shplemini.hpp

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// === AUDIT STATUS ===
// internal:    { status: not started, auditors: [], date: YYYY-MM-DD }
// external_1:  { status: not started, auditors: [], date: YYYY-MM-DD }
// external_2:  { status: not started, auditors: [], date: YYYY-MM-DD }
// =====================
 
#pragma once
#include "barretenberg/commitment_schemes/claim.hpp"
#include "barretenberg/commitment_schemes/claim_batcher.hpp"
#include "barretenberg/commitment_schemes/commitment_key.hpp"
#include "barretenberg/commitment_schemes/gemini/gemini_impl.hpp"
#include "barretenberg/commitment_schemes/shplonk/shplonk.hpp"
#include "barretenberg/commitment_schemes/small_subgroup_ipa/small_subgroup_ipa.hpp"
#include "barretenberg/commitment_schemes/verification_key.hpp"
#include "barretenberg/flavor/repeated_commitments_data.hpp"
#include "barretenberg/sumcheck/zk_sumcheck_data.hpp"
#include "barretenberg/transcript/transcript.hpp"
 
namespace bb {
 
template <typename Curve> class ShpleminiProver_ {
  public:
    using FF = typename Curve::ScalarField;
    using GroupElement = typename Curve::Element;
    using Commitment = typename Curve::AffineElement;
    using Polynomial = bb::Polynomial<FF>;
    using OpeningClaim = ProverOpeningClaim<Curve>;
 
    using VK = CommitmentKey<Curve>;
    using ShplonkProver = ShplonkProver_<Curve>;
    using GeminiProver = GeminiProver_<Curve>;
    using PolynomialBatcher = GeminiProver::PolynomialBatcher;
 
    template <typename Transcript>
    static OpeningClaim prove(const FF circuit_size,
                              PolynomialBatcher& polynomial_batcher,
                              std::span<FF> multilinear_challenge,
                              const CommitmentKey<Curve>& commitment_key,
                              const std::shared_ptr<Transcript>& transcript,
                              const std::array<Polynomial, NUM_SMALL_IPA_EVALUATIONS>& libra_polynomials = {},
                              const std::vector<Polynomial>& sumcheck_round_univariates = {},
                              const std::vector<std::array<FF, 3>>& sumcheck_round_evaluations = {})
    {
        // While Shplemini is not templated on Flavor, we derive ZK flag this way
        const bool has_zk = (libra_polynomials[0].size() > 0);
 
        // When padding is enabled, the size of the multilinear challenge may be bigger than the log of `circuit_size`.
        const size_t virtual_log_n = multilinear_challenge.size();
 
        std::vector<OpeningClaim> opening_claims = GeminiProver::prove(
            circuit_size, polynomial_batcher, multilinear_challenge, commitment_key, transcript, has_zk);
        // Create opening claims for Libra masking univariates and Sumcheck Round Univariates
        std::vector<OpeningClaim> libra_opening_claims;
 
        if (has_zk) {
            const auto gemini_r = opening_claims[0].opening_pair.challenge;
            libra_opening_claims = compute_libra_opening_claims(gemini_r, libra_polynomials, transcript);
        }
 
        // Currently, only used in ECCVM.
        std::vector<OpeningClaim> sumcheck_round_claims;
 
        if (!sumcheck_round_univariates.empty()) {
            sumcheck_round_claims = compute_sumcheck_round_claims(
                circuit_size, multilinear_challenge, sumcheck_round_univariates, sumcheck_round_evaluations);
        }
 
        return ShplonkProver::prove(
            commitment_key, opening_claims, transcript, libra_opening_claims, sumcheck_round_claims, virtual_log_n);
    };
 
    template <typename Transcript>
    static std::vector<OpeningClaim> compute_libra_opening_claims(
        const FF gemini_r,
        const std::array<Polynomial, NUM_SMALL_IPA_EVALUATIONS>& libra_polynomials,
        const std::shared_ptr<Transcript>& transcript)
    {
        OpeningClaim new_claim;
 
        std::vector<OpeningClaim> libra_opening_claims = {};
 
        static constexpr FF subgroup_generator = Curve::subgroup_generator;
 
        std::array<std::string, NUM_SMALL_IPA_EVALUATIONS> libra_eval_labels = {
            "Libra:concatenation_eval", "Libra:shifted_grand_sum_eval", "Libra:grand_sum_eval", "Libra:quotient_eval"
        };
        const std::array<FF, NUM_SMALL_IPA_EVALUATIONS> evaluation_points = {
            gemini_r, gemini_r * subgroup_generator, gemini_r, gemini_r
        };
        for (size_t idx = 0; idx < NUM_SMALL_IPA_EVALUATIONS; idx++) {
            new_claim.polynomial = std::move(libra_polynomials[idx]);
            new_claim.opening_pair.challenge = evaluation_points[idx];
            new_claim.opening_pair.evaluation = new_claim.polynomial.evaluate(evaluation_points[idx]);
            transcript->send_to_verifier(libra_eval_labels[idx], new_claim.opening_pair.evaluation);
            libra_opening_claims.push_back(new_claim);
        }
 
        return libra_opening_claims;
    }
 
    static std::vector<OpeningClaim> compute_sumcheck_round_claims(
        const FF circuit_size,
        std::span<FF> multilinear_challenge,
        const std::vector<Polynomial>& sumcheck_round_univariates,
        const std::vector<std::array<FF, 3>>& sumcheck_round_evaluations)
    {
        OpeningClaim new_claim;
        std::vector<OpeningClaim> sumcheck_round_claims = {};
 
        const size_t log_n = numeric::get_msb(static_cast<uint32_t>(circuit_size));
        for (size_t idx = 0; idx < log_n; idx++) {
            const std::vector<FF> evaluation_points = { FF(0), FF(1), multilinear_challenge[idx] };
            size_t eval_idx = 0;
            new_claim.polynomial = std::move(sumcheck_round_univariates[idx]);
 
            for (auto& eval_point : evaluation_points) {
                new_claim.opening_pair.challenge = eval_point;
                new_claim.opening_pair.evaluation = sumcheck_round_evaluations[idx][eval_idx];
                sumcheck_round_claims.push_back(new_claim);
                eval_idx++;
            }
        }
 
        return sumcheck_round_claims;
    }
};
template <typename Curve> class ShpleminiVerifier_ {
    using Fr = typename Curve::ScalarField;
    using GroupElement = typename Curve::Element;
    using Commitment = typename Curve::AffineElement;
    using VK = VerifierCommitmentKey<Curve>;
    using ShplonkVerifier = ShplonkVerifier_<Curve>;
    using GeminiVerifier = GeminiVerifier_<Curve>;
    using ClaimBatcher = ClaimBatcher_<Curve>;
 
  public:
    template <typename Transcript>
    static BatchOpeningClaim<Curve> compute_batch_opening_claim(
        std::span<const Fr> padding_indicator_array,
        ClaimBatcher& claim_batcher,
        const std::vector<Fr>& multivariate_challenge,
        const Commitment& g1_identity,
        const std::shared_ptr<Transcript>& transcript,
        const RepeatedCommitmentsData& repeated_commitments = {},
        const bool has_zk = false,
        bool* consistency_checked = nullptr, // TODO(https://github.com/AztecProtocol/barretenberg/issues/1191).
                                             // Shplemini Refactoring: Remove bool pointer
        const std::array<Commitment, NUM_LIBRA_COMMITMENTS>& libra_commitments = {},
        const Fr& libra_univariate_evaluation = Fr{ 0 },
        const std::vector<Commitment>& sumcheck_round_commitments = {},
        const std::vector<std::array<Fr, 3>>& sumcheck_round_evaluations = {})
 
    {
        // TODO(https://github.com/AztecProtocol/barretenberg/issues/1463): Investigate whether this function can be
        // simplified using the new Shplonk api
        const size_t virtual_log_n = multivariate_challenge.size();
 
        const bool committed_sumcheck = !sumcheck_round_evaluations.empty();
 
        Fr batched_evaluation = Fr{ 0 };
 
        // While Shplemini is not templated on Flavor, we derive ZK flag this way
        Commitment hiding_polynomial_commitment;
        if (has_zk) {
            hiding_polynomial_commitment =
                transcript->template receive_from_prover<Commitment>("Gemini:masking_poly_comm");
            batched_evaluation = transcript->template receive_from_prover<Fr>("Gemini:masking_poly_eval");
        }
 
        // Get the challenge ρ to batch commitments to multilinear polynomials and their shifts
        const Fr gemini_batching_challenge = transcript->template get_challenge<Fr>("rho");
 
        // Process Gemini transcript data:
        // - Get Gemini commitments (com(A₁), com(A₂), … , com(Aₙ₋₁))
        const std::vector<Commitment> fold_commitments =
            GeminiVerifier::get_fold_commitments(virtual_log_n, transcript);
        // - Get Gemini evaluation challenge for Aᵢ, i = 0, … , d−1
        const Fr gemini_evaluation_challenge = transcript->template get_challenge<Fr>("Gemini:r");
 
        // - Get evaluations (A₀(−r), A₁(−r²), ... , Aₙ₋₁(−r²⁽ⁿ⁻¹⁾))
        const std::vector<Fr> gemini_fold_neg_evaluations =
            GeminiVerifier::get_gemini_evaluations(virtual_log_n, transcript);
 
        // Get evaluations of partially evaluated batched interleaved polynomials P₊(rˢ) and P₋((-r)ˢ)
        Fr p_pos = Fr(0);
        Fr p_neg = Fr(0);
        if (claim_batcher.interleaved) {
            p_pos = transcript->template receive_from_prover<Fr>("Gemini:P_pos");
            p_neg = transcript->template receive_from_prover<Fr>("Gemini:P_neg");
        }
 
        // - Compute vector (r, r², ... , r^{2^{d-1}}), where d = log_n
        const std::vector<Fr> gemini_eval_challenge_powers =
            gemini::powers_of_evaluation_challenge(gemini_evaluation_challenge, virtual_log_n);
 
        std::array<Fr, NUM_SMALL_IPA_EVALUATIONS> libra_evaluations;
        if (has_zk) {
            libra_evaluations[0] = transcript->template receive_from_prover<Fr>("Libra:concatenation_eval");
            libra_evaluations[1] = transcript->template receive_from_prover<Fr>("Libra:shifted_grand_sum_eval");
            libra_evaluations[2] = transcript->template receive_from_prover<Fr>("Libra:grand_sum_eval");
            libra_evaluations[3] = transcript->template receive_from_prover<Fr>("Libra:quotient_eval");
        }
 
        // Process Shplonk transcript data:
        // - Get Shplonk batching challenge
        const Fr shplonk_batching_challenge = transcript->template get_challenge<Fr>("Shplonk:nu");
 
        // Compute the powers of ν that are required for batching Gemini, SmallSubgroupIPA, and committed sumcheck
        // univariate opening claims.
        const std::vector<Fr> shplonk_batching_challenge_powers = compute_shplonk_batching_challenge_powers(
            shplonk_batching_challenge, virtual_log_n, has_zk, committed_sumcheck);
        // - Get the quotient commitment for the Shplonk batching of Gemini opening claims
        const auto Q_commitment = transcript->template receive_from_prover<Commitment>("Shplonk:Q");
 
        // Start populating the vector (Q, f₀, ... , fₖ₋₁, g₀, ... , gₘ₋₁, com(A₁), ... , com(A_{d-1}), [1]₁) where fᵢ
        // are the k commitments to unshifted polynomials and gⱼ are the m commitments to shifted polynomials
        std::vector<Commitment> commitments{ Q_commitment };
 
        // Get Shplonk opening point z
        const Fr shplonk_evaluation_challenge = transcript->template get_challenge<Fr>("Shplonk:z");
 
        // Start computing the scalar to be multiplied by [1]₁
        Fr constant_term_accumulator = Fr(0);
 
        // Initialize the vector of scalars placing the scalar 1 correposnding to Q_commitment
        std::vector<Fr> scalars;
 
        scalars.emplace_back(Fr(1));
 
        // Compute 1/(z − r), 1/(z + r), 1/(z - r²),  1/(z + r²), … , 1/(z - r^{2^{d-1}}), 1/(z + r^{2^{d-1}})
        // These represent the denominators of the summand terms in Shplonk partially evaluated polynomial Q_z
        const std::vector<Fr> inverse_vanishing_evals = ShplonkVerifier::compute_inverted_gemini_denominators(
            shplonk_evaluation_challenge, gemini_eval_challenge_powers);
 
        // Compute the additional factors to be multiplied with unshifted and shifted commitments when lazily
        // reconstructing the commitment of Q_z
        claim_batcher.compute_scalars_for_each_batch(
            inverse_vanishing_evals, shplonk_batching_challenge, gemini_evaluation_challenge);
 
        if (has_zk) {
            commitments.emplace_back(hiding_polynomial_commitment);
            scalars.emplace_back(-claim_batcher.get_unshifted_batch_scalar()); // corresponds to ρ⁰
        }
 
        // Place the commitments to prover polynomials in the commitments vector. Compute the evaluation of the
        // batched multilinear polynomial. Populate the vector of scalars for the final batch mul
 
        Fr gemini_batching_challenge_power = Fr(1);
        if (has_zk) {
            // ρ⁰ is used to batch the hiding polynomial which has already been added to the commitments vector
            gemini_batching_challenge_power *= gemini_batching_challenge;
        }
 
        // Compute the Shplonk batching power for the interleaved claims. This is \nu^{d+1} where d = log_n as the
        // interleaved claims are sent after the rest of Gemini fold claims. Add the evaluations of (P₊(rˢ) ⋅ ν^{d+1}) /
        // (z − r^s) and (P₋(rˢ) ⋅ ν^{d+2})/(z − r^s) to the constant term accumulator
        Fr shplonk_batching_pos = Fr{ 0 };
        Fr shplonk_batching_neg = Fr{ 0 };
        if (claim_batcher.interleaved) {
            // Currently, the prover places the Interleaving claims before the Gemini dummy claims.
            // TODO(https://github.com/AztecProtocol/barretenberg/issues/1293): Decouple Gemini from Interleaving.
            const size_t interleaved_pos_index = 2 * virtual_log_n;
            const size_t interleaved_neg_index = interleaved_pos_index + 1;
            shplonk_batching_pos = shplonk_batching_challenge_powers[interleaved_pos_index];
            shplonk_batching_neg = shplonk_batching_challenge_powers[interleaved_neg_index];
            constant_term_accumulator += claim_batcher.interleaved->shplonk_denominator *
                                         (p_pos * shplonk_batching_pos + p_neg * shplonk_batching_neg);
        }
        // Update the commitments and scalars vectors as well as the batched evaluation given the present batches
        claim_batcher.update_batch_mul_inputs_and_batched_evaluation(commitments,
                                                                     scalars,
                                                                     batched_evaluation,
                                                                     gemini_batching_challenge,
                                                                     gemini_batching_challenge_power,
                                                                     shplonk_batching_pos,
                                                                     shplonk_batching_neg);
 
        // Reconstruct Aᵢ(r²ⁱ) for i=0, ..., d - 1 from the batched evaluation of the multilinear polynomials and
        // Aᵢ(−r²ⁱ) for i = 0, ..., d - 1. In the case of interleaving, we compute A₀(r) as A₀₊(r) + P₊(r^s).
        const std::vector<Fr> gemini_fold_pos_evaluations =
            GeminiVerifier_<Curve>::compute_fold_pos_evaluations(padding_indicator_array,
                                                                 batched_evaluation,
                                                                 multivariate_challenge,
                                                                 gemini_eval_challenge_powers,
                                                                 gemini_fold_neg_evaluations,
                                                                 p_neg);
 
        // Place the commitments to Gemini fold polynomials Aᵢ in the vector of batch_mul commitments, compute the
        // contributions from Aᵢ(−r²ⁱ) for i=1, … , d − 1 to the constant term accumulator, add corresponding scalars
        // for the batch mul
        batch_gemini_claims_received_from_prover(padding_indicator_array,
                                                 fold_commitments,
                                                 gemini_fold_neg_evaluations,
                                                 gemini_fold_pos_evaluations,
                                                 inverse_vanishing_evals,
                                                 shplonk_batching_challenge_powers,
                                                 commitments,
                                                 scalars,
                                                 constant_term_accumulator);
        const Fr full_a_0_pos = gemini_fold_pos_evaluations[0];
        // Retrieve  the contribution without P₊(r^s)
        Fr a_0_pos = full_a_0_pos - p_pos;
        // Add contributions from A₀₊(r) and  A₀₋(-r) to constant_term_accumulator:
        //  Add  A₀₊(r)/(z−r) to the constant term accumulator
        constant_term_accumulator += a_0_pos * inverse_vanishing_evals[0];
        // Add  A₀₋(-r)/(z+r) to the constant term accumulator
        constant_term_accumulator +=
            gemini_fold_neg_evaluations[0] * shplonk_batching_challenge * inverse_vanishing_evals[1];
 
        remove_repeated_commitments(commitments, scalars, repeated_commitments, has_zk);
 
        // For ZK flavors, the sumcheck output contains the evaluations of Libra univariates that submitted to the
        // ShpleminiVerifier, otherwise this argument is set to be empty
        if (has_zk) {
            add_zk_data(virtual_log_n,
                        commitments,
                        scalars,
                        constant_term_accumulator,
                        libra_commitments,
                        libra_evaluations,
                        gemini_evaluation_challenge,
                        shplonk_batching_challenge_powers,
                        shplonk_evaluation_challenge);
 
            *consistency_checked = SmallSubgroupIPAVerifier<Curve>::check_libra_evaluations_consistency(
                libra_evaluations, gemini_evaluation_challenge, multivariate_challenge, libra_univariate_evaluation);
        }
 
        // Currently, only used in ECCVM
        if (committed_sumcheck) {
            batch_sumcheck_round_claims(commitments,
                                        scalars,
                                        constant_term_accumulator,
                                        multivariate_challenge,
                                        shplonk_batching_challenge_powers,
                                        shplonk_evaluation_challenge,
                                        sumcheck_round_commitments,
                                        sumcheck_round_evaluations);
        }
 
        // Finalize the batch opening claim
        commitments.emplace_back(g1_identity);
        scalars.emplace_back(constant_term_accumulator);
 
        return { commitments, scalars, shplonk_evaluation_challenge };
    };
 
    static void batch_gemini_claims_received_from_prover(std::span<const Fr> padding_indicator_array,
                                                         const std::vector<Commitment>& fold_commitments,
                                                         std::span<const Fr> gemini_neg_evaluations,
                                                         std::span<const Fr> gemini_pos_evaluations,
                                                         std::span<const Fr> inverse_vanishing_evals,
                                                         std::span<const Fr> shplonk_batching_challenge_powers,
                                                         std::vector<Commitment>& commitments,
                                                         std::vector<Fr>& scalars,
                                                         Fr& constant_term_accumulator)
    {
        const size_t virtual_log_n = gemini_neg_evaluations.size();
        // Start from 1, because the commitment to A_0 is reconstructed from the commitments to the multilinear
        // polynomials. The corresponding evaluations are also handled separately.
        for (size_t j = 1; j < virtual_log_n; ++j) {
            // The index of 1/ (z - r^{2^{j}}) in the vector of inverted Gemini denominators
            const size_t pos_index = 2 * j;
            // The index of 1/ (z + r^{2^{j}}) in the vector of inverted Gemini denominators
            const size_t neg_index = 2 * j + 1;
 
            // Compute the "positive" scaling factor  (ν^{2j}) / (z - r^{2^{j}})
            Fr scaling_factor_pos = shplonk_batching_challenge_powers[pos_index] * inverse_vanishing_evals[pos_index];
            // Compute the "negative" scaling factor  (ν^{2j+1}) / (z + r^{2^{j}})
            Fr scaling_factor_neg = shplonk_batching_challenge_powers[neg_index] * inverse_vanishing_evals[neg_index];
 
            // Accumulate the const term contribution given by
            // v^{2j} * A_j(r^{2^j}) /(z - r^{2^j}) + v^{2j+1} * A_j(-r^{2^j}) /(z+ r^{2^j})
            constant_term_accumulator +=
                scaling_factor_neg * gemini_neg_evaluations[j] + scaling_factor_pos * gemini_pos_evaluations[j];
 
            // Place the scaling factor to the 'scalars' vector
            scalars.emplace_back(-padding_indicator_array[j] * (scaling_factor_neg + scaling_factor_pos));
            // Move com(Aᵢ) to the 'commitments' vector
            commitments.emplace_back(std::move(fold_commitments[j - 1]));
        }
    }
 
    // TODO(https://github.com/AztecProtocol/barretenberg/issues/1151) Avoid erasing vector elements.
    static void remove_repeated_commitments(std::vector<Commitment>& commitments,
                                            std::vector<Fr>& scalars,
                                            const RepeatedCommitmentsData& repeated_commitments,
                                            bool has_zk)
    {
        // We started populating commitments and scalars by adding Shplonk:Q commitmment and the corresponding scalar
        // factor 1. In the case of ZK, we also added Gemini:masking_poly_comm before populating the vector with
        // commitments to prover polynomials
        const size_t offset = has_zk ? 2 : 1;
 
        // Extract the indices from the container, which is normally created in a given Flavor
        const size_t& first_range_to_be_shifted_start = repeated_commitments.first_range_to_be_shifted_start + offset;
        const size_t& first_range_shifted_start = repeated_commitments.first_range_shifted_start + offset;
        const size_t& first_range_size = repeated_commitments.first_range_size;
 
        const size_t& second_range_to_be_shifted_start = repeated_commitments.second_range_to_be_shifted_start + offset;
        const size_t& second_range_shifted_start = repeated_commitments.second_range_shifted_start + offset;
        const size_t& second_range_size = repeated_commitments.second_range_size;
 
        // Iterate over the first range of to-be-shifted scalars and their shifted counterparts
        for (size_t i = 0; i < first_range_size; i++) {
            size_t idx_to_be_shifted = i + first_range_to_be_shifted_start;
            size_t idx_shifted = i + first_range_shifted_start;
            scalars[idx_to_be_shifted] = scalars[idx_to_be_shifted] + scalars[idx_shifted];
        }
 
        // Iterate over the second range of to-be-shifted precomputed scalars and their shifted counterparts (if
        // provided)
        for (size_t i = 0; i < second_range_size; i++) {
            size_t idx_to_be_shifted = i + second_range_to_be_shifted_start;
            size_t idx_shifted = i + second_range_shifted_start;
            scalars[idx_to_be_shifted] = scalars[idx_to_be_shifted] + scalars[idx_shifted];
        }
 
        if (second_range_shifted_start > first_range_shifted_start) {
            // Erase the shifted scalars and commitments from the second range (if provided)
            for (size_t i = 0; i < second_range_size; ++i) {
                scalars.erase(scalars.begin() + static_cast<std::ptrdiff_t>(second_range_shifted_start));
                commitments.erase(commitments.begin() + static_cast<std::ptrdiff_t>(second_range_shifted_start));
            }
 
            // Erase the shifted scalars and commitments from the first range
            for (size_t i = 0; i < first_range_size; ++i) {
                scalars.erase(scalars.begin() + static_cast<std::ptrdiff_t>(first_range_shifted_start));
                commitments.erase(commitments.begin() + static_cast<std::ptrdiff_t>(first_range_shifted_start));
            }
        } else {
            // Erase the shifted scalars and commitments from the first range
            for (size_t i = 0; i < first_range_size; ++i) {
                scalars.erase(scalars.begin() + static_cast<std::ptrdiff_t>(first_range_shifted_start));
                commitments.erase(commitments.begin() + static_cast<std::ptrdiff_t>(first_range_shifted_start));
            }
            // Erase the shifted scalars and commitments from the second range (if provided)
            for (size_t i = 0; i < second_range_size; ++i) {
                scalars.erase(scalars.begin() + static_cast<std::ptrdiff_t>(second_range_shifted_start));
                commitments.erase(commitments.begin() + static_cast<std::ptrdiff_t>(second_range_shifted_start));
            }
        }
    }
 
    static void add_zk_data(const size_t virtual_log_n,
                            std::vector<Commitment>& commitments,
                            std::vector<Fr>& scalars,
                            Fr& constant_term_accumulator,
                            const std::array<Commitment, NUM_LIBRA_COMMITMENTS>& libra_commitments,
                            const std::array<Fr, NUM_SMALL_IPA_EVALUATIONS>& libra_evaluations,
                            const Fr& gemini_evaluation_challenge,
                            const std::vector<Fr>& shplonk_batching_challenge_powers,
                            const Fr& shplonk_evaluation_challenge)
 
    {
        // add Libra commitments to the vector of commitments
        for (size_t idx = 0; idx < libra_commitments.size(); idx++) {
            commitments.push_back(libra_commitments[idx]);
        }
 
        // compute corresponding scalars and the correction to the constant term
        std::array<Fr, NUM_SMALL_IPA_EVALUATIONS> denominators;
        std::array<Fr, NUM_SMALL_IPA_EVALUATIONS> batching_scalars;
        // compute Shplonk denominators and invert them
        denominators[0] = Fr(1) / (shplonk_evaluation_challenge - gemini_evaluation_challenge);
        denominators[1] =
            Fr(1) / (shplonk_evaluation_challenge - Fr(Curve::subgroup_generator) * gemini_evaluation_challenge);
        denominators[2] = denominators[0];
        denominators[3] = denominators[0];
 
        // compute the scalars to be multiplied against the commitments [libra_concatenated], [grand_sum], [grand_sum],
        // and [libra_quotient]
        for (size_t idx = 0; idx < NUM_SMALL_IPA_EVALUATIONS; idx++) {
            Fr scaling_factor = denominators[idx] *
                                shplonk_batching_challenge_powers[2 * virtual_log_n + NUM_INTERLEAVING_CLAIMS + idx];
            batching_scalars[idx] = -scaling_factor;
            constant_term_accumulator += scaling_factor * libra_evaluations[idx];
        }
 
        // to save a scalar mul, add the sum of the batching scalars corresponding to the big sum evaluations
        scalars.push_back(batching_scalars[0]);
        scalars.push_back(batching_scalars[1] + batching_scalars[2]);
        scalars.push_back(batching_scalars[3]);
    }
 
    static void batch_sumcheck_round_claims(std::vector<Commitment>& commitments,
                                            std::vector<Fr>& scalars,
                                            Fr& constant_term_accumulator,
                                            const std::vector<Fr>& multilinear_challenge,
                                            const std::vector<Fr>& shplonk_batching_challenge_powers,
                                            const Fr& shplonk_evaluation_challenge,
                                            const std::vector<Commitment>& sumcheck_round_commitments,
                                            const std::vector<std::array<Fr, 3>>& sumcheck_round_evaluations)
    {
 
        std::vector<Fr> denominators = {};
 
        // The number of Gemini claims is equal to `2 * log_n` and `log_n` is equal to the size of
        // `multilinear_challenge`, as this method is never used with padding.
        const size_t num_gemini_claims = 2 * multilinear_challenge.size();
        // Denominators for the opening claims at 0 and 1. Need to be computed only once as opposed to the claims at the
        // sumcheck round challenges.
        std::array<Fr, 2> const_denominators;
 
        const_denominators[0] = Fr(1) / (shplonk_evaluation_challenge);
        const_denominators[1] = Fr(1) / (shplonk_evaluation_challenge - Fr{ 1 });
 
        // Compute the denominators corresponding to the evaluation claims at the round challenges and add the
        // commitments to the sumcheck round univariates to the vector of commitments
        for (const auto& [challenge, comm] : zip_view(multilinear_challenge, sumcheck_round_commitments)) {
            denominators.push_back(shplonk_evaluation_challenge - challenge);
            commitments.push_back(comm);
        }
 
        // Invert denominators
        if constexpr (!Curve::is_stdlib_type) {
            Fr::batch_invert(denominators);
        } else {
            for (auto& denominator : denominators) {
                denominator = Fr{ 1 } / denominator;
            }
        }
 
        // Each commitment to a sumcheck round univariate [S_i] is multiplied by the sum of three scalars corresponding
        // to the evaluations at 0, 1, and the round challenge u_i.
        // Compute the power of `shplonk_batching_challenge` to add sumcheck univariate commitments and evaluations to
        // the batch.
        size_t power = num_gemini_claims + NUM_INTERLEAVING_CLAIMS + NUM_SMALL_IPA_EVALUATIONS;
        for (const auto& [eval_array, denominator] : zip_view(sumcheck_round_evaluations, denominators)) {
            // Initialize batched_scalar corresponding to 3 evaluations claims
            Fr batched_scalar = Fr(0);
            Fr const_term_contribution = Fr(0);
            // Compute the contribution from the evaluations at 0 and 1
            for (size_t idx = 0; idx < 2; idx++) {
                Fr current_scaling_factor = const_denominators[idx] * shplonk_batching_challenge_powers[power++];
                batched_scalar -= current_scaling_factor;
                const_term_contribution += current_scaling_factor * eval_array[idx];
            }
 
            // Compute the contribution from the evaluation at the challenge u_i
            Fr current_scaling_factor = denominator * shplonk_batching_challenge_powers[power++];
            batched_scalar -= current_scaling_factor;
            const_term_contribution += current_scaling_factor * eval_array[2];
 
            // Update Shplonk constant term accumualator
            constant_term_accumulator += const_term_contribution;
            scalars.push_back(batched_scalar);
        }
    };
};
} // namespace bb