diff --git a/willow/src/zk/rlwe_relation.rs b/willow/src/zk/rlwe_relation.rs
index 3aa2cef..a77cde1 100644
--- a/willow/src/zk/rlwe_relation.rs
+++ b/willow/src/zk/rlwe_relation.rs
@@ -327,28 +327,41 @@ pub fn generate_challenge_matrix(
     result
 }
 
-// Multiplies a 128 by n matrix m and a length n vector v.
-// m is a binary matrix each column of which has entries given by the bits of a single entry in the
-// input vector m.
-// Both the output and v are vectors of 128 bit signed integers.
-pub fn multiply_by_challenge_matrix(
+// Applies a challenge matrix R1-R2 to a vector v and checks if the result satisfies the conditions
+// for not needing to be rejected. An internal error is returned in the event of rejection otherwise
+// the resulting vector z is returned.
+//
+// To understand the rejection conditions see the comment for generate_range_product.
+pub fn try_matrices_and_compute_z(
     v: &[i128],
-    m: &[u128],
+    R1: &[u128],
+    R2: &[u128],
+    y: &[i128],
+    half_loose_bound: i128,
 ) -> Result<Vec<i128>, status::StatusError> {
     let n = v.len();
-    if m.len() != n {
-        return Err(status::failed_precondition("m and v have different lengths".to_string()));
+    if n != R1.len() || n != R2.len() {
+        return Err(status::failed_precondition(
+            "R1, R2, and v must have the same length".to_string(),
+        ));
     }
-
-    let mut result = vec![0 as i128; 128];
-    for i in 0..n {
-        for j in 0..128 {
-            if m[i] & (1u128 << j) != 0 {
-                result[j] += v[i];
+    let mut z = vec![0 as i128; 128];
+    for j in 0..128 {
+        let mut u = 0i128;
+        for i in 0..n {
+            if R1[i] & (1u128 << j) != 0 {
+                u += v[i];
             }
+            if R2[i] & (1u128 << j) != 0 {
+                u -= v[i];
+            }
+        }
+        z[j] = u + y[j];
+        if u.abs() > half_loose_bound / 128 || z[j].abs() > half_loose_bound {
+            return Err(status::internal("Sample Rejected"));
         }
     }
-    Ok(result)
+    Ok(z)
 }
 
 // Linearly combines the 128 vector challenges of a challenge matrix into a single vector challenge
@@ -459,7 +472,6 @@ fn generate_range_product(
     let mut z = vec![0 as i128; 128];
     let mut attempts = 0;
     loop {
-        let mut done = true;
         attempts += 1;
         y = (0..128).map(|_| (rng.gen_range(0..possible_y) as i128)).collect();
         for i in 0..128 {
@@ -474,21 +486,9 @@ fn generate_range_product(
         // subtracting the other we get a challenge matrix with the correct distribution.
         R1 = generate_challenge_matrix(transcript, challenge_label, v.len());
         R2 = generate_challenge_matrix(transcript, challenge_label, v.len());
-        let u1 = multiply_by_challenge_matrix(v, &R1)?;
-        let u2 = multiply_by_challenge_matrix(v, &R2)?;
-        for i in 0..128 {
-            let u = u1[i] - u2[i];
-            if u.abs() > half_loose_bound / 128 {
-                done = false;
-                break;
-            }
-            z[i] = u + y[i];
-            if z[i].abs() > half_loose_bound {
-                done = false;
-                break;
-            }
-        }
-        if done {
+        let z_or_error = try_matrices_and_compute_z(v, &R1, &R2, &y, half_loose_bound);
+        if z_or_error.is_ok() {
+            z = z_or_error.unwrap();
             break;
         }
         if attempts > 1000 {
@@ -1244,16 +1244,44 @@ mod tests {
     }
 
     #[test]
-    fn test_multiply_by_challenge_matrix_basic_case() -> googletest::Result<()> {
-        let v = &[10i128, 20i128];
-        let m = &[(1u128 << 0) | (1u128 << 2), (1u128 << 1) | (1u128 << 2)];
+    fn test_try_matrices_and_compute_z_valid() -> googletest::Result<()> {
+        let v = [1, -2, 3, -4];
+        let R1 = [1, 2, 3, 4];
+        let R2 = [4, 3, 2, 1];
+        let y = [1; 128];
+        let half_loose_bound = 10000;
+        let result = try_matrices_and_compute_z(&v, &R1, &R2, &y, half_loose_bound)?;
+        let mut expected_z = vec![1; 128];
+        expected_z[0] += 10;
+        expected_z[1] += 0;
+        expected_z[2] += -5;
+        verify_eq!(result, expected_z)?;
+        Ok(())
+    }
 
-        let mut expected_result = vec![0i128; 128];
-        expected_result[0] = 10;
-        expected_result[1] = 20;
-        expected_result[2] = 30;
+    #[test]
+    fn test_try_matrices_and_compute_z_mismatched_lengths() -> googletest::Result<()> {
+        let v = [1, -2, 3, -4];
+        let R1 = [1, 2, 3];
+        let R2 = [4, 3, 2, 1];
+        let y = [1; 128];
+        let half_loose_bound = 1000;
+        let result = try_matrices_and_compute_z(&v, &R1, &R2, &y, half_loose_bound);
+        assert!(result.is_err());
+        verify_eq!(result.unwrap_err().message(), "R1, R2, and v must have the same length")?;
+        Ok(())
+    }
 
-        assert_eq!(multiply_by_challenge_matrix(v, m).unwrap(), expected_result);
+    #[test]
+    fn test_try_matrices_and_compute_z_sample_rejected() -> googletest::Result<()> {
+        let v = [1000, -2000, 3000, -4000];
+        let R1 = [1, 2, 3, 4];
+        let R2 = [4, 3, 2, 1];
+        let y = [1; 128];
+        let half_loose_bound = 100000;
+        let result = try_matrices_and_compute_z(&v, &R1, &R2, &y, half_loose_bound);
+        assert!(result.is_err());
+        verify_eq!(result.unwrap_err().message(), "Sample Rejected")?;
         Ok(())
     }