cycle_group.cpp

Go to the documentation of this file.

// === AUDIT STATUS ===
// internal:    { status: Complete, auditors: [Luke], commit: a48c205d6dcd4338f5b83b4fda18bff6015be07b}
// external_1:  { status: Complete, auditors: [Sherlock], commit: 13c1b927c6c }
// external_2:  { status: not started, auditors: [], commit: }
// =====================
 
#include "../field/field.hpp"
#include "../field/field_utils.hpp"
#include "barretenberg/common/assert.hpp"
#include "barretenberg/common/thread.hpp"
#include "barretenberg/common/zip_view.hpp"
#include "barretenberg/crypto/pedersen_commitment/pedersen.hpp"
#include "barretenberg/ecc/curves/grumpkin/grumpkin.hpp"
 
#include "./cycle_group.hpp"
#include "barretenberg/numeric/general/general.hpp"
#include "barretenberg/stdlib/primitives/plookup/plookup.hpp"
#include "barretenberg/stdlib_circuit_builders/plookup_tables/fixed_base/fixed_base.hpp"
#include "barretenberg/stdlib_circuit_builders/plookup_tables/types.hpp"
#include "barretenberg/transcript/origin_tag.hpp"
namespace bb::stdlib {
 
template <typename Builder> cycle_group<Builder>::cycle_group(Builder* _context)
{
    *this = constant_infinity(_context);
}
 
template <typename Builder>
cycle_group<Builder>::cycle_group(const field_t& x, const field_t& y, bool assert_on_curve)
    : _x(x)
    , _y(y)
    , context(nullptr)
{
    if (_x.get_context() != nullptr) {
        context = _x.get_context();
    } else if (_y.get_context() != nullptr) {
        context = _y.get_context();
    }
 
    // Auto-detect infinity: point is at infinity iff both coordinates are zero
    // we can check this efficiently using the observation that:
    // (x^2 + 5 * y^2 = 0) has no non-trivial solutions in fr, since fr modulus p == 2 mod 5
    const field_t x_sqr = _x.sqr();
    const field_t five_y = _y * bb::fr(5);
    _is_infinity = (_y.madd(five_y, x_sqr).is_zero());
 
    // For the simplicity of methods in this class, we ensure that the coordinates of a point always have the same
    // constancy. If they don't, we convert the non-constant coordinate to a fixed witness. Should be rare.
    if (_x.is_constant() != _y.is_constant()) {
        if (_x.is_constant()) {
            _x.convert_constant_to_fixed_witness(context);
        } else {
            _y.convert_constant_to_fixed_witness(context);
        }
    }
 
    // Elements are always expected to be on the curve but may or may not be constrained as such.
    BB_ASSERT(get_value().on_curve(), "cycle_group: Point is not on curve");
    if (assert_on_curve) {
        validate_on_curve();
    }
}
 
template <typename Builder>
cycle_group<Builder>::cycle_group(const field_t& x, const field_t& y, bool_t is_infinity, bool assert_on_curve)
    : _x(x)
    , _y(y)
    , _is_infinity(is_infinity)
{
    if (_x.get_context() != nullptr) {
        context = _x.get_context();
    } else if (_y.get_context() != nullptr) {
        context = _y.get_context();
    } else {
        context = is_infinity.get_context();
    }
 
    if (is_infinity.is_constant() && is_infinity.get_value()) {
        *this = constant_infinity(this->context);
    }
 
    // For the simplicity of methods in this class, we ensure that the coordinates of a point always have the same
    // constancy. If they don't, we convert the non-constant coordinate to a fixed witness. Should be rare.
    if (_x.is_constant() != _y.is_constant()) {
        if (_x.is_constant()) {
            _x.convert_constant_to_fixed_witness(context);
        } else {
            _y.convert_constant_to_fixed_witness(context);
        }
    }
 
    // If both coordinates are constant, is_infinity must also be constant.
    BB_ASSERT(!(_x.is_constant() && _y.is_constant() && !_is_infinity.is_constant()),
              "cycle_group: constant coordinates with non-constant infinity flag");
 
    // Elements are always expected to be on the curve but may or may not be constrained as such.
    BB_ASSERT(get_value().on_curve(), "cycle_group: Point is not on curve");
    if (assert_on_curve) {
        validate_on_curve();
    }
}
 
template <typename Builder>
cycle_group<Builder>::cycle_group(const AffineElement& _in)
    : _x(_in.is_point_at_infinity() ? 0 : _in.x)
    , _y(_in.is_point_at_infinity() ? 0 : _in.y)
    , _is_infinity(_in.is_point_at_infinity())
    , context(nullptr)
{}
 
template <typename Builder> cycle_group<Builder> cycle_group<Builder>::one(Builder* _context)
{
    field_t x(_context, Group::one.x);
    field_t y(_context, Group::one.y);
    // Generator point is known to be on the curve
    return cycle_group<Builder>(x, y, /*assert_on_curve=*/false);
}
 
template <typename Builder> cycle_group<Builder> cycle_group<Builder>::constant_infinity(Builder* _context)
{
    // Use the AffineElement constructor with an infinity point
    // This properly sets (0, 0) coordinates and is_infinity = true
    cycle_group result(AffineElement::infinity());
 
    // If context provided, create field_t/bool_t with that context
    if (_context != nullptr) {
        result._x = field_t(_context, bb::fr(0));
        result._y = field_t(_context, bb::fr(0));
        result._is_infinity = bool_t(_context, true);
        result.context = _context;
    }
 
    return result;
}
 
template <typename Builder>
cycle_group<Builder> cycle_group<Builder>::from_witness(Builder* _context, const AffineElement& _in)
{
    // By convention we set the coordinates of the point at infinity to (0,0).
    field_t x_val = _in.is_point_at_infinity() ? field_t::from_witness(_context, bb::fr::zero())
                                               : field_t::from_witness(_context, _in.x);
    field_t y_val = _in.is_point_at_infinity() ? field_t::from_witness(_context, bb::fr::zero())
                                               : field_t::from_witness(_context, _in.y);
    // The constructor auto-detects infinity from coordinates and validates on curve.
    cycle_group result(x_val, y_val, /*assert_on_curve=*/true);
    result.set_free_witness_tag();
    return result;
}
 
template <typename Builder>
cycle_group<Builder> cycle_group<Builder>::from_constant_witness(Builder* _context, const AffineElement& _in)
{
    cycle_group result(_context);
 
    if (_in.is_point_at_infinity()) {
        result = constant_infinity(_context);
    } else {
        result._x = field_t::from_witness(_context, _in.x);
        result._y = field_t::from_witness(_context, _in.y);
        result._x.assert_equal(result._x.get_value());
        result._y.assert_equal(result._y.get_value());
    }
    // point at infinity is circuit constant
    result._is_infinity = _in.is_point_at_infinity();
    result.unset_free_witness_tag();
    return result;
}
 
template <typename Builder> Builder* cycle_group<Builder>::get_context(const cycle_group& other) const
{
    if (get_context() != nullptr) {
        return get_context();
    }
    return other.get_context();
}
 
template <typename Builder> typename cycle_group<Builder>::AffineElement cycle_group<Builder>::get_value() const
{
    AffineElement result(x().get_value(), y().get_value());
    if (is_point_at_infinity().get_value()) {
        result.self_set_infinity();
    }
    return result;
}
 
template <typename Builder> void cycle_group<Builder>::validate_on_curve() const
{
    // This class is for short Weierstrass curves only!
    static_assert(Group::curve_a == 0);
    auto xx = _x * _x;
    auto xxx = xx * _x;
    auto res = _y.madd(_y, -xxx - Group::curve_b);
    // If this is the point at infinity, then res is changed to 0, otherwise it remains unchanged
    res *= !is_point_at_infinity();
    res.assert_is_zero();
}
 
template <typename Builder> void cycle_group<Builder>::standardize()
{
    // Set coordinates to (0, 0) if point is at infinity for canonical representation
    this->_x = field_t::conditional_assign(this->_is_infinity, 0, this->_x);
    this->_y = field_t::conditional_assign(this->_is_infinity, 0, this->_y);
    // Mark witnesses as used to prevent boomerang detection false positives when
    // this is the final operation (e.g., final batch_mul output)
    mark_witness_as_used(this->_x);
    mark_witness_as_used(this->_y);
}
 
template <typename Builder>
cycle_group<Builder> cycle_group<Builder>::dbl(const std::optional<AffineElement> hint) const
{
    // If this is a constant point at infinity, return early
    if (this->is_constant_point_at_infinity()) {
        return *this;
    }
 
    // To support the point at infinity, we conditionally modify y to be 1 to avoid division by zero in the
    // doubling formula
    auto modified_y = field_t::conditional_assign(is_point_at_infinity(), 1, _y);
 
    // Compute the doubled point coordinates (either from hint or by native calculation)
    bb::fr x3;
    bb::fr y3;
    if (hint.has_value()) {
        x3 = hint.value().x;
        y3 = hint.value().y;
    } else {
        const bb::fr x1 = _x.get_value();
        const bb::fr y1 = modified_y.get_value();
 
        // N.B. the formula to derive the witness value for x3 mirrors the formula in elliptic_relation.hpp
        // Specifically, we derive x^4 via the Short Weierstrass curve formula y^2 = x^3 + b i.e. x^4 = x * (y^2 - b)
        // We must follow this pattern exactly to support the edge-case where the input is the point at infinity.
        const bb::fr y_pow_2 = y1.sqr();
        const bb::fr x_pow_4 = x1 * (y_pow_2 - Group::curve_b);
        const bb::fr lambda_squared = (x_pow_4 * 9) / (y_pow_2 * 4);
        const bb::fr lambda = (x1 * x1 * 3) / (y1 + y1);
        x3 = lambda_squared - x1 - x1;
        y3 = lambda * (x1 - x3) - y1;
    }
 
    // Construct the doubled point based on whether this is a constant or witness
    cycle_group result;
    if (is_constant()) {
        result = cycle_group(x3, y3, is_point_at_infinity(), /*assert_on_curve=*/false);
        // Propagate the origin tag as-is
        result.set_origin_tag(get_origin_tag());
    } else {
        // Create result witness and construct ECC double constraint
        result = cycle_group(
            witness_t(context, x3), witness_t(context, y3), is_point_at_infinity(), /*assert_on_curve=*/false);
 
        context->create_ecc_dbl_gate(bb::ecc_dbl_gate_<bb::fr>{
            .x1 = _x.get_witness_index(),
            .y1 = modified_y.get_witness_index(),
            .x3 = result._x.get_witness_index(),
            .y3 = result._y.get_witness_index(),
        });
 
        // Merge the x and y origin tags since the output depends on both
        result._x.set_origin_tag(OriginTag(_x.get_origin_tag(), _y.get_origin_tag()));
        result._y.set_origin_tag(OriginTag(_x.get_origin_tag(), _y.get_origin_tag()));
    }
 
    return result;
}
 
template <typename Builder>
cycle_group<Builder> cycle_group<Builder>::_unconditional_add_or_subtract(const cycle_group& other,
                                                                          bool is_addition,
                                                                          const std::optional<AffineElement> hint) const
{
    // Reject point-at-infinity operands: the incomplete ecc_add_gate is degenerate at (0, 0) (the
    // canonical witness representation of infinity) and admits forged outputs there.
    this->is_point_at_infinity().assert_equal(
        false, "cycle_group::_unconditional_add_or_subtract called on point at infinity");
    other.is_point_at_infinity().assert_equal(
        false, "cycle_group::_unconditional_add_or_subtract called on point at infinity");
 
    auto context = get_context(other);
 
    // If one point is a witness and the other is a constant, convert the constant to a fixed witness then call this
    // method again so we can use the ecc_add gate
    const bool lhs_constant = is_constant();
    const bool rhs_constant = other.is_constant();
 
    if (lhs_constant && !rhs_constant) {
        auto lhs = cycle_group::from_constant_witness(context, get_value());
        lhs.set_origin_tag(get_origin_tag()); // Maintain the origin tag
        return lhs._unconditional_add_or_subtract(other, is_addition, hint);
    }
    if (!lhs_constant && rhs_constant) {
        auto rhs = cycle_group::from_constant_witness(context, other.get_value());
        rhs.set_origin_tag(other.get_origin_tag()); // Maintain the origin tag
        return _unconditional_add_or_subtract(rhs, is_addition, hint);
    }
 
    // Compute the result coordinates (either from hint or by native calculation)
    bb::fr x3;
    bb::fr y3;
    if (hint.has_value()) {
        x3 = hint.value().x;
        y3 = hint.value().y;
    } else {
        const AffineElement p1 = get_value();
        const AffineElement p2 = other.get_value();
        AffineElement p3 = is_addition ? (Element(p1) + Element(p2)) : (Element(p1) - Element(p2));
        x3 = p3.x;
        y3 = p3.y;
    }
 
    // Construct the result based on whether inputs are constant or witness
    // Note: this code path is for non-infinity points, so result cannot be at infinity
    // We use the private 4-arg constructor to explicitly set is_infinity=false, avoiding the
    // auto-detection gates that the 2-arg constructor would add.
    cycle_group result;
    if (lhs_constant && rhs_constant) {
        result = cycle_group(x3, y3, /*is_infinity=*/bool_t(false), /*assert_on_curve=*/false);
    } else {
        // Both points are witnesses - create result witness and construct ECC add constraint
        result = cycle_group(
            witness_t(context, x3), witness_t(context, y3), /*is_infinity=*/false, /*assert_on_curve=*/false);
 
        context->create_ecc_add_gate(bb::ecc_add_gate_{
            .x1 = _x.get_witness_index(),
            .y1 = _y.get_witness_index(),
            .x2 = other._x.get_witness_index(),
            .y2 = other._y.get_witness_index(),
            .x3 = result._x.get_witness_index(),
            .y3 = result._y.get_witness_index(),
            .is_addition = is_addition,
        });
    }
 
    // Merge the origin tags from both inputs
    result.set_origin_tag(OriginTag(get_origin_tag(), other.get_origin_tag()));
 
    return result;
}
 
template <typename Builder>
cycle_group<Builder> cycle_group<Builder>::unconditional_add(const cycle_group& other,
                                                             const std::optional<AffineElement> hint) const
{
    return _unconditional_add_or_subtract(other, /*is_addition=*/true, hint);
}
 
template <typename Builder>
cycle_group<Builder> cycle_group<Builder>::unconditional_subtract(const cycle_group& other,
                                                                  const std::optional<AffineElement> hint) const
{
    return _unconditional_add_or_subtract(other, /*is_addition=*/false, hint);
}
 
template <typename Builder>
cycle_group<Builder> cycle_group<Builder>::checked_unconditional_add(const cycle_group& other,
                                                                     const std::optional<AffineElement> hint) const
{
    const field_t x_delta = this->_x - other._x;
    if (x_delta.is_constant()) {
        BB_ASSERT(x_delta.get_value() != 0);
    } else {
        x_delta.assert_is_not_zero("cycle_group::checked_unconditional_add, x-coordinate collision");
    }
    return unconditional_add(other, hint);
}
 
template <typename Builder>
cycle_group<Builder> cycle_group<Builder>::checked_unconditional_subtract(const cycle_group& other,
                                                                          const std::optional<AffineElement> hint) const
{
    const field_t x_delta = this->_x - other._x;
    if (x_delta.is_constant()) {
        BB_ASSERT(x_delta.get_value() != 0);
    } else {
        x_delta.assert_is_not_zero("cycle_group::checked_unconditional_subtract, x-coordinate collision");
    }
    return unconditional_subtract(other, hint);
}
 
template <typename Builder> cycle_group<Builder> cycle_group<Builder>::operator+(const cycle_group& other) const
{
    // If lhs is constant point at infinity, return the rhs and vice versa
    if (this->is_constant_point_at_infinity()) {
        return other;
    }
    if (other.is_constant_point_at_infinity()) {
        return *this;
    }
 
    const bool_t x_coordinates_match = (_x == other._x);
    const bool_t y_coordinates_match = (_y == other._y);
 
    const field_t x1 = _x;
    const field_t y1 = _y;
    const field_t x2 = other._x;
    const field_t y2 = other._y;
 
    // Execute point addition with modified lambda = (y2 - y1)/(x2 - x1 + x_coordinates_match) to avoid the possibility
    // of division by zero.
    const field_t x_diff = x2.add_two(-x1, x_coordinates_match);
    // Compute lambda in one of two ways depending on whether either numerator or denominator is constant or not
    field_t lambda;
    if ((y1.is_constant() && y2.is_constant()) || x_diff.is_constant()) {
        lambda = (y2 - y1).divide_no_zero_check(x_diff);
    } else {
        // Note: branch saves one gate vs just using divide_no_zero_check because we avoid computing y2 - y1 in circuit
        Builder* context = get_context(other);
        lambda = field_t::from_witness(context, (y2.get_value() - y1.get_value()) / x_diff.get_value());
        // We need to manually propagate the origin tag
        lambda.set_origin_tag(OriginTag(x_diff.get_origin_tag(), y1.get_origin_tag(), y2.get_origin_tag()));
        // Constrain x_diff * lambda = y2 - y1
        field_t::evaluate_polynomial_identity(x_diff, lambda, -y2, y1);
    }
    const field_t add_result_x = lambda.madd(lambda, -(x2 + x1));     // x3 = lambda^2 - x1 - x2
    const field_t add_result_y = lambda.madd(x1 - add_result_x, -y1); // y3 = lambda * (x1 - x3) - y1
 
    // Compute the doubling result
    const cycle_group dbl_result = dbl();
 
    // If the addition amounts to a doubling then the result is the doubling result, else the addition result.
    const bool_t double_predicate = (x_coordinates_match && y_coordinates_match);
    auto result_x = field_t::conditional_assign(double_predicate, dbl_result._x, add_result_x);
    auto result_y = field_t::conditional_assign(double_predicate, dbl_result._y, add_result_y);
 
    // If the lhs is the point at infinity, return rhs
    const bool_t lhs_infinity = is_point_at_infinity();
    result_x = field_t::conditional_assign(lhs_infinity, other._x, result_x);
    result_y = field_t::conditional_assign(lhs_infinity, other._y, result_y);
 
    // If the rhs is the point at infinity, return lhs
    const bool_t rhs_infinity = other.is_point_at_infinity();
    result_x = field_t::conditional_assign(rhs_infinity, _x, result_x);
    result_y = field_t::conditional_assign(rhs_infinity, _y, result_y);
 
    // The result is the point at infinity if:
    // (lhs._x, lhs._y) == (rhs._x, -rhs._y) and neither are infinity, OR both are the point at infinity
    const bool_t infinity_predicate = (x_coordinates_match && !y_coordinates_match);
    bool_t result_is_infinity = infinity_predicate && (!lhs_infinity && !rhs_infinity);
    result_is_infinity = result_is_infinity || (lhs_infinity && rhs_infinity);
 
    return cycle_group(result_x, result_y, /*is_infinity=*/result_is_infinity, /*assert_on_curve=*/false);
}
 
template <typename Builder> cycle_group<Builder> cycle_group<Builder>::operator-(const cycle_group& other) const
{
    // If lhs is constant point at infinity, return -rhs
    if (this->is_constant_point_at_infinity()) {
        return -other;
    }
    // If rhs is constant point at infinity, return the lhs
    if (other.is_constant_point_at_infinity()) {
        return *this;
    }
 
    Builder* context = get_context(other);
 
    const bool_t x_coordinates_match = (_x == other._x);
    const bool_t y_coordinates_match = (_y == other._y);
 
    const field_t x1 = _x;
    const field_t y1 = _y;
    const field_t x2 = other._x;
    const field_t y2 = other._y;
 
    // Execute point addition with modified lambda = (-y2 - y1)/(x2 - x1 + x_coordinates_match) to avoid the possibility
    // of division by zero.
    const field_t x_diff = x2.add_two(-x1, x_coordinates_match);
    // Compute lambda in one of two ways depending on whether either numerator or denominator is constant or not
    field_t lambda;
    if ((y1.is_constant() && y2.is_constant()) || x_diff.is_constant()) {
        lambda = (-y2 - y1).divide_no_zero_check(x_diff);
    } else {
        // Note: branch saves one gate vs using divide_no_zero_check because we avoid computing (-y2 - y1) in circuit
        lambda = field_t::from_witness(context, (-y2.get_value() - y1.get_value()) / x_diff.get_value());
        // We need to manually propagate the origin tag
        lambda.set_origin_tag(OriginTag(x_diff.get_origin_tag(), y1.get_origin_tag(), y2.get_origin_tag()));
        // Constrain x_diff * lambda = -y2 - y1
        field_t::evaluate_polynomial_identity(x_diff, lambda, y2, y1);
    }
    const field_t sub_result_x = lambda.madd(lambda, -(x2 + x1));     // x3 = lambda^2 - x1 - x2
    const field_t sub_result_y = lambda.madd(x1 - sub_result_x, -y1); // y3 = lambda * (x1 - x3) - y1
 
    // Compute the doubling result
    const cycle_group dbl_result = dbl();
 
    // If the subtraction amounts to a doubling then the result is the doubling result, else the subtraction result.
    // Note: The assumption here is that x1 == x2 && y1 != y2 implies y1 == -y2 which is true assuming that the points
    // are both on the curve.
    const bool_t double_predicate = (x_coordinates_match && !y_coordinates_match);
    auto result_x = field_t::conditional_assign(double_predicate, dbl_result._x, sub_result_x);
    auto result_y = field_t::conditional_assign(double_predicate, dbl_result._y, sub_result_y);
 
    mark_witness_as_used(result_x);
    mark_witness_as_used(result_y);
 
    // If the lhs is the point at infinity, return -rhs
    const bool_t lhs_infinity = is_point_at_infinity();
    result_x = field_t::conditional_assign(lhs_infinity, other._x, result_x);
    result_y = field_t::conditional_assign(lhs_infinity, (-other._y), result_y);
 
    // If the rhs is the point at infinity, return lhs
    const bool_t rhs_infinity = other.is_point_at_infinity();
    result_x = field_t::conditional_assign(rhs_infinity, _x, result_x);
    result_y = field_t::conditional_assign(rhs_infinity, _y, result_y);
 
    // The result is the point at infinity if:
    // (lhs.x, lhs.y) == (rhs.x, rhs.y) and neither are infinity, OR both are the point at infinity
    const bool_t infinity_predicate = (x_coordinates_match && y_coordinates_match);
    mark_witness_as_used(field_t(infinity_predicate));
    bool_t result_is_infinity = infinity_predicate && (!lhs_infinity && !rhs_infinity);
    result_is_infinity = result_is_infinity || (lhs_infinity && rhs_infinity);
 
    return cycle_group(result_x, result_y, /*is_infinity=*/result_is_infinity, /*assert_on_curve=*/false);
}
 
template <typename Builder> cycle_group<Builder> cycle_group<Builder>::operator-() const
{
    cycle_group result(*this);
    // We have to normalize immediately. All the methods, related to
    // elliptic curve operations, assume that the coordinates are in normalized form and
    // don't perform any extra normalizations
    result._y = (-_y).normalize();
    return result;
}
 
template <typename Builder> cycle_group<Builder>& cycle_group<Builder>::operator+=(const cycle_group& other)
{
    *this = *this + other;
    return *this;
}
 
template <typename Builder> cycle_group<Builder>& cycle_group<Builder>::operator-=(const cycle_group& other)
{
    *this = *this - other;
    return *this;
}
 
template <typename Builder>
typename cycle_group<Builder>::batch_mul_internal_output cycle_group<Builder>::_variable_base_batch_mul_internal(
    const std::span<cycle_scalar> scalars,
    const std::span<cycle_group> base_points,
    const std::span<AffineElement const> offset_generators,
    const bool unconditional_add)
{
    BB_ASSERT_EQ(!scalars.empty(), true, "Empty scalars provided to variable base batch mul!");
    BB_ASSERT_EQ(scalars.size(), base_points.size(), "Points/scalars size mismatch in variable base batch mul!");
    BB_ASSERT_EQ(offset_generators.size(), base_points.size() + 1, "Too few offset generators provided!");
    const size_t num_points = scalars.size();
 
    // Extract the circuit context from any scalar or point
    Builder* context = nullptr;
    for (auto [scalar, point] : zip_view(scalars, base_points)) {
        if (context = scalar.get_context(); context != nullptr) {
            break;
        }
        if (context = point.get_context(); context != nullptr) {
            break;
        }
    }
 
    // All cycle_scalars are guaranteed to be 254 bits
    constexpr size_t num_rounds = numeric::ceil_div(cycle_scalar::NUM_BITS, ROM_TABLE_BITS);
 
    // Decompose each scalar into 4-bit slices. Note: This operation enforces range constraints on the lo/hi limbs of
    // each scalar (LO_BITS and (num_bits - LO_BITS) respectively).
    std::vector<straus_scalar_slices> scalar_slices;
    scalar_slices.reserve(num_points);
    for (const auto& scalar : scalars) {
        scalar_slices.emplace_back(context, scalar, ROM_TABLE_BITS);
    }
 
    // Compute the result of each point addition involved in the Straus MSM algorithm natively so they can be used as
    // "hints" in the in-circuit Straus algorithm. This includes the additions needed to construct the point tables and
    // those needed to compute the MSM via Straus. Points are computed as Element types with a Z-coordinate then
    // batch-converted to AffineElement types. This avoids the need to compute modular inversions for every group
    // operation, which dramatically reduces witness generation times.
    std::vector<Element> operation_transcript;
    Element offset_generator_accumulator = offset_generators[0];
    {
        // For each point, construct native straus lookup table of the form {G , G + [1]P, G + [2]P, ... , G + [15]P}
        std::vector<std::vector<Element>> native_straus_tables;
        for (size_t i = 0; i < num_points; ++i) {
            AffineElement point = base_points[i].get_value();
            AffineElement offset = offset_generators[i + 1];
            std::vector<Element> table = straus_lookup_table::compute_native_table(point, offset, ROM_TABLE_BITS);
            // Copy all but the first entry (the offset generator) into the operation transcript for use as hints
            std::copy(table.begin() + 1, table.end(), std::back_inserter(operation_transcript));
            native_straus_tables.emplace_back(std::move(table));
        }
 
        // Perform the Straus algorithm natively to generate the witness values (hints) for all intermediate points
        Element accumulator = offset_generators[0];
        for (size_t i = 0; i < num_rounds; ++i) {
            if (i != 0) {
                // Perform doublings of the accumulator and offset generator accumulator
                for (size_t j = 0; j < ROM_TABLE_BITS; ++j) {
                    accumulator = accumulator.dbl();
                    operation_transcript.push_back(accumulator);
                    offset_generator_accumulator = offset_generator_accumulator.dbl();
                }
            }
            for (size_t j = 0; j < num_points; ++j) {
                // Look up and accumulate the appropriate point for this scalar slice
                auto slice_value = static_cast<size_t>(scalar_slices[j].slices_native[num_rounds - i - 1]);
                const Element point = native_straus_tables[j][slice_value];
                accumulator += point;
 
                // Populate hint and update offset generator accumulator
                operation_transcript.push_back(accumulator);
                offset_generator_accumulator += Element(offset_generators[j + 1]);
            }
        }
    }
 
    // Normalize the computed witness points and convert them into AffineElements
    Element::batch_normalize(&operation_transcript[0], operation_transcript.size());
    std::vector<AffineElement> operation_hints;
    operation_hints.reserve(operation_transcript.size());
    for (const Element& element : operation_transcript) {
        operation_hints.emplace_back(element.x, element.y);
    }
 
    // Construct an in-circuit Straus lookup table for each point
    std::vector<straus_lookup_table> point_tables;
    point_tables.reserve(num_points);
    const size_t hints_per_table = (1ULL << ROM_TABLE_BITS) - 1;
    for (size_t i = 0; i < num_points; ++i) {
        // Construct Straus table
        std::span<AffineElement> table_hints(&operation_hints[i * hints_per_table], hints_per_table);
        straus_lookup_table table(context, base_points[i], offset_generators[i + 1], ROM_TABLE_BITS, table_hints);
        point_tables.push_back(std::move(table));
    }
 
    // Initialize pointer to the precomputed Straus algorithm hints (stored just after the table construction hints)
    AffineElement* hint_ptr = &operation_hints[num_points * hints_per_table];
    cycle_group accumulator = offset_generators[0];
 
    // Execute Straus algorithm in-circuit using the precomputed hints.
    // If unconditional_add == false, accumulate x-coordinate differences to batch-validate no collisions.
    // If unconditional_add == true (constant inputs), no per-slice check is added; soundness relies on the
    // offset-generator domain separator keeping per-slice operands distinct.
    field_t coordinate_check_product = 1;
    for (size_t i = 0; i < num_rounds; ++i) {
        // Double the accumulator ROM_TABLE_BITS times (except in first round)
        if (i != 0) {
            for (size_t j = 0; j < ROM_TABLE_BITS; ++j) {
                accumulator = accumulator.dbl(*hint_ptr);
                hint_ptr++;
            }
        }
        // Add the contribution from each point's scalar slice for this round
        for (size_t j = 0; j < num_points; ++j) {
            const field_t scalar_slice = scalar_slices[j][num_rounds - i - 1];
            BB_ASSERT_EQ(scalar_slice.get_value(), scalar_slices[j].slices_native[num_rounds - i - 1]); // Sanity check
            const cycle_group point = point_tables[j].read(scalar_slice);
            if (!unconditional_add) {
                coordinate_check_product *= point._x - accumulator._x;
            }
            accumulator = accumulator.unconditional_add(point, *hint_ptr);
            hint_ptr++;
        }
    }
 
    // Batch-validate no x-coordinate collisions occurred. We batch because each assert_is_not_zero
    // requires an expensive modular inversion during witness generation.
    if (!unconditional_add) {
        coordinate_check_product.assert_is_not_zero("_variable_base_batch_mul_internal x-coordinate collision");
    }
 
    // Set CONSTANT tag on accumulator to avoid tag conflicts during intermediate operations
    // The final tag will be set in batch_mul from result_tag
    accumulator.set_origin_tag(OriginTag::constant());
 
    // Note: offset_generator_accumulator represents the sum of all the offset generator terms present in `accumulator`.
    // We don't subtract it off yet as we may be able to combine it with other constant terms in `batch_mul` before
    // performing the subtraction.
    return { accumulator, AffineElement(offset_generator_accumulator) };
}
 
template <typename Builder>
typename cycle_group<Builder>::batch_mul_internal_output cycle_group<Builder>::_fixed_base_batch_mul_internal(
    const std::span<cycle_scalar> scalars, const std::span<AffineElement> base_points)
{
    BB_ASSERT_EQ(scalars.size(), base_points.size(), "Points/scalars size mismatch in fixed-base batch mul");
 
    using MultiTableId = plookup::MultiTableId;
    using ColumnIdx = plookup::ColumnIdx;
    using plookup::fixed_base::table;
 
    std::vector<MultiTableId> multitable_ids;
    std::vector<field_t> scalar_limbs;
 
    for (const auto [point, scalar] : zip_view(base_points, scalars)) {
        std::array<MultiTableId, 2> table_id = table::get_lookup_table_ids_for_point(point);
        multitable_ids.push_back(table_id[0]);
        multitable_ids.push_back(table_id[1]);
        scalar_limbs.push_back(scalar.lo());
        scalar_limbs.push_back(scalar.hi());
    }
 
    // Look up the multiples of each slice of each lo/hi scalar limb in the corresponding plookup table.
    std::vector<cycle_group> lookup_points;
    Element offset_generator_accumulator = Group::point_at_infinity;
    for (const auto [table_id, scalar] : zip_view(multitable_ids, scalar_limbs)) {
        // Each lookup returns multiple EC points corresponding to different bit slices of the scalar.
        // For a scalar slice s_i at bit position (table_bits*i), the table stores the point:
        //     P_i = [offset_generator_i] + (s_i * 2^(table_bits*i)) * [base_point]
        plookup::ReadData<field_t> lookup_data = plookup_read<Builder>::get_lookup_accumulators(table_id, scalar);
        for (size_t j = 0; j < lookup_data[ColumnIdx::C2].size(); ++j) {
            const field_t x = lookup_data[ColumnIdx::C2][j];
            const field_t y = lookup_data[ColumnIdx::C3][j];
            // Lookup table points are never at infinity (they are precomputed points on the curve).
            // Use private constructor to avoid auto-detection gates.
            auto is_infinity = bool_t(x.get_context(), false);
            lookup_points.push_back(cycle_group(x, y, is_infinity, /*assert_on_curve=*/false));
        }
        // Update offset accumulator with the total offset for the corresponding multitable
        offset_generator_accumulator += table::get_generator_offset_for_table_id(table_id);
    }
 
    // Compute the witness values of the batch_mul algorithm natively, as Element types with a Z-coordinate.
    std::vector<Element> operation_transcript;
    {
        Element accumulator = lookup_points[0].get_value();
        for (size_t i = 1; i < lookup_points.size(); ++i) {
            accumulator += (lookup_points[i].get_value());
            operation_transcript.push_back(accumulator);
        }
    }
    // Batch-convert to AffineElement types, and feed these points as "hints" into the in-circuit addition. This avoids
    // the need to compute modular inversions for every group operation, which dramatically reduces witness generation
    // times.
    Element::batch_normalize(&operation_transcript[0], operation_transcript.size());
    std::vector<AffineElement> operation_hints;
    operation_hints.reserve(operation_transcript.size());
    for (const Element& element : operation_transcript) {
        operation_hints.emplace_back(element.x, element.y);
    }
 
    // Perform the in-circuit point additions sequentially. Each addition costs 1 gate iff additions are chained such
    // that the output of each addition is the input to the next. Otherwise, each addition costs 2 gates.
    // Set all lookup_points to have CONSTANT tags to avoid tag conflicts during intermediate operations
    // The final tag will be set in batch_mul from result_tag
    auto const_tag = OriginTag::constant();
    for (auto& point : lookup_points) {
        point.set_origin_tag(const_tag);
    }
    cycle_group accumulator = lookup_points[0];
    for (size_t i = 1; i < lookup_points.size(); ++i) {
        accumulator = accumulator.unconditional_add(lookup_points[i], operation_hints[i - 1]);
    }
 
    // The offset_generator_accumulator represents the sum of all the offset generator terms present in `accumulator`.
    // We don't subtract off yet, as we may be able to combine `offset_generator_accumulator` with other constant
    // terms in `batch_mul` before performing the subtraction.
    return { accumulator, offset_generator_accumulator };
}
 
template <typename Builder>
typename cycle_group<Builder>::batch_mul_internal_output cycle_group<Builder>::_fixed_base_plookup_batch_mul_internal(
    const std::span<cycle_scalar> scalars,
    const std::span<AffineElement const> base_points,
    const std::span<AffineElement const> offset_generators,
    const size_t table_bits)
{
    BB_ASSERT_EQ(!scalars.empty(), true, "Empty scalars provided to fixed base plookup batch mul!");
    BB_ASSERT_EQ(scalars.size(), base_points.size(), "Points/scalars size mismatch in fixed base plookup batch mul!");
    BB_ASSERT_EQ(offset_generators.size(), base_points.size() + 1, "Too few offset generators provided!");
    const size_t num_points = scalars.size();
 
    Builder* context = nullptr;
    for (const auto& scalar : scalars) {
        if (context = scalar.get_context(); context != nullptr) {
            break;
        }
    }
    BB_ASSERT(context != nullptr);
    BB_ASSERT_EQ(cycle_scalar::LO_BITS % table_bits,
                 0UL,
                 "table_bits must evenly divide cycle_scalar::LO_BITS. The Straus algorithm splits the scalar "
                 "into lo/hi limbs and decomposes each separately; if LO_BITS is not a multiple of table_bits, "
                 "the hi-limb slices start at the wrong bit-offset and the MSM result is incorrect. "
                 "Valid values for table_bits (given LO_BITS=128) are: 1, 2, 4, 8, 16, 32, 64, 128.");
 
    const size_t num_rounds = numeric::ceil_div(cycle_scalar::NUM_BITS, table_bits);
 
    // Decompose each scalar into table_bits-bit slices (also enforces range constraints)
    std::vector<straus_scalar_slices> scalar_slices;
    scalar_slices.reserve(num_points);
    for (const auto& scalar : scalars) {
        scalar_slices.emplace_back(context, scalar, table_bits);
    }
 
    // Create plookup tables for each constant base point (zero gate cost).
    // Phase 1 (parallel): compute native table entries and BasicTable column data — no builder access.
    // Phase 2 (serial):   register each BasicTable with the builder (builder is not thread-safe).
    std::vector<typename straus_plookup_table::PrecomputedData> precomputed_tables(num_points);
    parallel_for(num_points, [&](size_t i) {
        precomputed_tables[i] =
            straus_plookup_table::build_precomputed_data(base_points[i], offset_generators[i + 1], table_bits);
    });
    std::vector<straus_plookup_table> point_tables;
    point_tables.reserve(num_points);
    for (size_t i = 0; i < num_points; ++i) {
        point_tables.emplace_back(context, std::move(precomputed_tables[i]));
    }
 
    // Compute all intermediate points natively for use as hints in the in-circuit Straus algorithm.
    // Using projective coordinates + batch normalize to avoid per-operation modular inversions.
    // The per-point lookup tables (point_tables) already hold the precomputed affine entries; we reuse
    // them directly rather than rebuilding projective copies.
    std::vector<Element> operation_transcript;
    Element offset_generator_accumulator = offset_generators[0];
    {
        // Perform Straus algorithm natively
        Element accumulator = offset_generators[0];
        for (size_t i = 0; i < num_rounds; ++i) {
            if (i != 0) {
                for (size_t j = 0; j < table_bits; ++j) {
                    accumulator = accumulator.dbl();
                    operation_transcript.push_back(accumulator);
                    offset_generator_accumulator = offset_generator_accumulator.dbl();
                }
            }
            for (size_t j = 0; j < num_points; ++j) {
                auto slice_value = static_cast<size_t>(scalar_slices[j].slices_native[num_rounds - i - 1]);
                const Element point(point_tables[j].get_native_table()[slice_value]);
                accumulator += point;
                operation_transcript.push_back(accumulator);
                offset_generator_accumulator += Element(offset_generators[j + 1]);
            }
        }
    }
 
    // Batch-normalize all hint points
    Element::batch_normalize(operation_transcript.data(), operation_transcript.size());
    std::vector<AffineElement> operation_hints;
    operation_hints.reserve(operation_transcript.size());
    for (const Element& element : operation_transcript) {
        operation_hints.emplace_back(element.x, element.y);
    }
 
    // Execute Straus algorithm in-circuit using plookup reads and precomputed hints
    AffineElement* hint_ptr = operation_hints.data();
    cycle_group accumulator = offset_generators[0];
 
    for (size_t i = 0; i < num_rounds; ++i) {
        if (i != 0) {
            for (size_t j = 0; j < table_bits; ++j) {
                accumulator = accumulator.dbl(*hint_ptr);
                hint_ptr++;
            }
        }
        for (size_t j = 0; j < num_points; ++j) {
            const field_t scalar_slice = scalar_slices[j][num_rounds - i - 1];
            const cycle_group point = point_tables[j].read(scalar_slice);
            // No per-slice x-collision check; soundness relies on the offset-generator domain separator
            // keeping `accumulator` distinct from any plookup table entry across the loop.
            accumulator = accumulator.unconditional_add(point, *hint_ptr);
            hint_ptr++;
        }
    }
 
    accumulator.set_origin_tag(OriginTag::constant());
    return { accumulator, AffineElement(offset_generator_accumulator) };
}
 
template <typename Builder>
cycle_group<Builder> cycle_group<Builder>::fixed_batch_mul(const std::vector<cycle_group>& constant_points,
                                                           const std::vector<cycle_scalar>& scalars,
                                                           const GeneratorContext& context,
                                                           const size_t table_bits)
{
    BB_ASSERT_EQ(scalars.size(), constant_points.size(), "Points/scalars size mismatch in fixed_batch_mul!");
 
    if (scalars.empty()) {
        return cycle_group{ Group::point_at_infinity };
    }
 
    // Merge all tags
    OriginTag result_tag = OriginTag::constant();
    for (auto [point, scalar] : zip_view(constant_points, scalars)) {
        result_tag = OriginTag(result_tag, OriginTag(point.get_origin_tag(), scalar.get_origin_tag()));
    }
 
    std::vector<cycle_scalar> plookup_scalars;
    std::vector<AffineElement> plookup_points;
    bool has_non_constant_component = false;
    Element constant_acc = Group::point_at_infinity;
 
    for (const auto [point, scalar] : zip_view(constant_points, scalars)) {
        BB_ASSERT(point.is_constant());
        if (scalar.is_constant()) {
            // Both constant: compute natively
            constant_acc += point.get_value() * scalar.get_value();
        } else {
            if (point.get_value().is_point_at_infinity()) {
                // Constant infinity contributes nothing, but still need range constraints on scalar
                auto* ctx = scalar.get_context();
                ctx->create_limbed_range_constraint(scalar.lo().get_witness_index(),
                                                    cycle_scalar::LO_BITS,
                                                    table_bits,
                                                    "fixed_batch_mul: lo range constraint for scalar with constant "
                                                    "infinity");
                ctx->create_limbed_range_constraint(scalar.hi().get_witness_index(),
                                                    cycle_scalar::HI_BITS,
                                                    table_bits,
                                                    "fixed_batch_mul: hi range constraint for scalar with constant "
                                                    "infinity");
                continue;
            }
            plookup_scalars.push_back(scalar);
            plookup_points.push_back(point.get_value());
            has_non_constant_component = true;
        }
    }
 
    if (!has_non_constant_component) {
        auto result = cycle_group(constant_acc);
        result.set_origin_tag(result_tag);
        return result;
    }
 
    // Compute offset generators
    const size_t num_offset_generators = plookup_points.size() + 1;
    const std::span<AffineElement const> offset_generators =
        context.generators->get(num_offset_generators, 0, OFFSET_GENERATOR_DOMAIN_SEPARATOR);
 
    // Run the plookup-based Straus algorithm
    Element offset_accumulator = -constant_acc;
    const auto [accumulator, offset_generator_delta] =
        _fixed_base_plookup_batch_mul_internal(plookup_scalars, plookup_points, offset_generators, table_bits);
    offset_accumulator += offset_generator_delta;
 
    // Note: No collision check on the offset subtraction; soundness relies on the offset-generator domain
    // separator keeping aggregate `offset_accumulator` distinct from `accumulator`.
    cycle_group result;
    if (!constant_acc.is_point_at_infinity()) {
        result = accumulator.unconditional_add(AffineElement(-offset_accumulator));
    } else {
        result = accumulator - cycle_group(AffineElement(offset_accumulator));
    }
 
    result.set_origin_tag(result_tag);
    return result;
}
 
template <typename Builder>
cycle_group<Builder> cycle_group<Builder>::batch_mul(const std::vector<cycle_group>& base_points,
                                                     const std::vector<cycle_scalar>& scalars,
                                                     const GeneratorContext& context)
{
    BB_ASSERT_EQ(scalars.size(), base_points.size(), "Points/scalars size mismatch in batch mul!");
 
    if (scalars.empty()) {
        cycle_group result{ Group::point_at_infinity };
        return result;
    }
 
    std::vector<cycle_scalar> variable_base_scalars;
    std::vector<cycle_group> variable_base_points;
    std::vector<cycle_scalar> fixed_base_scalars;
    std::vector<AffineElement> fixed_base_points;
 
    // Merge all tags
    OriginTag result_tag = OriginTag::constant(); // Initialize as CONSTANT so merging with input tags works correctly
    for (auto [point, scalar] : zip_view(base_points, scalars)) {
        result_tag = OriginTag(result_tag, OriginTag(point.get_origin_tag(), scalar.get_origin_tag()));
    }
 
    // We can unconditionally add in the variable-base algorithm iff all of the input points are fixed-base points (i.e.
    // we are doing fixed-base mul over points not present in our plookup tables)
    bool can_unconditional_add = true;
    bool has_non_constant_component = false;
    Element constant_acc = Group::point_at_infinity;
    for (const auto [point, scalar] : zip_view(base_points, scalars)) {
        if (scalar.is_constant() && point.is_constant()) {
            // Case 1: both point and scalar are constant; update constant accumulator without adding gates
            constant_acc += (point.get_value()) * (scalar.get_value());
        } else if (!scalar.is_constant() && point.is_constant()) {
            if (point.get_value().is_point_at_infinity()) {
                // oi mate, why are you creating a circuit that multiplies a known point at infinity?
#ifndef FUZZING_DISABLE_WARNINGS
                info("Warning: Performing batch mul with constant point at infinity!");
#endif
                // Constant infinity * witness scalar contributes nothing to the result, however, we must still apply
                // the range constraints that the cycle_scalar constructor defers to this method.
                auto* ctx = scalar.get_context();
                ctx->create_limbed_range_constraint(scalar.lo().get_witness_index(),
                                                    cycle_scalar::LO_BITS,
                                                    ROM_TABLE_BITS,
                                                    "batch_mul: lo range constraint for scalar with constant "
                                                    "infinity");
                ctx->create_limbed_range_constraint(scalar.hi().get_witness_index(),
                                                    cycle_scalar::HI_BITS,
                                                    ROM_TABLE_BITS,
                                                    "batch_mul: hi range constraint for scalar with constant "
                                                    "infinity");
                continue;
            }
            if (plookup::fixed_base::table::lookup_table_exists_for_point(point.get_value())) {
                // Case 2A: constant point is one of two for which we have plookup tables; use fixed-base Straus
                fixed_base_scalars.push_back(scalar);
                fixed_base_points.push_back(point.get_value());
            } else {
                // Case 2B: constant point but no precomputed lookup tables; use variable-base Straus with ROM tables
                variable_base_scalars.push_back(scalar);
                variable_base_points.push_back(point);
            }
            has_non_constant_component = true;
        } else {
            // Case 3: point is a witness; use variable-base Straus with ROM tables
            variable_base_scalars.push_back(scalar);
            variable_base_points.push_back(point);
            can_unconditional_add = false;
            has_non_constant_component = true;
        }
    }
 
    // If all inputs are constant, return the computed constant component and call it a day.
    if (!has_non_constant_component) {
        auto result = cycle_group(constant_acc);
        result.set_origin_tag(result_tag);
        return result;
    }
 
    // Add the constant component into our offset accumulator. (Note: we'll subtract `offset_accumulator` from the MSM
    // output later on so we negate here to counter that future negation).
    Element offset_accumulator = -constant_acc;
    const bool has_variable_points = !variable_base_points.empty();
    const bool has_fixed_points = !fixed_base_points.empty();
 
    cycle_group result;
    if (has_fixed_points) {
        // Compute the result of the fixed-base portion of the MSM and update the offset accumulator
        const auto [fixed_accumulator, offset_generator_delta] =
            _fixed_base_batch_mul_internal(fixed_base_scalars, fixed_base_points);
        offset_accumulator += offset_generator_delta;
        result = fixed_accumulator;
    }
 
    if (has_variable_points) {
        // Compute required offset generators; one per point plus one extra for the initial accumulator
        const size_t num_offset_generators = variable_base_points.size() + 1;
        const std::span<AffineElement const> offset_generators =
            context.generators->get(num_offset_generators, 0, OFFSET_GENERATOR_DOMAIN_SEPARATOR);
 
        // Compute the result of the variable-base portion of the MSM and update the offset accumulator
        const auto [variable_accumulator, offset_generator_delta] = _variable_base_batch_mul_internal(
            variable_base_scalars, variable_base_points, offset_generators, can_unconditional_add);
        offset_accumulator += offset_generator_delta;
 
        // Combine the variable-base result with the fixed-base result (if present)
        if (has_fixed_points) {
            result = can_unconditional_add ? result.unconditional_add(variable_accumulator)
                                           : result.checked_unconditional_add(variable_accumulator);
        } else {
            result = variable_accumulator;
        }
    }
 
    // Update `result` to remove the offset generator terms, and add in any constant terms from `constant_acc`.
    // We have two potential modes here:
    // 1. All inputs are fixed-base and constant_acc is not the point at infinity
    // 2. Everything else.
    // Case 1 is a special case, as we *know* we cannot hit incomplete addition edge cases, under the assumption that
    // all input points are linearly independent of one another. Because constant_acc is not the point at infinity we
    // know that at least 1 input scalar was not zero, i.e. the output will not be the point at infinity. We also know
    // that, under case 1, we won't trigger the doubling formula either, as every point is linearly independent of every
    // other point (including offset generators).
    if (!constant_acc.is_point_at_infinity() && can_unconditional_add) {
        result = result.unconditional_add(AffineElement(-offset_accumulator));
    } else {
        // For case 2, we must use a full subtraction operation that handles all possible edge cases, as the output
        // point may be the point at infinity.
        // Note about optimizations for posterity: An honest prover might hit the point at infinity, but won't
        // trigger the doubling edge case (since doubling edge case implies input points are also the offset generator
        // points). We could do the following which would be slightly cheaper than operator-:
        // 1. If x-coords match, assert y-coords do not match
        // 2. If x-coords match, return point at infinity, else unconditionally compute result - offset_accumulator.
        result = result - cycle_group(AffineElement(offset_accumulator));
    }
    // Ensure the tag of the result is a union of all inputs
    result.set_origin_tag(result_tag);
    return result;
}
 
template <typename Builder> cycle_group<Builder> cycle_group<Builder>::operator*(const cycle_scalar& scalar) const
{
    return batch_mul({ *this }, { scalar });
}
 
template <typename Builder> cycle_group<Builder>& cycle_group<Builder>::operator*=(const cycle_scalar& scalar)
{
    *this = operator*(scalar);
    return *this;
}
 
template <typename Builder> cycle_group<Builder> cycle_group<Builder>::operator*(const BigScalarField& scalar) const
{
    return batch_mul({ *this }, { scalar });
}
 
template <typename Builder> cycle_group<Builder>& cycle_group<Builder>::operator*=(const BigScalarField& scalar)
{
    *this = operator*(scalar);
    return *this;
}
 
template <typename Builder> bool_t<Builder> cycle_group<Builder>::operator==(cycle_group& other)
{
    this->standardize();
    other.standardize();
    const auto equal = (_x == other._x) && (_y == other._y) && (this->_is_infinity == other._is_infinity);
    return equal;
}
 
template <typename Builder> void cycle_group<Builder>::assert_equal(cycle_group& other, std::string const& msg)
{
    this->standardize();
    other.standardize();
    _x.assert_equal(other._x, msg);
    _y.assert_equal(other._y, msg);
    this->_is_infinity.assert_equal(other._is_infinity);
}
 
template <typename Builder>
cycle_group<Builder> cycle_group<Builder>::conditional_assign(const bool_t& predicate,
                                                              const cycle_group& lhs,
                                                              const cycle_group& rhs)
{
    auto x_res = field_t::conditional_assign(predicate, lhs._x, rhs._x);
    auto y_res = field_t::conditional_assign(predicate, lhs._y, rhs._y);
    auto _is_infinity_res =
        bool_t::conditional_assign(predicate, lhs.is_point_at_infinity(), rhs.is_point_at_infinity());
 
    cycle_group<Builder> result(x_res, y_res, _is_infinity_res, /*assert_on_curve=*/false);
    return result;
};
 
template class cycle_group<bb::UltraCircuitBuilder>;
template class cycle_group<bb::MegaCircuitBuilder>;
 
} // namespace bb::stdlib