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feat: translator on Shplemini (#9329)
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In this PR:
* implement concatenation trick (to work for both Gemini and Shplemini),
try to document it and fix some other documentation in Shplemini
* switch Translator to Shplemini

The Translator VM works on many many small polynomials (whose length is
determined by a "minicircuit size"). To avoid the permutation relation
having a very high degree, these small polynomials are split into
groups, and each group is concatenated into a single polynomial. We want
the prover to avoid having to commit to these extra concatenation
polynomials (as they will likely not be sparse at all) but rather reuse
the commitments to the polynomials in its corresponding concatenation
group, also showing they are correctly related in the opening protocol.
Briefly, in Gemini, this is achieved by adding the contributibution to
the batched concatenated polynomials when computing the fold polynomials
(A_0, A_1, ..., A_(logn -1)) but computing A_0- and A_0+ using the
polyinomials in the batched concatenated groups. As the verifier only
receives commitments to A_1, .., A_(logn-1) and has to compute the
commitments to A_0- and A_0+ , it can then do this using the commitments
of the polynomials in concatenation groups.
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@maramihali
maramihali authored Oct 23, 2024
1 parent 9dda91e commit 21fa3cf
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161 changes: 90 additions & 71 deletions barretenberg/cpp/src/barretenberg/commitment_schemes/gemini/gemini.hpp
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Original file line number Diff line number Diff line change
Expand Up @@ -103,19 +103,25 @@ template <typename Curve> class GeminiProver_ {
static std::vector<Polynomial> compute_fold_polynomials(const size_t log_N,
std::span<const Fr> multilinear_challenge,
Polynomial&& batched_unshifted,
Polynomial&& batched_to_be_shifted);
Polynomial&& batched_to_be_shifted,
Polynomial&& batched_concatenated = {});

static std::vector<Claim> compute_fold_polynomial_evaluations(const size_t log_N,
std::vector<Polynomial>&& fold_polynomials,
const Fr& r_challenge);
static std::vector<Claim> compute_fold_polynomial_evaluations(
const size_t log_N,
std::vector<Polynomial>&& fold_polynomials,
const Fr& r_challenge,
std::vector<Polynomial>&& batched_groups_to_be_concatenated = {});

template <typename Transcript>
static std::vector<Claim> prove(const Fr circuit_size,
RefSpan<Polynomial> f_polynomials,
RefSpan<Polynomial> g_polynomials,
std::span<Fr> multilinear_challenge,
const std::shared_ptr<CommitmentKey<Curve>>& commitment_key,
const std::shared_ptr<Transcript>& transcript);
const std::shared_ptr<Transcript>& transcript,
RefSpan<Polynomial> concatenated_polynomials = {},
const std::vector<RefVector<Polynomial>>& groups_to_be_concatenated = {});

}; // namespace bb

template <typename Curve> class GeminiVerifier_ {
Expand All @@ -136,32 +142,38 @@ template <typename Curve> class GeminiVerifier_ {
* @return Fold polynomial opening claims: (r, A₀(r), C₀₊), (-r, A₀(-r), C₀₋), and
* (Cⱼ, Aⱼ(-r^{2ʲ}), -r^{2}), j = [1, ..., m-1]
*/
static std::vector<OpeningClaim<Curve>> reduce_verification(std::span<Fr> multilinear_challenge,
std::span<Fr> multilinear_evaluations,
RefSpan<GroupElement> unshifted_commitments,
RefSpan<GroupElement> to_be_shifted_commitments,
auto& transcript)
static std::vector<OpeningClaim<Curve>> reduce_verification(
std::span<Fr> multilinear_challenge,
RefSpan<Fr> unshifted_evaluations,
RefSpan<Fr> shifted_evaluations,
RefSpan<Commitment> unshifted_commitments,
RefSpan<Commitment> to_be_shifted_commitments,
auto& transcript,
const std::vector<RefVector<Commitment>>& concatenation_group_commitments = {},
RefSpan<Fr> concatenated_evaluations = {})

{
const size_t num_variables = multilinear_challenge.size();

const size_t N = 1 << num_variables;

Fr rho = transcript->template get_challenge<Fr>("rho");
std::vector<Fr> rhos = gemini::powers_of_rho(rho, multilinear_evaluations.size());

GroupElement batched_commitment_unshifted = GroupElement::zero();
GroupElement batched_commitment_to_be_shifted = GroupElement::zero();

Fr batched_evaluation = Fr::zero();
for (size_t i = 0; i < multilinear_evaluations.size(); ++i) {
batched_evaluation += multilinear_evaluations[i] * rhos[i];
Fr batched_evaluation = Fr(0);
Fr batching_scalar = Fr(1);
for (auto [eval, comm] : zip_view(unshifted_evaluations, unshifted_commitments)) {
batched_evaluation += eval * batching_scalar;
batched_commitment_unshifted += comm * batching_scalar;
batching_scalar *= rho;
}

const size_t num_unshifted = unshifted_commitments.size();
const size_t num_to_be_shifted = to_be_shifted_commitments.size();
for (size_t i = 0; i < num_unshifted; i++) {
batched_commitment_unshifted += unshifted_commitments[i] * rhos[i];
}
for (size_t i = 0; i < num_to_be_shifted; i++) {
batched_commitment_to_be_shifted += to_be_shifted_commitments[i] * rhos[num_unshifted + i];
for (auto [eval, comm] : zip_view(shifted_evaluations, to_be_shifted_commitments)) {
batched_evaluation += eval * batching_scalar;
batched_commitment_to_be_shifted += comm * batching_scalar;
batching_scalar *= rho;
}

// Get polynomials Fold_i, i = 1,...,m-1 from transcript
Expand All @@ -173,22 +185,71 @@ template <typename Curve> class GeminiVerifier_ {

// Get evaluations a_i, i = 0,...,m-1 from transcript
const std::vector<Fr> evaluations = get_gemini_evaluations(num_variables, transcript);

// C₀_r_pos = ∑ⱼ ρʲ⋅[fⱼ] + r⁻¹⋅∑ⱼ ρᵏ⁺ʲ [gⱼ], the commitment to A₀₊
// C₀_r_neg = ∑ⱼ ρʲ⋅[fⱼ] - r⁻¹⋅∑ⱼ ρᵏ⁺ʲ [gⱼ], the commitment to A₀₋
GroupElement C0_r_pos = batched_commitment_unshifted;
GroupElement C0_r_neg = batched_commitment_unshifted;

Fr r_inv = r.invert();
if (!batched_commitment_to_be_shifted.is_point_at_infinity()) {
batched_commitment_to_be_shifted = batched_commitment_to_be_shifted * r_inv;
C0_r_pos += batched_commitment_to_be_shifted;
C0_r_neg -= batched_commitment_to_be_shifted;
}

// If verifying the opening for the translator VM, we reconstruct the commitment of the batched concatenated
// polynomials, partially evaluated in r and -r, using the commitments in the concatenation groups and add their
// contribution as well to to C₀_r_pos and C₀_r_neg
ASSERT(concatenated_evaluations.size() == concatenation_group_commitments.size());
if (!concatenation_group_commitments.empty()) {
size_t concatenation_group_size = concatenation_group_commitments[0].size();
// The "real" size of polynomials in concatenation groups (i.e. the number of non-zero values)
const size_t mini_circuit_size = N / concatenation_group_size;
Fr current_r_shift_pos = Fr(1);
Fr current_r_shift_neg = Fr(1);
const Fr r_pow_minicircuit = r.pow(mini_circuit_size);
const Fr r_neg_pow_minicircuit = (-r).pow(mini_circuit_size);
std::vector<Fr> r_shifts_pos;
std::vector<Fr> r_shifts_neg;
for (size_t i = 0; i < concatenation_group_size; ++i) {
r_shifts_pos.emplace_back(current_r_shift_pos);
r_shifts_neg.emplace_back(current_r_shift_neg);
current_r_shift_pos *= r_pow_minicircuit;
current_r_shift_neg *= r_neg_pow_minicircuit;
}
size_t j = 0;
GroupElement batched_concatenated_pos = GroupElement::zero();
GroupElement batched_concatenated_neg = GroupElement::zero();
for (auto& concatenation_group_commitment : concatenation_group_commitments) {
// Compute the contribution from each group j of commitments Gⱼ = {C₀, C₁, C₂, C₃, ...}
// where s = mini_circuit_size as
// C₀_r_pos += ∑ᵢ ρᵏ⁺ᵐ⁺ʲ⋅ rⁱˢ ⋅ Cᵢ
// C₀_r_neg += ∑ᵢ ρᵏ⁺ᵐ⁺ʲ⋅ (-r)ⁱˢ ⋅ Cᵢ
for (size_t i = 0; i < concatenation_group_size; ++i) {
batched_concatenated_pos += concatenation_group_commitment[i] * batching_scalar * r_shifts_pos[i];
batched_concatenated_neg += concatenation_group_commitment[i] * batching_scalar * r_shifts_neg[i];
}
batched_evaluation += concatenated_evaluations[j] * batching_scalar;
batching_scalar *= rho;
j++;
}

// Add the contributions from concatenation groups to get the final [A₀₊] and [A₀₋]
C0_r_pos += batched_concatenated_pos;
C0_r_neg += batched_concatenated_neg;
}

// Compute evaluation A₀(r)
auto a_0_pos = compute_gemini_batched_univariate_evaluation(
num_variables, batched_evaluation, multilinear_challenge, r_squares, evaluations);

// C₀_r_pos = ∑ⱼ ρʲ⋅[fⱼ] + r⁻¹⋅∑ⱼ ρᵏ⁺ʲ [gⱼ]
// C₀_r_pos = ∑ⱼ ρʲ⋅[fⱼ] - r⁻¹⋅∑ⱼ ρᵏ⁺ʲ [gⱼ]
auto [c0_r_pos, c0_r_neg] =
compute_simulated_commitments(batched_commitment_unshifted, batched_commitment_to_be_shifted, r);

std::vector<OpeningClaim<Curve>> fold_polynomial_opening_claims;
fold_polynomial_opening_claims.reserve(num_variables + 1);

// ( [A₀₊], r, A₀(r) )
fold_polynomial_opening_claims.emplace_back(OpeningClaim<Curve>{ { r, a_0_pos }, c0_r_pos });
fold_polynomial_opening_claims.emplace_back(OpeningClaim<Curve>{ { r, a_0_pos }, C0_r_pos });
// ( [A₀₋], -r, A₀(-r) )
fold_polynomial_opening_claims.emplace_back(OpeningClaim<Curve>{ { -r, evaluations[0] }, c0_r_neg });
fold_polynomial_opening_claims.emplace_back(OpeningClaim<Curve>{ { -r, evaluations[0] }, C0_r_neg });
for (size_t l = 0; l < num_variables - 1; ++l) {
// ([A₀₋], −r^{2ˡ}, Aₗ(−r^{2ˡ}) )
fold_polynomial_opening_claims.emplace_back(
Expand Down Expand Up @@ -283,48 +344,6 @@ template <typename Curve> class GeminiVerifier_ {

return batched_eval_accumulator;
}

/**
* @brief Computes two commitments to A₀ partially evaluated in r and -r.
*
* @param batched_commitment_unshifted batched commitment to non-shifted polynomials
* @param batched_commitment_to_be_shifted batched commitment to to-be-shifted polynomials
* @param r evaluation point at which we have partially evaluated A₀ at r and -r.
* @return std::pair<Commitment, Commitment> c0_r_pos, c0_r_neg
*/
static std::pair<GroupElement, GroupElement> compute_simulated_commitments(
GroupElement& batched_commitment_unshifted, GroupElement& batched_commitment_to_be_shifted, Fr r)
{
// C₀ᵣ₊ = [F] + r⁻¹⋅[G]
GroupElement C0_r_pos;
// C₀ᵣ₋ = [F] - r⁻¹⋅[G]
GroupElement C0_r_neg;
Fr r_inv = r.invert(); // r⁻¹

// If in a recursive setting, perform a batch mul. Otherwise, accumulate directly.
// TODO(#673): The following if-else represents the stldib/native code paths. Once the "native" verifier is
// achieved through a builder Simulator, the stdlib codepath should become the only codepath.
if constexpr (Curve::is_stdlib_type) {
std::vector<GroupElement> commitments = { batched_commitment_unshifted, batched_commitment_to_be_shifted };
auto builder = r.get_context();
auto one = Fr(builder, 1);
// TODO(#707): these batch muls include the use of 1 as a scalar. This is handled appropriately as a non-mul
// (add-accumulate) in the goblin batch_mul but is done inefficiently as a scalar mul in the conventional
// emulated batch mul.
C0_r_pos = GroupElement::batch_mul(commitments, { one, r_inv });
C0_r_neg = GroupElement::batch_mul(commitments, { one, -r_inv });
} else {
C0_r_pos = batched_commitment_unshifted;
C0_r_neg = batched_commitment_unshifted;
if (!batched_commitment_to_be_shifted.is_point_at_infinity()) {
batched_commitment_to_be_shifted = batched_commitment_to_be_shifted * r_inv;
C0_r_pos += batched_commitment_to_be_shifted;
C0_r_neg -= batched_commitment_to_be_shifted;
}
}

return { C0_r_pos, C0_r_neg };
}
};

} // namespace bb
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