Improve sparse polynomial evaluation algorithm (#317)

* time-optimal algorithm for sparse polynomial evaluation

* update version

src/spartan/batched.rs

@@ -490,17 +490,7 @@ impl<E: Engine, EE: EvaluationEngineTrait<E>> BatchedRelaxedR1CSSNARKTrait<E>
     let evals_Z = zip_with!(iter, (self.evals_W, U, r_y), |eval_W, U, r_y| {
       let eval_X = {
         // constant term
-        let poly_X = iter::once((0, U.u))
-          .chain(
-            //remaining inputs
-            U.X
-            .iter()
-            .enumerate()
-            // filter_map uses the sparsity of the polynomial, if irrelevant
-            // we should replace by UniPoly
-            .filter_map(|(i, x_i)| (!x_i.is_zero_vartime()).then_some((i + 1, *x_i))),
-          )
-          .collect();
+        let poly_X = iter::once(U.u).chain(U.X.iter().cloned()).collect();
         SparsePolynomial::new(r_y.len() - 1, poly_X).evaluate(&r_y[1..])
       };
       (E::Scalar::ONE - r_y[0]) * eval_W + r_y[0] * eval_X

src/spartan/batched_ppsnark.rs

@@ -927,15 +927,7 @@ impl<E: Engine, EE: EvaluationEngineTrait<E>> BatchedRelaxedR1CSSNARKTrait<E>
 
           let X = {
             // constant term
-            let poly_X = std::iter::once((0, U.u))
-              .chain(
-                //remaining inputs
-                (0..U.X.len())
-                // filter_map uses the sparsity of the polynomial, if irrelevant
-                // we should replace by UniPoly
-                .filter_map(|i| (!U.X[i].is_zero_vartime()).then_some((i + 1, U.X[i]))),
-              )
-              .collect();
+            let poly_X = std::iter::once(U.u).chain(U.X.iter().cloned()).collect();
             SparsePolynomial::new(num_vars_log, poly_X).evaluate(&rand_sc_unpad[1..])
           };
 

src/spartan/mod.rs

@@ -23,7 +23,6 @@ use crate::{
 };
 use ff::Field;
 use itertools::Itertools as _;
-use polys::multilinear::SparsePolynomial;
 use rayon::{iter::IntoParallelRefIterator, prelude::*};
 use rayon_scan::ScanParallelIterator as _;
 use ref_cast::RefCast;

src/spartan/polys/multilinear.rs

@@ -8,8 +8,7 @@ use ff::PrimeField;
 use itertools::Itertools as _;
 use rand_core::{CryptoRng, RngCore};
 use rayon::prelude::{
-  IndexedParallelIterator, IntoParallelIterator, IntoParallelRefIterator,
-  IntoParallelRefMutIterator, ParallelIterator,
+  IndexedParallelIterator, IntoParallelRefIterator, IntoParallelRefMutIterator, ParallelIterator,
 };
 use serde::{Deserialize, Serialize};
 
@@ -130,47 +129,36 @@ impl<Scalar: PrimeField> Index<usize> for MultilinearPolynomial<Scalar> {
 }
 
 /// Sparse multilinear polynomial, which means the $Z(\cdot)$ is zero at most points.
-/// So we do not have to store every evaluations of $Z(\cdot)$, only store the non-zero points.
-///
-/// For example, the evaluations are [0, 0, 0, 1, 0, 1, 0, 2].
-/// The sparse polynomial only store the non-zero values, [(3, 1), (5, 1), (7, 2)].
-/// In the tuple, the first is index, the second is value.
+/// In our context, sparse polynomials are non-zeros over the hypercube at locations that map to "small" integers
+/// We exploit this property to implement a time-optimal algorithm
 pub(crate) struct SparsePolynomial<Scalar> {
   num_vars: usize,
-  Z: Vec<(usize, Scalar)>,
+  Z: Vec<Scalar>,
 }
 
 impl<Scalar: PrimeField> SparsePolynomial<Scalar> {
-  pub fn new(num_vars: usize, Z: Vec<(usize, Scalar)>) -> Self {
+  pub fn new(num_vars: usize, Z: Vec<Scalar>) -> Self {
     Self { num_vars, Z }
   }
 
-  /// Computes the $\tilde{eq}$ extension polynomial.
-  /// return 1 when a == r, otherwise return 0.
-  fn compute_chi(a: &[bool], r: &[Scalar]) -> Scalar {
-    assert_eq!(a.len(), r.len());
-    let mut chi_i = Scalar::ONE;
-    for j in 0..r.len() {
-      if a[j] {
-        chi_i *= r[j];
-      } else {
-        chi_i *= Scalar::ONE - r[j];
-      }
-    }
-    chi_i
-  }
-
-  // Takes O(m log n) where m is the number of non-zero evaluations and n is the number of variables.
+  // a time-optimal algorithm to evaluate sparse polynomials
   pub fn evaluate(&self, r: &[Scalar]) -> Scalar {
     assert_eq!(self.num_vars, r.len());
 
-    (0..self.Z.len())
-      .into_par_iter()
-      .map(|i| {
-        let bits = (self.Z[i].0).get_bits(r.len());
-        Self::compute_chi(&bits, r) * self.Z[i].1
-      })
-      .sum()
+    let num_vars_z = self.Z.len().next_power_of_two().log_2();
+    let chis = EqPolynomial::evals_from_points(&r[self.num_vars - 1 - num_vars_z..]);
+    let eval_partial: Scalar = self
+      .Z
+      .iter()
+      .zip_eq(chis.iter())
+      .map(|(z, chi)| *z * *chi)
+      .sum();
+
+    let common = (0..self.num_vars - 1 - num_vars_z)
+      .map(|i| (Scalar::ONE - r[i]))
+      .product::<Scalar>();
+
+    common * eval_partial
   }
 }
 
@@ -232,18 +220,21 @@ mod tests {
   }
 
   fn test_sparse_polynomial_with<F: PrimeField>() {
-    // Let the polynomial have 3 variables, p(x_1, x_2, x_3) = (x_1 + x_2) * x_3
-    // Evaluations of the polynomial at boolean cube are [0, 0, 0, 1, 0, 1, 0, 2].
+    // Let the polynomial have 4 variables, but is non-zero at only 3 locations (out of 2^4 = 16) over the hypercube
+    let mut Z = vec![F::ONE, F::ONE, F::from(2)];
+    let m_poly = SparsePolynomial::<F>::new(4, Z.clone());
 
-    let TWO = F::from(2);
-    let Z = vec![(3, F::ONE), (5, F::ONE), (7, TWO)];
-    let m_poly = SparsePolynomial::<F>::new(3, Z);
+    Z.resize(16, F::ZERO); // append with zeros to make it a dense polynomial
+    let m_poly_dense = MultilinearPolynomial::new(Z);
 
-    let x = vec![F::ONE, F::ONE, F::ONE];
-    assert_eq!(m_poly.evaluate(x.as_slice()), TWO);
+    // evaluation point
+    let x = vec![F::from(5), F::from(8), F::from(5), F::from(3)];
 
-    let x = vec![F::ONE, F::ZERO, F::ONE];
-    assert_eq!(m_poly.evaluate(x.as_slice()), F::ONE);
+    // check evaluations
+    assert_eq!(
+      m_poly.evaluate(x.as_slice()),
+      m_poly_dense.evaluate(x.as_slice())
+    );
   }
 
   #[test]

src/spartan/ppsnark.rs

@@ -24,7 +24,7 @@ use crate::{
       },
       SumcheckProof,
     },
-    PolyEvalInstance, PolyEvalWitness, SparsePolynomial,
+    PolyEvalInstance, PolyEvalWitness,
   },
   traits::{
     commitment::{CommitmentEngineTrait, CommitmentTrait, Len},
@@ -42,7 +42,7 @@ use rayon::prelude::*;
 use serde::{Deserialize, Serialize};
 use std::sync::Arc;
 
-use super::polys::masked_eq::MaskedEqPolynomial;
+use super::polys::{masked_eq::MaskedEqPolynomial, multilinear::SparsePolynomial};
 
 fn padded<E: Engine>(v: &[E::Scalar], n: usize, e: &E::Scalar) -> Vec<E::Scalar> {
   let mut v_padded = vec![*e; n];
@@ -930,17 +930,15 @@ impl<E: Engine, EE: EvaluationEngineTrait<E>> RelaxedR1CSSNARKTrait<E> for Relax
           };
 
           let eval_X = {
-            // constant term
-            let poly_X = std::iter::once((0, U.u))
-              .chain(
-                //remaining inputs
-                (0..U.X.len())
-                // filter_map uses the sparsity of the polynomial, if irrelevant
-                // we should replace by UniPoly
-                .filter_map(|i| (!U.X[i].is_zero_vartime()).then_some((i + 1, U.X[i]))),
-              )
-              .collect();
-            SparsePolynomial::new(vk.num_vars.log_2(), poly_X).evaluate(&rand_sc_unpad[1..])
+            // public IO is (u, X)
+            let X = vec![U.u]
+              .into_iter()
+              .chain(U.X.iter().cloned())
+              .collect::<Vec<E::Scalar>>();
+
+            // evaluate the sparse polynomial at rand_sc_unpad[1..]
+            let poly_X = SparsePolynomial::new(rand_sc_unpad.len() - 1, X);
+            poly_X.evaluate(&rand_sc_unpad[1..])
           };
 
           self.eval_W + factor * rand_sc_unpad[0] * eval_X

src/spartan/snark.rs

@@ -32,7 +32,7 @@ use itertools::Itertools as _;
 use once_cell::sync::OnceCell;
 use rayon::prelude::*;
 use serde::{Deserialize, Serialize};
-use std::{iter, sync::Arc};
+use std::sync::Arc;
 
 /// A type that represents the prover's key
 #[derive(Debug, Clone)]
@@ -328,17 +328,12 @@ impl<E: Engine, EE: EvaluationEngineTrait<E>> RelaxedR1CSSNARKTrait<E> for Relax
     // verify claim_inner_final
     let eval_Z = {
       let eval_X = {
-        // constant term
-        let poly_X = iter::once((0, U.u))
-          .chain(
-            //remaining inputs
-            (0..U.X.len())
-          // filter_map uses the sparsity of the polynomial, if irrelevant
-          // we should replace by UniPoly
-          .filter_map(|i| (!U.X[i].is_zero_vartime()).then_some((i + 1, U.X[i]))),
-          )
-          .collect();
-        SparsePolynomial::new(usize::try_from(vk.S.num_vars.ilog2()).unwrap(), poly_X)
+        // public IO is (u, X)
+        let X = vec![U.u]
+          .into_iter()
+          .chain(U.X.iter().cloned())
+          .collect::<Vec<E::Scalar>>();
+        SparsePolynomial::new(usize::try_from(vk.S.num_vars.ilog2()).unwrap(), X)
           .evaluate(&r_y[1..])
       };
       (E::Scalar::ONE - r_y[0]) * self.eval_W + r_y[0] * eval_X