non_native_field_relation.hpp

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// === AUDIT STATUS ===
// internal:    { status: Complete, auditors: [Luke, Raju], commit: }
// external_1:  { status: not started, auditors: [], commit: }
// external_2:  { status: not started, auditors: [], commit: }
// =====================
 
#pragma once
#include "barretenberg/numeric/uint256/uint256.hpp"
#include "barretenberg/relations/relation_types.hpp"
 
namespace bb {
 
template <typename FF_> class NonNativeFieldRelationImpl {
  public:
    using FF = FF_;
 
    static constexpr std::array<size_t, 1> SUBRELATION_PARTIAL_LENGTHS{
        6 // combined non-native field sub-relation
    };
 
    template <typename AllEntities> inline static bool skip(const AllEntities& in)
    {
        return in[AllEntities::EntityId::q_nnf].is_zero();
    }
 
    template <typename ContainerOverSubrelations, typename AllEntities, typename Parameters>
    static void accumulate(ContainerOverSubrelations& accumulators,
                           const AllEntities& in,
                           BB_UNUSED const Parameters& params,
                           const FF& scaling_factor)
    {
        // all accumulators are of the same length, so we set our accumulator type to (arbitrarily) be the first one.
        // if there were one that were shorter, we could also profitably use a `ShortAccumulator` type. however,
        // that is not the case here.
        using Accumulator = typename std::tuple_element_t<0, ContainerOverSubrelations>;
        using CoefficientAccumulator = typename Accumulator::CoefficientAccumulator;
 
        auto w_1_m = CoefficientAccumulator(in[AllEntities::EntityId::w_l]);
        auto w_2_m = CoefficientAccumulator(in[AllEntities::EntityId::w_r]);
        auto w_3_m = CoefficientAccumulator(in[AllEntities::EntityId::w_o]);
        auto w_4_m = CoefficientAccumulator(in[AllEntities::EntityId::w_4]);
        auto w_1_shift_m = CoefficientAccumulator(in[AllEntities::EntityId::w_l_shift]);
        auto w_2_shift_m = CoefficientAccumulator(in[AllEntities::EntityId::w_r_shift]);
        auto w_3_shift_m = CoefficientAccumulator(in[AllEntities::EntityId::w_o_shift]);
        auto w_4_shift_m = CoefficientAccumulator(in[AllEntities::EntityId::w_4_shift]);
 
        auto q_2_m = CoefficientAccumulator(in[AllEntities::EntityId::q_r]);
        auto q_3_m = CoefficientAccumulator(in[AllEntities::EntityId::q_o]);
        auto q_4_m = CoefficientAccumulator(in[AllEntities::EntityId::q_4]);
        auto q_m_m = CoefficientAccumulator(in[AllEntities::EntityId::q_m]);
 
        auto q_nnf_m = CoefficientAccumulator(in[AllEntities::EntityId::q_nnf]);
        const FF LIMB_SIZE(uint256_t(1) << 68);
        const FF SUBLIMB_SHIFT(uint256_t(1) << 14);
 
        // Bigfield Product Gate 2 (selected by q_2 * q_4):
        // Computes cross-term contributions in limb multiplication.
        // Formula: (w_1 * w_2') + (w_1' * w_2) + (w_1 * w_4 + w_2 * w_3 - w_3') * 2^68 - w_4' = 0
        // where primed values (') denote shifted wires from the next row.
        auto limb_subproduct = w_1_m * w_2_shift_m + w_1_shift_m * w_2_m;
        auto non_native_field_gate_2_m = (w_1_m * w_4_m + w_2_m * w_3_m - w_3_shift_m);
        non_native_field_gate_2_m *= LIMB_SIZE;
        non_native_field_gate_2_m -= w_4_shift_m;
        non_native_field_gate_2_m += limb_subproduct;
        auto non_native_field_gate_2 = Accumulator(non_native_field_gate_2_m) * Accumulator(q_4_m);
 
        // Bigfield Product Gate 1 (selected by q_2 * q_3):
        // Accumulates limb products with 2^68 scaling for high-order terms.
        // Formula: (w_1 * w_2' + w_1' * w_2) * 2^68 + (w_1' * w_2') - w_3 - w_4 = 0
        limb_subproduct *= LIMB_SIZE;
        limb_subproduct += (w_1_shift_m * w_2_shift_m);
        auto non_native_field_gate_1_m = limb_subproduct;
        non_native_field_gate_1_m -= (w_3_m + w_4_m);
        // Transform to Accumulator for degree > 2 (CoefficientAccumulator limited to degree 2)
        auto non_native_field_gate_1 = Accumulator(non_native_field_gate_1_m) * Accumulator(q_3_m);
 
        // Bigfield Product Gate 3 (selected by q_2 * q_m):
        // Handles remaining cross-terms in the limb product expansion.
        // Formula: limb_subproduct + w_4 - w_3' - w_4' = 0
        auto non_native_field_gate_3_m = limb_subproduct;
        non_native_field_gate_3_m += w_4_m;
        non_native_field_gate_3_m -= (w_3_shift_m + w_4_shift_m);
        auto non_native_field_gate_3 = Accumulator(non_native_field_gate_3_m) * Accumulator(q_m_m);
 
        auto non_native_field_identity = non_native_field_gate_1 + non_native_field_gate_2 + non_native_field_gate_3;
        non_native_field_identity *= Accumulator(q_2_m);
 
        // Limb Accumulation Gate 1 (selected by q_3 * q_4):
        // Reconstructs a 68-bit limb from five 14-bit sublimbs stored across wires.
        // Constraint: w_2' * 2^56 + w_1' * 2^42 + w_3 * 2^28 + w_2 * 2^14 + w_1 - w_4 = 0
        // Horner form: ((((w_2' * 2^14 + w_1') * 2^14 + w_3) * 2^14 + w_2) * 2^14 + w_1) - w_4 = 0
        auto limb_accumulator_1_m = w_2_shift_m * SUBLIMB_SHIFT;
        limb_accumulator_1_m += w_1_shift_m;
        limb_accumulator_1_m *= SUBLIMB_SHIFT;
        limb_accumulator_1_m += w_3_m;
        limb_accumulator_1_m *= SUBLIMB_SHIFT;
        limb_accumulator_1_m += w_2_m;
        limb_accumulator_1_m *= SUBLIMB_SHIFT;
        limb_accumulator_1_m += w_1_m;
        limb_accumulator_1_m -= w_4_m;
        auto limb_accumulator_1_m_full = limb_accumulator_1_m * q_4_m;
 
        // Limb Accumulation Gate 2 (selected by q_3 * q_m):
        // Reconstructs a second 68-bit limb from five 14-bit sublimbs.
        // Constraint: w_3' * 2^56 + w_2' * 2^42 + w_1' * 2^28 + w_4 * 2^14 + w_3 - w_4' = 0
        // Horner form: ((((w_3' * 2^14 + w_2') * 2^14 + w_1') * 2^14 + w_4) * 2^14 + w_3) - w_4' = 0
        auto limb_accumulator_2_m = w_3_shift_m * SUBLIMB_SHIFT;
        limb_accumulator_2_m += w_2_shift_m;
        limb_accumulator_2_m *= SUBLIMB_SHIFT;
        limb_accumulator_2_m += w_1_shift_m;
        limb_accumulator_2_m *= SUBLIMB_SHIFT;
        limb_accumulator_2_m += w_4_m;
        limb_accumulator_2_m *= SUBLIMB_SHIFT;
        limb_accumulator_2_m += w_3_m;
        limb_accumulator_2_m -= w_4_shift_m;
        auto limb_accumulator_2_m_full = limb_accumulator_2_m * q_m_m;
 
        auto limb_accumulator_identity_m = limb_accumulator_1_m_full + limb_accumulator_2_m_full;
        Accumulator limb_accumulator_identity(limb_accumulator_identity_m);
        limb_accumulator_identity *= q_3_m; //  deg 3
 
        auto q_nnf_by_scaling_m = q_nnf_m * scaling_factor; // deg 1
        auto q_nnf_by_scaling = Accumulator(q_nnf_by_scaling_m);
 
        auto nnf_identity = non_native_field_identity + limb_accumulator_identity;
        nnf_identity *= q_nnf_by_scaling;          // deg
        std::get<0>(accumulators) += nnf_identity; // deg
    };
};
 
template <typename FF> using NonNativeFieldRelation = Relation<NonNativeFieldRelationImpl<FF>>;
} // namespace bb