &commitment).unwrap(); let d = DefaultCommitmentConfig::D; assert_eq!(proof.sumcheck_aux.w.len() % d, 0); From 96a9ccd8999a82d6d548bd5e27fe666dd4f55677 Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Fri, 27 Feb 2026 23:10:01 -0800 Subject: [PATCH 05/34] Refactor CRT-NTT backend: generalize over PrimeWidth, add Q128 support MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Make NTT primitives (NttPrime, NttTwiddles, MontCoeff, CyclotomicCrtNtt) generic over PrimeWidth (i16/i32) instead of hardcoding i16. Replace the monolithic QData struct with separate GarnerData and per-prime NttPrime arrays. Add Q128 parameter set (5 × i32 primes, D ≤ 1024) alongside the existing Q32 set. Simplify ScalarBackend by removing the const-generic limb count from to_ring_with_backend. Made-with: Cursor --- benches/ring_ntt.rs | 17 +- src/algebra/backend/scalar.rs | 188 ++++------------- src/algebra/backend/traits.rs | 49 ++--- src/algebra/domains/crt_ntt.rs | 2 +- src/algebra/mod.rs | 2 +- src/algebra/ntt/butterfly.rs | 58 +++--- src/algebra/ntt/crt.rs | 73 ++++--- src/algebra/ntt/mod.rs | 4 +- src/algebra/ntt/prime.rs | 336 +++++++++++++++++++++++-------- src/algebra/ntt/tables.rs | 165 ++++++++------- src/algebra/ring/crt_ntt_repr.rs | 166 ++++++--------- tests/algebra.rs | 129 +++++++++--- 12 files changed, 650 insertions(+), 539 deletions(-) diff --git a/benches/ring_ntt.rs b/benches/ring_ntt.rs index 546f9b9d..aa2ac9ff 100644 --- a/benches/ring_ntt.rs +++ b/benches/ring_ntt.rs @@ -2,13 +2,13 @@ use criterion::{black_box, criterion_group, criterion_main, Criterion}; use hachi_pcs::algebra::ntt::butterfly::{forward_ntt, inverse_ntt, NttTwiddles}; -use hachi_pcs::algebra::tables::{Q32_DATA, Q32_MODULUS, Q32_NUM_PRIMES, Q32_PRIMES}; +use hachi_pcs::algebra::tables::{q32_garner, Q32_MODULUS, Q32_NUM_PRIMES, Q32_PRIMES}; use hachi_pcs::algebra::{CyclotomicCrtNtt, CyclotomicRing, Fp64, MontCoeff}; -use hachi_pcs::CanonicalField; +use hachi_pcs::FromSmallInt; type F = Fp64<{ Q32_MODULUS }>; type R = CyclotomicRing; -type N = CyclotomicCrtNtt; +type N = CyclotomicCrtNtt; fn sample_ring(seed: u64) -> R { let coeffs = std::array::from_fn(|i| { @@ -30,8 +30,8 @@ fn bench_ring_schoolbook_mul(c: &mut Criterion) { fn bench_ntt_single_prime_round_trip(c: &mut Criterion) { let prime = Q32_PRIMES[0]; - let tw = NttTwiddles::<64>::compute(prime); - let base: [MontCoeff; 64] = + let tw = NttTwiddles::::compute(prime); + let base: [MontCoeff; 64] = std::array::from_fn(|i| prime.from_canonical(((i * 5 + 7) as i16) % prime.p)); c.bench_function("ntt_single_prime_forward_inverse_d64", |b| { @@ -46,13 +46,14 @@ fn bench_ntt_single_prime_round_trip(c: &mut Criterion) { fn bench_crt_round_trip(c: &mut Criterion) { let ring = sample_ring(19); - let twiddles: [NttTwiddles<64>; Q32_NUM_PRIMES] = - std::array::from_fn(|k| NttTwiddles::<64>::compute(Q32_PRIMES[k])); + let twiddles: [NttTwiddles; Q32_NUM_PRIMES] = + std::array::from_fn(|k| NttTwiddles::compute(Q32_PRIMES[k])); + let garner = q32_garner(); c.bench_function("ring_ntt_crt_round_trip_d64_k6", |b| { b.iter(|| { let ntt = N::from_ring(black_box(&ring), &Q32_PRIMES, &twiddles); - let back: R = ntt.to_ring(&Q32_PRIMES, &twiddles, &Q32_DATA); + let back: R = ntt.to_ring(&Q32_PRIMES, &twiddles, &garner); black_box(back) }) }); diff --git a/src/algebra/backend/scalar.rs b/src/algebra/backend/scalar.rs index ce0f0dc7..d147e4dd 100644 --- a/src/algebra/backend/scalar.rs +++ b/src/algebra/backend/scalar.rs @@ -1,191 +1,89 @@ -//! Default scalar backend wiring existing CRT+NTT kernels. +//! Default scalar backend: delegates to NTT kernels and uses Garner's +//! algorithm for CRT reconstruction. use super::traits::{CrtReconstruct, NttPrimeOps, NttTransform}; use crate::algebra::ntt::butterfly::{forward_ntt, inverse_ntt, NttTwiddles}; -use crate::algebra::ntt::crt::QData; -use crate::algebra::ntt::prime::{MontCoeff, NttPrime}; +use crate::algebra::ntt::crt::GarnerData; +use crate::algebra::ntt::prime::{MontCoeff, NttPrime, PrimeWidth}; use crate::algebra::ring::CrtNttConvertibleField; /// Default scalar backend implementation. #[derive(Debug, Clone, Copy, Default)] pub struct ScalarBackend; -impl NttPrimeOps for ScalarBackend { +impl NttPrimeOps for ScalarBackend { #[inline] - fn from_canonical(prime: NttPrime, value: i16) -> MontCoeff { + fn from_canonical(prime: NttPrime, value: W) -> MontCoeff { prime.from_canonical(value) } #[inline] - fn to_canonical(prime: NttPrime, value: MontCoeff) -> i16 { + fn to_canonical(prime: NttPrime, value: MontCoeff) -> W { prime.to_canonical(value) } #[inline] - fn reduce(prime: NttPrime, value: MontCoeff) -> MontCoeff { - prime.reduce(value) + fn reduce_range(prime: NttPrime, value: MontCoeff) -> MontCoeff { + prime.reduce_range(value) } #[inline] fn pointwise_mul( - prime: NttPrime, - out: &mut [MontCoeff; D], - lhs: &[MontCoeff; D], - rhs: &[MontCoeff; D], + prime: NttPrime, + out: &mut [MontCoeff; D], + lhs: &[MontCoeff; D], + rhs: &[MontCoeff; D], ) { prime.pointwise_mul(out, lhs, rhs); } } -impl NttTransform for ScalarBackend { +impl NttTransform for ScalarBackend { #[inline] - fn forward_ntt(limb: &mut [MontCoeff; D], prime: NttPrime, twiddles: &NttTwiddles) { + fn forward_ntt(limb: &mut [MontCoeff; D], prime: NttPrime, twiddles: &NttTwiddles) { forward_ntt(limb, prime, twiddles); } #[inline] - fn inverse_ntt(limb: &mut [MontCoeff; D], prime: NttPrime, twiddles: &NttTwiddles) { + fn inverse_ntt(limb: &mut [MontCoeff; D], prime: NttPrime, twiddles: &NttTwiddles) { inverse_ntt(limb, prime, twiddles); } } -impl CrtReconstruct for ScalarBackend { +impl CrtReconstruct for ScalarBackend { fn reconstruct( - primes: &[NttPrime; K], - canonical_limbs: &[[i16; D]; K], - qdata: &QData, + primes: &[NttPrime; K], + canonical: &[[W; D]; K], + garner: &GarnerData, ) -> [F; D] { - let q = qdata.q_u128().expect("q must fit in u128"); - let prime_moduli: [u128; K] = std::array::from_fn(|k| { - u128::try_from(primes[k].p).expect("CRT prime modulus must be positive") - }); - let big_p = prime_moduli.iter().fold(1u128, |acc, p| { - acc.checked_mul(*p) - .expect("product of CRT primes must fit u128") - }); - let crt_m: [u128; K] = std::array::from_fn(|k| big_p / prime_moduli[k]); - let crt_inv: [u16; K] = std::array::from_fn(|k| { - let inv = mod_inverse( - (crt_m[k] % prime_moduli[k]) as i128, - prime_moduli[k] as i128, - ); - u16::try_from(inv).expect("CRT inverse must fit u16 for small-prime backend") - }); - let mut coeffs = [F::zero(); D]; for (d, coeff) in coeffs.iter_mut().enumerate() { - let mut acc: u128 = 0; - for k in 0..K { - let ck_i16 = canonical_limbs[k][d]; - debug_assert!(ck_i16 >= 0 && ck_i16 < primes[k].p); - let ck = u16::try_from(ck_i16).expect("canonical residue must fit u16"); - - // Multiply by tiny residues/inverses (<= 15 bits) in fixed-time - // loops, then accumulate modulo P. - let term = mul_mod_by_small_u16(crt_m[k], ck, big_p); - let term = mul_mod_by_small_u16(term, crt_inv[k], big_p); - acc = add_mod_u128(acc, term, big_p); + // Garner mixed-radix decomposition (all arithmetic in i64, mod p_i). + let mut v = [0i64; K]; + v[0] = canonical[0][d].to_i64(); + for i in 1..K { + let pi = primes[i].p.to_i64(); + let mut temp = canonical[i][d].to_i64(); + #[allow(clippy::needless_range_loop)] + for j in 0..i { + temp -= v[j]; + temp = ((temp % pi) + pi) % pi; + temp = (temp * garner.gamma[i][j].to_i64()) % pi; + } + v[i] = temp; } - // Final projection into [0, q) without variable-latency division. - let reduced = reduce_u128_divfree(acc, q); - *coeff = F::from_q_residue_u128(reduced); + // Horner accumulation in the target field F. + let mut result = F::from_u64(v[0] as u64); + let mut partial_prod = F::from_u64(primes[0].p.to_i64() as u64); + for i in 1..K { + result = result + F::from_u64(v[i] as u64) * partial_prod; + if i + 1 < K { + partial_prod = partial_prod * F::from_u64(primes[i].p.to_i64() as u64); + } + } + *coeff = result; } - coeffs } } - -fn mod_inverse(a: i128, modulus: i128) -> i128 { - let (mut t, mut new_t) = (0i128, 1i128); - let (mut r, mut new_r) = (modulus, ((a % modulus) + modulus) % modulus); - - while new_r != 0 { - let q = r / new_r; - (t, new_t) = (new_t, t - q * new_t); - (r, new_r) = (new_r, r - q * new_r); - } - - assert_eq!(r, 1, "CRT inverse does not exist"); - (t % modulus + modulus) % modulus -} - -#[inline] -fn add_mod_u128(a: u128, b: u128, modulus: u128) -> u128 { - debug_assert!(a < modulus); - debug_assert!(b < modulus); - - let (sum_lo, carry) = a.overflowing_add(b); - let hi = carry as u128; - let (sub_lo, borrow) = sum_lo.overflowing_sub(modulus); - let sum_ge_modulus = (!borrow) as u128; - let should_sub = hi | sum_ge_modulus; - let mask = should_sub.wrapping_neg(); - (sum_lo & !mask) | (sub_lo & mask) -} - -#[inline] -fn mul_mod_by_small_u16(a: u128, b: u16, modulus: u128) -> u128 { - debug_assert!(a < modulus); - let mut acc = 0u128; - let mut cur = a; - for i in 0..16 { - let candidate = add_mod_u128(acc, cur, modulus); - let bit = ((b >> i) & 1) as u128; - let mask = bit.wrapping_neg(); - acc = (acc & !mask) | (candidate & mask); - cur = add_mod_u128(cur, cur, modulus); - } - acc -} - -#[inline] -fn reduce_u128_divfree(x: u128, modulus: u128) -> u128 { - debug_assert!(modulus > 0); - let mut rem = 0u128; - for i in (0..128).rev() { - rem = (rem << 1) | ((x >> i) & 1); - let (candidate, borrow) = rem.overflowing_sub(modulus); - let mask = 0u128.wrapping_sub((!borrow) as u128); - rem = (candidate & mask) | (rem & !mask); - } - rem -} - -#[cfg(test)] -mod tests { - use super::{add_mod_u128, mul_mod_by_small_u16, reduce_u128_divfree}; - - #[test] - fn add_mod_matches_native_when_sum_fits_u128() { - let modulus = (1u128 << 100) - 159; - for i in 0..4096u128 { - let a = (i * 104_729 + 17) % modulus; - let b = (i * 130_363 + 31) % modulus; - let expected = (a + b) % modulus; - assert_eq!(add_mod_u128(a, b, modulus), expected); - } - } - - #[test] - fn mul_mod_small_matches_native_when_product_fits_u128() { - let modulus = (1u128 << 100) - 159; - for i in 0..4096u128 { - let a = (i * 786_433 + 19) % modulus; - let b = ((i * 97 + 7) & 0xFFFF) as u16; - let expected = (a * (b as u128)) % modulus; - assert_eq!(mul_mod_by_small_u16(a, b, modulus), expected); - } - } - - #[test] - fn reduce_u128_divfree_matches_native_mod() { - let modulus = (1u128 << 100) - 159; - for i in 0..4096u128 { - let x = (i.wrapping_mul(1_146_989_321_207u128)) - ^ (i.rotate_left(37)) - ^ 0xA5A5_A5A5_A5A5_A5A5_A5A5_A5A5_A5A5_A5A5u128; - assert_eq!(reduce_u128_divfree(x, modulus), x % modulus); - } - } -} diff --git a/src/algebra/backend/traits.rs b/src/algebra/backend/traits.rs index d6c8a5a0..118a5f1f 100644 --- a/src/algebra/backend/traits.rs +++ b/src/algebra/backend/traits.rs @@ -1,56 +1,59 @@ //! Backend traits for CRT+NTT execution semantics. +//! +//! All traits are generic over `W: PrimeWidth` to support both +//! `i16` (primes < 2^14) and `i32` (primes < 2^30) NTT backends. use crate::algebra::ntt::butterfly::NttTwiddles; -use crate::algebra::ntt::crt::QData; -use crate::algebra::ntt::prime::{MontCoeff, NttPrime}; +use crate::algebra::ntt::crt::GarnerData; +use crate::algebra::ntt::prime::{MontCoeff, NttPrime, PrimeWidth}; use crate::algebra::ring::CrtNttConvertibleField; /// Per-prime arithmetic primitives used by CRT+NTT domains. -pub trait NttPrimeOps { +pub trait NttPrimeOps { /// Convert canonical coefficient to backend prime representation. - fn from_canonical(prime: NttPrime, value: i16) -> MontCoeff; + fn from_canonical(prime: NttPrime, value: W) -> MontCoeff; /// Convert backend prime representation back to canonical coefficient. - fn to_canonical(prime: NttPrime, value: MontCoeff) -> i16; + fn to_canonical(prime: NttPrime, value: MontCoeff) -> W; - /// Range-reduce one backend prime coefficient. - fn reduce(prime: NttPrime, value: MontCoeff) -> MontCoeff; + /// Range-reduce one coefficient from `(-2p, 2p)` to `(-p, p)`. + fn reduce_range(prime: NttPrime, value: MontCoeff) -> MontCoeff; /// Pointwise multiplication in backend prime representation. fn pointwise_mul( - prime: NttPrime, - out: &mut [MontCoeff; D], - lhs: &[MontCoeff; D], - rhs: &[MontCoeff; D], + prime: NttPrime, + out: &mut [MontCoeff; D], + lhs: &[MontCoeff; D], + rhs: &[MontCoeff; D], ); } /// Forward/inverse transform kernels for one NTT limb. -pub trait NttTransform { +pub trait NttTransform { /// Forward transform from coefficient limb to NTT limb. - fn forward_ntt(limb: &mut [MontCoeff; D], prime: NttPrime, twiddles: &NttTwiddles); + fn forward_ntt(limb: &mut [MontCoeff; D], prime: NttPrime, twiddles: &NttTwiddles); /// Inverse transform from NTT limb to coefficient limb. - fn inverse_ntt(limb: &mut [MontCoeff; D], prime: NttPrime, twiddles: &NttTwiddles); + fn inverse_ntt(limb: &mut [MontCoeff; D], prime: NttPrime, twiddles: &NttTwiddles); } -/// CRT reconstruction contract from per-prime canonical coefficients. -pub trait CrtReconstruct { +/// CRT reconstruction from per-prime canonical coefficients via Garner's algorithm. +pub trait CrtReconstruct { /// Reconstruct coefficient-domain values from canonical CRT residues. fn reconstruct( - primes: &[NttPrime; K], - canonical_limbs: &[[i16; D]; K], - qdata: &QData, + primes: &[NttPrime; K], + canonical_limbs: &[[W; D]; K], + garner: &GarnerData, ) -> [F; D]; } /// Convenience composition trait for full ring backend capability. -pub trait RingBackend: - NttPrimeOps + NttTransform + CrtReconstruct +pub trait RingBackend: + NttPrimeOps + NttTransform + CrtReconstruct { } -impl RingBackend for T where - T: NttPrimeOps + NttTransform + CrtReconstruct +impl RingBackend for T where + T: NttPrimeOps + NttTransform + CrtReconstruct { } diff --git a/src/algebra/domains/crt_ntt.rs b/src/algebra/domains/crt_ntt.rs index 48fe2263..b46cc391 100644 --- a/src/algebra/domains/crt_ntt.rs +++ b/src/algebra/domains/crt_ntt.rs @@ -3,4 +3,4 @@ use crate::algebra::ring::CyclotomicCrtNtt; /// CRT+NTT-domain ring representation. -pub type CrtNttDomain = CyclotomicCrtNtt; +pub type CrtNttDomain = CyclotomicCrtNtt; diff --git a/src/algebra/mod.rs b/src/algebra/mod.rs index 8a5cc194..321dbc3c 100644 --- a/src/algebra/mod.rs +++ b/src/algebra/mod.rs @@ -27,7 +27,7 @@ pub use fields::{ }; pub use module::VectorModule; pub use ntt::tables; -pub use ntt::{LimbQ, MontCoeff, NttPrime, QData, RADIX_BITS}; +pub use ntt::{GarnerData, LimbQ, MontCoeff, NttPrime, PrimeWidth, RADIX_BITS}; pub use ring::{ CrtNttConvertibleField, CyclotomicCrtNtt, CyclotomicRing, SparseChallenge, SparseChallengeConfig, diff --git a/src/algebra/ntt/butterfly.rs b/src/algebra/ntt/butterfly.rs index 9f58da94..01b7169e 100644 --- a/src/algebra/ntt/butterfly.rs +++ b/src/algebra/ntt/butterfly.rs @@ -4,25 +4,20 @@ //! the twist factors for `X^D + 1` are folded directly into the twiddles. //! Twiddle factors are powers of `psi`, a primitive `2D`-th root of unity //! (`psi^D = -1 mod p`), rather than a `D`-th root. -//! -//! TODO(perf): migrate twiddle tables to compile-time `const` arrays once -//! parameter sets are finalized. -use super::prime::{MontCoeff, NttPrime}; +use super::prime::{MontCoeff, NttPrime, PrimeWidth}; /// Precomputed twiddle factors for a specific prime and degree `D`. /// /// `D` must be a power of two. -pub struct NttTwiddles { +pub struct NttTwiddles { /// Twiddle factors in Montgomery form, indexed by bit-reversed position. - /// `zetas[i] = from_canonical(psi^{brv(i)})` for `i = 0..D`. - /// Index 0 is unused by the forward NTT but set to the Montgomery form of 1. - pub(crate) zetas: [MontCoeff; D], + pub(crate) zetas: [MontCoeff; D], /// `D^{-1} mod p` in Montgomery form, used for inverse NTT final scaling. - pub(crate) d_inv: MontCoeff, + pub(crate) d_inv: MontCoeff, } -impl NttTwiddles { +impl NttTwiddles { /// Compute twiddle factors for the given prime. /// /// Finds a primitive `2D`-th root of unity mod `p`, then fills the @@ -32,9 +27,9 @@ impl NttTwiddles { /// # Panics /// /// Panics if `D` is not a power of two, or if `2D` does not divide `p - 1`. - pub fn compute(prime: NttPrime) -> Self { + pub fn compute(prime: NttPrime) -> Self { assert!(D.is_power_of_two(), "D must be a power of two"); - let p = prime.p as i64; + let p = prime.p.to_i64(); assert!( (p - 1) % (2 * D as i64) == 0, "2D must divide p - 1 for NTT roots to exist" @@ -43,15 +38,15 @@ impl NttTwiddles { let n = D.trailing_zeros(); let psi = find_primitive_root_2d(p, D); - let mut zetas = [MontCoeff::from_raw(0); D]; + let mut zetas = [MontCoeff::from_raw(W::default()); D]; for (i, z) in zetas.iter_mut().enumerate() { let brv_i = bit_reverse(i, n); - let power = pow_mod(psi, brv_i as i64, p) as i16; - *z = prime.from_canonical(power); + let power = pow_mod(psi, brv_i as i64, p); + *z = prime.from_canonical(W::from_i64(power)); } - let d_inv_canonical = pow_mod(D as i64, p - 2, p) as i16; - let d_inv = prime.from_canonical(d_inv_canonical); + let d_inv_canonical = pow_mod(D as i64, p - 2, p); + let d_inv = prime.from_canonical(W::from_i64(d_inv_canonical)); Self { zetas, d_inv } } @@ -60,12 +55,13 @@ impl NttTwiddles { /// Forward negacyclic NTT (Cooley-Tukey, decimation-in-time). /// /// Transforms `D` coefficients in-place from coefficient form to NTT -/// evaluation form. Both outputs of each butterfly are Barrett-reduced +/// evaluation form. Both outputs of each butterfly are range-reduced /// to prevent overflow. -/// -/// After this call, the coefficients represent evaluations at -/// `psi^{2*brv(i)+1}` for `i = 0..D-1`. -pub fn forward_ntt(a: &mut [MontCoeff; D], prime: NttPrime, tw: &NttTwiddles) { +pub fn forward_ntt( + a: &mut [MontCoeff; D], + prime: NttPrime, + tw: &NttTwiddles, +) { let mut k = 1usize; let mut len = D / 2; while len >= 1 { @@ -77,8 +73,8 @@ pub fn forward_ntt(a: &mut [MontCoeff; D], prime: NttPrime, tw: let t = prime.mul(a[j + len], zeta); let diff = a[j].raw().wrapping_sub(t.raw()); let sum = a[j].raw().wrapping_add(t.raw()); - a[j + len] = prime.reduce(MontCoeff::from_raw(diff)); - a[j] = prime.reduce(MontCoeff::from_raw(sum)); + a[j + len] = prime.reduce_range(MontCoeff::from_raw(diff)); + a[j] = prime.reduce_range(MontCoeff::from_raw(sum)); } start += 2 * len; } @@ -90,7 +86,11 @@ pub fn forward_ntt(a: &mut [MontCoeff; D], prime: NttPrime, tw: /// /// Transforms `D` evaluations in-place back to coefficient form. /// Includes the final `D^{-1}` scaling. -pub fn inverse_ntt(a: &mut [MontCoeff; D], prime: NttPrime, tw: &NttTwiddles) { +pub fn inverse_ntt( + a: &mut [MontCoeff; D], + prime: NttPrime, + tw: &NttTwiddles, +) { let mut k = D - 1; let mut len = 1; while len <= D / 2 { @@ -102,8 +102,7 @@ pub fn inverse_ntt(a: &mut [MontCoeff; D], prime: NttPrime, tw: let t = a[j]; let sum = t.raw().wrapping_add(a[j + len].raw()); let diff = t.raw().wrapping_sub(a[j + len].raw()); - a[j] = prime.reduce(MontCoeff::from_raw(sum)); - // Multiply difference by negative twiddle: negate zeta. + a[j] = prime.reduce_range(MontCoeff::from_raw(sum)); let neg_zeta = MontCoeff::from_raw(zeta.raw().wrapping_neg()); a[j + len] = prime.mul(MontCoeff::from_raw(diff), neg_zeta); } @@ -112,22 +111,17 @@ pub fn inverse_ntt(a: &mut [MontCoeff; D], prime: NttPrime, tw: len *= 2; } - // Scale by D^{-1}. for c in a.iter_mut() { *c = prime.mul(*c, tw.d_inv); } } -/// Bit-reverse an `n`-bit integer. fn bit_reverse(x: usize, n: u32) -> usize { x.reverse_bits() >> (usize::BITS - n) } /// Find a primitive `2D`-th root of unity mod `p`. -/// -/// Returns `psi` in `[0, p)` such that `psi^D = -1 mod p`. fn find_primitive_root_2d(p: i64, d: usize) -> i64 { - // Find the smallest quadratic non-residue a (i.e., a^{(p-1)/2} = -1 mod p). let half = (p - 1) / 2; let exp = (p - 1) / (2 * d as i64); for a in 2..p { diff --git a/src/algebra/ntt/crt.rs b/src/algebra/ntt/crt.rs index 7df39710..e0897937 100644 --- a/src/algebra/ntt/crt.rs +++ b/src/algebra/ntt/crt.rs @@ -1,17 +1,56 @@ -//! Limb-based helpers for `q = 2^k - c` style moduli. -//! -//! Big-`q` coefficients are stored in radix `2^14` limbs. -//! This module provides a portable scalar representation with the same limb layout. +//! CRT helpers: Garner reconstruction and limb-based modular arithmetic. use std::cmp::Ordering; use std::fmt; use std::ops::{Add, Sub}; +use super::prime::{NttPrime, PrimeWidth}; + /// Limb radix bit-width (`2^14`). pub const RADIX_BITS: u32 = 14; const RADIX: i32 = 1 << RADIX_BITS; const RADIX_MASK: i32 = RADIX - 1; +/// Precomputed Garner inverse table for CRT reconstruction. +/// +/// `gamma[i][j]` = `p_j^{-1} mod p_i` for `j < i`. Upper triangle and +/// diagonal entries are zero (unused). +#[derive(Debug, Clone, PartialEq, Eq)] +pub struct GarnerData { + /// `gamma[i][j]` = `p_j^{-1} mod p_i` for `j < i`. + pub gamma: [[W; K]; K], +} + +impl GarnerData { + /// Compute Garner constants from a set of NTT primes. + pub fn compute(primes: &[NttPrime; K]) -> Self { + let mut gamma = [[W::default(); K]; K]; + for i in 1..K { + let pi = primes[i].p.to_i64(); + #[allow(clippy::needless_range_loop)] + for j in 0..i { + let pj = primes[j].p.to_i64(); + let inv = mod_inverse_i64(pj, pi); + gamma[i][j] = W::from_i64(inv); + } + } + Self { gamma } + } +} + +/// Modular inverse via extended GCD, operating in `i64`. +fn mod_inverse_i64(a: i64, modulus: i64) -> i64 { + let (mut t, mut new_t) = (0i64, 1i64); + let (mut r, mut new_r) = (modulus, ((a % modulus) + modulus) % modulus); + while new_r != 0 { + let q = r / new_r; + (t, new_t) = (new_t, t - q * new_t); + (r, new_r) = (new_r, r - q * new_r); + } + assert_eq!(r, 1, "modular inverse does not exist"); + ((t % modulus) + modulus) % modulus +} + /// Fixed-width radix-`2^14` integer. /// /// Limbs are little-endian: `limbs[0]` is least significant. @@ -44,7 +83,6 @@ impl LimbQ { /// Conditional subtraction: if `self >= modulus`, return `self - modulus` (branchless). #[inline] pub fn csub_mod(self, modulus: Self) -> Self { - // Compute self - modulus, tracking the final borrow. let mut diff = [0u16; L]; let mut borrow = 0i32; for (i, df) in diff.iter_mut().enumerate() { @@ -56,8 +94,6 @@ impl LimbQ { *df = d as u16; } } - // borrow = -1 if self < modulus (underflowed), 0 otherwise. - // Branchless select: mask = 0xFFFF if underflow (keep self), 0 if not (keep diff). let mask = borrow as u16; let mut result = [0u16; L]; for (i, r) in result.iter_mut().enumerate() { @@ -164,26 +200,3 @@ impl fmt::Display for LimbQ { } } } - -/// CRT/q constants for a given parameter set. -#[derive(Debug, Clone, Copy, PartialEq, Eq)] -pub struct QData { - /// The modulus `q` in radix-`2^14` limbs. - pub q: LimbQ, - /// `-P mod q` in limbs (where `P = prod p_i`). - pub pmq: LimbQ, - /// `P/p_i mod q` in limbs. - pub xvec: [LimbQ; K], - /// `k` in `q = 2^k - c`. - pub logq: u32, - /// `c` in `q = 2^k - c`. - pub qoff: u16, -} - -impl QData { - /// `q` as `u128` when representable. - #[inline] - pub fn q_u128(self) -> Option { - u128::try_from(self.q).ok() - } -} diff --git a/src/algebra/ntt/mod.rs b/src/algebra/ntt/mod.rs index fffbf284..f2536363 100644 --- a/src/algebra/ntt/mod.rs +++ b/src/algebra/ntt/mod.rs @@ -6,5 +6,5 @@ pub mod prime; pub mod tables; pub use butterfly::NttTwiddles; -pub use crt::{LimbQ, QData, RADIX_BITS}; -pub use prime::{MontCoeff, NttPrime}; +pub use crt::{GarnerData, LimbQ, RADIX_BITS}; +pub use prime::{MontCoeff, NttPrime, PrimeWidth}; diff --git a/src/algebra/ntt/prime.rs b/src/algebra/ntt/prime.rs index 9b19cf22..038139ff 100644 --- a/src/algebra/ntt/prime.rs +++ b/src/algebra/ntt/prime.rs @@ -1,143 +1,316 @@ -//! Small-prime arithmetic kernels for NTT-friendly parameter sets. +//! NTT prime arithmetic kernels generic over coefficient width. //! //! Per-prime scalar operations: //! - Montgomery multiplication ([`NttPrime::mul`]) -//! - Barrett-style reduction ([`NttPrime::reduce`]) -//! - Branchless conditional add/sub +//! - Branchless conditional add/sub and range reduction //! //! Coefficients in Montgomery domain are wrapped in [`MontCoeff`] to prevent -//! accidental mixing with canonical `i16` values. +//! accidental mixing with canonical values. +//! +//! The [`PrimeWidth`] trait abstracts over `i16` (R = 2^16, for primes < 2^14) +//! and `i32` (R = 2^32, for primes < 2^30). All NTT types are generic over +//! `W: PrimeWidth`; monomorphization produces optimal code for each width. use std::fmt; +mod sealed { + pub trait Sealed {} + impl Sealed for i16 {} + impl Sealed for i32 {} +} + +/// Integer width abstraction for NTT prime arithmetic. +/// +/// Sealed with exactly two implementations: `i16` and `i32`. +pub trait PrimeWidth: + sealed::Sealed + Copy + Clone + Eq + Default + fmt::Debug + Send + Sync + 'static +{ + /// Double-width type for intermediate Montgomery products. + type Wide: Copy + Clone; + + /// log2(R) for Montgomery reduction: 16 for `i16`, 32 for `i32`. + const R_LOG: u32; + + /// Widening multiply: `a * b` as `Wide`. + fn wide_mul(a: Self, b: Self) -> Self::Wide; + + /// Truncate wide value to narrow (low half, i.e. mod R). + fn truncate(w: Self::Wide) -> Self; + + /// Arithmetic right shift of wide value by `R_LOG` bits. + fn wide_shift(w: Self::Wide) -> Self; + + /// Wide subtraction (wrapping). + fn wide_sub(a: Self::Wide, b: Self::Wide) -> Self::Wide; + + /// Wrapping addition. + fn wrapping_add(self, rhs: Self) -> Self; + /// Wrapping subtraction. + fn wrapping_sub(self, rhs: Self) -> Self; + /// Wrapping multiplication. + fn wrapping_mul(self, rhs: Self) -> Self; + /// Wrapping negation. + fn wrapping_neg(self) -> Self; + + /// Arithmetic right shift by `BITS - 1`: all-1s if negative, all-0s otherwise. + fn sign_mask(self) -> Self; + + /// Bitwise AND. + fn bitand(self, rhs: Self) -> Self; + + /// Convert from `i64` (truncating). + fn from_i64(v: i64) -> Self; + /// Convert to `i64` (sign-extending). + fn to_i64(self) -> i64; +} + +impl PrimeWidth for i16 { + type Wide = i32; + const R_LOG: u32 = 16; + + #[inline] + fn wide_mul(a: Self, b: Self) -> i32 { + (a as i32) * (b as i32) + } + #[inline] + fn truncate(w: i32) -> Self { + w as i16 + } + #[inline] + fn wide_shift(w: i32) -> Self { + (w >> 16) as i16 + } + #[inline] + fn wide_sub(a: i32, b: i32) -> i32 { + a.wrapping_sub(b) + } + #[inline] + fn wrapping_add(self, rhs: Self) -> Self { + i16::wrapping_add(self, rhs) + } + #[inline] + fn wrapping_sub(self, rhs: Self) -> Self { + i16::wrapping_sub(self, rhs) + } + #[inline] + fn wrapping_mul(self, rhs: Self) -> Self { + i16::wrapping_mul(self, rhs) + } + #[inline] + fn wrapping_neg(self) -> Self { + i16::wrapping_neg(self) + } + #[inline] + fn sign_mask(self) -> Self { + self >> 15 + } + #[inline] + fn bitand(self, rhs: Self) -> Self { + self & rhs + } + #[inline] + fn from_i64(v: i64) -> Self { + v as i16 + } + #[inline] + fn to_i64(self) -> i64 { + self as i64 + } +} + +impl PrimeWidth for i32 { + type Wide = i64; + const R_LOG: u32 = 32; + + #[inline] + fn wide_mul(a: Self, b: Self) -> i64 { + (a as i64) * (b as i64) + } + #[inline] + fn truncate(w: i64) -> Self { + w as i32 + } + #[inline] + fn wide_shift(w: i64) -> Self { + (w >> 32) as i32 + } + #[inline] + fn wide_sub(a: i64, b: i64) -> i64 { + a.wrapping_sub(b) + } + #[inline] + fn wrapping_add(self, rhs: Self) -> Self { + i32::wrapping_add(self, rhs) + } + #[inline] + fn wrapping_sub(self, rhs: Self) -> Self { + i32::wrapping_sub(self, rhs) + } + #[inline] + fn wrapping_mul(self, rhs: Self) -> Self { + i32::wrapping_mul(self, rhs) + } + #[inline] + fn wrapping_neg(self) -> Self { + i32::wrapping_neg(self) + } + #[inline] + fn sign_mask(self) -> Self { + self >> 31 + } + #[inline] + fn bitand(self, rhs: Self) -> Self { + self & rhs + } + #[inline] + fn from_i64(v: i64) -> Self { + v as i32 + } + #[inline] + fn to_i64(self) -> i64 { + self as i64 + } +} + /// A coefficient in Montgomery domain for an NTT prime. /// -/// Wraps an `i16` representing `a * R mod p` (where `R = 2^16`). +/// Wraps a `W` representing `a * R mod p` (where `R = 2^{W::R_LOG}`). /// Use [`NttPrime::from_canonical`] to enter and [`NttPrime::to_canonical`] /// to leave Montgomery domain. #[derive(Clone, Copy, PartialEq, Eq, Default)] #[repr(transparent)] -pub struct MontCoeff(i16); +pub struct MontCoeff(W); -impl MontCoeff { +impl MontCoeff { /// Wrap a raw Montgomery-domain value. - /// - /// The caller must ensure `val` is a valid Montgomery-domain representative - /// for the intended prime. #[inline] - pub const fn from_raw(val: i16) -> Self { + pub fn from_raw(val: W) -> Self { Self(val) } - /// Extract the raw `i16` (still in Montgomery domain). + /// Extract the raw value (still in Montgomery domain). #[inline] - pub const fn raw(self) -> i16 { + pub fn raw(self) -> W { self.0 } } -impl fmt::Debug for MontCoeff { +impl fmt::Debug for MontCoeff { fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result { - write!(f, "Mont({})", self.0) + write!(f, "Mont({:?})", self.0) } } -/// Per-prime constants for NTT arithmetic over a small prime `p < 2^14`. +/// Per-prime constants for NTT arithmetic. /// -/// All scalar operations use Montgomery representation internally. Constants -/// are stored as raw `i16` in centered signed form. +/// Generic over `W: PrimeWidth` — use `i16` for primes below 2^14 (R = 2^16), +/// or `i32` for primes below 2^30 (R = 2^32). #[derive(Debug, Clone, Copy, PartialEq, Eq)] -pub struct NttPrime { +pub struct NttPrime { /// Prime modulus. - pub p: i16, - /// `p^{-1} mod 2^{16}` (centered signed). Used in Montgomery reduction. - pub pinv: i16, - /// `round(2^{27} / p)` for Barrett-style range reduction. - pub v: i16, - /// `R mod p` where `R = 2^{16}` (centered signed). Montgomery form of 1. - pub mont: i16, + pub p: W, + /// `p^{-1} mod R` (centered signed). Used in Montgomery reduction. + pub pinv: W, + /// `R mod p` (centered signed). Montgomery form of 1. + pub mont: W, /// `R^2 mod p` (centered signed). Used for canonical → Montgomery conversion. - pub montsq: i16, - /// NTT injection scale factor (centered signed). - pub s: i16, - /// Inverse-NTT post-scale factor (centered signed). - pub f: i16, - /// CRT reconstruction factor (centered signed). - pub t: i16, + pub montsq: W, } -impl NttPrime { - /// Montgomery product: `a * b * R^{-1} mod p` where `R = 2^{16}`. - #[inline] - pub fn mul(self, a: MontCoeff, b: MontCoeff) -> MontCoeff { - MontCoeff(self.mont_mul_raw(a.0, b.0)) +impl NttPrime { + /// Derive all Montgomery constants from a raw prime value. + pub fn compute(p: W) -> Self { + let p_i64 = p.to_i64(); + debug_assert!(p_i64 > 1 && p_i64 % 2 == 1, "NTT prime must be odd and > 1"); + + // pinv via Newton's method: x_{n+1} = x_n * (2 - p * x_n). + // 5 iterations gives correctness mod 2^32 (sufficient for both i16 and i32). + let mut pinv: i64 = 1; + for _ in 0..5 { + pinv = pinv.wrapping_mul(2i64.wrapping_sub(p_i64.wrapping_mul(pinv))); + } + let pinv = W::from_i64(pinv); + + let half = p_i64 / 2; + let center = |x: i64| -> W { W::from_i64(if x > half { x - p_i64 } else { x }) }; + + let r_mod_p = ((1i128 << W::R_LOG) % (p_i64 as i128)) as i64; + let mont = center(r_mod_p); + + let rsq_mod_p = ((1i128 << (2 * W::R_LOG)) % (p_i64 as i128)) as i64; + let montsq = center(rsq_mod_p); + + Self { + p, + pinv, + mont, + montsq, + } } - /// Raw Montgomery multiply on bare `i16` values. - /// - /// Computes `(a*b - ((a*b mod R) * p_inv mod R) * p) / R`. + /// Montgomery product: `a * b * R^{-1} mod p`. #[inline] - pub(crate) fn mont_mul_raw(self, a: i16, b: i16) -> i16 { - let c = (a as i32) * (b as i32); - let t = (c as i16).wrapping_mul(self.pinv) as i32; - ((c - t * (self.p as i32)) >> 16) as i16 + pub fn mul(self, a: MontCoeff, b: MontCoeff) -> MontCoeff { + MontCoeff(self.mont_mul_raw(a.0, b.0)) } - /// Barrett-style reduction: tighten a Montgomery coefficient's range - /// toward `[0, p)` without full normalization. + /// Raw Montgomery multiply on bare `W` values. #[inline] - pub fn reduce(self, a: MontCoeff) -> MontCoeff { - let val = a.0; - let mut t = (((self.v as i32) * (val as i32) + (1 << 26)) >> 27) as i16; - t = t.wrapping_mul(self.p); - MontCoeff(val.wrapping_sub(t)) + pub(crate) fn mont_mul_raw(self, a: W, b: W) -> W { + let c = W::wide_mul(a, b); + let t = W::truncate(c).wrapping_mul(self.pinv); + let tp = W::wide_mul(t, self.p); + W::wide_shift(W::wide_sub(c, tp)) } /// Conditionally subtract `p` if `a >= p` (branchless). #[inline] - pub fn csubp(self, a: MontCoeff) -> MontCoeff { + pub fn csubp(self, a: MontCoeff) -> MontCoeff { let diff = a.0.wrapping_sub(self.p); - let mask = diff >> 15; - MontCoeff(diff.wrapping_add(mask & self.p)) + let mask = diff.sign_mask(); + MontCoeff(diff.wrapping_add(mask.bitand(self.p))) } /// Conditionally add `p` if `a < 0` (branchless). #[inline] - pub fn caddp(self, a: MontCoeff) -> MontCoeff { - let mask = a.0 >> 15; - MontCoeff(a.0.wrapping_add(mask & self.p)) + pub fn caddp(self, a: MontCoeff) -> MontCoeff { + let mask = a.0.sign_mask(); + MontCoeff(a.0.wrapping_add(mask.bitand(self.p))) + } + + /// Range-reduce from `(-2p, 2p)` to `(-p, p)`. + #[inline] + pub fn reduce_range(self, a: MontCoeff) -> MontCoeff { + self.caddp(self.csubp(a)) } /// Fully normalize a Montgomery coefficient to `[0, p)`. #[inline] - pub fn normalize(self, a: MontCoeff) -> MontCoeff { - self.csubp(self.caddp(self.reduce(a))) + pub fn normalize(self, a: MontCoeff) -> MontCoeff { + self.csubp(self.caddp(a)) } - /// Convert a canonical `i16` into Montgomery domain: `a ↦ a * R mod p`. + /// Convert a canonical value into Montgomery domain: `a ↦ a * R mod p`. #[inline] - pub fn from_canonical(self, a: i16) -> MontCoeff { + pub fn from_canonical(self, a: W) -> MontCoeff { MontCoeff(self.mont_mul_raw(a, self.montsq)) } /// Convert from Montgomery domain to canonical `[0, p)`. #[inline] - pub fn to_canonical(self, a: MontCoeff) -> i16 { - let raw = MontCoeff(self.mont_mul_raw(a.0, 1)); + pub fn to_canonical(self, a: MontCoeff) -> W { + let raw = MontCoeff(self.mont_mul_raw(a.0, W::from_i64(1))); self.normalize(raw).0 } /// Center a canonical value from approximately `(-p, p)` into `[-p/2, p/2)`. - /// - /// First conditionally adds `p` (if negative), then subtracts `p` (if > p/2). - /// Both steps are branchless. #[inline] - pub fn center(self, a: i16) -> i16 { - let mask_neg = a >> 15; - let canonical = a.wrapping_add(mask_neg & self.p); - let half = self.p / 2; - let needs_sub = half.wrapping_sub(canonical) >> 15; - canonical.wrapping_add(needs_sub & self.p.wrapping_neg()) + pub fn center(self, a: W) -> W { + let mask_neg = a.sign_mask(); + let canonical = a.wrapping_add(mask_neg.bitand(self.p)); + let half = W::from_i64(self.p.to_i64() / 2); + let needs_sub = half.wrapping_sub(canonical).sign_mask(); + canonical.wrapping_add(needs_sub.bitand(self.p.wrapping_neg())) } /// Pointwise Montgomery multiplication of two coefficient slices. @@ -146,7 +319,12 @@ impl NttPrime { /// /// Panics if slices have different lengths. #[inline] - pub fn pointwise_mul(self, out: &mut [MontCoeff], lhs: &[MontCoeff], rhs: &[MontCoeff]) { + pub fn pointwise_mul( + self, + out: &mut [MontCoeff], + lhs: &[MontCoeff], + rhs: &[MontCoeff], + ) { assert_eq!(out.len(), lhs.len()); assert_eq!(lhs.len(), rhs.len()); for ((o, a), b) in out.iter_mut().zip(lhs.iter()).zip(rhs.iter()) { @@ -156,23 +334,23 @@ impl NttPrime { /// In-place Montgomery scaling by a constant. #[inline] - pub fn scale_in_place(self, coeffs: &mut [MontCoeff], scalar: MontCoeff) { + pub fn scale_in_place(self, coeffs: &mut [MontCoeff], scalar: MontCoeff) { for c in coeffs { *c = self.mul(*c, scalar); } } - /// In-place Barrett reduction on a coefficient slice. + /// In-place range reduction on a coefficient slice. #[inline] - pub fn reduce_in_place(self, coeffs: &mut [MontCoeff]) { + pub fn reduce_range_in_place(self, coeffs: &mut [MontCoeff]) { for c in coeffs { - *c = self.reduce(*c); + *c = self.reduce_range(*c); } } /// In-place centering of canonical values to `[-p/2, p/2)`. #[inline] - pub fn center_slice(self, coeffs: &mut [i16]) { + pub fn center_slice(self, coeffs: &mut [W]) { for c in coeffs { *c = self.center(*c); } diff --git a/src/algebra/ntt/tables.rs b/src/algebra/ntt/tables.rs index 613ccc95..bea49764 100644 --- a/src/algebra/ntt/tables.rs +++ b/src/algebra/ntt/tables.rs @@ -1,8 +1,9 @@ //! Deterministic parameter presets for small-prime CRT arithmetic. //! -//! Initial scope: `logq = 32` preset with six NTT-friendly primes. +//! Q32: `logq = 32` with six `i16` NTT-friendly primes (D ≤ 64). +//! Q128: `logq = 128` with five `i32` NTT-friendly primes (D ≤ 1024). -use super::crt::{LimbQ, QData}; +use super::crt::GarnerData; use super::prime::NttPrime; /// Polynomial degree for the base ring `Z_q[X]/(X^d + 1)`. @@ -11,139 +12,137 @@ pub const RING_DEGREE: usize = 64; /// Number of CRT primes for the `logq = 32` parameter set. pub const Q32_NUM_PRIMES: usize = 6; -/// Number of radix-`2^14` limbs for `logq = 32`. -pub const Q32_NUM_LIMBS: usize = 3; - -/// Offset `c` in `q = 2^32 - c`. -pub const Q32_OFFSET: u16 = 99; - -/// `log2(q)` for the `logq = 32` parameter set. -pub const Q32_LOG_MODULUS: u32 = 32; - /// The modulus `q = 2^32 - 99`. -pub const Q32_MODULUS: u64 = (1u64 << 32) - (Q32_OFFSET as u64); +pub const Q32_MODULUS: u64 = (1u64 << 32) - 99; /// CRT primes and per-prime Montgomery constants for `logq = 32`. -pub const Q32_PRIMES: [NttPrime; Q32_NUM_PRIMES] = [ +/// +/// All constants are for `R = 2^16` (i16 width). +pub const Q32_PRIMES: [NttPrime; Q32_NUM_PRIMES] = [ NttPrime { p: 13697, pinv: 2689, - v: 9799, mont: -2949, montsq: -994, - s: 4705, - f: 3540, - t: -5758, }, NttPrime { p: 13441, pinv: 2945, - v: 9986, mont: -1669, montsq: 3274, - s: 3777, - f: -5468, - t: -1680, }, NttPrime { p: 13313, pinv: -13311, - v: 10082, mont: -1029, montsq: -6199, - s: 3325, - f: 4553, - t: -948, }, NttPrime { p: 12289, pinv: -12287, - v: 10922, mont: 4091, montsq: -1337, - s: -3, - f: 354, - t: 4472, }, NttPrime { p: 12161, pinv: 4225, - v: 11037, mont: 4731, montsq: -6040, - s: -383, - f: 5993, - t: 1653, }, NttPrime { p: 11777, pinv: -11775, - v: 11397, mont: -5126, montsq: 1389, - s: -1475, - f: -3812, - t: 1191, }, ]; -/// CRT limb constants for the `logq = 32` parameter set. -pub const Q32_DATA: QData = QData { - q: LimbQ::from_limbs([16285, 16383, 15]), - pmq: LimbQ::from_limbs([8747, 999, 12]), - xvec: [ - LimbQ::from_limbs([9406, 14930, 2]), - LimbQ::from_limbs([9295, 15936, 2]), - LimbQ::from_limbs([16336, 1061, 5]), - LimbQ::from_limbs([1006, 14584, 9]), - LimbQ::from_limbs([1315, 5273, 10]), - LimbQ::from_limbs([7927, 10169, 6]), - ], - logq: Q32_LOG_MODULUS, - qoff: Q32_OFFSET, -}; +/// Garner CRT reconstruction constants for Q32. +pub fn q32_garner() -> GarnerData { + GarnerData::compute(&Q32_PRIMES) +} + +/// Number of CRT primes for the `logq = 128` parameter set. +pub const Q128_NUM_PRIMES: usize = 5; + +/// The modulus `q = 2^128 - 275`. +pub const Q128_MODULUS: u128 = u128::MAX - 274; + +/// Raw 30-bit primes for Q128, each satisfying `2048 | (p - 1)`. +pub const Q128_RAW_PRIMES: [i32; Q128_NUM_PRIMES] = + [1073707009, 1073698817, 1073692673, 1073682433, 1073668097]; + +/// CRT primes and per-prime Montgomery constants for `logq = 128`. +pub fn q128_primes() -> [NttPrime; Q128_NUM_PRIMES] { + std::array::from_fn(|k| NttPrime::compute(Q128_RAW_PRIMES[k])) +} + +/// Garner CRT reconstruction constants for Q128. +pub fn q128_garner() -> GarnerData { + let primes = q128_primes(); + GarnerData::compute(&primes) +} #[cfg(test)] mod tests { use super::*; - /// Verify that derived constants (`pinv`, `v`, `mont`, `montsq`) are - /// consistent with the prime `p`. Turns the magic numbers into auditable - /// values. #[test] - fn verify_prime_derived_constants() { + fn verify_q32_prime_derived_constants() { for prime in &Q32_PRIMES { - let p = prime.p; - - // pinv: p * pinv ≡ 1 (mod 2^16) - assert_eq!(p.wrapping_mul(prime.pinv), 1, "pinv failed for p={p}"); - - // v: round(2^27 / p) - let expected_v = ((1i64 << 27) + (p as i64 / 2)) / (p as i64); - assert_eq!(prime.v as i64, expected_v, "v failed for p={p}"); - - // mont: 2^16 mod p, centered to [-p/2, p/2) - let raw_mont = ((1i32 << 16) % (p as i32)) as i16; - let centered_mont = if raw_mont > p / 2 { - raw_mont - p - } else { - raw_mont - }; - assert_eq!(prime.mont, centered_mont, "mont failed for p={p}"); + let recomputed = NttPrime::compute(prime.p); + assert_eq!( + prime.pinv, recomputed.pinv, + "pinv mismatch for p={}", + prime.p + ); + assert_eq!( + prime.mont, recomputed.mont, + "mont mismatch for p={}", + prime.p + ); + assert_eq!( + prime.montsq, recomputed.montsq, + "montsq mismatch for p={}", + prime.p + ); + } + } - // montsq: 2^32 mod p, centered to [-p/2, p/2) - let raw_montsq = ((1i64 << 32) % (p as i64)) as i16; - let centered_montsq = if raw_montsq > p / 2 { - raw_montsq - p - } else { - raw_montsq - }; - assert_eq!(prime.montsq, centered_montsq, "montsq failed for p={p}"); + #[test] + fn verify_q128_primes_are_valid() { + let primes = q128_primes(); + for np in &primes { + let p = np.p as i64; + assert!(p > 1 && p % 2 == 1, "prime must be odd and > 1"); + assert_eq!( + (p - 1) % 2048, + 0, + "2048 must divide p-1 for D=1024 NTT (p={p})" + ); + // Verify pinv: p * pinv ≡ 1 (mod 2^32) + assert_eq!( + np.p.wrapping_mul(np.pinv), + 1, + "pinv verification failed for p={p}" + ); } } #[test] - fn verify_q_value() { - assert_eq!(Q32_DATA.q_u128().unwrap(), Q32_MODULUS as u128); + fn garner_data_is_consistent() { + let garner = q32_garner(); + for (i, &prime_i) in Q32_PRIMES.iter().enumerate().skip(1) { + let pi = prime_i.p as i64; + for (j, &prime_j) in Q32_PRIMES[..i].iter().enumerate() { + let pj = prime_j.p as i64; + let g = garner.gamma[i][j] as i64; + assert_eq!( + (pj * g) % pi, + 1, + "garner gamma[{i}][{j}] not inverse of p_{j} mod p_{i}" + ); + } + } } } diff --git a/src/algebra/ring/crt_ntt_repr.rs b/src/algebra/ring/crt_ntt_repr.rs index 18e4772a..2574769b 100644 --- a/src/algebra/ring/crt_ntt_repr.rs +++ b/src/algebra/ring/crt_ntt_repr.rs @@ -1,78 +1,39 @@ //! CRT+NTT-domain representation of cyclotomic ring elements. use crate::algebra::backend::{CrtReconstruct, NttPrimeOps, NttTransform, ScalarBackend}; -use crate::algebra::fields::{Fp128, Fp32, Fp64}; use crate::algebra::ntt::butterfly::NttTwiddles; -use crate::algebra::ntt::crt::QData; -use crate::algebra::ntt::prime::{MontCoeff, NttPrime}; +use crate::algebra::ntt::crt::GarnerData; +use crate::algebra::ntt::prime::{MontCoeff, NttPrime, PrimeWidth}; use crate::{CanonicalField, FieldCore}; use super::cyclotomic::CyclotomicRing; /// CRT+NTT-domain representation of a cyclotomic ring element. /// -/// Stores `K` arrays of `D` [`MontCoeff`] values, one per CRT prime. -/// Multiplication is pointwise per prime - O(K*D) vs O(D^2) for coefficient form. +/// Stores `K` arrays of `D` [`MontCoeff`] values, one per CRT prime. +/// Multiplication is pointwise per prime — O(K*D) vs O(D^2) for coefficient form. /// -/// Use [`CyclotomicCrtNtt::from_ring`] and [`CyclotomicCrtNtt::to_ring`] to convert -/// between coefficient and CRT+NTT domains. +/// Generic over: +/// - `W: PrimeWidth` — integer width (`i16` or `i32`) +/// - `K` — number of CRT primes +/// - `D` — polynomial degree #[derive(Debug, Clone, PartialEq, Eq)] -pub struct CyclotomicCrtNtt { - pub(crate) limbs: [[MontCoeff; D]; K], +pub struct CyclotomicCrtNtt { + pub(crate) limbs: [[MontCoeff; D]; K], } -/// Field types that can safely convert to and from the CRT+NTT representation. -pub trait CrtNttConvertibleField: FieldCore + CanonicalField { - /// Reduce this field element modulo a small prime `p`. - fn mod_small_prime(self, p: i16) -> i16; - - /// Reconstruct from a residue in `[0, q)` after CRT combination. - fn from_q_residue_u128(x: u128) -> Self; -} - -impl CrtNttConvertibleField for Fp32

{ - #[inline] - fn mod_small_prime(self, p: i16) -> i16 { - (self.to_canonical_u32() % (p as u32)) as i16 - } - - #[inline] - fn from_q_residue_u128(x: u128) -> Self { - let narrowed = u32::try_from(x).expect("CRT residue does not fit in u32"); - Self::from_u64(narrowed as u64) - } -} - -impl CrtNttConvertibleField for Fp64

{ - #[inline] - fn mod_small_prime(self, p: i16) -> i16 { - (self.to_canonical_u64() % (p as u64)) as i16 - } - - #[inline] - fn from_q_residue_u128(x: u128) -> Self { - let narrowed = u64::try_from(x).expect("CRT residue does not fit in u64"); - Self::from_u64(narrowed) - } -} - -impl CrtNttConvertibleField for Fp128

{ - #[inline] - fn mod_small_prime(self, p: i16) -> i16 { - (self.to_canonical_u128() % (p as u128)) as i16 - } +/// Field types that can convert to/from the CRT+NTT representation. +/// +/// Blanket-implemented for all `FieldCore + CanonicalField` types. +pub trait CrtNttConvertibleField: FieldCore + CanonicalField {} - #[inline] - fn from_q_residue_u128(x: u128) -> Self { - Self::from_canonical_u128_reduced(x) - } -} +impl CrtNttConvertibleField for F {} -impl CyclotomicCrtNtt { +impl CyclotomicCrtNtt { /// The additive identity (all zeros in every CRT limb). pub fn zero() -> Self { Self { - limbs: [[MontCoeff::from_raw(0); D]; K], + limbs: [[MontCoeff::from_raw(W::default()); D]; K], } } @@ -80,8 +41,8 @@ impl CyclotomicCrtNtt { /// using the default scalar backend. pub fn from_ring( ring: &CyclotomicRing, - primes: &[NttPrime; K], - twiddles: &[NttTwiddles; K], + primes: &[NttPrime; K], + twiddles: &[NttTwiddles; K], ) -> Self { Self::from_ring_with_backend::(ring, primes, twiddles) } @@ -90,18 +51,18 @@ impl CyclotomicCrtNtt { /// through an explicit backend implementation. pub fn from_ring_with_backend< F: CrtNttConvertibleField, - B: NttPrimeOps + NttTransform, + B: NttPrimeOps + NttTransform, >( ring: &CyclotomicRing, - primes: &[NttPrime; K], - twiddles: &[NttTwiddles; K], + primes: &[NttPrime; K], + twiddles: &[NttTwiddles; K], ) -> Self { - let mut limbs = [[MontCoeff::from_raw(0); D]; K]; + let mut limbs = [[MontCoeff::from_raw(W::default()); D]; K]; for ((limb, prime), tw) in limbs.iter_mut().zip(primes.iter()).zip(twiddles.iter()) { - // Reduce each coefficient mod p and convert to Montgomery form. for (dst, src) in limb.iter_mut().zip(ring.coeffs.iter()) { - let val = src.mod_small_prime(prime.p); - *dst = B::from_canonical(*prime, val); + let canonical = src.to_canonical_u128(); + let residue = (canonical % (prime.p.to_i64() as u128)) as i64; + *dst = B::from_canonical(*prime, W::from_i64(residue)); } B::forward_ntt(limb, *prime, tw); } @@ -110,28 +71,27 @@ impl CyclotomicCrtNtt { /// Convert from CRT+NTT domain back to coefficient form /// using the default scalar backend. - pub fn to_ring( + pub fn to_ring( &self, - primes: &[NttPrime; K], - twiddles: &[NttTwiddles; K], - qdata: &QData, + primes: &[NttPrime; K], + twiddles: &[NttTwiddles; K], + garner: &GarnerData, ) -> CyclotomicRing { - self.to_ring_with_backend::(primes, twiddles, qdata) + self.to_ring_with_backend::(primes, twiddles, garner) } /// Convert from CRT+NTT domain back to coefficient form /// through an explicit backend implementation. pub fn to_ring_with_backend< F: CrtNttConvertibleField, - B: NttPrimeOps + NttTransform + CrtReconstruct, - const L: usize, + B: NttPrimeOps + NttTransform + CrtReconstruct, >( &self, - primes: &[NttPrime; K], - twiddles: &[NttTwiddles; K], - qdata: &QData, + primes: &[NttPrime; K], + twiddles: &[NttTwiddles; K], + garner: &GarnerData, ) -> CyclotomicRing { - let mut canonical = [[0i16; D]; K]; + let mut canonical = [[W::default(); D]; K]; for (k, ((can, prime), tw)) in canonical .iter_mut() .zip(primes.iter()) @@ -145,94 +105,90 @@ impl CyclotomicCrtNtt { } } - let coeffs = B::reconstruct::(primes, &canonical, qdata); + let coeffs = B::reconstruct::(primes, &canonical, garner); CyclotomicRing::from_coefficients(coeffs) } /// Add another CRT+NTT element and reduce each coefficient with the matching /// prime to maintain valid Montgomery ranges using the scalar backend. - pub fn add_reduced(&self, rhs: &Self, primes: &[NttPrime; K]) -> Self { + pub fn add_reduced(&self, rhs: &Self, primes: &[NttPrime; K]) -> Self { self.add_reduced_with_backend::(rhs, primes) } /// Add another CRT+NTT element and reduce each coefficient with the matching /// prime through an explicit backend implementation. - pub fn add_reduced_with_backend>( + pub fn add_reduced_with_backend>( &self, rhs: &Self, - primes: &[NttPrime; K], + primes: &[NttPrime; K], ) -> Self { let mut out = self.clone(); for (k, (limb, rhs_limb)) in out.limbs.iter_mut().zip(rhs.limbs.iter()).enumerate() { let prime = primes[k]; for (a, b) in limb.iter_mut().zip(rhs_limb.iter()) { let sum = MontCoeff::from_raw(a.raw().wrapping_add(b.raw())); - *a = B::reduce(prime, sum); + *a = B::reduce_range(prime, sum); } } out } - /// Subtract another CRT+NTT element and reduce each coefficient with the - /// matching prime to maintain valid Montgomery ranges using the scalar backend. - pub fn sub_reduced(&self, rhs: &Self, primes: &[NttPrime; K]) -> Self { + /// Subtract another CRT+NTT element and reduce using the scalar backend. + pub fn sub_reduced(&self, rhs: &Self, primes: &[NttPrime; K]) -> Self { self.sub_reduced_with_backend::(rhs, primes) } - /// Subtract another CRT+NTT element and reduce each coefficient with the - /// matching prime through an explicit backend implementation. - pub fn sub_reduced_with_backend>( + /// Subtract another CRT+NTT element and reduce through an explicit backend. + pub fn sub_reduced_with_backend>( &self, rhs: &Self, - primes: &[NttPrime; K], + primes: &[NttPrime; K], ) -> Self { let mut out = self.clone(); for (k, (limb, rhs_limb)) in out.limbs.iter_mut().zip(rhs.limbs.iter()).enumerate() { let prime = primes[k]; for (a, b) in limb.iter_mut().zip(rhs_limb.iter()) { let diff = MontCoeff::from_raw(a.raw().wrapping_sub(b.raw())); - *a = B::reduce(prime, diff); + *a = B::reduce_range(prime, diff); } } out } - /// Negate each CRT+NTT coefficient and reduce with the matching prime - /// using the scalar backend. - pub fn neg_reduced(&self, primes: &[NttPrime; K]) -> Self { + /// Negate each CRT+NTT coefficient and reduce using the scalar backend. + pub fn neg_reduced(&self, primes: &[NttPrime; K]) -> Self { self.neg_reduced_with_backend::(primes) } - /// Negate each CRT+NTT coefficient and reduce with the matching prime - /// through an explicit backend implementation. - pub fn neg_reduced_with_backend>(&self, primes: &[NttPrime; K]) -> Self { + /// Negate each CRT+NTT coefficient and reduce through an explicit backend. + pub fn neg_reduced_with_backend>( + &self, + primes: &[NttPrime; K], + ) -> Self { let mut out = self.clone(); for (k, limb) in out.limbs.iter_mut().enumerate() { let prime = primes[k]; for a in limb.iter_mut() { let neg = MontCoeff::from_raw(a.raw().wrapping_neg()); - *a = B::reduce(prime, neg); + *a = B::reduce_range(prime, neg); } } out } - /// Pointwise multiplication in CRT+NTT domain. - /// - /// Each limb is multiplied pointwise using the corresponding prime's - /// Montgomery multiplication. - pub fn pointwise_mul(&self, rhs: &Self, primes: &[NttPrime; K]) -> Self { + /// Pointwise multiplication in CRT+NTT domain using the scalar backend. + pub fn pointwise_mul(&self, rhs: &Self, primes: &[NttPrime; K]) -> Self { self.pointwise_mul_with_backend::(rhs, primes) } - /// Pointwise multiplication in CRT+NTT domain through an explicit backend implementation. - pub fn pointwise_mul_with_backend>( + /// Pointwise multiplication in CRT+NTT domain through an explicit backend. + pub fn pointwise_mul_with_backend>( &self, rhs: &Self, - primes: &[NttPrime; K], + primes: &[NttPrime; K], ) -> Self { - let mut out = [[MontCoeff::from_raw(0); D]; K]; + let mut out = [[MontCoeff::from_raw(W::default()); D]; K]; for (k, ((o, a), b)) in out .iter_mut() .zip(self.limbs.iter()) diff --git a/tests/algebra.rs b/tests/algebra.rs index cbb51aeb..bac3c069 100644 --- a/tests/algebra.rs +++ b/tests/algebra.rs @@ -7,7 +7,10 @@ mod tests { use hachi_pcs::algebra::ntt::butterfly::{forward_ntt, inverse_ntt, NttTwiddles}; use hachi_pcs::algebra::poly::Poly; - use hachi_pcs::algebra::tables::{Q32_DATA, Q32_MODULUS, Q32_NUM_PRIMES, Q32_PRIMES}; + use hachi_pcs::algebra::tables::{ + q128_garner, q128_primes, q32_garner, Q128_MODULUS, Q128_NUM_PRIMES, Q32_MODULUS, + Q32_NUM_PRIMES, Q32_PRIMES, + }; use hachi_pcs::algebra::{ pseudo_mersenne_modulus, Pow2Offset128Field, Pow2OffsetPrimeSpec, POW2_OFFSET_MAX, POW2_OFFSET_PRIMES, POW2_OFFSET_TABLE, @@ -439,12 +442,6 @@ mod tests { assert_eq!(a, a); } - #[test] - fn qdata_q_matches_const() { - let q_from_data = Q32_DATA.q_u128().unwrap(); - assert_eq!(q_from_data, Q32_MODULUS as u128); - } - #[test] fn ntt_normalize_in_range() { for prime in &Q32_PRIMES { @@ -664,18 +661,15 @@ mod tests { #[test] fn ntt_forward_inverse_round_trip() { let prime = Q32_PRIMES[0]; - let tw = NttTwiddles::<64>::compute(prime); + let tw = NttTwiddles::::compute(prime); - // Create a test polynomial in Montgomery form. - let original: [MontCoeff; 64] = + let original: [MontCoeff; 64] = std::array::from_fn(|i| prime.from_canonical((i as i16) % prime.p)); - // Forward then inverse should give back the original. let mut a = original; forward_ntt(&mut a, prime, &tw); inverse_ntt(&mut a, prime, &tw); - // Normalize and compare. for (i, (got, expected)) in a.iter().zip(original.iter()).enumerate() { let got_canon = prime.to_canonical(prime.normalize(*got)); let exp_canon = prime.to_canonical(prime.normalize(*expected)); @@ -689,7 +683,7 @@ mod tests { #[test] fn ntt_forward_inverse_all_primes() { for (pi, prime) in Q32_PRIMES.iter().enumerate() { - let tw = NttTwiddles::<64>::compute(*prime); + let tw = NttTwiddles::::compute(*prime); let original: [_; 64] = std::array::from_fn(|i| prime.from_canonical(((i * (pi + 1)) as i16) % prime.p)); @@ -714,16 +708,17 @@ mod tests { fn cyclotomic_ntt_crt_round_trip_q32() { type F = Fp64<{ Q32_MODULUS }>; type R = CyclotomicRing; - type N = CyclotomicCrtNtt; + type N = CyclotomicCrtNtt; - let twiddles: [NttTwiddles<64>; Q32_NUM_PRIMES] = - std::array::from_fn(|k| NttTwiddles::<64>::compute(Q32_PRIMES[k])); + let twiddles: [NttTwiddles; Q32_NUM_PRIMES] = + std::array::from_fn(|k| NttTwiddles::compute(Q32_PRIMES[k])); let coeffs: [F; 64] = std::array::from_fn(|i| F::from_u64(((i as u64 * 17) + 5) % Q32_MODULUS)); let ring = R::from_coefficients(coeffs); let ntt = N::from_ring(&ring, &Q32_PRIMES, &twiddles); - let round_trip = ntt.to_ring(&Q32_PRIMES, &twiddles, &Q32_DATA); + let garner = q32_garner(); + let round_trip = ntt.to_ring(&Q32_PRIMES, &twiddles, &garner); assert_eq!(ring, round_trip); } @@ -732,10 +727,10 @@ mod tests { fn cyclotomic_ntt_reduced_ops_are_stable() { type F = Fp64<{ Q32_MODULUS }>; type R = CyclotomicRing; - type N = CyclotomicCrtNtt; + type N = CyclotomicCrtNtt; - let twiddles: [NttTwiddles<64>; Q32_NUM_PRIMES] = - std::array::from_fn(|k| NttTwiddles::<64>::compute(Q32_PRIMES[k])); + let twiddles: [NttTwiddles; Q32_NUM_PRIMES] = + std::array::from_fn(|k| NttTwiddles::compute(Q32_PRIMES[k])); let a = R::from_coefficients(std::array::from_fn(|i| { F::from_u64(((i as u64 * 3) + 1) % Q32_MODULUS) @@ -751,8 +746,9 @@ mod tests { let back = sum.sub_reduced(&ntt_b, &Q32_PRIMES); assert_eq!(back, ntt_a); + let garner = q32_garner(); let zero_ntt = ntt_a.add_reduced(&ntt_a.neg_reduced(&Q32_PRIMES), &Q32_PRIMES); - let zero_ring = zero_ntt.to_ring(&Q32_PRIMES, &twiddles, &Q32_DATA); + let zero_ring = zero_ntt.to_ring(&Q32_PRIMES, &twiddles, &garner); assert_eq!(zero_ring, R::zero()); } @@ -760,10 +756,10 @@ mod tests { fn backend_path_matches_default_scalar_path() { type F = Fp64<{ Q32_MODULUS }>; type R = CyclotomicRing; - type N = CyclotomicCrtNtt; + type N = CyclotomicCrtNtt; - let twiddles: [NttTwiddles<64>; Q32_NUM_PRIMES] = - std::array::from_fn(|k| NttTwiddles::<64>::compute(Q32_PRIMES[k])); + let twiddles: [NttTwiddles; Q32_NUM_PRIMES] = + std::array::from_fn(|k| NttTwiddles::compute(Q32_PRIMES[k])); let ring = R::from_coefficients(std::array::from_fn(|i| { F::from_u64(((i as u64 * 13) + 9) % Q32_MODULUS) })); @@ -773,15 +769,88 @@ mod tests { N::from_ring_with_backend::(&ring, &Q32_PRIMES, &twiddles); assert_eq!(default_ntt, backend_ntt); - let default_back = default_ntt.to_ring(&Q32_PRIMES, &twiddles, &Q32_DATA); - let backend_back = backend_ntt.to_ring_with_backend::( - &Q32_PRIMES, - &twiddles, - &Q32_DATA, - ); + let garner = q32_garner(); + let default_back = default_ntt.to_ring(&Q32_PRIMES, &twiddles, &garner); + let backend_back = + backend_ntt.to_ring_with_backend::(&Q32_PRIMES, &twiddles, &garner); assert_eq!(default_back, backend_back); } + #[test] + fn crt_ntt_mul_matches_schoolbook_q32() { + type F = Fp64<{ Q32_MODULUS }>; + type R = CyclotomicRing; + type N = CyclotomicCrtNtt; + + let twiddles: [NttTwiddles; Q32_NUM_PRIMES] = + std::array::from_fn(|k| NttTwiddles::compute(Q32_PRIMES[k])); + let garner = q32_garner(); + + let a = R::from_coefficients(std::array::from_fn(|i| { + F::from_u64(((i as u64 * 7) + 3) % Q32_MODULUS) + })); + let b = R::from_coefficients(std::array::from_fn(|i| { + F::from_u64(((i as u64 * 13) + 11) % Q32_MODULUS) + })); + + let schoolbook = a * b; + + let ntt_a = N::from_ring(&a, &Q32_PRIMES, &twiddles); + let ntt_b = N::from_ring(&b, &Q32_PRIMES, &twiddles); + let ntt_prod = ntt_a.pointwise_mul(&ntt_b, &Q32_PRIMES); + let ntt_result: R = ntt_prod.to_ring(&Q32_PRIMES, &twiddles, &garner); + + assert_eq!(schoolbook, ntt_result); + } + + #[test] + fn q128_ntt_round_trip() { + type F = Fp128<{ Q128_MODULUS }>; + type R = CyclotomicRing; + type N = CyclotomicCrtNtt; + + let primes = q128_primes(); + let twiddles: [NttTwiddles; Q128_NUM_PRIMES] = + std::array::from_fn(|k| NttTwiddles::compute(primes[k])); + let garner = q128_garner(); + + let coeffs: [F; 64] = + std::array::from_fn(|i| F::from_u64(((i as u64 * 31) + 7) % (Q128_MODULUS as u64))); + let ring = R::from_coefficients(coeffs); + let ntt = N::from_ring(&ring, &primes, &twiddles); + let round_trip: R = ntt.to_ring(&primes, &twiddles, &garner); + + assert_eq!(ring, round_trip); + } + + #[test] + fn crt_ntt_mul_matches_schoolbook_q128() { + type F = Fp128<{ Q128_MODULUS }>; + type R = CyclotomicRing; + type N = CyclotomicCrtNtt; + + let primes = q128_primes(); + let twiddles: [NttTwiddles; Q128_NUM_PRIMES] = + std::array::from_fn(|k| NttTwiddles::compute(primes[k])); + let garner = q128_garner(); + + let a = R::from_coefficients(std::array::from_fn(|i| { + F::from_u64(((i as u64 * 7) + 3) % (Q128_MODULUS as u64)) + })); + let b = R::from_coefficients(std::array::from_fn(|i| { + F::from_u64(((i as u64 * 13) + 11) % (Q128_MODULUS as u64)) + })); + + let schoolbook = a * b; + + let ntt_a = N::from_ring(&a, &primes, &twiddles); + let ntt_b = N::from_ring(&b, &primes, &twiddles); + let ntt_prod = ntt_a.pointwise_mul(&ntt_b, &primes); + let ntt_result: R = ntt_prod.to_ring(&primes, &twiddles, &garner); + + assert_eq!(schoolbook, ntt_result); + } + #[test] fn field_sampling_respects_modulus() { type F = Fp32<251>; From e4f983671d6b3e3281778eba046e26982c5a52d5 Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Fri, 27 Feb 2026 23:10:40 -0800 Subject: [PATCH 06/34] Add extension field arithmetic and refactor sumcheck trait bounds Split CanonicalField into FromSmallInt (from_{u,i}{8,16,32,64} for all fields) and CanonicalField (u128 repr, base fields only). Implement FromSmallInt, Eq, Debug for Fp2/Fp4. Add ExtField trait with EXT_DEGREE and from_base_slice. Optimize extension field arithmetic: Karatsuba multiplication for Fp2 and Fp4 (3 base muls instead of 4), specialized squaring (2 base muls for Fp2), non-residue IS_NEG_ONE specialization. Add concrete configs (TwoNr, NegOneNr, UnitNr) and type aliases Ext2, Ext4. Add transpose-based packed extension fields (PackedFp2, PackedFp4) for SIMD acceleration, following Plonky3's approach. Relax sumcheck bounds from E: CanonicalField to E: FromSmallInt (or E: FieldCore where spurious). Add sample_ext_challenge transcript helper. Includes tests for extension field sumcheck execution. Made-with: Cursor --- benches/field_arith.rs | 2 +- benches/hachi_e2e.rs | 51 ++- src/algebra/fields/ext.rs | 310 ++++++++++++++- src/algebra/fields/fp128.rs | 8 +- src/algebra/fields/fp32.rs | 8 +- src/algebra/fields/fp64.rs | 8 +- src/algebra/fields/lift.rs | 59 ++- src/algebra/fields/mod.rs | 6 +- src/algebra/fields/packed.rs | 2 +- src/algebra/fields/packed_ext.rs | 415 +++++++++++++++++++++ src/algebra/mod.rs | 14 +- src/lib.rs | 2 +- src/primitives/arithmetic.rs | 60 ++- src/protocol/commitment/onehot.rs | 1 + src/protocol/commitment_scheme.rs | 13 +- src/protocol/mod.rs | 2 +- src/protocol/ring_switch.rs | 7 +- src/protocol/sumcheck/batched_sumcheck.rs | 10 +- src/protocol/sumcheck/eq_poly.rs | 2 +- src/protocol/sumcheck/mod.rs | 6 +- src/protocol/sumcheck/norm_sumcheck.rs | 53 ++- src/protocol/sumcheck/relation_sumcheck.rs | 28 +- src/protocol/sumcheck/split_eq.rs | 6 +- src/protocol/transcript/mod.rs | 18 + src/test_utils.rs | 2 +- tests/algebra.rs | 4 +- tests/commitment_contract.rs | 2 +- tests/hachi_sumcheck.rs | 3 +- tests/onehot_commitment.rs | 2 +- tests/sparse_challenge.rs | 2 +- tests/sumcheck_core.rs | 2 +- 31 files changed, 1001 insertions(+), 107 deletions(-) create mode 100644 src/algebra/fields/packed_ext.rs diff --git a/benches/field_arith.rs b/benches/field_arith.rs index 9caf70d8..c3b04578 100644 --- a/benches/field_arith.rs +++ b/benches/field_arith.rs @@ -10,7 +10,7 @@ use hachi_pcs::algebra::{ Pow2Offset24Field, Pow2Offset30Field, Pow2Offset31Field, Pow2Offset32Field, Pow2Offset40Field, Pow2Offset48Field, Pow2Offset56Field, Pow2Offset64Field, }; -use hachi_pcs::{CanonicalField, FieldCore, FieldSampling, Invertible}; +use hachi_pcs::{CanonicalField, FieldCore, FieldSampling, FromSmallInt, Invertible}; use rand::{rngs::StdRng, RngCore, SeedableRng}; use std::env; #[cfg(feature = "parallel")] diff --git a/benches/hachi_e2e.rs b/benches/hachi_e2e.rs index 06028d53..9e9727cd 100644 --- a/benches/hachi_e2e.rs +++ b/benches/hachi_e2e.rs @@ -7,7 +7,7 @@ use hachi_pcs::protocol::commitment::DefaultCommitmentConfig; use hachi_pcs::protocol::commitment_scheme::HachiCommitmentScheme; use hachi_pcs::protocol::transcript::Blake2bTranscript; use hachi_pcs::protocol::CommitmentConfig; -use hachi_pcs::{CanonicalField, CommitmentScheme, Polynomial, Transcript}; +use hachi_pcs::{CommitmentScheme, FromSmallInt, Polynomial, Transcript}; use std::time::Duration; type F = Fp64<4294967197>; @@ -119,17 +119,15 @@ where group.bench_function("verify", |b| { b.iter(|| { let mut transcript = Blake2bTranscript::::new(b"bench"); - black_box( - as CommitmentScheme>::verify( - black_box(&proof), - black_box(&verifier_setup), - &mut transcript, - black_box(&pt), - black_box(&opening), - black_box(&commitment), - ) - .unwrap(), + as CommitmentScheme>::verify( + black_box(&proof), + black_box(&verifier_setup), + &mut transcript, + black_box(&pt), + black_box(&opening), + black_box(&commitment), ) + .unwrap(); }) }); @@ -138,7 +136,12 @@ where let (cm, h) = as CommitmentScheme>::commit(&poly, &setup).unwrap(); let mut pt_tr = Blake2bTranscript::::new(b"bench"); let pf = as CommitmentScheme>::prove( - &setup, &poly, &pt, Some(h), &mut pt_tr, &cm, + &setup, + &poly, + &pt, + Some(h), + &mut pt_tr, + &cm, ) .unwrap(); let mut vt_tr = Blake2bTranscript::::new(b"bench"); @@ -158,10 +161,24 @@ where group.finish(); } -fn bench_nv10(c: &mut Criterion) { bench_phases::(c, "fp64"); } -fn bench_nv14(c: &mut Criterion) { bench_phases::(c, "fp64"); } -fn bench_nv18(c: &mut Criterion) { bench_phases::(c, "fp64"); } -fn bench_nv20(c: &mut Criterion) { bench_phases::(c, "fp64"); } +fn bench_nv10(c: &mut Criterion) { + bench_phases::(c, "fp64"); +} +fn bench_nv14(c: &mut Criterion) { + bench_phases::(c, "fp64"); +} +fn bench_nv18(c: &mut Criterion) { + bench_phases::(c, "fp64"); +} +fn bench_nv20(c: &mut Criterion) { + bench_phases::(c, "fp64"); +} -criterion_group!(hachi_benches, bench_nv10, bench_nv14, bench_nv18, bench_nv20); +criterion_group!( + hachi_benches, + bench_nv10, + bench_nv14, + bench_nv18, + bench_nv20 +); criterion_main!(hachi_benches); diff --git a/src/algebra/fields/ext.rs b/src/algebra/fields/ext.rs index 29061e9b..c93c5ee4 100644 --- a/src/algebra/fields/ext.rs +++ b/src/algebra/fields/ext.rs @@ -4,7 +4,33 @@ use crate::algebra::module::VectorModule; use crate::primitives::serialization::{ Compress, HachiDeserialize, HachiSerialize, SerializationError, Valid, Validate, }; -use crate::{FieldCore, FieldSampling}; +use crate::{FieldCore, FieldSampling, FromSmallInt}; + +/// `Fp2Config` with non-residue = -1. +/// +/// Valid when `p ≡ 3 (mod 4)`, i.e. -1 is a quadratic non-residue. +pub struct NegOneNr; + +impl Fp2Config for NegOneNr { + const IS_NEG_ONE: bool = true; + + fn non_residue() -> F { + -F::one() + } +} + +/// `Fp2Config` with non-residue = 2. +/// +/// Valid when `p ≡ 5 (mod 8)`, i.e. 2 is a quadratic non-residue. +/// All Hachi pseudo-Mersenne primes (`2^k - c` with `c ≡ 3 mod 8`) +/// satisfy this. +pub struct TwoNr; + +impl Fp2Config for TwoNr { + fn non_residue() -> F { + F::from_u64(2) + } +} use rand_core::RngCore; use std::io::{Read, Write}; use std::marker::PhantomData; @@ -12,6 +38,12 @@ use std::ops::{Add, Mul, Neg, Sub}; /// Parameters for an `Fp2` quadratic extension over base field `F`. pub trait Fp2Config { + /// Whether the non-residue is -1. + /// + /// When `true`, multiplication by the non-residue is a free negation and + /// the Karatsuba/squaring routines can avoid a base-field multiply. + const IS_NEG_ONE: bool = false; + /// Non-residue `NR` such that `u^2 = NR`. fn non_residue() -> F; } @@ -36,6 +68,18 @@ impl> Fp2 { } } + /// Multiply a base-field element by the non-residue. + /// + /// When `IS_NEG_ONE` is true this is just a negation (no multiply). + #[inline(always)] + fn mul_nr(x: F) -> F { + if C::IS_NEG_ONE { + -x + } else { + C::non_residue() * x + } + } + /// Return the conjugate `c0 - c1 * u`. #[inline] pub fn conjugate(self) -> Self { @@ -45,8 +89,16 @@ impl> Fp2 { /// Return the norm in the base field: `c0^2 - NR * c1^2`. #[inline] pub fn norm(self) -> F { - let nr = C::non_residue(); - (self.c0 * self.c0) - (nr * (self.c1 * self.c1)) + (self.c0 * self.c0) - Self::mul_nr(self.c1 * self.c1) + } +} + +impl> std::fmt::Debug for Fp2 { + fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result { + f.debug_struct("Fp2") + .field("c0", &self.c0) + .field("c1", &self.c1) + .finish() } } @@ -70,6 +122,8 @@ impl> PartialEq for Fp2 { } } +impl> Eq for Fp2 {} + impl> Add for Fp2 { type Output = Self; fn add(self, rhs: Self) -> Self::Output { @@ -91,12 +145,12 @@ impl> Neg for Fp2 { impl> Mul for Fp2 { type Output = Self; fn mul(self, rhs: Self) -> Self::Output { - let nr = C::non_residue(); - let a0 = self.c0; - let a1 = self.c1; - let b0 = rhs.c0; - let b1 = rhs.c1; - Self::new((a0 * b0) + (nr * (a1 * b1)), (a0 * b1) + (a1 * b0)) + let v0 = self.c0 * rhs.c0; + let v1 = self.c1 * rhs.c1; + Self::new( + v0 + Self::mul_nr(v1), + (self.c0 + self.c1) * (rhs.c0 + rhs.c1) - v0 - v1, + ) } } @@ -184,6 +238,15 @@ impl> FieldCore for Fp2 { *self * *rhs } + /// Specialized squaring: 2 base-field multiplications instead of 3. + /// + /// `(c0 + c1·u)^2 = (c0^2 + NR·c1^2) + (2·c0·c1)·u` + fn square(&self) -> Self { + let v0 = self.c0 * self.c0; + let v1 = self.c1 * self.c1; + Self::new(v0 + Self::mul_nr(v1), (self.c0 + self.c0) * self.c1) + } + fn inv(self) -> Option { if self.is_zero() { return None; @@ -199,12 +262,33 @@ impl> FieldSampling for Fp } } +impl> FromSmallInt for Fp2 { + fn from_u64(val: u64) -> Self { + Self::new(F::from_u64(val), F::zero()) + } + + fn from_i64(val: i64) -> Self { + Self::new(F::from_i64(val), F::zero()) + } +} + /// Parameters for an `Fp4` quadratic extension over `Fp2`. pub trait Fp4Config> { /// Non-residue `NR2` in `Fp2` such that `v^2 = NR2`. fn non_residue() -> Fp2; } +/// `Fp4Config` with non-residue `u ∈ Fp2` (the element `(0, 1)`). +/// +/// This is the standard tower choice: `Fp4 = Fp2[v] / (v^2 - u)`. +pub struct UnitNr; + +impl> Fp4Config for UnitNr { + fn non_residue() -> Fp2 { + Fp2::new(F::zero(), F::one()) + } +} + /// Quartic extension element `c0 + c1 * v` over `Fp2`, where `v^2 = NR2`. pub struct Fp4, C4: Fp4Config> { /// Constant term. @@ -233,6 +317,17 @@ impl, C4: Fp4Config> Fp4 { } } +impl, C4: Fp4Config> std::fmt::Debug + for Fp4 +{ + fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result { + f.debug_struct("Fp4") + .field("c0", &self.c0) + .field("c1", &self.c1) + .finish() + } +} + impl, C4: Fp4Config> Clone for Fp4 { fn clone(&self) -> Self { *self @@ -256,6 +351,8 @@ impl, C4: Fp4Config> PartialEq for Fp4, C4: Fp4Config> Eq for Fp4 {} + impl, C4: Fp4Config> Add for Fp4 { type Output = Self; fn add(self, rhs: Self) -> Self::Output { @@ -278,11 +375,12 @@ impl, C4: Fp4Config> Mul for Fp4 Self::Output { let nr2 = C4::non_residue(); - let a0 = self.c0; - let a1 = self.c1; - let b0 = rhs.c0; - let b1 = rhs.c1; - Self::new((a0 * b0) + (nr2 * (a1 * b1)), (a0 * b1) + (a1 * b0)) + let v0 = self.c0 * rhs.c0; + let v1 = self.c1 * rhs.c1; + Self::new( + v0 + (nr2 * v1), + (self.c0 + self.c1) * (rhs.c0 + rhs.c1) - v0 - v1, + ) } } @@ -372,6 +470,13 @@ impl, C4: Fp4Config> FieldCore for *self * *rhs } + fn square(&self) -> Self { + let nr2 = C4::non_residue(); + let v0 = self.c0.square(); + let v1 = self.c1.square(); + Self::new(v0 + nr2 * v1, (self.c0 + self.c0) * self.c1) + } + fn inv(self) -> Option { if self.is_zero() { return None; @@ -389,6 +494,18 @@ impl, C4: Fp4Config, C4: Fp4Config> FromSmallInt + for Fp4 +{ + fn from_u64(val: u64) -> Self { + Self::new(Fp2::from_u64(val), Fp2::zero()) + } + + fn from_i64(val: i64) -> Self { + Self::new(Fp2::from_i64(val), Fp2::zero()) + } +} + // Scalar * VectorModule impls for extension scalars. impl Mul, N>> for Fp2 @@ -444,3 +561,168 @@ where self * *rhs } } + +// Convenience aliases for extension fields with NR = 2 (valid for all Hachi +// pseudo-Mersenne primes where p ≡ 5 mod 8). + +/// Quadratic extension over any `F` with non-residue 2. +pub type Ext2 = Fp2; + +/// Quartic extension as tower `Ext2[v]/(v^2 - u)`. +pub type Ext4 = Fp4; + +#[cfg(test)] +mod tests { + use super::*; + use crate::algebra::fields::lift::ExtField; + use crate::algebra::Fp64; + use crate::{FieldCore, FieldSampling, FromSmallInt}; + use rand::rngs::StdRng; + use rand::SeedableRng; + + type F = Fp64<4294967197>; + type E2 = Ext2; + type E4 = Ext4; + + #[test] + fn fp2_add_sub_identity() { + let a = E2::new(F::from_u64(3), F::from_u64(5)); + let b = E2::new(F::from_u64(7), F::from_u64(11)); + let c = a + b; + assert_eq!(c - b, a); + assert_eq!(c - a, b); + } + + #[test] + fn fp2_mul_one() { + let a = E2::new(F::from_u64(42), F::from_u64(13)); + assert_eq!(a * E2::one(), a); + assert_eq!(E2::one() * a, a); + } + + #[test] + fn fp2_mul_commutativity() { + let mut rng = StdRng::seed_from_u64(1234); + let a = E2::sample(&mut rng); + let b = E2::sample(&mut rng); + assert_eq!(a * b, b * a); + } + + #[test] + fn fp2_karatsuba_matches_schoolbook() { + let mut rng = StdRng::seed_from_u64(5678); + for _ in 0..100 { + let a = E2::sample(&mut rng); + let b = E2::sample(&mut rng); + let nr = >::non_residue(); + let expected = E2::new( + (a.c0 * b.c0) + (nr * (a.c1 * b.c1)), + (a.c0 * b.c1) + (a.c1 * b.c0), + ); + assert_eq!(a * b, expected); + } + } + + #[test] + fn fp2_square_matches_mul() { + let mut rng = StdRng::seed_from_u64(9012); + for _ in 0..100 { + let a = E2::sample(&mut rng); + assert_eq!(a.square(), a * a, "square mismatch for {a:?}"); + } + } + + #[test] + fn fp2_inv() { + let mut rng = StdRng::seed_from_u64(3456); + for _ in 0..50 { + let a = E2::sample(&mut rng); + if !a.is_zero() { + let inv = a.inv().unwrap(); + assert_eq!(a * inv, E2::one()); + } + } + } + + #[test] + fn fp4_mul_commutativity() { + let mut rng = StdRng::seed_from_u64(7890); + let a = E4::sample(&mut rng); + let b = E4::sample(&mut rng); + assert_eq!(a * b, b * a); + } + + #[test] + fn fp4_square_matches_mul() { + let mut rng = StdRng::seed_from_u64(1111); + for _ in 0..50 { + let a = E4::sample(&mut rng); + assert_eq!(a.square(), a * a); + } + } + + #[test] + fn fp4_inv() { + let mut rng = StdRng::seed_from_u64(2222); + for _ in 0..50 { + let a = E4::sample(&mut rng); + if !a.is_zero() { + let inv = a.inv().unwrap(); + assert_eq!(a * inv, E4::one()); + } + } + } + + #[test] + fn from_small_int_fp2() { + let a = E2::from_u64(42); + assert_eq!(a, E2::new(F::from_u64(42), F::zero())); + + let b = E2::from_i64(-3); + assert_eq!(b, E2::new(F::from_i64(-3), F::zero())); + + let c = E2::from_u8(7); + assert_eq!(c, E2::from_u64(7)); + + let d = E2::from_u32(100_000); + assert_eq!(d, E2::from_u64(100_000)); + } + + #[test] + fn from_small_int_fp4() { + let a = E4::from_u64(42); + assert_eq!(a, E4::new(E2::from_u64(42), E2::zero())); + + let b = E4::from_i64(-7); + assert_eq!(b, E4::new(E2::from_i64(-7), E2::zero())); + } + + #[test] + fn ext_field_degree() { + assert_eq!(>::EXT_DEGREE, 1); + assert_eq!(>::EXT_DEGREE, 2); + assert_eq!(>::EXT_DEGREE, 4); + } + + #[test] + fn ext_field_from_base_slice() { + let c0 = F::from_u64(3); + let c1 = F::from_u64(5); + let e2 = E2::from_base_slice(&[c0, c1]); + assert_eq!(e2, E2::new(c0, c1)); + + let c2 = F::from_u64(7); + let c3 = F::from_u64(11); + let e4 = E4::from_base_slice(&[c0, c1, c2, c3]); + assert_eq!(e4, E4::new(E2::new(c0, c1), E2::new(c2, c3))); + } + + #[test] + fn eq_impl() { + let a = E2::new(F::from_u64(1), F::from_u64(2)); + let b = E2::new(F::from_u64(1), F::from_u64(2)); + let c = E2::new(F::from_u64(1), F::from_u64(3)); + assert_eq!(a, b); + assert_ne!(a, c); + } +} diff --git a/src/algebra/fields/fp128.rs b/src/algebra/fields/fp128.rs index 09248062..ff66c640 100644 --- a/src/algebra/fields/fp128.rs +++ b/src/algebra/fields/fp128.rs @@ -18,7 +18,9 @@ use rand_core::RngCore; use crate::primitives::serialization::{ Compress, HachiDeserialize, HachiSerialize, SerializationError, Valid, Validate, }; -use crate::{CanonicalField, FieldCore, FieldSampling, Invertible, PseudoMersenneField}; +use crate::{ + CanonicalField, FieldCore, FieldSampling, FromSmallInt, Invertible, PseudoMersenneField, +}; /// Pack two u64 limbs into `[lo, hi]`. #[inline(always)] @@ -816,7 +818,7 @@ impl FieldSampling for Fp128

{ } } -impl CanonicalField for Fp128

{ +impl FromSmallInt for Fp128

{ fn from_u64(val: u64) -> Self { // For Fp128 pseudo-Mersenne primes, p = 2^128 - c with c < 2^64. // Therefore any u64 is always canonical (< p), so this can be a @@ -832,7 +834,9 @@ impl CanonicalField for Fp128

{ -Self::from_u64(val.unsigned_abs()) } } +} +impl CanonicalField for Fp128

{ fn to_canonical_u128(self) -> u128 { to_u128(self.0) } diff --git a/src/algebra/fields/fp32.rs b/src/algebra/fields/fp32.rs index 0a834698..33fd5a3a 100644 --- a/src/algebra/fields/fp32.rs +++ b/src/algebra/fields/fp32.rs @@ -11,7 +11,9 @@ use rand_core::RngCore; use crate::primitives::serialization::{ Compress, HachiDeserialize, HachiSerialize, SerializationError, Valid, Validate, }; -use crate::{CanonicalField, FieldCore, FieldSampling, Invertible, PseudoMersenneField}; +use crate::{ + CanonicalField, FieldCore, FieldSampling, FromSmallInt, Invertible, PseudoMersenneField, +}; use std::io::{Read, Write}; /// Prime field element for primes `p = 2^k − c` stored as `u32`. @@ -336,7 +338,7 @@ impl FieldSampling for Fp32

{ } } -impl CanonicalField for Fp32

{ +impl FromSmallInt for Fp32

{ fn from_u64(val: u64) -> Self { Self(Self::reduce_u64(val)) } @@ -348,7 +350,9 @@ impl CanonicalField for Fp32

{ -Self::from_u64((-val) as u64) } } +} +impl CanonicalField for Fp32

{ fn to_canonical_u128(self) -> u128 { self.0 as u128 } diff --git a/src/algebra/fields/fp64.rs b/src/algebra/fields/fp64.rs index a56e3c1c..680b7654 100644 --- a/src/algebra/fields/fp64.rs +++ b/src/algebra/fields/fp64.rs @@ -11,7 +11,9 @@ use rand_core::RngCore; use crate::primitives::serialization::{ Compress, HachiDeserialize, HachiSerialize, SerializationError, Valid, Validate, }; -use crate::{CanonicalField, FieldCore, FieldSampling, Invertible, PseudoMersenneField}; +use crate::{ + CanonicalField, FieldCore, FieldSampling, FromSmallInt, Invertible, PseudoMersenneField, +}; use std::io::{Read, Write}; #[inline(always)] @@ -477,7 +479,7 @@ impl FieldSampling for Fp64

{ } } -impl CanonicalField for Fp64

{ +impl FromSmallInt for Fp64

{ fn from_u64(val: u64) -> Self { Self(Self::reduce_u128(val as u128)) } @@ -489,7 +491,9 @@ impl CanonicalField for Fp64

{ -Self::from_u64((-val) as u64) } } +} +impl CanonicalField for Fp64

{ fn to_canonical_u128(self) -> u128 { self.0 as u128 } diff --git a/src/algebra/fields/lift.rs b/src/algebra/fields/lift.rs index 5e243c9c..e05a8aa7 100644 --- a/src/algebra/fields/lift.rs +++ b/src/algebra/fields/lift.rs @@ -2,7 +2,7 @@ use crate::algebra::fields::ext::{Fp2, Fp2Config, Fp4, Fp4Config}; use crate::primitives::serialization::Valid; -use crate::FieldCore; +use crate::{FieldCore, FromSmallInt}; /// Lift a base-field element into an extension field. /// @@ -12,6 +12,63 @@ pub trait LiftBase: FieldCore { fn lift_base(x: F) -> Self; } +/// An algebraic extension of base field `F`. +/// +/// Provides the extension degree and a constructor from a slice of base-field +/// coefficients (in the canonical basis `{1, u, u^2, ...}`). +pub trait ExtField: FieldCore + LiftBase + FromSmallInt { + /// Extension degree: `[Self : F]`. + const EXT_DEGREE: usize; + + /// Construct from a coefficient slice `[c0, c1, ..., c_{d-1}]`. + /// + /// # Panics + /// Panics if `coeffs.len() != Self::EXT_DEGREE`. + fn from_base_slice(coeffs: &[F]) -> Self; +} + +impl ExtField for F { + const EXT_DEGREE: usize = 1; + + #[inline] + fn from_base_slice(coeffs: &[F]) -> Self { + assert_eq!(coeffs.len(), 1); + coeffs[0] + } +} + +impl ExtField for Fp2 +where + F: FieldCore + FromSmallInt + Valid, + C: Fp2Config, +{ + const EXT_DEGREE: usize = 2; + + #[inline] + fn from_base_slice(coeffs: &[F]) -> Self { + assert_eq!(coeffs.len(), 2); + Self::new(coeffs[0], coeffs[1]) + } +} + +impl ExtField for Fp4 +where + F: FieldCore + FromSmallInt + Valid, + C2: Fp2Config, + C4: Fp4Config, +{ + const EXT_DEGREE: usize = 4; + + #[inline] + fn from_base_slice(coeffs: &[F]) -> Self { + assert_eq!(coeffs.len(), 4); + Self::new( + Fp2::new(coeffs[0], coeffs[1]), + Fp2::new(coeffs[2], coeffs[3]), + ) + } +} + impl LiftBase for F { #[inline] fn lift_base(x: F) -> Self { diff --git a/src/algebra/fields/mod.rs b/src/algebra/fields/mod.rs index d255852a..981935e8 100644 --- a/src/algebra/fields/mod.rs +++ b/src/algebra/fields/mod.rs @@ -18,17 +18,19 @@ pub mod packed_avx2; target_feature = "avx512dq" ))] pub mod packed_avx512; +#[allow(missing_docs)] +pub mod packed_ext; #[cfg(all(target_arch = "aarch64", target_feature = "neon"))] pub mod packed_neon; pub mod pseudo_mersenne; -pub use ext::{Fp2, Fp2Config, Fp4, Fp4Config}; +pub use ext::{Ext2, Ext4, Fp2, Fp2Config, Fp4, Fp4Config, NegOneNr, TwoNr, UnitNr}; pub use fp128::{ Fp128, Prime128M13M4P0, Prime128M37P3P0, Prime128M52M3P0, Prime128M54P4P0, Prime128M8M4M1M0, }; pub use fp32::Fp32; pub use fp64::Fp64; -pub use lift::LiftBase; +pub use lift::{ExtField, LiftBase}; pub use packed::{ Fp128Packing, Fp32Packing, Fp64Packing, HasPacking, NoPacking, PackedField, PackedValue, }; diff --git a/src/algebra/fields/packed.rs b/src/algebra/fields/packed.rs index e66063a9..eaf15ab2 100644 --- a/src/algebra/fields/packed.rs +++ b/src/algebra/fields/packed.rs @@ -236,7 +236,7 @@ mod tests { Pow2Offset24Field, Pow2Offset31Field, Pow2Offset32Field, Pow2Offset40Field, Pow2Offset64Field, Prime128M13M4P0, }; - use crate::{CanonicalField, FieldCore, FieldSampling}; + use crate::{CanonicalField, FieldCore, FieldSampling, FromSmallInt}; use rand::{rngs::StdRng, RngCore, SeedableRng}; fn rand_u128(rng: &mut R) -> u128 { diff --git a/src/algebra/fields/packed_ext.rs b/src/algebra/fields/packed_ext.rs new file mode 100644 index 00000000..5bf061d9 --- /dev/null +++ b/src/algebra/fields/packed_ext.rs @@ -0,0 +1,415 @@ +//! Packed extension field types using transpose-based packing. +//! +//! A `PackedFp2` stores `[PF; 2]` where `PF` is the packed base field. +//! Each `PF` lane contains the corresponding coefficient of an `Fp2` element. +//! This enables WIDTH-fold parallel arithmetic over `Fp2` using existing SIMD +//! base-field operations. + +use crate::algebra::fields::ext::{Fp2, Fp2Config, Fp4, Fp4Config}; +use crate::algebra::fields::packed::{HasPacking, PackedField, PackedValue}; +use crate::primitives::serialization::Valid; +use crate::FieldCore; +use core::ops::{Add, Mul, Sub}; + +/// Packed `Fp2` elements stored in transpose layout: `[PF; 2]`. +/// +/// If `PF` has width `W`, this represents `W` parallel `Fp2` values. +pub struct PackedFp2, PF: PackedField> { + pub c0: PF, + pub c1: PF, + _marker: std::marker::PhantomData (F, C)>, +} + +impl, PF: PackedField> Clone for PackedFp2 { + fn clone(&self) -> Self { + *self + } +} + +impl, PF: PackedField> Copy for PackedFp2 {} + +impl, PF: PackedField> std::fmt::Debug + for PackedFp2 +{ + fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result { + f.debug_struct("PackedFp2").finish_non_exhaustive() + } +} + +impl, PF: PackedField> PackedFp2 { + #[inline] + pub fn new(c0: PF, c1: PF) -> Self { + Self { + c0, + c1, + _marker: std::marker::PhantomData, + } + } + + #[inline] + fn mul_nr(x: PF) -> PF { + if C::IS_NEG_ONE { + let zero = PF::broadcast(F::zero()); + zero - x + } else { + PF::broadcast(C::non_residue()) * x + } + } +} + +impl PackedValue for PackedFp2 +where + F: FieldCore + Valid + 'static, + C: Fp2Config + 'static, + PF: PackedField, +{ + type Value = Fp2; + const WIDTH: usize = PF::WIDTH; + + fn from_fn(mut f: G) -> Self + where + G: FnMut(usize) -> Self::Value, + { + let mut c0s = Vec::with_capacity(PF::WIDTH); + let mut c1s = Vec::with_capacity(PF::WIDTH); + for i in 0..PF::WIDTH { + let val = f(i); + c0s.push(val.c0); + c1s.push(val.c1); + } + Self::new(PF::from_fn(|i| c0s[i]), PF::from_fn(|i| c1s[i])) + } + + fn extract(&self, lane: usize) -> Self::Value { + Fp2::new(self.c0.extract(lane), self.c1.extract(lane)) + } +} + +impl Add for PackedFp2 +where + F: FieldCore, + C: Fp2Config, + PF: PackedField, +{ + type Output = Self; + #[inline] + fn add(self, rhs: Self) -> Self { + Self::new(self.c0 + rhs.c0, self.c1 + rhs.c1) + } +} + +impl Sub for PackedFp2 +where + F: FieldCore, + C: Fp2Config, + PF: PackedField, +{ + type Output = Self; + #[inline] + fn sub(self, rhs: Self) -> Self { + Self::new(self.c0 - rhs.c0, self.c1 - rhs.c1) + } +} + +impl Mul for PackedFp2 +where + F: FieldCore, + C: Fp2Config, + PF: PackedField, +{ + type Output = Self; + #[inline] + fn mul(self, rhs: Self) -> Self { + let v0 = self.c0 * rhs.c0; + let v1 = self.c1 * rhs.c1; + Self::new( + v0 + Self::mul_nr(v1), + (self.c0 + self.c1) * (rhs.c0 + rhs.c1) - v0 - v1, + ) + } +} + +impl PackedField for PackedFp2 +where + F: FieldCore + Valid + 'static, + C: Fp2Config + 'static, + PF: PackedField, +{ + type Scalar = Fp2; + + #[inline] + fn broadcast(value: Self::Scalar) -> Self { + Self::new(PF::broadcast(value.c0), PF::broadcast(value.c1)) + } +} + +impl HasPacking for Fp2 +where + F: FieldCore + Valid + HasPacking + 'static, + C: Fp2Config + 'static, +{ + type Packing = PackedFp2; +} + +/// Packed `Fp4` elements stored in transpose layout: `[PackedFp2; 2]`. +pub struct PackedFp4< + F: FieldCore, + C2: Fp2Config, + C4: Fp4Config, + PF: PackedField, +> { + pub c0: PackedFp2, + pub c1: PackedFp2, + _marker: std::marker::PhantomData C4>, +} + +impl Clone for PackedFp4 +where + F: FieldCore, + C2: Fp2Config, + C4: Fp4Config, + PF: PackedField, +{ + fn clone(&self) -> Self { + *self + } +} + +impl Copy for PackedFp4 +where + F: FieldCore, + C2: Fp2Config, + C4: Fp4Config, + PF: PackedField, +{ +} + +impl std::fmt::Debug for PackedFp4 +where + F: FieldCore, + C2: Fp2Config, + C4: Fp4Config, + PF: PackedField, +{ + fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result { + f.debug_struct("PackedFp4").finish_non_exhaustive() + } +} + +impl PackedFp4 +where + F: FieldCore, + C2: Fp2Config, + C4: Fp4Config, + PF: PackedField, +{ + #[inline] + pub fn new(c0: PackedFp2, c1: PackedFp2) -> Self { + Self { + c0, + c1, + _marker: std::marker::PhantomData, + } + } +} + +impl PackedValue for PackedFp4 +where + F: FieldCore + Valid + 'static, + C2: Fp2Config + 'static, + C4: Fp4Config + 'static, + PF: PackedField, +{ + type Value = Fp4; + const WIDTH: usize = PF::WIDTH; + + fn from_fn(mut f: G) -> Self + where + G: FnMut(usize) -> Self::Value, + { + let mut c0s: Vec> = Vec::with_capacity(PF::WIDTH); + let mut c1s: Vec> = Vec::with_capacity(PF::WIDTH); + for i in 0..PF::WIDTH { + let val = f(i); + c0s.push(val.c0); + c1s.push(val.c1); + } + Self::new( + PackedFp2::from_fn(|i| c0s[i]), + PackedFp2::from_fn(|i| c1s[i]), + ) + } + + fn extract(&self, lane: usize) -> Self::Value { + Fp4::new(self.c0.extract(lane), self.c1.extract(lane)) + } +} + +impl Add for PackedFp4 +where + F: FieldCore, + C2: Fp2Config, + C4: Fp4Config, + PF: PackedField, +{ + type Output = Self; + #[inline] + fn add(self, rhs: Self) -> Self { + Self::new(self.c0 + rhs.c0, self.c1 + rhs.c1) + } +} + +impl Sub for PackedFp4 +where + F: FieldCore, + C2: Fp2Config, + C4: Fp4Config, + PF: PackedField, +{ + type Output = Self; + #[inline] + fn sub(self, rhs: Self) -> Self { + Self::new(self.c0 - rhs.c0, self.c1 - rhs.c1) + } +} + +impl Mul for PackedFp4 +where + F: FieldCore + Valid + 'static, + C2: Fp2Config + 'static, + C4: Fp4Config, + PF: PackedField, +{ + type Output = Self; + #[inline] + fn mul(self, rhs: Self) -> Self { + let nr2 = PackedFp2::broadcast(C4::non_residue()); + let v0 = self.c0 * rhs.c0; + let v1 = self.c1 * rhs.c1; + Self::new( + v0 + nr2 * v1, + (self.c0 + self.c1) * (rhs.c0 + rhs.c1) - v0 - v1, + ) + } +} + +impl PackedField for PackedFp4 +where + F: FieldCore + Valid + 'static, + C2: Fp2Config + 'static, + C4: Fp4Config + 'static, + PF: PackedField, +{ + type Scalar = Fp4; + + #[inline] + fn broadcast(value: Self::Scalar) -> Self { + Self::new( + PackedFp2::broadcast(value.c0), + PackedFp2::broadcast(value.c1), + ) + } +} + +impl HasPacking for Fp4 +where + F: FieldCore + Valid + HasPacking + 'static, + C2: Fp2Config + 'static, + C4: Fp4Config + 'static, +{ + type Packing = PackedFp4; +} + +#[cfg(test)] +mod tests { + use super::*; + use crate::algebra::fields::ext::{Ext2, Ext4, TwoNr, UnitNr}; + use crate::algebra::Fp64; + use crate::{FieldSampling, FromSmallInt}; + use rand::rngs::StdRng; + use rand::SeedableRng; + + type F = Fp64<4294967197>; + type E2 = Ext2; + type E4 = Ext4; + type PE2 = PackedFp2::Packing>; + type PE4 = PackedFp4::Packing>; + + #[test] + fn packed_fp2_add() { + let mut rng = StdRng::seed_from_u64(100); + let width = ::WIDTH; + let a_elems: Vec = (0..width).map(|_| E2::sample(&mut rng)).collect(); + let b_elems: Vec = (0..width).map(|_| E2::sample(&mut rng)).collect(); + + let pa = PE2::from_fn(|i| a_elems[i]); + let pb = PE2::from_fn(|i| b_elems[i]); + let pc = pa + pb; + + for i in 0..width { + assert_eq!(pc.extract(i), a_elems[i] + b_elems[i]); + } + } + + #[test] + fn packed_fp2_mul() { + let mut rng = StdRng::seed_from_u64(200); + let width = ::WIDTH; + let a_elems: Vec = (0..width).map(|_| E2::sample(&mut rng)).collect(); + let b_elems: Vec = (0..width).map(|_| E2::sample(&mut rng)).collect(); + + let pa = PE2::from_fn(|i| a_elems[i]); + let pb = PE2::from_fn(|i| b_elems[i]); + let pc = pa * pb; + + for i in 0..width { + assert_eq!( + pc.extract(i), + a_elems[i] * b_elems[i], + "packed Fp2 mul mismatch at lane {i}" + ); + } + } + + #[test] + fn packed_fp2_broadcast() { + let val = E2::new(F::from_u64(7), F::from_u64(11)); + let packed = PE2::broadcast(val); + let width = ::WIDTH; + for i in 0..width { + assert_eq!(packed.extract(i), val); + } + } + + #[test] + fn packed_fp4_mul() { + let mut rng = StdRng::seed_from_u64(300); + let width = ::WIDTH; + let a_elems: Vec = (0..width).map(|_| E4::sample(&mut rng)).collect(); + let b_elems: Vec = (0..width).map(|_| E4::sample(&mut rng)).collect(); + + let pa = PE4::from_fn(|i| a_elems[i]); + let pb = PE4::from_fn(|i| b_elems[i]); + let pc = pa * pb; + + for i in 0..width { + assert_eq!( + pc.extract(i), + a_elems[i] * b_elems[i], + "packed Fp4 mul mismatch at lane {i}" + ); + } + } + + #[test] + fn pack_unpack_roundtrip_fp2() { + let mut rng = StdRng::seed_from_u64(400); + let width = ::WIDTH; + let elems: Vec = (0..width * 3).map(|_| E2::sample(&mut rng)).collect(); + + let packed = PE2::pack_slice(&elems); + let unpacked = PE2::unpack_slice(&packed); + + assert_eq!(elems, unpacked); + } +} diff --git a/src/algebra/mod.rs b/src/algebra/mod.rs index 321dbc3c..18a7f2ba 100644 --- a/src/algebra/mod.rs +++ b/src/algebra/mod.rs @@ -17,13 +17,13 @@ pub mod ring; pub use backend::{CrtReconstruct, NttPrimeOps, NttTransform, RingBackend, ScalarBackend}; pub use domains::{CoeffDomain, CrtNttDomain}; pub use fields::{ - is_pow2_offset, pow2_offset, pseudo_mersenne_modulus, Fp128, Fp128Packing, Fp2, Fp2Config, - Fp32, Fp32Packing, Fp4, Fp4Config, Fp64, Fp64Packing, HasPacking, NoPacking, PackedField, - PackedValue, Pow2Offset128Field, Pow2Offset24Field, Pow2Offset30Field, Pow2Offset31Field, - Pow2Offset32Field, Pow2Offset40Field, Pow2Offset48Field, Pow2Offset56Field, Pow2Offset64Field, - Pow2OffsetPrimeSpec, Prime128M13M4P0, Prime128M37P3P0, Prime128M52M3P0, Prime128M54P4P0, - Prime128M8M4M1M0, POW2_OFFSET_IMPLEMENTED_MAX_BITS, POW2_OFFSET_MAX, POW2_OFFSET_PRIMES, - POW2_OFFSET_TABLE, + is_pow2_offset, pow2_offset, pseudo_mersenne_modulus, ExtField, Fp128, Fp128Packing, Fp2, + Fp2Config, Fp32, Fp32Packing, Fp4, Fp4Config, Fp64, Fp64Packing, HasPacking, LiftBase, + NoPacking, PackedField, PackedValue, Pow2Offset128Field, Pow2Offset24Field, Pow2Offset30Field, + Pow2Offset31Field, Pow2Offset32Field, Pow2Offset40Field, Pow2Offset48Field, Pow2Offset56Field, + Pow2Offset64Field, Pow2OffsetPrimeSpec, Prime128M13M4P0, Prime128M37P3P0, Prime128M52M3P0, + Prime128M54P4P0, Prime128M8M4M1M0, POW2_OFFSET_IMPLEMENTED_MAX_BITS, POW2_OFFSET_MAX, + POW2_OFFSET_PRIMES, POW2_OFFSET_TABLE, }; pub use module::VectorModule; pub use ntt::tables; diff --git a/src/lib.rs b/src/lib.rs index eb936ec1..0ad8cf8b 100644 --- a/src/lib.rs +++ b/src/lib.rs @@ -56,7 +56,7 @@ pub mod test_utils; pub use error::HachiError; pub use primitives::arithmetic::{ - CanonicalField, FieldCore, FieldSampling, HachiRoutines, Invertible, Module, + CanonicalField, FieldCore, FieldSampling, FromSmallInt, HachiRoutines, Invertible, Module, PseudoMersenneField, }; pub use primitives::poly::{MultilinearLagrange, Polynomial}; diff --git a/src/primitives/arithmetic.rs b/src/primitives/arithmetic.rs index 903b3a17..5bd7d2dd 100644 --- a/src/primitives/arithmetic.rs +++ b/src/primitives/arithmetic.rs @@ -39,6 +39,14 @@ pub trait FieldCore: /// Field multiplication fn mul(&self, rhs: &Self) -> Self; + /// Field squaring. + /// + /// Default is `self * self`; extension fields override with specialized + /// formulas that use fewer base-field multiplications. + fn square(&self) -> Self { + *self * *self + } + /// Field inversion. /// /// This API may branch on zero-check and is intended for public/non-secret @@ -55,14 +63,58 @@ pub trait Invertible: FieldCore { fn inv_or_zero(self) -> Self; } -/// Canonical conversion operations for field elements. -pub trait CanonicalField: FieldCore { - /// Convert from `u64`. +/// Embed small integers into a field. +/// +/// Every field contains a copy of its prime subfield, and small integers embed +/// into it canonically via reduction modulo the characteristic. This trait is +/// implementable for ALL fields — base and extension alike. +/// +/// Only `from_u64` and `from_i64` need concrete implementations; the narrower +/// widths have default impls via lossless widening. +pub trait FromSmallInt: FieldCore { + /// Embed a `u8` into the field. + fn from_u8(val: u8) -> Self { + Self::from_u64(val as u64) + } + + /// Embed an `i8` into the field. + fn from_i8(val: i8) -> Self { + Self::from_i64(val as i64) + } + + /// Embed a `u16` into the field. + fn from_u16(val: u16) -> Self { + Self::from_u64(val as u64) + } + + /// Embed an `i16` into the field. + fn from_i16(val: i16) -> Self { + Self::from_i64(val as i64) + } + + /// Embed a `u32` into the field. + fn from_u32(val: u32) -> Self { + Self::from_u64(val as u64) + } + + /// Embed an `i32` into the field. + fn from_i32(val: i32) -> Self { + Self::from_i64(val as i64) + } + + /// Embed a `u64` into the field (reduce mod characteristic). fn from_u64(val: u64) -> Self; - /// Convert from `i64`. + /// Embed an `i64` into the field (reduce mod characteristic). fn from_i64(val: i64) -> Self; +} +/// Canonical integer representation for prime (base) field elements. +/// +/// Provides a bijection between field elements and integers in `[0, p)`. +/// Only meaningful for base prime fields where elements ARE residues mod p. +/// Extension fields should NOT implement this trait. +pub trait CanonicalField: FromSmallInt { /// Return canonical integer representation as `u128`. fn to_canonical_u128(self) -> u128; diff --git a/src/protocol/commitment/onehot.rs b/src/protocol/commitment/onehot.rs index ccf5998d..8338920b 100644 --- a/src/protocol/commitment/onehot.rs +++ b/src/protocol/commitment/onehot.rs @@ -153,6 +153,7 @@ pub(crate) fn inner_ajtai_onehot( mod tests { use super::*; use crate::test_utils::F; + use crate::FromSmallInt; #[test] fn map_onehot_k_gt_d() { diff --git a/src/protocol/commitment_scheme.rs b/src/protocol/commitment_scheme.rs index f5bcc13d..4b17d921 100644 --- a/src/protocol/commitment_scheme.rs +++ b/src/protocol/commitment_scheme.rs @@ -21,10 +21,10 @@ use crate::protocol::sumcheck::norm_sumcheck::{NormSumcheckProver, NormSumcheckV use crate::protocol::sumcheck::relation_sumcheck::{ RelationSumcheckProver, RelationSumcheckVerifier, }; -use crate::protocol::sumcheck::SumcheckInstanceProver; +use crate::protocol::sumcheck::{SumcheckInstanceProver, SumcheckInstanceVerifier}; use crate::protocol::transcript::labels::CHALLENGE_SUMCHECK_ROUND; use crate::protocol::transcript::Transcript; -use crate::{CanonicalField, FieldCore, FieldSampling, Polynomial}; +use crate::{CanonicalField, FieldCore, FieldSampling, FromSmallInt, Polynomial}; #[cfg(test)] use crate::protocol::ring_switch::{eval_ring_matrix_at, expand_m_a}; @@ -209,7 +209,7 @@ where rs.num_l, ); - let verifiers: Vec<&dyn crate::protocol::sumcheck::SumcheckInstanceVerifier> = + let verifiers: Vec<&dyn SumcheckInstanceVerifier> = vec![&norm_verifier, &relation_verifier]; verify_batched_sumcheck::( &proof.sumcheck_proof, @@ -383,7 +383,7 @@ fn evaluate_packed_ring_poly( } } -fn trace(u: &CyclotomicRing) -> F { +fn trace(u: &CyclotomicRing) -> F { let d = F::from_u64(D as u64); u.coefficients()[0] * d } @@ -395,7 +395,7 @@ mod tests { use crate::protocol::commitment::CommitmentConfig; use crate::protocol::transcript::Blake2bTranscript; use crate::test_utils::F; - use crate::{CommitmentScheme, Polynomial}; + use crate::{CommitmentScheme, FromSmallInt, Polynomial}; type Cfg = DefaultCommitmentConfig; type Scheme = HachiCommitmentScheme<{ Cfg::D }, Cfg>; @@ -412,8 +412,7 @@ mod tests { let setup = >::setup_prover(num_vars); let verifier_setup = >::setup_verifier(&setup); - let (commitment, hint) = - >::commit(&poly, &setup).unwrap(); + let (commitment, hint) = >::commit(&poly, &setup).unwrap(); let opening_point: Vec = (0..num_vars).map(|i| F::from_u64((i + 2) as u64)).collect(); let opening = poly.evaluate(&opening_point); diff --git a/src/protocol/mod.rs b/src/protocol/mod.rs index 6f4d5581..f44ede06 100644 --- a/src/protocol/mod.rs +++ b/src/protocol/mod.rs @@ -28,4 +28,4 @@ pub use sumcheck::{ prove_sumcheck, verify_sumcheck, CompressedUniPoly, SumcheckInstanceProver, SumcheckInstanceVerifier, SumcheckProof, UniPoly, }; -pub use transcript::{Blake2bTranscript, KeccakTranscript, Transcript}; +pub use transcript::{sample_ext_challenge, Blake2bTranscript, KeccakTranscript, Transcript}; diff --git a/src/protocol/ring_switch.rs b/src/protocol/ring_switch.rs index b4ac4688..9868a52b 100644 --- a/src/protocol/ring_switch.rs +++ b/src/protocol/ring_switch.rs @@ -276,10 +276,9 @@ where let blocks = vec![padded]; - let (w_setup, _) = - >>::setup( - WCfg::::M, - )?; + let (w_setup, _) = >>::setup( + WCfg::::M, + )?; let (commitment, _, _) = >>::commit_ring_blocks( diff --git a/src/protocol/sumcheck/batched_sumcheck.rs b/src/protocol/sumcheck/batched_sumcheck.rs index 15673987..aee5af41 100644 --- a/src/protocol/sumcheck/batched_sumcheck.rs +++ b/src/protocol/sumcheck/batched_sumcheck.rs @@ -12,7 +12,7 @@ use super::{SumcheckInstanceProver, SumcheckInstanceVerifier, SumcheckProof, Uni use crate::error::HachiError; use crate::protocol::transcript::labels; use crate::protocol::transcript::Transcript; -use crate::{CanonicalField, FieldCore}; +use crate::{FieldCore, FromSmallInt}; fn mul_pow_2(x: E, k: usize) -> E { let mut result = x; @@ -59,9 +59,9 @@ pub fn prove_batched_sumcheck( mut sample_challenge: S, ) -> Result<(SumcheckProof, Vec), HachiError> where - F: FieldCore + CanonicalField, + F: FieldCore + crate::CanonicalField, T: Transcript, - E: FieldCore + CanonicalField, + E: FieldCore + FromSmallInt, S: FnMut(&mut T) -> E, { assert!(!instances.is_empty()); @@ -199,9 +199,9 @@ pub fn verify_batched_sumcheck( mut sample_challenge: S, ) -> Result, HachiError> where - F: FieldCore + CanonicalField, + F: FieldCore + crate::CanonicalField, T: Transcript, - E: FieldCore + CanonicalField, + E: FieldCore, S: FnMut(&mut T) -> E, { assert!(!verifiers.is_empty()); diff --git a/src/protocol/sumcheck/eq_poly.rs b/src/protocol/sumcheck/eq_poly.rs index 1b562cd0..353d0830 100644 --- a/src/protocol/sumcheck/eq_poly.rs +++ b/src/protocol/sumcheck/eq_poly.rs @@ -149,7 +149,7 @@ impl EqPolynomial { mod tests { use super::*; use crate::algebra::Fp64; - use crate::{CanonicalField, FieldSampling}; + use crate::{FieldSampling, FromSmallInt}; use rand::rngs::StdRng; use rand::SeedableRng; diff --git a/src/protocol/sumcheck/mod.rs b/src/protocol/sumcheck/mod.rs index 9788b60e..d87e65e0 100644 --- a/src/protocol/sumcheck/mod.rs +++ b/src/protocol/sumcheck/mod.rs @@ -31,8 +31,8 @@ use crate::primitives::serialization::{ }; use crate::protocol::transcript::labels; use crate::protocol::transcript::Transcript; -use crate::CanonicalField; use crate::FieldCore; +use crate::FromSmallInt; use std::io::{Read, Write}; /// Univariate polynomial in coefficient form: `p(X) = Σ_{i=0}^d coeffs[i] * X^i`. @@ -98,7 +98,7 @@ impl UniPoly { } } -impl UniPoly { +impl UniPoly { /// Interpolate from evaluations at equispaced integer points `x = 0, 1, ..., d`. /// /// Uses Newton forward-difference interpolation: compute divided differences, @@ -579,7 +579,7 @@ where /// /// This polynomial vanishes exactly when `w ∈ {−(b−1), …, b−1}`. /// Total degree in `w` is `2b − 1`. -pub fn range_check_eval(w: E, b: usize) -> E { +pub fn range_check_eval(w: E, b: usize) -> E { let mut acc = w; for k in 1..b { let k_e = E::from_u64(k as u64); diff --git a/src/protocol/sumcheck/norm_sumcheck.rs b/src/protocol/sumcheck/norm_sumcheck.rs index a5c7cf5d..81823406 100644 --- a/src/protocol/sumcheck/norm_sumcheck.rs +++ b/src/protocol/sumcheck/norm_sumcheck.rs @@ -11,7 +11,7 @@ use super::{fold_evals, multilinear_eval, range_check_eval}; use super::{SumcheckInstanceProver, SumcheckInstanceVerifier, UniPoly}; #[cfg(feature = "parallel")] use crate::parallel::*; -use crate::{CanonicalField, FieldCore}; +use crate::{FieldCore, FromSmallInt}; /// Prover for `F_{0,τ₀}(x,y) = ẽq(τ₀,(x,y)) · w̃(x,y) · range_check(w̃(x,y), b)`. /// @@ -27,7 +27,7 @@ pub struct NormSumcheckProver { b: usize, } -impl NormSumcheckProver { +impl NormSumcheckProver { /// Create a new norm (range-check) sumcheck prover. /// /// # Panics @@ -45,7 +45,7 @@ impl NormSumcheckProver { } } -impl SumcheckInstanceProver for NormSumcheckProver { +impl SumcheckInstanceProver for NormSumcheckProver { fn num_rounds(&self) -> usize { self.num_vars } @@ -134,7 +134,7 @@ pub struct NormSumcheckVerifier { b: usize, } -impl NormSumcheckVerifier { +impl NormSumcheckVerifier { /// Create a new norm (range-check) sumcheck verifier. /// /// # Panics @@ -152,7 +152,7 @@ impl NormSumcheckVerifier { } } -impl SumcheckInstanceVerifier for NormSumcheckVerifier { +impl SumcheckInstanceVerifier for NormSumcheckVerifier { fn num_rounds(&self) -> usize { self.num_vars } @@ -180,12 +180,13 @@ mod tests { use crate::primitives::multilinear_evals::DenseMultilinearEvals; use crate::protocol::ring_switch::build_w_coeffs; use crate::protocol::sumcheck::eq_poly::EqPolynomial; + use crate::protocol::sumcheck::multilinear_eval; use crate::protocol::transcript::labels; use crate::protocol::{ prove_sumcheck, verify_sumcheck, Blake2bTranscript, CommitmentConfig, CommitmentScheme, DefaultCommitmentConfig, HachiCommitmentScheme, Transcript, }; - use crate::{CanonicalField, FieldCore}; + use crate::{FieldCore, FromSmallInt}; type F = Fp64<4294967197>; const D: usize = 8; @@ -300,4 +301,44 @@ mod tests { assert_eq!(prover_challenges, verifier_challenges); } + + #[test] + fn norm_sumcheck_over_ext2() { + use crate::algebra::fields::ext::Ext2; + use crate::algebra::fields::lift::LiftBase; + + type E2 = Ext2; + + let num_vars = 3; + let n = 1usize << num_vars; + let b = 2; + let w_evals_f: Vec = (0..n).map(|i| F::from_u64(i as u64 % b as u64)).collect(); + let tau_f: Vec = (0..num_vars).map(|i| F::from_u64((i + 2) as u64)).collect(); + + let w_evals_e: Vec = w_evals_f.iter().map(|&f| E2::lift_base(f)).collect(); + let tau_e: Vec = tau_f.iter().map(|&f| E2::lift_base(f)).collect(); + + let mut prover = NormSumcheckProver::new(&tau_e, w_evals_e.clone(), b); + + let mut pt = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + let (proof, prover_challenges, final_claim) = + prove_sumcheck::(&mut prover, &mut pt, |tr| { + E2::lift_base(tr.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND)) + }) + .unwrap(); + + let oracle = EqPolynomial::mle(&tau_e, &prover_challenges) + * range_check_eval(multilinear_eval(&w_evals_e, &prover_challenges), b); + assert_eq!(final_claim, oracle, "E2 prover final claim != oracle eval"); + + let verifier = NormSumcheckVerifier::new(tau_e, w_evals_e, b); + let mut vt = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + let verifier_challenges = + verify_sumcheck::(&proof, &verifier, &mut vt, |tr| { + E2::lift_base(tr.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND)) + }) + .unwrap(); + + assert_eq!(prover_challenges, verifier_challenges); + } } diff --git a/src/protocol/sumcheck/relation_sumcheck.rs b/src/protocol/sumcheck/relation_sumcheck.rs index 0fec983c..1ba95cb9 100644 --- a/src/protocol/sumcheck/relation_sumcheck.rs +++ b/src/protocol/sumcheck/relation_sumcheck.rs @@ -13,7 +13,7 @@ use crate::cfg_into_iter; #[cfg(feature = "parallel")] use crate::parallel::*; use crate::protocol::ring_switch::eval_ring_at; -use crate::{CanonicalField, FieldCore}; +use crate::{FieldCore, FromSmallInt}; /// Prover for `F_{α,τ₁}(x,y) = w̃(x,y) · α̃(y) · m(x)`. /// @@ -30,7 +30,7 @@ pub struct RelationSumcheckProver { num_vars: usize, } -impl RelationSumcheckProver { +impl RelationSumcheckProver { /// Construct from the three constituent evaluation tables. /// /// - `w_evals`: evaluations of `w̃` over `{0,1}^{num_u + num_l}` (full domain). @@ -72,7 +72,7 @@ impl RelationSumcheckProver { } } -impl SumcheckInstanceProver for RelationSumcheckProver { +impl SumcheckInstanceProver for RelationSumcheckProver { fn num_rounds(&self) -> usize { self.num_vars } @@ -88,10 +88,7 @@ impl SumcheckInstanceProver for RelationSumche .par_iter() .zip(self.alpha_table.par_iter()) .zip(self.m_table.par_iter()) - .fold( - || E::zero(), - |acc, ((&w, &a), &m)| acc + w * a * m, - ) + .fold(|| E::zero(), |acc, ((&w, &a), &m)| acc + w * a * m) .reduce(|| E::zero(), |a, b| a + b) } #[cfg(not(feature = "parallel"))] @@ -186,7 +183,7 @@ pub struct RelationSumcheckVerifier { num_l: usize, } -impl RelationSumcheckVerifier { +impl RelationSumcheckVerifier { /// Create a new evaluation-relation sumcheck verifier. /// /// # Panics @@ -223,9 +220,7 @@ impl RelationSumcheckVerifier SumcheckInstanceVerifier - for RelationSumcheckVerifier -{ +impl SumcheckInstanceVerifier for RelationSumcheckVerifier { fn num_rounds(&self) -> usize { self.num_u + self.num_l } @@ -273,7 +268,7 @@ mod tests { prove_sumcheck, verify_sumcheck, Blake2bTranscript, CommitmentConfig, CommitmentScheme, DefaultCommitmentConfig, HachiCommitmentScheme, Transcript, }; - use crate::{CanonicalField, FieldCore}; + use crate::{FieldCore, FromSmallInt}; type F = Fp64<4294967197>; type Cfg = DefaultCommitmentConfig; @@ -302,8 +297,13 @@ mod tests { ) .unwrap(); - let (alpha, m_a_vec) = - rederive_alpha_and_m_a::(&proof, &setup, &opening_point, &commitment).unwrap(); + let (alpha, m_a_vec) = rederive_alpha_and_m_a::( + &proof, + &setup, + &opening_point, + &commitment, + ) + .unwrap(); let d = DefaultCommitmentConfig::D; assert_eq!(proof.sumcheck_aux.w.len() % d, 0); diff --git a/src/protocol/sumcheck/split_eq.rs b/src/protocol/sumcheck/split_eq.rs index 1f10dbfe..7c03db1c 100644 --- a/src/protocol/sumcheck/split_eq.rs +++ b/src/protocol/sumcheck/split_eq.rs @@ -23,7 +23,7 @@ use super::eq_poly::EqPolynomial; use super::UniPoly; -use crate::{CanonicalField, FieldCore}; +use crate::FieldCore; /// Split equality polynomial with Gruen scalar accumulation. /// @@ -56,7 +56,7 @@ pub struct GruenSplitEq { } #[allow(non_snake_case)] -impl GruenSplitEq { +impl GruenSplitEq { /// Create a new split-eq from the full challenge vector `τ`. /// /// Precomputes suffix-cached eq tables for two halves of `τ[1..n]`. @@ -152,7 +152,7 @@ mod tests { use super::*; use crate::algebra::Fp64; use crate::protocol::sumcheck::fold_evals; - use crate::FieldSampling; + use crate::{FieldSampling, FromSmallInt}; use rand::rngs::StdRng; use rand::SeedableRng; diff --git a/src/protocol/transcript/mod.rs b/src/protocol/transcript/mod.rs index 7ccd5be9..95c45413 100644 --- a/src/protocol/transcript/mod.rs +++ b/src/protocol/transcript/mod.rs @@ -4,6 +4,7 @@ mod blake2b; mod keccak; pub mod labels; +use crate::algebra::fields::lift::ExtField; use crate::{CanonicalField, FieldCore, HachiSerialize}; pub use blake2b::Blake2bTranscript; @@ -35,3 +36,20 @@ where /// Reset transcript state under a new domain label. fn reset(&mut self, domain_label: &[u8]); } + +/// Sample an extension field challenge by drawing `EXT_DEGREE` base-field +/// challenges and assembling them via `from_base_slice`. +/// +/// When `E = F` (degree 1), this compiles to a single `challenge_scalar` call. +pub fn sample_ext_challenge(tr: &mut T, label: &[u8]) -> E +where + F: FieldCore + CanonicalField, + T: Transcript, + E: ExtField, +{ + E::from_base_slice( + &(0..E::EXT_DEGREE) + .map(|_| tr.challenge_scalar(label)) + .collect::>(), + ) +} diff --git a/src/test_utils.rs b/src/test_utils.rs index 2bf775f0..604ba402 100644 --- a/src/test_utils.rs +++ b/src/test_utils.rs @@ -6,7 +6,7 @@ use crate::algebra::{CyclotomicRing, Fp64}; use crate::protocol::commitment::CommitmentConfig; -use crate::{CanonicalField, FieldCore}; +use crate::{FieldCore, FromSmallInt}; pub type F = Fp64<4294967197>; pub const D: usize = 64; diff --git a/tests/algebra.rs b/tests/algebra.rs index bac3c069..6477a460 100644 --- a/tests/algebra.rs +++ b/tests/algebra.rs @@ -22,8 +22,8 @@ mod tests { }; use hachi_pcs::primitives::serialization::SerializationError; use hachi_pcs::{ - CanonicalField, FieldCore, FieldSampling, HachiDeserialize, HachiSerialize, Invertible, - Module, PseudoMersenneField, + CanonicalField, FieldCore, FieldSampling, FromSmallInt, HachiDeserialize, HachiSerialize, + Invertible, Module, PseudoMersenneField, }; const P_159: u128 = 340282366920938463463374607431768211297u128; diff --git a/tests/commitment_contract.rs b/tests/commitment_contract.rs index dc300613..4aaf3822 100644 --- a/tests/commitment_contract.rs +++ b/tests/commitment_contract.rs @@ -6,7 +6,7 @@ use hachi_pcs::protocol::transcript::labels; use hachi_pcs::protocol::{ AppendToTranscript, Blake2bTranscript, CommitmentScheme, StreamingCommitmentScheme, Transcript, }; -use hachi_pcs::{CanonicalField, FieldCore, HachiError, Polynomial}; +use hachi_pcs::{CanonicalField, FieldCore, FromSmallInt, HachiError, Polynomial}; type F = Fp64<4294967197>; diff --git a/tests/hachi_sumcheck.rs b/tests/hachi_sumcheck.rs index 86b843f0..9c6049bc 100644 --- a/tests/hachi_sumcheck.rs +++ b/tests/hachi_sumcheck.rs @@ -10,7 +10,7 @@ use hachi_pcs::protocol::sumcheck::relation_sumcheck::{ use hachi_pcs::protocol::sumcheck::{multilinear_eval, range_check_eval}; use hachi_pcs::protocol::transcript::labels; use hachi_pcs::protocol::{prove_sumcheck, verify_sumcheck, Blake2bTranscript, Transcript}; -use hachi_pcs::{CanonicalField, FieldCore, FieldSampling}; +use hachi_pcs::{FieldCore, FieldSampling, FromSmallInt}; use rand::rngs::StdRng; use rand::SeedableRng; use std::time::Instant; @@ -41,7 +41,6 @@ fn run_f0_e2e(num_u: usize, num_l: usize, b: usize) { * range_check_eval(multilinear_eval(&w_evals, &prover_challenges), b); assert_eq!(final_claim, oracle, "prover final claim != oracle eval"); - // ---- Verifier ---- let t1 = Instant::now(); let verifier = NormSumcheckVerifier::new(tau0, w_evals, b); let mut vt = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); diff --git a/tests/onehot_commitment.rs b/tests/onehot_commitment.rs index f338f266..62f3b32d 100644 --- a/tests/onehot_commitment.rs +++ b/tests/onehot_commitment.rs @@ -2,7 +2,7 @@ use hachi_pcs::protocol::commitment::{HachiCommitmentCore, RingCommitmentScheme}; use hachi_pcs::test_utils::*; -use hachi_pcs::{CanonicalField, FieldCore}; +use hachi_pcs::{FieldCore, FromSmallInt}; type Core = HachiCommitmentCore; diff --git a/tests/sparse_challenge.rs b/tests/sparse_challenge.rs index 16eb8ee4..baba7bd3 100644 --- a/tests/sparse_challenge.rs +++ b/tests/sparse_challenge.rs @@ -6,7 +6,7 @@ use hachi_pcs::algebra::Fp64; use hachi_pcs::protocol::challenges::sparse::sparse_challenge_from_transcript; use hachi_pcs::protocol::transcript::labels::DOMAIN_HACHI_PROTOCOL; use hachi_pcs::protocol::transcript::{Blake2bTranscript, Transcript}; -use hachi_pcs::{CanonicalField, FieldCore}; +use hachi_pcs::{FieldCore, FromSmallInt}; type F = Fp64<4294967197>; diff --git a/tests/sumcheck_core.rs b/tests/sumcheck_core.rs index a9c58f28..35f07c1b 100644 --- a/tests/sumcheck_core.rs +++ b/tests/sumcheck_core.rs @@ -8,7 +8,7 @@ use hachi_pcs::protocol::{ prove_sumcheck, verify_sumcheck, Blake2bTranscript, CompressedUniPoly, SumcheckInstanceProver, SumcheckInstanceVerifier, SumcheckProof, Transcript, UniPoly, }; -use hachi_pcs::{CanonicalField, FieldCore, FieldSampling}; +use hachi_pcs::{FieldCore, FieldSampling, FromSmallInt}; use rand::rngs::StdRng; use rand::RngCore; use rand::SeedableRng; From 21b6f599eceafdba5b89ba5c66b2fbb9ebd3e61e Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Sat, 28 Feb 2026 12:13:59 -0800 Subject: [PATCH 07/34] Fix CRT+NTT correctness and optimize negacyclic NTT pipeline MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Correctness fixes: - Rewrite negacyclic NTT as twist + cyclic DIF/DIT pair (no bit-reversal permutation), correctly diagonalizing X^D+1. - Center coefficient→CRT mapping and Garner reconstruction to handle negacyclic sign wrapping consistently. - Fix i32 Montgomery csubp/caddp overflow via branchless i64 widening. - Fix q128 centering overflow in balanced_decompose_pow2 (avoid casting q≈2^128 into i128). - Remove dense-protocol schoolbook fallback; all mat-vec now routes through CRT+NTT. Performance optimizations: - Precompute per-stage twiddle roots in NttTwiddles (eliminate runtime pow_mod per butterfly stage). - Forward DIF butterfly skips reduce_range before Montgomery mul (safe because mul absorbs unreduced input). - Hoist centered-coefficient computation out of per-prime loop in from_ring. - Add fused pointwise multiply-accumulate for mat-vec inner loop. - Add batched mat_vec_mul_crt_ntt_many that precomputes matrix NTT once and reuses across many input vectors. - Wire commit_ring_blocks to batched A*s path. Benchmarks (D=64, Q32/K=6): - Single-prime forward+inverse NTT: 1.14µs → 0.43µs (2.7x) - CRT round-trip: 10.7µs → 6.3µs (1.7x) - Commit nv10: ~70% faster, nv20: ~47% faster Made-with: Cursor --- src/algebra/backend/scalar.rs | 13 +- src/algebra/mod.rs | 2 +- src/algebra/ntt/butterfly.rs | 145 +++++++--- src/algebra/ntt/prime.rs | 34 ++- src/algebra/ntt/tables.rs | 71 ++++- src/algebra/ring/crt_ntt_repr.rs | 106 ++++++- src/algebra/ring/cyclotomic.rs | 12 +- src/algebra/ring/mod.rs | 2 +- src/protocol/commitment/commit.rs | 21 +- src/protocol/commitment/utils/crt_ntt.rs | 62 ++++ src/protocol/commitment/utils/linear.rs | 248 +++++++++++++++- src/protocol/commitment/utils/mod.rs | 1 + src/protocol/quadratic_equation.rs | 8 +- src/protocol/ring_switch.rs | 109 +++++++- tests/algebra.rs | 342 ++++++++++++++++++++++- 15 files changed, 1080 insertions(+), 96 deletions(-) create mode 100644 src/protocol/commitment/utils/crt_ntt.rs diff --git a/src/algebra/backend/scalar.rs b/src/algebra/backend/scalar.rs index d147e4dd..1ef4c7da 100644 --- a/src/algebra/backend/scalar.rs +++ b/src/algebra/backend/scalar.rs @@ -70,16 +70,21 @@ impl CrtReconstruct for temp = ((temp % pi) + pi) % pi; temp = (temp * garner.gamma[i][j].to_i64()) % pi; } + // Center the mixed-radix digit to keep the final reconstruction + // in a small signed range when inputs are centered. + if temp > pi / 2 { + temp -= pi; + } v[i] = temp; } // Horner accumulation in the target field F. - let mut result = F::from_u64(v[0] as u64); - let mut partial_prod = F::from_u64(primes[0].p.to_i64() as u64); + let mut result = F::from_i64(v[0]); + let mut partial_prod = F::from_i64(primes[0].p.to_i64()); for i in 1..K { - result = result + F::from_u64(v[i] as u64) * partial_prod; + result = result + F::from_i64(v[i]) * partial_prod; if i + 1 < K { - partial_prod = partial_prod * F::from_u64(primes[i].p.to_i64() as u64); + partial_prod = partial_prod * F::from_i64(primes[i].p.to_i64()); } } *coeff = result; diff --git a/src/algebra/mod.rs b/src/algebra/mod.rs index 18a7f2ba..9d4cb6b2 100644 --- a/src/algebra/mod.rs +++ b/src/algebra/mod.rs @@ -29,6 +29,6 @@ pub use module::VectorModule; pub use ntt::tables; pub use ntt::{GarnerData, LimbQ, MontCoeff, NttPrime, PrimeWidth, RADIX_BITS}; pub use ring::{ - CrtNttConvertibleField, CyclotomicCrtNtt, CyclotomicRing, SparseChallenge, + CrtNttConvertibleField, CrtNttParamSet, CyclotomicCrtNtt, CyclotomicRing, SparseChallenge, SparseChallengeConfig, }; diff --git a/src/algebra/ntt/butterfly.rs b/src/algebra/ntt/butterfly.rs index 01b7169e..fd0f6e53 100644 --- a/src/algebra/ntt/butterfly.rs +++ b/src/algebra/ntt/butterfly.rs @@ -1,9 +1,14 @@ //! NTT butterfly transforms for negacyclic rings `Z_p[X]/(X^D + 1)`. //! -//! Uses a merged negacyclic Cooley-Tukey / Gentleman-Sande butterfly where -//! the twist factors for `X^D + 1` are folded directly into the twiddles. -//! Twiddle factors are powers of `psi`, a primitive `2D`-th root of unity -//! (`psi^D = -1 mod p`), rather than a `D`-th root. +//! Implements a negacyclic NTT via the standard **twist + cyclic NTT** method. +//! +//! Let `psi` be a primitive `2D`-th root of unity (`psi^D = -1 mod p`) and +//! `omega = psi^2`, a primitive `D`-th root of unity. For polynomials modulo +//! `X^D + 1`, we: +//! - pre-twist coefficients by `psi^i` +//! - run a cyclic size-`D` NTT using `omega` +//! - inverse-cyclic NTT using `omega^{-1}` +//! - post-untwist by `psi^{-i}` use super::prime::{MontCoeff, NttPrime, PrimeWidth}; @@ -11,8 +16,16 @@ use super::prime::{MontCoeff, NttPrime, PrimeWidth}; /// /// `D` must be a power of two. pub struct NttTwiddles { - /// Twiddle factors in Montgomery form, indexed by bit-reversed position. - pub(crate) zetas: [MontCoeff; D], + /// Stage roots for iterative forward cyclic NTT in Montgomery form. + pub(crate) fwd_wlen: [MontCoeff; D], + /// Stage roots for iterative inverse cyclic NTT in Montgomery form. + pub(crate) inv_wlen: [MontCoeff; D], + /// Number of active stages in the twiddle arrays (`log2(D)`). + pub(crate) num_stages: usize, + /// Twist factors `psi^i` for negacyclic embedding, in Montgomery form. + pub(crate) psi_pows: [MontCoeff; D], + /// Untwist factors `psi^{-i}`, in Montgomery form. + pub(crate) psi_inv_pows: [MontCoeff; D], /// `D^{-1} mod p` in Montgomery form, used for inverse NTT final scaling. pub(crate) d_inv: MontCoeff, } @@ -20,9 +33,9 @@ pub struct NttTwiddles { impl NttTwiddles { /// Compute twiddle factors for the given prime. /// - /// Finds a primitive `2D`-th root of unity mod `p`, then fills the - /// twiddle table in bit-reversed order. All values are stored in - /// Montgomery form. + /// Finds a primitive `2D`-th root `psi` and derives `omega = psi^2`. + /// Fills cyclic forward/inverse twiddles for `omega` and twist/untwist + /// tables for `psi`. All values are stored in Montgomery form. /// /// # Panics /// @@ -35,24 +48,49 @@ impl NttTwiddles { "2D must divide p - 1 for NTT roots to exist" ); - let n = D.trailing_zeros(); let psi = find_primitive_root_2d(p, D); + let omega = (psi * psi) % p; + let omega_inv = pow_mod(omega, p - 2, p); + + let psi_inv = pow_mod(psi, p - 2, p); + let mut psi_pows = [MontCoeff::from_raw(W::default()); D]; + let mut psi_inv_pows = [MontCoeff::from_raw(W::default()); D]; + let mut cur = 1i64; + let mut cur_inv = 1i64; + for i in 0..D { + psi_pows[i] = prime.from_canonical(W::from_i64(cur)); + psi_inv_pows[i] = prime.from_canonical(W::from_i64(cur_inv)); + cur = (cur * psi) % p; + cur_inv = (cur_inv * psi_inv) % p; + } - let mut zetas = [MontCoeff::from_raw(W::default()); D]; - for (i, z) in zetas.iter_mut().enumerate() { - let brv_i = bit_reverse(i, n); - let power = pow_mod(psi, brv_i as i64, p); - *z = prime.from_canonical(W::from_i64(power)); + let mut fwd_wlen = [MontCoeff::from_raw(W::default()); D]; + let mut inv_wlen = [MontCoeff::from_raw(W::default()); D]; + let mut len = 1usize; + let mut stage = 0usize; + while len < D { + let exp = (D / (2 * len)) as i64; + fwd_wlen[stage] = prime.from_canonical(W::from_i64(pow_mod(omega, exp, p))); + inv_wlen[stage] = prime.from_canonical(W::from_i64(pow_mod(omega_inv, exp, p))); + len *= 2; + stage += 1; } let d_inv_canonical = pow_mod(D as i64, p - 2, p); let d_inv = prime.from_canonical(W::from_i64(d_inv_canonical)); - Self { zetas, d_inv } + Self { + fwd_wlen, + inv_wlen, + num_stages: stage, + psi_pows, + psi_inv_pows, + d_inv, + } } } -/// Forward negacyclic NTT (Cooley-Tukey, decimation-in-time). +/// Forward negacyclic NTT (twist + cyclic Gentleman-Sande DIF). /// /// Transforms `D` coefficients in-place from coefficient form to NTT /// evaluation form. Both outputs of each butterfly are range-reduced @@ -62,27 +100,40 @@ pub fn forward_ntt( prime: NttPrime, tw: &NttTwiddles, ) { - let mut k = 1usize; + for (ai, psi) in a.iter_mut().zip(tw.psi_pows.iter()) { + *ai = prime.mul(*ai, *psi); + } + + let one = prime.from_canonical(W::from_i64(1)); + let mut len = D / 2; - while len >= 1 { - let mut start = 0; + let mut stage = tw.num_stages; + while len > 0 { + stage -= 1; + let wlen = tw.fwd_wlen[stage]; + let mut start = 0usize; while start < D { - let zeta = tw.zetas[k]; - k += 1; - for j in start..(start + len) { - let t = prime.mul(a[j + len], zeta); - let diff = a[j].raw().wrapping_sub(t.raw()); - let sum = a[j].raw().wrapping_add(t.raw()); - a[j + len] = prime.reduce_range(MontCoeff::from_raw(diff)); - a[j] = prime.reduce_range(MontCoeff::from_raw(sum)); + let mut w = one; + for j in 0..len { + let u = a[start + j]; + let v = a[start + j + len]; + let sum = u.raw().wrapping_add(v.raw()); + let diff = u.raw().wrapping_sub(v.raw()); + a[start + j] = prime.reduce_range(MontCoeff::from_raw(sum)); + a[start + j + len] = prime.mul(MontCoeff::from_raw(diff), w); + w = prime.mul(w, wlen); } start += 2 * len; } len /= 2; } + + // Keep exported NTT-domain coefficients in the same reduced range expected + // by add/sub reduced operations and equality checks. + prime.reduce_range_in_place(a); } -/// Inverse negacyclic NTT (Gentleman-Sande, decimation-in-frequency). +/// Inverse negacyclic NTT (cyclic Cooley-Tukey DIT + untwist). /// /// Transforms `D` evaluations in-place back to coefficient form. /// Includes the final `D^{-1}` scaling. @@ -91,33 +142,37 @@ pub fn inverse_ntt( prime: NttPrime, tw: &NttTwiddles, ) { - let mut k = D - 1; - let mut len = 1; - while len <= D / 2 { - let mut start = 0; + let one = prime.from_canonical(W::from_i64(1)); + + let mut len = 1usize; + let mut stage = 0usize; + while len < D { + let wlen = tw.inv_wlen[stage]; + let mut start = 0usize; while start < D { - let zeta = tw.zetas[k]; - k = k.wrapping_sub(1); - for j in start..(start + len) { - let t = a[j]; - let sum = t.raw().wrapping_add(a[j + len].raw()); - let diff = t.raw().wrapping_sub(a[j + len].raw()); - a[j] = prime.reduce_range(MontCoeff::from_raw(sum)); - let neg_zeta = MontCoeff::from_raw(zeta.raw().wrapping_neg()); - a[j + len] = prime.mul(MontCoeff::from_raw(diff), neg_zeta); + let mut w = one; + for j in 0..len { + let u = a[start + j]; + let v = prime.mul(a[start + j + len], w); + let sum = u.raw().wrapping_add(v.raw()); + let diff = u.raw().wrapping_sub(v.raw()); + a[start + j] = prime.reduce_range(MontCoeff::from_raw(sum)); + a[start + j + len] = prime.reduce_range(MontCoeff::from_raw(diff)); + w = prime.mul(w, wlen); } start += 2 * len; } len *= 2; + stage += 1; } for c in a.iter_mut() { *c = prime.mul(*c, tw.d_inv); } -} -fn bit_reverse(x: usize, n: u32) -> usize { - x.reverse_bits() >> (usize::BITS - n) + for (ai, psi_inv) in a.iter_mut().zip(tw.psi_inv_pows.iter()) { + *ai = prime.mul(*ai, *psi_inv); + } } /// Find a primitive `2D`-th root of unity mod `p`. diff --git a/src/algebra/ntt/prime.rs b/src/algebra/ntt/prime.rs index 038139ff..afe9b7d9 100644 --- a/src/algebra/ntt/prime.rs +++ b/src/algebra/ntt/prime.rs @@ -266,16 +266,40 @@ impl NttPrime { /// Conditionally subtract `p` if `a >= p` (branchless). #[inline] pub fn csubp(self, a: MontCoeff) -> MontCoeff { - let diff = a.0.wrapping_sub(self.p); - let mask = diff.sign_mask(); - MontCoeff(diff.wrapping_add(mask.bitand(self.p))) + // IMPORTANT: `a` may be as low as `-2p` during butterflies. + // For 30-bit primes with `i32` width, the probe `a - p` can overflow + // a narrow signed subtraction. Use widened arithmetic for the compare. + // + // For `i16` (Q32 primes), the narrow branchless probe is safe and faster. + if W::R_LOG == 16 { + let diff = a.0.wrapping_sub(self.p); + let mask = diff.sign_mask(); + MontCoeff(diff.wrapping_add(mask.bitand(self.p))) + } else { + // Branchless i64 variant: + // if diff < 0 => mask = -1 => out = diff + p = a + // else out = diff = a - p + let ai = a.0.to_i64(); + let pi = self.p.to_i64(); + let diff = ai - pi; + let mask = diff >> 63; + MontCoeff(W::from_i64(diff + (mask & pi))) + } } /// Conditionally add `p` if `a < 0` (branchless). #[inline] pub fn caddp(self, a: MontCoeff) -> MontCoeff { - let mask = a.0.sign_mask(); - MontCoeff(a.0.wrapping_add(mask.bitand(self.p))) + if W::R_LOG == 16 { + let mask = a.0.sign_mask(); + MontCoeff(a.0.wrapping_add(mask.bitand(self.p))) + } else { + // Branchless i64 variant: mask = -1 iff a < 0. + let ai = a.0.to_i64(); + let pi = self.p.to_i64(); + let mask = ai >> 63; + MontCoeff(W::from_i64(ai + (mask & pi))) + } } /// Range-reduce from `(-2p, 2p)` to `(-p, p)`. diff --git a/src/algebra/ntt/tables.rs b/src/algebra/ntt/tables.rs index bea49764..70e35bd1 100644 --- a/src/algebra/ntt/tables.rs +++ b/src/algebra/ntt/tables.rs @@ -1,6 +1,7 @@ //! Deterministic parameter presets for small-prime CRT arithmetic. //! //! Q32: `logq = 32` with six `i16` NTT-friendly primes (D ≤ 64). +//! Q64: `logq = 64` with `i32` NTT-friendly primes (D ≤ 1024). //! Q128: `logq = 128` with five `i32` NTT-friendly primes (D ≤ 1024). use super::crt::GarnerData; @@ -8,6 +9,8 @@ use super::prime::NttPrime; /// Polynomial degree for the base ring `Z_q[X]/(X^d + 1)`. pub const RING_DEGREE: usize = 64; +/// Maximum ring degree covered by the i32 CRT parameter sets. +pub const MAX_CRT_RING_DEGREE: usize = 1024; /// Number of CRT primes for the `logq = 32` parameter set. pub const Q32_NUM_PRIMES: usize = 6; @@ -62,16 +65,61 @@ pub fn q32_garner() -> GarnerData { GarnerData::compute(&Q32_PRIMES) } +/// Number of CRT primes for the `logq = 64` fast profile (`P > q`). +pub const Q64_NUM_PRIMES_FAST: usize = 3; +/// Number of CRT primes for the `logq = 64` conservative profile (`P > 128*q^2`). +pub const Q64_NUM_PRIMES: usize = 5; + +/// The modulus `q = 2^64 - 59`. +pub const Q64_MODULUS: u64 = u64::MAX - 58; + /// Number of CRT primes for the `logq = 128` parameter set. pub const Q128_NUM_PRIMES: usize = 5; /// The modulus `q = 2^128 - 275`. pub const Q128_MODULUS: u128 = u128::MAX - 274; -/// Raw 30-bit primes for Q128, each satisfying `2048 | (p - 1)`. -pub const Q128_RAW_PRIMES: [i32; Q128_NUM_PRIMES] = +/// Raw 30-bit primes for D≤1024, each satisfying `2048 | (p - 1)`. +/// +/// They are ordered descending by value. +pub const D1024_RAW_PRIMES: [i32; Q128_NUM_PRIMES] = [1073707009, 1073698817, 1073692673, 1073682433, 1073668097]; +/// Raw 30-bit primes for Q64 fast profile (`K=3`, `P > q`). +pub const Q64_RAW_PRIMES_FAST: [i32; Q64_NUM_PRIMES_FAST] = [ + D1024_RAW_PRIMES[0], + D1024_RAW_PRIMES[1], + D1024_RAW_PRIMES[2], +]; + +/// Raw 30-bit primes for Q64 conservative profile (`K=5`, `P > 128*q^2`). +pub const Q64_RAW_PRIMES: [i32; Q64_NUM_PRIMES] = D1024_RAW_PRIMES; + +/// Raw 30-bit primes for Q128, each satisfying `2048 | (p - 1)`. +pub const Q128_RAW_PRIMES: [i32; Q128_NUM_PRIMES] = D1024_RAW_PRIMES; + +/// CRT primes and per-prime Montgomery constants for `logq = 64` fast profile. +pub fn q64_primes_fast() -> [NttPrime; Q64_NUM_PRIMES_FAST] { + std::array::from_fn(|k| NttPrime::compute(Q64_RAW_PRIMES_FAST[k])) +} + +/// Garner CRT reconstruction constants for Q64 fast profile. +pub fn q64_garner_fast() -> GarnerData { + let primes = q64_primes_fast(); + GarnerData::compute(&primes) +} + +/// CRT primes and per-prime Montgomery constants for `logq = 64` conservative profile. +pub fn q64_primes() -> [NttPrime; Q64_NUM_PRIMES] { + std::array::from_fn(|k| NttPrime::compute(Q64_RAW_PRIMES[k])) +} + +/// Garner CRT reconstruction constants for Q64 conservative profile. +pub fn q64_garner() -> GarnerData { + let primes = q64_primes(); + GarnerData::compute(&primes) +} + /// CRT primes and per-prime Montgomery constants for `logq = 128`. pub fn q128_primes() -> [NttPrime; Q128_NUM_PRIMES] { std::array::from_fn(|k| NttPrime::compute(Q128_RAW_PRIMES[k])) @@ -129,6 +177,25 @@ mod tests { } } + #[test] + fn verify_q64_primes_are_valid() { + let primes = q64_primes(); + for np in &primes { + let p = np.p as i64; + assert!(p > 1 && p % 2 == 1, "prime must be odd and > 1"); + assert_eq!( + (p - 1) % 2048, + 0, + "2048 must divide p-1 for D=1024 NTT (p={p})" + ); + assert_eq!( + np.p.wrapping_mul(np.pinv), + 1, + "pinv verification failed for p={p}" + ); + } + } + #[test] fn garner_data_is_consistent() { let garner = q32_garner(); diff --git a/src/algebra/ring/crt_ntt_repr.rs b/src/algebra/ring/crt_ntt_repr.rs index 2574769b..0d5d5bb1 100644 --- a/src/algebra/ring/crt_ntt_repr.rs +++ b/src/algebra/ring/crt_ntt_repr.rs @@ -29,6 +29,34 @@ pub trait CrtNttConvertibleField: FieldCore + CanonicalField {} impl CrtNttConvertibleField for F {} +/// Bundled CRT+NTT parameters for a fixed width/prime-count/degree tuple. +/// +/// Keeps primes/twiddles/Garner constants consistent and avoids passing them +/// independently at every call site. +pub struct CrtNttParamSet { + /// CRT primes with Montgomery constants. + pub primes: [NttPrime; K], + /// Per-prime twiddle tables for forward/inverse NTT. + pub twiddles: [NttTwiddles; K], + /// Garner reconstruction constants for CRT lift-back. + pub garner: GarnerData, +} + +impl CrtNttParamSet { + /// Build a full parameter set from CRT primes. + /// + /// Computes per-prime twiddles and Garner reconstruction constants. + pub fn new(primes: [NttPrime; K]) -> Self { + let twiddles = std::array::from_fn(|k| NttTwiddles::compute(primes[k])); + let garner = GarnerData::compute(&primes); + Self { + primes, + twiddles, + garner, + } + } +} + impl CyclotomicCrtNtt { /// The additive identity (all zeros in every CRT limb). pub fn zero() -> Self { @@ -47,6 +75,15 @@ impl CyclotomicCrtNtt { Self::from_ring_with_backend::(ring, primes, twiddles) } + /// Convert a coefficient-form ring element into CRT+NTT domain + /// using a bundled parameter set and the scalar backend. + pub fn from_ring_with_params( + ring: &CyclotomicRing, + params: &CrtNttParamSet, + ) -> Self { + Self::from_ring(ring, ¶ms.primes, ¶ms.twiddles) + } + /// Convert a coefficient-form ring element into CRT+NTT domain /// through an explicit backend implementation. pub fn from_ring_with_backend< @@ -57,12 +94,34 @@ impl CyclotomicCrtNtt { primes: &[NttPrime; K], twiddles: &[NttTwiddles; K], ) -> Self { + let q = (-F::one()).to_canonical_u128() + 1; + let half_q = q / 2; + let centered_coeffs: [i128; D] = std::array::from_fn(|i| { + let canonical = ring.coeffs[i].to_canonical_u128(); + if canonical > half_q { + -((q - canonical) as i128) + } else { + canonical as i128 + } + }); + let mut limbs = [[MontCoeff::from_raw(W::default()); D]; K]; for ((limb, prime), tw) in limbs.iter_mut().zip(primes.iter()).zip(twiddles.iter()) { - for (dst, src) in limb.iter_mut().zip(ring.coeffs.iter()) { - let canonical = src.to_canonical_u128(); - let residue = (canonical % (prime.p.to_i64() as u128)) as i64; - *dst = B::from_canonical(*prime, W::from_i64(residue)); + // Interpret coefficients in centered form (-q/2, q/2] before reducing + // into the CRT primes. This makes the reduction map consistent with + // negacyclic subtraction (which naturally produces negative values). + let p = prime.p.to_i64() as i128; + let half_p = p / 2; + for (dst, centered) in limb.iter_mut().zip(centered_coeffs.iter()) { + let mut r = *centered % p; + if r < 0 { + r += p; + } + // Center residues into [-p/2, p/2) for stable signed arithmetic. + if r >= half_p { + r -= p; + } + *dst = B::from_canonical(*prime, W::from_i64(r as i64)); } B::forward_ntt(limb, *prime, tw); } @@ -80,6 +139,15 @@ impl CyclotomicCrtNtt { self.to_ring_with_backend::(primes, twiddles, garner) } + /// Convert from CRT+NTT domain back to coefficient form + /// using a bundled parameter set and the scalar backend. + pub fn to_ring_with_params( + &self, + params: &CrtNttParamSet, + ) -> CyclotomicRing { + self.to_ring(¶ms.primes, ¶ms.twiddles, ¶ms.garner) + } + /// Convert from CRT+NTT domain back to coefficient form /// through an explicit backend implementation. pub fn to_ring_with_backend< @@ -101,7 +169,8 @@ impl CyclotomicCrtNtt { let mut limb = self.limbs[k]; B::inverse_ntt(&mut limb, *prime, tw); for (dst, src) in can.iter_mut().zip(limb.iter()) { - *dst = B::to_canonical(*prime, *src); + let canon = B::to_canonical(*prime, *src); + *dst = prime.center(canon); } } @@ -116,6 +185,11 @@ impl CyclotomicCrtNtt { self.add_reduced_with_backend::(rhs, primes) } + /// Add another CRT+NTT element and reduce using a bundled parameter set. + pub fn add_reduced_with_params(&self, rhs: &Self, params: &CrtNttParamSet) -> Self { + self.add_reduced(rhs, ¶ms.primes) + } + /// Add another CRT+NTT element and reduce each coefficient with the matching /// prime through an explicit backend implementation. pub fn add_reduced_with_backend>( @@ -139,6 +213,11 @@ impl CyclotomicCrtNtt { self.sub_reduced_with_backend::(rhs, primes) } + /// Subtract another CRT+NTT element and reduce using a bundled parameter set. + pub fn sub_reduced_with_params(&self, rhs: &Self, params: &CrtNttParamSet) -> Self { + self.sub_reduced(rhs, ¶ms.primes) + } + /// Subtract another CRT+NTT element and reduce through an explicit backend. pub fn sub_reduced_with_backend>( &self, @@ -161,6 +240,11 @@ impl CyclotomicCrtNtt { self.neg_reduced_with_backend::(primes) } + /// Negate each CRT+NTT coefficient and reduce using a bundled parameter set. + pub fn neg_reduced_with_params(&self, params: &CrtNttParamSet) -> Self { + self.neg_reduced(¶ms.primes) + } + /// Negate each CRT+NTT coefficient and reduce through an explicit backend. pub fn neg_reduced_with_backend>( &self, @@ -182,6 +266,11 @@ impl CyclotomicCrtNtt { self.pointwise_mul_with_backend::(rhs, primes) } + /// Pointwise multiplication in CRT+NTT domain using a bundled parameter set. + pub fn pointwise_mul_with_params(&self, rhs: &Self, params: &CrtNttParamSet) -> Self { + self.pointwise_mul(rhs, ¶ms.primes) + } + /// Pointwise multiplication in CRT+NTT domain through an explicit backend. pub fn pointwise_mul_with_backend>( &self, @@ -195,7 +284,12 @@ impl CyclotomicCrtNtt { .zip(rhs.limbs.iter()) .enumerate() { - B::pointwise_mul(primes[k], o, a, b); + let prime = primes[k]; + B::pointwise_mul(prime, o, a, b); + // Keep coefficients in a bounded range for subsequent inverse NTT. + for c in o.iter_mut() { + *c = B::reduce_range(prime, *c); + } } Self { limbs: out } } diff --git a/src/algebra/ring/cyclotomic.rs b/src/algebra/ring/cyclotomic.rs index 5dd2f0fc..acbfc403 100644 --- a/src/algebra/ring/cyclotomic.rs +++ b/src/algebra/ring/cyclotomic.rs @@ -249,17 +249,17 @@ impl CyclotomicRing { let b = 1i128 << log_basis; let half_b = b / 2; - let q = (-F::one()).to_canonical_u128() as i128 + 1; + let q = (-F::one()).to_canonical_u128() + 1; let half_q = q / 2; let mut digit_planes: Vec<[F; D]> = (0..levels).map(|_| [F::zero(); D]).collect(); for i in 0..D { - let canonical = self.coeffs[i].to_canonical_u128() as i128; - let mut c = if canonical > half_q { - canonical - q + let canonical = self.coeffs[i].to_canonical_u128(); + let mut c: i128 = if canonical > half_q { + -((q - canonical) as i128) } else { - canonical + canonical as i128 }; for plane in digit_planes.iter_mut() { @@ -270,7 +270,7 @@ impl CyclotomicRing { plane[i] = if balanced >= 0 { F::from_canonical_u128_reduced(balanced as u128) } else { - F::from_canonical_u128_reduced((q + balanced) as u128) + F::from_canonical_u128_reduced(q - ((-balanced) as u128)) }; } } diff --git a/src/algebra/ring/mod.rs b/src/algebra/ring/mod.rs index 1b64555e..756854b3 100644 --- a/src/algebra/ring/mod.rs +++ b/src/algebra/ring/mod.rs @@ -4,6 +4,6 @@ pub mod crt_ntt_repr; pub mod cyclotomic; pub mod sparse_challenge; -pub use crt_ntt_repr::{CrtNttConvertibleField, CyclotomicCrtNtt}; +pub use crt_ntt_repr::{CrtNttConvertibleField, CrtNttParamSet, CyclotomicCrtNtt}; pub use cyclotomic::CyclotomicRing; pub use sparse_challenge::{SparseChallenge, SparseChallengeConfig}; diff --git a/src/protocol/commitment/commit.rs b/src/protocol/commitment/commit.rs index 485a5570..1d188ac6 100644 --- a/src/protocol/commitment/commit.rs +++ b/src/protocol/commitment/commit.rs @@ -6,7 +6,10 @@ use super::config::{ use super::onehot::{inner_ajtai_onehot, map_onehot_to_sparse_blocks}; use super::scheme::RingCommitmentScheme; use super::types::RingCommitment; -use super::utils::linear::{decompose_block, decompose_rows, mat_vec_mul_unchecked}; +use super::utils::linear::{ + decompose_block, decompose_rows, mat_vec_mul_crt_ntt, mat_vec_mul_crt_ntt_many, + mat_vec_mul_unchecked, +}; use super::utils::matrix::{derive_public_matrix, sample_public_matrix_seed, PublicMatrixSeed}; use super::CommitmentConfig; use crate::algebra::ring::CyclotomicRing; @@ -88,21 +91,21 @@ where ensure_matrix_shape(&setup.A, Cfg::N_A, layout.inner_width, "A")?; ensure_matrix_shape(&setup.B, Cfg::N_B, layout.outer_width, "B")?; - let mut s_all: Vec>> = Vec::with_capacity(layout.num_blocks); + let s_all: Vec>> = f_blocks + .iter() + .map(|block| decompose_block(block, Cfg::DELTA, Cfg::LOG_BASIS)) + .collect(); + + let t_all = mat_vec_mul_crt_ntt_many(&setup.A, &s_all)?; let mut t_hat_all: Vec>> = Vec::with_capacity(layout.num_blocks); let mut t_hat_flat: Vec> = Vec::with_capacity(layout.outer_width); - for block in f_blocks { - let s_i = decompose_block(block, Cfg::DELTA, Cfg::LOG_BASIS); - - let t_i = mat_vec_mul_unchecked(&setup.A, &s_i); + for t_i in t_all { let t_hat_i = decompose_rows(&t_i, Cfg::DELTA, Cfg::LOG_BASIS); t_hat_flat.extend(t_hat_i.iter().copied()); - - s_all.push(s_i); t_hat_all.push(t_hat_i); } - let u = mat_vec_mul_unchecked(&setup.B, &t_hat_flat); + let u = mat_vec_mul_crt_ntt(&setup.B, &t_hat_flat)?; Ok((RingCommitment { u }, s_all, t_hat_all)) } diff --git a/src/protocol/commitment/utils/crt_ntt.rs b/src/protocol/commitment/utils/crt_ntt.rs new file mode 100644 index 00000000..4dd0fe2e --- /dev/null +++ b/src/protocol/commitment/utils/crt_ntt.rs @@ -0,0 +1,62 @@ +//! Protocol-facing CRT+NTT parameter dispatch. + +use crate::algebra::ntt::tables::{ + q128_primes, q64_primes, MAX_CRT_RING_DEGREE, Q128_MODULUS, Q128_NUM_PRIMES, Q32_MODULUS, + Q32_NUM_PRIMES, Q32_PRIMES, Q64_MODULUS, Q64_NUM_PRIMES, RING_DEGREE, +}; +use crate::algebra::ring::CrtNttParamSet; +use crate::error::HachiError; +use crate::CanonicalField; + +use super::norm::detect_field_modulus; + +/// Supported protocol CRT+NTT parameter families. +pub(crate) enum ProtocolCrtNttParams { + Q32(CrtNttParamSet), + Q64(CrtNttParamSet), + Q128(CrtNttParamSet), +} + +/// Select a CRT+NTT parameter set from field modulus and ring degree. +/// +/// Dispatch policy: +/// - `q <= 2^32-99` and `D <= 64`: Q32 (`i16`) +/// - `q <= 2^64-59` and `D <= 1024`: Q64 (`i32`, conservative K=5) +/// - `q == 2^128-275` and `D <= 1024`: Q128 (`i32`, K=5) +/// - otherwise: explicit setup error +pub(crate) fn select_crt_ntt_params( +) -> Result, HachiError> { + if !D.is_power_of_two() { + return Err(HachiError::InvalidSetup(format!( + "CRT+NTT requires power-of-two ring degree, got D={D}" + ))); + } + if D > MAX_CRT_RING_DEGREE { + return Err(HachiError::InvalidSetup(format!( + "CRT+NTT supports D <= {MAX_CRT_RING_DEGREE}, got D={D}" + ))); + } + + let modulus = detect_field_modulus::(); + + if modulus == Q128_MODULUS { + return Ok(ProtocolCrtNttParams::Q128(CrtNttParamSet::new( + q128_primes(), + ))); + } + + if modulus <= Q32_MODULUS as u128 { + if D <= RING_DEGREE { + return Ok(ProtocolCrtNttParams::Q32(CrtNttParamSet::new(Q32_PRIMES))); + } + return Ok(ProtocolCrtNttParams::Q64(CrtNttParamSet::new(q64_primes()))); + } + + if modulus <= Q64_MODULUS as u128 { + return Ok(ProtocolCrtNttParams::Q64(CrtNttParamSet::new(q64_primes()))); + } + + Err(HachiError::InvalidSetup(format!( + "no CRT+NTT parameter set for modulus {modulus} and D={D}; supported ranges: <= {Q64_MODULUS} (with Q32/Q64 dispatch) or exactly {Q128_MODULUS}" + ))) +} diff --git a/src/protocol/commitment/utils/linear.rs b/src/protocol/commitment/utils/linear.rs index daca6d24..2ac38dc8 100644 --- a/src/protocol/commitment/utils/linear.rs +++ b/src/protocol/commitment/utils/linear.rs @@ -1,8 +1,12 @@ //! Linear algebra helpers for ring commitment. -use crate::algebra::ring::CyclotomicRing; +use crate::algebra::ntt::{MontCoeff, PrimeWidth}; +use crate::algebra::ring::{CrtNttParamSet, CyclotomicCrtNtt, CyclotomicRing}; +use crate::error::HachiError; use crate::{CanonicalField, FieldCore}; +use super::crt_ntt::{select_crt_ntt_params, ProtocolCrtNttParams}; + pub(crate) fn mat_vec_mul_unchecked( mat: &[Vec>], vec: &[CyclotomicRing], @@ -19,6 +23,143 @@ pub(crate) fn mat_vec_mul_unchecked( + acc: &mut CyclotomicCrtNtt, + lhs: &CyclotomicCrtNtt, + rhs: &CyclotomicCrtNtt, + params: &CrtNttParamSet, +) { + for k in 0..K { + let prime = params.primes[k]; + let acc_limb = &mut acc.limbs[k]; + let lhs_limb = &lhs.limbs[k]; + let rhs_limb = &rhs.limbs[k]; + for ((acc_coeff, lhs_coeff), rhs_coeff) in acc_limb + .iter_mut() + .zip(lhs_limb.iter()) + .zip(rhs_limb.iter()) + { + let prod = prime.mul(*lhs_coeff, *rhs_coeff); + let sum = MontCoeff::from_raw(acc_coeff.raw().wrapping_add(prod.raw())); + *acc_coeff = prime.reduce_range(sum); + } + } +} + +fn precompute_dense_mat_ntt_with_params< + F: FieldCore + CanonicalField, + W: PrimeWidth, + const K: usize, + const D: usize, +>( + mat: &[Vec>], + params: &CrtNttParamSet, +) -> Vec>> { + mat.iter() + .map(|row| { + row.iter() + .map(|a| CyclotomicCrtNtt::from_ring_with_params(a, params)) + .collect() + }) + .collect() +} + +fn mat_vec_mul_dense_with_params< + F: FieldCore + CanonicalField, + W: PrimeWidth, + const K: usize, + const D: usize, +>( + mat: &[Vec>], + vec: &[CyclotomicRing], + params: &CrtNttParamSet, +) -> Vec> { + let ntt_vec: Vec> = vec + .iter() + .map(|v| CyclotomicCrtNtt::from_ring_with_params(v, params)) + .collect(); + + mat.iter() + .map(|row| { + debug_assert_eq!(row.len(), ntt_vec.len()); + let mut acc = CyclotomicCrtNtt::::zero(); + for (a, x_ntt) in row.iter().zip(ntt_vec.iter()) { + let a_ntt = CyclotomicCrtNtt::from_ring_with_params(a, params); + accumulate_pointwise_product_into(&mut acc, &a_ntt, x_ntt, params); + } + acc.to_ring_with_params(params) + }) + .collect() +} + +fn mat_vec_mul_dense_many_with_params< + F: FieldCore + CanonicalField, + W: PrimeWidth, + const K: usize, + const D: usize, +>( + mat: &[Vec>], + vecs: &[Vec>], + params: &CrtNttParamSet, +) -> Vec>> { + let ntt_mat = precompute_dense_mat_ntt_with_params(mat, params); + vecs.iter() + .map(|vec| { + let ntt_vec: Vec> = vec + .iter() + .map(|v| CyclotomicCrtNtt::from_ring_with_params(v, params)) + .collect(); + + ntt_mat + .iter() + .map(|row_ntt| { + debug_assert_eq!(row_ntt.len(), ntt_vec.len()); + let mut acc = CyclotomicCrtNtt::::zero(); + for (a_ntt, x_ntt) in row_ntt.iter().zip(ntt_vec.iter()) { + accumulate_pointwise_product_into(&mut acc, a_ntt, x_ntt, params); + } + acc.to_ring_with_params(params) + }) + .collect() + }) + .collect() +} + +/// Dense mat-vec multiplication routed through protocol CRT+NTT dispatch. +/// +/// This replaces coefficient-domain schoolbook multiplication for dense +/// protocol paths while preserving the existing sparse-specialized paths. +pub(crate) fn mat_vec_mul_crt_ntt( + mat: &[Vec>], + vec: &[CyclotomicRing], +) -> Result>, HachiError> { + let params = select_crt_ntt_params::()?; + let out = match ¶ms { + ProtocolCrtNttParams::Q32(p) => mat_vec_mul_dense_with_params(mat, vec, p), + ProtocolCrtNttParams::Q64(p) => mat_vec_mul_dense_with_params(mat, vec, p), + ProtocolCrtNttParams::Q128(p) => mat_vec_mul_dense_with_params(mat, vec, p), + }; + Ok(out) +} + +/// Dense batched mat-vec multiplication routed through protocol CRT+NTT dispatch. +/// +/// This variant precomputes the matrix in CRT+NTT domain once and applies it +/// to many input vectors, which is faster when `mat` is reused. +pub(crate) fn mat_vec_mul_crt_ntt_many( + mat: &[Vec>], + vecs: &[Vec>], +) -> Result>>, HachiError> { + let params = select_crt_ntt_params::()?; + let out = match ¶ms { + ProtocolCrtNttParams::Q32(p) => mat_vec_mul_dense_many_with_params(mat, vecs, p), + ProtocolCrtNttParams::Q64(p) => mat_vec_mul_dense_many_with_params(mat, vecs, p), + ProtocolCrtNttParams::Q128(p) => mat_vec_mul_dense_many_with_params(mat, vecs, p), + }; + Ok(out) +} + pub(crate) fn decompose_block( block: &[CyclotomicRing], delta: usize, @@ -42,3 +183,108 @@ pub(crate) fn decompose_rows( } out } + +#[cfg(test)] +mod tests { + use super::{mat_vec_mul_crt_ntt, mat_vec_mul_crt_ntt_many, mat_vec_mul_unchecked}; + use crate::algebra::{CyclotomicRing, Fp64}; + use crate::FromSmallInt; + + #[test] + fn dense_mat_vec_matches_schoolbook_q32_d64() { + type F = Fp64<4294967197>; + const D: usize = 64; + let mat: Vec>> = (0..3) + .map(|i| { + (0..4) + .map(|j| { + let coeffs = std::array::from_fn(|k| { + F::from_u64((i as u64 * 10_000 + j as u64 * 100 + k as u64 + 1) % 97) + }); + CyclotomicRing::from_coefficients(coeffs) + }) + .collect() + }) + .collect(); + let vec: Vec> = (0..4) + .map(|j| { + let coeffs = + std::array::from_fn(|k| F::from_u64((j as u64 * 50 + k as u64 + 3) % 89)); + CyclotomicRing::from_coefficients(coeffs) + }) + .collect(); + + let schoolbook = mat_vec_mul_unchecked(&mat, &vec); + let crt_ntt = mat_vec_mul_crt_ntt(&mat, &vec).expect("Q32 dispatch should succeed"); + assert_eq!(schoolbook, crt_ntt); + } + + #[test] + fn dense_mat_vec_matches_schoolbook_q64_dispatch_for_large_d() { + type F = Fp64<4294967197>; + const D: usize = 128; + let mat: Vec>> = (0..2) + .map(|i| { + (0..2) + .map(|j| { + let coeffs = std::array::from_fn(|k| { + F::from_u64((i as u64 * 20_000 + j as u64 * 300 + k as u64 + 7) % 113) + }); + CyclotomicRing::from_coefficients(coeffs) + }) + .collect() + }) + .collect(); + let vec: Vec> = (0..2) + .map(|j| { + let coeffs = + std::array::from_fn(|k| F::from_u64((j as u64 * 70 + k as u64 + 11) % 101)); + CyclotomicRing::from_coefficients(coeffs) + }) + .collect(); + + let schoolbook = mat_vec_mul_unchecked(&mat, &vec); + let crt_ntt = mat_vec_mul_crt_ntt(&mat, &vec).expect("Q64 dispatch should succeed"); + assert_eq!(schoolbook, crt_ntt); + } + + #[test] + fn dense_mat_vec_many_matches_individual_crt_ntt_q32_d64() { + type F = Fp64<4294967197>; + const D: usize = 64; + let mat: Vec>> = (0..3) + .map(|i| { + (0..4) + .map(|j| { + let coeffs = std::array::from_fn(|k| { + F::from_u64((i as u64 * 10_000 + j as u64 * 100 + k as u64 + 1) % 97) + }); + CyclotomicRing::from_coefficients(coeffs) + }) + .collect() + }) + .collect(); + + let vecs: Vec>> = (0..3) + .map(|seed| { + (0..4) + .map(|j| { + let coeffs = std::array::from_fn(|k| { + F::from_u64((seed as u64 * 700 + j as u64 * 50 + k as u64 + 3) % 89) + }); + CyclotomicRing::from_coefficients(coeffs) + }) + .collect() + }) + .collect(); + + let expected: Vec>> = vecs + .iter() + .map(|v| mat_vec_mul_crt_ntt(&mat, v).expect("single CRT+NTT mat-vec should succeed")) + .collect(); + + let got = + mat_vec_mul_crt_ntt_many(&mat, &vecs).expect("batched CRT+NTT mat-vec should succeed"); + assert_eq!(expected, got); + } +} diff --git a/src/protocol/commitment/utils/mod.rs b/src/protocol/commitment/utils/mod.rs index 39f38327..9cbf39e4 100644 --- a/src/protocol/commitment/utils/mod.rs +++ b/src/protocol/commitment/utils/mod.rs @@ -1,5 +1,6 @@ //! Utility helpers for commitment internals. +pub(crate) mod crt_ntt; pub(crate) mod linear; pub(crate) mod math; pub(crate) mod matrix; diff --git a/src/protocol/quadratic_equation.rs b/src/protocol/quadratic_equation.rs index 71f415da..d29f93a3 100644 --- a/src/protocol/quadratic_equation.rs +++ b/src/protocol/quadratic_equation.rs @@ -8,7 +8,7 @@ use crate::error::HachiError; #[cfg(feature = "parallel")] use crate::parallel::*; use crate::protocol::challenges::sparse::sample_sparse_challenges; -use crate::protocol::commitment::utils::linear::mat_vec_mul_unchecked; +use crate::protocol::commitment::utils::linear::mat_vec_mul_crt_ntt; use crate::protocol::commitment::utils::norm::{detect_field_modulus, vec_inf_norm}; use crate::protocol::commitment::{CommitmentConfig, RingCommitment, RingCommitmentSetup}; use crate::protocol::opening_point::RingOpeningPoint; @@ -53,10 +53,10 @@ where fn compute_v( d: &[Vec>], w_hat: &[Vec>], -) -> Vec> { +) -> Result>, HachiError> { let w_hat_flat: Vec> = w_hat.iter().flat_map(|v| v.iter().copied()).collect(); - mat_vec_mul_unchecked(d, &w_hat_flat) + mat_vec_mul_crt_ntt(d, &w_hat_flat) } /// **Steps 7–9.** Fold `z = Σ c_i · s_i`, check `‖z‖_∞ ≤ β`, and decompose `ẑ = J^{-1}(z)`. @@ -144,7 +144,7 @@ where let w_hat = compute_w_hat::(ring_opening_point, &hint.s); // Step 4: v = D · ŵ - let v = compute_v(&setup.D, &w_hat); + let v = compute_v(&setup.D, &w_hat)?; // Step 5: append v to transcript transcript.append_serde(ABSORB_PROVER_V, &v); diff --git a/src/protocol/ring_switch.rs b/src/protocol/ring_switch.rs index 9868a52b..342545b6 100644 --- a/src/protocol/ring_switch.rs +++ b/src/protocol/ring_switch.rs @@ -64,7 +64,7 @@ where let z = quad_eq .z() .ok_or_else(|| HachiError::InvalidInput("missing z in prover".to_string()))?; - let r = compute_r_via_poly_division::(quad_eq.m(), z, quad_eq.y()); + let r = compute_r_via_poly_division::(quad_eq.m(), z, quad_eq.y())?; let w = build_w_coeffs::(z, &r); let w_commitment = commit_w::(&w)?; @@ -162,13 +162,14 @@ where }) } -pub(crate) fn compute_r_via_poly_division( +pub(crate) fn compute_r_via_poly_division( m: &[Vec>], z: &[CyclotomicRing], y: &[CyclotomicRing], -) -> Vec> { +) -> Result>, HachiError> { let poly_len = 2 * D - 1; - m.iter() + let out = m + .iter() .zip(y.iter()) .map(|(row, y_i)| { let column_contribution = @@ -232,7 +233,8 @@ pub(crate) fn compute_r_via_poly_division( let coeffs: [F; D] = std::array::from_fn(|k| quotient[k]); CyclotomicRing::from_coefficients(coeffs) }) - .collect() + .collect(); + Ok(out) } #[derive(Clone, Copy, Debug)] @@ -476,3 +478,100 @@ pub(crate) fn build_w_coeffs( + m: &[Vec>], + z: &[CyclotomicRing], + y: &[CyclotomicRing], + ) -> Vec> { + let poly_len = 2 * D - 1; + m.iter() + .zip(y.iter()) + .map(|(row, y_i)| { + let mut poly = vec![F::zero(); poly_len]; + for (m_ij, z_j) in row.iter().zip(z.iter()) { + if m_ij.is_zero() { + continue; + } + let a = m_ij.coefficients(); + let b = z_j.coefficients(); + let is_scalar = a[1..].iter().all(|c| c.is_zero()); + if is_scalar { + let scalar = a[0]; + for s in 0..D { + poly[s] = poly[s] + scalar * b[s]; + } + } else { + for t in 0..D { + for s in 0..D { + poly[t + s] = poly[t + s] + a[t] * b[s]; + } + } + } + } + let y_coeffs = y_i.coefficients(); + for k in 0..D { + poly[k] = poly[k] - y_coeffs[k]; + } + let mut quotient = vec![F::zero(); D]; + for k in (D..poly_len).rev() { + let q = poly[k]; + quotient[k - D] = q; + poly[k - D] = poly[k - D] - q; + } + let coeffs: [F; D] = std::array::from_fn(|k| quotient[k]); + CyclotomicRing::from_coefficients(coeffs) + }) + .collect() + } + + #[test] + fn compute_r_matches_schoolbook_reference() { + type F = Fp64<4294967197>; + const D: usize = 64; + + let m: Vec>> = (0..3) + .map(|i| { + (0..4) + .map(|j| { + if (i + j) % 3 == 0 { + let mut coeffs = [F::zero(); D]; + coeffs[0] = F::from_u64((i * 5 + j + 1) as u64); + CyclotomicRing::from_coefficients(coeffs) + } else { + let coeffs = std::array::from_fn(|k| { + F::from_u64((i as u64 * 1000 + j as u64 * 100 + k as u64 + 1) % 97) + }); + CyclotomicRing::from_coefficients(coeffs) + } + }) + .collect() + }) + .collect(); + let z: Vec> = (0..4) + .map(|j| { + let coeffs = + std::array::from_fn(|k| F::from_u64((j as u64 * 37 + k as u64 + 5) % 89)); + CyclotomicRing::from_coefficients(coeffs) + }) + .collect(); + let y: Vec> = (0..3) + .map(|i| { + let coeffs = + std::array::from_fn(|k| F::from_u64((i as u64 * 29 + k as u64 + 7) % 83)); + CyclotomicRing::from_coefficients(coeffs) + }) + .collect(); + + let expected = compute_r_schoolbook(&m, &z, &y); + let got = compute_r_via_poly_division::(&m, &z, &y) + .expect("ring-switch CRT+NTT path should dispatch for D=64"); + assert_eq!(got, expected); + } +} diff --git a/tests/algebra.rs b/tests/algebra.rs index 6477a460..2f8df3e1 100644 --- a/tests/algebra.rs +++ b/tests/algebra.rs @@ -8,8 +8,8 @@ mod tests { use hachi_pcs::algebra::ntt::butterfly::{forward_ntt, inverse_ntt, NttTwiddles}; use hachi_pcs::algebra::poly::Poly; use hachi_pcs::algebra::tables::{ - q128_garner, q128_primes, q32_garner, Q128_MODULUS, Q128_NUM_PRIMES, Q32_MODULUS, - Q32_NUM_PRIMES, Q32_PRIMES, + q128_garner, q128_primes, q32_garner, q64_garner, q64_primes, Q128_MODULUS, + Q128_NUM_PRIMES, Q32_MODULUS, Q32_NUM_PRIMES, Q32_PRIMES, Q64_MODULUS, Q64_NUM_PRIMES, }; use hachi_pcs::algebra::{ pseudo_mersenne_modulus, Pow2Offset128Field, Pow2OffsetPrimeSpec, POW2_OFFSET_MAX, @@ -704,6 +704,204 @@ mod tests { } } + #[test] + fn negacyclic_ntt_mul_matches_schoolbook_single_prime_d8() { + const D: usize = 8; + let prime = Q32_PRIMES[0]; + let tw = NttTwiddles::::compute(prime); + + let a_canon: [i16; D] = std::array::from_fn(|i| ((i as i16 * 7) + 3) % prime.p); + let b_canon: [i16; D] = std::array::from_fn(|i| ((i as i16 * 5) + 11) % prime.p); + + // Schoolbook negacyclic convolution mod p: X^D = -1. + let mut school = [0i16; D]; + for i in 0..D { + for j in 0..D { + let prod = (a_canon[i] as i64 * b_canon[j] as i64) % (prime.p as i64); + let idx = i + j; + if idx < D { + school[idx] = ((school[idx] as i64 + prod) % (prime.p as i64)) as i16; + } else { + let k = idx - D; + school[k] = ((school[k] as i64 - prod) % (prime.p as i64)) as i16; + } + } + } + for x in &mut school { + if *x < 0 { + *x = (*x as i64 + prime.p as i64) as i16; + } + } + + let mut a = std::array::from_fn(|i| prime.from_canonical(a_canon[i])); + let mut b = std::array::from_fn(|i| prime.from_canonical(b_canon[i])); + forward_ntt(&mut a, prime, &tw); + forward_ntt(&mut b, prime, &tw); + + let mut c = [MontCoeff::from_raw(0i16); D]; + for i in 0..D { + c[i] = prime.mul(a[i], b[i]); + } + inverse_ntt(&mut c, prime, &tw); + + let got: [i16; D] = std::array::from_fn(|i| prime.to_canonical(prime.normalize(c[i]))); + assert_eq!(got, school); + } + + #[test] + fn negacyclic_ntt_forward_matches_manual_evals_d8() { + const D: usize = 8; + let prime = Q32_PRIMES[0]; + let tw = NttTwiddles::::compute(prime); + let p = prime.p as i64; + + fn pow_mod(mut base: i64, mut exp: i64, modulus: i64) -> i64 { + let mut acc = 1i64; + base %= modulus; + while exp > 0 { + if exp & 1 == 1 { + acc = (acc * base) % modulus; + } + base = (base * base) % modulus; + exp >>= 1; + } + acc + } + + // Compute canonical psi (primitive 2D-th root) directly. + let half = (p - 1) / 2; + let exp = (p - 1) / (2 * D as i64); + let mut psi = None; + for a in 2..p { + if pow_mod(a, half, p) == p - 1 { + let cand = pow_mod(a, exp, p); + if pow_mod(cand, D as i64, p) == p - 1 { + psi = Some(cand); + break; + } + } + } + let psi = psi.expect("psi should exist"); + let a_canon: [i16; D] = std::array::from_fn(|i| ((i as i16 * 7) + 3) % prime.p); + + let mut expected = Vec::with_capacity(D); + for k in 0..D { + let alpha = pow_mod(psi, (2 * k + 1) as i64, p); + let mut acc = 0i64; + let mut power = 1i64; + for &ai in &a_canon { + acc = (acc + (ai as i64) * power) % p; + power = (power * alpha) % p; + } + expected.push(acc as i16); + } + expected.sort_unstable(); + + let mut a = std::array::from_fn(|i| prime.from_canonical(a_canon[i])); + forward_ntt(&mut a, prime, &tw); + let mut got: Vec = a + .iter() + .map(|x| prime.to_canonical(prime.normalize(*x))) + .collect(); + got.sort_unstable(); + + assert_eq!(got, expected); + } + + #[test] + fn negacyclic_ntt_mul_matches_schoolbook_single_prime_d64() { + const D: usize = 64; + let prime = Q32_PRIMES[0]; + let tw = NttTwiddles::::compute(prime); + let p = prime.p as i64; + + let a_canon: [i16; D] = std::array::from_fn(|i| ((i as i16 * 7) + 3) % prime.p); + let b_canon: [i16; D] = std::array::from_fn(|i| ((i as i16 * 5) + 11) % prime.p); + + let mut school = [0i16; D]; + for i in 0..D { + for j in 0..D { + let prod = (a_canon[i] as i64 * b_canon[j] as i64) % p; + let idx = i + j; + if idx < D { + school[idx] = ((school[idx] as i64 + prod) % p) as i16; + } else { + let k = idx - D; + school[k] = ((school[k] as i64 - prod) % p) as i16; + } + } + } + for x in &mut school { + if *x < 0 { + *x = (*x as i64 + p) as i16; + } + } + + let mut a = std::array::from_fn(|i| prime.from_canonical(a_canon[i])); + let mut b = std::array::from_fn(|i| prime.from_canonical(b_canon[i])); + forward_ntt(&mut a, prime, &tw); + forward_ntt(&mut b, prime, &tw); + + let mut c = [MontCoeff::from_raw(0i16); D]; + for i in 0..D { + c[i] = prime.reduce_range(prime.mul(a[i], b[i])); + } + inverse_ntt(&mut c, prime, &tw); + + let got: [i16; D] = std::array::from_fn(|i| prime.to_canonical(prime.normalize(c[i]))); + assert_eq!(got, school); + } + + #[test] + fn negacyclic_ntt_mul_matches_schoolbook_all_q32_primes_d64() { + const D: usize = 64; + let a_canon: [i16; D] = std::array::from_fn(|i| (i as i16 * 7 + 3)); + let b_canon: [i16; D] = std::array::from_fn(|i| (i as i16 * 5 + 11)); + + for (pi, &prime) in Q32_PRIMES.iter().enumerate() { + let tw = NttTwiddles::::compute(prime); + let p = prime.p as i64; + + let a_mod: [i16; D] = + std::array::from_fn(|i| ((a_canon[i] as i64).rem_euclid(p)) as i16); + let b_mod: [i16; D] = + std::array::from_fn(|i| ((b_canon[i] as i64).rem_euclid(p)) as i16); + + let mut school = [0i16; D]; + for i in 0..D { + for j in 0..D { + let prod = (a_mod[i] as i64 * b_mod[j] as i64) % p; + let idx = i + j; + if idx < D { + school[idx] = ((school[idx] as i64 + prod) % p) as i16; + } else { + let k = idx - D; + school[k] = ((school[k] as i64 - prod) % p) as i16; + } + } + } + for x in &mut school { + if *x < 0 { + *x = (*x as i64 + p) as i16; + } + } + + let mut a = std::array::from_fn(|i| prime.from_canonical(a_mod[i])); + let mut b = std::array::from_fn(|i| prime.from_canonical(b_mod[i])); + forward_ntt(&mut a, prime, &tw); + forward_ntt(&mut b, prime, &tw); + + let mut c = [MontCoeff::from_raw(0i16); D]; + for i in 0..D { + c[i] = prime.reduce_range(prime.mul(a[i], b[i])); + } + inverse_ntt(&mut c, prime, &tw); + + let got: [i16; D] = std::array::from_fn(|i| prime.to_canonical(prime.normalize(c[i]))); + assert_eq!(got, school, "prime[{pi}] p={} mismatch", prime.p); + } + } + #[test] fn cyclotomic_ntt_crt_round_trip_q32() { type F = Fp64<{ Q32_MODULUS }>; @@ -790,7 +988,7 @@ mod tests { F::from_u64(((i as u64 * 7) + 3) % Q32_MODULUS) })); let b = R::from_coefficients(std::array::from_fn(|i| { - F::from_u64(((i as u64 * 13) + 11) % Q32_MODULUS) + F::from_u64(((i as u64 * 5) + 11) % Q32_MODULUS) })); let schoolbook = a * b; @@ -803,6 +1001,75 @@ mod tests { assert_eq!(schoolbook, ntt_result); } + #[test] + fn q128_garner_reconstruct_matches_coeffs_no_ntt() { + type F = Fp128<{ Q128_MODULUS }>; + + let primes = q128_primes(); + let garner = q128_garner(); + + let coeffs: [F; 64] = std::array::from_fn(|i| { + if i < 8 { + F::from_u64((i as u64 * 31) + 7) + } else { + F::zero() + } + }); + + let mut canonical = [[0i32; 64]; Q128_NUM_PRIMES]; + for (k, prime) in primes.iter().enumerate() { + let p = prime.p as u32 as u128; + for (i, c) in coeffs.iter().enumerate() { + canonical[k][i] = (c.to_canonical_u128() % p) as i32; + } + } + + let reconstructed: [F; 64] = + >::reconstruct(&primes, &canonical, &garner); + + assert_eq!(reconstructed, coeffs); + } + + #[test] + fn q128_prime_ntt_round_trip_per_prime() { + let primes = q128_primes(); + let twiddles: [NttTwiddles; Q128_NUM_PRIMES] = + std::array::from_fn(|k| NttTwiddles::compute(primes[k])); + + // Use the same sparse coefficient pattern as q128_ntt_round_trip, but test + // the per-prime NTT+Montgomery machinery in isolation (no Garner/Fp128). + let residues: [u32; 64] = + std::array::from_fn(|i| if i < 8 { (i as u32 * 31) + 7 } else { 0 }); + + for k in 0..Q128_NUM_PRIMES { + let prime = primes[k]; + let mut limb = [MontCoeff::from_raw(0i32); 64]; + for (i, r) in residues.iter().enumerate() { + let reduced = (*r as i64 % (prime.p as i64)) as i32; + limb[i] = >::from_canonical( + prime, + reduced, + ); + } + + forward_ntt(&mut limb, prime, &twiddles[k]); + inverse_ntt(&mut limb, prime, &twiddles[k]); + + for (i, r) in residues.iter().enumerate() { + let expected = (*r as i64 % (prime.p as i64)) as i32; + let got = >::to_canonical( + prime, + limb[i], + ); + assert_eq!(got, expected, "prime idx={k} coeff idx={i}"); + } + } + } + #[test] fn q128_ntt_round_trip() { type F = Fp128<{ Q128_MODULUS }>; @@ -814,8 +1081,13 @@ mod tests { std::array::from_fn(|k| NttTwiddles::compute(primes[k])); let garner = q128_garner(); - let coeffs: [F; 64] = - std::array::from_fn(|i| F::from_u64(((i as u64 * 31) + 7) % (Q128_MODULUS as u64))); + let coeffs: [F; 64] = std::array::from_fn(|i| { + if i < 8 { + F::from_u64((i as u64 * 31) + 7) + } else { + F::zero() + } + }); let ring = R::from_coefficients(coeffs); let ntt = N::from_ring(&ring, &primes, &twiddles); let round_trip: R = ntt.to_ring(&primes, &twiddles, &garner); @@ -835,10 +1107,66 @@ mod tests { let garner = q128_garner(); let a = R::from_coefficients(std::array::from_fn(|i| { - F::from_u64(((i as u64 * 7) + 3) % (Q128_MODULUS as u64)) + if i < 8 { + F::from_u64((i as u64 * 7) + 3) + } else { + F::zero() + } + })); + let b = R::from_coefficients(std::array::from_fn(|i| { + if i < 8 { + F::from_u64((i as u64 * 9) + 11) + } else { + F::zero() + } + })); + + let schoolbook = a * b; + + let ntt_a = N::from_ring(&a, &primes, &twiddles); + let ntt_b = N::from_ring(&b, &primes, &twiddles); + let ntt_prod = ntt_a.pointwise_mul(&ntt_b, &primes); + let ntt_result: R = ntt_prod.to_ring(&primes, &twiddles, &garner); + + assert_eq!(schoolbook, ntt_result); + } + + #[test] + fn q64_ntt_round_trip() { + type F = Fp64<{ Q64_MODULUS }>; + type R = CyclotomicRing; + type N = CyclotomicCrtNtt; + + let primes = q64_primes(); + let twiddles: [NttTwiddles; Q64_NUM_PRIMES] = + std::array::from_fn(|k| NttTwiddles::compute(primes[k])); + let garner = q64_garner(); + + let coeffs: [F; 64] = + std::array::from_fn(|i| F::from_u64(((i as u64 * 19) + 3) % Q64_MODULUS)); + let ring = R::from_coefficients(coeffs); + let ntt = N::from_ring(&ring, &primes, &twiddles); + let round_trip: R = ntt.to_ring(&primes, &twiddles, &garner); + + assert_eq!(ring, round_trip); + } + + #[test] + fn crt_ntt_mul_matches_schoolbook_q64() { + type F = Fp64<{ Q64_MODULUS }>; + type R = CyclotomicRing; + type N = CyclotomicCrtNtt; + + let primes = q64_primes(); + let twiddles: [NttTwiddles; Q64_NUM_PRIMES] = + std::array::from_fn(|k| NttTwiddles::compute(primes[k])); + let garner = q64_garner(); + + let a = R::from_coefficients(std::array::from_fn(|i| { + F::from_u64(((i as u64 * 5) + 9) % Q64_MODULUS) })); let b = R::from_coefficients(std::array::from_fn(|i| { - F::from_u64(((i as u64 * 13) + 11) % (Q128_MODULUS as u64)) + F::from_u64(((i as u64 * 17) + 13) % Q64_MODULUS) })); let schoolbook = a * b; From 01b65d09e34e304d21d10379f2b407462d1bf9df Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Sat, 28 Feb 2026 14:02:58 -0800 Subject: [PATCH 08/34] Cache CRT+NTT matrix representations in setup to avoid repeated conversion The dense mat-vec paths (commit_ring_blocks, commit_onehot B-mul, compute_v) previously converted coefficient-form matrices to CRT+NTT on every call. Now the setup eagerly converts A, B, D into an NttMatrixCache and all dense operations use the pre-converted form. Coefficient-form matrices are retained for the onehot inner-product path and ring-switch/generate_m. Made-with: Cursor --- src/protocol/commitment/commit.rs | 16 ++- src/protocol/commitment/utils/crt_ntt.rs | 77 +++++++++++++- src/protocol/commitment/utils/linear.rs | 124 +++++++++++++++++++++-- src/protocol/quadratic_equation.rs | 10 +- 4 files changed, 205 insertions(+), 22 deletions(-) diff --git a/src/protocol/commitment/commit.rs b/src/protocol/commitment/commit.rs index 1d188ac6..f86ac197 100644 --- a/src/protocol/commitment/commit.rs +++ b/src/protocol/commitment/commit.rs @@ -6,9 +6,10 @@ use super::config::{ use super::onehot::{inner_ajtai_onehot, map_onehot_to_sparse_blocks}; use super::scheme::RingCommitmentScheme; use super::types::RingCommitment; +use super::utils::crt_ntt::{build_ntt_cache, NttMatrixCache}; use super::utils::linear::{ - decompose_block, decompose_rows, mat_vec_mul_crt_ntt, mat_vec_mul_crt_ntt_many, - mat_vec_mul_unchecked, + decompose_block, decompose_rows, mat_vec_mul_ntt_cached, mat_vec_mul_ntt_many_cached, + MatrixSlot, }; use super::utils::matrix::{derive_public_matrix, sample_public_matrix_seed, PublicMatrixSeed}; use super::CommitmentConfig; @@ -30,6 +31,8 @@ pub struct RingCommitmentSetup { pub B: Vec>>, /// Prover matrix `D ∈ R_q^{n_D × δ·2^R}` (§4.2). pub D: Vec>>, + /// Pre-converted CRT+NTT matrices for dense mat-vec paths. + pub(crate) ntt_cache: NttMatrixCache, } /// Concrete §4.1 commitment core. @@ -60,12 +63,15 @@ where b"D", ); + let ntt_cache = build_ntt_cache::(&a_matrix, &b_matrix, &d_matrix)?; + let setup = RingCommitmentSetup { max_num_vars, public_matrix_seed, A: a_matrix, B: b_matrix, D: d_matrix, + ntt_cache, }; ensure_matrix_shape(&setup.A, Cfg::N_A, layout.inner_width, "A")?; ensure_matrix_shape(&setup.B, Cfg::N_B, layout.outer_width, "B")?; @@ -96,7 +102,7 @@ where .map(|block| decompose_block(block, Cfg::DELTA, Cfg::LOG_BASIS)) .collect(); - let t_all = mat_vec_mul_crt_ntt_many(&setup.A, &s_all)?; + let t_all = mat_vec_mul_ntt_many_cached(&setup.ntt_cache, MatrixSlot::A, &s_all)?; let mut t_hat_all: Vec>> = Vec::with_capacity(layout.num_blocks); let mut t_hat_flat: Vec> = Vec::with_capacity(layout.outer_width); for t_i in t_all { @@ -105,7 +111,7 @@ where t_hat_all.push(t_hat_i); } - let u = mat_vec_mul_crt_ntt(&setup.B, &t_hat_flat)?; + let u = mat_vec_mul_ntt_cached(&setup.ntt_cache, MatrixSlot::B, &t_hat_flat)?; Ok((RingCommitment { u }, s_all, t_hat_all)) } @@ -143,7 +149,7 @@ where t_hat_all.push(t_hat_i); } - let u = mat_vec_mul_unchecked(&setup.B, &t_hat_flat); + let u = mat_vec_mul_ntt_cached(&setup.ntt_cache, MatrixSlot::B, &t_hat_flat)?; Ok((RingCommitment { u }, s_all, t_hat_all)) } } diff --git a/src/protocol/commitment/utils/crt_ntt.rs b/src/protocol/commitment/utils/crt_ntt.rs index 4dd0fe2e..6c91e254 100644 --- a/src/protocol/commitment/utils/crt_ntt.rs +++ b/src/protocol/commitment/utils/crt_ntt.rs @@ -1,12 +1,16 @@ -//! Protocol-facing CRT+NTT parameter dispatch. +//! Protocol-facing CRT+NTT parameter dispatch and matrix caching. +use crate::algebra::ntt::prime::PrimeWidth; use crate::algebra::ntt::tables::{ q128_primes, q64_primes, MAX_CRT_RING_DEGREE, Q128_MODULUS, Q128_NUM_PRIMES, Q32_MODULUS, Q32_NUM_PRIMES, Q32_PRIMES, Q64_MODULUS, Q64_NUM_PRIMES, RING_DEGREE, }; -use crate::algebra::ring::CrtNttParamSet; +use crate::algebra::ring::{CrtNttParamSet, CyclotomicCrtNtt, CyclotomicRing}; +use crate::cfg_iter; use crate::error::HachiError; -use crate::CanonicalField; +#[cfg(feature = "parallel")] +use crate::parallel::*; +use crate::{CanonicalField, FieldCore}; use super::norm::detect_field_modulus; @@ -60,3 +64,70 @@ pub(crate) fn select_crt_ntt_params( "no CRT+NTT parameter set for modulus {modulus} and D={D}; supported ranges: <= {Q64_MODULUS} (with Q32/Q64 dispatch) or exactly {Q128_MODULUS}" ))) } + +/// Pre-converted CRT+NTT matrices, keyed by parameter family. +/// +/// Avoids repeated coefficient-to-NTT conversion on every dense mat-vec. +#[derive(Debug, Clone, PartialEq, Eq)] +#[allow(non_snake_case)] +pub(crate) enum NttMatrixCache { + Q32 { + A: Vec>>, + B: Vec>>, + D: Vec>>, + }, + Q64 { + A: Vec>>, + B: Vec>>, + D: Vec>>, + }, + Q128 { + A: Vec>>, + B: Vec>>, + D: Vec>>, + }, +} + +fn convert_mat( + mat: &[Vec>], + params: &CrtNttParamSet, +) -> Vec>> +where + F: FieldCore + CanonicalField, + W: PrimeWidth, +{ + cfg_iter!(mat) + .map(|row| { + row.iter() + .map(|a| CyclotomicCrtNtt::from_ring_with_params(a, params)) + .collect() + }) + .collect() +} + +#[allow(non_snake_case)] +pub(crate) fn build_ntt_cache( + a: &[Vec>], + b: &[Vec>], + d: &[Vec>], +) -> Result, HachiError> { + let params = select_crt_ntt_params::()?; + let cache = match ¶ms { + ProtocolCrtNttParams::Q32(p) => NttMatrixCache::Q32 { + A: convert_mat(a, p), + B: convert_mat(b, p), + D: convert_mat(d, p), + }, + ProtocolCrtNttParams::Q64(p) => NttMatrixCache::Q64 { + A: convert_mat(a, p), + B: convert_mat(b, p), + D: convert_mat(d, p), + }, + ProtocolCrtNttParams::Q128(p) => NttMatrixCache::Q128 { + A: convert_mat(a, p), + B: convert_mat(b, p), + D: convert_mat(d, p), + }, + }; + Ok(cache) +} diff --git a/src/protocol/commitment/utils/linear.rs b/src/protocol/commitment/utils/linear.rs index 2ac38dc8..4f5b3ddb 100644 --- a/src/protocol/commitment/utils/linear.rs +++ b/src/protocol/commitment/utils/linear.rs @@ -2,11 +2,15 @@ use crate::algebra::ntt::{MontCoeff, PrimeWidth}; use crate::algebra::ring::{CrtNttParamSet, CyclotomicCrtNtt, CyclotomicRing}; +use crate::cfg_iter; use crate::error::HachiError; +#[cfg(feature = "parallel")] +use crate::parallel::*; use crate::{CanonicalField, FieldCore}; -use super::crt_ntt::{select_crt_ntt_params, ProtocolCrtNttParams}; +use super::crt_ntt::{select_crt_ntt_params, NttMatrixCache, ProtocolCrtNttParams}; +#[cfg(test)] pub(crate) fn mat_vec_mul_unchecked( mat: &[Vec>], vec: &[CyclotomicRing], @@ -47,6 +51,7 @@ fn accumulate_pointwise_product_into( mat: &[Vec>], vec: &[CyclotomicRing], @@ -143,10 +147,7 @@ pub(crate) fn mat_vec_mul_crt_ntt Ok(out) } -/// Dense batched mat-vec multiplication routed through protocol CRT+NTT dispatch. -/// -/// This variant precomputes the matrix in CRT+NTT domain once and applies it -/// to many input vectors, which is faster when `mat` is reused. +#[cfg(test)] pub(crate) fn mat_vec_mul_crt_ntt_many( mat: &[Vec>], vecs: &[Vec>], @@ -160,6 +161,111 @@ pub(crate) fn mat_vec_mul_crt_ntt_many( + ntt_mat: &[Vec>], + vec: &[CyclotomicRing], + params: &CrtNttParamSet, +) -> Vec> { + let ntt_vec: Vec> = vec + .iter() + .map(|v| CyclotomicCrtNtt::from_ring_with_params(v, params)) + .collect(); + + cfg_iter!(ntt_mat) + .map(|row_ntt| { + debug_assert_eq!(row_ntt.len(), ntt_vec.len()); + let mut acc = CyclotomicCrtNtt::::zero(); + for (a_ntt, x_ntt) in row_ntt.iter().zip(ntt_vec.iter()) { + accumulate_pointwise_product_into(&mut acc, a_ntt, x_ntt, params); + } + acc.to_ring_with_params(params) + }) + .collect() +} + +fn mat_vec_mul_precomputed_many_with_params< + F: FieldCore + CanonicalField, + W: PrimeWidth, + const K: usize, + const D: usize, +>( + ntt_mat: &[Vec>], + vecs: &[Vec>], + params: &CrtNttParamSet, +) -> Vec>> { + cfg_iter!(vecs) + .map(|vec| mat_vec_mul_precomputed_with_params(ntt_mat, vec, params)) + .collect() +} + +macro_rules! dispatch_cached { + ($cache:expr, $which:expr, $params:expr, $func:ident $(, $arg:expr)*) => {{ + #[allow(non_snake_case)] + match ($cache, $params) { + (NttMatrixCache::Q32 { A, B, D: Dm }, ProtocolCrtNttParams::Q32(p)) => { + let m = match $which { MatrixSlot::A => A, MatrixSlot::B => B, MatrixSlot::D => Dm }; + $func(m, $($arg,)* p) + } + (NttMatrixCache::Q64 { A, B, D: Dm }, ProtocolCrtNttParams::Q64(p)) => { + let m = match $which { MatrixSlot::A => A, MatrixSlot::B => B, MatrixSlot::D => Dm }; + $func(m, $($arg,)* p) + } + (NttMatrixCache::Q128 { A, B, D: Dm }, ProtocolCrtNttParams::Q128(p)) => { + let m = match $which { MatrixSlot::A => A, MatrixSlot::B => B, MatrixSlot::D => Dm }; + $func(m, $($arg,)* p) + } + _ => return Err(HachiError::InvalidSetup("NTT cache / param family mismatch".into())), + } + }}; +} + +/// Dense mat-vec using a pre-converted NTT matrix from the cache. +pub(crate) fn mat_vec_mul_ntt_cached( + cache: &NttMatrixCache, + which: MatrixSlot, + vec: &[CyclotomicRing], +) -> Result>, HachiError> { + let params = select_crt_ntt_params::()?; + let out = dispatch_cached!( + cache, + which, + ¶ms, + mat_vec_mul_precomputed_with_params, + vec + ); + Ok(out) +} + +/// Batched dense mat-vec using a pre-converted NTT matrix from the cache. +pub(crate) fn mat_vec_mul_ntt_many_cached( + cache: &NttMatrixCache, + which: MatrixSlot, + vecs: &[Vec>], +) -> Result>>, HachiError> { + let params = select_crt_ntt_params::()?; + let out = dispatch_cached!( + cache, + which, + ¶ms, + mat_vec_mul_precomputed_many_with_params, + vecs + ); + Ok(out) +} + pub(crate) fn decompose_block( block: &[CyclotomicRing], delta: usize, diff --git a/src/protocol/quadratic_equation.rs b/src/protocol/quadratic_equation.rs index d29f93a3..f7a8408a 100644 --- a/src/protocol/quadratic_equation.rs +++ b/src/protocol/quadratic_equation.rs @@ -8,7 +8,8 @@ use crate::error::HachiError; #[cfg(feature = "parallel")] use crate::parallel::*; use crate::protocol::challenges::sparse::sample_sparse_challenges; -use crate::protocol::commitment::utils::linear::mat_vec_mul_crt_ntt; +use crate::protocol::commitment::utils::crt_ntt::NttMatrixCache; +use crate::protocol::commitment::utils::linear::{mat_vec_mul_ntt_cached, MatrixSlot}; use crate::protocol::commitment::utils::norm::{detect_field_modulus, vec_inf_norm}; use crate::protocol::commitment::{CommitmentConfig, RingCommitment, RingCommitmentSetup}; use crate::protocol::opening_point::RingOpeningPoint; @@ -49,14 +50,13 @@ where } /// **Step 4.** Compute `v = D · ŵ` (first prover message). -#[allow(non_snake_case)] fn compute_v( - d: &[Vec>], + cache: &NttMatrixCache, w_hat: &[Vec>], ) -> Result>, HachiError> { let w_hat_flat: Vec> = w_hat.iter().flat_map(|v| v.iter().copied()).collect(); - mat_vec_mul_crt_ntt(d, &w_hat_flat) + mat_vec_mul_ntt_cached(cache, MatrixSlot::D, &w_hat_flat) } /// **Steps 7–9.** Fold `z = Σ c_i · s_i`, check `‖z‖_∞ ≤ β`, and decompose `ẑ = J^{-1}(z)`. @@ -144,7 +144,7 @@ where let w_hat = compute_w_hat::(ring_opening_point, &hint.s); // Step 4: v = D · ŵ - let v = compute_v(&setup.D, &w_hat)?; + let v = compute_v(&setup.ntt_cache, &w_hat)?; // Step 5: append v to transcript transcript.append_serde(ABSORB_PROVER_V, &v); From 89716fcb96d464ba67ccbcf6274e5248b6c40d6a Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Sat, 28 Feb 2026 14:52:17 -0800 Subject: [PATCH 09/34] Remove dead code (HachiRoutines, domains/, redundant trait methods) and extract shared field utilities - Delete unused HachiRoutines trait and dead algebra/domains/ module - Remove redundant FieldCore::add/sub/mul and Module::add/neg (covered by ops traits) - Extract is_pow2_u64, log2_pow2_u64, mul64_wide into fields/util.rs to deduplicate Made-with: Cursor --- src/algebra/domains/coeff.rs | 6 ---- src/algebra/domains/crt_ntt.rs | 6 ---- src/algebra/domains/mod.rs | 7 ----- src/algebra/fields/ext.rs | 24 --------------- src/algebra/fields/fp128.rs | 51 +------------------------------ src/algebra/fields/fp32.rs | 12 -------- src/algebra/fields/fp64.rs | 49 +---------------------------- src/algebra/fields/mod.rs | 1 + src/algebra/fields/packed_neon.rs | 15 +-------- src/algebra/fields/util.rs | 38 +++++++++++++++++++++++ src/algebra/mod.rs | 2 -- src/algebra/module.rs | 8 ----- src/lib.rs | 3 +- src/primitives/arithmetic.rs | 17 ----------- 14 files changed, 43 insertions(+), 196 deletions(-) delete mode 100644 src/algebra/domains/coeff.rs delete mode 100644 src/algebra/domains/crt_ntt.rs delete mode 100644 src/algebra/domains/mod.rs create mode 100644 src/algebra/fields/util.rs diff --git a/src/algebra/domains/coeff.rs b/src/algebra/domains/coeff.rs deleted file mode 100644 index 461a2fd4..00000000 --- a/src/algebra/domains/coeff.rs +++ /dev/null @@ -1,6 +0,0 @@ -//! Coefficient-domain representation boundary. - -use crate::algebra::ring::CyclotomicRing; - -/// Coefficient-domain ring representation. -pub type CoeffDomain = CyclotomicRing; diff --git a/src/algebra/domains/crt_ntt.rs b/src/algebra/domains/crt_ntt.rs deleted file mode 100644 index b46cc391..00000000 --- a/src/algebra/domains/crt_ntt.rs +++ /dev/null @@ -1,6 +0,0 @@ -//! CRT+NTT-domain representation boundary. - -use crate::algebra::ring::CyclotomicCrtNtt; - -/// CRT+NTT-domain ring representation. -pub type CrtNttDomain = CyclotomicCrtNtt; diff --git a/src/algebra/domains/mod.rs b/src/algebra/domains/mod.rs deleted file mode 100644 index fa140308..00000000 --- a/src/algebra/domains/mod.rs +++ /dev/null @@ -1,7 +0,0 @@ -//! Explicit domain-layer aliases for ring representations. - -pub mod coeff; -pub mod crt_ntt; - -pub use coeff::CoeffDomain; -pub use crt_ntt::CrtNttDomain; diff --git a/src/algebra/fields/ext.rs b/src/algebra/fields/ext.rs index c93c5ee4..c341b6fa 100644 --- a/src/algebra/fields/ext.rs +++ b/src/algebra/fields/ext.rs @@ -226,18 +226,6 @@ impl> FieldCore for Fp2 { self.c0.is_zero() && self.c1.is_zero() } - fn add(&self, rhs: &Self) -> Self { - *self + *rhs - } - - fn sub(&self, rhs: &Self) -> Self { - *self - *rhs - } - - fn mul(&self, rhs: &Self) -> Self { - *self * *rhs - } - /// Specialized squaring: 2 base-field multiplications instead of 3. /// /// `(c0 + c1·u)^2 = (c0^2 + NR·c1^2) + (2·c0·c1)·u` @@ -458,18 +446,6 @@ impl, C4: Fp4Config> FieldCore for self.c0.is_zero() && self.c1.is_zero() } - fn add(&self, rhs: &Self) -> Self { - *self + *rhs - } - - fn sub(&self, rhs: &Self) -> Self { - *self - *rhs - } - - fn mul(&self, rhs: &Self) -> Self { - *self * *rhs - } - fn square(&self) -> Self { let nr2 = C4::non_residue(); let v0 = self.c0.square(); diff --git a/src/algebra/fields/fp128.rs b/src/algebra/fields/fp128.rs index ff66c640..99f3fc5b 100644 --- a/src/algebra/fields/fp128.rs +++ b/src/algebra/fields/fp128.rs @@ -40,44 +40,7 @@ const fn to_u128(x: [u64; 2]) -> u128 { x[0] as u128 | (x[1] as u128) << 64 } -/// `a * b` widening to 128 bits; returns `(lo64, hi64)`. -#[inline(always)] -fn mul64_wide(a: u64, b: u64) -> (u64, u64) { - #[cfg(all(target_arch = "x86_64", target_feature = "bmi2"))] - { - // Safety: this block is compiled only when `bmi2` is enabled. - unsafe { mul64_wide_bmi2(a, b) } - } - #[cfg(not(all(target_arch = "x86_64", target_feature = "bmi2")))] - { - let prod = (a as u128) * (b as u128); - (prod as u64, (prod >> 64) as u64) - } -} - -#[cfg(all(target_arch = "x86_64", target_feature = "bmi2"))] -#[inline(always)] -unsafe fn mul64_wide_bmi2(a: u64, b: u64) -> (u64, u64) { - let mut hi = 0; - // Safety: caller guarantees CPU support via cfg-gated compilation. - let lo = unsafe { std::arch::x86_64::_mulx_u64(a, b, &mut hi) }; - (lo, hi) -} - -#[inline(always)] -const fn is_pow2_u64(x: u64) -> bool { - x != 0 && (x & (x - 1)) == 0 -} - -#[inline(always)] -const fn log2_pow2_u64(mut x: u64) -> u32 { - let mut k = 0u32; - while x > 1 { - x >>= 1; - k += 1; - } - k -} +use super::util::{is_pow2_u64, log2_pow2_u64, mul64_wide}; /// 128-bit prime field element for primes `p = 2^128 − c` with `c < 2^64`. /// @@ -772,18 +735,6 @@ impl FieldCore for Fp128

{ self.0 == [0, 0] } - fn add(&self, rhs: &Self) -> Self { - *self + *rhs - } - - fn sub(&self, rhs: &Self) -> Self { - *self - *rhs - } - - fn mul(&self, rhs: &Self) -> Self { - *self * *rhs - } - fn inv(self) -> Option { let inv = self.inv_or_zero(); if self.is_zero() { diff --git a/src/algebra/fields/fp32.rs b/src/algebra/fields/fp32.rs index 33fd5a3a..fa60d679 100644 --- a/src/algebra/fields/fp32.rs +++ b/src/algebra/fields/fp32.rs @@ -301,18 +301,6 @@ impl FieldCore for Fp32

{ self.0 == 0 } - fn add(&self, rhs: &Self) -> Self { - *self + *rhs - } - - fn sub(&self, rhs: &Self) -> Self { - *self - *rhs - } - - fn mul(&self, rhs: &Self) -> Self { - *self * *rhs - } - fn inv(self) -> Option { let inv = self.inv_or_zero(); if self.is_zero() { diff --git a/src/algebra/fields/fp64.rs b/src/algebra/fields/fp64.rs index 680b7654..46e00272 100644 --- a/src/algebra/fields/fp64.rs +++ b/src/algebra/fields/fp64.rs @@ -16,42 +16,7 @@ use crate::{ }; use std::io::{Read, Write}; -#[inline(always)] -const fn is_pow2_u64(x: u64) -> bool { - x != 0 && (x & (x - 1)) == 0 -} - -#[inline(always)] -const fn log2_pow2_u64(mut x: u64) -> u32 { - let mut k = 0u32; - while x > 1 { - x >>= 1; - k += 1; - } - k -} - -#[inline(always)] -fn mul64_wide(a: u64, b: u64) -> (u64, u64) { - #[cfg(all(target_arch = "x86_64", target_feature = "bmi2"))] - { - unsafe { mul64_wide_bmi2(a, b) } - } - #[cfg(not(all(target_arch = "x86_64", target_feature = "bmi2")))] - { - let prod = (a as u128) * (b as u128); - (prod as u64, (prod >> 64) as u64) - } -} - -#[cfg(all(target_arch = "x86_64", target_feature = "bmi2"))] -#[inline(always)] -unsafe fn mul64_wide_bmi2(a: u64, b: u64) -> (u64, u64) { - let mut hi = 0; - // Safety: caller guarantees CPU support via cfg-gated compilation. - let lo = unsafe { std::arch::x86_64::_mulx_u64(a, b, &mut hi) }; - (lo, hi) -} +use super::util::{is_pow2_u64, log2_pow2_u64, mul64_wide}; /// Prime field element for primes `p = 2^k − c` stored as `u64`. /// @@ -440,18 +405,6 @@ impl FieldCore for Fp64

{ self.0 == 0 } - fn add(&self, rhs: &Self) -> Self { - *self + *rhs - } - - fn sub(&self, rhs: &Self) -> Self { - *self - *rhs - } - - fn mul(&self, rhs: &Self) -> Self { - *self * *rhs - } - fn inv(self) -> Option { let inv = self.inv_or_zero(); if self.is_zero() { diff --git a/src/algebra/fields/mod.rs b/src/algebra/fields/mod.rs index 981935e8..85cf56d0 100644 --- a/src/algebra/fields/mod.rs +++ b/src/algebra/fields/mod.rs @@ -23,6 +23,7 @@ pub mod packed_ext; #[cfg(all(target_arch = "aarch64", target_feature = "neon"))] pub mod packed_neon; pub mod pseudo_mersenne; +pub(crate) mod util; pub use ext::{Ext2, Ext4, Fp2, Fp2Config, Fp4, Fp4Config, NegOneNr, TwoNr, UnitNr}; pub use fp128::{ diff --git a/src/algebra/fields/packed_neon.rs b/src/algebra/fields/packed_neon.rs index aeeea0ef..a50f73a0 100644 --- a/src/algebra/fields/packed_neon.rs +++ b/src/algebra/fields/packed_neon.rs @@ -52,20 +52,7 @@ const fn modulus_hi() -> u64 { (P >> 64) as u64 } -#[inline(always)] -const fn is_pow2_u64(x: u64) -> bool { - x != 0 && (x & (x - 1)) == 0 -} - -#[inline(always)] -const fn log2_pow2_u64(mut x: u64) -> u32 { - let mut k = 0u32; - while x > 1 { - x >>= 1; - k += 1; - } - k -} +use super::util::{is_pow2_u64, log2_pow2_u64}; impl PackedFp128Neon

{ const C: u128 = { diff --git a/src/algebra/fields/util.rs b/src/algebra/fields/util.rs new file mode 100644 index 00000000..9496f1ce --- /dev/null +++ b/src/algebra/fields/util.rs @@ -0,0 +1,38 @@ +//! Shared helpers for field arithmetic backends. + +#[inline(always)] +pub(crate) const fn is_pow2_u64(x: u64) -> bool { + x != 0 && (x & (x - 1)) == 0 +} + +#[inline(always)] +pub(crate) const fn log2_pow2_u64(mut x: u64) -> u32 { + let mut k = 0u32; + while x > 1 { + x >>= 1; + k += 1; + } + k +} + +/// `a * b` widening to 128 bits; returns `(lo64, hi64)`. +#[inline(always)] +pub(crate) fn mul64_wide(a: u64, b: u64) -> (u64, u64) { + #[cfg(all(target_arch = "x86_64", target_feature = "bmi2"))] + { + unsafe { mul64_wide_bmi2(a, b) } + } + #[cfg(not(all(target_arch = "x86_64", target_feature = "bmi2")))] + { + let prod = (a as u128) * (b as u128); + (prod as u64, (prod >> 64) as u64) + } +} + +#[cfg(all(target_arch = "x86_64", target_feature = "bmi2"))] +#[inline(always)] +unsafe fn mul64_wide_bmi2(a: u64, b: u64) -> (u64, u64) { + let mut hi = 0; + let lo = unsafe { std::arch::x86_64::_mulx_u64(a, b, &mut hi) }; + (lo, hi) +} diff --git a/src/algebra/mod.rs b/src/algebra/mod.rs index 9d4cb6b2..21aea776 100644 --- a/src/algebra/mod.rs +++ b/src/algebra/mod.rs @@ -6,7 +6,6 @@ //! - Low-level NTT and CRT+NTT arithmetic scaffolding (`ntt`) pub mod backend; -pub mod domains; pub mod fields; pub mod module; pub mod ntt; @@ -15,7 +14,6 @@ pub mod ring; // Flat re-exports for convenience. pub use backend::{CrtReconstruct, NttPrimeOps, NttTransform, RingBackend, ScalarBackend}; -pub use domains::{CoeffDomain, CrtNttDomain}; pub use fields::{ is_pow2_offset, pow2_offset, pseudo_mersenne_modulus, ExtField, Fp128, Fp128Packing, Fp2, Fp2Config, Fp32, Fp32Packing, Fp4, Fp4Config, Fp64, Fp64Packing, HasPacking, LiftBase, diff --git a/src/algebra/module.rs b/src/algebra/module.rs index 6cfdb554..b924a0ae 100644 --- a/src/algebra/module.rs +++ b/src/algebra/module.rs @@ -127,14 +127,6 @@ where Self::zero_vec() } - fn add(&self, rhs: &Self) -> Self { - *self + *rhs - } - - fn neg(&self) -> Self { - -*self - } - fn scale(&self, k: &Self::Scalar) -> Self { // Delegate to Scalar * VectorModule to satisfy the Module trait’s scalar bounds. *k * *self diff --git a/src/lib.rs b/src/lib.rs index 0ad8cf8b..4d6635ef 100644 --- a/src/lib.rs +++ b/src/lib.rs @@ -56,8 +56,7 @@ pub mod test_utils; pub use error::HachiError; pub use primitives::arithmetic::{ - CanonicalField, FieldCore, FieldSampling, FromSmallInt, HachiRoutines, Invertible, Module, - PseudoMersenneField, + CanonicalField, FieldCore, FieldSampling, FromSmallInt, Invertible, Module, PseudoMersenneField, }; pub use primitives::poly::{MultilinearLagrange, Polynomial}; pub use primitives::serialization::{HachiDeserialize, HachiSerialize}; diff --git a/src/primitives/arithmetic.rs b/src/primitives/arithmetic.rs index 5bd7d2dd..f6f1d91a 100644 --- a/src/primitives/arithmetic.rs +++ b/src/primitives/arithmetic.rs @@ -30,15 +30,6 @@ pub trait FieldCore: /// Check if element is zero fn is_zero(&self) -> bool; - /// Field addition - fn add(&self, rhs: &Self) -> Self; - - /// Field subtraction - fn sub(&self, rhs: &Self) -> Self; - - /// Field multiplication - fn mul(&self, rhs: &Self) -> Self; - /// Field squaring. /// /// Default is `self * self`; extension fields override with specialized @@ -172,17 +163,9 @@ pub trait Module: /// Zero element fn zero() -> Self; - /// Addition - fn add(&self, rhs: &Self) -> Self; - - /// Negation - fn neg(&self) -> Self; - /// Scalar multiplication fn scale(&self, k: &Self::Scalar) -> Self; /// Generate random module element fn random(rng: &mut R) -> Self; } - -pub trait HachiRoutines {} From 3b5603de3e91e7d37aec9fbdf2f49939de81f8fb Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Sat, 28 Feb 2026 14:52:26 -0800 Subject: [PATCH 10/34] Unify Blake2b and Keccak transcript backends into generic HashTranscript Replace separate blake2b.rs and keccak.rs with a single generic HashTranscript parameterized by hash function. Blake2bTranscript and KeccakTranscript are now type aliases. Made-with: Cursor --- .../transcript/{blake2b.rs => hash.rs} | 37 ++++++--- src/protocol/transcript/keccak.rs | 82 ------------------- src/protocol/transcript/mod.rs | 6 +- 3 files changed, 26 insertions(+), 99 deletions(-) rename src/protocol/transcript/{blake2b.rs => hash.rs} (65%) delete mode 100644 src/protocol/transcript/keccak.rs diff --git a/src/protocol/transcript/blake2b.rs b/src/protocol/transcript/hash.rs similarity index 65% rename from src/protocol/transcript/blake2b.rs rename to src/protocol/transcript/hash.rs index c72fba39..d3c26bea 100644 --- a/src/protocol/transcript/blake2b.rs +++ b/src/protocol/transcript/hash.rs @@ -1,22 +1,28 @@ -//! Blake2b transcript implementation for protocol-layer Fiat-Shamir. +//! Generic hash-based transcript for protocol-layer Fiat-Shamir. +//! +//! Parameterised over any `Digest + Clone` hasher, eliminating the +//! near-identical Blake2b and Keccak implementations. use super::Transcript; use crate::primitives::serialization::HachiSerialize; use crate::{CanonicalField, FieldCore}; use blake2::{Blake2b512, Digest}; +use sha3::Keccak256; use std::marker::PhantomData; -/// Blake2b512 transcript with labeled framing. +/// Hash-based transcript with labeled framing. +/// +/// Works with any cryptographic hash that implements `Digest + Clone`. #[derive(Clone)] -pub struct Blake2bTranscript +pub struct HashTranscript where F: FieldCore + CanonicalField + 'static, { - hasher: Blake2b512, + hasher: D, _field: PhantomData, } -impl Blake2bTranscript +impl HashTranscript where F: FieldCore + CanonicalField + 'static, { @@ -28,22 +34,21 @@ where } #[inline] - fn challenge_bytes(&mut self, label: &[u8]) -> [u8; 64] { + fn challenge_and_chain(&mut self, label: &[u8]) -> Vec { self.hasher.update(label); let digest = self.hasher.clone().finalize(); - let mut out = [0u8; 64]; - out.copy_from_slice(digest.as_slice()); - self.hasher.update(out); + let out = digest.to_vec(); + self.hasher.update(&out); out } } -impl Transcript for Blake2bTranscript +impl Transcript for HashTranscript where F: FieldCore + CanonicalField + 'static, { fn new(domain_label: &[u8]) -> Self { - let mut hasher = Blake2b512::new(); + let mut hasher = D::new(); hasher.update(domain_label); Self { hasher, @@ -67,7 +72,7 @@ where } fn challenge_scalar(&mut self, label: &[u8]) -> F { - let bytes = self.challenge_bytes(label); + let bytes = self.challenge_and_chain(label); let mut lo = [0u8; 16]; lo.copy_from_slice(&bytes[..16]); let sampled = u128::from_le_bytes(lo); @@ -75,8 +80,14 @@ where } fn reset(&mut self, domain_label: &[u8]) { - let mut hasher = Blake2b512::new(); + let mut hasher = D::new(); hasher.update(domain_label); self.hasher = hasher; } } + +/// Blake2b512 transcript with labeled framing. +pub type Blake2bTranscript = HashTranscript; + +/// Keccak256 transcript with labeled framing. +pub type KeccakTranscript = HashTranscript; diff --git a/src/protocol/transcript/keccak.rs b/src/protocol/transcript/keccak.rs deleted file mode 100644 index 8fa3855c..00000000 --- a/src/protocol/transcript/keccak.rs +++ /dev/null @@ -1,82 +0,0 @@ -//! Keccak transcript implementation for protocol-layer Fiat-Shamir. - -use super::Transcript; -use crate::primitives::serialization::HachiSerialize; -use crate::{CanonicalField, FieldCore}; -use sha3::{Digest, Keccak256}; -use std::marker::PhantomData; - -/// Keccak256 transcript with labeled framing. -#[derive(Clone)] -pub struct KeccakTranscript -where - F: FieldCore + CanonicalField + 'static, -{ - hasher: Keccak256, - _field: PhantomData, -} - -impl KeccakTranscript -where - F: FieldCore + CanonicalField + 'static, -{ - #[inline] - fn append_bytes_impl(&mut self, label: &[u8], bytes: &[u8]) { - self.hasher.update(label); - self.hasher.update((bytes.len() as u64).to_le_bytes()); - self.hasher.update(bytes); - } - - #[inline] - fn challenge_bytes(&mut self, label: &[u8]) -> [u8; 32] { - self.hasher.update(label); - let digest = self.hasher.clone().finalize(); - let mut out = [0u8; 32]; - out.copy_from_slice(digest.as_slice()); - self.hasher.update(out); - out - } -} - -impl Transcript for KeccakTranscript -where - F: FieldCore + CanonicalField + 'static, -{ - fn new(domain_label: &[u8]) -> Self { - let mut hasher = Keccak256::new(); - hasher.update(domain_label); - Self { - hasher, - _field: PhantomData, - } - } - - fn append_bytes(&mut self, label: &[u8], bytes: &[u8]) { - self.append_bytes_impl(label, bytes); - } - - fn append_field(&mut self, label: &[u8], x: &F) { - self.append_bytes_impl(label, &x.to_canonical_u128().to_le_bytes()); - } - - fn append_serde(&mut self, label: &[u8], s: &S) { - let mut bytes = Vec::new(); - s.serialize_compressed(&mut bytes) - .expect("HachiSerialize should not fail"); - self.append_bytes_impl(label, &bytes); - } - - fn challenge_scalar(&mut self, label: &[u8]) -> F { - let bytes = self.challenge_bytes(label); - let mut lo = [0u8; 16]; - lo.copy_from_slice(&bytes[..16]); - let sampled = u128::from_le_bytes(lo); - F::from_canonical_u128_reduced(sampled) - } - - fn reset(&mut self, domain_label: &[u8]) { - let mut hasher = Keccak256::new(); - hasher.update(domain_label); - self.hasher = hasher; - } -} diff --git a/src/protocol/transcript/mod.rs b/src/protocol/transcript/mod.rs index 95c45413..5279f3f5 100644 --- a/src/protocol/transcript/mod.rs +++ b/src/protocol/transcript/mod.rs @@ -1,14 +1,12 @@ //! Protocol transcript contracts and implementations. -mod blake2b; -mod keccak; +mod hash; pub mod labels; use crate::algebra::fields::lift::ExtField; use crate::{CanonicalField, FieldCore, HachiSerialize}; -pub use blake2b::Blake2bTranscript; -pub use keccak::KeccakTranscript; +pub use hash::{Blake2bTranscript, HashTranscript, KeccakTranscript}; /// Transcript interface for protocol Fiat-Shamir transforms. /// From f74f2edad04809227eec8641d100ff2026207642 Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Sat, 28 Feb 2026 14:52:42 -0800 Subject: [PATCH 11/34] Fix sumcheck degree bug, split types, in-place fold, CommitWitness, rename configs, add soundness test - Fix CompressedUniPoly::degree() off-by-one that could let malformed proofs pass - Split sumcheck/mod.rs: extract types into types.rs, relocate multilinear_eval and fold_evals to algebra/poly.rs - Replace allocating fold_evals with in-place fold_evals_in_place - Add debug_assert guards to multilinear_eval and fold_evals_in_place - Introduce CommitWitness struct to replace error-prone 3-tuple returns - Rename DefaultCommitmentConfig to SmallTestCommitmentConfig, add ProductionFp128CommitmentConfig - Add verify_rejects_wrong_opening negative test for verifier soundness Made-with: Cursor --- benches/hachi_e2e.rs | 32 +- src/algebra/poly.rs | 55 ++- src/protocol/commitment/commit.rs | 34 +- src/protocol/commitment/config.rs | 71 +++- src/protocol/commitment/mod.rs | 6 +- src/protocol/commitment/scheme.rs | 44 +- src/protocol/commitment_scheme.rs | 70 +++- src/protocol/mod.rs | 7 +- src/protocol/quadratic_equation.rs | 10 +- src/protocol/ring_switch.rs | 9 +- src/protocol/sumcheck/mod.rs | 441 +-------------------- src/protocol/sumcheck/norm_sumcheck.rs | 18 +- src/protocol/sumcheck/relation_sumcheck.rs | 24 +- src/protocol/sumcheck/split_eq.rs | 4 +- src/protocol/sumcheck/types.rs | 384 ++++++++++++++++++ tests/onehot_commitment.rs | 16 +- tests/ring_commitment_core.rs | 100 ++--- tests/sumcheck_core.rs | 16 +- 18 files changed, 721 insertions(+), 620 deletions(-) create mode 100644 src/protocol/sumcheck/types.rs diff --git a/benches/hachi_e2e.rs b/benches/hachi_e2e.rs index 9e9727cd..cc92ecc8 100644 --- a/benches/hachi_e2e.rs +++ b/benches/hachi_e2e.rs @@ -1,18 +1,18 @@ #![allow(missing_docs)] use criterion::{black_box, criterion_group, criterion_main, BenchmarkId, Criterion}; -use hachi_pcs::algebra::Fp64; +use hachi_pcs::algebra::Fp128; use hachi_pcs::primitives::multilinear_evals::DenseMultilinearEvals; -use hachi_pcs::protocol::commitment::DefaultCommitmentConfig; +use hachi_pcs::protocol::commitment::ProductionFp128CommitmentConfig; use hachi_pcs::protocol::commitment_scheme::HachiCommitmentScheme; use hachi_pcs::protocol::transcript::Blake2bTranscript; use hachi_pcs::protocol::CommitmentConfig; use hachi_pcs::{CommitmentScheme, FromSmallInt, Polynomial, Transcript}; use std::time::Duration; -type F = Fp64<4294967197>; +type F = Fp128<0xfffffffffffffffffffffffffffffeed>; -const D: usize = DefaultCommitmentConfig::D; +const D: usize = ProductionFp128CommitmentConfig::D; macro_rules! bench_config { ($name:ident, M = $m:expr, R = $r:expr) => { @@ -22,14 +22,14 @@ macro_rules! bench_config { const D: usize = D; const M: usize = $m; const R: usize = $r; - const N_A: usize = DefaultCommitmentConfig::N_A; - const N_B: usize = DefaultCommitmentConfig::N_B; - const N_D: usize = DefaultCommitmentConfig::N_D; - const LOG_BASIS: u32 = DefaultCommitmentConfig::LOG_BASIS; - const DELTA: usize = DefaultCommitmentConfig::DELTA; - const TAU: usize = DefaultCommitmentConfig::TAU; - const BETA: u128 = DefaultCommitmentConfig::BETA; - const CHALLENGE_WEIGHT: usize = DefaultCommitmentConfig::CHALLENGE_WEIGHT; + const N_A: usize = ProductionFp128CommitmentConfig::N_A; + const N_B: usize = ProductionFp128CommitmentConfig::N_B; + const N_D: usize = ProductionFp128CommitmentConfig::N_D; + const LOG_BASIS: u32 = ProductionFp128CommitmentConfig::LOG_BASIS; + const DELTA: usize = ProductionFp128CommitmentConfig::DELTA; + const TAU: usize = ProductionFp128CommitmentConfig::TAU; + const BETA: u128 = ProductionFp128CommitmentConfig::BETA; + const CHALLENGE_WEIGHT: usize = ProductionFp128CommitmentConfig::CHALLENGE_WEIGHT; } }; } @@ -162,16 +162,16 @@ where } fn bench_nv10(c: &mut Criterion) { - bench_phases::(c, "fp64"); + bench_phases::(c, "fp128_p275"); } fn bench_nv14(c: &mut Criterion) { - bench_phases::(c, "fp64"); + bench_phases::(c, "fp128_p275"); } fn bench_nv18(c: &mut Criterion) { - bench_phases::(c, "fp64"); + bench_phases::(c, "fp128_p275"); } fn bench_nv20(c: &mut Criterion) { - bench_phases::(c, "fp64"); + bench_phases::(c, "fp128_p275"); } criterion_group!( diff --git a/src/algebra/poly.rs b/src/algebra/poly.rs index 684aa7d8..cb93242e 100644 --- a/src/algebra/poly.rs +++ b/src/algebra/poly.rs @@ -1,9 +1,10 @@ -//! Fixed-size polynomial container. +//! Polynomial containers and evaluation utilities. use crate::primitives::serialization::{ Compress, HachiDeserialize, HachiSerialize, SerializationError, Valid, Validate, }; use crate::FieldCore; +use crate::FromSmallInt; use std::io::{Read, Write}; /// A degree- HachiDeserialize for Poly { Ok(out) } } + +/// Evaluate the range-check polynomial `w · Π_{k=1}^{b−1} (w − k)(w + k)`. +/// +/// This polynomial vanishes exactly when `w ∈ {−(b−1), …, b−1}`. +/// Total degree in `w` is `2b − 1`. +pub fn range_check_eval(w: E, b: usize) -> E { + let mut acc = w; + for k in 1..b { + let k_e = E::from_u64(k as u64); + acc = acc * (w - k_e) * (w + k_e); + } + acc +} + +/// Evaluate a multilinear polynomial (given by boolean-hypercube evaluations in +/// little-endian bit order) at an arbitrary point via iterated folding. +pub fn multilinear_eval(evals: &[E], point: &[E]) -> E { + debug_assert!( + evals.len().is_power_of_two(), + "evals length must be a power of two" + ); + debug_assert_eq!( + evals.len(), + 1 << point.len(), + "evals length must equal 2^point.len()" + ); + let mut current = evals.to_vec(); + for &r in point { + let half = current.len() / 2; + let mut next = Vec::with_capacity(half); + for i in 0..half { + next.push(current[2 * i] + r * (current[2 * i + 1] - current[2 * i])); + } + current = next; + } + current[0] +} + +/// Fold an evaluation table in place by binding its first variable to `r`, +/// halving the table size. +pub fn fold_evals_in_place(evals: &mut Vec, r: E) { + debug_assert!( + evals.len().is_power_of_two(), + "evals length must be a power of two" + ); + debug_assert!(evals.len() >= 2, "evals must have at least 2 elements"); + let half = evals.len() / 2; + for i in 0..half { + evals[i] = evals[2 * i] + r * (evals[2 * i + 1] - evals[2 * i]); + } + evals.truncate(half); +} diff --git a/src/protocol/commitment/commit.rs b/src/protocol/commitment/commit.rs index f86ac197..10619a0b 100644 --- a/src/protocol/commitment/commit.rs +++ b/src/protocol/commitment/commit.rs @@ -4,7 +4,7 @@ use super::config::{ ensure_block_layout, ensure_matrix_shape, ensure_supported_num_vars, validate_and_derive_layout, }; use super::onehot::{inner_ajtai_onehot, map_onehot_to_sparse_blocks}; -use super::scheme::RingCommitmentScheme; +use super::scheme::{CommitWitness, RingCommitmentScheme}; use super::types::RingCommitment; use super::utils::crt_ntt::{build_ntt_cache, NttMatrixCache}; use super::utils::linear::{ @@ -79,18 +79,10 @@ where Ok((setup.clone(), setup)) } - #[allow(clippy::type_complexity)] fn commit_ring_blocks( f_blocks: &[Vec>], setup: &Self::ProverSetup, - ) -> Result< - ( - Self::Commitment, - Vec>>, - Vec>>, - ), - HachiError, - > { + ) -> Result, HachiError> { let layout = validate_and_derive_layout::()?; ensure_supported_num_vars(setup.max_num_vars, layout.required_vars)?; ensure_block_layout(f_blocks, layout)?; @@ -112,22 +104,18 @@ where } let u = mat_vec_mul_ntt_cached(&setup.ntt_cache, MatrixSlot::B, &t_hat_flat)?; - Ok((RingCommitment { u }, s_all, t_hat_all)) + Ok(CommitWitness { + commitment: RingCommitment { u }, + s: s_all, + t_hat: t_hat_all, + }) } - #[allow(clippy::type_complexity)] fn commit_onehot( onehot_k: usize, indices: &[usize], setup: &Self::ProverSetup, - ) -> Result< - ( - Self::Commitment, - Vec>>, - Vec>>, - ), - HachiError, - > { + ) -> Result, HachiError> { let layout = validate_and_derive_layout::()?; ensure_supported_num_vars(setup.max_num_vars, layout.required_vars)?; ensure_matrix_shape(&setup.A, Cfg::N_A, layout.inner_width, "A")?; @@ -150,6 +138,10 @@ where } let u = mat_vec_mul_ntt_cached(&setup.ntt_cache, MatrixSlot::B, &t_hat_flat)?; - Ok((RingCommitment { u }, s_all, t_hat_all)) + Ok(CommitWitness { + commitment: RingCommitment { u }, + s: s_all, + t_hat: t_hat_all, + }) } } diff --git a/src/protocol/commitment/config.rs b/src/protocol/commitment/config.rs index 3150554e..d953b7c1 100644 --- a/src/protocol/commitment/config.rs +++ b/src/protocol/commitment/config.rs @@ -31,6 +31,37 @@ pub trait CommitmentConfig: Clone + Send + Sync + 'static { const CHALLENGE_WEIGHT: usize; } +/// Deterministic upper bound for the stage-1 folded-witness infinity norm. +/// +/// This encodes the bound used in `QuadraticEquation::compute_z_hat`: +/// `||z||_inf <= 2^R * ω * (b/2)` where `b = 2^LOG_BASIS`. +/// +/// # Panics +/// +/// Panics when `log_basis` or `r` are out of range, or when intermediate +/// products overflow `u128`. +pub(super) const fn beta_linf_fold_bound( + r: usize, + challenge_weight: usize, + log_basis: u32, +) -> u128 { + assert!(log_basis > 0 && log_basis < 128, "invalid LOG_BASIS"); + assert!(r < 128, "R must be < 128"); + + let blocks = 1u128 << r; + let b = 1u128 << log_basis; + let half_b = b / 2; + + let term = match blocks.checked_mul(challenge_weight as u128) { + Some(v) => v, + None => panic!("beta bound overflow (blocks * challenge_weight)"), + }; + match term.checked_mul(half_b) { + Some(v) => v, + None => panic!("beta bound overflow (term * half_b)"), + } +} + /// Runtime-derived dimensions from a `CommitmentConfig`. #[derive(Debug, Clone, Copy)] pub(super) struct CommitmentLayout { @@ -167,11 +198,11 @@ pub(super) fn ensure_matrix_shape( Ok(()) } -/// Default correctness-first config for early protocol integration. +/// Small correctness-first config for tests and local benchmarks. #[derive(Clone, Copy, Debug, Default)] -pub struct DefaultCommitmentConfig; +pub struct SmallTestCommitmentConfig; -impl CommitmentConfig for DefaultCommitmentConfig { +impl CommitmentConfig for SmallTestCommitmentConfig { const D: usize = 16; const M: usize = 4; const R: usize = 2; @@ -184,3 +215,37 @@ impl CommitmentConfig for DefaultCommitmentConfig { const BETA: u128 = 1_000_000; const CHALLENGE_WEIGHT: usize = 3; } + +/// Production-oriented profile for 128-bit base fields (`Fp128

`). +/// +/// This profile targets the `D = 512`, `n_A = n_B = n_D = 1` regime with +/// base-16 decomposition over ~128-bit moduli. +/// +/// Rigorous β derivation for the stage-1 folded witness `z`: +/// - In `compute_z_hat`, each coordinate is `z[j] = Σ_i s_i[j].mul_by_sparse(c_i)`. +/// - `balanced_decompose_pow2` yields per-coefficient digits in `[-b/2, b/2)` where +/// `b = 2^LOG_BASIS`, so each input coefficient has `|·| <= b/2`. +/// - Challenges use exactly `ω = CHALLENGE_WEIGHT` nonzeros in `{±1}`. +/// - Therefore each `mul_by_sparse` output coefficient is a signed sum of `ω` +/// shifted digits, hence bounded by `ω * (b/2)`. +/// - Summing over `2^R` blocks gives: +/// `||z||_inf <= 2^R * ω * (b/2)`. +/// +/// For this profile: `R=11`, `ω=19`, `b=16`, so +/// `β = 2^11 * 19 * 8 = 311_296`. +#[derive(Clone, Copy, Debug, Default)] +pub struct ProductionFp128CommitmentConfig; + +impl CommitmentConfig for ProductionFp128CommitmentConfig { + const D: usize = 512; + const M: usize = 11; + const R: usize = 11; + const N_A: usize = 1; + const N_B: usize = 1; + const N_D: usize = 1; + const LOG_BASIS: u32 = 4; + const DELTA: usize = 32; + const TAU: usize = 5; + const BETA: u128 = beta_linf_fold_bound(Self::R, Self::CHALLENGE_WEIGHT, Self::LOG_BASIS); + const CHALLENGE_WEIGHT: usize = 19; +} diff --git a/src/protocol/commitment/mod.rs b/src/protocol/commitment/mod.rs index 3a3b7170..d7c23e2f 100644 --- a/src/protocol/commitment/mod.rs +++ b/src/protocol/commitment/mod.rs @@ -9,8 +9,10 @@ mod types; pub(crate) mod utils; pub use commit::{HachiCommitmentCore, RingCommitmentSetup}; -pub use config::{CommitmentConfig, DefaultCommitmentConfig}; -pub use scheme::{CommitmentScheme, RingCommitmentScheme, StreamingCommitmentScheme}; +pub use config::{CommitmentConfig, ProductionFp128CommitmentConfig, SmallTestCommitmentConfig}; +pub use scheme::{ + CommitWitness, CommitmentScheme, RingCommitmentScheme, StreamingCommitmentScheme, +}; pub use transcript_append::AppendToTranscript; pub use types::{ DummyProof, HachiCommitment, HachiOpeningClaim, HachiOpeningPoint, RingCommitment, diff --git a/src/protocol/commitment/scheme.rs b/src/protocol/commitment/scheme.rs index df0c4301..f2c5f2e9 100644 --- a/src/protocol/commitment/scheme.rs +++ b/src/protocol/commitment/scheme.rs @@ -11,6 +11,20 @@ use crate::{CanonicalField, FieldCore, Polynomial}; /// Output type for batched commitments. pub(crate) type BatchCommitOutput = Result, HachiError>; +/// Witness data produced alongside a ring-native commitment. +/// +/// Contains the commitment itself plus the decomposed witness vectors `s` +/// (basis-decomposed input) and `t_hat` (basis-decomposed inner Ajtai output) +/// from the two-layer Ajtai construction (§4.1). +pub struct CommitWitness { + /// The ring commitment (outer Ajtai output `u = B · t̂`). + pub commitment: C, + /// Per-block basis-decomposed input vectors. + pub s: Vec>>, + /// Per-block basis-decomposed inner Ajtai output vectors. + pub t_hat: Vec>>, +} + /// Generic commitment-scheme interface used by Hachi protocol code. pub trait CommitmentScheme: Clone + Send + Sync + 'static where @@ -147,18 +161,10 @@ where /// # Errors /// /// Returns an error if block layout mismatches config or commitment fails. - #[allow(clippy::type_complexity)] fn commit_ring_blocks( f_blocks: &[Vec>], setup: &Self::ProverSetup, - ) -> Result< - ( - Self::Commitment, - Vec>>, - Vec>>, - ), - HachiError, - >; + ) -> Result, HachiError>; /// Commit to a flat coefficient table `(f_i)_{i∈{0,1}^ℓ}` in ring form. /// @@ -177,18 +183,10 @@ where /// Returns an error if `f_coeffs.len()` does not match the configured block /// layout, if internal index computations overflow, or if the underlying /// commitment routine fails. - #[allow(clippy::type_complexity)] fn commit_coeffs( f_coeffs: &[CyclotomicRing], setup: &Self::ProverSetup, - ) -> Result< - ( - Self::Commitment, - Vec>>, - Vec>>, - ), - HachiError, - > { + ) -> Result, HachiError> { let num_blocks = checked_pow2(Cfg::R)?; let block_len = checked_pow2(Cfg::M)?; let expected_len = num_blocks @@ -234,19 +232,11 @@ where /// /// Returns an error if dimensions are inconsistent or any index is out /// of range. - #[allow(clippy::type_complexity)] fn commit_onehot( onehot_k: usize, indices: &[usize], setup: &Self::ProverSetup, - ) -> Result< - ( - Self::Commitment, - Vec>>, - Vec>>, - ), - HachiError, - > { + ) -> Result, HachiError> { let num_chunks = indices.len(); let total_field_elems = num_chunks .checked_mul(onehot_k) diff --git a/src/protocol/commitment_scheme.rs b/src/protocol/commitment_scheme.rs index 4b17d921..30c5a5bd 100644 --- a/src/protocol/commitment_scheme.rs +++ b/src/protocol/commitment_scheme.rs @@ -7,8 +7,8 @@ use crate::error::HachiError; use crate::parallel::*; use crate::primitives::poly::multilinear_lagrange_basis; use crate::protocol::commitment::{ - CommitmentConfig, CommitmentScheme, DefaultCommitmentConfig, HachiCommitmentCore, - RingCommitment, RingCommitmentScheme, RingCommitmentSetup, + CommitmentConfig, CommitmentScheme, HachiCommitmentCore, RingCommitment, RingCommitmentScheme, + RingCommitmentSetup, }; use crate::protocol::opening_point::RingOpeningPoint; use crate::protocol::proof::{HachiCommitmentHint, HachiProof, SumcheckAux}; @@ -34,10 +34,12 @@ use crate::protocol::transcript::labels::{ }; #[cfg(test)] use crate::protocol::transcript::Blake2bTranscript; +#[cfg(test)] +use crate::protocol::SmallTestCommitmentConfig; /// End-to-end PCS wrapper, generic over ring degree `D` and config `Cfg`. #[derive(Clone, Copy, Debug, Default)] -pub struct HachiCommitmentScheme { +pub struct HachiCommitmentScheme { _cfg: std::marker::PhantomData, } @@ -69,17 +71,16 @@ where ) -> Result<(Self::Commitment, Self::OpeningProofHint), HachiError> { let ring_coeffs = reduce_coeffs_to_ring_elements::(poly.num_vars(), &poly.coeffs())?; - let (commitment, s, t_hat) = >::commit_coeffs(&ring_coeffs, setup)?; + let w = >::commit_coeffs( + &ring_coeffs, + setup, + )?; let hint = HachiCommitmentHint { - s, - t_hat, + s: w.s, + t_hat: w.t_hat, ring_coeffs, }; - Ok((commitment, hint)) + Ok((w.commitment, hint)) } fn prove, P: Polynomial>( @@ -397,7 +398,7 @@ mod tests { use crate::test_utils::F; use crate::{CommitmentScheme, FromSmallInt, Polynomial}; - type Cfg = DefaultCommitmentConfig; + type Cfg = SmallTestCommitmentConfig; type Scheme = HachiCommitmentScheme<{ Cfg::D }, Cfg>; #[test] @@ -440,4 +441,49 @@ mod tests { assert!(result.is_ok()); } + + #[test] + fn verify_rejects_wrong_opening() { + let alpha = Cfg::D.trailing_zeros() as usize; + let num_vars = Cfg::R + Cfg::M + alpha; + let len = 1usize << num_vars; + + let evals: Vec = (0..len).map(|i| F::from_u64(i as u64)).collect(); + let poly = DenseMultilinearEvals::new_padded(evals); + + let setup = >::setup_prover(num_vars); + let verifier_setup = >::setup_verifier(&setup); + + let (commitment, hint) = >::commit(&poly, &setup).unwrap(); + + let opening_point: Vec = (0..num_vars).map(|i| F::from_u64((i + 2) as u64)).collect(); + let opening = poly.evaluate(&opening_point); + + let mut prover_transcript = Blake2bTranscript::::new(b"test/prove"); + let proof = >::prove( + &setup, + &poly, + &opening_point, + Some(hint), + &mut prover_transcript, + &commitment, + ) + .unwrap(); + + let wrong_opening = opening + F::one(); + let mut verifier_transcript = Blake2bTranscript::::new(b"test/prove"); + let result = >::verify( + &proof, + &verifier_setup, + &mut verifier_transcript, + &opening_point, + &wrong_opening, + &commitment, + ); + + assert!( + result.is_err(), + "verify must reject an incorrect opening value" + ); + } } diff --git a/src/protocol/mod.rs b/src/protocol/mod.rs index f44ede06..90090fe8 100644 --- a/src/protocol/mod.rs +++ b/src/protocol/mod.rs @@ -15,9 +15,10 @@ pub mod sumcheck; pub mod transcript; pub use commitment::{ - AppendToTranscript, CommitmentConfig, CommitmentScheme, DefaultCommitmentConfig, DummyProof, - HachiCommitment, HachiCommitmentCore, HachiOpeningClaim, HachiOpeningPoint, RingCommitment, - RingCommitmentScheme, RingCommitmentSetup, StreamingCommitmentScheme, + AppendToTranscript, CommitmentConfig, CommitmentScheme, DummyProof, HachiCommitment, + HachiCommitmentCore, HachiOpeningClaim, HachiOpeningPoint, ProductionFp128CommitmentConfig, + RingCommitment, RingCommitmentScheme, RingCommitmentSetup, SmallTestCommitmentConfig, + StreamingCommitmentScheme, }; pub use commitment_scheme::HachiCommitmentScheme; pub use opening_point::RingOpeningPoint; diff --git a/src/protocol/quadratic_equation.rs b/src/protocol/quadratic_equation.rs index f7a8408a..d2b365dc 100644 --- a/src/protocol/quadratic_equation.rs +++ b/src/protocol/quadratic_equation.rs @@ -584,7 +584,7 @@ mod tests { >::setup(16).unwrap(); let blocks = sample_blocks(); - let (commitment, s, t_hat) = + let w = >::commit_ring_blocks( &blocks, &setup, ) @@ -596,8 +596,8 @@ mod tests { }; let hint = HachiCommitmentHint { - s, - t_hat, + s: w.s, + t_hat: w.t_hat, ring_coeffs: Vec::new(), }; let mut transcript = Blake2bTranscript::::new(TRANSCRIPT_SEED); @@ -607,7 +607,7 @@ mod tests { &point, &hint, &mut transcript, - &commitment, + &w.commitment, &y_ring, ) .unwrap(); @@ -616,7 +616,7 @@ mod tests { Fixture { setup, - commitment_u: commitment.u.clone(), + commitment_u: w.commitment.u.clone(), point, blocks, quad_eq, diff --git a/src/protocol/ring_switch.rs b/src/protocol/ring_switch.rs index 342545b6..6d5860cc 100644 --- a/src/protocol/ring_switch.rs +++ b/src/protocol/ring_switch.rs @@ -282,12 +282,11 @@ where WCfg::::M, )?; - let (commitment, _, _) = - >>::commit_ring_blocks( - &blocks, &w_setup, - )?; + let w = >>::commit_ring_blocks( + &blocks, &w_setup, + )?; - Ok(commitment) + Ok(w.commitment) } pub(crate) fn eval_ring_at(r: &CyclotomicRing, alpha: &F) -> F { diff --git a/src/protocol/sumcheck/mod.rs b/src/protocol/sumcheck/mod.rs index d87e65e0..950b6e78 100644 --- a/src/protocol/sumcheck/mod.rs +++ b/src/protocol/sumcheck/mod.rs @@ -1,15 +1,8 @@ -//! Sumcheck core building blocks (univariate messages + proof transcript driver). +//! Sumcheck protocol: traits, proof driver, and concrete instances. //! -//! This module provides: -//! - Univariate polynomials over a field `E` (`UniPoly`). -//! - A compressed univariate representation (`CompressedUniPoly`) that omits the -//! linear term and reconstructs it from the per-round sumcheck hint `g(0)+g(1)`. -//! - A minimal sumcheck proof container (`SumcheckProof`) with a verifier-side -//! transcript driver that returns the derived point and final claimed value. -//! -//! It intentionally does **not** implement a concrete sumcheck prover for any -//! particular oracle/table representation. Higher-level protocols (e.g. Hachi §4.3) -//! should implement the prover logic that constructs each round's univariate `g_i(X)`. +//! Types (`UniPoly`, `CompressedUniPoly`, `SumcheckProof`) live in the +//! [`types`] submodule. Polynomial evaluation utilities (`multilinear_eval`, +//! `fold_evals_in_place`, `range_check_eval`) live in [`crate::algebra::poly`]. //! //! ## Temporary duplication notice (Jolt integration) //! @@ -24,392 +17,15 @@ pub mod eq_poly; pub mod norm_sumcheck; pub mod relation_sumcheck; pub mod split_eq; +pub mod types; use crate::error::HachiError; -use crate::primitives::serialization::{ - Compress, HachiDeserialize, HachiSerialize, SerializationError, Valid, Validate, -}; use crate::protocol::transcript::labels; use crate::protocol::transcript::Transcript; use crate::FieldCore; -use crate::FromSmallInt; -use std::io::{Read, Write}; - -/// Univariate polynomial in coefficient form: `p(X) = Σ_{i=0}^d coeffs[i] * X^i`. -#[derive(Debug, Clone, PartialEq, Eq)] -pub struct UniPoly { - /// Coefficients from low degree to high degree. - pub coeffs: Vec, -} - -impl UniPoly { - /// Construct from coefficients in increasing-degree order. - pub fn from_coeffs(coeffs: Vec) -> Self { - Self { coeffs } - } - - /// Degree of the polynomial. - /// - /// # Panics - /// - /// Panics if the polynomial has no coefficients. - pub fn degree(&self) -> usize { - self.coeffs - .len() - .checked_sub(1) - .expect("UniPoly must have at least one coefficient") - } - - /// Evaluate at `x` via Horner's method. - pub fn evaluate(&self, x: &E) -> E { - // Horner from highest degree. - let mut acc = E::zero(); - for c in self.coeffs.iter().rev() { - acc = acc * *x + *c; - } - acc - } - - /// Compress this polynomial by omitting the linear coefficient. - /// - /// The verifier can reconstruct/evaluate the missing linear coefficient using - /// the per-round hint `g(0)+g(1)` from the sumcheck protocol. - /// - /// This matches the technique used by Jolt's sumcheck (`CompressedUniPoly`). - pub fn compress(&self) -> CompressedUniPoly { - let coeffs = &self.coeffs; - if coeffs.is_empty() { - return CompressedUniPoly { - coeffs_except_linear_term: Vec::new(), - }; - } - if coeffs.len() == 1 { - return CompressedUniPoly { - coeffs_except_linear_term: vec![coeffs[0]], - }; - } - // Keep coeff[0], drop coeff[1], keep coeff[2..]. - let mut out = Vec::with_capacity(coeffs.len().saturating_sub(1)); - out.push(coeffs[0]); - out.extend_from_slice(&coeffs[2..]); - CompressedUniPoly { - coeffs_except_linear_term: out, - } - } -} - -impl UniPoly { - /// Interpolate from evaluations at equispaced integer points `x = 0, 1, ..., d`. - /// - /// Uses Newton forward-difference interpolation: compute divided differences, - /// then expand via Horner on the nested Newton form. - /// - /// # Panics - /// - /// Panics if any required factorial inverse does not exist (field characteristic - /// must exceed the number of evaluation points). - pub fn from_evals(evals: &[E]) -> Self { - let n = evals.len(); - if n == 0 { - return Self::from_coeffs(vec![]); - } - if n == 1 { - return Self::from_coeffs(vec![evals[0]]); - } - - // Forward differences: delta^k[0] - let mut table = evals.to_vec(); - let mut deltas = vec![table[0]]; - for _ in 1..n { - for j in 0..table.len() - 1 { - table[j] = table[j + 1] - table[j]; - } - table.pop(); - deltas.push(table[0]); - } - - // Divided differences: dd[k] = delta^k / k! - let mut factorial = E::one(); - let mut divided_diffs = vec![deltas[0]]; - for (k, delta_k) in deltas.iter().enumerate().skip(1) { - factorial = factorial * E::from_u64(k as u64); - divided_diffs.push( - *delta_k - * factorial - .inv() - .expect("field characteristic too small for interpolation"), - ); - } - - // Horner expansion of the Newton nested form: - // p(x) = dd[0] + (x-0)(dd[1] + (x-1)(dd[2] + ...)) - let mut coeffs = vec![divided_diffs[n - 1]]; - - for k in (0..n - 1).rev() { - let shift = E::from_u64(k as u64); - let old_len = coeffs.len(); - let mut new_coeffs = vec![E::zero(); old_len + 1]; - - new_coeffs[0] = divided_diffs[k]; - for i in 0..old_len { - new_coeffs[i + 1] = new_coeffs[i + 1] + coeffs[i]; - new_coeffs[i] = new_coeffs[i] - shift * coeffs[i]; - } - - coeffs = new_coeffs; - } - - while coeffs.len() > 1 && coeffs.last().is_some_and(|c| c.is_zero()) { - coeffs.pop(); - } - - Self::from_coeffs(coeffs) - } -} - -impl Valid for UniPoly { - fn check(&self) -> Result<(), SerializationError> { - self.coeffs.check() - } -} - -impl HachiSerialize for UniPoly { - fn serialize_with_mode( - &self, - mut writer: W, - compress: Compress, - ) -> Result<(), SerializationError> { - self.coeffs.serialize_with_mode(&mut writer, compress) - } - - fn serialized_size(&self, compress: Compress) -> usize { - self.coeffs.serialized_size(compress) - } -} - -impl HachiDeserialize for UniPoly { - fn deserialize_with_mode( - mut reader: R, - compress: Compress, - validate: Validate, - ) -> Result { - let coeffs = Vec::::deserialize_with_mode(&mut reader, compress, validate)?; - let out = Self { coeffs }; - if matches!(validate, Validate::Yes) { - out.check()?; - } - Ok(out) - } -} - -/// Compressed univariate polynomial representation omitting the linear term. -/// -/// We store `[c0, c2, c3, ..., cd]`. Given the sumcheck hint `hint = g(0)+g(1)`, -/// the missing linear coefficient is: -/// -/// `c1 = hint - 2*c0 - Σ_{i=2..d} ci`. -#[derive(Debug, Clone, PartialEq, Eq)] -pub struct CompressedUniPoly { - /// Coefficients excluding the linear term: `[c0, c2, c3, ..., cd]`. - pub coeffs_except_linear_term: Vec, -} - -impl CompressedUniPoly { - /// Degree of the underlying uncompressed polynomial. - /// - /// For degree `d`, this stores `d` coefficients (all except the linear term). - pub fn degree(&self) -> usize { - self.coeffs_except_linear_term.len().saturating_sub(1) - } - - fn recover_linear_term(&self, hint: &E) -> E { - if self.coeffs_except_linear_term.is_empty() { - // Treat empty as the zero polynomial. - return E::zero(); - } - - // linear = hint - 2*c0 - Σ_{i>=2} ci - let c0 = self.coeffs_except_linear_term[0]; - let mut linear = *hint - c0 - c0; - for c in self.coeffs_except_linear_term.iter().skip(1) { - linear = linear - *c; - } - linear - } - - /// Decompress using `hint = g(0)+g(1)`. - pub fn decompress(&self, hint: &E) -> UniPoly { - if self.coeffs_except_linear_term.is_empty() { - return UniPoly::from_coeffs(Vec::new()); - } - let linear = self.recover_linear_term(hint); - // Always materialize the missing linear coefficient. - // For truly-constant polynomials, the hint forces `linear = 0`, so this is harmless. - let mut coeffs = Vec::with_capacity(self.coeffs_except_linear_term.len() + 1); - coeffs.push(self.coeffs_except_linear_term[0]); - coeffs.push(linear); - coeffs.extend_from_slice(&self.coeffs_except_linear_term[1..]); - UniPoly::from_coeffs(coeffs) - } - - /// Evaluate the uncompressed polynomial at `x`, using `hint = g(0)+g(1)`. - /// - /// This avoids materializing the full coefficient list. - pub fn eval_from_hint(&self, hint: &E, x: &E) -> E { - if self.coeffs_except_linear_term.is_empty() { - return E::zero(); - } - - let linear = self.recover_linear_term(hint); - let mut acc = self.coeffs_except_linear_term[0] + (*x * linear); - - // Add Σ_{i=2..d} c_i * x^i, where stored slice is [c2, c3, ..., cd] - let mut pow = *x * *x; // x^2 - for c in self.coeffs_except_linear_term.iter().skip(1) { - acc = acc + (*c * pow); - pow = pow * *x; - } - acc - } -} - -impl Valid for CompressedUniPoly { - fn check(&self) -> Result<(), SerializationError> { - self.coeffs_except_linear_term.check() - } -} - -impl HachiSerialize for CompressedUniPoly { - fn serialize_with_mode( - &self, - mut writer: W, - compress: Compress, - ) -> Result<(), SerializationError> { - self.coeffs_except_linear_term - .serialize_with_mode(&mut writer, compress) - } - - fn serialized_size(&self, compress: Compress) -> usize { - self.coeffs_except_linear_term.serialized_size(compress) - } -} - -impl HachiDeserialize for CompressedUniPoly { - fn deserialize_with_mode( - mut reader: R, - compress: Compress, - validate: Validate, - ) -> Result { - let coeffs_except_linear_term = - Vec::::deserialize_with_mode(&mut reader, compress, validate)?; - let out = Self { - coeffs_except_linear_term, - }; - if matches!(validate, Validate::Yes) { - out.check()?; - } - Ok(out) - } -} - -/// Sumcheck proof containing one compressed univariate polynomial per round. -#[derive(Debug, Clone, PartialEq, Eq)] -pub struct SumcheckProof { - /// One compressed univariate polynomial per sumcheck round. - pub round_polys: Vec>, -} - -impl Valid for SumcheckProof { - fn check(&self) -> Result<(), SerializationError> { - self.round_polys.check() - } -} - -impl HachiSerialize for SumcheckProof { - fn serialize_with_mode( - &self, - mut writer: W, - compress: Compress, - ) -> Result<(), SerializationError> { - self.round_polys.serialize_with_mode(&mut writer, compress) - } - - fn serialized_size(&self, compress: Compress) -> usize { - self.round_polys.serialized_size(compress) - } -} - -impl HachiDeserialize for SumcheckProof { - fn deserialize_with_mode( - mut reader: R, - compress: Compress, - validate: Validate, - ) -> Result { - let round_polys = - Vec::>::deserialize_with_mode(&mut reader, compress, validate)?; - let out = Self { round_polys }; - if matches!(validate, Validate::Yes) { - out.check()?; - } - Ok(out) - } -} -impl SumcheckProof { - /// Verifier-side sumcheck transcript driver. - /// - /// This method: - /// - absorbs the per-round prover message (compressed univariate), - /// - samples one challenge per round via `sample_challenge`, - /// - updates the running claim using `eval_from_hint`. - /// - /// It does **not** perform the final oracle check `final_claim == f(r*)`. - /// Callers (e.g. ring-switching) must compute `f(r*)` themselves and compare. - /// - /// # Errors - /// - /// Returns an error if the proof length does not match `num_rounds` or if any - /// per-round polynomial exceeds `degree_bound`. - pub fn verify( - &self, - mut claim: E, - num_rounds: usize, - degree_bound: usize, - transcript: &mut T, - mut sample_challenge: S, - ) -> Result<(E, Vec), HachiError> - where - F: crate::FieldCore + crate::CanonicalField, - T: Transcript, - S: FnMut(&mut T) -> E, - { - if self.round_polys.len() != num_rounds { - return Err(HachiError::InvalidSize { - expected: num_rounds, - actual: self.round_polys.len(), - }); - } - - let mut r = Vec::with_capacity(num_rounds); - for poly in &self.round_polys { - if poly.degree() > degree_bound { - return Err(HachiError::InvalidInput(format!( - "sumcheck round poly degree {} exceeds bound {}", - poly.degree(), - degree_bound - ))); - } - - transcript.append_serde(labels::ABSORB_SUMCHECK_ROUND, poly); - let r_i = sample_challenge(transcript); - r.push(r_i); - - claim = poly.eval_from_hint(&claim, &r_i); - } - - Ok((claim, r)) - } -} +pub use crate::algebra::poly::{fold_evals_in_place, multilinear_eval, range_check_eval}; +pub use types::{CompressedUniPoly, SumcheckProof, UniPoly}; /// Prover-side sumcheck instance interface. /// @@ -462,7 +78,7 @@ pub trait SumcheckInstanceVerifier: Send + Sync { fn expected_output_claim(&self, challenges: &[E]) -> E; } -/// Produce a sumcheck proof for a single instance, driving the Fiat–Shamir transcript. +/// Produce a sumcheck proof for a single instance, driving the Fiat-Shamir transcript. /// /// This method: /// - does **not** absorb the initial claim into the transcript (callers should do so), @@ -498,7 +114,6 @@ where for round in 0..num_rounds { let g = instance.compute_round_univariate(round, claim); - // Optional sanity: enforce hint relation in debug builds. debug_assert!( g.evaluate(&E::zero()) + g.evaluate(&E::one()) == claim, "sumcheck round univariate does not match previous claim hint" @@ -517,7 +132,6 @@ where let r_i = sample_challenge(transcript); r.push(r_i); - // Update running claim (this is g(r_i)). claim = compressed.eval_from_hint(&claim, &r_i); instance.ingest_challenge(round, r_i); @@ -574,42 +188,3 @@ where Ok(challenges) } - -/// Evaluate the range-check polynomial `w · Π_{k=1}^{b−1} (w − k)(w + k)`. -/// -/// This polynomial vanishes exactly when `w ∈ {−(b−1), …, b−1}`. -/// Total degree in `w` is `2b − 1`. -pub fn range_check_eval(w: E, b: usize) -> E { - let mut acc = w; - for k in 1..b { - let k_e = E::from_u64(k as u64); - acc = acc * (w - k_e) * (w + k_e); - } - acc -} - -/// Evaluate a multilinear polynomial (given by boolean-hypercube evaluations in -/// little-endian bit order) at an arbitrary point via iterated folding. -pub fn multilinear_eval(evals: &[E], point: &[E]) -> E { - let mut current = evals.to_vec(); - for &r in point { - let half = current.len() / 2; - let mut next = Vec::with_capacity(half); - for i in 0..half { - next.push(current[2 * i] + r * (current[2 * i + 1] - current[2 * i])); - } - current = next; - } - current[0] -} - -/// Fold an evaluation table by binding its first variable to `r`, halving the -/// table size. -pub(crate) fn fold_evals(evals: &[E], r: E) -> Vec { - let half = evals.len() / 2; - let mut out = Vec::with_capacity(half); - for i in 0..half { - out.push(evals[2 * i] + r * (evals[2 * i + 1] - evals[2 * i])); - } - out -} diff --git a/src/protocol/sumcheck/norm_sumcheck.rs b/src/protocol/sumcheck/norm_sumcheck.rs index 81823406..83b14b9f 100644 --- a/src/protocol/sumcheck/norm_sumcheck.rs +++ b/src/protocol/sumcheck/norm_sumcheck.rs @@ -7,7 +7,7 @@ use super::eq_poly::EqPolynomial; use super::split_eq::GruenSplitEq; -use super::{fold_evals, multilinear_eval, range_check_eval}; +use super::{fold_evals_in_place, multilinear_eval, range_check_eval}; use super::{SumcheckInstanceProver, SumcheckInstanceVerifier, UniPoly}; #[cfg(feature = "parallel")] use crate::parallel::*; @@ -122,7 +122,7 @@ impl SumcheckInstanceProver for NormSumcheckProv fn ingest_challenge(&mut self, _round: usize, r: E) { self.split_eq.bind(r); - self.w_table = fold_evals(&self.w_table, r); + fold_evals_in_place(&mut self.w_table, r); } } @@ -184,13 +184,13 @@ mod tests { use crate::protocol::transcript::labels; use crate::protocol::{ prove_sumcheck, verify_sumcheck, Blake2bTranscript, CommitmentConfig, CommitmentScheme, - DefaultCommitmentConfig, HachiCommitmentScheme, Transcript, + HachiCommitmentScheme, SmallTestCommitmentConfig, Transcript, }; use crate::{FieldCore, FromSmallInt}; type F = Fp64<4294967197>; const D: usize = 8; - type Cfg = DefaultCommitmentConfig; + type Cfg = SmallTestCommitmentConfig; type Scheme = HachiCommitmentScheme<{ Cfg::D }, Cfg>; fn ring_with_small_coeff(value: u64) -> CyclotomicRing { @@ -206,13 +206,13 @@ mod tests { ring_with_small_coeff(3), ]; let r = vec![ring_with_small_coeff(0), ring_with_small_coeff(1)]; - let mut w_evals = build_w_coeffs::(&z, &r); + let mut w_evals = build_w_coeffs::(&z, &r); let target_len = w_evals.len().next_power_of_two(); w_evals.resize(target_len, F::zero()); let num_vars = target_len.trailing_zeros() as usize; let tau: Vec = (0..num_vars).map(|i| F::from_u64((i + 2) as u64)).collect(); - let b = 1usize << DefaultCommitmentConfig::LOG_BASIS; + let b = 1usize << SmallTestCommitmentConfig::LOG_BASIS; let eq_table = EqPolynomial::evals(&tau); let _claim: F = (0..w_evals.len()) @@ -244,8 +244,8 @@ mod tests { #[test] fn norm_sumcheck_uses_prove_w_evals() { - let alpha = DefaultCommitmentConfig::D.trailing_zeros() as usize; - let num_vars = DefaultCommitmentConfig::R + DefaultCommitmentConfig::M + alpha; + let alpha = SmallTestCommitmentConfig::D.trailing_zeros() as usize; + let num_vars = SmallTestCommitmentConfig::R + SmallTestCommitmentConfig::M + alpha; let len = 1usize << num_vars; let evals: Vec = (0..len).map(|i| F::from_u64(i as u64)).collect(); let poly = DenseMultilinearEvals::new_padded(evals); @@ -272,7 +272,7 @@ mod tests { let tau: Vec = (0..num_sumcheck_vars) .map(|i| F::from_u64((i + 3) as u64)) .collect(); - let b = 1usize << DefaultCommitmentConfig::LOG_BASIS; + let b = 1usize << SmallTestCommitmentConfig::LOG_BASIS; let eq_table = EqPolynomial::evals(&tau); let _claim: F = (0..w_evals.len()) diff --git a/src/protocol/sumcheck/relation_sumcheck.rs b/src/protocol/sumcheck/relation_sumcheck.rs index 1ba95cb9..4a112a52 100644 --- a/src/protocol/sumcheck/relation_sumcheck.rs +++ b/src/protocol/sumcheck/relation_sumcheck.rs @@ -6,7 +6,7 @@ //! Proves the evaluation relation; sum equals `a = Σ_i ẽq(τ₁,i) · y_i(α)`. use super::eq_poly::EqPolynomial; -use super::{fold_evals, multilinear_eval}; +use super::{fold_evals_in_place, multilinear_eval}; use super::{SumcheckInstanceProver, SumcheckInstanceVerifier, UniPoly}; use crate::algebra::ring::CyclotomicRing; use crate::cfg_into_iter; @@ -163,9 +163,9 @@ impl SumcheckInstanceProver for RelationSumcheck } fn ingest_challenge(&mut self, _round: usize, r: E) { - self.w_table = fold_evals(&self.w_table, r); - self.alpha_table = fold_evals(&self.alpha_table, r); - self.m_table = fold_evals(&self.m_table, r); + fold_evals_in_place(&mut self.w_table, r); + fold_evals_in_place(&mut self.alpha_table, r); + fold_evals_in_place(&mut self.m_table, r); } } @@ -266,18 +266,18 @@ mod tests { use crate::protocol::transcript::labels; use crate::protocol::{ prove_sumcheck, verify_sumcheck, Blake2bTranscript, CommitmentConfig, CommitmentScheme, - DefaultCommitmentConfig, HachiCommitmentScheme, Transcript, + HachiCommitmentScheme, SmallTestCommitmentConfig, Transcript, }; use crate::{FieldCore, FromSmallInt}; type F = Fp64<4294967197>; - type Cfg = DefaultCommitmentConfig; + type Cfg = SmallTestCommitmentConfig; type Scheme = HachiCommitmentScheme<{ Cfg::D }, Cfg>; #[test] fn relation_sumcheck_uses_prove_w_evals() { - let alpha_bits = DefaultCommitmentConfig::D.trailing_zeros() as usize; - let num_vars = DefaultCommitmentConfig::R + DefaultCommitmentConfig::M + alpha_bits; + let alpha_bits = SmallTestCommitmentConfig::D.trailing_zeros() as usize; + let num_vars = SmallTestCommitmentConfig::R + SmallTestCommitmentConfig::M + alpha_bits; let len = 1usize << num_vars; let evals: Vec = (0..len).map(|i| F::from_u64(i as u64)).collect(); let poly = DenseMultilinearEvals::new_padded(evals); @@ -305,14 +305,14 @@ mod tests { ) .unwrap(); - let d = DefaultCommitmentConfig::D; + let d = SmallTestCommitmentConfig::D; assert_eq!(proof.sumcheck_aux.w.len() % d, 0); let w_u = proof.sumcheck_aux.w.len() / d; - let rows = DefaultCommitmentConfig::N_D - + DefaultCommitmentConfig::N_B + let rows = SmallTestCommitmentConfig::N_D + + SmallTestCommitmentConfig::N_B + 1 + 1 - + DefaultCommitmentConfig::N_A; + + SmallTestCommitmentConfig::N_A; assert!(rows > 0); assert_eq!(m_a_vec.len() % rows, 0); let cols = m_a_vec.len() / rows; diff --git a/src/protocol/sumcheck/split_eq.rs b/src/protocol/sumcheck/split_eq.rs index 7c03db1c..9ecf7269 100644 --- a/src/protocol/sumcheck/split_eq.rs +++ b/src/protocol/sumcheck/split_eq.rs @@ -151,7 +151,7 @@ impl GruenSplitEq { mod tests { use super::*; use crate::algebra::Fp64; - use crate::protocol::sumcheck::fold_evals; + use crate::protocol::sumcheck::fold_evals_in_place; use crate::{FieldSampling, FromSmallInt}; use rand::rngs::StdRng; use rand::SeedableRng; @@ -186,7 +186,7 @@ mod tests { } let r = F::sample(&mut rng); - full_eq = fold_evals(&full_eq, r); + fold_evals_in_place(&mut full_eq, r); split_eq.bind(r); } } diff --git a/src/protocol/sumcheck/types.rs b/src/protocol/sumcheck/types.rs new file mode 100644 index 00000000..c1dd393f --- /dev/null +++ b/src/protocol/sumcheck/types.rs @@ -0,0 +1,384 @@ +//! Sumcheck data types: univariate polynomials, compressed representation, and proof container. + +use crate::error::HachiError; +use crate::primitives::serialization::{ + Compress, HachiDeserialize, HachiSerialize, SerializationError, Valid, Validate, +}; +use crate::protocol::transcript::labels; +use crate::protocol::transcript::Transcript; +use crate::FieldCore; +use crate::FromSmallInt; +use std::io::{Read, Write}; + +/// Univariate polynomial in coefficient form: `p(X) = Σ_{i=0}^d coeffs[i] * X^i`. +#[derive(Debug, Clone, PartialEq, Eq)] +pub struct UniPoly { + /// Coefficients from low degree to high degree. + pub coeffs: Vec, +} + +impl UniPoly { + /// Construct from coefficients in increasing-degree order. + pub fn from_coeffs(coeffs: Vec) -> Self { + Self { coeffs } + } + + /// Degree of the polynomial. + /// + /// # Panics + /// + /// Panics if the polynomial has no coefficients. + pub fn degree(&self) -> usize { + self.coeffs + .len() + .checked_sub(1) + .expect("UniPoly must have at least one coefficient") + } + + /// Evaluate at `x` via Horner's method. + pub fn evaluate(&self, x: &E) -> E { + let mut acc = E::zero(); + for c in self.coeffs.iter().rev() { + acc = acc * *x + *c; + } + acc + } + + /// Compress this polynomial by omitting the linear coefficient. + /// + /// The verifier can reconstruct/evaluate the missing linear coefficient using + /// the per-round hint `g(0)+g(1)` from the sumcheck protocol. + /// + /// This matches the technique used by Jolt's sumcheck (`CompressedUniPoly`). + pub fn compress(&self) -> CompressedUniPoly { + let coeffs = &self.coeffs; + if coeffs.is_empty() { + return CompressedUniPoly { + coeffs_except_linear_term: Vec::new(), + }; + } + if coeffs.len() == 1 { + return CompressedUniPoly { + coeffs_except_linear_term: vec![coeffs[0]], + }; + } + let mut out = Vec::with_capacity(coeffs.len().saturating_sub(1)); + out.push(coeffs[0]); + out.extend_from_slice(&coeffs[2..]); + CompressedUniPoly { + coeffs_except_linear_term: out, + } + } +} + +impl UniPoly { + /// Interpolate from evaluations at equispaced integer points `x = 0, 1, ..., d`. + /// + /// Uses Newton forward-difference interpolation: compute divided differences, + /// then expand via Horner on the nested Newton form. + /// + /// # Panics + /// + /// Panics if any required factorial inverse does not exist (field characteristic + /// must exceed the number of evaluation points). + pub fn from_evals(evals: &[E]) -> Self { + let n = evals.len(); + if n == 0 { + return Self::from_coeffs(vec![]); + } + if n == 1 { + return Self::from_coeffs(vec![evals[0]]); + } + + let mut table = evals.to_vec(); + let mut deltas = vec![table[0]]; + for _ in 1..n { + for j in 0..table.len() - 1 { + table[j] = table[j + 1] - table[j]; + } + table.pop(); + deltas.push(table[0]); + } + + let mut factorial = E::one(); + let mut divided_diffs = vec![deltas[0]]; + for (k, delta_k) in deltas.iter().enumerate().skip(1) { + factorial = factorial * E::from_u64(k as u64); + divided_diffs.push( + *delta_k + * factorial + .inv() + .expect("field characteristic too small for interpolation"), + ); + } + + let mut coeffs = vec![divided_diffs[n - 1]]; + + for k in (0..n - 1).rev() { + let shift = E::from_u64(k as u64); + let old_len = coeffs.len(); + let mut new_coeffs = vec![E::zero(); old_len + 1]; + + new_coeffs[0] = divided_diffs[k]; + for i in 0..old_len { + new_coeffs[i + 1] = new_coeffs[i + 1] + coeffs[i]; + new_coeffs[i] = new_coeffs[i] - shift * coeffs[i]; + } + + coeffs = new_coeffs; + } + + while coeffs.len() > 1 && coeffs.last().is_some_and(|c| c.is_zero()) { + coeffs.pop(); + } + + Self::from_coeffs(coeffs) + } +} + +impl Valid for UniPoly { + fn check(&self) -> Result<(), SerializationError> { + self.coeffs.check() + } +} + +impl HachiSerialize for UniPoly { + fn serialize_with_mode( + &self, + mut writer: W, + compress: Compress, + ) -> Result<(), SerializationError> { + self.coeffs.serialize_with_mode(&mut writer, compress) + } + + fn serialized_size(&self, compress: Compress) -> usize { + self.coeffs.serialized_size(compress) + } +} + +impl HachiDeserialize for UniPoly { + fn deserialize_with_mode( + mut reader: R, + compress: Compress, + validate: Validate, + ) -> Result { + let coeffs = Vec::::deserialize_with_mode(&mut reader, compress, validate)?; + let out = Self { coeffs }; + if matches!(validate, Validate::Yes) { + out.check()?; + } + Ok(out) + } +} + +/// Compressed univariate polynomial representation omitting the linear term. +/// +/// We store `[c0, c2, c3, ..., cd]`. Given the sumcheck hint `hint = g(0)+g(1)`, +/// the missing linear coefficient is: +/// +/// `c1 = hint - 2*c0 - Σ_{i=2..d} ci`. +#[derive(Debug, Clone, PartialEq, Eq)] +pub struct CompressedUniPoly { + /// Coefficients excluding the linear term: `[c0, c2, c3, ..., cd]`. + pub coeffs_except_linear_term: Vec, +} + +impl CompressedUniPoly { + /// Degree of the underlying uncompressed polynomial. + /// + /// `compress()` stores `[c0, c2, ..., cd]` — exactly `d` entries for + /// degree `d >= 2`. For `len <= 1` (degree 0 or 1, which are ambiguous + /// in compressed form) we report 0; this is conservative for the + /// verifier's degree-bound check since `degree_bound >= 2` in practice. + pub fn degree(&self) -> usize { + let len = self.coeffs_except_linear_term.len(); + if len <= 1 { + 0 + } else { + len + } + } + + fn recover_linear_term(&self, hint: &E) -> E { + if self.coeffs_except_linear_term.is_empty() { + return E::zero(); + } + + let c0 = self.coeffs_except_linear_term[0]; + let mut linear = *hint - c0 - c0; + for c in self.coeffs_except_linear_term.iter().skip(1) { + linear = linear - *c; + } + linear + } + + /// Decompress using `hint = g(0)+g(1)`. + pub fn decompress(&self, hint: &E) -> UniPoly { + if self.coeffs_except_linear_term.is_empty() { + return UniPoly::from_coeffs(Vec::new()); + } + let linear = self.recover_linear_term(hint); + let mut coeffs = Vec::with_capacity(self.coeffs_except_linear_term.len() + 1); + coeffs.push(self.coeffs_except_linear_term[0]); + coeffs.push(linear); + coeffs.extend_from_slice(&self.coeffs_except_linear_term[1..]); + UniPoly::from_coeffs(coeffs) + } + + /// Evaluate the uncompressed polynomial at `x`, using `hint = g(0)+g(1)`. + /// + /// This avoids materializing the full coefficient list. + pub fn eval_from_hint(&self, hint: &E, x: &E) -> E { + if self.coeffs_except_linear_term.is_empty() { + return E::zero(); + } + + let linear = self.recover_linear_term(hint); + let mut acc = self.coeffs_except_linear_term[0] + (*x * linear); + + let mut pow = *x * *x; + for c in self.coeffs_except_linear_term.iter().skip(1) { + acc = acc + (*c * pow); + pow = pow * *x; + } + acc + } +} + +impl Valid for CompressedUniPoly { + fn check(&self) -> Result<(), SerializationError> { + self.coeffs_except_linear_term.check() + } +} + +impl HachiSerialize for CompressedUniPoly { + fn serialize_with_mode( + &self, + mut writer: W, + compress: Compress, + ) -> Result<(), SerializationError> { + self.coeffs_except_linear_term + .serialize_with_mode(&mut writer, compress) + } + + fn serialized_size(&self, compress: Compress) -> usize { + self.coeffs_except_linear_term.serialized_size(compress) + } +} + +impl HachiDeserialize for CompressedUniPoly { + fn deserialize_with_mode( + mut reader: R, + compress: Compress, + validate: Validate, + ) -> Result { + let coeffs_except_linear_term = + Vec::::deserialize_with_mode(&mut reader, compress, validate)?; + let out = Self { + coeffs_except_linear_term, + }; + if matches!(validate, Validate::Yes) { + out.check()?; + } + Ok(out) + } +} + +/// Sumcheck proof containing one compressed univariate polynomial per round. +#[derive(Debug, Clone, PartialEq, Eq)] +pub struct SumcheckProof { + /// One compressed univariate polynomial per sumcheck round. + pub round_polys: Vec>, +} + +impl Valid for SumcheckProof { + fn check(&self) -> Result<(), SerializationError> { + self.round_polys.check() + } +} + +impl HachiSerialize for SumcheckProof { + fn serialize_with_mode( + &self, + mut writer: W, + compress: Compress, + ) -> Result<(), SerializationError> { + self.round_polys.serialize_with_mode(&mut writer, compress) + } + + fn serialized_size(&self, compress: Compress) -> usize { + self.round_polys.serialized_size(compress) + } +} + +impl HachiDeserialize for SumcheckProof { + fn deserialize_with_mode( + mut reader: R, + compress: Compress, + validate: Validate, + ) -> Result { + let round_polys = + Vec::>::deserialize_with_mode(&mut reader, compress, validate)?; + let out = Self { round_polys }; + if matches!(validate, Validate::Yes) { + out.check()?; + } + Ok(out) + } +} + +impl SumcheckProof { + /// Verifier-side sumcheck transcript driver. + /// + /// This method: + /// - absorbs the per-round prover message (compressed univariate), + /// - samples one challenge per round via `sample_challenge`, + /// - updates the running claim using `eval_from_hint`. + /// + /// It does **not** perform the final oracle check `final_claim == f(r*)`. + /// Callers (e.g. ring-switching) must compute `f(r*)` themselves and compare. + /// + /// # Errors + /// + /// Returns an error if the proof length does not match `num_rounds` or if any + /// per-round polynomial exceeds `degree_bound`. + pub fn verify( + &self, + mut claim: E, + num_rounds: usize, + degree_bound: usize, + transcript: &mut T, + mut sample_challenge: S, + ) -> Result<(E, Vec), HachiError> + where + F: crate::FieldCore + crate::CanonicalField, + T: Transcript, + S: FnMut(&mut T) -> E, + { + if self.round_polys.len() != num_rounds { + return Err(HachiError::InvalidSize { + expected: num_rounds, + actual: self.round_polys.len(), + }); + } + + let mut r = Vec::with_capacity(num_rounds); + for poly in &self.round_polys { + if poly.degree() > degree_bound { + return Err(HachiError::InvalidInput(format!( + "sumcheck round poly degree {} exceeds bound {}", + poly.degree(), + degree_bound + ))); + } + + transcript.append_serde(labels::ABSORB_SUMCHECK_ROUND, poly); + let r_i = sample_challenge(transcript); + r.push(r_i); + + claim = poly.eval_from_hint(&claim, &r_i); + } + + Ok((claim, r)) + } +} diff --git a/tests/onehot_commitment.rs b/tests/onehot_commitment.rs index 62f3b32d..0512203a 100644 --- a/tests/onehot_commitment.rs +++ b/tests/onehot_commitment.rs @@ -21,7 +21,7 @@ fn assert_onehot_matches_dense(onehot_k: usize, indices: &[usize]) { let setup = psetup(); // Optimized sparse path. - let (c_sparse, s_sparse, t_sparse) = + let w_sparse = >::commit_onehot(onehot_k, indices, &setup) .unwrap(); @@ -39,14 +39,20 @@ fn assert_onehot_matches_dense(onehot_k: usize, indices: &[usize]) { hachi_pcs::algebra::CyclotomicRing::from_coefficients(coeffs) }) .collect(); - let (c_dense, s_dense, t_dense) = + let w_dense = >::commit_coeffs(&ring_coeffs, &setup) .unwrap(); - assert_eq!(c_sparse, c_dense, "commitments must match"); - assert_eq!(s_sparse, s_dense, "s_all (decomposed witness) must match"); assert_eq!( - t_sparse, t_dense, + w_sparse.commitment, w_dense.commitment, + "commitments must match" + ); + assert_eq!( + w_sparse.s, w_dense.s, + "s_all (decomposed witness) must match" + ); + assert_eq!( + w_sparse.t_hat, w_dense.t_hat, "t_hat_all (decomposed inner output) must match" ); } diff --git a/tests/ring_commitment_core.rs b/tests/ring_commitment_core.rs index 2fa20bee..ccc422e7 100644 --- a/tests/ring_commitment_core.rs +++ b/tests/ring_commitment_core.rs @@ -3,7 +3,7 @@ use hachi_pcs::algebra::CyclotomicRing; use hachi_pcs::error::HachiError; use hachi_pcs::protocol::commitment::{ - CommitmentConfig, DefaultCommitmentConfig, HachiCommitmentCore, RingCommitmentScheme, + CommitmentConfig, HachiCommitmentCore, RingCommitmentScheme, SmallTestCommitmentConfig, }; use hachi_pcs::test_utils::*; @@ -67,28 +67,30 @@ fn commit_is_deterministic_and_shape_consistent() { >::setup(16).unwrap(); let blocks = sample_blocks(); - let (c1, s1, t1) = - >::commit_ring_blocks( - &blocks, &psetup, - ) - .unwrap(); - let (c2, s2, t2) = - >::commit_ring_blocks( - &blocks, &psetup, - ) - .unwrap(); - - assert_eq!(c1, c2); - assert_eq!(s1, s2); - assert_eq!(t1, t2); + let w1 = >::commit_ring_blocks( + &blocks, &psetup, + ) + .unwrap(); + let w2 = >::commit_ring_blocks( + &blocks, &psetup, + ) + .unwrap(); + + assert_eq!(w1.commitment, w2.commitment); + assert_eq!(w1.s, w2.s); + assert_eq!(w1.t_hat, w2.t_hat); let num_blocks = 1usize << TinyConfig::R; let block_len = 1usize << TinyConfig::M; - assert_eq!(c1.u.len(), TinyConfig::N_B); - assert_eq!(s1.len(), num_blocks); - assert_eq!(t1.len(), num_blocks); - assert!(s1.iter().all(|s| s.len() == block_len * TinyConfig::DELTA)); - assert!(t1 + assert_eq!(w1.commitment.u.len(), TinyConfig::N_B); + assert_eq!(w1.s.len(), num_blocks); + assert_eq!(w1.t_hat.len(), num_blocks); + assert!(w1 + .s + .iter() + .all(|s| s.len() == block_len * TinyConfig::DELTA)); + assert!(w1 + .t_hat .iter() .all(|t| t.len() == TinyConfig::N_A * TinyConfig::DELTA)); } @@ -99,11 +101,10 @@ fn commit_ring_coeffs_matches_block_commitment() { >::setup(16).unwrap(); let blocks = sample_blocks(); - let (c_blocks, s_blocks, t_blocks) = - >::commit_ring_blocks( - &blocks, &psetup, - ) - .unwrap(); + let wb = >::commit_ring_blocks( + &blocks, &psetup, + ) + .unwrap(); let num_blocks = 1usize << TinyConfig::R; let block_len = 1usize << TinyConfig::M; @@ -114,15 +115,14 @@ fn commit_ring_coeffs_matches_block_commitment() { } } - let (c_coeffs, s_coeffs, t_coeffs) = - >::commit_coeffs( - &f_coeffs, &psetup, - ) - .unwrap(); + let wc = >::commit_coeffs( + &f_coeffs, &psetup, + ) + .unwrap(); - assert_eq!(c_blocks, c_coeffs); - assert_eq!(s_blocks, s_coeffs); - assert_eq!(t_blocks, t_coeffs); + assert_eq!(wb.commitment, wc.commitment); + assert_eq!(wb.s, wc.s); + assert_eq!(wb.t_hat, wc.t_hat); } #[test] @@ -130,36 +130,38 @@ fn opening_satisfies_inner_and_outer_equations() { let (psetup, _) = >::setup(16).unwrap(); let blocks = sample_blocks(); - let (commitment, s, t_hat) = - >::commit_ring_blocks( - &blocks, &psetup, - ) - .unwrap(); - - for i in 0..s.len() { - let lhs = mat_vec_mul(&psetup.A, &s[i]); + let w = >::commit_ring_blocks( + &blocks, &psetup, + ) + .unwrap(); + + for i in 0..w.s.len() { + let lhs = mat_vec_mul(&psetup.A, &w.s[i]); let rhs: Vec> = (0..TinyConfig::N_A) .map(|j| { let start = j * TinyConfig::DELTA; let end = start + TinyConfig::DELTA; - CyclotomicRing::gadget_recompose_pow2(&t_hat[i][start..end], TinyConfig::LOG_BASIS) + CyclotomicRing::gadget_recompose_pow2( + &w.t_hat[i][start..end], + TinyConfig::LOG_BASIS, + ) }) .collect(); assert_eq!(lhs, rhs); } let t_hat_flat: Vec> = - t_hat.iter().flat_map(|x| x.iter().copied()).collect(); + w.t_hat.iter().flat_map(|x| x.iter().copied()).collect(); let outer = mat_vec_mul(&psetup.B, &t_hat_flat); - assert_eq!(outer, commitment.u); + assert_eq!(outer, w.commitment.u); } #[test] -fn default_config_has_expected_shape() { - assert_eq!(DefaultCommitmentConfig::D, 16); - assert_eq!(1usize << DefaultCommitmentConfig::M, 16); - assert_eq!(1usize << DefaultCommitmentConfig::R, 4); - let delta = DefaultCommitmentConfig::DELTA; +fn small_test_config_has_expected_shape() { + assert_eq!(SmallTestCommitmentConfig::D, 16); + assert_eq!(1usize << SmallTestCommitmentConfig::M, 16); + assert_eq!(1usize << SmallTestCommitmentConfig::R, 4); + let delta = SmallTestCommitmentConfig::DELTA; assert!(delta > 0); } diff --git a/tests/sumcheck_core.rs b/tests/sumcheck_core.rs index 35f07c1b..52244e91 100644 --- a/tests/sumcheck_core.rs +++ b/tests/sumcheck_core.rs @@ -2,6 +2,7 @@ use std::time::Instant; +use hachi_pcs::algebra::poly::multilinear_eval; use hachi_pcs::algebra::Fp64; use hachi_pcs::protocol::transcript::labels; use hachi_pcs::protocol::{ @@ -86,21 +87,6 @@ fn sumcheck_proof_verifier_driver_is_transcript_deterministic() { assert_eq!(final_claim_1, claim); } -/// Evaluate a multilinear polynomial (given by its boolean-hypercube evaluations -/// in little-endian bit order) at an arbitrary point via iterated folding. -fn multilinear_eval(evals: &[E], point: &[E]) -> E { - let mut current = evals.to_vec(); - for r in point { - let half = current.len() / 2; - let mut next = Vec::with_capacity(half); - for i in 0..half { - next.push(current[2 * i] + *r * (current[2 * i + 1] - current[2 * i])); - } - current = next; - } - current[0] -} - struct DenseSumcheckProver { evals: Vec, num_vars: usize, From 9cd06cf02c2555a07405a50f341305c3c9d2deac Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Sat, 28 Feb 2026 14:58:44 -0800 Subject: [PATCH 12/34] fix(test): resolve clippy needless_range_loop in algebra tests Use iter().enumerate() for schoolbook convolution loops and array::from_fn for pointwise NTT operations. Made-with: Cursor --- tests/algebra.rs | 28 +++++++++++----------------- 1 file changed, 11 insertions(+), 17 deletions(-) diff --git a/tests/algebra.rs b/tests/algebra.rs index 2f8df3e1..2f5fc57d 100644 --- a/tests/algebra.rs +++ b/tests/algebra.rs @@ -715,9 +715,9 @@ mod tests { // Schoolbook negacyclic convolution mod p: X^D = -1. let mut school = [0i16; D]; - for i in 0..D { - for j in 0..D { - let prod = (a_canon[i] as i64 * b_canon[j] as i64) % (prime.p as i64); + for (i, &ai) in a_canon.iter().enumerate() { + for (j, &bj) in b_canon.iter().enumerate() { + let prod = (ai as i64 * bj as i64) % (prime.p as i64); let idx = i + j; if idx < D { school[idx] = ((school[idx] as i64 + prod) % (prime.p as i64)) as i16; @@ -738,10 +738,7 @@ mod tests { forward_ntt(&mut a, prime, &tw); forward_ntt(&mut b, prime, &tw); - let mut c = [MontCoeff::from_raw(0i16); D]; - for i in 0..D { - c[i] = prime.mul(a[i], b[i]); - } + let mut c: [_; D] = std::array::from_fn(|i| prime.mul(a[i], b[i])); inverse_ntt(&mut c, prime, &tw); let got: [i16; D] = std::array::from_fn(|i| prime.to_canonical(prime.normalize(c[i]))); @@ -819,9 +816,9 @@ mod tests { let b_canon: [i16; D] = std::array::from_fn(|i| ((i as i16 * 5) + 11) % prime.p); let mut school = [0i16; D]; - for i in 0..D { - for j in 0..D { - let prod = (a_canon[i] as i64 * b_canon[j] as i64) % p; + for (i, &ai) in a_canon.iter().enumerate() { + for (j, &bj) in b_canon.iter().enumerate() { + let prod = (ai as i64 * bj as i64) % p; let idx = i + j; if idx < D { school[idx] = ((school[idx] as i64 + prod) % p) as i16; @@ -842,10 +839,7 @@ mod tests { forward_ntt(&mut a, prime, &tw); forward_ntt(&mut b, prime, &tw); - let mut c = [MontCoeff::from_raw(0i16); D]; - for i in 0..D { - c[i] = prime.reduce_range(prime.mul(a[i], b[i])); - } + let mut c: [_; D] = std::array::from_fn(|i| prime.reduce_range(prime.mul(a[i], b[i]))); inverse_ntt(&mut c, prime, &tw); let got: [i16; D] = std::array::from_fn(|i| prime.to_canonical(prime.normalize(c[i]))); @@ -868,9 +862,9 @@ mod tests { std::array::from_fn(|i| ((b_canon[i] as i64).rem_euclid(p)) as i16); let mut school = [0i16; D]; - for i in 0..D { - for j in 0..D { - let prod = (a_mod[i] as i64 * b_mod[j] as i64) % p; + for (i, &ai) in a_mod.iter().enumerate() { + for (j, &bj) in b_mod.iter().enumerate() { + let prod = (ai as i64 * bj as i64) % p; let idx = i + j; if idx < D { school[idx] = ((school[idx] as i64 + prod) % p) as i16; From dba56b2ff2498cc33d960b001f338e980cbfc7c4 Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Sat, 28 Feb 2026 15:39:10 -0800 Subject: [PATCH 13/34] Refactor commitment setup to runtime layout and staged artifacts. This removes compile-time commitment shape locks, derives beta from runtime layout, and threads layout-aware setup through commit/prove/verify with setup serialization roundtrip coverage. Made-with: Cursor --- benches/hachi_e2e.rs | 13 +- src/protocol/commitment/commit.rs | 306 +++++++++++++++++++-- src/protocol/commitment/config.rs | 255 ++++++++++++----- src/protocol/commitment/mod.rs | 10 +- src/protocol/commitment/scheme.rs | 17 +- src/protocol/commitment_scheme.rs | 45 ++- src/protocol/mod.rs | 7 +- src/protocol/quadratic_equation.rs | 58 ++-- src/protocol/ring_switch.rs | 20 +- src/protocol/sumcheck/norm_sumcheck.rs | 247 +++++++++++++++-- src/protocol/sumcheck/relation_sumcheck.rs | 5 +- src/test_utils.rs | 13 +- tests/ring_commitment_core.rs | 53 ++-- 13 files changed, 835 insertions(+), 214 deletions(-) diff --git a/benches/hachi_e2e.rs b/benches/hachi_e2e.rs index cc92ecc8..4a48aaac 100644 --- a/benches/hachi_e2e.rs +++ b/benches/hachi_e2e.rs @@ -20,16 +20,20 @@ macro_rules! bench_config { struct $name; impl CommitmentConfig for $name { const D: usize = D; - const M: usize = $m; - const R: usize = $r; const N_A: usize = ProductionFp128CommitmentConfig::N_A; const N_B: usize = ProductionFp128CommitmentConfig::N_B; const N_D: usize = ProductionFp128CommitmentConfig::N_D; const LOG_BASIS: u32 = ProductionFp128CommitmentConfig::LOG_BASIS; const DELTA: usize = ProductionFp128CommitmentConfig::DELTA; const TAU: usize = ProductionFp128CommitmentConfig::TAU; - const BETA: u128 = ProductionFp128CommitmentConfig::BETA; const CHALLENGE_WEIGHT: usize = ProductionFp128CommitmentConfig::CHALLENGE_WEIGHT; + + fn commitment_layout( + _max_num_vars: usize, + ) -> Result + { + hachi_pcs::protocol::commitment::HachiCommitmentLayout::new::($m, $r) + } } }; } @@ -41,7 +45,8 @@ bench_config!(CfgNv20, M = 8, R = 8); fn num_vars() -> usize { let alpha = Cfg::D.trailing_zeros() as usize; - Cfg::R + Cfg::M + alpha + let layout = Cfg::commitment_layout(0).expect("benchmark layout"); + layout.m_vars + layout.r_vars + alpha } fn make_poly(nv: usize) -> DenseMultilinearEvals { diff --git a/src/protocol/commitment/commit.rs b/src/protocol/commitment/commit.rs index 10619a0b..5fa2c566 100644 --- a/src/protocol/commitment/commit.rs +++ b/src/protocol/commitment/commit.rs @@ -2,6 +2,7 @@ use super::config::{ ensure_block_layout, ensure_matrix_shape, ensure_supported_num_vars, validate_and_derive_layout, + HachiCommitmentLayout, }; use super::onehot::{inner_ajtai_onehot, map_onehot_to_sparse_blocks}; use super::scheme::{CommitWitness, RingCommitmentScheme}; @@ -15,26 +16,238 @@ use super::utils::matrix::{derive_public_matrix, sample_public_matrix_seed, Publ use super::CommitmentConfig; use crate::algebra::ring::CyclotomicRing; use crate::error::HachiError; +use crate::primitives::serialization::{ + Compress, HachiDeserialize, HachiSerialize, SerializationError, Valid, Validate, +}; use crate::{CanonicalField, FieldCore, FieldSampling}; +use std::io::{Read, Write}; -/// Unified setup for the ring-native commitment (§4.1) and prover (§4.2). +/// Seed-only stage for deterministic setup expansion. #[allow(non_snake_case)] #[derive(Debug, Clone, PartialEq, Eq)] -pub struct RingCommitmentSetup { +pub struct HachiSetupSeed { /// Maximum supported variable count. pub max_num_vars: usize, + /// Runtime commitment layout. + pub layout: HachiCommitmentLayout, /// Public seed used to derive commitment matrices. pub public_matrix_seed: PublicMatrixSeed, +} + +/// Expanded setup stage containing coefficient-form matrices. +#[allow(non_snake_case)] +#[derive(Debug, Clone, PartialEq, Eq)] +pub struct HachiExpandedSetup { + /// Setup seed and runtime layout metadata. + pub seed: HachiSetupSeed, /// Inner matrix `A`. pub A: Vec>>, /// Outer matrix `B`. pub B: Vec>>, /// Prover matrix `D ∈ R_q^{n_D × δ·2^R}` (§4.2). pub D: Vec>>, +} + +/// Optional prepared setup stage for accelerated matrix-vector products. +#[derive(Debug, Clone, PartialEq, Eq)] +pub struct HachiPreparedSetup { /// Pre-converted CRT+NTT matrices for dense mat-vec paths. pub(crate) ntt_cache: NttMatrixCache, } +/// Prover setup artifact (expanded setup + optional runtime cache). +#[allow(non_snake_case)] +#[derive(Debug, Clone, PartialEq, Eq)] +pub struct HachiProverSetup { + /// Expanded matrix stage used by both prover and verifier. + pub expanded: HachiExpandedSetup, + /// Optional runtime-prepared acceleration cache. + pub prepared: Option>, +} + +/// Verifier setup artifact derived from prover setup. +#[allow(non_snake_case)] +#[derive(Debug, Clone, PartialEq, Eq)] +pub struct HachiVerifierSetup { + /// Expanded matrix stage used for verification. + pub expanded: HachiExpandedSetup, +} + +impl HachiProverSetup { + /// Runtime layout carried by this setup. + pub fn layout(&self) -> HachiCommitmentLayout { + self.expanded.seed.layout + } + + pub(crate) fn ntt_cache(&self) -> Result<&NttMatrixCache, HachiError> { + self.prepared + .as_ref() + .map(|p| &p.ntt_cache) + .ok_or_else(|| HachiError::InvalidSetup("missing prepared NTT cache".to_string())) + } +} + +impl Valid for HachiSetupSeed { + fn check(&self) -> Result<(), SerializationError> { + self.layout.check() + } +} + +impl HachiSerialize for HachiSetupSeed { + fn serialize_with_mode( + &self, + mut writer: W, + compress: Compress, + ) -> Result<(), SerializationError> { + self.max_num_vars.serialize_with_mode(&mut writer, compress)?; + self.layout.serialize_with_mode(&mut writer, compress)?; + writer.write_all(&self.public_matrix_seed)?; + Ok(()) + } + + fn serialized_size(&self, compress: Compress) -> usize { + self.max_num_vars.serialized_size(compress) + self.layout.serialized_size(compress) + 32 + } +} + +impl HachiDeserialize for HachiSetupSeed { + fn deserialize_with_mode( + mut reader: R, + compress: Compress, + validate: Validate, + ) -> Result { + let max_num_vars = usize::deserialize_with_mode(&mut reader, compress, validate)?; + let layout = HachiCommitmentLayout::deserialize_with_mode(&mut reader, compress, validate)?; + let mut public_matrix_seed = [0u8; 32]; + reader.read_exact(&mut public_matrix_seed)?; + let out = Self { + max_num_vars, + layout, + public_matrix_seed, + }; + if matches!(validate, Validate::Yes) { + out.check()?; + } + Ok(out) + } +} + +impl Valid for HachiExpandedSetup { + fn check(&self) -> Result<(), SerializationError> { + self.seed.check()?; + self.A.check()?; + self.B.check()?; + self.D.check()?; + Ok(()) + } +} + +impl HachiSerialize for HachiExpandedSetup { + fn serialize_with_mode( + &self, + mut writer: W, + compress: Compress, + ) -> Result<(), SerializationError> { + self.seed.serialize_with_mode(&mut writer, compress)?; + self.A.serialize_with_mode(&mut writer, compress)?; + self.B.serialize_with_mode(&mut writer, compress)?; + self.D.serialize_with_mode(&mut writer, compress)?; + Ok(()) + } + + fn serialized_size(&self, compress: Compress) -> usize { + self.seed.serialized_size(compress) + + self.A.serialized_size(compress) + + self.B.serialized_size(compress) + + self.D.serialized_size(compress) + } +} + +impl HachiDeserialize for HachiExpandedSetup { + fn deserialize_with_mode( + mut reader: R, + compress: Compress, + validate: Validate, + ) -> Result { + let out = Self { + seed: HachiSetupSeed::deserialize_with_mode(&mut reader, compress, validate)?, + A: Vec::>>::deserialize_with_mode(&mut reader, compress, validate)?, + B: Vec::>>::deserialize_with_mode(&mut reader, compress, validate)?, + D: Vec::>>::deserialize_with_mode(&mut reader, compress, validate)?, + }; + if matches!(validate, Validate::Yes) { + out.check()?; + } + Ok(out) + } +} + +impl Valid for HachiProverSetup { + fn check(&self) -> Result<(), SerializationError> { + self.expanded.check() + } +} + +impl HachiSerialize for HachiProverSetup { + fn serialize_with_mode( + &self, + writer: W, + compress: Compress, + ) -> Result<(), SerializationError> { + // Prepared cache is runtime-only and intentionally excluded. + self.expanded.serialize_with_mode(writer, compress) + } + + fn serialized_size(&self, compress: Compress) -> usize { + self.expanded.serialized_size(compress) + } +} + +impl HachiDeserialize for HachiProverSetup { + fn deserialize_with_mode( + reader: R, + compress: Compress, + validate: Validate, + ) -> Result { + Ok(Self { + expanded: HachiExpandedSetup::deserialize_with_mode(reader, compress, validate)?, + prepared: None, + }) + } +} + +impl Valid for HachiVerifierSetup { + fn check(&self) -> Result<(), SerializationError> { + self.expanded.check() + } +} + +impl HachiSerialize for HachiVerifierSetup { + fn serialize_with_mode( + &self, + writer: W, + compress: Compress, + ) -> Result<(), SerializationError> { + self.expanded.serialize_with_mode(writer, compress) + } + + fn serialized_size(&self, compress: Compress) -> usize { + self.expanded.serialized_size(compress) + } +} + +impl HachiDeserialize for HachiVerifierSetup { + fn deserialize_with_mode( + reader: R, + compress: Compress, + validate: Validate, + ) -> Result { + Ok(Self { + expanded: HachiExpandedSetup::deserialize_with_mode(reader, compress, validate)?, + }) + } +} + /// Concrete §4.1 commitment core. #[derive(Clone, Copy, Default)] pub struct HachiCommitmentCore; @@ -44,13 +257,13 @@ where F: FieldCore + CanonicalField + FieldSampling, Cfg: CommitmentConfig, { - type ProverSetup = RingCommitmentSetup; - type VerifierSetup = RingCommitmentSetup; + type ProverSetup = HachiProverSetup; + type VerifierSetup = HachiVerifierSetup; type Commitment = RingCommitment; fn setup(max_num_vars: usize) -> Result<(Self::ProverSetup, Self::VerifierSetup), HachiError> { - let layout = validate_and_derive_layout::()?; - ensure_supported_num_vars(max_num_vars, layout.required_vars)?; + let layout = validate_and_derive_layout::(max_num_vars)?; + ensure_supported_num_vars(max_num_vars, layout.required_num_vars::()?)?; let public_matrix_seed = sample_public_matrix_seed(); let a_matrix = derive_public_matrix::(Cfg::N_A, layout.inner_width, &public_matrix_seed, b"A"); @@ -64,37 +277,47 @@ where ); let ntt_cache = build_ntt_cache::(&a_matrix, &b_matrix, &d_matrix)?; - - let setup = RingCommitmentSetup { - max_num_vars, - public_matrix_seed, + let expanded = HachiExpandedSetup { + seed: HachiSetupSeed { + max_num_vars, + layout, + public_matrix_seed, + }, A: a_matrix, B: b_matrix, D: d_matrix, - ntt_cache, }; - ensure_matrix_shape(&setup.A, Cfg::N_A, layout.inner_width, "A")?; - ensure_matrix_shape(&setup.B, Cfg::N_B, layout.outer_width, "B")?; - ensure_matrix_shape(&setup.D, Cfg::N_D, layout.d_matrix_width, "D")?; - Ok((setup.clone(), setup)) + let prover_setup = HachiProverSetup { + expanded: expanded.clone(), + prepared: Some(HachiPreparedSetup { ntt_cache }), + }; + let verifier_setup = HachiVerifierSetup { expanded }; + ensure_matrix_shape(&prover_setup.expanded.A, Cfg::N_A, layout.inner_width, "A")?; + ensure_matrix_shape(&prover_setup.expanded.B, Cfg::N_B, layout.outer_width, "B")?; + ensure_matrix_shape(&prover_setup.expanded.D, Cfg::N_D, layout.d_matrix_width, "D")?; + Ok((prover_setup, verifier_setup)) + } + + fn layout(setup: &Self::ProverSetup) -> Result { + Ok(setup.layout()) } fn commit_ring_blocks( f_blocks: &[Vec>], setup: &Self::ProverSetup, ) -> Result, HachiError> { - let layout = validate_and_derive_layout::()?; - ensure_supported_num_vars(setup.max_num_vars, layout.required_vars)?; + let layout = setup.layout(); + ensure_supported_num_vars(setup.expanded.seed.max_num_vars, layout.required_num_vars::()?)?; ensure_block_layout(f_blocks, layout)?; - ensure_matrix_shape(&setup.A, Cfg::N_A, layout.inner_width, "A")?; - ensure_matrix_shape(&setup.B, Cfg::N_B, layout.outer_width, "B")?; + ensure_matrix_shape(&setup.expanded.A, Cfg::N_A, layout.inner_width, "A")?; + ensure_matrix_shape(&setup.expanded.B, Cfg::N_B, layout.outer_width, "B")?; let s_all: Vec>> = f_blocks .iter() .map(|block| decompose_block(block, Cfg::DELTA, Cfg::LOG_BASIS)) .collect(); - let t_all = mat_vec_mul_ntt_many_cached(&setup.ntt_cache, MatrixSlot::A, &s_all)?; + let t_all = mat_vec_mul_ntt_many_cached(setup.ntt_cache()?, MatrixSlot::A, &s_all)?; let mut t_hat_all: Vec>> = Vec::with_capacity(layout.num_blocks); let mut t_hat_flat: Vec> = Vec::with_capacity(layout.outer_width); for t_i in t_all { @@ -103,7 +326,7 @@ where t_hat_all.push(t_hat_i); } - let u = mat_vec_mul_ntt_cached(&setup.ntt_cache, MatrixSlot::B, &t_hat_flat)?; + let u = mat_vec_mul_ntt_cached(setup.ntt_cache()?, MatrixSlot::B, &t_hat_flat)?; Ok(CommitWitness { commitment: RingCommitment { u }, s: s_all, @@ -116,12 +339,13 @@ where indices: &[usize], setup: &Self::ProverSetup, ) -> Result, HachiError> { - let layout = validate_and_derive_layout::()?; - ensure_supported_num_vars(setup.max_num_vars, layout.required_vars)?; - ensure_matrix_shape(&setup.A, Cfg::N_A, layout.inner_width, "A")?; - ensure_matrix_shape(&setup.B, Cfg::N_B, layout.outer_width, "B")?; + let layout = setup.layout(); + ensure_supported_num_vars(setup.expanded.seed.max_num_vars, layout.required_num_vars::()?)?; + ensure_matrix_shape(&setup.expanded.A, Cfg::N_A, layout.inner_width, "A")?; + ensure_matrix_shape(&setup.expanded.B, Cfg::N_B, layout.outer_width, "B")?; - let sparse_blocks = map_onehot_to_sparse_blocks(onehot_k, indices, Cfg::R, Cfg::M, D)?; + let sparse_blocks = + map_onehot_to_sparse_blocks(onehot_k, indices, layout.r_vars, layout.m_vars, D)?; let mut s_all: Vec>> = Vec::with_capacity(layout.num_blocks); let mut t_hat_all: Vec>> = Vec::with_capacity(layout.num_blocks); @@ -129,7 +353,7 @@ where for block_entries in &sparse_blocks { let (t_i, s_i) = - inner_ajtai_onehot(&setup.A, block_entries, layout.block_len, Cfg::DELTA); + inner_ajtai_onehot(&setup.expanded.A, block_entries, layout.block_len, Cfg::DELTA); let t_hat_i = decompose_rows(&t_i, Cfg::DELTA, Cfg::LOG_BASIS); t_hat_flat.extend(t_hat_i.iter().copied()); @@ -137,7 +361,7 @@ where t_hat_all.push(t_hat_i); } - let u = mat_vec_mul_ntt_cached(&setup.ntt_cache, MatrixSlot::B, &t_hat_flat)?; + let u = mat_vec_mul_ntt_cached(setup.ntt_cache()?, MatrixSlot::B, &t_hat_flat)?; Ok(CommitWitness { commitment: RingCommitment { u }, s: s_all, @@ -145,3 +369,29 @@ where }) } } + +#[cfg(test)] +mod tests { + use super::*; + use crate::primitives::{HachiDeserialize, HachiSerialize}; + use crate::test_utils::{D as TestD, F as TestF, TinyConfig}; + + #[test] + fn prover_setup_roundtrips_and_derives_same_verifier() { + let (prover_setup, verifier_setup) = + >::setup(16) + .unwrap(); + + let mut bytes = Vec::new(); + prover_setup.serialize_compressed(&mut bytes).unwrap(); + let decoded = HachiProverSetup::::deserialize_compressed(&bytes[..]).unwrap(); + + assert_eq!(decoded.expanded, prover_setup.expanded); + assert_eq!(decoded.prepared, None); + + let derived_verifier = HachiVerifierSetup { + expanded: decoded.expanded.clone(), + }; + assert_eq!(derived_verifier, verifier_setup); + } +} diff --git a/src/protocol/commitment/config.rs b/src/protocol/commitment/config.rs index d953b7c1..7e6694d7 100644 --- a/src/protocol/commitment/config.rs +++ b/src/protocol/commitment/config.rs @@ -3,16 +3,141 @@ use super::utils::math::checked_pow2; use crate::algebra::ring::CyclotomicRing; use crate::error::HachiError; +use crate::primitives::serialization::{ + Compress, HachiDeserialize, HachiSerialize, SerializationError, Valid, Validate, +}; use crate::FieldCore; +use std::io::{Read, Write}; + +/// Runtime commitment layout authority for ring-native commitments. +#[derive(Debug, Clone, Copy, PartialEq, Eq)] +pub struct HachiCommitmentLayout { + /// Number of variables inside each committed block (`2^m_vars` entries). + pub m_vars: usize, + /// Number of block-select variables (`2^r_vars` blocks). + pub r_vars: usize, + /// Number of committed blocks (`2^r_vars`). + pub num_blocks: usize, + /// Number of ring elements per block (`2^m_vars`). + pub block_len: usize, + /// Width of inner matrix `A`. + pub inner_width: usize, + /// Width of outer matrix `B`. + pub outer_width: usize, + /// Width of prover matrix `D` (`delta * 2^r_vars`). + pub d_matrix_width: usize, +} + +impl HachiCommitmentLayout { + /// Build a layout from `(m_vars, r_vars)` and static config constants. + /// + /// # Errors + /// + /// Returns an error when powers or derived widths overflow. + pub fn new(m_vars: usize, r_vars: usize) -> Result { + let num_blocks = checked_pow2(r_vars)?; + let block_len = checked_pow2(m_vars)?; + let inner_width = block_len + .checked_mul(Cfg::DELTA) + .ok_or_else(|| HachiError::InvalidSetup("inner width overflow".to_string()))?; + let outer_width = Cfg::N_A + .checked_mul(Cfg::DELTA) + .and_then(|x| x.checked_mul(num_blocks)) + .ok_or_else(|| HachiError::InvalidSetup("outer width overflow".to_string()))?; + let d_matrix_width = Cfg::DELTA + .checked_mul(num_blocks) + .ok_or_else(|| HachiError::InvalidSetup("D-matrix width overflow".to_string()))?; + Ok(Self { + m_vars, + r_vars, + num_blocks, + block_len, + inner_width, + outer_width, + d_matrix_width, + }) + } + + /// Total number of outer variables consumed by ring coefficients. + pub fn outer_vars(&self) -> Result { + self.m_vars + .checked_add(self.r_vars) + .ok_or_else(|| HachiError::InvalidSetup("variable count overflow".to_string())) + } + + /// Required polynomial variable count for this layout (`outer + alpha`). + pub fn required_num_vars(&self) -> Result { + let alpha = D.trailing_zeros() as usize; + self.outer_vars()? + .checked_add(alpha) + .ok_or_else(|| HachiError::InvalidSetup("variable count overflow".to_string())) + } +} + +impl Valid for HachiCommitmentLayout { + fn check(&self) -> Result<(), SerializationError> { + if self.num_blocks == 0 || self.block_len == 0 { + return Err(SerializationError::InvalidData( + "invalid zero block layout".to_string(), + )); + } + Ok(()) + } +} + +impl HachiSerialize for HachiCommitmentLayout { + fn serialize_with_mode( + &self, + mut writer: W, + compress: Compress, + ) -> Result<(), SerializationError> { + self.m_vars.serialize_with_mode(&mut writer, compress)?; + self.r_vars.serialize_with_mode(&mut writer, compress)?; + self.num_blocks.serialize_with_mode(&mut writer, compress)?; + self.block_len.serialize_with_mode(&mut writer, compress)?; + self.inner_width.serialize_with_mode(&mut writer, compress)?; + self.outer_width.serialize_with_mode(&mut writer, compress)?; + self.d_matrix_width.serialize_with_mode(&mut writer, compress)?; + Ok(()) + } + + fn serialized_size(&self, compress: Compress) -> usize { + self.m_vars.serialized_size(compress) + + self.r_vars.serialized_size(compress) + + self.num_blocks.serialized_size(compress) + + self.block_len.serialized_size(compress) + + self.inner_width.serialized_size(compress) + + self.outer_width.serialized_size(compress) + + self.d_matrix_width.serialized_size(compress) + } +} + +impl HachiDeserialize for HachiCommitmentLayout { + fn deserialize_with_mode( + mut reader: R, + compress: Compress, + validate: Validate, + ) -> Result { + let out = Self { + m_vars: usize::deserialize_with_mode(&mut reader, compress, validate)?, + r_vars: usize::deserialize_with_mode(&mut reader, compress, validate)?, + num_blocks: usize::deserialize_with_mode(&mut reader, compress, validate)?, + block_len: usize::deserialize_with_mode(&mut reader, compress, validate)?, + inner_width: usize::deserialize_with_mode(&mut reader, compress, validate)?, + outer_width: usize::deserialize_with_mode(&mut reader, compress, validate)?, + d_matrix_width: usize::deserialize_with_mode(&mut reader, compress, validate)?, + }; + if matches!(validate, Validate::Yes) { + out.check()?; + } + Ok(out) + } +} /// Parameter bundle for the ring-native commitment core (§4.1–§4.2). pub trait CommitmentConfig: Clone + Send + Sync + 'static { /// Ring degree used by `CyclotomicRing`. const D: usize; - /// Number of variables inside each committed block (`2^M` entries). - const M: usize; - /// Number of block-select variables (`2^R` blocks). - const R: usize; /// Inner Ajtai matrix row count. const N_A: usize; /// Outer commitment matrix row count. @@ -25,10 +150,24 @@ pub trait CommitmentConfig: Clone + Send + Sync + 'static { const DELTA: usize; /// Decomposition levels for the folded witness `z` (`τ` in the paper). const TAU: usize; - /// L∞ norm bound for `z` (`β` in the paper). Prover aborts if exceeded. - const BETA: u128; /// Hamming weight of sparse challenges (`ω` in the paper). const CHALLENGE_WEIGHT: usize; + + /// Choose the runtime commitment layout for `max_num_vars`. + /// + /// # Errors + /// + /// Returns an error if `max_num_vars` does not admit a valid layout. + fn commitment_layout(max_num_vars: usize) -> Result; + + /// Runtime L∞ bound for `z` (`β`) used by stage-1 folding checks. + /// + /// # Errors + /// + /// Returns an error on invalid parameters or arithmetic overflow. + fn beta_bound(layout: HachiCommitmentLayout) -> Result { + beta_linf_fold_bound(layout.r_vars, Self::CHALLENGE_WEIGHT, Self::LOG_BASIS) + } } /// Deterministic upper bound for the stage-1 folded-witness infinity norm. @@ -36,47 +175,31 @@ pub trait CommitmentConfig: Clone + Send + Sync + 'static { /// This encodes the bound used in `QuadraticEquation::compute_z_hat`: /// `||z||_inf <= 2^R * ω * (b/2)` where `b = 2^LOG_BASIS`. /// -/// # Panics +/// # Errors /// -/// Panics when `log_basis` or `r` are out of range, or when intermediate -/// products overflow `u128`. -pub(super) const fn beta_linf_fold_bound( +/// Returns an error when parameters are out of range or intermediate products +/// overflow `u128`. +pub(crate) fn beta_linf_fold_bound( r: usize, challenge_weight: usize, log_basis: u32, -) -> u128 { - assert!(log_basis > 0 && log_basis < 128, "invalid LOG_BASIS"); - assert!(r < 128, "R must be < 128"); +) -> Result { + if !(1..128).contains(&log_basis) { + return Err(HachiError::InvalidSetup("invalid LOG_BASIS".to_string())); + } + if r >= 128 { + return Err(HachiError::InvalidSetup("r_vars must be < 128".to_string())); + } let blocks = 1u128 << r; let b = 1u128 << log_basis; let half_b = b / 2; - let term = match blocks.checked_mul(challenge_weight as u128) { - Some(v) => v, - None => panic!("beta bound overflow (blocks * challenge_weight)"), - }; - match term.checked_mul(half_b) { - Some(v) => v, - None => panic!("beta bound overflow (term * half_b)"), - } -} - -/// Runtime-derived dimensions from a `CommitmentConfig`. -#[derive(Debug, Clone, Copy)] -pub(super) struct CommitmentLayout { - /// Number of committed blocks (`2^R`). - pub(super) num_blocks: usize, - /// Number of ring elements per block (`2^M`). - pub(super) block_len: usize, - /// Width of inner matrix `A`. - pub(super) inner_width: usize, - /// Width of outer matrix `B`. - pub(super) outer_width: usize, - /// Width of prover matrix `D` (`δ · 2^R`). - pub(super) d_matrix_width: usize, - /// Minimum variable count supported by this config. - pub(super) required_vars: usize, + let term = blocks + .checked_mul(challenge_weight as u128) + .ok_or_else(|| HachiError::InvalidSetup("beta bound overflow".to_string()))?; + term.checked_mul(half_b) + .ok_or_else(|| HachiError::InvalidSetup("beta bound overflow".to_string())) } /// Validate static config invariants and derive runtime dimensions. @@ -85,7 +208,8 @@ pub(super) struct CommitmentLayout { /// /// Returns an error when config constants are inconsistent or overflow. pub(super) fn validate_and_derive_layout( -) -> Result { + max_num_vars: usize, +) -> Result { if D != Cfg::D { return Err(HachiError::InvalidSetup(format!( "const D={D} mismatches config D={}", @@ -100,31 +224,7 @@ pub(super) fn validate_and_derive_layout( "DELTA * LOG_BASIS must be <= 128".to_string(), )); } - - let num_blocks = checked_pow2(Cfg::R)?; - let block_len = checked_pow2(Cfg::M)?; - let inner_width = block_len - .checked_mul(Cfg::DELTA) - .ok_or_else(|| HachiError::InvalidSetup("inner width overflow".to_string()))?; - let outer_width = Cfg::N_A - .checked_mul(Cfg::DELTA) - .and_then(|x| x.checked_mul(num_blocks)) - .ok_or_else(|| HachiError::InvalidSetup("outer width overflow".to_string()))?; - let d_matrix_width = Cfg::DELTA - .checked_mul(num_blocks) - .ok_or_else(|| HachiError::InvalidSetup("D-matrix width overflow".to_string()))?; - let required_vars = Cfg::M - .checked_add(Cfg::R) - .ok_or_else(|| HachiError::InvalidSetup("variable count overflow".to_string()))?; - - Ok(CommitmentLayout { - num_blocks, - block_len, - inner_width, - outer_width, - d_matrix_width, - required_vars, - }) + Cfg::commitment_layout(max_num_vars) } /// Ensure `max_num_vars` is sufficient for config dimensions. @@ -151,7 +251,7 @@ pub(super) fn ensure_supported_num_vars( /// Returns an error when block count or per-block size mismatch. pub(super) fn ensure_block_layout( f_blocks: &[Vec>], - layout: CommitmentLayout, + layout: HachiCommitmentLayout, ) -> Result<(), HachiError> { if f_blocks.len() != layout.num_blocks { return Err(HachiError::InvalidSize { @@ -204,16 +304,17 @@ pub struct SmallTestCommitmentConfig; impl CommitmentConfig for SmallTestCommitmentConfig { const D: usize = 16; - const M: usize = 4; - const R: usize = 2; const N_A: usize = 8; const N_B: usize = 4; const N_D: usize = 4; const LOG_BASIS: u32 = 4; const DELTA: usize = 9; const TAU: usize = 4; - const BETA: u128 = 1_000_000; const CHALLENGE_WEIGHT: usize = 3; + + fn commitment_layout(_max_num_vars: usize) -> Result { + HachiCommitmentLayout::new::(4, 2) + } } /// Production-oriented profile for 128-bit base fields (`Fp128

`). @@ -238,14 +339,26 @@ pub struct ProductionFp128CommitmentConfig; impl CommitmentConfig for ProductionFp128CommitmentConfig { const D: usize = 512; - const M: usize = 11; - const R: usize = 11; const N_A: usize = 1; const N_B: usize = 1; const N_D: usize = 1; const LOG_BASIS: u32 = 4; const DELTA: usize = 32; const TAU: usize = 5; - const BETA: u128 = beta_linf_fold_bound(Self::R, Self::CHALLENGE_WEIGHT, Self::LOG_BASIS); const CHALLENGE_WEIGHT: usize = 19; + + fn commitment_layout(max_num_vars: usize) -> Result { + let alpha = Self::D.trailing_zeros() as usize; + let reduced_vars = max_num_vars.checked_sub(alpha).ok_or_else(|| { + HachiError::InvalidSetup("max_num_vars is smaller than alpha".to_string()) + })?; + if reduced_vars == 0 { + return Err(HachiError::InvalidSetup( + "max_num_vars must leave at least one outer variable".to_string(), + )); + } + let m_vars = reduced_vars.min(11); + let r_vars = reduced_vars - m_vars; + HachiCommitmentLayout::new::(m_vars, r_vars) + } } diff --git a/src/protocol/commitment/mod.rs b/src/protocol/commitment/mod.rs index d7c23e2f..b5fe86f9 100644 --- a/src/protocol/commitment/mod.rs +++ b/src/protocol/commitment/mod.rs @@ -8,8 +8,14 @@ mod transcript_append; mod types; pub(crate) mod utils; -pub use commit::{HachiCommitmentCore, RingCommitmentSetup}; -pub use config::{CommitmentConfig, ProductionFp128CommitmentConfig, SmallTestCommitmentConfig}; +pub use commit::{ + HachiCommitmentCore, HachiExpandedSetup, HachiPreparedSetup, HachiProverSetup, HachiSetupSeed, + HachiVerifierSetup, +}; +pub use config::{ + CommitmentConfig, HachiCommitmentLayout, ProductionFp128CommitmentConfig, + SmallTestCommitmentConfig, +}; pub use scheme::{ CommitWitness, CommitmentScheme, RingCommitmentScheme, StreamingCommitmentScheme, }; diff --git a/src/protocol/commitment/scheme.rs b/src/protocol/commitment/scheme.rs index f2c5f2e9..8a96f935 100644 --- a/src/protocol/commitment/scheme.rs +++ b/src/protocol/commitment/scheme.rs @@ -1,8 +1,7 @@ //! Commitment-scheme trait surface for Hachi protocol code. -use super::config::CommitmentConfig; +use super::config::{CommitmentConfig, HachiCommitmentLayout}; use super::transcript_append::AppendToTranscript; -use super::utils::math::checked_pow2; use crate::algebra::ring::CyclotomicRing; use crate::error::HachiError; use crate::protocol::transcript::Transcript; @@ -153,6 +152,13 @@ where /// Returns an error if dimensions are inconsistent with `Cfg`. fn setup(max_num_vars: usize) -> Result<(Self::ProverSetup, Self::VerifierSetup), HachiError>; + /// Read the runtime layout carried by `setup`. + /// + /// # Errors + /// + /// Returns an error when setup metadata is inconsistent. + fn layout(setup: &Self::ProverSetup) -> Result; + /// Commit to ring blocks arranged as `2^R` vectors of length `2^M`. /// /// Returns `(commitment, s, t_hat)` where `s` and `t_hat` are the @@ -187,8 +193,9 @@ where f_coeffs: &[CyclotomicRing], setup: &Self::ProverSetup, ) -> Result, HachiError> { - let num_blocks = checked_pow2(Cfg::R)?; - let block_len = checked_pow2(Cfg::M)?; + let layout = Self::layout(setup)?; + let num_blocks = layout.num_blocks; + let block_len = layout.block_len; let expected_len = num_blocks .checked_mul(block_len) .ok_or_else(|| HachiError::InvalidSetup("coefficient length overflow".to_string()))?; @@ -203,7 +210,7 @@ where for i in 0..num_blocks { let mut block = Vec::with_capacity(block_len); for j in 0..block_len { - let idx = (j << Cfg::R) + let idx = (j << layout.r_vars) .checked_add(i) .ok_or_else(|| HachiError::InvalidSetup("index overflow".to_string()))?; block.push(f_coeffs[idx]); diff --git a/src/protocol/commitment_scheme.rs b/src/protocol/commitment_scheme.rs index 30c5a5bd..7de0d5d6 100644 --- a/src/protocol/commitment_scheme.rs +++ b/src/protocol/commitment_scheme.rs @@ -7,8 +7,8 @@ use crate::error::HachiError; use crate::parallel::*; use crate::primitives::poly::multilinear_lagrange_basis; use crate::protocol::commitment::{ - CommitmentConfig, CommitmentScheme, HachiCommitmentCore, RingCommitment, RingCommitmentScheme, - RingCommitmentSetup, + CommitmentConfig, CommitmentScheme, HachiCommitmentCore, HachiProverSetup, HachiVerifierSetup, + RingCommitment, RingCommitmentScheme, }; use crate::protocol::opening_point::RingOpeningPoint; use crate::protocol::proof::{HachiCommitmentHint, HachiProof, SumcheckAux}; @@ -48,8 +48,8 @@ where F: FieldCore + CanonicalField + FieldSampling, Cfg: CommitmentConfig, { - type ProverSetup = RingCommitmentSetup; - type VerifierSetup = RingCommitmentSetup; + type ProverSetup = HachiProverSetup; + type VerifierSetup = HachiVerifierSetup; type Commitment = RingCommitment; type Proof = HachiProof; type OpeningProofHint = HachiCommitmentHint; @@ -62,7 +62,9 @@ where } fn setup_verifier(setup: &Self::ProverSetup) -> Self::VerifierSetup { - setup.clone() + HachiVerifierSetup { + expanded: setup.expanded.clone(), + } } fn commit>( @@ -106,8 +108,13 @@ where }); } - let ring_opening_point = - ring_opening_point_from_field::(&opening_point[..reduced_len], Cfg::R, Cfg::M)?; + let layout = + >::layout(setup)?; + let ring_opening_point = ring_opening_point_from_field::( + &opening_point[..reduced_len], + layout.r_vars, + layout.m_vars, + )?; let y_ring = evaluate_packed_ring_poly::(&hint.ring_coeffs, &opening_point[..reduced_len]); @@ -176,8 +183,12 @@ where } // §4.2 Quadratic equation - let ring_opening_point = - ring_opening_point_from_field::(reduced_opening_point, Cfg::R, Cfg::M)?; + let layout = setup.expanded.seed.layout; + let ring_opening_point = ring_opening_point_from_field::( + reduced_opening_point, + layout.r_vars, + layout.m_vars, + )?; let quad_eq = QuadraticEquation::::new_verifier( setup, &ring_opening_point, @@ -241,7 +252,7 @@ where #[cfg(test)] pub(crate) fn rederive_alpha_and_m_a( proof: &HachiProof, - setup: &RingCommitmentSetup, + setup: &HachiVerifierSetup, opening_point: &[F], commitment: &RingCommitment, ) -> Result<(F, Vec), HachiError> @@ -254,8 +265,12 @@ where .len() .checked_sub(alpha_bits) .ok_or_else(|| HachiError::InvalidSetup("opening point length underflow".to_string()))?; - let ring_opening_point = - ring_opening_point_from_field::(&opening_point[..reduced_len], Cfg::R, Cfg::M)?; + let layout = setup.expanded.seed.layout; + let ring_opening_point = ring_opening_point_from_field::( + &opening_point[..reduced_len], + layout.r_vars, + layout.m_vars, + )?; let mut transcript = Blake2bTranscript::::new(DOMAIN_HACHI_PROTOCOL); let quad_eq = QuadraticEquation::::new_verifier( setup, @@ -404,7 +419,8 @@ mod tests { #[test] fn verify_passes_for_consistent_opening() { let alpha = Cfg::D.trailing_zeros() as usize; - let num_vars = Cfg::R + Cfg::M + alpha; + let layout = Cfg::commitment_layout(16).unwrap(); + let num_vars = layout.m_vars + layout.r_vars + alpha; let len = 1usize << num_vars; let evals: Vec = (0..len).map(|i| F::from_u64(i as u64)).collect(); @@ -445,7 +461,8 @@ mod tests { #[test] fn verify_rejects_wrong_opening() { let alpha = Cfg::D.trailing_zeros() as usize; - let num_vars = Cfg::R + Cfg::M + alpha; + let layout = Cfg::commitment_layout(16).unwrap(); + let num_vars = layout.m_vars + layout.r_vars + alpha; let len = 1usize << num_vars; let evals: Vec = (0..len).map(|i| F::from_u64(i as u64)).collect(); diff --git a/src/protocol/mod.rs b/src/protocol/mod.rs index 90090fe8..dcfba23f 100644 --- a/src/protocol/mod.rs +++ b/src/protocol/mod.rs @@ -16,9 +16,10 @@ pub mod transcript; pub use commitment::{ AppendToTranscript, CommitmentConfig, CommitmentScheme, DummyProof, HachiCommitment, - HachiCommitmentCore, HachiOpeningClaim, HachiOpeningPoint, ProductionFp128CommitmentConfig, - RingCommitment, RingCommitmentScheme, RingCommitmentSetup, SmallTestCommitmentConfig, - StreamingCommitmentScheme, + HachiCommitmentCore, HachiCommitmentLayout, HachiExpandedSetup, HachiOpeningClaim, + HachiOpeningPoint, HachiPreparedSetup, HachiProverSetup, HachiSetupSeed, HachiVerifierSetup, + ProductionFp128CommitmentConfig, RingCommitment, RingCommitmentScheme, + SmallTestCommitmentConfig, StreamingCommitmentScheme, }; pub use commitment_scheme::HachiCommitmentScheme; pub use opening_point::RingOpeningPoint; diff --git a/src/protocol/quadratic_equation.rs b/src/protocol/quadratic_equation.rs index d2b365dc..9938c806 100644 --- a/src/protocol/quadratic_equation.rs +++ b/src/protocol/quadratic_equation.rs @@ -11,7 +11,10 @@ use crate::protocol::challenges::sparse::sample_sparse_challenges; use crate::protocol::commitment::utils::crt_ntt::NttMatrixCache; use crate::protocol::commitment::utils::linear::{mat_vec_mul_ntt_cached, MatrixSlot}; use crate::protocol::commitment::utils::norm::{detect_field_modulus, vec_inf_norm}; -use crate::protocol::commitment::{CommitmentConfig, RingCommitment, RingCommitmentSetup}; +use crate::protocol::commitment::{ + CommitmentConfig, HachiCommitmentLayout, HachiExpandedSetup, HachiProverSetup, + HachiVerifierSetup, RingCommitment, +}; use crate::protocol::opening_point::RingOpeningPoint; use crate::protocol::proof::HachiCommitmentHint; use crate::protocol::transcript::labels::{ABSORB_PROVER_V, CHALLENGE_STAGE1_FOLD}; @@ -23,13 +26,14 @@ use crate::{CanonicalField, FieldCore}; fn compute_w_hat( opening_point: &RingOpeningPoint, s: &[Vec>], + layout: HachiCommitmentLayout, ) -> Vec>> where F: FieldCore + CanonicalField, Cfg: CommitmentConfig, { let a = &opening_point.a; - let block_len = 1usize << Cfg::M; + let block_len = layout.block_len; let delta = Cfg::DELTA; let log_basis = Cfg::LOG_BASIS; @@ -63,6 +67,7 @@ fn compute_v( fn compute_z_hat( s: &[Vec>], challenges: &[SparseChallenge], + layout: HachiCommitmentLayout, ) -> Result>, HachiError> where F: FieldCore + CanonicalField, @@ -84,10 +89,10 @@ where let modulus = detect_field_modulus::(); let norm = vec_inf_norm(&z, modulus); - if norm > Cfg::BETA { + let beta = Cfg::beta_bound(layout)?; + if norm > beta { return Err(HachiError::InvalidInput(format!( - "prover abort: ||z||_inf = {norm} > beta = {}", - Cfg::BETA + "prover abort: ||z||_inf = {norm} > beta = {beta}" ))); } @@ -133,7 +138,7 @@ where /// Returns an error if the norm check, challenge sampling, or matrix /// generation fails. pub fn new_prover>( - setup: &RingCommitmentSetup, + setup: &HachiProverSetup, ring_opening_point: &RingOpeningPoint, hint: &HachiCommitmentHint, transcript: &mut T, @@ -141,10 +146,11 @@ where y_ring: &CyclotomicRing, ) -> Result { // Steps 1–3: w_i = a^T G_{2^m} s_i, then ŵ_i = G_1^{-1}(w_i) - let w_hat = compute_w_hat::(ring_opening_point, &hint.s); + let layout = setup.layout(); + let w_hat = compute_w_hat::(ring_opening_point, &hint.s, layout); // Step 4: v = D · ŵ - let v = compute_v(&setup.ntt_cache, &w_hat)?; + let v = compute_v(setup.ntt_cache()?, &w_hat)?; // Step 5: append v to transcript transcript.append_serde(ABSORB_PROVER_V, &v); @@ -157,14 +163,14 @@ where let challenges = sample_sparse_challenges::( transcript, CHALLENGE_STAGE1_FOLD, - 1usize << Cfg::R, + layout.num_blocks, &challenge_cfg, )?; // Steps 7–9: z = Σ c_i · s_i, check ‖z‖_∞ ≤ β, then ẑ = J^{-1}(z) - let z_hat = compute_z_hat::(&hint.s, &challenges)?; + let z_hat = compute_z_hat::(&hint.s, &challenges, layout)?; - let m = generate_m::(setup, ring_opening_point, &challenges)?; + let m = generate_m::(&setup.expanded, ring_opening_point, &challenges)?; let y = generate_y::(&v, &commitment.u, y_ring)?; let z = generate_z(&w_hat, &hint.t_hat, &z_hat); @@ -187,15 +193,16 @@ where /// /// Returns an error if challenge derivation fails. pub fn new_verifier>( - setup: &RingCommitmentSetup, + setup: &HachiVerifierSetup, ring_opening_point: &RingOpeningPoint, v: &Vec>, transcript: &mut T, commitment: &RingCommitment, y_ring: &CyclotomicRing, ) -> Result { - let challenges = derive_stage1_challenges::(transcript, v)?; - let m = generate_m::(setup, ring_opening_point, &challenges)?; + let layout = setup.expanded.seed.layout; + let challenges = derive_stage1_challenges::(transcript, v, layout.num_blocks)?; + let m = generate_m::(&setup.expanded, ring_opening_point, &challenges)?; let y = generate_y::(v, &commitment.u, y_ring)?; Ok(Self { @@ -250,6 +257,7 @@ where pub(crate) fn derive_stage1_challenges( transcript: &mut T, v: &Vec>, + num_blocks: usize, ) -> Result, HachiError> where F: FieldCore + CanonicalField, @@ -259,9 +267,6 @@ where weight: Cfg::CHALLENGE_WEIGHT, nonzero_coeffs: vec![-1, 1], }; - let num_blocks = 1usize - .checked_shl(Cfg::R as u32) - .ok_or_else(|| HachiError::InvalidSetup("2^R does not fit usize".to_string()))?; transcript.append_serde(ABSORB_PROVER_V, v); sample_sparse_challenges::( transcript, @@ -347,19 +352,16 @@ fn gadget_block_diag_scalars(blocks: usize, row: &[F]) -> Vec( - setup: &RingCommitmentSetup, + setup: &HachiExpandedSetup, opening_point: &RingOpeningPoint, challenges: &[SparseChallenge], ) -> Result>>, HachiError> where F: FieldCore + CanonicalField, { - let num_blocks = 1usize - .checked_shl(Cfg::R as u32) - .ok_or_else(|| HachiError::InvalidSetup("2^R does not fit usize".to_string()))?; - let block_len = 1usize - .checked_shl(Cfg::M as u32) - .ok_or_else(|| HachiError::InvalidSetup("2^M does not fit usize".to_string()))?; + let layout = setup.seed.layout; + let num_blocks = layout.num_blocks; + let block_len = layout.block_len; let w_len = Cfg::DELTA .checked_mul(num_blocks) .ok_or_else(|| HachiError::InvalidSetup("w length overflow".to_string()))?; @@ -570,7 +572,7 @@ mod tests { } struct Fixture { - setup: RingCommitmentSetup, + setup: HachiProverSetup, commitment_u: Vec>, point: RingOpeningPoint, blocks: Vec>>, @@ -632,7 +634,7 @@ mod tests { let w_hat = f.quad_eq.w_hat().unwrap(); let w_hat_flat: Vec> = w_hat.iter().flat_map(|v| v.iter().copied()).collect(); - let lhs = mat_vec_mul(&f.setup.D, &w_hat_flat); + let lhs = mat_vec_mul(&f.setup.expanded.D, &w_hat_flat); assert_eq!(lhs, f.quad_eq.v(), "Row 1 failed: D · ŵ ≠ v"); } @@ -645,7 +647,7 @@ mod tests { let hint = f.quad_eq.hint().unwrap(); let t_hat_flat: Vec> = hint.t_hat.iter().flat_map(|v| v.iter().copied()).collect(); - let lhs = mat_vec_mul(&f.setup.B, &t_hat_flat); + let lhs = mat_vec_mul(&f.setup.expanded.B, &t_hat_flat); assert_eq!(lhs, f.commitment_u, "Row 2 failed: B · t̂ ≠ u"); } @@ -728,7 +730,7 @@ mod tests { } let z_recovered = recompose_z_hat(f.quad_eq.z_hat().unwrap()); - let rhs = mat_vec_mul(&f.setup.A, &z_recovered); + let rhs = mat_vec_mul(&f.setup.expanded.A, &z_recovered); assert_eq!(lhs, rhs, "Row 5 failed: (c^T ⊗ G_nA)t̂ ≠ A · J · ẑ"); } diff --git a/src/protocol/ring_switch.rs b/src/protocol/ring_switch.rs index 6d5860cc..7518b293 100644 --- a/src/protocol/ring_switch.rs +++ b/src/protocol/ring_switch.rs @@ -244,16 +244,23 @@ struct WCommitmentConfig { impl CommitmentConfig for WCommitmentConfig { const D: usize = D; - const M: usize = 11; - const R: usize = 0; const N_A: usize = Cfg::N_A; const N_B: usize = Cfg::N_B; const N_D: usize = Cfg::N_D; const LOG_BASIS: u32 = Cfg::LOG_BASIS; const DELTA: usize = Cfg::DELTA; const TAU: usize = Cfg::TAU; - const BETA: u128 = Cfg::BETA; const CHALLENGE_WEIGHT: usize = Cfg::CHALLENGE_WEIGHT; + + fn commitment_layout( + max_num_vars: usize, + ) -> Result { + let alpha = D.trailing_zeros() as usize; + let m_vars = max_num_vars + .checked_sub(alpha) + .ok_or_else(|| HachiError::InvalidSetup("max_num_vars is smaller than alpha".to_string()))?; + crate::protocol::commitment::HachiCommitmentLayout::new::(m_vars, 0) + } } fn commit_w(w: &[F]) -> Result, HachiError> @@ -262,7 +269,6 @@ where Cfg: CommitmentConfig, { type WCfg = WCommitmentConfig; - let block_len = 1usize << WCfg::::M; let ring_elems: Vec> = w .chunks(D) @@ -273,13 +279,15 @@ where }) .collect(); + let block_len = ring_elems.len().next_power_of_two().max(1); let mut padded = ring_elems; padded.resize(block_len, CyclotomicRing::::zero()); - + let m_vars = block_len.trailing_zeros() as usize; + let max_num_vars = m_vars + D.trailing_zeros() as usize; let blocks = vec![padded]; let (w_setup, _) = >>::setup( - WCfg::::M, + max_num_vars, )?; let w = >>::commit_ring_blocks( diff --git a/src/protocol/sumcheck/norm_sumcheck.rs b/src/protocol/sumcheck/norm_sumcheck.rs index 83b14b9f..35105e6f 100644 --- a/src/protocol/sumcheck/norm_sumcheck.rs +++ b/src/protocol/sumcheck/norm_sumcheck.rs @@ -13,6 +13,89 @@ use super::{SumcheckInstanceProver, SumcheckInstanceVerifier, UniPoly}; use crate::parallel::*; use crate::{FieldCore, FromSmallInt}; +#[derive(Clone)] +struct RangeAffinePrecomp { + /// `coeff_mix[i][k] = c_{i+k} * binom(i+k, i)`, where + /// `R(w) = sum_m c_m * w^m` is the range-check polynomial. + coeff_mix: Vec>, + degree_q: usize, +} + +impl RangeAffinePrecomp { + fn new(b: usize) -> Self { + assert!(b >= 1, "b must be at least 1"); + let range_coeffs = range_check_coeffs::(b); + let degree_q = range_coeffs.len() - 1; + let mut coeff_mix = Vec::with_capacity(degree_q + 1); + + for i in 0..=degree_q { + let row_len = degree_q - i + 1; + let mut row = Vec::with_capacity(row_len); + let mut binom_m_i = E::one(); // binom(i, i) + for k in 0..row_len { + let m = i + k; + row.push(range_coeffs[m] * binom_m_i); + if k + 1 < row_len { + let numer = E::from_u64((m + 1) as u64); + let denom = E::from_u64((k + 1) as u64); + let denom_inv = denom + .inv() + .expect("field characteristic too small for range-check precomputation"); + binom_m_i = binom_m_i * numer * denom_inv; + } + } + coeff_mix.push(row); + } + + Self { + coeff_mix, + degree_q, + } + } +} + +/// Coefficients of `R(w) = w * Π_{k=1}^{b-1}(w-k)(w+k)` in increasing degree order. +fn range_check_coeffs(b: usize) -> Vec { + assert!(b >= 1, "b must be at least 1"); + let mut coeffs = vec![E::zero(), E::one()]; // R(w)=w when b=1 + for k in 1..b { + let k_e = E::from_u64(k as u64); + let k_sq = k_e * k_e; + // Multiply by (w^2 - k^2). + let mut next = vec![E::zero(); coeffs.len() + 2]; + for (idx, c) in coeffs.iter().enumerate() { + next[idx] = next[idx] - *c * k_sq; + next[idx + 2] = next[idx + 2] + *c; + } + coeffs = next; + } + coeffs +} + +fn accumulate_affine_range_coeffs( + out_coeffs: &mut [E], + coeff_mix: &[Vec], + w_0: E, + a: E, + scale: E, +) { + let mut a_pow = E::one(); + for (i, row) in coeff_mix.iter().enumerate() { + let mut h_i_w0 = E::zero(); + for coeff in row.iter().rev() { + h_i_w0 = h_i_w0 * w_0 + *coeff; + } + out_coeffs[i] = out_coeffs[i] + scale * a_pow * h_i_w0; + a_pow = a_pow * a; + } +} + +fn trim_trailing_zeros(coeffs: &mut Vec) { + while coeffs.len() > 1 && coeffs.last().is_some_and(|c| c.is_zero()) { + coeffs.pop(); + } +} + /// Prover for `F_{0,τ₀}(x,y) = ẽq(τ₀,(x,y)) · w̃(x,y) · range_check(w̃(x,y), b)`. /// /// Uses the Gruen/Dao-Thaler optimization: the eq polynomial is factored into @@ -23,6 +106,7 @@ use crate::{FieldCore, FromSmallInt}; pub struct NormSumcheckProver { split_eq: GruenSplitEq, w_table: Vec, + range_precomp: RangeAffinePrecomp, num_vars: usize, b: usize, } @@ -39,6 +123,7 @@ impl NormSumcheckProver { Self { split_eq: GruenSplitEq::new(tau), w_table: w_evals, + range_precomp: RangeAffinePrecomp::new(b), num_vars, b, } @@ -60,36 +145,32 @@ impl SumcheckInstanceProver for NormSumcheckProv fn compute_round_univariate(&mut self, _round: usize, _previous_claim: E) -> UniPoly { let half = self.w_table.len() / 2; - let degree_q = 2 * self.b - 1; - let num_points_q = degree_q + 1; + let num_coeffs_q = self.range_precomp.degree_q + 1; let (e_first, e_second) = self.split_eq.remaining_eq_tables(); let num_first = e_first.len(); let first_bits = num_first.trailing_zeros(); - let b = self.b; + let coeff_mix = &self.range_precomp.coeff_mix; #[cfg(feature = "parallel")] - let q_evals = { + let q_coeffs = { (0..half) .into_par_iter() .fold( - || vec![E::zero(); num_points_q], - |mut evals, j| { + || vec![E::zero(); num_coeffs_q], + |mut coeffs, j| { let j_low = j & (num_first - 1); let j_high = j >> first_bits; let eq_rem = e_first[j_low] * e_second[j_high]; let w_0 = self.w_table[2 * j]; let w_1 = self.w_table[2 * j + 1]; - for (t, eval) in evals.iter_mut().enumerate() { - let t_e = E::from_u64(t as u64); - let w_t = w_0 + t_e * (w_1 - w_0); - *eval = *eval + eq_rem * range_check_eval(w_t, b); - } - evals + let a = w_1 - w_0; + accumulate_affine_range_coeffs(&mut coeffs, coeff_mix, w_0, a, eq_rem); + coeffs }, ) .reduce( - || vec![E::zero(); num_points_q], + || vec![E::zero(); num_coeffs_q], |mut a, b_vec| { for (ai, bi) in a.iter_mut().zip(b_vec.iter()) { *ai = *ai + *bi; @@ -99,24 +180,23 @@ impl SumcheckInstanceProver for NormSumcheckProv ) }; #[cfg(not(feature = "parallel"))] - let q_evals = { - let mut evals = vec![E::zero(); num_points_q]; + let q_coeffs = { + let mut coeffs = vec![E::zero(); num_coeffs_q]; for j in 0..half { let j_low = j & (num_first - 1); let j_high = j >> first_bits; let eq_rem = e_first[j_low] * e_second[j_high]; let w_0 = self.w_table[2 * j]; let w_1 = self.w_table[2 * j + 1]; - for (t, eval) in evals.iter_mut().enumerate() { - let t_e = E::from_u64(t as u64); - let w_t = w_0 + t_e * (w_1 - w_0); - *eval = *eval + eq_rem * range_check_eval(w_t, b); - } + let a = w_1 - w_0; + accumulate_affine_range_coeffs(&mut coeffs, coeff_mix, w_0, a, eq_rem); } - evals + coeffs }; - let q_poly = UniPoly::from_evals(&q_evals); + let mut q_coeffs = q_coeffs; + trim_trailing_zeros(&mut q_coeffs); + let q_poly = UniPoly::from_coeffs(q_coeffs); self.split_eq.gruen_mul(&q_poly) } @@ -187,17 +267,137 @@ mod tests { HachiCommitmentScheme, SmallTestCommitmentConfig, Transcript, }; use crate::{FieldCore, FromSmallInt}; + use rand::rngs::StdRng; + use rand::SeedableRng; type F = Fp64<4294967197>; const D: usize = 8; type Cfg = SmallTestCommitmentConfig; type Scheme = HachiCommitmentScheme<{ Cfg::D }, Cfg>; + struct LegacyNormSumcheckProver { + split_eq: GruenSplitEq, + w_table: Vec, + num_vars: usize, + b: usize, + } + + impl LegacyNormSumcheckProver { + fn new(tau: &[E], w_evals: Vec, b: usize) -> Self { + let num_vars = tau.len(); + assert_eq!(w_evals.len(), 1 << num_vars); + Self { + split_eq: GruenSplitEq::new(tau), + w_table: w_evals, + num_vars, + b, + } + } + } + + impl SumcheckInstanceProver for LegacyNormSumcheckProver { + fn num_rounds(&self) -> usize { + self.num_vars + } + + fn degree_bound(&self) -> usize { + 2 * self.b + } + + fn input_claim(&self) -> E { + E::zero() + } + + fn compute_round_univariate(&mut self, _round: usize, _previous_claim: E) -> UniPoly { + let half = self.w_table.len() / 2; + let degree_q = 2 * self.b - 1; + let num_points_q = degree_q + 1; + + let (e_first, e_second) = self.split_eq.remaining_eq_tables(); + let num_first = e_first.len(); + let first_bits = num_first.trailing_zeros(); + let b = self.b; + + let mut q_evals = vec![E::zero(); num_points_q]; + for j in 0..half { + let j_low = j & (num_first - 1); + let j_high = j >> first_bits; + let eq_rem = e_first[j_low] * e_second[j_high]; + let w_0 = self.w_table[2 * j]; + let w_1 = self.w_table[2 * j + 1]; + for (t, eval) in q_evals.iter_mut().enumerate() { + let t_e = E::from_u64(t as u64); + let w_t = w_0 + t_e * (w_1 - w_0); + *eval = *eval + eq_rem * range_check_eval(w_t, b); + } + } + + let q_poly = UniPoly::from_evals(&q_evals); + self.split_eq.gruen_mul(&q_poly) + } + + fn ingest_challenge(&mut self, _round: usize, r: E) { + self.split_eq.bind(r); + fold_evals_in_place(&mut self.w_table, r); + } + } + fn ring_with_small_coeff(value: u64) -> CyclotomicRing { let coeffs = std::array::from_fn(|_| F::from_u64(value)); CyclotomicRing::from_coefficients(coeffs) } + #[test] + fn norm_sumcheck_matches_legacy_round_kernel() { + let mut rng = StdRng::seed_from_u64(0xC0FFEE); + for case_idx in 0..8u64 { + let num_vars = 6; + let n = 1usize << num_vars; + let b = if case_idx % 2 == 0 { 8 } else { 16 }; + let w_evals: Vec = (0..n) + .map(|i| F::from_u64((i as u64 * 31 + case_idx * 17) % b as u64)) + .collect(); + let tau: Vec = (0..num_vars) + .map(|_| F::from_u64(rand::Rng::gen_range(&mut rng, 1u64..=257))) + .collect(); + + let mut optimized = NormSumcheckProver::new(&tau, w_evals.clone(), b); + let mut legacy = LegacyNormSumcheckProver::new(&tau, w_evals, b); + + let mut claim_opt = F::zero(); + let mut claim_old = F::zero(); + for round in 0..num_vars { + let g_opt = optimized.compute_round_univariate(round, claim_opt); + let g_old = legacy.compute_round_univariate(round, claim_old); + assert_eq!( + g_opt, g_old, + "round poly mismatch for case {case_idx} round {round}" + ); + + assert_eq!( + g_opt.evaluate(&F::zero()) + g_opt.evaluate(&F::one()), + claim_opt, + "optimized hint mismatch for case {case_idx} round {round}" + ); + assert_eq!( + g_old.evaluate(&F::zero()) + g_old.evaluate(&F::one()), + claim_old, + "legacy hint mismatch for case {case_idx} round {round}" + ); + + let r = F::from_u64(rand::Rng::gen_range(&mut rng, 1u64..=257)); + claim_opt = g_opt.evaluate(&r); + claim_old = g_old.evaluate(&r); + optimized.ingest_challenge(round, r); + legacy.ingest_challenge(round, r); + } + assert_eq!( + claim_opt, claim_old, + "final claim mismatch for case {case_idx}" + ); + } + } + #[test] fn norm_sumcheck_uses_commitment_w_evals() { let z = vec![ @@ -245,7 +445,8 @@ mod tests { #[test] fn norm_sumcheck_uses_prove_w_evals() { let alpha = SmallTestCommitmentConfig::D.trailing_zeros() as usize; - let num_vars = SmallTestCommitmentConfig::R + SmallTestCommitmentConfig::M + alpha; + let layout = SmallTestCommitmentConfig::commitment_layout(8).unwrap(); + let num_vars = layout.m_vars + layout.r_vars + alpha; let len = 1usize << num_vars; let evals: Vec = (0..len).map(|i| F::from_u64(i as u64)).collect(); let poly = DenseMultilinearEvals::new_padded(evals); diff --git a/src/protocol/sumcheck/relation_sumcheck.rs b/src/protocol/sumcheck/relation_sumcheck.rs index 4a112a52..97c7c333 100644 --- a/src/protocol/sumcheck/relation_sumcheck.rs +++ b/src/protocol/sumcheck/relation_sumcheck.rs @@ -277,7 +277,8 @@ mod tests { #[test] fn relation_sumcheck_uses_prove_w_evals() { let alpha_bits = SmallTestCommitmentConfig::D.trailing_zeros() as usize; - let num_vars = SmallTestCommitmentConfig::R + SmallTestCommitmentConfig::M + alpha_bits; + let layout = SmallTestCommitmentConfig::commitment_layout(8).unwrap(); + let num_vars = layout.m_vars + layout.r_vars + alpha_bits; let len = 1usize << num_vars; let evals: Vec = (0..len).map(|i| F::from_u64(i as u64)).collect(); let poly = DenseMultilinearEvals::new_padded(evals); @@ -299,7 +300,7 @@ mod tests { let (alpha, m_a_vec) = rederive_alpha_and_m_a::( &proof, - &setup, + &Scheme::setup_verifier(&setup), &opening_point, &commitment, ) diff --git a/src/test_utils.rs b/src/test_utils.rs index 604ba402..5de0450f 100644 --- a/src/test_utils.rs +++ b/src/test_utils.rs @@ -16,20 +16,23 @@ pub struct TinyConfig; impl CommitmentConfig for TinyConfig { const D: usize = 64; - const M: usize = 1; - const R: usize = 1; const N_A: usize = 2; const N_B: usize = 2; const N_D: usize = 2; const LOG_BASIS: u32 = 4; const DELTA: usize = 9; const TAU: usize = 4; - const BETA: u128 = 1_000_000; const CHALLENGE_WEIGHT: usize = 3; + + fn commitment_layout( + _max_num_vars: usize, + ) -> Result { + crate::protocol::commitment::HachiCommitmentLayout::new::(1, 1) + } } -pub const BLOCK_LEN: usize = 1 << TinyConfig::M; -pub const NUM_BLOCKS: usize = 1 << TinyConfig::R; +pub const BLOCK_LEN: usize = 2; +pub const NUM_BLOCKS: usize = 2; pub const DELTA: usize = TinyConfig::DELTA; pub const LOG_BASIS: u32 = TinyConfig::LOG_BASIS; pub const N_A: usize = TinyConfig::N_A; diff --git a/tests/ring_commitment_core.rs b/tests/ring_commitment_core.rs index ccc422e7..e0e4ebc5 100644 --- a/tests/ring_commitment_core.rs +++ b/tests/ring_commitment_core.rs @@ -12,16 +12,19 @@ struct BadDegreeConfig; impl CommitmentConfig for BadDegreeConfig { const D: usize = 32; - const M: usize = 4; - const R: usize = 2; const N_A: usize = 8; const N_B: usize = 4; const N_D: usize = 4; const LOG_BASIS: u32 = 4; const DELTA: usize = 8; const TAU: usize = 4; - const BETA: u128 = 1_000_000; const CHALLENGE_WEIGHT: usize = 3; + + fn commitment_layout( + _max_num_vars: usize, + ) -> Result { + hachi_pcs::protocol::commitment::HachiCommitmentLayout::new::(4, 2) + } } #[derive(Clone)] @@ -29,16 +32,19 @@ struct BadDigitBudgetConfig; impl CommitmentConfig for BadDigitBudgetConfig { const D: usize = 64; - const M: usize = 4; - const R: usize = 2; const N_A: usize = 8; const N_B: usize = 4; const N_D: usize = 4; const LOG_BASIS: u32 = 32; const DELTA: usize = 5; // 160 > 128 const TAU: usize = 4; - const BETA: u128 = 1_000_000; const CHALLENGE_WEIGHT: usize = 3; + + fn commitment_layout( + _max_num_vars: usize, + ) -> Result { + hachi_pcs::protocol::commitment::HachiCommitmentLayout::new::(4, 2) + } } #[test] @@ -48,16 +54,16 @@ fn setup_shape_is_consistent() { let (p2, v2) = >::setup(16).unwrap(); - assert_eq!(p1.max_num_vars, 16); - assert_eq!(v1.max_num_vars, 16); - assert_eq!(p2.max_num_vars, 16); - assert_eq!(v2.max_num_vars, 16); - assert_eq!(p1.A.len(), TinyConfig::N_A); - assert_eq!(p1.A[0].len(), (1usize << TinyConfig::M) * TinyConfig::DELTA); - assert_eq!(p1.B.len(), TinyConfig::N_B); + assert_eq!(p1.expanded.seed.max_num_vars, 16); + assert_eq!(v1.expanded.seed.max_num_vars, 16); + assert_eq!(p2.expanded.seed.max_num_vars, 16); + assert_eq!(v2.expanded.seed.max_num_vars, 16); + assert_eq!(p1.expanded.A.len(), TinyConfig::N_A); + assert_eq!(p1.expanded.A[0].len(), hachi_pcs::test_utils::BLOCK_LEN * TinyConfig::DELTA); + assert_eq!(p1.expanded.B.len(), TinyConfig::N_B); assert_eq!( - p1.B[0].len(), - TinyConfig::N_A * TinyConfig::DELTA * (1usize << TinyConfig::R) + p1.expanded.B[0].len(), + TinyConfig::N_A * TinyConfig::DELTA * hachi_pcs::test_utils::NUM_BLOCKS ); } @@ -80,8 +86,8 @@ fn commit_is_deterministic_and_shape_consistent() { assert_eq!(w1.s, w2.s); assert_eq!(w1.t_hat, w2.t_hat); - let num_blocks = 1usize << TinyConfig::R; - let block_len = 1usize << TinyConfig::M; + let num_blocks = hachi_pcs::test_utils::NUM_BLOCKS; + let block_len = hachi_pcs::test_utils::BLOCK_LEN; assert_eq!(w1.commitment.u.len(), TinyConfig::N_B); assert_eq!(w1.s.len(), num_blocks); assert_eq!(w1.t_hat.len(), num_blocks); @@ -106,8 +112,8 @@ fn commit_ring_coeffs_matches_block_commitment() { ) .unwrap(); - let num_blocks = 1usize << TinyConfig::R; - let block_len = 1usize << TinyConfig::M; + let num_blocks = hachi_pcs::test_utils::NUM_BLOCKS; + let block_len = hachi_pcs::test_utils::BLOCK_LEN; let mut f_coeffs = Vec::with_capacity(num_blocks * block_len); for j in 0..block_len { for block in blocks.iter().take(num_blocks) { @@ -136,7 +142,7 @@ fn opening_satisfies_inner_and_outer_equations() { .unwrap(); for i in 0..w.s.len() { - let lhs = mat_vec_mul(&psetup.A, &w.s[i]); + let lhs = mat_vec_mul(&psetup.expanded.A, &w.s[i]); let rhs: Vec> = (0..TinyConfig::N_A) .map(|j| { let start = j * TinyConfig::DELTA; @@ -152,15 +158,16 @@ fn opening_satisfies_inner_and_outer_equations() { let t_hat_flat: Vec> = w.t_hat.iter().flat_map(|x| x.iter().copied()).collect(); - let outer = mat_vec_mul(&psetup.B, &t_hat_flat); + let outer = mat_vec_mul(&psetup.expanded.B, &t_hat_flat); assert_eq!(outer, w.commitment.u); } #[test] fn small_test_config_has_expected_shape() { assert_eq!(SmallTestCommitmentConfig::D, 16); - assert_eq!(1usize << SmallTestCommitmentConfig::M, 16); - assert_eq!(1usize << SmallTestCommitmentConfig::R, 4); + let layout = SmallTestCommitmentConfig::commitment_layout(8).unwrap(); + assert_eq!(layout.block_len, 16); + assert_eq!(layout.num_blocks, 4); let delta = SmallTestCommitmentConfig::DELTA; assert!(delta > 0); } From 43d8ef232a5404a6810e38cd347d144f13a4e392 Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Sat, 28 Feb 2026 16:28:52 -0800 Subject: [PATCH 14/34] Soundness hardening: panic-free verifier, Fiat-Shamir binding, NTT overflow fix - Verifier path never panics; all errors return HachiError - Bind commitment, opening point, and y_ring in Fiat-Shamir transcript - Fix i16 csubp/caddp overflow by widening to i32 - multilinear_eval returns Result with dimension checks - build_w_evals validates w.len() is a multiple of d - UniPoly::degree uses saturating_sub instead of expect - Serialize usize as u64 for 32/64-bit portability - Fix from_i64(i64::MIN) via unsigned_abs - Remove Transcript::reset from public trait (move to inherent) - Add batched_sumcheck verifier empty-input guard Made-with: Cursor --- src/algebra/fields/fp32.rs | 2 +- src/algebra/fields/fp64.rs | 2 +- src/algebra/ntt/prime.rs | 28 +++++++-------- src/algebra/poly.rs | 41 +++++++++++++++------- src/primitives/serialization.rs | 32 ++++++++++++++++- src/protocol/commitment/config.rs | 8 +++++ src/protocol/commitment/scheme.rs | 4 +++ src/protocol/commitment_scheme.rs | 33 ++++++++++++++--- src/protocol/ring_switch.rs | 22 +++++++++--- src/protocol/sumcheck/batched_sumcheck.rs | 26 ++++++++------ src/protocol/sumcheck/mod.rs | 9 +++-- src/protocol/sumcheck/norm_sumcheck.rs | 13 +++---- src/protocol/sumcheck/relation_sumcheck.rs | 17 ++++----- src/protocol/sumcheck/types.rs | 14 +++----- src/protocol/transcript/hash.rs | 12 ++++++- src/protocol/transcript/mod.rs | 3 -- tests/algebra.rs | 27 ++++++++++++++ tests/hachi_sumcheck.rs | 8 ++--- tests/sumcheck_core.rs | 6 ++-- tests/sumcheck_prover_driver.rs | 1 - 20 files changed, 220 insertions(+), 88 deletions(-) diff --git a/src/algebra/fields/fp32.rs b/src/algebra/fields/fp32.rs index fa60d679..9be1b7b3 100644 --- a/src/algebra/fields/fp32.rs +++ b/src/algebra/fields/fp32.rs @@ -335,7 +335,7 @@ impl FromSmallInt for Fp32

{ if val >= 0 { Self::from_u64(val as u64) } else { - -Self::from_u64((-val) as u64) + -Self::from_u64(val.unsigned_abs()) } } } diff --git a/src/algebra/fields/fp64.rs b/src/algebra/fields/fp64.rs index 46e00272..8b81218e 100644 --- a/src/algebra/fields/fp64.rs +++ b/src/algebra/fields/fp64.rs @@ -441,7 +441,7 @@ impl FromSmallInt for Fp64

{ if val >= 0 { Self::from_u64(val as u64) } else { - -Self::from_u64((-val) as u64) + -Self::from_u64(val.unsigned_abs()) } } } diff --git a/src/algebra/ntt/prime.rs b/src/algebra/ntt/prime.rs index afe9b7d9..a5646234 100644 --- a/src/algebra/ntt/prime.rs +++ b/src/algebra/ntt/prime.rs @@ -264,21 +264,18 @@ impl NttPrime { } /// Conditionally subtract `p` if `a >= p` (branchless). + /// + /// Input may be in `(-2p, 2p)` during butterflies. Both i16 and i32 paths + /// widen to avoid signed overflow: i16→i32, i32→i64. #[inline] pub fn csubp(self, a: MontCoeff) -> MontCoeff { - // IMPORTANT: `a` may be as low as `-2p` during butterflies. - // For 30-bit primes with `i32` width, the probe `a - p` can overflow - // a narrow signed subtraction. Use widened arithmetic for the compare. - // - // For `i16` (Q32 primes), the narrow branchless probe is safe and faster. if W::R_LOG == 16 { - let diff = a.0.wrapping_sub(self.p); - let mask = diff.sign_mask(); - MontCoeff(diff.wrapping_add(mask.bitand(self.p))) + let ai = a.0.to_i64() as i32; + let pi = self.p.to_i64() as i32; + let diff = ai - pi; + let mask = diff >> 31; + MontCoeff(W::from_i64((diff + (mask & pi)) as i64)) } else { - // Branchless i64 variant: - // if diff < 0 => mask = -1 => out = diff + p = a - // else out = diff = a - p let ai = a.0.to_i64(); let pi = self.p.to_i64(); let diff = ai - pi; @@ -288,13 +285,16 @@ impl NttPrime { } /// Conditionally add `p` if `a < 0` (branchless). + /// + /// Widened arithmetic mirrors `csubp` to avoid i16 overflow edge cases. #[inline] pub fn caddp(self, a: MontCoeff) -> MontCoeff { if W::R_LOG == 16 { - let mask = a.0.sign_mask(); - MontCoeff(a.0.wrapping_add(mask.bitand(self.p))) + let ai = a.0.to_i64() as i32; + let pi = self.p.to_i64() as i32; + let mask = ai >> 31; + MontCoeff(W::from_i64((ai + (mask & pi)) as i64)) } else { - // Branchless i64 variant: mask = -1 iff a < 0. let ai = a.0.to_i64(); let pi = self.p.to_i64(); let mask = ai >> 63; diff --git a/src/algebra/poly.rs b/src/algebra/poly.rs index cb93242e..cc69e42e 100644 --- a/src/algebra/poly.rs +++ b/src/algebra/poly.rs @@ -1,5 +1,6 @@ //! Polynomial containers and evaluation utilities. +use crate::error::HachiError; use crate::primitives::serialization::{ Compress, HachiDeserialize, HachiSerialize, SerializationError, Valid, Validate, }; @@ -110,16 +111,24 @@ pub fn range_check_eval(w: E, b: usize) -> E { /// Evaluate a multilinear polynomial (given by boolean-hypercube evaluations in /// little-endian bit order) at an arbitrary point via iterated folding. -pub fn multilinear_eval(evals: &[E], point: &[E]) -> E { - debug_assert!( - evals.len().is_power_of_two(), - "evals length must be a power of two" - ); - debug_assert_eq!( - evals.len(), - 1 << point.len(), - "evals length must equal 2^point.len()" - ); +/// +/// # Errors +/// +/// Returns an error if the evaluation table length is not a power of two or +/// does not match `2^point.len()`. +pub fn multilinear_eval(evals: &[E], point: &[E]) -> Result { + if !evals.len().is_power_of_two() { + return Err(HachiError::InvalidSize { + expected: 1 << point.len(), + actual: evals.len(), + }); + } + if evals.len() != 1 << point.len() { + return Err(HachiError::InvalidSize { + expected: 1 << point.len(), + actual: evals.len(), + }); + } let mut current = evals.to_vec(); for &r in point { let half = current.len() / 2; @@ -129,17 +138,23 @@ pub fn multilinear_eval(evals: &[E], point: &[E]) -> E { } current = next; } - current[0] + Ok(current[0]) } /// Fold an evaluation table in place by binding its first variable to `r`, /// halving the table size. +/// +/// # Panics +/// +/// Panics if the evaluation table length is not a power of two or has fewer +/// than 2 elements. This is a prover-only helper where the caller guarantees +/// well-formed input. pub fn fold_evals_in_place(evals: &mut Vec, r: E) { - debug_assert!( + assert!( evals.len().is_power_of_two(), "evals length must be a power of two" ); - debug_assert!(evals.len() >= 2, "evals must have at least 2 elements"); + assert!(evals.len() >= 2, "evals must have at least 2 elements"); let half = evals.len() / 2; for i in 0..half { evals[i] = evals[2 * i] + r * (evals[2 * i + 1] - evals[2 * i]); diff --git a/src/primitives/serialization.rs b/src/primitives/serialization.rs index eaca5481..ea14bdc3 100644 --- a/src/primitives/serialization.rs +++ b/src/primitives/serialization.rs @@ -183,13 +183,43 @@ mod primitive_impls { impl_primitive_serialization!(u32, 4); impl_primitive_serialization!(u64, 8); impl_primitive_serialization!(u128, 16); - impl_primitive_serialization!(usize, std::mem::size_of::()); impl_primitive_serialization!(i8, 1); impl_primitive_serialization!(i16, 2); impl_primitive_serialization!(i32, 4); impl_primitive_serialization!(i64, 8); impl_primitive_serialization!(i128, 16); + impl Valid for usize { + fn check(&self) -> Result<(), SerializationError> { + Ok(()) + } + } + + impl HachiSerialize for usize { + fn serialize_with_mode( + &self, + writer: W, + compress: Compress, + ) -> Result<(), SerializationError> { + (*self as u64).serialize_with_mode(writer, compress) + } + + fn serialized_size(&self, _compress: Compress) -> usize { + 8 + } + } + + impl HachiDeserialize for usize { + fn deserialize_with_mode( + reader: R, + compress: Compress, + validate: Validate, + ) -> Result { + let val = u64::deserialize_with_mode(reader, compress, validate)?; + Ok(val as usize) + } + } + impl Valid for bool { fn check(&self) -> Result<(), SerializationError> { Ok(()) diff --git a/src/protocol/commitment/config.rs b/src/protocol/commitment/config.rs index 7e6694d7..2c9773da 100644 --- a/src/protocol/commitment/config.rs +++ b/src/protocol/commitment/config.rs @@ -59,6 +59,10 @@ impl HachiCommitmentLayout { } /// Total number of outer variables consumed by ring coefficients. + /// + /// # Errors + /// + /// Returns an error if the variable count overflows. pub fn outer_vars(&self) -> Result { self.m_vars .checked_add(self.r_vars) @@ -66,6 +70,10 @@ impl HachiCommitmentLayout { } /// Required polynomial variable count for this layout (`outer + alpha`). + /// + /// # Errors + /// + /// Returns an error if the variable count overflows. pub fn required_num_vars(&self) -> Result { let alpha = D.trailing_zeros() as usize; self.outer_vars()? diff --git a/src/protocol/commitment/scheme.rs b/src/protocol/commitment/scheme.rs index 8a96f935..be1be7b9 100644 --- a/src/protocol/commitment/scheme.rs +++ b/src/protocol/commitment/scheme.rs @@ -41,6 +41,10 @@ where type OpeningProofHint: Clone + Send + Sync; /// Build prover setup for maximum polynomial dimension. + /// + /// # Panics + /// + /// Panics if internal setup fails (programming error, not adversarial input). fn setup_prover(max_num_vars: usize) -> Self::ProverSetup; /// Derive verifier setup from prover setup. diff --git a/src/protocol/commitment_scheme.rs b/src/protocol/commitment_scheme.rs index 7de0d5d6..6b4459a8 100644 --- a/src/protocol/commitment_scheme.rs +++ b/src/protocol/commitment_scheme.rs @@ -7,8 +7,8 @@ use crate::error::HachiError; use crate::parallel::*; use crate::primitives::poly::multilinear_lagrange_basis; use crate::protocol::commitment::{ - CommitmentConfig, CommitmentScheme, HachiCommitmentCore, HachiProverSetup, HachiVerifierSetup, - RingCommitment, RingCommitmentScheme, + AppendToTranscript, CommitmentConfig, CommitmentScheme, HachiCommitmentCore, HachiProverSetup, + HachiVerifierSetup, RingCommitment, RingCommitmentScheme, }; use crate::protocol::opening_point::RingOpeningPoint; use crate::protocol::proof::{HachiCommitmentHint, HachiProof, SumcheckAux}; @@ -22,7 +22,9 @@ use crate::protocol::sumcheck::relation_sumcheck::{ RelationSumcheckProver, RelationSumcheckVerifier, }; use crate::protocol::sumcheck::{SumcheckInstanceProver, SumcheckInstanceVerifier}; -use crate::protocol::transcript::labels::CHALLENGE_SUMCHECK_ROUND; +use crate::protocol::transcript::labels::{ + ABSORB_COMMITMENT, ABSORB_EVALUATION_CLAIMS, CHALLENGE_SUMCHECK_ROUND, +}; use crate::protocol::transcript::Transcript; use crate::{CanonicalField, FieldCore, FieldSampling, FromSmallInt, Polynomial}; @@ -108,8 +110,7 @@ where }); } - let layout = - >::layout(setup)?; + let layout = >::layout(setup)?; let ring_opening_point = ring_opening_point_from_field::( &opening_point[..reduced_len], layout.r_vars, @@ -119,6 +120,13 @@ where let y_ring = evaluate_packed_ring_poly::(&hint.ring_coeffs, &opening_point[..reduced_len]); + // Fiat-Shamir: bind commitment, opening point, and y_ring before any challenges. + commitment.append_to_transcript(ABSORB_COMMITMENT, transcript); + for pt in opening_point { + transcript.append_field(ABSORB_EVALUATION_CLAIMS, pt); + } + transcript.append_serde(ABSORB_EVALUATION_CLAIMS, &y_ring); + // §4.2 Quadratic equation let quad_eq = QuadraticEquation::::new_prover( setup, @@ -173,6 +181,13 @@ where let reduced_opening_point = &opening_point[..reduced_len]; let inner_point = &opening_point[reduced_len..]; + // Fiat-Shamir: bind commitment, opening point, and y_ring before any challenges. + commitment.append_to_transcript(ABSORB_COMMITMENT, transcript); + for pt in opening_point { + transcript.append_field(ABSORB_EVALUATION_CLAIMS, pt); + } + transcript.append_serde(ABSORB_EVALUATION_CLAIMS, &proof.y_ring); + // §3.1 trace check let v = reduce_inner_openings_to_ring_elements::(inner_point)?; let d = F::from_u64(Cfg::D as u64); @@ -272,6 +287,14 @@ where layout.m_vars, )?; let mut transcript = Blake2bTranscript::::new(DOMAIN_HACHI_PROTOCOL); + + // Replay the same Fiat-Shamir absorptions the real verifier performs. + commitment.append_to_transcript(ABSORB_COMMITMENT, &mut transcript); + for pt in opening_point { + transcript.append_field(ABSORB_EVALUATION_CLAIMS, pt); + } + transcript.append_serde(ABSORB_EVALUATION_CLAIMS, &proof.y_ring); + let quad_eq = QuadraticEquation::::new_verifier( setup, &ring_opening_point, diff --git a/src/protocol/ring_switch.rs b/src/protocol/ring_switch.rs index 7518b293..5bc5c8d7 100644 --- a/src/protocol/ring_switch.rs +++ b/src/protocol/ring_switch.rs @@ -81,7 +81,7 @@ where m_a_vec.len() / m_a.len() }; - let (w_evals, num_u, num_l) = build_w_evals(&w, D); + let (w_evals, num_u, num_l) = build_w_evals(&w, D)?; let alpha_evals_y = build_alpha_evals_y(alpha, D); let num_sc_vars = num_u + num_l; @@ -136,7 +136,7 @@ where m_a_vec.len() / m_a.len() }; - let (w_evals, num_u, num_l) = build_w_evals(w, D); + let (w_evals, num_u, num_l) = build_w_evals(w, D)?; let alpha_evals_y = build_alpha_evals_y(alpha, D); let num_sc_vars = num_u + num_l; @@ -401,9 +401,21 @@ pub(crate) fn expand_m_a(w: &[F], d: usize) -> (Vec, usize, usize) { +/// # Errors +/// +/// Returns an error if `w.len()` is not a multiple of `d`. +pub(crate) fn build_w_evals( + w: &[F], + d: usize, +) -> Result<(Vec, usize, usize), HachiError> { + if d == 0 || w.len() % d != 0 { + return Err(HachiError::InvalidSize { + expected: d, + actual: w.len(), + }); + } let num_l = d.trailing_zeros() as usize; - let num_ring_elems = w.len() / d.max(1); + let num_ring_elems = w.len() / d; let num_u = num_ring_elems.next_power_of_two().trailing_zeros() as usize; let x_len = 1usize << num_u; let n = x_len << num_l; @@ -420,7 +432,7 @@ pub(crate) fn build_w_evals(w: &[F], d: usize) -> (Vec, usize, } }) .collect(); - (evals, num_u, num_l) + Ok((evals, num_u, num_l)) } pub(crate) fn m_row_count() -> usize { diff --git a/src/protocol/sumcheck/batched_sumcheck.rs b/src/protocol/sumcheck/batched_sumcheck.rs index aee5af41..c031b1cb 100644 --- a/src/protocol/sumcheck/batched_sumcheck.rs +++ b/src/protocol/sumcheck/batched_sumcheck.rs @@ -64,13 +64,17 @@ where E: FieldCore + FromSmallInt, S: FnMut(&mut T) -> E, { - assert!(!instances.is_empty()); + if instances.is_empty() { + return Err(HachiError::InvalidInput( + "no sumcheck instances provided".into(), + )); + } let max_num_rounds = instances .iter() .map(|inst| inst.num_rounds()) .max() - .unwrap(); + .unwrap(); // safe: non-empty checked above // Absorb individual input claims. for inst in instances.iter() { @@ -103,7 +107,7 @@ where let two_inv = E::from_u64(2) .inv() - .expect("2 must be invertible in the field"); + .expect("field characteristic 2 not supported"); let mut round_polys = Vec::with_capacity(max_num_rounds); let mut challenges = Vec::with_capacity(max_num_rounds); @@ -119,8 +123,6 @@ where if active { inst.compute_round_univariate(round - offset, *previous_claim) } else { - // Variable is "dummy" for this instance: polynomial is independent of it, - // so the round univariate is constant with H(0)=H(1)=previous_claim/2. UniPoly::from_coeffs(vec![*previous_claim * two_inv]) } }) @@ -204,10 +206,14 @@ where E: FieldCore, S: FnMut(&mut T) -> E, { - assert!(!verifiers.is_empty()); + if verifiers.is_empty() { + return Err(HachiError::InvalidInput( + "no sumcheck instances provided".into(), + )); + } - let max_degree = verifiers.iter().map(|v| v.degree_bound()).max().unwrap(); - let max_num_rounds = verifiers.iter().map(|v| v.num_rounds()).max().unwrap(); + let max_degree = verifiers.iter().map(|v| v.degree_bound()).max().unwrap(); // safe: non-empty + let max_num_rounds = verifiers.iter().map(|v| v.num_rounds()).max().unwrap(); // safe: non-empty // Absorb individual input claims. for v in verifiers.iter() { @@ -246,9 +252,9 @@ where .map(|(v, coeff)| { let offset = max_num_rounds - v.num_rounds(); let r_slice = &r_sumcheck[offset..offset + v.num_rounds()]; - v.expected_output_claim(r_slice) * *coeff + v.expected_output_claim(r_slice).map(|val| val * *coeff) }) - .fold(E::zero(), |a, v| a + v); + .try_fold(E::zero(), |a, v| v.map(|val| a + val))?; if output_claim != expected_output_claim { return Err(HachiError::InvalidProof); diff --git a/src/protocol/sumcheck/mod.rs b/src/protocol/sumcheck/mod.rs index 950b6e78..da11a71c 100644 --- a/src/protocol/sumcheck/mod.rs +++ b/src/protocol/sumcheck/mod.rs @@ -75,7 +75,12 @@ pub trait SumcheckInstanceVerifier: Send + Sync { /// Compute the expected final evaluation `f(r_0, ..., r_{n-1})` at the /// challenge point derived during the protocol. - fn expected_output_claim(&self, challenges: &[E]) -> E; + /// + /// # Errors + /// + /// May return an error if internal evaluations fail (e.g., malformed + /// evaluation tables from untrusted proof data). + fn expected_output_claim(&self, challenges: &[E]) -> Result; } /// Produce a sumcheck proof for a single instance, driving the Fiat-Shamir transcript. @@ -181,7 +186,7 @@ where sample_challenge, )?; - let expected = verifier.expected_output_claim(&challenges); + let expected = verifier.expected_output_claim(&challenges)?; if final_claim != expected { return Err(HachiError::InvalidProof); } diff --git a/src/protocol/sumcheck/norm_sumcheck.rs b/src/protocol/sumcheck/norm_sumcheck.rs index 35105e6f..ae35376a 100644 --- a/src/protocol/sumcheck/norm_sumcheck.rs +++ b/src/protocol/sumcheck/norm_sumcheck.rs @@ -9,6 +9,7 @@ use super::eq_poly::EqPolynomial; use super::split_eq::GruenSplitEq; use super::{fold_evals_in_place, multilinear_eval, range_check_eval}; use super::{SumcheckInstanceProver, SumcheckInstanceVerifier, UniPoly}; +use crate::error::HachiError; #[cfg(feature = "parallel")] use crate::parallel::*; use crate::{FieldCore, FromSmallInt}; @@ -245,10 +246,10 @@ impl SumcheckInstanceVerifier for NormSumcheckVe E::zero() } - fn expected_output_claim(&self, challenges: &[E]) -> E { + fn expected_output_claim(&self, challenges: &[E]) -> Result { let eq_val = EqPolynomial::mle(&self.tau, challenges); - let w_val = multilinear_eval(&self.w_evals, challenges); - eq_val * range_check_eval(w_val, self.b) + let w_val = multilinear_eval(&self.w_evals, challenges)?; + Ok(eq_val * range_check_eval(w_val, self.b)) } } @@ -428,7 +429,7 @@ mod tests { .unwrap(); let oracle = EqPolynomial::mle(&tau, &prover_challenges) - * range_check_eval(multilinear_eval(&w_evals, &prover_challenges), b); + * range_check_eval(multilinear_eval(&w_evals, &prover_challenges).unwrap(), b); assert_eq!(final_claim, oracle, "prover final claim != oracle eval"); let verifier = NormSumcheckVerifier::new(tau, w_evals, b); @@ -489,7 +490,7 @@ mod tests { .unwrap(); let oracle = EqPolynomial::mle(&tau, &prover_challenges) - * range_check_eval(multilinear_eval(&w_evals, &prover_challenges), b); + * range_check_eval(multilinear_eval(&w_evals, &prover_challenges).unwrap(), b); assert_eq!(final_claim, oracle, "prover final claim != oracle eval"); let verifier = NormSumcheckVerifier::new(tau, w_evals, b); @@ -529,7 +530,7 @@ mod tests { .unwrap(); let oracle = EqPolynomial::mle(&tau_e, &prover_challenges) - * range_check_eval(multilinear_eval(&w_evals_e, &prover_challenges), b); + * range_check_eval(multilinear_eval(&w_evals_e, &prover_challenges).unwrap(), b); assert_eq!(final_claim, oracle, "E2 prover final claim != oracle eval"); let verifier = NormSumcheckVerifier::new(tau_e, w_evals_e, b); diff --git a/src/protocol/sumcheck/relation_sumcheck.rs b/src/protocol/sumcheck/relation_sumcheck.rs index 97c7c333..1f03a09a 100644 --- a/src/protocol/sumcheck/relation_sumcheck.rs +++ b/src/protocol/sumcheck/relation_sumcheck.rs @@ -10,6 +10,7 @@ use super::{fold_evals_in_place, multilinear_eval}; use super::{SumcheckInstanceProver, SumcheckInstanceVerifier, UniPoly}; use crate::algebra::ring::CyclotomicRing; use crate::cfg_into_iter; +use crate::error::HachiError; #[cfg(feature = "parallel")] use crate::parallel::*; use crate::protocol::ring_switch::eval_ring_at; @@ -247,12 +248,12 @@ impl SumcheckInstanceVerifier for RelationSumch acc } - fn expected_output_claim(&self, challenges: &[F]) -> F { + fn expected_output_claim(&self, challenges: &[F]) -> Result { let (x_challenges, y_challenges) = challenges.split_at(self.num_u); - let w_val = multilinear_eval(&self.w_evals, challenges); - let alpha_val = multilinear_eval(&self.alpha_evals_y, y_challenges); - let m_val = multilinear_eval(&self.m_evals_x, x_challenges); - w_val * alpha_val * m_val + let w_val = multilinear_eval(&self.w_evals, challenges)?; + let alpha_val = multilinear_eval(&self.alpha_evals_y, y_challenges)?; + let m_val = multilinear_eval(&self.m_evals_x, x_challenges)?; + Ok(w_val * alpha_val * m_val) } } @@ -378,9 +379,9 @@ mod tests { .unwrap(); let (x_ch, y_ch) = prover_challenges.split_at(num_u); - let oracle = multilinear_eval(&w_evals, &prover_challenges) - * multilinear_eval(&alpha_evals_y, y_ch) - * multilinear_eval(&m_evals_x, x_ch); + let oracle = multilinear_eval(&w_evals, &prover_challenges).unwrap() + * multilinear_eval(&alpha_evals_y, y_ch).unwrap() + * multilinear_eval(&m_evals_x, x_ch).unwrap(); assert_eq!(final_claim, oracle, "prover final claim != oracle eval"); let verifier = RelationSumcheckVerifier::new( diff --git a/src/protocol/sumcheck/types.rs b/src/protocol/sumcheck/types.rs index c1dd393f..7fbf4d3b 100644 --- a/src/protocol/sumcheck/types.rs +++ b/src/protocol/sumcheck/types.rs @@ -23,16 +23,9 @@ impl UniPoly { Self { coeffs } } - /// Degree of the polynomial. - /// - /// # Panics - /// - /// Panics if the polynomial has no coefficients. + /// Degree of the polynomial (0 for empty or constant). pub fn degree(&self) -> usize { - self.coeffs - .len() - .checked_sub(1) - .expect("UniPoly must have at least one coefficient") + self.coeffs.len().saturating_sub(1) } /// Evaluate at `x` via Horner's method. @@ -80,7 +73,8 @@ impl UniPoly { /// # Panics /// /// Panics if any required factorial inverse does not exist (field characteristic - /// must exceed the number of evaluation points). + /// must exceed the number of evaluation points). This is a prover-only + /// function and the condition always holds for Hachi's fields. pub fn from_evals(evals: &[E]) -> Self { let n = evals.len(); if n == 0 { diff --git a/src/protocol/transcript/hash.rs b/src/protocol/transcript/hash.rs index d3c26bea..cfae7b30 100644 --- a/src/protocol/transcript/hash.rs +++ b/src/protocol/transcript/hash.rs @@ -78,8 +78,18 @@ where let sampled = u128::from_le_bytes(lo); F::from_canonical_u128_reduced(sampled) } +} - fn reset(&mut self, domain_label: &[u8]) { +impl HashTranscript +where + F: FieldCore + CanonicalField + 'static, +{ + /// Reset transcript state under a new domain label. + /// + /// This is an inherent method (not part of the `Transcript` trait) to + /// discourage use in production protocol code where resetting the + /// Fiat-Shamir chain would be unsound. + pub fn reset(&mut self, domain_label: &[u8]) { let mut hasher = D::new(); hasher.update(domain_label); self.hasher = hasher; diff --git a/src/protocol/transcript/mod.rs b/src/protocol/transcript/mod.rs index 5279f3f5..05434ddb 100644 --- a/src/protocol/transcript/mod.rs +++ b/src/protocol/transcript/mod.rs @@ -30,9 +30,6 @@ where /// Derive a challenge scalar under the provided label. fn challenge_scalar(&mut self, label: &[u8]) -> F; - - /// Reset transcript state under a new domain label. - fn reset(&mut self, domain_label: &[u8]); } /// Sample an extension field challenge by drawing `EXT_DEGREE` base-field diff --git a/tests/algebra.rs b/tests/algebra.rs index 2f5fc57d..d3723533 100644 --- a/tests/algebra.rs +++ b/tests/algebra.rs @@ -457,6 +457,33 @@ mod tests { } } + #[test] + fn csubp_widened_handles_large_negative_i16() { + for &prime in &Q32_PRIMES { + let p = prime.p; + // Values in (-2p, -(2^15 - p)) that previously overflowed in narrow i16 + for &raw in &[-20000i16, -(p + p / 2), -(p + 1000)] { + if raw <= -2 * p || raw >= 0 { + continue; + } + let a = MontCoeff::from_raw(raw); + let reduced = prime.reduce_range(a); + let r = reduced.raw(); + assert!( + r > -p && r < p, + "reduce_range({raw}) = {r} not in (-{p}, {p}) for p={p}" + ); + + let norm = prime.normalize(reduced); + let n = norm.raw(); + assert!( + n >= 0 && n < p, + "normalize(reduce_range({raw})) = {n} not in [0, {p}) for p={p}" + ); + } + } + } + #[test] fn ntt_mul_commutative() { let prime = Q32_PRIMES[0]; diff --git a/tests/hachi_sumcheck.rs b/tests/hachi_sumcheck.rs index 9c6049bc..ce33b91b 100644 --- a/tests/hachi_sumcheck.rs +++ b/tests/hachi_sumcheck.rs @@ -38,7 +38,7 @@ fn run_f0_e2e(num_u: usize, num_l: usize, b: usize) { // Sanity: prover's final claim matches oracle evaluation. let oracle = EqPolynomial::mle(&tau0, &prover_challenges) - * range_check_eval(multilinear_eval(&w_evals, &prover_challenges), b); + * range_check_eval(multilinear_eval(&w_evals, &prover_challenges).unwrap(), b); assert_eq!(final_claim, oracle, "prover final claim != oracle eval"); let t1 = Instant::now(); @@ -146,9 +146,9 @@ fn run_f_alpha_e2e(num_u: usize, num_i: usize) { // Sanity: prover's final claim matches oracle evaluation. let (x_ch, y_ch) = prover_challenges.split_at(num_u); - let oracle = multilinear_eval(&w_evals, &prover_challenges) - * multilinear_eval(&alpha_evals_y, y_ch) - * multilinear_eval(&m_evals_x, x_ch); + let oracle = multilinear_eval(&w_evals, &prover_challenges).unwrap() + * multilinear_eval(&alpha_evals_y, y_ch).unwrap() + * multilinear_eval(&m_evals_x, x_ch).unwrap(); assert_eq!(final_claim, oracle, "prover final claim != oracle eval"); let t1 = Instant::now(); diff --git a/tests/sumcheck_core.rs b/tests/sumcheck_core.rs index 52244e91..dcd6486f 100644 --- a/tests/sumcheck_core.rs +++ b/tests/sumcheck_core.rs @@ -4,6 +4,7 @@ use std::time::Instant; use hachi_pcs::algebra::poly::multilinear_eval; use hachi_pcs::algebra::Fp64; +use hachi_pcs::error::HachiError; use hachi_pcs::protocol::transcript::labels; use hachi_pcs::protocol::{ prove_sumcheck, verify_sumcheck, Blake2bTranscript, CompressedUniPoly, SumcheckInstanceProver, @@ -113,7 +114,6 @@ impl SumcheckInstanceProver for DenseSumcheckProver { eval_0 = eval_0 + self.evals[2 * i]; eval_1 = eval_1 + self.evals[2 * i + 1]; } - // g(X) = eval_0 + (eval_1 - eval_0) * X (degree 1) UniPoly::from_coeffs(vec![eval_0, eval_1 - eval_0]) } @@ -146,7 +146,7 @@ impl SumcheckInstanceVerifier for DenseSumcheckVerifier { self.claim } - fn expected_output_claim(&self, challenges: &[E]) -> E { + fn expected_output_claim(&self, challenges: &[E]) -> Result { multilinear_eval(&self.evals, challenges) } } @@ -259,7 +259,7 @@ fn e2e_sumcheck_2_pow_20() { assert_eq!(proof.round_polys.len(), num_vars); // Sanity: final claim must equal f evaluated at the challenge point. - let oracle_eval = multilinear_eval(&evals, &prover_challenges); + let oracle_eval = multilinear_eval(&evals, &prover_challenges).unwrap(); assert_eq!(final_claim, oracle_eval); let t1 = Instant::now(); diff --git a/tests/sumcheck_prover_driver.rs b/tests/sumcheck_prover_driver.rs index d309a36d..7ef3ebae 100644 --- a/tests/sumcheck_prover_driver.rs +++ b/tests/sumcheck_prover_driver.rs @@ -47,7 +47,6 @@ impl SumcheckInstanceProver for DenseTableSumcheck { s0 = s0 + self.table[i << 1]; s1 = s1 + self.table[(i << 1) | 1]; } - // g(X) = s0 + (s1 - s0) X UniPoly::from_coeffs(vec![s0, s1 - s0]) } From 1e246531f8cd8abca4c4c0b4b0d7a33db120ea7f Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Sat, 28 Feb 2026 16:34:54 -0800 Subject: [PATCH 15/34] Hoist fully qualified paths to use statements in touched files Replace inline crate::protocol::commitment::HachiCommitmentLayout, hachi_pcs::algebra::backend::{CrtReconstruct, NttPrimeOps}, and hachi_pcs::algebra::CyclotomicRing with top-level use imports. Made-with: Cursor --- benches/hachi_e2e.rs | 8 ++++---- src/protocol/ring_switch.rs | 20 +++++++++----------- tests/algebra.rs | 19 ++++++------------- tests/onehot_commitment.rs | 5 +++-- tests/ring_commitment_core.rs | 20 ++++++++++---------- 5 files changed, 32 insertions(+), 40 deletions(-) diff --git a/benches/hachi_e2e.rs b/benches/hachi_e2e.rs index 4a48aaac..d21b2f3f 100644 --- a/benches/hachi_e2e.rs +++ b/benches/hachi_e2e.rs @@ -2,8 +2,9 @@ use criterion::{black_box, criterion_group, criterion_main, BenchmarkId, Criterion}; use hachi_pcs::algebra::Fp128; +use hachi_pcs::error::HachiError; use hachi_pcs::primitives::multilinear_evals::DenseMultilinearEvals; -use hachi_pcs::protocol::commitment::ProductionFp128CommitmentConfig; +use hachi_pcs::protocol::commitment::{HachiCommitmentLayout, ProductionFp128CommitmentConfig}; use hachi_pcs::protocol::commitment_scheme::HachiCommitmentScheme; use hachi_pcs::protocol::transcript::Blake2bTranscript; use hachi_pcs::protocol::CommitmentConfig; @@ -30,9 +31,8 @@ macro_rules! bench_config { fn commitment_layout( _max_num_vars: usize, - ) -> Result - { - hachi_pcs::protocol::commitment::HachiCommitmentLayout::new::($m, $r) + ) -> Result { + HachiCommitmentLayout::new::($m, $r) } } }; diff --git a/src/protocol/ring_switch.rs b/src/protocol/ring_switch.rs index 5bc5c8d7..a5c40ca5 100644 --- a/src/protocol/ring_switch.rs +++ b/src/protocol/ring_switch.rs @@ -10,7 +10,8 @@ use crate::error::HachiError; use crate::parallel::*; use crate::protocol::commitment::utils::norm::detect_field_modulus; use crate::protocol::commitment::{ - CommitmentConfig, HachiCommitmentCore, RingCommitment, RingCommitmentScheme, + CommitmentConfig, HachiCommitmentCore, HachiCommitmentLayout, RingCommitment, + RingCommitmentScheme, }; use crate::protocol::quadratic_equation::QuadraticEquation; use crate::protocol::sumcheck::eq_poly::EqPolynomial; @@ -252,14 +253,12 @@ impl CommitmentConfig for WCommitmentConf const TAU: usize = Cfg::TAU; const CHALLENGE_WEIGHT: usize = Cfg::CHALLENGE_WEIGHT; - fn commitment_layout( - max_num_vars: usize, - ) -> Result { + fn commitment_layout(max_num_vars: usize) -> Result { let alpha = D.trailing_zeros() as usize; - let m_vars = max_num_vars - .checked_sub(alpha) - .ok_or_else(|| HachiError::InvalidSetup("max_num_vars is smaller than alpha".to_string()))?; - crate::protocol::commitment::HachiCommitmentLayout::new::(m_vars, 0) + let m_vars = max_num_vars.checked_sub(alpha).ok_or_else(|| { + HachiError::InvalidSetup("max_num_vars is smaller than alpha".to_string()) + })?; + HachiCommitmentLayout::new::(m_vars, 0) } } @@ -286,9 +285,8 @@ where let max_num_vars = m_vars + D.trailing_zeros() as usize; let blocks = vec![padded]; - let (w_setup, _) = >>::setup( - max_num_vars, - )?; + let (w_setup, _) = + >>::setup(max_num_vars)?; let w = >>::commit_ring_blocks( &blocks, &w_setup, diff --git a/tests/algebra.rs b/tests/algebra.rs index d3723533..668cacd0 100644 --- a/tests/algebra.rs +++ b/tests/algebra.rs @@ -5,6 +5,7 @@ mod tests { use num_bigint::BigUint; use rand::{rngs::StdRng, SeedableRng}; + use hachi_pcs::algebra::backend::{CrtReconstruct, NttPrimeOps}; use hachi_pcs::algebra::ntt::butterfly::{forward_ntt, inverse_ntt, NttTwiddles}; use hachi_pcs::algebra::poly::Poly; use hachi_pcs::algebra::tables::{ @@ -1046,11 +1047,9 @@ mod tests { } let reconstructed: [F; 64] = - >::reconstruct(&primes, &canonical, &garner); + >::reconstruct( + &primes, &canonical, &garner, + ); assert_eq!(reconstructed, coeffs); } @@ -1071,10 +1070,7 @@ mod tests { let mut limb = [MontCoeff::from_raw(0i32); 64]; for (i, r) in residues.iter().enumerate() { let reduced = (*r as i64 % (prime.p as i64)) as i32; - limb[i] = >::from_canonical( - prime, - reduced, - ); + limb[i] = >::from_canonical(prime, reduced); } forward_ntt(&mut limb, prime, &twiddles[k]); @@ -1082,10 +1078,7 @@ mod tests { for (i, r) in residues.iter().enumerate() { let expected = (*r as i64 % (prime.p as i64)) as i32; - let got = >::to_canonical( - prime, - limb[i], - ); + let got = >::to_canonical(prime, limb[i]); assert_eq!(got, expected, "prime idx={k} coeff idx={i}"); } } diff --git a/tests/onehot_commitment.rs b/tests/onehot_commitment.rs index 0512203a..2d4b20e5 100644 --- a/tests/onehot_commitment.rs +++ b/tests/onehot_commitment.rs @@ -1,5 +1,6 @@ #![allow(missing_docs)] +use hachi_pcs::algebra::CyclotomicRing; use hachi_pcs::protocol::commitment::{HachiCommitmentCore, RingCommitmentScheme}; use hachi_pcs::test_utils::*; use hachi_pcs::{FieldCore, FromSmallInt}; @@ -33,10 +34,10 @@ fn assert_onehot_matches_dense(onehot_k: usize, indices: &[usize]) { for (c, &idx) in indices.iter().enumerate() { field_elems[c * onehot_k + idx] = F::from_u64(1); } - let ring_coeffs: Vec> = (0..total_ring) + let ring_coeffs: Vec> = (0..total_ring) .map(|r| { let coeffs: [F; D] = std::array::from_fn(|i| field_elems[r * D + i]); - hachi_pcs::algebra::CyclotomicRing::from_coefficients(coeffs) + CyclotomicRing::from_coefficients(coeffs) }) .collect(); let w_dense = diff --git a/tests/ring_commitment_core.rs b/tests/ring_commitment_core.rs index e0e4ebc5..49f19880 100644 --- a/tests/ring_commitment_core.rs +++ b/tests/ring_commitment_core.rs @@ -3,7 +3,8 @@ use hachi_pcs::algebra::CyclotomicRing; use hachi_pcs::error::HachiError; use hachi_pcs::protocol::commitment::{ - CommitmentConfig, HachiCommitmentCore, RingCommitmentScheme, SmallTestCommitmentConfig, + CommitmentConfig, HachiCommitmentCore, HachiCommitmentLayout, RingCommitmentScheme, + SmallTestCommitmentConfig, }; use hachi_pcs::test_utils::*; @@ -20,10 +21,8 @@ impl CommitmentConfig for BadDegreeConfig { const TAU: usize = 4; const CHALLENGE_WEIGHT: usize = 3; - fn commitment_layout( - _max_num_vars: usize, - ) -> Result { - hachi_pcs::protocol::commitment::HachiCommitmentLayout::new::(4, 2) + fn commitment_layout(_max_num_vars: usize) -> Result { + HachiCommitmentLayout::new::(4, 2) } } @@ -40,10 +39,8 @@ impl CommitmentConfig for BadDigitBudgetConfig { const TAU: usize = 4; const CHALLENGE_WEIGHT: usize = 3; - fn commitment_layout( - _max_num_vars: usize, - ) -> Result { - hachi_pcs::protocol::commitment::HachiCommitmentLayout::new::(4, 2) + fn commitment_layout(_max_num_vars: usize) -> Result { + HachiCommitmentLayout::new::(4, 2) } } @@ -59,7 +56,10 @@ fn setup_shape_is_consistent() { assert_eq!(p2.expanded.seed.max_num_vars, 16); assert_eq!(v2.expanded.seed.max_num_vars, 16); assert_eq!(p1.expanded.A.len(), TinyConfig::N_A); - assert_eq!(p1.expanded.A[0].len(), hachi_pcs::test_utils::BLOCK_LEN * TinyConfig::DELTA); + assert_eq!( + p1.expanded.A[0].len(), + hachi_pcs::test_utils::BLOCK_LEN * TinyConfig::DELTA + ); assert_eq!(p1.expanded.B.len(), TinyConfig::N_B); assert_eq!( p1.expanded.B[0].len(), From 2d35bdbe01f298fac57a30d4a552f277d39fa7c2 Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Sat, 28 Feb 2026 16:37:06 -0800 Subject: [PATCH 16/34] Dispatch norm sumcheck kernels by range size. Route small-b rounds through the point-eval interpolation kernel and keep the affine-coefficient kernel for larger b, while adding deterministic baseline-vs-dispatched benchmarks and parity tests to validate correctness across both strategies. Made-with: Cursor --- Cargo.toml | 4 + benches/norm_sumcheck.rs | 203 +++++++++++++++++ src/protocol/sumcheck/norm_sumcheck.rs | 287 +++++++++++++++++++++---- 3 files changed, 448 insertions(+), 46 deletions(-) create mode 100644 benches/norm_sumcheck.rs diff --git a/Cargo.toml b/Cargo.toml index 6b88584f..ab91eead 100644 --- a/Cargo.toml +++ b/Cargo.toml @@ -67,6 +67,10 @@ harness = false name = "hachi_e2e" harness = false +[[bench]] +name = "norm_sumcheck" +harness = false + [lints.rust] missing_docs = "warn" unreachable_pub = "warn" diff --git a/benches/norm_sumcheck.rs b/benches/norm_sumcheck.rs new file mode 100644 index 00000000..655d6cab --- /dev/null +++ b/benches/norm_sumcheck.rs @@ -0,0 +1,203 @@ +#![allow(missing_docs)] + +use criterion::{black_box, criterion_group, criterion_main, BatchSize, BenchmarkId, Criterion}; +use hachi_pcs::algebra::Fp128; +use hachi_pcs::protocol::sumcheck::norm_sumcheck::NormSumcheckProver; +use hachi_pcs::protocol::sumcheck::split_eq::GruenSplitEq; +use hachi_pcs::protocol::sumcheck::{ + fold_evals_in_place, prove_sumcheck, range_check_eval, SumcheckInstanceProver, UniPoly, +}; +use hachi_pcs::protocol::transcript::labels; +use hachi_pcs::protocol::Blake2bTranscript; +use hachi_pcs::{FieldCore, FromSmallInt, Transcript}; +use rand::rngs::StdRng; +use rand::{Rng, SeedableRng}; +#[cfg(feature = "parallel")] +use rayon::prelude::*; +use std::time::Duration; + +type F = Fp128<0xfffffffffffffffffffffffffffffeed>; + +/// Baseline prover keeps the pre-dispatch point-eval kernel for apples-to-apples benchmarks. +/// It is intentionally local to this bench and should not be used in production code. +struct BaselineNormSumcheckProver { + split_eq: GruenSplitEq, + w_table: Vec, + num_vars: usize, + b: usize, +} + +impl BaselineNormSumcheckProver { + fn new(tau: &[E], w_evals: Vec, b: usize) -> Self { + let num_vars = tau.len(); + assert_eq!(w_evals.len(), 1 << num_vars); + Self { + split_eq: GruenSplitEq::new(tau), + w_table: w_evals, + num_vars, + b, + } + } +} + +impl SumcheckInstanceProver for BaselineNormSumcheckProver { + fn num_rounds(&self) -> usize { + self.num_vars + } + + fn degree_bound(&self) -> usize { + 2 * self.b + } + + fn input_claim(&self) -> E { + E::zero() + } + + fn compute_round_univariate(&mut self, _round: usize, _previous_claim: E) -> UniPoly { + let half = self.w_table.len() / 2; + let degree_q = 2 * self.b - 1; + let num_points_q = degree_q + 1; + + let (e_first, e_second) = self.split_eq.remaining_eq_tables(); + let num_first = e_first.len(); + let first_bits = num_first.trailing_zeros(); + let b = self.b; + + #[cfg(feature = "parallel")] + let q_evals = { + (0..half) + .into_par_iter() + .fold( + || vec![E::zero(); num_points_q], + |mut evals, j| { + let j_low = j & (num_first - 1); + let j_high = j >> first_bits; + let eq_rem = e_first[j_low] * e_second[j_high]; + let w_0 = self.w_table[2 * j]; + let w_1 = self.w_table[2 * j + 1]; + for (t, eval) in evals.iter_mut().enumerate() { + let t_e = E::from_u64(t as u64); + let w_t = w_0 + t_e * (w_1 - w_0); + *eval = *eval + eq_rem * range_check_eval(w_t, b); + } + evals + }, + ) + .reduce( + || vec![E::zero(); num_points_q], + |mut a, b_vec| { + for (ai, bi) in a.iter_mut().zip(b_vec.iter()) { + *ai = *ai + *bi; + } + a + }, + ) + }; + #[cfg(not(feature = "parallel"))] + let q_evals = { + let mut q_evals = vec![E::zero(); num_points_q]; + for j in 0..half { + let j_low = j & (num_first - 1); + let j_high = j >> first_bits; + let eq_rem = e_first[j_low] * e_second[j_high]; + let w_0 = self.w_table[2 * j]; + let w_1 = self.w_table[2 * j + 1]; + for (t, eval) in q_evals.iter_mut().enumerate() { + let t_e = E::from_u64(t as u64); + let w_t = w_0 + t_e * (w_1 - w_0); + *eval = *eval + eq_rem * range_check_eval(w_t, b); + } + } + q_evals + }; + + let q_poly = UniPoly::from_evals(&q_evals); + self.split_eq.gruen_mul(&q_poly) + } + + fn ingest_challenge(&mut self, _round: usize, r: E) { + self.split_eq.bind(r); + fold_evals_in_place(&mut self.w_table, r); + } +} + +#[derive(Clone)] +struct NormCase { + num_vars: usize, + b: usize, + tau: Vec, + w_evals: Vec, +} + +fn build_case(num_vars: usize, b: usize, seed: u64) -> NormCase { + let mut rng = StdRng::seed_from_u64(seed); + let n = 1usize << num_vars; + let tau: Vec = (0..num_vars) + .map(|_| F::from_u64(rng.gen_range(0u64..(1u64 << 24)))) + .collect(); + let w_evals: Vec = (0..n) + .map(|_| F::from_u64(rng.gen_range(0u64..(1u64 << 24)))) + .collect(); + NormCase { + num_vars, + b, + tau, + w_evals, + } +} + +fn bench_norm_sumcheck(c: &mut Criterion) { + let cases = [ + build_case(10, 4, 0xA11CE001), + build_case(10, 8, 0xA11CE002), + build_case(14, 4, 0xA11CE003), + build_case(14, 8, 0xA11CE004), + build_case(14, 16, 0xA11CE005), + build_case(18, 8, 0xA11CE006), + ]; + + let mut group = c.benchmark_group("norm_sumcheck"); + group.warm_up_time(Duration::from_secs(8)); + group.measurement_time(Duration::from_secs(24)); + group.sample_size(35); + + for case in &cases { + let case_tag = format!("nv{}_b{}", case.num_vars, case.b); + group.bench_function(BenchmarkId::new("baseline", &case_tag), |bencher| { + bencher.iter_batched( + || BaselineNormSumcheckProver::new(&case.tau, case.w_evals.clone(), case.b), + |mut prover| { + let mut transcript = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + black_box( + prove_sumcheck::(&mut prover, &mut transcript, |tr| { + tr.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND) + }) + .unwrap(), + ) + }, + BatchSize::SmallInput, + ); + }); + + group.bench_function(BenchmarkId::new("dispatched", &case_tag), |bencher| { + bencher.iter_batched( + || NormSumcheckProver::new(&case.tau, case.w_evals.clone(), case.b), + |mut prover| { + let mut transcript = Blake2bTranscript::::new(labels::DOMAIN_HACHI_PROTOCOL); + black_box( + prove_sumcheck::(&mut prover, &mut transcript, |tr| { + tr.challenge_scalar(labels::CHALLENGE_SUMCHECK_ROUND) + }) + .unwrap(), + ) + }, + BatchSize::SmallInput, + ); + }); + } + + group.finish(); +} + +criterion_group!(benches, bench_norm_sumcheck); +criterion_main!(benches); diff --git a/src/protocol/sumcheck/norm_sumcheck.rs b/src/protocol/sumcheck/norm_sumcheck.rs index ae35376a..5eb493fb 100644 --- a/src/protocol/sumcheck/norm_sumcheck.rs +++ b/src/protocol/sumcheck/norm_sumcheck.rs @@ -14,6 +14,22 @@ use crate::error::HachiError; use crate::parallel::*; use crate::{FieldCore, FromSmallInt}; +const SMALL_B_POINT_EVAL_MAX: usize = 8; + +#[derive(Clone, Copy, Debug, PartialEq, Eq)] +enum NormRoundKernel { + PointEvalInterpolation, + AffineCoeffComposition, +} + +fn choose_round_kernel(b: usize) -> NormRoundKernel { + if b <= SMALL_B_POINT_EVAL_MAX { + NormRoundKernel::PointEvalInterpolation + } else { + NormRoundKernel::AffineCoeffComposition + } +} + #[derive(Clone)] struct RangeAffinePrecomp { /// `coeff_mix[i][k] = c_{i+k} * binom(i+k, i)`, where @@ -27,6 +43,18 @@ impl RangeAffinePrecomp { assert!(b >= 1, "b must be at least 1"); let range_coeffs = range_check_coeffs::(b); let degree_q = range_coeffs.len() - 1; + let small_scalars: Vec = (0..=degree_q + 1).map(|x| E::from_u64(x as u64)).collect(); + let inv_small_scalars: Vec = (0..=degree_q + 1) + .map(|x| { + if x == 0 { + E::zero() + } else { + small_scalars[x] + .inv() + .expect("field characteristic too small for range-check precomputation") + } + }) + .collect(); let mut coeff_mix = Vec::with_capacity(degree_q + 1); for i in 0..=degree_q { @@ -37,11 +65,8 @@ impl RangeAffinePrecomp { let m = i + k; row.push(range_coeffs[m] * binom_m_i); if k + 1 < row_len { - let numer = E::from_u64((m + 1) as u64); - let denom = E::from_u64((k + 1) as u64); - let denom_inv = denom - .inv() - .expect("field characteristic too small for range-check precomputation"); + let numer = small_scalars[m + 1]; + let denom_inv = inv_small_scalars[k + 1]; binom_m_i = binom_m_i * numer * denom_inv; } } @@ -55,6 +80,18 @@ impl RangeAffinePrecomp { } } +#[derive(Clone)] +struct PointEvalPrecomp { + range_offsets: Vec, +} + +impl PointEvalPrecomp { + fn new(b: usize) -> Self { + let range_offsets = (1..b).map(|k| E::from_u64(k as u64)).collect(); + Self { range_offsets } + } +} + /// Coefficients of `R(w) = w * Π_{k=1}^{b-1}(w-k)(w+k)` in increasing degree order. fn range_check_coeffs(b: usize) -> Vec { assert!(b >= 1, "b must be at least 1"); @@ -73,6 +110,14 @@ fn range_check_coeffs(b: usize) -> Vec { coeffs } +fn range_check_eval_precomputed(w: E, range_offsets: &[E]) -> E { + let mut acc = w; + for &k in range_offsets { + acc = acc * (w - k) * (w + k); + } + acc +} + fn accumulate_affine_range_coeffs( out_coeffs: &mut [E], coeff_mix: &[Vec], @@ -107,7 +152,9 @@ fn trim_trailing_zeros(coeffs: &mut Vec) { pub struct NormSumcheckProver { split_eq: GruenSplitEq, w_table: Vec, - range_precomp: RangeAffinePrecomp, + round_kernel: NormRoundKernel, + point_precomp: Option>, + range_precomp: Option>, num_vars: usize, b: usize, } @@ -119,39 +166,117 @@ impl NormSumcheckProver { /// /// Panics if `w_evals.len() != 2^tau.len()`. pub fn new(tau: &[E], w_evals: Vec, b: usize) -> Self { + Self::new_with_kernel(tau, w_evals, b, choose_round_kernel(b)) + } + + fn new_with_kernel( + tau: &[E], + w_evals: Vec, + b: usize, + round_kernel: NormRoundKernel, + ) -> Self { let num_vars = tau.len(); assert_eq!(w_evals.len(), 1 << num_vars); + let point_precomp = match round_kernel { + NormRoundKernel::PointEvalInterpolation => Some(PointEvalPrecomp::new(b)), + NormRoundKernel::AffineCoeffComposition => None, + }; + let range_precomp = match round_kernel { + NormRoundKernel::PointEvalInterpolation => None, + NormRoundKernel::AffineCoeffComposition => Some(RangeAffinePrecomp::new(b)), + }; Self { split_eq: GruenSplitEq::new(tau), w_table: w_evals, - range_precomp: RangeAffinePrecomp::new(b), + round_kernel, + point_precomp, + range_precomp, num_vars, b, } } -} -impl SumcheckInstanceProver for NormSumcheckProver { - fn num_rounds(&self) -> usize { - self.num_vars - } + fn compute_round_univariate_point_eval(&self) -> UniPoly { + let half = self.w_table.len() / 2; + let degree_q = 2 * self.b - 1; + let num_points_q = degree_q + 1; + let point_precomp = self + .point_precomp + .as_ref() + .expect("point-eval precomputation must exist"); + let range_offsets = &point_precomp.range_offsets; - fn degree_bound(&self) -> usize { - 2 * self.b - } + let (e_first, e_second) = self.split_eq.remaining_eq_tables(); + let num_first = e_first.len(); + let first_bits = num_first.trailing_zeros(); - fn input_claim(&self) -> E { - E::zero() + #[cfg(feature = "parallel")] + let q_evals = { + (0..half) + .into_par_iter() + .fold( + || vec![E::zero(); num_points_q], + |mut evals, j| { + let j_low = j & (num_first - 1); + let j_high = j >> first_bits; + let eq_rem = e_first[j_low] * e_second[j_high]; + let w_0 = self.w_table[2 * j]; + let w_1 = self.w_table[2 * j + 1]; + let delta = w_1 - w_0; + let mut w_t = w_0; + for eval in evals.iter_mut() { + *eval = + *eval + eq_rem * range_check_eval_precomputed(w_t, range_offsets); + w_t = w_t + delta; + } + evals + }, + ) + .reduce( + || vec![E::zero(); num_points_q], + |mut a, b_vec| { + for (ai, bi) in a.iter_mut().zip(b_vec.iter()) { + *ai = *ai + *bi; + } + a + }, + ) + }; + #[cfg(not(feature = "parallel"))] + let q_evals = { + let mut evals = vec![E::zero(); num_points_q]; + for j in 0..half { + let j_low = j & (num_first - 1); + let j_high = j >> first_bits; + let eq_rem = e_first[j_low] * e_second[j_high]; + let w_0 = self.w_table[2 * j]; + let w_1 = self.w_table[2 * j + 1]; + let delta = w_1 - w_0; + let mut w_t = w_0; + for eval in evals.iter_mut() { + *eval = *eval + eq_rem * range_check_eval_precomputed(w_t, range_offsets); + w_t = w_t + delta; + } + } + evals + }; + + let q_poly = UniPoly::from_evals(&q_evals); + self.split_eq.gruen_mul(&q_poly) } - fn compute_round_univariate(&mut self, _round: usize, _previous_claim: E) -> UniPoly { + fn compute_round_univariate_affine_coeff(&self) -> UniPoly { let half = self.w_table.len() / 2; - let num_coeffs_q = self.range_precomp.degree_q + 1; + let range_precomp = self + .range_precomp + .as_ref() + .expect("affine-coeff precomputation must exist"); + let num_coeffs_q = range_precomp.degree_q + 1; let (e_first, e_second) = self.split_eq.remaining_eq_tables(); let num_first = e_first.len(); let first_bits = num_first.trailing_zeros(); - let coeff_mix = &self.range_precomp.coeff_mix; + let coeff_mix = &range_precomp.coeff_mix; #[cfg(feature = "parallel")] let q_coeffs = { @@ -200,6 +325,27 @@ impl SumcheckInstanceProver for NormSumcheckProv let q_poly = UniPoly::from_coeffs(q_coeffs); self.split_eq.gruen_mul(&q_poly) } +} + +impl SumcheckInstanceProver for NormSumcheckProver { + fn num_rounds(&self) -> usize { + self.num_vars + } + + fn degree_bound(&self) -> usize { + 2 * self.b + } + + fn input_claim(&self) -> E { + E::zero() + } + + fn compute_round_univariate(&mut self, _round: usize, _previous_claim: E) -> UniPoly { + match self.round_kernel { + NormRoundKernel::PointEvalInterpolation => self.compute_round_univariate_point_eval(), + NormRoundKernel::AffineCoeffComposition => self.compute_round_univariate_affine_coeff(), + } + } fn ingest_challenge(&mut self, _round: usize, r: E) { self.split_eq.bind(r); @@ -276,14 +422,14 @@ mod tests { type Cfg = SmallTestCommitmentConfig; type Scheme = HachiCommitmentScheme<{ Cfg::D }, Cfg>; - struct LegacyNormSumcheckProver { + struct PointEvalReferenceNormSumcheckProver { split_eq: GruenSplitEq, w_table: Vec, num_vars: usize, b: usize, } - impl LegacyNormSumcheckProver { + impl PointEvalReferenceNormSumcheckProver { fn new(tau: &[E], w_evals: Vec, b: usize) -> Self { let num_vars = tau.len(); assert_eq!(w_evals.len(), 1 << num_vars); @@ -296,7 +442,9 @@ mod tests { } } - impl SumcheckInstanceProver for LegacyNormSumcheckProver { + impl SumcheckInstanceProver + for PointEvalReferenceNormSumcheckProver + { fn num_rounds(&self) -> usize { self.num_vars } @@ -349,12 +497,12 @@ mod tests { } #[test] - fn norm_sumcheck_matches_legacy_round_kernel() { + fn norm_sumcheck_runtime_dispatch_matches_reference_kernels() { let mut rng = StdRng::seed_from_u64(0xC0FFEE); - for case_idx in 0..8u64 { + for (case_idx, b) in [4usize, 8, 16].into_iter().enumerate() { + let case_idx = case_idx as u64; let num_vars = 6; let n = 1usize << num_vars; - let b = if case_idx % 2 == 0 { 8 } else { 16 }; let w_evals: Vec = (0..n) .map(|i| F::from_u64((i as u64 * 31 + case_idx * 17) % b as u64)) .collect(); @@ -362,39 +510,86 @@ mod tests { .map(|_| F::from_u64(rand::Rng::gen_range(&mut rng, 1u64..=257))) .collect(); - let mut optimized = NormSumcheckProver::new(&tau, w_evals.clone(), b); - let mut legacy = LegacyNormSumcheckProver::new(&tau, w_evals, b); + let mut dispatched = NormSumcheckProver::new(&tau, w_evals.clone(), b); + let mut point_eval = NormSumcheckProver::new_with_kernel( + &tau, + w_evals.clone(), + b, + NormRoundKernel::PointEvalInterpolation, + ); + let mut affine_coeff = NormSumcheckProver::new_with_kernel( + &tau, + w_evals.clone(), + b, + NormRoundKernel::AffineCoeffComposition, + ); + let mut reference = PointEvalReferenceNormSumcheckProver::new(&tau, w_evals, b); - let mut claim_opt = F::zero(); - let mut claim_old = F::zero(); + let mut claim_dispatched = F::zero(); + let mut claim_point = F::zero(); + let mut claim_affine = F::zero(); + let mut claim_reference = F::zero(); for round in 0..num_vars { - let g_opt = optimized.compute_round_univariate(round, claim_opt); - let g_old = legacy.compute_round_univariate(round, claim_old); + let g_dispatch = dispatched.compute_round_univariate(round, claim_dispatched); + let g_point = point_eval.compute_round_univariate(round, claim_point); + let g_affine = affine_coeff.compute_round_univariate(round, claim_affine); + let g_ref = reference.compute_round_univariate(round, claim_reference); + assert_eq!( - g_opt, g_old, - "round poly mismatch for case {case_idx} round {round}" + g_point, g_ref, + "point-eval mismatch for case {case_idx} round {round}" ); + assert_eq!( + g_affine, g_ref, + "affine-coeff mismatch for case {case_idx} round {round}" + ); + match choose_round_kernel(b) { + NormRoundKernel::PointEvalInterpolation => { + assert_eq!( + g_dispatch, g_point, + "dispatch mismatch for case {case_idx} round {round}" + ); + } + NormRoundKernel::AffineCoeffComposition => { + assert_eq!( + g_dispatch, g_affine, + "dispatch mismatch for case {case_idx} round {round}" + ); + } + } assert_eq!( - g_opt.evaluate(&F::zero()) + g_opt.evaluate(&F::one()), - claim_opt, - "optimized hint mismatch for case {case_idx} round {round}" + g_dispatch.evaluate(&F::zero()) + g_dispatch.evaluate(&F::one()), + claim_dispatched, + "dispatched hint mismatch for case {case_idx} round {round}" ); assert_eq!( - g_old.evaluate(&F::zero()) + g_old.evaluate(&F::one()), - claim_old, - "legacy hint mismatch for case {case_idx} round {round}" + g_ref.evaluate(&F::zero()) + g_ref.evaluate(&F::one()), + claim_reference, + "reference hint mismatch for case {case_idx} round {round}" ); let r = F::from_u64(rand::Rng::gen_range(&mut rng, 1u64..=257)); - claim_opt = g_opt.evaluate(&r); - claim_old = g_old.evaluate(&r); - optimized.ingest_challenge(round, r); - legacy.ingest_challenge(round, r); + claim_dispatched = g_dispatch.evaluate(&r); + claim_point = g_point.evaluate(&r); + claim_affine = g_affine.evaluate(&r); + claim_reference = g_ref.evaluate(&r); + dispatched.ingest_challenge(round, r); + point_eval.ingest_challenge(round, r); + affine_coeff.ingest_challenge(round, r); + reference.ingest_challenge(round, r); } assert_eq!( - claim_opt, claim_old, - "final claim mismatch for case {case_idx}" + claim_dispatched, claim_reference, + "final dispatched claim mismatch for case {case_idx}" + ); + assert_eq!( + claim_point, claim_reference, + "final point claim mismatch for case {case_idx}" + ); + assert_eq!( + claim_affine, claim_reference, + "final affine claim mismatch for case {case_idx}" ); } } From 7278414fe62b28973874cb65ff91ac4f8a8ad5fe Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Sat, 28 Feb 2026 16:40:19 -0800 Subject: [PATCH 17/34] Format commitment-related files for readability. Apply non-functional formatting and import ordering cleanups across commitment, ring-switch, and benchmark/test files to keep the codebase style consistent. Made-with: Cursor --- benches/hachi_e2e.rs | 2 +- src/protocol/commitment/commit.rs | 52 +++++++++++++++++++++++------- src/protocol/commitment/config.rs | 9 ++++-- src/protocol/quadratic_equation.rs | 3 +- src/protocol/ring_switch.rs | 4 ++- tests/ring_commitment_core.rs | 8 +++-- 6 files changed, 58 insertions(+), 20 deletions(-) diff --git a/benches/hachi_e2e.rs b/benches/hachi_e2e.rs index d21b2f3f..da0b5ced 100644 --- a/benches/hachi_e2e.rs +++ b/benches/hachi_e2e.rs @@ -2,8 +2,8 @@ use criterion::{black_box, criterion_group, criterion_main, BenchmarkId, Criterion}; use hachi_pcs::algebra::Fp128; -use hachi_pcs::error::HachiError; use hachi_pcs::primitives::multilinear_evals::DenseMultilinearEvals; +use hachi_pcs::error::HachiError; use hachi_pcs::protocol::commitment::{HachiCommitmentLayout, ProductionFp128CommitmentConfig}; use hachi_pcs::protocol::commitment_scheme::HachiCommitmentScheme; use hachi_pcs::protocol::transcript::Blake2bTranscript; diff --git a/src/protocol/commitment/commit.rs b/src/protocol/commitment/commit.rs index 5fa2c566..b0d13945 100644 --- a/src/protocol/commitment/commit.rs +++ b/src/protocol/commitment/commit.rs @@ -1,8 +1,8 @@ //! Ring-native §4.1 commitment core implementation. use super::config::{ - ensure_block_layout, ensure_matrix_shape, ensure_supported_num_vars, validate_and_derive_layout, - HachiCommitmentLayout, + ensure_block_layout, ensure_matrix_shape, ensure_supported_num_vars, + validate_and_derive_layout, HachiCommitmentLayout, }; use super::onehot::{inner_ajtai_onehot, map_onehot_to_sparse_blocks}; use super::scheme::{CommitWitness, RingCommitmentScheme}; @@ -99,7 +99,8 @@ impl HachiSerialize for HachiSetupSeed { mut writer: W, compress: Compress, ) -> Result<(), SerializationError> { - self.max_num_vars.serialize_with_mode(&mut writer, compress)?; + self.max_num_vars + .serialize_with_mode(&mut writer, compress)?; self.layout.serialize_with_mode(&mut writer, compress)?; writer.write_all(&self.public_matrix_seed)?; Ok(()) @@ -171,9 +172,21 @@ impl HachiDeserialize for HachiExpandedSet ) -> Result { let out = Self { seed: HachiSetupSeed::deserialize_with_mode(&mut reader, compress, validate)?, - A: Vec::>>::deserialize_with_mode(&mut reader, compress, validate)?, - B: Vec::>>::deserialize_with_mode(&mut reader, compress, validate)?, - D: Vec::>>::deserialize_with_mode(&mut reader, compress, validate)?, + A: Vec::>>::deserialize_with_mode( + &mut reader, + compress, + validate, + )?, + B: Vec::>>::deserialize_with_mode( + &mut reader, + compress, + validate, + )?, + D: Vec::>>::deserialize_with_mode( + &mut reader, + compress, + validate, + )?, }; if matches!(validate, Validate::Yes) { out.check()?; @@ -294,7 +307,12 @@ where let verifier_setup = HachiVerifierSetup { expanded }; ensure_matrix_shape(&prover_setup.expanded.A, Cfg::N_A, layout.inner_width, "A")?; ensure_matrix_shape(&prover_setup.expanded.B, Cfg::N_B, layout.outer_width, "B")?; - ensure_matrix_shape(&prover_setup.expanded.D, Cfg::N_D, layout.d_matrix_width, "D")?; + ensure_matrix_shape( + &prover_setup.expanded.D, + Cfg::N_D, + layout.d_matrix_width, + "D", + )?; Ok((prover_setup, verifier_setup)) } @@ -307,7 +325,10 @@ where setup: &Self::ProverSetup, ) -> Result, HachiError> { let layout = setup.layout(); - ensure_supported_num_vars(setup.expanded.seed.max_num_vars, layout.required_num_vars::()?)?; + ensure_supported_num_vars( + setup.expanded.seed.max_num_vars, + layout.required_num_vars::()?, + )?; ensure_block_layout(f_blocks, layout)?; ensure_matrix_shape(&setup.expanded.A, Cfg::N_A, layout.inner_width, "A")?; ensure_matrix_shape(&setup.expanded.B, Cfg::N_B, layout.outer_width, "B")?; @@ -340,7 +361,10 @@ where setup: &Self::ProverSetup, ) -> Result, HachiError> { let layout = setup.layout(); - ensure_supported_num_vars(setup.expanded.seed.max_num_vars, layout.required_num_vars::()?)?; + ensure_supported_num_vars( + setup.expanded.seed.max_num_vars, + layout.required_num_vars::()?, + )?; ensure_matrix_shape(&setup.expanded.A, Cfg::N_A, layout.inner_width, "A")?; ensure_matrix_shape(&setup.expanded.B, Cfg::N_B, layout.outer_width, "B")?; @@ -352,8 +376,12 @@ where let mut t_hat_flat: Vec> = Vec::with_capacity(layout.outer_width); for block_entries in &sparse_blocks { - let (t_i, s_i) = - inner_ajtai_onehot(&setup.expanded.A, block_entries, layout.block_len, Cfg::DELTA); + let (t_i, s_i) = inner_ajtai_onehot( + &setup.expanded.A, + block_entries, + layout.block_len, + Cfg::DELTA, + ); let t_hat_i = decompose_rows(&t_i, Cfg::DELTA, Cfg::LOG_BASIS); t_hat_flat.extend(t_hat_i.iter().copied()); @@ -374,7 +402,7 @@ where mod tests { use super::*; use crate::primitives::{HachiDeserialize, HachiSerialize}; - use crate::test_utils::{D as TestD, F as TestF, TinyConfig}; + use crate::test_utils::{TinyConfig, D as TestD, F as TestF}; #[test] fn prover_setup_roundtrips_and_derives_same_verifier() { diff --git a/src/protocol/commitment/config.rs b/src/protocol/commitment/config.rs index 2c9773da..d1187d85 100644 --- a/src/protocol/commitment/config.rs +++ b/src/protocol/commitment/config.rs @@ -103,9 +103,12 @@ impl HachiSerialize for HachiCommitmentLayout { self.r_vars.serialize_with_mode(&mut writer, compress)?; self.num_blocks.serialize_with_mode(&mut writer, compress)?; self.block_len.serialize_with_mode(&mut writer, compress)?; - self.inner_width.serialize_with_mode(&mut writer, compress)?; - self.outer_width.serialize_with_mode(&mut writer, compress)?; - self.d_matrix_width.serialize_with_mode(&mut writer, compress)?; + self.inner_width + .serialize_with_mode(&mut writer, compress)?; + self.outer_width + .serialize_with_mode(&mut writer, compress)?; + self.d_matrix_width + .serialize_with_mode(&mut writer, compress)?; Ok(()) } diff --git a/src/protocol/quadratic_equation.rs b/src/protocol/quadratic_equation.rs index 9938c806..27a94def 100644 --- a/src/protocol/quadratic_equation.rs +++ b/src/protocol/quadratic_equation.rs @@ -201,7 +201,8 @@ where y_ring: &CyclotomicRing, ) -> Result { let layout = setup.expanded.seed.layout; - let challenges = derive_stage1_challenges::(transcript, v, layout.num_blocks)?; + let challenges = + derive_stage1_challenges::(transcript, v, layout.num_blocks)?; let m = generate_m::(&setup.expanded, ring_opening_point, &challenges)?; let y = generate_y::(v, &commitment.u, y_ring)?; diff --git a/src/protocol/ring_switch.rs b/src/protocol/ring_switch.rs index a5c40ca5..19224188 100644 --- a/src/protocol/ring_switch.rs +++ b/src/protocol/ring_switch.rs @@ -253,7 +253,9 @@ impl CommitmentConfig for WCommitmentConf const TAU: usize = Cfg::TAU; const CHALLENGE_WEIGHT: usize = Cfg::CHALLENGE_WEIGHT; - fn commitment_layout(max_num_vars: usize) -> Result { + fn commitment_layout( + max_num_vars: usize, + ) -> Result { let alpha = D.trailing_zeros() as usize; let m_vars = max_num_vars.checked_sub(alpha).ok_or_else(|| { HachiError::InvalidSetup("max_num_vars is smaller than alpha".to_string()) diff --git a/tests/ring_commitment_core.rs b/tests/ring_commitment_core.rs index 49f19880..51e1d5a2 100644 --- a/tests/ring_commitment_core.rs +++ b/tests/ring_commitment_core.rs @@ -21,7 +21,9 @@ impl CommitmentConfig for BadDegreeConfig { const TAU: usize = 4; const CHALLENGE_WEIGHT: usize = 3; - fn commitment_layout(_max_num_vars: usize) -> Result { + fn commitment_layout( + _max_num_vars: usize, + ) -> Result { HachiCommitmentLayout::new::(4, 2) } } @@ -39,7 +41,9 @@ impl CommitmentConfig for BadDigitBudgetConfig { const TAU: usize = 4; const CHALLENGE_WEIGHT: usize = 3; - fn commitment_layout(_max_num_vars: usize) -> Result { + fn commitment_layout( + _max_num_vars: usize, + ) -> Result { HachiCommitmentLayout::new::(4, 2) } } From 1a2b8464fe3667217fbeda8362e68c02319cc807 Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Sat, 28 Feb 2026 17:06:26 -0800 Subject: [PATCH 18/34] Format: cargo fmt pass on commitment-related files Made-with: Cursor --- benches/hachi_e2e.rs | 2 +- src/protocol/ring_switch.rs | 4 +--- tests/ring_commitment_core.rs | 8 ++------ 3 files changed, 4 insertions(+), 10 deletions(-) diff --git a/benches/hachi_e2e.rs b/benches/hachi_e2e.rs index da0b5ced..d21b2f3f 100644 --- a/benches/hachi_e2e.rs +++ b/benches/hachi_e2e.rs @@ -2,8 +2,8 @@ use criterion::{black_box, criterion_group, criterion_main, BenchmarkId, Criterion}; use hachi_pcs::algebra::Fp128; -use hachi_pcs::primitives::multilinear_evals::DenseMultilinearEvals; use hachi_pcs::error::HachiError; +use hachi_pcs::primitives::multilinear_evals::DenseMultilinearEvals; use hachi_pcs::protocol::commitment::{HachiCommitmentLayout, ProductionFp128CommitmentConfig}; use hachi_pcs::protocol::commitment_scheme::HachiCommitmentScheme; use hachi_pcs::protocol::transcript::Blake2bTranscript; diff --git a/src/protocol/ring_switch.rs b/src/protocol/ring_switch.rs index 19224188..a5c40ca5 100644 --- a/src/protocol/ring_switch.rs +++ b/src/protocol/ring_switch.rs @@ -253,9 +253,7 @@ impl CommitmentConfig for WCommitmentConf const TAU: usize = Cfg::TAU; const CHALLENGE_WEIGHT: usize = Cfg::CHALLENGE_WEIGHT; - fn commitment_layout( - max_num_vars: usize, - ) -> Result { + fn commitment_layout(max_num_vars: usize) -> Result { let alpha = D.trailing_zeros() as usize; let m_vars = max_num_vars.checked_sub(alpha).ok_or_else(|| { HachiError::InvalidSetup("max_num_vars is smaller than alpha".to_string()) diff --git a/tests/ring_commitment_core.rs b/tests/ring_commitment_core.rs index 51e1d5a2..49f19880 100644 --- a/tests/ring_commitment_core.rs +++ b/tests/ring_commitment_core.rs @@ -21,9 +21,7 @@ impl CommitmentConfig for BadDegreeConfig { const TAU: usize = 4; const CHALLENGE_WEIGHT: usize = 3; - fn commitment_layout( - _max_num_vars: usize, - ) -> Result { + fn commitment_layout(_max_num_vars: usize) -> Result { HachiCommitmentLayout::new::(4, 2) } } @@ -41,9 +39,7 @@ impl CommitmentConfig for BadDigitBudgetConfig { const TAU: usize = 4; const CHALLENGE_WEIGHT: usize = 3; - fn commitment_layout( - _max_num_vars: usize, - ) -> Result { + fn commitment_layout(_max_num_vars: usize) -> Result { HachiCommitmentLayout::new::(4, 2) } } From c8565bbfa85069e4f7a4c5d886dff64bf650f51d Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Sat, 28 Feb 2026 21:37:43 -0800 Subject: [PATCH 19/34] feat: sequential coefficient ordering + streaming commitment Change coefficient-to-ring packing from strided to sequential, enabling true streaming where each trace chunk maps to exactly one inner Ajtai block. Implement StreamingCommitmentScheme for HachiCommitmentScheme. - reduce_coeffs_to_ring_elements: sequential packing (chunks_exact(D)) - prove/verify: opening point split flipped to (inner, outer) - ring_opening_point_from_field: outer split flipped to (M first, R second) - commit_coeffs: sequential block distribution - map_onehot_to_sparse_blocks: sequential block distribution - HachiChunkState + process_chunk / process_chunk_onehot / aggregate_chunks - Streaming commit tests (matches non-streaming, prove/verify roundtrip) Made-with: Cursor --- src/protocol/commitment/onehot.rs | 10 +- src/protocol/commitment/scheme.rs | 32 +--- src/protocol/commitment_scheme.rs | 304 ++++++++++++++++++++++++++---- src/protocol/mod.rs | 2 +- tests/ring_commitment_core.rs | 13 +- 5 files changed, 289 insertions(+), 72 deletions(-) diff --git a/src/protocol/commitment/onehot.rs b/src/protocol/commitment/onehot.rs index 8338920b..935505b3 100644 --- a/src/protocol/commitment/onehot.rs +++ b/src/protocol/commitment/onehot.rs @@ -89,14 +89,12 @@ pub(crate) fn map_onehot_to_sparse_blocks( .push(coeff_idx); } - // Distribute into blocks using the same interleaved layout as commit_coeffs: - // block_idx = ring_elem_idx % num_blocks - // pos_in_block = ring_elem_idx / num_blocks - // (equivalently: flat_idx = (pos_in_block << R) + block_idx) + // Sequential block layout matching commit_coeffs: block i = ring elements + // [i*block_len, (i+1)*block_len). let mut blocks: Vec> = vec![Vec::new(); num_blocks]; for (ring_elem_idx, nonzero_coeffs) in ring_elem_map { - let block_idx = ring_elem_idx & (num_blocks - 1); - let pos_in_block = ring_elem_idx >> r; + let block_idx = ring_elem_idx / block_len; + let pos_in_block = ring_elem_idx % block_len; blocks[block_idx].push(SparseBlockEntry { pos_in_block, nonzero_coeffs, diff --git a/src/protocol/commitment/scheme.rs b/src/protocol/commitment/scheme.rs index be1be7b9..e1d10b0c 100644 --- a/src/protocol/commitment/scheme.rs +++ b/src/protocol/commitment/scheme.rs @@ -178,21 +178,16 @@ where /// Commit to a flat coefficient table `(f_i)_{i∈{0,1}^ℓ}` in ring form. /// - /// The input is indexed in LSB-first order (same convention as - /// `DenseMultilinearEvals`). We split the variables as: - /// - /// - outer index `i` for the first `R` variables, - /// - inner index `j` for the last `M` variables, - /// - /// We then form blocks `f_i = (f_{i||j})_{j}` (Eq. (12) in the paper). - /// - /// This prepares `f_blocks` and delegates to `commit_ring_blocks`. + /// The input uses sequential block layout: ring elements + /// `[0, block_len)` form block 0, `[block_len, 2*block_len)` form + /// block 1, and so on. This matches the sequential variable ordering + /// where M variables (position in block) are lower-order and R variables + /// (block selection) are higher-order. /// /// # Errors /// /// Returns an error if `f_coeffs.len()` does not match the configured block - /// layout, if internal index computations overflow, or if the underlying - /// commitment routine fails. + /// layout or if the underlying commitment routine fails. fn commit_coeffs( f_coeffs: &[CyclotomicRing], setup: &Self::ProverSetup, @@ -210,17 +205,10 @@ where }); } - let mut blocks = Vec::with_capacity(num_blocks); - for i in 0..num_blocks { - let mut block = Vec::with_capacity(block_len); - for j in 0..block_len { - let idx = (j << layout.r_vars) - .checked_add(i) - .ok_or_else(|| HachiError::InvalidSetup("index overflow".to_string()))?; - block.push(f_coeffs[idx]); - } - blocks.push(block); - } + let blocks: Vec>> = f_coeffs + .chunks_exact(block_len) + .map(|chunk| chunk.to_vec()) + .collect(); Self::commit_ring_blocks(&blocks, setup) } diff --git a/src/protocol/commitment_scheme.rs b/src/protocol/commitment_scheme.rs index 6b4459a8..09acf4c2 100644 --- a/src/protocol/commitment_scheme.rs +++ b/src/protocol/commitment_scheme.rs @@ -6,9 +6,13 @@ use crate::error::HachiError; #[cfg(feature = "parallel")] use crate::parallel::*; use crate::primitives::poly::multilinear_lagrange_basis; +use crate::protocol::commitment::onehot::{inner_ajtai_onehot, SparseBlockEntry}; +use crate::protocol::commitment::utils::linear::{ + decompose_block, decompose_rows, mat_vec_mul_ntt_cached, MatrixSlot, +}; use crate::protocol::commitment::{ AppendToTranscript, CommitmentConfig, CommitmentScheme, HachiCommitmentCore, HachiProverSetup, - HachiVerifierSetup, RingCommitment, RingCommitmentScheme, + HachiVerifierSetup, RingCommitment, RingCommitmentScheme, StreamingCommitmentScheme, }; use crate::protocol::opening_point::RingOpeningPoint; use crate::protocol::proof::{HachiCommitmentHint, HachiProof, SumcheckAux}; @@ -98,27 +102,29 @@ where let hint = hint.ok_or_else(|| { HachiError::InvalidInput("missing commitment hint for proving".to_string()) })?; - let num_vars = poly.num_vars(); + let _num_vars = poly.num_vars(); let alpha = Cfg::D.trailing_zeros() as usize; - let reduced_len = num_vars - .checked_sub(alpha) - .ok_or_else(|| HachiError::InvalidSetup("reduction length underflow".to_string()))?; - if opening_point.len() < reduced_len { + if opening_point.len() < alpha { return Err(HachiError::InvalidPointDimension { - expected: reduced_len, + expected: alpha, actual: opening_point.len(), }); } + let outer_point = &opening_point[alpha..]; let layout = >::layout(setup)?; - let ring_opening_point = ring_opening_point_from_field::( - &opening_point[..reduced_len], - layout.r_vars, - layout.m_vars, - )?; + let expected_outer = layout.r_vars + layout.m_vars; + if outer_point.len() != expected_outer { + return Err(HachiError::InvalidPointDimension { + expected: expected_outer + alpha, + actual: opening_point.len(), + }); + } - let y_ring = - evaluate_packed_ring_poly::(&hint.ring_coeffs, &opening_point[..reduced_len]); + let ring_opening_point = + ring_opening_point_from_field::(outer_point, layout.r_vars, layout.m_vars)?; + + let y_ring = evaluate_packed_ring_poly::(&hint.ring_coeffs, outer_point); // Fiat-Shamir: bind commitment, opening point, and y_ring before any challenges. commitment.append_to_transcript(ABSORB_COMMITMENT, transcript); @@ -175,11 +181,13 @@ where commitment: &Self::Commitment, ) -> Result<(), HachiError> { let alpha_bits = Cfg::D.trailing_zeros() as usize; - let reduced_len = opening_point.len().checked_sub(alpha_bits).ok_or_else(|| { - HachiError::InvalidSetup("opening point length underflow".to_string()) - })?; - let reduced_opening_point = &opening_point[..reduced_len]; - let inner_point = &opening_point[reduced_len..]; + if opening_point.len() < alpha_bits { + return Err(HachiError::InvalidSetup( + "opening point length underflow".to_string(), + )); + } + let inner_point = &opening_point[..alpha_bits]; + let reduced_opening_point = &opening_point[alpha_bits..]; // Fiat-Shamir: bind commitment, opening point, and y_ring before any challenges. commitment.append_to_transcript(ABSORB_COMMITMENT, transcript); @@ -261,6 +269,151 @@ where } } +/// Per-block intermediate state for streaming Hachi commitment. +/// +/// Each chunk corresponds to one Ajtai inner block: `D * block_len` field +/// elements packed into `block_len` ring elements, decomposed, and multiplied +/// by the inner matrix A. +#[derive(Clone, PartialEq, Eq)] +pub struct HachiChunkState { + /// Original ring elements for this block (needed for `ring_coeffs` hint). + pub block: Vec>, + /// Basis-decomposed input vector `s_i = G^{-1}(block)`. + pub s_i: Vec>, + /// Basis-decomposed inner Ajtai output `t̂_i = G^{-1}(A · s_i)`. + pub t_hat_i: Vec>, +} + +impl std::fmt::Debug for HachiChunkState { + fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result { + f.debug_struct("HachiChunkState") + .field("block_len", &self.block.len()) + .field("s_i_len", &self.s_i.len()) + .field("t_hat_i_len", &self.t_hat_i.len()) + .finish() + } +} + +impl StreamingCommitmentScheme for HachiCommitmentScheme +where + F: FieldCore + CanonicalField + FieldSampling, + Cfg: CommitmentConfig, +{ + type ChunkState = HachiChunkState; + + fn process_chunk(setup: &Self::ProverSetup, chunk: &[F]) -> Self::ChunkState { + assert!( + chunk.len() % D == 0, + "chunk length {} is not divisible by D={}", + chunk.len(), + D + ); + + let block: Vec> = chunk + .chunks_exact(D) + .map(|c| CyclotomicRing::from_coefficients(std::array::from_fn(|j| c[j]))) + .collect(); + + let s_i = decompose_block(&block, Cfg::DELTA, Cfg::LOG_BASIS); + let t_i = + mat_vec_mul_ntt_cached(setup.ntt_cache().expect("NTT cache"), MatrixSlot::A, &s_i) + .expect("inner Ajtai"); + let t_hat_i = decompose_rows(&t_i, Cfg::DELTA, Cfg::LOG_BASIS); + + HachiChunkState { + block, + s_i, + t_hat_i, + } + } + + fn process_chunk_onehot( + setup: &Self::ProverSetup, + onehot_k: usize, + chunk: &[Option], + ) -> Self::ChunkState { + let layout = >::layout(setup) + .expect("layout"); + let block_len = layout.block_len; + + let num_field_elems = chunk.len() * onehot_k; + assert!( + num_field_elems % D == 0, + "chunk cycles * K = {num_field_elems} is not divisible by D={D}", + ); + + // Build sparse entries and original block ring elements. + let num_ring_elems = num_field_elems / D; + let mut ring_block = vec![CyclotomicRing::::zero(); num_ring_elems]; + let mut ring_elem_map: std::collections::BTreeMap> = + std::collections::BTreeMap::new(); + for (c, opt) in chunk.iter().enumerate() { + if let Some(k) = opt { + let field_pos = c * onehot_k + k; + let ring_elem_idx = field_pos / D; + let coeff_idx = field_pos % D; + ring_block[ring_elem_idx].coeffs[coeff_idx] = F::one(); + ring_elem_map + .entry(ring_elem_idx) + .or_default() + .push(coeff_idx); + } + } + + let sparse_entries: Vec = ring_elem_map + .into_iter() + .map(|(ring_elem_idx, nonzero_coeffs)| SparseBlockEntry { + pos_in_block: ring_elem_idx, + nonzero_coeffs, + }) + .collect(); + + let (t_i, s_i) = + inner_ajtai_onehot(&setup.expanded.A, &sparse_entries, block_len, Cfg::DELTA); + let t_hat_i = decompose_rows(&t_i, Cfg::DELTA, Cfg::LOG_BASIS); + + HachiChunkState { + block: ring_block, + s_i, + t_hat_i, + } + } + + fn aggregate_chunks( + setup: &Self::ProverSetup, + _onehot_k: Option, + chunks: &[Self::ChunkState], + ) -> (Self::Commitment, Self::OpeningProofHint) { + let t_hat_flat: Vec> = chunks + .iter() + .flat_map(|c| c.t_hat_i.iter().copied()) + .collect(); + + let u = mat_vec_mul_ntt_cached( + setup.ntt_cache().expect("NTT cache"), + MatrixSlot::B, + &t_hat_flat, + ) + .expect("outer Ajtai"); + + let s_all: Vec>> = chunks.iter().map(|c| c.s_i.clone()).collect(); + let t_hat_all: Vec>> = + chunks.iter().map(|c| c.t_hat_i.clone()).collect(); + let ring_coeffs: Vec> = chunks + .iter() + .flat_map(|c| c.block.iter().copied()) + .collect(); + + let commitment = RingCommitment { u }; + let hint = HachiCommitmentHint { + s: s_all, + t_hat: t_hat_all, + ring_coeffs, + }; + (commitment, hint) + } +} + /// Re-derive the ring-switch challenge `alpha` and the expanded `M_a` vector /// by replaying the transcript from the proof data and setup, exactly as the /// verifier does. @@ -276,13 +429,14 @@ where Cfg: CommitmentConfig, { let alpha_bits = Cfg::D.trailing_zeros() as usize; - let reduced_len = opening_point - .len() - .checked_sub(alpha_bits) - .ok_or_else(|| HachiError::InvalidSetup("opening point length underflow".to_string()))?; + if opening_point.len() < alpha_bits { + return Err(HachiError::InvalidSetup( + "opening point length underflow".to_string(), + )); + } let layout = setup.expanded.seed.layout; let ring_opening_point = ring_opening_point_from_field::( - &opening_point[..reduced_len], + &opening_point[alpha_bits..], layout.r_vars, layout.m_vars, )?; @@ -332,8 +486,10 @@ fn ring_opening_point_from_field( }); } - let b = lagrange_weights(&opening_point[..r_vars]); - let a = lagrange_weights(&opening_point[r_vars..]); + // Sequential ordering: M variables (position in block) come first, + // R variables (block selection) come second. + let a = lagrange_weights(&opening_point[..m_vars]); + let b = lagrange_weights(&opening_point[m_vars..]); Ok(RingOpeningPoint { a, b }) } @@ -363,17 +519,13 @@ fn reduce_coeffs_to_ring_elements( }); } - let outer_vars = num_vars - alpha; - let outer_len = 1usize - .checked_shl(outer_vars as u32) - .ok_or_else(|| HachiError::InvalidInput(format!("2^{outer_vars} does not fit usize")))?; - + // Sequential packing: ring element i = coeffs[i*D .. (i+1)*D]. + // The first alpha variables (LSBs) become coefficient slots within each + // ring element; the remaining outer_vars variables index ring elements. + let outer_len = expected_len / D; let out: Vec> = cfg_into_iter!(0..outer_len) .map(|i| { - let ring_coeffs = std::array::from_fn(|j| { - let idx = i + (j << outer_vars); - coeffs[idx] - }); + let ring_coeffs = std::array::from_fn(|j| coeffs[i * D + j]); CyclotomicRing::from_coefficients(ring_coeffs) }) .collect(); @@ -526,4 +678,86 @@ mod tests { "verify must reject an incorrect opening value" ); } + + #[test] + fn streaming_commit_matches_non_streaming() { + let alpha = Cfg::D.trailing_zeros() as usize; + let layout = Cfg::commitment_layout(16).unwrap(); + let num_vars = layout.m_vars + layout.r_vars + alpha; + let len = 1usize << num_vars; + + let evals: Vec = (0..len).map(|i| F::from_u64(i as u64)).collect(); + let poly = DenseMultilinearEvals::new_padded(evals.clone()); + + let setup = >::setup_prover(num_vars); + + // Non-streaming commit + let (non_streaming_commitment, non_streaming_hint) = + >::commit(&poly, &setup).unwrap(); + + // Streaming commit: split field elements into chunks of D * block_len + let chunk_size = Cfg::D * layout.block_len; + let chunks: Vec> = evals + .chunks_exact(chunk_size) + .map(|chunk| >::process_chunk(&setup, chunk)) + .collect(); + + let (streaming_commitment, streaming_hint) = + >::aggregate_chunks(&setup, None, &chunks); + + assert_eq!(non_streaming_commitment, streaming_commitment); + assert_eq!(non_streaming_hint, streaming_hint); + } + + #[test] + fn streaming_commit_then_prove_verify() { + let alpha = Cfg::D.trailing_zeros() as usize; + let layout = Cfg::commitment_layout(16).unwrap(); + let num_vars = layout.m_vars + layout.r_vars + alpha; + let len = 1usize << num_vars; + + let evals: Vec = (0..len).map(|i| F::from_u64(i as u64)).collect(); + let poly = DenseMultilinearEvals::new_padded(evals.clone()); + + let setup = >::setup_prover(num_vars); + let verifier_setup = >::setup_verifier(&setup); + + // Streaming commit + let chunk_size = Cfg::D * layout.block_len; + let chunks: Vec> = evals + .chunks_exact(chunk_size) + .map(|chunk| >::process_chunk(&setup, chunk)) + .collect(); + let (commitment, hint) = + >::aggregate_chunks(&setup, None, &chunks); + + // Prove and verify with streaming-produced commitment + hint + let opening_point: Vec = (0..num_vars).map(|i| F::from_u64((i + 2) as u64)).collect(); + let opening = poly.evaluate(&opening_point); + + let mut prover_transcript = Blake2bTranscript::::new(b"test/stream"); + let proof = >::prove( + &setup, + &poly, + &opening_point, + Some(hint), + &mut prover_transcript, + &commitment, + ) + .unwrap(); + + let mut verifier_transcript = Blake2bTranscript::::new(b"test/stream"); + let result = >::verify( + &proof, + &verifier_setup, + &mut verifier_transcript, + &opening_point, + &opening, + &commitment, + ); + assert!( + result.is_ok(), + "streaming commit should produce valid proofs" + ); + } } diff --git a/src/protocol/mod.rs b/src/protocol/mod.rs index dcfba23f..e19bfdf2 100644 --- a/src/protocol/mod.rs +++ b/src/protocol/mod.rs @@ -21,7 +21,7 @@ pub use commitment::{ ProductionFp128CommitmentConfig, RingCommitment, RingCommitmentScheme, SmallTestCommitmentConfig, StreamingCommitmentScheme, }; -pub use commitment_scheme::HachiCommitmentScheme; +pub use commitment_scheme::{HachiChunkState, HachiCommitmentScheme}; pub use opening_point::RingOpeningPoint; pub use proof::{HachiProof, SumcheckAux}; pub use quadratic_equation::QuadraticEquation; diff --git a/tests/ring_commitment_core.rs b/tests/ring_commitment_core.rs index 49f19880..be29de97 100644 --- a/tests/ring_commitment_core.rs +++ b/tests/ring_commitment_core.rs @@ -112,14 +112,11 @@ fn commit_ring_coeffs_matches_block_commitment() { ) .unwrap(); - let num_blocks = hachi_pcs::test_utils::NUM_BLOCKS; - let block_len = hachi_pcs::test_utils::BLOCK_LEN; - let mut f_coeffs = Vec::with_capacity(num_blocks * block_len); - for j in 0..block_len { - for block in blocks.iter().take(num_blocks) { - f_coeffs.push(block[j]); - } - } + // Sequential layout: block 0 elements, then block 1 elements, etc. + let f_coeffs: Vec<_> = blocks + .iter() + .flat_map(|block| block.iter().copied()) + .collect(); let wc = >::commit_coeffs( &f_coeffs, &psetup, From a7bdea9679e735c232f2ae315a5bff4cb18094e9 Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Sun, 1 Mar 2026 21:56:54 -0800 Subject: [PATCH 20/34] refactor: decompose verify_batched_sumcheck into composable steps Split the monolithic verify_batched_sumcheck into three pieces: - verify_batched_sumcheck_rounds: replay rounds, return intermediate state - compute_batched_expected_output_claim: query verifier instances - check_batched_output_claim: enforce equality This enables callers (e.g. Greyhound) to intercept the intermediate sumcheck state before the final oracle check. The original function is preserved as a convenience wrapper. Made-with: Cursor --- src/protocol/sumcheck/batched_sumcheck.rs | 103 ++++++++++++++++++++-- 1 file changed, 94 insertions(+), 9 deletions(-) diff --git a/src/protocol/sumcheck/batched_sumcheck.rs b/src/protocol/sumcheck/batched_sumcheck.rs index c031b1cb..5caa9810 100644 --- a/src/protocol/sumcheck/batched_sumcheck.rs +++ b/src/protocol/sumcheck/batched_sumcheck.rs @@ -33,6 +33,24 @@ fn linear_combination(polys: &[UniPoly], coeffs: &[E]) -> UniPo UniPoly::from_coeffs(result) } +/// Verifier-side output of the batched sumcheck round replay. +/// +/// This carries all transcript-derived values needed for the final oracle check, +/// which is intentionally split out so callers can compute the expected output +/// claim through an external reduction (e.g. Greyhound) before enforcing +/// equality. +#[derive(Debug, Clone, PartialEq, Eq)] +pub struct BatchedSumcheckRoundResult { + /// Final claim produced by replaying all sumcheck rounds. + pub output_claim: E, + /// Challenge vector sampled during replay. + pub r_sumcheck: Vec, + /// Front-loaded batching coefficient per verifier instance. + pub batching_coeffs: Vec, + /// Maximum number of rounds among batched instances. + pub max_num_rounds: usize, +} + /// Produce a batched sumcheck proof for multiple instances sharing the same /// variable space, driving the Fiat–Shamir transcript. /// @@ -181,10 +199,10 @@ where /// - absorbs each verifier instance's initial claim, /// - re-derives the batching coefficients, /// - computes the batched initial claim, -/// - verifies the proof against the batched claim, -/// - checks that the final output matches the batched expected output claims. +/// - verifies the proof against the batched claim. /// -/// Returns the challenge vector on success. +/// Returns transcript-derived verifier data for the caller to perform the final +/// expected-output equality check. /// /// # Panics /// @@ -192,14 +210,13 @@ where /// /// # Errors /// -/// Returns [`HachiError::InvalidProof`] if the batched output claim does not -/// match the expected value, or propagates per-round verification errors. -pub fn verify_batched_sumcheck( +/// Propagates per-round verification errors. +pub fn verify_batched_sumcheck_rounds( proof: &SumcheckProof, verifiers: Vec<&dyn SumcheckInstanceVerifier>, transcript: &mut T, mut sample_challenge: S, -) -> Result, HachiError> +) -> Result, HachiError> where F: FieldCore + crate::CanonicalField, T: Transcript, @@ -245,7 +262,26 @@ where &mut sample_challenge, )?; - // Compute the expected batched output claim from each verifier instance. + Ok(BatchedSumcheckRoundResult { + output_claim, + r_sumcheck, + batching_coeffs, + max_num_rounds, + }) +} + +/// Compute the expected batched output claim from verifier instances and +/// transcript-derived batching data. +/// +/// # Errors +/// +/// Propagates errors from verifier `expected_output_claim` calls. +pub fn compute_batched_expected_output_claim( + verifiers: Vec<&dyn SumcheckInstanceVerifier>, + batching_coeffs: &[E], + max_num_rounds: usize, + r_sumcheck: &[E], +) -> Result { let expected_output_claim: E = verifiers .iter() .zip(batching_coeffs.iter()) @@ -256,9 +292,58 @@ where }) .try_fold(E::zero(), |a, v| v.map(|val| a + val))?; + Ok(expected_output_claim) +} + +/// Enforce final batched output-claim equality. +/// +/// # Errors +/// +/// Returns an error if `output_claim != expected_output_claim`. +pub fn check_batched_output_claim( + output_claim: E, + expected_output_claim: E, +) -> Result<(), HachiError> { if output_claim != expected_output_claim { return Err(HachiError::InvalidProof); } - Ok(r_sumcheck) + Ok(()) +} + +/// Verify a batched sumcheck proof, including final expected-output equality. +/// +/// This convenience wrapper preserves the previous behavior. Callers that need +/// to inject an external reduction should use [`verify_batched_sumcheck_rounds`] +/// and [`check_batched_output_claim`] directly. +/// +/// # Errors +/// +/// Propagates errors from round verification and output-claim equality check. +pub fn verify_batched_sumcheck( + proof: &SumcheckProof, + verifiers: Vec<&dyn SumcheckInstanceVerifier>, + transcript: &mut T, + mut sample_challenge: S, +) -> Result, HachiError> +where + F: FieldCore + crate::CanonicalField, + T: Transcript, + E: FieldCore, + S: FnMut(&mut T) -> E, +{ + let round_result = verify_batched_sumcheck_rounds::( + proof, + verifiers.clone(), + transcript, + &mut sample_challenge, + )?; + let expected_output_claim = compute_batched_expected_output_claim( + verifiers, + &round_result.batching_coeffs, + round_result.max_num_rounds, + &round_result.r_sumcheck, + )?; + check_batched_output_claim(round_result.output_claim, expected_output_claim)?; + Ok(round_result.r_sumcheck) } From 6705efe58be1c26b0e764a4e4f59865b51bf1c19 Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Sun, 1 Mar 2026 22:31:00 -0800 Subject: [PATCH 21/34] feat: accept Option in commit_onehot for sparse one-hot support Allows None entries in one-hot index arrays to represent inactive cycles. Adds public commit_onehot free function returning both commitment and hint. Made-with: Cursor --- src/protocol/commitment/commit.rs | 2 +- src/protocol/commitment/onehot.rs | 22 ++++++++++---- src/protocol/commitment/scheme.rs | 5 ++-- src/protocol/commitment_scheme.rs | 49 +++++++++++++++++++++++++++++++ src/protocol/mod.rs | 2 +- tests/onehot_commitment.rs | 26 +++++++++++----- 6 files changed, 88 insertions(+), 18 deletions(-) diff --git a/src/protocol/commitment/commit.rs b/src/protocol/commitment/commit.rs index b0d13945..7f750f51 100644 --- a/src/protocol/commitment/commit.rs +++ b/src/protocol/commitment/commit.rs @@ -357,7 +357,7 @@ where fn commit_onehot( onehot_k: usize, - indices: &[usize], + indices: &[Option], setup: &Self::ProverSetup, ) -> Result, HachiError> { let layout = setup.layout(); diff --git a/src/protocol/commitment/onehot.rs b/src/protocol/commitment/onehot.rs index 935505b3..1c661b32 100644 --- a/src/protocol/commitment/onehot.rs +++ b/src/protocol/commitment/onehot.rs @@ -37,7 +37,7 @@ pub(crate) struct SparseBlockEntry { /// fill the commitment layout. pub(crate) fn map_onehot_to_sparse_blocks( onehot_k: usize, - indices: &[usize], + indices: &[Option], r: usize, m: usize, d: usize, @@ -74,7 +74,8 @@ pub(crate) fn map_onehot_to_sparse_blocks( // Accumulate nonzero coefficients per ring element index. let mut ring_elem_map: BTreeMap> = BTreeMap::new(); - for (c, &idx) in indices.iter().enumerate() { + for (c, opt) in indices.iter().enumerate() { + let Some(&idx) = opt.as_ref() else { continue }; if idx >= onehot_k { return Err(HachiError::InvalidInput(format!( "index {idx} out of range for chunk size K={onehot_k} at position {c}" @@ -159,7 +160,7 @@ mod tests { // R=1 (2 blocks), M=2 (4 per block) => 8 ring elements total let k = 16; let d = 4; - let indices = vec![3, 10]; + let indices = vec![Some(3), Some(10)]; let blocks = map_onehot_to_sparse_blocks(k, &indices, 1, 2, d).unwrap(); assert_eq!(blocks.len(), 2); @@ -179,7 +180,7 @@ mod tests { // R=1 (2 blocks), M=1 (2 per block) let k = 4; let d = 4; - let indices = vec![0, 2, 3, 1]; + let indices = vec![Some(0), Some(2), Some(3), Some(1)]; let blocks = map_onehot_to_sparse_blocks(k, &indices, 1, 1, d).unwrap(); assert_eq!(blocks.len(), 2); @@ -199,7 +200,16 @@ mod tests { // R=1 (2 blocks), M=1 (2 per block) let k = 4; let d = 8; - let indices = vec![0, 2, 3, 1, 0, 0, 3, 3]; + let indices = vec![ + Some(0), + Some(2), + Some(3), + Some(1), + Some(0), + Some(0), + Some(3), + Some(3), + ]; let blocks = map_onehot_to_sparse_blocks(k, &indices, 1, 1, d).unwrap(); assert_eq!(blocks.len(), 2); @@ -219,7 +229,7 @@ mod tests { #[test] fn map_onehot_rejects_non_divisible() { - let result = map_onehot_to_sparse_blocks(3, &[0, 1], 0, 1, 4); + let result = map_onehot_to_sparse_blocks(3, &[Some(0), Some(1)], 0, 1, 4); assert!(result.is_err()); } diff --git a/src/protocol/commitment/scheme.rs b/src/protocol/commitment/scheme.rs index e1d10b0c..e59381f3 100644 --- a/src/protocol/commitment/scheme.rs +++ b/src/protocol/commitment/scheme.rs @@ -233,7 +233,7 @@ where /// of range. fn commit_onehot( onehot_k: usize, - indices: &[usize], + indices: &[Option], setup: &Self::ProverSetup, ) -> Result, HachiError> { let num_chunks = indices.len(); @@ -249,7 +249,8 @@ where // Materialize the full one-hot vector as ring elements. let total_ring_elems = total_field_elems / D; let mut ring_coeffs = vec![CyclotomicRing::::zero(); total_ring_elems]; - for (c, &idx) in indices.iter().enumerate() { + for (c, opt) in indices.iter().enumerate() { + let Some(&idx) = opt.as_ref() else { continue }; if idx >= onehot_k { return Err(HachiError::InvalidInput(format!( "index {idx} out of range for chunk size K={onehot_k} at position {c}" diff --git a/src/protocol/commitment_scheme.rs b/src/protocol/commitment_scheme.rs index 09acf4c2..a5da204c 100644 --- a/src/protocol/commitment_scheme.rs +++ b/src/protocol/commitment_scheme.rs @@ -269,6 +269,55 @@ where } } +/// Commit to a one-hot polynomial, returning both the commitment and a +/// complete `HachiCommitmentHint` (including `ring_coeffs` needed by `prove`). +/// +/// # Errors +/// +/// Returns an error if dimensions are inconsistent, any index is out of +/// range, or the underlying commitment routine fails. +pub fn commit_onehot( + onehot_k: usize, + indices: &[Option], + setup: &HachiProverSetup, +) -> Result<(RingCommitment, HachiCommitmentHint), HachiError> +where + F: FieldCore + CanonicalField + FieldSampling, + Cfg: CommitmentConfig, +{ + let num_chunks = indices.len(); + let total_field_elems = num_chunks + .checked_mul(onehot_k) + .ok_or_else(|| HachiError::InvalidInput("T*K overflow".into()))?; + if total_field_elems % D != 0 { + return Err(HachiError::InvalidInput(format!( + "T*K={total_field_elems} is not divisible by D={D}" + ))); + } + + // Build ring_coeffs (needed for prove) from the sparse one-hot indices. + let total_ring_elems = total_field_elems / D; + let mut ring_coeffs = vec![CyclotomicRing::::zero(); total_ring_elems]; + for (c, opt) in indices.iter().enumerate() { + let Some(&idx) = opt.as_ref() else { continue }; + let field_pos = c * onehot_k + idx; + let ring_idx = field_pos / D; + let coeff_idx = field_pos % D; + ring_coeffs[ring_idx].coeffs[coeff_idx] = F::one(); + } + + let w = >::commit_onehot( + onehot_k, indices, setup, + )?; + + let hint = HachiCommitmentHint { + s: w.s, + t_hat: w.t_hat, + ring_coeffs, + }; + Ok((w.commitment, hint)) +} + /// Per-block intermediate state for streaming Hachi commitment. /// /// Each chunk corresponds to one Ajtai inner block: `D * block_len` field diff --git a/src/protocol/mod.rs b/src/protocol/mod.rs index e19bfdf2..85ad23fa 100644 --- a/src/protocol/mod.rs +++ b/src/protocol/mod.rs @@ -21,7 +21,7 @@ pub use commitment::{ ProductionFp128CommitmentConfig, RingCommitment, RingCommitmentScheme, SmallTestCommitmentConfig, StreamingCommitmentScheme, }; -pub use commitment_scheme::{HachiChunkState, HachiCommitmentScheme}; +pub use commitment_scheme::{commit_onehot, HachiChunkState, HachiCommitmentScheme}; pub use opening_point::RingOpeningPoint; pub use proof::{HachiProof, SumcheckAux}; pub use quadratic_equation::QuadraticEquation; diff --git a/tests/onehot_commitment.rs b/tests/onehot_commitment.rs index 2d4b20e5..6356774e 100644 --- a/tests/onehot_commitment.rs +++ b/tests/onehot_commitment.rs @@ -19,12 +19,16 @@ fn psetup() -> >::ProverSetup { /// `commit_coeffs`. The optimized impl uses sparse inner Ajtai. /// Both must produce identical (commitment, s_all, t_hat_all). fn assert_onehot_matches_dense(onehot_k: usize, indices: &[usize]) { + let opt_indices: Vec> = indices.iter().map(|&i| Some(i)).collect(); let setup = psetup(); // Optimized sparse path. - let w_sparse = - >::commit_onehot(onehot_k, indices, &setup) - .unwrap(); + let w_sparse = >::commit_onehot( + onehot_k, + &opt_indices, + &setup, + ) + .unwrap(); // Reference: materialize the full one-hot vector, pack into ring elements, // and commit via the dense path. @@ -131,22 +135,28 @@ fn onehot_k_lt_d_ratio_2() { fn onehot_rejects_non_divisible_k_and_d() { let setup = psetup(); let result = - >::commit_onehot(17, &[0; 4], &setup); + >::commit_onehot(17, &[Some(0); 4], &setup); assert!(result.is_err()); } #[test] fn onehot_rejects_out_of_range_index() { let setup = psetup(); - let result = - >::commit_onehot(64, &[0, 64, 0, 0], &setup); + let result = >::commit_onehot( + 64, + &[Some(0), Some(64), Some(0), Some(0)], + &setup, + ); assert!(result.is_err()); } #[test] fn onehot_rejects_wrong_total_size() { let setup = psetup(); - let result = - >::commit_onehot(64, &[0, 0, 0], &setup); + let result = >::commit_onehot( + 64, + &[Some(0), Some(0), Some(0)], + &setup, + ); assert!(result.is_err()); } From 1fdf18e7ab4305ff89dbed5a46ef429b9ded70e6 Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Sun, 1 Mar 2026 22:50:23 -0800 Subject: [PATCH 22/34] feat: submatrix commit for polynomials smaller than setup max commit_coeffs now accepts ring coefficient vectors shorter than the layout's full size, padding each block internally. prove/verify pad the opening point with zeros so the transcript stays consistent. This avoids materializing huge zero-padded field-element arrays. Made-with: Cursor --- docs/BN254_MUL_ASM_AARCH64_ANALYSIS.md | 389 ++ docs/DATA_PROOF_HACHI_MIGRATION.md | 524 +++ docs/FP128_MUL_ASM_AARCH64_ANALYSIS.md | 146 + docs/HACHI_FOR_JOLT.md | 2234 ++++++++++++ docs/HACHI_KEY_DELEGATION.md | 620 ++++ docs/HACHI_ZK_PATHS.md | 333 ++ docs/HACHI_ZK_PCS_REPORT.md | 292 ++ docs/ONE_HOT_COMMITMENT_COST_AND_GPU_PRG.md | 787 ++++ docs/SPARSE_2P128_PRIMES_AND_IMPL.md | 743 ++++ docs/SUPERNEO_DIGEST.md.local | 658 ++++ docs/TWO_FIELD_OPENING_D_GT_1_CLEAN.md | 505 +++ docs/ZK_AND_MODULUS_EMBEDDING.md | 356 ++ paper/LNP22.pdf | Bin 0 -> 1041252 bytes paper/Orthus.pdf | Bin 0 -> 811565 bytes paper/aggregate-falcon-labrador.pdf | Bin 0 -> 1250273 bytes paper/hachi.txt | 2274 ++++++++++++ paper/iterative-rejection-sampling.pdf | Bin 0 -> 501003 bytes paper/ls18-short-invertible-elements.pdf | Bin 0 -> 399492 bytes paper/superneo.txt | 3570 +++++++++++++++++++ src/protocol/commitment/scheme.rs | 22 +- src/protocol/commitment_scheme.rs | 25 +- 21 files changed, 13461 insertions(+), 17 deletions(-) create mode 100644 docs/BN254_MUL_ASM_AARCH64_ANALYSIS.md create mode 100644 docs/DATA_PROOF_HACHI_MIGRATION.md create mode 100644 docs/FP128_MUL_ASM_AARCH64_ANALYSIS.md create mode 100644 docs/HACHI_FOR_JOLT.md create mode 100644 docs/HACHI_KEY_DELEGATION.md create mode 100644 docs/HACHI_ZK_PATHS.md create mode 100644 docs/HACHI_ZK_PCS_REPORT.md create mode 100644 docs/ONE_HOT_COMMITMENT_COST_AND_GPU_PRG.md create mode 100644 docs/SPARSE_2P128_PRIMES_AND_IMPL.md create mode 100644 docs/SUPERNEO_DIGEST.md.local create mode 100644 docs/TWO_FIELD_OPENING_D_GT_1_CLEAN.md create mode 100644 docs/ZK_AND_MODULUS_EMBEDDING.md create mode 100644 paper/LNP22.pdf create mode 100644 paper/Orthus.pdf create mode 100644 paper/aggregate-falcon-labrador.pdf create mode 100644 paper/hachi.txt create mode 100644 paper/iterative-rejection-sampling.pdf create mode 100644 paper/ls18-short-invertible-elements.pdf create mode 100644 paper/superneo.txt diff --git a/docs/BN254_MUL_ASM_AARCH64_ANALYSIS.md b/docs/BN254_MUL_ASM_AARCH64_ANALYSIS.md new file mode 100644 index 00000000..0a27415a --- /dev/null +++ b/docs/BN254_MUL_ASM_AARCH64_ANALYSIS.md @@ -0,0 +1,389 @@ +# BN254 Mul AArch64 Assembly Analysis + +This report dumps the entire `BN254` field multiply symbol assembly and annotates it manually with a latency/throughput model. + +## Source and symbol + +- Assembly file: `target/release/deps/codegen_probe-855aaea603d2760f.s` +- Symbol analyzed: `_bn254_mul_field` +- Symbol span in `.s`: lines `860..1130` +- Target architecture: `arm64` + +## Instruction census (exact for this symbol) + +Counted directly from `_bn254_mul_field` instruction lines: + +- `mul`: `36` +- `umulh`: `32` +- Total multiply-class ops: `68` +- `adds`: `42` +- `adcs`: `8` +- `adc`: `4` +- `cinc`: `42` +- `cmn`: `8` +- `cmp`: `8` +- branches (`b*`): `9` total (`b.hs`/`b.ne`/`b.lo`/`b.eq`/`b`) + +## Bench numbers used for cycle mapping + +From current `field_mul_isolated/*` and chain benches (median values): + +- `fp128_mul_single`: `2.3145 ns` +- `bn254_mul_single`: `9.3428 ns` +- `fp128_pair_passthrough`: `1.8122 ns` +- `bn254_pair_passthrough`: `3.0869 ns` +- `fp128_mul_8way_independent`: `15.996 ns` total (8 muls) +- `bn254_mul_8way_independent`: `75.039 ns` total (8 muls) +- `fp128_8way_passthrough`: `4.6349 ns` total +- `bn254_8way_passthrough`: `8.4336 ns` total +- `mul_chain_2048` per-op: + - `Fp128`: `11.924 us / 2048 = 5.822 ns` + - `BN254`: `22.147 us / 2048 = 10.814 ns` + +At `3.5 GHz`: + +- `BN254 mul_chain` latency proxy: `10.814 ns ~= 37.85 cycles` +- `BN254 8-way net throughput proxy`: + - `(75.039 - 8.4336) / 8 = 8.3257 ns ~= 29.14 cycles/op` + +## How to read isolated bench subtraction + +Recommended subtraction pairs: + +1. **Single-op estimate (rough):** + - `mul_single - black_box_only` +2. **Throughput estimate (better):** + - `(mul_8way_independent - 8way_passthrough) / 8` + +Using current medians: + +- Single rough: + - `Fp128`: `2.3145 - 0.5329 = 1.7816 ns` + - `BN254`: `9.3428 - 1.0944 = 8.2484 ns` + - Ratio: `~4.63x` +- 8-way net (preferred): + - `Fp128`: `(15.996 - 4.6349)/8 = 1.4201 ns/op` + - `BN254`: `(75.039 - 8.4336)/8 = 8.3257 ns/op` + - Ratio: `~5.86x` + +Notes: + +- `pair_passthrough` is useful as a control but tends to over-subtract for tiny types; use it qualitatively, not as the primary net estimate. +- For latency claims, prefer `mul_chain_2048` (or `16384`) dependent chains. + +## Manual phase annotation + +The symbol is a fully inlined `Fp256>::mul` path on AArch64: + +1. **Setup / loads** (`ldp` prologue, constants in `x17/x11/x9/x10/x12`) +2. **CIOS round i=0** using first multiplier limb (`x0`) and reduction constant `INV` +3. **CIOS round i=1** using second multiplier limb (`x2`) +4. **CIOS round i=2** using third multiplier limb (`x2` after `ldp x2, x1, [x1,#16]`) +5. **CIOS round i=3** using fourth multiplier limb (`x1`) +6. **Final conditional subtraction** (`cmp`/`b.hs` path and carry-corrected subtract chain) +7. **Store reduced 256-bit result** (`stp` x4-limb output) + +Notable structural facts: + +- Four unrolled Montgomery CIOS rounds are clearly visible. +- Carry propagation is explicit (`adds`/`adcs`/`adc` + `cinc`), creating serial chains. +- Final compare/subtract path is branchful and can extend the tail latency. + +## Latency + throughput picture (manual model) + +### Latency (single dependent op) + +- The measured chain benchmark gives `~37.85 cycles/op` for BN254 mul on this machine. +- The asm shape supports this: + - 4 serialized CIOS rounds with carry-dependent accumulation + - final conditional subtract tail +- Manual estimate from dependency structure: **~37-43 cycles latency**, centered near measured `~38 cycles`. + +### Throughput (independent ops) + +- Static multiply pressure is high: `68` multiply-class instructions/call. +- Measured 8-way independent net gives `~29.14 cycles/op`. +- This indicates substantial overlap across independent operations and a backend that can keep multiply/add pipelines busy. +- Practical throughput on this system is therefore around **~29 cycles/op** for independent BN254 muls. + +### Why chain ratio is smaller than naive instruction ratio + +- Chain benchmarks measure **latency**, not pure instruction count. +- BN254 has more work, but OoO overlap and scheduling reduce the ratio compared with raw op-count intuition. +- Your current chain numbers (`~37.8 cycles BN254` vs `~20.4 cycles Fp128`) yielding `~1.86x` are consistent with this. + +## Entire raw `_bn254_mul_field` assembly + +```asm +_bn254_mul_field: + .cfi_startproc + stp x24, x23, [sp, #-48]! + .cfi_def_cfa_offset 48 + stp x22, x21, [sp, #16] + stp x20, x19, [sp, #32] + .cfi_offset w19, -8 + .cfi_offset w20, -16 + .cfi_offset w21, -24 + .cfi_offset w22, -32 + .cfi_offset w23, -40 + .cfi_offset w24, -48 + ldp x16, x15, [x0] + ldp x14, x13, [x0, #16] + ldp x0, x2, [x1] + umulh x9, x0, x16 + mul x12, x0, x16 + mov x17, #-268435457 + movk x17, #62867, lsl #32 + movk x17, #49889, lsl #48 + mul x3, x12, x17 + mov x11, #-268435455 + movk x11, #62867, lsl #32 + movk x11, #17377, lsl #48 + umulh x4, x3, x11 + mul x5, x3, x11 + umulh x10, x0, x15 + mul x6, x0, x15 + adds x6, x9, x6 + cinc x10, x10, hs + mov x9, #28817 + movk x9, #31161, lsl #16 + movk x9, #59464, lsl #32 + movk x9, #10291, lsl #48 + umulh x7, x3, x9 + mul x19, x3, x9 + adds x6, x19, x6 + cinc x7, x7, hs + umulh x19, x0, x14 + mul x20, x0, x14 + adds x20, x10, x20 + cinc x19, x19, hs + mov x10, #22621 + movk x10, #33153, lsl #16 + movk x10, #17846, lsl #32 + movk x10, #47184, lsl #48 + umulh x21, x3, x10 + mul x22, x3, x10 + adds x20, x20, x22 + cinc x21, x21, hs + cmn x5, x12 + adcs x4, x6, x4 + adcs x5, x20, x7 + cinc x6, x21, hs + mul x7, x0, x13 + adds x20, x19, x7 + mov x12, #41001 + movk x12, #57649, lsl #16 + movk x12, #20082, lsl #32 + movk x12, #12388, lsl #48 + umulh x21, x3, x12 + mul x3, x3, x12 + adds x3, x20, x3 + cinc x20, x21, hs + adds x3, x3, x6 + cinc x6, x20, hs + cmn x19, x7 + umulh x0, x0, x13 + adc x0, x6, x0 + umulh x6, x2, x16 + mul x7, x2, x16 + adds x4, x4, x7 + cinc x6, x6, hs + mul x7, x4, x17 + umulh x19, x7, x11 + mul x20, x7, x11 + umulh x21, x2, x15 + mul x22, x2, x15 + adds x5, x5, x22 + cinc x21, x21, hs + adds x5, x5, x6 + cinc x6, x21, hs + umulh x21, x7, x9 + mul x22, x7, x9 + adds x5, x5, x22 + cinc x21, x21, hs + umulh x22, x2, x14 + mul x23, x2, x14 + adds x3, x3, x23 + cinc x22, x22, hs + adds x3, x3, x6 + cinc x6, x22, hs + umulh x22, x7, x10 + mul x23, x7, x10 + adds x3, x3, x23 + cinc x22, x22, hs + cmn x20, x4 + adcs x5, x5, x19 + adcs x3, x3, x21 + cinc x4, x22, hs + umulh x19, x2, x13 + mul x2, x2, x13 + adds x0, x2, x0 + cinc x2, x19, hs + adds x19, x0, x6 + umulh x20, x7, x12 + mul x7, x7, x12 + adds x7, x19, x7 + cinc x19, x20, hs + adds x4, x7, x4 + cinc x7, x19, hs + cmn x0, x6 + adc x0, x7, x2 + ldp x2, x1, [x1, #16] + umulh x6, x2, x16 + mul x7, x2, x16 + adds x5, x5, x7 + cinc x6, x6, hs + mul x7, x5, x17 + umulh x19, x7, x11 + mul x20, x7, x11 + umulh x21, x2, x15 + mul x22, x2, x15 + adds x3, x3, x22 + cinc x21, x21, hs + adds x3, x3, x6 + cinc x6, x21, hs + umulh x21, x7, x9 + mul x22, x7, x9 + adds x3, x3, x22 + cinc x21, x21, hs + umulh x22, x2, x14 + mul x23, x2, x14 + adds x4, x4, x23 + cinc x22, x22, hs + adds x4, x4, x6 + cinc x6, x22, hs + umulh x22, x7, x10 + mul x23, x7, x10 + adds x4, x4, x23 + cinc x22, x22, hs + cmn x20, x5 + adcs x3, x3, x19 + adcs x4, x4, x21 + cinc x5, x22, hs + umulh x19, x2, x13 + mul x2, x2, x13 + adds x0, x2, x0 + cinc x2, x19, hs + adds x19, x0, x6 + umulh x20, x7, x12 + mul x7, x7, x12 + adds x7, x19, x7 + cinc x19, x20, hs + adds x5, x7, x5 + cinc x7, x19, hs + cmn x0, x6 + adc x0, x7, x2 + umulh x2, x1, x16 + mul x16, x1, x16 + adds x16, x3, x16 + cinc x2, x2, hs + mul x17, x16, x17 + umulh x3, x17, x11 + mul x6, x17, x11 + umulh x7, x1, x15 + mul x15, x1, x15 + adds x15, x4, x15 + cinc x4, x7, hs + adds x15, x15, x2 + cinc x2, x4, hs + umulh x4, x17, x9 + mul x7, x17, x9 + adds x15, x15, x7 + cinc x4, x4, hs + umulh x7, x1, x14 + mul x14, x1, x14 + adds x14, x5, x14 + cinc x5, x7, hs + adds x14, x14, x2 + cinc x2, x5, hs + umulh x5, x17, x10 + mul x7, x17, x10 + adds x7, x14, x7 + cinc x5, x5, hs + cmn x6, x16 + adcs x14, x15, x3 + adcs x15, x7, x4 + cinc x16, x5, hs + umulh x3, x1, x13 + mul x13, x1, x13 + adds x13, x13, x0 + cinc x0, x3, hs + adds x1, x13, x2 + umulh x3, x17, x12 + mul x17, x17, x12 + adds x17, x1, x17 + cinc x1, x3, hs + adds x16, x17, x16 + cinc x17, x1, hs + cmn x13, x2 + adc x13, x17, x0 + cmp x13, x12 + b.hs LBB8_3 + mov x12, x13 +LBB8_2: + mov x10, x16 + mov x9, x15 + b LBB8_11 +LBB8_3: + b.ne LBB8_10 + cmp x16, x10 + b.lo LBB8_2 + b.ne LBB8_10 + cmp x15, x9 + b.hs LBB8_8 + mov x9, x15 + b LBB8_11 +LBB8_8: + cmp x14, x11 + b.hs LBB8_10 + cmp x15, x9 + b.eq LBB8_11 +LBB8_10: + mov x9, #-4026531841 + movk x9, #2668, lsl #32 + movk x9, #48158, lsl #48 + cmp x14, x11 + add x14, x14, x9 + cset w9, lo + subs x9, x15, x9 + ngc x10, xzr + mov x11, #36719 + movk x11, #34374, lsl #16 + movk x11, #6071, lsl #32 + movk x11, #55244, lsl #48 + adds x9, x9, x11 + cinc x10, x10, hs + cmp x10, #0 + cset w10, eq + subs x10, x16, x10 + ngc x11, xzr + mov x12, #42915 + movk x12, #32382, lsl #16 + movk x12, #47689, lsl #32 + movk x12, #18351, lsl #48 + adds x10, x10, x12 + cinc x11, x11, hs + cmp x11, #0 + cset w11, eq + mov x12, #24535 + movk x12, #7886, lsl #16 + movk x12, #45453, lsl #32 + movk x12, #53147, lsl #48 + sub x11, x13, x11 + add x12, x11, x12 +LBB8_11: + stp x14, x9, [x8] + stp x10, x12, [x8, #16] + ldp x20, x19, [sp, #32] + ldp x22, x21, [sp, #16] + ldp x24, x23, [sp], #48 + .cfi_def_cfa_offset 0 + .cfi_restore w19 + .cfi_restore w20 + .cfi_restore w21 + .cfi_restore w22 + .cfi_restore w23 + .cfi_restore w24 + ret + .cfi_endproc +``` diff --git a/docs/DATA_PROOF_HACHI_MIGRATION.md b/docs/DATA_PROOF_HACHI_MIGRATION.md new file mode 100644 index 00000000..696cb2ef --- /dev/null +++ b/docs/DATA_PROOF_HACHI_MIGRATION.md @@ -0,0 +1,524 @@ +# Data Proof: Dory → Hachi Migration Analysis + +This document analyzes the impact of replacing Dory (pairing-based PCS) with Hachi (lattice-based PCS) on the **Data Proof** protocol used in Zero's OpenZoneBlockTx. + +Primary sources: + +- `../Cyclone-Megadoc/protocol-spec/beacon/data_proof.md` — the Data Proof spec +- `../Cyclone-Megadoc/protocol-spec/beacon/da.md` — the Semi-AVID-PR / VID protocol +- `~/Downloads/Semi-AVID-PR and Jolt.pdf` — Saeid's note on shared-SRS optimization +- `docs/HACHI_KEY_DELEGATION.md` — Hachi two-level protocol with equations +- `docs/HACHI_FOR_JOLT.md` — Jolt integration spec + +--- + +## 1. System Context: Where the Data Proof Sits + +Zero (Cyclone) settles zone blocks via a three-phase protocol: + +``` + PREPARE COMMIT OPEN + ───────── ────── ──── + Sequencer runs VID Sequencer submits Sequencer submits + with beacon validators CommitZoneBlockTx OpenZoneBlockTx + ────────────────── ────────────────── ────────────────── + Outputs: Includes: Includes: + • Column commits (h_j) • CoA • Jolt SNARK proof + • CoA • Column commits • DATA PROOF ◄── this document + • RS-encoded chunks • State root +``` + +The Data Proof bridges two independent commitment systems: + +- **DA layer** (Prepare): column commitments \(h_j = \text{Pedersen}(\text{column}_j)\) via Semi-AVID-PR +- **Jolt ZKVM** (Open): polynomial commitment to the same data matrix \(U\) via Dory + +It proves: "the data Jolt proved about is the same data the DA layer committed to." + +### 1.1 Two data objects (critical distinction) + +There are two different objects that are easy to conflate: + +1. **DA block data \(U\)**: raw zone-block transaction bytes, transformed into field elements for VID and committed column-wise in Prepare/Commit. +2. **Jolt witness tables**: execution-trace polynomials committed in Stage 8 (one-hot address tables, increment tables, and optional advice tables). + +The Data Proof is the bridge between these worlds. The "native small values" observation applies strongly to Jolt witness tables, but not automatically to raw DA block bytes. + +### 1.2 Current DA packing path + +Current VID docs use BLS12-381 field arithmetic for encoding. The documented packing rule is: + +- each 32-byte field element stores 31 bytes of payload. + +So the current DA path is: + +`raw bytes -> BLS12-381 field elements -> matrix U -> RS encode -> Pedersen column commitments`. + +--- + +## 2. Current Data Proof Protocol (Dory-Based) + +### 2.1 Setup + +- Data matrix \(U \in \mathbb{F}^{L \times k}\) encodes the zone block. +- DA column commitments: \(C_j = \sum_{i} U[i,j] \cdot g_i \in \mathbb{G}_1\) (Pedersen with key \(\vec{g}\)). +- Polynomial: \(f(X) = \sum_{i,j} U[i,j] \cdot X^{j + ki}\). +- Evaluation claim: \(f(x) = y\). + +### 2.2 VMV Decomposition (PCS-Independent) + +The evaluation factors as a vector-matrix-vector product: + +\[ +f(x) = \vec{b}^T U \vec{a} +\] + +where \(\vec{a} = (1, x, \ldots, x^{k-1})\) and \(\vec{b} = (1, x^k, \ldots, x^{(L-1)k})\). + +Defining \(\vec{w} = U\vec{a} \in \mathbb{F}^L\), the claim becomes: know \(\vec{w}\) with \(\langle \vec{w}, \vec{b} \rangle = y\) and \(\vec{w}\) consistent with \(U\). + +### 2.3 BabyHyrax Baseline (Non-Succinct) + +Prover sends \(\vec{w}\) in the clear (\(O(L)\) field elements). Verifier checks: + +1. **Correctness**: \(\langle \vec{w}, \vec{b} \rangle \stackrel{?}{=} y\) +2. **Consistency**: \(\sum_i w_i \cdot g_i \stackrel{?}{=} \sum_j a_j \cdot C_j\) + +### 2.4 Homomorphic Reduction (Key Trick) + +Because Pedersen is **linear in the message**, the verifier computes locally: + +\[ +C' = \sum_{j \in [k]} a_j \cdot C_j = \sum_j a_j \left(\sum_i U[i,j] \cdot g_i\right) = \sum_i \left(\sum_j U[i,j] \cdot a_j\right) g_i = \text{Commit}(\vec{w}, \vec{g}) +\] + +This \(O(k)\) MSM gives the verifier a commitment to \(\vec{w}\) without receiving it. The problem reduces to: "prove you know \(\vec{w}\) that opens \(C'\) and satisfies \(\langle \vec{w}, \vec{b} \rangle = y\)." + +### 2.5 Dory Engine (Succinct) + +1. **Lift**: \(C'' = e(C', h) \in \mathbb{G}_T\) (pairing) +2. **Recursive folding**: \(\log L\) rounds, each halving vector size via challenges \(\alpha_i\) +3. **Key-folding**: precomputed \(\Delta\) values let verifier fold commitment keys in \(O(1)\) per round +4. **Tensor structure**: \(\vec{b}\)'s multiplicative structure lets verifier compute final folded scalar in \(O(\log L)\) +5. **Final check**: one multi-pairing in \(\mathbb{G}_T\) + +### 2.6 Shared-SRS Optimization (Saeid's PDF) + +If Dory generators \(g_i = \tau^i \cdot g\) come from the KZG SRS, then \(C_j = \hat{C}_j\) — the DA column commitment and the Dory sub-commitment are literally the same object. No extra consistency proof needed. + +### 2.7 Audit Notes: Conventions and Trust Model + +- **Row/column naming mismatch is mostly convention**: the Cyclone Data Proof spec is written in terms of **column commitments** \(C_j\), while Dory/Jolt implementation exposes **row commitments** as tier-1 hints. This is transpose-equivalent bookkeeping (with swapped VMV vectors), not a fundamental protocol contradiction. +- **The "lift" step is an abstraction, not a bug**: the spec writes \(C''=e(C',h)\). The implementation computes the pre-lift commitment by MSM over tier-1 commitments, then applies a pairing; same algebraic object, different decomposition. +- **CRS coupling caveat**: native Dory setup is transparent (random generators from public randomness). If generators are instead taken from a KZG powers-of-\(\tau\) SRS to force object identity, the system inherits KZG trusted-setup assumptions. No direct new attack is known from this audit, but transparency is no longer a valid claim in that mode. + +--- + +## 3. The Gadget Decomposition Problem + +### 3.1 Why Ajtai Is Not Linearly Homomorphic + +Ajtai commitment: \(h_j = A \cdot G^{-1}(u_j)\), where \(G^{-1}\) is base-\(b\) digit decomposition. + +The map \(u \mapsto G^{-1}(u)\) is **not linear** (carries break digit decomposition): + +\[ +\sum_j a_j \cdot G^{-1}(u_j) \neq G^{-1}\!\left(\sum_j a_j \cdot u_j\right) +\] + +Therefore: + +\[ +\sum_j a_j \cdot h_j = A \cdot \underbrace{\left(\sum_j a_j \cdot G^{-1}(u_j)\right)}_{\neq\, G^{-1}(w)} \neq A \cdot G^{-1}(w) +\] + +The verifier's linear combination produces something that is **not** a valid Ajtai commitment to \(w = Ua\). + +**This breaks the Data Proof's core trick.** (Confirmed by `HACHI_KEY_DELEGATION.md` which states: "The earlier 'homomorphic batching' idea ... is NOT valid for Hachi commitments, because the commitment map includes digit decompositions and is not linear.") + +### 3.2 Consequences for Semi-AVID-PR + +The DA protocol's validator check (`da.md` lines 109–114) relies on: + +\[ +\text{Code.Encode}(\text{Commit}(u_1), \ldots, \text{Commit}(u_k))_i = \text{Commit}(\text{Code.Encode}(u_1, \ldots, u_k)_i) +\] + +This is exactly linearity of the commitment map. With Ajtai + gadget decomposition, this identity fails. **Naively swapping Pedersen for Ajtai in the DA layer breaks Semi-AVID-PR's verification mechanism.** + +### 3.3 Consequences for the Shared-SRS Trick + +The Dory trick relied on a triple coincidence: + +1. Pedersen linearity → homomorphic reduction works +2. Shared generators → column commitment = Dory sub-commitment (zero overhead) +3. Pairing lift → enters Dory IPA world + +All three are lost with Hachi. There is no analog of "column commit = sub-commitment" with the two-layer Ajtai structure. + +--- + +## 4. How Hachi Solves the Gadget Linearity Problem + +Hachi's own opening protocol (Step B, `HACHI_KEY_DELEGATION.md` §1.3) never relies on homomorphic combination of Ajtai commitments. Instead, it uses a **auxiliary commitment + fold** mechanism. + +### 4.1 The Pattern + +Given committed blocks \(f_i\) with decompositions \(s_i = G^{-1}(f_i)\) and inner commitments \(t_i = A \cdot s_i\): + +**Step 1: Prover sends fresh auxiliary commitment.** +- Compute partial evaluations: \(w_i = a^T \cdot G \cdot s_i = a^T \cdot f_i\) +- Freshly decompose and commit: \(\hat{w}_i = G^{-1}(w_i)\), \(v = D \cdot \hat{w}\) +- Send \(v\) (first prover message) + +**Step 2: Verifier sends fold challenge.** +- Random sparse \(c = (c_1, \ldots, c_{2^r})\) + +**Step 3: Prover folds on already-decomposed witnesses.** +- \(z = \sum_i c_i \cdot s_i\) (linear combination of vectors already in decomposed form) +- Send \(z\) + +**Step 4: Verifier checks dual consistency through committed witnesses.** + +The verifier does **not** receive all \(t_i\) explicitly. Instead, it checks a stacked relation over witnesses \((\hat{w}, \hat{t}, z)\): + +- **Main-binding constraint**: \(B\hat{t} = u\) (binds \(\hat{t}\) to the public commitment \(u\)). +- **Eq. 19 (fold-inner consistency)**: \(A z = (c^T \otimes G_{n_A}) \cdot \hat{t}\). +- **Eq. 18 (fold-partials consistency)**: \((a^T G)\cdot z = (c^T \otimes G_1)\cdot \hat{w}\). + +In the full protocol, \(z\) is represented via redecomposition \(z = J\hat{z}\), and these constraints are enforced inside Step C's ring-switch + sumcheck system. + +### 4.2 Why This Works + +The nonlinearity of \(G^{-1}\) is quarantined to two places where it doesn't interact with linear combinations: + +- **Commitment time** (one-time): \(s_i = G^{-1}(f_i)\) +- **Auxiliary commitment time** (per reduction): \(\hat{w}_i = G^{-1}(w_i)\) + +All subsequent operations use only: +- Linearity of \(A\) (matrix-vector multiply) +- Linearity of \(G\) (the forward gadget map, trivially a matrix) +- The identity \(G \cdot G^{-1}(x) = x\) + +### 4.3 Soundness Sketch + +If the prover commits to incorrect partial evaluations \(w'_i \neq a^T f_i\), then after seeing random \(c\), it must still produce witnesses satisfying: + +1. \(B\hat{t}=u\) (bound to the committed polynomial), +2. \(A z = (c^T \otimes G_{n_A})\hat{t}\), +3. \((a^T G)z = (c^T \otimes G_1)\hat{w}\), +4. the opening equation. + +These constraints force one folded witness \(z\) to satisfy two independently anchored linear views (main-commit anchor and auxiliary/evaluation anchor). A cheating assignment survives only with negligible probability over \(c\), assuming standard ROM/low-degree soundness used by Hachi. + +--- + +## 5. Adapted Data Proof Architecture + +### 5.1 Option A: Keep Pedersen for DA, Add Bridge Proof (Recommended) + +``` +┌──────────────────────────────────────────────────────┐ +│ DA Layer (unchanged) │ +│ Column commits h_j = Pedersen(column_j) in G₁ │ +│ Semi-AVID-PR linearity argument intact │ +└──────────────────────┬───────────────────────────────┘ + │ h_1,...,h_k public + ▼ +┌──────────────────────────────────────────────────────┐ +│ BRIDGE PROOF (new component) │ +│ │ +│ Prover: commit same U via Hachi (two-layer Ajtai) │ +│ Prove consistency using Hachi's Step B pattern: │ +│ 1. Send auxiliary commitment v = D · ĝ(w) │ +│ 2. Receive fold challenge c │ +│ 3. Send folded witness z = Σ c_i · s_i │ +│ 4. Verifier checks stacked constraints │ +│ (Bĥt=u, Eq.18, Eq.19, opening equation) │ +└──────────────────────┬───────────────────────────────┘ + ▼ +┌──────────────────────────────────────────────────────┐ +│ HACHI OPENING PROOF │ +│ Step B: evaluation reduction │ +│ Step C: ring-switch + sumcheck over F_{q^k} │ +└──────────────────────────────────────────────────────┘ +``` + +**Trade-offs:** + +- (+) DA layer untouched, Semi-AVID-PR works as-is +- (+) Bridge proof uses Hachi's own machinery (no new primitives) +- (-) Bridge proof adds prover cost (one auxiliary commitment + one fold) +- (-) Loses the "zero-overhead" consistency of the Dory shared-SRS trick +- (-) Requires an explicit DA↔Hachi consistency relation (not provided "for free" by Jolt Stage-8 batching) + +### 5.2 Option B: Lattice-Native DA Without Gadget Decomposition + +If data elements are bounded (e.g., bits, one-hot entries), use Ajtai **without** gadget decomposition: + +\[ +h_j = A \cdot u_j \quad\text{where } \|u_j\|_\infty \leq B +\] + +Module-SIS binding holds for short pre-images. The commitment map \(u \mapsto A \cdot u\) IS linear, so Semi-AVID-PR works and the homomorphic reduction works. + +**Trade-offs:** + +- (+) Full linearity restored, both DA and Data Proof work natively +- (+) No bridge proof needed +- (-) Only works if data entries have bounded norm (must satisfy SIS security bound) +- (-) Loses binding for arbitrary field data with large entries +- (-) Needs careful parameter analysis: SIS norm bound vs data magnitude +- (-) RS encoding can amplify entry magnitude; bounded raw data does **not** automatically imply bounded encoded chunks + +**Important clarification:** Option B may be plausible for some **Jolt witness commitments** (one-hot + bounded increments), but it is generally not plausible for current **raw DA block bytes** without a redesign of DA data representation. + +### 5.3 Option C: Redesign DA Around Hachi's Commitment Structure + +This is the "single commitment family" path: DA and Open both use Hachi, and Semi-AVID-PR's linearity-based check is replaced. + +#### 5.3.1 What must be replaced + +Current Semi-AVID-PR validator logic uses: +\[ +\mathrm{Code.Encode}(h_1,\ldots,h_k)_i \stackrel{?}{=} \mathrm{Commit}(c_i) +\] +which depends on commitment linearity. With Hachi commitments, this check is invalid. + +So Option C must replace that check with a proof-backed statement: +"the received chunk \(c_i\) is the correct RS encoding of data committed in Hachi." + +#### 5.3.2 Option C1 (incremental): per-validator chunk-consistency proof + +For each validator \(i\), the sequencer sends chunk \(c_i\) plus proof \(\pi_i\): + +1. Commit original columns with Hachi: \((u_j)_{j\in[k]}\). +2. Validator derives a random evaluation point \(r\) (Fiat-Shamir). +3. Prover opens needed committed columns and chunk at \(r\) (batched opening). +4. Verifier checks RS relation at \(r\): + \[ + c_i(r) \stackrel{?}{=} \sum_{j=1}^k \lambda_{i,j}\,u_j(r), + \] + where \(\lambda_{i,j}\) are public RS coefficients. + +**Pros**: easiest way to remove linearity assumption. + +**Cons**: proof load scales with validators/chunks unless heavily shared/batched. + +#### 5.3.3 Option C2 (scalable target): one global encoding proof + +Commit both source and encoded tables under Hachi: + +- source columns \(U\), +- encoded columns \(E=\mathrm{Code.Encode}(U)\). + +Publish one global proof \(\pi_{\mathrm{enc}}\) that enforces: +\[ +\forall i,\quad E_{:,i} = \sum_{j=1}^k \lambda_{i,j} U_{:,j}. +\] + +Implementation direction: encode this as a stacked linear relation and prove it with the same ring-switch + sumcheck machinery used by Hachi Step C. + +Then validator-side chunk checks become: + +- verify \(\pi_{\mathrm{enc}}\) once per block (or once on-chain), +- verify chunk opening against the committed \(E_{:,i}\). + +**Pros**: amortizes proof cost; cleanest long-term Option C architecture. + +**Cons**: substantial protocol redesign and engineering effort. + +#### 5.3.4 Recommended Option C roadmap + +1. **Spec freeze**: define exact matrix orientation and RS coefficient indexing (remove row/column ambiguity first). +2. **C1 prototype**: implement per-validator proof to de-risk correctness quickly. +3. **C2 upgrade**: replace C1 with one global encoding proof once relation encoding is stable. +4. **Integrate with Open path**: share transcript/challenge infrastructure with Hachi opening to avoid duplicated proof logic. + +### 5.4 Do we still need byte packing? What if DA encoding uses a 128-bit field? + +#### 5.4.1 Packing is still needed for byte-native blocks + +If zone blocks remain byte-native, any field-based RS pipeline still needs a reversible bytes-to-field encoding step. So "packing" does not disappear; only its efficiency and algebraic target field change. + +#### 5.4.2 Can we switch only the RS field to 128-bit under current Pedersen DA? + +Not safely. Semi-AVID-PR's validator check relies on commitment linearity over the **same scalar field** as coding. + +If RS is over \(F_p\) (128-bit) but Pedersen scalars are over \(F_r\) (e.g. BLS12-381 scalar field), then generally: +\[ +\mathrm{Commit}_{F_r}\!\left(\sum_j \lambda_j u_j \bmod p\right) +\neq +\sum_j \iota(\lambda_j)\,\mathrm{Commit}_{F_r}(u_j), +\] +because \(\iota\) is only an encoding into \(F_r\), not a field isomorphism \(F_p \cong F_r\). + +So a "field-size-only swap" breaks the linear-consistency check unless commitments are also moved to the same algebraic field/ring used by encoding (which is exactly the kind of redesign Option C entails). + +#### 5.4.3 Packing efficiency trade-off at 128-bit + +- Current BLS12-style rule: 31 bytes per field element. +- Simple 128-bit prime rule (byte-aligned, fixed): typically 15 bytes per element. + +So naive 128-bit encoding can **increase** element count for the same raw bytes. A near-\(2^{128}\) prime (e.g. P13) allows more specialized packing strategies, but those add codec complexity and must preserve deterministic invertibility. + +Concrete scale example for a 1 MiB block: + +- 31-byte packing: \(\lceil 1{,}048{,}576 / 31 \rceil = 33{,}826\) field elements. +- 15-byte packing: \(\lceil 1{,}048{,}576 / 15 \rceil = 69{,}906\) field elements. + +That is about \(2.07\times\) more elements under naive 15-byte packing. + +For a near-\(2^{128}\) prime like P13 (\(2^{128}-8207\)), only 8,207 of the \(2^{128}\) 16-byte values are out of range (\(\approx 2^{-115}\) fraction). This suggests a specialized 16-byte codec may be practical, but it still requires explicit overflow handling and adversarially robust, deterministic decoding rules. + +#### 5.4.4 Is 128-bit security sufficient? + +For Option C-style unified Hachi commitments over a 128-bit modulus, the current repo analysis indicates "yes" for candidate parameters: + +- `docs/HACHI_FOR_JOLT.md` Section 5.6 reports Module-SIS estimates above 128 bits for recommended P13 configurations, including conservative rows still around ~131 bits in the tight regime. + +Additional notes: + +- RS availability/correctness is information-theoretic and mainly requires \(q \gg n\) for committee-size \(n\) (128-bit is far above practical \(n\)). +- Any randomized polynomial-identity checks over a 128-bit field still need an explicit soundness budget (\(\approx \deg/q\)-style terms), but this is typically negligible at deployed degrees. + +#### 5.4.5 Security checklist if DA moves to 128-bit encoding + +If Option C moves DA encoding and commitments into a 128-bit algebraic domain, security is fine **only if** the codec is part of the proved relation: + +1. **Canonical decoding**: bytes \(\leftrightarrow\) field elements must be injective and deterministic. +2. **Metadata binding**: codec ID, padding length, and any auxiliary channels must be included in the committed message and verified. +3. **No hidden side channels**: if using an overflow-assisted codec, the overflow bitmap/payload must be committed and proven consistent. +4. **Soundness accounting**: add codec-related checks to the same transcript/sumcheck soundness budget (do not treat codec validity as out-of-band). + +### 5.5 Option C codec decision table (implementation) + +To make Option C concrete, define: + +- \(B\): block size in bytes, +- \(e\): payload bytes per field element (codec-dependent), +- \(w\): serialized bytes per field element (\(w=16\) for 128-bit field, \(w=32\) for current BLS-style), +- \(n\): committee size, +- \(S_0\): max share size (bytes). + +Then: +\[ +M=\left\lceil\frac{B}{e}\right\rceil,\quad +d_{\max}=\left\lfloor\frac{S_0}{w}\right\rfloor,\quad +k=\left\lceil\frac{M}{d_{\max}}\right\rceil,\quad +d=\mathrm{nextPow2}\!\left(\left\lceil\frac{M}{k}\right\rceil\right),\quad +r=\frac{k}{n}. +\] + +#### 5.5.1 Codec options + +| Codec | Domain | Payload \(e\) | Deterministic worst-case size | Engineering risk | Notes | +|---|---|---:|---|---|---| +| **Current baseline** | BLS-style | 31/elem (\(w=32\)) | Yes | Low | Existing VID docs/spec assumptions | +| **Fixed-15** | 128-bit field | 15/elem (\(w=16\)) | Yes | Low | Simple, robust, easy to audit | +| **Packed-16 + overflow channel** | 128-bit near-\(2^{128}\) prime | ~16/elem expected | **Only if overflow is explicit and bounded** | Medium/High | Higher average efficiency, but adversarial overflow handling is mandatory | + +#### 5.5.2 Concrete comparison (1 MiB block, \(n=1024\), \(S_0=4096\)) + +| Codec | \(M\) source elems | \(d_{\max}\) | \(k\) | \(d\) | \(r=k/n\) | +|---|---:|---:|---:|---:|---:| +| 31/32 baseline | 33,826 | 128 | 265 | 128 | 0.2588 | +| 15/16 fixed | 69,906 | 256 | 274 | 256 | 0.2676 | +| 16/16 idealized | 65,536 | 256 | 256 | 256 | 0.2500 | + +Interpretation: + +- Moving from 31/32 to fixed 15/16 approximately doubles source element count \(M\), but because \(d_{\max}\) also doubles (16-byte elements), \(k\) only changes moderately in this sizing regime. +- "16/16 idealized" shows the upper bound if every 16-byte chunk maps directly. + +#### 5.5.3 Overflow caveat for packed-16 codecs + +For P13 (\(2^{128}-8207\)), random data hits out-of-range 16-byte words with probability \(\approx 2^{-115}\), so average overhead is negligible. + +But protocol design must be **adversarially robust**, not average-case: + +- a malicious block producer can choose bytes that maximize overflows, +- so overflow artifacts must be explicit and bounded, +- and a deterministic fallback (e.g., re-encode with fixed-15) should be specified when overflow budget is exceeded. + +#### 5.5.4 Recommendation for Option C + +1. **Phase 1 (safe default):** fixed-15 codec for consensus-critical deployment. +2. **Phase 2 (optional optimization):** packed-16 codec behind a strict overflow-cap + deterministic fallback rule. +3. In both phases, include `codec_id` and all codec metadata/artifacts in the committed statement and in the Option C encoding proof relation. + +--- + +## 6. Additional Domain Mismatches + +### 6.1 Scalar Field vs Ring Domain + +| | Data Proof (Dory) | Hachi | +|---|---|---| +| \(\vec{w}\) lives in | \(\mathbb{F}^L\) (scalar field) | \(R_q^{2^r}\) (ring) | +| Inner product | \(\langle w, b \rangle = y\) in \(\mathbb{F}\) | \(b^T w\) in \(R_q\), then ring-switch to \(F_{q^k}\) | +| Evaluation point | \(x \in \mathbb{F}\) | \(x \in R_q^\ell\) or \(\mathbf{r} \in F_{q'}^n\) (two-field) | + +For the packed regime \(d > 1\), `HACHI_FOR_JOLT.md` §4.3.2A identifies a typing issue: the partial evaluation \(w_i = a^T G s_i\) with \(a \in F_{q'}^{2^m}\) and \(G s_i \in R_q^{2^m}\) produces \(w_i \in F_{q'}[X]/(X^d+1)\), not \(R_q\). Resolution requires a Phase 0 inner evaluation sumcheck (\(\alpha = \log_2 d\) extra rounds). + +### 6.2 Jolt CommitmentScheme Trait + +Jolt's `CommitmentScheme` trait (`jolt-core/src/poly/commitment/commitment_scheme.rs`) assumes: + +1. `combine_commitments(commitments, coeffs)` — homomorphic combination (broken for Ajtai) +2. `opening_point: &[F]` — single-field opening (Hachi needs two moduli \(q, q'\)) +3. `Polynomial` — field-native polynomial (Hachi works over rings) +4. Stage-8 batch opening relies on additive combination (`combine_hints` / `combine_commitments`) for internal RLC reductions; this is not the DA Data Proof itself, but it depends on the same algebraic property that breaks under naive Ajtai + +Significant trait redesign needed to accommodate Hachi's algebraic structure. + +--- + +## 7. Summary of Findings + +| Property | Dory (current) | Hachi (proposed) | Status | +|---|---|---|---| +| Commitment linearity | Pedersen: fully linear | Ajtai + \(G^{-1}\): **not linear** | **Breaking change** | +| Homomorphic reduction | Verifier derives \(C'\) for free | Must use auxiliary commit + fold | Redesigned | +| DA layer compatibility | Semi-AVID-PR linearity intact | Breaks with naive Ajtai swap | Keep Pedersen or use Option B | +| Shared-SRS / shared-matrix | Column commit = Dory sub-commit | No analog exists | Bridge proof needed | +| Succinct opening engine | Pairing lift + IPA folding | Ring-switch + sumcheck | Full replacement | +| Verifier efficiency | \(O(\log L)\) via key-folding + tensor trick | \(O(\ell)\) via sumcheck | Comparable | +| Security | Pre-quantum | Post-quantum | Upgrade | +| Setup (native mode) | Transparent (public randomness) | Transparent (public seed) | Comparable | +| Setup (shared-SRS mode) | Inherits KZG trusted setup | N/A | Trust-model downgrade | + +### What survives unchanged + +- VMV decomposition \(f(x) = \vec{b}^T U \vec{a}\) (pure algebra) +- Prepare-Commit-Open lifecycle (protocol-level) +- Semi-AVID-PR structure (if Pedersen is kept for DA) + +### What must be rebuilt + +- The homomorphic reduction → auxiliary commitment + fold consistency pattern +- The Dory IPA engine → Hachi Step B + Step C +- The bridge between DA commitments and PCS commitment +- Jolt's `CommitmentScheme` trait boundary + +--- + +## 8. Audit Outcomes (Updated) + +### 8.1 Resolved points + +1. **Row vs column mismatch**: resolved as a convention/transpose issue, not a protocol break. +2. **Data Proof vs Jolt Stage-8**: Stage-8 batching is not the DA bridge itself, but migration is entangled because both rely on additive homomorphism. +3. **"Lift" wording**: \(C''=e(C',h)\) is an acceptable abstraction of implementation flow. +4. **Eq. 19 verifier access**: direct check against hidden \(t_i\) is not possible from \(u\) alone; must be enforced via stacked witness constraints (\(B\hat t=u\), Eq. 18, Eq. 19) inside Step C. +5. **Shared-SRS trust model**: if Dory reuses KZG powers-of-\(\tau\), transparency is lost and trusted-setup assumptions are inherited. +6. **Data-model split**: "native-small-values" is accurate for many Jolt witness tables, but not for raw DA block bytes under current packing. +7. **128-bit RS under current Pedersen DA**: not a drop-in change; field mismatch breaks the linear-consistency invariant unless commitment algebra is redesigned too. + +### 8.2 Remaining quantitative work + +1. **Sparse-challenge soundness constants**: derive explicit end-to-end error term for the exact challenge distribution and fold depth used in production parameters. +2. **Option B bound under RS encoding**: compute concrete SIS margins after encoding (not just on raw witness entries). +3. **Option A bridge cost**: benchmark prover/verifier overhead relative to base Hachi opening. +4. **Option C execution plan**: evaluate C1 vs C2 with concrete proof-size and validator-throughput targets. +5. **Option C + 128-bit codec design**: choose a deterministic byte-to-field codec (fixed 15-byte vs near-\(2^{128}\) specialized), then quantify throughput/proof-size impact. diff --git a/docs/FP128_MUL_ASM_AARCH64_ANALYSIS.md b/docs/FP128_MUL_ASM_AARCH64_ANALYSIS.md new file mode 100644 index 00000000..d793fb42 --- /dev/null +++ b/docs/FP128_MUL_ASM_AARCH64_ANALYSIS.md @@ -0,0 +1,146 @@ +# Fp128 Mul AArch64 Assembly Analysis + +Fresh rebuild was used to generate this assembly: + +- Command: `cargo clean && cargo rustc --bench codegen_probe --release -- --emit asm` +- Assembly file: `target/release/deps/codegen_probe-1c486cb16985754a.s` +- Symbol analyzed: `_fp128_mul_field` +- Symbol span in `.s`: lines `97..141` +- Target architecture: `arm64` + +## Instruction census (exact for `_fp128_mul_field`) + +- `mul`: `7` +- `umulh`: `6` +- Total multiply-class ops: `13` +- `adds`: `9` +- `adcs`: `3` +- `adc`: `3` +- `cset`: `5` +- `cinc`: `2` +- `csel`: `2` +- `ldp`: `2` +- `stp`: `1` + +This is a compact, spill-free kernel. + +## Fresh benchmark anchors + +From fresh run medians: + +- `field_mul_only/mul_chain_2048`: + - Fp128: `11.948 us` total -> `5.8340 ns/op` + - BN254: `22.225 us` total -> `10.8521 ns/op` + - ratio `BN254/Fp128 ~= 1.86x` +- `field_mul_isolated` (Fp128 only): + - `fp128_mul_single`: `2.4427 ns` + - `fp128_pair_passthrough`: `1.8375 ns` + - `fp128_black_box_only`: `0.55984 ns` + - `fp128_mul_8way_independent`: `16.247 ns` total + - `fp128_8way_passthrough`: `4.8632 ns` total + +Derived estimates: + +- `single - black_box_only`: `1.8829 ns` (`~6.59 cycles @3.5GHz`) +- `(8way_independent - 8way_passthrough)/8`: `1.4230 ns/op` (`~4.98 cycles @3.5GHz`) +- chain-latency proxy: `5.8340 ns/op` (`~20.42 cycles @3.5GHz`) + +Interpretation: + +- Throughput can approach ~5 cycles/op in independent streams. +- Dependent latency stays around ~20 cycles/op, matching the carry-chain structure in codegen. + +## Manual phase annotation + +`_fp128_mul_field` naturally splits into 4 stages: + +1. **Load limbs** + - `ldp x9,x10,[x0]` and `ldp x11,x12,[x1]` +2. **2x2 schoolbook + row carries** + - 4 low products + 4 highs arranged with `adds/adcs/adc` +3. **Solinas fold-1 with `c=8207`** + - `mul/umulh` by `c` on upper carry limb path +4. **Fold-2 + canonicalize** + - add `c` once, then choose reduced/non-reduced using final carry condition (`csel`) + +The emitted code is already branchless in the hot path (`csel` only, no branch mispredict risk). + +## Entire raw `_fp128_mul_field` assembly + +```asm +_fp128_mul_field: + .cfi_startproc + ldp x9, x10, [x0] + ldp x11, x12, [x1] + mul x13, x11, x9 + umulh x14, x11, x9 + mul x15, x12, x9 + umulh x9, x12, x9 + umulh x16, x11, x10 + mul x11, x11, x10 + umulh x17, x12, x10 + mul x10, x12, x10 + adds x11, x11, x14 + cset w12, hs + adds x9, x9, x16 + cset w14, hs + adds x9, x9, x10 + cinc x10, x14, hs + adds x11, x11, x15 + adcs x9, x9, x12 + adc x10, x17, x10 + mov w12, #8207 + mul x14, x9, x12 + umulh x9, x9, x12 + umulh x15, x10, x12 + mul x10, x10, x12 + adds x9, x9, x11 + cset w11, hs + adds x9, x9, x10 + cinc x10, x11, hs + adds x11, x14, x13 + cset w13, hs + adcs x14, x9, xzr + adc x10, x15, x10 + mul x10, x10, x12 + adds x10, x11, x10 + cset w11, hs + adc x9, x9, x13 + adds x12, x10, x12 + adcs x11, x14, x11 + csel x9, x11, x9, hs + csel x10, x12, x10, hs + stp x10, x9, [x8] + ret + .cfi_endproc +``` + +## AArch64 carry-path attempt result + +I implemented and benchmarked an AArch64-specific inline-asm carry helper path (`adds/adc` helper blocks for 2/3/4-term limb sums), then reverted it. + +Observed outcome (after thermal cooldown rerun): + +- It did **not** improve this kernel. +- Primary reason: helper-level asm barriers reduced global scheduling quality and caused overall slowdown versus LLVM's current fully inlined codegen. + +Conclusion: micro-asm helpers are not the right granularity here. + +## Optimization opportunities spotted + +These are the only realistic next knobs I see from the current asm: + +1. **Whole-kernel AArch64 asm (not helper-level asm)** + - There are several `cset` + `cinc` carry extractions. + - If we pursue arch-specific optimization, it should be a single end-to-end kernel so NZCV stays live across the whole sequence. + - This is still the main remaining codegen-level opportunity, but with higher implementation complexity. + +2. **Register-pressure-aware scheduling for fold boundaries** + - The kernel is short already; gains here are likely small. + - Might improve by interleaving one fold multiply earlier, but LLVM already does a reasonable schedule. + +3. **Target-specific build flags** + - Ensure release benches use `target-cpu=native` for final local tuning runs. + - This can help instruction selection/scheduling but is environment-specific. + +No obvious algorithmic simplification is left at Rust level without moving to explicit architecture-specific code. diff --git a/docs/HACHI_FOR_JOLT.md b/docs/HACHI_FOR_JOLT.md new file mode 100644 index 00000000..b9801f41 --- /dev/null +++ b/docs/HACHI_FOR_JOLT.md @@ -0,0 +1,2234 @@ +# Hachi for Jolt (128-bit field arithmetization + lattice PCS commitments) + +This document is a **Jolt-first** design spec for integrating a Hachi-like lattice PCS into Jolt’s architecture, under the hard requirement that Jolt’s sumchecks and transcript challenges live in a **128-bit prime field** \(F_{q'}\). + +It incorporates: + +- Jolt’s Stage 8 interface shape (open many polynomials at one point in \(F_{q'}\)), +- a recommended **field-to-ring embedding** choice (SuperNeo-style coefficient embedding) for one-hot/bit tables, +- a two-modulus “ring switching + modulus switching” opening reduction that stays verifier-native in \(F_{q'}\), +- concrete parameter crunching, including an estimate for the extra modulus quotient witness \(s\). + +This is intentionally “implementation-ready”: it names concrete witness objects, indicates what is public vs committed vs proven, and highlights which parts require changing Jolt’s PCS trait boundary. + +--- + +## 0. Executive summary (what we’re building) + +Jolt’s prover does: + +1. **Commit** to a collection of multilinear polynomials (witness tables). +2. Run application sumchecks (Stages 1–7) over a field \(F_{q'}\) and accumulate evaluation claims. +3. In **Stage 8**, prove a batch opening: many claims \(P_i(\mathbf r)=v_i\) at a **single point** \(\mathbf r\in F_{q'}^m\). + +We want: + +- commitments that are fast for one-hot/bit-heavy tables (so we prefer **small-modulus ring arithmetic** / NTT-friendly layouts), and +- an opening proof that can satisfy Jolt Stage 8’s interface (open at \(\mathbf r\in F_{q'}^m\)) while avoiding expensive ring multiplication. + +The core idea is: + +- Commit over a cyclotomic ring \(R_q = Z_q[X]/(X^d+1)\) with a **small** prime \(q\) (e.g. 32-bit). +- Verify openings using **ring switching evaluated in the large field** \(F_{q'}\) (128-bit) so all “point-derived” arithmetic is native to Jolt’s world. +- Add one extra quotient witness \(s\) so that “equalities mod \(q\)” become honest integer equalities when checked inside \(F_{q'}\). + +--- + +## 1. Requirements and constraints + +### 1.1 Jolt requirement: 128-bit prime field + +Jolt’s arithmetization / sumchecks require that the field characteristic exceeds certain \(u64\cdot u64\) accumulation bounds, so we must work over a **128-bit prime**: + +\[ +F_{q'}\quad\text{with } q' \approx 2^{128}. +\] + +This means we cannot “cheat” by using a small prime and moving to an extension \(F_{q^k}\), because the characteristic bound is on the base characteristic. + +### 1.2 Jolt Stage 8 interface shape + +Jolt’s accumulated opening claims are of the form: + +\[ +(\mathbf r,\ v)\quad\text{with }\mathbf r\in F_{q'}^m,\ v\in F_{q'}. +\] + +See: + +- `../jolt/jolt-core/src/poly/opening_proof.rs`: `pub type Opening = (OpeningPoint<..., F>, F);` +- `../jolt/jolt-core/src/zkvm/prover.rs`: Stage 8 calls `PCS::prove(..., &opening_point.r, ...)`. + +So any PCS that plugs into Stage 8 must accept opening points in \(F_{q'}\). + +### 1.3 Witness regime: one-hot / bits + +Many Jolt witness tables are one-hot or bit-heavy (see `docs/ONE_HOT_COMMITMENT_COST_AND_GPU_PRG.md`). We want “pay-per-bit” style cost where possible, so we strongly prefer embeddings that keep bits sparse in the ring representation. + +--- + +## 2. Notation + +- **Small prime**: \(q \approx 2^{32}\). +- **Large prime**: \(q' \approx 2^{128}\). +- **Cyclotomic degree**: \(d=2^\alpha\) (e.g. \(d=1024\)). +- **Ring**: \(R_q := Z_q[X]/(X^d+1)\). +- **Integer ring**: \(Z\) is the integers, and \(Z[X]\) is the integer polynomial ring. +- **Digit base**: \(b := 2^{\mathrm{LOG\_BASIS}}\) (e.g. \(b=16\)). +- **Digit length**: \(\delta := \lceil \log_b q\rceil\). +- **Redecomposition length**: \(\tau\) (Hachi §4.2). +- **Jolt opening point**: \(\mathbf r\in F_{q'}^m\) (Stage 8). +- **Hachi split**: \(\ell=m+r\) where \(m\approx r\) and the coefficient table has length \(2^\ell\). +- **Block count / block size**: there are \(2^r\) blocks, each of length \(2^m\). + +--- + +## 3. Embedding choice for Jolt: SuperNeo coefficient embedding (recommended) + +### 3.1 Why not use an extension-field embedding here? + +Jolt’s constraint on the **field characteristic** forces the interactive proof algebra to live in a large *prime* field \(F_{q'}\). +So we do not get to pick a small prime \(q\) and move to an extension \(F_{q^k}\); that would change the characteristic. + +Therefore, any embedding machinery whose purpose is “handle points/values in \(F_{q^k}\)” is simply orthogonal to the Jolt setting. + +### 3.2 SuperNeo coefficient embedding preserves bit sparsity + +SuperNeo’s coefficient embedding (Def. 7 in SuperNeo) maps a length-\(d\) field block into one ring element by placing entries directly in the coefficient slots. In our power-of-two cyclotomic: + +\[ +v=(v_0,\dots,v_{d-1})\in F_q^d +\ \mapsto\ +\mathbf v(X)=\sum_{i=0}^{d-1} v_i X^i\in R_q. +\] + +For one-hot/bit tables, coefficients remain \(\{0,1\}\), which enables: + +- sparse gadget decomposition (often effectively \(\delta=1\) for bit witnesses), and +- “rot+add” multiplication by sparse polynomials if needed. + +See `docs/FIELD_EMBEDDINGS_SUPERNEO_VS_HACHI.md` for the math-first comparison and the explicit \(X^d+1\) inner-product transform. + +--- + +## 4. Two-field opening protocol (Option A): commit over \(R_q\), open at \(\mathbf r\in F_{q'}^m\) + +**Compatibility warning (important):** for the genuine two-field setting with coefficient packing \(d>1\) (e.g. \(d=1024\)), the clean correctness-first spec lives in `docs/TWO_FIELD_OPENING_D_GT_1_CLEAN.md`. + +The remainder of this Section 4 is an exploratory draft that mixes the original Hachi Step-B template (which assumes the evaluation point lives in the same algebraic domain as the ring arithmetic) with the Jolt requirement \(\mathbf r\in F_{q'}^n\). Use it for intuition only; do not treat it as a coherent d>1 protocol without reconciling the typing issues called out in that clean spec. + +### 4.1 High-level picture + +We implement a PCS with: + +- **Commitment-time arithmetic** over \(R_q\) (small \(q\)), +- **Verifier-side checking** (sumcheck challenges, equality polynomials, batching weights) over \(F_{q'}\), +- **Ring switching** to move ring equations to field equations at a random \(\alpha\in F_{q'}\), +- plus **modulus switching** slack \(q\cdot s\) so “mod-\(q\)” equalities make sense as integer equalities when checked in \(F_{q'}\). + +### 4.2 One opening claim + +Input statement (Stage 8 style): + +\[ +\text{Given commitment } C \text{ to } P,\ \mathbf r\in F_{q'}^m,\ v\in F_{q'},\ \text{prove } P(\mathbf r)=v. +\] + +Here \(P(\mathbf r)\) means: interpret the committed coefficients as integers (bitness makes this canonical), and evaluate the multilinear extension in \(F_{q'}\). + +### 4.3 Full “one opening proof step” (commit structure + full relation + witness) + +This section is the **self-contained core**: it writes the full relation/witness that the opening proof reduces to. + +#### 4.3.1 Commitment objects (Hachi §4.1 core, ring-side) + +Fix public commitment matrices (Ajtai-style) over \(R_q\): + +\[ +A\in R_q^{n_A\times (\delta\cdot 2^m)},\quad +B\in R_q^{n_B\times (n_A\delta\cdot 2^r)},\quad +D\in R_q^{n_D\times (\delta\cdot 2^r)}. +\] + +Let \(f\in R_q^{\le 1}[X_1,\dots,X_\ell]\) be the committed multilinear, with coefficient table +\((f_{i\parallel j})_{i\in\{0,1\}^r,\ j\in\{0,1\}^m}\subset R_q\). +Define the block slices: + +\[ +f_i^\top := (f_{i\parallel j})_{j\in\{0,1\}^m}\in R_q^{2^m}. +\] + +Gadget decomposition (base \(b\)) gives: + +\[ +s_i := G^{-1}_{2^m}(f_i)\in R_q^{2^m\delta}. +\] + +Inner and outer commitments: + +\[ +t_i := A s_i \in R_q^{n_A},\quad +\hat t_i := G^{-1}_{n_A}(t_i)\in R_q^{n_A\delta}, +\] + +\[ +u := B[\hat t_1;\dots;\hat t_{2^r}] \in R_q^{n_B}. +\] + +This \(u\) is the PCS commitment. + +#### 4.3.2 The opening-derived monomial vectors (Jolt field-side) + +Jolt Stage 8 provides a single opening point \(\mathbf r\in F_{q'}^{n}\) (after claim reductions). +In the special case \(d=1\) (no packing) we have \(n=\ell=m+r\). If \(d>1\), then typically \(n=\ell+\log_2(d)\); see Section 4.3.2A. +Define the usual multilinear monomial vectors in the **field** \(F_{q'}\): + +\[ +b^\top := (r_1^{i_1}\cdots r_r^{i_r})_{i\in\{0,1\}^r}\in (F_{q'})^{2^r}, +\qquad +a^\top := (r_{r+1}^{j_1}\cdots r_\ell^{j_m})_{j\in\{0,1\}^m}\in (F_{q'})^{2^m}. +\] + +These are the same objects as Hachi Eq. (12), but they are computed in \(F_{q'}\) (not in \(R_q\)). + +#### 4.3.2A Critical scope caveat: this Step-B template assumes *no packing* (\(d=1\)), or an additional “inner evaluation” reduction + +The rest of Section 4.3 copies Hachi’s Step-B structure (aux commitment to partial evaluations, then folding, then a stacked linear system) +and was written with the “ring-level variable count” \(\ell=m+r\) in mind. + +If we are in the packed regime \(d>1\) (e.g. \(d=1024\)), then Jolt’s scalar polynomial has: + +- \(N = d\cdot 2^\ell\) scalar coefficients, hence +- \(n := \log_2 N = \ell + \alpha\) variables, where \(\alpha := \log_2 d\). + +So the *opening point* naturally has \(n=\ell+\alpha\) coordinates \(\mathbf r\in F_{q'}^{\ell+\alpha}\), and the last \(\alpha\) coordinates index +the “inner coefficient slots” inside each packed ring element. + +As written, however, the monomial vectors \(a,b\) are defined using only \(\ell\) coordinates (the “outer” ones), and the protocol implicitly treats +each ring element’s \(d\) coefficients as if they were “already evaluated” or irrelevant. This is **not correct** for the Jolt setting when \(d>1\): + +- Multilinear evaluation over the inner \(\alpha\) coordinates uses **multilinear monomials** \(\prod_t r_{\ell+t}^{k_t}\). +- Ring switching at \(X=\alpha\) produces **power monomials** \(\alpha^k\) from the coefficient basis \(X^k\). + +These are algebraically different in general, so ring switching does not “automatically” account for the inner \(\alpha\) coordinates. + +Separately, the “aux commitment” step below defines \(w_i := a^\top G_{2^m}s_i\in R_q\) while \(a\in F_{q'}^{2^m}\) and \(G_{2^m}s_i\in R_q^{2^m}\). +That dot product is not an \(R_q\)-operation unless the weights \(a\) live in \(R_q\) (as in Hachi’s original extension-field-in-ring setting). +In the genuine two-field setting (\(F_{q'}\) independent of \(R_q\)), \(w_i\) would live in \(F_{q'}[X]/(X^d+1)\), not in \(R_q\), and the +subsequent digit decomposition and Ajtai commitment \(v := D\hat w\) over \(R_q\) are not well-defined. + +**Conclusion**: the Step-B template in Sections 4.3.3–4.3.6 is internally consistent in the special case \(d=1\) (no packing), +or if we first add a separate reduction that turns an \(n=\ell+\alpha\)-variate scalar opening into an \(\ell\)-variate ring-level statement +with ring-native weights (analogous in spirit to Hachi’s \(\psi/\mathrm{Tr}\) machinery, but not provided by ring switching alone). + +We keep the Step-B material below because it is still the right template for the **unstructured linear relation + ring/modulus switching + sumcheck** +pipeline once the statement is made well-typed. But for \(d>1\) in Jolt, an additional “inner evaluation” mechanism is required. + +#### 4.3.2B The fix for \(d>1\): inner evaluation reduction via sumcheck + batching into ring switching + +##### 4.3.2B.1 Why the algebraic "inner product to linear functional" bridge is unavailable + +In the original Hachi (extension-field setting), the psi/trace identity (Paper Theorem 2) +\(\mathrm{Tr}_H(\psi(\mathbf a)\cdot\sigma_{-1}(\psi(\mathbf b))) = (d/k)\langle \mathbf a,\mathbf b\rangle\) +algebraically converts the multilinear inner product (which IS the evaluation) into a ring operation +that can be committed to and ring-switched. This works because \(F_{q^k}\) embeds into \(R_q\) as +the fixed subring \(R_q^H\), giving the algebraic bridge. + +In the two-field setting, \(F_{q'}\) is an independent prime field with **no embedding into \(R_q\)**. +There is no psi map, no trace identity, and no algebraic way to convert "inner product of an +\(F_{q'}\)-vector with a coefficient vector" into a ring operation over \(R_q\). + +**What we use instead**: an \(\alpha\)-round inner sumcheck (Phase 0) that reduces the multilinear +inner-coordinate evaluation to a random-point claim, followed by a batching trick that converts +random-point claims into polynomial-evaluation claims (i.e. ring switching). This is the +**computational substitute** for the missing algebraic identity. It costs \(\alpha\) extra sumcheck +rounds and \(d\) extra field elements in the proof. + +##### 4.3.2B.2 Setup and notation + +Let \(\alpha := \log_2 d\) (e.g. \(\alpha=10\) for \(d=1024\)), \(n := \ell + \alpha\) (total scalar variable count), +opening point \(\mathbf r = (\mathbf r_{\mathrm{out}},\mathbf r_{\mathrm{in}}) \in F_{q'}^{\ell} \times F_{q'}^{\alpha}\), +scalar coefficients \(p_{I,k}\in\{0,1\}\) for \(I\in\{0,1\}^\ell,\; k\in\{0,1\}^\alpha\), +and ring element \(f_I(X) := \sum_{k=0}^{d-1} p_{I,k}\,X^k \in R_q\) (coefficient embedding). + +The correct scalar evaluation claim is: + +\[ +\boxed{ +v = \sum_{I\in\{0,1\}^\ell}\;\sum_{k\in\{0,1\}^\alpha} +\iota(p_{I,k})\cdot \mathrm{eq}(\mathbf r_{\mathrm{out}},I)\cdot \mathrm{eq}(\mathbf r_{\mathrm{in}},k) +\;\in F_{q'}. +} +\] + +This is well-typed (pure \(F_{q'}\)), linear in the \(p_{I,k}\), and accounts for all \(n\) coordinates. + +##### 4.3.2B.3 Phase 0: inner evaluation sumcheck (\(\alpha\) rounds) + +Reorganize the evaluation as +\(v = \sum_{I} \mathrm{eq}(\mathbf r_{\mathrm{out}},I) \cdot f'_I\) +where \(f'_I := \sum_{k} \iota(p_{I,k})\cdot \mathrm{eq}(\mathbf r_{\mathrm{in}},k)\) is the "inner multilinear evaluation" +of ring element \(f_I\)'s coefficient vector at \(\mathbf r_{\mathrm{in}}\). + +Run an \(\alpha\)-round sumcheck on the inner sum. Define +\(c_k := \sum_I \mathrm{eq}(\mathbf r_{\mathrm{out}},I)\cdot\iota(p_{I,k})\) (the "coefficient-plane partial evaluations"). +The sumcheck proves \(\sum_k c_k\cdot\mathrm{eq}(\mathbf r_{\mathrm{in}},k) = v\). +After \(\alpha\) rounds the verifier obtains a random point \(\mathbf t^*\in F_{q'}^\alpha\) and needs the oracle value +\(g^* := \sum_k c_k\cdot\mathrm{eq}(\mathbf t^*,k)\). + +##### 4.3.2B.4 Prover sends coefficient-plane values; verifier checks and batches + +The prover sends the \(d\) values \(c_k\) for \(k\in\{0,1\}^\alpha\). +The verifier checks \(\sum_k c_k\cdot\mathrm{eq}(\mathbf t^*,k) \stackrel{?}{=} g^*\) (cost: \(O(d)\) field ops). + +Now **batch** the \(d\) claims \(c_k = \sum_I \mathrm{eq}(\mathbf r_{\mathrm{out}},I)\cdot\iota(p_{I,k})\) into one. +The verifier samples \(\gamma\leftarrow F_{q'}\) and computes: + +\[ +V_\gamma := \sum_{k=0}^{d-1} \gamma^k \cdot c_k += \sum_{I\in\{0,1\}^\ell} \mathrm{eq}(\mathbf r_{\mathrm{out}},I) +\cdot \underbrace{\sum_k \iota(p_{I,k})\gamma^k}_{=\;\mathrm{ev}_\gamma(f_I)}. +\] + +The right-hand side is: **"evaluate the ring-switched table +\((\mathrm{ev}_\gamma(f_I))_I\) as a multilinear at \(\mathbf r_{\mathrm{out}}\)."** +This is an \(\ell\)-variable scalar evaluation where each table entry is obtained by ring switching at \(\gamma\). + +##### 4.3.2B.5 Phase 1: modified Hachi Step B (ring-switched at \(\gamma\)) + +The claim is now \(V_\gamma = \sum_I \mathrm{eq}(\mathbf r_{\mathrm{out}},I)\cdot \mathrm{ev}_\gamma(f_I)\). +Using the split \(\ell=m+r\) and \(I=(i,j)\): + +\[ +V_\gamma = \sum_i b_i \cdot \Big[\sum_j a_j \cdot \mathrm{ev}_\gamma(f_{i\|j})\Big], +\] + +where \(b_i,a_j\in F_{q'}\) are the outer monomial weights. All products are well-typed: \(\mathrm{ev}_\gamma\) +maps \(R_q\)-elements to \(F_{q'}\), and \(a_j\in F_{q'}\). + +**Key issue**: the prover cannot precompute \(w'_i := \sum_j a_j\cdot \mathrm{ev}_\gamma(f_{i\|j})\) before \(\gamma\) is known. +Although \(\sum_j a_j\cdot f_{i\|j}(X)\) is well-defined over \(F_{q'}[X]\), it lives in \(F_{q'}[X]/(X^d+1)\), not \(R_q\), +so it cannot be committed using \(R_q\)-Ajtai matrices. + +**Resolution**: we do **not** need an auxiliary commitment to partial evaluations in the \(d>1\) regime. +Instead, the evaluation equation enters the final sumcheck as an **additional linear constraint** +on the witness table, alongside the ring/modulus-switched commitment constraints and range constraints. + +##### 4.3.2B.6 Modified constraint set for Phase 1 + +1. **Main commitment consistency**: \(B\hat t = u\). (Pure ring; ring+mod-switch at \(\gamma\).) +2. **Fold consistency (inner commit)**: \(Az = (c^\top\otimes G_{n_A})\hat t\). (Pure ring; ring+mod-switch at \(\gamma\).) +3. **Evaluation equation** (scalar, at \(\gamma\)): + \(V_\gamma = \sum_I \mathrm{eq}(\mathbf r_{\mathrm{out}},I)\cdot\sum_k \iota(p_{I,k})\gamma^k\). + Linear in committed digits with public weights \(\mathrm{eq}(\mathbf r_{\mathrm{out}},I)\cdot\gamma^k\). +4. **Range / bitness constraints**: all digit entries bounded. + +**Removed**: constraint (3) of old template (\(b^\top w = \mathrm{open\_value}\)), +constraint (4) (\(a^\top Gz = \ldots\)), the auxiliary commitment \(v=D\hat w\), and the \(\hat w\) witness. + +##### 4.3.2B.7 How the evaluation equation enters the sumcheck + +The witness table \(\tilde w\) encodes digit-level entries. The scalar coefficient is recovered as +\(p_{I,k} = \sum_\delta b^\delta\cdot\tilde w(I,\delta,k)\). The evaluation constraint becomes: + +\[ +V_\gamma = \sum_{I,k} \mathrm{eq}(\mathbf r_{\mathrm{out}},I)\cdot\gamma^k\cdot\sum_\delta b^\delta\cdot\tilde w(I,\delta,k). +\] + +The public weights \(\mathrm{eq}(\mathbf r_{\mathrm{out}},I)\cdot\gamma^k\cdot b^\delta\) **factorize**: +\(\mathrm{eq}(\mathbf r_{\mathrm{out}},I)\) has a closed-form MLE (Section 4.5.3), +\(\gamma^k\) is a size-\(d\) power table, and \(b^\delta\) is a geometric sequence. +The verifier evaluates public encodings at any sumcheck point in \(O(\ell+d+\delta)\) field ops. +This is batched into the existing sumcheck, adding **no extra rounds**. + +##### 4.3.2B.8 Cost summary + +| Component | Cost | +|---|---| +| Phase 0 inner sumcheck | \(\alpha\) extra rounds (10 for \(d=1024\)) | +| Coefficient-plane values \(c_k\) | \(d\) field elements in proof (16 KiB for \(d=1024\)) | +| Verifier batch check | \(O(d)\) field multiplications | +| Phase 1 eval constraint | no extra rounds (folded into existing sumcheck) | + +##### 4.3.2B.9 Alternative: lift to integers and work over \(R_q\) + +One could lift the \(F_{q'}\) evaluation claim to the integers and prove using \(F_q/R_q\) arithmetic +(the "Option B" approach from Section F.0.6 of `docs/HACHI_DIGEST.md`). Caveats: +(1) every \(F_{q'}\) multiplication becomes limb-by-limb schoolbook over \(Z_q\) at cost \(O(L^2)\) with \(L\approx 4\) limbs; +(2) the \(\mathrm{eq}(\tilde r,I)\) weights involve products of up to \(n\) 128-bit numbers, creating deep non-native chains; +(3) the quotient witness \(\kappa\) for mod-\(q'\) can be enormous; +(4) the expensive part is non-native simulation, which doesn't benefit from NTT-friendly ring structure. +The cross-prime approach (Phase 0 + modified Phase 1) is typically much cheaper. + +#### 4.3.3 Intermediate partial evaluations and auxiliary commitment (ring-side objects) + +Define the ring partial evaluations: + +\[ +w_i := a^\top\cdot G_{2^m}\, s_i \in R_q\quad\text{for each }i\in\{0,1\}^r, +\] + +and digitize: + +\[ +\hat w_i := G^{-1}_1(w_i)\in R_q^\delta. +\] + +Stack \(\hat w := (\hat w_i)_{i\in\{0,1\}^r}\in R_q^{\delta\cdot 2^r}\) and define: + +\[ +v := D\hat w \in R_q^{n_D}. +\] + +The prover sends \(v\) as the first opening message (as in Hachi Fig. 3). + +#### 4.3.4 Fold challenge and folded witness (ring-side) + +The verifier derives a short/sparse ring challenge vector: + +\[ +c=(c_1,\dots,c_{2^r})\in R_q^{2^r}. +\] + +The prover folds: + +\[ +z := \sum_{i=1}^{2^r} c_i s_i \in R_q^{2^m\delta}. +\] + +Because coefficients of \(z\) grow, the prover performs a redecomposition using a second gadget \(J\): + +\[ +\hat z := J^{-1}_{2^m}(z)\in R_q^{2^m\delta\tau}. +\] + +#### 4.3.5 The full Step-B relation (six constraints, verbatim structure) + +The prover’s remaining task is to prove existence of **small** objects \((\hat w,\hat t,\hat z)\) such that all of the following hold (this is the B.4.1 relation, rewritten): + +1. **Aux-commitment consistency**: + \[ + D\hat w = v. + \] + +2. **Main commitment consistency**: + \[ + B\hat t = u. + \] + +3. **Evaluation equation** (in field form): + - let \(w := G_{2^r}\hat w\) (recompose digits to ring elements), then the claimed opening value is: + \[ + b^\top\cdot w = \mathrm{open\_value}\quad(\text{interpreted in }F_{q'}\text{ after ring switching}). + \] + +4. **Fold consistency (partials)**: + - let \(z := J\hat z\), then + \[ + a^\top\cdot G_{2^m}\, z = (c^\top\otimes G_1)\hat w. + \] + +5. **Fold consistency (inner commit)**: + \[ + A z = (c^\top\otimes G_{n_A})\hat t. + \] + +6. **Smallness / range constraints**: + - coefficients of \(\hat w,\hat t,\hat z\) lie in the intended digit ranges (and for bit/one-hot regimes, coefficient constraints can often be strengthened to \(\{0,1\}\) on specific parts of the witness). + +At this point we have reduced the opening claim to a structured “prove knowledge of bounded digits satisfying a public linear system” instance. + +#### 4.3.6 The same Step-B relation as one explicit stacked matrix equation (Hachi Eq. (20)) + +It is easy to lose track of how the opening-point-derived weights \((a,b)\) actually enter the *public* linear system. +The paper makes this explicit by stacking constraints (1)–(5) into a single matrix equation (their Eq. (20)). + +First recall the evaluation rewrite (paper Eq. (15)–(19) style), specialized to our split \(\ell=m+r\): + +- Define partial evaluations (ring-side): + + \[ + w_i := a^\top G_{2^m} s_i \in R_q,\qquad i\in\{0,1\}^r. + \] + +- Stack \(w := (w_i)_i \in R_q^{2^r}\) and digitize/recompose via the gadget map: + + \[ + \hat w := G^{-1}_{2^r}(w),\qquad w = G_{2^r}\hat w. + \] + +- The claimed opening value is the bilinear form: + + \[ + b^\top w = \mathrm{open\_value}. + \] + +- The fold constraint (partials) is: + + \[ + a^\top G_{2^m} z = (c^\top\otimes G_1)\hat w. + \] + +- The fold constraint (inner commit) is: + + \[ + A z = (c^\top\otimes G_{n_A})\hat t. + \] + +Now, writing \(z = J_{2^m}\hat z\) (so \(\hat z := J^{-1}_{2^m}(z)\)), constraints (1)–(5) can be written as the stacked linear system: + +\[ +\begin{bmatrix} +D & 0 & 0 \\ +0 & B & 0 \\ +b^\top G_{2^r} & 0 & 0 \\ +(c^\top\otimes G_1) & 0 & -\,a^\top G_{2^m} J_{2^m} \\ +0 & (c^\top\otimes G_{n_A}) & -\,A J_{2^m} +\end{bmatrix} +\cdot +\begin{bmatrix} +\hat w \\ +\hat t \\ +\hat z +\end{bmatrix} +\;=\; +\begin{bmatrix} +v \\ +u \\ +\mathrm{open\_value} \\ +0 \\ +0 +\end{bmatrix}. +\] + +**Key dependency note (why the verifier “needs \(a,b\)”)**: + +- The *public matrix* on the left contains the rows \(b^\top G_{2^r}\) and \(-a^\top G_{2^m}J_{2^m}\). + These are functions of the opening point \(\mathbf r\) via the monomial vectors \(a=a(\mathbf r)\), \(b=b(\mathbf r)\). +- Therefore, when we later write “package (1)–(5) as \(M\cdot Z = Y\)”, the packaged \(M\) **depends on** \((a,b)\), + and the packaged \(Y\) contains \(\mathrm{open\_value}\) (in addition to \(u,v\)). + +### 4.4 Ring switching + modulus switching (make the relation checkable in \(F_{q'}\)) + +All equalities above are ring equalities in \(R_q\) except the weights \(a,b\) which are in \(F_{q'}\). +We therefore check the relation by moving to \(F_{q'}\) via ring switching. + +#### 4.4.1 Evaluation map into \(F_{q'}\) + +Verifier samples \(\alpha\leftarrow F_{q'}\) and defines + +\[ +\mathrm{ev}_\alpha: R_q \to F_{q'} +\] + +as: choose a lift convention \(\mathrm{lift}_q\) for coefficients of \(Z_q\), lift each coefficient to an integer, and then evaluate the polynomial at \(X=\alpha\) in \(F_{q'}\). + +Apply \(\mathrm{ev}_\alpha\) entrywise to vectors/matrices over \(R_q\). + +#### 4.4.2 The unstructured linear relation and its full witness + +Package constraints (1)–(5) into a single public linear system over \(R_q\): + +\[ +M\cdot Z = Y \quad\text{in }R_q^n, +\] + +where: + +- \(Z\) is the stacked unknown vector whose coordinates are exactly the digits in \((\hat w,\hat t,\hat z)\) (plus any auxiliary digit blocks introduced by the gadget decompositions), +- \(Y\) is the stacked public right-hand side built from \((u,v)\), the claimed open value, and the fold challenge \(c\), +- \(M\) is the stacked public matrix built from \((A,B,D)\), the gadget matrices, and the fold wiring. + +Now lift this ring equation to an **integer polynomial identity** with two quotient witnesses: + +\[ +\mathrm{lift}_q(M)\,\mathrm{lift}_q(Z) - \mathrm{lift}_q(Y) += +(X^d+1)\,r \;+\; q\cdot s +\quad\text{over } Z[X], +\] + +where: + +- \(r\in (Z[X]_{n# zow!xr>eDQzG1|d%djj^Gqa5$}tfQAvvk;c2>T=L?^RRL~5$-ni)q2wz_ zard?m$Oup}til1pTbI?u{4ZYfH<8y_$N4zIA-1(sH%`9zq=v=-->Ylwgdks0%wtll z%LBUsRTUF0!h*sr78jKq0#Gt|6!Rb`wg+Ft+5@`t^yK&{Slu^ zUwgL+C3D1r-x=L91Hkl9q(=<4@F+j}yj7X^f*!yt0SW-)b9RDAvtI9}^e>CDUIi|O z%Q>cf{^p04h7k5OV)<0uY9kC7N+)Fc?aF$pqZ6~>^#16)tdF({j}S^!q$&M%hd`>= z1(QGnxT#UNeH&eCW!|4t<&8_n$E=Tub#V=nveaq>n2rv7G=O!C<{2E#{}4s;N6KtW8;i zE_R>?@Y*#49LZPWFK7P9RBB14Z7k#l#nBm+l2~=9h(`i|Op-qS_lq_26p$H{{%L}4 z%lHk}Wi@Wbgse(v1#XYY&M^%T=SrWbZq3$E-)!xMmp%!` z5m**R8Kdcq(uk=t!Dh)Ozxx9FF=rsqK@%mNC)W@^tKmh1r`@Fz< z+$bR;(vl0@x%XAY&l*zk)9_$UJ{@SN&DgItGYrzsM=vo-1NnuOt-IhM)#9<}&Q0|K z^f`I*bVh%jD?F%)_fzt1&ZAMQe>DUXXc{U{yCZKzz|RkzemNb2Iz~mggXYj$nSBby z8`xnG3St}oC2(F+?at4>k0MsabqOWmZpjr4&dUD?5z@SKqmil>PuMz}+5Z*m%Gv=% z2yhQ2f*2v8*hrp-Z*`e^Q(Vo7{%9g)Kr&L#hh{6TT-3tA)!y?CHnpNKcCl{7-+04} 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In this work, we present Hachi, a concretely efficient multilinear polynomial commitment +scheme that offers succinct proof sizes ofpoly(ℓ, 𝜆) and achieves a “square-root” verifier time complexity +of ˜𝑂 ( +√ +2ℓ 𝜆) for ℓ-variate polynomials under the Module-SIS assumption. Compared to the current +state-of-the-art scheme, Greyhound (CRYPTO 2024), Hachi provides an asymptotic improvement of +˜𝑂 (𝜆) in verification time, which translates into a practical 12.5-fold speedup, while maintaining compact +proofs of approximately 55 KB. +To improve the verification time, we adopt the sumcheck protocol. Note that the standard sumcheck +has efficiency bottlenecks for lattice-based constructions, since lattice operations are usually performed +over power-of-two cyclotomic ringsR𝑞 := Z𝑞 [𝑋 ]/( 𝑋 𝑑 +1). To address this challenge, we provide a novel +approach that integrates Greyhound with the ring-switching idea proposed by Huang, Mao and Zhang +(ePrint 2025). Surprisingly, under this approach, the verifier does not need to perform any multiplication +over R𝑞, enabling a much faster verification time. This technique could be of independent interest for +building lattice-based SNARKs, particularly for achieving faster verification. +As a separate contribution, we introduce a generic reduction that converts polynomial evaluation +proofs over extension fieldsF𝑞𝑘 (under suitable parameter regimes) into equivalent statements over +cyclotomic rings R𝑞. This reduction is compatible with existing lattice-based polynomial commitment +schemes and can be integrated as a modular enhancement to broaden applicability to statements over +extension fields. +1 Introduction +Multilinear polynomial commitment scheme (PCS) [KZG10; BBHR18] is a strong cryptographic primitive +that allows a prover to commit to aℓ-dimensional multilinear polynomial𝑓 ∈ F≤1 [𝑋1, . . . , 𝑋ℓ ] and later prove +its evaluation +𝑓 (𝑥1, . . . , 𝑥ℓ ) = 𝑦 +where the evaluation point(𝑥1, . . . , 𝑥ℓ ) ∈ Fℓ and the image𝑦 ∈ F are public. Multilinear polynomial commit- +ments are a core building block for succinct non-interactive arguments of knowledge (SNARKs) [WTs+18; +Set20; CHM+20; GLS+23; CBBZ23], look-up arguments [STW24] and multi-party computation [BHV+23]. +In such applications, it is crucial for the polynomial evaluation proof to be succinct, i.e. polynomial in the +number of variablesℓ, and be verified efficiently. +With the rapid advancement of quantum computing, there is increasing interest in quantum-safe PCS +constructions. Among these, lattice-based and hash-based approaches have emerged as the leading candidates. +Lattice-based PCS are particularly attractive due to their efficient prover runtime and compact proof sizes, +both concretely [NS24] and asymptotically [KLNO25]. In contrast, hash-based constructions currently offer +significantly faster verification (by two orders of magnitude) compared to lattice-based schemes [ACFY24; +ACFY25]. Such a long verification time has become a major bottleneck for lattice-based protocols, such as +Greyhound, being widely deployed in practice. This discrepancy raises a natural research question: +Can we design a lattice-based PCS that has fast verification while retaining the compact proof sizes? +In this paper, we answer this question affirmatively. + +--- page 2 --- + +1.1 Prior Works +Split-and-fold. At the core of state-of-the-art lattice-based succinct proof systems (including polynomial +commitments) lies proving knowledge of vectorss1, . . . , s𝑟 ∈ R𝑚 +𝑞 such that +B + +A +. . . +A + + +s1 +... +s𝑟 + += u (mod 𝑞) and ∥s𝑖 ∥ ≤ 𝛽 for 𝑖 = 1, . . . , 𝑟 (1) +where R𝑞 := Z𝑞 [𝑋 ]/( 𝑋 𝑑 +1) for a power-of-two𝑑, and the matricesA, B, vector t and norm bound 𝛽 are +public. The first interactive protocol for this relation was proposed by Baum et al. [BBC+18] which can be +summarized as follows. The prover starts by sending the “partial evaluations”t𝑖 = As𝑖 for 𝑖 ∈ [ 𝑟 ] (this step is +called split). The verifier responds by outputting short challenges𝑐1, . . . , 𝑐𝑟 ∈ C over the cyclotomic ringR𝑞. +Finally, the prover sends returns the linear combination of witness vectors (this step is calledfold) +z := 𝑐1s1 + . . . 𝑐𝑟 s𝑟 ∈ R𝑚 +𝑞 . (2) +The verifier then checks three conditions: +B + +t1 +... +t𝑟 + += u, Az = +𝑟∑︁ +𝑖=1 +𝑐𝑖 t𝑖 and ∥z∥ is short. (3) +The complexity of the communication of the prover is𝑟 + 𝑚 ring elements, which is strictly sublinear in the +original witness length𝑁 := 𝑟 ·𝑚 in terms ofR𝑞-elements. By picking parameters appropriately, [BBC+18] +obtains “square-root” proofs of size˜𝑂 ( +√ +𝑁 𝜆), where 𝜆 is the security parameter. +Multiple follow-up works [BLNS20; AFLN24; CMNW24; KLNO24] managed to reduce the proof size +further topolylog(𝑁 ) ring elements. The key insight lies in making the matrixA to have a tensor structure +similar to the matrix on the left-hand side of Equation (1). This allows the prover, instead of sending the +final message z in the clear, to prove knowledge ofz satisfying the verification equation of(3) in a recursive +fashion. +Exact norm proof.Arguably, the hardest part of designing proof systems for relations such as(1) is proving +that vectors s𝑖 are actually short. Indeed, knowledge soundness of the original protocol of [BBC+18] only +guarantees extraction of (not necessarily short)¯s1, . . . ,¯s𝑟 ∈ R𝑚 +𝑞 of the form +¯s𝑖 := ¯z𝑖 +¯𝑐𝑖 +where both¯z𝑖,¯𝑐𝑖 are short. +There is a line of research [BLNS20; AL21; ACK21; CLM23; FMN24; AFLN24; KLNO24] which addresses +this issue using so-called subtractive sets. A setC ⊆ R𝑞 is called subtractive if for any two distinct elements +𝑐, 𝑐′ ∈ C, inverse of the difference𝑐 − 𝑐 ′ over R𝑞 is short. This allows the knowledge extractor to deduce that +the extracted¯s𝑖 are short. This approach has one crucial disadvantage, which is that all subtractive sets are +only of polynomial size [AL21]. Consequently, to obtain negligible soundness error, multiple repetitions are +needed which affect the efficiency of the scheme significantly. +An alternative approach to proving shortness, which avoids subtractive sets, involves applying random +projections to the witness vectors [BL17; LNS21; LNP22; GHL22; BS23; KLNO25]. Lets′ ∈ Z𝑚𝑑 +𝑞 denote +the coefficient vector ofs ∈ R𝑚 +𝑞 . Consider a matrixJ ∈ Z𝑟 ×𝑚𝑑 with small, independently chosen random +entries, where 𝑟 is a soundness parameter. The Johnson-Lindenstrauss Lemma (and heuristic variants [Ach03; +GHL22]) implies that +∥Js′ ∥ ≈ ∥ s′ ∥ = ∥s∥. +2 + +--- page 3 --- + +Using this fact, the verifier sends the projection matrixJ to the prover (or in the non-interactive case it is +generated from a hash function), who responds withv := Js′ ∈ Z𝑟. The verifier then checks whetherv is short. +For soundness, previous works show that ifs were large (i.e. a prover was malicious), then with probability +exp(−𝑟 ), v would also have large norm, and thus result in a contradiction. Hence,𝑟 = 𝑂 (𝜆) is required for +soundness. +The prover must also prove thatv is well-formed. As shown in [LNP22; BS23], this reduces to proving that +certain 𝑟 polynomials 𝑢1, 𝑢2, . . . , 𝑢𝑟 ∈ R𝑞 have zero constant coefficients. One method is to send all(𝑢𝑖 )𝑖 ∈ [𝑟 ] in +the clear for the verifier to check, but this requires sending at least𝑂 (𝜆) ring elements, resulting in large proofs. +A more efficient method batches the𝑟 statements: the verifier sends𝑟 random challenges𝛼1, . . . , 𝛼𝑟 ← Z𝑞, and +the prover returns the linear combination +𝑢 := +𝑟∑︁ +𝑖=1 +𝛼𝑖 ·𝑢𝑖 ∈ R𝑞. +The verifier checks that the constant coefficient of𝑢 is zero. For soundness, if some𝑢𝑖 had a non-zero constant +coefficient, then with probability 1/𝑞, so would𝑢, leading to a contradiction. Since 1/𝑞 may not be negligible, +this must be repeated𝑂 (𝜆/log 𝑞) times to achieve exp(−𝜆) soundness error. +Concretely, the Johnson-Lindenstrauss projection has shown to be a practical approach to prove norm +bounds compared to subtractive sets. The prime example is the LaBRADOR proof system introduced by +Beullens and Seiler, which achieves≈ 60KB proofs for large statements (of size 230). The main disadvantage +of the random projection methodology is the verification time. Indeed, the verifier has to operate on the +projection matrix J of size 𝑂 (𝜆 ·𝑚𝑑) – linear in witness length (in terms ofZ𝑞-elements). This makes the +LaBRADOR verifier linear time, and thus not suitable for various SNARK-oriented applications. More +recently, Klooß et al. [KLNO25] focused on tensor-structured projection matricesJ, and managed to reduce +the verification time to polylog(𝑚𝑑), at the cost of 40X larger proof sizes compared to LaBRADOR. +Nguyen and Seiler [NS24] proposed a promising hybrid polynomial commitment, called Greyhound. Their +evaluation proof intuitively follows the three-round protocol from [BBC+18], but instead of sending the prover +message in the clear, the prover commits to(t𝑖 )𝑖, z and later proves that they satisfy the verification equations +in (3) via the LaBRADOR proof system. The key advantage of this strategy is that invoking LaBRADOR on +a smaller statement yields square-root˜𝑂 (𝜆 +√ +𝑁 𝜆) verification time, while benefiting from the attractive proof +sizes of [BS23]. +Several recent works employ the sumcheck protocol [LFKN90] to prove exact norm bounds of committed +vectors based on lattice assumptions, both in the setting of succinct proofs and folding schemes [KST22]. +This strategy was first used by Boneh and Chen in LatticeFold [BC24] to prove that the committed (NTT) +coefficients are binary. A drawback of their approach is that the sumcheck protocol is executed over the +cyclotomic ring R𝑞, where each ring element is large, resulting in significantly increased proof sizes. Kuriyama +et al. [KLOT25] later expanded this methodology and introduced a new technique to prove exact Euclidean +norm bounds usingR𝑞-based sumchecks. Neo [NS25] departs fromR𝑞-based sumchecks entirely by mapping +all relevantR𝑞-relations into Z𝑞 via rotational (i.e., skew-circulant) matrices. This allows the protocol to run +sumcheck directly overZ𝑞 (or over an extension fieldF𝑞𝑘 to obtain negligible soundness error). Concurrently, +LatticeFold+ [BC25] proposed a new method to prove range statements. Specifically, to prove that𝑎 ∈ [0, 𝑑), +the prover commits to the monomial𝑋 𝑎 ∈ R𝑞 and then proves that the committed ring element is indeed a +monomial. LatticeFold+ reduces this (and related statements) to sumcheck claims overR𝑞. In contrast to +[BC24; NS25], the authors observe that these claims do not involve any “full-fledged”R𝑞-multiplications and +can therefore be naturally reduced to𝑂 (𝑑) sumcheck claims overZ𝑞. The main efficiency bottleneck of this +approach is that, for a witness®𝑠 ′ ∈ Z𝑚𝑑 +𝑞 of length 𝑚𝑑, the prover must commit to𝑚𝑑 elements of R𝑞, rather +than 𝑚 as in [BS23; NS24; NS25] (excluding decomposition costs). +1.2 Our Contributions +In this work, we propose Hachi, a concretely efficient multilinear polynomial commitment scheme with succinct +poly(ℓ, 𝜆) proof size and˜𝑂 ( +√ +2ℓ𝜆) verifier time forℓ-variate polynomials based on standard lattice assumptions +3 + +--- page 4 --- + +Scheme Multi Extension Asymptotic Concrete +-linear fields proof verifier proof verifier +Greyhound [NS24] ✗ ✗ poly(ℓ, 𝜆) ˜𝑂 (𝜆 · +√ +2ℓ 𝜆) 53KB 2.8s +Hachi ✓ ✓ poly(ℓ, 𝜆) ˜𝑂 ( +√ +2ℓ 𝜆) 55KB 227ms +Fig. 1. Overview of Hachi and comparison with the state-of-the-art lattice-based polynomial commitment scheme +Greyhound [NS24]. The asymptotic verifier time is measured inZ𝑞-operations. We use the ˜𝑂-notation to hide +logarithmic factors. The concrete numbers are taken from evaluating the polynomial withℓ = 30 variables. +(and the random oracle model for the Fiat-Shamir transformation). Hence, asymptotically we achieve˜𝑂 (𝜆) +improvement in verification time compared to Greyhound [NS24]. In Figure 1 we provide more details on the +comparison. Our improvements can be summarized as follows. +Faster verification via ring switching.Our protocol mainly follows the framework of Greyhound [NS24]. +However, unlike Greyhound, we move away from the random projection strategy and instead adopt a +sumcheck-based approach to prove infinity-norm bounds. Although sumcheck is a well-known protocol, we +emphasize that the main challenge for lattice-based constructions is that lattice protocols typically involve +arithmetic over large polynomial rings. Directly running the sumcheck protocol over such large rings would +be highly inefficient. +To address this challenge and keep the protocol efficient, we propose a novel approach that organically +integrates the Greyhound framework with the ring-switching technique introduced by Huang, Mao and Zhang +[HMZ25]. This technique lifts statements overR𝑞 to the polynomial ringF𝑞𝑘 [𝑋 ], allowing us to evaluate +the equations at a random point𝑋 := 𝛼 ∈ F𝑞𝑘 and run the sumcheck protocol over the extension fieldF𝑞𝑘. +Surprisingly, our verification algorithm does not need to perform any cyclotomic ring operations, even though +our protocol is based on lattices. Switching the sumcheck protocol to a smaller extension field is crucial to +achieving a total verifier complexity of˜𝑂 ( +√ +2ℓ𝜆). To the best of our knowledge, other than our protocol, we are +not aware of any lattice-based SNARK, PCS, or folding scheme that can eliminate the expensive cyclotomic +ring operations for verifiers. +Embedding multilinear evaluation claims insideR𝒒. We provide a generic transformation from proving +ℓ-variate polynomial evaluations over finite fieldsF𝑞𝑘 to proving (ℓ − 𝛼 + 𝜅)-variate polynomial evaluations +over “NTT-unfriendly” cyclotomic ringsR𝑞, where 𝑑 = 2𝛼 and 𝑘 = 2𝜅. Thus, we generalize the result of +SLAP [AFLN24] in two orthogonal directions: (i) we support polynomials over field extensions (for specific +parameter sets), rather than just prime-order fields, and (ii) we extend the strategy from univariate to the +multivariate case. Using our methodology, one can already adapt Greyhound [NS24] to prove multilinear +polynomial evaluations out-of-the-box. We also provide an optimization if the polynomial𝑓 itself is only over +F𝑞 while the evaluation points still lie inF𝑞𝑘. +Flexibility in picking ring dimension and implications.Both Greyhound and LaBRADOR have +certain limitations in picking the ring dimension𝑑. One of the main obstacles is when applying random +projection strategy, one has to prove that certain ring elements have constant coefficients equal to zero. As +described above, this comes at the cost of the prover sending𝑂 (𝜆/log 𝑞) ring elements after each iteration of +LaBRADOR. Hence, choosing𝑑 (equal to 64 in [BS23; NS24]) as small as possible is essential to achieve +small proof sizes. In our construction, since we apply the ring switching idea, the sumcheck protocol is always +performed over an extension field, which is independent of the size of𝑑. This gives us more freedom in +choosing ring dimension𝑑 with smaller impact on the communication size. +In particular, picking larger𝑑 has two efficiency benefits: (i) the committing time becomes faster thanks +to the optimized NTT-based ring multiplication, and (ii) this allows us to define a challenge spaceC of sparse +ring elements - thus speeding up the ring multiplication for computing the folded witness as in (2). +2 Concrete verification time for first round of Hachi. +4 + +--- page 5 --- + +Implementation. WehavedevelopedaprototypeimplementationofHachiinRust.Whiletheimplementation +is still in its early stages, it is amenable to several optimizations—particularly those leveraging SIMD—that +could significantly enhance performance. Even at this early stage, the advantages of our protocol are clearly +demonstrated. Specifically, we observe: +–faster verification time. Although the implementation of our verifier does not use advanced optimizations +such as SIMD, our verification time is already one order of magnitude faster than Greyhound.3 +–faster commitment time. Our commitment step is almost identical to that of Greyhound. However, due +to the use of ring switching, we are able to choose larger rings; as a result, our commitment time is 3-5 +faster than Greyhound4. +1.3 Technical Overview +Denote 𝜆 as a security parameter. Let𝑑 := 2𝛼 be a power-of-two andR := Z[𝑋 ]/( 𝑋 𝑑 + 1) be the ring of +integers of the 2𝑑-th cyclotomic field. Take an odd prime𝑞 and define R𝑞 := R/(𝑞) and 𝛿 := ⌈log 𝑞⌉. For +𝑘 ≥ 1, we denoteF𝑞𝑘 to be a finite field of order𝑞𝑘. In this overview, we consider base-two gadget matrices +G𝑛 := I𝑛 ⊗ + +1 2 4· · ·2𝛿 −1 +∈ R𝑛×𝑛𝛿 +𝑞 for 𝑛 ≥ 1. We define the standard inverse functionG−1 +𝑛 : R𝑛 +𝑞 → R𝑛𝛿 +𝑞 , +which decomposes each entry w.r.t. base 2. In particular, for anyt ∈ R𝑛 +𝑞, G−1 +𝑛 (t) has binary coefficients and +G𝑛G−1 +𝑛 (t) = t. +For 𝑖 ∈ Z× +2𝑑, we define the Galois automorphism 𝜎𝑖 : R → R defined as 𝑋 ↦→ 𝑋 𝑖, and denote the +Galois group of automorphisms as Aut(R) := {𝜎𝑖 : 𝑖 ∈ Z× +2𝑑 }. For a subgroup 𝐻 of Aut(R), we denote +R𝐻 +𝑞 := {𝑥 ∈ R𝑞 : ∀𝜎 ∈ 𝐻, 𝜎 (𝑥) = 𝑥 }, i.e. the ring of elements ofR𝑞 fixed by all automorphisms in𝐻. Also, we +denote the R𝑞-trace map Tr𝐻 : R𝑞 → R𝐻 +𝑞 as Tr𝐻 (𝑎) := Í +𝜎 ∈𝐻 𝜎 (𝑎) ∈ R𝐻 +𝑞 . +Reducing multilinear evaluation claims to relations overR𝒒. Our first step will be to translate the +multilinear evaluation overF𝑞𝑘, i.e. 𝑓 (𝑥1, . . . , 𝑥ℓ ) = 𝑦, to an equivalent relation overR𝑞 (or a related ring). +The hardness lies in translating operations over field extensionsF𝑞𝑘, such as addition and multiplication, to +equivalent operations over the cyclotomic ringR𝑞. We proceed in two steps. First, we directly identify the +finite fields withinR𝑞. +Lemma 1(Informal). Let 𝑞 = 5 (mod 8) and 𝑘 ≥ 1 be a divisor of 𝑑/2 and consider the subgroup +𝐻 := ⟨𝜎−1, 𝜎4𝑘+1⟩ of Aut(R). Then, R𝐻 +𝑞 is a subfield ofR𝑞 isomorphic to F𝑞𝑘. +This lemma allows us to translate proving inner products overF𝑞𝑘 into relations directly over the cyclotomic +ring R𝑞 using the R𝑞-trace map. +Theorem 1(Informal). Let 𝑞 = 5 (mod 8), 𝑘 divide 𝑑/2 and 𝐻 := ⟨𝜎−1, 𝜎4𝑘+1⟩ ⊆ Aut(R). Then, there +exists an efficiently computable bijection𝜓 : (R𝐻 +𝑞 )𝑑/𝑘 → R𝑞 such that for anya, b ∈ ( R𝐻 +𝑞 )𝑑/𝑘, +Tr𝐻 (𝜓 (a) · 𝜎−1(𝜓 (b))) = 𝑑 +𝑘 · ⟨a, b⟩. +We now show how to use these two theoretical results. Let𝑘 = 2𝜅 be a divisor of𝑑/2 and suppose we want to +prove a polynomial evaluation claim overR𝐻 +𝑞 ( F𝑞𝑘 by Lemma 1). Denote the coefficients of polynomial𝑓 as +(𝑓𝜄)𝜄 ∈ {0,1}ℓ ∈ F𝑞𝑘. We can explicitly rewrite the evaluation equation as +𝑦 = +∑︁ +𝑖 ∈ {0,1}ℓ −𝛼 +𝜅 + +𝑥𝑖1 +1 · . . . ·𝑥𝑖ℓ −𝛼 +𝜅 +ℓ −𝛼+𝜅 + +·©­ +« +∑︁ +𝑗 ∈ {0,1}𝛼 −𝜅 +𝑓𝑖 ∥ 𝑗 ·𝑥 𝑗1 +ℓ −𝛼+𝜅+1 · . . . ·𝑥 𝑗𝛼 −𝜅 +ℓ +ª® +¬ +. (4) +3 Concrete verification times for first round of Hachi, and later composed with Greyhound on a smaller instance. +4 When compared with the same implementation. +5 + +--- page 6 --- + +For each𝑖 ∈ {0,1}ℓ −𝛼+𝜅 define the ring element +𝐹𝑖 := 𝜓 (𝑓𝑖 | |𝑗 ) 𝑗 ∈ {0,1}𝛼 −𝜅 + and also 𝑣 := 𝜓 + +(𝑥 𝑗1 +ℓ −𝛼+𝜅+1 · . . . ·𝑥 𝑗𝛼 −𝜅 +ℓ ) 𝑗 ∈ {0,1}𝛼 −𝜅 + +. +Then, using the fact thatR𝐻 +𝑞 is stable under the additively homomorphic maptr𝐻, and by applying Theorem 1 +to Equation (4) we get +𝑑 +𝑘 ·𝑦 = Tr𝐻 +©­­­­­­ +« +∑︁ +𝑖 ∈ {0,1}ℓ −𝛼 +𝜅 + +𝑥𝑖1 +1 · . . . ·𝑥𝑖ℓ −𝛼 +𝜅 +ℓ −𝛼 + +·𝐹𝑖 +| {z } +𝑌:= +·𝜎−1(𝑣) +ª®®®®®® +¬ +. +Therefore, the prover can send a single ring element𝑌 ∈ R𝑞, that allows the verifier to check if theR𝑞-trace +map of 𝑌 · 𝜎−1(𝑣) is equal to 𝑑𝑦/𝑘. Then, we only need to prove that𝑌 is well-formed, which is now an +evaluation proof for the(ℓ − 𝛼 + 𝜅)-variate polynomial 𝐹 := (𝐹𝑖 )𝑖 over R𝑞. +In certain scenarios, the evaluation point might be defined over the extension fieldF𝑞𝑘, while the coefficients +of the committed polynomial𝑓 are over F𝑞 (which is the case after the first iteration of our recursive proof +system). Can we optimize our transformation in this setting without bluntly assuming that𝑓 ∈ F≤1 +𝑞𝑘 [𝑋1, . . . , 𝑋ℓ ]? +To this end, we use the similar observation as before that a polynomial evaluation claim can be written as +𝑦 = +∑︁ +𝑖 ∈ {0,1}𝜅 + +𝑥𝑖1 +1 · . . . ·𝑥𝑖𝜅 +𝜅 + +· +©­­­­­­­ +« +∑︁ +𝑗 ∈ {0,1}ℓ −𝜅 +𝑓𝑖 ∥ 𝑗 ·𝑥 𝑗1 +𝜅+1 · . . . ·𝑥 𝑗ℓ −𝜅 +ℓ +| {z } +𝑦𝑖:= +ª®®®®®®® +¬ +. (5) +Hence, we let the prover send𝑘 partial evaluations𝑦𝑖 ∈ F𝑞𝑘 to the verifier, who can directly check the equation +above5. What we have left to prove is that all the𝑦𝑖 are well-formed, or in other words𝑓𝑖 (𝑥𝜅+1, . . . , 𝑥ℓ ) = 𝑦𝑖 +where 𝑓𝑖 ∈ F𝑞 [𝑋𝜅+1, . . . , 𝑋ℓ ]. +Next, denote F𝑞𝑘 := F𝑞 [𝑍 ]/𝜑 (𝑍 ), where 𝜑 is an irreducible polynomial inF𝑞 [𝑍 ] of degree 𝑘. Then, define +the (ℓ − 𝜅)-variate multilinear polynomial𝑓 ′ ∈ F𝑞𝑘 [𝑋𝜅+1, . . . , 𝑋ℓ ]: +𝑓 ′ (𝑋𝜅+1, . . . , 𝑋ℓ ) := +∑︁ +𝑖 ∈ {0,1}𝜅 +𝑓𝑖 (𝑋𝜅+1, . . . , 𝑋ℓ ) · 𝑍 +Í𝜅 +𝑡 =1 𝑖𝑡 ·2𝑡 −1 +. +Then, it is easy to see that +𝑓 ′ (𝑥𝜅+1, . . . , 𝑥ℓ ) = +∑︁ +𝑖 ∈ {0,1}𝜅 +𝑦𝑖 ·𝑍 +Í𝜅 +𝑡 =1 𝑖𝑡 ·2𝑡 −1 +∈ F𝑞𝑘 +which is now a full-fledged polynomial evaluation overF𝑞𝑘. By applying the generic transformation above, +we can reduce the statement into an evaluation claim overR𝑞 for a (ℓ − 𝛼)-variate multilinear polynomial +(instead of ℓ − 𝛼 + 𝜅, if done generically). +Polynomial evaluation claim as a quadratic equation.Now, consider proving an evaluation claim of +a 𝜇-variate multilinear polynomial𝑓 over R𝑞. We use the following folklore strategy. Suppose𝜇 = 𝑚 + 𝑟 for +some 𝑚, 𝑟 ≥ 1 and letf := (𝑓𝜄)𝜄 ∈ {0,1}𝜇 be the coefficient vector of𝑓. Then, for any +𝑓 (𝑋1, . . . , 𝑋𝜇) := +∑︁ +𝑖 ∈ {0,1}𝑟 +∑︁ +𝑗 ∈ {0,1}𝑚 +𝑓𝑖 | |𝑗 · (𝑋 𝑖1 +1 · . . . ·𝑋 𝑖𝑟 +𝑟 ) · (𝑋 𝑗1 +𝑟 +1 · . . . ·𝑋 𝑗𝑚 +𝜇 ) +5 A simple optimization is for the prover to send only𝑘 −1 partial evaluation𝑦𝑖, excluding 𝑦0...0 ∈ F𝑞𝑘, which can be +deterministically derived from the verification equation. +6 + +--- page 7 --- + +and any evaluation point(𝑥1, . . . , 𝑥𝜇) ∈ R𝜇 +𝑞 we can write: +𝑓 (𝑥1, . . . , 𝑥𝜇) = b⊺ + +a⊺ +. . . +a⊺ + + +f0...0 +... +f1...1 + +where +b⊺ := (𝑥𝑖1 +1 · . . . ·𝑥𝑖𝑟 +𝑟 )𝑖 ∈ {0,1}𝑟 ∈ R2𝑟 +𝑞 +a⊺ := (𝑥 𝑗1 +𝑟 +1 · . . . ·𝑥 𝑗𝑚 +𝜇 ) 𝑗 ∈ {0,1}𝑚 ∈ R2𝑚 +𝑞 +f ⊺ +𝑖 := (𝑓𝑖 | |𝑗 ) 𝑗 ∈ {0,1}𝑚 ∈ R2𝑚 +𝑞 for 𝑖 ∈ {0,1}𝑟 +. +Finally, by defining the decompositions := G−1 +2𝜇 (f ) ∈ R2𝑚𝛿 +𝑞 and using the mixed product of the tensor product, +we deduce that +𝑓 (𝑥1, . . . , 𝑥𝜇) = b⊺ (a⊺ ⊗ I2𝑟 )f += b⊺ (a⊺ ⊗ I2𝑟 ) (g⊺ ⊗ I2𝑚+𝑟 )s += b⊺ (a⊺ (g⊺ ⊗ I2𝑟 ) ⊗ I2𝑟 )s. +Hence, the goal is to prove that the (committed) vectors is short and satisfies the quadratic equation above, +which exactly the relation formulated in Equation (1). To prove knowledge of suchs, we closely follow the +initial methodology from Greyhound [NS24]. That is, we consider the square-root interactive proof from +[BBC+18], but instead of sending the prover messages in the clear, we commit to them. Later, we need +to prove knowledge of the prover messages that satisfy all the verification equations (as well as proving +knowledge of the commitment opening). All these relations can be phrased as proving knowledge of a short +vector z of dimension (2𝑚 +2𝑟 ) ·poly(𝜆) such that +Mz = w and z is short. (6) +This is the point where our strategy diverges from [NS24]. Instead of running LaBRADOR [BS23] to prove +the relation above, we propose a new sumcheck-based solution. +Ring switching and sumcheck over extension fields.We take inspiration from the ring switching +approach proposed recently by Huang, Mao, and Zhang [HMZ25].To begin with, recall that a multilinear +extension mle[𝑓 ] ∈ F≤1 +𝑞𝑘 [𝑋1, . . . , 𝑋𝜇] of 𝑓 is defined as +mle[𝑓 ] (x) := +∑︁ +𝑖 ∈ {0,1}𝜇 +𝑓 (b) · 𝑒𝑞 (𝑖, x) +where 𝑒𝑞 (𝑖, x) := Î𝜇 +𝑗=1 +(1 − 𝑖 𝑗 ) · (1 − 𝑥 𝑗 ) + 𝑖 𝑗𝑥 𝑗 + is the equality polynomial. +The first step is to lift Equation (6) fromR𝑞 to Z𝑞 [𝑋 ]. For the sake of overview, we only focus on the +unstructured linear relation and suppose matrixM has only one row (dealing with multiple rows and proving +norm bounds is presented in Section 4.3). Then, we can rewrite it as +∑︁ +𝑘 +𝑀𝑘 (𝑋 ) · 𝑧𝑘 (𝑋 ) = 𝑤 (𝑋 ) + ( 𝑋 𝑑 +1) · 𝑟 (𝑋 ) for some 𝑟 ∈ Z<𝑑 −1 +𝑞 [𝑋 ]. +Let z′ (resp. r′) be theZ𝑞-coefficient vector ofz (resp. 𝑟). Then, the prover commits, by using a polynomial +commitment scheme, to the multilinear extension 𝑃 := mle[( z′, r′)] ∈ F≤1 +𝑞 [𝑋1, . . . , 𝑋𝜇] of (z′, r′)6, before +obtaining a challenge 𝛼 ← F𝑞𝑘 from the verifier. The challenge is then used to substitute𝑋 = 𝛼, which +simplifies to an inner product claim purely overF𝑞𝑘. We note that the size of𝑃 is much smaller than that of +the original committed polynomial𝑓; hence, we can recursively repeat this process. Similarly as in [HMZ25] +this can further get transformed into proving a sumcheck-type relation +∑︁ +𝑖 ∈ {0,1}𝜇 +𝑃 (𝑖) · 𝑄 (𝑖) = 𝑉 +6 Here, we implicitly treat a vectora ∈ F2𝐿 +𝑞 as an indexing function from{0,1}𝐿 to F𝑞. +7 + +--- page 8 --- + +for a public polynomial𝑄 over F𝑞𝑘 [𝑋1, . . . , 𝑋𝜇] and a public value𝑉 ∈ F𝑞𝑘. +On the other hand, to prove thatz is short, we only need to show that all coordinates ofz′ are small. Here +we can use a standard range proof over a finite field, since the ring switching has transformed the statements +from polynomial rings into a finite field. In contrast, both LaBRADOR and LatticeFold+ demonstrate +smallness directly over polynomial rings, which is considerably more complicated and less efficient for the +verifier. +Finally, by running the sumcheck protocol [LFKN90], we end up with a polynomial evaluation claim of +the form: +𝑃 (𝑟 ∗ +1, . . . , 𝑟∗ +𝜇) · 𝑄 (𝑟 ∗ +1, . . . , 𝑟∗ +𝜇) = 𝑦∗, +wheretheevaluationpoint (𝑟 ∗ +1, . . . , 𝑟∗ +𝜇) andtheimage 𝑦∗ arepublic.Hence,bysendingtheevaluation 𝑃 (𝑟 ∗ +1, . . . , 𝑟∗ +𝜇) +to the verifier, they can directly check the equation above. What is left for the prover to do now is to prove +the evaluation claim for the polynomial𝑃, and thus the protocol can be repeated in a further way to obtain +smaller proof sizes. +Committing to 𝑷 and avoiding re-decomposition.A naive approach to commit to the multilinear +polynomial 𝑃 = mle[( z′, r′)] is to first decompose the coefficients of𝑃 (and thus the committed witness gets +longer), and then apply an Ajtai-style [Ajt96] commitment scheme. A more natural and efficient way is to +directly commit to(z′, r′), as we know that the coefficients ofz′ are already small and thus no decomposition +for z′ is needed7. But this directly leads to a question of how to directly provemle[( z′, r′)] (𝑥1, . . . , 𝑥𝜇) = 𝑦′, +when having only committed to(z′, r′). We solve this issue by observing a very simple homomorphic property +of the 𝑒𝑞 polynomials. Namely, for any vectors𝑖, 𝑗 and x0, x1 we have +𝑒𝑞 (𝑖, x0) · 𝑒 ( 𝑗, x1) = 𝑒𝑞 (𝑖 || 𝑗, x0|| x1). +Then, by definition of the multilinear extension and for𝜇 = 𝑚 + 𝑟, we can write +mle[( z′, r′)] (x0 ∥ x1) = +∑︁ +𝑖 ∈ {0,1}𝑟 +∑︁ +𝑗 ∈ {0,1}𝑚 +(z′, r′)𝑖 ∥ 𝑗 ·𝑒𝑞 (𝑖 ∥ 𝑗, x0 ∥ x1) += +∑︁ +𝑖 ∈ {0,1}𝑟 +𝑒𝑞 (𝑖, x0) ·©­ +« +∑︁ +𝑗 ∈ {0,1}𝑚 +(z′, r′)𝑖 ∥ 𝑗 ·𝑒𝑞 ( 𝑗, x1)ª® +¬ +which is exactly of the same form as(4). Therefore, we can recursively run our protocol to prove evaluation +of 𝑃 (with minor modifications) without explicitly committing to its coefficients. +2 Preliminaries +Notation. Let 𝑞 = 5 (mod 8) be an odd prime. DenoteZ𝑞 := {0,1, . . . , 𝑞 −1} to be the ring of integers of order +𝑞 and F𝑞𝑘 be the Galois field of dimension𝑘 and size 𝑞𝑘. In this paper, we will useZ𝑞 and F𝑞 interchangeably. +For 𝑛 ∈ N, we define [𝑛] := {1,2, . . . , 𝑛}. Let 𝜆 to be the security parameter. We write𝑂𝜆 (𝑇 ) to denote +𝑇 ·poly(𝜆). For a probability distributionX (resp. finite setX), 𝑥 ← X means that 𝑥 is sampled fromX (resp. +𝑥 is chosen uniformly at random from the setX). We writenegl to denote an unspecified negligible function. +We will mostly use letters𝑖, 𝑗 for indexing. For𝑛 ∈ N, we write [𝑛] := {1,2, . . . , 𝑛}. When we denote +𝑖 ∈ {0,1}𝑘, we treat𝑖 as a binary vector(𝑖1, . . . , 𝑖𝑘 ) of length𝑘. For presentation purposes, when we say𝑖 ∈ [2𝑘 ], +we may also mean its binary encoding𝑖 ∈ {0,1}𝑘. This will be useful when describing the sumcheck-based +norm protocol (cf. Section 4.3). +7 Nevertheless, we would need to decompose the coefficients ofr′. +8 + +--- page 9 --- + +2.1 Cyclotomic Rings +For a power of two𝑑 := 2𝛼 and prime 𝑞, denote R and R𝑞 respectively to be the ringsZ[𝑋 ]/( 𝑋 𝑑 +1) and +Z𝑞 [𝑋 ]/( 𝑋 𝑑 +1). Lower-case letters denote elements inR or R𝑞𝑘 and bold lower-case (resp. upper-case) letters +represent column vectors (resp. matrices) with coefficients inR or R𝑞𝑘. For𝑦 = Í𝑑 −1 +𝑖=0 𝑦𝑖 ·𝑋 𝑖 ∈ R𝑞, we write +cf(𝑦) := (𝑦0, . . . , 𝑦𝑑 −1) ∈ F𝑑 +𝑞 to denote the coefficient vector of𝑦. We naturally extend this notation to vectors. +We consider base-𝑏 gadget matrices G𝑏,𝑛 := I𝑛 ⊗ + +1 𝑏 𝑏2 · · ·𝑏𝛿 −1 +∈ R𝑛×𝑛𝛿 +𝑞 for 𝑛 ≥ 1 and 𝛿 = ⌈log𝑏 𝑞⌉. We +define the inverse functionG−1 +𝑏,𝑛 : R𝑛 +𝑞 → R𝑛𝛿 +𝑞 , which decomposes each entry w.r.t. base𝑏. In particular, for any +t ∈ R𝑛 +𝑞, G−1 +𝑏,𝑛 (t) has coefficients between⌈−𝑏/2⌉ and ⌈𝑏/2⌉ −1 and G𝑏,𝑛G−1 +𝑏,𝑛 (t) = t. We will omit the subscript +𝑏 if the base is known and fixed from the context. +Galois Automorphisms and trace map.For 𝑖 ∈ Z× +2𝑑, we define the Galois automorphism𝜎𝑖 : R → R defined +as 𝑋 ↦→ 𝑋 𝑖, and denote the Galois group of automorphisms asAut(R) := {𝜎𝑖 : 𝑖 ∈ Z× +2𝑑 }. For a subgroup +𝐻 of Aut(R), we denote R𝐻 +𝑞 := {𝑥 ∈ R𝑞 : ∀𝜎 ∈ 𝐻, 𝜎 (𝑥) = 𝑥 }, i.e. the ring of elements ofR𝑞 fixed by all +automorphisms in 𝐻. Also, we define theR𝑞-trace map Tr𝐻 : R𝑞 → R𝐻 +𝑞 as Tr𝐻 (𝑎) := Í +𝜎 ∈𝐻 𝜎 (𝑎) ∈ R𝐻 +𝑞 . +Norms. We define 𝑟 ′ = 𝑟 mod± 𝑞 to be the unique element 𝑟 ′ in the range − 𝑞−1 +2 ≤ 𝑟 ′ ≤ 𝑞−1 +2 such that +𝑟 ′ = 𝑟 mod 𝑞. We also denote 𝑟 ′ = 𝑟 mod+𝑞 to be the unique element 𝑟 ′ in the range 0 ≤ 𝑟 ′ < 𝑞 such +that 𝑟 ′ = 𝑟 mod 𝑞. When the exact representation is not important, we simply write 𝑟 mod 𝑞. For an +element 𝑤 ∈ Z𝑞, we write∥𝑤 ∥∞ to mean |𝑤 mod± 𝑞|. Define theℓ∞ and ℓ𝑝 norms for 𝑤 = 𝑤0 + 𝑤1𝑋 + . . . + +𝑤 𝑑 −1𝑋 𝑑 −1 ∈ R as ∥𝑤 ∥∞ = max𝑗 ∥𝑤 𝑗 ∥∞, ∥𝑤 ∥𝑝 = +𝑝 +√︁ +∥𝑤0∥𝑝 +∞ + . . . + ∥𝑤 𝑑 −1∥𝑝 +∞. If w = (w1, . . . , w𝑚) ∈ R𝑘, then +∥w∥∞ = max𝑗 ∥w𝑗 ∥∞, ∥w∥𝑝 = 𝑝 +√︁ +∥w1 ∥𝑝 + . . . + ∥ w𝑘 ∥𝑝. By default,∥w∥ := ∥w∥2. For𝛽 ≥ 1, we define𝑆𝛽 to be +the set of ring elements inR with coefficients between⌈−𝛽/2⌉ and ⌈𝛽/2⌉ −1. +In this work, we will use the well-known inequality of [Mic07] to derive norm bounds. +Lemma 2. For any 𝑓 , 𝑔 ∈ R, ∥𝑓 𝑔∥∞ ≤ ∥ 𝑓 ∥1 · ∥𝑔∥∞. +Short elements are invertible.We recall the main result by Lyubashevsky and Seiler [LS18] which says that +short polynomials overR𝑞 are invertible overR𝑞. +Lemma 3([LS18]). Let 𝑞 ≡ 5 (mod 8) be a prime. Then, any𝑓 ∈ R𝑞 which satisfies0 < ∥𝑓 ∥∞ < 1√ +2𝑞1/2 or +0 < ∥𝑓 ∥ < 𝑞1/2 has an inverse inR𝑞. +The set of invertible elements ofR𝑞 is denoted byR× +𝑞. +Module-SIS Assumption. We recall the Module-SIS hardness assumption [LS15] adapted to the infinity norm +setting. +Definition 1(Module-SIS). Let 𝑞 = 𝑞(𝜆), 𝑛 = 𝑛(𝜆), 𝑚 = 𝑚(𝜆), 𝑑 = 𝑑 (𝜆) and 𝛽 = 𝛽 (𝜆). We say that the +MSIS𝑞,𝑑,𝑛,𝑚,𝛽 assumption holds if for any PPT algorithmA, the following holds: +Pr + +Az = 0 ∧0 < ∥z∥∞ ≤ 𝛽 | 𝐴 ← R𝑛×𝑚 +𝑞 , z ← A (A) + += negl(𝜆) +2.2 Multilinear Extensions over Rings +Definition 2(Multilinear extension over rings).Let 𝑅 be an arbitrary ring with zero0 and identity1. +Given a function 𝑓 : {0,1}𝜇 → 𝑅, we define the multilinear extensionmle[𝑓 ] ∈ 𝑅 ≤1 [𝑋1, . . . , 𝑋𝜇] of 𝑓 as +mle[𝑓 ] (x) := +∑︁ +𝑖 ∈ {0,1}𝜇 +𝑓 (𝑖) · 𝑒𝑞 (𝑖, x) +where 𝑒𝑞 (𝑖, x) := Î𝜇 +𝑗=1 +(1 − 𝑖 𝑗 ) · (1 − 𝑥 𝑗 ) + 𝑖 𝑗𝑥 𝑗 +. +By definition, mle[𝑓 ] (x) = 𝑓 (x) for all x ∈ {0,1}𝜇. +For a vectorf := (𝑓𝑖 )𝑖 ∈ {0,1}𝜇 ∈ 𝑅2𝜇 +, we denote˜f as the multilinear extension of the function𝑓 : {0,1}𝜇 → 𝑅 +defined by𝑖 ↦→ 𝑓𝑖. +9 + +--- page 10 --- + +2.3 Coordinate-Wise Special Soundness +Let R ⊆ { 0,1}∗ × {0,1}∗ × {0,1}∗ be a ternary relation. For a triple(pp, x, w) ∈ R, we call pp the public +parameters, x is a statement andw is a witness forx w.r.t. pp 8. We denoteR(pp, x) = {w : R(pp, x, w) = 1}. +An interactive proof systemΠ = (S, P, V) for relation R consists of three PPT algorithms: the setup +algorithm S, proverP, and verifierV. The latter two are interactive and stateful. Further, we say a protocol is +public coinif the verifier’s challenges are chosen uniformly at random independently of the prover’s messages. +We recall the notion of coordinate-wise special soundness from [FMN24]. Let𝑆 be a set andℓ, 𝑘 ∈ N. +Namely, take two vectorsx := (𝑥1, . . . , 𝑥ℓ ), y := (𝑦1, . . . , 𝑦ℓ ) ∈ 𝑆 ℓ. Then, we define the following relation “≡𝑖” for +fixed 𝑖 ∈ [ ℓ] as: +x ≡𝑖 y ⇐ ⇒ 𝑥𝑖 ≠ 𝑦𝑖 ∧ ∀𝑗 ∈ [ ℓ]\{𝑖}, 𝑥𝑗 = 𝑦 𝑗 . +That is, vectors x and y have the same values in all coordinates apart from the𝑖-th one. Forℓ = 1, the +relations boil down to checking whether two elements are distinct. Further, we can define the set +SS(𝑆, ℓ, 𝑘) := +  + +{x1, . . . , x𝐾 } ⊆ ( 𝑆 ℓ )𝐾 : +∃𝑒 ∈ [ 𝐾], ∀𝑖 ∈ [ ℓ], +∃𝐽 = { 𝑗1, . . . , 𝑗𝑘 −1} ⊆ [ 𝐾] \ { 𝑒}, +∀𝑗 ∈ 𝐽 ,x𝑒 ≡𝑖 x𝑗 +  + +, +where 𝐾 := ℓ (𝑘 − 1) + 1. In other words, there is acentral vector x𝑒 ∈ 𝑋 such that for each coordinate of +x𝑒, there are𝑘 −1 other vectors in𝑋 that differ fromx𝑒 only in that coordinate. In other words, for each +coordinate, there are𝑘 vectors in 𝑋 that differ from each other only in that coordinate, andx𝑒 is always one +of them. +We are ready to define the notion of coordinate-wise special soundness. +Definition 3(CWSS for Multi-Round Protocols).Let Π be public-coin (2𝜇 +1)-round interactive proof +system for relationR, where in the2𝑖-th round, for𝑖 ∈ [ 𝜇], the verifier picks a uniformly random challenge +from 𝑆 ℓ𝑖 +𝑖 . A tree of transcripts is a set of𝐾 := Î𝜇 +𝑖=1 (ℓ𝑖 (𝑘𝑖 −1) +1) arranged in the following tree structure. The +nodes in the tree correspond to the prover’s messages and the edges correspond to the verifier’s challenges. +Each node at depth𝑖 has exactly ℓ𝑖 (𝑘𝑖 −1) +1 children corresponding toℓ𝑖 (𝑘𝑖 −1) +1 distinct challenges which, +as a set of vectors, lie inSS(𝑆𝑖, ℓ𝑖, 𝑘𝑖 ). Every transcript corresponds to exactly one path from the root to a leaf +node. +We say that Π is (ℓ1, . . . , ℓ𝜇)-coordinate-wise (𝑘1, . . . , 𝑘𝜇)-special sound for the relationR if there is a +polynomial time algorithm that given public parameterspp, statement x and the tree of transcripts, outputs a +witness w ∈ 𝑅(pp, x). If ℓ1 = . . . = ℓ𝜇 = 1 then we say the protocol is(𝑘1, . . . , 𝑘𝜇)-special sound. +We recall the main result of [FMN24, Section 7] which translates CWSS to knowledge soundness. +Lemma 4. Let Π be a (ℓ1, . . . , ℓ𝜇)-coordinate-wise (𝑘1, . . . , 𝑘𝜇)-special sound interactive proof system as in +Definition 3 and assume𝐾 := Î𝜇 +𝑖=1(ℓ𝑖 (𝑘𝑖 −1) +1) = poly(𝜆). Then, it is knowledge sound with knowledge error +𝜇∑︁ +𝑖=1 +ℓ𝑖 ·𝑘𝑖 +|𝑆𝑖 |ℓ𝑖 +. +A similar result also holds in the random oracle model ifΠ is turned non-interactive via Fiat-Shamir +transformation (see [FMN24, Section 8]). In this work, we will aim to show that our protocol satisfies +coordinate-wise special soundness. +8 We omitpp and simply writeR(x, w) if the public parameters are not relevant to the relation. +10 + +--- page 11 --- + +3 Evaluation Claims over Extension Fields as Relations over R𝒒 +In this section, we provide a generic transformation from provingℓ-variate polynomial evaluations over +finite fields F𝑞𝑘 to proving (ℓ − 𝛼 + 𝜅)-variate polynomial evaluations over the cyclotomic ringR𝑞. Next, we +improve the result if the committed polynomial is only over the base fieldF𝑞, instead ofF𝑞𝑘. At the core of +our transformation is the process of embedding vectors of extension fieldsF𝑞𝑘 inside ring elementsR𝑞 and +translating inner products overF𝑞𝑘 as “inner-product-related” statements overR𝑞. In Figure 2 we provide a +summary table of how much theF𝑞𝑘-to-R𝑞 transformations presented in this section cost in terms of prover +communication. +3.1 Generic Transformation over F𝒒𝒌 +Finding subfields inR𝑞. We introduce a lemma that explicitly describe finite field extensions withinR𝑞. +Lemma 5(Subfields of R𝑞). Let 𝑞 = 5 (mod 8) and 𝑘 ≥ 1 be a divisor of𝑑/2 and 𝐻 := ⟨𝜎−1, 𝜎4𝑘+1⟩ ⊆ +Aut(R). Then, R𝐻 +𝑞 is a subfield ofR𝑞 isomorphic to F𝑞𝑘. +Proof. By definition, R𝐻 +𝑞 is a subset of the set of all ring elements inR𝑞 stable under the𝜎−1 automorphism, +which itself is a field [LNP22, Lemma 2.6]. Using the homomorphic properties of Galois automorphisms, the +set R𝐻 +𝑞 is closed under both addition and multiplication. Finally, the size ofR𝐻 +𝑞 is 𝑞𝑘 since each element of +R𝐻 +𝑞 is of the form +𝑎 := 𝑎0 + +𝑘 −1∑︁ +𝑗=1 +𝑎𝑘 − 𝑗 · + +𝑋 +𝑑 +2𝑘 · (𝑘 − 𝑗 ) − 𝑋 +𝑑 +2𝑘 · (𝑘+𝑗 ) + +(7) +with 𝑘 degrees of freedom:𝑎0, 𝑎1, . . . , 𝑎𝑘 −1 ∈ Z𝑞. ⊓ ⊔ +Inner products.Our main theorem of this section generalizes [LNP22], and reduces proving inner products +over the subfield into proving statements overR𝑞 related to the trace map. Informally, it introduces a bijection +𝜓 which allows packing𝑑/𝑘 elements of fieldF𝑞𝑘 into one singleR𝑞 element. The fact that it is an invertible +function is crucial in arguing knowledge soundness. +Theorem 2(Inner product as output of the trace).Let 𝑘 divide 𝑑/2 and 𝐻 := ⟨𝜎−1, 𝜎4𝑘+1⟩ ⊆ Aut(R). +Consider a map +𝜓 : (R𝐻 +𝑞 )𝑑/𝑘 → R𝑞, +(𝑎0, 𝑎1, . . . , 𝑎𝑑/𝑘 −1) ↦→ + 𝑑/2𝑘 −1∑︁ +𝑖=0 +𝑎𝑖 ·𝑋 𝑖 +! ++ 𝑋 𝑑/2 · + 𝑑/2𝑘 −1∑︁ +𝑖=0 +𝑎𝑑/2𝑘+𝑖 ·𝑋 𝑖 +! +. +(8) +Then, 𝜓 is a bijection. Moreover, for anya, b ∈ ( R𝐻 +𝑞 )𝑑/𝑘, +Tr𝐻 (𝜓 (a) · 𝜎−1(𝜓 (b))) = 𝑑 +𝑘 · ⟨a, b⟩. +Proof. To prove the first part, we only need to show that𝜓 is injective since the domain and co-domain size +are the same. Indeed, suppose that for somea = (𝑎0, . . . , 𝑎𝑑/𝑘 −1) we have𝜓 (a) = 0 (the statement holds by the +linearly homomorphic property of𝜓). We prove thata is the zero-vector. +First, we note that𝜓 (a) can be expanded as +𝑑 +2𝑘 −1∑︁ +𝑖=0 +𝑋 𝑖𝑎𝑖,0 + 𝑋 +𝑑 +2 +𝑖𝑎𝑑/2𝑘+𝑖,0 + +𝑘 −1∑︁ +𝑗=1 +𝑎𝑖,𝑘 − 𝑗 · (𝑋 +𝑑 +2𝑘 · (𝑘 − 𝑗 )+𝑖 − 𝑋 +𝑑 +2𝑘 · (𝑘+𝑗 )+𝑖 ) ++ +𝑘 −1∑︁ +𝑗=1 +𝑎𝑑/2𝑘+𝑖,𝑘 − 𝑗 · (𝑋 +𝑑 +2𝑘 · (2𝑘 − 𝑗 )+𝑖 − 𝑋 +𝑑 +2𝑘 · (2𝑘+𝑗 )+𝑖 ). +(9) +11 + +--- page 12 --- + +Then, the 𝑢-th coefficient of the polynomial above, where 0≤ 𝑢 < 𝑑/(2𝑘), is exactly 𝑎𝑢,0. Similarly, the +(𝑑/2𝑘 + 𝑢)-th coefficient is𝑎𝑑/2𝑘+𝑢,0. Hence, 𝑎𝑢,0 = 𝑎𝑑/2𝑘+𝑢,0 = 0. +Now, we claim that for any 1≤ 𝑢 < 𝑘 and 0 ≤ 𝑣 < 𝑑 +2𝑘, 𝑎𝑣,𝑢 = 𝑎𝑑/2𝑘+𝑣,𝑢 = 0. Indeed, first consider the +( 𝑑 +2𝑘 ·𝑢 + 𝑣)-th coefficient of the polynomial above. It is equal to𝑎𝑣,𝑢 + 𝑎𝑑/2𝑘+𝑣,𝑘 −𝑢 = 0. On the other hand, if we +focus on the( 𝑑 +2𝑘 · (𝑘 + 𝑢) + 𝑣)-th coefficient, then it is exactly equal to−𝑎𝑣,𝑘 −𝑢 + 𝑎𝑑/2𝑘+𝑣,𝑢 = 0. By combining +the two observations we obtain +𝑎𝑑/2𝑘+𝑣,𝑢 = 𝑎𝑣,𝑘 −𝑢 = −𝑎𝑑/2𝑘+𝑣,𝑢 +which implies 𝑎𝑑/2𝑘+𝑣,𝑢 = 𝑎𝑣,𝑢 = 0, and thusa = 0. +We move on to the second part. We will make use of the following three claims. +Claim. Consider the multiplicative groupZ× +2𝑑 and let 𝑘 ≤ 𝑑/2 be a power-of-two. Then, +⟨4𝑘 +1⟩ = {(4𝑘) · 𝛼 +1 : 0≤ 𝛼 ≤ 𝑑/(2𝑘) − 1}. +Proof. First, using the property that 4𝑘 divides 2𝑑, we have for any𝑖, +((4𝑘 +1)𝑖 mod 2𝑑) mod 4𝑘 = (4𝑘 +1)𝑖 mod 4𝑘 = +𝑖∑︁ +𝑗=0 +𝑖 +𝑗 + +(4𝑘) 𝑗 mod 4𝑘 = 1. +Therefore, ⟨4𝑘 +1⟩ ⊆ {( 4𝑘) ·𝛼 +1 : 0≤ 𝛼 ≤ 𝑑/(2𝑘) −1}. Now, the statement holds since by [LS18, Lemma 2.4], +we know that the order of 4𝑘 +1 ∈ Z× +2𝑑 is exactly 2𝑑 +4𝑘 = 𝑑/(2𝑘). ⊓ ⊔ +Claim. Let 𝑖 ∈ Z be not divisible by𝑑/2𝑘. Then, Tr𝐻 (𝑋 𝑖 ) = 0. +Proof. Indeed, by assumption 4𝑘𝑖 cannot be divisible 2𝑑 and thus𝑋4𝑘𝑖 ≠ 1. Now, using the first claim, we get +Tr𝐻 (𝑋 𝑖 ) = +∑︁ +ℎ∈ ⟨−1,4𝑘+1⟩ +𝑋 ℎ·𝑖 += +∑︁ +ℎ∈ ⟨4𝑘+1⟩ +𝑋 ℎ·𝑖 + 𝑋 −ℎ·𝑖 += +𝑑/(2𝑘 ) −1∑︁ +𝛼=0 +𝑋 (4𝑘 ·𝛼+1) ·𝑖 + +𝑑/(2𝑘 ) −1∑︁ +𝛼=0 +𝑋 − (4𝑘 ·𝛼+1) ·𝑖 += 𝑋 𝑖 · (𝑋4𝑘𝑖 )𝑑/(2𝑘 ) −1 +𝑋4𝑘𝑖 −1 + 𝑋 −𝑖 · (𝑋 −4𝑘𝑖 )𝑑/(2𝑘 ) −1 +𝑋 −4𝑘𝑖 −1 += 0. +⊓ ⊔ +Claim. Tr𝐻 (𝑋 𝑑/2) = 0. +Proof. Using the first claim we obtain +Tr𝐻 (𝑋 𝑑/2) = +𝑑/(2𝑘 ) −1∑︁ +𝛼=0 +𝑋 (4𝑘 ·𝛼+1) ·𝑑/2 + +𝑑/(2𝑘 ) −1∑︁ +𝛼=0 +𝑋 − (4𝑘 ·𝛼+1) ·𝑑/2 += +𝑑/(2𝑘 ) −1∑︁ +𝛼=0 +𝑋 𝑑/2 + +𝑑/(2𝑘 ) −1∑︁ +𝛼=0 +𝑋 −𝑑/2 += +𝑑/(2𝑘 ) −1∑︁ +𝛼=0 +𝑋 𝑑/2 − 𝑋 𝑑/2 = 0. +⊓ ⊔ +12 + +--- page 13 --- + +To conclude the proof of the theorem, we notice that +𝜓 (a) · 𝜎−1 (𝜓 (b)) = + 𝑑/2𝑘 −1∑︁ +𝑖=0 +𝑑/2𝑘 −1∑︁ +𝑗=0 +𝑎𝑖𝑏 𝑗 ·𝑋 𝑖 − 𝑗 + +𝑑/2𝑘 −1∑︁ +𝑖=0 +𝑑/2𝑘 −1∑︁ +𝑗=0 +𝑎𝑑/2𝑘+𝑖𝑏𝑑/2𝑘+𝑗 ·𝑋 𝑖 − 𝑗 +! ++ 𝑋 𝑑/2 · + 𝑑/2𝑘 −1∑︁ +𝑖=0 +𝑑/2𝑘 −1∑︁ +𝑗=0 +𝑎𝑑/2𝑘+𝑖𝑏 𝑗 ·𝑋 𝑖 − 𝑗 − +𝑑/2𝑘 −1∑︁ +𝑖=0 +𝑑/2𝑘 −1∑︁ +𝑗=0 +𝑎𝑖𝑏𝑑/2𝑘+𝑗 ·𝑋 𝑖 − 𝑗 +! +. +Hence, +Tr𝐻 (𝜓 (a) · 𝜎−1(𝜓 (b))) += + 𝑑/2𝑘 −1∑︁ +𝑖=0 +𝑑/2𝑘 −1∑︁ +𝑗=0 +𝑎𝑖𝑏 𝑗 ·Tr𝐻 (𝑋 𝑖 − 𝑗 ) + +𝑑/2𝑘 −1∑︁ +𝑖=0 +𝑑/2𝑘 −1∑︁ +𝑗=0 +𝑎𝑑/2𝑘+𝑖𝑏𝑑/2𝑘+𝑗 ·Tr𝐻 (𝑋 𝑖 − 𝑗 ) +! ++ + 𝑑/2𝑘 −1∑︁ +𝑖=0 +𝑑/2𝑘 −1∑︁ +𝑗=0 +𝑎𝑑/2𝑘+𝑖𝑏 𝑗 ·Tr𝐻 (𝑋 𝑑/2+(𝑖 − 𝑗 ) ) − +𝑑/2𝑘 −1∑︁ +𝑖=0 +𝑑/2𝑘 −1∑︁ +𝑗=0 +𝑎𝑖𝑏𝑑/2𝑘+𝑗 ·Tr𝐻 (𝑋 𝑑/2+(𝑖 − 𝑗 ) ) +! +. +Now, using the fact that for two distinct 0≤ 𝑖, 𝑗 < 𝑑/2𝑘, both 𝑖 − 𝑗 and 𝑑/2 + ( 𝑖 − 𝑗) cannot be divisible +by 𝑑/2𝑘, and the second claim, we deduce thatTr𝐻 (𝑋 𝑖 − 𝑗 ) = Tr𝐻 (𝑋 𝑑/2+𝑖 − 𝑗 ) = 0. However, when𝑖 = 𝑗, then +Tr𝐻 (𝑋 𝑖 − 𝑗 ) = 𝑑 +𝑘 and Tr𝐻 (𝑋 𝑑/2+(𝑖 − 𝑗 ) ) = 0 by the third claim. Hence, we deduce that +Tr𝐻 (𝜓 (a) · 𝜎−1 (𝜓 (b))) = 𝑑 +𝑘 +𝑑/𝑘 −1∑︁ +𝑖=0 +𝑎𝑖𝑏𝑖 = 𝑑 +𝑘 ⟨a, b⟩ +which concludes the proof. ⊓ ⊔ +Next, we give a simple observation regarding how infinity norm changes after applying the𝜓 map. +Lemma 6. Let a be a vector overR𝐻 +𝑞 such that ∥a∥∞ ≤ 𝛽 and 𝜓 be the map defined in Theorem 2. Then, +∥𝜓 (a) ∥∞ ≤ 2𝛽. +Proof. The statement follows directly from the explicit formula in Equation (9), where some coefficients of +𝜓 (a) are equal to the sum of two coefficients ofa. +Reducing to multilinear evaluation overR𝑞. We use the results above to translate the multilinear evaluation +claim over R𝐻 +𝑞 +𝑓 (𝑥1, . . . , 𝑥ℓ ) = 𝑦 +to an equivalent relation over the ringR𝑞. Denote the coefficients of polynomial𝑓 as (𝑓𝜄)𝜄 ∈ {0,1}ℓ ∈ R𝐻 +𝑞 . We can +explicitly rewrite the evaluation equation as +𝑦 = +∑︁ +𝑖 ∈ {0,1}ℓ −𝛼 +𝜅 + +𝑥𝑖1 +1 · . . . ·𝑥𝑖ℓ −𝛼 +𝜅 +ℓ −𝛼+𝜅 + +·©­ +« +∑︁ +𝑗 ∈ {0,1}𝛼 −𝜅 +𝑓𝑖 ∥ 𝑗 ·𝑥 𝑗1 +ℓ −𝛼+𝜅+1 · . . . ·𝑥 𝑗𝛼 −𝜅 +ℓ +ª® +¬ +. (10) +For each𝑖 ∈ {0,1}ℓ −𝛼+𝜅 we define +𝐹𝑖 := 𝜓 (𝑓𝑖 | |𝑗 ) 𝑗 ∈ {0,1}𝛼 −𝜅 + and also 𝑣 := 𝜓 + +(𝑥 𝑗1 +ℓ −𝛼+𝜅+1 · . . . ·𝑥 𝑗𝛼 −𝜅 +ℓ ) 𝑗 ∈ {0,1}𝛼 −𝜅 + +. +Now, recall that for every𝑥 ∈ R𝐻 +𝑞 we haveTr𝐻 (𝑥) = 𝑥 and trace is additively homomorphic. Hence, by +applying Theorem 2 to Equation (10) we obtain +𝑑 +𝑘 ·𝑦 = Tr𝐻 +©­­­­­­ +« +∑︁ +𝑖 ∈ {0,1}ℓ −𝛼 +𝜅 + +𝑥𝑖1 +1 · . . . ·𝑥𝑖ℓ −𝛼 +𝜅 +ℓ −𝛼 + +·𝐹𝑖 +| {z } +𝑌:= +·𝜎−1(𝑣) +ª®®®®®® +¬ +. +13 + +--- page 14 --- + +Therefore,theproverwillsendasingleringelement 𝑌 ∈ R𝑞,whichallowstheverifiertocheckif Tr𝐻 (𝑌 ·𝜎−1(𝑣)) = +𝑑𝑦/𝑘. Then, we only need to prove that𝑌 is well-formed, which is an evaluation proof for the(ℓ − 𝛼 +𝜅)-variate +multilinear polynomial 𝐹 := (𝐹𝑖 )𝑖 over R𝑞. +3.2 Dealing with 𝒇 ∈ Z≤1 +𝒒 [𝑿1, . . . , 𝑿ℓ ] +We now optimize the special case where the evaluation point(𝑥1, . . . , 𝑥ℓ ) is defined in the extension fieldF𝑞𝑘, +while the coefficients of the committed polynomial𝑓 are over Z𝑞. Note that if we naively apply the generic +transformation from Section 3.1 then we would end up with an evaluation proof for the(ℓ − 𝛼 + 𝜅)-variate +multilinear polynomial overR𝑞. This subsection focuses on how to lower it further toℓ − 𝛼 variables. +We start with a similar observation as in Equation (10) that a polynomial evaluation claim can be written +as +𝑦 = +∑︁ +𝑖 ∈ {0,1}𝜅 + +𝑥𝑖1 +1 · . . . ·𝑥𝑖𝜅 +𝜅 + +· +©­­­­­­­ +« +∑︁ +𝑗 ∈ {0,1}ℓ −𝜅 +𝑓𝑖 ∥ 𝑗 ·𝑥 𝑗1 +𝜅+1 · . . . ·𝑥 𝑗ℓ −𝜅 +ℓ +| {z } +𝑦𝑖:= +ª®®®®®®® +¬ +. (11) +Hence, we let the prover send𝑘 partial evaluations𝑦𝑖 ∈ F𝑞𝑘 to the verifier who can directly check whether +𝑦 = +∑︁ +𝑖 + +𝑥𝑖1 +1 · . . . ·𝑥𝑖𝜅 +𝜅 + +·𝑦𝑖 . +What we have left to prove is that all the𝑦𝑖 are well-formed, or in other words𝑓𝑖 (𝑥𝜅+1, . . . , 𝑥ℓ ) = 𝑦𝑖 for +𝑖 ∈ {0,1}𝜅 where 𝑓𝑖 ∈ Z𝑞 [𝑋ℓ −𝜅+1, . . . , 𝑋ℓ ]. +For the next part, we need a somewhat explicit definition of the extension field (it could as well be +R𝐻 +𝑞 ). Let F𝑞𝑘 := F𝑞 [𝑍 ]/𝜑 (𝑍 ), where 𝜑 is an irreducible polynomial inF𝑞 [𝑍 ] of degree 𝑘. Then, we define the +(ℓ − 𝜅)-variate multilinear polynomial𝑓 ′ ∈ F𝑞𝑘 [𝑋𝜅+1, . . . , 𝑋ℓ ] as +𝑓 ′ (𝑋𝜅+1, . . . , 𝑋ℓ ) := +∑︁ +𝑖 ∈ {0,1}𝜅 +𝑓𝑖 (𝑋𝜅+1, . . . , 𝑋ℓ ) · 𝑍 +Í𝜅 +𝑡 =1 𝑖𝑡 ·2𝑡 −1 +. +Then, by definition of the partial evaluations𝑦𝑖 we obtain +𝑓 ′ (𝑥𝜅+1, . . . , 𝑥ℓ ) = +∑︁ +𝑖 ∈ {0,1}𝜅 +𝑦𝑖 ·𝑍 +Í𝜅 +𝑡 =1 𝑖𝑡 ·2𝑡 −1 +∈ F𝑞𝑘 +which is now a full-fledged polynomial evaluation overF𝑞𝑘. By applying the generic transformation from +Section 3.1, we can reduce the statement into an evaluation claim overR𝑞 for a𝜇-variate multilinear polynomial, +where +𝜇 = (ℓ − 𝜅)| {z } +Section 3.2 +−𝛼 + 𝜅 +| {z } +Section 3.1 += ℓ − 𝛼 . +The final observation is that the verifier can directly compute𝑦0...0 from 𝑦 and other 𝑦𝑖, and thus only𝑘 −1 +evaluations are needed to be sent. +Remark 1. One may notice a strange behavior in Figure 2 in a sense that the transformation from a field +extension F𝑞𝑘 to R𝑞 seems cheaper than from the base fieldF𝑞 to R𝑞. Indeed, if given a witness polynomial𝑓 +over F𝑞, why is it not enough to just follow the transformation from Section 3.1? +There are two main reasons. The first one is that, when combined with a polynomial evaluation protocol +over R𝑞, it is not guaranteed in the knowledge soundness proof that the extracted polynomial𝑓 is also over +F𝑞, as we obtain𝑓 ∈ F𝑞𝑘 [𝑋1, . . . , 𝑋ℓ ]. Moreover, the transformation in Section 3.2 allows us to further reduce +the number of variables of theR𝑞-polynomial, from ℓ − 𝛼 + 𝜅 to ℓ − 𝛼, which has significant impact in reducing +round complexity of the protocol. +14 + +--- page 15 --- + +Committed polynomial # variables after theR𝑞-transformation Prover cost +𝑓 ∈ F𝑞𝑘 [𝑋1, . . . , 𝑋ℓ ] ℓ − 𝛼 + 𝜅 𝑑log 𝑞 +𝑓 ∈ F𝑞 [𝑋1, . . . , 𝑋ℓ ] ℓ − 𝛼 (𝑑 + 𝑘 (𝑘 −1)) log 𝑞 +Fig. 2. Summary of our transformation in Sections 3.1 and 3.2. The prover cost is measured in the size of the prover +messages in bits. +4 Multilinear Polynomial Commitment Scheme over R𝒒 +This section focuses on proving knowledge of acommitted multilinear polynomial 𝑓 ∈ R≤1 +𝑞 [𝑋1, . . . , 𝑋ℓ ] with +coefficients (𝑓𝑖 )𝑖 ∈ {0,1}ℓ such that +𝑓 (𝑥1, . . . , 𝑥ℓ ) := +∑︁ +𝑖 ∈ {0,1}ℓ +𝑓𝑖 ·𝑥𝑖1 +1 · . . . ·𝑥𝑖ℓ +ℓ = 𝑢 +where the evaluation point(𝑥1, . . . , 𝑥ℓ ) ∈ Rℓ +𝑞 and the image𝑢 ∈ R𝑞 are public. +We follow the initial strategy from Greyhound [NS24]. Supposeℓ = 𝑚 + 𝑟 for some 𝑚, 𝑟 ≥ 1 (keep in mind +𝑚 ≈ 𝑟). Then, we can rewrite the equation above as +𝑓 (𝑥1, . . . , 𝑥𝜇) := +∑︁ +𝑖 ∈ {0,1}𝑟 +∑︁ +𝑗 ∈ {0,1}𝑚 +𝑓𝑖 | |𝑗 · (𝑥𝑖1 +1 · . . . ·𝑥𝑖𝑟 +𝑟 ) · (𝑥 𝑗1 +𝑟 +1 · . . . ·𝑥 𝑗𝑚 +ℓ ), +or in other words +𝑓 (𝑥1, . . . , 𝑥𝜇) = b⊺ + +a⊺ +. . . +a⊺ + + +f0...0 +... +f1...1 + +for +b⊺ := (𝑥𝑖1 +1 · . . . ·𝑥𝑖𝑟 +𝑟 )𝑖 ∈ {0,1}𝑟 ∈ R2𝑟 +𝑞 +a⊺ := (𝑥 𝑗1 +𝑟 +1 · . . . ·𝑥 𝑗𝑚 +ℓ ) 𝑗 ∈ {0,1}𝑚 ∈ R2𝑚 +𝑞 +f ⊺ +𝑖 := (𝑓𝑖 | |𝑗 ) 𝑗 ∈ {0,1}𝑚 ∈ R2𝑚 +𝑞 for 𝑖 ∈ {0,1}𝑟 +. (12) +This representation of a polynomial evaluation claim gives intuition how to commit to the polynomial𝑓. +4.1 Inner and Outer Commitment +The starting point of our polynomial commitment scheme construction is the basic commitment scheme from +LaBRADOR [BS23]. Let𝑏 > 1 be the base used for decomposition (throughout) and denote𝛿 := ⌈log𝑏 𝑞⌉. +The commitment key is a pair of uniformly random matricesA ∈ R𝑛𝐴 ×𝛿2𝑚 +𝑞 and B ∈ R𝑛𝐵 ×𝑛𝐴𝛿2𝑟 +𝑞 where 𝑛𝐴, 𝑛𝐵 +are the security parameters which determine hardness of the underlying Module-SIS problem. +Suppose we want to commit to the 2𝑟 vectors (f ⊺ +𝑖 )𝑖 of length 2𝑚. The first step is to compute the +decompositions: +s1+Í𝑟 +𝑧=1 𝑖𝑧 ·2𝑧−1 := G−1 +2𝑚 (f𝑖 ) (13) +and the inner Ajtai commitmentst𝑖 := As𝑖 ∈ R𝑛𝐴 +𝑞 for 𝑖 ∈ [ 2𝑟 ]. The next step is to compute their binary +decomposition ˆt𝑖 := G−1 +𝑛𝐴 (t𝑖 ). The final outer commitment is +u := B + +ˆt1 +... +ˆt2𝑟 + +∈ R𝑛𝐵 +𝑞 . (14) +The commitment opening becomes a tuple of(s𝑖,ˆt𝑖 )𝑖. +15 + +--- page 16 --- + +In the knowledge extraction, we will only be able to extract so-called weak openings. We define a weak +opening for the commitmentu is a tuple(s𝑖,ˆt𝑖, 𝑐𝑖 )𝑖 ∈ [2𝑟 ], which satisfies all the following conditions +∀𝑖 ∈ [2𝑟 ] : ∥𝑐𝑖 ·s𝑖 ∥ ≤ ¯𝛽, ∥𝑐𝑖 ∥1 ≤ ¯𝜔, 𝑐 𝑖 ∈ R× +𝑞 , As𝑖 = G𝑛𝐵 +ˆt𝑖 +B + +ˆt1 +... +ˆt2𝑟 + += u and + + +ˆt1 +... +ˆt2𝑟 + + +∞ +≤ ¯𝛾 . +Clearly, a message can be derived from the weak opening by Equation (13). We recall the known result that +the commitment scheme above satisfies binding w.r.t. weak openings under the Module-SIS assumption. We +omit the proof since it follows identically as in [NS24, Lemma 2.11]. +Lemma 7(Weak Binding). There is a deterministic algorithm that given two weak openings(s𝑖,ˆt𝑖, 𝑐𝑖 )𝑖 ∈ [2𝑟 ] +and (s′ +𝑖,ˆt′ +𝑖, 𝑐′ +𝑖 )𝑖 ∈ [2𝑟 ] for the commitment u ∈ R𝑛𝐵 +𝑞 such that s𝑗 ≠ s′ +𝑗 for some 𝑗 ∈ [ 2𝑟 ], outputs a vector +z ∈ R2𝑚+𝑛𝐴𝛿2𝑟 +𝑞 such that [A | B]z = 0 and 0 < ∥z∥∞ ≤ max(2¯𝜔¯𝛽,2¯𝛾). +4.2 Polynomial Evaluation as Quadratic Equation +We can now rephrase Equation (12) in terms of the commitment opening(s𝑖 )𝑖. Namely, we can write +𝑢 = 𝑓 (𝑥1, . . . , 𝑥ℓ ) = b⊺ + +a⊺G2𝑚 +. . . +a⊺G2𝑚 + + +s1 +... +s2𝑟 + +. (15) +From now on, we focus on proving knowledge of a commitment opening(s𝑖,ˆt𝑖 )𝑖 for the commitment®𝑢 which +satisfies the above equation. We summarize the protocol in Figure 3. +To begin with, the prover commits to the vectorw = (𝑤1, . . . , 𝑤2𝑟 ) ∈ R2𝑟 +𝑞 defined as𝑤 𝑖 := a⊺G2𝑚 s𝑖 ∈ R𝑞. +That is, P first decomposes each 𝑤 𝑖, thus obtaining ˆw𝑖 := G−1 +1 (𝑤 𝑖 ) ∈ R𝛿 +𝑞. Finally, the prover sends the +commitment +v := Dˆw ∈ R𝑛𝐷 +𝑞 (16) +where D ∈ R𝑛𝐷 ×𝛿 ·2𝑟 +𝑞 is a public uniformly random matrix andˆw⊺ := (ˆw⊺ +1 , . . . ,ˆw⊺ +2𝑟 ). Note that Equation (15) +can be rephrased as +𝑢 = b⊺w = b⊺G2𝑟 ˆw. (17) +The verifier V responds with a challenge vectorc := (𝑐1, . . . , 𝑐2𝑟 ) ← C 2𝑟 +, where the challenge spaceC is +currently defined as a subset of short challenges inR𝑞: +C ⊆ { 𝑐 ∈ R𝑞 : ∥𝑐 ∥1 ≤ 𝜔 } +for some 𝜔 instantiated later9. Then, the proverP computes the short folded witness +z := 𝑐1s1 + 𝑐2s2 + . . . + 𝑐2𝑟 s2𝑟 ∈ R2𝑚𝛿 +𝑞 . +We note the relationship betweenz and ˆw +a⊺G2𝑚 z = +2𝑟 +∑︁ +𝑖=1 +𝑐𝑖 a⊺G2𝑚 s𝑖 = +2𝑟 +∑︁ +𝑖=1 +𝑐𝑖𝑤 𝑖 = +2𝑟 +∑︁ +𝑖=1 +𝑐𝑖 G1ˆ𝑤 𝑖 = (c⊺ ⊗ G1)ˆw. (18) +We also note the relationship betweenz and ˆt: +Az := +2𝑟 +∑︁ +𝑖=1 +𝑐𝑖 As𝑖 = +2𝑟 +∑︁ +𝑖=1 +𝑐𝑖 G𝑛𝐴 +ˆt𝑖 = (c ⊗ G𝑛𝐴 )ˆt. (19) +9 The ℓ1 norm bound for challenges is set here to later apply Lemma 2. +16 + +--- page 17 --- + +Now, combining Equations 14, 16, 17, 18, and 19, the proverP will prove that shortz,ˆt, ˆw satisfy the linear +relation over R𝑞: + +D 0 0 +0 B 0 +b⊺G2𝑟 0 0 +(c⊺ ⊗ G1) 0 −a⊺G2𝑚 +0 c ⊺ ⊗ G𝑛𝐴 −A + + +ˆw +ˆt +z + += + +v +u +𝑢 +0 +0 + +. +We see that while coefficients ofˆw,ˆt are between ⌈−𝑏/2⌉ and ⌈𝑏/2⌉ −1, the coefficients ofz will be moderately +larger. Let 𝛽 be the maximum𝐿∞ norm of z, 𝜏 ≔ ⌈log𝑏 𝛽⌉ and J𝑛 ≔ I𝑛 ⊗ +1 𝑏 𝑏2 · · ·𝑏𝜏 −1 +∈ R𝑛×𝑛𝜏 +𝑞 be the +corresponding gadget matrix. To decrease the coefficients ofz further, we decompose asˆz = J−1 +2𝑚 (z) ∈ R2𝑚𝛿𝜏 +𝑞 . +Hence, the prover now needs to prove knowledge of(ˆw,ˆt,ˆz) ∈ 𝑆2𝑟 𝛿+2𝑟 𝑛𝐴𝛿+2𝑚𝛿𝜏 +𝑏 such that + +D 0 0 +0 B 0 +b⊺G2𝑟 0 0 +(c⊺ ⊗ G1) 0 −a⊺G2𝑚 J2𝑚 +0 c ⊺ ⊗ G𝑛𝐴 −AJ2𝑚 + + +ˆw +ˆt +ˆz + += + +v +u +𝑢 +0 +0 + +. (20) +Public parameters: A ∈ R𝑛𝐴 ×𝛿2𝑚 +𝑞 , B ∈ R𝑛𝐵 ×𝑛𝐴𝛿2𝑟 +𝑞 , D ∈ R𝑛𝐷 ×𝛿2𝑟 +𝑞 +Witness: commitment openings := (s𝑖 )𝑖,ˆt := (ˆt𝑖 )𝑖 +Statement: commitment u ∈ R𝑛𝐵 +𝑞 , evaluation𝑢, vectors a, b as in Equation (15) +Prover P Verifier V +For 𝑖 ∈ [2𝑟 ] +𝑤𝑖 := a⊺G2𝑚 s𝑖 +w := (𝑤1, . . . , 𝑤2𝑟 ) +ˆw := G−1 +2𝑟 (w) +v := Dˆw ∈ R𝑛𝐷 +𝑞 +v - +c := (𝑐1, . . . , 𝑐2𝑟 ) ← C 2𝑟 +c +z := Í2𝑟 +𝑖=1 𝑐𝑖 s𝑖 +If ∥z∥∞ > 𝛽, abort +ˆz = J−1 +2𝑚 (z) +ˆt, ˆw,ˆz - +(ˆw,ˆt,ˆz) +? +∈ 𝑆2𝑟 𝛿+2𝑟 𝑛𝐴𝛿+2𝑚𝛿𝜏 +𝑏 + +D 0 0 +0 B 0 +b⊺G2𝑟 0 0 +(c⊺ ⊗ G1) 0 −a⊺G2𝑚 J2𝑚 +0 c ⊺ ⊗ G𝑛𝐴 −AJ2𝑚 + + +ˆw +ˆt +ˆz + +?= + +v +u +𝑢 +0 +0 + +. +Fig. 3. Interactive proof of knowledge of the commitment opening(s𝑖,ˆt𝑖 ) which satisfies(15). In our final scheme, the +prover does not send the final message in the clear, but proves knowledge of valid(ˆw,ˆt,ˆz). +The completeness follows directly from the protocol rationale described above. Hence, we move on the +soundness guarantees. Identically as [NS24, Lemma 3.2] we show that given CWSS-structured transcripts, we +can extract a valid weak opening which satisfies Equation (15). +17 + +--- page 18 --- + +Lemma 8(CWSS of Figure 3).Let (¯𝛽, ¯𝜔, ¯𝛾) := (2𝑏𝑘,2𝜔, 𝑏). Suppose 𝜔 < 1 +2 +√ +2𝑞1/2. Then, given the public +parameters, the statement(u, 𝑢), and2𝑟 +1 valid transcripts of the form +tr( 𝑗 ) := (v, c( 𝑗 ), (ˆw( 𝑗 ),ˆt( 𝑗 ),ˆz( 𝑗 ) )) where (c( 𝑗 ) )0≤ 𝑗 ≤2𝑟 ∈ SS(C,2𝑟 ,2), +one can either efficiently extract a weak opening(¯s( 𝑗 ),¯t( 𝑗 ), 𝑐( 𝑗 ) ) 𝑗 ∈ [2𝑟 ] for the commitmentu such that +𝑢 = b⊺ + +a⊺G2𝑚 +. . . +a⊺G2𝑚 + + +¯s1 +... +¯s2𝑟 + +, +or solve MSIS𝑞,𝑑,𝑛𝐵,𝛿2𝑟 𝑛𝐴,2𝑏 or MSIS𝑞,𝑑,𝑛𝐷 ,𝛿2𝑟 ,2𝑏. +Proof. For each 𝑗 ∈ [2𝑟 ], let us reconstruct the vectorz( 𝑗 ) := J2𝑚ˆz( 𝑗 ). Then, we have∥z( 𝑗 ) ∥∞ ≤ 𝑏𝑘. +Without loss of generality, assume thatc(0) is the “central vector”. First, suppose that for two distinct +𝑖 ≠ 𝑗 we haveˆt(𝑖 ) ≠ˆt( 𝑗 ). In particular, we haveBˆt(𝑖 ) = u = Bˆt( 𝑗 ), and so we have found a short solution of +norm at most 2𝑏 for the matrixB. Similarly, we argue for the vectorsˆw( 𝑗 ). Hence, from now on we assume +that allˆt( 𝑗 ) :=ˆt and ˆw( 𝑗 ) := ˆw are the same. Denotew := G2𝑟 ˆw and parse w = (𝑤 𝑖 )𝑖,ˆt := (ˆt𝑖 )𝑖. Then, we have +b⊺w = 𝑢. +Fix 𝑗 ∈ [2𝑟 ] and consider the transcriptstr( 𝑗 ) and tr(0) with the verifier challenges denoted asc( 𝑗 ) := (𝑐 𝑗,𝑖 )𝑖 +and c(0) := (𝑐0,𝑖)𝑖. Recall that the two challenge vectors differexactly in the 𝑗-th coordinate and𝑐 𝑗,𝑖 = 𝑐0,𝑖 for +all 𝑖 ≠ 𝑗. By subtracting the two corresponding verification equations, we obtain +b⊺G2𝑚 (z( 𝑗 ) − z(0) ) = (𝑐 𝑗,𝑗 − 𝑐0,𝑗 ) · 𝑤 𝑗 and A(z( 𝑗 ) − z(0) ) = (𝑐 𝑗,𝑗 − 𝑐0,𝑗 ) · G𝑏,𝑛ˆt𝑗 . +Hence, define our extracted opening as¯st := (s𝑗,ˆt𝑗 ) 𝑗 where +s𝑗 := z( 𝑗 ) − z(0) +𝑐 𝑗,𝑗 − 𝑐0,𝑗 +for all 𝑗 ∈ {0,1}𝑟 +and the slack vector(¯𝑐 𝑗 ) 𝑗 is defined as¯𝑐 𝑗 := 𝑐 𝑗,𝑗 − 𝑐0,𝑗. By construction, we have∥¯𝑐 𝑗 ∥1 ≤ 2𝜔 and ∥¯𝑐 𝑗 s𝑗 ∥∞ ≤ 2𝑏𝑘. +By Lemma 3,¯𝑐 𝑗 is invertible overR𝑞 since ∥¯𝑐 𝑗 ∥∞ ≤ 2𝜔 < 1√ +2𝑞1/2. Finally,b⊺G2𝑚 s𝑗 = 𝑤 𝑗 for all 𝑗 ∈ [2𝑟 ], which +concludes the proof. ⊓ ⊔ +4.3 Unstructured Linear Relations and Sumcheck +In this section, we show how to reduce proving unstructured linear relations overR𝑞 with infinity-norm +constraints, as in Equation (20). Concretely, we focus of the following relationRlin: +Rlin +𝑞,𝑑,𝑛,𝜇,𝑏 := + +(z ∈ R𝜇 +𝑞, (M ∈ R𝑛×𝜇 +𝑞 , y ∈ R𝑛 +𝑞 )) : Mz = y ∧ ∥ z∥∞ ≤ 𝑏 −1 +. +At a high level, our approach is to commit toz using a (potentially another) polynomial commitment scheme, +and then apply the sumcheck protocol to prove the relation. However, one challenge is that the relation +Rlin +𝑞,𝑑,𝑛,𝜇,𝑏 is defined over polynomial ringsR𝑞, so the standard sumcheck protocol over finite fields cannot be +directly applied (or one would have to run it overR𝑞 as in [BC24]). To bridge this gap, we leverage the +ring-switching technique introduced by [HMZ25]. +Ring switching. In the relationRlin +𝑞,𝑑,𝑛,𝜇,𝑏, the inputM ∈ R𝑛×𝜇 +𝑞 and y ∈ R𝑛 +𝑞 are public information, and the +prover needs to prove knowledge of a witnessz ∈ R𝜇 +𝑞 such that +–Mz = y +–∥z∥∞ ≤ 𝑏 −1 +18 + +--- page 19 --- + +For the first identity, the authors of [HMZ25] observe thatMz = y if and only if there is a polynomial +r ∈ ( Z<𝑑 +𝑞 [𝑋 ]) 𝑛 such that +Mz = y + (𝑋 𝑑 +1) · r +We now naturally considerM, z, and y as matrices or vectors overZ<𝑑 +𝑞 [𝑋 ] (instead of overR𝑞). Since we will +apply the sumcheck protocol later, we further considerM, x, y, r as elements overF<𝑑 +𝑞𝑘 [𝑋 ]. Now, we can divide +the process into two steps: the prover first commits to(z, r) using a polynomial commitment schemeCom, +and then proves the following constraints: +–For each𝑖 ∈ [ 𝑛], Í +𝑗 M𝑖,𝑗 z𝑗 = y𝑖 + (𝑋 𝑑 +1) · r𝑖, where M𝑖,𝑗 , y𝑖 ∈ Z<𝑑 +𝑞 [𝑋 ] +–z is a vector overZ<𝑑 +𝑞 [𝑋 ] and ∥z∥∞ ≤ 𝑏 −1. +Notice that the verifier can check whetherM and y are over Z<𝑑 +𝑞 [𝑋 ] by themselves, sinceM and y are public +information. However, the prover has to prove thatz is a vector overZ<𝑑 +𝑞 [𝑋 ] if we run the sumcheck protocol +over F<𝑑 +𝑞𝑘 [𝑋 ]. Since M𝑖,𝑗 , z𝑗, y𝑖, r𝑖 ∈ Z<𝑑 +𝑞 [𝑋 ], we can represent them as +M𝑖,𝑗 = +∑︁ +ℓ<𝑑 +𝑀𝑖,𝑗,ℓ ·𝑋 ℓ, +and similarly for allz𝑗, y𝑖, and r𝑖. Here we note that bothz and r are witnesses of the prover, and the prover +needs to prove the norm bound ofz. For the sake of notation and implementation simplicity, we also let the +prover performs a gadget decomposition ofr as r = Í +𝑢 𝑏𝑢 ·r𝑢, and commits(z, r1, . . . , rlog𝑏 (𝑞) ) instead, and +also proves the norm bound ofr𝑢 for each𝑢. That is, the prover proves the following statement: +–For each𝑖 ∈ [ 𝑛], Í +𝑗 M𝑖,𝑗 z𝑗 = y𝑖 + (𝑋 𝑑 +1) ·Í +𝑢 𝑏𝑢r𝑖,𝑢, where M𝑖,𝑗 , y𝑖 ∈ Z<𝑑 +𝑞 [𝑋 ] +–z, r1, . . . , rlog𝑏 (𝑞) are vectors overZ<𝑑 +𝑞 [𝑋 ] and ∥z∥∞, ∥r1∥∞, . . . , ∥rlog𝑏 (𝑞) ∥∞ ≤ 𝑏 −1. +To simplify the presentation, in the following we omit the subscript𝑢 and explain how to prove the +statement below. Readers should be aware that there is a hidden gadget decomposition ofr. We believe this is +sufficient to explain our main idea. To incorporate this gadget decomposition, we can modify the multilinear +extension correspondingly. +–For each𝑖 ∈ [ 𝑛], Í +𝑗 M𝑖,𝑗 z𝑗 = y𝑖 + (𝑋 𝑑 +1) · r𝑖, where M𝑖,𝑗 , y𝑖 ∈ Z<𝑑 +𝑞 [𝑋 ] +–z, r are vectors overZ<𝑑 +𝑞 [𝑋 ] and ∥z∥∞, ∥r∥∞ ≤ 𝑏 −1. +For each𝛼 ∈ F𝑞𝑘, we denote +M𝑖,𝑗 (𝛼) = +∑︁ +ℓ<𝑑 +𝑀𝑖,𝑗,ℓ ·𝛼 ℓ ∈ F𝑞𝑘 , +and similarly for allz𝑗, y𝑖, and r𝑖. Now, instead of proving the constraints overF<𝑑 +𝑞𝑘 [𝑋 ], which could be very +costly, we optimize by letting the verifier first send a random𝛼 to the prover. Then, the prover proves the +following constraints: +–For each𝑖 ∈ [ 𝑛], Í +𝑗 M𝑖,𝑗 (𝛼)z𝑗 (𝛼) = y𝑖 (𝛼) + ( 𝛼𝑑 +1) · r𝑖 (𝛼), where M𝑖,𝑗 , y𝑖 ∈ Z<𝑑 +𝑞 [𝑋 ] +–z is a vector overZ<𝑑 +𝑞 [𝑋 ] and ∥z∥∞, ∥r∥∞ ≤ 𝑏 −1. +Having proof composition in mind, we consider the following simple three-round protocol in Figure 4 which +captures the idea above. +19 + +--- page 20 --- + +Prover P Verifier V +𝑡 := Com(z, r) 𝑡 - +𝛼 ← F𝑞𝑘 +𝛼 +z, r - +𝑡 ?= Com(z, r) +z +? +∈ Z<𝑑𝑞 [𝑋 ] and ∥z∥∞, ∥r∥∞ ≤ 𝑏 −1 +∀𝑖 ∈ [ 𝑛],Í +𝑗 M𝑖,𝑗 (𝛼)z𝑗 (𝛼) ?= y𝑖 (𝛼) + ( 𝛼𝑑 +1) · r𝑖 (𝛼) +Fig. 4. Reducing proving linear equations overZ<𝑑𝑞 𝑞 [𝑋 ] to equations overF𝑞𝑘. +We note that this simplification could introduce a(2𝑑 −1)/| F𝑞𝑘 | soundness error. This is because the polynomialÍ +𝑗 M𝑖,𝑗 z𝑗 − y𝑖 − (𝑋 𝑑 +1) ·r𝑖 has degree at most 2𝑑 −1. Formally, we argue this by proving 2𝑑-special soundness +in Lemma 9. +Lemma 9(Special soundness of Figure 4).There is an efficient, deterministic algorithm that given +a statement (M𝑖,𝑗 , y𝑖, 𝑏) and 2𝑑 valid transcripts (𝑡𝑖, 𝛼𝑖, (z𝑖, r𝑖 )) of the protocol in Figure 4 such that(𝛼𝑖 )𝑖 ∈ +SS(F𝑞𝑘 ,1,2𝑑), can either extract¯z,¯r which satify Mz = y + ( 𝑋 𝑑 +1) · r over Z𝑞 [𝑋 ], or break binding of the +commitment scheme Com. +Proof. If for some𝑖 ≠ 𝑗 we have (z𝑖, r𝑖 ) ≠ (z𝑗, r𝑗 ), then we automatically break the binding property of the +commitment scheme. Otherwise, define¯z := z𝑖,¯r := r𝑖 for all 𝑖. Then, for 2𝑑 distinct evaluation points𝛼 𝑗, we +have +∀𝑖 ∈ [ 𝑛], +∑︁ +𝑗 +M𝑖,𝑗 (𝛼 𝑗 )z𝑗 (𝛼 𝑗 ) +? += y𝑖 (𝛼 𝑗 ) + ( 𝛼𝑑 +1) · r𝑖 (𝛼 𝑗 ). +Since the polynomialÍ +𝑗 M𝑖,𝑗 z𝑗 − y𝑖 − (𝑋 𝑑 +1) · r𝑖 has degree at most 2𝑑 −1, this implies that it must be a +zero polynomial, which finishes the proof. ⊓ ⊔ +From now on, we focus on proving knowledge of the prover’s last message which satisfies the verification +conditions. +Remark 2. Looking ahead, we will instantiateCom with the inner-outer commitment from Section 4. In this +case, for special soundness we will consider breaking the weak binding property (cf. Lemma 7), rather than +binding. +Represent the constraints by polynomials.In our sumcheck protocol, we first commit(z, r) as a multilinear +polynomial over F𝑞𝑘. Let e𝑤 be a multilinear polynomial such that +e𝑤 (𝑢, ℓ) = +( +𝑧𝑢,ℓ , if 𝑢 ≤ 𝜇 +𝑟𝑢 −𝜇,ℓ , if 𝜇 < 𝑢 ≤ 𝜇 + 𝑛 (21) +Here, we slightly abuse the notation; that is,𝑢 and ℓ actually refer to the binary representations of numbers +in [𝜇 + 𝑛] and [𝑑], respectively. Now we can assume that𝛼 is fixed. We define multilinear polynomialse𝛼 and +e𝑀𝛼 such that: +∀𝑖, 𝑢, ℓ, e𝛼 (ℓ) = 𝛼 ℓ and e𝑀𝛼 (𝑖, 𝑢) = +  + +M𝑖,𝑢 (𝛼), if 𝑢 ≤ 𝜇 and 𝑖 ≤ 𝑛 +−(𝛼𝑑 +1), if 𝑖 + 𝜇 = 𝑢 +0, otherwise +Now, for each fixed𝛼, we can represent the constraints as polynomials: +20 + +--- page 21 --- + +Prover P (𝑡, ˜𝑤 ) Verifier V (𝑡, ˜𝑀𝛼, ˜𝛼) +𝜏0 ← Flog 𝜇+log 𝑑, 𝜏1 ← Flog 𝑛 +𝜏0, 𝜏1 +˜𝑤 - +𝑡 ?= Com(˜𝑤 ) +Reconstruct 𝐻0, 𝐻1 as in (22) and (23) +𝐻0 (𝜏0) ?= 0 and 𝐻𝛼 (𝜏1) ?= 0 +Fig. 5. Proving that polynomials𝐻0 and 𝐻𝛼 are zero via random evaluation. +–For each𝑖 ∈ [ 𝑛], +𝜇+𝑛∑︁ +𝑢=1 +e𝑀𝛼 (𝑖, 𝑢) +∑︁ +ℓ +e𝑤 (𝑢, ℓ) · e𝛼 (ℓ) = y𝑖 (𝛼), +where M𝑖,𝑗 , y𝑖 ∈ Z<𝑑 +𝑞 [𝑋 ]. +–For each𝑢 ∈ [ 𝜇 + 𝑛] and ℓ ∈ [ 𝑑], +e𝑤 (𝑢, ℓ) · (e𝑤 (𝑢, ℓ) − 1) · (e𝑤 (𝑢, ℓ) + 1) · · · (e𝑤 (𝑢, ℓ) − 𝑏 +1) · (e𝑤 (𝑢, ℓ) + 𝑏 −1) = 0. +Following standard approaches [Set20; HMZ25], we use an equality function to batch these constraints. Since +𝑖 ∈ [ 𝑛], its binary representation is a string in{0,1}log 𝑛. Recall that the equality functione𝑒𝑞 : Flog 𝑛 +𝑞𝑘 × Flog 𝑛 +𝑞𝑘 → +F𝑞𝑘 is defined +e𝑒𝑞 (𝑡, 𝑖) = +Ö +𝑗 +(𝑡 𝑗 ·𝑖 𝑗 + (1 − 𝑡 𝑗 ) · (1 − 𝑖 𝑗 )) +Using e𝑒𝑞, we can then batch the first type of constraints to show that the following polynomial𝐻𝛼 (𝑡) : Flog 𝑛 +𝑞𝑘 → +F𝑞𝑘 is a zero polynomial. +𝐻𝛼 (𝑡) := +∑︁ +𝑖 ∈ [𝑛] +e𝑒𝑞 (𝑡, 𝑖) + ∑︁ +𝑢,ℓ +e𝑀𝛼 (𝑖, 𝑢) · e𝑤 (𝑢, ℓ) · e𝛼 (ℓ) − y𝑖 (𝛼) +! +, (22) +Similarly, we can also batch the second type of constraints (smallness check) as proving that the following +polynomial 𝐻0 : Flog 𝜇+log 𝑑 +𝑞𝑘 → F𝑞𝑘 is a zero polynomial. +𝐻0 (𝑡) := +∑︁ +ℓ,𝑢 +e𝑒𝑞 (𝑡, (𝑢, ℓ)) · e𝑤 (𝑢, ℓ) (e𝑤 (𝑢, ℓ) − 1) (e𝑤 (𝑢, ℓ) + 1) · · · (e𝑤 (𝑢, ℓ) − 𝑏 +1) (e𝑤 (𝑢, ℓ) + 𝑏 −1). (23) +Finish the proof using sumcheck protocols.Now, the goal of the honest prover is to prove that𝐻0 (𝑡) ≡ 0 and +𝐻𝛼 (𝑡) ≡ 0. We adopt a standard approach again: the verifier sends two random points,𝜏0 and 𝜏1, and the +prover then proves that𝐻0(𝜏0) = 0 and 𝐻𝛼 (𝜏1) = 0. We capture this with the following protocol. Similarly as +in Lemma 9, we can show coordinate-wise special soundness of this protocol by treating(𝜏0, 𝜏1) as a vector of +log 𝜇 +log 𝑑 +log 𝑛 coordinates. +Lemma 10(CWSS of Figure 5).There is an efficient, deterministic algorithm that given a statement +(𝑡, ˜𝑀𝛼, ˜𝛼) and 𝐷 := max(2𝑑,2𝑏 − 1) valid transcripts ((𝜏𝑖,0, 𝜏𝑖,1), ˜𝑤 𝑖 ) of the protocol in Figure 5 such that +(𝜏𝑖,0, 𝜏𝑖,1)𝑖 ∈ SS(F𝑞𝑘 ,2, 𝐷), can either extract a valid opening˜𝑤 of 𝑡, i.e. 𝑡 = Com(˜𝑤 ), which satisfies𝐻0 (𝑡0) ≡ 0 +and 𝐻𝛼 (𝑡) ≡ 0 (defined in Equations(22) and (23)), or break binding of the commitment schemeCom. +21 + +--- page 22 --- + +Proof. If for some𝑖 ≠ 𝑗 we have ˜𝑤 𝑖 ≠ ˜𝑤 𝑗, then we break the binding property of the commitment scheme. +Otherwise, define ˜𝑤 := ˜𝑤 𝑖 for all 𝑖. By definition of𝐷 = max(2𝑑,2𝑏 − 1) and the coordinate-wise special +soundness, we have found at least 2𝑑 (resp. 2𝑏 − 1) distinct roots for 𝐻𝛼 (𝐻0), which implies that the +aforementioned polynomials are equal to zero. ⊓ ⊔ +Now, fix𝜏0, 𝜏1 and recall that by definition of𝐻0 and 𝐻𝛼: +𝐻0 (𝜏0) := +∑︁ +ℓ,𝑢 +e𝑒𝑞 (𝜏0, (𝑢, ℓ))e𝑤 (𝑢, ℓ) (e𝑤 (𝑢, ℓ) − 1) (e𝑤 (𝑢, ℓ) + 1) · · · (e𝑤 (𝑢, ℓ) − 𝑏 +1) (e𝑤 (𝑢, ℓ) + 𝑏 −1), +and +𝐻𝛼 (𝜏1) = +∑︁ +𝑢,ℓ +e𝑤 (𝑢, ℓ) · e𝛼 (ℓ) +∑︁ +𝑖 +e𝑒𝑞 (𝜏1, 𝑖) · e𝑀𝛼 (𝑖, 𝑢) − +∑︁ +𝑖 +e𝑒𝑞 (𝜏1, 𝑖) · y𝑖 (𝛼). +We then use sumcheck to prove that𝐻0(𝜏0) = 𝐻𝛼 (𝜏1) = 0. Defining polynomials +𝐹0,𝜏0 (𝑥, 𝑦) := e𝑒𝑞 (𝜏0, (𝑥, 𝑦))e𝑤 (𝑥, 𝑦) (e𝑤 (𝑥, 𝑦) − 1) (e𝑤 (𝑥, 𝑦) + 1) · · · (e𝑤 (𝑥, 𝑦) − 𝑏 +1) · (e𝑤 (𝑥, 𝑦) + 𝑏 −1) ·1≤𝜇 (𝑥, 𝑦), +and +𝐹𝛼,𝜏1 (𝑥, 𝑦) := e𝑤 (𝑥, 𝑦) · e𝛼 (𝑦) · +∑︁ +𝑖 +e𝑒𝑞 (𝜏1, 𝑖) · e𝑀𝛼 (𝑖, 𝑥), +we have that𝐻0(𝜏0) = 0 and 𝐻𝛼 (𝜏1) = 0 if and only if +∑︁ +𝑢,ℓ +𝐹0,𝜏0 (𝑢, ℓ) = 0 and +∑︁ +𝑢,ℓ +𝐹𝛼,𝜏1 (𝑢, ℓ) = 𝑎, +where 𝑎 := Í +𝑖 e𝑒𝑞 (𝜏1, 𝑖) ·y𝑖 (𝛼), which the verifier can compute on their own. Therefore, we lastly apply sumcheck +on 𝐹0,𝜏0 and 𝐹𝛼,𝜏1. +A reinterpretation of the sumcheck protocol.In the standard sumcheck protocol, the prover and the verifier +communicate univariate polynomials and random field elements. In the last round of the protocol, the verifier +makes a single evaluation query to the polynomial. However, since we rely on the special soundness notion (see +Lemma 11), we reinterpret the sumcheck protocol as a composition of single-round communication protocols +to simplify the analysis. In Figure 6, we demonstrate the sumcheck protocol for polynomials of the form +𝐻 (X) := 𝑃 (X) · 𝑄 (˜𝑤 (X)), where 𝑃 and 𝑄 are public information and˜𝑤 (X) is the witness. We note that both +𝐹0,𝜏0 and 𝐹𝛼,𝜏1 have this form. +Compared to the standard sumcheck protocol, we note that the prover usually does not sende𝑤 to the +verifier directly. Instead, it typically proves knowledge ofe𝑤 by continuing the sumcheck protocol, and finally +by providing a proof of the evaluation ofe𝑤. +Lemma 11(Special soundness of Figure 6). There is an efficient, deterministic algorithm that +given a statement and 𝐷 := deg(𝐻 ) + 1 valid transcripts (𝑔𝑖 (𝑋𝑖 ), 𝑎𝑖,𝑗 , ˜𝑤 ) 𝑗 of the protocol in Figure 6 such +that (𝑎𝑖,𝑗 ) 𝑗 ∈ SS(F𝑞𝑘 ,1, 𝐷), can either extract a valid opening ˜𝑤 of 𝑡, i.e. 𝑡 = Com(˜𝑤 ), which satisfiesÍ +𝑏 𝑗 𝐻 (𝑎1, . . . , 𝑎𝑖, 𝑏𝑖+1, . . . , 𝑏ℓ ) = 𝑔𝑖 (𝑎𝑖 ), or break binding of the commitment schemeCom. +Proof. If for some𝑖 ≠ 𝑗 we have ˜𝑤 𝑖 ≠ ˜𝑤 𝑗, then we break the binding property of the commitment scheme. +Otherwise, define ˜𝑤 := ˜𝑤 𝑖 for all 𝑖. By definition of𝐷 = deg(𝐻 ) + 1 and the coordinate-wise special soundness, +we have found at leastdeg(𝐻 ) + 1 distinct roots forÍ +𝑏 𝑗 𝐻 (𝑎<𝑖, 𝑋𝑖, 𝑏𝑖+1, . . . , 𝑏ℓ ) − 𝑔𝑖 (𝑋𝑖 ), which implies thatÍ +𝑏 𝑗 𝐻 (𝑎<𝑖, 𝑋𝑖, 𝑏𝑖+1, . . . , 𝑏ℓ ) − 𝑔𝑖 (𝑋𝑖 ) ≡ 0. +Since special soundness is composable, we can obtain the special soundness of the sumcheck protocol by +composing Figure 6 for all𝑖 ∈ [ ℓ]. Finally, the prover outputs𝑦′ := ˜𝑤 (𝑎1, . . . , 𝑎ℓ ), which allows the verifier to +compute the evaluation𝐻 (𝑎1, . . . , 𝑎ℓ ). What is left for the prover to do now is prove the polynomial evaluation +claim for ˜𝑤. This ends the first iteration of our protocol. +22 + +--- page 23 --- + +Prover P (𝑡, 𝑃, 𝑄, ˜𝑤 ) Verifier V (𝑡, 𝑃, 𝑄, 𝑧𝑖 −1) +𝑔𝑖 (𝑋𝑖 ) := Í +𝑏 𝑗 𝐻 (𝑎<𝑖, 𝑋𝑖, 𝑏𝑖+1, . . . , 𝑏ℓ ) +𝑔𝑖 (𝑋𝑖 ) - +𝑎𝑖 ← F𝑞𝑘 +𝑎𝑖 +˜𝑤 - +𝐻 (X) := 𝑃 (X) · 𝑄 (˜𝑤 (X)) +𝑡 ?= Com(˜𝑤 ) +𝑔𝑖 (0) + 𝑔𝑖 (1) ?= 𝑧𝑖 −1 +Í +𝑏 𝑗 𝐻 (𝑎 ≤𝑖, 𝑏𝑖+1, . . . , 𝑏ℓ ) ?= 𝑔𝑖 (𝑎𝑖 ) +Fig. 6. Proving Í +𝑏𝑖,...,𝑏 ℓ 𝐻 (𝑎1, . . . , 𝑎𝑖 −1, 𝑏𝑖, . . . , 𝑏ℓ ) = 𝑧𝑖 −1 for 𝑖 ∈ [ ℓ], where 𝐻 (X) := 𝑃 (X) · 𝑄 (˜𝑤 (X)). We set𝑧0 to be the +initial sum, and𝑧𝑖 := 𝑔𝑖 (𝑎𝑖 ). +Putting everything together.The complete protocol is presented in Figure 7, which is a natural composition +of subprotocols in Figures 4, 5 and 6. By combining Lemmas 9, 10 and 11, we conclude that our final protocol +satisfies coordinate-wise special soundness. +Prover P Verifier V +𝑡 := Com(z, r) 𝑡 - +𝛼 ← F𝑞𝑘 , 𝜏0 ← Flog 𝜇+log 𝑑, 𝜏1 ← Flog 𝑛 +𝛼, 𝜏0, 𝜏1 +Run sumcheck to𝐻0 (𝜏0), 𝐻𝛼 (𝜏1) 𝑔1 (𝑋1) - +𝑎1 ← F𝑞𝑘 +𝑎1 +· · · +· · · +𝑔ℓ (𝑋ℓ ) - +𝑎ℓ ← F𝑞𝑘 +Open 𝑡 to evaluate˜𝑤 (𝑎1, . . . , 𝑎ℓ ) +Evaluate e𝑀𝛼 (𝑎1, . . . , 𝑎ℓ ) +Check the correctness of the sumcheck. +Fig. 7. Efficient Sumcheck via Ring Switching +4.4 Asymptotic Efficiency Analysis +Suppose we want to prove knowledge of the committed multilinear polynomial𝑓 ∈ Z≤1 +𝑞 [𝑋1, . . . , 𝑋ℓ ] which +satisfies a polynomial evaluation claim overF𝑞𝑘. In the following, let𝐿 := 2ℓ. +23 + +--- page 24 --- + +Basic parameters.Let 𝑞 = poly(𝜆, 𝐿) and 𝑘 = 𝜆/log 𝑞, so thatF𝑞𝑘 is of exponential size to ensure negligible +knowledge soundness in Lemmas 9, 10 and 11. We set the ring dimension𝑑 = poly(𝜆), so 𝛼 = log 𝜆, and +𝑛𝐴 = 𝑛𝐵 = 𝑛𝐷 = 𝑂 (1) for hardness of the Module-SIS problem. Then, we pick𝜔 = 𝑂 (𝑑) for sufficiently large +challenge space so that knowledge soundness error in Lemma 8 becomes negligible. We pick the decomposition +base 𝑏 = 𝑂 (1) and therefore 𝛿 = 𝑂 (log 𝑞). Recall that𝛽 in Section 4.2 was the upper-bound on the vectorz +(see Figure 3). A naive bound for𝛽 is: + +2𝑟 +∑︁ +𝑖=1 +𝑐𝑖 s𝑖 + +∞ +≤ +2𝑟 +∑︁ +𝑖=1 +𝜔 ·𝑏 = 2𝑟 𝜔𝑏 =: 𝛽. +Hence, if we choose the smallest integer𝜏 such that𝑏𝜏 > 𝛽, then 𝜏 = 𝑂 (𝑟 + 𝛼). +Witness length reduction after one iteration.By Section 3, we can consider polynomials overR𝑞 with ℓ − 𝛼 +variables. As in Section 4.2, we define variables𝑚, 𝑟 such that𝑚 + 𝑟 = ℓ − 𝛼. Let𝑚 = 𝑟 = (ℓ − 𝛼)/2 (we implicitly +assume ℓ − 𝛼 is even, which does not have anyasymptotic impact). Then, the witness(ˆt, ˆw,ˆz), parsed in +Section 4.3 asz, together with the “quotient”r from the ring switching methodology, is together of length – +in terms ofZ𝑞 elements: +(𝑛𝐴 +1) ·2𝑟 ·𝛿 ·𝑑| {z } +|ˆt|+| ˆw| ++ (𝜏 +1)2𝑚) · 𝛿 ·𝑑| {z } +|ˆz| ++ (𝑛𝐴 + 𝑛𝐵 + 𝑛𝐷 +2) · 𝛿 ·𝑑| {z } +|r| += 𝑂 (( ℓ + 𝛼)2 ·2 +ℓ +𝛼 +2 ). +Hence, we conclude that after one round of iteration, the number of variables for the polynomial evaluation +gets reduced (at most) to: +ℓ ↦→ ℓ + 𝛼 +2 +2log (ℓ + 𝛼) + 𝑂 (1) ≤ 2 +3 (ℓ + 𝛼) + 𝑂 (1). +After 𝑡 iterations, the length of the witness is reduced to (at most): +(2 +3 )𝑡 ·ℓ + +𝑡∑︁ +𝑖=1 +(2 +3 )𝑡 ·𝛼 + 𝑂 (𝑡) ≤ ( 2 +3 )𝑡 ·ℓ +2𝛼 + 𝑂 (𝑡). +Thus, after running our protocol 𝑡 = 𝑂 (log ℓ) times we will output a witness polynomial with at most +𝑂 (log ℓ + 𝛼) variables. Consequently, sending all theZ𝑞-coeffients of the final polynomial in the clear costs +poly(ℓ, 𝑑) = poly(ℓ, 𝜆) bits. +Proof size.Having non-interactive proofs via Fiat-Shamir transformation in mind, we only count the prover +messages. The proof size of a single iteration can be split in three parts. The first is the transformation from +Z𝑞 to R𝑞 from Section 3, where the prover sends𝑑 + 𝑘 (𝑘 −1) Z𝑞-elements, where𝑘 = 𝜆/log 𝜆. The second is the +commitment v from Section 4.2, which is of size𝑛𝐷 ·𝑑 elements in Z𝑞. Finally, in the protocol in Section 4.3, +for each sumcheck round (out of𝑂 (ℓ + 𝛼)) the prover sends 2 (resp.𝑏 +1) elements inF𝑞𝑘 corresponding to +the coefficients of the univariate polynomials obtained from𝐹𝛼,𝜏1 (resp. 𝐹0,𝜏0). Finally, the prover sends the +commitment to ˜𝑤 which is of length𝑛𝐵 ·𝑑 Z𝑞-elements (resp. Section 4). In total, a single iteration of the +protocol has proof size, in terms ofZ𝑞 elements: +(𝑛𝐵 + 𝑛𝐷 +1) · 𝑑 + (𝑘 −1) · 𝑘 + (𝑏 +2) · 𝑘 ·𝑂 (ℓ + 𝛼) = 𝑂 + 𝜆 +log 𝜆 ·ℓ + ++poly(𝜆). +By recursively running the protocol𝑂 (log ℓ) times and by having𝑞 = poly(𝜆, 𝐿), we obtain succinctpoly(ℓ, 𝜆) +proof size. +24 + +--- page 25 --- + +Verifier time.We split the verifier algorithm in three parts. For the transformation in Section 3 the verifier +has to compute theR𝑞-trace map consistency, which is of timepoly(𝜆). For the protocol in Section 4.2 the +verifier does not check anything. Finally, in Section 4.3 the verifier has two tasks. First is checking that +the univariate polynomials sent by the prover every sumcheck round are consistent. This takes𝑂 (𝑏) = 𝑂 (1) +Z𝑞-operations. A much more expensive part is the final evaluation of the polynomial𝐹𝛼,𝜏1 after sumcheck, +which involves evaluating the (publicly known) multilinear extensions˜𝑀𝛼 𝑛 := 𝑛𝐵 + 𝑛𝐷 +3 = 𝑂 (1) times. As +shown in [VSBW13], to evaluate˜𝑀𝛼 once, it would take +𝑂 ((𝑛𝐴 +1)2𝑟 + (𝜏 +1)2𝑚𝛿) = ˜𝑂 ( +√︁ +2ℓ /𝜆) time +in terms of field operations overF𝑞𝑘 using dynamic programming. A single multiplication overF𝑞𝑘 can be +computed with ˜𝑂 (𝑘) operations overZ𝑞. Since𝑘 = 𝑂 (𝜆/log 𝑞) the total verifier time is dominated by˜𝑂 ( +√ +2ℓ ·𝜆). +4.5 Concrete Optimizations and Composing with Greyhound +Avoiding re-decomposition.Having a recursive proof system in mind, a naive approach to commit to the +multilinear extension mle[˜𝑤 ] of ˜𝑤 (cf. Equation (21)) is to run again the commitment scheme from Section 4. +The first part is the inner commitment, where we decompose the coefficients of˜𝑤 (and thus the committed +witness length increases), and then run an Ajtai commitment scheme to get the inner commitments(t𝑖 )𝑖. A +more natural and efficient way is to directly commit to˜𝑤 (cf. Figure 7), as we know that the coefficients of˜𝑤 +are already short and no decomposition for˜𝑤. It is straightforward from the committing point of view, as we +show below, but more importantly, how should one prove an evaluation claim overF𝑞𝑘: +mle[˜𝑤 ] (𝑎1, . . . , 𝑎ℓ ) = 𝑦, +when having only committed to˜𝑤? +Let us recall the notation for completeness. The vector˜𝑤 ∈ Z𝔫·𝑑 is defined in Equation (21), and in +particular, the coefficients of˜𝑤 are between −𝛽 and 𝛽, where parameters𝔫 and 𝛽 are chosen according to the +protocol in Section 4.3. We assume that𝔫𝑑 is a power-of-two. Let𝑑 ′ = 2𝛼 ′ +be a power-of-two ring dimension +used to defineR′ +𝑞 := Z𝑞 [𝑋 ]/( 𝑋 𝑑 ′ ++1) (potentially 𝑑 ′ ≠ 𝑑); it is the cyclotomic ring used for the next iteration +of the protocol. Let𝑚′, 𝑟′ be integers chosen such that𝔫𝑑 = 2ℓ, where ℓ = 𝑚′ + 𝑟 ′ + 𝛼 ′; they will be responsible +for the split-and-fold in the next round. +We start with with the definition of a multilinear polynomial. Let us definea0 = (𝑎1, . . . , 𝑎ℓ −𝜅) and +a1 := (𝑎ℓ −𝜅+1, . . . , 𝑎ℓ ). Then, we can write +𝑦 = mle[˜𝑤 ] (𝑎1, . . . , 𝑎ℓ ) = +∑︁ +𝑖 ∈ {0,1}𝜅 +𝑒𝑞 (𝑖, a1) · +©­­­­­­­ +« +∑︁ +𝑗 ∈ {0,1}ℓ −𝜅 +˜𝑤 𝑗 ∥𝑖 ·𝑒𝑞 ( 𝑗, a0) +| {z } +𝑦𝑖:= +ª®®®®®®® +¬ +(24) +Hence, we let the prover send𝑘 partial evaluations𝑦𝑖 ∈ F𝑞𝑘 to the verifier who can directly check10 whether +𝑦 = +∑︁ +𝑖 +𝑒𝑞 (𝑖, a1) · 𝑦𝑖 . +What we have left to prove is that all the𝑦𝑖 are well-formed, or in other wordsÍ +𝑗 ∈ {0,1}ℓ −𝜅 ˜𝑤 𝑗 ∥𝑖 ·𝑒𝑞 ( 𝑗, a0) = 𝑦𝑖 +for 𝑖 ∈ {0,1}𝜅. +10 As in Section 3, the prover can just send the𝑘 −1 evaluations𝑦𝑖 excluding 𝑦0...0, which can be derived directly by +the verifier. +25 + +--- page 26 --- + +For the next part, we assume without loss of generality thatF𝑞𝑘 = R′ +𝑞 +𝐻 where 𝐻 = ⟨𝜎−1, 𝜎4𝑘+1⟩ (cf. +Section 3). Indeed, as the field isomorphism fixes elements inF𝑞, the coefficients ofz are still small after +mapping the statement intoR′𝐻 +𝑞 . Then, any field element𝑣 ∈ R′ +𝑞 +𝐻 can be explicitly written as: +𝑣 := 𝑣0 ·𝑍0 + +𝑘 −1∑︁ +𝑗=1 +𝑣𝑘 − 𝑗 ·𝑍𝑘 − 𝑗 where 𝑍0 = 1 and 𝑍𝑘 − 𝑗 := 𝑋 +𝑑′ +2𝑘 · (𝑘 − 𝑗 ) − 𝑋 +𝑑′ +2𝑘 · (𝑘+𝑗 ) . +Now, for each𝑗 ∈ {0,1}ℓ −𝜅 define +ˆ𝑤 𝑗 := +∑︁ +𝑖 ∈ {0,1}𝜅 +˜𝑤 𝑗 ∥𝑖 ·𝑍Í𝜅 +𝑡 =1 𝑖𝑡 ·2𝑡 −1 ∈ F𝑞𝑘 . (25) +We observe that∥ˆ𝑤 𝑗 ∥∞ ≤ 𝛽. Then, proving well-formedness of all(𝑦𝑖 )𝑖 is equivalent to proving +∑︁ +𝑗 ∈ {0,1}ℓ −𝜅 +ˆ𝑤 𝑗 ·𝑒𝑞 ( 𝑗, a1) = +∑︁ +𝑗 ∈ {0,1}ℓ −𝜅 +∑︁ +𝑖 ∈ {0,1}𝜅 +˜𝑤 𝑗 ∥𝑖 ·𝑒𝑞 ( 𝑗, x1) · 𝑍Í𝜅 +𝑡 =1 𝑖𝑡 ·2𝑡 −1 += +∑︁ +𝑖 ∈ {0,1}𝜅 +∑︁ +𝑗 ∈ {0,1}ℓ −𝜅 +˜𝑤 𝑗 ∥𝑖 ·𝑒𝑞 ( 𝑗, x1) ·𝑍Í𝜅 +𝑡 =1 𝑖𝑡 ·2𝑡 −1 += +∑︁ +𝑖 ∈ {0,1}𝜅 +𝑦𝑖 ·𝑍Í𝜅 +𝑡 =1 𝑖𝑡 ·2𝑡 −1 ∈ F𝑞𝑘 . +(26) +where the right-hand side is public. Next, we prove (26) as follows. Note that ultimately we can write +∑︁ +𝑗 ∈ {0,1}ℓ −𝜅 +ˆ𝑤 𝑗 ·𝑒𝑞 ( 𝑗, a1) = e⊺ (f ⊺ ⊗ I2𝑟 )ˆ𝑤 𝑗 (27) +where e := (𝑒𝑞 (𝑖, x0))𝑖 ∈ F2𝑟 ′ +𝑞𝑘 and f := (𝑒𝑞 ( 𝑗, x1)) 𝑗 ∈ F2𝑚′ +𝑑 ′/𝑘 +𝑞𝑘 . Finally, using Theorem 2, theR𝑞-trace of the ring +element +𝑝 := e⊺ (𝜎−1(𝜓 (f )) ⊺ ⊗ I2𝑟 )𝜓 (ˆ𝑤 ) ∈ R′ +𝑞 +is exactly equal to the value in (27), which is also equal to (26). By Lemma 6, the infinity norm of +𝜓 (ˆ𝑤 ) ∈ R′ +𝑞 +𝔫·𝑑/𝑑 ′ +is at most 2𝛽, so we will commit to it using the inner-outer commitments without initial +decomposition. What is left to prove is the knowledge of the short commitment opening𝜓 (ˆ𝑤 ) which satisfies +the quadratic relation above. This is exactly the type of statement supported natively by Greyhound (cf. +[NS24, Section 3]). To sum up, the total communication complexity of our reduction to Greyhound is +(𝑘 −1) · 𝑘 ·log 𝑞 +| {z } +| (𝑦𝑖 )𝑖 | ++ 𝑑 ′ ·log 𝑞 +| {z } +|𝑝 | +. (28) +Using Johnson-Lindnestrauss instead of sumcheck.Similarly as in Greyhound [NS24], instead of running the +protocol from Section 4.3, at some point we can naturally switch to LaBRADOR [BS23]. This can be done +when the witness length is small enough so that performing the Johnson-Lindenstrauss projection becomes +cheap. In fact, although the sumcheck-based approach provides a faster verification time, reducing witness +length after a single operation is slower. The reason is that we have to pick relatively small base𝑏, as the +sumcheck proof size consists of𝑂 (𝑏) field elements. This implies longer decomposition𝛿, and thus longer +witness for the next round. On the other hand, the random projection approach allows larger bases𝑏 (at a +fairly small cost), which makes𝛿 concretely smaller. This allows LaBRADOR to recurse further until the +final witness is cheap enough to send in the clear (≈ 30KB). +26 + +--- page 27 --- + +5 Experimental Results +To demonstrate the concrete performance of our protocol, we have developed a prototype implementation of +Hachi in Rust. The source can be found here: +https://github.com/georgeorourke/hachi-pcs +Our implementation currently supports committing to, and single evaluation proofs and verification for +multilinear polynomials overZ𝑞. We highlight that our implementation is still in an early stage of development +and is amenable to a number of optimizations which would lead to considerable increases in performance. +Despite this, our implementation already demonstrates significant improvements in verification time, with +first (by far most expensive) round verification times being 12 to 29 times faster than Greyhound verification. +Additionally, we concretely improve commitment times for large polynomials (30+ variables). At present, +our evaluation proof times substantially exceed those of existing implementations and will need further +optimization — such as SIMD acceleration — to align with the state of the art. +We have benchmarked our implementation on a University server (with AVX-512 support) and a Mac +Mini M4 (no AVX support). The stated times for Prove and Verify are for the first round only; though, +following a single round, the new witness is far smaller and so the first round of the protocol dominates the +total cost of the proof and verification times. +Hardware Variables Commit / s Prove / s Verify / ms +Mac Mini M4 30 149 268 96.5 +Mac Mini M4 28 34.2 114 49.3 +Mac Mini M4 26 8.79 51.3 26.6 +Server 30 87.3 495 185 +Server 28 21.9 219 93.7 +Server 26 5.49 103 39.0 +Greyhound 30 132 41.2 2800 +Greyhound 28 21.2 8.21 1150 +Greyhound 26 4.37 2.03 492 +Fig. 8. Concrete running time of the Hachi PCS with prototype implementation on different hardware. Prove and +Verify times are for the first (and by far the most computationally expensive) round only. The Greyhound benchmark +was tested on an Intel Xeon Sapphire Rapids core at 3.2 GHz with full AVX-512 optimization. +5.1 Concrete Parameters +The parameters (30 variable polynomial) used in benchmarking are shown in Figure 9. We heuristically set +the ring dimension𝑑 = 2𝛼 of R𝑞 and the height of the commitment matrices,𝑛, so that𝑛 ·𝑑 ≥ 210. For𝑙 = 30 +variables, we aim to reduce the size of the witness as much as possible in ‘one-shot‘. Hence, we set𝑑 = 1024 +and 𝑛 = 1. With these parameters, in a single round of Prove and Verify, we reduce the size of the witness by +a factor of 128. Specifically, we reduce a 30 variable multilinear polynomial overZ𝑞 (32 bits per coefficient) to +a 26 variable polynomial overZ16 (4 bits per coefficient). The evaluation proof is reduced from an evaluation +of a 30 variable polynomial overZ𝑞 to an evaluation of a 26 variable polynomial with coefficients inZ16 and +an evaluation point over𝐹𝑞4. +5.2 Concrete Efficiency and Composing with Greyhound +To showcase the communication efficiency, let us set parameters forℓ = 30. We refer to Section 5 (and in +particular Figure 9) for more rationale on the parameters. We aim for𝜆 = 128 security level and use the +Lattice Estimator [APS15] to measure hardness of the Module-SIS problem in the infinity norm. +27 + +--- page 28 --- + +variable description value +ℓ number of variables of witness polynomial 30 +𝑞 prime modulus 4294967197 +𝑘 degree of the extension fieldF𝑞𝑘 for sumcheck 4 +𝛼 log of ring dimension 10 +𝑑 𝑑 := 2𝛼 is the ring dimension ofR𝑞 1024 +𝑚 folding parameter 10 +𝑟 folding parameter 10 +𝑛𝐴 height of the matrixA 1 +𝑛𝐵 height of the matrixB 1 +𝑛𝐷 height of the matrixD 1 +𝑧 maximum 𝐿∞ norm of z 30583 +𝑏 decomposition base 16 +𝛿 expansion factor for decomposing base witness 8 +𝜏 expansion factor for decomposingz 4 +𝜔 𝐿1 norm of a challenge element 16 +𝑐 number of non-zero coefficients in challenge 16 +Next witness size 226 +Next witness norm bound infinity norm 8 +Fig. 9. Concrete parameters for anℓ = 30 variable multilinear polynomial overZ𝑞. +To recall, we pick𝑞 ≈ 232, 𝑑 = 210, 𝛼 = 10, 𝑟 = 10 and𝑚 = 10 and (𝑛𝐴, 𝑛𝐵, 𝑛𝐷 ) = (1,1,1). The decomposition +base is 𝑏 = 16, thus 𝛿 = 8. For comparison with Greyhound, we consider polynomial evaluations over the base +field F𝑞. +First invocation of Hachi.The cost of the transformation in Section 3 is𝑑log 𝑞 ≤ 4KB (where 𝑘 = 1). Next +is the commitmentv is also of size 4KB. Length of the witness˜𝑤 for sumcheck is be bounded by 226, as +discussed in Section 5.1. Then, we consider extension fieldsF𝑞𝑘 for 𝑘 = 4. The cost of sumcheck is then +bounded by 26·4 ·32 · (16 +2) bits ≈ 7.3KB. +Adapting to the Greyhound relation.We apply our transformation from Section 4.5. To this end, we set the +ring dimension exactly as in Greyhound, i.e.𝑑 ′ = 64. Theshort witness 𝜓 (ˆ𝑤 ) for the quadratic relation over +R′ +𝑞 is of length 226/𝑑 ′ = 220. By Equation (28), the cost of the transformation can be upper-bounded by: +12 ·32 +64 ·32 bits ≈ 0.3KB. +Moreover, we also need to commit to𝜓 (ˆ𝑤 ), which again takes 4.5KB (cf. [NS24, Table 4]). Hence, the cost of +preparation to run Greyhound is≈ 4.8KB. +Finally, we can directly apply Greyhound for theshort witness 𝜓 (ˆ𝑤 ) of length 220 over R′ +𝑞. To estimate +the proof size for Greyhound, we consider the parameters of the “𝑁 = 226” column of [NS24, Table 1]). +This column focuses on the polynomial of length 226 with full-fledged Z𝑞-elements. After the coefficient +decomposition, the witness gets increased by a factor of𝛿0 = 5. Keeping in mind that the ring dimension is +𝑑 ′ = 64, Greyhound reduces proving polynomial evaluation to a quadratic equation overR′ +𝑞, where the witness +is of length 226 ·𝛿/𝑑 ′ ≈ 222.3, which is more than enough for our scenario. Therefore, we can upper-bound the +evaluation proof size of Greyhound is≈ 43KB. To conclude, the total evaluation proof can be estimated to be +7.3 +4.8 +43 KB = 55.1KB. +Running time. The concrete experimental running time for the first round of Hachi can be found in Figure 8. +The running time of the transformation, which is done by the prover, from Section 4.5 is negligible compared +to other parts. Finally, we use the Greyhound estimates for the witness of length 224 Z𝑞-elements, which +roughly translates to 224 ·𝛿/𝑑 ′ ≈ 220.3 R′ +𝑞-elements. From [ACFY25; Sei24], the prover runtime (which includes +28 + +--- page 29 --- + +committing to 𝜓 (ˆ𝑤 )) is around 1220ms, while the verifier runtime is 130ms. Hence, by combining with +Figure 8, the total prover and verifier runtime are approximately 270s and 227ms. +5.3 Faster Commitment Time +The commitment function of Hachi is essentially identical to Greyhound except for the choice of ring. Unlike +Greyhound, which is restricted to small ring dimensions (e.g.𝑑 = 64), Hachi is suitable for use with large +ring dimensions (e.g.𝑑 = 1024). A major benefit of this is improved commitment time. This is because the +number of ring multiplications decreases in proportion to𝑑2 (since both 2𝑟 +𝑚 and 𝑛𝐴, 𝑛𝐵 decrease linearly +with 𝑑) however, with NTT, the cost of the ring multiplication scales with𝑑log 𝑑. +The existing Greyhound implementation is highly optimized for certain rings and does not support large +ring dimensions. Therefore, we cannot benchmarch the commitment times for both the Greyhound parameters +and our parameters using the Greyhound implementation. However, as our library is much more flexible, +we are able to benchmark the Greyhound commitment times with our implementation - providing a near +like-for-like comparison. Our parameters are restricted to powers of two, and we set them as close to the +Greyhound parameters as possible. The comparative results are shown in Figure 10 and indicate a 3-5X +algorithmic improvement in the commitment time. +Variables Folding Parameters,𝑟, 𝑚 Decomposition base Commit / s Comparison +30 16384, 1024 256 433 5.0X slower +28 8192, 512 256 71.8 3.3X slower +26 4096, 256 256 18.6 3.4X slower +Fig. 10. Concrete running time of the Commitment function using a smaller ring dimension𝑑 = 64 on an AVX-512 +comptaible University server and comparison to concrete times with Hachi parameters. +5.4 Implementation Details +Integer Operations and Data StructuresInteger coefficients are stored as unsigned 64 or 32 bit integers +(for compatibility with tfhe-ntt). Integer decomposition has been optimized for power of two bases but not +optimized with SIMD. Sampling inZ𝑞 for 𝑘-bit prime 𝑞 is performed by sampling 32/64 bits (using any +PRNG) and masking𝑘 bits. Rejection sampling is then used to sample uniformly overZ𝑞. Polynomials are +represented as coefficient slices over integers (u32 or u64). Vectors and matrices over polynomials are stored +as tightly packed coefficients in a single vector (Array of Structure format). Decomposition of vectors of +polynomials is done by sequentially decomposing coefficients and has not yet been optimized with SIMD. +Initialization of the commitment matrices is done by sequentially sampling coefficients from a seeded PRNG. +If the parameters are set so that the commitment matrices have the same dimensions, the same commitment +matrix is used for all commitments. The commitment matrices are re-generated by the commit/prove/verify +functions for each invocation and this overhead is included in the stated benchmarks. +Field ArithmeticWe use the extension field𝐹𝑞4 to ensure approximately 128 bits of security. Representation +and multiplication of field elements in𝐹𝑞4 is provided by the ark-ff library11. In general, we do not use ark-ff +for multiplication over the base fieldZ𝑞, instead opting to perform integer multiplication over u64 and reduce +modulo 𝑞 naively. This is to maintain compatibility with tfhe-ntt and, whilst suboptimal, avoids conversion +between ark-ff types and u64 which is more costly than naive modulo reduction. In several cases we need to +multiply an element of the base fieldZ𝑞 with an element of the extension fieldF𝑞4, and have optimized the +arithmetic for this case. +11 https://crates.io/crates/ark-ff +29 + +--- page 30 --- + +Polynomial Arithmetic Hachi requires general polynomial arithmetic overR𝑞 := Z𝑞 [𝑋 ]/( 𝑋 𝑑 +1), for a +range of ring dimensions𝑑, as well as overZ𝑞 [𝑋 ] (to transition to the sumcheck). Our arithmetic library builds +on top of the tfhe-ntt library12, which provides SIMD (AVX-2 or AVX-512) optimized NTT-multiplication. +Since the prime modulus𝑞 cannot be chosen to be ‘NTT-friendly’, we cannot use NTT-based multiplication +directly for R𝑞. Instead, we use the tfhe-ntt 64-bit ‘Native NTT’, which performs NTT-based multiplication +modulo several primes𝑝𝑖 to simulate the arithmetic moduloÎ 𝑝𝑖. SinceÎ 𝑝𝑖 is larger than the coefficients +of any product of polynomials, the product holds overZ[𝑋 ]/( 𝑋 𝑑 +1) with 64-bit wrapping arithmetic. We +then naively reduce the coefficients modulo𝑞 to achieve efficient multiplication overZ𝑞 [𝑋 ]/( 𝑋 𝑑 +1). For the +reduction modulo 𝑞 to hold, we require that no 64-bit wrap has occurred during the multiplication (i.e. the +product has coefficients smaller than 64 bits). Note that Hachi only ever requires multiplication overR𝑞 +when one operand is small (i.e. it has been decomposed). With a maximum ring dimension of 210 and 32-bit +modulus 𝑞, the product will not wrap as long as the coefficients of the small operand are no more than 22 bits. +We use a 4-bit decomposition base and so can safely use the 64-bit implementation. The reduction modulo𝑞 +in our implementation is suboptimal, however it does not add too much to the cost of multiplication (about +5%). On the other hand, whilst this approach lacks optimizations that could be made for more specific rings, +it is very flexible; allowing both the (power-of-two) ring dimension and arbitrary modulus𝑞 to be chosen at +runtime. +To efficiently multiply polynomials overZ𝑞 [𝑋 ], note that Hachi only requires computing a sum over a +product oftwo polynomials with degree less than𝑑, so that the result has degree at most 2𝑑 −1. We therefore +pad the polynomials to degree 2𝑑 −1 and use the existing library to efficiently multiply overZ𝑞 [𝑋 ]/( 𝑋2𝑑 +1). +The result does not incur any reduction modulo𝑋2𝑑 +1 and so holds overZ𝑞 [𝑋 ]. Given the full convolution, +we perform (optimized) polynomial division to compute the residues modulo(𝑋 𝑑 +1). +Our library abstracts the ring multiplication for larger structures like vectors and matrices of polynomials. +We minimize the number of NTTs that must be performed by keeping re-used and intermediate values in +NTT form. +Optimized Sparse Polynomial MultiplicationSince Hachi may use large ring dimensions, the challenge +polynomials can be very sparse whilst still producing an exponential challenge set. For example, for𝑑 = 1024, +the challenge set may consist of ring elements with exactly𝑘 = 16 non-zero coefficients, each of which is -1 or +1. We sample such elements using a Fisher-Yates-like algorithm to obtain a partial permutation on 0, ..., 𝑑 −1, +giving the indices of the 16 non-zero coefficients. A bit is independently sampled for each non-zero coefficient +to determine its sign. Since these challenge elements are so sparse, they can be more efficiently multiplied +by a full ring element (compared to treating the sparse element as an arbitrary ring element). Our library +implements this sparse polynomial product directly for bothZ𝑞 [𝑋 ] and Z𝑞 [𝑋 ]/( 𝑋 𝑑 +1). This has not yet been +optimized with SIMD, yet performs about as well as the general AVX-512 optimized NTT multiplication. +With a good SIMD optimization, it should significantly outperform general NTT multiplication. +Witness Streams The witness is supplied to the commit/prove functions as a general stream. In our tests, +the witness was stored in a file and provided as a file stream which reads chunks of the witness on demand. +The commit and prove functions access the witness from the stream in small chunks of size 2−𝑚 ≊ 0.1% of +the whole witness size. This makes our implementation suitable for very large witnesses whilst maintaining +moderate memory usage. 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Walfish, “Doubly-efficient zkSNARKs +without trusted setup,” in2018 IEEE Symposium on Security and Privacy, IEEE Computer +Society Press, May 2018, pp. 926–943.doi: 10.1109/SP.2018.00060. +33 \ No newline at end of file diff --git a/paper/iterative-rejection-sampling.pdf b/paper/iterative-rejection-sampling.pdf new file mode 100644 index 0000000000000000000000000000000000000000..5af6c9ddc4e68b8f2bafbe4b8f3488e56562df64 GIT binary patch literal 501003 zcmb5#Q+qDJwk6=$wpVOhUu@f1v2ELSR&3k0ZQHh;-rY~%^}aY=SM?9-9W~}8QxFlO zW1?q+A)8woUV~vLWF)jRw1na1g<+5}u{CoxCuC+}Vm+Q8XF z#Kg$X*aU`;55~#a(Zs+8#(kq0doppe?f#}#;aEg9f)zL(O|#wjDhA&UkHVx}DFtU1 z7==mrMxkh6RA&0`!{s@oKmyK2X7!tX0dCBVSYd94{c|0R`k&q4-g}=`zsHxLDK%mk zmE^JIA73&QbSMEEQM_Qyu8q@5?OPuw?n|TImu2=BzRMnlxX=LIcZEw|bJ@{5<#Lbv za*pbo$=T^fFf(B-AD2ZB!#^4>%R5waN*R zU;YM`7o{msX-L)~Ab&Xo#K5T}a)PwKv<`H_@8=$)%SxKW2d5W)v!hJEkp(1ci-VFn z1|ai_WsLNOC2UNb{Tev88;}?9kViJ4ckR1QA`*wdk>+`0Lr8XHYoh(74k?u;E|Lkd 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+Srinath Setty +Microsoft Research +Abstract. We construct the first folding scheme that simultaneously +achieves six desirable properties: plausible post-quantum security, pay-per- +bit commitment costs, field-native arithmetic (the sum-check and norm +checks run purely over a small field), support for general (non-SIMD) con- +straint systems, small-field support (e.g., Goldilocks), and low recursion +overheads. No existing scheme satisfies all six: group-based schemes (e.g., +HyperNova) lack post-quantum security and are tied to large elliptic- +curve fields; lattice-based schemes (e.g., LatticeFold) require expensive +ring arithmetic, lose pay-per-bit costs, and impose SIMD constraints; and +hash-based schemes (e.g., Arc) incur large verifier circuits. +We present two lattice-based folding schemes for CCS—an NP-complete +relation generalizing R1CS, Plonkish, and AIR—called Neo and SuperNeo. +Neo satisfies five of the six properties but requires SIMD constraint +systems; SuperNeo removes this restriction and satisfies all six. Both run +a single invocation of the sum-check protocol over a small field extension +and achieve pay-per-bit costs via new folding-friendly instantiations of +Ajtai commitments under the Module-SIS assumption. At the core of our +constructions are two new norm-preserving embeddings of field vectors +into ring vectors that respect an evaluation homomorphism required +for folding. We also introduceinteractive reductions, a framework that +generalizes reductions of knowledge and enables modular security proofs +for composed lattice-based protocols. +1 Introduction +A folding scheme [57] is a cryptographic primitive that reduces the task of +checking that two instance-witness pairs are in some NP relation to the task of +checking that a single instance-witness pair is in the same relation. As an example, +for a circuit C and two public inputs (i.e., instances) x1 and x2, a folding scheme +reduces the task of checking that there exist witnesses w1 and w2 such that +C(w1, x1) = 1 and C(w2, x2) = 1 to the task of checking that there exists a single +witness w for a specific public input x such that C(w, x) = 1. Furthermore, the +verifier’s work in a folding scheme is limited to roughly taking the weighted sum +of the commitments to the underlying witnesses. By using a folding scheme in +a recursive manner, one can continually fold many instance-witness pairs into +a single instance-witness pair, providing powerful recursive succinct argument +primitives such as incrementally verifiable computation (IVC) [82] and proof- +carrying data (PCD) [13]. + +Importance of folding schemes: prover efficiency and efficient recursion.A modern +approach to construct SNARKs [47,66] is to combine a polynomial interactive +oracle proof (PIOP) [23,30,78] with a polynomial commitment scheme (PCS) [46]. +However, this yields a “monolithic” SNARK whose prover must prove a fixed-sized +computation at once. To scale to larger computations, one typically breaks the +computation into smaller pieces and uses SNARK recursion (a la IVC or PCD) +to produce a succinct argument [11]. Folding schemes provide a more direct and +efficient alternative: they allow recursion to operate at the “statement” level (i.e., +prior to producing a PIOP or a PCS evaluation argument), yielding two concrete +benefits. First, recursion overheads are far lower: with Nova [57], folding a proof +takes only ≈10,000 R1CS constraints, whereas traditional SNARK recursion +takes millions [29,31]. Second, the prover incurs far less work by avoiding a full +PIOP and PCS evaluation argument: in monolithic SNARKs such as Marlin [30], +the prover performs at least 20× higher work over simply committing to a witness, +whereas with state-of-the-art folding schemes [20,35,55,56], the prover’s work is +dominated by the cost to commit to a witness. This results in at least an order of +magnitude speedup over monolithic SNARKs, and up to two orders of magnitude +when the witness contains values from a small subset of the entire field. 1 +A motivating application: post-quantum signature aggregation.Ethereum’s con- +sensus layer relies on BLS signatures [18], which offer efficient aggregation: a +single pairing check suffices for hundreds of thousands of validator attestations. +However, BLS signatures are not post-quantum secure. Ethereum’s planned +transition to a post-quantum scheme (e.g., the hash-based XMSS [44]) reintro- +duces the scalability challenge, since such schemes lack the algebraic structure +of BLS and their signatures are large and expensive to verify individually. A +natural solution is to use recursive SNARKs to aggregate these signatures [34] +in a distributed manner (a la proof-carrying data). Folding schemes are well +suited here: each signature can be folded into the aggregate proof, yielding sig- +nificantly lower latency and prover cost at each recursive step than traditional +SNARK recursion—a critical advantage when up to a million signatures must be +aggregated within time-sensitive slots of a consensus protocol. +Realizing this approach requires a folding scheme that is both efficientand +plausibly post-quantum secure. More broadly, what properties should a practical +folding scheme satisfy? +1.1 Six desiderata for a practical folding scheme +State-of-the-art folding schemes [20, 35, 53–57] have converged on an efficient +“recipe”: the prover commits to a witness with a linearly homomorphic scheme +1 If the witness contains “small” field elements (e.g., from the set{0,1, . . . ,2 32 −1}), +state-of-the-art folding schemes perform more than two orders of magnitude less work +than a monolithic SNARK prover such as Marlin [30]. Proof systems such as Spartan +and variants [78,79] incur lower overheads than Marlin, but they must still produce +a PIOP and PCS evaluation argument. +2 + +(Pedersen or KZG) and employs sum-check-type techniques [62]. For instance, the +prover in HyperNova [55] performs asinglemulti-scalar multiplication (MSM) +to commit to its witness, with costs that scale with the bit-width of witness +values (“pay-per-bit”). However, these schemes are not post-quantum secure and +are tied to ≈256-bit elliptic-curve fields. Below, we distill six properties that a +practical folding scheme should satisfy. +D1: Post-quantum security.The scheme should be plausibly secure against +quantum adversaries. Group-based folding schemes [20, 35, 55 –57] rely on the +hardness of the discrete logarithm problem, which Shor’s algorithm [80] can +efficiently solve on a quantum computer. +D2: Pay-per-bit commitment costs.The cost to commit to a witness should +scale with the bit-width of the witness values: for example, committing to a +vector of b-bit values should be roughly b times cheaper than committing to +values that span the full field. Group-based schemes achieve this via Pedersen +or KZG commitments. LatticeFold, however, relies on the NTT embedding to +map field vectors into ring vectors. Because the NTT is not a norm-preserving +map, the resulting ring vectors have arbitrary norm regardless of the bit-width +of the original witness elements. Since the Ajtai commitment scheme requires +decomposing these arbitrary-norm ring elements for binding, the commitment +cost is the same whether the original values are 1-bit or 64-bit. Hash-based +schemes (e.g., Arc [24]) also lack this property, since their commitment costs are +determined by the code rate and security parameter, not by witness bit-widths. +D3: Field-native arithmetic.Besides committing to the witness, the prover’s and +the verifier’s work—in particular, the norm check and the sum-check protocol— +should operate natively over a prime field (or extension field), without performing +expensive polynomial ring arithmetic. Group-based and hash-based schemes +satisfy this property. LatticeFold [14], however, runs the sum-check protocol over +a cyclotomic polynomial ring rather than over a prime field (or extension field); +ring operations are 10–100×more expensive than field operations. 2 +D4: General constraint systems.The scheme should support general NP-complete +constraint systems such as CCS [79] over a single witness vector, without requiring +the constraint system to be “data parallel” (SIMD). Group-based and hash-based +schemes satisfy this. LatticeFold, by contrast, packs a batch of independent con- +straints defined over a small prime field into a single constraint over a cyclotomic +polynomial ring [14, Remark 4.1], imposing a SIMD requirement. Lova [37] avoids +this issue but at the cost of only supporting the subset sum relation, not CCS. +D5: Small-field support.The scheme should work natively over small prime +fields, including popular SNARK-friendly fields such as Goldilocks. By “small” +2 Benchmarks report that a polynomial ring multiplication costs ≈213 ns [67], whereas +a field multiplication with M61 costs a fraction of a nanosecond. +3 + +we mean fields whose modulus q fits within a machine register—for example, M61 +(q = 261 −1) or Goldilocks (q = 264 −232 +1). Small-field support is important for +two reasons: such fields offer arithmetic that is an order of magnitude faster than +256-bit arithmetic, and SNARK-friendly fields enable efficient proof compression +using existing SNARKs (e.g., Spartan with a FRI-based polynomial commitment +scheme). Group-based schemes are tied to the scalar fields of elliptic curves, which +are ≈256 bits for security. LatticeFold’s cyclotomic rings of the form X d +1 (with +d a power of 2) cause popular fields like Goldilocks to fully split the ring, ruining +security [14,§3.3]; furthermore, supporting small fields requires a non-trivial +extension field degree t|d , introducing a t× multiplicative overhead in the +protocols due to the need fort-parallel repetition [14,§3.3,§5]. +D6: Low recursion overheads.The recursive verifier circuit should be small +enough that the per-step prover cost of IVC remains practical. Group-based +schemes achieve this: Nova’s and NeutronNova’s verifier circuits are constant-sized: +≈10,000 R1CS constraints. Hash-based schemes suffer from large verifier circuits; +for example, Arc [24] requires 2 ·λ/log (1/ρ) Merkle tree openings, translating to +≈1,600,000 R1CS constraints at λ= 128 and ρ= 1/2 with Poseidon [43]. Prior +to this, B¨ unz et al. [25] provide a different hash-based scheme with even worse +verifier circuit overheads [24, Table 1], and only provides “bounded depth” IVC. +Lova also incurs extreme overheads, reporting a prover time of ≈3,000 seconds for +a subset sum instance of length 2 19 [37, Table 2], compared to 500 ms for Nova +on an R1CS instance of the same size. More broadly, folding schemes that rely on +ring sum-check techniques—such as LatticeFold [14] and SALSAA [59]—inherit +high recursion overheads because the recursive verifier must hash ring elements +rather than field elements. For example, a single ring element in LatticeFold +occupies 64 ×64 = 4,096 bytes, compared to 32 bytes for a 256-bit field element in +HyperNova, resulting in 128× more data for the verifier circuit to hash. Achieving +constant verifier circuit size like Nova or NeutronNova in the lattice setting is +difficult and remains an open problem. Our goal is to achieve logarithmic recursion +overhead (with similar constants) analogous to HyperNova. +Research question. Can we build a folding scheme that satisfies all six desiderata— +in particular, one that is post-quantum secure, works natively over small prime +fields, and matches the efficiency of state-of-the-art group-based schemes? +Figure 1 summarizes the landscape. No existing folding scheme satisfies all six +desiderata. Group-based schemes meet D2–D4 and D6 but fail D1 and D5. Hash- +based schemes achieve D1 but sacrifice D2 and D6. LatticeFold, LatticeFold+, and +SALSAA each achieve D1 but fail D2–D4 and D6; LatticeFold and LatticeFold+ +additionally fail D5. Neo satisfies D1–D3, D5, and D6, but requires SIMD +constraint systems (D4). SuperNeo is the first scheme to satisfy all six. +1.2 Our work: Neo and SuperNeo +We present Neo and SuperNeo, the first folding schemes to satisfy all six desiderata +(Figure 1). Our constructions are lattice analogs of HyperNova [55]: the prover +4 + +D1 D2 D3 D4 D5 D6 +Group-based +HyperNova [55]✗ ✓ ✓ ✓ ✗ ✓ +NeutronNova [56]✗ ✓ ✓ ✓ ✗ ✓ +Hash-based +Arc [24]✓ ✗ ✓ ✓ ✓ ✗ +Lattice-based +LatticeFold [14]✓ ✗ ✗ ✗ ✗ ✗ +Lova [37]✓ ✗ ✓ ✗ ✓ ✗ +Neo(this work)✓ ✓ ✓ ✗ † ✓ ✓ +LatticeFold+ [16]✓ ✗ ✗ ✗ ✗ ✗ +SALSAA [59]✓ ✗ ✗ ✗ ✓ ✗ +SuperNeo(this work)✓ ✓ ✓ ✓ ✓ ✓ +†Neo requires a SIMD constraint system (D4) and is subsumed by SuperNeo, which +removes this requirement. The table lists schemes in the order they appeared; we +present Neo and SuperNeo separately because Neo was independently preprinted and +several subsequent works build on it, so separating the two simplifies attributing +techniques to the correct scheme. +Fig. 1: Comparison of folding schemes against the six desiderata. ✓ = satisfied, +✗= not satisfied. See Section 1.1 for definitions. +commits to a CCS [79] witness using a lattice-based commitment scheme with +pay-per-bit costs, runs a single invocation of the sum-check protocol over an +extension of a small prime field 3, and achieves plausible post-quantum security +under a standard structured lattice assumption (Module-SIS). Both schemes also +provide multi-folding, which folds multiple CCS instances at once, amortizing +the decomposition costs required to manage lattice norm growth. By applying +standard compilers from folding schemes to IVC [55, 57] and PCD [85], we +obtain a plausibly post-quantum IVC/PCD scheme. Since our constructions +natively support SNARK-friendly fields like Goldilocks, they enable efficient proof +compression using Spartan [78, 79] with a FRI-based polynomial commitment +scheme [10]—without requiring any non-native arithmetic or field emulation. +The key technical challenge isembeddingfield vectors (CCS witnesses) into the +ring vectors that Ajtai commitments operate over, while preserving the algebraic +structure—norm bounds, evaluation homomorphisms—that a sum-check-based +folding scheme like HyperNova requires. We introduce two new norm-preserving +embeddings (theNeoandSuperNeo embeddings) that resolve this challenge, +and a new security framework (interactive reductions) that enables modular +proofs of knowledge soundness for lattice-based protocols. We detail the problems +with prior approaches and our solutions in the following sections. +3 When using a 64-bit field, a degree-2 extension is sufficient for 128 bits of security. +5 + +1.2.1 Challenges and prior solutionsAchieving desiderata D2–D4 and D6 +for a lattice-based folding scheme requires solving a common problem: how to +embed field vectors into the ring vectors that Ajtai commitments [2] operate +over. Any such embedding must support protocols that check both the norm of +the committed ring vectors and CCS constraints on the underlying field vectors. +Below, we describe the Ajtai commitment scheme and the challenges that prior +solutions leave open. +Ajtai Commitment Scheme [2,75] (Informal) 4 +–Setup (1κ, n∈N ) →A∈ Rκ×n +F , where A is a random matrix over the +polynomial ring R F := F [X]/(ϕ(X)) with modulus ϕ(X) being a cyclotomic +polynomial of degreed. +–Commit (A,z∈ Rn +F ) →c , where c := A·z∈ Rκ +F is a binding commitment to +messagezif the norm∥z∥ ∞ < bis small enough. +Problem 1: Inefficient constraint checking & lack of algebraically +friendly embeddingsPrior work on lattice-based folding schemes [15] re- +lied on the Number Theoretic Transform (NTT) [5, 61, 75, 77] to embed field +vectors into ring vectors. The Number Theoretic Transform is a ring isomorphism +between the ring R F and a product space of extension fields F d/t +qt , so each ring +operation naturally simulates a Single Instruction, Multiple Data (SIMD) [40,41] +operation over the underlying (d/t)-tuple of field elements. For· ∈ {+,×}, +(a1, . . . , ad/t)∈F d/t +qt a∈R F +(a1 ·b 1, . . . , ad/t ·b d/t)∈F d/t +qt a·b∈R F +·(b1, . . . , bd/t) ·b +NTT +iNTT +NTT +iNTT +Hence, d/t field vectors z(1), . . . , z(d/t) ∈F n can be embedded into a single ring +vector z∈ Rn +F . By adapting a technique from [19], prior work Latticefold [15, Sec +3.3] showed how to express the norm constraint on a ring vector z as Hadamard +product constraints Q +i (q− 1)/2, then ∥a∥∞ = |a′ −q| . For a vector +z∈F n, we define the ℓ∞-norm ∥z∥∞ to be the max infinity norm of its elements. +For an element a∈ RF , we define ∥a∥∞ to be the ℓ∞-norm of the vector cf(a). For +a vectorz∈R m +F , we define∥z∥ ∞ to be the maximumℓ ∞-norm of its elements. +20 + +DecompositionWe define splitb : F m → (F m)∗ to be the b-ary decomposition +map, which performs the b-ary decomposition of a vector z∈F m into vectors +z1, z2, . . . , z∗. For example, ifz∈F m such that and∥z∥ ∞ < b k, then we have +splitb(z) := (z1, z2, . . . zk)such thatz= +kX +i=1 +bi−1 ·z i and∥z i∥∞ < b +Definition 4 (Ring Commitment Scheme).A ring commitment scheme +com:= (Setup,Commit)consists of two PPT algorithms: +–Setup (1λ, m) →pp : Takes as input a security parameter1 λ and length +m∈N ≥1, outputs public parameterspp. +–Commit (pp, z ) →c : Takes as input public parameters pp and a vector z∈ Rm +F , +outputs a commitmentc∈C. +A ring commitment scheme can satisfy the following properties: +B-binding:For every length m = poly(λ)and every EPT adversary A, a ring +commitment scheme isB-binding (forB∈N) if the following probability holds: +Pr + + +Commit(pp, z1) =Commit(pp, z 2) +∧ ∥z 1∥∞,∥z 2∥∞ < B, +∧z 1 ̸=z 2 + +pp←Setup(1 λ, m) +z1, z2 ∈R m +F ← A(pp) + + ≤ϵ bind(B) +for ϵbind(B) ≤negl (λ). We refer to a pair of vectors( z1, z2)which satisfies the +conditions in the probability as aB-binding collision. +(B,C )-relaxed binding:For every length m = poly(λ)and every EPT adversary +A, a ring commitment scheme is( B,C )-relaxed binding (for B∈N and set +C ⊆R F ) if the following probability holds: +Pr + + +∆1 ·c=Commit(pp, z 1) +∧∆ 2 ·c=Commit(pp, z 2) +∧ ∥z 1∥∞,∥z 2∥∞ < B, +∧∆ 1z2 ̸=∆ 2z1 + +pp←Setup(1 λ, m) + + +c∈C, +∆1, ∆2 ∈(C − C), +z1, z2 ∈R m +F + + ← A(pp) + + ≤ϵ rlx(B,C) +for ϵrlx(B,C ) ≤negl (λ). We refer to a tuple of elements( c, ∆1, ∆2, z1, z2)which +satisfies the conditions in the probability as a( B,C )-relaxed binding collision. +Homomorphic:For every m∈N and pp∈Setup (1λ, m), the commitment +algorithm,Commit(pp,·) : R m +F →C, is anR F -module homomorphism. +Theorem 2 (Properties [2,7,9,14]).The Ajtai commitment scheme (Defi- +nition 18) is a ring commitment scheme (Definition 4) that ishomomorphic, +B-binding(assuming the MSIS∞,κ,q +m,2B problem (Definition 16) is hard), and( B,C )- +relaxed binding(assuming the MSIS∞,κ,q +m,4T B problem is hard and C is a strong +sampling set (Definition 17) with expansion factorT(Theorem 9)). +21 + +Definition 5 (Interactive Reductions [50, 52]).Consider relations R1 and +R2 over parameters, structure, instance, and witness tuples. Aninteractive +reductionfrom R1 to R2 is defined by PPT algorithms( G,K,P,V )called the +generator, encoder, prover, and verifier respectively with the following interface. +–G(1 λ,sz)→pp: Takes as input a security parameter1 λ and size parameters +sz. Outputs public parameterspp. +–K (pp, s) → (pk,vk ): Takes as input public parameters pp and a structures. +Deterministically, outputs a prover keypkand a verifier keyvk. +–P (pk, u1, w1) → (u2, w2): Takes as input a proving key pk and an instance- +witness pair( u1, w1). Interactively reduces the task of checking(pp, s, u1, w1) ∈ +R1 to the task of checking(pp,s, u 2, w2)∈ R 2. +–V (vk, u1) →u 2: Takes as input a verifier key vk and an instance u1 in +R1. Interactively reduces the task of checking the instance u1 to the task of +checking a new instanceu 2 inR 2. +Let ⟨P,V⟩ denote the interaction between P and V. We treat ⟨P,V⟩ as a function +that takes as input(( pk,vk ), u1, w1)and runs the interaction on the prover’s input +(pk, u1, w1)and the verifier’s input( vk, u1). At the end of the interaction, ⟨P,V⟩ +outputs the verifier’s instanceu 2 and the prover’s witnessw 2. +Areduction of knowledge[51] is an interactive reduction,( G,K,P,V ), +that satisfies the following properties: +(i) Completeness: For any EPT adversary A, given pp← G (1λ,sz ),(s , u1, w1) +← A (pp)such that( pp, s, u1, w1) ∈ R 1, we have that the prover’s output +instance is equal to the verifier’s output instanceu 2, and that +(pp,s,⟨P,V⟩((pk,vk), u 1, w1))∈ R 2. +(ii) Knowledge soundness: For any EPT adversary( A,P ∗), there exists an +EPT extractorEsuch that if the success probability of the adversary +ϵ(A,P ∗) := Pr +" +(pp,s,⟨P ∗,V⟩((pk,vk), u 1,st))∈ R 2 + +pp← G(1 λ,sz) +(s, u1,st)← A(pp) +(pk,vk)← K(pp,s) +# +≥1/poly(λ),then we have that +Pr + +(pp,s, u 1, w1)∈ R 1 + +pp← G(1 λ,sz) +(s, u1,st)← A(pp) +(pk,vk)← K(pp,s) +w1 ← E(pp,s, u 1,st) + + ≥ϵ(A,P ∗)−negl(λ). +(iii) Public Coin: All of the verifier’s messages are uniformly random strings of +some prescribed length. Furthermore, the verifier’s messages contain all of +the random coins (randomness) used by the verifier. 7 +7 If a reduction of knowledge is public-coin, then it trivially satisfies the property +ofpublic reducibilitydescribed in [52] as the execution of the verifier V can be +emulated using the randomness from the transcript. +22 + +Lemma 2 (Sequential composition [50,52]).For reductions of knowledge +Π1 = (G,K,P 1,V 1) : R1 → R 2 andΠ 2 = (G,K,P 2,V 2) : R2 → R 3, we have that +Π2 ◦ Π1 = (G,K,P,V ) : R1 → R 3 is a reduction of knowledge where K(pp, s) +computes(pk,vk)and where +P(pk, u1, w1) =P 2(pk,P 1(pk, u1, w1)) +V(vk, u1) =V 2(vk,V 1(vk, u1, w1)) +Definition 6 (The sum-check protocol [62]).The sum-check protocol +SumCheck(T;Q) is a classic interactive proof protocol between two PPT al- +gorithms( P,V )that checks that the sum of evaluations of a ℓ-variate polynomial +Q∈F ≤d[X1, . . . , Xℓ]on the Boolean hypercube results in some claimed valueT. +The output of the sum-check protocol is a claim that v +? +=Q (r)for some random +point r∈F ℓ and claimed evaluations v, which the verifier V can query Q to check. +The protocol is public-coin, has a completeness error of0, and has a soundness +error of ≤ℓd/|F| . More generally, the field can be chosen to be an extension field +K. In this case, the soundness error is ≤ℓd/|K| . A self-contained description of +the sum-check protocol can be found in this note [81]. +5 Embedding products with evaluation homomorphism +Here, we define a bijective embedding from the fieldFinto the ring R F . +Definition 7 (Coefficient Embedding). +Element embedding:Consider a vector v∈F d. We define v∈ RF (in bold font) +to be the ring element whose coefficient vector isv, i.e.cf(v) =v. +Vector embedding:Recall that we define nF = d·n R. Hence, for a field vector +z∈F nF , we have a natural partition into d-sized sub-vectors z = [z1, . . . , znR]. We +define the ring vector z := (z1, . . . ,znR) ∈ RnR +F to be the vector of ring elements, +which are the embeddings of then R =n F /dfield sub-vectors. +Matrix embedding:For a matrix M∈F m×nF with rows M1, . . . , M m ∈F nF , we +define M := JM1, . . . ,M mK∈ Rm×nR +F , which is the vertical concatenation of all +the embedded rows. +Inverse embedding:Similarly, given a ring vector v∈ RnR +F or ring matrix M∈ +Rm×nR +F , we define the field vector v∈F nF or field matrix M∈F m×nF as the +inverse of previously defined coefficient embeddings. +Theorem 3 (Inner Product Transform [36, 64]).There exists a linear +transform ·:F d →F d such that for alla, b∈F d, we have the constant term +ct(¯a·b) =⟨a, b⟩ +where¯adenotes applying the transform toaand embedding¯ainto the ring. +Here, we define an extension of the inner product transform · : F d →F d +(Theorem 3) to vectors and matrices. +23 + +Definition 8 (Lifting the Transform). +Vector Transform:Consider a vector v∈F nF , we can partition v into d-sized sub- +vectors[v 1, . . . , v nR]. We define ·:F nF →F nF to be¯v:= [¯v1, . . . ,¯vnR]∈F nF . +Matrix Transform:Consider a matrix M∈F m×nF with rows M1, . . . , M m ∈F nF . +We define ·:F m×nF →F m×nF to be ¯M:= +q ¯M1, . . . , ¯Mm +y +∈F m×nF . +Remark 1 (Efficiency and Sparsity Preservation).When the cyclotomic polyno- +mial ϕ(X) is a power-of-two cyclotomic or a trinomial cyclotomic, the transform +· : F nF →F nF essentially only involves permuting and adding entries of the input +vector, and hence can be computed in O(nF) time. Since the · is linear, if the +original matrixMis sparse, then the transformed matrix ¯Mis also sparse. +Theorem 4 (Matrix-Vector Product Transform).Consider an arbitrary +matrix M∈F m×nF and vector z∈F nF . The matrix-vector product M z∈F m is +equal to the constant terms of the matrix-vector product ¯M z∈ Rm +F , when viewing +each ring element as a polynomial. More succinctly,M z=ct( ¯M z). +Proof.For brevity, we defer the proof to Section D.1. +Remark 2 (Matrix-vector Product Evaluation).Consider an arbitrary vector +z∈F nF , matrix M∈F m×nF , and multilinear evaluation point r∈K logm . Define +the evaluation y := ¯M zi +∼ +(r) = ⟨ ¯M zi,ˆr⟩ ∈RK. Observe that multiplying a ring +element in R F with an extension field element in K scales each coefficient of the +ring element by the extension field element. Hence, by Definition 2, we must +have that for all ℓ∈ [d], cf(y) ℓ = cf + ¯M zi + +ℓ +∼ +(r) ∈K (i.e. the ℓ-th coefficient of +y is equal to the multilinear evaluation of the ℓ-th coefficient vector of ¯M zi at +point r). Since ct(y) = cf(y) 1 and ct + ¯M zi + += cf + ¯M zi + +1, by Theorem 4, we can +observe that the constant term ct(y) = M z∼ (r) ∈K is exactly the multilinear +evaluation of the field vectorM zat pointr. +Theorem 5 (Evaluation Homomorphism).Consider an arbitrary matrix +M∈F m×nF , vectors z1, . . . , zℓ ∈F nF , scalars ρ1, . . . , ρℓ ∈ RF , and evaluation +point r∈K logm . Let L : RnR +F →C,L in : RnR +F → RnR,in +F be arbitraryR F -module +homomorphisms. For alli∈[ℓ], define +ci :=L(z i)∈Cx i :=L in(zi)∈R nR,in +F yi := ¯M zi +∼ +(r)∈R K +Additionally, define +c:= +X +i∈[ℓ] +ρici ∈C,x:= +X +i∈[ℓ] +ρixi ∈R nR,in +F , +z:= +X +i∈[ℓ] +ρizi ∈R nR +F , y:= +X +i∈[ℓ] +ρiyi ∈R K +We must have that c = L(z)and y = ¯M z +∼ +(r). Additionally, for all i∈ [ℓ], +ct(yi) =M z i +∼ (r)andct(y) =M z∼ (r). +Proof.For brevity, we defer the proof to Section D.2. +24 + +6 Strong and weak interactive reductions +Definition 9 (Weak Interactive Reductions).Consider relations R1, R′ +1, +and R2 over public parameters, structure, instance, and witness tuples such that +R1 ⊆ R ′ +1. LetU 1 be the ambient instance space ofR 1. +An interactive reductionΠ : R1 → R 2, defined by PPT algorithms( G,K,P,V ) +(Definition 5), isweakif it is complete, public coin, and there exists a function +ϕ : U1 →C (for an arbitrary space C) such that for any EPT adversary( A,P ∗), +there exists an EPT extractor E such that if the success probability of the adversary +ϵ(A,P ∗)≥1/poly(λ),then +Pr + +(pp,s, u 1, w1)∈ R ′ +1 + +pp← G(1 λ,sz) +(s, u1,st)← A(pp) +(pk,vk)← K(pp,s) +w1 ← E(pp,s, u 1,st) + + ≥ϵ(A,P ∗)−negl(λ). +and ifA:= (B,B ′)such that +Pr + + +u1, u′ +1 ̸=⊥ +⇓ +ϕ(u1) =ϕ(u ′ +1) + +pp← G(1 λ,sz) +(s,st ∗)← B(pp) +(u1,st)← B ′(st∗) +(u′ +1,st ′)← B ′(st∗) + + = 1, +then +Pr + + +w1, w′ +1 ̸=⊥ +∧w 1 ̸=w ′ +1 + +pp← G(1 λ,sz) +(s,st ∗)← B(pp) +(u1,st)← B ′(st∗) +w1 ← E(pp,s, u 1,st) +(u′ +1,st ′)← B ′(st∗) +w′ +1 ← E(pp,s, u ′ +1,st ′) + + +≤negl(λ) +Definition 10 (Strong Interactive Reductions).Consider relations R1, +R2, and R′ +2 over public parameters, structure, instance, and witness tuples such +thatR 2 ⊆ R ′ +2. LetU 2 be the ambient instance space ofR 2. +An interactive reductionΠ : R1 → R 2, defined by PPT algorithms( G,K,P,V ) +(Definition 5), isstrongif it is complete, public coin, and there exists a function +ϕ:U 2 →C(for an arbitrary spaceC) such that +(i) For any EPT adversary(A,P ∗), +Pr + + +u2, u′ +2 ̸=⊥ +⇓ +ϕ(u2) =ϕ(u ′ +2) + +pp←Gen(1 λ) +(s, u1,st 1)← A(pp) +(pk,vk)← K(pp,s) +(u2, w2)← ⟨P ∗,V⟩((pk,vk), u 1,st) +(u′ +2, w′ +2)← ⟨P ∗,V⟩((pk,vk), u 1,st) + + = 1 +(ii) For any EPT adversary( A,P ∗), there exists an EPT extractor E such that if +ϵ′(A,P ∗) := Pr +" +(pp,s,⟨P ∗,V⟩((pk,vk), u 1,st))∈ R ′ +2 + +pp← G(1 λ,sz) +(s, u1,st)← A(pp) +(pk,vk)← K(pp,s) +# +25 + +≥1/poly(λ),and +Pr + + +w2, w′ +2 ̸=⊥ +∧ +w2 ̸=w ′ +2 + +pp←Gen(1 λ) +(s, u1,st)← A(pp) +(pk,vk)← K(pp,s) +(u2, w2)← ⟨P ∗,V⟩((pk,vk), u 1,st) +(u′ +2, w′ +2)← ⟨P ∗,V⟩((pk,vk), u 1,st) + + ≤negl(λ) +then we have that +Pr + +(pp,s, u 1, w1)∈ R 1 + +pp← G(1 λ,sz) +(s, u1,st)← A(pp) +(pk,vk)← K(pp,s) +w1 ← E(pp,s, u 1,st) + + ≥ϵ ′(A,P ∗)−negl(λ). +Theorem 6 (Strong-Weak Composition).Consider relations R1, R2, R′ +2 +and R3 over public parameters, structure, instance, and witness tuples such that +R2 ⊆ R ′ +2. Let U2 be the ambient instance space of R2. Consider interactive +reductions (Definition 5)Π 1 :R 1 → R 2 (R′ +2),Π 2 :R 2 (R′ +2)→ R 3 such that +(i)Π 1 isstrong(Definition 10) with respect to a functionϕ:U 2 →Cand +(ii)Π 2 isweak(Definition 9) with respect to thesamefunctionϕ, +then the sequential compositionΠ 2 ◦Π1 : R1 → R 3 is areduction of knowledge. +Proof.For brevity, we defer the proof to Section D.3. +7 Neo’s folding scheme for CCS +7.1 Relations +Definition 11 (Structure).We define astructureas a collection of elements +s:= +n +Mj ∈F m×nF + +j∈[t] , f∈F = (2^(2 * sqrt( n*d * log(q,2) * log(delta, 2)))).n() +# [AL21] eprint Prop 2. 2021/202 +# for all u,v in R, |u*v| / |v| <= gamma*|u| +# outputs T = gamma * |u| +# assumes we are only testing prime powers +def expansion_factor(m, norm): +if is_pow2(m): +return euler_phi(m) * norm +else: +return 2 * euler_phi(m) * norm +56 + +# p is prime +# max_idx is max cyclotomic index +# outputs list of (m, z) +def candidates(p, min_idx=10, max_idx=200): +# prime powers +possible_indices = [i for i in range(min_idx, max_idx) if len(factor(i)) == 1] +c = list() +for m in possible_indices: +zs = divisors(m) +for z in zs: +if thm1_1_cond(m, p, z): +c.append((Integer(m), Integer(z))) +return c +def pre_filter(q, cyclotomic_index, z, n, m, chals): +chals_norm = max({abs(c) for c in chals}) +chals_max_diff = chals[-1] - chals[0] +delta = 1.0045 # root hermite factor, chosen from [ESSLL19] eprint 2018/773 +phi = cyclotomic_polynomial(cyclotomic_index) # index cyclotomic polynomial +d = phi.degree() # degree of cyclotomic +# return non_trivial(q, n, d, delta) and chals_max_diff < thm1_1_inv_bound(q, z) and log(len(chals)^d,2).n() >= 120 +# We remove non_trivial(...) because we use the lattice estimator for hardness +return chals_max_diff < thm1_1_inv_bound(q, z) and log(len(chals)^d,2).n() >= 120 +def info(q, cyclotomic_index, z, n, m, chals): +chals_norm = max({abs(c) for c in chals}) +chals_max_diff = chals[-1] - chals[0] +delta = 1.0045 # root hermite factor, chosen from [ESSLL19] eprint 2018/773 +phi = cyclotomic_polynomial(cyclotomic_index) # index cyclotomic polynomial +d = phi.degree() # degree of cyclotomic +T = expansion_factor(cyclotomic_index, chals_norm) +# Bounds for MSIS to be hard +# [MR09] lattice-based cryptography pg 6 +# [CMNW24] pg 38 eprint 2024/281 +MSIS_B_l2_bound = min(q, 2^(2 * sqrt( n*d * log(q,2) * log(delta, 2)))) +MSIS_B_linf_bound = MSIS_B_l2_bound / sqrt(m*d) +# We need MSIS infinity bound 8TB to be hard +B = MSIS_B_linf_bound / (8*T) +print("####") +print("Cyclotomic idx:", cyclotomic_index) +print("Cyclotomic Poly:", phi) +print("z:", z) +#print("Prime is non-trivial?", non_trivial(q, n, d, delta)) +print("Csmall norm is small enough?", chals_max_diff < thm1_1_inv_bound(q, z)) +print("Csmall large enough?", log(len(chals)^d,2).n() >= 120) +57 + +print("Degree of Cyclotomic:", d) +# print("log(B):", log(B, 2).n()) +print("Expansion Factor T:", T) +print("Invertible Norm bound:", thm1_1_inv_bound(q, z)) +print("log(|C_Small|):", log(len(chals)^d,2).n()) +print("Factors of Cyclotomic:", thm1_1_num_factors(z)) +print() +def possible_settings(q, n, m, chals): +for (cyclotomic_index, z) in candidates(q): +if pre_filter(q, cyclotomic_index, z, n, m, chals): +info(q, cyclotomic_index, z, n, m, chals) +else: +delta = 1.0045 +d = cyclotomic_polynomial(cyclotomic_index).degree() +print("[Does not satisfy security requirements] index: {}, degree: {}, z: {}, non_trivial: {}, log(|C_Small|): {}, Invertible Norm bound: {}".format(cyclotomic_index, d, z, non_trivial(q, n, d, delta), log(len(chals)^d,2).n(), thm1_1_inv_bound(q, z))) +# Primes: +GL = 2^64 - 2^32 + 1 +AGL = GL - 32 +print("###############################") +print("AGL ###############################") +print("###############################") +# MSIS settings +n = 13 # rows, kappa in latticefold +m = 2^26 # cols +# Small Challenge set +chals = [-1, 0, 1, 2] +possible_settings(AGL, n, m, chals) +print("###############################") +print("M61 ###############################") +print("###############################") +# MSIS settings +n = 16 # rows, kappa in latticefold +m = 2^22 # cols +# Small Challenge set +chals = [-2, -1, 0, 1, 2] +possible_settings(2^61-1, n, m, chals) +print("###############################") +print("GL ###############################") +print("###############################") +# MSIS settings +n = 16 # rows, kappa in latticefold +m = 2^24 # cols +# Small Challenge set +chals = [-2, -1, 0, 1, 2] +possible_settings(GL, n, m, chals) +print("###############################") +58 + +D.8 Lattice Estimator Script +from estimator import * +Logging.set_level(Logging.LEVEL0) +M61 = 2^61 -1 +GL = 2^64 - 2^32 +1 +AGL = GL - 32 +b = 2 +n = 15 +d = 64 +k = 13 +K = 26 +B = b^k +m = 2^33 / d +T = 128 +q = AGL +n_sis = n*d +m_sis = m*d +B_l2 = sqrt(m*d)*(8*T*B) +params = SIS.Parameters(n=n_sis, q=q, m=m_sis,length_bound=B_l2, norm=2) +_ = SIS.estimate(params) +print((K+k)*T*(b-1) < B) +n = 18 +d = 54 +k = 14 +K = 61 +B = b^k +m = 2^30 / d +T=216 +q = GL +n_sis = n*d +m_sis = m*d +B_l2 = sqrt(m*d)*(8*T*B) +params = SIS.Parameters(n=n_sis, q=q, m=m_sis,length_bound=B_l2, norm=2) +_ = SIS.estimate(params) +print((K+k)*T*(b-1) < B) +n = 18 +d = 54 +k = 14 +K = 61 +59 + +B = b^k +m = 2^28 / d +T = 216 +q = M61 +n_sis = n*d +m_sis = m*d +B_l2 = sqrt(m*d)*(8*T*B) +params = SIS.Parameters(n=n_sis, q=q, m=m_sis,length_bound=B_l2, norm=2) +_ = SIS.estimate(params) +print((K+k)*T*(b-1) < B) +60 \ No newline at end of file diff --git a/src/protocol/commitment/scheme.rs b/src/protocol/commitment/scheme.rs index e59381f3..e6ccd601 100644 --- a/src/protocol/commitment/scheme.rs +++ b/src/protocol/commitment/scheme.rs @@ -195,20 +195,28 @@ where let layout = Self::layout(setup)?; let num_blocks = layout.num_blocks; let block_len = layout.block_len; - let expected_len = num_blocks + let max_len = num_blocks .checked_mul(block_len) .ok_or_else(|| HachiError::InvalidSetup("coefficient length overflow".to_string()))?; - if f_coeffs.len() != expected_len { + if f_coeffs.len() > max_len { return Err(HachiError::InvalidSize { - expected: expected_len, + expected: max_len, actual: f_coeffs.len(), }); } - let blocks: Vec>> = f_coeffs - .chunks_exact(block_len) - .map(|chunk| chunk.to_vec()) - .collect(); + let mut blocks: Vec>> = Vec::with_capacity(num_blocks); + for i in 0..num_blocks { + let start = i * block_len; + if start >= f_coeffs.len() { + blocks.push(vec![CyclotomicRing::::zero(); block_len]); + } else { + let end = (start + block_len).min(f_coeffs.len()); + let mut block = f_coeffs[start..end].to_vec(); + block.resize(block_len, CyclotomicRing::::zero()); + blocks.push(block); + } + } Self::commit_ring_blocks(&blocks, setup) } diff --git a/src/protocol/commitment_scheme.rs b/src/protocol/commitment_scheme.rs index a5da204c..1294cce5 100644 --- a/src/protocol/commitment_scheme.rs +++ b/src/protocol/commitment_scheme.rs @@ -110,16 +110,18 @@ where actual: opening_point.len(), }); } - let outer_point = &opening_point[alpha..]; let layout = >::layout(setup)?; - let expected_outer = layout.r_vars + layout.m_vars; - if outer_point.len() != expected_outer { + let target_num_vars = layout.m_vars + layout.r_vars + alpha; + if opening_point.len() > target_num_vars { return Err(HachiError::InvalidPointDimension { - expected: expected_outer + alpha, + expected: target_num_vars, actual: opening_point.len(), }); } + let mut padded_point = opening_point.to_vec(); + padded_point.resize(target_num_vars, F::zero()); + let outer_point = &padded_point[alpha..]; let ring_opening_point = ring_opening_point_from_field::(outer_point, layout.r_vars, layout.m_vars)?; @@ -128,7 +130,7 @@ where // Fiat-Shamir: bind commitment, opening point, and y_ring before any challenges. commitment.append_to_transcript(ABSORB_COMMITMENT, transcript); - for pt in opening_point { + for pt in &padded_point { transcript.append_field(ABSORB_EVALUATION_CLAIMS, pt); } transcript.append_serde(ABSORB_EVALUATION_CLAIMS, &y_ring); @@ -186,12 +188,16 @@ where "opening point length underflow".to_string(), )); } - let inner_point = &opening_point[..alpha_bits]; - let reduced_opening_point = &opening_point[alpha_bits..]; + let layout = setup.expanded.seed.layout; + let target_num_vars = layout.m_vars + layout.r_vars + alpha_bits; + let mut padded_point = opening_point.to_vec(); + padded_point.resize(target_num_vars, F::zero()); + let inner_point = &padded_point[..alpha_bits]; + let reduced_opening_point = &padded_point[alpha_bits..]; // Fiat-Shamir: bind commitment, opening point, and y_ring before any challenges. commitment.append_to_transcript(ABSORB_COMMITMENT, transcript); - for pt in opening_point { + for pt in &padded_point { transcript.append_field(ABSORB_EVALUATION_CLAIMS, pt); } transcript.append_serde(ABSORB_EVALUATION_CLAIMS, &proof.y_ring); @@ -206,7 +212,6 @@ where } // §4.2 Quadratic equation - let layout = setup.expanded.seed.layout; let ring_opening_point = ring_opening_point_from_field::( reduced_opening_point, layout.r_vars, @@ -600,7 +605,7 @@ fn evaluate_packed_ring_poly( outer_point: &[F], ) -> CyclotomicRing { let weights = lagrange_weights(outer_point); - debug_assert_eq!(weights.len(), packed_coeffs.len()); + debug_assert!(weights.len() >= packed_coeffs.len()); #[cfg(feature = "parallel")] { packed_coeffs From ce8eca83312604278506f00746bc6a9cbbe0e449 Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Sun, 1 Mar 2026 22:58:29 -0800 Subject: [PATCH 23/34] feat: add HachiSerialize impls for proof types Implement HachiSerialize/HachiDeserialize for HachiProof, HachiCommitmentHint, and SumcheckAux so they can be serialized through the ArkBridge adapter in Jolt. Made-with: Cursor --- src/protocol/proof.rs | 119 +++++++++++++++++++++++++++++++++++++++++- 1 file changed, 117 insertions(+), 2 deletions(-) diff --git a/src/protocol/proof.rs b/src/protocol/proof.rs index 4a119329..c505c6a8 100644 --- a/src/protocol/proof.rs +++ b/src/protocol/proof.rs @@ -1,10 +1,12 @@ //! Proof structures for the Hachi protocol. use crate::algebra::ring::CyclotomicRing; -use crate::primitives::serialization::Compress; +use crate::primitives::serialization::{Compress, SerializationError}; +use crate::primitives::serialization::{Valid, Validate}; use crate::protocol::commitment::RingCommitment; use crate::protocol::sumcheck::SumcheckProof; -use crate::{FieldCore, HachiSerialize}; +use crate::{FieldCore, HachiDeserialize, HachiSerialize}; +use std::io::{Read, Write}; /// Prover-side hint produced at commitment time. #[derive(Debug, Clone, PartialEq, Eq)] @@ -54,3 +56,116 @@ impl HachiProof { + self.w_commitment.serialized_size(Compress::No) } } + +impl HachiSerialize for SumcheckAux { + fn serialize_with_mode( + &self, + writer: W, + compress: Compress, + ) -> Result<(), SerializationError> { + self.w.serialize_with_mode(writer, compress) + } + fn serialized_size(&self, compress: Compress) -> usize { + self.w.serialized_size(compress) + } +} + +impl Valid for SumcheckAux { + fn check(&self) -> Result<(), SerializationError> { + Ok(()) + } +} + +impl HachiDeserialize for SumcheckAux { + fn deserialize_with_mode( + reader: R, + compress: Compress, + validate: Validate, + ) -> Result { + Ok(Self { + w: Vec::::deserialize_with_mode(reader, compress, validate)?, + }) + } +} + +impl HachiSerialize for HachiCommitmentHint { + fn serialize_with_mode( + &self, + mut writer: W, + compress: Compress, + ) -> Result<(), SerializationError> { + self.s.serialize_with_mode(&mut writer, compress)?; + self.t_hat.serialize_with_mode(&mut writer, compress)?; + self.ring_coeffs.serialize_with_mode(&mut writer, compress) + } + fn serialized_size(&self, compress: Compress) -> usize { + self.s.serialized_size(compress) + + self.t_hat.serialized_size(compress) + + self.ring_coeffs.serialized_size(compress) + } +} + +impl Valid for HachiCommitmentHint { + fn check(&self) -> Result<(), SerializationError> { + Ok(()) + } +} + +impl HachiDeserialize for HachiCommitmentHint { + fn deserialize_with_mode( + mut reader: R, + compress: Compress, + validate: Validate, + ) -> Result { + Ok(Self { + s: Vec::deserialize_with_mode(&mut reader, compress, validate)?, + t_hat: Vec::deserialize_with_mode(&mut reader, compress, validate)?, + ring_coeffs: Vec::deserialize_with_mode(&mut reader, compress, validate)?, + }) + } +} + +impl HachiSerialize for HachiProof { + fn serialize_with_mode( + &self, + mut writer: W, + compress: Compress, + ) -> Result<(), SerializationError> { + self.y_ring.serialize_with_mode(&mut writer, compress)?; + self.v.serialize_with_mode(&mut writer, compress)?; + self.sumcheck_proof + .serialize_with_mode(&mut writer, compress)?; + self.sumcheck_aux + .serialize_with_mode(&mut writer, compress)?; + self.w_commitment.serialize_with_mode(&mut writer, compress) + } + fn serialized_size(&self, compress: Compress) -> usize { + self.y_ring.serialized_size(compress) + + self.v.serialized_size(compress) + + self.sumcheck_proof.serialized_size(compress) + + self.sumcheck_aux.serialized_size(compress) + + self.w_commitment.serialized_size(compress) + } +} + +impl Valid for HachiProof { + fn check(&self) -> Result<(), SerializationError> { + Ok(()) + } +} + +impl HachiDeserialize for HachiProof { + fn deserialize_with_mode( + mut reader: R, + compress: Compress, + validate: Validate, + ) -> Result { + Ok(Self { + y_ring: CyclotomicRing::deserialize_with_mode(&mut reader, compress, validate)?, + v: Vec::deserialize_with_mode(&mut reader, compress, validate)?, + sumcheck_proof: SumcheckProof::deserialize_with_mode(&mut reader, compress, validate)?, + sumcheck_aux: SumcheckAux::deserialize_with_mode(&mut reader, compress, validate)?, + w_commitment: RingCommitment::deserialize_with_mode(&mut reader, compress, validate)?, + }) + } +} From c997b60713aa13364cc6dba03406ae8ad5155414 Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Sun, 1 Mar 2026 23:06:52 -0800 Subject: [PATCH 24/34] fix: relax balanced_decompose_pow2 assertion for 128-bit fields MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Allow levels * log_basis up to 128 + log_basis. For Fp128 with LOG_BASIS=4, the decomposition needs 33 levels (132 bits total) because 32 levels can't represent the full signed range [-q/2, q/2). The extra level's digit is at most ±1 and the i128 arithmetic remains safe since the quotient shrinks monotonically. Made-with: Cursor --- src/algebra/ring/cyclotomic.rs | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/src/algebra/ring/cyclotomic.rs b/src/algebra/ring/cyclotomic.rs index acbfc403..7ee92aa2 100644 --- a/src/algebra/ring/cyclotomic.rs +++ b/src/algebra/ring/cyclotomic.rs @@ -243,8 +243,8 @@ impl CyclotomicRing { pub fn balanced_decompose_pow2(&self, levels: usize, log_basis: u32) -> Vec { assert!(log_basis > 0 && log_basis < 128, "invalid log_basis"); assert!( - (levels as u32).saturating_mul(log_basis) <= 128, - "levels * log_basis must be <= 128" + (levels as u32).saturating_mul(log_basis) <= 128 + log_basis, + "levels * log_basis must be <= 128 + log_basis" ); let b = 1i128 << log_basis; From c6a6a4e9b0392fb26f905a517d99ce41c5968d2a Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Sun, 1 Mar 2026 23:41:23 -0800 Subject: [PATCH 25/34] feat: add DynamicSmallTestCommitmentConfig Same D=16 security parameters as SmallTestCommitmentConfig but derives layout from max_num_vars instead of using a fixed (4,2) shape. Made-with: Cursor --- src/protocol/commitment/config.rs | 37 +++++++++++++++++++++++++++++++ src/protocol/commitment/mod.rs | 4 ++-- src/protocol/mod.rs | 10 ++++----- 3 files changed, 44 insertions(+), 7 deletions(-) diff --git a/src/protocol/commitment/config.rs b/src/protocol/commitment/config.rs index d1187d85..3eeba2eb 100644 --- a/src/protocol/commitment/config.rs +++ b/src/protocol/commitment/config.rs @@ -310,6 +310,9 @@ pub(super) fn ensure_matrix_shape( } /// Small correctness-first config for tests and local benchmarks. +/// +/// Fixed layout (m_vars=4, r_vars=2) for fast test iteration. For larger +/// polynomials, use [`DynamicSmallTestCommitmentConfig`] instead. #[derive(Clone, Copy, Debug, Default)] pub struct SmallTestCommitmentConfig; @@ -328,6 +331,40 @@ impl CommitmentConfig for SmallTestCommitmentConfig { } } +/// D=16 config with dynamic layout that adapts to polynomial size. +/// +/// Uses the same D=16 ring dimension as [`SmallTestCommitmentConfig`] but +/// derives `m_vars`/`r_vars` from `max_num_vars`, so it can commit +/// polynomials with an arbitrary number of variables. +#[derive(Clone, Copy, Debug, Default)] +pub struct DynamicSmallTestCommitmentConfig; + +impl CommitmentConfig for DynamicSmallTestCommitmentConfig { + const D: usize = 16; + const N_A: usize = 8; + const N_B: usize = 4; + const N_D: usize = 4; + const LOG_BASIS: u32 = 4; + const DELTA: usize = 9; + const TAU: usize = 4; + const CHALLENGE_WEIGHT: usize = 3; + + fn commitment_layout(max_num_vars: usize) -> Result { + let alpha = Self::D.trailing_zeros() as usize; + let reduced_vars = max_num_vars.checked_sub(alpha).ok_or_else(|| { + HachiError::InvalidSetup("max_num_vars is smaller than alpha".to_string()) + })?; + if reduced_vars == 0 { + return Err(HachiError::InvalidSetup( + "max_num_vars must leave at least one outer variable".to_string(), + )); + } + let m_vars = reduced_vars.min(11); + let r_vars = reduced_vars - m_vars; + HachiCommitmentLayout::new::(m_vars, r_vars) + } +} + /// Production-oriented profile for 128-bit base fields (`Fp128

`). /// /// This profile targets the `D = 512`, `n_A = n_B = n_D = 1` regime with diff --git a/src/protocol/commitment/mod.rs b/src/protocol/commitment/mod.rs index b5fe86f9..ab276414 100644 --- a/src/protocol/commitment/mod.rs +++ b/src/protocol/commitment/mod.rs @@ -13,8 +13,8 @@ pub use commit::{ HachiVerifierSetup, }; pub use config::{ - CommitmentConfig, HachiCommitmentLayout, ProductionFp128CommitmentConfig, - SmallTestCommitmentConfig, + CommitmentConfig, DynamicSmallTestCommitmentConfig, HachiCommitmentLayout, + ProductionFp128CommitmentConfig, SmallTestCommitmentConfig, }; pub use scheme::{ CommitWitness, CommitmentScheme, RingCommitmentScheme, StreamingCommitmentScheme, diff --git a/src/protocol/mod.rs b/src/protocol/mod.rs index 85ad23fa..27b5ac70 100644 --- a/src/protocol/mod.rs +++ b/src/protocol/mod.rs @@ -15,11 +15,11 @@ pub mod sumcheck; pub mod transcript; pub use commitment::{ - AppendToTranscript, CommitmentConfig, CommitmentScheme, DummyProof, HachiCommitment, - HachiCommitmentCore, HachiCommitmentLayout, HachiExpandedSetup, HachiOpeningClaim, - HachiOpeningPoint, HachiPreparedSetup, HachiProverSetup, HachiSetupSeed, HachiVerifierSetup, - ProductionFp128CommitmentConfig, RingCommitment, RingCommitmentScheme, - SmallTestCommitmentConfig, StreamingCommitmentScheme, + AppendToTranscript, CommitmentConfig, CommitmentScheme, DummyProof, + DynamicSmallTestCommitmentConfig, HachiCommitment, HachiCommitmentCore, HachiCommitmentLayout, + HachiExpandedSetup, HachiOpeningClaim, HachiOpeningPoint, HachiPreparedSetup, HachiProverSetup, + HachiSetupSeed, HachiVerifierSetup, ProductionFp128CommitmentConfig, RingCommitment, + RingCommitmentScheme, SmallTestCommitmentConfig, StreamingCommitmentScheme, }; pub use commitment_scheme::{commit_onehot, HachiChunkState, HachiCommitmentScheme}; pub use opening_point::RingOpeningPoint; From 5bf876240f285dc5a7089f3d22f89520a5870a2f Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Sun, 1 Mar 2026 23:57:49 -0800 Subject: [PATCH 26/34] =?UTF-8?q?perf:=20true=20submatrix=20in=20commit=5F?= =?UTF-8?q?coeffs=20=E2=80=94=20skip=20zero=20blocks?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Short polynomials no longer pad to block_len. commit_coeffs accepts fewer ring elements than num_blocks * block_len, decomposes only the non-zero blocks, and fills remaining entries with zero s/t_hat without allocation or mat-vec multiplication. Also relax debug_assert in mat_vec_mul_precomputed to >= (zip handles the shorter vector correctly). Made-with: Cursor --- src/protocol/commitment/commit.rs | 51 +++++++++++++++++++++++++ src/protocol/commitment/scheme.rs | 22 ++++------- src/protocol/commitment/utils/linear.rs | 2 +- 3 files changed, 59 insertions(+), 16 deletions(-) diff --git a/src/protocol/commitment/commit.rs b/src/protocol/commitment/commit.rs index 7f750f51..ec547b96 100644 --- a/src/protocol/commitment/commit.rs +++ b/src/protocol/commitment/commit.rs @@ -355,6 +355,57 @@ where }) } + fn commit_coeffs( + f_coeffs: &[CyclotomicRing], + setup: &Self::ProverSetup, + ) -> Result, HachiError> { + let layout = setup.layout(); + let num_blocks = layout.num_blocks; + let block_len = layout.block_len; + let max_len = num_blocks + .checked_mul(block_len) + .ok_or_else(|| HachiError::InvalidSetup("coefficient length overflow".to_string()))?; + if f_coeffs.len() > max_len { + return Err(HachiError::InvalidSize { + expected: max_len, + actual: f_coeffs.len(), + }); + } + + let zero_s = vec![CyclotomicRing::::zero(); Cfg::DELTA]; + let zero_t_hat = + vec![CyclotomicRing::::zero(); Cfg::N_A.checked_mul(Cfg::DELTA).unwrap()]; + + let mut s_all: Vec>> = Vec::with_capacity(num_blocks); + let mut t_hat_all: Vec>> = Vec::with_capacity(num_blocks); + let mut t_hat_flat: Vec> = Vec::with_capacity(layout.outer_width); + + for i in 0..num_blocks { + let start = i * block_len; + if start >= f_coeffs.len() { + s_all.push(zero_s.clone()); + t_hat_flat.extend(zero_t_hat.iter().copied()); + t_hat_all.push(zero_t_hat.clone()); + } else { + let end = (start + block_len).min(f_coeffs.len()); + let block = &f_coeffs[start..end]; + let s_i = decompose_block(block, Cfg::DELTA, Cfg::LOG_BASIS); + let t_i = mat_vec_mul_ntt_cached(setup.ntt_cache()?, MatrixSlot::A, &s_i)?; + let t_hat_i = decompose_rows(&t_i, Cfg::DELTA, Cfg::LOG_BASIS); + t_hat_flat.extend(t_hat_i.iter().copied()); + s_all.push(s_i); + t_hat_all.push(t_hat_i); + } + } + + let u = mat_vec_mul_ntt_cached(setup.ntt_cache()?, MatrixSlot::B, &t_hat_flat)?; + Ok(CommitWitness { + commitment: RingCommitment { u }, + s: s_all, + t_hat: t_hat_all, + }) + } + fn commit_onehot( onehot_k: usize, indices: &[Option], diff --git a/src/protocol/commitment/scheme.rs b/src/protocol/commitment/scheme.rs index e6ccd601..e59381f3 100644 --- a/src/protocol/commitment/scheme.rs +++ b/src/protocol/commitment/scheme.rs @@ -195,28 +195,20 @@ where let layout = Self::layout(setup)?; let num_blocks = layout.num_blocks; let block_len = layout.block_len; - let max_len = num_blocks + let expected_len = num_blocks .checked_mul(block_len) .ok_or_else(|| HachiError::InvalidSetup("coefficient length overflow".to_string()))?; - if f_coeffs.len() > max_len { + if f_coeffs.len() != expected_len { return Err(HachiError::InvalidSize { - expected: max_len, + expected: expected_len, actual: f_coeffs.len(), }); } - let mut blocks: Vec>> = Vec::with_capacity(num_blocks); - for i in 0..num_blocks { - let start = i * block_len; - if start >= f_coeffs.len() { - blocks.push(vec![CyclotomicRing::::zero(); block_len]); - } else { - let end = (start + block_len).min(f_coeffs.len()); - let mut block = f_coeffs[start..end].to_vec(); - block.resize(block_len, CyclotomicRing::::zero()); - blocks.push(block); - } - } + let blocks: Vec>> = f_coeffs + .chunks_exact(block_len) + .map(|chunk| chunk.to_vec()) + .collect(); Self::commit_ring_blocks(&blocks, setup) } diff --git a/src/protocol/commitment/utils/linear.rs b/src/protocol/commitment/utils/linear.rs index 4f5b3ddb..33134302 100644 --- a/src/protocol/commitment/utils/linear.rs +++ b/src/protocol/commitment/utils/linear.rs @@ -186,7 +186,7 @@ fn mat_vec_mul_precomputed_with_params< cfg_iter!(ntt_mat) .map(|row_ntt| { - debug_assert_eq!(row_ntt.len(), ntt_vec.len()); + debug_assert!(row_ntt.len() >= ntt_vec.len()); let mut acc = CyclotomicCrtNtt::::zero(); for (a_ntt, x_ntt) in row_ntt.iter().zip(ntt_vec.iter()) { accumulate_pointwise_product_into(&mut acc, a_ntt, x_ntt, params); From 2b03961a0c4c0b6772a844be7ceb28a612846cdb Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Mon, 2 Mar 2026 00:02:53 -0800 Subject: [PATCH 27/34] fix: use inner_width for zero_s in commit_coeffs/commit_onehot prove expects s[i] to have inner_width entries. Use the correct length for zero blocks to match the dense path's decompose_block output size. Made-with: Cursor --- src/protocol/commitment/commit.rs | 35 ++++++++++++++++++++----------- 1 file changed, 23 insertions(+), 12 deletions(-) diff --git a/src/protocol/commitment/commit.rs b/src/protocol/commitment/commit.rs index ec547b96..15fcfeaa 100644 --- a/src/protocol/commitment/commit.rs +++ b/src/protocol/commitment/commit.rs @@ -372,7 +372,8 @@ where }); } - let zero_s = vec![CyclotomicRing::::zero(); Cfg::DELTA]; + let inner_width = layout.inner_width; + let zero_s = vec![CyclotomicRing::::zero(); inner_width]; let zero_t_hat = vec![CyclotomicRing::::zero(); Cfg::N_A.checked_mul(Cfg::DELTA).unwrap()]; @@ -426,18 +427,28 @@ where let mut t_hat_all: Vec>> = Vec::with_capacity(layout.num_blocks); let mut t_hat_flat: Vec> = Vec::with_capacity(layout.outer_width); - for block_entries in &sparse_blocks { - let (t_i, s_i) = inner_ajtai_onehot( - &setup.expanded.A, - block_entries, - layout.block_len, - Cfg::DELTA, - ); - let t_hat_i = decompose_rows(&t_i, Cfg::DELTA, Cfg::LOG_BASIS); - t_hat_flat.extend(t_hat_i.iter().copied()); + let inner_width = layout.inner_width; + let zero_s = vec![CyclotomicRing::::zero(); inner_width]; + let zero_t_hat = + vec![CyclotomicRing::::zero(); Cfg::N_A.checked_mul(Cfg::DELTA).unwrap()]; - s_all.push(s_i); - t_hat_all.push(t_hat_i); + for block_entries in &sparse_blocks { + if block_entries.is_empty() { + s_all.push(zero_s.clone()); + t_hat_flat.extend(zero_t_hat.iter().copied()); + t_hat_all.push(zero_t_hat.clone()); + } else { + let (t_i, s_i) = inner_ajtai_onehot( + &setup.expanded.A, + block_entries, + layout.block_len, + Cfg::DELTA, + ); + let t_hat_i = decompose_rows(&t_i, Cfg::DELTA, Cfg::LOG_BASIS); + t_hat_flat.extend(t_hat_i.iter().copied()); + s_all.push(s_i); + t_hat_all.push(t_hat_i); + } } let u = mat_vec_mul_ntt_cached(setup.ntt_cache()?, MatrixSlot::B, &t_hat_flat)?; From 06d70950e54749e5b897a29fccdf6ab61034d157 Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Mon, 2 Mar 2026 00:45:26 -0800 Subject: [PATCH 28/34] fix: configure rayon with 64MB stack for D>=512 ring elements CRT-NTT conversion puts ~28KB on the stack per ring element ([[MontCoeff; D]; K] + [i128; D]). With D=512 and the commit call chain depth, rayon's default thread stack overflows. ensure_large_thread_stack() is called from setup() and is safe to call multiple times (only the first configures the pool). Made-with: Cursor --- src/lib.rs | 20 ++++++++++++++++++++ src/protocol/commitment/commit.rs | 1 + 2 files changed, 21 insertions(+) diff --git a/src/lib.rs b/src/lib.rs index 4d6635ef..1fd4ea6f 100644 --- a/src/lib.rs +++ b/src/lib.rs @@ -61,3 +61,23 @@ pub use primitives::arithmetic::{ pub use primitives::poly::{MultilinearLagrange, Polynomial}; pub use primitives::serialization::{HachiDeserialize, HachiSerialize}; pub use protocol::{CommitmentScheme, StreamingCommitmentScheme, Transcript}; + +/// Minimum rayon thread stack size for large ring degrees (D >= 512). +/// +/// CRT-NTT conversion puts `[[MontCoeff; D]; K]` on the stack per ring element, +/// which at D=512, K=5 is ~20 KB per frame. Rayon's default thread stack is +/// too small for the call depth commit → decompose → mat-vec → NTT. +const MIN_THREAD_STACK: usize = 64 * 1024 * 1024; + +/// Install the global rayon thread pool with a stack large enough for D=512+. +/// +/// Safe to call multiple times — only the first call configures the pool. +/// Subsequent calls are silently ignored. +pub fn ensure_large_thread_stack() { + #[cfg(feature = "parallel")] + { + let _ = rayon::ThreadPoolBuilder::new() + .stack_size(MIN_THREAD_STACK) + .build_global(); + } +} diff --git a/src/protocol/commitment/commit.rs b/src/protocol/commitment/commit.rs index 15fcfeaa..b29ea2a2 100644 --- a/src/protocol/commitment/commit.rs +++ b/src/protocol/commitment/commit.rs @@ -275,6 +275,7 @@ where type Commitment = RingCommitment; fn setup(max_num_vars: usize) -> Result<(Self::ProverSetup, Self::VerifierSetup), HachiError> { + crate::ensure_large_thread_stack(); let layout = validate_and_derive_layout::(max_num_vars)?; ensure_supported_num_vars(max_num_vars, layout.required_num_vars::()?)?; let public_matrix_seed = sample_public_matrix_seed(); From 8962dcf9dfaf7a8d59b521b824606279402fac56 Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Mon, 2 Mar 2026 00:57:49 -0800 Subject: [PATCH 29/34] feat: add commit_mixed for mega-polynomial commitment Exposes MegaPolyBlock enum (Dense/OneHot/Zero) and commit_mixed() which processes heterogeneous blocks in a single commitment. This lets Jolt pack all witness polynomials into one Hachi commitment (one block per polynomial) instead of N independent commitments. Also makes SparseBlockEntry and map_onehot_to_sparse_blocks public so callers can construct one-hot block descriptors. Made-with: Cursor --- src/protocol/commitment/commit.rs | 106 +++++++++++++++++++++++++++++- src/protocol/commitment/mod.rs | 5 +- src/protocol/commitment/onehot.rs | 4 +- 3 files changed, 110 insertions(+), 5 deletions(-) diff --git a/src/protocol/commitment/commit.rs b/src/protocol/commitment/commit.rs index b29ea2a2..8ce24011 100644 --- a/src/protocol/commitment/commit.rs +++ b/src/protocol/commitment/commit.rs @@ -4,7 +4,7 @@ use super::config::{ ensure_block_layout, ensure_matrix_shape, ensure_supported_num_vars, validate_and_derive_layout, HachiCommitmentLayout, }; -use super::onehot::{inner_ajtai_onehot, map_onehot_to_sparse_blocks}; +use super::onehot::{inner_ajtai_onehot, map_onehot_to_sparse_blocks, SparseBlockEntry}; use super::scheme::{CommitWitness, RingCommitmentScheme}; use super::types::RingCommitment; use super::utils::crt_ntt::{build_ntt_cache, NttMatrixCache}; @@ -461,6 +461,110 @@ where } } +/// Describes one block of a mega-polynomial commitment. +/// +/// A mega-polynomial packs multiple heterogeneous polynomials into a single +/// Hachi commitment by assigning each polynomial to its own block. Blocks +/// can be dense (arbitrary ring coefficients), sparse one-hot, or zero. +pub enum MegaPolyBlock<'a, F: FieldCore, const D: usize> { + /// Dense block: full ring coefficients (length ≤ block_len). + Dense(&'a [CyclotomicRing]), + /// One-hot block: sparse entries within this block. + OneHot(&'a [SparseBlockEntry]), + /// Empty block: all coefficients are zero (no allocation or computation). + Zero, +} + +impl HachiCommitmentCore { + /// Commit a mega-polynomial composed of heterogeneous blocks. + /// + /// Each block occupies `block_len` ring elements. Dense blocks are + /// decomposed via `balanced_decompose_pow2`; one-hot blocks use sparse + /// inner Ajtai; zero blocks are free. + /// + /// The number of blocks must equal `layout.num_blocks` (power of 2). + /// + /// # Errors + /// + /// Returns `HachiError` if the number of blocks doesn't match the layout + /// or if matrix shapes are inconsistent. + /// + /// # Panics + /// + /// Panics if `Cfg::N_A * Cfg::DELTA` overflows. + #[allow(non_snake_case)] + pub fn commit_mixed( + blocks: &[MegaPolyBlock<'_, F, D>], + setup: &HachiProverSetup, + ) -> Result, F, D>, HachiError> + where + F: FieldCore + CanonicalField + FieldSampling, + Cfg: CommitmentConfig, + { + let layout = setup.layout(); + if blocks.len() != layout.num_blocks { + return Err(HachiError::InvalidSize { + expected: layout.num_blocks, + actual: blocks.len(), + }); + } + ensure_matrix_shape(&setup.expanded.A, Cfg::N_A, layout.inner_width, "A")?; + ensure_matrix_shape(&setup.expanded.B, Cfg::N_B, layout.outer_width, "B")?; + + let inner_width = layout.inner_width; + let zero_s = vec![CyclotomicRing::::zero(); inner_width]; + let zero_t_hat = + vec![CyclotomicRing::::zero(); Cfg::N_A.checked_mul(Cfg::DELTA).unwrap()]; + + let mut s_all: Vec>> = Vec::with_capacity(layout.num_blocks); + let mut t_hat_all: Vec>> = Vec::with_capacity(layout.num_blocks); + let mut t_hat_flat: Vec> = Vec::with_capacity(layout.outer_width); + + for block in blocks { + match block { + MegaPolyBlock::Zero => { + s_all.push(zero_s.clone()); + t_hat_flat.extend(zero_t_hat.iter().copied()); + t_hat_all.push(zero_t_hat.clone()); + } + MegaPolyBlock::Dense(coeffs) => { + let s_i = decompose_block(coeffs, Cfg::DELTA, Cfg::LOG_BASIS); + let t_i = mat_vec_mul_ntt_cached(setup.ntt_cache()?, MatrixSlot::A, &s_i)?; + let t_hat_i = decompose_rows(&t_i, Cfg::DELTA, Cfg::LOG_BASIS); + t_hat_flat.extend(t_hat_i.iter().copied()); + s_all.push(s_i); + t_hat_all.push(t_hat_i); + } + MegaPolyBlock::OneHot(sparse_entries) => { + if sparse_entries.is_empty() { + s_all.push(zero_s.clone()); + t_hat_flat.extend(zero_t_hat.iter().copied()); + t_hat_all.push(zero_t_hat.clone()); + } else { + let (t_i, s_i) = inner_ajtai_onehot( + &setup.expanded.A, + sparse_entries, + layout.block_len, + Cfg::DELTA, + ); + let t_hat_i = decompose_rows(&t_i, Cfg::DELTA, Cfg::LOG_BASIS); + t_hat_flat.extend(t_hat_i.iter().copied()); + s_all.push(s_i); + t_hat_all.push(t_hat_i); + } + } + } + } + + let u = mat_vec_mul_ntt_cached(setup.ntt_cache()?, MatrixSlot::B, &t_hat_flat)?; + Ok(CommitWitness { + commitment: RingCommitment { u }, + s: s_all, + t_hat: t_hat_all, + }) + } +} + #[cfg(test)] mod tests { use super::*; diff --git a/src/protocol/commitment/mod.rs b/src/protocol/commitment/mod.rs index ab276414..72b89716 100644 --- a/src/protocol/commitment/mod.rs +++ b/src/protocol/commitment/mod.rs @@ -2,7 +2,7 @@ mod commit; mod config; -pub(crate) mod onehot; +pub mod onehot; mod scheme; mod transcript_append; mod types; @@ -10,12 +10,13 @@ pub(crate) mod utils; pub use commit::{ HachiCommitmentCore, HachiExpandedSetup, HachiPreparedSetup, HachiProverSetup, HachiSetupSeed, - HachiVerifierSetup, + HachiVerifierSetup, MegaPolyBlock, }; pub use config::{ CommitmentConfig, DynamicSmallTestCommitmentConfig, HachiCommitmentLayout, ProductionFp128CommitmentConfig, SmallTestCommitmentConfig, }; +pub use onehot::{map_onehot_to_sparse_blocks, SparseBlockEntry}; pub use scheme::{ CommitWitness, CommitmentScheme, RingCommitmentScheme, StreamingCommitmentScheme, }; diff --git a/src/protocol/commitment/onehot.rs b/src/protocol/commitment/onehot.rs index 1c661b32..18b8c1dc 100644 --- a/src/protocol/commitment/onehot.rs +++ b/src/protocol/commitment/onehot.rs @@ -12,7 +12,7 @@ use crate::{CanonicalField, FieldCore}; /// Describes a nonzero ring element within one block of the commitment layout. #[derive(Debug, Clone)] -pub(crate) struct SparseBlockEntry { +pub struct SparseBlockEntry { /// Position within the block (0..2^M). pub pos_in_block: usize, /// Coefficient indices that are 1 within this ring element. @@ -35,7 +35,7 @@ pub(crate) struct SparseBlockEntry { /// Returns an error if K and D are not "nicely matched" (one must divide /// the other), if any index is out of range, or if the dimensions don't /// fill the commitment layout. -pub(crate) fn map_onehot_to_sparse_blocks( +pub fn map_onehot_to_sparse_blocks( onehot_k: usize, indices: &[Option], r: usize, From 5dad74f772cf2833f54ce3436a8f70e39613ae83 Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Mon, 2 Mar 2026 08:00:59 -0800 Subject: [PATCH 30/34] perf: drop s vectors from CommitWitness and HachiCommitmentHint MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit The basis-decomposed s_i vectors (one per block, each block_len*delta ring elements) were stored in both CommitWitness and HachiCommitmentHint. At production parameters (D=512, block_len=2048, delta=32), each s_i is 512 MB — storing all 64 of them consumed ~32 GB. Instead, recompute s_i on the fly in compute_w_hat and compute_z_hat from ring_coeffs using decompose_block. Peak memory drops from O(blocks * block_len * delta) to O(block_len * delta) per thread. Also adds setup_with_layout for caller-specified HachiCommitmentLayout, and makes decompose_block, SparseBlockEntry, map_onehot_to_sparse_blocks public for downstream (Jolt) mega-polynomial integration. Made-with: Cursor --- src/protocol/commitment/commit.rs | 101 ++++++++++++++++-------- src/protocol/commitment/mod.rs | 2 +- src/protocol/commitment/scheme.rs | 9 +-- src/protocol/commitment/utils/linear.rs | 3 +- src/protocol/commitment/utils/mod.rs | 2 +- src/protocol/commitment_scheme.rs | 4 - src/protocol/proof.rs | 13 ++- src/protocol/quadratic_equation.rs | 83 ++++++++++++------- tests/onehot_commitment.rs | 4 - tests/ring_commitment_core.rs | 17 ++-- 10 files changed, 144 insertions(+), 94 deletions(-) diff --git a/src/protocol/commitment/commit.rs b/src/protocol/commitment/commit.rs index 8ce24011..628eec66 100644 --- a/src/protocol/commitment/commit.rs +++ b/src/protocol/commitment/commit.rs @@ -8,10 +8,7 @@ use super::onehot::{inner_ajtai_onehot, map_onehot_to_sparse_blocks, SparseBlock use super::scheme::{CommitWitness, RingCommitmentScheme}; use super::types::RingCommitment; use super::utils::crt_ntt::{build_ntt_cache, NttMatrixCache}; -use super::utils::linear::{ - decompose_block, decompose_rows, mat_vec_mul_ntt_cached, mat_vec_mul_ntt_many_cached, - MatrixSlot, -}; +use super::utils::linear::{decompose_block, decompose_rows, mat_vec_mul_ntt_cached, MatrixSlot}; use super::utils::matrix::{derive_public_matrix, sample_public_matrix_seed, PublicMatrixSeed}; use super::CommitmentConfig; use crate::algebra::ring::CyclotomicRing; @@ -334,15 +331,11 @@ where ensure_matrix_shape(&setup.expanded.A, Cfg::N_A, layout.inner_width, "A")?; ensure_matrix_shape(&setup.expanded.B, Cfg::N_B, layout.outer_width, "B")?; - let s_all: Vec>> = f_blocks - .iter() - .map(|block| decompose_block(block, Cfg::DELTA, Cfg::LOG_BASIS)) - .collect(); - - let t_all = mat_vec_mul_ntt_many_cached(setup.ntt_cache()?, MatrixSlot::A, &s_all)?; let mut t_hat_all: Vec>> = Vec::with_capacity(layout.num_blocks); let mut t_hat_flat: Vec> = Vec::with_capacity(layout.outer_width); - for t_i in t_all { + for block in f_blocks { + let s_i = decompose_block(block, Cfg::DELTA, Cfg::LOG_BASIS); + let t_i = mat_vec_mul_ntt_cached(setup.ntt_cache()?, MatrixSlot::A, &s_i)?; let t_hat_i = decompose_rows(&t_i, Cfg::DELTA, Cfg::LOG_BASIS); t_hat_flat.extend(t_hat_i.iter().copied()); t_hat_all.push(t_hat_i); @@ -351,7 +344,6 @@ where let u = mat_vec_mul_ntt_cached(setup.ntt_cache()?, MatrixSlot::B, &t_hat_flat)?; Ok(CommitWitness { commitment: RingCommitment { u }, - s: s_all, t_hat: t_hat_all, }) } @@ -373,19 +365,15 @@ where }); } - let inner_width = layout.inner_width; - let zero_s = vec![CyclotomicRing::::zero(); inner_width]; let zero_t_hat = vec![CyclotomicRing::::zero(); Cfg::N_A.checked_mul(Cfg::DELTA).unwrap()]; - let mut s_all: Vec>> = Vec::with_capacity(num_blocks); let mut t_hat_all: Vec>> = Vec::with_capacity(num_blocks); let mut t_hat_flat: Vec> = Vec::with_capacity(layout.outer_width); for i in 0..num_blocks { let start = i * block_len; if start >= f_coeffs.len() { - s_all.push(zero_s.clone()); t_hat_flat.extend(zero_t_hat.iter().copied()); t_hat_all.push(zero_t_hat.clone()); } else { @@ -395,7 +383,6 @@ where let t_i = mat_vec_mul_ntt_cached(setup.ntt_cache()?, MatrixSlot::A, &s_i)?; let t_hat_i = decompose_rows(&t_i, Cfg::DELTA, Cfg::LOG_BASIS); t_hat_flat.extend(t_hat_i.iter().copied()); - s_all.push(s_i); t_hat_all.push(t_hat_i); } } @@ -403,7 +390,6 @@ where let u = mat_vec_mul_ntt_cached(setup.ntt_cache()?, MatrixSlot::B, &t_hat_flat)?; Ok(CommitWitness { commitment: RingCommitment { u }, - s: s_all, t_hat: t_hat_all, }) } @@ -424,22 +410,18 @@ where let sparse_blocks = map_onehot_to_sparse_blocks(onehot_k, indices, layout.r_vars, layout.m_vars, D)?; - let mut s_all: Vec>> = Vec::with_capacity(layout.num_blocks); let mut t_hat_all: Vec>> = Vec::with_capacity(layout.num_blocks); let mut t_hat_flat: Vec> = Vec::with_capacity(layout.outer_width); - let inner_width = layout.inner_width; - let zero_s = vec![CyclotomicRing::::zero(); inner_width]; let zero_t_hat = vec![CyclotomicRing::::zero(); Cfg::N_A.checked_mul(Cfg::DELTA).unwrap()]; for block_entries in &sparse_blocks { if block_entries.is_empty() { - s_all.push(zero_s.clone()); t_hat_flat.extend(zero_t_hat.iter().copied()); t_hat_all.push(zero_t_hat.clone()); } else { - let (t_i, s_i) = inner_ajtai_onehot( + let (t_i, _s_i) = inner_ajtai_onehot( &setup.expanded.A, block_entries, layout.block_len, @@ -447,7 +429,6 @@ where ); let t_hat_i = decompose_rows(&t_i, Cfg::DELTA, Cfg::LOG_BASIS); t_hat_flat.extend(t_hat_i.iter().copied()); - s_all.push(s_i); t_hat_all.push(t_hat_i); } } @@ -455,12 +436,72 @@ where let u = mat_vec_mul_ntt_cached(setup.ntt_cache()?, MatrixSlot::B, &t_hat_flat)?; Ok(CommitWitness { commitment: RingCommitment { u }, - s: s_all, t_hat: t_hat_all, }) } } +impl HachiCommitmentCore { + /// Create a setup with a caller-specified layout, bypassing + /// `CommitmentConfig::commitment_layout`. + /// + /// Use this when the desired `(m_vars, r_vars)` split differs from what + /// the config's heuristic would choose (e.g. mega-polynomial commitments + /// where each sub-polynomial occupies one block). + /// + /// # Errors + /// + /// Returns `HachiError` on invalid layout or matrix generation failures. + #[allow(non_snake_case)] + pub fn setup_with_layout( + layout: HachiCommitmentLayout, + ) -> Result<(HachiProverSetup, HachiVerifierSetup), HachiError> + where + F: FieldCore + CanonicalField + FieldSampling, + Cfg: CommitmentConfig, + { + crate::ensure_large_thread_stack(); + let max_num_vars = layout.required_num_vars::()?; + let public_matrix_seed = sample_public_matrix_seed(); + let a_matrix = + derive_public_matrix::(Cfg::N_A, layout.inner_width, &public_matrix_seed, b"A"); + let b_matrix = + derive_public_matrix::(Cfg::N_B, layout.outer_width, &public_matrix_seed, b"B"); + let d_matrix = derive_public_matrix::( + Cfg::N_D, + layout.d_matrix_width, + &public_matrix_seed, + b"D", + ); + + let ntt_cache = build_ntt_cache::(&a_matrix, &b_matrix, &d_matrix)?; + let expanded = HachiExpandedSetup { + seed: HachiSetupSeed { + max_num_vars, + layout, + public_matrix_seed, + }, + A: a_matrix, + B: b_matrix, + D: d_matrix, + }; + let prover_setup = HachiProverSetup { + expanded: expanded.clone(), + prepared: Some(HachiPreparedSetup { ntt_cache }), + }; + let verifier_setup = HachiVerifierSetup { expanded }; + ensure_matrix_shape(&prover_setup.expanded.A, Cfg::N_A, layout.inner_width, "A")?; + ensure_matrix_shape(&prover_setup.expanded.B, Cfg::N_B, layout.outer_width, "B")?; + ensure_matrix_shape( + &prover_setup.expanded.D, + Cfg::N_D, + layout.d_matrix_width, + "D", + )?; + Ok((prover_setup, verifier_setup)) + } +} + /// Describes one block of a mega-polynomial commitment. /// /// A mega-polynomial packs multiple heterogeneous polynomials into a single @@ -511,19 +552,15 @@ impl HachiCommitmentCore { ensure_matrix_shape(&setup.expanded.A, Cfg::N_A, layout.inner_width, "A")?; ensure_matrix_shape(&setup.expanded.B, Cfg::N_B, layout.outer_width, "B")?; - let inner_width = layout.inner_width; - let zero_s = vec![CyclotomicRing::::zero(); inner_width]; let zero_t_hat = vec![CyclotomicRing::::zero(); Cfg::N_A.checked_mul(Cfg::DELTA).unwrap()]; - let mut s_all: Vec>> = Vec::with_capacity(layout.num_blocks); let mut t_hat_all: Vec>> = Vec::with_capacity(layout.num_blocks); let mut t_hat_flat: Vec> = Vec::with_capacity(layout.outer_width); for block in blocks { match block { MegaPolyBlock::Zero => { - s_all.push(zero_s.clone()); t_hat_flat.extend(zero_t_hat.iter().copied()); t_hat_all.push(zero_t_hat.clone()); } @@ -532,16 +569,14 @@ impl HachiCommitmentCore { let t_i = mat_vec_mul_ntt_cached(setup.ntt_cache()?, MatrixSlot::A, &s_i)?; let t_hat_i = decompose_rows(&t_i, Cfg::DELTA, Cfg::LOG_BASIS); t_hat_flat.extend(t_hat_i.iter().copied()); - s_all.push(s_i); t_hat_all.push(t_hat_i); } MegaPolyBlock::OneHot(sparse_entries) => { if sparse_entries.is_empty() { - s_all.push(zero_s.clone()); t_hat_flat.extend(zero_t_hat.iter().copied()); t_hat_all.push(zero_t_hat.clone()); } else { - let (t_i, s_i) = inner_ajtai_onehot( + let (t_i, _s_i) = inner_ajtai_onehot( &setup.expanded.A, sparse_entries, layout.block_len, @@ -549,7 +584,6 @@ impl HachiCommitmentCore { ); let t_hat_i = decompose_rows(&t_i, Cfg::DELTA, Cfg::LOG_BASIS); t_hat_flat.extend(t_hat_i.iter().copied()); - s_all.push(s_i); t_hat_all.push(t_hat_i); } } @@ -559,7 +593,6 @@ impl HachiCommitmentCore { let u = mat_vec_mul_ntt_cached(setup.ntt_cache()?, MatrixSlot::B, &t_hat_flat)?; Ok(CommitWitness { commitment: RingCommitment { u }, - s: s_all, t_hat: t_hat_all, }) } diff --git a/src/protocol/commitment/mod.rs b/src/protocol/commitment/mod.rs index 72b89716..2801d510 100644 --- a/src/protocol/commitment/mod.rs +++ b/src/protocol/commitment/mod.rs @@ -6,7 +6,7 @@ pub mod onehot; mod scheme; mod transcript_append; mod types; -pub(crate) mod utils; +pub mod utils; pub use commit::{ HachiCommitmentCore, HachiExpandedSetup, HachiPreparedSetup, HachiProverSetup, HachiSetupSeed, diff --git a/src/protocol/commitment/scheme.rs b/src/protocol/commitment/scheme.rs index e59381f3..c3d3fb95 100644 --- a/src/protocol/commitment/scheme.rs +++ b/src/protocol/commitment/scheme.rs @@ -12,14 +12,13 @@ pub(crate) type BatchCommitOutput = Result, HachiError>; /// Witness data produced alongside a ring-native commitment. /// -/// Contains the commitment itself plus the decomposed witness vectors `s` -/// (basis-decomposed input) and `t_hat` (basis-decomposed inner Ajtai output) -/// from the two-layer Ajtai construction (§4.1). +/// Contains the commitment itself plus `t_hat` (basis-decomposed inner Ajtai +/// output) from the two-layer Ajtai construction (§4.1). The decomposed input +/// vectors `s` are NOT stored; they are recomputed from `ring_coeffs` during +/// proving to avoid multi-GB memory usage at production parameters. pub struct CommitWitness { /// The ring commitment (outer Ajtai output `u = B · t̂`). pub commitment: C, - /// Per-block basis-decomposed input vectors. - pub s: Vec>>, /// Per-block basis-decomposed inner Ajtai output vectors. pub t_hat: Vec>>, } diff --git a/src/protocol/commitment/utils/linear.rs b/src/protocol/commitment/utils/linear.rs index 33134302..5e70ef96 100644 --- a/src/protocol/commitment/utils/linear.rs +++ b/src/protocol/commitment/utils/linear.rs @@ -266,7 +266,8 @@ pub(crate) fn mat_vec_mul_ntt_many_cached( +/// Basis-decompose a block of ring elements into `block.len() * delta` gadget components. +pub fn decompose_block( block: &[CyclotomicRing], delta: usize, log_basis: u32, diff --git a/src/protocol/commitment/utils/mod.rs b/src/protocol/commitment/utils/mod.rs index 9cbf39e4..f7035759 100644 --- a/src/protocol/commitment/utils/mod.rs +++ b/src/protocol/commitment/utils/mod.rs @@ -1,7 +1,7 @@ //! Utility helpers for commitment internals. pub(crate) mod crt_ntt; -pub(crate) mod linear; +pub mod linear; pub(crate) mod math; pub(crate) mod matrix; pub(crate) mod norm; diff --git a/src/protocol/commitment_scheme.rs b/src/protocol/commitment_scheme.rs index 1294cce5..2a418f58 100644 --- a/src/protocol/commitment_scheme.rs +++ b/src/protocol/commitment_scheme.rs @@ -84,7 +84,6 @@ where setup, )?; let hint = HachiCommitmentHint { - s: w.s, t_hat: w.t_hat, ring_coeffs, }; @@ -316,7 +315,6 @@ where )?; let hint = HachiCommitmentHint { - s: w.s, t_hat: w.t_hat, ring_coeffs, }; @@ -450,7 +448,6 @@ where ) .expect("outer Ajtai"); - let s_all: Vec>> = chunks.iter().map(|c| c.s_i.clone()).collect(); let t_hat_all: Vec>> = chunks.iter().map(|c| c.t_hat_i.clone()).collect(); let ring_coeffs: Vec> = chunks @@ -460,7 +457,6 @@ where let commitment = RingCommitment { u }; let hint = HachiCommitmentHint { - s: s_all, t_hat: t_hat_all, ring_coeffs, }; diff --git a/src/protocol/proof.rs b/src/protocol/proof.rs index c505c6a8..89efd9e8 100644 --- a/src/protocol/proof.rs +++ b/src/protocol/proof.rs @@ -9,10 +9,13 @@ use crate::{FieldCore, HachiDeserialize, HachiSerialize}; use std::io::{Read, Write}; /// Prover-side hint produced at commitment time. +/// +/// Stores the ring-level coefficients and the decomposed inner-Ajtai outputs +/// `t̂_i`. The basis-decomposed inputs `s_i` are NOT stored; they are +/// recomputed from `ring_coeffs` during proving to avoid multi-GB memory +/// usage at production parameters. #[derive(Debug, Clone, PartialEq, Eq)] pub struct HachiCommitmentHint { - /// Decomposed `s_i` blocks from the commitment phase. - pub s: Vec>>, /// Decomposed `t̂_i` blocks from the commitment phase. pub t_hat: Vec>>, /// Ring coefficients from the §3.1 reduction (evaluation table). @@ -94,14 +97,11 @@ impl HachiSerialize for HachiCommitmentHint mut writer: W, compress: Compress, ) -> Result<(), SerializationError> { - self.s.serialize_with_mode(&mut writer, compress)?; self.t_hat.serialize_with_mode(&mut writer, compress)?; self.ring_coeffs.serialize_with_mode(&mut writer, compress) } fn serialized_size(&self, compress: Compress) -> usize { - self.s.serialized_size(compress) - + self.t_hat.serialized_size(compress) - + self.ring_coeffs.serialized_size(compress) + self.t_hat.serialized_size(compress) + self.ring_coeffs.serialized_size(compress) } } @@ -118,7 +118,6 @@ impl HachiDeserialize for HachiCommitmentH validate: Validate, ) -> Result { Ok(Self { - s: Vec::deserialize_with_mode(&mut reader, compress, validate)?, t_hat: Vec::deserialize_with_mode(&mut reader, compress, validate)?, ring_coeffs: Vec::deserialize_with_mode(&mut reader, compress, validate)?, }) diff --git a/src/protocol/quadratic_equation.rs b/src/protocol/quadratic_equation.rs index 27a94def..22b23cde 100644 --- a/src/protocol/quadratic_equation.rs +++ b/src/protocol/quadratic_equation.rs @@ -4,12 +4,15 @@ //! the quadratic equation components M, y, z, and v. use crate::algebra::ring::{CyclotomicRing, SparseChallenge, SparseChallengeConfig}; +use crate::cfg_iter; use crate::error::HachiError; #[cfg(feature = "parallel")] use crate::parallel::*; use crate::protocol::challenges::sparse::sample_sparse_challenges; use crate::protocol::commitment::utils::crt_ntt::NttMatrixCache; -use crate::protocol::commitment::utils::linear::{mat_vec_mul_ntt_cached, MatrixSlot}; +use crate::protocol::commitment::utils::linear::{ + decompose_block, mat_vec_mul_ntt_cached, MatrixSlot, +}; use crate::protocol::commitment::utils::norm::{detect_field_modulus, vec_inf_norm}; use crate::protocol::commitment::{ CommitmentConfig, HachiCommitmentLayout, HachiExpandedSetup, HachiProverSetup, @@ -19,13 +22,16 @@ use crate::protocol::opening_point::RingOpeningPoint; use crate::protocol::proof::HachiCommitmentHint; use crate::protocol::transcript::labels::{ABSORB_PROVER_V, CHALLENGE_STAGE1_FOLD}; use crate::protocol::transcript::Transcript; -use crate::{cfg_into_iter, cfg_iter}; use crate::{CanonicalField, FieldCore}; /// **Steps 1–3.** Compute `w_i = a^T G_{2^m} s_i` and decompose: `ŵ_i = G_1^{-1}(w_i)`. +/// +/// Recomputes each block's `s_i` from `ring_coeffs` on the fly to avoid +/// storing all `s_i` simultaneously (which can be tens of GB at production +/// parameters). fn compute_w_hat( opening_point: &RingOpeningPoint, - s: &[Vec>], + ring_coeffs: &[CyclotomicRing], layout: HachiCommitmentLayout, ) -> Vec>> where @@ -39,8 +45,21 @@ where debug_assert_eq!(a.len(), block_len); - cfg_iter!(s) - .map(|s_i| { + let blocks: Vec<&[CyclotomicRing]> = (0..layout.num_blocks) + .map(|i| { + let start = i * block_len; + let end = (start + block_len).min(ring_coeffs.len()); + if start < ring_coeffs.len() { + &ring_coeffs[start..end] + } else { + &[] as &[CyclotomicRing] + } + }) + .collect(); + + cfg_iter!(blocks) + .map(|block| { + let s_i = decompose_block(block, delta, log_basis); let mut w_i = CyclotomicRing::::zero(); for (j, a_j) in a.iter().enumerate().take(block_len) { let start = j * delta; @@ -64,8 +83,12 @@ fn compute_v( } /// **Steps 7–9.** Fold `z = Σ c_i · s_i`, check `‖z‖_∞ ≤ β`, and decompose `ẑ = J^{-1}(z)`. +/// +/// Recomputes each block's `s_i` from `ring_coeffs` one block at a time, +/// accumulating `z[j] += c_i * s_i[j]` across blocks. Peak memory is one +/// block's `s_i` rather than the full `s` tensor. fn compute_z_hat( - s: &[Vec>], + ring_coeffs: &[CyclotomicRing], challenges: &[SparseChallenge], layout: HachiCommitmentLayout, ) -> Result>, HachiError> @@ -73,19 +96,28 @@ where F: FieldCore + CanonicalField, Cfg: CommitmentConfig, { - debug_assert_eq!(challenges.len(), s.len()); - let len = s[0].len(); - - let z: Vec> = cfg_into_iter!(0..len) - .map(|j| { - challenges - .iter() - .zip(s.iter()) - .fold(CyclotomicRing::::zero(), |acc, (c_i, s_i)| { - acc + s_i[j].mul_by_sparse(c_i) - }) - }) - .collect(); + let block_len = layout.block_len; + let delta = Cfg::DELTA; + let log_basis = Cfg::LOG_BASIS; + let inner_width = block_len * delta; + + debug_assert_eq!(challenges.len(), layout.num_blocks); + + let mut z = vec![CyclotomicRing::::zero(); inner_width]; + + for (i, c_i) in challenges.iter().enumerate() { + let start = i * block_len; + let end = (start + block_len).min(ring_coeffs.len()); + let block = if start < ring_coeffs.len() { + &ring_coeffs[start..end] + } else { + &[] as &[CyclotomicRing] + }; + let s_i = decompose_block(block, delta, log_basis); + for (j, z_j) in z.iter_mut().enumerate() { + *z_j += s_i[j].mul_by_sparse(c_i); + } + } let modulus = detect_field_modulus::(); let norm = vec_inf_norm(&z, modulus); @@ -145,11 +177,8 @@ where commitment: &RingCommitment, y_ring: &CyclotomicRing, ) -> Result { - // Steps 1–3: w_i = a^T G_{2^m} s_i, then ŵ_i = G_1^{-1}(w_i) let layout = setup.layout(); - let w_hat = compute_w_hat::(ring_opening_point, &hint.s, layout); - - // Step 4: v = D · ŵ + let w_hat = compute_w_hat::(ring_opening_point, &hint.ring_coeffs, layout); let v = compute_v(setup.ntt_cache()?, &w_hat)?; // Step 5: append v to transcript @@ -167,8 +196,7 @@ where &challenge_cfg, )?; - // Steps 7–9: z = Σ c_i · s_i, check ‖z‖_∞ ≤ β, then ẑ = J^{-1}(z) - let z_hat = compute_z_hat::(&hint.s, &challenges, layout)?; + let z_hat = compute_z_hat::(&hint.ring_coeffs, &challenges, layout)?; let m = generate_m::(&setup.expanded, ring_opening_point, &challenges)?; let y = generate_y::(&v, &commitment.u, y_ring)?; @@ -598,10 +626,11 @@ mod tests { b: sample_b(), }; + let ring_coeffs: Vec> = + blocks.iter().flat_map(|b| b.iter().copied()).collect(); let hint = HachiCommitmentHint { - s: w.s, t_hat: w.t_hat, - ring_coeffs: Vec::new(), + ring_coeffs, }; let mut transcript = Blake2bTranscript::::new(TRANSCRIPT_SEED); let y_ring = CyclotomicRing::::zero(); diff --git a/tests/onehot_commitment.rs b/tests/onehot_commitment.rs index 6356774e..e3f6c109 100644 --- a/tests/onehot_commitment.rs +++ b/tests/onehot_commitment.rs @@ -52,10 +52,6 @@ fn assert_onehot_matches_dense(onehot_k: usize, indices: &[usize]) { w_sparse.commitment, w_dense.commitment, "commitments must match" ); - assert_eq!( - w_sparse.s, w_dense.s, - "s_all (decomposed witness) must match" - ); assert_eq!( w_sparse.t_hat, w_dense.t_hat, "t_hat_all (decomposed inner output) must match" diff --git a/tests/ring_commitment_core.rs b/tests/ring_commitment_core.rs index be29de97..4b09501e 100644 --- a/tests/ring_commitment_core.rs +++ b/tests/ring_commitment_core.rs @@ -83,18 +83,11 @@ fn commit_is_deterministic_and_shape_consistent() { .unwrap(); assert_eq!(w1.commitment, w2.commitment); - assert_eq!(w1.s, w2.s); assert_eq!(w1.t_hat, w2.t_hat); let num_blocks = hachi_pcs::test_utils::NUM_BLOCKS; - let block_len = hachi_pcs::test_utils::BLOCK_LEN; assert_eq!(w1.commitment.u.len(), TinyConfig::N_B); - assert_eq!(w1.s.len(), num_blocks); assert_eq!(w1.t_hat.len(), num_blocks); - assert!(w1 - .s - .iter() - .all(|s| s.len() == block_len * TinyConfig::DELTA)); assert!(w1 .t_hat .iter() @@ -124,7 +117,6 @@ fn commit_ring_coeffs_matches_block_commitment() { .unwrap(); assert_eq!(wb.commitment, wc.commitment); - assert_eq!(wb.s, wc.s); assert_eq!(wb.t_hat, wc.t_hat); } @@ -138,8 +130,13 @@ fn opening_satisfies_inner_and_outer_equations() { ) .unwrap(); - for i in 0..w.s.len() { - let lhs = mat_vec_mul(&psetup.expanded.A, &w.s[i]); + for (i, block) in blocks.iter().enumerate() { + let s_i = hachi_pcs::protocol::commitment::utils::linear::decompose_block( + block, + TinyConfig::DELTA, + TinyConfig::LOG_BASIS, + ); + let lhs = mat_vec_mul(&psetup.expanded.A, &s_i); let rhs: Vec> = (0..TinyConfig::N_A) .map(|j| { let start = j * TinyConfig::DELTA; From ce940d033d87ce88de3e4f8616900045755d46a3 Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Mon, 2 Mar 2026 09:08:35 -0800 Subject: [PATCH 31/34] chore: untrack docs/ and paper/ from version control Keep these files locally for reference but remove from the committed tree. They can be selectively re-added later. Made-with: Cursor --- docs/BN254_MUL_ASM_AARCH64_ANALYSIS.md | 389 -- docs/DATA_PROOF_HACHI_MIGRATION.md | 524 --- docs/FP128_MUL_ASM_AARCH64_ANALYSIS.md | 146 - docs/HACHI_FOR_JOLT.md | 2234 ------------ docs/HACHI_KEY_DELEGATION.md | 620 ---- docs/HACHI_ZK_PATHS.md | 333 -- docs/HACHI_ZK_PCS_REPORT.md | 292 -- docs/ONE_HOT_COMMITMENT_COST_AND_GPU_PRG.md | 787 ---- docs/SPARSE_2P128_PRIMES_AND_IMPL.md | 743 ---- docs/SUPERNEO_DIGEST.md.local | 658 ---- docs/TWO_FIELD_OPENING_D_GT_1_CLEAN.md | 505 --- docs/ZK_AND_MODULUS_EMBEDDING.md | 356 -- paper/LNP22.pdf | Bin 1041252 -> 0 bytes paper/Orthus.pdf | Bin 811565 -> 0 bytes paper/aggregate-falcon-labrador.pdf | Bin 1250273 -> 0 bytes paper/hachi.txt | 2274 ------------ paper/iterative-rejection-sampling.pdf | Bin 501003 -> 0 bytes paper/ls18-short-invertible-elements.pdf | Bin 399492 -> 0 bytes paper/superneo.txt | 3570 ------------------- 19 files changed, 13431 deletions(-) delete mode 100644 docs/BN254_MUL_ASM_AARCH64_ANALYSIS.md delete mode 100644 docs/DATA_PROOF_HACHI_MIGRATION.md delete mode 100644 docs/FP128_MUL_ASM_AARCH64_ANALYSIS.md delete mode 100644 docs/HACHI_FOR_JOLT.md delete mode 100644 docs/HACHI_KEY_DELEGATION.md delete mode 100644 docs/HACHI_ZK_PATHS.md delete mode 100644 docs/HACHI_ZK_PCS_REPORT.md delete mode 100644 docs/ONE_HOT_COMMITMENT_COST_AND_GPU_PRG.md delete mode 100644 docs/SPARSE_2P128_PRIMES_AND_IMPL.md delete mode 100644 docs/SUPERNEO_DIGEST.md.local delete mode 100644 docs/TWO_FIELD_OPENING_D_GT_1_CLEAN.md delete mode 100644 docs/ZK_AND_MODULUS_EMBEDDING.md delete mode 100644 paper/LNP22.pdf delete mode 100644 paper/Orthus.pdf delete mode 100644 paper/aggregate-falcon-labrador.pdf delete mode 100644 paper/hachi.txt delete mode 100644 paper/iterative-rejection-sampling.pdf delete mode 100644 paper/ls18-short-invertible-elements.pdf delete mode 100644 paper/superneo.txt diff --git a/docs/BN254_MUL_ASM_AARCH64_ANALYSIS.md b/docs/BN254_MUL_ASM_AARCH64_ANALYSIS.md deleted file mode 100644 index 0a27415a..00000000 --- a/docs/BN254_MUL_ASM_AARCH64_ANALYSIS.md +++ /dev/null @@ -1,389 +0,0 @@ -# BN254 Mul AArch64 Assembly Analysis - -This report dumps the entire `BN254` field multiply symbol assembly and annotates it manually with a latency/throughput model. - -## Source and symbol - -- Assembly file: `target/release/deps/codegen_probe-855aaea603d2760f.s` -- Symbol analyzed: `_bn254_mul_field` -- Symbol span in `.s`: lines `860..1130` -- Target architecture: `arm64` - -## Instruction census (exact for this symbol) - -Counted directly from `_bn254_mul_field` instruction lines: - -- `mul`: `36` -- `umulh`: `32` -- Total multiply-class ops: `68` -- `adds`: `42` -- `adcs`: `8` -- `adc`: `4` -- `cinc`: `42` -- `cmn`: `8` -- `cmp`: `8` -- branches (`b*`): `9` total (`b.hs`/`b.ne`/`b.lo`/`b.eq`/`b`) - -## Bench numbers used for cycle mapping - -From current `field_mul_isolated/*` and chain benches (median values): - -- `fp128_mul_single`: `2.3145 ns` -- `bn254_mul_single`: `9.3428 ns` -- `fp128_pair_passthrough`: `1.8122 ns` -- `bn254_pair_passthrough`: `3.0869 ns` -- `fp128_mul_8way_independent`: `15.996 ns` total (8 muls) -- `bn254_mul_8way_independent`: `75.039 ns` total (8 muls) -- `fp128_8way_passthrough`: `4.6349 ns` total -- `bn254_8way_passthrough`: `8.4336 ns` total -- `mul_chain_2048` per-op: - - `Fp128`: `11.924 us / 2048 = 5.822 ns` - - `BN254`: `22.147 us / 2048 = 10.814 ns` - -At `3.5 GHz`: - -- `BN254 mul_chain` latency proxy: `10.814 ns ~= 37.85 cycles` -- `BN254 8-way net throughput proxy`: - - `(75.039 - 8.4336) / 8 = 8.3257 ns ~= 29.14 cycles/op` - -## How to read isolated bench subtraction - -Recommended subtraction pairs: - -1. **Single-op estimate (rough):** - - `mul_single - black_box_only` -2. **Throughput estimate (better):** - - `(mul_8way_independent - 8way_passthrough) / 8` - -Using current medians: - -- Single rough: - - `Fp128`: `2.3145 - 0.5329 = 1.7816 ns` - - `BN254`: `9.3428 - 1.0944 = 8.2484 ns` - - Ratio: `~4.63x` -- 8-way net (preferred): - - `Fp128`: `(15.996 - 4.6349)/8 = 1.4201 ns/op` - - `BN254`: `(75.039 - 8.4336)/8 = 8.3257 ns/op` - - Ratio: `~5.86x` - -Notes: - -- `pair_passthrough` is useful as a control but tends to over-subtract for tiny types; use it qualitatively, not as the primary net estimate. -- For latency claims, prefer `mul_chain_2048` (or `16384`) dependent chains. - -## Manual phase annotation - -The symbol is a fully inlined `Fp256>::mul` path on AArch64: - -1. **Setup / loads** (`ldp` prologue, constants in `x17/x11/x9/x10/x12`) -2. **CIOS round i=0** using first multiplier limb (`x0`) and reduction constant `INV` -3. **CIOS round i=1** using second multiplier limb (`x2`) -4. **CIOS round i=2** using third multiplier limb (`x2` after `ldp x2, x1, [x1,#16]`) -5. **CIOS round i=3** using fourth multiplier limb (`x1`) -6. **Final conditional subtraction** (`cmp`/`b.hs` path and carry-corrected subtract chain) -7. **Store reduced 256-bit result** (`stp` x4-limb output) - -Notable structural facts: - -- Four unrolled Montgomery CIOS rounds are clearly visible. -- Carry propagation is explicit (`adds`/`adcs`/`adc` + `cinc`), creating serial chains. -- Final compare/subtract path is branchful and can extend the tail latency. - -## Latency + throughput picture (manual model) - -### Latency (single dependent op) - -- The measured chain benchmark gives `~37.85 cycles/op` for BN254 mul on this machine. -- The asm shape supports this: - - 4 serialized CIOS rounds with carry-dependent accumulation - - final conditional subtract tail -- Manual estimate from dependency structure: **~37-43 cycles latency**, centered near measured `~38 cycles`. - -### Throughput (independent ops) - -- Static multiply pressure is high: `68` multiply-class instructions/call. -- Measured 8-way independent net gives `~29.14 cycles/op`. -- This indicates substantial overlap across independent operations and a backend that can keep multiply/add pipelines busy. -- Practical throughput on this system is therefore around **~29 cycles/op** for independent BN254 muls. - -### Why chain ratio is smaller than naive instruction ratio - -- Chain benchmarks measure **latency**, not pure instruction count. -- BN254 has more work, but OoO overlap and scheduling reduce the ratio compared with raw op-count intuition. -- Your current chain numbers (`~37.8 cycles BN254` vs `~20.4 cycles Fp128`) yielding `~1.86x` are consistent with this. - -## Entire raw `_bn254_mul_field` assembly - -```asm -_bn254_mul_field: - .cfi_startproc - stp x24, x23, [sp, #-48]! - .cfi_def_cfa_offset 48 - stp x22, x21, [sp, #16] - stp x20, x19, [sp, #32] - .cfi_offset w19, -8 - .cfi_offset w20, -16 - .cfi_offset w21, -24 - .cfi_offset w22, -32 - .cfi_offset w23, -40 - .cfi_offset w24, -48 - ldp x16, x15, [x0] - ldp x14, x13, [x0, #16] - ldp x0, x2, [x1] - umulh x9, x0, x16 - mul x12, x0, x16 - mov x17, #-268435457 - movk x17, #62867, lsl #32 - movk x17, #49889, lsl #48 - mul x3, x12, x17 - mov x11, #-268435455 - movk x11, #62867, lsl #32 - movk x11, #17377, lsl #48 - umulh x4, x3, x11 - mul x5, x3, x11 - umulh x10, x0, x15 - mul x6, x0, x15 - adds x6, x9, x6 - cinc x10, x10, hs - mov x9, #28817 - movk x9, #31161, lsl #16 - movk x9, #59464, lsl #32 - movk x9, #10291, lsl #48 - umulh x7, x3, x9 - mul x19, x3, x9 - adds x6, x19, x6 - cinc x7, x7, hs - umulh x19, x0, x14 - mul x20, x0, x14 - adds x20, x10, x20 - cinc x19, x19, hs - mov x10, #22621 - movk x10, #33153, lsl #16 - movk x10, #17846, lsl #32 - movk x10, #47184, lsl #48 - umulh x21, x3, x10 - mul x22, x3, x10 - adds x20, x20, x22 - cinc x21, x21, hs - cmn x5, x12 - adcs x4, x6, x4 - adcs x5, x20, x7 - cinc x6, x21, hs - mul x7, x0, x13 - adds x20, x19, x7 - mov x12, #41001 - movk x12, #57649, lsl #16 - movk x12, #20082, lsl #32 - movk x12, #12388, lsl #48 - umulh x21, x3, x12 - mul x3, x3, x12 - adds x3, x20, x3 - cinc x20, x21, hs - adds x3, x3, x6 - cinc x6, x20, hs - cmn x19, x7 - umulh x0, x0, x13 - adc x0, x6, x0 - umulh x6, x2, x16 - mul x7, x2, x16 - adds x4, x4, x7 - cinc x6, x6, hs - mul x7, x4, x17 - umulh x19, x7, x11 - mul x20, x7, x11 - umulh x21, x2, x15 - mul x22, x2, x15 - adds x5, x5, x22 - cinc x21, x21, hs - adds x5, x5, x6 - cinc x6, x21, hs - umulh x21, x7, x9 - mul x22, x7, x9 - adds x5, x5, x22 - cinc x21, x21, hs - umulh x22, x2, x14 - mul x23, x2, x14 - adds x3, x3, x23 - cinc x22, x22, hs - adds x3, x3, x6 - cinc x6, x22, hs - umulh x22, x7, x10 - mul x23, x7, x10 - adds x3, x3, x23 - cinc x22, x22, hs - cmn x20, x4 - adcs x5, x5, x19 - adcs x3, x3, x21 - cinc x4, x22, hs - umulh x19, x2, x13 - mul x2, x2, x13 - adds x0, x2, x0 - cinc x2, x19, hs - adds x19, x0, x6 - umulh x20, x7, x12 - mul x7, x7, x12 - adds x7, x19, x7 - cinc x19, x20, hs - adds x4, x7, x4 - cinc x7, x19, hs - cmn x0, x6 - adc x0, x7, x2 - ldp x2, x1, [x1, #16] - umulh x6, x2, x16 - mul x7, x2, x16 - adds x5, x5, x7 - cinc x6, x6, hs - mul x7, x5, x17 - umulh x19, x7, x11 - mul x20, x7, x11 - umulh x21, x2, x15 - mul x22, x2, x15 - adds x3, x3, x22 - cinc x21, x21, hs - adds x3, x3, x6 - cinc x6, x21, hs - umulh x21, x7, x9 - mul x22, x7, x9 - adds x3, x3, x22 - cinc x21, x21, hs - umulh x22, x2, x14 - mul x23, x2, x14 - adds x4, x4, x23 - cinc x22, x22, hs - adds x4, x4, x6 - cinc x6, x22, hs - umulh x22, x7, x10 - mul x23, x7, x10 - adds x4, x4, x23 - cinc x22, x22, hs - cmn x20, x5 - adcs x3, x3, x19 - adcs x4, x4, x21 - cinc x5, x22, hs - umulh x19, x2, x13 - mul x2, x2, x13 - adds x0, x2, x0 - cinc x2, x19, hs - adds x19, x0, x6 - umulh x20, x7, x12 - mul x7, x7, x12 - adds x7, x19, x7 - cinc x19, x20, hs - adds x5, x7, x5 - cinc x7, x19, hs - cmn x0, x6 - adc x0, x7, x2 - umulh x2, x1, x16 - mul x16, x1, x16 - adds x16, x3, x16 - cinc x2, x2, hs - mul x17, x16, x17 - umulh x3, x17, x11 - mul x6, x17, x11 - umulh x7, x1, x15 - mul x15, x1, x15 - adds x15, x4, x15 - cinc x4, x7, hs - adds x15, x15, x2 - cinc x2, x4, hs - umulh x4, x17, x9 - mul x7, x17, x9 - adds x15, x15, x7 - cinc x4, x4, hs - umulh x7, x1, x14 - mul x14, x1, x14 - adds x14, x5, x14 - cinc x5, x7, hs - adds x14, x14, x2 - cinc x2, x5, hs - umulh x5, x17, x10 - mul x7, x17, x10 - adds x7, x14, x7 - cinc x5, x5, hs - cmn x6, x16 - adcs x14, x15, x3 - adcs x15, x7, x4 - cinc x16, x5, hs - umulh x3, x1, x13 - mul x13, x1, x13 - adds x13, x13, x0 - cinc x0, x3, hs - adds x1, x13, x2 - umulh x3, x17, x12 - mul x17, x17, x12 - adds x17, x1, x17 - cinc x1, x3, hs - adds x16, x17, x16 - cinc x17, x1, hs - cmn x13, x2 - adc x13, x17, x0 - cmp x13, x12 - b.hs LBB8_3 - mov x12, x13 -LBB8_2: - mov x10, x16 - mov x9, x15 - b LBB8_11 -LBB8_3: - b.ne LBB8_10 - cmp x16, x10 - b.lo LBB8_2 - b.ne LBB8_10 - cmp x15, x9 - b.hs LBB8_8 - mov x9, x15 - b LBB8_11 -LBB8_8: - cmp x14, x11 - b.hs LBB8_10 - cmp x15, x9 - b.eq LBB8_11 -LBB8_10: - mov x9, #-4026531841 - movk x9, #2668, lsl #32 - movk x9, #48158, lsl #48 - cmp x14, x11 - add x14, x14, x9 - cset w9, lo - subs x9, x15, x9 - ngc x10, xzr - mov x11, #36719 - movk x11, #34374, lsl #16 - movk x11, #6071, lsl #32 - movk x11, #55244, lsl #48 - adds x9, x9, x11 - cinc x10, x10, hs - cmp x10, #0 - cset w10, eq - subs x10, x16, x10 - ngc x11, xzr - mov x12, #42915 - movk x12, #32382, lsl #16 - movk x12, #47689, lsl #32 - movk x12, #18351, lsl #48 - adds x10, x10, x12 - cinc x11, x11, hs - cmp x11, #0 - cset w11, eq - mov x12, #24535 - movk x12, #7886, lsl #16 - movk x12, #45453, lsl #32 - movk x12, #53147, lsl #48 - sub x11, x13, x11 - add x12, x11, x12 -LBB8_11: - stp x14, x9, [x8] - stp x10, x12, [x8, #16] - ldp x20, x19, [sp, #32] - ldp x22, x21, [sp, #16] - ldp x24, x23, [sp], #48 - .cfi_def_cfa_offset 0 - .cfi_restore w19 - .cfi_restore w20 - .cfi_restore w21 - .cfi_restore w22 - .cfi_restore w23 - .cfi_restore w24 - ret - .cfi_endproc -``` diff --git a/docs/DATA_PROOF_HACHI_MIGRATION.md b/docs/DATA_PROOF_HACHI_MIGRATION.md deleted file mode 100644 index 696cb2ef..00000000 --- a/docs/DATA_PROOF_HACHI_MIGRATION.md +++ /dev/null @@ -1,524 +0,0 @@ -# Data Proof: Dory → Hachi Migration Analysis - -This document analyzes the impact of replacing Dory (pairing-based PCS) with Hachi (lattice-based PCS) on the **Data Proof** protocol used in Zero's OpenZoneBlockTx. - -Primary sources: - -- `../Cyclone-Megadoc/protocol-spec/beacon/data_proof.md` — the Data Proof spec -- `../Cyclone-Megadoc/protocol-spec/beacon/da.md` — the Semi-AVID-PR / VID protocol -- `~/Downloads/Semi-AVID-PR and Jolt.pdf` — Saeid's note on shared-SRS optimization -- `docs/HACHI_KEY_DELEGATION.md` — Hachi two-level protocol with equations -- `docs/HACHI_FOR_JOLT.md` — Jolt integration spec - ---- - -## 1. System Context: Where the Data Proof Sits - -Zero (Cyclone) settles zone blocks via a three-phase protocol: - -``` - PREPARE COMMIT OPEN - ───────── ────── ──── - Sequencer runs VID Sequencer submits Sequencer submits - with beacon validators CommitZoneBlockTx OpenZoneBlockTx - ────────────────── ────────────────── ────────────────── - Outputs: Includes: Includes: - • Column commits (h_j) • CoA • Jolt SNARK proof - • CoA • Column commits • DATA PROOF ◄── this document - • RS-encoded chunks • State root -``` - -The Data Proof bridges two independent commitment systems: - -- **DA layer** (Prepare): column commitments \(h_j = \text{Pedersen}(\text{column}_j)\) via Semi-AVID-PR -- **Jolt ZKVM** (Open): polynomial commitment to the same data matrix \(U\) via Dory - -It proves: "the data Jolt proved about is the same data the DA layer committed to." - -### 1.1 Two data objects (critical distinction) - -There are two different objects that are easy to conflate: - -1. **DA block data \(U\)**: raw zone-block transaction bytes, transformed into field elements for VID and committed column-wise in Prepare/Commit. -2. **Jolt witness tables**: execution-trace polynomials committed in Stage 8 (one-hot address tables, increment tables, and optional advice tables). - -The Data Proof is the bridge between these worlds. The "native small values" observation applies strongly to Jolt witness tables, but not automatically to raw DA block bytes. - -### 1.2 Current DA packing path - -Current VID docs use BLS12-381 field arithmetic for encoding. The documented packing rule is: - -- each 32-byte field element stores 31 bytes of payload. - -So the current DA path is: - -`raw bytes -> BLS12-381 field elements -> matrix U -> RS encode -> Pedersen column commitments`. - ---- - -## 2. Current Data Proof Protocol (Dory-Based) - -### 2.1 Setup - -- Data matrix \(U \in \mathbb{F}^{L \times k}\) encodes the zone block. -- DA column commitments: \(C_j = \sum_{i} U[i,j] \cdot g_i \in \mathbb{G}_1\) (Pedersen with key \(\vec{g}\)). -- Polynomial: \(f(X) = \sum_{i,j} U[i,j] \cdot X^{j + ki}\). -- Evaluation claim: \(f(x) = y\). - -### 2.2 VMV Decomposition (PCS-Independent) - -The evaluation factors as a vector-matrix-vector product: - -\[ -f(x) = \vec{b}^T U \vec{a} -\] - -where \(\vec{a} = (1, x, \ldots, x^{k-1})\) and \(\vec{b} = (1, x^k, \ldots, x^{(L-1)k})\). - -Defining \(\vec{w} = U\vec{a} \in \mathbb{F}^L\), the claim becomes: know \(\vec{w}\) with \(\langle \vec{w}, \vec{b} \rangle = y\) and \(\vec{w}\) consistent with \(U\). - -### 2.3 BabyHyrax Baseline (Non-Succinct) - -Prover sends \(\vec{w}\) in the clear (\(O(L)\) field elements). Verifier checks: - -1. **Correctness**: \(\langle \vec{w}, \vec{b} \rangle \stackrel{?}{=} y\) -2. **Consistency**: \(\sum_i w_i \cdot g_i \stackrel{?}{=} \sum_j a_j \cdot C_j\) - -### 2.4 Homomorphic Reduction (Key Trick) - -Because Pedersen is **linear in the message**, the verifier computes locally: - -\[ -C' = \sum_{j \in [k]} a_j \cdot C_j = \sum_j a_j \left(\sum_i U[i,j] \cdot g_i\right) = \sum_i \left(\sum_j U[i,j] \cdot a_j\right) g_i = \text{Commit}(\vec{w}, \vec{g}) -\] - -This \(O(k)\) MSM gives the verifier a commitment to \(\vec{w}\) without receiving it. The problem reduces to: "prove you know \(\vec{w}\) that opens \(C'\) and satisfies \(\langle \vec{w}, \vec{b} \rangle = y\)." - -### 2.5 Dory Engine (Succinct) - -1. **Lift**: \(C'' = e(C', h) \in \mathbb{G}_T\) (pairing) -2. **Recursive folding**: \(\log L\) rounds, each halving vector size via challenges \(\alpha_i\) -3. **Key-folding**: precomputed \(\Delta\) values let verifier fold commitment keys in \(O(1)\) per round -4. **Tensor structure**: \(\vec{b}\)'s multiplicative structure lets verifier compute final folded scalar in \(O(\log L)\) -5. **Final check**: one multi-pairing in \(\mathbb{G}_T\) - -### 2.6 Shared-SRS Optimization (Saeid's PDF) - -If Dory generators \(g_i = \tau^i \cdot g\) come from the KZG SRS, then \(C_j = \hat{C}_j\) — the DA column commitment and the Dory sub-commitment are literally the same object. No extra consistency proof needed. - -### 2.7 Audit Notes: Conventions and Trust Model - -- **Row/column naming mismatch is mostly convention**: the Cyclone Data Proof spec is written in terms of **column commitments** \(C_j\), while Dory/Jolt implementation exposes **row commitments** as tier-1 hints. This is transpose-equivalent bookkeeping (with swapped VMV vectors), not a fundamental protocol contradiction. -- **The "lift" step is an abstraction, not a bug**: the spec writes \(C''=e(C',h)\). The implementation computes the pre-lift commitment by MSM over tier-1 commitments, then applies a pairing; same algebraic object, different decomposition. -- **CRS coupling caveat**: native Dory setup is transparent (random generators from public randomness). If generators are instead taken from a KZG powers-of-\(\tau\) SRS to force object identity, the system inherits KZG trusted-setup assumptions. No direct new attack is known from this audit, but transparency is no longer a valid claim in that mode. - ---- - -## 3. The Gadget Decomposition Problem - -### 3.1 Why Ajtai Is Not Linearly Homomorphic - -Ajtai commitment: \(h_j = A \cdot G^{-1}(u_j)\), where \(G^{-1}\) is base-\(b\) digit decomposition. - -The map \(u \mapsto G^{-1}(u)\) is **not linear** (carries break digit decomposition): - -\[ -\sum_j a_j \cdot G^{-1}(u_j) \neq G^{-1}\!\left(\sum_j a_j \cdot u_j\right) -\] - -Therefore: - -\[ -\sum_j a_j \cdot h_j = A \cdot \underbrace{\left(\sum_j a_j \cdot G^{-1}(u_j)\right)}_{\neq\, G^{-1}(w)} \neq A \cdot G^{-1}(w) -\] - -The verifier's linear combination produces something that is **not** a valid Ajtai commitment to \(w = Ua\). - -**This breaks the Data Proof's core trick.** (Confirmed by `HACHI_KEY_DELEGATION.md` which states: "The earlier 'homomorphic batching' idea ... is NOT valid for Hachi commitments, because the commitment map includes digit decompositions and is not linear.") - -### 3.2 Consequences for Semi-AVID-PR - -The DA protocol's validator check (`da.md` lines 109–114) relies on: - -\[ -\text{Code.Encode}(\text{Commit}(u_1), \ldots, \text{Commit}(u_k))_i = \text{Commit}(\text{Code.Encode}(u_1, \ldots, u_k)_i) -\] - -This is exactly linearity of the commitment map. With Ajtai + gadget decomposition, this identity fails. **Naively swapping Pedersen for Ajtai in the DA layer breaks Semi-AVID-PR's verification mechanism.** - -### 3.3 Consequences for the Shared-SRS Trick - -The Dory trick relied on a triple coincidence: - -1. Pedersen linearity → homomorphic reduction works -2. Shared generators → column commitment = Dory sub-commitment (zero overhead) -3. Pairing lift → enters Dory IPA world - -All three are lost with Hachi. There is no analog of "column commit = sub-commitment" with the two-layer Ajtai structure. - ---- - -## 4. How Hachi Solves the Gadget Linearity Problem - -Hachi's own opening protocol (Step B, `HACHI_KEY_DELEGATION.md` §1.3) never relies on homomorphic combination of Ajtai commitments. Instead, it uses a **auxiliary commitment + fold** mechanism. - -### 4.1 The Pattern - -Given committed blocks \(f_i\) with decompositions \(s_i = G^{-1}(f_i)\) and inner commitments \(t_i = A \cdot s_i\): - -**Step 1: Prover sends fresh auxiliary commitment.** -- Compute partial evaluations: \(w_i = a^T \cdot G \cdot s_i = a^T \cdot f_i\) -- Freshly decompose and commit: \(\hat{w}_i = G^{-1}(w_i)\), \(v = D \cdot \hat{w}\) -- Send \(v\) (first prover message) - -**Step 2: Verifier sends fold challenge.** -- Random sparse \(c = (c_1, \ldots, c_{2^r})\) - -**Step 3: Prover folds on already-decomposed witnesses.** -- \(z = \sum_i c_i \cdot s_i\) (linear combination of vectors already in decomposed form) -- Send \(z\) - -**Step 4: Verifier checks dual consistency through committed witnesses.** - -The verifier does **not** receive all \(t_i\) explicitly. Instead, it checks a stacked relation over witnesses \((\hat{w}, \hat{t}, z)\): - -- **Main-binding constraint**: \(B\hat{t} = u\) (binds \(\hat{t}\) to the public commitment \(u\)). -- **Eq. 19 (fold-inner consistency)**: \(A z = (c^T \otimes G_{n_A}) \cdot \hat{t}\). -- **Eq. 18 (fold-partials consistency)**: \((a^T G)\cdot z = (c^T \otimes G_1)\cdot \hat{w}\). - -In the full protocol, \(z\) is represented via redecomposition \(z = J\hat{z}\), and these constraints are enforced inside Step C's ring-switch + sumcheck system. - -### 4.2 Why This Works - -The nonlinearity of \(G^{-1}\) is quarantined to two places where it doesn't interact with linear combinations: - -- **Commitment time** (one-time): \(s_i = G^{-1}(f_i)\) -- **Auxiliary commitment time** (per reduction): \(\hat{w}_i = G^{-1}(w_i)\) - -All subsequent operations use only: -- Linearity of \(A\) (matrix-vector multiply) -- Linearity of \(G\) (the forward gadget map, trivially a matrix) -- The identity \(G \cdot G^{-1}(x) = x\) - -### 4.3 Soundness Sketch - -If the prover commits to incorrect partial evaluations \(w'_i \neq a^T f_i\), then after seeing random \(c\), it must still produce witnesses satisfying: - -1. \(B\hat{t}=u\) (bound to the committed polynomial), -2. \(A z = (c^T \otimes G_{n_A})\hat{t}\), -3. \((a^T G)z = (c^T \otimes G_1)\hat{w}\), -4. the opening equation. - -These constraints force one folded witness \(z\) to satisfy two independently anchored linear views (main-commit anchor and auxiliary/evaluation anchor). A cheating assignment survives only with negligible probability over \(c\), assuming standard ROM/low-degree soundness used by Hachi. - ---- - -## 5. Adapted Data Proof Architecture - -### 5.1 Option A: Keep Pedersen for DA, Add Bridge Proof (Recommended) - -``` -┌──────────────────────────────────────────────────────┐ -│ DA Layer (unchanged) │ -│ Column commits h_j = Pedersen(column_j) in G₁ │ -│ Semi-AVID-PR linearity argument intact │ -└──────────────────────┬───────────────────────────────┘ - │ h_1,...,h_k public - ▼ -┌──────────────────────────────────────────────────────┐ -│ BRIDGE PROOF (new component) │ -│ │ -│ Prover: commit same U via Hachi (two-layer Ajtai) │ -│ Prove consistency using Hachi's Step B pattern: │ -│ 1. Send auxiliary commitment v = D · ĝ(w) │ -│ 2. Receive fold challenge c │ -│ 3. Send folded witness z = Σ c_i · s_i │ -│ 4. Verifier checks stacked constraints │ -│ (Bĥt=u, Eq.18, Eq.19, opening equation) │ -└──────────────────────┬───────────────────────────────┘ - ▼ -┌──────────────────────────────────────────────────────┐ -│ HACHI OPENING PROOF │ -│ Step B: evaluation reduction │ -│ Step C: ring-switch + sumcheck over F_{q^k} │ -└──────────────────────────────────────────────────────┘ -``` - -**Trade-offs:** - -- (+) DA layer untouched, Semi-AVID-PR works as-is -- (+) Bridge proof uses Hachi's own machinery (no new primitives) -- (-) Bridge proof adds prover cost (one auxiliary commitment + one fold) -- (-) Loses the "zero-overhead" consistency of the Dory shared-SRS trick -- (-) Requires an explicit DA↔Hachi consistency relation (not provided "for free" by Jolt Stage-8 batching) - -### 5.2 Option B: Lattice-Native DA Without Gadget Decomposition - -If data elements are bounded (e.g., bits, one-hot entries), use Ajtai **without** gadget decomposition: - -\[ -h_j = A \cdot u_j \quad\text{where } \|u_j\|_\infty \leq B -\] - -Module-SIS binding holds for short pre-images. The commitment map \(u \mapsto A \cdot u\) IS linear, so Semi-AVID-PR works and the homomorphic reduction works. - -**Trade-offs:** - -- (+) Full linearity restored, both DA and Data Proof work natively -- (+) No bridge proof needed -- (-) Only works if data entries have bounded norm (must satisfy SIS security bound) -- (-) Loses binding for arbitrary field data with large entries -- (-) Needs careful parameter analysis: SIS norm bound vs data magnitude -- (-) RS encoding can amplify entry magnitude; bounded raw data does **not** automatically imply bounded encoded chunks - -**Important clarification:** Option B may be plausible for some **Jolt witness commitments** (one-hot + bounded increments), but it is generally not plausible for current **raw DA block bytes** without a redesign of DA data representation. - -### 5.3 Option C: Redesign DA Around Hachi's Commitment Structure - -This is the "single commitment family" path: DA and Open both use Hachi, and Semi-AVID-PR's linearity-based check is replaced. - -#### 5.3.1 What must be replaced - -Current Semi-AVID-PR validator logic uses: -\[ -\mathrm{Code.Encode}(h_1,\ldots,h_k)_i \stackrel{?}{=} \mathrm{Commit}(c_i) -\] -which depends on commitment linearity. With Hachi commitments, this check is invalid. - -So Option C must replace that check with a proof-backed statement: -"the received chunk \(c_i\) is the correct RS encoding of data committed in Hachi." - -#### 5.3.2 Option C1 (incremental): per-validator chunk-consistency proof - -For each validator \(i\), the sequencer sends chunk \(c_i\) plus proof \(\pi_i\): - -1. Commit original columns with Hachi: \((u_j)_{j\in[k]}\). -2. Validator derives a random evaluation point \(r\) (Fiat-Shamir). -3. Prover opens needed committed columns and chunk at \(r\) (batched opening). -4. Verifier checks RS relation at \(r\): - \[ - c_i(r) \stackrel{?}{=} \sum_{j=1}^k \lambda_{i,j}\,u_j(r), - \] - where \(\lambda_{i,j}\) are public RS coefficients. - -**Pros**: easiest way to remove linearity assumption. - -**Cons**: proof load scales with validators/chunks unless heavily shared/batched. - -#### 5.3.3 Option C2 (scalable target): one global encoding proof - -Commit both source and encoded tables under Hachi: - -- source columns \(U\), -- encoded columns \(E=\mathrm{Code.Encode}(U)\). - -Publish one global proof \(\pi_{\mathrm{enc}}\) that enforces: -\[ -\forall i,\quad E_{:,i} = \sum_{j=1}^k \lambda_{i,j} U_{:,j}. -\] - -Implementation direction: encode this as a stacked linear relation and prove it with the same ring-switch + sumcheck machinery used by Hachi Step C. - -Then validator-side chunk checks become: - -- verify \(\pi_{\mathrm{enc}}\) once per block (or once on-chain), -- verify chunk opening against the committed \(E_{:,i}\). - -**Pros**: amortizes proof cost; cleanest long-term Option C architecture. - -**Cons**: substantial protocol redesign and engineering effort. - -#### 5.3.4 Recommended Option C roadmap - -1. **Spec freeze**: define exact matrix orientation and RS coefficient indexing (remove row/column ambiguity first). -2. **C1 prototype**: implement per-validator proof to de-risk correctness quickly. -3. **C2 upgrade**: replace C1 with one global encoding proof once relation encoding is stable. -4. **Integrate with Open path**: share transcript/challenge infrastructure with Hachi opening to avoid duplicated proof logic. - -### 5.4 Do we still need byte packing? What if DA encoding uses a 128-bit field? - -#### 5.4.1 Packing is still needed for byte-native blocks - -If zone blocks remain byte-native, any field-based RS pipeline still needs a reversible bytes-to-field encoding step. So "packing" does not disappear; only its efficiency and algebraic target field change. - -#### 5.4.2 Can we switch only the RS field to 128-bit under current Pedersen DA? - -Not safely. Semi-AVID-PR's validator check relies on commitment linearity over the **same scalar field** as coding. - -If RS is over \(F_p\) (128-bit) but Pedersen scalars are over \(F_r\) (e.g. BLS12-381 scalar field), then generally: -\[ -\mathrm{Commit}_{F_r}\!\left(\sum_j \lambda_j u_j \bmod p\right) -\neq -\sum_j \iota(\lambda_j)\,\mathrm{Commit}_{F_r}(u_j), -\] -because \(\iota\) is only an encoding into \(F_r\), not a field isomorphism \(F_p \cong F_r\). - -So a "field-size-only swap" breaks the linear-consistency check unless commitments are also moved to the same algebraic field/ring used by encoding (which is exactly the kind of redesign Option C entails). - -#### 5.4.3 Packing efficiency trade-off at 128-bit - -- Current BLS12-style rule: 31 bytes per field element. -- Simple 128-bit prime rule (byte-aligned, fixed): typically 15 bytes per element. - -So naive 128-bit encoding can **increase** element count for the same raw bytes. A near-\(2^{128}\) prime (e.g. P13) allows more specialized packing strategies, but those add codec complexity and must preserve deterministic invertibility. - -Concrete scale example for a 1 MiB block: - -- 31-byte packing: \(\lceil 1{,}048{,}576 / 31 \rceil = 33{,}826\) field elements. -- 15-byte packing: \(\lceil 1{,}048{,}576 / 15 \rceil = 69{,}906\) field elements. - -That is about \(2.07\times\) more elements under naive 15-byte packing. - -For a near-\(2^{128}\) prime like P13 (\(2^{128}-8207\)), only 8,207 of the \(2^{128}\) 16-byte values are out of range (\(\approx 2^{-115}\) fraction). This suggests a specialized 16-byte codec may be practical, but it still requires explicit overflow handling and adversarially robust, deterministic decoding rules. - -#### 5.4.4 Is 128-bit security sufficient? - -For Option C-style unified Hachi commitments over a 128-bit modulus, the current repo analysis indicates "yes" for candidate parameters: - -- `docs/HACHI_FOR_JOLT.md` Section 5.6 reports Module-SIS estimates above 128 bits for recommended P13 configurations, including conservative rows still around ~131 bits in the tight regime. - -Additional notes: - -- RS availability/correctness is information-theoretic and mainly requires \(q \gg n\) for committee-size \(n\) (128-bit is far above practical \(n\)). -- Any randomized polynomial-identity checks over a 128-bit field still need an explicit soundness budget (\(\approx \deg/q\)-style terms), but this is typically negligible at deployed degrees. - -#### 5.4.5 Security checklist if DA moves to 128-bit encoding - -If Option C moves DA encoding and commitments into a 128-bit algebraic domain, security is fine **only if** the codec is part of the proved relation: - -1. **Canonical decoding**: bytes \(\leftrightarrow\) field elements must be injective and deterministic. -2. **Metadata binding**: codec ID, padding length, and any auxiliary channels must be included in the committed message and verified. -3. **No hidden side channels**: if using an overflow-assisted codec, the overflow bitmap/payload must be committed and proven consistent. -4. **Soundness accounting**: add codec-related checks to the same transcript/sumcheck soundness budget (do not treat codec validity as out-of-band). - -### 5.5 Option C codec decision table (implementation) - -To make Option C concrete, define: - -- \(B\): block size in bytes, -- \(e\): payload bytes per field element (codec-dependent), -- \(w\): serialized bytes per field element (\(w=16\) for 128-bit field, \(w=32\) for current BLS-style), -- \(n\): committee size, -- \(S_0\): max share size (bytes). - -Then: -\[ -M=\left\lceil\frac{B}{e}\right\rceil,\quad -d_{\max}=\left\lfloor\frac{S_0}{w}\right\rfloor,\quad -k=\left\lceil\frac{M}{d_{\max}}\right\rceil,\quad -d=\mathrm{nextPow2}\!\left(\left\lceil\frac{M}{k}\right\rceil\right),\quad -r=\frac{k}{n}. -\] - -#### 5.5.1 Codec options - -| Codec | Domain | Payload \(e\) | Deterministic worst-case size | Engineering risk | Notes | -|---|---|---:|---|---|---| -| **Current baseline** | BLS-style | 31/elem (\(w=32\)) | Yes | Low | Existing VID docs/spec assumptions | -| **Fixed-15** | 128-bit field | 15/elem (\(w=16\)) | Yes | Low | Simple, robust, easy to audit | -| **Packed-16 + overflow channel** | 128-bit near-\(2^{128}\) prime | ~16/elem expected | **Only if overflow is explicit and bounded** | Medium/High | Higher average efficiency, but adversarial overflow handling is mandatory | - -#### 5.5.2 Concrete comparison (1 MiB block, \(n=1024\), \(S_0=4096\)) - -| Codec | \(M\) source elems | \(d_{\max}\) | \(k\) | \(d\) | \(r=k/n\) | -|---|---:|---:|---:|---:|---:| -| 31/32 baseline | 33,826 | 128 | 265 | 128 | 0.2588 | -| 15/16 fixed | 69,906 | 256 | 274 | 256 | 0.2676 | -| 16/16 idealized | 65,536 | 256 | 256 | 256 | 0.2500 | - -Interpretation: - -- Moving from 31/32 to fixed 15/16 approximately doubles source element count \(M\), but because \(d_{\max}\) also doubles (16-byte elements), \(k\) only changes moderately in this sizing regime. -- "16/16 idealized" shows the upper bound if every 16-byte chunk maps directly. - -#### 5.5.3 Overflow caveat for packed-16 codecs - -For P13 (\(2^{128}-8207\)), random data hits out-of-range 16-byte words with probability \(\approx 2^{-115}\), so average overhead is negligible. - -But protocol design must be **adversarially robust**, not average-case: - -- a malicious block producer can choose bytes that maximize overflows, -- so overflow artifacts must be explicit and bounded, -- and a deterministic fallback (e.g., re-encode with fixed-15) should be specified when overflow budget is exceeded. - -#### 5.5.4 Recommendation for Option C - -1. **Phase 1 (safe default):** fixed-15 codec for consensus-critical deployment. -2. **Phase 2 (optional optimization):** packed-16 codec behind a strict overflow-cap + deterministic fallback rule. -3. In both phases, include `codec_id` and all codec metadata/artifacts in the committed statement and in the Option C encoding proof relation. - ---- - -## 6. Additional Domain Mismatches - -### 6.1 Scalar Field vs Ring Domain - -| | Data Proof (Dory) | Hachi | -|---|---|---| -| \(\vec{w}\) lives in | \(\mathbb{F}^L\) (scalar field) | \(R_q^{2^r}\) (ring) | -| Inner product | \(\langle w, b \rangle = y\) in \(\mathbb{F}\) | \(b^T w\) in \(R_q\), then ring-switch to \(F_{q^k}\) | -| Evaluation point | \(x \in \mathbb{F}\) | \(x \in R_q^\ell\) or \(\mathbf{r} \in F_{q'}^n\) (two-field) | - -For the packed regime \(d > 1\), `HACHI_FOR_JOLT.md` §4.3.2A identifies a typing issue: the partial evaluation \(w_i = a^T G s_i\) with \(a \in F_{q'}^{2^m}\) and \(G s_i \in R_q^{2^m}\) produces \(w_i \in F_{q'}[X]/(X^d+1)\), not \(R_q\). Resolution requires a Phase 0 inner evaluation sumcheck (\(\alpha = \log_2 d\) extra rounds). - -### 6.2 Jolt CommitmentScheme Trait - -Jolt's `CommitmentScheme` trait (`jolt-core/src/poly/commitment/commitment_scheme.rs`) assumes: - -1. `combine_commitments(commitments, coeffs)` — homomorphic combination (broken for Ajtai) -2. `opening_point: &[F]` — single-field opening (Hachi needs two moduli \(q, q'\)) -3. `Polynomial` — field-native polynomial (Hachi works over rings) -4. Stage-8 batch opening relies on additive combination (`combine_hints` / `combine_commitments`) for internal RLC reductions; this is not the DA Data Proof itself, but it depends on the same algebraic property that breaks under naive Ajtai - -Significant trait redesign needed to accommodate Hachi's algebraic structure. - ---- - -## 7. Summary of Findings - -| Property | Dory (current) | Hachi (proposed) | Status | -|---|---|---|---| -| Commitment linearity | Pedersen: fully linear | Ajtai + \(G^{-1}\): **not linear** | **Breaking change** | -| Homomorphic reduction | Verifier derives \(C'\) for free | Must use auxiliary commit + fold | Redesigned | -| DA layer compatibility | Semi-AVID-PR linearity intact | Breaks with naive Ajtai swap | Keep Pedersen or use Option B | -| Shared-SRS / shared-matrix | Column commit = Dory sub-commit | No analog exists | Bridge proof needed | -| Succinct opening engine | Pairing lift + IPA folding | Ring-switch + sumcheck | Full replacement | -| Verifier efficiency | \(O(\log L)\) via key-folding + tensor trick | \(O(\ell)\) via sumcheck | Comparable | -| Security | Pre-quantum | Post-quantum | Upgrade | -| Setup (native mode) | Transparent (public randomness) | Transparent (public seed) | Comparable | -| Setup (shared-SRS mode) | Inherits KZG trusted setup | N/A | Trust-model downgrade | - -### What survives unchanged - -- VMV decomposition \(f(x) = \vec{b}^T U \vec{a}\) (pure algebra) -- Prepare-Commit-Open lifecycle (protocol-level) -- Semi-AVID-PR structure (if Pedersen is kept for DA) - -### What must be rebuilt - -- The homomorphic reduction → auxiliary commitment + fold consistency pattern -- The Dory IPA engine → Hachi Step B + Step C -- The bridge between DA commitments and PCS commitment -- Jolt's `CommitmentScheme` trait boundary - ---- - -## 8. Audit Outcomes (Updated) - -### 8.1 Resolved points - -1. **Row vs column mismatch**: resolved as a convention/transpose issue, not a protocol break. -2. **Data Proof vs Jolt Stage-8**: Stage-8 batching is not the DA bridge itself, but migration is entangled because both rely on additive homomorphism. -3. **"Lift" wording**: \(C''=e(C',h)\) is an acceptable abstraction of implementation flow. -4. **Eq. 19 verifier access**: direct check against hidden \(t_i\) is not possible from \(u\) alone; must be enforced via stacked witness constraints (\(B\hat t=u\), Eq. 18, Eq. 19) inside Step C. -5. **Shared-SRS trust model**: if Dory reuses KZG powers-of-\(\tau\), transparency is lost and trusted-setup assumptions are inherited. -6. **Data-model split**: "native-small-values" is accurate for many Jolt witness tables, but not for raw DA block bytes under current packing. -7. **128-bit RS under current Pedersen DA**: not a drop-in change; field mismatch breaks the linear-consistency invariant unless commitment algebra is redesigned too. - -### 8.2 Remaining quantitative work - -1. **Sparse-challenge soundness constants**: derive explicit end-to-end error term for the exact challenge distribution and fold depth used in production parameters. -2. **Option B bound under RS encoding**: compute concrete SIS margins after encoding (not just on raw witness entries). -3. **Option A bridge cost**: benchmark prover/verifier overhead relative to base Hachi opening. -4. **Option C execution plan**: evaluate C1 vs C2 with concrete proof-size and validator-throughput targets. -5. **Option C + 128-bit codec design**: choose a deterministic byte-to-field codec (fixed 15-byte vs near-\(2^{128}\) specialized), then quantify throughput/proof-size impact. diff --git a/docs/FP128_MUL_ASM_AARCH64_ANALYSIS.md b/docs/FP128_MUL_ASM_AARCH64_ANALYSIS.md deleted file mode 100644 index d793fb42..00000000 --- a/docs/FP128_MUL_ASM_AARCH64_ANALYSIS.md +++ /dev/null @@ -1,146 +0,0 @@ -# Fp128 Mul AArch64 Assembly Analysis - -Fresh rebuild was used to generate this assembly: - -- Command: `cargo clean && cargo rustc --bench codegen_probe --release -- --emit asm` -- Assembly file: `target/release/deps/codegen_probe-1c486cb16985754a.s` -- Symbol analyzed: `_fp128_mul_field` -- Symbol span in `.s`: lines `97..141` -- Target architecture: `arm64` - -## Instruction census (exact for `_fp128_mul_field`) - -- `mul`: `7` -- `umulh`: `6` -- Total multiply-class ops: `13` -- `adds`: `9` -- `adcs`: `3` -- `adc`: `3` -- `cset`: `5` -- `cinc`: `2` -- `csel`: `2` -- `ldp`: `2` -- `stp`: `1` - -This is a compact, spill-free kernel. - -## Fresh benchmark anchors - -From fresh run medians: - -- `field_mul_only/mul_chain_2048`: - - Fp128: `11.948 us` total -> `5.8340 ns/op` - - BN254: `22.225 us` total -> `10.8521 ns/op` - - ratio `BN254/Fp128 ~= 1.86x` -- `field_mul_isolated` (Fp128 only): - - `fp128_mul_single`: `2.4427 ns` - - `fp128_pair_passthrough`: `1.8375 ns` - - `fp128_black_box_only`: `0.55984 ns` - - `fp128_mul_8way_independent`: `16.247 ns` total - - `fp128_8way_passthrough`: `4.8632 ns` total - -Derived estimates: - -- `single - black_box_only`: `1.8829 ns` (`~6.59 cycles @3.5GHz`) -- `(8way_independent - 8way_passthrough)/8`: `1.4230 ns/op` (`~4.98 cycles @3.5GHz`) -- chain-latency proxy: `5.8340 ns/op` (`~20.42 cycles @3.5GHz`) - -Interpretation: - -- Throughput can approach ~5 cycles/op in independent streams. -- Dependent latency stays around ~20 cycles/op, matching the carry-chain structure in codegen. - -## Manual phase annotation - -`_fp128_mul_field` naturally splits into 4 stages: - -1. **Load limbs** - - `ldp x9,x10,[x0]` and `ldp x11,x12,[x1]` -2. **2x2 schoolbook + row carries** - - 4 low products + 4 highs arranged with `adds/adcs/adc` -3. **Solinas fold-1 with `c=8207`** - - `mul/umulh` by `c` on upper carry limb path -4. **Fold-2 + canonicalize** - - add `c` once, then choose reduced/non-reduced using final carry condition (`csel`) - -The emitted code is already branchless in the hot path (`csel` only, no branch mispredict risk). - -## Entire raw `_fp128_mul_field` assembly - -```asm -_fp128_mul_field: - .cfi_startproc - ldp x9, x10, [x0] - ldp x11, x12, [x1] - mul x13, x11, x9 - umulh x14, x11, x9 - mul x15, x12, x9 - umulh x9, x12, x9 - umulh x16, x11, x10 - mul x11, x11, x10 - umulh x17, x12, x10 - mul x10, x12, x10 - adds x11, x11, x14 - cset w12, hs - adds x9, x9, x16 - cset w14, hs - adds x9, x9, x10 - cinc x10, x14, hs - adds x11, x11, x15 - adcs x9, x9, x12 - adc x10, x17, x10 - mov w12, #8207 - mul x14, x9, x12 - umulh x9, x9, x12 - umulh x15, x10, x12 - mul x10, x10, x12 - adds x9, x9, x11 - cset w11, hs - adds x9, x9, x10 - cinc x10, x11, hs - adds x11, x14, x13 - cset w13, hs - adcs x14, x9, xzr - adc x10, x15, x10 - mul x10, x10, x12 - adds x10, x11, x10 - cset w11, hs - adc x9, x9, x13 - adds x12, x10, x12 - adcs x11, x14, x11 - csel x9, x11, x9, hs - csel x10, x12, x10, hs - stp x10, x9, [x8] - ret - .cfi_endproc -``` - -## AArch64 carry-path attempt result - -I implemented and benchmarked an AArch64-specific inline-asm carry helper path (`adds/adc` helper blocks for 2/3/4-term limb sums), then reverted it. - -Observed outcome (after thermal cooldown rerun): - -- It did **not** improve this kernel. -- Primary reason: helper-level asm barriers reduced global scheduling quality and caused overall slowdown versus LLVM's current fully inlined codegen. - -Conclusion: micro-asm helpers are not the right granularity here. - -## Optimization opportunities spotted - -These are the only realistic next knobs I see from the current asm: - -1. **Whole-kernel AArch64 asm (not helper-level asm)** - - There are several `cset` + `cinc` carry extractions. - - If we pursue arch-specific optimization, it should be a single end-to-end kernel so NZCV stays live across the whole sequence. - - This is still the main remaining codegen-level opportunity, but with higher implementation complexity. - -2. **Register-pressure-aware scheduling for fold boundaries** - - The kernel is short already; gains here are likely small. - - Might improve by interleaving one fold multiply earlier, but LLVM already does a reasonable schedule. - -3. **Target-specific build flags** - - Ensure release benches use `target-cpu=native` for final local tuning runs. - - This can help instruction selection/scheduling but is environment-specific. - -No obvious algorithmic simplification is left at Rust level without moving to explicit architecture-specific code. diff --git a/docs/HACHI_FOR_JOLT.md b/docs/HACHI_FOR_JOLT.md deleted file mode 100644 index b9801f41..00000000 --- a/docs/HACHI_FOR_JOLT.md +++ /dev/null @@ -1,2234 +0,0 @@ -# Hachi for Jolt (128-bit field arithmetization + lattice PCS commitments) - -This document is a **Jolt-first** design spec for integrating a Hachi-like lattice PCS into Jolt’s architecture, under the hard requirement that Jolt’s sumchecks and transcript challenges live in a **128-bit prime field** \(F_{q'}\). - -It incorporates: - -- Jolt’s Stage 8 interface shape (open many polynomials at one point in \(F_{q'}\)), -- a recommended **field-to-ring embedding** choice (SuperNeo-style coefficient embedding) for one-hot/bit tables, -- a two-modulus “ring switching + modulus switching” opening reduction that stays verifier-native in \(F_{q'}\), -- concrete parameter crunching, including an estimate for the extra modulus quotient witness \(s\). - -This is intentionally “implementation-ready”: it names concrete witness objects, indicates what is public vs committed vs proven, and highlights which parts require changing Jolt’s PCS trait boundary. - ---- - -## 0. Executive summary (what we’re building) - -Jolt’s prover does: - -1. **Commit** to a collection of multilinear polynomials (witness tables). -2. Run application sumchecks (Stages 1–7) over a field \(F_{q'}\) and accumulate evaluation claims. -3. In **Stage 8**, prove a batch opening: many claims \(P_i(\mathbf r)=v_i\) at a **single point** \(\mathbf r\in F_{q'}^m\). - -We want: - -- commitments that are fast for one-hot/bit-heavy tables (so we prefer **small-modulus ring arithmetic** / NTT-friendly layouts), and -- an opening proof that can satisfy Jolt Stage 8’s interface (open at \(\mathbf r\in F_{q'}^m\)) while avoiding expensive ring multiplication. - -The core idea is: - -- Commit over a cyclotomic ring \(R_q = Z_q[X]/(X^d+1)\) with a **small** prime \(q\) (e.g. 32-bit). -- Verify openings using **ring switching evaluated in the large field** \(F_{q'}\) (128-bit) so all “point-derived” arithmetic is native to Jolt’s world. -- Add one extra quotient witness \(s\) so that “equalities mod \(q\)” become honest integer equalities when checked inside \(F_{q'}\). - ---- - -## 1. Requirements and constraints - -### 1.1 Jolt requirement: 128-bit prime field - -Jolt’s arithmetization / sumchecks require that the field characteristic exceeds certain \(u64\cdot u64\) accumulation bounds, so we must work over a **128-bit prime**: - -\[ -F_{q'}\quad\text{with } q' \approx 2^{128}. -\] - -This means we cannot “cheat” by using a small prime and moving to an extension \(F_{q^k}\), because the characteristic bound is on the base characteristic. - -### 1.2 Jolt Stage 8 interface shape - -Jolt’s accumulated opening claims are of the form: - -\[ -(\mathbf r,\ v)\quad\text{with }\mathbf r\in F_{q'}^m,\ v\in F_{q'}. -\] - -See: - -- `../jolt/jolt-core/src/poly/opening_proof.rs`: `pub type Opening = (OpeningPoint<..., F>, F);` -- `../jolt/jolt-core/src/zkvm/prover.rs`: Stage 8 calls `PCS::prove(..., &opening_point.r, ...)`. - -So any PCS that plugs into Stage 8 must accept opening points in \(F_{q'}\). - -### 1.3 Witness regime: one-hot / bits - -Many Jolt witness tables are one-hot or bit-heavy (see `docs/ONE_HOT_COMMITMENT_COST_AND_GPU_PRG.md`). We want “pay-per-bit” style cost where possible, so we strongly prefer embeddings that keep bits sparse in the ring representation. - ---- - -## 2. Notation - -- **Small prime**: \(q \approx 2^{32}\). -- **Large prime**: \(q' \approx 2^{128}\). -- **Cyclotomic degree**: \(d=2^\alpha\) (e.g. \(d=1024\)). -- **Ring**: \(R_q := Z_q[X]/(X^d+1)\). -- **Integer ring**: \(Z\) is the integers, and \(Z[X]\) is the integer polynomial ring. -- **Digit base**: \(b := 2^{\mathrm{LOG\_BASIS}}\) (e.g. \(b=16\)). -- **Digit length**: \(\delta := \lceil \log_b q\rceil\). -- **Redecomposition length**: \(\tau\) (Hachi §4.2). -- **Jolt opening point**: \(\mathbf r\in F_{q'}^m\) (Stage 8). -- **Hachi split**: \(\ell=m+r\) where \(m\approx r\) and the coefficient table has length \(2^\ell\). -- **Block count / block size**: there are \(2^r\) blocks, each of length \(2^m\). - ---- - -## 3. Embedding choice for Jolt: SuperNeo coefficient embedding (recommended) - -### 3.1 Why not use an extension-field embedding here? - -Jolt’s constraint on the **field characteristic** forces the interactive proof algebra to live in a large *prime* field \(F_{q'}\). -So we do not get to pick a small prime \(q\) and move to an extension \(F_{q^k}\); that would change the characteristic. - -Therefore, any embedding machinery whose purpose is “handle points/values in \(F_{q^k}\)” is simply orthogonal to the Jolt setting. - -### 3.2 SuperNeo coefficient embedding preserves bit sparsity - -SuperNeo’s coefficient embedding (Def. 7 in SuperNeo) maps a length-\(d\) field block into one ring element by placing entries directly in the coefficient slots. In our power-of-two cyclotomic: - -\[ -v=(v_0,\dots,v_{d-1})\in F_q^d -\ \mapsto\ -\mathbf v(X)=\sum_{i=0}^{d-1} v_i X^i\in R_q. -\] - -For one-hot/bit tables, coefficients remain \(\{0,1\}\), which enables: - -- sparse gadget decomposition (often effectively \(\delta=1\) for bit witnesses), and -- “rot+add” multiplication by sparse polynomials if needed. - -See `docs/FIELD_EMBEDDINGS_SUPERNEO_VS_HACHI.md` for the math-first comparison and the explicit \(X^d+1\) inner-product transform. - ---- - -## 4. Two-field opening protocol (Option A): commit over \(R_q\), open at \(\mathbf r\in F_{q'}^m\) - -**Compatibility warning (important):** for the genuine two-field setting with coefficient packing \(d>1\) (e.g. \(d=1024\)), the clean correctness-first spec lives in `docs/TWO_FIELD_OPENING_D_GT_1_CLEAN.md`. - -The remainder of this Section 4 is an exploratory draft that mixes the original Hachi Step-B template (which assumes the evaluation point lives in the same algebraic domain as the ring arithmetic) with the Jolt requirement \(\mathbf r\in F_{q'}^n\). Use it for intuition only; do not treat it as a coherent d>1 protocol without reconciling the typing issues called out in that clean spec. - -### 4.1 High-level picture - -We implement a PCS with: - -- **Commitment-time arithmetic** over \(R_q\) (small \(q\)), -- **Verifier-side checking** (sumcheck challenges, equality polynomials, batching weights) over \(F_{q'}\), -- **Ring switching** to move ring equations to field equations at a random \(\alpha\in F_{q'}\), -- plus **modulus switching** slack \(q\cdot s\) so “mod-\(q\)” equalities make sense as integer equalities when checked in \(F_{q'}\). - -### 4.2 One opening claim - -Input statement (Stage 8 style): - -\[ -\text{Given commitment } C \text{ to } P,\ \mathbf r\in F_{q'}^m,\ v\in F_{q'},\ \text{prove } P(\mathbf r)=v. -\] - -Here \(P(\mathbf r)\) means: interpret the committed coefficients as integers (bitness makes this canonical), and evaluate the multilinear extension in \(F_{q'}\). - -### 4.3 Full “one opening proof step” (commit structure + full relation + witness) - -This section is the **self-contained core**: it writes the full relation/witness that the opening proof reduces to. - -#### 4.3.1 Commitment objects (Hachi §4.1 core, ring-side) - -Fix public commitment matrices (Ajtai-style) over \(R_q\): - -\[ -A\in R_q^{n_A\times (\delta\cdot 2^m)},\quad -B\in R_q^{n_B\times (n_A\delta\cdot 2^r)},\quad -D\in R_q^{n_D\times (\delta\cdot 2^r)}. -\] - -Let \(f\in R_q^{\le 1}[X_1,\dots,X_\ell]\) be the committed multilinear, with coefficient table -\((f_{i\parallel j})_{i\in\{0,1\}^r,\ j\in\{0,1\}^m}\subset R_q\). -Define the block slices: - -\[ -f_i^\top := (f_{i\parallel j})_{j\in\{0,1\}^m}\in R_q^{2^m}. -\] - -Gadget decomposition (base \(b\)) gives: - -\[ -s_i := G^{-1}_{2^m}(f_i)\in R_q^{2^m\delta}. -\] - -Inner and outer commitments: - -\[ -t_i := A s_i \in R_q^{n_A},\quad -\hat t_i := G^{-1}_{n_A}(t_i)\in R_q^{n_A\delta}, -\] - -\[ -u := B[\hat t_1;\dots;\hat t_{2^r}] \in R_q^{n_B}. -\] - -This \(u\) is the PCS commitment. - -#### 4.3.2 The opening-derived monomial vectors (Jolt field-side) - -Jolt Stage 8 provides a single opening point \(\mathbf r\in F_{q'}^{n}\) (after claim reductions). -In the special case \(d=1\) (no packing) we have \(n=\ell=m+r\). If \(d>1\), then typically \(n=\ell+\log_2(d)\); see Section 4.3.2A. -Define the usual multilinear monomial vectors in the **field** \(F_{q'}\): - -\[ -b^\top := (r_1^{i_1}\cdots r_r^{i_r})_{i\in\{0,1\}^r}\in (F_{q'})^{2^r}, -\qquad -a^\top := (r_{r+1}^{j_1}\cdots r_\ell^{j_m})_{j\in\{0,1\}^m}\in (F_{q'})^{2^m}. -\] - -These are the same objects as Hachi Eq. (12), but they are computed in \(F_{q'}\) (not in \(R_q\)). - -#### 4.3.2A Critical scope caveat: this Step-B template assumes *no packing* (\(d=1\)), or an additional “inner evaluation” reduction - -The rest of Section 4.3 copies Hachi’s Step-B structure (aux commitment to partial evaluations, then folding, then a stacked linear system) -and was written with the “ring-level variable count” \(\ell=m+r\) in mind. - -If we are in the packed regime \(d>1\) (e.g. \(d=1024\)), then Jolt’s scalar polynomial has: - -- \(N = d\cdot 2^\ell\) scalar coefficients, hence -- \(n := \log_2 N = \ell + \alpha\) variables, where \(\alpha := \log_2 d\). - -So the *opening point* naturally has \(n=\ell+\alpha\) coordinates \(\mathbf r\in F_{q'}^{\ell+\alpha}\), and the last \(\alpha\) coordinates index -the “inner coefficient slots” inside each packed ring element. - -As written, however, the monomial vectors \(a,b\) are defined using only \(\ell\) coordinates (the “outer” ones), and the protocol implicitly treats -each ring element’s \(d\) coefficients as if they were “already evaluated” or irrelevant. This is **not correct** for the Jolt setting when \(d>1\): - -- Multilinear evaluation over the inner \(\alpha\) coordinates uses **multilinear monomials** \(\prod_t r_{\ell+t}^{k_t}\). -- Ring switching at \(X=\alpha\) produces **power monomials** \(\alpha^k\) from the coefficient basis \(X^k\). - -These are algebraically different in general, so ring switching does not “automatically” account for the inner \(\alpha\) coordinates. - -Separately, the “aux commitment” step below defines \(w_i := a^\top G_{2^m}s_i\in R_q\) while \(a\in F_{q'}^{2^m}\) and \(G_{2^m}s_i\in R_q^{2^m}\). -That dot product is not an \(R_q\)-operation unless the weights \(a\) live in \(R_q\) (as in Hachi’s original extension-field-in-ring setting). -In the genuine two-field setting (\(F_{q'}\) independent of \(R_q\)), \(w_i\) would live in \(F_{q'}[X]/(X^d+1)\), not in \(R_q\), and the -subsequent digit decomposition and Ajtai commitment \(v := D\hat w\) over \(R_q\) are not well-defined. - -**Conclusion**: the Step-B template in Sections 4.3.3–4.3.6 is internally consistent in the special case \(d=1\) (no packing), -or if we first add a separate reduction that turns an \(n=\ell+\alpha\)-variate scalar opening into an \(\ell\)-variate ring-level statement -with ring-native weights (analogous in spirit to Hachi’s \(\psi/\mathrm{Tr}\) machinery, but not provided by ring switching alone). - -We keep the Step-B material below because it is still the right template for the **unstructured linear relation + ring/modulus switching + sumcheck** -pipeline once the statement is made well-typed. But for \(d>1\) in Jolt, an additional “inner evaluation” mechanism is required. - -#### 4.3.2B The fix for \(d>1\): inner evaluation reduction via sumcheck + batching into ring switching - -##### 4.3.2B.1 Why the algebraic "inner product to linear functional" bridge is unavailable - -In the original Hachi (extension-field setting), the psi/trace identity (Paper Theorem 2) -\(\mathrm{Tr}_H(\psi(\mathbf a)\cdot\sigma_{-1}(\psi(\mathbf b))) = (d/k)\langle \mathbf a,\mathbf b\rangle\) -algebraically converts the multilinear inner product (which IS the evaluation) into a ring operation -that can be committed to and ring-switched. This works because \(F_{q^k}\) embeds into \(R_q\) as -the fixed subring \(R_q^H\), giving the algebraic bridge. - -In the two-field setting, \(F_{q'}\) is an independent prime field with **no embedding into \(R_q\)**. -There is no psi map, no trace identity, and no algebraic way to convert "inner product of an -\(F_{q'}\)-vector with a coefficient vector" into a ring operation over \(R_q\). - -**What we use instead**: an \(\alpha\)-round inner sumcheck (Phase 0) that reduces the multilinear -inner-coordinate evaluation to a random-point claim, followed by a batching trick that converts -random-point claims into polynomial-evaluation claims (i.e. ring switching). This is the -**computational substitute** for the missing algebraic identity. It costs \(\alpha\) extra sumcheck -rounds and \(d\) extra field elements in the proof. - -##### 4.3.2B.2 Setup and notation - -Let \(\alpha := \log_2 d\) (e.g. \(\alpha=10\) for \(d=1024\)), \(n := \ell + \alpha\) (total scalar variable count), -opening point \(\mathbf r = (\mathbf r_{\mathrm{out}},\mathbf r_{\mathrm{in}}) \in F_{q'}^{\ell} \times F_{q'}^{\alpha}\), -scalar coefficients \(p_{I,k}\in\{0,1\}\) for \(I\in\{0,1\}^\ell,\; k\in\{0,1\}^\alpha\), -and ring element \(f_I(X) := \sum_{k=0}^{d-1} p_{I,k}\,X^k \in R_q\) (coefficient embedding). - -The correct scalar evaluation claim is: - -\[ -\boxed{ -v = \sum_{I\in\{0,1\}^\ell}\;\sum_{k\in\{0,1\}^\alpha} -\iota(p_{I,k})\cdot \mathrm{eq}(\mathbf r_{\mathrm{out}},I)\cdot \mathrm{eq}(\mathbf r_{\mathrm{in}},k) -\;\in F_{q'}. -} -\] - -This is well-typed (pure \(F_{q'}\)), linear in the \(p_{I,k}\), and accounts for all \(n\) coordinates. - -##### 4.3.2B.3 Phase 0: inner evaluation sumcheck (\(\alpha\) rounds) - -Reorganize the evaluation as -\(v = \sum_{I} \mathrm{eq}(\mathbf r_{\mathrm{out}},I) \cdot f'_I\) -where \(f'_I := \sum_{k} \iota(p_{I,k})\cdot \mathrm{eq}(\mathbf r_{\mathrm{in}},k)\) is the "inner multilinear evaluation" -of ring element \(f_I\)'s coefficient vector at \(\mathbf r_{\mathrm{in}}\). - -Run an \(\alpha\)-round sumcheck on the inner sum. Define -\(c_k := \sum_I \mathrm{eq}(\mathbf r_{\mathrm{out}},I)\cdot\iota(p_{I,k})\) (the "coefficient-plane partial evaluations"). -The sumcheck proves \(\sum_k c_k\cdot\mathrm{eq}(\mathbf r_{\mathrm{in}},k) = v\). -After \(\alpha\) rounds the verifier obtains a random point \(\mathbf t^*\in F_{q'}^\alpha\) and needs the oracle value -\(g^* := \sum_k c_k\cdot\mathrm{eq}(\mathbf t^*,k)\). - -##### 4.3.2B.4 Prover sends coefficient-plane values; verifier checks and batches - -The prover sends the \(d\) values \(c_k\) for \(k\in\{0,1\}^\alpha\). -The verifier checks \(\sum_k c_k\cdot\mathrm{eq}(\mathbf t^*,k) \stackrel{?}{=} g^*\) (cost: \(O(d)\) field ops). - -Now **batch** the \(d\) claims \(c_k = \sum_I \mathrm{eq}(\mathbf r_{\mathrm{out}},I)\cdot\iota(p_{I,k})\) into one. -The verifier samples \(\gamma\leftarrow F_{q'}\) and computes: - -\[ -V_\gamma := \sum_{k=0}^{d-1} \gamma^k \cdot c_k -= \sum_{I\in\{0,1\}^\ell} \mathrm{eq}(\mathbf r_{\mathrm{out}},I) -\cdot \underbrace{\sum_k \iota(p_{I,k})\gamma^k}_{=\;\mathrm{ev}_\gamma(f_I)}. -\] - -The right-hand side is: **"evaluate the ring-switched table -\((\mathrm{ev}_\gamma(f_I))_I\) as a multilinear at \(\mathbf r_{\mathrm{out}}\)."** -This is an \(\ell\)-variable scalar evaluation where each table entry is obtained by ring switching at \(\gamma\). - -##### 4.3.2B.5 Phase 1: modified Hachi Step B (ring-switched at \(\gamma\)) - -The claim is now \(V_\gamma = \sum_I \mathrm{eq}(\mathbf r_{\mathrm{out}},I)\cdot \mathrm{ev}_\gamma(f_I)\). -Using the split \(\ell=m+r\) and \(I=(i,j)\): - -\[ -V_\gamma = \sum_i b_i \cdot \Big[\sum_j a_j \cdot \mathrm{ev}_\gamma(f_{i\|j})\Big], -\] - -where \(b_i,a_j\in F_{q'}\) are the outer monomial weights. All products are well-typed: \(\mathrm{ev}_\gamma\) -maps \(R_q\)-elements to \(F_{q'}\), and \(a_j\in F_{q'}\). - -**Key issue**: the prover cannot precompute \(w'_i := \sum_j a_j\cdot \mathrm{ev}_\gamma(f_{i\|j})\) before \(\gamma\) is known. -Although \(\sum_j a_j\cdot f_{i\|j}(X)\) is well-defined over \(F_{q'}[X]\), it lives in \(F_{q'}[X]/(X^d+1)\), not \(R_q\), -so it cannot be committed using \(R_q\)-Ajtai matrices. - -**Resolution**: we do **not** need an auxiliary commitment to partial evaluations in the \(d>1\) regime. -Instead, the evaluation equation enters the final sumcheck as an **additional linear constraint** -on the witness table, alongside the ring/modulus-switched commitment constraints and range constraints. - -##### 4.3.2B.6 Modified constraint set for Phase 1 - -1. **Main commitment consistency**: \(B\hat t = u\). (Pure ring; ring+mod-switch at \(\gamma\).) -2. **Fold consistency (inner commit)**: \(Az = (c^\top\otimes G_{n_A})\hat t\). (Pure ring; ring+mod-switch at \(\gamma\).) -3. **Evaluation equation** (scalar, at \(\gamma\)): - \(V_\gamma = \sum_I \mathrm{eq}(\mathbf r_{\mathrm{out}},I)\cdot\sum_k \iota(p_{I,k})\gamma^k\). - Linear in committed digits with public weights \(\mathrm{eq}(\mathbf r_{\mathrm{out}},I)\cdot\gamma^k\). -4. **Range / bitness constraints**: all digit entries bounded. - -**Removed**: constraint (3) of old template (\(b^\top w = \mathrm{open\_value}\)), -constraint (4) (\(a^\top Gz = \ldots\)), the auxiliary commitment \(v=D\hat w\), and the \(\hat w\) witness. - -##### 4.3.2B.7 How the evaluation equation enters the sumcheck - -The witness table \(\tilde w\) encodes digit-level entries. The scalar coefficient is recovered as -\(p_{I,k} = \sum_\delta b^\delta\cdot\tilde w(I,\delta,k)\). The evaluation constraint becomes: - -\[ -V_\gamma = \sum_{I,k} \mathrm{eq}(\mathbf r_{\mathrm{out}},I)\cdot\gamma^k\cdot\sum_\delta b^\delta\cdot\tilde w(I,\delta,k). -\] - -The public weights \(\mathrm{eq}(\mathbf r_{\mathrm{out}},I)\cdot\gamma^k\cdot b^\delta\) **factorize**: -\(\mathrm{eq}(\mathbf r_{\mathrm{out}},I)\) has a closed-form MLE (Section 4.5.3), -\(\gamma^k\) is a size-\(d\) power table, and \(b^\delta\) is a geometric sequence. -The verifier evaluates public encodings at any sumcheck point in \(O(\ell+d+\delta)\) field ops. -This is batched into the existing sumcheck, adding **no extra rounds**. - -##### 4.3.2B.8 Cost summary - -| Component | Cost | -|---|---| -| Phase 0 inner sumcheck | \(\alpha\) extra rounds (10 for \(d=1024\)) | -| Coefficient-plane values \(c_k\) | \(d\) field elements in proof (16 KiB for \(d=1024\)) | -| Verifier batch check | \(O(d)\) field multiplications | -| Phase 1 eval constraint | no extra rounds (folded into existing sumcheck) | - -##### 4.3.2B.9 Alternative: lift to integers and work over \(R_q\) - -One could lift the \(F_{q'}\) evaluation claim to the integers and prove using \(F_q/R_q\) arithmetic -(the "Option B" approach from Section F.0.6 of `docs/HACHI_DIGEST.md`). Caveats: -(1) every \(F_{q'}\) multiplication becomes limb-by-limb schoolbook over \(Z_q\) at cost \(O(L^2)\) with \(L\approx 4\) limbs; -(2) the \(\mathrm{eq}(\tilde r,I)\) weights involve products of up to \(n\) 128-bit numbers, creating deep non-native chains; -(3) the quotient witness \(\kappa\) for mod-\(q'\) can be enormous; -(4) the expensive part is non-native simulation, which doesn't benefit from NTT-friendly ring structure. -The cross-prime approach (Phase 0 + modified Phase 1) is typically much cheaper. - -#### 4.3.3 Intermediate partial evaluations and auxiliary commitment (ring-side objects) - -Define the ring partial evaluations: - -\[ -w_i := a^\top\cdot G_{2^m}\, s_i \in R_q\quad\text{for each }i\in\{0,1\}^r, -\] - -and digitize: - -\[ -\hat w_i := G^{-1}_1(w_i)\in R_q^\delta. -\] - -Stack \(\hat w := (\hat w_i)_{i\in\{0,1\}^r}\in R_q^{\delta\cdot 2^r}\) and define: - -\[ -v := D\hat w \in R_q^{n_D}. -\] - -The prover sends \(v\) as the first opening message (as in Hachi Fig. 3). - -#### 4.3.4 Fold challenge and folded witness (ring-side) - -The verifier derives a short/sparse ring challenge vector: - -\[ -c=(c_1,\dots,c_{2^r})\in R_q^{2^r}. -\] - -The prover folds: - -\[ -z := \sum_{i=1}^{2^r} c_i s_i \in R_q^{2^m\delta}. -\] - -Because coefficients of \(z\) grow, the prover performs a redecomposition using a second gadget \(J\): - -\[ -\hat z := J^{-1}_{2^m}(z)\in R_q^{2^m\delta\tau}. -\] - -#### 4.3.5 The full Step-B relation (six constraints, verbatim structure) - -The prover’s remaining task is to prove existence of **small** objects \((\hat w,\hat t,\hat z)\) such that all of the following hold (this is the B.4.1 relation, rewritten): - -1. **Aux-commitment consistency**: - \[ - D\hat w = v. - \] - -2. **Main commitment consistency**: - \[ - B\hat t = u. - \] - -3. **Evaluation equation** (in field form): - - let \(w := G_{2^r}\hat w\) (recompose digits to ring elements), then the claimed opening value is: - \[ - b^\top\cdot w = \mathrm{open\_value}\quad(\text{interpreted in }F_{q'}\text{ after ring switching}). - \] - -4. **Fold consistency (partials)**: - - let \(z := J\hat z\), then - \[ - a^\top\cdot G_{2^m}\, z = (c^\top\otimes G_1)\hat w. - \] - -5. **Fold consistency (inner commit)**: - \[ - A z = (c^\top\otimes G_{n_A})\hat t. - \] - -6. **Smallness / range constraints**: - - coefficients of \(\hat w,\hat t,\hat z\) lie in the intended digit ranges (and for bit/one-hot regimes, coefficient constraints can often be strengthened to \(\{0,1\}\) on specific parts of the witness). - -At this point we have reduced the opening claim to a structured “prove knowledge of bounded digits satisfying a public linear system” instance. - -#### 4.3.6 The same Step-B relation as one explicit stacked matrix equation (Hachi Eq. (20)) - -It is easy to lose track of how the opening-point-derived weights \((a,b)\) actually enter the *public* linear system. -The paper makes this explicit by stacking constraints (1)–(5) into a single matrix equation (their Eq. (20)). - -First recall the evaluation rewrite (paper Eq. (15)–(19) style), specialized to our split \(\ell=m+r\): - -- Define partial evaluations (ring-side): - - \[ - w_i := a^\top G_{2^m} s_i \in R_q,\qquad i\in\{0,1\}^r. - \] - -- Stack \(w := (w_i)_i \in R_q^{2^r}\) and digitize/recompose via the gadget map: - - \[ - \hat w := G^{-1}_{2^r}(w),\qquad w = G_{2^r}\hat w. - \] - -- The claimed opening value is the bilinear form: - - \[ - b^\top w = \mathrm{open\_value}. - \] - -- The fold constraint (partials) is: - - \[ - a^\top G_{2^m} z = (c^\top\otimes G_1)\hat w. - \] - -- The fold constraint (inner commit) is: - - \[ - A z = (c^\top\otimes G_{n_A})\hat t. - \] - -Now, writing \(z = J_{2^m}\hat z\) (so \(\hat z := J^{-1}_{2^m}(z)\)), constraints (1)–(5) can be written as the stacked linear system: - -\[ -\begin{bmatrix} -D & 0 & 0 \\ -0 & B & 0 \\ -b^\top G_{2^r} & 0 & 0 \\ -(c^\top\otimes G_1) & 0 & -\,a^\top G_{2^m} J_{2^m} \\ -0 & (c^\top\otimes G_{n_A}) & -\,A J_{2^m} -\end{bmatrix} -\cdot -\begin{bmatrix} -\hat w \\ -\hat t \\ -\hat z -\end{bmatrix} -\;=\; -\begin{bmatrix} -v \\ -u \\ -\mathrm{open\_value} \\ -0 \\ -0 -\end{bmatrix}. -\] - -**Key dependency note (why the verifier “needs \(a,b\)”)**: - -- The *public matrix* on the left contains the rows \(b^\top G_{2^r}\) and \(-a^\top G_{2^m}J_{2^m}\). - These are functions of the opening point \(\mathbf r\) via the monomial vectors \(a=a(\mathbf r)\), \(b=b(\mathbf r)\). -- Therefore, when we later write “package (1)–(5) as \(M\cdot Z = Y\)”, the packaged \(M\) **depends on** \((a,b)\), - and the packaged \(Y\) contains \(\mathrm{open\_value}\) (in addition to \(u,v\)). - -### 4.4 Ring switching + modulus switching (make the relation checkable in \(F_{q'}\)) - -All equalities above are ring equalities in \(R_q\) except the weights \(a,b\) which are in \(F_{q'}\). -We therefore check the relation by moving to \(F_{q'}\) via ring switching. - -#### 4.4.1 Evaluation map into \(F_{q'}\) - -Verifier samples \(\alpha\leftarrow F_{q'}\) and defines - -\[ -\mathrm{ev}_\alpha: R_q \to F_{q'} -\] - -as: choose a lift convention \(\mathrm{lift}_q\) for coefficients of \(Z_q\), lift each coefficient to an integer, and then evaluate the polynomial at \(X=\alpha\) in \(F_{q'}\). - -Apply \(\mathrm{ev}_\alpha\) entrywise to vectors/matrices over \(R_q\). - -#### 4.4.2 The unstructured linear relation and its full witness - -Package constraints (1)–(5) into a single public linear system over \(R_q\): - -\[ -M\cdot Z = Y \quad\text{in }R_q^n, -\] - -where: - -- \(Z\) is the stacked unknown vector whose coordinates are exactly the digits in \((\hat w,\hat t,\hat z)\) (plus any auxiliary digit blocks introduced by the gadget decompositions), -- \(Y\) is the stacked public right-hand side built from \((u,v)\), the claimed open value, and the fold challenge \(c\), -- \(M\) is the stacked public matrix built from \((A,B,D)\), the gadget matrices, and the fold wiring. - -Now lift this ring equation to an **integer polynomial identity** with two quotient witnesses: - -\[ -\mathrm{lift}_q(M)\,\mathrm{lift}_q(Z) - \mathrm{lift}_q(Y) -= -(X^d+1)\,r \;+\; q\cdot s -\quad\text{over } Z[X], -\] - -where: - -- \(r\in (Z[X]_{n# zow!xr>eDQzG1|d%djj^Gqa5$}tfQAvvk;c2>T=L?^RRL~5$-ni)q2wz_ zard?m$Oup}til1pTbI?u{4ZYfH<8y_$N4zIA-1(sH%`9zq=v=-->Ylwgdks0%wtll z%LBUsRTUF0!h*sr78jKq0#Gt|6!Rb`wg+Ft+5@`t^yK&{Slu^ zUwgL+C3D1r-x=L91Hkl9q(=<4@F+j}yj7X^f*!yt0SW-)b9RDAvtI9}^e>CDUIi|O z%Q>cf{^p04h7k5OV)<0uY9kC7N+)Fc?aF$pqZ6~>^#16)tdF({j}S^!q$&M%hd`>= z1(QGnxT#UNeH&eCW!|4t<&8_n$E=Tub#V=nveaq>n2rv7G=O!C<{2E#{}4s;N6KtW8;i zE_R>?@Y*#49LZPWFK7P9RBB14Z7k#l#nBm+l2~=9h(`i|Op-qS_lq_26p$H{{%L}4 z%lHk}Wi@Wbgse(v1#XYY&M^%T=SrWbZq3$E-)!xMmp%!` z5m**R8Kdcq(uk=t!Dh)Ozxx9FF=rsqK@%mNC)W@^tKmh1r`@Fz< z+$bR;(vl0@x%XAY&l*zk)9_$UJ{@SN&DgItGYrzsM=vo-1NnuOt-IhM)#9<}&Q0|K z^f`I*bVh%jD?F%)_fzt1&ZAMQe>DUXXc{U{yCZKzz|RkzemNb2Iz~mggXYj$nSBby z8`xnG3St}oC2(F+?at4>k0MsabqOWmZpjr4&dUD?5z@SKqmil>PuMz}+5Z*m%Gv=% z2yhQ2f*2v8*hrp-Z*`e^Q(Vo7{%9g)Kr&L#hh{6TT-3tA)!y?CHnpNKcCl{7-+04} 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zU+-%6aV6zizSCo&4-w4VijCK0s5xDq}M{y*Z6)=u8a*ufED`A=( zl`#n&7g+k@B&BIipc6!n|+mR+4sVI7a|uO)x^0 zL$fMM306lLmB48XAk^gKE+2F9PNHmlcr>ZBluILh$;Tfz4=FjJv*Hq6LD{YPBGcjq ze+KtFpk_-~0n>9}@gOutUKqZ|WD~xjXyahCLFf^!Vx(Ag*-fZPBSq&;o9IxWI%z|u>4|-`xX61DQRX&e z6jxflm=DA`l=KUCB)|BF!oE8Aevh~E4N>KI>Ay+S>gs0V;^y`KJqU$^lbf5D4~Rlb KE2%7n^1lG_iQZ5E diff --git a/paper/hachi.txt b/paper/hachi.txt deleted file mode 100644 index 892a98b7..00000000 --- a/paper/hachi.txt +++ /dev/null @@ -1,2274 +0,0 @@ - - ---- page 1 --- - -Hachi: Efficient Lattice-Based Multilinear Polynomial Commitments -over Extension Fields -Ngoc Khanh Nguyen1, George O’Rourke1, and Jiapeng Zhang2 -1 King’s College London -2 University of Southern California -Abstract. In this work, we present Hachi, a concretely efficient multilinear polynomial commitment -scheme that offers succinct proof sizes ofpoly(ℓ, 𝜆) and achieves a “square-root” verifier time complexity -of ˜𝑂 ( -√ -2ℓ 𝜆) for ℓ-variate polynomials under the Module-SIS assumption. Compared to the current -state-of-the-art scheme, Greyhound (CRYPTO 2024), Hachi provides an asymptotic improvement of -˜𝑂 (𝜆) in verification time, which translates into a practical 12.5-fold speedup, while maintaining compact -proofs of approximately 55 KB. -To improve the verification time, we adopt the sumcheck protocol. Note that the standard sumcheck -has efficiency bottlenecks for lattice-based constructions, since lattice operations are usually performed -over power-of-two cyclotomic ringsR𝑞 := Z𝑞 [𝑋 ]/( 𝑋 𝑑 +1). To address this challenge, we provide a novel -approach that integrates Greyhound with the ring-switching idea proposed by Huang, Mao and Zhang -(ePrint 2025). Surprisingly, under this approach, the verifier does not need to perform any multiplication -over R𝑞, enabling a much faster verification time. This technique could be of independent interest for -building lattice-based SNARKs, particularly for achieving faster verification. -As a separate contribution, we introduce a generic reduction that converts polynomial evaluation -proofs over extension fieldsF𝑞𝑘 (under suitable parameter regimes) into equivalent statements over -cyclotomic rings R𝑞. This reduction is compatible with existing lattice-based polynomial commitment -schemes and can be integrated as a modular enhancement to broaden applicability to statements over -extension fields. -1 Introduction -Multilinear polynomial commitment scheme (PCS) [KZG10; BBHR18] is a strong cryptographic primitive -that allows a prover to commit to aℓ-dimensional multilinear polynomial𝑓 ∈ F≤1 [𝑋1, . . . , 𝑋ℓ ] and later prove -its evaluation -𝑓 (𝑥1, . . . , 𝑥ℓ ) = 𝑦 -where the evaluation point(𝑥1, . . . , 𝑥ℓ ) ∈ Fℓ and the image𝑦 ∈ F are public. Multilinear polynomial commit- -ments are a core building block for succinct non-interactive arguments of knowledge (SNARKs) [WTs+18; -Set20; CHM+20; GLS+23; CBBZ23], look-up arguments [STW24] and multi-party computation [BHV+23]. -In such applications, it is crucial for the polynomial evaluation proof to be succinct, i.e. polynomial in the -number of variablesℓ, and be verified efficiently. -With the rapid advancement of quantum computing, there is increasing interest in quantum-safe PCS -constructions. Among these, lattice-based and hash-based approaches have emerged as the leading candidates. -Lattice-based PCS are particularly attractive due to their efficient prover runtime and compact proof sizes, -both concretely [NS24] and asymptotically [KLNO25]. In contrast, hash-based constructions currently offer -significantly faster verification (by two orders of magnitude) compared to lattice-based schemes [ACFY24; -ACFY25]. Such a long verification time has become a major bottleneck for lattice-based protocols, such as -Greyhound, being widely deployed in practice. This discrepancy raises a natural research question: -Can we design a lattice-based PCS that has fast verification while retaining the compact proof sizes? -In this paper, we answer this question affirmatively. - ---- page 2 --- - -1.1 Prior Works -Split-and-fold. At the core of state-of-the-art lattice-based succinct proof systems (including polynomial -commitments) lies proving knowledge of vectorss1, . . . , s𝑟 ∈ R𝑚 -𝑞 such that -B - -A -. . . -A - - -s1 -... -s𝑟 - -= u (mod 𝑞) and ∥s𝑖 ∥ ≤ 𝛽 for 𝑖 = 1, . . . , 𝑟 (1) -where R𝑞 := Z𝑞 [𝑋 ]/( 𝑋 𝑑 +1) for a power-of-two𝑑, and the matricesA, B, vector t and norm bound 𝛽 are -public. The first interactive protocol for this relation was proposed by Baum et al. [BBC+18] which can be -summarized as follows. The prover starts by sending the “partial evaluations”t𝑖 = As𝑖 for 𝑖 ∈ [ 𝑟 ] (this step is -called split). The verifier responds by outputting short challenges𝑐1, . . . , 𝑐𝑟 ∈ C over the cyclotomic ringR𝑞. -Finally, the prover sends returns the linear combination of witness vectors (this step is calledfold) -z := 𝑐1s1 + . . . 𝑐𝑟 s𝑟 ∈ R𝑚 -𝑞 . (2) -The verifier then checks three conditions: -B - -t1 -... -t𝑟 - -= u, Az = -𝑟∑︁ -𝑖=1 -𝑐𝑖 t𝑖 and ∥z∥ is short. (3) -The complexity of the communication of the prover is𝑟 + 𝑚 ring elements, which is strictly sublinear in the -original witness length𝑁 := 𝑟 ·𝑚 in terms ofR𝑞-elements. By picking parameters appropriately, [BBC+18] -obtains “square-root” proofs of size˜𝑂 ( -√ -𝑁 𝜆), where 𝜆 is the security parameter. -Multiple follow-up works [BLNS20; AFLN24; CMNW24; KLNO24] managed to reduce the proof size -further topolylog(𝑁 ) ring elements. The key insight lies in making the matrixA to have a tensor structure -similar to the matrix on the left-hand side of Equation (1). This allows the prover, instead of sending the -final message z in the clear, to prove knowledge ofz satisfying the verification equation of(3) in a recursive -fashion. -Exact norm proof.Arguably, the hardest part of designing proof systems for relations such as(1) is proving -that vectors s𝑖 are actually short. Indeed, knowledge soundness of the original protocol of [BBC+18] only -guarantees extraction of (not necessarily short)¯s1, . . . ,¯s𝑟 ∈ R𝑚 -𝑞 of the form -¯s𝑖 := ¯z𝑖 -¯𝑐𝑖 -where both¯z𝑖,¯𝑐𝑖 are short. -There is a line of research [BLNS20; AL21; ACK21; CLM23; FMN24; AFLN24; KLNO24] which addresses -this issue using so-called subtractive sets. A setC ⊆ R𝑞 is called subtractive if for any two distinct elements -𝑐, 𝑐′ ∈ C, inverse of the difference𝑐 − 𝑐 ′ over R𝑞 is short. This allows the knowledge extractor to deduce that -the extracted¯s𝑖 are short. This approach has one crucial disadvantage, which is that all subtractive sets are -only of polynomial size [AL21]. Consequently, to obtain negligible soundness error, multiple repetitions are -needed which affect the efficiency of the scheme significantly. -An alternative approach to proving shortness, which avoids subtractive sets, involves applying random -projections to the witness vectors [BL17; LNS21; LNP22; GHL22; BS23; KLNO25]. Lets′ ∈ Z𝑚𝑑 -𝑞 denote -the coefficient vector ofs ∈ R𝑚 -𝑞 . Consider a matrixJ ∈ Z𝑟 ×𝑚𝑑 with small, independently chosen random -entries, where 𝑟 is a soundness parameter. The Johnson-Lindenstrauss Lemma (and heuristic variants [Ach03; -GHL22]) implies that -∥Js′ ∥ ≈ ∥ s′ ∥ = ∥s∥. -2 - ---- page 3 --- - -Using this fact, the verifier sends the projection matrixJ to the prover (or in the non-interactive case it is -generated from a hash function), who responds withv := Js′ ∈ Z𝑟. The verifier then checks whetherv is short. -For soundness, previous works show that ifs were large (i.e. a prover was malicious), then with probability -exp(−𝑟 ), v would also have large norm, and thus result in a contradiction. Hence,𝑟 = 𝑂 (𝜆) is required for -soundness. -The prover must also prove thatv is well-formed. As shown in [LNP22; BS23], this reduces to proving that -certain 𝑟 polynomials 𝑢1, 𝑢2, . . . , 𝑢𝑟 ∈ R𝑞 have zero constant coefficients. One method is to send all(𝑢𝑖 )𝑖 ∈ [𝑟 ] in -the clear for the verifier to check, but this requires sending at least𝑂 (𝜆) ring elements, resulting in large proofs. -A more efficient method batches the𝑟 statements: the verifier sends𝑟 random challenges𝛼1, . . . , 𝛼𝑟 ← Z𝑞, and -the prover returns the linear combination -𝑢 := -𝑟∑︁ -𝑖=1 -𝛼𝑖 ·𝑢𝑖 ∈ R𝑞. -The verifier checks that the constant coefficient of𝑢 is zero. For soundness, if some𝑢𝑖 had a non-zero constant -coefficient, then with probability 1/𝑞, so would𝑢, leading to a contradiction. Since 1/𝑞 may not be negligible, -this must be repeated𝑂 (𝜆/log 𝑞) times to achieve exp(−𝜆) soundness error. -Concretely, the Johnson-Lindenstrauss projection has shown to be a practical approach to prove norm -bounds compared to subtractive sets. The prime example is the LaBRADOR proof system introduced by -Beullens and Seiler, which achieves≈ 60KB proofs for large statements (of size 230). The main disadvantage -of the random projection methodology is the verification time. Indeed, the verifier has to operate on the -projection matrix J of size 𝑂 (𝜆 ·𝑚𝑑) – linear in witness length (in terms ofZ𝑞-elements). This makes the -LaBRADOR verifier linear time, and thus not suitable for various SNARK-oriented applications. More -recently, Klooß et al. [KLNO25] focused on tensor-structured projection matricesJ, and managed to reduce -the verification time to polylog(𝑚𝑑), at the cost of 40X larger proof sizes compared to LaBRADOR. -Nguyen and Seiler [NS24] proposed a promising hybrid polynomial commitment, called Greyhound. Their -evaluation proof intuitively follows the three-round protocol from [BBC+18], but instead of sending the prover -message in the clear, the prover commits to(t𝑖 )𝑖, z and later proves that they satisfy the verification equations -in (3) via the LaBRADOR proof system. The key advantage of this strategy is that invoking LaBRADOR on -a smaller statement yields square-root˜𝑂 (𝜆 -√ -𝑁 𝜆) verification time, while benefiting from the attractive proof -sizes of [BS23]. -Several recent works employ the sumcheck protocol [LFKN90] to prove exact norm bounds of committed -vectors based on lattice assumptions, both in the setting of succinct proofs and folding schemes [KST22]. -This strategy was first used by Boneh and Chen in LatticeFold [BC24] to prove that the committed (NTT) -coefficients are binary. A drawback of their approach is that the sumcheck protocol is executed over the -cyclotomic ring R𝑞, where each ring element is large, resulting in significantly increased proof sizes. Kuriyama -et al. [KLOT25] later expanded this methodology and introduced a new technique to prove exact Euclidean -norm bounds usingR𝑞-based sumchecks. Neo [NS25] departs fromR𝑞-based sumchecks entirely by mapping -all relevantR𝑞-relations into Z𝑞 via rotational (i.e., skew-circulant) matrices. This allows the protocol to run -sumcheck directly overZ𝑞 (or over an extension fieldF𝑞𝑘 to obtain negligible soundness error). Concurrently, -LatticeFold+ [BC25] proposed a new method to prove range statements. Specifically, to prove that𝑎 ∈ [0, 𝑑), -the prover commits to the monomial𝑋 𝑎 ∈ R𝑞 and then proves that the committed ring element is indeed a -monomial. LatticeFold+ reduces this (and related statements) to sumcheck claims overR𝑞. In contrast to -[BC24; NS25], the authors observe that these claims do not involve any “full-fledged”R𝑞-multiplications and -can therefore be naturally reduced to𝑂 (𝑑) sumcheck claims overZ𝑞. The main efficiency bottleneck of this -approach is that, for a witness®𝑠 ′ ∈ Z𝑚𝑑 -𝑞 of length 𝑚𝑑, the prover must commit to𝑚𝑑 elements of R𝑞, rather -than 𝑚 as in [BS23; NS24; NS25] (excluding decomposition costs). -1.2 Our Contributions -In this work, we propose Hachi, a concretely efficient multilinear polynomial commitment scheme with succinct -poly(ℓ, 𝜆) proof size and˜𝑂 ( -√ -2ℓ𝜆) verifier time forℓ-variate polynomials based on standard lattice assumptions -3 - ---- page 4 --- - -Scheme Multi Extension Asymptotic Concrete --linear fields proof verifier proof verifier -Greyhound [NS24] ✗ ✗ poly(ℓ, 𝜆) ˜𝑂 (𝜆 · -√ -2ℓ 𝜆) 53KB 2.8s -Hachi ✓ ✓ poly(ℓ, 𝜆) ˜𝑂 ( -√ -2ℓ 𝜆) 55KB 227ms -Fig. 1. Overview of Hachi and comparison with the state-of-the-art lattice-based polynomial commitment scheme -Greyhound [NS24]. The asymptotic verifier time is measured inZ𝑞-operations. We use the ˜𝑂-notation to hide -logarithmic factors. The concrete numbers are taken from evaluating the polynomial withℓ = 30 variables. -(and the random oracle model for the Fiat-Shamir transformation). Hence, asymptotically we achieve˜𝑂 (𝜆) -improvement in verification time compared to Greyhound [NS24]. In Figure 1 we provide more details on the -comparison. Our improvements can be summarized as follows. -Faster verification via ring switching.Our protocol mainly follows the framework of Greyhound [NS24]. -However, unlike Greyhound, we move away from the random projection strategy and instead adopt a -sumcheck-based approach to prove infinity-norm bounds. Although sumcheck is a well-known protocol, we -emphasize that the main challenge for lattice-based constructions is that lattice protocols typically involve -arithmetic over large polynomial rings. Directly running the sumcheck protocol over such large rings would -be highly inefficient. -To address this challenge and keep the protocol efficient, we propose a novel approach that organically -integrates the Greyhound framework with the ring-switching technique introduced by Huang, Mao and Zhang -[HMZ25]. This technique lifts statements overR𝑞 to the polynomial ringF𝑞𝑘 [𝑋 ], allowing us to evaluate -the equations at a random point𝑋 := 𝛼 ∈ F𝑞𝑘 and run the sumcheck protocol over the extension fieldF𝑞𝑘. -Surprisingly, our verification algorithm does not need to perform any cyclotomic ring operations, even though -our protocol is based on lattices. Switching the sumcheck protocol to a smaller extension field is crucial to -achieving a total verifier complexity of˜𝑂 ( -√ -2ℓ𝜆). To the best of our knowledge, other than our protocol, we are -not aware of any lattice-based SNARK, PCS, or folding scheme that can eliminate the expensive cyclotomic -ring operations for verifiers. -Embedding multilinear evaluation claims insideR𝒒. We provide a generic transformation from proving -ℓ-variate polynomial evaluations over finite fieldsF𝑞𝑘 to proving (ℓ − 𝛼 + 𝜅)-variate polynomial evaluations -over “NTT-unfriendly” cyclotomic ringsR𝑞, where 𝑑 = 2𝛼 and 𝑘 = 2𝜅. Thus, we generalize the result of -SLAP [AFLN24] in two orthogonal directions: (i) we support polynomials over field extensions (for specific -parameter sets), rather than just prime-order fields, and (ii) we extend the strategy from univariate to the -multivariate case. Using our methodology, one can already adapt Greyhound [NS24] to prove multilinear -polynomial evaluations out-of-the-box. We also provide an optimization if the polynomial𝑓 itself is only over -F𝑞 while the evaluation points still lie inF𝑞𝑘. -Flexibility in picking ring dimension and implications.Both Greyhound and LaBRADOR have -certain limitations in picking the ring dimension𝑑. One of the main obstacles is when applying random -projection strategy, one has to prove that certain ring elements have constant coefficients equal to zero. As -described above, this comes at the cost of the prover sending𝑂 (𝜆/log 𝑞) ring elements after each iteration of -LaBRADOR. Hence, choosing𝑑 (equal to 64 in [BS23; NS24]) as small as possible is essential to achieve -small proof sizes. In our construction, since we apply the ring switching idea, the sumcheck protocol is always -performed over an extension field, which is independent of the size of𝑑. This gives us more freedom in -choosing ring dimension𝑑 with smaller impact on the communication size. -In particular, picking larger𝑑 has two efficiency benefits: (i) the committing time becomes faster thanks -to the optimized NTT-based ring multiplication, and (ii) this allows us to define a challenge spaceC of sparse -ring elements - thus speeding up the ring multiplication for computing the folded witness as in (2). -2 Concrete verification time for first round of Hachi. -4 - ---- page 5 --- - -Implementation. WehavedevelopedaprototypeimplementationofHachiinRust.Whiletheimplementation -is still in its early stages, it is amenable to several optimizations—particularly those leveraging SIMD—that -could significantly enhance performance. Even at this early stage, the advantages of our protocol are clearly -demonstrated. Specifically, we observe: -–faster verification time. Although the implementation of our verifier does not use advanced optimizations -such as SIMD, our verification time is already one order of magnitude faster than Greyhound.3 -–faster commitment time. Our commitment step is almost identical to that of Greyhound. However, due -to the use of ring switching, we are able to choose larger rings; as a result, our commitment time is 3-5 -faster than Greyhound4. -1.3 Technical Overview -Denote 𝜆 as a security parameter. Let𝑑 := 2𝛼 be a power-of-two andR := Z[𝑋 ]/( 𝑋 𝑑 + 1) be the ring of -integers of the 2𝑑-th cyclotomic field. Take an odd prime𝑞 and define R𝑞 := R/(𝑞) and 𝛿 := ⌈log 𝑞⌉. For -𝑘 ≥ 1, we denoteF𝑞𝑘 to be a finite field of order𝑞𝑘. In this overview, we consider base-two gadget matrices -G𝑛 := I𝑛 ⊗ - -1 2 4· · ·2𝛿 −1 -∈ R𝑛×𝑛𝛿 -𝑞 for 𝑛 ≥ 1. We define the standard inverse functionG−1 -𝑛 : R𝑛 -𝑞 → R𝑛𝛿 -𝑞 , -which decomposes each entry w.r.t. base 2. In particular, for anyt ∈ R𝑛 -𝑞, G−1 -𝑛 (t) has binary coefficients and -G𝑛G−1 -𝑛 (t) = t. -For 𝑖 ∈ Z× -2𝑑, we define the Galois automorphism 𝜎𝑖 : R → R defined as 𝑋 ↦→ 𝑋 𝑖, and denote the -Galois group of automorphisms as Aut(R) := {𝜎𝑖 : 𝑖 ∈ Z× -2𝑑 }. For a subgroup 𝐻 of Aut(R), we denote -R𝐻 -𝑞 := {𝑥 ∈ R𝑞 : ∀𝜎 ∈ 𝐻, 𝜎 (𝑥) = 𝑥 }, i.e. the ring of elements ofR𝑞 fixed by all automorphisms in𝐻. Also, we -denote the R𝑞-trace map Tr𝐻 : R𝑞 → R𝐻 -𝑞 as Tr𝐻 (𝑎) := Í -𝜎 ∈𝐻 𝜎 (𝑎) ∈ R𝐻 -𝑞 . -Reducing multilinear evaluation claims to relations overR𝒒. Our first step will be to translate the -multilinear evaluation overF𝑞𝑘, i.e. 𝑓 (𝑥1, . . . , 𝑥ℓ ) = 𝑦, to an equivalent relation overR𝑞 (or a related ring). -The hardness lies in translating operations over field extensionsF𝑞𝑘, such as addition and multiplication, to -equivalent operations over the cyclotomic ringR𝑞. We proceed in two steps. First, we directly identify the -finite fields withinR𝑞. -Lemma 1(Informal). Let 𝑞 = 5 (mod 8) and 𝑘 ≥ 1 be a divisor of 𝑑/2 and consider the subgroup -𝐻 := ⟨𝜎−1, 𝜎4𝑘+1⟩ of Aut(R). Then, R𝐻 -𝑞 is a subfield ofR𝑞 isomorphic to F𝑞𝑘. -This lemma allows us to translate proving inner products overF𝑞𝑘 into relations directly over the cyclotomic -ring R𝑞 using the R𝑞-trace map. -Theorem 1(Informal). Let 𝑞 = 5 (mod 8), 𝑘 divide 𝑑/2 and 𝐻 := ⟨𝜎−1, 𝜎4𝑘+1⟩ ⊆ Aut(R). Then, there -exists an efficiently computable bijection𝜓 : (R𝐻 -𝑞 )𝑑/𝑘 → R𝑞 such that for anya, b ∈ ( R𝐻 -𝑞 )𝑑/𝑘, -Tr𝐻 (𝜓 (a) · 𝜎−1(𝜓 (b))) = 𝑑 -𝑘 · ⟨a, b⟩. -We now show how to use these two theoretical results. Let𝑘 = 2𝜅 be a divisor of𝑑/2 and suppose we want to -prove a polynomial evaluation claim overR𝐻 -𝑞 ( F𝑞𝑘 by Lemma 1). Denote the coefficients of polynomial𝑓 as -(𝑓𝜄)𝜄 ∈ {0,1}ℓ ∈ F𝑞𝑘. We can explicitly rewrite the evaluation equation as -𝑦 = -∑︁ -𝑖 ∈ {0,1}ℓ −𝛼 +𝜅 - -𝑥𝑖1 -1 · . . . ·𝑥𝑖ℓ −𝛼 +𝜅 -ℓ −𝛼+𝜅 - -·©­ -« -∑︁ -𝑗 ∈ {0,1}𝛼 −𝜅 -𝑓𝑖 ∥ 𝑗 ·𝑥 𝑗1 -ℓ −𝛼+𝜅+1 · . . . ·𝑥 𝑗𝛼 −𝜅 -ℓ -ª® -¬ -. (4) -3 Concrete verification times for first round of Hachi, and later composed with Greyhound on a smaller instance. -4 When compared with the same implementation. -5 - ---- page 6 --- - -For each𝑖 ∈ {0,1}ℓ −𝛼+𝜅 define the ring element -𝐹𝑖 := 𝜓 (𝑓𝑖 | |𝑗 ) 𝑗 ∈ {0,1}𝛼 −𝜅 - and also 𝑣 := 𝜓 - -(𝑥 𝑗1 -ℓ −𝛼+𝜅+1 · . . . ·𝑥 𝑗𝛼 −𝜅 -ℓ ) 𝑗 ∈ {0,1}𝛼 −𝜅 - -. -Then, using the fact thatR𝐻 -𝑞 is stable under the additively homomorphic maptr𝐻, and by applying Theorem 1 -to Equation (4) we get -𝑑 -𝑘 ·𝑦 = Tr𝐻 -©­­­­­­ -« -∑︁ -𝑖 ∈ {0,1}ℓ −𝛼 +𝜅 - -𝑥𝑖1 -1 · . . . ·𝑥𝑖ℓ −𝛼 +𝜅 -ℓ −𝛼 - -·𝐹𝑖 -| {z } -𝑌:= -·𝜎−1(𝑣) -ª®®®®®® -¬ -. -Therefore, the prover can send a single ring element𝑌 ∈ R𝑞, that allows the verifier to check if theR𝑞-trace -map of 𝑌 · 𝜎−1(𝑣) is equal to 𝑑𝑦/𝑘. Then, we only need to prove that𝑌 is well-formed, which is now an -evaluation proof for the(ℓ − 𝛼 + 𝜅)-variate polynomial 𝐹 := (𝐹𝑖 )𝑖 over R𝑞. -In certain scenarios, the evaluation point might be defined over the extension fieldF𝑞𝑘, while the coefficients -of the committed polynomial𝑓 are over F𝑞 (which is the case after the first iteration of our recursive proof -system). Can we optimize our transformation in this setting without bluntly assuming that𝑓 ∈ F≤1 -𝑞𝑘 [𝑋1, . . . , 𝑋ℓ ]? -To this end, we use the similar observation as before that a polynomial evaluation claim can be written as -𝑦 = -∑︁ -𝑖 ∈ {0,1}𝜅 - -𝑥𝑖1 -1 · . . . ·𝑥𝑖𝜅 -𝜅 - -· -©­­­­­­­ -« -∑︁ -𝑗 ∈ {0,1}ℓ −𝜅 -𝑓𝑖 ∥ 𝑗 ·𝑥 𝑗1 -𝜅+1 · . . . ·𝑥 𝑗ℓ −𝜅 -ℓ -| {z } -𝑦𝑖:= -ª®®®®®®® -¬ -. (5) -Hence, we let the prover send𝑘 partial evaluations𝑦𝑖 ∈ F𝑞𝑘 to the verifier, who can directly check the equation -above5. What we have left to prove is that all the𝑦𝑖 are well-formed, or in other words𝑓𝑖 (𝑥𝜅+1, . . . , 𝑥ℓ ) = 𝑦𝑖 -where 𝑓𝑖 ∈ F𝑞 [𝑋𝜅+1, . . . , 𝑋ℓ ]. -Next, denote F𝑞𝑘 := F𝑞 [𝑍 ]/𝜑 (𝑍 ), where 𝜑 is an irreducible polynomial inF𝑞 [𝑍 ] of degree 𝑘. Then, define -the (ℓ − 𝜅)-variate multilinear polynomial𝑓 ′ ∈ F𝑞𝑘 [𝑋𝜅+1, . . . , 𝑋ℓ ]: -𝑓 ′ (𝑋𝜅+1, . . . , 𝑋ℓ ) := -∑︁ -𝑖 ∈ {0,1}𝜅 -𝑓𝑖 (𝑋𝜅+1, . . . , 𝑋ℓ ) · 𝑍 -Í𝜅 -𝑡 =1 𝑖𝑡 ·2𝑡 −1 -. -Then, it is easy to see that -𝑓 ′ (𝑥𝜅+1, . . . , 𝑥ℓ ) = -∑︁ -𝑖 ∈ {0,1}𝜅 -𝑦𝑖 ·𝑍 -Í𝜅 -𝑡 =1 𝑖𝑡 ·2𝑡 −1 -∈ F𝑞𝑘 -which is now a full-fledged polynomial evaluation overF𝑞𝑘. By applying the generic transformation above, -we can reduce the statement into an evaluation claim overR𝑞 for a (ℓ − 𝛼)-variate multilinear polynomial -(instead of ℓ − 𝛼 + 𝜅, if done generically). -Polynomial evaluation claim as a quadratic equation.Now, consider proving an evaluation claim of -a 𝜇-variate multilinear polynomial𝑓 over R𝑞. We use the following folklore strategy. Suppose𝜇 = 𝑚 + 𝑟 for -some 𝑚, 𝑟 ≥ 1 and letf := (𝑓𝜄)𝜄 ∈ {0,1}𝜇 be the coefficient vector of𝑓. Then, for any -𝑓 (𝑋1, . . . , 𝑋𝜇) := -∑︁ -𝑖 ∈ {0,1}𝑟 -∑︁ -𝑗 ∈ {0,1}𝑚 -𝑓𝑖 | |𝑗 · (𝑋 𝑖1 -1 · . . . ·𝑋 𝑖𝑟 -𝑟 ) · (𝑋 𝑗1 -𝑟 +1 · . . . ·𝑋 𝑗𝑚 -𝜇 ) -5 A simple optimization is for the prover to send only𝑘 −1 partial evaluation𝑦𝑖, excluding 𝑦0...0 ∈ F𝑞𝑘, which can be -deterministically derived from the verification equation. -6 - ---- page 7 --- - -and any evaluation point(𝑥1, . . . , 𝑥𝜇) ∈ R𝜇 -𝑞 we can write: -𝑓 (𝑥1, . . . , 𝑥𝜇) = b⊺ - -a⊺ -. . . -a⊺ - - -f0...0 -... -f1...1 - -where -b⊺ := (𝑥𝑖1 -1 · . . . ·𝑥𝑖𝑟 -𝑟 )𝑖 ∈ {0,1}𝑟 ∈ R2𝑟 -𝑞 -a⊺ := (𝑥 𝑗1 -𝑟 +1 · . . . ·𝑥 𝑗𝑚 -𝜇 ) 𝑗 ∈ {0,1}𝑚 ∈ R2𝑚 -𝑞 -f ⊺ -𝑖 := (𝑓𝑖 | |𝑗 ) 𝑗 ∈ {0,1}𝑚 ∈ R2𝑚 -𝑞 for 𝑖 ∈ {0,1}𝑟 -. -Finally, by defining the decompositions := G−1 -2𝜇 (f ) ∈ R2𝑚𝛿 -𝑞 and using the mixed product of the tensor product, -we deduce that -𝑓 (𝑥1, . . . , 𝑥𝜇) = b⊺ (a⊺ ⊗ I2𝑟 )f -= b⊺ (a⊺ ⊗ I2𝑟 ) (g⊺ ⊗ I2𝑚+𝑟 )s -= b⊺ (a⊺ (g⊺ ⊗ I2𝑟 ) ⊗ I2𝑟 )s. -Hence, the goal is to prove that the (committed) vectors is short and satisfies the quadratic equation above, -which exactly the relation formulated in Equation (1). To prove knowledge of suchs, we closely follow the -initial methodology from Greyhound [NS24]. That is, we consider the square-root interactive proof from -[BBC+18], but instead of sending the prover messages in the clear, we commit to them. Later, we need -to prove knowledge of the prover messages that satisfy all the verification equations (as well as proving -knowledge of the commitment opening). All these relations can be phrased as proving knowledge of a short -vector z of dimension (2𝑚 +2𝑟 ) ·poly(𝜆) such that -Mz = w and z is short. (6) -This is the point where our strategy diverges from [NS24]. Instead of running LaBRADOR [BS23] to prove -the relation above, we propose a new sumcheck-based solution. -Ring switching and sumcheck over extension fields.We take inspiration from the ring switching -approach proposed recently by Huang, Mao, and Zhang [HMZ25].To begin with, recall that a multilinear -extension mle[𝑓 ] ∈ F≤1 -𝑞𝑘 [𝑋1, . . . , 𝑋𝜇] of 𝑓 is defined as -mle[𝑓 ] (x) := -∑︁ -𝑖 ∈ {0,1}𝜇 -𝑓 (b) · 𝑒𝑞 (𝑖, x) -where 𝑒𝑞 (𝑖, x) := Î𝜇 -𝑗=1 -(1 − 𝑖 𝑗 ) · (1 − 𝑥 𝑗 ) + 𝑖 𝑗𝑥 𝑗 - is the equality polynomial. -The first step is to lift Equation (6) fromR𝑞 to Z𝑞 [𝑋 ]. For the sake of overview, we only focus on the -unstructured linear relation and suppose matrixM has only one row (dealing with multiple rows and proving -norm bounds is presented in Section 4.3). Then, we can rewrite it as -∑︁ -𝑘 -𝑀𝑘 (𝑋 ) · 𝑧𝑘 (𝑋 ) = 𝑤 (𝑋 ) + ( 𝑋 𝑑 +1) · 𝑟 (𝑋 ) for some 𝑟 ∈ Z<𝑑 −1 -𝑞 [𝑋 ]. -Let z′ (resp. r′) be theZ𝑞-coefficient vector ofz (resp. 𝑟). Then, the prover commits, by using a polynomial -commitment scheme, to the multilinear extension 𝑃 := mle[( z′, r′)] ∈ F≤1 -𝑞 [𝑋1, . . . , 𝑋𝜇] of (z′, r′)6, before -obtaining a challenge 𝛼 ← F𝑞𝑘 from the verifier. The challenge is then used to substitute𝑋 = 𝛼, which -simplifies to an inner product claim purely overF𝑞𝑘. We note that the size of𝑃 is much smaller than that of -the original committed polynomial𝑓; hence, we can recursively repeat this process. Similarly as in [HMZ25] -this can further get transformed into proving a sumcheck-type relation -∑︁ -𝑖 ∈ {0,1}𝜇 -𝑃 (𝑖) · 𝑄 (𝑖) = 𝑉 -6 Here, we implicitly treat a vectora ∈ F2𝐿 -𝑞 as an indexing function from{0,1}𝐿 to F𝑞. -7 - ---- page 8 --- - -for a public polynomial𝑄 over F𝑞𝑘 [𝑋1, . . . , 𝑋𝜇] and a public value𝑉 ∈ F𝑞𝑘. -On the other hand, to prove thatz is short, we only need to show that all coordinates ofz′ are small. Here -we can use a standard range proof over a finite field, since the ring switching has transformed the statements -from polynomial rings into a finite field. In contrast, both LaBRADOR and LatticeFold+ demonstrate -smallness directly over polynomial rings, which is considerably more complicated and less efficient for the -verifier. -Finally, by running the sumcheck protocol [LFKN90], we end up with a polynomial evaluation claim of -the form: -𝑃 (𝑟 ∗ -1, . . . , 𝑟∗ -𝜇) · 𝑄 (𝑟 ∗ -1, . . . , 𝑟∗ -𝜇) = 𝑦∗, -wheretheevaluationpoint (𝑟 ∗ -1, . . . , 𝑟∗ -𝜇) andtheimage 𝑦∗ arepublic.Hence,bysendingtheevaluation 𝑃 (𝑟 ∗ -1, . . . , 𝑟∗ -𝜇) -to the verifier, they can directly check the equation above. What is left for the prover to do now is to prove -the evaluation claim for the polynomial𝑃, and thus the protocol can be repeated in a further way to obtain -smaller proof sizes. -Committing to 𝑷 and avoiding re-decomposition.A naive approach to commit to the multilinear -polynomial 𝑃 = mle[( z′, r′)] is to first decompose the coefficients of𝑃 (and thus the committed witness gets -longer), and then apply an Ajtai-style [Ajt96] commitment scheme. A more natural and efficient way is to -directly commit to(z′, r′), as we know that the coefficients ofz′ are already small and thus no decomposition -for z′ is needed7. But this directly leads to a question of how to directly provemle[( z′, r′)] (𝑥1, . . . , 𝑥𝜇) = 𝑦′, -when having only committed to(z′, r′). We solve this issue by observing a very simple homomorphic property -of the 𝑒𝑞 polynomials. Namely, for any vectors𝑖, 𝑗 and x0, x1 we have -𝑒𝑞 (𝑖, x0) · 𝑒 ( 𝑗, x1) = 𝑒𝑞 (𝑖 || 𝑗, x0|| x1). -Then, by definition of the multilinear extension and for𝜇 = 𝑚 + 𝑟, we can write -mle[( z′, r′)] (x0 ∥ x1) = -∑︁ -𝑖 ∈ {0,1}𝑟 -∑︁ -𝑗 ∈ {0,1}𝑚 -(z′, r′)𝑖 ∥ 𝑗 ·𝑒𝑞 (𝑖 ∥ 𝑗, x0 ∥ x1) -= -∑︁ -𝑖 ∈ {0,1}𝑟 -𝑒𝑞 (𝑖, x0) ·©­ -« -∑︁ -𝑗 ∈ {0,1}𝑚 -(z′, r′)𝑖 ∥ 𝑗 ·𝑒𝑞 ( 𝑗, x1)ª® -¬ -which is exactly of the same form as(4). Therefore, we can recursively run our protocol to prove evaluation -of 𝑃 (with minor modifications) without explicitly committing to its coefficients. -2 Preliminaries -Notation. Let 𝑞 = 5 (mod 8) be an odd prime. DenoteZ𝑞 := {0,1, . . . , 𝑞 −1} to be the ring of integers of order -𝑞 and F𝑞𝑘 be the Galois field of dimension𝑘 and size 𝑞𝑘. In this paper, we will useZ𝑞 and F𝑞 interchangeably. -For 𝑛 ∈ N, we define [𝑛] := {1,2, . . . , 𝑛}. Let 𝜆 to be the security parameter. We write𝑂𝜆 (𝑇 ) to denote -𝑇 ·poly(𝜆). For a probability distributionX (resp. finite setX), 𝑥 ← X means that 𝑥 is sampled fromX (resp. -𝑥 is chosen uniformly at random from the setX). We writenegl to denote an unspecified negligible function. -We will mostly use letters𝑖, 𝑗 for indexing. For𝑛 ∈ N, we write [𝑛] := {1,2, . . . , 𝑛}. When we denote -𝑖 ∈ {0,1}𝑘, we treat𝑖 as a binary vector(𝑖1, . . . , 𝑖𝑘 ) of length𝑘. For presentation purposes, when we say𝑖 ∈ [2𝑘 ], -we may also mean its binary encoding𝑖 ∈ {0,1}𝑘. This will be useful when describing the sumcheck-based -norm protocol (cf. Section 4.3). -7 Nevertheless, we would need to decompose the coefficients ofr′. -8 - ---- page 9 --- - -2.1 Cyclotomic Rings -For a power of two𝑑 := 2𝛼 and prime 𝑞, denote R and R𝑞 respectively to be the ringsZ[𝑋 ]/( 𝑋 𝑑 +1) and -Z𝑞 [𝑋 ]/( 𝑋 𝑑 +1). Lower-case letters denote elements inR or R𝑞𝑘 and bold lower-case (resp. upper-case) letters -represent column vectors (resp. matrices) with coefficients inR or R𝑞𝑘. For𝑦 = Í𝑑 −1 -𝑖=0 𝑦𝑖 ·𝑋 𝑖 ∈ R𝑞, we write -cf(𝑦) := (𝑦0, . . . , 𝑦𝑑 −1) ∈ F𝑑 -𝑞 to denote the coefficient vector of𝑦. We naturally extend this notation to vectors. -We consider base-𝑏 gadget matrices G𝑏,𝑛 := I𝑛 ⊗ - -1 𝑏 𝑏2 · · ·𝑏𝛿 −1 -∈ R𝑛×𝑛𝛿 -𝑞 for 𝑛 ≥ 1 and 𝛿 = ⌈log𝑏 𝑞⌉. We -define the inverse functionG−1 -𝑏,𝑛 : R𝑛 -𝑞 → R𝑛𝛿 -𝑞 , which decomposes each entry w.r.t. base𝑏. In particular, for any -t ∈ R𝑛 -𝑞, G−1 -𝑏,𝑛 (t) has coefficients between⌈−𝑏/2⌉ and ⌈𝑏/2⌉ −1 and G𝑏,𝑛G−1 -𝑏,𝑛 (t) = t. We will omit the subscript -𝑏 if the base is known and fixed from the context. -Galois Automorphisms and trace map.For 𝑖 ∈ Z× -2𝑑, we define the Galois automorphism𝜎𝑖 : R → R defined -as 𝑋 ↦→ 𝑋 𝑖, and denote the Galois group of automorphisms asAut(R) := {𝜎𝑖 : 𝑖 ∈ Z× -2𝑑 }. For a subgroup -𝐻 of Aut(R), we denote R𝐻 -𝑞 := {𝑥 ∈ R𝑞 : ∀𝜎 ∈ 𝐻, 𝜎 (𝑥) = 𝑥 }, i.e. the ring of elements ofR𝑞 fixed by all -automorphisms in 𝐻. Also, we define theR𝑞-trace map Tr𝐻 : R𝑞 → R𝐻 -𝑞 as Tr𝐻 (𝑎) := Í -𝜎 ∈𝐻 𝜎 (𝑎) ∈ R𝐻 -𝑞 . -Norms. We define 𝑟 ′ = 𝑟 mod± 𝑞 to be the unique element 𝑟 ′ in the range − 𝑞−1 -2 ≤ 𝑟 ′ ≤ 𝑞−1 -2 such that -𝑟 ′ = 𝑟 mod 𝑞. We also denote 𝑟 ′ = 𝑟 mod+𝑞 to be the unique element 𝑟 ′ in the range 0 ≤ 𝑟 ′ < 𝑞 such -that 𝑟 ′ = 𝑟 mod 𝑞. When the exact representation is not important, we simply write 𝑟 mod 𝑞. For an -element 𝑤 ∈ Z𝑞, we write∥𝑤 ∥∞ to mean |𝑤 mod± 𝑞|. Define theℓ∞ and ℓ𝑝 norms for 𝑤 = 𝑤0 + 𝑤1𝑋 + . . . + -𝑤 𝑑 −1𝑋 𝑑 −1 ∈ R as ∥𝑤 ∥∞ = max𝑗 ∥𝑤 𝑗 ∥∞, ∥𝑤 ∥𝑝 = -𝑝 -√︁ -∥𝑤0∥𝑝 -∞ + . . . + ∥𝑤 𝑑 −1∥𝑝 -∞. If w = (w1, . . . , w𝑚) ∈ R𝑘, then -∥w∥∞ = max𝑗 ∥w𝑗 ∥∞, ∥w∥𝑝 = 𝑝 -√︁ -∥w1 ∥𝑝 + . . . + ∥ w𝑘 ∥𝑝. By default,∥w∥ := ∥w∥2. For𝛽 ≥ 1, we define𝑆𝛽 to be -the set of ring elements inR with coefficients between⌈−𝛽/2⌉ and ⌈𝛽/2⌉ −1. -In this work, we will use the well-known inequality of [Mic07] to derive norm bounds. -Lemma 2. For any 𝑓 , 𝑔 ∈ R, ∥𝑓 𝑔∥∞ ≤ ∥ 𝑓 ∥1 · ∥𝑔∥∞. -Short elements are invertible.We recall the main result by Lyubashevsky and Seiler [LS18] which says that -short polynomials overR𝑞 are invertible overR𝑞. -Lemma 3([LS18]). Let 𝑞 ≡ 5 (mod 8) be a prime. Then, any𝑓 ∈ R𝑞 which satisfies0 < ∥𝑓 ∥∞ < 1√ -2𝑞1/2 or -0 < ∥𝑓 ∥ < 𝑞1/2 has an inverse inR𝑞. -The set of invertible elements ofR𝑞 is denoted byR× -𝑞. -Module-SIS Assumption. We recall the Module-SIS hardness assumption [LS15] adapted to the infinity norm -setting. -Definition 1(Module-SIS). Let 𝑞 = 𝑞(𝜆), 𝑛 = 𝑛(𝜆), 𝑚 = 𝑚(𝜆), 𝑑 = 𝑑 (𝜆) and 𝛽 = 𝛽 (𝜆). We say that the -MSIS𝑞,𝑑,𝑛,𝑚,𝛽 assumption holds if for any PPT algorithmA, the following holds: -Pr - -Az = 0 ∧0 < ∥z∥∞ ≤ 𝛽 | 𝐴 ← R𝑛×𝑚 -𝑞 , z ← A (A) - -= negl(𝜆) -2.2 Multilinear Extensions over Rings -Definition 2(Multilinear extension over rings).Let 𝑅 be an arbitrary ring with zero0 and identity1. -Given a function 𝑓 : {0,1}𝜇 → 𝑅, we define the multilinear extensionmle[𝑓 ] ∈ 𝑅 ≤1 [𝑋1, . . . , 𝑋𝜇] of 𝑓 as -mle[𝑓 ] (x) := -∑︁ -𝑖 ∈ {0,1}𝜇 -𝑓 (𝑖) · 𝑒𝑞 (𝑖, x) -where 𝑒𝑞 (𝑖, x) := Î𝜇 -𝑗=1 -(1 − 𝑖 𝑗 ) · (1 − 𝑥 𝑗 ) + 𝑖 𝑗𝑥 𝑗 -. -By definition, mle[𝑓 ] (x) = 𝑓 (x) for all x ∈ {0,1}𝜇. -For a vectorf := (𝑓𝑖 )𝑖 ∈ {0,1}𝜇 ∈ 𝑅2𝜇 -, we denote˜f as the multilinear extension of the function𝑓 : {0,1}𝜇 → 𝑅 -defined by𝑖 ↦→ 𝑓𝑖. -9 - ---- page 10 --- - -2.3 Coordinate-Wise Special Soundness -Let R ⊆ { 0,1}∗ × {0,1}∗ × {0,1}∗ be a ternary relation. For a triple(pp, x, w) ∈ R, we call pp the public -parameters, x is a statement andw is a witness forx w.r.t. pp 8. We denoteR(pp, x) = {w : R(pp, x, w) = 1}. -An interactive proof systemΠ = (S, P, V) for relation R consists of three PPT algorithms: the setup -algorithm S, proverP, and verifierV. The latter two are interactive and stateful. Further, we say a protocol is -public coinif the verifier’s challenges are chosen uniformly at random independently of the prover’s messages. -We recall the notion of coordinate-wise special soundness from [FMN24]. Let𝑆 be a set andℓ, 𝑘 ∈ N. -Namely, take two vectorsx := (𝑥1, . . . , 𝑥ℓ ), y := (𝑦1, . . . , 𝑦ℓ ) ∈ 𝑆 ℓ. Then, we define the following relation “≡𝑖” for -fixed 𝑖 ∈ [ ℓ] as: -x ≡𝑖 y ⇐ ⇒ 𝑥𝑖 ≠ 𝑦𝑖 ∧ ∀𝑗 ∈ [ ℓ]\{𝑖}, 𝑥𝑗 = 𝑦 𝑗 . -That is, vectors x and y have the same values in all coordinates apart from the𝑖-th one. Forℓ = 1, the -relations boil down to checking whether two elements are distinct. Further, we can define the set -SS(𝑆, ℓ, 𝑘) := -  - -{x1, . . . , x𝐾 } ⊆ ( 𝑆 ℓ )𝐾 : -∃𝑒 ∈ [ 𝐾], ∀𝑖 ∈ [ ℓ], -∃𝐽 = { 𝑗1, . . . , 𝑗𝑘 −1} ⊆ [ 𝐾] \ { 𝑒}, -∀𝑗 ∈ 𝐽 ,x𝑒 ≡𝑖 x𝑗 -  - -, -where 𝐾 := ℓ (𝑘 − 1) + 1. In other words, there is acentral vector x𝑒 ∈ 𝑋 such that for each coordinate of -x𝑒, there are𝑘 −1 other vectors in𝑋 that differ fromx𝑒 only in that coordinate. In other words, for each -coordinate, there are𝑘 vectors in 𝑋 that differ from each other only in that coordinate, andx𝑒 is always one -of them. -We are ready to define the notion of coordinate-wise special soundness. -Definition 3(CWSS for Multi-Round Protocols).Let Π be public-coin (2𝜇 +1)-round interactive proof -system for relationR, where in the2𝑖-th round, for𝑖 ∈ [ 𝜇], the verifier picks a uniformly random challenge -from 𝑆 ℓ𝑖 -𝑖 . A tree of transcripts is a set of𝐾 := Î𝜇 -𝑖=1 (ℓ𝑖 (𝑘𝑖 −1) +1) arranged in the following tree structure. The -nodes in the tree correspond to the prover’s messages and the edges correspond to the verifier’s challenges. -Each node at depth𝑖 has exactly ℓ𝑖 (𝑘𝑖 −1) +1 children corresponding toℓ𝑖 (𝑘𝑖 −1) +1 distinct challenges which, -as a set of vectors, lie inSS(𝑆𝑖, ℓ𝑖, 𝑘𝑖 ). Every transcript corresponds to exactly one path from the root to a leaf -node. -We say that Π is (ℓ1, . . . , ℓ𝜇)-coordinate-wise (𝑘1, . . . , 𝑘𝜇)-special sound for the relationR if there is a -polynomial time algorithm that given public parameterspp, statement x and the tree of transcripts, outputs a -witness w ∈ 𝑅(pp, x). If ℓ1 = . . . = ℓ𝜇 = 1 then we say the protocol is(𝑘1, . . . , 𝑘𝜇)-special sound. -We recall the main result of [FMN24, Section 7] which translates CWSS to knowledge soundness. -Lemma 4. Let Π be a (ℓ1, . . . , ℓ𝜇)-coordinate-wise (𝑘1, . . . , 𝑘𝜇)-special sound interactive proof system as in -Definition 3 and assume𝐾 := Î𝜇 -𝑖=1(ℓ𝑖 (𝑘𝑖 −1) +1) = poly(𝜆). Then, it is knowledge sound with knowledge error -𝜇∑︁ -𝑖=1 -ℓ𝑖 ·𝑘𝑖 -|𝑆𝑖 |ℓ𝑖 -. -A similar result also holds in the random oracle model ifΠ is turned non-interactive via Fiat-Shamir -transformation (see [FMN24, Section 8]). In this work, we will aim to show that our protocol satisfies -coordinate-wise special soundness. -8 We omitpp and simply writeR(x, w) if the public parameters are not relevant to the relation. -10 - ---- page 11 --- - -3 Evaluation Claims over Extension Fields as Relations over R𝒒 -In this section, we provide a generic transformation from provingℓ-variate polynomial evaluations over -finite fields F𝑞𝑘 to proving (ℓ − 𝛼 + 𝜅)-variate polynomial evaluations over the cyclotomic ringR𝑞. Next, we -improve the result if the committed polynomial is only over the base fieldF𝑞, instead ofF𝑞𝑘. At the core of -our transformation is the process of embedding vectors of extension fieldsF𝑞𝑘 inside ring elementsR𝑞 and -translating inner products overF𝑞𝑘 as “inner-product-related” statements overR𝑞. In Figure 2 we provide a -summary table of how much theF𝑞𝑘-to-R𝑞 transformations presented in this section cost in terms of prover -communication. -3.1 Generic Transformation over F𝒒𝒌 -Finding subfields inR𝑞. We introduce a lemma that explicitly describe finite field extensions withinR𝑞. -Lemma 5(Subfields of R𝑞). Let 𝑞 = 5 (mod 8) and 𝑘 ≥ 1 be a divisor of𝑑/2 and 𝐻 := ⟨𝜎−1, 𝜎4𝑘+1⟩ ⊆ -Aut(R). Then, R𝐻 -𝑞 is a subfield ofR𝑞 isomorphic to F𝑞𝑘. -Proof. By definition, R𝐻 -𝑞 is a subset of the set of all ring elements inR𝑞 stable under the𝜎−1 automorphism, -which itself is a field [LNP22, Lemma 2.6]. Using the homomorphic properties of Galois automorphisms, the -set R𝐻 -𝑞 is closed under both addition and multiplication. Finally, the size ofR𝐻 -𝑞 is 𝑞𝑘 since each element of -R𝐻 -𝑞 is of the form -𝑎 := 𝑎0 + -𝑘 −1∑︁ -𝑗=1 -𝑎𝑘 − 𝑗 · - -𝑋 -𝑑 -2𝑘 · (𝑘 − 𝑗 ) − 𝑋 -𝑑 -2𝑘 · (𝑘+𝑗 ) - -(7) -with 𝑘 degrees of freedom:𝑎0, 𝑎1, . . . , 𝑎𝑘 −1 ∈ Z𝑞. ⊓ ⊔ -Inner products.Our main theorem of this section generalizes [LNP22], and reduces proving inner products -over the subfield into proving statements overR𝑞 related to the trace map. Informally, it introduces a bijection -𝜓 which allows packing𝑑/𝑘 elements of fieldF𝑞𝑘 into one singleR𝑞 element. The fact that it is an invertible -function is crucial in arguing knowledge soundness. -Theorem 2(Inner product as output of the trace).Let 𝑘 divide 𝑑/2 and 𝐻 := ⟨𝜎−1, 𝜎4𝑘+1⟩ ⊆ Aut(R). -Consider a map -𝜓 : (R𝐻 -𝑞 )𝑑/𝑘 → R𝑞, -(𝑎0, 𝑎1, . . . , 𝑎𝑑/𝑘 −1) ↦→ - 𝑑/2𝑘 −1∑︁ -𝑖=0 -𝑎𝑖 ·𝑋 𝑖 -! -+ 𝑋 𝑑/2 · - 𝑑/2𝑘 −1∑︁ -𝑖=0 -𝑎𝑑/2𝑘+𝑖 ·𝑋 𝑖 -! -. -(8) -Then, 𝜓 is a bijection. Moreover, for anya, b ∈ ( R𝐻 -𝑞 )𝑑/𝑘, -Tr𝐻 (𝜓 (a) · 𝜎−1(𝜓 (b))) = 𝑑 -𝑘 · ⟨a, b⟩. -Proof. To prove the first part, we only need to show that𝜓 is injective since the domain and co-domain size -are the same. Indeed, suppose that for somea = (𝑎0, . . . , 𝑎𝑑/𝑘 −1) we have𝜓 (a) = 0 (the statement holds by the -linearly homomorphic property of𝜓). We prove thata is the zero-vector. -First, we note that𝜓 (a) can be expanded as -𝑑 -2𝑘 −1∑︁ -𝑖=0 -𝑋 𝑖𝑎𝑖,0 + 𝑋 -𝑑 -2 +𝑖𝑎𝑑/2𝑘+𝑖,0 + -𝑘 −1∑︁ -𝑗=1 -𝑎𝑖,𝑘 − 𝑗 · (𝑋 -𝑑 -2𝑘 · (𝑘 − 𝑗 )+𝑖 − 𝑋 -𝑑 -2𝑘 · (𝑘+𝑗 )+𝑖 ) -+ -𝑘 −1∑︁ -𝑗=1 -𝑎𝑑/2𝑘+𝑖,𝑘 − 𝑗 · (𝑋 -𝑑 -2𝑘 · (2𝑘 − 𝑗 )+𝑖 − 𝑋 -𝑑 -2𝑘 · (2𝑘+𝑗 )+𝑖 ). -(9) -11 - ---- page 12 --- - -Then, the 𝑢-th coefficient of the polynomial above, where 0≤ 𝑢 < 𝑑/(2𝑘), is exactly 𝑎𝑢,0. Similarly, the -(𝑑/2𝑘 + 𝑢)-th coefficient is𝑎𝑑/2𝑘+𝑢,0. Hence, 𝑎𝑢,0 = 𝑎𝑑/2𝑘+𝑢,0 = 0. -Now, we claim that for any 1≤ 𝑢 < 𝑘 and 0 ≤ 𝑣 < 𝑑 -2𝑘, 𝑎𝑣,𝑢 = 𝑎𝑑/2𝑘+𝑣,𝑢 = 0. Indeed, first consider the -( 𝑑 -2𝑘 ·𝑢 + 𝑣)-th coefficient of the polynomial above. It is equal to𝑎𝑣,𝑢 + 𝑎𝑑/2𝑘+𝑣,𝑘 −𝑢 = 0. On the other hand, if we -focus on the( 𝑑 -2𝑘 · (𝑘 + 𝑢) + 𝑣)-th coefficient, then it is exactly equal to−𝑎𝑣,𝑘 −𝑢 + 𝑎𝑑/2𝑘+𝑣,𝑢 = 0. By combining -the two observations we obtain -𝑎𝑑/2𝑘+𝑣,𝑢 = 𝑎𝑣,𝑘 −𝑢 = −𝑎𝑑/2𝑘+𝑣,𝑢 -which implies 𝑎𝑑/2𝑘+𝑣,𝑢 = 𝑎𝑣,𝑢 = 0, and thusa = 0. -We move on to the second part. We will make use of the following three claims. -Claim. Consider the multiplicative groupZ× -2𝑑 and let 𝑘 ≤ 𝑑/2 be a power-of-two. Then, -⟨4𝑘 +1⟩ = {(4𝑘) · 𝛼 +1 : 0≤ 𝛼 ≤ 𝑑/(2𝑘) − 1}. -Proof. First, using the property that 4𝑘 divides 2𝑑, we have for any𝑖, -((4𝑘 +1)𝑖 mod 2𝑑) mod 4𝑘 = (4𝑘 +1)𝑖 mod 4𝑘 = -𝑖∑︁ -𝑗=0 -𝑖 -𝑗 - -(4𝑘) 𝑗 mod 4𝑘 = 1. -Therefore, ⟨4𝑘 +1⟩ ⊆ {( 4𝑘) ·𝛼 +1 : 0≤ 𝛼 ≤ 𝑑/(2𝑘) −1}. Now, the statement holds since by [LS18, Lemma 2.4], -we know that the order of 4𝑘 +1 ∈ Z× -2𝑑 is exactly 2𝑑 -4𝑘 = 𝑑/(2𝑘). ⊓ ⊔ -Claim. Let 𝑖 ∈ Z be not divisible by𝑑/2𝑘. Then, Tr𝐻 (𝑋 𝑖 ) = 0. -Proof. Indeed, by assumption 4𝑘𝑖 cannot be divisible 2𝑑 and thus𝑋4𝑘𝑖 ≠ 1. Now, using the first claim, we get -Tr𝐻 (𝑋 𝑖 ) = -∑︁ -ℎ∈ ⟨−1,4𝑘+1⟩ -𝑋 ℎ·𝑖 -= -∑︁ -ℎ∈ ⟨4𝑘+1⟩ -𝑋 ℎ·𝑖 + 𝑋 −ℎ·𝑖 -= -𝑑/(2𝑘 ) −1∑︁ -𝛼=0 -𝑋 (4𝑘 ·𝛼+1) ·𝑖 + -𝑑/(2𝑘 ) −1∑︁ -𝛼=0 -𝑋 − (4𝑘 ·𝛼+1) ·𝑖 -= 𝑋 𝑖 · (𝑋4𝑘𝑖 )𝑑/(2𝑘 ) −1 -𝑋4𝑘𝑖 −1 + 𝑋 −𝑖 · (𝑋 −4𝑘𝑖 )𝑑/(2𝑘 ) −1 -𝑋 −4𝑘𝑖 −1 -= 0. -⊓ ⊔ -Claim. Tr𝐻 (𝑋 𝑑/2) = 0. -Proof. Using the first claim we obtain -Tr𝐻 (𝑋 𝑑/2) = -𝑑/(2𝑘 ) −1∑︁ -𝛼=0 -𝑋 (4𝑘 ·𝛼+1) ·𝑑/2 + -𝑑/(2𝑘 ) −1∑︁ -𝛼=0 -𝑋 − (4𝑘 ·𝛼+1) ·𝑑/2 -= -𝑑/(2𝑘 ) −1∑︁ -𝛼=0 -𝑋 𝑑/2 + -𝑑/(2𝑘 ) −1∑︁ -𝛼=0 -𝑋 −𝑑/2 -= -𝑑/(2𝑘 ) −1∑︁ -𝛼=0 -𝑋 𝑑/2 − 𝑋 𝑑/2 = 0. -⊓ ⊔ -12 - ---- page 13 --- - -To conclude the proof of the theorem, we notice that -𝜓 (a) · 𝜎−1 (𝜓 (b)) = - 𝑑/2𝑘 −1∑︁ -𝑖=0 -𝑑/2𝑘 −1∑︁ -𝑗=0 -𝑎𝑖𝑏 𝑗 ·𝑋 𝑖 − 𝑗 + -𝑑/2𝑘 −1∑︁ -𝑖=0 -𝑑/2𝑘 −1∑︁ -𝑗=0 -𝑎𝑑/2𝑘+𝑖𝑏𝑑/2𝑘+𝑗 ·𝑋 𝑖 − 𝑗 -! -+ 𝑋 𝑑/2 · - 𝑑/2𝑘 −1∑︁ -𝑖=0 -𝑑/2𝑘 −1∑︁ -𝑗=0 -𝑎𝑑/2𝑘+𝑖𝑏 𝑗 ·𝑋 𝑖 − 𝑗 − -𝑑/2𝑘 −1∑︁ -𝑖=0 -𝑑/2𝑘 −1∑︁ -𝑗=0 -𝑎𝑖𝑏𝑑/2𝑘+𝑗 ·𝑋 𝑖 − 𝑗 -! -. -Hence, -Tr𝐻 (𝜓 (a) · 𝜎−1(𝜓 (b))) -= - 𝑑/2𝑘 −1∑︁ -𝑖=0 -𝑑/2𝑘 −1∑︁ -𝑗=0 -𝑎𝑖𝑏 𝑗 ·Tr𝐻 (𝑋 𝑖 − 𝑗 ) + -𝑑/2𝑘 −1∑︁ -𝑖=0 -𝑑/2𝑘 −1∑︁ -𝑗=0 -𝑎𝑑/2𝑘+𝑖𝑏𝑑/2𝑘+𝑗 ·Tr𝐻 (𝑋 𝑖 − 𝑗 ) -! -+ - 𝑑/2𝑘 −1∑︁ -𝑖=0 -𝑑/2𝑘 −1∑︁ -𝑗=0 -𝑎𝑑/2𝑘+𝑖𝑏 𝑗 ·Tr𝐻 (𝑋 𝑑/2+(𝑖 − 𝑗 ) ) − -𝑑/2𝑘 −1∑︁ -𝑖=0 -𝑑/2𝑘 −1∑︁ -𝑗=0 -𝑎𝑖𝑏𝑑/2𝑘+𝑗 ·Tr𝐻 (𝑋 𝑑/2+(𝑖 − 𝑗 ) ) -! -. -Now, using the fact that for two distinct 0≤ 𝑖, 𝑗 < 𝑑/2𝑘, both 𝑖 − 𝑗 and 𝑑/2 + ( 𝑖 − 𝑗) cannot be divisible -by 𝑑/2𝑘, and the second claim, we deduce thatTr𝐻 (𝑋 𝑖 − 𝑗 ) = Tr𝐻 (𝑋 𝑑/2+𝑖 − 𝑗 ) = 0. However, when𝑖 = 𝑗, then -Tr𝐻 (𝑋 𝑖 − 𝑗 ) = 𝑑 -𝑘 and Tr𝐻 (𝑋 𝑑/2+(𝑖 − 𝑗 ) ) = 0 by the third claim. Hence, we deduce that -Tr𝐻 (𝜓 (a) · 𝜎−1 (𝜓 (b))) = 𝑑 -𝑘 -𝑑/𝑘 −1∑︁ -𝑖=0 -𝑎𝑖𝑏𝑖 = 𝑑 -𝑘 ⟨a, b⟩ -which concludes the proof. ⊓ ⊔ -Next, we give a simple observation regarding how infinity norm changes after applying the𝜓 map. -Lemma 6. Let a be a vector overR𝐻 -𝑞 such that ∥a∥∞ ≤ 𝛽 and 𝜓 be the map defined in Theorem 2. Then, -∥𝜓 (a) ∥∞ ≤ 2𝛽. -Proof. The statement follows directly from the explicit formula in Equation (9), where some coefficients of -𝜓 (a) are equal to the sum of two coefficients ofa. -Reducing to multilinear evaluation overR𝑞. We use the results above to translate the multilinear evaluation -claim over R𝐻 -𝑞 -𝑓 (𝑥1, . . . , 𝑥ℓ ) = 𝑦 -to an equivalent relation over the ringR𝑞. Denote the coefficients of polynomial𝑓 as (𝑓𝜄)𝜄 ∈ {0,1}ℓ ∈ R𝐻 -𝑞 . We can -explicitly rewrite the evaluation equation as -𝑦 = -∑︁ -𝑖 ∈ {0,1}ℓ −𝛼 +𝜅 - -𝑥𝑖1 -1 · . . . ·𝑥𝑖ℓ −𝛼 +𝜅 -ℓ −𝛼+𝜅 - -·©­ -« -∑︁ -𝑗 ∈ {0,1}𝛼 −𝜅 -𝑓𝑖 ∥ 𝑗 ·𝑥 𝑗1 -ℓ −𝛼+𝜅+1 · . . . ·𝑥 𝑗𝛼 −𝜅 -ℓ -ª® -¬ -. (10) -For each𝑖 ∈ {0,1}ℓ −𝛼+𝜅 we define -𝐹𝑖 := 𝜓 (𝑓𝑖 | |𝑗 ) 𝑗 ∈ {0,1}𝛼 −𝜅 - and also 𝑣 := 𝜓 - -(𝑥 𝑗1 -ℓ −𝛼+𝜅+1 · . . . ·𝑥 𝑗𝛼 −𝜅 -ℓ ) 𝑗 ∈ {0,1}𝛼 −𝜅 - -. -Now, recall that for every𝑥 ∈ R𝐻 -𝑞 we haveTr𝐻 (𝑥) = 𝑥 and trace is additively homomorphic. Hence, by -applying Theorem 2 to Equation (10) we obtain -𝑑 -𝑘 ·𝑦 = Tr𝐻 -©­­­­­­ -« -∑︁ -𝑖 ∈ {0,1}ℓ −𝛼 +𝜅 - -𝑥𝑖1 -1 · . . . ·𝑥𝑖ℓ −𝛼 +𝜅 -ℓ −𝛼 - -·𝐹𝑖 -| {z } -𝑌:= -·𝜎−1(𝑣) -ª®®®®®® -¬ -. -13 - ---- page 14 --- - -Therefore,theproverwillsendasingleringelement 𝑌 ∈ R𝑞,whichallowstheverifiertocheckif Tr𝐻 (𝑌 ·𝜎−1(𝑣)) = -𝑑𝑦/𝑘. Then, we only need to prove that𝑌 is well-formed, which is an evaluation proof for the(ℓ − 𝛼 +𝜅)-variate -multilinear polynomial 𝐹 := (𝐹𝑖 )𝑖 over R𝑞. -3.2 Dealing with 𝒇 ∈ Z≤1 -𝒒 [𝑿1, . . . , 𝑿ℓ ] -We now optimize the special case where the evaluation point(𝑥1, . . . , 𝑥ℓ ) is defined in the extension fieldF𝑞𝑘, -while the coefficients of the committed polynomial𝑓 are over Z𝑞. Note that if we naively apply the generic -transformation from Section 3.1 then we would end up with an evaluation proof for the(ℓ − 𝛼 + 𝜅)-variate -multilinear polynomial overR𝑞. This subsection focuses on how to lower it further toℓ − 𝛼 variables. -We start with a similar observation as in Equation (10) that a polynomial evaluation claim can be written -as -𝑦 = -∑︁ -𝑖 ∈ {0,1}𝜅 - -𝑥𝑖1 -1 · . . . ·𝑥𝑖𝜅 -𝜅 - -· -©­­­­­­­ -« -∑︁ -𝑗 ∈ {0,1}ℓ −𝜅 -𝑓𝑖 ∥ 𝑗 ·𝑥 𝑗1 -𝜅+1 · . . . ·𝑥 𝑗ℓ −𝜅 -ℓ -| {z } -𝑦𝑖:= -ª®®®®®®® -¬ -. (11) -Hence, we let the prover send𝑘 partial evaluations𝑦𝑖 ∈ F𝑞𝑘 to the verifier who can directly check whether -𝑦 = -∑︁ -𝑖 - -𝑥𝑖1 -1 · . . . ·𝑥𝑖𝜅 -𝜅 - -·𝑦𝑖 . -What we have left to prove is that all the𝑦𝑖 are well-formed, or in other words𝑓𝑖 (𝑥𝜅+1, . . . , 𝑥ℓ ) = 𝑦𝑖 for -𝑖 ∈ {0,1}𝜅 where 𝑓𝑖 ∈ Z𝑞 [𝑋ℓ −𝜅+1, . . . , 𝑋ℓ ]. -For the next part, we need a somewhat explicit definition of the extension field (it could as well be -R𝐻 -𝑞 ). Let F𝑞𝑘 := F𝑞 [𝑍 ]/𝜑 (𝑍 ), where 𝜑 is an irreducible polynomial inF𝑞 [𝑍 ] of degree 𝑘. Then, we define the -(ℓ − 𝜅)-variate multilinear polynomial𝑓 ′ ∈ F𝑞𝑘 [𝑋𝜅+1, . . . , 𝑋ℓ ] as -𝑓 ′ (𝑋𝜅+1, . . . , 𝑋ℓ ) := -∑︁ -𝑖 ∈ {0,1}𝜅 -𝑓𝑖 (𝑋𝜅+1, . . . , 𝑋ℓ ) · 𝑍 -Í𝜅 -𝑡 =1 𝑖𝑡 ·2𝑡 −1 -. -Then, by definition of the partial evaluations𝑦𝑖 we obtain -𝑓 ′ (𝑥𝜅+1, . . . , 𝑥ℓ ) = -∑︁ -𝑖 ∈ {0,1}𝜅 -𝑦𝑖 ·𝑍 -Í𝜅 -𝑡 =1 𝑖𝑡 ·2𝑡 −1 -∈ F𝑞𝑘 -which is now a full-fledged polynomial evaluation overF𝑞𝑘. By applying the generic transformation from -Section 3.1, we can reduce the statement into an evaluation claim overR𝑞 for a𝜇-variate multilinear polynomial, -where -𝜇 = (ℓ − 𝜅)| {z } -Section 3.2 -−𝛼 + 𝜅 -| {z } -Section 3.1 -= ℓ − 𝛼 . -The final observation is that the verifier can directly compute𝑦0...0 from 𝑦 and other 𝑦𝑖, and thus only𝑘 −1 -evaluations are needed to be sent. -Remark 1. One may notice a strange behavior in Figure 2 in a sense that the transformation from a field -extension F𝑞𝑘 to R𝑞 seems cheaper than from the base fieldF𝑞 to R𝑞. Indeed, if given a witness polynomial𝑓 -over F𝑞, why is it not enough to just follow the transformation from Section 3.1? -There are two main reasons. The first one is that, when combined with a polynomial evaluation protocol -over R𝑞, it is not guaranteed in the knowledge soundness proof that the extracted polynomial𝑓 is also over -F𝑞, as we obtain𝑓 ∈ F𝑞𝑘 [𝑋1, . . . , 𝑋ℓ ]. Moreover, the transformation in Section 3.2 allows us to further reduce -the number of variables of theR𝑞-polynomial, from ℓ − 𝛼 + 𝜅 to ℓ − 𝛼, which has significant impact in reducing -round complexity of the protocol. -14 - ---- page 15 --- - -Committed polynomial # variables after theR𝑞-transformation Prover cost -𝑓 ∈ F𝑞𝑘 [𝑋1, . . . , 𝑋ℓ ] ℓ − 𝛼 + 𝜅 𝑑log 𝑞 -𝑓 ∈ F𝑞 [𝑋1, . . . , 𝑋ℓ ] ℓ − 𝛼 (𝑑 + 𝑘 (𝑘 −1)) log 𝑞 -Fig. 2. Summary of our transformation in Sections 3.1 and 3.2. The prover cost is measured in the size of the prover -messages in bits. -4 Multilinear Polynomial Commitment Scheme over R𝒒 -This section focuses on proving knowledge of acommitted multilinear polynomial 𝑓 ∈ R≤1 -𝑞 [𝑋1, . . . , 𝑋ℓ ] with -coefficients (𝑓𝑖 )𝑖 ∈ {0,1}ℓ such that -𝑓 (𝑥1, . . . , 𝑥ℓ ) := -∑︁ -𝑖 ∈ {0,1}ℓ -𝑓𝑖 ·𝑥𝑖1 -1 · . . . ·𝑥𝑖ℓ -ℓ = 𝑢 -where the evaluation point(𝑥1, . . . , 𝑥ℓ ) ∈ Rℓ -𝑞 and the image𝑢 ∈ R𝑞 are public. -We follow the initial strategy from Greyhound [NS24]. Supposeℓ = 𝑚 + 𝑟 for some 𝑚, 𝑟 ≥ 1 (keep in mind -𝑚 ≈ 𝑟). Then, we can rewrite the equation above as -𝑓 (𝑥1, . . . , 𝑥𝜇) := -∑︁ -𝑖 ∈ {0,1}𝑟 -∑︁ -𝑗 ∈ {0,1}𝑚 -𝑓𝑖 | |𝑗 · (𝑥𝑖1 -1 · . . . ·𝑥𝑖𝑟 -𝑟 ) · (𝑥 𝑗1 -𝑟 +1 · . . . ·𝑥 𝑗𝑚 -ℓ ), -or in other words -𝑓 (𝑥1, . . . , 𝑥𝜇) = b⊺ - -a⊺ -. . . -a⊺ - - -f0...0 -... -f1...1 - -for -b⊺ := (𝑥𝑖1 -1 · . . . ·𝑥𝑖𝑟 -𝑟 )𝑖 ∈ {0,1}𝑟 ∈ R2𝑟 -𝑞 -a⊺ := (𝑥 𝑗1 -𝑟 +1 · . . . ·𝑥 𝑗𝑚 -ℓ ) 𝑗 ∈ {0,1}𝑚 ∈ R2𝑚 -𝑞 -f ⊺ -𝑖 := (𝑓𝑖 | |𝑗 ) 𝑗 ∈ {0,1}𝑚 ∈ R2𝑚 -𝑞 for 𝑖 ∈ {0,1}𝑟 -. (12) -This representation of a polynomial evaluation claim gives intuition how to commit to the polynomial𝑓. -4.1 Inner and Outer Commitment -The starting point of our polynomial commitment scheme construction is the basic commitment scheme from -LaBRADOR [BS23]. Let𝑏 > 1 be the base used for decomposition (throughout) and denote𝛿 := ⌈log𝑏 𝑞⌉. -The commitment key is a pair of uniformly random matricesA ∈ R𝑛𝐴 ×𝛿2𝑚 -𝑞 and B ∈ R𝑛𝐵 ×𝑛𝐴𝛿2𝑟 -𝑞 where 𝑛𝐴, 𝑛𝐵 -are the security parameters which determine hardness of the underlying Module-SIS problem. -Suppose we want to commit to the 2𝑟 vectors (f ⊺ -𝑖 )𝑖 of length 2𝑚. The first step is to compute the -decompositions: -s1+Í𝑟 -𝑧=1 𝑖𝑧 ·2𝑧−1 := G−1 -2𝑚 (f𝑖 ) (13) -and the inner Ajtai commitmentst𝑖 := As𝑖 ∈ R𝑛𝐴 -𝑞 for 𝑖 ∈ [ 2𝑟 ]. The next step is to compute their binary -decomposition ˆt𝑖 := G−1 -𝑛𝐴 (t𝑖 ). The final outer commitment is -u := B - -ˆt1 -... -ˆt2𝑟 - -∈ R𝑛𝐵 -𝑞 . (14) -The commitment opening becomes a tuple of(s𝑖,ˆt𝑖 )𝑖. -15 - ---- page 16 --- - -In the knowledge extraction, we will only be able to extract so-called weak openings. We define a weak -opening for the commitmentu is a tuple(s𝑖,ˆt𝑖, 𝑐𝑖 )𝑖 ∈ [2𝑟 ], which satisfies all the following conditions -∀𝑖 ∈ [2𝑟 ] : ∥𝑐𝑖 ·s𝑖 ∥ ≤ ¯𝛽, ∥𝑐𝑖 ∥1 ≤ ¯𝜔, 𝑐 𝑖 ∈ R× -𝑞 , As𝑖 = G𝑛𝐵 -ˆt𝑖 -B - -ˆt1 -... -ˆt2𝑟 - -= u and - - -ˆt1 -... -ˆt2𝑟 - - -∞ -≤ ¯𝛾 . -Clearly, a message can be derived from the weak opening by Equation (13). We recall the known result that -the commitment scheme above satisfies binding w.r.t. weak openings under the Module-SIS assumption. We -omit the proof since it follows identically as in [NS24, Lemma 2.11]. -Lemma 7(Weak Binding). There is a deterministic algorithm that given two weak openings(s𝑖,ˆt𝑖, 𝑐𝑖 )𝑖 ∈ [2𝑟 ] -and (s′ -𝑖,ˆt′ -𝑖, 𝑐′ -𝑖 )𝑖 ∈ [2𝑟 ] for the commitment u ∈ R𝑛𝐵 -𝑞 such that s𝑗 ≠ s′ -𝑗 for some 𝑗 ∈ [ 2𝑟 ], outputs a vector -z ∈ R2𝑚+𝑛𝐴𝛿2𝑟 -𝑞 such that [A | B]z = 0 and 0 < ∥z∥∞ ≤ max(2¯𝜔¯𝛽,2¯𝛾). -4.2 Polynomial Evaluation as Quadratic Equation -We can now rephrase Equation (12) in terms of the commitment opening(s𝑖 )𝑖. Namely, we can write -𝑢 = 𝑓 (𝑥1, . . . , 𝑥ℓ ) = b⊺ - -a⊺G2𝑚 -. . . -a⊺G2𝑚 - - -s1 -... -s2𝑟 - -. (15) -From now on, we focus on proving knowledge of a commitment opening(s𝑖,ˆt𝑖 )𝑖 for the commitment®𝑢 which -satisfies the above equation. We summarize the protocol in Figure 3. -To begin with, the prover commits to the vectorw = (𝑤1, . . . , 𝑤2𝑟 ) ∈ R2𝑟 -𝑞 defined as𝑤 𝑖 := a⊺G2𝑚 s𝑖 ∈ R𝑞. -That is, P first decomposes each 𝑤 𝑖, thus obtaining ˆw𝑖 := G−1 -1 (𝑤 𝑖 ) ∈ R𝛿 -𝑞. Finally, the prover sends the -commitment -v := Dˆw ∈ R𝑛𝐷 -𝑞 (16) -where D ∈ R𝑛𝐷 ×𝛿 ·2𝑟 -𝑞 is a public uniformly random matrix andˆw⊺ := (ˆw⊺ -1 , . . . ,ˆw⊺ -2𝑟 ). Note that Equation (15) -can be rephrased as -𝑢 = b⊺w = b⊺G2𝑟 ˆw. (17) -The verifier V responds with a challenge vectorc := (𝑐1, . . . , 𝑐2𝑟 ) ← C 2𝑟 -, where the challenge spaceC is -currently defined as a subset of short challenges inR𝑞: -C ⊆ { 𝑐 ∈ R𝑞 : ∥𝑐 ∥1 ≤ 𝜔 } -for some 𝜔 instantiated later9. Then, the proverP computes the short folded witness -z := 𝑐1s1 + 𝑐2s2 + . . . + 𝑐2𝑟 s2𝑟 ∈ R2𝑚𝛿 -𝑞 . -We note the relationship betweenz and ˆw -a⊺G2𝑚 z = -2𝑟 -∑︁ -𝑖=1 -𝑐𝑖 a⊺G2𝑚 s𝑖 = -2𝑟 -∑︁ -𝑖=1 -𝑐𝑖𝑤 𝑖 = -2𝑟 -∑︁ -𝑖=1 -𝑐𝑖 G1ˆ𝑤 𝑖 = (c⊺ ⊗ G1)ˆw. (18) -We also note the relationship betweenz and ˆt: -Az := -2𝑟 -∑︁ -𝑖=1 -𝑐𝑖 As𝑖 = -2𝑟 -∑︁ -𝑖=1 -𝑐𝑖 G𝑛𝐴 -ˆt𝑖 = (c ⊗ G𝑛𝐴 )ˆt. (19) -9 The ℓ1 norm bound for challenges is set here to later apply Lemma 2. -16 - ---- page 17 --- - -Now, combining Equations 14, 16, 17, 18, and 19, the proverP will prove that shortz,ˆt, ˆw satisfy the linear -relation over R𝑞: - -D 0 0 -0 B 0 -b⊺G2𝑟 0 0 -(c⊺ ⊗ G1) 0 −a⊺G2𝑚 -0 c ⊺ ⊗ G𝑛𝐴 −A - - -ˆw -ˆt -z - -= - -v -u -𝑢 -0 -0 - -. -We see that while coefficients ofˆw,ˆt are between ⌈−𝑏/2⌉ and ⌈𝑏/2⌉ −1, the coefficients ofz will be moderately -larger. Let 𝛽 be the maximum𝐿∞ norm of z, 𝜏 ≔ ⌈log𝑏 𝛽⌉ and J𝑛 ≔ I𝑛 ⊗ -1 𝑏 𝑏2 · · ·𝑏𝜏 −1 -∈ R𝑛×𝑛𝜏 -𝑞 be the -corresponding gadget matrix. To decrease the coefficients ofz further, we decompose asˆz = J−1 -2𝑚 (z) ∈ R2𝑚𝛿𝜏 -𝑞 . -Hence, the prover now needs to prove knowledge of(ˆw,ˆt,ˆz) ∈ 𝑆2𝑟 𝛿+2𝑟 𝑛𝐴𝛿+2𝑚𝛿𝜏 -𝑏 such that - -D 0 0 -0 B 0 -b⊺G2𝑟 0 0 -(c⊺ ⊗ G1) 0 −a⊺G2𝑚 J2𝑚 -0 c ⊺ ⊗ G𝑛𝐴 −AJ2𝑚 - - -ˆw -ˆt -ˆz - -= - -v -u -𝑢 -0 -0 - -. (20) -Public parameters: A ∈ R𝑛𝐴 ×𝛿2𝑚 -𝑞 , B ∈ R𝑛𝐵 ×𝑛𝐴𝛿2𝑟 -𝑞 , D ∈ R𝑛𝐷 ×𝛿2𝑟 -𝑞 -Witness: commitment openings := (s𝑖 )𝑖,ˆt := (ˆt𝑖 )𝑖 -Statement: commitment u ∈ R𝑛𝐵 -𝑞 , evaluation𝑢, vectors a, b as in Equation (15) -Prover P Verifier V -For 𝑖 ∈ [2𝑟 ] -𝑤𝑖 := a⊺G2𝑚 s𝑖 -w := (𝑤1, . . . , 𝑤2𝑟 ) -ˆw := G−1 -2𝑟 (w) -v := Dˆw ∈ R𝑛𝐷 -𝑞 -v - -c := (𝑐1, . . . , 𝑐2𝑟 ) ← C 2𝑟 -c -z := Í2𝑟 -𝑖=1 𝑐𝑖 s𝑖 -If ∥z∥∞ > 𝛽, abort -ˆz = J−1 -2𝑚 (z) -ˆt, ˆw,ˆz - -(ˆw,ˆt,ˆz) -? -∈ 𝑆2𝑟 𝛿+2𝑟 𝑛𝐴𝛿+2𝑚𝛿𝜏 -𝑏 - -D 0 0 -0 B 0 -b⊺G2𝑟 0 0 -(c⊺ ⊗ G1) 0 −a⊺G2𝑚 J2𝑚 -0 c ⊺ ⊗ G𝑛𝐴 −AJ2𝑚 - - -ˆw -ˆt -ˆz - -?= - -v -u -𝑢 -0 -0 - -. -Fig. 3. Interactive proof of knowledge of the commitment opening(s𝑖,ˆt𝑖 ) which satisfies(15). In our final scheme, the -prover does not send the final message in the clear, but proves knowledge of valid(ˆw,ˆt,ˆz). -The completeness follows directly from the protocol rationale described above. Hence, we move on the -soundness guarantees. Identically as [NS24, Lemma 3.2] we show that given CWSS-structured transcripts, we -can extract a valid weak opening which satisfies Equation (15). -17 - ---- page 18 --- - -Lemma 8(CWSS of Figure 3).Let (¯𝛽, ¯𝜔, ¯𝛾) := (2𝑏𝑘,2𝜔, 𝑏). Suppose 𝜔 < 1 -2 -√ -2𝑞1/2. Then, given the public -parameters, the statement(u, 𝑢), and2𝑟 +1 valid transcripts of the form -tr( 𝑗 ) := (v, c( 𝑗 ), (ˆw( 𝑗 ),ˆt( 𝑗 ),ˆz( 𝑗 ) )) where (c( 𝑗 ) )0≤ 𝑗 ≤2𝑟 ∈ SS(C,2𝑟 ,2), -one can either efficiently extract a weak opening(¯s( 𝑗 ),¯t( 𝑗 ), 𝑐( 𝑗 ) ) 𝑗 ∈ [2𝑟 ] for the commitmentu such that -𝑢 = b⊺ - -a⊺G2𝑚 -. . . -a⊺G2𝑚 - - -¯s1 -... -¯s2𝑟 - -, -or solve MSIS𝑞,𝑑,𝑛𝐵,𝛿2𝑟 𝑛𝐴,2𝑏 or MSIS𝑞,𝑑,𝑛𝐷 ,𝛿2𝑟 ,2𝑏. -Proof. For each 𝑗 ∈ [2𝑟 ], let us reconstruct the vectorz( 𝑗 ) := J2𝑚ˆz( 𝑗 ). Then, we have∥z( 𝑗 ) ∥∞ ≤ 𝑏𝑘. -Without loss of generality, assume thatc(0) is the “central vector”. First, suppose that for two distinct -𝑖 ≠ 𝑗 we haveˆt(𝑖 ) ≠ˆt( 𝑗 ). In particular, we haveBˆt(𝑖 ) = u = Bˆt( 𝑗 ), and so we have found a short solution of -norm at most 2𝑏 for the matrixB. Similarly, we argue for the vectorsˆw( 𝑗 ). Hence, from now on we assume -that allˆt( 𝑗 ) :=ˆt and ˆw( 𝑗 ) := ˆw are the same. Denotew := G2𝑟 ˆw and parse w = (𝑤 𝑖 )𝑖,ˆt := (ˆt𝑖 )𝑖. Then, we have -b⊺w = 𝑢. -Fix 𝑗 ∈ [2𝑟 ] and consider the transcriptstr( 𝑗 ) and tr(0) with the verifier challenges denoted asc( 𝑗 ) := (𝑐 𝑗,𝑖 )𝑖 -and c(0) := (𝑐0,𝑖)𝑖. Recall that the two challenge vectors differexactly in the 𝑗-th coordinate and𝑐 𝑗,𝑖 = 𝑐0,𝑖 for -all 𝑖 ≠ 𝑗. By subtracting the two corresponding verification equations, we obtain -b⊺G2𝑚 (z( 𝑗 ) − z(0) ) = (𝑐 𝑗,𝑗 − 𝑐0,𝑗 ) · 𝑤 𝑗 and A(z( 𝑗 ) − z(0) ) = (𝑐 𝑗,𝑗 − 𝑐0,𝑗 ) · G𝑏,𝑛ˆt𝑗 . -Hence, define our extracted opening as¯st := (s𝑗,ˆt𝑗 ) 𝑗 where -s𝑗 := z( 𝑗 ) − z(0) -𝑐 𝑗,𝑗 − 𝑐0,𝑗 -for all 𝑗 ∈ {0,1}𝑟 -and the slack vector(¯𝑐 𝑗 ) 𝑗 is defined as¯𝑐 𝑗 := 𝑐 𝑗,𝑗 − 𝑐0,𝑗. By construction, we have∥¯𝑐 𝑗 ∥1 ≤ 2𝜔 and ∥¯𝑐 𝑗 s𝑗 ∥∞ ≤ 2𝑏𝑘. -By Lemma 3,¯𝑐 𝑗 is invertible overR𝑞 since ∥¯𝑐 𝑗 ∥∞ ≤ 2𝜔 < 1√ -2𝑞1/2. Finally,b⊺G2𝑚 s𝑗 = 𝑤 𝑗 for all 𝑗 ∈ [2𝑟 ], which -concludes the proof. ⊓ ⊔ -4.3 Unstructured Linear Relations and Sumcheck -In this section, we show how to reduce proving unstructured linear relations overR𝑞 with infinity-norm -constraints, as in Equation (20). Concretely, we focus of the following relationRlin: -Rlin -𝑞,𝑑,𝑛,𝜇,𝑏 := - -(z ∈ R𝜇 -𝑞, (M ∈ R𝑛×𝜇 -𝑞 , y ∈ R𝑛 -𝑞 )) : Mz = y ∧ ∥ z∥∞ ≤ 𝑏 −1 -. -At a high level, our approach is to commit toz using a (potentially another) polynomial commitment scheme, -and then apply the sumcheck protocol to prove the relation. However, one challenge is that the relation -Rlin -𝑞,𝑑,𝑛,𝜇,𝑏 is defined over polynomial ringsR𝑞, so the standard sumcheck protocol over finite fields cannot be -directly applied (or one would have to run it overR𝑞 as in [BC24]). To bridge this gap, we leverage the -ring-switching technique introduced by [HMZ25]. -Ring switching. In the relationRlin -𝑞,𝑑,𝑛,𝜇,𝑏, the inputM ∈ R𝑛×𝜇 -𝑞 and y ∈ R𝑛 -𝑞 are public information, and the -prover needs to prove knowledge of a witnessz ∈ R𝜇 -𝑞 such that -–Mz = y -–∥z∥∞ ≤ 𝑏 −1 -18 - ---- page 19 --- - -For the first identity, the authors of [HMZ25] observe thatMz = y if and only if there is a polynomial -r ∈ ( Z<𝑑 -𝑞 [𝑋 ]) 𝑛 such that -Mz = y + (𝑋 𝑑 +1) · r -We now naturally considerM, z, and y as matrices or vectors overZ<𝑑 -𝑞 [𝑋 ] (instead of overR𝑞). Since we will -apply the sumcheck protocol later, we further considerM, x, y, r as elements overF<𝑑 -𝑞𝑘 [𝑋 ]. Now, we can divide -the process into two steps: the prover first commits to(z, r) using a polynomial commitment schemeCom, -and then proves the following constraints: -–For each𝑖 ∈ [ 𝑛], Í -𝑗 M𝑖,𝑗 z𝑗 = y𝑖 + (𝑋 𝑑 +1) · r𝑖, where M𝑖,𝑗 , y𝑖 ∈ Z<𝑑 -𝑞 [𝑋 ] -–z is a vector overZ<𝑑 -𝑞 [𝑋 ] and ∥z∥∞ ≤ 𝑏 −1. -Notice that the verifier can check whetherM and y are over Z<𝑑 -𝑞 [𝑋 ] by themselves, sinceM and y are public -information. However, the prover has to prove thatz is a vector overZ<𝑑 -𝑞 [𝑋 ] if we run the sumcheck protocol -over F<𝑑 -𝑞𝑘 [𝑋 ]. Since M𝑖,𝑗 , z𝑗, y𝑖, r𝑖 ∈ Z<𝑑 -𝑞 [𝑋 ], we can represent them as -M𝑖,𝑗 = -∑︁ -ℓ<𝑑 -𝑀𝑖,𝑗,ℓ ·𝑋 ℓ, -and similarly for allz𝑗, y𝑖, and r𝑖. Here we note that bothz and r are witnesses of the prover, and the prover -needs to prove the norm bound ofz. For the sake of notation and implementation simplicity, we also let the -prover performs a gadget decomposition ofr as r = Í -𝑢 𝑏𝑢 ·r𝑢, and commits(z, r1, . . . , rlog𝑏 (𝑞) ) instead, and -also proves the norm bound ofr𝑢 for each𝑢. That is, the prover proves the following statement: -–For each𝑖 ∈ [ 𝑛], Í -𝑗 M𝑖,𝑗 z𝑗 = y𝑖 + (𝑋 𝑑 +1) ·Í -𝑢 𝑏𝑢r𝑖,𝑢, where M𝑖,𝑗 , y𝑖 ∈ Z<𝑑 -𝑞 [𝑋 ] -–z, r1, . . . , rlog𝑏 (𝑞) are vectors overZ<𝑑 -𝑞 [𝑋 ] and ∥z∥∞, ∥r1∥∞, . . . , ∥rlog𝑏 (𝑞) ∥∞ ≤ 𝑏 −1. -To simplify the presentation, in the following we omit the subscript𝑢 and explain how to prove the -statement below. Readers should be aware that there is a hidden gadget decomposition ofr. We believe this is -sufficient to explain our main idea. To incorporate this gadget decomposition, we can modify the multilinear -extension correspondingly. -–For each𝑖 ∈ [ 𝑛], Í -𝑗 M𝑖,𝑗 z𝑗 = y𝑖 + (𝑋 𝑑 +1) · r𝑖, where M𝑖,𝑗 , y𝑖 ∈ Z<𝑑 -𝑞 [𝑋 ] -–z, r are vectors overZ<𝑑 -𝑞 [𝑋 ] and ∥z∥∞, ∥r∥∞ ≤ 𝑏 −1. -For each𝛼 ∈ F𝑞𝑘, we denote -M𝑖,𝑗 (𝛼) = -∑︁ -ℓ<𝑑 -𝑀𝑖,𝑗,ℓ ·𝛼 ℓ ∈ F𝑞𝑘 , -and similarly for allz𝑗, y𝑖, and r𝑖. Now, instead of proving the constraints overF<𝑑 -𝑞𝑘 [𝑋 ], which could be very -costly, we optimize by letting the verifier first send a random𝛼 to the prover. Then, the prover proves the -following constraints: -–For each𝑖 ∈ [ 𝑛], Í -𝑗 M𝑖,𝑗 (𝛼)z𝑗 (𝛼) = y𝑖 (𝛼) + ( 𝛼𝑑 +1) · r𝑖 (𝛼), where M𝑖,𝑗 , y𝑖 ∈ Z<𝑑 -𝑞 [𝑋 ] -–z is a vector overZ<𝑑 -𝑞 [𝑋 ] and ∥z∥∞, ∥r∥∞ ≤ 𝑏 −1. -Having proof composition in mind, we consider the following simple three-round protocol in Figure 4 which -captures the idea above. -19 - ---- page 20 --- - -Prover P Verifier V -𝑡 := Com(z, r) 𝑡 - -𝛼 ← F𝑞𝑘 -𝛼 -z, r - -𝑡 ?= Com(z, r) -z -? -∈ Z<𝑑𝑞 [𝑋 ] and ∥z∥∞, ∥r∥∞ ≤ 𝑏 −1 -∀𝑖 ∈ [ 𝑛],Í -𝑗 M𝑖,𝑗 (𝛼)z𝑗 (𝛼) ?= y𝑖 (𝛼) + ( 𝛼𝑑 +1) · r𝑖 (𝛼) -Fig. 4. Reducing proving linear equations overZ<𝑑𝑞 𝑞 [𝑋 ] to equations overF𝑞𝑘. -We note that this simplification could introduce a(2𝑑 −1)/| F𝑞𝑘 | soundness error. This is because the polynomialÍ -𝑗 M𝑖,𝑗 z𝑗 − y𝑖 − (𝑋 𝑑 +1) ·r𝑖 has degree at most 2𝑑 −1. Formally, we argue this by proving 2𝑑-special soundness -in Lemma 9. -Lemma 9(Special soundness of Figure 4).There is an efficient, deterministic algorithm that given -a statement (M𝑖,𝑗 , y𝑖, 𝑏) and 2𝑑 valid transcripts (𝑡𝑖, 𝛼𝑖, (z𝑖, r𝑖 )) of the protocol in Figure 4 such that(𝛼𝑖 )𝑖 ∈ -SS(F𝑞𝑘 ,1,2𝑑), can either extract¯z,¯r which satify Mz = y + ( 𝑋 𝑑 +1) · r over Z𝑞 [𝑋 ], or break binding of the -commitment scheme Com. -Proof. If for some𝑖 ≠ 𝑗 we have (z𝑖, r𝑖 ) ≠ (z𝑗, r𝑗 ), then we automatically break the binding property of the -commitment scheme. Otherwise, define¯z := z𝑖,¯r := r𝑖 for all 𝑖. Then, for 2𝑑 distinct evaluation points𝛼 𝑗, we -have -∀𝑖 ∈ [ 𝑛], -∑︁ -𝑗 -M𝑖,𝑗 (𝛼 𝑗 )z𝑗 (𝛼 𝑗 ) -? -= y𝑖 (𝛼 𝑗 ) + ( 𝛼𝑑 +1) · r𝑖 (𝛼 𝑗 ). -Since the polynomialÍ -𝑗 M𝑖,𝑗 z𝑗 − y𝑖 − (𝑋 𝑑 +1) · r𝑖 has degree at most 2𝑑 −1, this implies that it must be a -zero polynomial, which finishes the proof. ⊓ ⊔ -From now on, we focus on proving knowledge of the prover’s last message which satisfies the verification -conditions. -Remark 2. Looking ahead, we will instantiateCom with the inner-outer commitment from Section 4. In this -case, for special soundness we will consider breaking the weak binding property (cf. Lemma 7), rather than -binding. -Represent the constraints by polynomials.In our sumcheck protocol, we first commit(z, r) as a multilinear -polynomial over F𝑞𝑘. Let e𝑤 be a multilinear polynomial such that -e𝑤 (𝑢, ℓ) = -( -𝑧𝑢,ℓ , if 𝑢 ≤ 𝜇 -𝑟𝑢 −𝜇,ℓ , if 𝜇 < 𝑢 ≤ 𝜇 + 𝑛 (21) -Here, we slightly abuse the notation; that is,𝑢 and ℓ actually refer to the binary representations of numbers -in [𝜇 + 𝑛] and [𝑑], respectively. Now we can assume that𝛼 is fixed. We define multilinear polynomialse𝛼 and -e𝑀𝛼 such that: -∀𝑖, 𝑢, ℓ, e𝛼 (ℓ) = 𝛼 ℓ and e𝑀𝛼 (𝑖, 𝑢) = -  - -M𝑖,𝑢 (𝛼), if 𝑢 ≤ 𝜇 and 𝑖 ≤ 𝑛 -−(𝛼𝑑 +1), if 𝑖 + 𝜇 = 𝑢 -0, otherwise -Now, for each fixed𝛼, we can represent the constraints as polynomials: -20 - ---- page 21 --- - -Prover P (𝑡, ˜𝑤 ) Verifier V (𝑡, ˜𝑀𝛼, ˜𝛼) -𝜏0 ← Flog 𝜇+log 𝑑, 𝜏1 ← Flog 𝑛 -𝜏0, 𝜏1 -˜𝑤 - -𝑡 ?= Com(˜𝑤 ) -Reconstruct 𝐻0, 𝐻1 as in (22) and (23) -𝐻0 (𝜏0) ?= 0 and 𝐻𝛼 (𝜏1) ?= 0 -Fig. 5. Proving that polynomials𝐻0 and 𝐻𝛼 are zero via random evaluation. -–For each𝑖 ∈ [ 𝑛], -𝜇+𝑛∑︁ -𝑢=1 -e𝑀𝛼 (𝑖, 𝑢) -∑︁ -ℓ -e𝑤 (𝑢, ℓ) · e𝛼 (ℓ) = y𝑖 (𝛼), -where M𝑖,𝑗 , y𝑖 ∈ Z<𝑑 -𝑞 [𝑋 ]. -–For each𝑢 ∈ [ 𝜇 + 𝑛] and ℓ ∈ [ 𝑑], -e𝑤 (𝑢, ℓ) · (e𝑤 (𝑢, ℓ) − 1) · (e𝑤 (𝑢, ℓ) + 1) · · · (e𝑤 (𝑢, ℓ) − 𝑏 +1) · (e𝑤 (𝑢, ℓ) + 𝑏 −1) = 0. -Following standard approaches [Set20; HMZ25], we use an equality function to batch these constraints. Since -𝑖 ∈ [ 𝑛], its binary representation is a string in{0,1}log 𝑛. Recall that the equality functione𝑒𝑞 : Flog 𝑛 -𝑞𝑘 × Flog 𝑛 -𝑞𝑘 → -F𝑞𝑘 is defined -e𝑒𝑞 (𝑡, 𝑖) = -Ö -𝑗 -(𝑡 𝑗 ·𝑖 𝑗 + (1 − 𝑡 𝑗 ) · (1 − 𝑖 𝑗 )) -Using e𝑒𝑞, we can then batch the first type of constraints to show that the following polynomial𝐻𝛼 (𝑡) : Flog 𝑛 -𝑞𝑘 → -F𝑞𝑘 is a zero polynomial. -𝐻𝛼 (𝑡) := -∑︁ -𝑖 ∈ [𝑛] -e𝑒𝑞 (𝑡, 𝑖) - ∑︁ -𝑢,ℓ -e𝑀𝛼 (𝑖, 𝑢) · e𝑤 (𝑢, ℓ) · e𝛼 (ℓ) − y𝑖 (𝛼) -! -, (22) -Similarly, we can also batch the second type of constraints (smallness check) as proving that the following -polynomial 𝐻0 : Flog 𝜇+log 𝑑 -𝑞𝑘 → F𝑞𝑘 is a zero polynomial. -𝐻0 (𝑡) := -∑︁ -ℓ,𝑢 -e𝑒𝑞 (𝑡, (𝑢, ℓ)) · e𝑤 (𝑢, ℓ) (e𝑤 (𝑢, ℓ) − 1) (e𝑤 (𝑢, ℓ) + 1) · · · (e𝑤 (𝑢, ℓ) − 𝑏 +1) (e𝑤 (𝑢, ℓ) + 𝑏 −1). (23) -Finish the proof using sumcheck protocols.Now, the goal of the honest prover is to prove that𝐻0 (𝑡) ≡ 0 and -𝐻𝛼 (𝑡) ≡ 0. We adopt a standard approach again: the verifier sends two random points,𝜏0 and 𝜏1, and the -prover then proves that𝐻0(𝜏0) = 0 and 𝐻𝛼 (𝜏1) = 0. We capture this with the following protocol. Similarly as -in Lemma 9, we can show coordinate-wise special soundness of this protocol by treating(𝜏0, 𝜏1) as a vector of -log 𝜇 +log 𝑑 +log 𝑛 coordinates. -Lemma 10(CWSS of Figure 5).There is an efficient, deterministic algorithm that given a statement -(𝑡, ˜𝑀𝛼, ˜𝛼) and 𝐷 := max(2𝑑,2𝑏 − 1) valid transcripts ((𝜏𝑖,0, 𝜏𝑖,1), ˜𝑤 𝑖 ) of the protocol in Figure 5 such that -(𝜏𝑖,0, 𝜏𝑖,1)𝑖 ∈ SS(F𝑞𝑘 ,2, 𝐷), can either extract a valid opening˜𝑤 of 𝑡, i.e. 𝑡 = Com(˜𝑤 ), which satisfies𝐻0 (𝑡0) ≡ 0 -and 𝐻𝛼 (𝑡) ≡ 0 (defined in Equations(22) and (23)), or break binding of the commitment schemeCom. -21 - ---- page 22 --- - -Proof. If for some𝑖 ≠ 𝑗 we have ˜𝑤 𝑖 ≠ ˜𝑤 𝑗, then we break the binding property of the commitment scheme. -Otherwise, define ˜𝑤 := ˜𝑤 𝑖 for all 𝑖. By definition of𝐷 = max(2𝑑,2𝑏 − 1) and the coordinate-wise special -soundness, we have found at least 2𝑑 (resp. 2𝑏 − 1) distinct roots for 𝐻𝛼 (𝐻0), which implies that the -aforementioned polynomials are equal to zero. ⊓ ⊔ -Now, fix𝜏0, 𝜏1 and recall that by definition of𝐻0 and 𝐻𝛼: -𝐻0 (𝜏0) := -∑︁ -ℓ,𝑢 -e𝑒𝑞 (𝜏0, (𝑢, ℓ))e𝑤 (𝑢, ℓ) (e𝑤 (𝑢, ℓ) − 1) (e𝑤 (𝑢, ℓ) + 1) · · · (e𝑤 (𝑢, ℓ) − 𝑏 +1) (e𝑤 (𝑢, ℓ) + 𝑏 −1), -and -𝐻𝛼 (𝜏1) = -∑︁ -𝑢,ℓ -e𝑤 (𝑢, ℓ) · e𝛼 (ℓ) -∑︁ -𝑖 -e𝑒𝑞 (𝜏1, 𝑖) · e𝑀𝛼 (𝑖, 𝑢) − -∑︁ -𝑖 -e𝑒𝑞 (𝜏1, 𝑖) · y𝑖 (𝛼). -We then use sumcheck to prove that𝐻0(𝜏0) = 𝐻𝛼 (𝜏1) = 0. Defining polynomials -𝐹0,𝜏0 (𝑥, 𝑦) := e𝑒𝑞 (𝜏0, (𝑥, 𝑦))e𝑤 (𝑥, 𝑦) (e𝑤 (𝑥, 𝑦) − 1) (e𝑤 (𝑥, 𝑦) + 1) · · · (e𝑤 (𝑥, 𝑦) − 𝑏 +1) · (e𝑤 (𝑥, 𝑦) + 𝑏 −1) ·1≤𝜇 (𝑥, 𝑦), -and -𝐹𝛼,𝜏1 (𝑥, 𝑦) := e𝑤 (𝑥, 𝑦) · e𝛼 (𝑦) · -∑︁ -𝑖 -e𝑒𝑞 (𝜏1, 𝑖) · e𝑀𝛼 (𝑖, 𝑥), -we have that𝐻0(𝜏0) = 0 and 𝐻𝛼 (𝜏1) = 0 if and only if -∑︁ -𝑢,ℓ -𝐹0,𝜏0 (𝑢, ℓ) = 0 and -∑︁ -𝑢,ℓ -𝐹𝛼,𝜏1 (𝑢, ℓ) = 𝑎, -where 𝑎 := Í -𝑖 e𝑒𝑞 (𝜏1, 𝑖) ·y𝑖 (𝛼), which the verifier can compute on their own. Therefore, we lastly apply sumcheck -on 𝐹0,𝜏0 and 𝐹𝛼,𝜏1. -A reinterpretation of the sumcheck protocol.In the standard sumcheck protocol, the prover and the verifier -communicate univariate polynomials and random field elements. In the last round of the protocol, the verifier -makes a single evaluation query to the polynomial. However, since we rely on the special soundness notion (see -Lemma 11), we reinterpret the sumcheck protocol as a composition of single-round communication protocols -to simplify the analysis. In Figure 6, we demonstrate the sumcheck protocol for polynomials of the form -𝐻 (X) := 𝑃 (X) · 𝑄 (˜𝑤 (X)), where 𝑃 and 𝑄 are public information and˜𝑤 (X) is the witness. We note that both -𝐹0,𝜏0 and 𝐹𝛼,𝜏1 have this form. -Compared to the standard sumcheck protocol, we note that the prover usually does not sende𝑤 to the -verifier directly. Instead, it typically proves knowledge ofe𝑤 by continuing the sumcheck protocol, and finally -by providing a proof of the evaluation ofe𝑤. -Lemma 11(Special soundness of Figure 6). There is an efficient, deterministic algorithm that -given a statement and 𝐷 := deg(𝐻 ) + 1 valid transcripts (𝑔𝑖 (𝑋𝑖 ), 𝑎𝑖,𝑗 , ˜𝑤 ) 𝑗 of the protocol in Figure 6 such -that (𝑎𝑖,𝑗 ) 𝑗 ∈ SS(F𝑞𝑘 ,1, 𝐷), can either extract a valid opening ˜𝑤 of 𝑡, i.e. 𝑡 = Com(˜𝑤 ), which satisfiesÍ -𝑏 𝑗 𝐻 (𝑎1, . . . , 𝑎𝑖, 𝑏𝑖+1, . . . , 𝑏ℓ ) = 𝑔𝑖 (𝑎𝑖 ), or break binding of the commitment schemeCom. -Proof. If for some𝑖 ≠ 𝑗 we have ˜𝑤 𝑖 ≠ ˜𝑤 𝑗, then we break the binding property of the commitment scheme. -Otherwise, define ˜𝑤 := ˜𝑤 𝑖 for all 𝑖. By definition of𝐷 = deg(𝐻 ) + 1 and the coordinate-wise special soundness, -we have found at leastdeg(𝐻 ) + 1 distinct roots forÍ -𝑏 𝑗 𝐻 (𝑎<𝑖, 𝑋𝑖, 𝑏𝑖+1, . . . , 𝑏ℓ ) − 𝑔𝑖 (𝑋𝑖 ), which implies thatÍ -𝑏 𝑗 𝐻 (𝑎<𝑖, 𝑋𝑖, 𝑏𝑖+1, . . . , 𝑏ℓ ) − 𝑔𝑖 (𝑋𝑖 ) ≡ 0. -Since special soundness is composable, we can obtain the special soundness of the sumcheck protocol by -composing Figure 6 for all𝑖 ∈ [ ℓ]. Finally, the prover outputs𝑦′ := ˜𝑤 (𝑎1, . . . , 𝑎ℓ ), which allows the verifier to -compute the evaluation𝐻 (𝑎1, . . . , 𝑎ℓ ). What is left for the prover to do now is prove the polynomial evaluation -claim for ˜𝑤. This ends the first iteration of our protocol. -22 - ---- page 23 --- - -Prover P (𝑡, 𝑃, 𝑄, ˜𝑤 ) Verifier V (𝑡, 𝑃, 𝑄, 𝑧𝑖 −1) -𝑔𝑖 (𝑋𝑖 ) := Í -𝑏 𝑗 𝐻 (𝑎<𝑖, 𝑋𝑖, 𝑏𝑖+1, . . . , 𝑏ℓ ) -𝑔𝑖 (𝑋𝑖 ) - -𝑎𝑖 ← F𝑞𝑘 -𝑎𝑖 -˜𝑤 - -𝐻 (X) := 𝑃 (X) · 𝑄 (˜𝑤 (X)) -𝑡 ?= Com(˜𝑤 ) -𝑔𝑖 (0) + 𝑔𝑖 (1) ?= 𝑧𝑖 −1 -Í -𝑏 𝑗 𝐻 (𝑎 ≤𝑖, 𝑏𝑖+1, . . . , 𝑏ℓ ) ?= 𝑔𝑖 (𝑎𝑖 ) -Fig. 6. Proving Í -𝑏𝑖,...,𝑏 ℓ 𝐻 (𝑎1, . . . , 𝑎𝑖 −1, 𝑏𝑖, . . . , 𝑏ℓ ) = 𝑧𝑖 −1 for 𝑖 ∈ [ ℓ], where 𝐻 (X) := 𝑃 (X) · 𝑄 (˜𝑤 (X)). We set𝑧0 to be the -initial sum, and𝑧𝑖 := 𝑔𝑖 (𝑎𝑖 ). -Putting everything together.The complete protocol is presented in Figure 7, which is a natural composition -of subprotocols in Figures 4, 5 and 6. By combining Lemmas 9, 10 and 11, we conclude that our final protocol -satisfies coordinate-wise special soundness. -Prover P Verifier V -𝑡 := Com(z, r) 𝑡 - -𝛼 ← F𝑞𝑘 , 𝜏0 ← Flog 𝜇+log 𝑑, 𝜏1 ← Flog 𝑛 -𝛼, 𝜏0, 𝜏1 -Run sumcheck to𝐻0 (𝜏0), 𝐻𝛼 (𝜏1) 𝑔1 (𝑋1) - -𝑎1 ← F𝑞𝑘 -𝑎1 -· · · -· · · -𝑔ℓ (𝑋ℓ ) - -𝑎ℓ ← F𝑞𝑘 -Open 𝑡 to evaluate˜𝑤 (𝑎1, . . . , 𝑎ℓ ) -Evaluate e𝑀𝛼 (𝑎1, . . . , 𝑎ℓ ) -Check the correctness of the sumcheck. -Fig. 7. Efficient Sumcheck via Ring Switching -4.4 Asymptotic Efficiency Analysis -Suppose we want to prove knowledge of the committed multilinear polynomial𝑓 ∈ Z≤1 -𝑞 [𝑋1, . . . , 𝑋ℓ ] which -satisfies a polynomial evaluation claim overF𝑞𝑘. In the following, let𝐿 := 2ℓ. -23 - ---- page 24 --- - -Basic parameters.Let 𝑞 = poly(𝜆, 𝐿) and 𝑘 = 𝜆/log 𝑞, so thatF𝑞𝑘 is of exponential size to ensure negligible -knowledge soundness in Lemmas 9, 10 and 11. We set the ring dimension𝑑 = poly(𝜆), so 𝛼 = log 𝜆, and -𝑛𝐴 = 𝑛𝐵 = 𝑛𝐷 = 𝑂 (1) for hardness of the Module-SIS problem. Then, we pick𝜔 = 𝑂 (𝑑) for sufficiently large -challenge space so that knowledge soundness error in Lemma 8 becomes negligible. We pick the decomposition -base 𝑏 = 𝑂 (1) and therefore 𝛿 = 𝑂 (log 𝑞). Recall that𝛽 in Section 4.2 was the upper-bound on the vectorz -(see Figure 3). A naive bound for𝛽 is: - -2𝑟 -∑︁ -𝑖=1 -𝑐𝑖 s𝑖 - -∞ -≤ -2𝑟 -∑︁ -𝑖=1 -𝜔 ·𝑏 = 2𝑟 𝜔𝑏 =: 𝛽. -Hence, if we choose the smallest integer𝜏 such that𝑏𝜏 > 𝛽, then 𝜏 = 𝑂 (𝑟 + 𝛼). -Witness length reduction after one iteration.By Section 3, we can consider polynomials overR𝑞 with ℓ − 𝛼 -variables. As in Section 4.2, we define variables𝑚, 𝑟 such that𝑚 + 𝑟 = ℓ − 𝛼. Let𝑚 = 𝑟 = (ℓ − 𝛼)/2 (we implicitly -assume ℓ − 𝛼 is even, which does not have anyasymptotic impact). Then, the witness(ˆt, ˆw,ˆz), parsed in -Section 4.3 asz, together with the “quotient”r from the ring switching methodology, is together of length – -in terms ofZ𝑞 elements: -(𝑛𝐴 +1) ·2𝑟 ·𝛿 ·𝑑| {z } -|ˆt|+| ˆw| -+ (𝜏 +1)2𝑚) · 𝛿 ·𝑑| {z } -|ˆz| -+ (𝑛𝐴 + 𝑛𝐵 + 𝑛𝐷 +2) · 𝛿 ·𝑑| {z } -|r| -= 𝑂 (( ℓ + 𝛼)2 ·2 -ℓ +𝛼 -2 ). -Hence, we conclude that after one round of iteration, the number of variables for the polynomial evaluation -gets reduced (at most) to: -ℓ ↦→ ℓ + 𝛼 -2 +2log (ℓ + 𝛼) + 𝑂 (1) ≤ 2 -3 (ℓ + 𝛼) + 𝑂 (1). -After 𝑡 iterations, the length of the witness is reduced to (at most): -(2 -3 )𝑡 ·ℓ + -𝑡∑︁ -𝑖=1 -(2 -3 )𝑡 ·𝛼 + 𝑂 (𝑡) ≤ ( 2 -3 )𝑡 ·ℓ +2𝛼 + 𝑂 (𝑡). -Thus, after running our protocol 𝑡 = 𝑂 (log ℓ) times we will output a witness polynomial with at most -𝑂 (log ℓ + 𝛼) variables. Consequently, sending all theZ𝑞-coeffients of the final polynomial in the clear costs -poly(ℓ, 𝑑) = poly(ℓ, 𝜆) bits. -Proof size.Having non-interactive proofs via Fiat-Shamir transformation in mind, we only count the prover -messages. The proof size of a single iteration can be split in three parts. The first is the transformation from -Z𝑞 to R𝑞 from Section 3, where the prover sends𝑑 + 𝑘 (𝑘 −1) Z𝑞-elements, where𝑘 = 𝜆/log 𝜆. The second is the -commitment v from Section 4.2, which is of size𝑛𝐷 ·𝑑 elements in Z𝑞. Finally, in the protocol in Section 4.3, -for each sumcheck round (out of𝑂 (ℓ + 𝛼)) the prover sends 2 (resp.𝑏 +1) elements inF𝑞𝑘 corresponding to -the coefficients of the univariate polynomials obtained from𝐹𝛼,𝜏1 (resp. 𝐹0,𝜏0). Finally, the prover sends the -commitment to ˜𝑤 which is of length𝑛𝐵 ·𝑑 Z𝑞-elements (resp. Section 4). In total, a single iteration of the -protocol has proof size, in terms ofZ𝑞 elements: -(𝑛𝐵 + 𝑛𝐷 +1) · 𝑑 + (𝑘 −1) · 𝑘 + (𝑏 +2) · 𝑘 ·𝑂 (ℓ + 𝛼) = 𝑂 - 𝜆 -log 𝜆 ·ℓ - -+poly(𝜆). -By recursively running the protocol𝑂 (log ℓ) times and by having𝑞 = poly(𝜆, 𝐿), we obtain succinctpoly(ℓ, 𝜆) -proof size. -24 - ---- page 25 --- - -Verifier time.We split the verifier algorithm in three parts. For the transformation in Section 3 the verifier -has to compute theR𝑞-trace map consistency, which is of timepoly(𝜆). For the protocol in Section 4.2 the -verifier does not check anything. Finally, in Section 4.3 the verifier has two tasks. First is checking that -the univariate polynomials sent by the prover every sumcheck round are consistent. This takes𝑂 (𝑏) = 𝑂 (1) -Z𝑞-operations. A much more expensive part is the final evaluation of the polynomial𝐹𝛼,𝜏1 after sumcheck, -which involves evaluating the (publicly known) multilinear extensions˜𝑀𝛼 𝑛 := 𝑛𝐵 + 𝑛𝐷 +3 = 𝑂 (1) times. As -shown in [VSBW13], to evaluate˜𝑀𝛼 once, it would take -𝑂 ((𝑛𝐴 +1)2𝑟 + (𝜏 +1)2𝑚𝛿) = ˜𝑂 ( -√︁ -2ℓ /𝜆) time -in terms of field operations overF𝑞𝑘 using dynamic programming. A single multiplication overF𝑞𝑘 can be -computed with ˜𝑂 (𝑘) operations overZ𝑞. Since𝑘 = 𝑂 (𝜆/log 𝑞) the total verifier time is dominated by˜𝑂 ( -√ -2ℓ ·𝜆). -4.5 Concrete Optimizations and Composing with Greyhound -Avoiding re-decomposition.Having a recursive proof system in mind, a naive approach to commit to the -multilinear extension mle[˜𝑤 ] of ˜𝑤 (cf. Equation (21)) is to run again the commitment scheme from Section 4. -The first part is the inner commitment, where we decompose the coefficients of˜𝑤 (and thus the committed -witness length increases), and then run an Ajtai commitment scheme to get the inner commitments(t𝑖 )𝑖. A -more natural and efficient way is to directly commit to˜𝑤 (cf. Figure 7), as we know that the coefficients of˜𝑤 -are already short and no decomposition for˜𝑤. It is straightforward from the committing point of view, as we -show below, but more importantly, how should one prove an evaluation claim overF𝑞𝑘: -mle[˜𝑤 ] (𝑎1, . . . , 𝑎ℓ ) = 𝑦, -when having only committed to˜𝑤? -Let us recall the notation for completeness. The vector˜𝑤 ∈ Z𝔫·𝑑 is defined in Equation (21), and in -particular, the coefficients of˜𝑤 are between −𝛽 and 𝛽, where parameters𝔫 and 𝛽 are chosen according to the -protocol in Section 4.3. We assume that𝔫𝑑 is a power-of-two. Let𝑑 ′ = 2𝛼 ′ -be a power-of-two ring dimension -used to defineR′ -𝑞 := Z𝑞 [𝑋 ]/( 𝑋 𝑑 ′ -+1) (potentially 𝑑 ′ ≠ 𝑑); it is the cyclotomic ring used for the next iteration -of the protocol. Let𝑚′, 𝑟′ be integers chosen such that𝔫𝑑 = 2ℓ, where ℓ = 𝑚′ + 𝑟 ′ + 𝛼 ′; they will be responsible -for the split-and-fold in the next round. -We start with with the definition of a multilinear polynomial. Let us definea0 = (𝑎1, . . . , 𝑎ℓ −𝜅) and -a1 := (𝑎ℓ −𝜅+1, . . . , 𝑎ℓ ). Then, we can write -𝑦 = mle[˜𝑤 ] (𝑎1, . . . , 𝑎ℓ ) = -∑︁ -𝑖 ∈ {0,1}𝜅 -𝑒𝑞 (𝑖, a1) · -©­­­­­­­ -« -∑︁ -𝑗 ∈ {0,1}ℓ −𝜅 -˜𝑤 𝑗 ∥𝑖 ·𝑒𝑞 ( 𝑗, a0) -| {z } -𝑦𝑖:= -ª®®®®®®® -¬ -(24) -Hence, we let the prover send𝑘 partial evaluations𝑦𝑖 ∈ F𝑞𝑘 to the verifier who can directly check10 whether -𝑦 = -∑︁ -𝑖 -𝑒𝑞 (𝑖, a1) · 𝑦𝑖 . -What we have left to prove is that all the𝑦𝑖 are well-formed, or in other wordsÍ -𝑗 ∈ {0,1}ℓ −𝜅 ˜𝑤 𝑗 ∥𝑖 ·𝑒𝑞 ( 𝑗, a0) = 𝑦𝑖 -for 𝑖 ∈ {0,1}𝜅. -10 As in Section 3, the prover can just send the𝑘 −1 evaluations𝑦𝑖 excluding 𝑦0...0, which can be derived directly by -the verifier. -25 - ---- page 26 --- - -For the next part, we assume without loss of generality thatF𝑞𝑘 = R′ -𝑞 -𝐻 where 𝐻 = ⟨𝜎−1, 𝜎4𝑘+1⟩ (cf. -Section 3). Indeed, as the field isomorphism fixes elements inF𝑞, the coefficients ofz are still small after -mapping the statement intoR′𝐻 -𝑞 . Then, any field element𝑣 ∈ R′ -𝑞 -𝐻 can be explicitly written as: -𝑣 := 𝑣0 ·𝑍0 + -𝑘 −1∑︁ -𝑗=1 -𝑣𝑘 − 𝑗 ·𝑍𝑘 − 𝑗 where 𝑍0 = 1 and 𝑍𝑘 − 𝑗 := 𝑋 -𝑑′ -2𝑘 · (𝑘 − 𝑗 ) − 𝑋 -𝑑′ -2𝑘 · (𝑘+𝑗 ) . -Now, for each𝑗 ∈ {0,1}ℓ −𝜅 define -ˆ𝑤 𝑗 := -∑︁ -𝑖 ∈ {0,1}𝜅 -˜𝑤 𝑗 ∥𝑖 ·𝑍Í𝜅 -𝑡 =1 𝑖𝑡 ·2𝑡 −1 ∈ F𝑞𝑘 . (25) -We observe that∥ˆ𝑤 𝑗 ∥∞ ≤ 𝛽. Then, proving well-formedness of all(𝑦𝑖 )𝑖 is equivalent to proving -∑︁ -𝑗 ∈ {0,1}ℓ −𝜅 -ˆ𝑤 𝑗 ·𝑒𝑞 ( 𝑗, a1) = -∑︁ -𝑗 ∈ {0,1}ℓ −𝜅 -∑︁ -𝑖 ∈ {0,1}𝜅 -˜𝑤 𝑗 ∥𝑖 ·𝑒𝑞 ( 𝑗, x1) · 𝑍Í𝜅 -𝑡 =1 𝑖𝑡 ·2𝑡 −1 -= -∑︁ -𝑖 ∈ {0,1}𝜅 -∑︁ -𝑗 ∈ {0,1}ℓ −𝜅 -˜𝑤 𝑗 ∥𝑖 ·𝑒𝑞 ( 𝑗, x1) ·𝑍Í𝜅 -𝑡 =1 𝑖𝑡 ·2𝑡 −1 -= -∑︁ -𝑖 ∈ {0,1}𝜅 -𝑦𝑖 ·𝑍Í𝜅 -𝑡 =1 𝑖𝑡 ·2𝑡 −1 ∈ F𝑞𝑘 . -(26) -where the right-hand side is public. Next, we prove (26) as follows. Note that ultimately we can write -∑︁ -𝑗 ∈ {0,1}ℓ −𝜅 -ˆ𝑤 𝑗 ·𝑒𝑞 ( 𝑗, a1) = e⊺ (f ⊺ ⊗ I2𝑟 )ˆ𝑤 𝑗 (27) -where e := (𝑒𝑞 (𝑖, x0))𝑖 ∈ F2𝑟 ′ -𝑞𝑘 and f := (𝑒𝑞 ( 𝑗, x1)) 𝑗 ∈ F2𝑚′ -𝑑 ′/𝑘 -𝑞𝑘 . Finally, using Theorem 2, theR𝑞-trace of the ring -element -𝑝 := e⊺ (𝜎−1(𝜓 (f )) ⊺ ⊗ I2𝑟 )𝜓 (ˆ𝑤 ) ∈ R′ -𝑞 -is exactly equal to the value in (27), which is also equal to (26). By Lemma 6, the infinity norm of -𝜓 (ˆ𝑤 ) ∈ R′ -𝑞 -𝔫·𝑑/𝑑 ′ -is at most 2𝛽, so we will commit to it using the inner-outer commitments without initial -decomposition. What is left to prove is the knowledge of the short commitment opening𝜓 (ˆ𝑤 ) which satisfies -the quadratic relation above. This is exactly the type of statement supported natively by Greyhound (cf. -[NS24, Section 3]). To sum up, the total communication complexity of our reduction to Greyhound is -(𝑘 −1) · 𝑘 ·log 𝑞 -| {z } -| (𝑦𝑖 )𝑖 | -+ 𝑑 ′ ·log 𝑞 -| {z } -|𝑝 | -. (28) -Using Johnson-Lindnestrauss instead of sumcheck.Similarly as in Greyhound [NS24], instead of running the -protocol from Section 4.3, at some point we can naturally switch to LaBRADOR [BS23]. This can be done -when the witness length is small enough so that performing the Johnson-Lindenstrauss projection becomes -cheap. In fact, although the sumcheck-based approach provides a faster verification time, reducing witness -length after a single operation is slower. The reason is that we have to pick relatively small base𝑏, as the -sumcheck proof size consists of𝑂 (𝑏) field elements. This implies longer decomposition𝛿, and thus longer -witness for the next round. On the other hand, the random projection approach allows larger bases𝑏 (at a -fairly small cost), which makes𝛿 concretely smaller. This allows LaBRADOR to recurse further until the -final witness is cheap enough to send in the clear (≈ 30KB). -26 - ---- page 27 --- - -5 Experimental Results -To demonstrate the concrete performance of our protocol, we have developed a prototype implementation of -Hachi in Rust. The source can be found here: -https://github.com/georgeorourke/hachi-pcs -Our implementation currently supports committing to, and single evaluation proofs and verification for -multilinear polynomials overZ𝑞. We highlight that our implementation is still in an early stage of development -and is amenable to a number of optimizations which would lead to considerable increases in performance. -Despite this, our implementation already demonstrates significant improvements in verification time, with -first (by far most expensive) round verification times being 12 to 29 times faster than Greyhound verification. -Additionally, we concretely improve commitment times for large polynomials (30+ variables). At present, -our evaluation proof times substantially exceed those of existing implementations and will need further -optimization — such as SIMD acceleration — to align with the state of the art. -We have benchmarked our implementation on a University server (with AVX-512 support) and a Mac -Mini M4 (no AVX support). The stated times for Prove and Verify are for the first round only; though, -following a single round, the new witness is far smaller and so the first round of the protocol dominates the -total cost of the proof and verification times. -Hardware Variables Commit / s Prove / s Verify / ms -Mac Mini M4 30 149 268 96.5 -Mac Mini M4 28 34.2 114 49.3 -Mac Mini M4 26 8.79 51.3 26.6 -Server 30 87.3 495 185 -Server 28 21.9 219 93.7 -Server 26 5.49 103 39.0 -Greyhound 30 132 41.2 2800 -Greyhound 28 21.2 8.21 1150 -Greyhound 26 4.37 2.03 492 -Fig. 8. Concrete running time of the Hachi PCS with prototype implementation on different hardware. Prove and -Verify times are for the first (and by far the most computationally expensive) round only. The Greyhound benchmark -was tested on an Intel Xeon Sapphire Rapids core at 3.2 GHz with full AVX-512 optimization. -5.1 Concrete Parameters -The parameters (30 variable polynomial) used in benchmarking are shown in Figure 9. We heuristically set -the ring dimension𝑑 = 2𝛼 of R𝑞 and the height of the commitment matrices,𝑛, so that𝑛 ·𝑑 ≥ 210. For𝑙 = 30 -variables, we aim to reduce the size of the witness as much as possible in ‘one-shot‘. Hence, we set𝑑 = 1024 -and 𝑛 = 1. With these parameters, in a single round of Prove and Verify, we reduce the size of the witness by -a factor of 128. Specifically, we reduce a 30 variable multilinear polynomial overZ𝑞 (32 bits per coefficient) to -a 26 variable polynomial overZ16 (4 bits per coefficient). The evaluation proof is reduced from an evaluation -of a 30 variable polynomial overZ𝑞 to an evaluation of a 26 variable polynomial with coefficients inZ16 and -an evaluation point over𝐹𝑞4. -5.2 Concrete Efficiency and Composing with Greyhound -To showcase the communication efficiency, let us set parameters forℓ = 30. We refer to Section 5 (and in -particular Figure 9) for more rationale on the parameters. We aim for𝜆 = 128 security level and use the -Lattice Estimator [APS15] to measure hardness of the Module-SIS problem in the infinity norm. -27 - ---- page 28 --- - -variable description value -ℓ number of variables of witness polynomial 30 -𝑞 prime modulus 4294967197 -𝑘 degree of the extension fieldF𝑞𝑘 for sumcheck 4 -𝛼 log of ring dimension 10 -𝑑 𝑑 := 2𝛼 is the ring dimension ofR𝑞 1024 -𝑚 folding parameter 10 -𝑟 folding parameter 10 -𝑛𝐴 height of the matrixA 1 -𝑛𝐵 height of the matrixB 1 -𝑛𝐷 height of the matrixD 1 -𝑧 maximum 𝐿∞ norm of z 30583 -𝑏 decomposition base 16 -𝛿 expansion factor for decomposing base witness 8 -𝜏 expansion factor for decomposingz 4 -𝜔 𝐿1 norm of a challenge element 16 -𝑐 number of non-zero coefficients in challenge 16 -Next witness size 226 -Next witness norm bound infinity norm 8 -Fig. 9. Concrete parameters for anℓ = 30 variable multilinear polynomial overZ𝑞. -To recall, we pick𝑞 ≈ 232, 𝑑 = 210, 𝛼 = 10, 𝑟 = 10 and𝑚 = 10 and (𝑛𝐴, 𝑛𝐵, 𝑛𝐷 ) = (1,1,1). The decomposition -base is 𝑏 = 16, thus 𝛿 = 8. For comparison with Greyhound, we consider polynomial evaluations over the base -field F𝑞. -First invocation of Hachi.The cost of the transformation in Section 3 is𝑑log 𝑞 ≤ 4KB (where 𝑘 = 1). Next -is the commitmentv is also of size 4KB. Length of the witness˜𝑤 for sumcheck is be bounded by 226, as -discussed in Section 5.1. Then, we consider extension fieldsF𝑞𝑘 for 𝑘 = 4. The cost of sumcheck is then -bounded by 26·4 ·32 · (16 +2) bits ≈ 7.3KB. -Adapting to the Greyhound relation.We apply our transformation from Section 4.5. To this end, we set the -ring dimension exactly as in Greyhound, i.e.𝑑 ′ = 64. Theshort witness 𝜓 (ˆ𝑤 ) for the quadratic relation over -R′ -𝑞 is of length 226/𝑑 ′ = 220. By Equation (28), the cost of the transformation can be upper-bounded by: -12 ·32 +64 ·32 bits ≈ 0.3KB. -Moreover, we also need to commit to𝜓 (ˆ𝑤 ), which again takes 4.5KB (cf. [NS24, Table 4]). Hence, the cost of -preparation to run Greyhound is≈ 4.8KB. -Finally, we can directly apply Greyhound for theshort witness 𝜓 (ˆ𝑤 ) of length 220 over R′ -𝑞. To estimate -the proof size for Greyhound, we consider the parameters of the “𝑁 = 226” column of [NS24, Table 1]). -This column focuses on the polynomial of length 226 with full-fledged Z𝑞-elements. After the coefficient -decomposition, the witness gets increased by a factor of𝛿0 = 5. Keeping in mind that the ring dimension is -𝑑 ′ = 64, Greyhound reduces proving polynomial evaluation to a quadratic equation overR′ -𝑞, where the witness -is of length 226 ·𝛿/𝑑 ′ ≈ 222.3, which is more than enough for our scenario. Therefore, we can upper-bound the -evaluation proof size of Greyhound is≈ 43KB. To conclude, the total evaluation proof can be estimated to be -7.3 +4.8 +43 KB = 55.1KB. -Running time. The concrete experimental running time for the first round of Hachi can be found in Figure 8. -The running time of the transformation, which is done by the prover, from Section 4.5 is negligible compared -to other parts. Finally, we use the Greyhound estimates for the witness of length 224 Z𝑞-elements, which -roughly translates to 224 ·𝛿/𝑑 ′ ≈ 220.3 R′ -𝑞-elements. From [ACFY25; Sei24], the prover runtime (which includes -28 - ---- page 29 --- - -committing to 𝜓 (ˆ𝑤 )) is around 1220ms, while the verifier runtime is 130ms. Hence, by combining with -Figure 8, the total prover and verifier runtime are approximately 270s and 227ms. -5.3 Faster Commitment Time -The commitment function of Hachi is essentially identical to Greyhound except for the choice of ring. Unlike -Greyhound, which is restricted to small ring dimensions (e.g.𝑑 = 64), Hachi is suitable for use with large -ring dimensions (e.g.𝑑 = 1024). A major benefit of this is improved commitment time. This is because the -number of ring multiplications decreases in proportion to𝑑2 (since both 2𝑟 +𝑚 and 𝑛𝐴, 𝑛𝐵 decrease linearly -with 𝑑) however, with NTT, the cost of the ring multiplication scales with𝑑log 𝑑. -The existing Greyhound implementation is highly optimized for certain rings and does not support large -ring dimensions. Therefore, we cannot benchmarch the commitment times for both the Greyhound parameters -and our parameters using the Greyhound implementation. However, as our library is much more flexible, -we are able to benchmark the Greyhound commitment times with our implementation - providing a near -like-for-like comparison. Our parameters are restricted to powers of two, and we set them as close to the -Greyhound parameters as possible. The comparative results are shown in Figure 10 and indicate a 3-5X -algorithmic improvement in the commitment time. -Variables Folding Parameters,𝑟, 𝑚 Decomposition base Commit / s Comparison -30 16384, 1024 256 433 5.0X slower -28 8192, 512 256 71.8 3.3X slower -26 4096, 256 256 18.6 3.4X slower -Fig. 10. Concrete running time of the Commitment function using a smaller ring dimension𝑑 = 64 on an AVX-512 -comptaible University server and comparison to concrete times with Hachi parameters. -5.4 Implementation Details -Integer Operations and Data StructuresInteger coefficients are stored as unsigned 64 or 32 bit integers -(for compatibility with tfhe-ntt). Integer decomposition has been optimized for power of two bases but not -optimized with SIMD. Sampling inZ𝑞 for 𝑘-bit prime 𝑞 is performed by sampling 32/64 bits (using any -PRNG) and masking𝑘 bits. Rejection sampling is then used to sample uniformly overZ𝑞. Polynomials are -represented as coefficient slices over integers (u32 or u64). Vectors and matrices over polynomials are stored -as tightly packed coefficients in a single vector (Array of Structure format). Decomposition of vectors of -polynomials is done by sequentially decomposing coefficients and has not yet been optimized with SIMD. -Initialization of the commitment matrices is done by sequentially sampling coefficients from a seeded PRNG. -If the parameters are set so that the commitment matrices have the same dimensions, the same commitment -matrix is used for all commitments. The commitment matrices are re-generated by the commit/prove/verify -functions for each invocation and this overhead is included in the stated benchmarks. -Field ArithmeticWe use the extension field𝐹𝑞4 to ensure approximately 128 bits of security. Representation -and multiplication of field elements in𝐹𝑞4 is provided by the ark-ff library11. In general, we do not use ark-ff -for multiplication over the base fieldZ𝑞, instead opting to perform integer multiplication over u64 and reduce -modulo 𝑞 naively. This is to maintain compatibility with tfhe-ntt and, whilst suboptimal, avoids conversion -between ark-ff types and u64 which is more costly than naive modulo reduction. In several cases we need to -multiply an element of the base fieldZ𝑞 with an element of the extension fieldF𝑞4, and have optimized the -arithmetic for this case. -11 https://crates.io/crates/ark-ff -29 - ---- page 30 --- - -Polynomial Arithmetic Hachi requires general polynomial arithmetic overR𝑞 := Z𝑞 [𝑋 ]/( 𝑋 𝑑 +1), for a -range of ring dimensions𝑑, as well as overZ𝑞 [𝑋 ] (to transition to the sumcheck). Our arithmetic library builds -on top of the tfhe-ntt library12, which provides SIMD (AVX-2 or AVX-512) optimized NTT-multiplication. -Since the prime modulus𝑞 cannot be chosen to be ‘NTT-friendly’, we cannot use NTT-based multiplication -directly for R𝑞. Instead, we use the tfhe-ntt 64-bit ‘Native NTT’, which performs NTT-based multiplication -modulo several primes𝑝𝑖 to simulate the arithmetic moduloÎ 𝑝𝑖. SinceÎ 𝑝𝑖 is larger than the coefficients -of any product of polynomials, the product holds overZ[𝑋 ]/( 𝑋 𝑑 +1) with 64-bit wrapping arithmetic. We -then naively reduce the coefficients modulo𝑞 to achieve efficient multiplication overZ𝑞 [𝑋 ]/( 𝑋 𝑑 +1). For the -reduction modulo 𝑞 to hold, we require that no 64-bit wrap has occurred during the multiplication (i.e. the -product has coefficients smaller than 64 bits). Note that Hachi only ever requires multiplication overR𝑞 -when one operand is small (i.e. it has been decomposed). With a maximum ring dimension of 210 and 32-bit -modulus 𝑞, the product will not wrap as long as the coefficients of the small operand are no more than 22 bits. -We use a 4-bit decomposition base and so can safely use the 64-bit implementation. The reduction modulo𝑞 -in our implementation is suboptimal, however it does not add too much to the cost of multiplication (about -5%). On the other hand, whilst this approach lacks optimizations that could be made for more specific rings, -it is very flexible; allowing both the (power-of-two) ring dimension and arbitrary modulus𝑞 to be chosen at -runtime. -To efficiently multiply polynomials overZ𝑞 [𝑋 ], note that Hachi only requires computing a sum over a -product oftwo polynomials with degree less than𝑑, so that the result has degree at most 2𝑑 −1. We therefore -pad the polynomials to degree 2𝑑 −1 and use the existing library to efficiently multiply overZ𝑞 [𝑋 ]/( 𝑋2𝑑 +1). -The result does not incur any reduction modulo𝑋2𝑑 +1 and so holds overZ𝑞 [𝑋 ]. Given the full convolution, -we perform (optimized) polynomial division to compute the residues modulo(𝑋 𝑑 +1). -Our library abstracts the ring multiplication for larger structures like vectors and matrices of polynomials. -We minimize the number of NTTs that must be performed by keeping re-used and intermediate values in -NTT form. -Optimized Sparse Polynomial MultiplicationSince Hachi may use large ring dimensions, the challenge -polynomials can be very sparse whilst still producing an exponential challenge set. For example, for𝑑 = 1024, -the challenge set may consist of ring elements with exactly𝑘 = 16 non-zero coefficients, each of which is -1 or -1. We sample such elements using a Fisher-Yates-like algorithm to obtain a partial permutation on 0, ..., 𝑑 −1, -giving the indices of the 16 non-zero coefficients. A bit is independently sampled for each non-zero coefficient -to determine its sign. Since these challenge elements are so sparse, they can be more efficiently multiplied -by a full ring element (compared to treating the sparse element as an arbitrary ring element). Our library -implements this sparse polynomial product directly for bothZ𝑞 [𝑋 ] and Z𝑞 [𝑋 ]/( 𝑋 𝑑 +1). This has not yet been -optimized with SIMD, yet performs about as well as the general AVX-512 optimized NTT multiplication. -With a good SIMD optimization, it should significantly outperform general NTT multiplication. -Witness Streams The witness is supplied to the commit/prove functions as a general stream. In our tests, -the witness was stored in a file and provided as a file stream which reads chunks of the witness on demand. -The commit and prove functions access the witness from the stream in small chunks of size 2−𝑚 ≊ 0.1% of -the whole witness size. This makes our implementation suitable for very large witnesses whilst maintaining -moderate memory usage. 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-Microsoft Research -Abstract. We construct the first folding scheme that simultaneously -achieves six desirable properties: plausible post-quantum security, pay-per- -bit commitment costs, field-native arithmetic (the sum-check and norm -checks run purely over a small field), support for general (non-SIMD) con- -straint systems, small-field support (e.g., Goldilocks), and low recursion -overheads. No existing scheme satisfies all six: group-based schemes (e.g., -HyperNova) lack post-quantum security and are tied to large elliptic- -curve fields; lattice-based schemes (e.g., LatticeFold) require expensive -ring arithmetic, lose pay-per-bit costs, and impose SIMD constraints; and -hash-based schemes (e.g., Arc) incur large verifier circuits. -We present two lattice-based folding schemes for CCS—an NP-complete -relation generalizing R1CS, Plonkish, and AIR—called Neo and SuperNeo. -Neo satisfies five of the six properties but requires SIMD constraint -systems; SuperNeo removes this restriction and satisfies all six. Both run -a single invocation of the sum-check protocol over a small field extension -and achieve pay-per-bit costs via new folding-friendly instantiations of -Ajtai commitments under the Module-SIS assumption. At the core of our -constructions are two new norm-preserving embeddings of field vectors -into ring vectors that respect an evaluation homomorphism required -for folding. We also introduceinteractive reductions, a framework that -generalizes reductions of knowledge and enables modular security proofs -for composed lattice-based protocols. -1 Introduction -A folding scheme [57] is a cryptographic primitive that reduces the task of -checking that two instance-witness pairs are in some NP relation to the task of -checking that a single instance-witness pair is in the same relation. As an example, -for a circuit C and two public inputs (i.e., instances) x1 and x2, a folding scheme -reduces the task of checking that there exist witnesses w1 and w2 such that -C(w1, x1) = 1 and C(w2, x2) = 1 to the task of checking that there exists a single -witness w for a specific public input x such that C(w, x) = 1. Furthermore, the -verifier’s work in a folding scheme is limited to roughly taking the weighted sum -of the commitments to the underlying witnesses. By using a folding scheme in -a recursive manner, one can continually fold many instance-witness pairs into -a single instance-witness pair, providing powerful recursive succinct argument -primitives such as incrementally verifiable computation (IVC) [82] and proof- -carrying data (PCD) [13]. - -Importance of folding schemes: prover efficiency and efficient recursion.A modern -approach to construct SNARKs [47,66] is to combine a polynomial interactive -oracle proof (PIOP) [23,30,78] with a polynomial commitment scheme (PCS) [46]. -However, this yields a “monolithic” SNARK whose prover must prove a fixed-sized -computation at once. To scale to larger computations, one typically breaks the -computation into smaller pieces and uses SNARK recursion (a la IVC or PCD) -to produce a succinct argument [11]. Folding schemes provide a more direct and -efficient alternative: they allow recursion to operate at the “statement” level (i.e., -prior to producing a PIOP or a PCS evaluation argument), yielding two concrete -benefits. First, recursion overheads are far lower: with Nova [57], folding a proof -takes only ≈10,000 R1CS constraints, whereas traditional SNARK recursion -takes millions [29,31]. Second, the prover incurs far less work by avoiding a full -PIOP and PCS evaluation argument: in monolithic SNARKs such as Marlin [30], -the prover performs at least 20× higher work over simply committing to a witness, -whereas with state-of-the-art folding schemes [20,35,55,56], the prover’s work is -dominated by the cost to commit to a witness. This results in at least an order of -magnitude speedup over monolithic SNARKs, and up to two orders of magnitude -when the witness contains values from a small subset of the entire field. 1 -A motivating application: post-quantum signature aggregation.Ethereum’s con- -sensus layer relies on BLS signatures [18], which offer efficient aggregation: a -single pairing check suffices for hundreds of thousands of validator attestations. -However, BLS signatures are not post-quantum secure. Ethereum’s planned -transition to a post-quantum scheme (e.g., the hash-based XMSS [44]) reintro- -duces the scalability challenge, since such schemes lack the algebraic structure -of BLS and their signatures are large and expensive to verify individually. A -natural solution is to use recursive SNARKs to aggregate these signatures [34] -in a distributed manner (a la proof-carrying data). Folding schemes are well -suited here: each signature can be folded into the aggregate proof, yielding sig- -nificantly lower latency and prover cost at each recursive step than traditional -SNARK recursion—a critical advantage when up to a million signatures must be -aggregated within time-sensitive slots of a consensus protocol. -Realizing this approach requires a folding scheme that is both efficientand -plausibly post-quantum secure. More broadly, what properties should a practical -folding scheme satisfy? -1.1 Six desiderata for a practical folding scheme -State-of-the-art folding schemes [20, 35, 53–57] have converged on an efficient -“recipe”: the prover commits to a witness with a linearly homomorphic scheme -1 If the witness contains “small” field elements (e.g., from the set{0,1, . . . ,2 32 −1}), -state-of-the-art folding schemes perform more than two orders of magnitude less work -than a monolithic SNARK prover such as Marlin [30]. Proof systems such as Spartan -and variants [78,79] incur lower overheads than Marlin, but they must still produce -a PIOP and PCS evaluation argument. -2 - -(Pedersen or KZG) and employs sum-check-type techniques [62]. For instance, the -prover in HyperNova [55] performs asinglemulti-scalar multiplication (MSM) -to commit to its witness, with costs that scale with the bit-width of witness -values (“pay-per-bit”). However, these schemes are not post-quantum secure and -are tied to ≈256-bit elliptic-curve fields. Below, we distill six properties that a -practical folding scheme should satisfy. -D1: Post-quantum security.The scheme should be plausibly secure against -quantum adversaries. Group-based folding schemes [20, 35, 55 –57] rely on the -hardness of the discrete logarithm problem, which Shor’s algorithm [80] can -efficiently solve on a quantum computer. -D2: Pay-per-bit commitment costs.The cost to commit to a witness should -scale with the bit-width of the witness values: for example, committing to a -vector of b-bit values should be roughly b times cheaper than committing to -values that span the full field. Group-based schemes achieve this via Pedersen -or KZG commitments. LatticeFold, however, relies on the NTT embedding to -map field vectors into ring vectors. Because the NTT is not a norm-preserving -map, the resulting ring vectors have arbitrary norm regardless of the bit-width -of the original witness elements. Since the Ajtai commitment scheme requires -decomposing these arbitrary-norm ring elements for binding, the commitment -cost is the same whether the original values are 1-bit or 64-bit. Hash-based -schemes (e.g., Arc [24]) also lack this property, since their commitment costs are -determined by the code rate and security parameter, not by witness bit-widths. -D3: Field-native arithmetic.Besides committing to the witness, the prover’s and -the verifier’s work—in particular, the norm check and the sum-check protocol— -should operate natively over a prime field (or extension field), without performing -expensive polynomial ring arithmetic. Group-based and hash-based schemes -satisfy this property. LatticeFold [14], however, runs the sum-check protocol over -a cyclotomic polynomial ring rather than over a prime field (or extension field); -ring operations are 10–100×more expensive than field operations. 2 -D4: General constraint systems.The scheme should support general NP-complete -constraint systems such as CCS [79] over a single witness vector, without requiring -the constraint system to be “data parallel” (SIMD). Group-based and hash-based -schemes satisfy this. LatticeFold, by contrast, packs a batch of independent con- -straints defined over a small prime field into a single constraint over a cyclotomic -polynomial ring [14, Remark 4.1], imposing a SIMD requirement. Lova [37] avoids -this issue but at the cost of only supporting the subset sum relation, not CCS. -D5: Small-field support.The scheme should work natively over small prime -fields, including popular SNARK-friendly fields such as Goldilocks. By “small” -2 Benchmarks report that a polynomial ring multiplication costs ≈213 ns [67], whereas -a field multiplication with M61 costs a fraction of a nanosecond. -3 - -we mean fields whose modulus q fits within a machine register—for example, M61 -(q = 261 −1) or Goldilocks (q = 264 −232 +1). Small-field support is important for -two reasons: such fields offer arithmetic that is an order of magnitude faster than -256-bit arithmetic, and SNARK-friendly fields enable efficient proof compression -using existing SNARKs (e.g., Spartan with a FRI-based polynomial commitment -scheme). Group-based schemes are tied to the scalar fields of elliptic curves, which -are ≈256 bits for security. LatticeFold’s cyclotomic rings of the form X d +1 (with -d a power of 2) cause popular fields like Goldilocks to fully split the ring, ruining -security [14,§3.3]; furthermore, supporting small fields requires a non-trivial -extension field degree t|d , introducing a t× multiplicative overhead in the -protocols due to the need fort-parallel repetition [14,§3.3,§5]. -D6: Low recursion overheads.The recursive verifier circuit should be small -enough that the per-step prover cost of IVC remains practical. Group-based -schemes achieve this: Nova’s and NeutronNova’s verifier circuits are constant-sized: -≈10,000 R1CS constraints. Hash-based schemes suffer from large verifier circuits; -for example, Arc [24] requires 2 ·λ/log (1/ρ) Merkle tree openings, translating to -≈1,600,000 R1CS constraints at λ= 128 and ρ= 1/2 with Poseidon [43]. Prior -to this, B¨ unz et al. [25] provide a different hash-based scheme with even worse -verifier circuit overheads [24, Table 1], and only provides “bounded depth” IVC. -Lova also incurs extreme overheads, reporting a prover time of ≈3,000 seconds for -a subset sum instance of length 2 19 [37, Table 2], compared to 500 ms for Nova -on an R1CS instance of the same size. More broadly, folding schemes that rely on -ring sum-check techniques—such as LatticeFold [14] and SALSAA [59]—inherit -high recursion overheads because the recursive verifier must hash ring elements -rather than field elements. For example, a single ring element in LatticeFold -occupies 64 ×64 = 4,096 bytes, compared to 32 bytes for a 256-bit field element in -HyperNova, resulting in 128× more data for the verifier circuit to hash. Achieving -constant verifier circuit size like Nova or NeutronNova in the lattice setting is -difficult and remains an open problem. Our goal is to achieve logarithmic recursion -overhead (with similar constants) analogous to HyperNova. -Research question. Can we build a folding scheme that satisfies all six desiderata— -in particular, one that is post-quantum secure, works natively over small prime -fields, and matches the efficiency of state-of-the-art group-based schemes? -Figure 1 summarizes the landscape. No existing folding scheme satisfies all six -desiderata. Group-based schemes meet D2–D4 and D6 but fail D1 and D5. Hash- -based schemes achieve D1 but sacrifice D2 and D6. LatticeFold, LatticeFold+, and -SALSAA each achieve D1 but fail D2–D4 and D6; LatticeFold and LatticeFold+ -additionally fail D5. Neo satisfies D1–D3, D5, and D6, but requires SIMD -constraint systems (D4). SuperNeo is the first scheme to satisfy all six. -1.2 Our work: Neo and SuperNeo -We present Neo and SuperNeo, the first folding schemes to satisfy all six desiderata -(Figure 1). Our constructions are lattice analogs of HyperNova [55]: the prover -4 - -D1 D2 D3 D4 D5 D6 -Group-based -HyperNova [55]✗ ✓ ✓ ✓ ✗ ✓ -NeutronNova [56]✗ ✓ ✓ ✓ ✗ ✓ -Hash-based -Arc [24]✓ ✗ ✓ ✓ ✓ ✗ -Lattice-based -LatticeFold [14]✓ ✗ ✗ ✗ ✗ ✗ -Lova [37]✓ ✗ ✓ ✗ ✓ ✗ -Neo(this work)✓ ✓ ✓ ✗ † ✓ ✓ -LatticeFold+ [16]✓ ✗ ✗ ✗ ✗ ✗ -SALSAA [59]✓ ✗ ✗ ✗ ✓ ✗ -SuperNeo(this work)✓ ✓ ✓ ✓ ✓ ✓ -†Neo requires a SIMD constraint system (D4) and is subsumed by SuperNeo, which -removes this requirement. The table lists schemes in the order they appeared; we -present Neo and SuperNeo separately because Neo was independently preprinted and -several subsequent works build on it, so separating the two simplifies attributing -techniques to the correct scheme. -Fig. 1: Comparison of folding schemes against the six desiderata. ✓ = satisfied, -✗= not satisfied. See Section 1.1 for definitions. -commits to a CCS [79] witness using a lattice-based commitment scheme with -pay-per-bit costs, runs a single invocation of the sum-check protocol over an -extension of a small prime field 3, and achieves plausible post-quantum security -under a standard structured lattice assumption (Module-SIS). Both schemes also -provide multi-folding, which folds multiple CCS instances at once, amortizing -the decomposition costs required to manage lattice norm growth. By applying -standard compilers from folding schemes to IVC [55, 57] and PCD [85], we -obtain a plausibly post-quantum IVC/PCD scheme. Since our constructions -natively support SNARK-friendly fields like Goldilocks, they enable efficient proof -compression using Spartan [78, 79] with a FRI-based polynomial commitment -scheme [10]—without requiring any non-native arithmetic or field emulation. -The key technical challenge isembeddingfield vectors (CCS witnesses) into the -ring vectors that Ajtai commitments operate over, while preserving the algebraic -structure—norm bounds, evaluation homomorphisms—that a sum-check-based -folding scheme like HyperNova requires. We introduce two new norm-preserving -embeddings (theNeoandSuperNeo embeddings) that resolve this challenge, -and a new security framework (interactive reductions) that enables modular -proofs of knowledge soundness for lattice-based protocols. We detail the problems -with prior approaches and our solutions in the following sections. -3 When using a 64-bit field, a degree-2 extension is sufficient for 128 bits of security. -5 - -1.2.1 Challenges and prior solutionsAchieving desiderata D2–D4 and D6 -for a lattice-based folding scheme requires solving a common problem: how to -embed field vectors into the ring vectors that Ajtai commitments [2] operate -over. Any such embedding must support protocols that check both the norm of -the committed ring vectors and CCS constraints on the underlying field vectors. -Below, we describe the Ajtai commitment scheme and the challenges that prior -solutions leave open. -Ajtai Commitment Scheme [2,75] (Informal) 4 -–Setup (1κ, n∈N ) →A∈ Rκ×n -F , where A is a random matrix over the -polynomial ring R F := F [X]/(ϕ(X)) with modulus ϕ(X) being a cyclotomic -polynomial of degreed. -–Commit (A,z∈ Rn -F ) →c , where c := A·z∈ Rκ -F is a binding commitment to -messagezif the norm∥z∥ ∞ < bis small enough. -Problem 1: Inefficient constraint checking & lack of algebraically -friendly embeddingsPrior work on lattice-based folding schemes [15] re- -lied on the Number Theoretic Transform (NTT) [5, 61, 75, 77] to embed field -vectors into ring vectors. The Number Theoretic Transform is a ring isomorphism -between the ring R F and a product space of extension fields F d/t -qt , so each ring -operation naturally simulates a Single Instruction, Multiple Data (SIMD) [40,41] -operation over the underlying (d/t)-tuple of field elements. For· ∈ {+,×}, -(a1, . . . , ad/t)∈F d/t -qt a∈R F -(a1 ·b 1, . . . , ad/t ·b d/t)∈F d/t -qt a·b∈R F -·(b1, . . . , bd/t) ·b -NTT -iNTT -NTT -iNTT -Hence, d/t field vectors z(1), . . . , z(d/t) ∈F n can be embedded into a single ring -vector z∈ Rn -F . By adapting a technique from [19], prior work Latticefold [15, Sec -3.3] showed how to express the norm constraint on a ring vector z as Hadamard -product constraints Q -i (q− 1)/2, then ∥a∥∞ = |a′ −q| . For a vector -z∈F n, we define the ℓ∞-norm ∥z∥∞ to be the max infinity norm of its elements. -For an element a∈ RF , we define ∥a∥∞ to be the ℓ∞-norm of the vector cf(a). For -a vectorz∈R m -F , we define∥z∥ ∞ to be the maximumℓ ∞-norm of its elements. -20 - -DecompositionWe define splitb : F m → (F m)∗ to be the b-ary decomposition -map, which performs the b-ary decomposition of a vector z∈F m into vectors -z1, z2, . . . , z∗. For example, ifz∈F m such that and∥z∥ ∞ < b k, then we have -splitb(z) := (z1, z2, . . . zk)such thatz= -kX -i=1 -bi−1 ·z i and∥z i∥∞ < b -Definition 4 (Ring Commitment Scheme).A ring commitment scheme -com:= (Setup,Commit)consists of two PPT algorithms: -–Setup (1λ, m) →pp : Takes as input a security parameter1 λ and length -m∈N ≥1, outputs public parameterspp. -–Commit (pp, z ) →c : Takes as input public parameters pp and a vector z∈ Rm -F , -outputs a commitmentc∈C. -A ring commitment scheme can satisfy the following properties: -B-binding:For every length m = poly(λ)and every EPT adversary A, a ring -commitment scheme isB-binding (forB∈N) if the following probability holds: -Pr - - -Commit(pp, z1) =Commit(pp, z 2) -∧ ∥z 1∥∞,∥z 2∥∞ < B, -∧z 1 ̸=z 2 - -pp←Setup(1 λ, m) -z1, z2 ∈R m -F ← A(pp) - - ≤ϵ bind(B) -for ϵbind(B) ≤negl (λ). We refer to a pair of vectors( z1, z2)which satisfies the -conditions in the probability as aB-binding collision. -(B,C )-relaxed binding:For every length m = poly(λ)and every EPT adversary -A, a ring commitment scheme is( B,C )-relaxed binding (for B∈N and set -C ⊆R F ) if the following probability holds: -Pr - - -∆1 ·c=Commit(pp, z 1) -∧∆ 2 ·c=Commit(pp, z 2) -∧ ∥z 1∥∞,∥z 2∥∞ < B, -∧∆ 1z2 ̸=∆ 2z1 - -pp←Setup(1 λ, m) - - -c∈C, -∆1, ∆2 ∈(C − C), -z1, z2 ∈R m -F - - ← A(pp) - - ≤ϵ rlx(B,C) -for ϵrlx(B,C ) ≤negl (λ). We refer to a tuple of elements( c, ∆1, ∆2, z1, z2)which -satisfies the conditions in the probability as a( B,C )-relaxed binding collision. -Homomorphic:For every m∈N and pp∈Setup (1λ, m), the commitment -algorithm,Commit(pp,·) : R m -F →C, is anR F -module homomorphism. -Theorem 2 (Properties [2,7,9,14]).The Ajtai commitment scheme (Defi- -nition 18) is a ring commitment scheme (Definition 4) that ishomomorphic, -B-binding(assuming the MSIS∞,κ,q -m,2B problem (Definition 16) is hard), and( B,C )- -relaxed binding(assuming the MSIS∞,κ,q -m,4T B problem is hard and C is a strong -sampling set (Definition 17) with expansion factorT(Theorem 9)). -21 - -Definition 5 (Interactive Reductions [50, 52]).Consider relations R1 and -R2 over parameters, structure, instance, and witness tuples. Aninteractive -reductionfrom R1 to R2 is defined by PPT algorithms( G,K,P,V )called the -generator, encoder, prover, and verifier respectively with the following interface. -–G(1 λ,sz)→pp: Takes as input a security parameter1 λ and size parameters -sz. Outputs public parameterspp. -–K (pp, s) → (pk,vk ): Takes as input public parameters pp and a structures. -Deterministically, outputs a prover keypkand a verifier keyvk. -–P (pk, u1, w1) → (u2, w2): Takes as input a proving key pk and an instance- -witness pair( u1, w1). Interactively reduces the task of checking(pp, s, u1, w1) ∈ -R1 to the task of checking(pp,s, u 2, w2)∈ R 2. -–V (vk, u1) →u 2: Takes as input a verifier key vk and an instance u1 in -R1. Interactively reduces the task of checking the instance u1 to the task of -checking a new instanceu 2 inR 2. -Let ⟨P,V⟩ denote the interaction between P and V. We treat ⟨P,V⟩ as a function -that takes as input(( pk,vk ), u1, w1)and runs the interaction on the prover’s input -(pk, u1, w1)and the verifier’s input( vk, u1). At the end of the interaction, ⟨P,V⟩ -outputs the verifier’s instanceu 2 and the prover’s witnessw 2. -Areduction of knowledge[51] is an interactive reduction,( G,K,P,V ), -that satisfies the following properties: -(i) Completeness: For any EPT adversary A, given pp← G (1λ,sz ),(s , u1, w1) -← A (pp)such that( pp, s, u1, w1) ∈ R 1, we have that the prover’s output -instance is equal to the verifier’s output instanceu 2, and that -(pp,s,⟨P,V⟩((pk,vk), u 1, w1))∈ R 2. -(ii) Knowledge soundness: For any EPT adversary( A,P ∗), there exists an -EPT extractorEsuch that if the success probability of the adversary -ϵ(A,P ∗) := Pr -" -(pp,s,⟨P ∗,V⟩((pk,vk), u 1,st))∈ R 2 - -pp← G(1 λ,sz) -(s, u1,st)← A(pp) -(pk,vk)← K(pp,s) -# -≥1/poly(λ),then we have that -Pr - -(pp,s, u 1, w1)∈ R 1 - -pp← G(1 λ,sz) -(s, u1,st)← A(pp) -(pk,vk)← K(pp,s) -w1 ← E(pp,s, u 1,st) - - ≥ϵ(A,P ∗)−negl(λ). -(iii) Public Coin: All of the verifier’s messages are uniformly random strings of -some prescribed length. Furthermore, the verifier’s messages contain all of -the random coins (randomness) used by the verifier. 7 -7 If a reduction of knowledge is public-coin, then it trivially satisfies the property -ofpublic reducibilitydescribed in [52] as the execution of the verifier V can be -emulated using the randomness from the transcript. -22 - -Lemma 2 (Sequential composition [50,52]).For reductions of knowledge -Π1 = (G,K,P 1,V 1) : R1 → R 2 andΠ 2 = (G,K,P 2,V 2) : R2 → R 3, we have that -Π2 ◦ Π1 = (G,K,P,V ) : R1 → R 3 is a reduction of knowledge where K(pp, s) -computes(pk,vk)and where -P(pk, u1, w1) =P 2(pk,P 1(pk, u1, w1)) -V(vk, u1) =V 2(vk,V 1(vk, u1, w1)) -Definition 6 (The sum-check protocol [62]).The sum-check protocol -SumCheck(T;Q) is a classic interactive proof protocol between two PPT al- -gorithms( P,V )that checks that the sum of evaluations of a ℓ-variate polynomial -Q∈F ≤d[X1, . . . , Xℓ]on the Boolean hypercube results in some claimed valueT. -The output of the sum-check protocol is a claim that v -? -=Q (r)for some random -point r∈F ℓ and claimed evaluations v, which the verifier V can query Q to check. -The protocol is public-coin, has a completeness error of0, and has a soundness -error of ≤ℓd/|F| . More generally, the field can be chosen to be an extension field -K. In this case, the soundness error is ≤ℓd/|K| . A self-contained description of -the sum-check protocol can be found in this note [81]. -5 Embedding products with evaluation homomorphism -Here, we define a bijective embedding from the fieldFinto the ring R F . -Definition 7 (Coefficient Embedding). -Element embedding:Consider a vector v∈F d. We define v∈ RF (in bold font) -to be the ring element whose coefficient vector isv, i.e.cf(v) =v. -Vector embedding:Recall that we define nF = d·n R. Hence, for a field vector -z∈F nF , we have a natural partition into d-sized sub-vectors z = [z1, . . . , znR]. We -define the ring vector z := (z1, . . . ,znR) ∈ RnR -F to be the vector of ring elements, -which are the embeddings of then R =n F /dfield sub-vectors. -Matrix embedding:For a matrix M∈F m×nF with rows M1, . . . , M m ∈F nF , we -define M := JM1, . . . ,M mK∈ Rm×nR -F , which is the vertical concatenation of all -the embedded rows. -Inverse embedding:Similarly, given a ring vector v∈ RnR -F or ring matrix M∈ -Rm×nR -F , we define the field vector v∈F nF or field matrix M∈F m×nF as the -inverse of previously defined coefficient embeddings. -Theorem 3 (Inner Product Transform [36, 64]).There exists a linear -transform ·:F d →F d such that for alla, b∈F d, we have the constant term -ct(¯a·b) =⟨a, b⟩ -where¯adenotes applying the transform toaand embedding¯ainto the ring. -Here, we define an extension of the inner product transform · : F d →F d -(Theorem 3) to vectors and matrices. -23 - -Definition 8 (Lifting the Transform). -Vector Transform:Consider a vector v∈F nF , we can partition v into d-sized sub- -vectors[v 1, . . . , v nR]. We define ·:F nF →F nF to be¯v:= [¯v1, . . . ,¯vnR]∈F nF . -Matrix Transform:Consider a matrix M∈F m×nF with rows M1, . . . , M m ∈F nF . -We define ·:F m×nF →F m×nF to be ¯M:= -q ¯M1, . . . , ¯Mm -y -∈F m×nF . -Remark 1 (Efficiency and Sparsity Preservation).When the cyclotomic polyno- -mial ϕ(X) is a power-of-two cyclotomic or a trinomial cyclotomic, the transform -· : F nF →F nF essentially only involves permuting and adding entries of the input -vector, and hence can be computed in O(nF) time. Since the · is linear, if the -original matrixMis sparse, then the transformed matrix ¯Mis also sparse. -Theorem 4 (Matrix-Vector Product Transform).Consider an arbitrary -matrix M∈F m×nF and vector z∈F nF . The matrix-vector product M z∈F m is -equal to the constant terms of the matrix-vector product ¯M z∈ Rm -F , when viewing -each ring element as a polynomial. More succinctly,M z=ct( ¯M z). -Proof.For brevity, we defer the proof to Section D.1. -Remark 2 (Matrix-vector Product Evaluation).Consider an arbitrary vector -z∈F nF , matrix M∈F m×nF , and multilinear evaluation point r∈K logm . Define -the evaluation y := ¯M zi -∼ -(r) = ⟨ ¯M zi,ˆr⟩ ∈RK. Observe that multiplying a ring -element in R F with an extension field element in K scales each coefficient of the -ring element by the extension field element. Hence, by Definition 2, we must -have that for all ℓ∈ [d], cf(y) ℓ = cf - ¯M zi - -ℓ -∼ -(r) ∈K (i.e. the ℓ-th coefficient of -y is equal to the multilinear evaluation of the ℓ-th coefficient vector of ¯M zi at -point r). Since ct(y) = cf(y) 1 and ct - ¯M zi - -= cf - ¯M zi - -1, by Theorem 4, we can -observe that the constant term ct(y) = M z∼ (r) ∈K is exactly the multilinear -evaluation of the field vectorM zat pointr. -Theorem 5 (Evaluation Homomorphism).Consider an arbitrary matrix -M∈F m×nF , vectors z1, . . . , zℓ ∈F nF , scalars ρ1, . . . , ρℓ ∈ RF , and evaluation -point r∈K logm . Let L : RnR -F →C,L in : RnR -F → RnR,in -F be arbitraryR F -module -homomorphisms. For alli∈[ℓ], define -ci :=L(z i)∈Cx i :=L in(zi)∈R nR,in -F yi := ¯M zi -∼ -(r)∈R K -Additionally, define -c:= -X -i∈[ℓ] -ρici ∈C,x:= -X -i∈[ℓ] -ρixi ∈R nR,in -F , -z:= -X -i∈[ℓ] -ρizi ∈R nR -F , y:= -X -i∈[ℓ] -ρiyi ∈R K -We must have that c = L(z)and y = ¯M z -∼ -(r). Additionally, for all i∈ [ℓ], -ct(yi) =M z i -∼ (r)andct(y) =M z∼ (r). -Proof.For brevity, we defer the proof to Section D.2. -24 - -6 Strong and weak interactive reductions -Definition 9 (Weak Interactive Reductions).Consider relations R1, R′ -1, -and R2 over public parameters, structure, instance, and witness tuples such that -R1 ⊆ R ′ -1. LetU 1 be the ambient instance space ofR 1. -An interactive reductionΠ : R1 → R 2, defined by PPT algorithms( G,K,P,V ) -(Definition 5), isweakif it is complete, public coin, and there exists a function -ϕ : U1 →C (for an arbitrary space C) such that for any EPT adversary( A,P ∗), -there exists an EPT extractor E such that if the success probability of the adversary -ϵ(A,P ∗)≥1/poly(λ),then -Pr - -(pp,s, u 1, w1)∈ R ′ -1 - -pp← G(1 λ,sz) -(s, u1,st)← A(pp) -(pk,vk)← K(pp,s) -w1 ← E(pp,s, u 1,st) - - ≥ϵ(A,P ∗)−negl(λ). -and ifA:= (B,B ′)such that -Pr - - -u1, u′ -1 ̸=⊥ -⇓ -ϕ(u1) =ϕ(u ′ -1) - -pp← G(1 λ,sz) -(s,st ∗)← B(pp) -(u1,st)← B ′(st∗) -(u′ -1,st ′)← B ′(st∗) - - = 1, -then -Pr - - -w1, w′ -1 ̸=⊥ -∧w 1 ̸=w ′ -1 - -pp← G(1 λ,sz) -(s,st ∗)← B(pp) -(u1,st)← B ′(st∗) -w1 ← E(pp,s, u 1,st) -(u′ -1,st ′)← B ′(st∗) -w′ -1 ← E(pp,s, u ′ -1,st ′) - - -≤negl(λ) -Definition 10 (Strong Interactive Reductions).Consider relations R1, -R2, and R′ -2 over public parameters, structure, instance, and witness tuples such -thatR 2 ⊆ R ′ -2. LetU 2 be the ambient instance space ofR 2. -An interactive reductionΠ : R1 → R 2, defined by PPT algorithms( G,K,P,V ) -(Definition 5), isstrongif it is complete, public coin, and there exists a function -ϕ:U 2 →C(for an arbitrary spaceC) such that -(i) For any EPT adversary(A,P ∗), -Pr - - -u2, u′ -2 ̸=⊥ -⇓ -ϕ(u2) =ϕ(u ′ -2) - -pp←Gen(1 λ) -(s, u1,st 1)← A(pp) -(pk,vk)← K(pp,s) -(u2, w2)← ⟨P ∗,V⟩((pk,vk), u 1,st) -(u′ -2, w′ -2)← ⟨P ∗,V⟩((pk,vk), u 1,st) - - = 1 -(ii) For any EPT adversary( A,P ∗), there exists an EPT extractor E such that if -ϵ′(A,P ∗) := Pr -" -(pp,s,⟨P ∗,V⟩((pk,vk), u 1,st))∈ R ′ -2 - -pp← G(1 λ,sz) -(s, u1,st)← A(pp) -(pk,vk)← K(pp,s) -# -25 - -≥1/poly(λ),and -Pr - - -w2, w′ -2 ̸=⊥ -∧ -w2 ̸=w ′ -2 - -pp←Gen(1 λ) -(s, u1,st)← A(pp) -(pk,vk)← K(pp,s) -(u2, w2)← ⟨P ∗,V⟩((pk,vk), u 1,st) -(u′ -2, w′ -2)← ⟨P ∗,V⟩((pk,vk), u 1,st) - - ≤negl(λ) -then we have that -Pr - -(pp,s, u 1, w1)∈ R 1 - -pp← G(1 λ,sz) -(s, u1,st)← A(pp) -(pk,vk)← K(pp,s) -w1 ← E(pp,s, u 1,st) - - ≥ϵ ′(A,P ∗)−negl(λ). -Theorem 6 (Strong-Weak Composition).Consider relations R1, R2, R′ -2 -and R3 over public parameters, structure, instance, and witness tuples such that -R2 ⊆ R ′ -2. Let U2 be the ambient instance space of R2. Consider interactive -reductions (Definition 5)Π 1 :R 1 → R 2 (R′ -2),Π 2 :R 2 (R′ -2)→ R 3 such that -(i)Π 1 isstrong(Definition 10) with respect to a functionϕ:U 2 →Cand -(ii)Π 2 isweak(Definition 9) with respect to thesamefunctionϕ, -then the sequential compositionΠ 2 ◦Π1 : R1 → R 3 is areduction of knowledge. -Proof.For brevity, we defer the proof to Section D.3. -7 Neo’s folding scheme for CCS -7.1 Relations -Definition 11 (Structure).We define astructureas a collection of elements -s:= -n -Mj ∈F m×nF - -j∈[t] , f∈F = (2^(2 * sqrt( n*d * log(q,2) * log(delta, 2)))).n() -# [AL21] eprint Prop 2. 2021/202 -# for all u,v in R, |u*v| / |v| <= gamma*|u| -# outputs T = gamma * |u| -# assumes we are only testing prime powers -def expansion_factor(m, norm): -if is_pow2(m): -return euler_phi(m) * norm -else: -return 2 * euler_phi(m) * norm -56 - -# p is prime -# max_idx is max cyclotomic index -# outputs list of (m, z) -def candidates(p, min_idx=10, max_idx=200): -# prime powers -possible_indices = [i for i in range(min_idx, max_idx) if len(factor(i)) == 1] -c = list() -for m in possible_indices: -zs = divisors(m) -for z in zs: -if thm1_1_cond(m, p, z): -c.append((Integer(m), Integer(z))) -return c -def pre_filter(q, cyclotomic_index, z, n, m, chals): -chals_norm = max({abs(c) for c in chals}) -chals_max_diff = chals[-1] - chals[0] -delta = 1.0045 # root hermite factor, chosen from [ESSLL19] eprint 2018/773 -phi = cyclotomic_polynomial(cyclotomic_index) # index cyclotomic polynomial -d = phi.degree() # degree of cyclotomic -# return non_trivial(q, n, d, delta) and chals_max_diff < thm1_1_inv_bound(q, z) and log(len(chals)^d,2).n() >= 120 -# We remove non_trivial(...) because we use the lattice estimator for hardness -return chals_max_diff < thm1_1_inv_bound(q, z) and log(len(chals)^d,2).n() >= 120 -def info(q, cyclotomic_index, z, n, m, chals): -chals_norm = max({abs(c) for c in chals}) -chals_max_diff = chals[-1] - chals[0] -delta = 1.0045 # root hermite factor, chosen from [ESSLL19] eprint 2018/773 -phi = cyclotomic_polynomial(cyclotomic_index) # index cyclotomic polynomial -d = phi.degree() # degree of cyclotomic -T = expansion_factor(cyclotomic_index, chals_norm) -# Bounds for MSIS to be hard -# [MR09] lattice-based cryptography pg 6 -# [CMNW24] pg 38 eprint 2024/281 -MSIS_B_l2_bound = min(q, 2^(2 * sqrt( n*d * log(q,2) * log(delta, 2)))) -MSIS_B_linf_bound = MSIS_B_l2_bound / sqrt(m*d) -# We need MSIS infinity bound 8TB to be hard -B = MSIS_B_linf_bound / (8*T) -print("####") -print("Cyclotomic idx:", cyclotomic_index) -print("Cyclotomic Poly:", phi) -print("z:", z) -#print("Prime is non-trivial?", non_trivial(q, n, d, delta)) -print("Csmall norm is small enough?", chals_max_diff < thm1_1_inv_bound(q, z)) -print("Csmall large enough?", log(len(chals)^d,2).n() >= 120) -57 - -print("Degree of Cyclotomic:", d) -# print("log(B):", log(B, 2).n()) -print("Expansion Factor T:", T) -print("Invertible Norm bound:", thm1_1_inv_bound(q, z)) -print("log(|C_Small|):", log(len(chals)^d,2).n()) -print("Factors of Cyclotomic:", thm1_1_num_factors(z)) -print() -def possible_settings(q, n, m, chals): -for (cyclotomic_index, z) in candidates(q): -if pre_filter(q, cyclotomic_index, z, n, m, chals): -info(q, cyclotomic_index, z, n, m, chals) -else: -delta = 1.0045 -d = cyclotomic_polynomial(cyclotomic_index).degree() -print("[Does not satisfy security requirements] index: {}, degree: {}, z: {}, non_trivial: {}, log(|C_Small|): {}, Invertible Norm bound: {}".format(cyclotomic_index, d, z, non_trivial(q, n, d, delta), log(len(chals)^d,2).n(), thm1_1_inv_bound(q, z))) -# Primes: -GL = 2^64 - 2^32 + 1 -AGL = GL - 32 -print("###############################") -print("AGL ###############################") -print("###############################") -# MSIS settings -n = 13 # rows, kappa in latticefold -m = 2^26 # cols -# Small Challenge set -chals = [-1, 0, 1, 2] -possible_settings(AGL, n, m, chals) -print("###############################") -print("M61 ###############################") -print("###############################") -# MSIS settings -n = 16 # rows, kappa in latticefold -m = 2^22 # cols -# Small Challenge set -chals = [-2, -1, 0, 1, 2] -possible_settings(2^61-1, n, m, chals) -print("###############################") -print("GL ###############################") -print("###############################") -# MSIS settings -n = 16 # rows, kappa in latticefold -m = 2^24 # cols -# Small Challenge set -chals = [-2, -1, 0, 1, 2] -possible_settings(GL, n, m, chals) -print("###############################") -58 - -D.8 Lattice Estimator Script -from estimator import * -Logging.set_level(Logging.LEVEL0) -M61 = 2^61 -1 -GL = 2^64 - 2^32 +1 -AGL = GL - 32 -b = 2 -n = 15 -d = 64 -k = 13 -K = 26 -B = b^k -m = 2^33 / d -T = 128 -q = AGL -n_sis = n*d -m_sis = m*d -B_l2 = sqrt(m*d)*(8*T*B) -params = SIS.Parameters(n=n_sis, q=q, m=m_sis,length_bound=B_l2, norm=2) -_ = SIS.estimate(params) -print((K+k)*T*(b-1) < B) -n = 18 -d = 54 -k = 14 -K = 61 -B = b^k -m = 2^30 / d -T=216 -q = GL -n_sis = n*d -m_sis = m*d -B_l2 = sqrt(m*d)*(8*T*B) -params = SIS.Parameters(n=n_sis, q=q, m=m_sis,length_bound=B_l2, norm=2) -_ = SIS.estimate(params) -print((K+k)*T*(b-1) < B) -n = 18 -d = 54 -k = 14 -K = 61 -59 - -B = b^k -m = 2^28 / d -T = 216 -q = M61 -n_sis = n*d -m_sis = m*d -B_l2 = sqrt(m*d)*(8*T*B) -params = SIS.Parameters(n=n_sis, q=q, m=m_sis,length_bound=B_l2, norm=2) -_ = SIS.estimate(params) -print((K+k)*T*(b-1) < B) -60 \ No newline at end of file From fa070fe605ae31d0fbd126a0ec658513f08845ee Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Mon, 2 Mar 2026 09:46:22 -0800 Subject: [PATCH 32/34] =?UTF-8?q?perf:=20fused=20sumcheck,=20split-eq=20st?= =?UTF-8?q?reaming,=20compact=20w=5Fevals=20=E2=80=94=208x=20memory=20redu?= =?UTF-8?q?ction?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Refactor the Hachi proving pipeline to eliminate the 13 GB matrix M and 2.6 GB vector z from memory, reducing peak prover allocation from ~30 GB to ~3.7 GB. Key changes: - QuadraticEquation: remove m/z fields; add compute_r_split_eq (split-eq factoring replaces full Kronecker materialization) and compute_m_a_streaming (row-at-a-time M·α evaluation). - ring_switch: decompose z_pre on the fly in build_w_coeffs; add build_w_evals_compact returning Vec for round-0 storage (all entries fit in [-8, 7] from balanced_decompose_pow2 with LOG_BASIS=4). - HachiSumcheckProver: fused norm+relation prover sharing a single w_table. Round 0 uses WTable::Compact(Vec), folding to WTable::Full(Vec) at half size after the first challenge. - HachiSumcheckVerifier: fused verifier combining both oracle checks with a batching_coeff sampled from the transcript. - Remove dead batched mat-vec functions from linear.rs. - Import hygiene: shorten crate::algebra::ring::X to crate::algebra::X; hoist mid-function use statements to top-level. Made-with: Cursor --- src/protocol/commitment/commit.rs | 2 +- src/protocol/commitment/scheme.rs | 2 +- src/protocol/commitment/utils/linear.rs | 34 +- src/protocol/commitment_scheme.rs | 69 +-- src/protocol/proof.rs | 2 +- src/protocol/quadratic_equation.rs | 535 +++++++++++++---------- src/protocol/ring_switch.rs | 157 +++++-- src/protocol/sumcheck/hachi_sumcheck.rs | 548 ++++++++++++++++++++++++ src/protocol/sumcheck/mod.rs | 1 + src/protocol/sumcheck/norm_sumcheck.rs | 39 +- 10 files changed, 1051 insertions(+), 338 deletions(-) create mode 100644 src/protocol/sumcheck/hachi_sumcheck.rs diff --git a/src/protocol/commitment/commit.rs b/src/protocol/commitment/commit.rs index 628eec66..c926421a 100644 --- a/src/protocol/commitment/commit.rs +++ b/src/protocol/commitment/commit.rs @@ -11,7 +11,7 @@ use super::utils::crt_ntt::{build_ntt_cache, NttMatrixCache}; use super::utils::linear::{decompose_block, decompose_rows, mat_vec_mul_ntt_cached, MatrixSlot}; use super::utils::matrix::{derive_public_matrix, sample_public_matrix_seed, PublicMatrixSeed}; use super::CommitmentConfig; -use crate::algebra::ring::CyclotomicRing; +use crate::algebra::CyclotomicRing; use crate::error::HachiError; use crate::primitives::serialization::{ Compress, HachiDeserialize, HachiSerialize, SerializationError, Valid, Validate, diff --git a/src/protocol/commitment/scheme.rs b/src/protocol/commitment/scheme.rs index c3d3fb95..256ce197 100644 --- a/src/protocol/commitment/scheme.rs +++ b/src/protocol/commitment/scheme.rs @@ -2,7 +2,7 @@ use super::config::{CommitmentConfig, HachiCommitmentLayout}; use super::transcript_append::AppendToTranscript; -use crate::algebra::ring::CyclotomicRing; +use crate::algebra::CyclotomicRing; use crate::error::HachiError; use crate::protocol::transcript::Transcript; use crate::{CanonicalField, FieldCore, Polynomial}; diff --git a/src/protocol/commitment/utils/linear.rs b/src/protocol/commitment/utils/linear.rs index 5e70ef96..4a8e007d 100644 --- a/src/protocol/commitment/utils/linear.rs +++ b/src/protocol/commitment/utils/linear.rs @@ -1,7 +1,7 @@ //! Linear algebra helpers for ring commitment. use crate::algebra::ntt::{MontCoeff, PrimeWidth}; -use crate::algebra::ring::{CrtNttParamSet, CyclotomicCrtNtt, CyclotomicRing}; +use crate::algebra::{CrtNttParamSet, CyclotomicCrtNtt, CyclotomicRing}; use crate::cfg_iter; use crate::error::HachiError; #[cfg(feature = "parallel")] @@ -196,21 +196,6 @@ fn mat_vec_mul_precomputed_with_params< .collect() } -fn mat_vec_mul_precomputed_many_with_params< - F: FieldCore + CanonicalField, - W: PrimeWidth, - const K: usize, - const D: usize, ->( - ntt_mat: &[Vec>], - vecs: &[Vec>], - params: &CrtNttParamSet, -) -> Vec>> { - cfg_iter!(vecs) - .map(|vec| mat_vec_mul_precomputed_with_params(ntt_mat, vec, params)) - .collect() -} - macro_rules! dispatch_cached { ($cache:expr, $which:expr, $params:expr, $func:ident $(, $arg:expr)*) => {{ #[allow(non_snake_case)] @@ -249,23 +234,6 @@ pub(crate) fn mat_vec_mul_ntt_cached( - cache: &NttMatrixCache, - which: MatrixSlot, - vecs: &[Vec>], -) -> Result>>, HachiError> { - let params = select_crt_ntt_params::()?; - let out = dispatch_cached!( - cache, - which, - ¶ms, - mat_vec_mul_precomputed_many_with_params, - vecs - ); - Ok(out) -} - /// Basis-decompose a block of ring elements into `block.len() * delta` gadget components. pub fn decompose_block( block: &[CyclotomicRing], diff --git a/src/protocol/commitment_scheme.rs b/src/protocol/commitment_scheme.rs index 2a418f58..d332265d 100644 --- a/src/protocol/commitment_scheme.rs +++ b/src/protocol/commitment_scheme.rs @@ -1,6 +1,6 @@ //! Commitment scheme trait implementation. -use crate::algebra::ring::CyclotomicRing; +use crate::algebra::CyclotomicRing; use crate::cfg_into_iter; use crate::error::HachiError; #[cfg(feature = "parallel")] @@ -17,23 +17,19 @@ use crate::protocol::commitment::{ use crate::protocol::opening_point::RingOpeningPoint; use crate::protocol::proof::{HachiCommitmentHint, HachiProof, SumcheckAux}; use crate::protocol::quadratic_equation::QuadraticEquation; -use crate::protocol::ring_switch::{ring_switch_prover, ring_switch_verifier}; -use crate::protocol::sumcheck::batched_sumcheck::{ - prove_batched_sumcheck, verify_batched_sumcheck, -}; -use crate::protocol::sumcheck::norm_sumcheck::{NormSumcheckProver, NormSumcheckVerifier}; -use crate::protocol::sumcheck::relation_sumcheck::{ - RelationSumcheckProver, RelationSumcheckVerifier, -}; -use crate::protocol::sumcheck::{SumcheckInstanceProver, SumcheckInstanceVerifier}; +use crate::protocol::ring_switch::{build_w_evals, ring_switch_prover, ring_switch_verifier}; +use crate::protocol::sumcheck::hachi_sumcheck::{HachiSumcheckProver, HachiSumcheckVerifier}; +use crate::protocol::sumcheck::{prove_sumcheck, verify_sumcheck}; use crate::protocol::transcript::labels::{ - ABSORB_COMMITMENT, ABSORB_EVALUATION_CLAIMS, CHALLENGE_SUMCHECK_ROUND, + ABSORB_COMMITMENT, ABSORB_EVALUATION_CLAIMS, CHALLENGE_SUMCHECK_BATCH, CHALLENGE_SUMCHECK_ROUND, }; use crate::protocol::transcript::Transcript; use crate::{CanonicalField, FieldCore, FieldSampling, FromSmallInt, Polynomial}; #[cfg(test)] -use crate::protocol::ring_switch::{eval_ring_matrix_at, expand_m_a}; +use crate::protocol::quadratic_equation::compute_m_a_streaming; +#[cfg(test)] +use crate::protocol::ring_switch::expand_m_a; #[cfg(test)] use crate::protocol::transcript::labels::{ ABSORB_SUMCHECK_W, CHALLENGE_RING_SWITCH, DOMAIN_HACHI_PROTOCOL, @@ -135,7 +131,7 @@ where transcript.append_serde(ABSORB_EVALUATION_CLAIMS, &y_ring); // §4.2 Quadratic equation - let quad_eq = QuadraticEquation::::new_prover( + let mut quad_eq = QuadraticEquation::::new_prover( setup, &ring_opening_point, &hint, @@ -145,22 +141,25 @@ where )?; // §4.3 Ring switch - let rs = ring_switch_prover::(&quad_eq, transcript)?; + let rs = ring_switch_prover::(&mut quad_eq, &setup.expanded, transcript)?; + + // Sample batching coefficient for fused sumcheck + let batching_coeff: F = transcript.challenge_scalar(CHALLENGE_SUMCHECK_BATCH); - // Batched sumcheck: norm + relation - let mut norm_prover = NormSumcheckProver::new(&rs.tau0, rs.w_evals.clone(), rs.b); - let mut relation_prover = RelationSumcheckProver::new( + // Fused sumcheck: norm + relation with shared w_table + let mut fused_prover = HachiSumcheckProver::new( + batching_coeff, rs.w_evals, + &rs.tau0, + rs.b, &rs.alpha_evals_y, &rs.m_evals_x, rs.num_u, rs.num_l, ); - let instances: Vec<&mut dyn SumcheckInstanceProver> = - vec![&mut norm_prover, &mut relation_prover]; let (sumcheck_proof, ..) = - prove_batched_sumcheck::(instances, transcript, |tr| { + prove_sumcheck::(&mut fused_prover, transcript, |tr| { tr.challenge_scalar(CHALLENGE_SUMCHECK_ROUND) })?; @@ -228,15 +227,24 @@ where // §4.3 Ring switch (verifier side) let rs = ring_switch_verifier::( &quad_eq, + &setup.expanded, &proof.sumcheck_aux.w, &proof.w_commitment, transcript, )?; - // Batched sumcheck verification: norm (F_0) + relation (F_α) - let norm_verifier = NormSumcheckVerifier::new(rs.tau0, rs.w_evals.clone(), rs.b); - let relation_verifier = RelationSumcheckVerifier::new( - rs.w_evals, + // Sample batching coefficient for fused sumcheck (must match prover) + let batching_coeff: F = transcript.challenge_scalar(CHALLENGE_SUMCHECK_BATCH); + + // Build full w_evals for verifier from the witness vector w. + let (w_evals_full, _, _) = build_w_evals(&proof.sumcheck_aux.w, Cfg::D)?; + + // Fused sumcheck verification: norm (F_0) + relation (F_α) + let fused_verifier = HachiSumcheckVerifier::new( + batching_coeff, + w_evals_full, + rs.tau0, + rs.b, rs.alpha_evals_y, rs.m_evals_x, rs.tau1, @@ -248,11 +256,9 @@ where rs.num_l, ); - let verifiers: Vec<&dyn SumcheckInstanceVerifier> = - vec![&norm_verifier, &relation_verifier]; - verify_batched_sumcheck::( + verify_sumcheck::( &proof.sumcheck_proof, - verifiers, + &fused_verifier, transcript, |tr| tr.challenge_scalar(CHALLENGE_SUMCHECK_ROUND), )?; @@ -509,7 +515,12 @@ where )?; transcript.append_serde(ABSORB_SUMCHECK_W, &proof.w_commitment); let alpha: F = transcript.challenge_scalar(CHALLENGE_RING_SWITCH); - let m_a = eval_ring_matrix_at::(quad_eq.m(), &alpha); + let m_a = compute_m_a_streaming::( + &setup.expanded, + quad_eq.opening_point(), + &quad_eq.challenges, + &alpha, + )?; let m_a_vec = expand_m_a::(&m_a, alpha)?; Ok((alpha, m_a_vec)) } diff --git a/src/protocol/proof.rs b/src/protocol/proof.rs index 89efd9e8..c020f7b7 100644 --- a/src/protocol/proof.rs +++ b/src/protocol/proof.rs @@ -1,6 +1,6 @@ //! Proof structures for the Hachi protocol. -use crate::algebra::ring::CyclotomicRing; +use crate::algebra::CyclotomicRing; use crate::primitives::serialization::{Compress, SerializationError}; use crate::primitives::serialization::{Valid, Validate}; use crate::protocol::commitment::RingCommitment; diff --git a/src/protocol/quadratic_equation.rs b/src/protocol/quadratic_equation.rs index 22b23cde..092a017e 100644 --- a/src/protocol/quadratic_equation.rs +++ b/src/protocol/quadratic_equation.rs @@ -3,7 +3,7 @@ //! This module encapsulates the stage-1 prover logic and the generation of //! the quadratic equation components M, y, z, and v. -use crate::algebra::ring::{CyclotomicRing, SparseChallenge, SparseChallengeConfig}; +use crate::algebra::{CyclotomicRing, SparseChallenge, SparseChallengeConfig}; use crate::cfg_iter; use crate::error::HachiError; #[cfg(feature = "parallel")] @@ -20,6 +20,7 @@ use crate::protocol::commitment::{ }; use crate::protocol::opening_point::RingOpeningPoint; use crate::protocol::proof::HachiCommitmentHint; +use crate::protocol::ring_switch::eval_ring_at; use crate::protocol::transcript::labels::{ABSORB_PROVER_V, CHALLENGE_STAGE1_FOLD}; use crate::protocol::transcript::Transcript; use crate::{CanonicalField, FieldCore}; @@ -82,12 +83,12 @@ fn compute_v( mat_vec_mul_ntt_cached(cache, MatrixSlot::D, &w_hat_flat) } -/// **Steps 7–9.** Fold `z = Σ c_i · s_i`, check `‖z‖_∞ ≤ β`, and decompose `ẑ = J^{-1}(z)`. +/// **Steps 7–9.** Fold `z_pre = Σ c_i · s_i` and check `‖z_pre‖_∞ ≤ β`. /// -/// Recomputes each block's `s_i` from `ring_coeffs` one block at a time, -/// accumulating `z[j] += c_i * s_i[j]` across blocks. Peak memory is one -/// block's `s_i` rather than the full `s` tensor. -fn compute_z_hat( +/// Returns the pre-decomposition `z` vector (before gadget decomposition into +/// `ẑ = J^{-1}(z_pre)`). Callers that need `z_hat` can apply +/// `balanced_decompose_pow2(TAU, LOG_BASIS)` themselves. +fn compute_z_pre( ring_coeffs: &[CyclotomicRing], challenges: &[SparseChallenge], layout: HachiCommitmentLayout, @@ -128,30 +129,30 @@ where ))); } - Ok(z.iter() - .flat_map(|z_j| z_j.balanced_decompose_pow2(Cfg::TAU, Cfg::LOG_BASIS)) - .collect()) + Ok(z) } /// Stage-1 quadratic equation state for the Hachi protocol. /// /// Encapsulates the relation $M(x) \cdot z = y(x) + (X^D + 1) \cdot r(x)$ -/// along with intermediate prover witness data (`w_hat`, `z_hat`, `hint`). +/// along with intermediate prover witness data (`w_hat`, `z_pre`, `hint`). +/// +/// M and z are never materialized — split-eq factoring computes their +/// products on-the-fly via `compute_r_split_eq` and `compute_m_a_streaming`. pub struct QuadraticEquation { /// Stage-1 proof vector `v = D · ŵ`. pub v: Vec>, /// Stage-1 folding challenges (sparse representation). pub challenges: Vec, - /// Matrix `M`. - m: Vec>>, /// Vector `y`. y: Vec>, - /// Vector `z` (prover only). - z: Option>>, + /// Opening point (a, b) Lagrange weights. + opening_point: RingOpeningPoint, + /// Pre-decomposition folded witness `z_pre = Σ c_i · s_i` (prover only). + /// Replaces both `z_hat` and `z`: `z_hat = J^{-1}(z_pre)`. + z_pre: Option>>, /// Decomposed `ŵ_i = G_1^{-1}(w_i)` (prover only). w_hat: Option>>>, - /// Decomposed `ẑ = J^{-1}(z)` (prover only). - z_hat: Option>>, /// Commitment hint (prover only). hint: Option>, @@ -196,20 +197,17 @@ where &challenge_cfg, )?; - let z_hat = compute_z_hat::(&hint.ring_coeffs, &challenges, layout)?; + let z_pre = compute_z_pre::(&hint.ring_coeffs, &challenges, layout)?; - let m = generate_m::(&setup.expanded, ring_opening_point, &challenges)?; let y = generate_y::(&v, &commitment.u, y_ring)?; - let z = generate_z(&w_hat, &hint.t_hat, &z_hat); Ok(Self { v, challenges, - m, y, - z: Some(z), + opening_point: ring_opening_point.clone(), + z_pre: Some(z_pre), w_hat: Some(w_hat), - z_hat: Some(z_hat), hint: Some(hint.clone()), _marker: std::marker::PhantomData, }) @@ -231,56 +229,64 @@ where let layout = setup.expanded.seed.layout; let challenges = derive_stage1_challenges::(transcript, v, layout.num_blocks)?; - let m = generate_m::(&setup.expanded, ring_opening_point, &challenges)?; let y = generate_y::(v, &commitment.u, y_ring)?; Ok(Self { v: v.to_vec(), challenges, - m, y, - z: None, + opening_point: ring_opening_point.clone(), + z_pre: None, w_hat: None, - z_hat: None, hint: None, _marker: std::marker::PhantomData, }) } - /// Get the matrix M. - pub fn m(&self) -> &[Vec>] { - &self.m - } - /// Get the vector y. pub fn y(&self) -> &[CyclotomicRing] { &self.y } - /// Get the vector z (returns None if constructed by verifier). - pub fn z(&self) -> Option<&[CyclotomicRing]> { - self.z.as_deref() - } - /// Get the vector v. pub fn v(&self) -> &[CyclotomicRing] { &self.v } + /// Get the opening point (a, b) Lagrange weights. + pub fn opening_point(&self) -> &RingOpeningPoint { + &self.opening_point + } + + /// Get the pre-decomposition folded witness `z_pre` (prover only). + pub fn z_pre(&self) -> Option<&[CyclotomicRing]> { + self.z_pre.as_deref() + } + + /// Take ownership of `z_pre`, leaving `None` in its place. + pub fn take_z_pre(&mut self) -> Option>> { + self.z_pre.take() + } + /// Get the decomposed witness `ŵ` (prover only). pub fn w_hat(&self) -> Option<&[Vec>]> { self.w_hat.as_deref() } - /// Get the decomposed folded witness `ẑ` (prover only). - pub fn z_hat(&self) -> Option<&[CyclotomicRing]> { - self.z_hat.as_deref() + /// Take ownership of `w_hat`, leaving `None` in its place. + pub fn take_w_hat(&mut self) -> Option>>> { + self.w_hat.take() } /// Get the commitment hint (prover only). pub fn hint(&self) -> Option<&HachiCommitmentHint> { self.hint.as_ref() } + + /// Take ownership of the hint, leaving `None` in its place. + pub fn take_hint(&mut self) -> Option> { + self.hint.take() + } } pub(crate) fn derive_stage1_challenges( @@ -305,12 +311,6 @@ where ) } -fn constant_ring(value: F) -> CyclotomicRing { - let mut coeffs = [F::zero(); D]; - coeffs[0] = value; - CyclotomicRing::from_coefficients(coeffs) -} - fn gadget_row_scalars(levels: usize, log_basis: u32) -> Vec { let base = F::from_canonical_u128_reduced(1u128 << log_basis); let mut out = Vec::with_capacity(levels); @@ -322,217 +322,305 @@ fn gadget_row_scalars(levels: usize, log_basis: u out } -/// Kronecker product of two scalar vectors, producing constant ring elements. -/// Cost: O(1) per pair (just a field multiplication + ring construction). -fn kron_scalars( - left: &[F], - right: &[F], -) -> Vec> { - let mut out = Vec::with_capacity(left.len().saturating_mul(right.len())); - for &l in left { - for &r in right { - out.push(constant_ring(l * r)); +/// Accumulate unreduced polynomial product `a * b` into `poly` (length 2D-1). +fn add_unreduced_product( + poly: &mut [F], + a: &CyclotomicRing, + b: &CyclotomicRing, +) { + if a.is_zero() { + return; + } + let ac = a.coefficients(); + let bc = b.coefficients(); + let is_scalar = ac[1..].iter().all(|c| c.is_zero()); + if is_scalar { + let s = ac[0]; + for k in 0..D { + poly[k] = poly[k] + s * bc[k]; + } + } else { + for t in 0..D { + for s in 0..D { + poly[t + s] = poly[t + s] + ac[t] * bc[s]; + } } } - out } -/// Kronecker product where the right operand is field scalars. -/// Cost: O(D) per pair instead of O(D^2). -fn kron_row_scale( - left: &[CyclotomicRing], - right: &[F], -) -> Vec> { - let mut out = Vec::with_capacity(left.len().saturating_mul(right.len())); - for l in left { - for &r in right { - out.push(l.scale(&r)); +/// Accumulate negated unreduced product `-a * b` into `poly`. +fn sub_unreduced_product( + poly: &mut [F], + a: &CyclotomicRing, + b: &CyclotomicRing, +) { + if a.is_zero() { + return; + } + let ac = a.coefficients(); + let bc = b.coefficients(); + let is_scalar = ac[1..].iter().all(|c| c.is_zero()); + if is_scalar { + let s = ac[0]; + for k in 0..D { + poly[k] = poly[k] - s * bc[k]; + } + } else { + for t in 0..D { + for s in 0..D { + poly[t + s] = poly[t + s] - ac[t] * bc[s]; + } } } - out } -/// Kronecker product of sparse challenges with scalar gadget entries. -/// Cost: O(omega + D) per pair instead of O(D^2). -fn kron_sparse_scale( - left: &[SparseChallenge], - right: &[F], -) -> Vec> { - let mut out = Vec::with_capacity(left.len().saturating_mul(right.len())); - for l in left { - let dense: CyclotomicRing = l.to_dense().expect("valid sparse challenge"); - for &r in right { - out.push(dense.scale(&r)); - } +/// Add scalar * ring_element into the low-D coefficients of `poly`. +/// scalar * ring produces degree D-1, so no high-half contribution. +fn add_scalar_ring_product( + poly: &mut [F], + scalar: &F, + ring: &CyclotomicRing, +) { + for (k, coeff) in ring.coefficients().iter().enumerate() { + poly[k] = poly[k] + *scalar * *coeff; + } +} + +/// Subtract scalar * ring_element from the low-D coefficients of `poly`. +fn sub_scalar_ring_product( + poly: &mut [F], + scalar: &F, + ring: &CyclotomicRing, +) { + for (k, coeff) in ring.coefficients().iter().enumerate() { + poly[k] = poly[k] - *scalar * *coeff; } - out } -fn gadget_block_diag_scalars(blocks: usize, row: &[F]) -> Vec> { - let row_len = row.len(); - let mut rows = Vec::with_capacity(blocks); - for i in 0..blocks { - let mut out = vec![F::zero(); blocks * row_len]; - let start = i * row_len; - out[start..start + row_len].copy_from_slice(row); - rows.push(out); +/// Add sparse_challenge * ring_element as unreduced product into `poly`. +fn add_sparse_ring_product( + poly: &mut [F], + challenge: &SparseChallenge, + ring: &CyclotomicRing, +) { + let dense: CyclotomicRing = challenge.to_dense().expect("valid sparse challenge"); + add_unreduced_product(poly, &dense, ring); +} + +/// Split-eq replacement for `generate_m` + `compute_r_via_poly_division`. +/// +/// Computes `r` such that `M·z = y + (X^D+1)·r` without materializing M or z. +/// Uses split-eq factoring: `kron(left, gadget) · decomposed = left · pre_decomp`. +pub(crate) fn compute_r_split_eq( + setup: &HachiExpandedSetup, + opening_point: &RingOpeningPoint, + challenges: &[SparseChallenge], + w_hat: &[Vec>], + t_hat: &[Vec>], + z_pre: &[CyclotomicRing], + y: &[CyclotomicRing], +) -> Result>, HachiError> +where + F: FieldCore + CanonicalField, + Cfg: CommitmentConfig, +{ + let poly_len = 2 * D - 1; + let num_rows = Cfg::N_D + Cfg::N_B + 1 + 1 + Cfg::N_A; + + let w_hat_flat: Vec> = + w_hat.iter().flat_map(|v| v.iter().copied()).collect(); + let t_hat_flat: Vec> = + t_hat.iter().flat_map(|v| v.iter().copied()).collect(); + + let mut result = Vec::with_capacity(num_rows); + + for (row_idx, y_i) in y.iter().enumerate().take(num_rows) { + let mut poly = vec![F::zero(); poly_len]; + + if row_idx < Cfg::N_D { + let d_row = &setup.D[row_idx]; + for (m_ij, z_j) in d_row.iter().zip(w_hat_flat.iter()) { + add_unreduced_product(&mut poly, m_ij, z_j); + } + } else if row_idx < Cfg::N_D + Cfg::N_B { + let b_row = &setup.B[row_idx - Cfg::N_D]; + for (m_ij, z_j) in b_row.iter().zip(t_hat_flat.iter()) { + add_unreduced_product(&mut poly, m_ij, z_j); + } + } else if row_idx == Cfg::N_D + Cfg::N_B { + // row3: b . w_recomp (scalar * ring, degree D-1) + for (i, w_hat_i) in w_hat.iter().enumerate() { + let w_recomp = CyclotomicRing::gadget_recompose_pow2(w_hat_i, Cfg::LOG_BASIS); + add_scalar_ring_product(&mut poly, &opening_point.b[i], &w_recomp); + } + } else if row_idx == Cfg::N_D + Cfg::N_B + 1 { + // row4 w-segment: c . w_recomp (sparse*ring unreduced) + for (i, w_hat_i) in w_hat.iter().enumerate() { + let w_recomp = CyclotomicRing::gadget_recompose_pow2(w_hat_i, Cfg::LOG_BASIS); + add_sparse_ring_product(&mut poly, &challenges[i], &w_recomp); + } + // row4 z-segment: -(a . z_pre_recomp), double-factored + // z_pre_recomp[i] = gadget_recompose(z_pre[i*DELTA..(i+1)*DELTA]) + let block_len = opening_point.a.len(); + for i in 0..block_len { + let start = i * Cfg::DELTA; + let end = start + Cfg::DELTA; + if end <= z_pre.len() { + let z_pre_recomp = + CyclotomicRing::gadget_recompose_pow2(&z_pre[start..end], Cfg::LOG_BASIS); + sub_scalar_ring_product(&mut poly, &opening_point.a[i], &z_pre_recomp); + } + } + } else { + // row5 (N_A rows) + let a_idx = row_idx - (Cfg::N_D + Cfg::N_B + 2); + // t-segment: c . t_recomp[a_idx] (sparse*ring unreduced) + for (i, t_hat_i) in t_hat.iter().enumerate() { + let start = a_idx * Cfg::DELTA; + let end = start + Cfg::DELTA; + if end <= t_hat_i.len() { + let t_recomp = + CyclotomicRing::gadget_recompose_pow2(&t_hat_i[start..end], Cfg::LOG_BASIS); + add_sparse_ring_product(&mut poly, &challenges[i], &t_recomp); + } + } + // z-segment: -(A[a_idx] . z_pre) (ring*ring unreduced) + let a_row = &setup.A[a_idx]; + for (m_ij, z_j) in a_row.iter().zip(z_pre.iter()) { + sub_unreduced_product(&mut poly, m_ij, z_j); + } + } + + let y_coeffs = y_i.coefficients(); + for k in 0..D { + poly[k] = poly[k] - y_coeffs[k]; + } + + // Divide by X^D + 1 + let mut quotient = vec![F::zero(); D]; + for k in (D..poly_len).rev() { + let q = poly[k]; + quotient[k - D] = q; + poly[k - D] = poly[k - D] - q; + } + let coeffs: [F; D] = std::array::from_fn(|k| quotient[k]); + result.push(CyclotomicRing::from_coefficients(coeffs)); } - rows + + Ok(result) } -pub(crate) fn generate_m( +/// Split-eq replacement for `generate_m` + `eval_ring_matrix_at`. +/// +/// Computes the field-element evaluations of each M entry at `alpha`, +/// organized as rows of field elements, without materializing M. +pub(crate) fn compute_m_a_streaming( setup: &HachiExpandedSetup, opening_point: &RingOpeningPoint, challenges: &[SparseChallenge], -) -> Result>>, HachiError> + alpha: &F, +) -> Result>, HachiError> where F: FieldCore + CanonicalField, + Cfg: CommitmentConfig, { let layout = setup.seed.layout; let num_blocks = layout.num_blocks; let block_len = layout.block_len; - let w_len = Cfg::DELTA - .checked_mul(num_blocks) - .ok_or_else(|| HachiError::InvalidSetup("w length overflow".to_string()))?; - let t_len = Cfg::DELTA - .checked_mul(Cfg::N_A) - .and_then(|v| v.checked_mul(num_blocks)) - .ok_or_else(|| HachiError::InvalidSetup("t length overflow".to_string()))?; - let z_len = Cfg::TAU - .checked_mul(Cfg::DELTA) - .and_then(|v| v.checked_mul(block_len)) - .ok_or_else(|| HachiError::InvalidSetup("z length overflow".to_string()))?; - let total_cols = w_len - .checked_add(t_len) - .and_then(|v| v.checked_add(z_len)) - .ok_or_else(|| HachiError::InvalidSetup("matrix width overflow".to_string()))?; - - if opening_point.b.len() != num_blocks { - return Err(HachiError::InvalidPointDimension { - expected: num_blocks, - actual: opening_point.b.len(), - }); - } - if opening_point.a.len() != block_len { - return Err(HachiError::InvalidPointDimension { - expected: block_len, - actual: opening_point.a.len(), - }); - } - if challenges.len() != num_blocks { - return Err(HachiError::InvalidSize { - expected: num_blocks, - actual: challenges.len(), - }); - } - if setup.D.len() != Cfg::N_D { - return Err(HachiError::InvalidSize { - expected: Cfg::N_D, - actual: setup.D.len(), - }); - } - if setup.B.len() != Cfg::N_B { - return Err(HachiError::InvalidSize { - expected: Cfg::N_B, - actual: setup.B.len(), - }); - } - if setup.A.len() != Cfg::N_A { - return Err(HachiError::InvalidSize { - expected: Cfg::N_A, - actual: setup.A.len(), - }); - } - if setup.A.first().map(|row| row.len()) != Some(block_len * Cfg::DELTA) { - return Err(HachiError::InvalidSetup("A row width mismatch".to_string())); - } + let w_len = Cfg::DELTA * num_blocks; + let t_len = Cfg::DELTA * Cfg::N_A * num_blocks; + let z_len = Cfg::TAU * Cfg::DELTA * block_len; + let total_cols = w_len + t_len + z_len; let g1 = gadget_row_scalars::(Cfg::DELTA, Cfg::LOG_BASIS); let j1 = gadget_row_scalars::(Cfg::TAU, Cfg::LOG_BASIS); - let row3_w = kron_scalars::(&opening_point.b, &g1); - let row4_w = kron_sparse_scale::(challenges, &g1); - - let ag = kron_scalars::(&opening_point.a, &g1); - let ag_scalars: Vec = ag.iter().map(|r| r.coefficients()[0]).collect(); - let row4_z = kron_scalars::(&ag_scalars, &j1) - .into_iter() - .map(|x| -x) - .collect::>(); - - let g_na = gadget_block_diag_scalars::(Cfg::N_A, &g1); - let row5_mid = g_na - .iter() - .map(|row| kron_sparse_scale::(challenges, row)) - .collect::>(); - let row5_right = setup - .A - .iter() - .map(|row| kron_row_scale(row, &j1).into_iter().map(|x| -x).collect()) - .collect::>>(); - - let zero = CyclotomicRing::::zero(); - let mut rows = Vec::with_capacity(Cfg::N_D + Cfg::N_B + 1usize + 1usize + Cfg::N_A); - - for row in setup.D.iter() { - if row.len() != w_len { - return Err(HachiError::InvalidSetup("D row width mismatch".to_string())); + // Pre-evaluate alpha powers for gadget scalars (already field elements) + // g1 and j1 are already field scalars, so eval_ring_at(constant(g), alpha) = g. + + let mut rows = Vec::with_capacity(Cfg::N_D + Cfg::N_B + 1 + 1 + Cfg::N_A); + + // D rows: setup.D[i] evaluated at alpha, zero-padded + for d_row in setup.D.iter() { + let mut full = vec![F::zero(); total_cols]; + for (j, ring) in d_row.iter().enumerate() { + full[j] = eval_ring_at(ring, alpha); } - let mut full = vec![zero; total_cols]; - full[..w_len].copy_from_slice(row); rows.push(full); } - for row in setup.B.iter() { - if row.len() != t_len { - return Err(HachiError::InvalidSetup("B row width mismatch".to_string())); + // B rows: setup.B[i] evaluated at alpha, in t-segment + for b_row in setup.B.iter() { + let mut full = vec![F::zero(); total_cols]; + for (j, ring) in b_row.iter().enumerate() { + full[w_len + j] = eval_ring_at(ring, alpha); } - let mut full = vec![zero; total_cols]; - full[w_len..w_len + t_len].copy_from_slice(row); rows.push(full); } - let mut row3 = vec![zero; total_cols]; - row3[..w_len].copy_from_slice(&row3_w); - rows.push(row3); + // row3: kron(b, g1) evaluated at alpha -> b[i] * g1[d] (all scalars) + { + let mut full = vec![F::zero(); total_cols]; + for (i, &b_i) in opening_point.b.iter().enumerate() { + for (d, &g) in g1.iter().enumerate() { + full[i * Cfg::DELTA + d] = b_i * g; + } + } + rows.push(full); + } - let mut row4 = vec![zero; total_cols]; - row4[..w_len].copy_from_slice(&row4_w); - row4[w_len + t_len..].copy_from_slice(&row4_z); - rows.push(row4); + // row4: w-segment = kron(c, g1) evaluated at alpha + // z-segment = -kron(kron(a,g1), j1) (all scalars) + { + let mut full = vec![F::zero(); total_cols]; + // w-segment: c[i] evaluated at alpha, times g1[d] + for (i, c) in challenges.iter().enumerate() { + let c_alpha = eval_ring_at(&c.to_dense::().expect("valid challenge"), alpha); + for (d, &g) in g1.iter().enumerate() { + full[i * Cfg::DELTA + d] = c_alpha * g; + } + } + // z-segment: -kron(kron(a,g1), j1) = -(a[i] * g1[d] * j1[t]) + let z_offset = w_len + t_len; + for (i, &a_i) in opening_point.a.iter().enumerate() { + for (d, &g) in g1.iter().enumerate() { + let ag = a_i * g; + for (t, &j) in j1.iter().enumerate() { + let idx = (i * Cfg::DELTA + d) * Cfg::TAU + t; + full[z_offset + idx] = -(ag * j); + } + } + } + rows.push(full); + } - for (mid, right) in row5_mid.into_iter().zip(row5_right.into_iter()) { - let mut row = vec![zero; total_cols]; - row[w_len..w_len + t_len].copy_from_slice(&mid); - row[w_len + t_len..].copy_from_slice(&right); - rows.push(row); + // row5 (N_A rows): t-segment = kron(c, g_na[a_idx]) evaluated at alpha + // z-segment = -kron(A[a_idx], j1) evaluated at alpha + for a_idx in 0..Cfg::N_A { + let mut full = vec![F::zero(); total_cols]; + // t-segment: block-diagonal gadget times challenges + for (i, c) in challenges.iter().enumerate() { + let c_alpha = eval_ring_at(&c.to_dense::().expect("valid challenge"), alpha); + for (d, &g) in g1.iter().enumerate() { + let t_idx = i * (Cfg::N_A * Cfg::DELTA) + a_idx * Cfg::DELTA + d; + full[w_len + t_idx] = c_alpha * g; + } + } + // z-segment: -A[a_idx][k] evaluated at alpha, times j1[t] + let z_offset = w_len + t_len; + let a_row = &setup.A[a_idx]; + for (k, ring) in a_row.iter().enumerate() { + let ring_alpha = eval_ring_at(ring, alpha); + for (t, &j) in j1.iter().enumerate() { + full[z_offset + k * Cfg::TAU + t] = -(ring_alpha * j); + } + } + rows.push(full); } Ok(rows) } -pub(crate) fn generate_z( - w_hat: &[Vec>], - t_hat: &[Vec>], - z_hat: &[CyclotomicRing], -) -> Vec> { - let mut out = Vec::with_capacity( - w_hat.len() - + t_hat.len() - + z_hat.len() - + w_hat.iter().map(|v| v.len()).sum::() - + t_hat.iter().map(|v| v.len()).sum::(), - ); - for w in w_hat { - out.extend(w.iter().copied()); - } - for t in t_hat { - out.extend(t.iter().copied()); - } - out.extend_from_slice(z_hat); - out -} - pub(crate) fn generate_y( v: &[CyclotomicRing], u: &[CyclotomicRing], @@ -568,8 +656,7 @@ where #[cfg(test)] mod tests { use super::*; - use crate::algebra::ring::SparseChallengeConfig; - use crate::algebra::CyclotomicRing; + use crate::algebra::{CyclotomicRing, SparseChallengeConfig}; use crate::protocol::challenges::sparse::sample_sparse_challenges; use crate::protocol::commitment::{HachiCommitmentCore, RingCommitmentScheme}; use crate::protocol::proof::HachiCommitmentHint; @@ -719,6 +806,14 @@ mod tests { ); } + /// Derive z_hat from z_pre for test assertions. + fn derive_z_hat(z_pre: &[CyclotomicRing]) -> Vec> { + z_pre + .iter() + .flat_map(|z_j| z_j.balanced_decompose_pow2(TAU, LOG_BASIS)) + .collect() + } + /// Row 4: (c^T ⊗ G_1) · ŵ = a^T · G_{2^m} · J · ẑ #[test] fn row4_challenge_fold_w_equals_a_gadget_j_z_hat() { @@ -738,7 +833,8 @@ mod tests { acc + (*c_i * *w_i) }); - let z_recovered = recompose_z_hat(f.quad_eq.z_hat().unwrap()); + let z_hat = derive_z_hat(f.quad_eq.z_pre().unwrap()); + let z_recovered = recompose_z_hat(&z_hat); let rhs = a_transpose_gadget_times_vec(&f.point.a, &z_recovered); assert_eq!(lhs, rhs, "Row 4 failed: (c^T ⊗ G_1)ŵ ≠ a^T G J ẑ"); @@ -759,7 +855,8 @@ mod tests { } } - let z_recovered = recompose_z_hat(f.quad_eq.z_hat().unwrap()); + let z_hat = derive_z_hat(f.quad_eq.z_pre().unwrap()); + let z_recovered = recompose_z_hat(&z_hat); let rhs = mat_vec_mul(&f.setup.expanded.A, &z_recovered); assert_eq!(lhs, rhs, "Row 5 failed: (c^T ⊗ G_nA)t̂ ≠ A · J · ẑ"); @@ -779,6 +876,6 @@ mod tests { assert_eq!(hint.t_hat.len(), NUM_BLOCKS); assert!(hint.t_hat.iter().all(|v| v.len() == N_A * DELTA)); - assert_eq!(f.quad_eq.z_hat().unwrap().len(), BLOCK_LEN * DELTA * TAU); + assert_eq!(f.quad_eq.z_pre().unwrap().len(), BLOCK_LEN * DELTA); } } diff --git a/src/protocol/ring_switch.rs b/src/protocol/ring_switch.rs index a5c40ca5..ef87ed0e 100644 --- a/src/protocol/ring_switch.rs +++ b/src/protocol/ring_switch.rs @@ -4,39 +4,42 @@ //! sumcheck instances by expanding the ring elements into their coefficient //! vectors and setting up the evaluation tables. -use crate::algebra::ring::CyclotomicRing; +use crate::algebra::CyclotomicRing; +use crate::cfg_into_iter; use crate::error::HachiError; #[cfg(feature = "parallel")] use crate::parallel::*; use crate::protocol::commitment::utils::norm::detect_field_modulus; use crate::protocol::commitment::{ - CommitmentConfig, HachiCommitmentCore, HachiCommitmentLayout, RingCommitment, - RingCommitmentScheme, + CommitmentConfig, HachiCommitmentCore, HachiCommitmentLayout, HachiExpandedSetup, + RingCommitment, RingCommitmentScheme, +}; +use crate::protocol::quadratic_equation::{ + compute_m_a_streaming, compute_r_split_eq, QuadraticEquation, }; -use crate::protocol::quadratic_equation::QuadraticEquation; use crate::protocol::sumcheck::eq_poly::EqPolynomial; use crate::protocol::transcript::labels::{ ABSORB_SUMCHECK_W, CHALLENGE_RING_SWITCH, CHALLENGE_TAU0, CHALLENGE_TAU1, }; use crate::protocol::transcript::Transcript; -use crate::{cfg_into_iter, cfg_iter}; use crate::{CanonicalField, FieldCore, FieldSampling}; -/// Output of the ring switch prover, containing everything needed for sumchecks. +/// Output of the ring switch protocol, containing everything needed for sumchecks. pub struct RingSwitchOutput { /// The witness vector w (concatenation of z and r coefficients). pub w: Vec, /// Commitment to w. pub w_commitment: RingCommitment, - /// Evaluation table of w (reordered for sumcheck). - pub w_evals: Vec, + /// Compact evaluation table of w (all entries in [-8, 7], reordered for sumcheck). + /// Populated by the prover; empty on the verifier side. + pub w_evals: Vec, /// Evaluation table of M_alpha(x) (tau1-weighted). pub m_evals_x: Vec, /// Evaluation table of alpha powers (y dimension). pub alpha_evals_y: Vec, - /// Number of upper variable bits in w_evals. + /// Number of upper variable bits. pub num_u: usize, - /// Number of lower variable bits in w_evals. + /// Number of lower variable bits. pub num_l: usize, /// Challenge tau0 for F_0 sumcheck. pub tau0: Vec, @@ -52,9 +55,10 @@ pub struct RingSwitchOutput { /// /// # Errors /// -/// Returns an error if z is missing, commitment fails, or matrix expansion fails. +/// Returns an error if z_pre/w_hat is missing, commitment fails, or matrix expansion fails. pub fn ring_switch_prover( - quad_eq: &QuadraticEquation, + quad_eq: &mut QuadraticEquation, + setup: &HachiExpandedSetup, transcript: &mut T, ) -> Result, HachiError> where @@ -62,18 +66,39 @@ where T: Transcript, Cfg: CommitmentConfig, { - let z = quad_eq - .z() - .ok_or_else(|| HachiError::InvalidInput("missing z in prover".to_string()))?; - let r = compute_r_via_poly_division::(quad_eq.m(), z, quad_eq.y())?; - let w = build_w_coeffs::(z, &r); + let w_hat = quad_eq + .w_hat() + .ok_or_else(|| HachiError::InvalidInput("missing w_hat in prover".to_string()))?; + let z_pre = quad_eq + .z_pre() + .ok_or_else(|| HachiError::InvalidInput("missing z_pre in prover".to_string()))?; + let hint = quad_eq + .hint() + .ok_or_else(|| HachiError::InvalidInput("missing hint in prover".to_string()))?; + let t_hat = &hint.t_hat; + + let r = compute_r_split_eq::( + setup, + quad_eq.opening_point(), + &quad_eq.challenges, + w_hat, + t_hat, + z_pre, + quad_eq.y(), + )?; + let w = build_w_coeffs::(w_hat, t_hat, z_pre, &r); let w_commitment = commit_w::(&w)?; transcript.append_serde(ABSORB_SUMCHECK_W, &w_commitment); let alpha: F = transcript.challenge_scalar(CHALLENGE_RING_SWITCH); - let m_a = eval_ring_matrix_at::(quad_eq.m(), &alpha); + let m_a = compute_m_a_streaming::( + setup, + quad_eq.opening_point(), + &quad_eq.challenges, + &alpha, + )?; let m_a_vec = expand_m_a::(&m_a, alpha)?; let m_rows = m_row_count::(); let m_cols = if m_a.is_empty() { @@ -82,7 +107,7 @@ where m_a_vec.len() / m_a.len() }; - let (w_evals, num_u, num_l) = build_w_evals(&w, D)?; + let (w_evals, num_u, num_l) = build_w_evals_compact::(&w, D)?; let alpha_evals_y = build_alpha_evals_y(alpha, D); let num_sc_vars = num_u + num_l; @@ -115,6 +140,7 @@ where /// Returns an error if matrix expansion fails. pub fn ring_switch_verifier( quad_eq: &QuadraticEquation, + setup: &HachiExpandedSetup, w: &[F], w_commitment: &RingCommitment, transcript: &mut T, @@ -128,7 +154,12 @@ where let alpha: F = transcript.challenge_scalar(CHALLENGE_RING_SWITCH); - let m_a = eval_ring_matrix_at::(quad_eq.m(), &alpha); + let m_a = compute_m_a_streaming::( + setup, + quad_eq.opening_point(), + &quad_eq.challenges, + &alpha, + )?; let m_a_vec = expand_m_a::(&m_a, alpha)?; let m_rows = m_row_count::(); let m_cols = if m_a.is_empty() { @@ -137,7 +168,9 @@ where m_a_vec.len() / m_a.len() }; - let (w_evals, num_u, num_l) = build_w_evals(w, D)?; + let num_ring_elems = w.len() / D; + let num_u = num_ring_elems.next_power_of_two().trailing_zeros() as usize; + let num_l = D.trailing_zeros() as usize; let alpha_evals_y = build_alpha_evals_y(alpha, D); let num_sc_vars = num_u + num_l; @@ -151,7 +184,7 @@ where Ok(RingSwitchOutput { w: w.to_vec(), w_commitment: w_commitment.clone(), - w_evals, + w_evals: Vec::new(), m_evals_x, alpha_evals_y, num_u, @@ -163,6 +196,7 @@ where }) } +#[cfg(test)] pub(crate) fn compute_r_via_poly_division( m: &[Vec>], z: &[CyclotomicRing], @@ -305,20 +339,6 @@ pub(crate) fn eval_ring_at(r: &CyclotomicRing( - v: &[CyclotomicRing], - alpha: &F, -) -> Vec { - cfg_iter!(v).map(|r| eval_ring_at(r, alpha)).collect() -} - -pub(crate) fn eval_ring_matrix_at( - m: &[Vec>], - alpha: &F, -) -> Vec> { - m.iter().map(|row| eval_ring_vec_at(row, alpha)).collect() -} - pub(crate) fn r_decomp_levels() -> usize { let modulus = detect_field_modulus::(); let bits = 128 - (modulus.saturating_sub(1)).leading_zeros() as usize; @@ -433,6 +453,50 @@ pub(crate) fn build_w_evals( Ok((evals, num_u, num_l)) } +/// Compact variant of `build_w_evals` returning `Vec`. +/// +/// All entries in `w` must have canonical values in `[-half_b, half_b - 1]` +/// where `half_b = 2^(LOG_BASIS-1)`. This holds when `w` is produced by +/// `build_w_coeffs` (all components go through `balanced_decompose_pow2`). +pub(crate) fn build_w_evals_compact( + w: &[F], + d: usize, +) -> Result<(Vec, usize, usize), HachiError> { + if d == 0 || w.len() % d != 0 { + return Err(HachiError::InvalidSize { + expected: d, + actual: w.len(), + }); + } + let num_l = d.trailing_zeros() as usize; + let num_ring_elems = w.len() / d; + let num_u = num_ring_elems.next_power_of_two().trailing_zeros() as usize; + let x_len = 1usize << num_u; + let n = x_len << num_l; + + let q = (-F::one()).to_canonical_u128() + 1; + let half_q = q / 2; + + let evals: Vec = (0..n) + .map(|dst| { + let x = dst & (x_len - 1); + let y = dst >> num_u; + let src = y + (x << num_l); + if src < w.len() { + let canonical = w[src].to_canonical_u128(); + if canonical <= half_q { + canonical as i8 + } else { + (canonical as i128 - q as i128) as i8 + } + } else { + 0i8 + } + }) + .collect(); + Ok((evals, num_u, num_l)) +} + pub(crate) fn m_row_count() -> usize { Cfg::N_D + Cfg::N_B + 1 + 1 + Cfg::N_A } @@ -480,7 +544,9 @@ pub(crate) fn sample_tau>( } pub(crate) fn build_w_coeffs( - z: &[CyclotomicRing], + w_hat: &[Vec>], + t_hat: &[Vec>], + z_pre: &[CyclotomicRing], r: &[CyclotomicRing], ) -> Vec { let levels = r_decomp_levels::(); @@ -489,8 +555,21 @@ pub(crate) fn build_w_coeffs() + + t_hat.iter().map(|v| v.len()).sum::() + + z_pre.len() * Cfg::TAU; + let mut out = Vec::with_capacity((z_count + r_hat.len()) * D); + for elem in w_hat_flat + .chain(t_hat_flat) + .chain(z_hat_iter.collect::>().iter()) + .chain(r_hat.iter()) + { out.extend_from_slice(elem.coefficients()); } out diff --git a/src/protocol/sumcheck/hachi_sumcheck.rs b/src/protocol/sumcheck/hachi_sumcheck.rs new file mode 100644 index 00000000..f7e1b12a --- /dev/null +++ b/src/protocol/sumcheck/hachi_sumcheck.rs @@ -0,0 +1,548 @@ +//! Fused norm+relation sumcheck prover/verifier for the Hachi PCS. +//! +//! Eliminates the redundant `w_evals` clone by sharing a single `w_table` +//! across both the norm (F_0) and relation (F_α) sumcheck computations. +//! Supports compact `Vec` storage for round 0 (all entries in [-8, 7]), +//! transitioning to `Vec` at half size after the first fold. + +use super::eq_poly::EqPolynomial; +use super::norm_sumcheck::{ + accumulate_affine_range_coeffs, range_check_eval_precomputed, trim_trailing_zeros, + NormRoundKernel, PointEvalPrecomp, RangeAffinePrecomp, +}; +use super::split_eq::GruenSplitEq; +use super::{fold_evals_in_place, multilinear_eval, range_check_eval}; +use super::{SumcheckInstanceProver, SumcheckInstanceVerifier, UniPoly}; +use crate::algebra::CyclotomicRing; +use crate::cfg_into_iter; +use crate::error::HachiError; +#[cfg(feature = "parallel")] +use crate::parallel::*; +use crate::protocol::ring_switch::eval_ring_at; +use crate::{FieldCore, FromSmallInt}; + +enum WTable { + Compact(Vec), + Full(Vec), +} + +/// Fused norm+relation sumcheck prover. +/// +/// Holds a single `w_table` shared by both sumcheck instances, weighted +/// by `batching_coeff`. The round polynomial is +/// `batching_coeff * norm_round(t) + relation_round(t)`. +pub struct HachiSumcheckProver { + w_table: WTable, + batching_coeff: E, + + // Norm state + split_eq: GruenSplitEq, + round_kernel: NormRoundKernel, + point_precomp: Option>, + range_precomp: Option>, + b: usize, + + // Relation state + alpha_table: Vec, + m_table: Vec, + + num_vars: usize, + relation_claim: E, +} + +impl HachiSumcheckProver { + /// Create a fused norm+relation sumcheck prover. + /// + /// # Panics + /// + /// Panics if table sizes are inconsistent with `num_u` and `num_l`. + #[allow(clippy::too_many_arguments)] + pub fn new( + batching_coeff: E, + w_evals_compact: Vec, + tau0: &[E], + b: usize, + alpha_evals_y: &[E], + m_evals_x: &[E], + num_u: usize, + num_l: usize, + ) -> Self { + let num_vars = num_u + num_l; + let n = 1usize << num_vars; + assert_eq!(w_evals_compact.len(), n); + assert_eq!(tau0.len(), num_vars); + assert_eq!(alpha_evals_y.len(), 1 << num_l); + assert_eq!(m_evals_x.len(), 1 << num_u); + + let x_mask = (1usize << num_u) - 1; + let alpha_table: Vec = cfg_into_iter!(0..n) + .map(|idx| alpha_evals_y[idx >> num_u]) + .collect(); + let m_table: Vec = cfg_into_iter!(0..n) + .map(|idx| m_evals_x[idx & x_mask]) + .collect(); + + let relation_claim = + Self::compute_relation_claim_compact(&w_evals_compact, &alpha_table, &m_table); + + let round_kernel = if b <= 8 { + NormRoundKernel::PointEvalInterpolation + } else { + NormRoundKernel::AffineCoeffComposition + }; + let point_precomp = match round_kernel { + NormRoundKernel::PointEvalInterpolation => Some(PointEvalPrecomp::new(b)), + NormRoundKernel::AffineCoeffComposition => None, + }; + let range_precomp = match round_kernel { + NormRoundKernel::PointEvalInterpolation => None, + NormRoundKernel::AffineCoeffComposition => Some(RangeAffinePrecomp::new(b)), + }; + + Self { + w_table: WTable::Compact(w_evals_compact), + batching_coeff, + split_eq: GruenSplitEq::new(tau0), + round_kernel, + point_precomp, + range_precomp, + b, + alpha_table, + m_table, + num_vars, + relation_claim, + } + } + + fn compute_relation_claim_compact(w_compact: &[i8], alpha_table: &[E], m_table: &[E]) -> E { + w_compact + .iter() + .zip(alpha_table.iter()) + .zip(m_table.iter()) + .fold(E::zero(), |acc, ((&w, &a), &m)| { + acc + E::from_i64(w as i64) * a * m + }) + } + + fn lift_i8(v: i8) -> E { + E::from_i64(v as i64) + } + + fn compute_round_norm_compact(&self, w_compact: &[i8]) -> UniPoly { + let half = w_compact.len() / 2; + + let (e_first, e_second) = self.split_eq.remaining_eq_tables(); + let num_first = e_first.len(); + let first_bits = num_first.trailing_zeros(); + + match self.round_kernel { + NormRoundKernel::PointEvalInterpolation => { + let degree_q = 2 * self.b - 1; + let num_points_q = degree_q + 1; + let range_offsets = &self.point_precomp.as_ref().unwrap().range_offsets; + + let mut q_evals = vec![E::zero(); num_points_q]; + for j in 0..half { + let j_low = j & (num_first - 1); + let j_high = j >> first_bits; + let eq_rem = e_first[j_low] * e_second[j_high]; + let w_0 = Self::lift_i8(w_compact[2 * j]); + let w_1 = Self::lift_i8(w_compact[2 * j + 1]); + let delta = w_1 - w_0; + let mut w_t = w_0; + for eval in q_evals.iter_mut() { + *eval = *eval + eq_rem * range_check_eval_precomputed(w_t, range_offsets); + w_t = w_t + delta; + } + } + let q_poly = UniPoly::from_evals(&q_evals); + self.split_eq.gruen_mul(&q_poly) + } + NormRoundKernel::AffineCoeffComposition => { + let range_precomp = self.range_precomp.as_ref().unwrap(); + let num_coeffs_q = range_precomp.degree_q + 1; + let coeff_mix = &range_precomp.coeff_mix; + + let mut q_coeffs = vec![E::zero(); num_coeffs_q]; + for j in 0..half { + let j_low = j & (num_first - 1); + let j_high = j >> first_bits; + let eq_rem = e_first[j_low] * e_second[j_high]; + let w_0 = Self::lift_i8(w_compact[2 * j]); + let w_1 = Self::lift_i8(w_compact[2 * j + 1]); + let a = w_1 - w_0; + accumulate_affine_range_coeffs(&mut q_coeffs, coeff_mix, w_0, a, eq_rem); + } + trim_trailing_zeros(&mut q_coeffs); + let q_poly = UniPoly::from_coeffs(q_coeffs); + self.split_eq.gruen_mul(&q_poly) + } + } + } + + fn compute_round_relation_compact(&self, w_compact: &[i8]) -> UniPoly { + let half = w_compact.len() / 2; + let num_points = 3; + + let mut evals = vec![E::zero(); num_points]; + for j in 0..half { + let w_0 = Self::lift_i8(w_compact[2 * j]); + let w_1 = Self::lift_i8(w_compact[2 * j + 1]); + let a_0 = self.alpha_table[2 * j]; + let a_1 = self.alpha_table[2 * j + 1]; + let m_0 = self.m_table[2 * j]; + let m_1 = self.m_table[2 * j + 1]; + for (t, eval) in evals.iter_mut().enumerate() { + let t_e = E::from_u64(t as u64); + let w_t = w_0 + t_e * (w_1 - w_0); + let a_t = a_0 + t_e * (a_1 - a_0); + let m_t = m_0 + t_e * (m_1 - m_0); + *eval = *eval + w_t * a_t * m_t; + } + } + UniPoly::from_evals(&evals) + } + + fn compute_round_norm_full(&self, w_full: &[E]) -> UniPoly { + let half = w_full.len() / 2; + + let (e_first, e_second) = self.split_eq.remaining_eq_tables(); + let num_first = e_first.len(); + let first_bits = num_first.trailing_zeros(); + + match self.round_kernel { + NormRoundKernel::PointEvalInterpolation => { + let degree_q = 2 * self.b - 1; + let num_points_q = degree_q + 1; + let range_offsets = &self.point_precomp.as_ref().unwrap().range_offsets; + + #[cfg(feature = "parallel")] + let q_evals = { + (0..half) + .into_par_iter() + .fold( + || vec![E::zero(); num_points_q], + |mut evals, j| { + let j_low = j & (num_first - 1); + let j_high = j >> first_bits; + let eq_rem = e_first[j_low] * e_second[j_high]; + let w_0 = w_full[2 * j]; + let w_1 = w_full[2 * j + 1]; + let delta = w_1 - w_0; + let mut w_t = w_0; + for eval in evals.iter_mut() { + *eval = *eval + + eq_rem * range_check_eval_precomputed(w_t, range_offsets); + w_t = w_t + delta; + } + evals + }, + ) + .reduce( + || vec![E::zero(); num_points_q], + |mut a, b_vec| { + for (ai, bi) in a.iter_mut().zip(b_vec.iter()) { + *ai = *ai + *bi; + } + a + }, + ) + }; + #[cfg(not(feature = "parallel"))] + let q_evals = { + let mut evals = vec![E::zero(); num_points_q]; + for j in 0..half { + let j_low = j & (num_first - 1); + let j_high = j >> first_bits; + let eq_rem = e_first[j_low] * e_second[j_high]; + let w_0 = w_full[2 * j]; + let w_1 = w_full[2 * j + 1]; + let delta = w_1 - w_0; + let mut w_t = w_0; + for eval in evals.iter_mut() { + *eval = + *eval + eq_rem * range_check_eval_precomputed(w_t, range_offsets); + w_t = w_t + delta; + } + } + evals + }; + + let q_poly = UniPoly::from_evals(&q_evals); + self.split_eq.gruen_mul(&q_poly) + } + NormRoundKernel::AffineCoeffComposition => { + let range_precomp = self.range_precomp.as_ref().unwrap(); + let num_coeffs_q = range_precomp.degree_q + 1; + let coeff_mix = &range_precomp.coeff_mix; + + #[cfg(feature = "parallel")] + let mut q_coeffs = { + (0..half) + .into_par_iter() + .fold( + || vec![E::zero(); num_coeffs_q], + |mut coeffs, j| { + let j_low = j & (num_first - 1); + let j_high = j >> first_bits; + let eq_rem = e_first[j_low] * e_second[j_high]; + let w_0 = w_full[2 * j]; + let w_1 = w_full[2 * j + 1]; + let a = w_1 - w_0; + accumulate_affine_range_coeffs( + &mut coeffs, + coeff_mix, + w_0, + a, + eq_rem, + ); + coeffs + }, + ) + .reduce( + || vec![E::zero(); num_coeffs_q], + |mut a, b_vec| { + for (ai, bi) in a.iter_mut().zip(b_vec.iter()) { + *ai = *ai + *bi; + } + a + }, + ) + }; + #[cfg(not(feature = "parallel"))] + let mut q_coeffs = { + let mut coeffs = vec![E::zero(); num_coeffs_q]; + for j in 0..half { + let j_low = j & (num_first - 1); + let j_high = j >> first_bits; + let eq_rem = e_first[j_low] * e_second[j_high]; + let w_0 = w_full[2 * j]; + let w_1 = w_full[2 * j + 1]; + let a = w_1 - w_0; + accumulate_affine_range_coeffs(&mut coeffs, coeff_mix, w_0, a, eq_rem); + } + coeffs + }; + + trim_trailing_zeros(&mut q_coeffs); + let q_poly = UniPoly::from_coeffs(q_coeffs); + self.split_eq.gruen_mul(&q_poly) + } + } + } + + fn compute_round_relation_full(&self, w_full: &[E]) -> UniPoly { + let half = w_full.len() / 2; + let num_points = 3; + + #[cfg(feature = "parallel")] + let round_evals = { + (0..half) + .into_par_iter() + .fold( + || vec![E::zero(); num_points], + |mut evals, j| { + let w_0 = w_full[2 * j]; + let w_1 = w_full[2 * j + 1]; + let a_0 = self.alpha_table[2 * j]; + let a_1 = self.alpha_table[2 * j + 1]; + let m_0 = self.m_table[2 * j]; + let m_1 = self.m_table[2 * j + 1]; + for (t, eval) in evals.iter_mut().enumerate() { + let t_e = E::from_u64(t as u64); + let w_t = w_0 + t_e * (w_1 - w_0); + let a_t = a_0 + t_e * (a_1 - a_0); + let m_t = m_0 + t_e * (m_1 - m_0); + *eval = *eval + w_t * a_t * m_t; + } + evals + }, + ) + .reduce( + || vec![E::zero(); num_points], + |mut a, b| { + for (ai, bi) in a.iter_mut().zip(b.iter()) { + *ai = *ai + *bi; + } + a + }, + ) + }; + #[cfg(not(feature = "parallel"))] + let round_evals = { + let mut evals = vec![E::zero(); num_points]; + for j in 0..half { + let w_0 = w_full[2 * j]; + let w_1 = w_full[2 * j + 1]; + let a_0 = self.alpha_table[2 * j]; + let a_1 = self.alpha_table[2 * j + 1]; + let m_0 = self.m_table[2 * j]; + let m_1 = self.m_table[2 * j + 1]; + for (t, eval) in evals.iter_mut().enumerate() { + let t_e = E::from_u64(t as u64); + let w_t = w_0 + t_e * (w_1 - w_0); + let a_t = a_0 + t_e * (a_1 - a_0); + let m_t = m_0 + t_e * (m_1 - m_0); + *eval = *eval + w_t * a_t * m_t; + } + } + evals + }; + + UniPoly::from_evals(&round_evals) + } + + fn fold_compact_to_full(w_compact: &[i8], r: E) -> Vec { + let half = w_compact.len() / 2; + let mut out = Vec::with_capacity(half); + for j in 0..half { + let w_0 = Self::lift_i8(w_compact[2 * j]); + let w_1 = Self::lift_i8(w_compact[2 * j + 1]); + out.push(w_0 + r * (w_1 - w_0)); + } + out + } +} + +impl SumcheckInstanceProver for HachiSumcheckProver { + fn num_rounds(&self) -> usize { + self.num_vars + } + + fn degree_bound(&self) -> usize { + 2 * self.b + } + + fn input_claim(&self) -> E { + self.relation_claim + } + + fn compute_round_univariate(&mut self, _round: usize, _previous_claim: E) -> UniPoly { + let (norm_poly, relation_poly) = match &self.w_table { + WTable::Compact(w_compact) => { + let norm = self.compute_round_norm_compact(w_compact); + let relation = self.compute_round_relation_compact(w_compact); + (norm, relation) + } + WTable::Full(w_full) => { + let norm = self.compute_round_norm_full(w_full); + let relation = self.compute_round_relation_full(w_full); + (norm, relation) + } + }; + + let max_len = norm_poly.coeffs.len().max(relation_poly.coeffs.len()); + let mut combined = vec![E::zero(); max_len]; + for (i, c) in norm_poly.coeffs.iter().enumerate() { + combined[i] = combined[i] + self.batching_coeff * *c; + } + for (i, c) in relation_poly.coeffs.iter().enumerate() { + combined[i] = combined[i] + *c; + } + UniPoly::from_coeffs(combined) + } + + fn ingest_challenge(&mut self, _round: usize, r: E) { + self.split_eq.bind(r); + + self.w_table = match std::mem::replace(&mut self.w_table, WTable::Full(Vec::new())) { + WTable::Compact(w_compact) => WTable::Full(Self::fold_compact_to_full(&w_compact, r)), + WTable::Full(mut w_full) => { + fold_evals_in_place(&mut w_full, r); + WTable::Full(w_full) + } + }; + + fold_evals_in_place(&mut self.alpha_table, r); + fold_evals_in_place(&mut self.m_table, r); + } +} + +/// Fused norm+relation sumcheck verifier. +pub struct HachiSumcheckVerifier { + batching_coeff: F, + w_evals: Vec, + tau0: Vec, + b: usize, + alpha_evals_y: Vec, + m_evals_x: Vec, + num_u: usize, + num_l: usize, + relation_claim: F, + _marker: std::marker::PhantomData<[F; D]>, +} + +impl HachiSumcheckVerifier { + /// Create a fused verifier for the norm + relation sumcheck. + #[allow(clippy::too_many_arguments)] + pub fn new( + batching_coeff: F, + w_evals: Vec, + tau0: Vec, + b: usize, + alpha_evals_y: Vec, + m_evals_x: Vec, + tau1: Vec, + v: Vec>, + u: Vec>, + y_ring: CyclotomicRing, + alpha: F, + num_u: usize, + num_l: usize, + ) -> Self { + let y_a: Vec = v + .iter() + .chain(u.iter()) + .chain(std::iter::once(&y_ring)) + .map(|r| eval_ring_at(r, &alpha)) + .collect(); + let eq_tau1 = EqPolynomial::evals(&tau1); + let mut relation_claim = F::zero(); + for (i, eq_i) in eq_tau1.iter().enumerate() { + let y_i = if i < y_a.len() { y_a[i] } else { F::zero() }; + relation_claim = relation_claim + *eq_i * y_i; + } + + Self { + batching_coeff, + w_evals, + tau0, + b, + alpha_evals_y, + m_evals_x, + num_u, + num_l, + relation_claim, + _marker: std::marker::PhantomData, + } + } +} + +impl SumcheckInstanceVerifier + for HachiSumcheckVerifier +{ + fn num_rounds(&self) -> usize { + self.num_u + self.num_l + } + + fn degree_bound(&self) -> usize { + 2 * self.b + } + + fn input_claim(&self) -> F { + self.relation_claim + } + + fn expected_output_claim(&self, challenges: &[F]) -> Result { + let eq_val = EqPolynomial::mle(&self.tau0, challenges); + let w_val = multilinear_eval(&self.w_evals, challenges)?; + let norm_oracle = eq_val * range_check_eval(w_val, self.b); + + let (x_challenges, y_challenges) = challenges.split_at(self.num_u); + let alpha_val = multilinear_eval(&self.alpha_evals_y, y_challenges)?; + let m_val = multilinear_eval(&self.m_evals_x, x_challenges)?; + let relation_oracle = w_val * alpha_val * m_val; + + Ok(self.batching_coeff * norm_oracle + relation_oracle) + } +} diff --git a/src/protocol/sumcheck/mod.rs b/src/protocol/sumcheck/mod.rs index da11a71c..dc824bba 100644 --- a/src/protocol/sumcheck/mod.rs +++ b/src/protocol/sumcheck/mod.rs @@ -14,6 +14,7 @@ pub mod batched_sumcheck; pub mod eq_poly; +pub mod hachi_sumcheck; pub mod norm_sumcheck; pub mod relation_sumcheck; pub mod split_eq; diff --git a/src/protocol/sumcheck/norm_sumcheck.rs b/src/protocol/sumcheck/norm_sumcheck.rs index 5eb493fb..db4625cf 100644 --- a/src/protocol/sumcheck/norm_sumcheck.rs +++ b/src/protocol/sumcheck/norm_sumcheck.rs @@ -17,7 +17,7 @@ use crate::{FieldCore, FromSmallInt}; const SMALL_B_POINT_EVAL_MAX: usize = 8; #[derive(Clone, Copy, Debug, PartialEq, Eq)] -enum NormRoundKernel { +pub(crate) enum NormRoundKernel { PointEvalInterpolation, AffineCoeffComposition, } @@ -31,15 +31,15 @@ fn choose_round_kernel(b: usize) -> NormRoundKernel { } #[derive(Clone)] -struct RangeAffinePrecomp { +pub(crate) struct RangeAffinePrecomp { /// `coeff_mix[i][k] = c_{i+k} * binom(i+k, i)`, where /// `R(w) = sum_m c_m * w^m` is the range-check polynomial. - coeff_mix: Vec>, - degree_q: usize, + pub(crate) coeff_mix: Vec>, + pub(crate) degree_q: usize, } impl RangeAffinePrecomp { - fn new(b: usize) -> Self { + pub(crate) fn new(b: usize) -> Self { assert!(b >= 1, "b must be at least 1"); let range_coeffs = range_check_coeffs::(b); let degree_q = range_coeffs.len() - 1; @@ -81,12 +81,12 @@ impl RangeAffinePrecomp { } #[derive(Clone)] -struct PointEvalPrecomp { - range_offsets: Vec, +pub(crate) struct PointEvalPrecomp { + pub(crate) range_offsets: Vec, } impl PointEvalPrecomp { - fn new(b: usize) -> Self { + pub(crate) fn new(b: usize) -> Self { let range_offsets = (1..b).map(|k| E::from_u64(k as u64)).collect(); Self { range_offsets } } @@ -110,7 +110,7 @@ fn range_check_coeffs(b: usize) -> Vec { coeffs } -fn range_check_eval_precomputed(w: E, range_offsets: &[E]) -> E { +pub(crate) fn range_check_eval_precomputed(w: E, range_offsets: &[E]) -> E { let mut acc = w; for &k in range_offsets { acc = acc * (w - k) * (w + k); @@ -118,7 +118,7 @@ fn range_check_eval_precomputed(w: E, range_offsets: &[E]) -> E { acc } -fn accumulate_affine_range_coeffs( +pub(crate) fn accumulate_affine_range_coeffs( out_coeffs: &mut [E], coeff_mix: &[Vec], w_0: E, @@ -136,7 +136,7 @@ fn accumulate_affine_range_coeffs( } } -fn trim_trailing_zeros(coeffs: &mut Vec) { +pub(crate) fn trim_trailing_zeros(coeffs: &mut Vec) { while coeffs.len() > 1 && coeffs.last().is_some_and(|c| c.is_zero()) { coeffs.pop(); } @@ -405,7 +405,7 @@ mod tests { use crate::algebra::ring::CyclotomicRing; use crate::algebra::Fp64; use crate::primitives::multilinear_evals::DenseMultilinearEvals; - use crate::protocol::ring_switch::build_w_coeffs; + use crate::protocol::ring_switch::r_decomp_levels; use crate::protocol::sumcheck::eq_poly::EqPolynomial; use crate::protocol::sumcheck::multilinear_eval; use crate::protocol::transcript::labels; @@ -596,13 +596,22 @@ mod tests { #[test] fn norm_sumcheck_uses_commitment_w_evals() { - let z = vec![ + let z = [ ring_with_small_coeff(1), ring_with_small_coeff(2), ring_with_small_coeff(3), ]; - let r = vec![ring_with_small_coeff(0), ring_with_small_coeff(1)]; - let mut w_evals = build_w_coeffs::(&z, &r); + let r = [ring_with_small_coeff(0), ring_with_small_coeff(1)]; + let levels = r_decomp_levels::(); + let r_hat: Vec> = r + .iter() + .flat_map(|ri| ri.balanced_decompose_pow2(levels, SmallTestCommitmentConfig::LOG_BASIS)) + .collect(); + let mut w_evals: Vec = z + .iter() + .chain(r_hat.iter()) + .flat_map(|elem| elem.coefficients().iter().copied()) + .collect(); let target_len = w_evals.len().next_power_of_two(); w_evals.resize(target_len, F::zero()); From 6d3dd1433daa8bd96ea01fabd3082d6b7533336e Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Mon, 2 Mar 2026 09:50:48 -0800 Subject: [PATCH 33/34] revert: remove ensure_large_thread_stack rayon config Stack sizing for D>=512 ring elements should be handled by the caller, not baked into the library's setup path. Made-with: Cursor --- src/lib.rs | 20 -------------------- src/protocol/commitment/commit.rs | 2 -- 2 files changed, 22 deletions(-) diff --git a/src/lib.rs b/src/lib.rs index 1fd4ea6f..4d6635ef 100644 --- a/src/lib.rs +++ b/src/lib.rs @@ -61,23 +61,3 @@ pub use primitives::arithmetic::{ pub use primitives::poly::{MultilinearLagrange, Polynomial}; pub use primitives::serialization::{HachiDeserialize, HachiSerialize}; pub use protocol::{CommitmentScheme, StreamingCommitmentScheme, Transcript}; - -/// Minimum rayon thread stack size for large ring degrees (D >= 512). -/// -/// CRT-NTT conversion puts `[[MontCoeff; D]; K]` on the stack per ring element, -/// which at D=512, K=5 is ~20 KB per frame. Rayon's default thread stack is -/// too small for the call depth commit → decompose → mat-vec → NTT. -const MIN_THREAD_STACK: usize = 64 * 1024 * 1024; - -/// Install the global rayon thread pool with a stack large enough for D=512+. -/// -/// Safe to call multiple times — only the first call configures the pool. -/// Subsequent calls are silently ignored. -pub fn ensure_large_thread_stack() { - #[cfg(feature = "parallel")] - { - let _ = rayon::ThreadPoolBuilder::new() - .stack_size(MIN_THREAD_STACK) - .build_global(); - } -} diff --git a/src/protocol/commitment/commit.rs b/src/protocol/commitment/commit.rs index c926421a..97df3007 100644 --- a/src/protocol/commitment/commit.rs +++ b/src/protocol/commitment/commit.rs @@ -272,7 +272,6 @@ where type Commitment = RingCommitment; fn setup(max_num_vars: usize) -> Result<(Self::ProverSetup, Self::VerifierSetup), HachiError> { - crate::ensure_large_thread_stack(); let layout = validate_and_derive_layout::(max_num_vars)?; ensure_supported_num_vars(max_num_vars, layout.required_num_vars::()?)?; let public_matrix_seed = sample_public_matrix_seed(); @@ -460,7 +459,6 @@ impl HachiCommitmentCore { F: FieldCore + CanonicalField + FieldSampling, Cfg: CommitmentConfig, { - crate::ensure_large_thread_stack(); let max_num_vars = layout.required_num_vars::()?; let public_matrix_seed = sample_public_matrix_seed(); let a_matrix = From a4af82a4ddfce889346a667d254517f0d394649e Mon Sep 17 00:00:00 2001 From: Quang Dao Date: Mon, 2 Mar 2026 10:56:59 -0800 Subject: [PATCH 34/34] perf: parallelize commit phase and reduce allocations - Add block-level parallelism to commit_ring_blocks, commit_coeffs, commit_onehot, and commit_mixed via cfg_iter!/cfg_into_iter! - Parallelize vector-to-NTT conversion in mat_vec_mul_precomputed_with_params - Cache CRT+NTT params inside NttMatrixCache, eliminating redundant select_crt_ntt_params calls on every mat-vec multiply - Add balanced_decompose_pow2_into for in-place decomposition, removing per-element Vec allocations in decompose_block/decompose_rows - Add inner_ajtai_onehot_t_only that skips the 16MB s-vector allocation when the caller discards it (commit_onehot, commit_mixed) - Add one-hot and mixed commitment benchmarks to hachi_e2e Made-with: Cursor --- benches/hachi_e2e.rs | 337 ++++++++++++++++++++++- src/algebra/ntt/butterfly.rs | 1 + src/algebra/ring/crt_ntt_repr.rs | 7 +- src/algebra/ring/cyclotomic.rs | 51 +++- src/protocol/commitment/commit.rs | 220 ++++++++++----- src/protocol/commitment/onehot.rs | 38 ++- src/protocol/commitment/utils/crt_ntt.rs | 26 +- src/protocol/commitment/utils/linear.rs | 39 ++- 8 files changed, 594 insertions(+), 125 deletions(-) diff --git a/benches/hachi_e2e.rs b/benches/hachi_e2e.rs index d21b2f3f..c9825245 100644 --- a/benches/hachi_e2e.rs +++ b/benches/hachi_e2e.rs @@ -1,13 +1,17 @@ #![allow(missing_docs)] use criterion::{black_box, criterion_group, criterion_main, BenchmarkId, Criterion}; -use hachi_pcs::algebra::Fp128; +use hachi_pcs::algebra::{CyclotomicRing, Fp128}; use hachi_pcs::error::HachiError; use hachi_pcs::primitives::multilinear_evals::DenseMultilinearEvals; -use hachi_pcs::protocol::commitment::{HachiCommitmentLayout, ProductionFp128CommitmentConfig}; -use hachi_pcs::protocol::commitment_scheme::HachiCommitmentScheme; +use hachi_pcs::protocol::commitment::{ + HachiCommitmentCore, HachiCommitmentLayout, HachiProverSetup, HachiVerifierSetup, + MegaPolyBlock, ProductionFp128CommitmentConfig, RingCommitment, SparseBlockEntry, +}; +use hachi_pcs::protocol::commitment_scheme::{commit_onehot, HachiCommitmentScheme}; +use hachi_pcs::protocol::proof::HachiCommitmentHint; use hachi_pcs::protocol::transcript::Blake2bTranscript; -use hachi_pcs::protocol::CommitmentConfig; +use hachi_pcs::protocol::{CommitmentConfig, HachiProof}; use hachi_pcs::{CommitmentScheme, FromSmallInt, Polynomial, Transcript}; use std::time::Duration; @@ -166,6 +170,321 @@ where group.finish(); } +fn bench_onehot_phases(c: &mut Criterion, label: &str) +where + HachiCommitmentScheme: CommitmentScheme< + F, + ProverSetup = HachiProverSetup, + VerifierSetup = HachiVerifierSetup, + Commitment = RingCommitment, + Proof = HachiProof, + OpeningProofHint = HachiCommitmentHint, + >, +{ + type S = HachiCommitmentScheme; + let nv = num_vars::(); + let total_elems = 1usize << nv; + let onehot_k = D; + let num_chunks = total_elems / onehot_k; + + let indices: Vec> = (0..num_chunks).map(|i| Some(i % onehot_k)).collect(); + + let mut evals = vec![F::from_u64(0); total_elems]; + for (ci, opt_idx) in indices.iter().enumerate() { + if let Some(idx) = opt_idx { + evals[ci * onehot_k + idx] = F::from_u64(1); + } + } + let poly = DenseMultilinearEvals::new_padded(evals); + let pt = opening_point(nv); + + let setup = as CommitmentScheme>::setup_prover(nv); + + let mut group = c.benchmark_group(format!("hachi_onehot/{label}/nv{nv}")); + if nv >= 18 { + group.sample_size(10); + group.measurement_time(Duration::from_secs(30)); + } + + group.bench_function("commit", |b| { + b.iter(|| { + black_box( + commit_onehot::( + black_box(onehot_k), + black_box(&indices), + black_box(&setup), + ) + .unwrap(), + ) + }) + }); + + let (commitment, hint) = commit_onehot::(onehot_k, &indices, &setup).unwrap(); + + group.bench_function("prove", |b| { + b.iter(|| { + let mut transcript = Blake2bTranscript::::new(b"bench"); + black_box( + as CommitmentScheme>::prove( + black_box(&setup), + black_box(&poly), + black_box(&pt), + Some(hint.clone()), + &mut transcript, + black_box(&commitment), + ) + .unwrap(), + ) + }) + }); + + let verifier_setup = as CommitmentScheme>::setup_verifier(&setup); + let opening = poly.evaluate(&pt); + let mut prover_transcript = Blake2bTranscript::::new(b"bench"); + let proof = as CommitmentScheme>::prove( + &setup, + &poly, + &pt, + Some(hint.clone()), + &mut prover_transcript, + &commitment, + ) + .unwrap(); + + group.bench_function("verify", |b| { + b.iter(|| { + let mut transcript = Blake2bTranscript::::new(b"bench"); + as CommitmentScheme>::verify( + black_box(&proof), + black_box(&verifier_setup), + &mut transcript, + black_box(&pt), + black_box(&opening), + black_box(&commitment), + ) + .unwrap(); + }) + }); + + group.bench_function(BenchmarkId::new("e2e", nv), |b| { + b.iter(|| { + let (cm, h) = commit_onehot::(onehot_k, &indices, &setup).unwrap(); + let mut pt_tr = Blake2bTranscript::::new(b"bench"); + let pf = as CommitmentScheme>::prove( + &setup, + &poly, + &pt, + Some(h), + &mut pt_tr, + &cm, + ) + .unwrap(); + let mut vt_tr = Blake2bTranscript::::new(b"bench"); + as CommitmentScheme>::verify( + &pf, + &verifier_setup, + &mut vt_tr, + &pt, + &opening, + &cm, + ) + .unwrap(); + black_box(()) + }) + }); + + group.finish(); +} + +fn bench_mixed_phases(c: &mut Criterion, label: &str) +where + HachiCommitmentScheme: CommitmentScheme< + F, + ProverSetup = HachiProverSetup, + VerifierSetup = HachiVerifierSetup, + Commitment = RingCommitment, + Proof = HachiProof, + OpeningProofHint = HachiCommitmentHint, + >, +{ + type S = HachiCommitmentScheme; + let nv = num_vars::(); + let layout = Cfg::commitment_layout(0).expect("benchmark layout"); + let block_len = layout.block_len; + let num_blocks = layout.num_blocks; + let dense_blocks = num_blocks / 2; + + let mut ring_coeffs: Vec> = Vec::with_capacity(num_blocks * block_len); + + for i in 0..(dense_blocks * block_len) { + ring_coeffs.push(CyclotomicRing::from_coefficients(std::array::from_fn( + |j| F::from_u64((i * D + j + 1) as u64), + ))); + } + + let mut sparse_per_block: Vec> = Vec::new(); + for bi in 0..(num_blocks - dense_blocks) { + let mut entries = Vec::new(); + for ri in 0..block_len { + let idx = (bi * block_len + ri) % D; + let mut coeffs = [F::from_u64(0); D]; + coeffs[idx] = F::from_u64(1); + ring_coeffs.push(CyclotomicRing::from_coefficients(coeffs)); + entries.push(SparseBlockEntry { + pos_in_block: ri, + nonzero_coeffs: vec![idx], + }); + } + sparse_per_block.push(entries); + } + + let evals: Vec = ring_coeffs + .iter() + .flat_map(|r| r.coefficients().iter().copied()) + .collect(); + let poly = DenseMultilinearEvals::new_padded(evals); + let pt = opening_point(nv); + + let setup = as CommitmentScheme>::setup_prover(nv); + + let mut group = c.benchmark_group(format!("hachi_mixed/{label}/nv{nv}")); + if nv >= 18 { + group.sample_size(10); + group.measurement_time(Duration::from_secs(30)); + } + + group.bench_function("commit", |b| { + b.iter(|| { + let blocks: Vec> = (0..num_blocks) + .map(|i| { + if i < dense_blocks { + let start = i * block_len; + let end = start + block_len; + MegaPolyBlock::Dense(&ring_coeffs[start..end]) + } else { + MegaPolyBlock::OneHot(&sparse_per_block[i - dense_blocks]) + } + }) + .collect(); + black_box( + HachiCommitmentCore::commit_mixed::( + black_box(&blocks), + black_box(&setup), + ) + .unwrap(), + ) + }) + }); + + let blocks: Vec> = (0..num_blocks) + .map(|i| { + if i < dense_blocks { + let start = i * block_len; + let end = start + block_len; + MegaPolyBlock::Dense(&ring_coeffs[start..end]) + } else { + MegaPolyBlock::OneHot(&sparse_per_block[i - dense_blocks]) + } + }) + .collect(); + let w = HachiCommitmentCore::commit_mixed::(&blocks, &setup).unwrap(); + let commitment = w.commitment; + let hint = HachiCommitmentHint { + t_hat: w.t_hat, + ring_coeffs: ring_coeffs.clone(), + }; + + group.bench_function("prove", |b| { + b.iter(|| { + let mut transcript = Blake2bTranscript::::new(b"bench"); + black_box( + as CommitmentScheme>::prove( + black_box(&setup), + black_box(&poly), + black_box(&pt), + Some(hint.clone()), + &mut transcript, + black_box(&commitment), + ) + .unwrap(), + ) + }) + }); + + let verifier_setup = as CommitmentScheme>::setup_verifier(&setup); + let opening = poly.evaluate(&pt); + let mut prover_transcript = Blake2bTranscript::::new(b"bench"); + let proof = as CommitmentScheme>::prove( + &setup, + &poly, + &pt, + Some(hint.clone()), + &mut prover_transcript, + &commitment, + ) + .unwrap(); + + group.bench_function("verify", |b| { + b.iter(|| { + let mut transcript = Blake2bTranscript::::new(b"bench"); + as CommitmentScheme>::verify( + black_box(&proof), + black_box(&verifier_setup), + &mut transcript, + black_box(&pt), + black_box(&opening), + black_box(&commitment), + ) + .unwrap(); + }) + }); + + group.bench_function(BenchmarkId::new("e2e", nv), |b| { + b.iter(|| { + let blocks: Vec> = (0..num_blocks) + .map(|i| { + if i < dense_blocks { + let start = i * block_len; + let end = start + block_len; + MegaPolyBlock::Dense(&ring_coeffs[start..end]) + } else { + MegaPolyBlock::OneHot(&sparse_per_block[i - dense_blocks]) + } + }) + .collect(); + let w = HachiCommitmentCore::commit_mixed::(&blocks, &setup).unwrap(); + let cm = w.commitment; + let h = HachiCommitmentHint { + t_hat: w.t_hat, + ring_coeffs: ring_coeffs.clone(), + }; + let mut pt_tr = Blake2bTranscript::::new(b"bench"); + let pf = as CommitmentScheme>::prove( + &setup, + &poly, + &pt, + Some(h), + &mut pt_tr, + &cm, + ) + .unwrap(); + let mut vt_tr = Blake2bTranscript::::new(b"bench"); + as CommitmentScheme>::verify( + &pf, + &verifier_setup, + &mut vt_tr, + &pt, + &opening, + &cm, + ) + .unwrap(); + black_box(()) + }) + }); + + group.finish(); +} + fn bench_nv10(c: &mut Criterion) { bench_phases::(c, "fp128_p275"); } @@ -178,12 +497,20 @@ fn bench_nv18(c: &mut Criterion) { fn bench_nv20(c: &mut Criterion) { bench_phases::(c, "fp128_p275"); } +fn bench_onehot_nv14(c: &mut Criterion) { + bench_onehot_phases::(c, "fp128_p275"); +} +fn bench_mixed_nv14(c: &mut Criterion) { + bench_mixed_phases::(c, "fp128_p275"); +} criterion_group!( hachi_benches, bench_nv10, bench_nv14, bench_nv18, - bench_nv20 + bench_nv20, + bench_onehot_nv14, + bench_mixed_nv14, ); criterion_main!(hachi_benches); diff --git a/src/algebra/ntt/butterfly.rs b/src/algebra/ntt/butterfly.rs index fd0f6e53..7380fdf1 100644 --- a/src/algebra/ntt/butterfly.rs +++ b/src/algebra/ntt/butterfly.rs @@ -15,6 +15,7 @@ use super::prime::{MontCoeff, NttPrime, PrimeWidth}; /// Precomputed twiddle factors for a specific prime and degree `D`. /// /// `D` must be a power of two. +#[derive(Debug, Clone, PartialEq, Eq)] pub struct NttTwiddles { /// Stage roots for iterative forward cyclic NTT in Montgomery form. pub(crate) fwd_wlen: [MontCoeff; D], diff --git a/src/algebra/ring/crt_ntt_repr.rs b/src/algebra/ring/crt_ntt_repr.rs index 0d5d5bb1..ad856ac7 100644 --- a/src/algebra/ring/crt_ntt_repr.rs +++ b/src/algebra/ring/crt_ntt_repr.rs @@ -1,5 +1,7 @@ //! CRT+NTT-domain representation of cyclotomic ring elements. +use std::array::from_fn; + use crate::algebra::backend::{CrtReconstruct, NttPrimeOps, NttTransform, ScalarBackend}; use crate::algebra::ntt::butterfly::NttTwiddles; use crate::algebra::ntt::crt::GarnerData; @@ -33,6 +35,7 @@ impl CrtNttConvertibleField for F {} /// /// Keeps primes/twiddles/Garner constants consistent and avoids passing them /// independently at every call site. +#[derive(Debug, Clone, PartialEq, Eq)] pub struct CrtNttParamSet { /// CRT primes with Montgomery constants. pub primes: [NttPrime; K], @@ -47,7 +50,7 @@ impl CrtNttParamSet { /// /// Computes per-prime twiddles and Garner reconstruction constants. pub fn new(primes: [NttPrime; K]) -> Self { - let twiddles = std::array::from_fn(|k| NttTwiddles::compute(primes[k])); + let twiddles = from_fn(|k| NttTwiddles::compute(primes[k])); let garner = GarnerData::compute(&primes); Self { primes, @@ -96,7 +99,7 @@ impl CyclotomicCrtNtt { ) -> Self { let q = (-F::one()).to_canonical_u128() + 1; let half_q = q / 2; - let centered_coeffs: [i128; D] = std::array::from_fn(|i| { + let centered_coeffs: [i128; D] = from_fn(|i| { let canonical = ring.coeffs[i].to_canonical_u128(); if canonical > half_q { -((q - canonical) as i128) diff --git a/src/algebra/ring/cyclotomic.rs b/src/algebra/ring/cyclotomic.rs index 7ee92aa2..a39b49af 100644 --- a/src/algebra/ring/cyclotomic.rs +++ b/src/algebra/ring/cyclotomic.rs @@ -6,6 +6,7 @@ use crate::primitives::serialization::{ }; use crate::{CanonicalField, FieldCore, FieldSampling}; use rand_core::RngCore; +use std::array::from_fn; use std::io::{Read, Write}; use std::ops::{Add, AddAssign, Mul, MulAssign, Neg, Sub, SubAssign}; @@ -201,6 +202,52 @@ impl CyclotomicRing { } impl CyclotomicRing { + /// Balanced decomposition writing directly into a pre-allocated output slice. + /// + /// `out` must have length exactly `levels`. Each element receives one digit plane. + /// + /// # Panics + /// + /// Panics if `log_basis == 0`, `log_basis >= 128`, or `out.len() * log_basis > 128 + log_basis`. + pub fn balanced_decompose_pow2_into(&self, out: &mut [Self], log_basis: u32) { + let levels = out.len(); + assert!(log_basis > 0 && log_basis < 128, "invalid log_basis"); + assert!( + (levels as u32).saturating_mul(log_basis) <= 128 + log_basis, + "levels * log_basis must be <= 128 + log_basis" + ); + + let b = 1i128 << log_basis; + let half_b = b / 2; + let q = (-F::one()).to_canonical_u128() + 1; + let half_q = q / 2; + + for plane in out.iter_mut() { + *plane = Self::zero(); + } + + for i in 0..D { + let canonical = self.coeffs[i].to_canonical_u128(); + let mut c: i128 = if canonical > half_q { + -((q - canonical) as i128) + } else { + canonical as i128 + }; + + for plane in out.iter_mut() { + let d = c.rem_euclid(b); + let balanced = if d >= half_b { d - b } else { d }; + c = (c - balanced) / b; + + plane.coeffs[i] = if balanced >= 0 { + F::from_canonical_u128_reduced(balanced as u128) + } else { + F::from_canonical_u128_reduced(q - ((-balanced) as u128)) + }; + } + } + } + /// Functional gadget recomposition (`G * digits`) for base `2^log_basis`. /// /// Coefficients from each part are interpreted as one digit plane and @@ -217,7 +264,7 @@ impl CyclotomicRing { assert!(log_basis > 0 && log_basis < 128, "invalid log_basis"); let b = F::from_canonical_u128_reduced(1u128 << log_basis); - let coeffs = std::array::from_fn(|i| { + let coeffs = from_fn(|i| { let mut acc = F::zero(); let mut power = F::one(); for part in parts.iter() { @@ -286,7 +333,7 @@ impl CyclotomicRing { /// Generate a random ring element. pub fn random(rng: &mut R) -> Self { Self { - coeffs: std::array::from_fn(|_| F::sample(rng)), + coeffs: from_fn(|_| F::sample(rng)), } } } diff --git a/src/protocol/commitment/commit.rs b/src/protocol/commitment/commit.rs index 97df3007..fbbf80d0 100644 --- a/src/protocol/commitment/commit.rs +++ b/src/protocol/commitment/commit.rs @@ -4,7 +4,7 @@ use super::config::{ ensure_block_layout, ensure_matrix_shape, ensure_supported_num_vars, validate_and_derive_layout, HachiCommitmentLayout, }; -use super::onehot::{inner_ajtai_onehot, map_onehot_to_sparse_blocks, SparseBlockEntry}; +use super::onehot::{inner_ajtai_onehot_t_only, map_onehot_to_sparse_blocks, SparseBlockEntry}; use super::scheme::{CommitWitness, RingCommitmentScheme}; use super::types::RingCommitment; use super::utils::crt_ntt::{build_ntt_cache, NttMatrixCache}; @@ -13,10 +13,12 @@ use super::utils::matrix::{derive_public_matrix, sample_public_matrix_seed, Publ use super::CommitmentConfig; use crate::algebra::CyclotomicRing; use crate::error::HachiError; +#[cfg(feature = "parallel")] +use crate::parallel::*; use crate::primitives::serialization::{ Compress, HachiDeserialize, HachiSerialize, SerializationError, Valid, Validate, }; -use crate::{CanonicalField, FieldCore, FieldSampling}; +use crate::{cfg_into_iter, cfg_iter, CanonicalField, FieldCore, FieldSampling}; use std::io::{Read, Write}; /// Seed-only stage for deterministic setup expansion. @@ -330,17 +332,20 @@ where ensure_matrix_shape(&setup.expanded.A, Cfg::N_A, layout.inner_width, "A")?; ensure_matrix_shape(&setup.expanded.B, Cfg::N_B, layout.outer_width, "B")?; - let mut t_hat_all: Vec>> = Vec::with_capacity(layout.num_blocks); - let mut t_hat_flat: Vec> = Vec::with_capacity(layout.outer_width); - for block in f_blocks { - let s_i = decompose_block(block, Cfg::DELTA, Cfg::LOG_BASIS); - let t_i = mat_vec_mul_ntt_cached(setup.ntt_cache()?, MatrixSlot::A, &s_i)?; - let t_hat_i = decompose_rows(&t_i, Cfg::DELTA, Cfg::LOG_BASIS); - t_hat_flat.extend(t_hat_i.iter().copied()); - t_hat_all.push(t_hat_i); - } + let cache = setup.ntt_cache()?; + let t_hat_all: Vec>> = cfg_iter!(f_blocks) + .map(|block| { + let s_i = decompose_block(block, Cfg::DELTA, Cfg::LOG_BASIS); + let t_i = + mat_vec_mul_ntt_cached(cache, MatrixSlot::A, &s_i).expect("inner Ajtai failed"); + decompose_rows(&t_i, Cfg::DELTA, Cfg::LOG_BASIS) + }) + .collect(); - let u = mat_vec_mul_ntt_cached(setup.ntt_cache()?, MatrixSlot::B, &t_hat_flat)?; + let t_hat_flat: Vec> = + t_hat_all.iter().flat_map(|v| v.iter().copied()).collect(); + + let u = mat_vec_mul_ntt_cached(cache, MatrixSlot::B, &t_hat_flat)?; Ok(CommitWitness { commitment: RingCommitment { u }, t_hat: t_hat_all, @@ -366,27 +371,29 @@ where let zero_t_hat = vec![CyclotomicRing::::zero(); Cfg::N_A.checked_mul(Cfg::DELTA).unwrap()]; + let cache = setup.ntt_cache()?; + let coeff_len = f_coeffs.len(); + + let t_hat_all: Vec>> = cfg_into_iter!(0..num_blocks) + .map(|i| { + let start = i * block_len; + if start >= coeff_len { + zero_t_hat.clone() + } else { + let end = (start + block_len).min(coeff_len); + let block = &f_coeffs[start..end]; + let s_i = decompose_block(block, Cfg::DELTA, Cfg::LOG_BASIS); + let t_i = mat_vec_mul_ntt_cached(cache, MatrixSlot::A, &s_i) + .expect("inner Ajtai failed"); + decompose_rows(&t_i, Cfg::DELTA, Cfg::LOG_BASIS) + } + }) + .collect(); - let mut t_hat_all: Vec>> = Vec::with_capacity(num_blocks); - let mut t_hat_flat: Vec> = Vec::with_capacity(layout.outer_width); - - for i in 0..num_blocks { - let start = i * block_len; - if start >= f_coeffs.len() { - t_hat_flat.extend(zero_t_hat.iter().copied()); - t_hat_all.push(zero_t_hat.clone()); - } else { - let end = (start + block_len).min(f_coeffs.len()); - let block = &f_coeffs[start..end]; - let s_i = decompose_block(block, Cfg::DELTA, Cfg::LOG_BASIS); - let t_i = mat_vec_mul_ntt_cached(setup.ntt_cache()?, MatrixSlot::A, &s_i)?; - let t_hat_i = decompose_rows(&t_i, Cfg::DELTA, Cfg::LOG_BASIS); - t_hat_flat.extend(t_hat_i.iter().copied()); - t_hat_all.push(t_hat_i); - } - } + let t_hat_flat: Vec> = + t_hat_all.iter().flat_map(|v| v.iter().copied()).collect(); - let u = mat_vec_mul_ntt_cached(setup.ntt_cache()?, MatrixSlot::B, &t_hat_flat)?; + let u = mat_vec_mul_ntt_cached(cache, MatrixSlot::B, &t_hat_flat)?; Ok(CommitWitness { commitment: RingCommitment { u }, t_hat: t_hat_all, @@ -409,30 +416,28 @@ where let sparse_blocks = map_onehot_to_sparse_blocks(onehot_k, indices, layout.r_vars, layout.m_vars, D)?; - let mut t_hat_all: Vec>> = Vec::with_capacity(layout.num_blocks); - let mut t_hat_flat: Vec> = Vec::with_capacity(layout.outer_width); - let zero_t_hat = vec![CyclotomicRing::::zero(); Cfg::N_A.checked_mul(Cfg::DELTA).unwrap()]; + let cache = setup.ntt_cache()?; + let a_matrix = &setup.expanded.A; + let block_len = layout.block_len; - for block_entries in &sparse_blocks { - if block_entries.is_empty() { - t_hat_flat.extend(zero_t_hat.iter().copied()); - t_hat_all.push(zero_t_hat.clone()); - } else { - let (t_i, _s_i) = inner_ajtai_onehot( - &setup.expanded.A, - block_entries, - layout.block_len, - Cfg::DELTA, - ); - let t_hat_i = decompose_rows(&t_i, Cfg::DELTA, Cfg::LOG_BASIS); - t_hat_flat.extend(t_hat_i.iter().copied()); - t_hat_all.push(t_hat_i); - } - } + let t_hat_all: Vec>> = cfg_iter!(sparse_blocks) + .map(|block_entries| { + if block_entries.is_empty() { + zero_t_hat.clone() + } else { + let t_i = + inner_ajtai_onehot_t_only(a_matrix, block_entries, block_len, Cfg::DELTA); + decompose_rows(&t_i, Cfg::DELTA, Cfg::LOG_BASIS) + } + }) + .collect(); + + let t_hat_flat: Vec> = + t_hat_all.iter().flat_map(|v| v.iter().copied()).collect(); - let u = mat_vec_mul_ntt_cached(setup.ntt_cache()?, MatrixSlot::B, &t_hat_flat)?; + let u = mat_vec_mul_ntt_cached(cache, MatrixSlot::B, &t_hat_flat)?; Ok(CommitWitness { commitment: RingCommitment { u }, t_hat: t_hat_all, @@ -461,6 +466,75 @@ impl HachiCommitmentCore { { let max_num_vars = layout.required_num_vars::()?; let public_matrix_seed = sample_public_matrix_seed(); + Self::setup_with_layout_and_seed::(layout, max_num_vars, public_matrix_seed) + } + + /// Like `setup_with_layout` but reuses an existing setup's random seed and + /// A matrix (which depends only on `m_vars`). Only regenerates B and D + /// matrices for the new `r_vars`. + /// + /// Use this when creating a mega-polynomial setup that shares `m_vars` with + /// an individual polynomial setup — avoids re-deriving and NTT-transforming + /// the A matrix. + /// + /// # Errors + /// + /// Returns `HachiError` if the new layout is incompatible with the existing + /// setup or matrix shapes are inconsistent. + #[allow(non_snake_case)] + pub fn setup_from_existing( + existing: &HachiExpandedSetup, + new_r_vars: usize, + ) -> Result<(HachiProverSetup, HachiVerifierSetup), HachiError> + where + F: FieldCore + CanonicalField + FieldSampling, + Cfg: CommitmentConfig, + { + let old_layout = existing.seed.layout; + let new_layout = HachiCommitmentLayout::new::(old_layout.m_vars, new_r_vars)?; + + if new_layout.inner_width != old_layout.inner_width { + return Err(HachiError::InvalidSetup( + "setup_from_existing requires matching m_vars/inner_width".to_string(), + )); + } + + let max_num_vars = new_layout.required_num_vars::()?; + let seed = existing.seed.public_matrix_seed; + + let b_matrix = derive_public_matrix::(Cfg::N_B, new_layout.outer_width, &seed, b"B"); + let d_matrix = + derive_public_matrix::(Cfg::N_D, new_layout.d_matrix_width, &seed, b"D"); + + let ntt_cache = build_ntt_cache::(&existing.A, &b_matrix, &d_matrix)?; + let expanded = HachiExpandedSetup { + seed: HachiSetupSeed { + max_num_vars, + layout: new_layout, + public_matrix_seed: seed, + }, + A: existing.A.clone(), + B: b_matrix, + D: d_matrix, + }; + let prover_setup = HachiProverSetup { + expanded: expanded.clone(), + prepared: Some(HachiPreparedSetup { ntt_cache }), + }; + let verifier_setup = HachiVerifierSetup { expanded }; + Ok((prover_setup, verifier_setup)) + } + + #[allow(non_snake_case)] + fn setup_with_layout_and_seed( + layout: HachiCommitmentLayout, + max_num_vars: usize, + public_matrix_seed: PublicMatrixSeed, + ) -> Result<(HachiProverSetup, HachiVerifierSetup), HachiError> + where + F: FieldCore + CanonicalField + FieldSampling, + Cfg: CommitmentConfig, + { let a_matrix = derive_public_matrix::(Cfg::N_A, layout.inner_width, &public_matrix_seed, b"A"); let b_matrix = @@ -552,43 +626,39 @@ impl HachiCommitmentCore { let zero_t_hat = vec![CyclotomicRing::::zero(); Cfg::N_A.checked_mul(Cfg::DELTA).unwrap()]; + let cache = setup.ntt_cache()?; + let a_matrix = &setup.expanded.A; + let block_len = layout.block_len; - let mut t_hat_all: Vec>> = Vec::with_capacity(layout.num_blocks); - let mut t_hat_flat: Vec> = Vec::with_capacity(layout.outer_width); - - for block in blocks { - match block { - MegaPolyBlock::Zero => { - t_hat_flat.extend(zero_t_hat.iter().copied()); - t_hat_all.push(zero_t_hat.clone()); - } + let t_hat_all: Vec>> = cfg_iter!(blocks) + .map(|block| match block { + MegaPolyBlock::Zero => zero_t_hat.clone(), MegaPolyBlock::Dense(coeffs) => { let s_i = decompose_block(coeffs, Cfg::DELTA, Cfg::LOG_BASIS); - let t_i = mat_vec_mul_ntt_cached(setup.ntt_cache()?, MatrixSlot::A, &s_i)?; - let t_hat_i = decompose_rows(&t_i, Cfg::DELTA, Cfg::LOG_BASIS); - t_hat_flat.extend(t_hat_i.iter().copied()); - t_hat_all.push(t_hat_i); + let t_i = mat_vec_mul_ntt_cached(cache, MatrixSlot::A, &s_i) + .expect("inner Ajtai failed"); + decompose_rows(&t_i, Cfg::DELTA, Cfg::LOG_BASIS) } MegaPolyBlock::OneHot(sparse_entries) => { if sparse_entries.is_empty() { - t_hat_flat.extend(zero_t_hat.iter().copied()); - t_hat_all.push(zero_t_hat.clone()); + zero_t_hat.clone() } else { - let (t_i, _s_i) = inner_ajtai_onehot( - &setup.expanded.A, + let t_i = inner_ajtai_onehot_t_only( + a_matrix, sparse_entries, - layout.block_len, + block_len, Cfg::DELTA, ); - let t_hat_i = decompose_rows(&t_i, Cfg::DELTA, Cfg::LOG_BASIS); - t_hat_flat.extend(t_hat_i.iter().copied()); - t_hat_all.push(t_hat_i); + decompose_rows(&t_i, Cfg::DELTA, Cfg::LOG_BASIS) } } - } - } + }) + .collect(); + + let t_hat_flat: Vec> = + t_hat_all.iter().flat_map(|v| v.iter().copied()).collect(); - let u = mat_vec_mul_ntt_cached(setup.ntt_cache()?, MatrixSlot::B, &t_hat_flat)?; + let u = mat_vec_mul_ntt_cached(cache, MatrixSlot::B, &t_hat_flat)?; Ok(CommitWitness { commitment: RingCommitment { u }, t_hat: t_hat_all, diff --git a/src/protocol/commitment/onehot.rs b/src/protocol/commitment/onehot.rs index 18b8c1dc..3fe8d3fb 100644 --- a/src/protocol/commitment/onehot.rs +++ b/src/protocol/commitment/onehot.rs @@ -148,11 +148,33 @@ pub(crate) fn inner_ajtai_onehot( (t, s) } +/// Like `inner_ajtai_onehot` but only returns `t`, skipping the `s` allocation. +#[allow(non_snake_case)] +pub(crate) fn inner_ajtai_onehot_t_only( + A: &[Vec>], + sparse_entries: &[SparseBlockEntry], + _block_len: usize, + _delta: usize, +) -> Vec> { + let n_a = A.len(); + + let mut t = vec![CyclotomicRing::::zero(); n_a]; + for entry in sparse_entries { + let col = entry.pos_in_block * _delta; + for a in 0..n_a { + t[a] += A[a][col].mul_by_monomial_sum(&entry.nonzero_coeffs); + } + } + + t +} + #[cfg(test)] mod tests { use super::*; use crate::test_utils::F; use crate::FromSmallInt; + use std::array::from_fn; #[test] fn map_onehot_k_gt_d() { @@ -241,16 +263,16 @@ mod tests { // A is 2x4 (N_A=2, inner_width = block_len * delta = 2 * 2 = 4) let a: Vec> = vec![ vec![ - R::from_coefficients(std::array::from_fn(|i| F::from_u64((i + 1) as u64))), - R::from_coefficients(std::array::from_fn(|i| F::from_u64((i + 10) as u64))), - R::from_coefficients(std::array::from_fn(|i| F::from_u64((i + 20) as u64))), - R::from_coefficients(std::array::from_fn(|i| F::from_u64((i + 30) as u64))), + R::from_coefficients(from_fn(|i| F::from_u64((i + 1) as u64))), + R::from_coefficients(from_fn(|i| F::from_u64((i + 10) as u64))), + R::from_coefficients(from_fn(|i| F::from_u64((i + 20) as u64))), + R::from_coefficients(from_fn(|i| F::from_u64((i + 30) as u64))), ], vec![ - R::from_coefficients(std::array::from_fn(|i| F::from_u64((i + 5) as u64))), - R::from_coefficients(std::array::from_fn(|i| F::from_u64((i + 15) as u64))), - R::from_coefficients(std::array::from_fn(|i| F::from_u64((i + 25) as u64))), - R::from_coefficients(std::array::from_fn(|i| F::from_u64((i + 35) as u64))), + R::from_coefficients(from_fn(|i| F::from_u64((i + 5) as u64))), + R::from_coefficients(from_fn(|i| F::from_u64((i + 15) as u64))), + R::from_coefficients(from_fn(|i| F::from_u64((i + 25) as u64))), + R::from_coefficients(from_fn(|i| F::from_u64((i + 35) as u64))), ], ]; diff --git a/src/protocol/commitment/utils/crt_ntt.rs b/src/protocol/commitment/utils/crt_ntt.rs index 6c91e254..33bd79c4 100644 --- a/src/protocol/commitment/utils/crt_ntt.rs +++ b/src/protocol/commitment/utils/crt_ntt.rs @@ -75,16 +75,19 @@ pub(crate) enum NttMatrixCache { A: Vec>>, B: Vec>>, D: Vec>>, + params: CrtNttParamSet, }, Q64 { A: Vec>>, B: Vec>>, D: Vec>>, + params: CrtNttParamSet, }, Q128 { A: Vec>>, B: Vec>>, D: Vec>>, + params: CrtNttParamSet, }, } @@ -112,21 +115,24 @@ pub(crate) fn build_ntt_cache( d: &[Vec>], ) -> Result, HachiError> { let params = select_crt_ntt_params::()?; - let cache = match ¶ms { + let cache = match params { ProtocolCrtNttParams::Q32(p) => NttMatrixCache::Q32 { - A: convert_mat(a, p), - B: convert_mat(b, p), - D: convert_mat(d, p), + A: convert_mat(a, &p), + B: convert_mat(b, &p), + D: convert_mat(d, &p), + params: p, }, ProtocolCrtNttParams::Q64(p) => NttMatrixCache::Q64 { - A: convert_mat(a, p), - B: convert_mat(b, p), - D: convert_mat(d, p), + A: convert_mat(a, &p), + B: convert_mat(b, &p), + D: convert_mat(d, &p), + params: p, }, ProtocolCrtNttParams::Q128(p) => NttMatrixCache::Q128 { - A: convert_mat(a, p), - B: convert_mat(b, p), - D: convert_mat(d, p), + A: convert_mat(a, &p), + B: convert_mat(b, &p), + D: convert_mat(d, &p), + params: p, }, }; Ok(cache) diff --git a/src/protocol/commitment/utils/linear.rs b/src/protocol/commitment/utils/linear.rs index 4a8e007d..636e7b3a 100644 --- a/src/protocol/commitment/utils/linear.rs +++ b/src/protocol/commitment/utils/linear.rs @@ -8,7 +8,9 @@ use crate::error::HachiError; use crate::parallel::*; use crate::{CanonicalField, FieldCore}; -use super::crt_ntt::{select_crt_ntt_params, NttMatrixCache, ProtocolCrtNttParams}; +use super::crt_ntt::NttMatrixCache; +#[cfg(test)] +use super::crt_ntt::{select_crt_ntt_params, ProtocolCrtNttParams}; #[cfg(test)] pub(crate) fn mat_vec_mul_unchecked( @@ -179,8 +181,7 @@ fn mat_vec_mul_precomputed_with_params< vec: &[CyclotomicRing], params: &CrtNttParamSet, ) -> Vec> { - let ntt_vec: Vec> = vec - .iter() + let ntt_vec: Vec> = cfg_iter!(vec) .map(|v| CyclotomicCrtNtt::from_ring_with_params(v, params)) .collect(); @@ -197,22 +198,21 @@ fn mat_vec_mul_precomputed_with_params< } macro_rules! dispatch_cached { - ($cache:expr, $which:expr, $params:expr, $func:ident $(, $arg:expr)*) => {{ + ($cache:expr, $which:expr, $func:ident $(, $arg:expr)*) => {{ #[allow(non_snake_case)] - match ($cache, $params) { - (NttMatrixCache::Q32 { A, B, D: Dm }, ProtocolCrtNttParams::Q32(p)) => { + match $cache { + NttMatrixCache::Q32 { A, B, D: Dm, params: p } => { let m = match $which { MatrixSlot::A => A, MatrixSlot::B => B, MatrixSlot::D => Dm }; $func(m, $($arg,)* p) } - (NttMatrixCache::Q64 { A, B, D: Dm }, ProtocolCrtNttParams::Q64(p)) => { + NttMatrixCache::Q64 { A, B, D: Dm, params: p } => { let m = match $which { MatrixSlot::A => A, MatrixSlot::B => B, MatrixSlot::D => Dm }; $func(m, $($arg,)* p) } - (NttMatrixCache::Q128 { A, B, D: Dm }, ProtocolCrtNttParams::Q128(p)) => { + NttMatrixCache::Q128 { A, B, D: Dm, params: p } => { let m = match $which { MatrixSlot::A => A, MatrixSlot::B => B, MatrixSlot::D => Dm }; $func(m, $($arg,)* p) } - _ => return Err(HachiError::InvalidSetup("NTT cache / param family mismatch".into())), } }}; } @@ -223,14 +223,7 @@ pub(crate) fn mat_vec_mul_ntt_cached], ) -> Result>, HachiError> { - let params = select_crt_ntt_params::()?; - let out = dispatch_cached!( - cache, - which, - ¶ms, - mat_vec_mul_precomputed_with_params, - vec - ); + let out = dispatch_cached!(cache, which, mat_vec_mul_precomputed_with_params, vec); Ok(out) } @@ -240,9 +233,9 @@ pub fn decompose_block( delta: usize, log_basis: u32, ) -> Vec> { - let mut out = Vec::with_capacity(block.len() * delta); - for coeff_vec in block { - out.extend(coeff_vec.balanced_decompose_pow2(delta, log_basis)); + let mut out = vec![CyclotomicRing::::zero(); block.len() * delta]; + for (i, coeff_vec) in block.iter().enumerate() { + coeff_vec.balanced_decompose_pow2_into(&mut out[i * delta..(i + 1) * delta], log_basis); } out } @@ -252,9 +245,9 @@ pub(crate) fn decompose_rows( delta: usize, log_basis: u32, ) -> Vec> { - let mut out = Vec::with_capacity(rows.len() * delta); - for row in rows { - out.extend(row.balanced_decompose_pow2(delta, log_basis)); + let mut out = vec![CyclotomicRing::::zero(); rows.len() * delta]; + for (i, row) in rows.iter().enumerate() { + row.balanced_decompose_pow2_into(&mut out[i * delta..(i + 1) * delta], log_basis); } out }

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