mnt6_g2.cpp

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#include <libff/algebra/curves/mnt/mnt6/mnt6_g2.hpp>
 
namespace libff {
 
#ifdef PROFILE_OP_COUNTS
long long mnt6_G2::add_cnt = 0;
long long mnt6_G2::dbl_cnt = 0;
#endif
 
std::vector<size_t> mnt6_G2::wnaf_window_table;
std::vector<size_t> mnt6_G2::fixed_base_exp_window_table;
mnt6_Fq3 mnt6_G2::twist;
mnt6_Fq3 mnt6_G2::coeff_a;
mnt6_Fq3 mnt6_G2::coeff_b;
mnt6_G2 mnt6_G2::G2_zero = {};
mnt6_G2 mnt6_G2::G2_one = {};
bool mnt6_G2::initialized = false;
 
mnt6_G2::mnt6_G2()
{
    if (mnt6_G2::initialized)
    {
        this->X = G2_zero.X;
        this->Y = G2_zero.Y;
        this->Z = G2_zero.Z;
    }
}
 
mnt6_Fq3 mnt6_G2::mul_by_a(const mnt6_Fq3 &elt)
{
    return mnt6_Fq3(mnt6_twist_mul_by_a_c0 * elt.c1, mnt6_twist_mul_by_a_c1 * elt.c2, mnt6_twist_mul_by_a_c2 * elt.c0);
}
 
mnt6_Fq3 mnt6_G2::mul_by_b(const mnt6_Fq3 &elt)
{
    return mnt6_Fq3(mnt6_twist_mul_by_b_c0 * elt.c0, mnt6_twist_mul_by_b_c1 * elt.c1, mnt6_twist_mul_by_b_c2 * elt.c2);
}
 
void mnt6_G2::print() const
{
    if (this->is_zero())
    {
        printf("O\n");
    }
    else
    {
        mnt6_G2 copy(*this);
        copy.to_affine_coordinates();
        gmp_printf("(%Nd*z^2 + %Nd*z + %Nd , %Nd*z^2 + %Nd*z + %Nd)\n",
                   copy.X.c2.as_bigint().data, mnt6_Fq::num_limbs,
                   copy.X.c1.as_bigint().data, mnt6_Fq::num_limbs,
                   copy.X.c0.as_bigint().data, mnt6_Fq::num_limbs,
                   copy.Y.c2.as_bigint().data, mnt6_Fq::num_limbs,
                   copy.Y.c1.as_bigint().data, mnt6_Fq::num_limbs,
                   copy.Y.c0.as_bigint().data, mnt6_Fq::num_limbs);
    }
}
 
void mnt6_G2::print_coordinates() const
{
    if (this->is_zero())
    {
        printf("O\n");
    }
    else
    {
        gmp_printf("(%Nd*z^2 + %Nd*z + %Nd : %Nd*z^2 + %Nd*z + %Nd : %Nd*z^2 + %Nd*z + %Nd)\n",
                   this->X.c2.as_bigint().data, mnt6_Fq::num_limbs,
                   this->X.c1.as_bigint().data, mnt6_Fq::num_limbs,
                   this->X.c0.as_bigint().data, mnt6_Fq::num_limbs,
                   this->Y.c2.as_bigint().data, mnt6_Fq::num_limbs,
                   this->Y.c1.as_bigint().data, mnt6_Fq::num_limbs,
                   this->Y.c0.as_bigint().data, mnt6_Fq::num_limbs,
                   this->Z.c2.as_bigint().data, mnt6_Fq::num_limbs,
                   this->Z.c1.as_bigint().data, mnt6_Fq::num_limbs,
                   this->Z.c0.as_bigint().data, mnt6_Fq::num_limbs);
    }
}
 
void mnt6_G2::to_affine_coordinates()
{
    if (this->is_zero())
    {
        this->X = mnt6_Fq3::zero();
        this->Y = mnt6_Fq3::one();
        this->Z = mnt6_Fq3::zero();
    }
    else
    {
        const mnt6_Fq3 Z_inv = Z.inverse();
        this->X = this->X * Z_inv;
        this->Y = this->Y * Z_inv;
        this->Z = mnt6_Fq3::one();
    }
}
 
void mnt6_G2::to_special()
{
    this->to_affine_coordinates();
}
 
bool mnt6_G2::is_special() const
{
    return (this->is_zero() || this->Z == mnt6_Fq3::one());
}
 
bool mnt6_G2::is_zero() const
{
    // TODO: use zero for here
    return (this->X.is_zero() && this->Z.is_zero());
}
 
bool mnt6_G2::operator==(const mnt6_G2 &other) const
{
    if (this->is_zero())
    {
        return other.is_zero();
    }
 
    if (other.is_zero())
    {
        return false;
    }
 
    /* now neither is O */
 
    // X1/Z1 = X2/Z2 <=> X1*Z2 = X2*Z1
    if ((this->X * other.Z) != (other.X * this->Z))
    {
        return false;
    }
 
    // Y1/Z1 = Y2/Z2 <=> Y1*Z2 = Y2*Z1
    if ((this->Y * other.Z) != (other.Y * this->Z))
    {
        return false;
    }
 
    return true;
}
 
bool mnt6_G2::operator!=(const mnt6_G2& other) const
{
    return !(operator==(other));
}
 
mnt6_G2 mnt6_G2::operator+(const mnt6_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)
 
    // handle double case, and then all the rest
    /*
      The code below is equivalent to (but faster than) the snippet below:
 
      if (this->operator==(other))
      {
      return this->dbl();
      }
      else
      {
      return this->add(other);
      }
    */
 
    const mnt6_Fq3 X1Z2 = (this->X) * (other.Z);        // X1Z2 = X1*Z2
    const mnt6_Fq3 X2Z1 = (this->Z) * (other.X);        // X2Z1 = X2*Z1
 
    // (used both in add and double checks)
 
    const mnt6_Fq3 Y1Z2 = (this->Y) * (other.Z);        // Y1Z2 = Y1*Z2
    const mnt6_Fq3 Y2Z1 = (this->Z) * (other.Y);        // Y2Z1 = Y2*Z1
 
    if (X1Z2 == X2Z1 && Y1Z2 == Y2Z1)
    {
        // perform dbl case
        const mnt6_Fq3 XX   = (this->X).squared();                   // XX  = X1^2
        const mnt6_Fq3 ZZ   = (this->Z).squared();                   // ZZ  = Z1^2
        const mnt6_Fq3 w    = mnt6_G2::mul_by_a(ZZ) + (XX + XX + XX); // w   = a*ZZ + 3*XX
        const mnt6_Fq3 Y1Z1 = (this->Y) * (this->Z);
        const mnt6_Fq3 s    = Y1Z1 + Y1Z1;                             // s   = 2*Y1*Z1
        const mnt6_Fq3 ss   = s.squared();                             // ss  = s^2
        const mnt6_Fq3 sss  = s * ss;                                  // sss = s*ss
        const mnt6_Fq3 R    = (this->Y) * s;                          // R   = Y1*s
        const mnt6_Fq3 RR   = R.squared();                             // RR  = R^2
        const mnt6_Fq3 B    = ((this->X)+R).squared()-XX-RR;          // B   = (X1+R)^2 - XX - RR
        const mnt6_Fq3 h    = w.squared() - (B+B);                     // h   = w^2 - 2*B
        const mnt6_Fq3 X3   = h * s;                                   // X3  = h*s
        const mnt6_Fq3 Y3   = w * (B-h)-(RR+RR);                       // Y3  = w*(B-h) - 2*RR
        const mnt6_Fq3 Z3   = sss;                                     // Z3  = sss
 
        return mnt6_G2(X3, Y3, Z3);
    }
 
    // if we have arrived here we are in the add case
    const mnt6_Fq3 Z1Z2 = (this->Z) * (other.Z);   // Z1Z2 = Z1*Z2
    const mnt6_Fq3 u    = Y2Z1 - Y1Z2;               // u    = Y2*Z1-Y1Z2
    const mnt6_Fq3 uu   = u.squared();               // uu   = u^2
    const mnt6_Fq3 v    = X2Z1 - X1Z2;               // v    = X2*Z1-X1Z2
    const mnt6_Fq3 vv   = v.squared();               // vv   = v^2
    const mnt6_Fq3 vvv  = v * vv;                    // vvv  = v*vv
    const mnt6_Fq3 R    = vv * X1Z2;                 // R    = vv*X1Z2
    const mnt6_Fq3 A    = uu * Z1Z2 - (vvv + R + R); // A    = uu*Z1Z2 - vvv - 2*R
    const mnt6_Fq3 X3   = v * A;                     // X3   = v*A
    const mnt6_Fq3 Y3   = u * (R-A) - vvv * Y1Z2;    // Y3   = u*(R-A) - vvv*Y1Z2
    const mnt6_Fq3 Z3   = vvv * Z1Z2;                // Z3   = vvv*Z1Z2
 
    return mnt6_G2(X3, Y3, Z3);
}
 
mnt6_G2 mnt6_G2::operator-() const
{
    return mnt6_G2(this->X, -(this->Y), this->Z);
}
 
 
mnt6_G2 mnt6_G2::operator-(const mnt6_G2 &other) const
{
    return (*this) + (-other);
}
 
mnt6_G2 mnt6_G2::add(const mnt6_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)
 
    // handle double case
    if (this->operator==(other))
    {
        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-projective.html#addition-add-1998-cmo-2
 
    const mnt6_Fq3 Y1Z2 = (this->Y) * (other.Z);        // Y1Z2 = Y1*Z2
    const mnt6_Fq3 X1Z2 = (this->X) * (other.Z);        // X1Z2 = X1*Z2
    const mnt6_Fq3 Z1Z2 = (this->Z) * (other.Z);        // Z1Z2 = Z1*Z2
    const mnt6_Fq3 u    = (other.Y) * (this->Z) - Y1Z2; // u    = Y2*Z1-Y1Z2
    const mnt6_Fq3 uu   = u.squared();                    // uu   = u^2
    const mnt6_Fq3 v    = (other.X) * (this->Z) - X1Z2; // v    = X2*Z1-X1Z2
    const mnt6_Fq3 vv   = v.squared();                    // vv   = v^2
    const mnt6_Fq3 vvv  = v * vv;                         // vvv  = v*vv
    const mnt6_Fq3 R    = vv * X1Z2;                      // R    = vv*X1Z2
    const mnt6_Fq3 A    = uu * Z1Z2 - (vvv + R + R);      // A    = uu*Z1Z2 - vvv - 2*R
    const mnt6_Fq3 X3   = v * A;                          // X3   = v*A
    const mnt6_Fq3 Y3   = u * (R-A) - vvv * Y1Z2;         // Y3   = u*(R-A) - vvv*Y1Z2
    const mnt6_Fq3 Z3   = vvv * Z1Z2;                     // Z3   = vvv*Z1Z2
 
    return mnt6_G2(X3, Y3, Z3);
}
 
mnt6_G2 mnt6_G2::mixed_add(const mnt6_G2 &other) const
{
#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-projective.html#addition-add-1998-cmo-2
    //assert(other.Z == mnt6_Fq3::one());
 
    if (this->is_zero())
    {
        return other;
    }
 
    if (other.is_zero())
    {
        return (*this);
    }
 
#ifdef DEBUG
    assert(other.is_special());
#endif
 
    const mnt6_Fq3 &X1Z2 = (this->X);                   // X1Z2 = X1*Z2 (but other is special and not zero)
    const mnt6_Fq3 X2Z1 = (this->Z) * (other.X);       // X2Z1 = X2*Z1
 
    // (used both in add and double checks)
 
    const mnt6_Fq3 &Y1Z2 = (this->Y);                   // Y1Z2 = Y1*Z2 (but other is special and not zero)
    const mnt6_Fq3 Y2Z1 = (this->Z) * (other.Y);       // Y2Z1 = Y2*Z1
 
    if (X1Z2 == X2Z1 && Y1Z2 == Y2Z1)
    {
        return this->dbl();
    }
 
    const mnt6_Fq3 u = Y2Z1 - this->Y;             // u = Y2*Z1-Y1
    const mnt6_Fq3 uu = u.squared();                // uu = u2
    const mnt6_Fq3 v = X2Z1 - this->X;             // v = X2*Z1-X1
    const mnt6_Fq3 vv = v.squared();                // vv = v2
    const mnt6_Fq3 vvv = v*vv;                      // vvv = v*vv
    const mnt6_Fq3 R = vv * this->X;               // R = vv*X1
    const mnt6_Fq3 A = uu * this->Z - vvv - R - R; // A = uu*Z1-vvv-2*R
    const mnt6_Fq3 X3 = v * A;                      // X3 = v*A
    const mnt6_Fq3 Y3 = u*(R-A) - vvv * this->Y;   // Y3 = u*(R-A)-vvv*Y1
    const mnt6_Fq3 Z3 = vvv * this->Z;             // Z3 = vvv*Z1
 
    return mnt6_G2(X3, Y3, Z3);
}
 
mnt6_G2 mnt6_G2::dbl() const
{
#ifdef PROFILE_OP_COUNTS
    this->dbl_cnt++;
#endif
    if (this->is_zero())
    {
        return (*this);
    }
    else
    {
        // 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
 
        const mnt6_Fq3 XX   = (this->X).squared();                   // XX  = X1^2
        const mnt6_Fq3 ZZ   = (this->Z).squared();                   // ZZ  = Z1^2
        const mnt6_Fq3 w    = mnt6_G2::mul_by_a(ZZ) + (XX + XX + XX); // w   = a*ZZ + 3*XX
        const mnt6_Fq3 Y1Z1 = (this->Y) * (this->Z);
        const mnt6_Fq3 s    = Y1Z1 + Y1Z1;                            // s   = 2*Y1*Z1
        const mnt6_Fq3 ss   = s.squared();                            // ss  = s^2
        const mnt6_Fq3 sss  = s * ss;                                 // sss = s*ss
        const mnt6_Fq3 R    = (this->Y) * s;                         // R   = Y1*s
        const mnt6_Fq3 RR   = R.squared();                            // RR  = R^2
        const mnt6_Fq3 B    = ((this->X)+R).squared()-XX-RR;         // B   = (X1+R)^2 - XX - RR
        const mnt6_Fq3 h    = w.squared() - (B+B);                    // h   = w^2-2*B
        const mnt6_Fq3 X3   = h * s;                                  // X3  = h*s
        const mnt6_Fq3 Y3   = w * (B-h)-(RR+RR);                      // Y3  = w*(B-h) - 2*RR
        const mnt6_Fq3 Z3   = sss;                                    // Z3  = sss
 
        return mnt6_G2(X3, Y3, Z3);
    }
}
 
mnt6_G2 mnt6_G2::mul_by_q() const
{
    return mnt6_G2(mnt6_twist_mul_by_q_X * (this->X).Frobenius_map(1),
                   mnt6_twist_mul_by_q_Y * (this->Y).Frobenius_map(1),
                   (this->Z).Frobenius_map(1));
}
 
bool mnt6_G2::is_well_formed() const
{
    if (this->is_zero())
    {
        return true;
    }
    else
    {
 
        /*
          y^2 = x^3 + ax + b
 
          We are using projective, so equation we need to check is actually
 
          (y/z)^2 = (x/z)^3 + a (x/z) + b
          z y^2 = x^3  + a z^2 x + b z^3
 
          z (y^2 - b z^2) = x ( x^2 + a z^2)
        */
        const mnt6_Fq3 X2 = this->X.squared();
        const mnt6_Fq3 Y2 = this->Y.squared();
        const mnt6_Fq3 Z2 = this->Z.squared();
        const mnt6_Fq3 aZ2 = mnt6_twist_coeff_a * Z2;
 
        return (this->Z * (Y2 - mnt6_twist_coeff_b * Z2) == this->X * (X2 + aZ2));
    }
}
 
mnt6_G2 mnt6_G2::zero()
{
    return G2_zero;
}
 
mnt6_G2 mnt6_G2::one()
{
    return G2_one;
}
 
mnt6_G2 mnt6_G2::random_element()
{
    return (mnt6_Fr::random_element().as_bigint()) * G2_one;
}
 
std::ostream& operator<<(std::ostream &out, const mnt6_G2 &g)
{
    mnt6_G2 copy(g);
    copy.to_affine_coordinates();
 
    out << (copy.is_zero() ? 1 : 0) << OUTPUT_SEPARATOR;
#ifdef NO_PT_COMPRESSION
    out << copy.X << OUTPUT_SEPARATOR << copy.Y;
#else
    /* storing LSB of Y */
    out << copy.X << OUTPUT_SEPARATOR << (copy.Y.c0.as_bigint().data[0] & 1);
#endif
 
    return out;
}
 
std::istream& operator>>(std::istream &in, mnt6_G2 &g)
{
    char is_zero;
    mnt6_Fq3 tX, tY;
 
#ifdef NO_PT_COMPRESSION
    in >> is_zero >> tX >> tY;
    is_zero -= '0';
#else
    in.read((char*)&is_zero, 1); // this reads is_zero;
    is_zero -= '0';
    consume_OUTPUT_SEPARATOR(in);
 
    unsigned char Y_lsb;
    in >> tX;
    consume_OUTPUT_SEPARATOR(in);
    in.read((char*)&Y_lsb, 1);
    Y_lsb -= '0';
 
    // y = +/- sqrt(x^3 + a*x + b)
    if (!is_zero)
    {
        const mnt6_Fq3 tX2 = tX.squared();
        const mnt6_Fq3 tY2 = (tX2 + mnt6_twist_coeff_a) * tX + mnt6_twist_coeff_b;
        tY = tY2.sqrt();
 
        if ((tY.c0.as_bigint().data[0] & 1) != Y_lsb)
        {
            tY = -tY;
        }
    }
#endif
    // using projective coordinates
    if (!is_zero)
    {
        g.X = tX;
        g.Y = tY;
        g.Z = mnt6_Fq3::one();
    }
    else
    {
        g = mnt6_G2::zero();
    }
 
    return in;
}
 
void mnt6_G2::batch_to_special_all_non_zeros(std::vector<mnt6_G2> &vec)
{
    std::vector<mnt6_Fq3> Z_vec;
    Z_vec.reserve(vec.size());
 
    for (auto &el: vec)
    {
        Z_vec.emplace_back(el.Z);
    }
    batch_invert<mnt6_Fq3>(Z_vec);
 
    const mnt6_Fq3 one = mnt6_Fq3::one();
 
    for (size_t i = 0; i < vec.size(); ++i)
    {
        vec[i] = mnt6_G2(vec[i].X * Z_vec[i], vec[i].Y * Z_vec[i], one);
    }
}
 
} // libff