From 431faccbdbee989300992a1d1d04a42bc2602b7e Mon Sep 17 00:00:00 2001
From: Daniel Lubarov <daniel@lubarov.com>
Date: Wed, 16 Feb 2022 22:37:20 -0800
Subject: [PATCH] Change `compute_permutation_z_polys` to batch permutation
 checks (#492)

* Change `compute_permutation_z_polys` to batch permutation checks

* feedback
---
 starky/src/permutation.rs | 77 +++++++++++++++++++++++++++------------
 1 file changed, 54 insertions(+), 23 deletions(-)

diff --git a/starky/src/permutation.rs b/starky/src/permutation.rs
index 1f7655b4..01cfa8bf 100644
--- a/starky/src/permutation.rs
+++ b/starky/src/permutation.rs
@@ -67,48 +67,46 @@ where
     // Before batching, each permutation pair leads to `num_challenges` permutation arguments, so we
     // start with the cartesian product of `permutation_pairs` and `0..num_challenges`. Then we
     // chunk these arguments based on our batch size.
-    let permutation_instances = permutation_pairs
+    let permutation_batches = permutation_pairs
         .iter()
         .cartesian_product(0..config.num_challenges)
         .chunks(stark.permutation_batch_size())
         .into_iter()
-        .flat_map(|batch| {
-            batch.enumerate().map(|(i, (pair, chal))| {
-                let challenge = permutation_challenge_sets[i].challenges[chal];
-                PermutationInstance { pair, challenge }
-            })
+        .map(|batch| {
+            batch
+                .enumerate()
+                .map(|(i, (pair, chal))| {
+                    let challenge = permutation_challenge_sets[i].challenges[chal];
+                    PermutationInstance { pair, challenge }
+                })
+                .collect_vec()
         })
         .collect_vec();
 
-    permutation_instances
+    permutation_batches
         .into_par_iter()
-        .map(|instance| compute_permutation_z_poly(instance, trace_poly_values))
+        .map(|instances| compute_permutation_z_poly(&instances, trace_poly_values))
         .collect()
 }
 
 /// Compute a single Z polynomial.
-// TODO: Change this to handle a batch of `PermutationInstance`s.
 fn compute_permutation_z_poly<F: Field>(
-    instance: PermutationInstance<F>,
+    instances: &[PermutationInstance<F>],
     trace_poly_values: &[PolynomialValues<F>],
 ) -> PolynomialValues<F> {
-    let PermutationInstance { pair, challenge } = instance;
-    let PermutationPair { column_pairs } = pair;
-    let PermutationChallenge { beta, gamma } = challenge;
-
     let degree = trace_poly_values[0].len();
-    let mut reduced_lhs = PolynomialValues::constant(gamma, degree);
-    let mut reduced_rhs = PolynomialValues::constant(gamma, degree);
+    let (reduced_lhs_polys, reduced_rhs_polys): (Vec<_>, Vec<_>) = instances
+        .iter()
+        .map(|instance| permutation_reduced_polys(instance, trace_poly_values, degree))
+        .unzip();
 
-    for ((lhs, rhs), weight) in column_pairs.iter().zip(beta.powers()) {
-        reduced_lhs.add_assign_scaled(&trace_poly_values[*lhs], weight);
-        reduced_rhs.add_assign_scaled(&trace_poly_values[*rhs], weight);
-    }
+    let numerator = poly_product_elementwise(reduced_lhs_polys.into_iter());
+    let denominator = poly_product_elementwise(reduced_rhs_polys.into_iter());
 
     // Compute the quotients.
-    let reduced_rhs_inverses = F::batch_multiplicative_inverse(&reduced_rhs.values);
-    let mut quotients = reduced_lhs.values;
-    batch_multiply_inplace(&mut quotients, &reduced_rhs_inverses);
+    let denominator_inverses = F::batch_multiplicative_inverse(&denominator.values);
+    let mut quotients = numerator.values;
+    batch_multiply_inplace(&mut quotients, &denominator_inverses);
 
     // Compute Z, which contains partial products of the quotients.
     let mut partial_products = Vec::with_capacity(degree);
@@ -120,6 +118,39 @@ fn compute_permutation_z_poly<F: Field>(
     PolynomialValues::new(partial_products)
 }
 
+/// Computes the reduced polynomial, `\sum beta^i f_i(x) + gamma`, for both the "left" and "right"
+/// sides of a given `PermutationPair`.
+fn permutation_reduced_polys<F: Field>(
+    instance: &PermutationInstance<F>,
+    trace_poly_values: &[PolynomialValues<F>],
+    degree: usize,
+) -> (PolynomialValues<F>, PolynomialValues<F>) {
+    let PermutationInstance {
+        pair: PermutationPair { column_pairs },
+        challenge: PermutationChallenge { beta, gamma },
+    } = instance;
+
+    let mut reduced_lhs = PolynomialValues::constant(*gamma, degree);
+    let mut reduced_rhs = PolynomialValues::constant(*gamma, degree);
+    for ((lhs, rhs), weight) in column_pairs.iter().zip(beta.powers()) {
+        reduced_lhs.add_assign_scaled(&trace_poly_values[*lhs], weight);
+        reduced_rhs.add_assign_scaled(&trace_poly_values[*rhs], weight);
+    }
+    (reduced_lhs, reduced_rhs)
+}
+
+/// Computes the elementwise product of a set of polynomials. Assumes that the set is non-empty and
+/// that each polynomial has the same length.
+fn poly_product_elementwise<F: Field>(
+    mut polys: impl Iterator<Item = PolynomialValues<F>>,
+) -> PolynomialValues<F> {
+    let mut product = polys.next().expect("Expected at least one polynomial");
+    for poly in polys {
+        batch_multiply_inplace(&mut product.values, &poly.values)
+    }
+    product
+}
+
 fn get_permutation_challenge<F: RichField, H: Hasher<F>>(
     challenger: &mut Challenger<F, H>,
 ) -> PermutationChallenge<F> {