biggroup_nafs.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/common/assert.hpp"
#include "barretenberg/ecc/curves/secp256k1/secp256k1.hpp"
#include "barretenberg/stdlib/primitives/biggroup/biggroup.hpp"
 
namespace bb::stdlib::element_default {
 
template <typename C, class Fq, class Fr, class G>
template <size_t wnaf_size>
std::pair<uint64_t, bool> element<C, Fq, Fr, G>::get_staggered_wnaf_fragment_value(const uint64_t fragment_u64,
                                                                                   const uint64_t stagger,
                                                                                   bool is_negative,
                                                                                   bool wnaf_skew)
{
    // If there is no stagger then there is no need to change anything
    if (stagger == 0) {
        return std::make_pair(0, wnaf_skew);
    }
 
    // Sanity check input fragment
    BB_ASSERT_LT(fragment_u64, (1ULL << stagger), "biggroup_nafs: fragment value ≥ 2^{stagger}");
 
    // Convert the fragment to signed int for easier manipulation
    int fragment = static_cast<int>(fragment_u64);
 
    // Inverse the fragment if it's negative
    if (is_negative) {
        fragment = -fragment;
    }
    // If the value is positive and there is a skew in wnaf, subtract 2^{stagger}.
    if (!is_negative && wnaf_skew) {
        fragment -= (1 << stagger);
    }
 
    // If the value is negative and there is a skew in wnaf, add 2^{stagger}.
    if (is_negative && wnaf_skew) {
        fragment += (1 << stagger);
    }
 
    // If the lowest bit is zero, then set final skew to 1 and
    // (i) add 1 to the absolute value of the fragment if it's positive
    // (ii) subtract 1 from the absolute value of the fragment if it's negative
    bool output_skew = (fragment_u64 & 1) == 0;
    if (!is_negative && output_skew) {
        fragment += 1;
    } else if (is_negative && output_skew) {
        fragment -= 1;
    }
 
    // Compute raw wnaf value: w = 2e + 1  =>  e = (w - 1) / 2  => e = ⌊w / 2⌋
    const int signed_wnaf_value = (fragment / 2);
    constexpr int wnaf_window_size = (1ULL << (wnaf_size - 1));
    uint64_t output_fragment = 0;
    if (fragment < 0) {
        output_fragment = static_cast<uint64_t>(wnaf_window_size + signed_wnaf_value - 1);
    } else {
        output_fragment = static_cast<uint64_t>(wnaf_window_size + signed_wnaf_value);
    }
 
    return std::make_pair(output_fragment, output_skew);
}
 
template <typename C, class Fq, class Fr, class G>
template <size_t wnaf_size>
std::vector<field_t<C>> element<C, Fq, Fr, G>::convert_wnaf_values_to_witnesses(
    C* builder, const uint64_t* wnaf_values, bool is_negative, size_t rounds, const bool range_constrain_wnaf)
{
    constexpr uint64_t wnaf_window_size = (1ULL << (wnaf_size - 1));
 
    std::vector<field_ct> wnaf_entries;
    for (size_t i = 0; i < rounds; ++i) {
        // Predicate == sign of current wnaf value
        const bool predicate = (wnaf_values[i] >> 31U) & 1U;            // sign bit (32nd bit)
        const uint64_t wnaf_magnitude = (wnaf_values[i] & 0x7fffffffU); // 31-bit magnitude
 
        // If the signs of current entry and the whole scalar are the same, then add the magnitude of the
        // wnaf value to the windows size to form an entry. Otherwise, subract the magnitude along with 1.
        // The extra 1 is needed to get a uniform representation of (2e' + 1) as explained in the README.
        uint64_t offset_wnaf_entry = 0;
        if ((!predicate && !is_negative) || (predicate && is_negative)) {
            offset_wnaf_entry = wnaf_window_size + wnaf_magnitude;
        } else {
            offset_wnaf_entry = wnaf_window_size - wnaf_magnitude - 1;
        }
        field_ct wnaf_entry(witness_ct(builder, offset_wnaf_entry));
 
        // In some cases we may want to skip range constraining the wnaf entries. For example when we use these
        // entries to lookup in a ROM or regular table, it implicitly enforces the range constraint.
        if (range_constrain_wnaf) {
            wnaf_entry.create_range_constraint(wnaf_size, "biggroup_nafs: wnaf_entry is not in range");
        }
        wnaf_entries.emplace_back(wnaf_entry);
    }
    return wnaf_entries;
}
 
template <typename C, class Fq, class Fr, class G>
template <size_t wnaf_size>
Fr element<C, Fq, Fr, G>::reconstruct_bigfield_from_wnaf(Builder* builder,
                                                         const std::vector<field_t<Builder>>& wnaf,
                                                         const bool_ct& positive_skew,
                                                         const bool_ct& negative_skew,
                                                         const field_t<Builder>& stagger_fragment,
                                                         const size_t stagger,
                                                         const size_t rounds)
{
    // Collect positive wnaf entries for accumulation
    std::vector<field_ct> accumulator;
    for (size_t i = 0; i < rounds; ++i) {
        field_ct entry = wnaf[rounds - 1 - i];
        entry *= field_ct(uint256_t(1) << (i * wnaf_size));
        accumulator.emplace_back(entry);
    }
 
    // Accumulate entries, shift by stagger and add the stagger itself
    field_ct sum = field_ct::accumulate(accumulator);
    sum = sum * field_ct(bb::fr(1ULL << stagger));
    sum += (stagger_fragment);
    sum = sum.normalize();
 
    // Convert this value to bigfield element
    Fr reconstructed_positive_part =
        Fr(sum, field_ct::from_witness_index(builder, builder->zero_idx()), /*can_overflow*/ false);
 
    // Double the final value and add the positive skew
    reconstructed_positive_part =
        (reconstructed_positive_part + reconstructed_positive_part)
            .add_to_lower_limb(field_t<Builder>(positive_skew), /*other_maximum_value*/ uint256_t(1));
 
    // Start reconstructing the negative part: start with wnaf constant 0xff...ff
    // See the README for explanation of this constant
    constexpr uint64_t wnaf_window_size = (1ULL << (wnaf_size - 1));
    uint256_t negative_constant_wnaf_offset(0);
    for (size_t i = 0; i < rounds; ++i) {
        negative_constant_wnaf_offset += uint256_t((wnaf_window_size * 2) - 1) * (uint256_t(1) << (i * wnaf_size));
    }
 
    // Shift by stagger
    negative_constant_wnaf_offset = negative_constant_wnaf_offset << stagger;
 
    // Add for stagger (if any)
    if (stagger > 0) {
        negative_constant_wnaf_offset += ((1ULL << wnaf_size) - 1ULL); // from stagger fragment
    }
 
    // Add the negative skew to the bigfield constant
    Fr reconstructed_negative_part =
        Fr(nullptr, negative_constant_wnaf_offset).add_to_lower_limb(field_t<Builder>(negative_skew), uint256_t(1));
 
    // output = x_pos - x_neg (x_pos and x_neg are both non-negative)
    Fr reconstructed = reconstructed_positive_part - reconstructed_negative_part;
 
    return reconstructed;
}
 
template <typename C, class Fq, class Fr, class G>
template <size_t num_bits, size_t wnaf_size, size_t lo_stagger, size_t hi_stagger>
std::pair<Fr, typename element<C, Fq, Fr, G>::secp256k1_wnaf> element<C, Fq, Fr, G>::compute_secp256k1_single_wnaf(
    C* builder,
    const secp256k1::fr& scalar,
    size_t stagger,
    bool is_negative,
    const bool range_constrain_wnaf,
    bool is_lo)
{
    // The number of rounds is the minimal required to cover the whole scalar with wnaf_size windows
    constexpr size_t num_rounds = ((num_bits + wnaf_size - 1) / wnaf_size);
 
    // Stagger mask is needed to retrieve the lowest bits that will not be used in montgomery ladder directly
    const uint64_t stagger_mask = (1ULL << stagger) - 1;
 
    // Stagger scalar represents the lower "staggered" bits that are not used in the ladder
    const uint64_t stagger_scalar = scalar.data[0] & stagger_mask;
 
    std::array<uint64_t, num_rounds> wnaf_values = { 0 };
    bool skew_without_stagger = false;
    uint256_t k_u256{ scalar.data[0], scalar.data[1], scalar.data[2], scalar.data[3] };
    k_u256 = k_u256 >> stagger;
    if (is_lo) {
        bb::wnaf::fixed_wnaf<num_bits - lo_stagger, 1, wnaf_size>(
            &k_u256.data[0], &wnaf_values[0], skew_without_stagger, 0);
    } else {
        bb::wnaf::fixed_wnaf<num_bits - hi_stagger, 1, wnaf_size>(
            &k_u256.data[0], &wnaf_values[0], skew_without_stagger, 0);
    }
 
    // Number of rounds that are needed to reconstruct the scalar without staggered bits
    const size_t num_rounds_excluding_stagger_bits = ((num_bits + wnaf_size - 1 - stagger) / wnaf_size);
 
    // Compute the stagger-related fragment and the final skew due to the same
    const auto [first_fragment, skew] =
        get_staggered_wnaf_fragment_value<wnaf_size>(stagger_scalar, stagger, is_negative, skew_without_stagger);
 
    // Get wnaf witnesses
    // Note that we only range constrain the wnaf entries if range_constrain_wnaf is set to true.
    std::vector<field_ct> wnaf = convert_wnaf_values_to_witnesses<wnaf_size>(
        builder, &wnaf_values[0], is_negative, num_rounds_excluding_stagger_bits, range_constrain_wnaf);
 
    // Compute and constrain skews
    bool_ct negative_skew(witness_ct(builder, is_negative ? 0 : skew), /*use_range_constraint*/ true);
    bool_ct positive_skew(witness_ct(builder, is_negative ? skew : 0), /*use_range_constraint*/ true);
 
    // Enforce that both positive_skew, negative_skew are not set at the same time
    bool_ct both_skews_cannot_be_one = !(positive_skew & negative_skew);
    both_skews_cannot_be_one.assert_equal(
        bool_ct(builder, true), "biggroup_nafs: both positive and negative skews cannot be set at the same time");
 
    // Initialize stagger witness
    field_ct stagger_fragment = witness_ct(builder, first_fragment);
 
    // We only range constrain the stagger fragment if range_constrain_wnaf is set. This is because in some cases
    // we may use the stagger fragment to lookup in a ROM/regular table, which implicitly enforces the range constraint.
    if (range_constrain_wnaf) {
        stagger_fragment.create_range_constraint(wnaf_size, "biggroup_nafs: stagger fragment is not in range");
    }
 
    // Reconstruct the bigfield scalar from (wnaf + stagger) representation
    Fr reconstructed = reconstruct_bigfield_from_wnaf<wnaf_size>(
        builder, wnaf, positive_skew, negative_skew, stagger_fragment, stagger, num_rounds_excluding_stagger_bits);
 
    secp256k1_wnaf wnaf_out{ .wnaf = wnaf,
                             .positive_skew = positive_skew,
                             .negative_skew = negative_skew,
                             .least_significant_wnaf_fragment = stagger_fragment,
                             .has_wnaf_fragment = (stagger > 0) };
 
    return std::make_pair(reconstructed, wnaf_out);
}
 
template <typename C, class Fq, class Fr, class G>
template <size_t wnaf_size, size_t lo_stagger, size_t hi_stagger>
typename element<C, Fq, Fr, G>::secp256k1_wnaf_pair element<C, Fq, Fr, G>::compute_secp256k1_endo_wnaf(
    const Fr& scalar, const bool range_constrain_wnaf)
{
    C* builder = scalar.get_context();
 
    constexpr size_t num_bits = 129;
 
    // Decomposes the scalar k into two 129-bit scalars klo, khi such that
    // k = klo + ζ * khi (mod n)
    //   = klo - λ * khi (mod n)
    // where ζ is the primitive sixth root of unity mod n, and λ is the primitive cube root of unity mod n
    // (note that ζ = -λ). We know that for any scalar k, such a decomposition exists and klo and khi are 128-bits long.
    secp256k1::fr k(uint256_t(scalar.get_value() % Fr::modulus_u512));
    secp256k1::fr klo(0);
    secp256k1::fr khi(0);
    bool klo_negative = false;
    bool khi_negative = false;
    secp256k1::fr::split_into_endomorphism_scalars(k.from_montgomery_form(), klo, khi);
 
    // The low and high scalars must be less than 2^129 in absolute value. In some cases, the khi value
    // is returned as negative, in which case we negate it and set a flag to indicate this. This is because
    // we decompose the scalar as:
    // k = klo + ζ * khi (mod n)
    //   = klo - λ * khi (mod n)
    // where λ is the cube root of unity. If khi is negative, then -λ * khi is positive, and vice versa.
    if (khi.uint256_t_no_montgomery_conversion().get_msb() >= 129) {
        khi_negative = true;
        khi = -khi;
    }
 
    BB_ASSERT_LT(klo.uint256_t_no_montgomery_conversion().get_msb(), 129ULL, "biggroup_nafs: klo > 129 bits");
    BB_ASSERT_LT(khi.uint256_t_no_montgomery_conversion().get_msb(), 129ULL, "biggroup_nafs: khi > 129 bits");
 
    const auto [klo_reconstructed, klo_out] =
        element<C, Fq, Fr, G>::compute_secp256k1_single_wnaf<num_bits, wnaf_size, lo_stagger, hi_stagger>(
            builder, klo, lo_stagger, klo_negative, range_constrain_wnaf, true);
 
    const auto [khi_reconstructed, khi_out] =
        element<C, Fq, Fr, G>::compute_secp256k1_single_wnaf<num_bits, wnaf_size, lo_stagger, hi_stagger>(
            builder, khi, hi_stagger, khi_negative, range_constrain_wnaf, false);
 
    uint256_t minus_lambda_val(-secp256k1::fr::cube_root_of_unity());
    Fr minus_lambda(bb::fr(minus_lambda_val.slice(0, 136)), bb::fr(minus_lambda_val.slice(136, 256)), false);
 
    Fr reconstructed_scalar = khi_reconstructed.madd(minus_lambda, { klo_reconstructed });
 
    // Constant scalars are always reduced mod n by design (scalar < n), however
    // the reconstructed_scalar may be larger than n as it's a witness. So we need to
    // reduce the reconstructed_scalar mod n explicitly to match the original scalar.
    // This is necessary for assert_equal to pass.
    if (scalar.is_constant()) {
        reconstructed_scalar.self_reduce();
    }
 
    // Validate that the reconstructed scalar matches the original scalar in circuit
    scalar.assert_equal(reconstructed_scalar, "biggroup_nafs: reconstructed scalar does not match reduced input");
 
    return { .klo = klo_out, .khi = khi_out };
}
 
template <typename C, class Fq, class Fr, class G>
std::vector<bool_t<C>> element<C, Fq, Fr, G>::compute_naf(const Fr& scalar, const size_t max_num_bits)
{
    // Get the circuit builder
    C* builder = scalar.get_context();
 
    // To compute the NAF representation, we first reduce the scalar modulo r (the scalar field modulus).
    uint512_t scalar_multiplier_512 = uint512_t(scalar.get_value()) % uint512_t(Fr::modulus);
    uint256_t scalar_multiplier = scalar_multiplier_512.lo;
 
    // Number of rounds is either the max_num_bits provided, or the full size of the scalar field modulus.
    // If the scalar is zero, we use the full size of the scalar field modulus as we use scalar = r in this case.
    const size_t num_rounds = (max_num_bits == 0 || scalar_multiplier == 0) ? Fr::modulus.get_msb() + 1 : max_num_bits;
 
    // NAF can't handle 0 so we set scalar = r in this case.
    if (scalar_multiplier == 0) {
        scalar_multiplier = Fr::modulus;
    }
 
    // NAF representation consists of num_rounds bits and a skew bit.
    // Given a scalar k, we compute the NAF representation as follows:
    //
    // k = -skew + ₀∑ⁿ⁻¹ (1 - 2 * naf_i) * 2^i
    //
    // where naf_i = (1 - k_{i + 1}) ∈ {0, 1} and k_{i + 1} is the (i + 1)-th bit of the scalar k.
    // If naf_i = 0, then the i-th NAF entry is +1, otherwise it is -1. See the README for more details.
    //
    std::vector<bool_ct> naf_entries(num_rounds + 1);
 
    // If the scalar is even, we set the skew flag to true and add 1 to the scalar.
    // Sidenote: we apply range constraints to the boolean witnesses instead of full 1-bit range gates.
    const bool skew_value = !scalar_multiplier.get_bit(0);
    scalar_multiplier += uint256_t(static_cast<uint64_t>(skew_value));
    naf_entries[num_rounds] = bool_ct(witness_ct(builder, skew_value), /*use_range_constraint*/ true);
 
    // We need to manually propagate the origin tag
    naf_entries[num_rounds].set_origin_tag(scalar.get_origin_tag());
 
    for (size_t i = 0; i < num_rounds - 1; ++i) {
        // If the next entry is false, we need to flip the sign of the current entry (naf_entry := (1 - next_bit)).
        // Apply a basic range constraint per bool, and not a full 1-bit range gate. Results in ~`num_rounds`/4 gates
        // per scalar.
        const bool next_entry = scalar_multiplier.get_bit(i + 1);
        naf_entries[num_rounds - i - 1] = bool_ct(witness_ct(builder, !next_entry), /*use_range_constraint*/ true);
 
        // We need to manually propagate the origin tag
        naf_entries[num_rounds - i - 1].set_origin_tag(scalar.get_origin_tag());
    }
 
    // The most significant NAF entry is always (+1) as we are working with scalars < 2^{max_num_bits}.
    // Recall that true represents (-1) and false represents (+1).
    naf_entries[0] = bool_ct(witness_ct(builder, false), /*use_range_constraint*/ true);
    naf_entries[0].set_origin_tag(scalar.get_origin_tag());
 
    // validate correctness of NAF
    if constexpr (!Fr::is_composite) {
        std::vector<Fr> accumulators;
        for (size_t i = 0; i < num_rounds; ++i) {
            // bit = 1 - 2 * naf
            Fr entry(naf_entries[num_rounds - i - 1]);
            entry *= -2;
            entry += 1;
            entry *= static_cast<Fr>(uint256_t(1) << (i));
            accumulators.emplace_back(entry);
        }
        accumulators.emplace_back(-Fr(naf_entries[num_rounds])); // -skew
        Fr accumulator_result = Fr::accumulate(accumulators);
        scalar.assert_equal(accumulator_result);
    } else {
        const auto reconstruct_half_naf = [](bool_ct* nafs, const size_t half_round_length) {
            field_ct negative_accumulator(0);
            field_ct positive_accumulator(0);
            for (size_t i = 0; i < half_round_length; ++i) {
                negative_accumulator = negative_accumulator + negative_accumulator + field_ct(nafs[i]);
                positive_accumulator = positive_accumulator + positive_accumulator + field_ct(1) - field_ct(nafs[i]);
            }
            return std::make_pair(positive_accumulator, negative_accumulator);
        };
 
        std::pair<field_ct, field_ct> hi_accumulators;
        std::pair<field_ct, field_ct> lo_accumulators;
 
        if (num_rounds > Fr::NUM_LIMB_BITS * 2) {
            const size_t midpoint = num_rounds - (Fr::NUM_LIMB_BITS * 2);
            hi_accumulators = reconstruct_half_naf(&naf_entries[0], midpoint);
            lo_accumulators = reconstruct_half_naf(&naf_entries[midpoint], num_rounds - midpoint);
        } else {
            // If the number of rounds is ≤ (2 * Fr::NUM_LIMB_BITS), the high bits of the resulting Fr element are 0.
            const field_ct zero = field_ct::from_witness_index(builder, builder->zero_idx());
            lo_accumulators = reconstruct_half_naf(&naf_entries[0], num_rounds);
            hi_accumulators = std::make_pair(zero, zero);
        }
 
        // Add the skew bit to the low accumulator's negative part
        lo_accumulators.second = lo_accumulators.second + field_ct(naf_entries[num_rounds]);
 
        Fr reconstructed_positive = Fr(lo_accumulators.first, hi_accumulators.first);
        Fr reconstructed_negative = Fr(lo_accumulators.second, hi_accumulators.second);
        Fr accumulator = reconstructed_positive - reconstructed_negative;
 
        // Constant scalars are always reduced mod n by design (scalar < n), however
        // the reconstructed accumulator may be larger than n as its a witness. So we need to
        // reduce the reconstructed accumulator mod n explicitly to match the original scalar.
        // This is necessary for assert_equal to pass.
        if (scalar.is_constant()) {
            accumulator.self_reduce();
        }
 
        // Validate that the reconstructed scalar matches the original scalar in circuit
        accumulator.assert_equal(scalar);
    }
 
    // Propagate tags to naf
    const auto original_tag = scalar.get_origin_tag();
    for (auto& naf_entry : naf_entries) {
        naf_entry.set_origin_tag(original_tag);
    }
    return naf_entries;
}
} // namespace bb::stdlib::element_default