//! Sumcheck based batch opening and verify commitment.
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// TODO: refactoring this code to somewhere else
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// currently IOP depends on PCS because perm check requires commitment.
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// The sumcheck based batch opening therefore cannot stay in the PCS repo --
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// which creates a cyclic dependency.
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use crate::{
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pcs::{
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multilinear_kzg::util::eq_eval,
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prelude::{Commitment, PCSError},
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PolynomialCommitmentScheme,
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},
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poly_iop::{prelude::SumCheck, PolyIOP},
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IOPProof,
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};
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use arithmetic::{
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build_eq_x_r_vec, fix_last_variables, DenseMultilinearExtension, VPAuxInfo, VirtualPolynomial,
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};
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use ark_ec::{AffineCurve, PairingEngine, ProjectiveCurve};
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use ark_std::{end_timer, log2, start_timer, One, Zero};
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use std::{marker::PhantomData, rc::Rc};
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use transcript::IOPTranscript;
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#[derive(Clone, Debug, Default, PartialEq, Eq)]
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pub struct BatchProof<E, PCS>
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where
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E: PairingEngine,
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PCS: PolynomialCommitmentScheme<E>,
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{
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/// A sum check proof proving tilde g's sum
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pub(crate) sum_check_proof: IOPProof<E::Fr>,
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/// f_i(point_i)
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pub f_i_eval_at_point_i: Vec<E::Fr>,
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/// proof for g'(a_2)
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pub(crate) g_prime_proof: PCS::Proof,
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}
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/// Steps:
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/// 1. get challenge point t from transcript
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/// 2. build eq(t,i) for i in [0..k]
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/// 3. build \tilde g(i, b) = eq(t, i) * f_i(b)
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/// 4. compute \tilde eq
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/// 5. run sumcheck on \tilde eq * \tilde g(i, b)
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/// 6. build g'(a2) where (a1, a2) is the sumcheck's point
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pub(crate) fn multi_open_internal<E, PCS>(
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prover_param: &PCS::ProverParam,
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polynomials: &[PCS::Polynomial],
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points: &[PCS::Point],
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evals: &[PCS::Evaluation],
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transcript: &mut IOPTranscript<E::Fr>,
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) -> Result<BatchProof<E, PCS>, PCSError>
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where
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E: PairingEngine,
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PCS: PolynomialCommitmentScheme<
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E,
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Polynomial = Rc<DenseMultilinearExtension<E::Fr>>,
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Point = Vec<E::Fr>,
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Evaluation = E::Fr,
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>,
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{
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let open_timer = start_timer!(|| format!("multi open {} points", points.len()));
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// TODO: sanity checks
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let num_var = polynomials[0].num_vars;
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let k = polynomials.len();
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let ell = log2(k) as usize;
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let merged_num_var = num_var + ell;
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// challenge point t
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let t = transcript.get_and_append_challenge_vectors("t".as_ref(), ell)?;
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// eq(t, i) for i in [0..k]
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let eq_t_i_list = build_eq_x_r_vec(t.as_ref())?;
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// \tilde g(i, b) = eq(t, i) * f_i(b)
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let timer = start_timer!(|| format!("compute tilde g for {} points", points.len()));
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let mut tilde_g_eval = vec![E::Fr::zero(); 1 << (ell + num_var)];
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let block_size = 1 << num_var;
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for (index, f_i) in polynomials.iter().enumerate() {
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for (j, &f_i_eval) in f_i.iter().enumerate() {
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tilde_g_eval[index * block_size + j] = f_i_eval * eq_t_i_list[index];
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}
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}
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let tilde_g = Rc::new(DenseMultilinearExtension::from_evaluations_vec(
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merged_num_var,
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tilde_g_eval,
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));
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end_timer!(timer);
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let timer = start_timer!(|| format!("compute tilde eq for {} points", points.len()));
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let mut tilde_eq_eval = vec![E::Fr::zero(); 1 << (ell + num_var)];
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for (index, point) in points.iter().enumerate() {
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let eq_b_zi = build_eq_x_r_vec(point)?;
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let start = index * block_size;
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tilde_eq_eval[start..start + block_size].copy_from_slice(eq_b_zi.as_slice());
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}
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let tilde_eq = Rc::new(DenseMultilinearExtension::from_evaluations_vec(
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merged_num_var,
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tilde_eq_eval,
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));
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end_timer!(timer);
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// built the virtual polynomial for SumCheck
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let timer = start_timer!(|| format!("sum check prove of {} variables", num_var + ell));
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let step = start_timer!(|| "add mle");
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let mut sum_check_vp = VirtualPolynomial::new(num_var + ell);
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sum_check_vp.add_mle_list([tilde_g.clone(), tilde_eq], E::Fr::one())?;
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end_timer!(step);
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let proof = match <PolyIOP<E::Fr> as SumCheck<E::Fr>>::prove(&sum_check_vp, transcript) {
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Ok(p) => p,
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Err(_e) => {
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// cannot wrap IOPError with PCSError due to cyclic dependency
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return Err(PCSError::InvalidProver(
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"Sumcheck in batch proving Failed".to_string(),
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));
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},
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};
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end_timer!(timer);
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// (a1, a2) := sumcheck's point
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let step = start_timer!(|| "open at a2");
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let a1 = &proof.point[num_var..];
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let a2 = &proof.point[..num_var];
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end_timer!(step);
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// build g'(a2)
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let step = start_timer!(|| "evaluate at a2");
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let g_prime = Rc::new(fix_last_variables(&tilde_g, a1));
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end_timer!(step);
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let step = start_timer!(|| "pcs open");
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let (g_prime_proof, _g_prime_eval) = PCS::open(prover_param, &g_prime, a2.to_vec().as_ref())?;
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// assert_eq!(g_prime_eval, tilde_g_eval);
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end_timer!(step);
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let step = start_timer!(|| "evaluate fi(pi)");
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end_timer!(step);
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end_timer!(open_timer);
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Ok(BatchProof {
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sum_check_proof: proof,
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f_i_eval_at_point_i: evals.to_vec(),
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g_prime_proof,
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})
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}
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/// Steps:
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/// 1. get challenge point t from transcript
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/// 2. build g' commitment
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/// 3. ensure \sum_i eq(t, <i>) * f_i_evals matches the sum via SumCheck
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/// verification 4. verify commitment
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pub(crate) fn batch_verify_internal<E, PCS>(
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verifier_param: &PCS::VerifierParam,
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f_i_commitments: &[Commitment<E>],
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points: &[PCS::Point],
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proof: &BatchProof<E, PCS>,
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transcript: &mut IOPTranscript<E::Fr>,
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) -> Result<bool, PCSError>
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where
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E: PairingEngine,
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PCS: PolynomialCommitmentScheme<
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E,
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Polynomial = Rc<DenseMultilinearExtension<E::Fr>>,
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Point = Vec<E::Fr>,
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Evaluation = E::Fr,
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Commitment = Commitment<E>,
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>,
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{
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let open_timer = start_timer!(|| "batch verification");
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// TODO: sanity checks
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let k = f_i_commitments.len();
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let ell = log2(k) as usize;
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let num_var = proof.sum_check_proof.point.len() - ell;
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// challenge point t
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let t = transcript.get_and_append_challenge_vectors("t".as_ref(), ell)?;
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// sum check point (a1, a2)
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let a1 = &proof.sum_check_proof.point[num_var..];
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let a2 = &proof.sum_check_proof.point[..num_var];
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// build g' commitment
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let eq_a1_list = build_eq_x_r_vec(a1)?;
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let eq_t_list = build_eq_x_r_vec(t.as_ref())?;
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let mut g_prime_commit = E::G1Affine::zero().into_projective();
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for i in 0..k {
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let tmp = eq_a1_list[i] * eq_t_list[i];
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g_prime_commit += &f_i_commitments[i].0.mul(tmp);
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}
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// ensure \sum_i eq(t, <i>) * f_i_evals matches the sum via SumCheck
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// verification
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let mut sum = E::Fr::zero();
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for (i, &e) in eq_t_list.iter().enumerate().take(k) {
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sum += e * proof.f_i_eval_at_point_i[i];
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}
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let aux_info = VPAuxInfo {
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max_degree: 2,
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num_variables: num_var + ell,
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phantom: PhantomData,
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};
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let subclaim = match <PolyIOP<E::Fr> as SumCheck<E::Fr>>::verify(
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sum,
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&proof.sum_check_proof,
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&aux_info,
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transcript,
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) {
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Ok(p) => p,
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Err(_e) => {
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// cannot wrap IOPError with PCSError due to cyclic dependency
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return Err(PCSError::InvalidProver(
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"Sumcheck in batch verification failed".to_string(),
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));
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},
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};
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let mut eq_tilde_eval = E::Fr::zero();
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for (point, &coef) in points.iter().zip(eq_a1_list.iter()) {
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eq_tilde_eval += coef * eq_eval(a2, point)?;
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}
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let tilde_g_eval = subclaim.expected_evaluation / eq_tilde_eval;
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// verify commitment
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let res = PCS::verify(
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verifier_param,
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&Commitment(g_prime_commit.into_affine()),
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a2.to_vec().as_ref(),
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&tilde_g_eval,
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&proof.g_prime_proof,
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)?;
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end_timer!(open_timer);
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Ok(res)
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}
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#[cfg(test)]
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mod tests {
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use super::*;
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use crate::pcs::{
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prelude::{MultilinearKzgPCS, MultilinearUniversalParams},
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StructuredReferenceString,
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};
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use arithmetic::get_batched_nv;
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use ark_bls12_381::Bls12_381 as E;
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use ark_ec::PairingEngine;
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use ark_poly::{DenseMultilinearExtension, MultilinearExtension};
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use ark_std::{
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rand::{CryptoRng, RngCore},
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test_rng,
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vec::Vec,
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UniformRand,
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};
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type Fr = <E as PairingEngine>::Fr;
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fn test_multi_open_helper<R: RngCore + CryptoRng>(
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ml_params: &MultilinearUniversalParams<E>,
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polys: &[Rc<DenseMultilinearExtension<Fr>>],
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rng: &mut R,
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) -> Result<(), PCSError> {
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let merged_nv = get_batched_nv(polys[0].num_vars(), polys.len());
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let (ml_ck, ml_vk) = ml_params.trim(merged_nv)?;
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let mut points = Vec::new();
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for poly in polys.iter() {
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let point = (0..poly.num_vars())
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.map(|_| Fr::rand(rng))
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.collect::<Vec<Fr>>();
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points.push(point);
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}
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let evals = polys
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.iter()
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.zip(points.iter())
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.map(|(f, p)| f.evaluate(p).unwrap())
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.collect::<Vec<_>>();
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let commitments = polys
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.iter()
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.map(|poly| MultilinearKzgPCS::commit(&ml_ck.clone(), poly).unwrap())
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.collect::<Vec<_>>();
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let mut transcript = IOPTranscript::new("test transcript".as_ref());
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transcript.append_field_element("init".as_ref(), &Fr::zero())?;
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let batch_proof = multi_open_internal::<E, MultilinearKzgPCS<E>>(
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&ml_ck,
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polys,
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&points,
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&evals,
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&mut transcript,
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)?;
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// good path
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let mut transcript = IOPTranscript::new("test transcript".as_ref());
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transcript.append_field_element("init".as_ref(), &Fr::zero())?;
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assert!(batch_verify_internal::<E, MultilinearKzgPCS<E>>(
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&ml_vk,
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&commitments,
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&points,
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&batch_proof,
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&mut transcript
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)?);
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Ok(())
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}
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#[test]
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fn test_multi_open_internal() -> Result<(), PCSError> {
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let mut rng = test_rng();
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let ml_params = MultilinearUniversalParams::<E>::gen_srs_for_testing(&mut rng, 20)?;
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for num_poly in 5..6 {
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for nv in 15..16 {
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let polys1: Vec<_> = (0..num_poly)
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.map(|_| Rc::new(DenseMultilinearExtension::rand(nv, &mut rng)))
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.collect();
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test_multi_open_helper(&ml_params, &polys1, &mut rng)?;
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}
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}
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Ok(())
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}
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}
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