optimize cross-term computation (#328)

* optimize computation of cross-term

* clippy fixes

* cargo fmt

examples/minroot.rs

@@ -49,7 +49,7 @@ impl<G: Group> MinRootIteration<G> {
     let mut x_i = *x_0;
     let mut y_i = *y_0;
     for _i in 0..num_iters {
-      let x_i_plus_1 = (x_i + y_i).pow_vartime(&exp.to_u64_digits()); // computes the fifth root of x_i + y_i
+      let x_i_plus_1 = (x_i + y_i).pow_vartime(exp.to_u64_digits()); // computes the fifth root of x_i + y_i
 
       // sanity check
       if cfg!(debug_assertions) {

src/provider/keccak.rs

@@ -87,7 +87,7 @@ impl<E: Engine> TranscriptEngineTrait<E> for Keccak256Transcript<E> {
 
   fn absorb<T: TranscriptReprTrait<E::GE>>(&mut self, label: &'static [u8], o: &T) {
     self.transcript.update(label);
-    self.transcript.update(&o.to_transcript_bytes());
+    self.transcript.update(o.to_transcript_bytes());
   }
 
   fn dom_sep(&mut self, bytes: &'static [u8]) {

src/r1cs/mod.rs

@@ -250,42 +250,26 @@ impl<E: Engine> R1CSShape<E> {
     U2: &R1CSInstance<E>,
     W2: &R1CSWitness<E>,
   ) -> Result<(Vec<E::Scalar>, Commitment<E>), NovaError> {
-    let (AZ_1, BZ_1, CZ_1) = {
-      let Z1 = [W1.W.clone(), vec![U1.u], U1.X.clone()].concat();
-      self.multiply_vec(&Z1)?
-    };
-
-    let (AZ_2, BZ_2, CZ_2) = {
-      let Z2 = [W2.W.clone(), vec![E::Scalar::ONE], U2.X.clone()].concat();
-      self.multiply_vec(&Z2)?
-    };
+    let Z1 = [W1.W.clone(), vec![U1.u], U1.X.clone()].concat();
+    let Z2 = [W2.W.clone(), vec![E::Scalar::ONE], U2.X.clone()].concat();
+
+    // The following code uses the optimization suggested in
+    // Section 5.2 of [Mova](https://eprint.iacr.org/2024/1220.pdf)
+    let Z = Z1
+      .into_par_iter()
+      .zip(Z2.into_par_iter())
+      .map(|(z1, z2)| z1 + z2)
+      .collect::<Vec<E::Scalar>>();
+    let u = U1.u + E::Scalar::ONE; // U2.u = 1
 
-    let (AZ_1_circ_BZ_2, AZ_2_circ_BZ_1, u_1_cdot_CZ_2, u_2_cdot_CZ_1) = {
-      let AZ_1_circ_BZ_2 = (0..AZ_1.len())
-        .into_par_iter()
-        .map(|i| AZ_1[i] * BZ_2[i])
-        .collect::<Vec<E::Scalar>>();
-      let AZ_2_circ_BZ_1 = (0..AZ_2.len())
-        .into_par_iter()
-        .map(|i| AZ_2[i] * BZ_1[i])
-        .collect::<Vec<E::Scalar>>();
-      let u_1_cdot_CZ_2 = (0..CZ_2.len())
-        .into_par_iter()
-        .map(|i| U1.u * CZ_2[i])
-        .collect::<Vec<E::Scalar>>();
-      let u_2_cdot_CZ_1 = (0..CZ_1.len())
-        .into_par_iter()
-        .map(|i| CZ_1[i])
-        .collect::<Vec<E::Scalar>>();
-      (AZ_1_circ_BZ_2, AZ_2_circ_BZ_1, u_1_cdot_CZ_2, u_2_cdot_CZ_1)
-    };
+    let (AZ, BZ, CZ) = self.multiply_vec(&Z)?;
 
-    let T = AZ_1_circ_BZ_2
+    let T = AZ
       .par_iter()
-      .zip(&AZ_2_circ_BZ_1)
-      .zip(&u_1_cdot_CZ_2)
-      .zip(&u_2_cdot_CZ_1)
-      .map(|(((a, b), c), d)| *a + *b - *c - *d)
+      .zip(BZ.par_iter())
+      .zip(CZ.par_iter())
+      .zip(W1.E.par_iter())
+      .map(|(((az, bz), cz), e)| *az * *bz - u * *cz - *e)
       .collect::<Vec<E::Scalar>>();
 
     let comm_T = CE::<E>::commit(ck, &T);

src/spartan/polys/masked_eq.rs

@@ -85,6 +85,7 @@ mod tests {
   use rand_chacha::ChaCha20Rng;
   use rand_core::{CryptoRng, RngCore, SeedableRng};
 
+  #[allow(clippy::needless_borrows_for_generic_args)]
   fn test_masked_eq_polynomial_with<F: PrimeField, R: RngCore + CryptoRng>(
     num_vars: usize,
     num_masked_vars: usize,

src/spartan/polys/multilinear.rs

@@ -278,6 +278,7 @@ mod tests {
   }
 
   /// Returns a random ML polynomial
+  #[allow(clippy::needless_borrows_for_generic_args)]
   fn random<R: RngCore + CryptoRng, Scalar: PrimeField>(
     num_vars: usize,
     mut rng: &mut R,