shplonk.hpp

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// === AUDIT STATUS ===
// internal:    { status: Complete, auditors: [Khashayar], commit: }
// external_1:  { status: not started, auditors: [], commit: }
// external_2:  { status: not started, auditors: [], commit: }
// =====================
 
#pragma once
#include "barretenberg/commitment_schemes/claim.hpp"
#include "barretenberg/commitment_schemes/commitment_key.hpp"
#include "barretenberg/commitment_schemes/verification_key.hpp"
#include "barretenberg/common/assert.hpp"
#include "barretenberg/stdlib/primitives/curves/bn254.hpp"
#include "barretenberg/transcript/transcript.hpp"
 
namespace bb {
 
template <typename Curve> class ShplonkProver_ {
    using Fr = typename Curve::ScalarField;
    using Polynomial = bb::Polynomial<Fr>;
 
  public:
    static Polynomial compute_batched_quotient(const size_t virtual_log_n,
                                               std::span<const ProverOpeningClaim<Curve>> opening_claims,
                                               const Fr& nu,
                                               std::span<Fr> gemini_fold_pos_evaluations,
                                               std::span<const ProverOpeningClaim<Curve>> libra_opening_claims,
                                               std::span<const ProverOpeningClaim<Curve>> sumcheck_round_claims)
    {
        // Find the maximum polynomial size among all claims to determine the dyadic size of the batched polynomial.
        size_t max_poly_size{ 0 };
 
        for (const auto& claim_set : { opening_claims, libra_opening_claims, sumcheck_round_claims }) {
            for (const auto& claim : claim_set) {
                max_poly_size = std::max(max_poly_size, claim.polynomial.size());
            }
        }
        // The polynomials in Sumcheck Round claims and Libra opening claims are generally not dyadic,
        // so we round up to the next power of 2.
        max_poly_size = numeric::round_up_power_2(max_poly_size);
 
        // Q(X) = ∑ⱼ νʲ ⋅ ( fⱼ(X) − vⱼ) / ( X − xⱼ )
        Polynomial Q(max_poly_size);
        Polynomial tmp(max_poly_size);
 
        Fr current_nu = Fr::one();
 
        size_t fold_idx = 0;
        for (const auto& claim : opening_claims) {
 
            // Gemini Fold Polynomials have to be opened at -r^{2^j} and r^{2^j}.
            if (claim.gemini_fold) {
                tmp = claim.polynomial;
                tmp.at(0) = tmp[0] - gemini_fold_pos_evaluations[fold_idx++];
                tmp.factor_roots(-claim.opening_pair.challenge);
                // Add the claim quotient to the batched quotient polynomial
                Q.add_scaled(tmp, current_nu);
                current_nu *= nu;
            }
 
            // Compute individual claim quotient tmp = ( fⱼ(X) − vⱼ) / ( X − xⱼ )
            tmp = claim.polynomial;
            tmp.at(0) = tmp[0] - claim.opening_pair.evaluation;
            tmp.factor_roots(claim.opening_pair.challenge);
            // Add the claim quotient to the batched quotient polynomial
            Q.add_scaled(tmp, current_nu);
            current_nu *= nu;
        }
        // We use the same batching challenge for Gemini and Libra opening claims. The number of the claims
        // batched before adding Libra commitments and evaluations is bounded by 2 * `virtual_log_n` + 2, where
        // 2 * `virtual_log_n` is the number of fold claims including the dummy ones, and +2 is reserved for
        // interleaving.
        if (!libra_opening_claims.empty()) {
            current_nu = nu.pow(2 * virtual_log_n + NUM_INTERLEAVING_CLAIMS);
        }
 
        for (const auto& claim : libra_opening_claims) {
            // Compute individual claim quotient tmp = ( fⱼ(X) − vⱼ) / ( X − xⱼ )
            tmp = claim.polynomial;
            tmp.at(0) = tmp[0] - claim.opening_pair.evaluation;
            tmp.factor_roots(claim.opening_pair.challenge);
 
            // Add the claim quotient to the batched quotient polynomial
            Q.add_scaled(tmp, current_nu);
            current_nu *= nu;
        }
 
        for (const auto& claim : sumcheck_round_claims) {
 
            // Compute individual claim quotient tmp = ( fⱼ(X) − vⱼ) / ( X − xⱼ )
            tmp = claim.polynomial;
            tmp.at(0) = tmp[0] - claim.opening_pair.evaluation;
            tmp.factor_roots(claim.opening_pair.challenge);
 
            // Add the claim quotient to the batched quotient polynomial
            Q.add_scaled(tmp, current_nu);
            current_nu *= nu;
        }
        // Return batched quotient polynomial Q(X)
        return Q;
    };
 
    static ProverOpeningClaim<Curve> compute_partially_evaluated_batched_quotient(
        const size_t virtual_log_n,
        std::span<ProverOpeningClaim<Curve>> opening_claims,
        Polynomial& batched_quotient_Q,
        const Fr& nu_challenge,
        const Fr& z_challenge,
        std::span<Fr> gemini_fold_pos_evaluations,
        std::span<ProverOpeningClaim<Curve>> libra_opening_claims = {},
        std::span<ProverOpeningClaim<Curve>> sumcheck_opening_claims = {})
    {
        // Our main use case is the opening of Gemini fold polynomials and each Gemini fold is opened at 2 points.
        const size_t num_gemini_opening_claims = 2 * opening_claims.size();
        const size_t num_opening_claims =
            num_gemini_opening_claims + libra_opening_claims.size() + sumcheck_opening_claims.size();
 
        // {ẑⱼ(z)}ⱼ , where ẑⱼ(r) = 1/zⱼ(z) = 1/(z - xⱼ)
        std::vector<Fr> inverse_vanishing_evals;
        inverse_vanishing_evals.reserve(num_opening_claims);
        for (const auto& claim : opening_claims) {
            if (claim.gemini_fold) {
                inverse_vanishing_evals.emplace_back(z_challenge + claim.opening_pair.challenge);
            }
            inverse_vanishing_evals.emplace_back(z_challenge - claim.opening_pair.challenge);
        }
 
        // Add the terms (z - uₖ) for k = 0, …, d−1 where d is the number of rounds in Sumcheck
        for (const auto& claim : libra_opening_claims) {
            inverse_vanishing_evals.emplace_back(z_challenge - claim.opening_pair.challenge);
        }
 
        for (const auto& claim : sumcheck_opening_claims) {
            inverse_vanishing_evals.emplace_back(z_challenge - claim.opening_pair.challenge);
        }
 
        Fr::batch_invert(inverse_vanishing_evals);
 
        // G(X) = Q(X) - Q_z(X) = Q(X) - ∑ⱼ νʲ ⋅ ( fⱼ(X) − vⱼ) / ( z − xⱼ ),
        // s.t. G(r) = 0
        Polynomial G(std::move(batched_quotient_Q)); // G(X) = Q(X)
 
        // G₀ = ∑ⱼ νʲ ⋅ vⱼ / ( z − xⱼ )
        Fr current_nu = Fr::one();
        Polynomial tmp(G.size());
        size_t idx = 0;
 
        size_t fold_idx = 0;
        for (auto& claim : opening_claims) {
 
            if (claim.gemini_fold) {
                tmp = claim.polynomial;
                tmp.at(0) = tmp[0] - gemini_fold_pos_evaluations[fold_idx++];
                Fr scaling_factor = current_nu * inverse_vanishing_evals[idx++]; // = νʲ / (z − xⱼ )
                // G -= νʲ ⋅ ( fⱼ(X) − vⱼ) / ( z − xⱼ )
                G.add_scaled(tmp, -scaling_factor);
 
                current_nu *= nu_challenge;
            }
            // tmp = νʲ ⋅ ( fⱼ(X) − vⱼ) / ( z − xⱼ )
            claim.polynomial.at(0) = claim.polynomial[0] - claim.opening_pair.evaluation;
            Fr scaling_factor = current_nu * inverse_vanishing_evals[idx++]; // = νʲ / (z − xⱼ )
 
            // G -= νʲ ⋅ ( fⱼ(X) − vⱼ) / ( z − xⱼ )
            G.add_scaled(claim.polynomial, -scaling_factor);
 
            current_nu *= nu_challenge;
        }
 
        // Take into account the constant proof size in Gemini
        if (!libra_opening_claims.empty()) {
            current_nu = nu_challenge.pow(2 * virtual_log_n + NUM_INTERLEAVING_CLAIMS);
        }
 
        for (auto& claim : libra_opening_claims) {
            // Compute individual claim quotient tmp = ( fⱼ(X) − vⱼ) / ( X − xⱼ )
            claim.polynomial.at(0) = claim.polynomial[0] - claim.opening_pair.evaluation;
            Fr scaling_factor = current_nu * inverse_vanishing_evals[idx++]; // = νʲ / (z − xⱼ )
 
            // Add the claim quotient to the batched quotient polynomial
            G.add_scaled(claim.polynomial, -scaling_factor);
            current_nu *= nu_challenge;
        }
 
        for (auto& claim : sumcheck_opening_claims) {
            claim.polynomial.at(0) = claim.polynomial[0] - claim.opening_pair.evaluation;
            Fr scaling_factor = current_nu * inverse_vanishing_evals[idx++]; // = νʲ / (z − xⱼ )
 
            // Add the claim quotient to the batched quotient polynomial
            G.add_scaled(claim.polynomial, -scaling_factor);
            current_nu *= nu_challenge;
        }
        // Return opening pair (z, 0) and polynomial G(X) = Q(X) - Q_z(X)
        return { .polynomial = G, .opening_pair = { .challenge = z_challenge, .evaluation = Fr::zero() } };
    };
    static std::vector<Fr> compute_gemini_fold_pos_evaluations(
        std::span<const ProverOpeningClaim<Curve>> opening_claims)
    {
        std::vector<Fr> gemini_fold_pos_evaluations;
        gemini_fold_pos_evaluations.reserve(opening_claims.size());
 
        for (const auto& claim : opening_claims) {
            if (claim.gemini_fold) {
                // -r^{2^i} is stored in the claim
                const Fr evaluation_point = -claim.opening_pair.challenge;
                // Compute Fold_i(r^{2^i})
                const Fr evaluation = claim.polynomial.evaluate(evaluation_point);
                gemini_fold_pos_evaluations.emplace_back(evaluation);
            }
        }
        return gemini_fold_pos_evaluations;
    }
 
    template <typename Transcript>
    static ProverOpeningClaim<Curve> prove(const CommitmentKey<Curve>& commitment_key,
                                           std::span<ProverOpeningClaim<Curve>> opening_claims,
                                           const std::shared_ptr<Transcript>& transcript,
                                           std::span<ProverOpeningClaim<Curve>> libra_opening_claims = {},
                                           std::span<ProverOpeningClaim<Curve>> sumcheck_round_claims = {},
                                           const size_t virtual_log_n = 0)
    {
        const Fr nu = transcript->template get_challenge<Fr>("Shplonk:nu");
 
        // Compute the evaluations Fold_i(r^{2^i}) for i>0.
        std::vector<Fr> gemini_fold_pos_evaluations = compute_gemini_fold_pos_evaluations(opening_claims);
 
        auto batched_quotient = compute_batched_quotient(virtual_log_n,
                                                         opening_claims,
                                                         nu,
                                                         gemini_fold_pos_evaluations,
                                                         libra_opening_claims,
                                                         sumcheck_round_claims);
        auto batched_quotient_commitment = commitment_key.commit(batched_quotient);
        transcript->send_to_verifier("Shplonk:Q", batched_quotient_commitment);
        const Fr z = transcript->template get_challenge<Fr>("Shplonk:z");
 
        return compute_partially_evaluated_batched_quotient(virtual_log_n,
                                                            opening_claims,
                                                            batched_quotient,
                                                            nu,
                                                            z,
                                                            gemini_fold_pos_evaluations,
                                                            libra_opening_claims,
                                                            sumcheck_round_claims);
    }
};
 
template <typename Curve> class ShplonkVerifier_ {
    using Fr = typename Curve::ScalarField;
    using GroupElement = typename Curve::Element;
    using Commitment = typename Curve::AffineElement;
    using VK = VerifierCommitmentKey<Curve>;
 
    // Random challenges
    std::vector<Fr> pows_of_nu;
    // Commitment to quotient polynomial
    Commitment quotient;
    // Partial evaluation challenge
    Fr z_challenge;
    // Commitments \f$[f_1], \dots, [f_n]\f$
    std::vector<Commitment> commitments;
    // Scalar coefficients of \f$[f_1], \dots, [f_n]\f$ in the MSM needed to compute the commitment to the partially
    // evaluated quotient
    std::vector<Fr> scalars;
    // Coefficient of the identity in partially evaluated quotient
    Fr identity_scalar_coefficient = Fr(0);
    // Target evaluation
    Fr evaluation = Fr(0);
 
  public:
    template <typename Transcript>
    ShplonkVerifier_(std::vector<Commitment>& polynomial_commitments,
                     std::shared_ptr<Transcript>& transcript,
                     const size_t num_claims)
        : pows_of_nu({ Fr(1), transcript->template get_challenge<Fr>("Shplonk:nu") })
        , quotient(transcript->template receive_from_prover<Commitment>("Shplonk:Q"))
        , z_challenge(transcript->template get_challenge<Fr>("Shplonk:z"))
        , commitments({ quotient })
        , scalars{ Fr{ 1 } }
    {
        BB_ASSERT_GT(num_claims, 1U, "Using Shplonk with just one claim. Should use batch reduction.");
        const size_t num_commitments = commitments.size();
        commitments.reserve(num_commitments);
        scalars.reserve(num_commitments);
        pows_of_nu.reserve(num_claims);
 
        commitments.insert(commitments.end(), polynomial_commitments.begin(), polynomial_commitments.end());
        scalars.insert(scalars.end(), commitments.size() - 1, Fr(0)); // Initialized as circuit constants
        // The first two powers of nu have already been initialized, we need another `num_claims - 2` powers to batch
        // all the claims
        for (size_t idx = 0; idx < num_claims - 2; idx++) {
            pows_of_nu.emplace_back(pows_of_nu.back() * pows_of_nu[1]);
        }
 
        if constexpr (Curve::is_stdlib_type) {
            evaluation.convert_constant_to_fixed_witness(pows_of_nu[1].get_context());
        }
    }
 
    OpeningClaim<Curve> finalize(const Commitment& g1_identity)
    {
        commitments.emplace_back(g1_identity);
        scalars.emplace_back(identity_scalar_coefficient);
        GroupElement result;
        if constexpr (Curve::is_stdlib_type) {
            result = GroupElement::batch_mul(commitments, scalars);
        } else {
            result = batch_mul_native<Curve>(commitments, scalars);
        }
 
        return { { z_challenge, evaluation }, result };
    }
 
    // TODO(https://github.com/AztecProtocol/barretenberg/issues/1475): Compute g1_identity inside the function body
    BatchOpeningClaim<Curve> export_batch_opening_claim(const Commitment& g1_identity)
    {
        commitments.emplace_back(g1_identity);
        scalars.emplace_back(identity_scalar_coefficient);
 
        return { commitments, scalars, z_challenge };
    }
 
    template <typename Transcript>
    static ShplonkVerifier_<Curve> reduce_verification_no_finalize(std::span<const OpeningClaim<Curve>> claims,
                                                                   std::shared_ptr<Transcript>& transcript)
    {
        // Initialize Shplonk verifier
        const size_t num_claims = claims.size();
        std::vector<Commitment> polynomial_commiments;
        polynomial_commiments.reserve(num_claims);
        for (const auto& claim : claims) {
            polynomial_commiments.emplace_back(claim.commitment);
        }
        ShplonkVerifier_<Curve> verifier(polynomial_commiments, transcript, num_claims);
 
        // Compute { 1 / (z - x_i) }
        std::vector<Fr> inverse_vanishing_evals;
        inverse_vanishing_evals.reserve(num_claims);
        if constexpr (Curve::is_stdlib_type) {
            for (const auto& claim : claims) {
                inverse_vanishing_evals.emplace_back((verifier.z_challenge - claim.opening_pair.challenge).invert());
            }
        } else {
            for (const auto& claim : claims) {
                inverse_vanishing_evals.emplace_back(verifier.z_challenge - claim.opening_pair.challenge);
            }
            Fr::batch_invert(inverse_vanishing_evals);
        }
 
        // Update the Shplonk verifier state with each claim
        // For each claim: s_i -= ν^i / (z - x_i) and θ += ν^i * v_i / (z - x_i)
        for (size_t idx = 0; idx < claims.size(); idx++) {
            // Compute ν^i / (z - x_i)
            auto scalar_factor = verifier.pows_of_nu[idx] * inverse_vanishing_evals[idx];
            // s_i -= ν^i / (z - x_i)
            verifier.scalars[idx + 1] -= scalar_factor;
            // θ += ν^i * v_i / (z - x_i)
            verifier.identity_scalar_coefficient += scalar_factor * claims[idx].opening_pair.evaluation;
        }
 
        return verifier;
    };
 
    template <typename Transcript>
    static OpeningClaim<Curve> reduce_verification(Commitment g1_identity,
                                                   std::span<const OpeningClaim<Curve>> claims,
                                                   std::shared_ptr<Transcript>& transcript)
    {
        auto verifier = ShplonkVerifier_::reduce_verification_no_finalize(claims, transcript);
        return verifier.finalize(g1_identity);
    };
 
    static std::vector<Fr> compute_inverted_gemini_denominators(const Fr& shplonk_eval_challenge,
                                                                const std::vector<Fr>& gemini_eval_challenge_powers)
    {
        std::vector<Fr> denominators;
        const size_t virtual_log_n = gemini_eval_challenge_powers.size();
        const size_t num_gemini_claims = 2 * virtual_log_n;
        denominators.reserve(num_gemini_claims);
 
        for (const auto& gemini_eval_challenge_power : gemini_eval_challenge_powers) {
            // Place 1/(z - r ^ {2^j})
            denominators.emplace_back(shplonk_eval_challenge - gemini_eval_challenge_power);
            // Place 1/(z + r ^ {2^j})
            denominators.emplace_back(shplonk_eval_challenge + gemini_eval_challenge_power);
        }
 
        if constexpr (!Curve::is_stdlib_type) {
            Fr::batch_invert(denominators);
        } else {
            for (auto& denominator : denominators) {
                denominator = denominator.invert();
            }
        }
        return denominators;
    }
};
 
template <typename Fr>
static std::vector<Fr> compute_shplonk_batching_challenge_powers(const Fr& shplonk_batching_challenge,
                                                                 const size_t virtual_log_n,
                                                                 bool has_zk = false,
                                                                 bool committed_sumcheck = false)
{
    // Minimum size of `denominators`
    size_t num_powers = 2 * virtual_log_n + NUM_INTERLEAVING_CLAIMS;
    // Each round univariate is opened at 0, 1, and a round challenge.
    static constexpr size_t NUM_COMMITTED_SUMCHECK_CLAIMS_PER_ROUND = 3;
 
    // Shplonk evaluation and batching challenges are re-used in SmallSubgroupIPA.
    if (has_zk) {
        num_powers += NUM_SMALL_IPA_EVALUATIONS;
    }
 
    // Commited sumcheck adds 3 claims per round.
    if (committed_sumcheck) {
        num_powers += NUM_COMMITTED_SUMCHECK_CLAIMS_PER_ROUND * virtual_log_n;
    }
 
    std::vector<Fr> result;
    result.reserve(num_powers);
    result.emplace_back(Fr{ 1 });
    for (size_t idx = 1; idx < num_powers; idx++) {
        result.emplace_back(result[idx - 1] * shplonk_batching_challenge);
    }
    return result;
}
} // namespace bb