gemini_impl.hpp

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// === AUDIT STATUS ===
// internal:    { status: Planned, auditors: [Khashayar], commit: }
// external_1:  { status: not started, auditors: [], commit: }
// external_2:  { status: not started, auditors: [], commit: }
// =====================
 
#pragma once
#include "barretenberg/common/bb_bench.hpp"
#include "barretenberg/common/thread.hpp"
#include "gemini.hpp"
 
namespace bb {
template <typename Curve>
template <typename Transcript>
std::vector<typename GeminiProver_<Curve>::Claim> GeminiProver_<Curve>::prove(
    size_t circuit_size,
    PolynomialBatcher& polynomial_batcher,
    std::span<Fr> multilinear_challenge,
    const CommitmentKey<Curve>& commitment_key,
    const std::shared_ptr<Transcript>& transcript,
    bool has_zk)
{
    // To achieve fixed proof size in Ultra and Mega, the multilinear opening challenge is be padded to a fixed size.
    const size_t virtual_log_n = multilinear_challenge.size();
    const size_t log_n = numeric::get_msb(circuit_size);
 
    // Get the batching challenge
    const Fr rho = transcript->template get_challenge<Fr>("rho");
 
    Polynomial A_0 = polynomial_batcher.compute_batched(rho);
 
    // Construct the d-1 Gemini foldings of A₀(X)
    std::vector<Polynomial> fold_polynomials = compute_fold_polynomials(log_n, multilinear_challenge, A_0, has_zk);
 
    // If virtual_log_n >= log_n, pad the fold commitments with dummy group elements [1]_1.
    for (size_t l = 0; l < virtual_log_n - 1; l++) {
        std::string label = "Gemini:FOLD_" + std::to_string(l + 1);
        // When has_zk is true, we are sending commitments to 0. Seems to work, but maybe brittle.
        transcript->send_to_verifier(label, commitment_key.commit(fold_polynomials[l]));
    }
    const Fr r_challenge = transcript->template get_challenge<Fr>("Gemini:r");
 
    const bool gemini_challenge_in_small_subgroup = (has_zk) && (r_challenge.pow(Curve::SUBGROUP_SIZE) == Fr(1));
 
    // If Gemini evaluation challenge lands in the multiplicative subgroup used by SmallSubgroupIPA protocol, the
    // evaluations of prover polynomials at this challenge would leak witness data.
    // TODO(https://github.com/AztecProtocol/barretenberg/issues/1194). Handle edge cases in PCS
    if (gemini_challenge_in_small_subgroup) {
        throw_or_abort("Gemini evaluation challenge is in the SmallSubgroup.");
    }
 
    // Compute polynomials A₀₊(X) = F(X) + G(X)/r and A₀₋(X) = F(X) - G(X)/r
    auto [A_0_pos, A_0_neg] = polynomial_batcher.compute_partially_evaluated_batch_polynomials(r_challenge);
    // Construct claims for the d + 1 univariate evaluations A₀₊(r), A₀₋(-r), and Foldₗ(−r^{2ˡ}), l = 1, ..., d-1
    std::vector<Claim> claims = construct_univariate_opening_claims(
        virtual_log_n, std::move(A_0_pos), std::move(A_0_neg), std::move(fold_polynomials), r_challenge);
 
    for (size_t l = 1; l <= virtual_log_n; l++) {
        std::string label = "Gemini:a_" + std::to_string(l);
        transcript->send_to_verifier(label, claims[l].opening_pair.evaluation);
    }
 
    // If running Gemini for the Translator VM polynomials, A₀(r) = A₀₊(r) + P₊(rˢ) and A₀(-r) = A₀₋(-r) + P₋(rˢ)
    // where s is the size of the interleaved group assumed even. The prover sends P₊(rˢ) and P₋(rˢ) to the verifier
    // so it can reconstruct the evaluation of A₀(r) and A₀(-r) respectively
    // TODO(https://github.com/AztecProtocol/barretenberg/issues/1282)
    if (polynomial_batcher.has_interleaved()) {
        auto [P_pos, P_neg] = polynomial_batcher.compute_partially_evaluated_interleaved_polynomial(r_challenge);
        Fr r_pow = r_challenge.pow(polynomial_batcher.get_group_size());
        Fr P_pos_eval = P_pos.evaluate(r_pow);
        Fr P_neg_eval = P_neg.evaluate(r_pow);
        claims.emplace_back(Claim{ std::move(P_pos), { r_pow, P_pos_eval } });
        transcript->send_to_verifier("Gemini:P_pos", P_pos_eval);
        claims.emplace_back(Claim{ std::move(P_neg), { r_pow, P_neg_eval } });
        transcript->send_to_verifier("Gemini:P_neg", P_neg_eval);
    }
 
    return claims;
};
 
template <typename Curve>
std::vector<typename GeminiProver_<Curve>::Polynomial> GeminiProver_<Curve>::compute_fold_polynomials(
    const size_t log_n, std::span<const Fr> multilinear_challenge, const Polynomial& A_0, const bool& has_zk)
{
    BB_BENCH_NAME("Gemini::compute_fold_polynomials");
    const size_t virtual_log_n = multilinear_challenge.size();
 
    // Cost per iteration: 1 subtraction + 1 multiplication + 1 addition
    constexpr size_t fold_iteration_cost =
        (2 * thread_heuristics::FF_ADDITION_COST) + thread_heuristics::FF_MULTIPLICATION_COST;
 
    // Reserve and allocate space for m-1 Fold polynomials, the foldings of the full batched polynomial A₀
    std::vector<Polynomial> fold_polynomials;
    fold_polynomials.reserve(virtual_log_n - 1);
    for (size_t l = 0; l < log_n - 1; ++l) {
        // size of the previous polynomial/2
        const size_t n_l = 1 << (log_n - l - 1);
 
        // A_l_fold = Aₗ₊₁(X) = (1-uₗ)⋅even(Aₗ)(X) + uₗ⋅odd(Aₗ)(X)
        fold_polynomials.emplace_back(Polynomial(n_l));
    }
 
    // A_l = Aₗ(X) is the polynomial being folded
    // in the first iteration, we take the batched polynomial
    // in the next iteration, it is the previously folded one
    auto A_l = A_0.data();
    for (size_t l = 0; l < log_n - 1; ++l) {
        // size of the previous polynomial/2
        const size_t n_l = 1 << (log_n - l - 1);
 
        // Opening point is the same for all
        const Fr u_l = multilinear_challenge[l];
 
        // A_l_fold = Aₗ₊₁(X) = (1-uₗ)⋅even(Aₗ)(X) + uₗ⋅odd(Aₗ)(X)
        auto A_l_fold = fold_polynomials[l].data();
 
        parallel_for_heuristic(
            n_l,
            [&](size_t j) {
                // fold(Aₗ)[j] = (1-uₗ)⋅even(Aₗ)[j] + uₗ⋅odd(Aₗ)[j]
                //            = (1-uₗ)⋅Aₗ[2j]      + uₗ⋅Aₗ[2j+1]
                //            = Aₗ₊₁[j]
                A_l_fold[j] = A_l[j << 1] + u_l * (A_l[(j << 1) + 1] - A_l[j << 1]);
            },
            fold_iteration_cost);
        // set Aₗ₊₁ = Aₗ for the next iteration
        A_l = A_l_fold;
    }
 
    // Perform virtual rounds.
    // After the first `log_n - 1` rounds, the prover's `fold` univariates stabilize. With ZK, the verifier multiplies
    // the evaluations by 0, otherwise, when `virtual_log_n > log_n`, the prover honestly computes and sends the
    // constant folds.
    const auto& last = fold_polynomials.back();
    const Fr u_last = multilinear_challenge[log_n - 1];
    const Fr final_eval = last.at(0) + u_last * (last.at(1) - last.at(0));
    Polynomial const_fold(1);
    // Temporary fix: when we're running a zk proof, the verifier uses a `padding_indicator_array`. So the evals in
    // rounds past `log_n - 1` will be ignored. Hence the prover also needs to ignore them, otherwise Shplonk will fail.
    const_fold.at(0) = final_eval * Fr(static_cast<int>(!has_zk));
    fold_polynomials.emplace_back(const_fold);
 
    // FOLD_{log_n+1}, ..., FOLD_{d_v-1}
    Fr tail = Fr(1);
    for (size_t k = log_n; k < virtual_log_n - 1; ++k) {
        tail *= (Fr(1) - multilinear_challenge[k]); // multiply by (1 - u_k)
        Polynomial next_const(1);
        next_const.at(0) = final_eval * tail * Fr(static_cast<int>(!has_zk));
        fold_polynomials.emplace_back(next_const);
    }
 
    return fold_polynomials;
};
 
template <typename Curve>
std::vector<typename GeminiProver_<Curve>::Claim> GeminiProver_<Curve>::construct_univariate_opening_claims(
    const size_t log_n,
    Polynomial&& A_0_pos,
    Polynomial&& A_0_neg,
    std::vector<Polynomial>&& fold_polynomials,
    const Fr& r_challenge)
{
    std::vector<Claim> claims;
 
    // Compute evaluation of partially evaluated batch polynomial (positive) A₀₊(r)
    Fr a_0_pos = A_0_pos.evaluate(r_challenge);
    claims.emplace_back(Claim{ std::move(A_0_pos), { r_challenge, a_0_pos } });
    // Compute evaluation of partially evaluated batch polynomial (negative) A₀₋(-r)
    Fr a_0_neg = A_0_neg.evaluate(-r_challenge);
    claims.emplace_back(Claim{ std::move(A_0_neg), { -r_challenge, a_0_neg } });
 
    // Compute univariate opening queries rₗ = r^{2ˡ} for l = 0, 1, ..., m-1
    std::vector<Fr> r_squares = gemini::powers_of_evaluation_challenge(r_challenge, log_n);
 
    // Each fold polynomial Aₗ has to be opened at −r^{2ˡ} and r^{2ˡ}. To avoid storing two copies of Aₗ for l = 1,...,
    // m-1, we use a flag that is processed by ShplonkProver.
    const bool gemini_fold = true;
 
    // Compute the remaining m opening pairs {−r^{2ˡ}, Aₗ(−r^{2ˡ})}, l = 1, ..., m-1.
    for (size_t l = 0; l < log_n - 1; ++l) {
        Fr evaluation = fold_polynomials[l].evaluate(-r_squares[l + 1]);
        claims.emplace_back(Claim{ std::move(fold_polynomials[l]), { -r_squares[l + 1], evaluation }, gemini_fold });
    }
 
    return claims;
};
 
} // namespace bb