w circ l function

pcs/src/commit.rs

@@ -165,28 +165,17 @@ impl<E: PairingEngine> MultilinearCommitmentScheme<E> for KZGMultilinearPC<E> {
         Ok(Proof { proofs })
     }
 
-    /// On input a polynomial `p` and a point `point`, outputs a proof for the
-    /// same. This function does not need to take the evaluation value as an
-    /// input.
-    ///
-    /// This function takes 2^{num_var +1} number of scalar multiplications over
-    /// G2:
-    /// - it proceeds with `num_var` number of rounds,
-    /// - at round i, we compute an MSM for `2^{num_var - i + 1}` number of G2
-    ///   elements.
+    /// TODO
     fn multi_open(
         prover_param: &Self::ProverParam,
         polynomials: &[impl MultilinearExtension<E::Fr>],
         point: &[E::Fr],
     ) -> Result<Self::Proof, PCSErrors> {
         let open_timer = start_timer!(|| "open");
 
-        let poly = merge_polynomials(polynomials);
-
-        let proof = Self::open(prover_param, &poly, point)?;
-
         end_timer!(open_timer);
-        Ok(proof)
+
+        todo!()
     }
 
     /// Verifies that `value` is the evaluation at `x` of the polynomial

pcs/src/lib.rs

@@ -1,6 +1,7 @@
 mod commit;
 mod errors;
 mod param;
+mod util;
 
 use ark_ec::PairingEngine;
 use ark_poly::MultilinearExtension;
@@ -47,11 +48,11 @@ pub trait MultilinearCommitmentScheme<E: PairingEngine> {
         point: &[E::Fr],
     ) -> Result<Self::Proof, PCSErrors>;
 
-    /// On input a polynomial `p` and a point `point`, outputs a proof for the
-    /// same.
+    /// On input a list of polynomials and a point `point`, outputs a proof for
+    /// the same.
     fn multi_open(
         prover_param: &Self::ProverParam,
-        polynomial: &[impl MultilinearExtension<E::Fr>],
+        polynomials: &[impl MultilinearExtension<E::Fr>],
         point: &[E::Fr],
     ) -> Result<Self::Proof, PCSErrors>;
 

pcs/src/util.rs

@@ -0,0 +1,130 @@
+//! Useful utilities for KZG PCS
+
+use crate::PCSErrors;
+use ark_ff::PrimeField;
+use ark_poly::{
+    univariate::DensePolynomial, DenseMultilinearExtension, EvaluationDomain, Evaluations,
+    MultilinearExtension, Polynomial, Radix2EvaluationDomain,
+};
+use ark_std::{end_timer, start_timer};
+
+/// Compute W \circ l.
+///
+/// Given an MLE W, and a list of univariate polynomials l, evaluate W at l.
+///
+/// Returns an error is l's length does not matches number of variables in W.
+pub(crate) fn compute_w_circ_l<F: PrimeField>(
+    w: &DenseMultilinearExtension<F>,
+    l: &[DensePolynomial<F>],
+) -> Result<DensePolynomial<F>, PCSErrors> {
+    let timer = start_timer!(|| "compute W \\circ l");
+
+    if w.num_vars != l.len() {
+        return Err(PCSErrors::InvalidParameters(
+            "l's length does not match num_variables".to_string(),
+        ));
+    }
+
+    let mut res_eval: Vec<F> = vec![];
+    let uni_degree = l.len() + 1;
+
+    let domain = match Radix2EvaluationDomain::<F>::new(uni_degree) {
+        Some(p) => p,
+        None => {
+            return Err(PCSErrors::InvalidParameters(
+                "failed to build radix 2 domain".to_string(),
+            ))
+        },
+    };
+
+    for point in domain.elements() {
+        let l_eval: Vec<F> = l.iter().map(|x| x.evaluate(&point)).collect();
+        res_eval.push(w.evaluate(l_eval.as_ref()).unwrap())
+    }
+    let evaluation = Evaluations::from_vec_and_domain(res_eval, domain);
+    let res = evaluation.interpolate();
+
+    end_timer!(timer);
+    Ok(res)
+}
+
+#[cfg(test)]
+mod test {
+    use super::*;
+    use ark_bls12_381::Fr;
+    use ark_poly::UVPolynomial;
+
+    #[test]
+    fn test_w_circ_l() -> Result<(), PCSErrors> {
+        test_w_circ_l_helper::<Fr>()
+    }
+
+    fn test_w_circ_l_helper<F: PrimeField>() -> Result<(), PCSErrors> {
+        {
+            // Example from page 53:
+            // W = 3x1x2 + 2x2 whose evaluations are
+            // 0, 0 |-> 0
+            // 0, 1 |-> 2
+            // 1, 0 |-> 0
+            // 1, 1 |-> 5
+            let w_eval = vec![F::zero(), F::from(2u64), F::zero(), F::from(5u64)];
+            let w = DenseMultilinearExtension::from_evaluations_vec(2, w_eval);
+
+            // l0 =   t + 2
+            // l1 = -2t + 4
+            let l0 = DensePolynomial::from_coefficients_vec(vec![F::from(2u64), F::one()]);
+            let l1 = DensePolynomial::from_coefficients_vec(vec![F::from(4u64), -F::from(2u64)]);
+
+            // res = -6t^2 - 4t + 32
+            let res = compute_w_circ_l(&w, [l1, l0].as_ref())?;
+            let res_rec = DensePolynomial::from_coefficients_vec(vec![
+                F::from(32u64),
+                -F::from(4u64),
+                -F::from(6u64),
+            ]);
+            assert_eq!(res, res_rec);
+        }
+        {
+            // A random example
+            // W = x1x2x3 - 2x1x2 + 3x2x3 - 4x1x3 + 5x1 - 6x2 + 7x3
+            // 0, 0, 0 |->  0
+            // 0, 0, 1 |->  7
+            // 0, 1, 0 |-> -6
+            // 0, 1, 1 |->  4
+            // 1, 0, 0 |->  5
+            // 1, 0, 1 |->  8
+            // 1, 1, 0 |-> -3
+            // 1, 1, 1 |->  4
+            let w_eval = vec![
+                F::zero(),
+                F::from(7u64),
+                -F::from(6u64),
+                F::from(4u64),
+                F::from(5u64),
+                F::from(8u64),
+                -F::from(3u64),
+                F::from(4u64),
+            ];
+            let w = DenseMultilinearExtension::from_evaluations_vec(3, w_eval);
+
+            // l0 =   t + 2
+            // l1 =  3t - 4
+            // l2 = -5t + 6
+            let l0 = DensePolynomial::from_coefficients_vec(vec![F::from(2u64), F::one()]);
+            let l1 = DensePolynomial::from_coefficients_vec(vec![-F::from(4u64), F::from(3u64)]);
+            let l2 = DensePolynomial::from_coefficients_vec(vec![F::from(6u64), -F::from(5u64)]);
+            let res = compute_w_circ_l(&w, [l2, l1, l0].as_ref())?;
+
+            // res = -15t^3 - 23t^2 + 130t - 76
+            let res_rec = DensePolynomial::from_coefficients_vec(vec![
+                -F::from(76u64),
+                F::from(130u64),
+                -F::from(23u64),
+                -F::from(15u64),
+            ]);
+
+            assert_eq!(res, res_rec);
+        }
+        Ok(())
+    }
+}