alt_bn128_g2.cpp

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#include <libff/algebra/curves/alt_bn128/alt_bn128_g2.hpp>
 
namespace libff
{
 
static const uint8_t G2_ZERO_FLAG = 1 << 0;
static const uint8_t G2_Y_LSB_FLAG = 1 << 1;
 
#ifdef PROFILE_OP_COUNTS
long long alt_bn128_G2::add_cnt = 0;
long long alt_bn128_G2::dbl_cnt = 0;
#endif
 
std::vector<size_t> alt_bn128_G2::wnaf_window_table;
std::vector<size_t> alt_bn128_G2::fixed_base_exp_window_table;
alt_bn128_G2 alt_bn128_G2::G2_zero;
alt_bn128_G2 alt_bn128_G2::G2_one;
alt_bn128_Fq2 alt_bn128_G2::coeff_a;
alt_bn128_Fq2 alt_bn128_G2::coeff_b;
bigint<alt_bn128_G2::h_limbs> alt_bn128_G2::h;
 
alt_bn128_G2::alt_bn128_G2()
{
    this->X = G2_zero.X;
    this->Y = G2_zero.Y;
    this->Z = G2_zero.Z;
}
 
alt_bn128_Fq2 alt_bn128_G2::mul_by_b(const alt_bn128_Fq2 &elt)
{
    return alt_bn128_Fq2(
        alt_bn128_twist_mul_by_b_c0 * elt.coeffs[0],
        alt_bn128_twist_mul_by_b_c1 * elt.coeffs[1]);
}
 
void alt_bn128_G2::print() const
{
    if (this->is_zero()) {
        printf("O\n");
    } else {
        alt_bn128_G2 copy(*this);
        copy.to_affine_coordinates();
        gmp_printf(
            "(%Nd*z + %Nd , %Nd*z + %Nd)\n",
            copy.X.coeffs[1].as_bigint().data,
            alt_bn128_Fq::num_limbs,
            copy.X.coeffs[0].as_bigint().data,
            alt_bn128_Fq::num_limbs,
            copy.Y.coeffs[1].as_bigint().data,
            alt_bn128_Fq::num_limbs,
            copy.Y.coeffs[0].as_bigint().data,
            alt_bn128_Fq::num_limbs);
    }
}
 
void alt_bn128_G2::print_coordinates() const
{
    if (this->is_zero()) {
        printf("O\n");
    } else {
        gmp_printf(
            "(%Nd*z + %Nd : %Nd*z + %Nd : %Nd*z + %Nd)\n",
            this->X.coeffs[1].as_bigint().data,
            alt_bn128_Fq::num_limbs,
            this->X.coeffs[0].as_bigint().data,
            alt_bn128_Fq::num_limbs,
            this->Y.coeffs[1].as_bigint().data,
            alt_bn128_Fq::num_limbs,
            this->Y.coeffs[0].as_bigint().data,
            alt_bn128_Fq::num_limbs,
            this->Z.coeffs[1].as_bigint().data,
            alt_bn128_Fq::num_limbs,
            this->Z.coeffs[0].as_bigint().data,
            alt_bn128_Fq::num_limbs);
    }
}
 
void alt_bn128_G2::to_affine_coordinates()
{
    if (this->is_zero()) {
        this->X = alt_bn128_Fq2::zero();
        this->Y = alt_bn128_Fq2::one();
        this->Z = alt_bn128_Fq2::zero();
    } else {
        const alt_bn128_Fq2 Z_inv = Z.inverse();
        const alt_bn128_Fq2 Z2_inv = Z_inv.squared();
        const alt_bn128_Fq2 Z3_inv = Z2_inv * Z_inv;
        this->X = this->X * Z2_inv;
        this->Y = this->Y * Z3_inv;
        this->Z = alt_bn128_Fq2::one();
    }
}
 
void alt_bn128_G2::to_special() { this->to_affine_coordinates(); }
 
bool alt_bn128_G2::is_special() const
{
    return (this->is_zero() || this->Z == alt_bn128_Fq2::one());
}
 
bool alt_bn128_G2::is_zero() const { return (this->Z.is_zero()); }
 
bool alt_bn128_G2::operator==(const alt_bn128_G2 &other) const
{
    if (this->is_zero()) {
        return other.is_zero();
    }
 
    if (other.is_zero()) {
        return false;
    }
 
    /* now neither is O */
 
    // using Jacobian coordinates so:
    //   (X1:Y1:Z1) = (X2:Y2:Z2)
    //   iff
    //   X1/Z1^2 == X2/Z2^2 and Y1/Z1^3 == Y2/Z2^3
    //   iff
    //   X1 * Z2^2 == X2 * Z1^2 and Y1 * Z2^3 == Y2 * Z1^3
 
    const alt_bn128_Fq2 Z1_squared = (this->Z).squared();
    const alt_bn128_Fq2 Z2_squared = (other.Z).squared();
 
    if ((this->X * Z2_squared) != (other.X * Z1_squared)) {
        return false;
    }
 
    const alt_bn128_Fq2 Z1_cubed = (this->Z) * Z1_squared;
    const alt_bn128_Fq2 Z2_cubed = (other.Z) * Z2_squared;
 
    if ((this->Y * Z2_cubed) != (other.Y * Z1_cubed)) {
        return false;
    }
 
    return true;
}
 
bool alt_bn128_G2::operator!=(const alt_bn128_G2 &other) const
{
    return !(operator==(other));
}
 
alt_bn128_G2 alt_bn128_G2::operator+(const alt_bn128_G2 &other) const
{
    return add(other);
}
 
alt_bn128_G2 alt_bn128_G2::operator-() const
{
    return alt_bn128_G2(this->X, -(this->Y), this->Z);
}
 
alt_bn128_G2 alt_bn128_G2::operator-(const alt_bn128_G2 &other) const
{
    return (*this) + (-other);
}
 
alt_bn128_G2 alt_bn128_G2::add(const alt_bn128_G2 &other) const
{
    // handle special cases having to do with O
    if (this->is_zero()) {
        return other;
    }
 
    if (other.is_zero()) {
        return *this;
    }
 
    // no need to handle points of order 2,4
    // (they cannot exist in a prime-order subgroup)
 
    // check for doubling case
 
    // using Jacobian coordinates so:
    //   (X1:Y1:Z1) = (X2:Y2:Z2)
    //   iff
    //   X1/Z1^2 == X2/Z2^2 and Y1/Z1^3 == Y2/Z2^3
    //   iff
    //   X1 * Z2^2 == X2 * Z1^2 and Y1 * Z2^3 == Y2 * Z1^3
 
    const alt_bn128_Fq2 Z1Z1 = (this->Z).squared();
    const alt_bn128_Fq2 Z2Z2 = (other.Z).squared();
 
    const alt_bn128_Fq2 U1 = this->X * Z2Z2;
    const alt_bn128_Fq2 U2 = other.X * Z1Z1;
 
    const alt_bn128_Fq2 Z1_cubed = (this->Z) * Z1Z1;
    const alt_bn128_Fq2 Z2_cubed = (other.Z) * Z2Z2;
 
    // S1 = Y1 * Z2 * Z2Z2
    const alt_bn128_Fq2 S1 = (this->Y) * Z2_cubed;
    // S2 = Y2 * Z1 * Z1Z1
    const alt_bn128_Fq2 S2 = (other.Y) * Z1_cubed;
 
    if (U1 == U2 && S1 == S2) {
        // dbl case; nothing of above can be reused
        return this->dbl();
    }
 
#ifdef PROFILE_OP_COUNTS
    this->add_cnt++;
#endif
 
    // rest of add case
    const alt_bn128_Fq2 H = U2 - U1; // H = U2-U1
    const alt_bn128_Fq2 S2_minus_S1 = S2 - S1;
    // I = (2 * H)^2
    const alt_bn128_Fq2 I = (H + H).squared();
    // J = H * I
    const alt_bn128_Fq2 J = H * I;
    // r = 2 * (S2-S1)
    const alt_bn128_Fq2 r = S2_minus_S1 + S2_minus_S1;
    // V = U1 * I
    const alt_bn128_Fq2 V = U1 * I;
    // X3 = r^2 - J - 2 * V
    const alt_bn128_Fq2 X3 = r.squared() - J - (V + V);
    const alt_bn128_Fq2 S1_J = S1 * J;
    // Y3 = r * (V-X3)-2 S1 J
    const alt_bn128_Fq2 Y3 = r * (V - X3) - (S1_J + S1_J);
    // Z3 = ((Z1+Z2)^2-Z1Z1-Z2Z2) * H
    const alt_bn128_Fq2 Z3 = ((this->Z + other.Z).squared() - Z1Z1 - Z2Z2) * H;
 
    return alt_bn128_G2(X3, Y3, Z3);
}
 
alt_bn128_G2 alt_bn128_G2::mixed_add(const alt_bn128_G2 &other) const
{
#ifdef DEBUG
    assert(other.is_special());
#endif
 
    // handle special cases having to do with O
    if (this->is_zero()) {
        return other;
    }
 
    if (other.is_zero()) {
        return *this;
    }
 
    // no need to handle points of order 2,4
    // (they cannot exist in a prime-order subgroup)
 
    // check for doubling case
 
    // using Jacobian coordinates so:
    //   (X1:Y1:Z1) = (X2:Y2:Z2)
    //   iff
    //   X1/Z1^2 == X2/Z2^2 and Y1/Z1^3 == Y2/Z2^3
    //   iff
    //   X1 * Z2^2 == X2 * Z1^2 and Y1 * Z2^3 == Y2 * Z1^3
 
    // we know that Z2 = 1
 
    const alt_bn128_Fq2 Z1Z1 = (this->Z).squared();
 
    const alt_bn128_Fq2 &U1 = this->X;
    const alt_bn128_Fq2 U2 = other.X * Z1Z1;
 
    const alt_bn128_Fq2 Z1_cubed = (this->Z) * Z1Z1;
 
    // S1 = Y1 * Z2 * Z2Z2
    const alt_bn128_Fq2 &S1 = (this->Y);
    // S2 = Y2 * Z1 * Z1Z1
    const alt_bn128_Fq2 S2 = (other.Y) * Z1_cubed;
 
    if (U1 == U2 && S1 == S2) {
        // dbl case; nothing of above can be reused
        return this->dbl();
    }
 
#ifdef PROFILE_OP_COUNTS
    this->add_cnt++;
#endif
 
    // NOTE: does not handle O and pts of order 2,4
    // http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-madd-2007-bl
    // H = U2-X1
    const alt_bn128_Fq2 H = U2 - (this->X);
    // HH = H&2
    const alt_bn128_Fq2 HH = H.squared();
    // I = 4*HH
    alt_bn128_Fq2 I = HH + HH;
    I = I + I;
    // J = H*I
    const alt_bn128_Fq2 J = H * I;
    // r = 2*(S2-Y1)
    alt_bn128_Fq2 r = S2 - (this->Y);
    r = r + r;
    // V = X1*I
    const alt_bn128_Fq2 V = (this->X) * I;
    // X3 = r^2-J-2*V
    const alt_bn128_Fq2 X3 = r.squared() - J - V - V;
    // Y3 = r*(V-X3)-2*Y1*J
    alt_bn128_Fq2 Y3 = (this->Y) * J;
    Y3 = r * (V - X3) - Y3 - Y3;
    // Z3 = (Z1+H)^2-Z1Z1-HH
    const alt_bn128_Fq2 Z3 = ((this->Z) + H).squared() - Z1Z1 - HH;
 
    return alt_bn128_G2(X3, Y3, Z3);
}
 
alt_bn128_G2 alt_bn128_G2::dbl() const
{
#ifdef PROFILE_OP_COUNTS
    this->dbl_cnt++;
#endif
    // handle point at infinity
    if (this->is_zero()) {
        return (*this);
    }
 
    // NOTE: does not handle O and pts of order 2,4
    // http://www.hyperelliptic.org/EFD/g1p/auto-shortw-projective.html#doubling-dbl-2007-bl
 
    // A = X1^2
    const alt_bn128_Fq2 A = (this->X).squared();
    // B = Y1^2
    const alt_bn128_Fq2 B = (this->Y).squared();
    // C = B^2
    const alt_bn128_Fq2 C = B.squared();
    alt_bn128_Fq2 D = (this->X + B).squared() - A - C;
    // D = 2 * ((X1 + B)^2 - A - C)
    D = D + D;
    // E = 3 * A
    const alt_bn128_Fq2 E = A + A + A;
    // F = E^2
    const alt_bn128_Fq2 F = E.squared();
    // X3 = F - 2 D
    const alt_bn128_Fq2 X3 = F - (D + D);
    alt_bn128_Fq2 eightC = C + C;
    eightC = eightC + eightC;
    eightC = eightC + eightC;
    // Y3 = E * (D - X3) - 8 * C
    const alt_bn128_Fq2 Y3 = E * (D - X3) - eightC;
    const alt_bn128_Fq2 Y1Z1 = (this->Y) * (this->Z);
    // Z3 = 2 * Y1 * Z1
    const alt_bn128_Fq2 Z3 = Y1Z1 + Y1Z1;
 
    return alt_bn128_G2(X3, Y3, Z3);
}
 
alt_bn128_G2 alt_bn128_G2::mul_by_q() const
{
    return alt_bn128_G2(
        alt_bn128_twist_mul_by_q_X * (this->X).Frobenius_map(1),
        alt_bn128_twist_mul_by_q_Y * (this->Y).Frobenius_map(1),
        (this->Z).Frobenius_map(1));
}
 
alt_bn128_G2 alt_bn128_G2::mul_by_cofactor() const
{
    return alt_bn128_G2::h * (*this);
}
 
bool alt_bn128_G2::is_well_formed() const
{
    if (this->is_zero()) {
        return true;
    } else {
        // y^2 = x^3 + b
        //
        // We are using Jacobian coordinates, so equation we need to check is
        // actually
        //
        // (y/z^3)^2 = (x/z^2)^3 + b
        // y^2 / z^6 = x^3 / z^6 + b
        // y^2 = x^3 + b z^6
        const alt_bn128_Fq2 X2 = this->X.squared();
        const alt_bn128_Fq2 Y2 = this->Y.squared();
        const alt_bn128_Fq2 Z2 = this->Z.squared();
 
        const alt_bn128_Fq2 X3 = this->X * X2;
        const alt_bn128_Fq2 Z3 = this->Z * Z2;
        const alt_bn128_Fq2 Z6 = Z3.squared();
 
        return (Y2 == X3 + alt_bn128_twist_coeff_b * Z6);
    }
}
 
bool alt_bn128_G2::is_in_safe_subgroup() const
{
    return zero() == scalar_field::mod * (*this);
}
 
const alt_bn128_G2 &alt_bn128_G2::zero() { return G2_zero; }
 
const alt_bn128_G2 &alt_bn128_G2::one() { return G2_one; }
 
alt_bn128_G2 alt_bn128_G2::random_element()
{
    return (alt_bn128_Fr::random_element().as_bigint()) * G2_one;
}
 
void alt_bn128_G2::write_uncompressed(std::ostream &out) const
{
    // <is_zero> | <x-coord> | <y-coord>
    alt_bn128_G2 copy(*this);
    copy.to_affine_coordinates();
    const char is_zero = copy.is_zero() ? '1' : '0';
    out.write(&is_zero, 1) << copy.X << copy.Y;
}
 
void alt_bn128_G2::write_compressed(std::ostream &out) const
{
    // <flags> | <x-coord>
    alt_bn128_G2 copy(*this);
    copy.to_affine_coordinates();
 
    const uint8_t flags =
        (copy.is_zero() ? G2_ZERO_FLAG : 0) |
        ((copy.Y.coeffs[0].as_bigint().data[0] & 1) ? G2_Y_LSB_FLAG : 0);
    const char flags_char = '0' + flags;
    out.write(&flags_char, 1) << copy.X;
}
 
void alt_bn128_G2::read_uncompressed(std::istream &in, alt_bn128_G2 &g)
{
    // <is_zero> | <x-coord> | <y-coord>
    char is_zero;
    in.read(&is_zero, 1) >> g.X >> g.Y;
    ;
    is_zero -= '0';
 
    if (!is_zero) {
        g.Z = alt_bn128_Fq2::one();
    } else {
        g = alt_bn128_G2::zero();
    }
}
 
void alt_bn128_G2::read_compressed(std::istream &in, alt_bn128_G2 &g)
{
    // <flags> | <x-coord>
    char flags_char;
    in.read(&flags_char, 1) >> g.X;
    const uint8_t flags = flags_char - '0';
 
    // alt_bn128_Fq2 tX;
 
    // y = +/- sqrt(x^3 + b)
    if (0 == (flags & G2_ZERO_FLAG)) {
        const uint8_t Y_lsb = (flags & G2_Y_LSB_FLAG) ? 1 : 0;
        const alt_bn128_Fq2 tX2 = g.X.squared();
        const alt_bn128_Fq2 tY2 = tX2 * g.X + alt_bn128_twist_coeff_b;
        g.Y = tY2.sqrt();
 
        if ((uint8_t)(g.Y.coeffs[0].as_bigint().data[0] & 1) != Y_lsb) {
            g.Y = -g.Y;
        }
 
        g.Z = alt_bn128_Fq2::one();
    } else {
        g = alt_bn128_G2::zero();
    }
}
 
std::ostream &operator<<(std::ostream &out, const alt_bn128_G2 &g)
{
#ifdef NO_PT_COMPRESSION
    g.write_uncompressed(out);
#else
    g.write_compressed(out);
#endif
    return out;
}
 
std::istream &operator>>(std::istream &in, alt_bn128_G2 &g)
{
#ifdef NO_PT_COMPRESSION
    alt_bn128_G2::read_uncompressed(in, g);
#else
    alt_bn128_G2::read_compressed(in, g);
#endif
    return in;
}
 
void alt_bn128_G2::batch_to_special_all_non_zeros(
    std::vector<alt_bn128_G2> &vec)
{
    std::vector<alt_bn128_Fq2> Z_vec;
    Z_vec.reserve(vec.size());
 
    for (auto &el : vec) {
        Z_vec.emplace_back(el.Z);
    }
    batch_invert<alt_bn128_Fq2>(Z_vec);
 
    const alt_bn128_Fq2 one = alt_bn128_Fq2::one();
 
    for (size_t i = 0; i < vec.size(); ++i) {
        const alt_bn128_Fq2 Z2 = Z_vec[i].squared();
        const alt_bn128_Fq2 Z3 = Z_vec[i] * Z2;
 
        vec[i].X = vec[i].X * Z2;
        vec[i].Y = vec[i].Y * Z3;
        vec[i].Z = one;
    }
}
 
} // namespace libff