libff::edwards_G2 Class Reference

#include <edwards_g2.hpp>

Collaboration diagram for libff::edwards_G2:

Public Types
typedef edwards_Fq	base_field
typedef edwards_Fq3	twist_field
typedef edwards_Fr	scalar_field

Public Member Functions
	edwards_G2 ()
	edwards_G2 (const edwards_Fq3 &X, const edwards_Fq3 &Y)
void	print () const
void	print_coordinates () const
void	to_affine_coordinates ()
void	to_special ()
bool	is_special () const
bool	is_zero () const
bool	operator== (const edwards_G2 &other) const
bool	operator!= (const edwards_G2 &other) const
edwards_G2	operator+ (const edwards_G2 &other) const
edwards_G2	operator- () const
edwards_G2	operator- (const edwards_G2 &other) const
edwards_G2	add (const edwards_G2 &other) const
edwards_G2	mixed_add (const edwards_G2 &other) const
edwards_G2	dbl () const
edwards_G2	mul_by_q () const
bool	is_well_formed () const

Static Public Member Functions
static edwards_Fq3	mul_by_a (const edwards_Fq3 &elt)
static edwards_Fq3	mul_by_d (const edwards_Fq3 &elt)
static edwards_G2	zero ()
static edwards_G2	one ()
static edwards_G2	random_element ()
static size_t	size_in_bits ()
static bigint< base_field::num_limbs >	base_field_char ()
static bigint< scalar_field::num_limbs >	order ()
static void	batch_to_special_all_non_zeros (std::vector< edwards_G2 > &vec)

Public Attributes
edwards_Fq3	X
edwards_Fq3	Y
edwards_Fq3	Z

Static Public Attributes
static std::vector< size_t >	wnaf_window_table
static std::vector< size_t >	fixed_base_exp_window_table
static edwards_G2	G2_zero = {}
static edwards_G2	G2_one = {}
static bool	initialized = false

Friends
std::ostream &	operator<< (std::ostream &out, const edwards_G2 &g)
std::istream &	operator>> (std::istream &in, edwards_G2 &g)

Detailed Description

Definition at line 22 of file edwards_g2.hpp.

Member Typedef Documentation

base_field

edwards_Fq libff::edwards_G2::base_field

Definition at line 42 of file edwards_g2.hpp.

scalar_field

edwards_Fr libff::edwards_G2::scalar_field

Definition at line 44 of file edwards_g2.hpp.

twist_field

edwards_Fq3 libff::edwards_G2::twist_field

Definition at line 43 of file edwards_g2.hpp.

Constructor & Destructor Documentation

edwards_G2() [1/2]

libff::edwards_G2::edwards_G2

(

)

Definition at line 24 of file edwards_g2.cpp.

{
    if (initialized)
    {
        this->X = G2_zero.X;
        this->Y = G2_zero.Y;
        this->Z = G2_zero.Z;
    }
}

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edwards_G2() [2/2]

libff::edwards_G2::edwards_G2 ( const edwards_Fq3 & X, const edwards_Fq3 & Y )

inline

Definition at line 47 of file edwards_g2.hpp.

: X(Y), Y(X), Z(X*Y) {};

Member Function Documentation

add()

edwards_G2 libff::edwards_G2::add

(

const edwards_G2 &

other

)

const

Definition at line 202 of file edwards_g2.cpp.

{
#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-twisted-inverted.html#addition-add-2008-bbjlp
 
    const edwards_Fq3 A = (this->Z) * (other.Z);                       // A = Z1*Z2
    const edwards_Fq3 B = edwards_G2::mul_by_d(A.squared());           // B = d*A^2
    const edwards_Fq3 C = (this->X) * (other.X);                       // C = X1*X2
    const edwards_Fq3 D = (this->Y) * (other.Y);                       // D = Y1*Y2
    const edwards_Fq3 E = C*D;                                         // E = C*D
    const edwards_Fq3 H = C - edwards_G2::mul_by_a(D);                 // H = C-a*D
    const edwards_Fq3 I = (this->X+this->Y)*(other.X+other.Y)-C-D;     // I = (X1+Y1)*(X2+Y2)-C-D
    const edwards_Fq3 X3 = (E+B)*H;                                    // X3 = (E+B)*H
    const edwards_Fq3 Y3 = (E-B)*I;                                    // Y3 = (E-B)*I
    const edwards_Fq3 Z3 = A*H*I;                                      // Z3 = A*H*I
 
    return edwards_G2(X3, Y3, Z3);
}

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base_field_char()

static bigint< base_field::num_limbs > libff::edwards_G2::base_field_char ( )

inlinestatic

Definition at line 77 of file edwards_g2.hpp.

{ return base_field::field_char(); }

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batch_to_special_all_non_zeros()

void libff::edwards_G2::batch_to_special_all_non_zeros ( std::vector< edwards_G2 > & vec )

static

Definition at line 396 of file edwards_g2.cpp.

{
    std::vector<edwards_Fq3> Z_vec;
    Z_vec.reserve(vec.size());
 
    for (auto &el: vec)
    {
        Z_vec.emplace_back(el.Z);
    }
    batch_invert<edwards_Fq3>(Z_vec);
 
    const edwards_Fq3 one = edwards_Fq3::one();
 
    for (size_t i = 0; i < vec.size(); ++i)
    {
        vec[i].X = vec[i].X * Z_vec[i];
        vec[i].Y = vec[i].Y * Z_vec[i];
        vec[i].Z = one;
    }
}

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dbl()

edwards_G2 libff::edwards_G2::dbl

(

)

const

Definition at line 261 of file edwards_g2.cpp.

{
#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-twisted-inverted.html#doubling-dbl-2008-bbjlp
 
        const edwards_Fq3 A = (this->X).squared();                      // A = X1^2
        const edwards_Fq3 B = (this->Y).squared();                      // B = Y1^2
        const edwards_Fq3 U = edwards_G2::mul_by_a(B);                  // U = a*B
        const edwards_Fq3 C = A+U;                                      // C = A+U
        const edwards_Fq3 D = A-U;                                      // D = A-U
        const edwards_Fq3 E = (this->X+this->Y).squared()-A-B;          // E = (X1+Y1)^2-A-B
        const edwards_Fq3 X3 = C*D;                                     // X3 = C*D
        const edwards_Fq3 dZZ = edwards_G2::mul_by_d(this->Z.squared());
        const edwards_Fq3 Y3 = E*(C-dZZ-dZZ);                           // Y3 = E*(C-2*d*Z1^2)
        const edwards_Fq3 Z3 = D*E;                                     // Z3 = D*E
 
        return edwards_G2(X3, Y3, Z3);
    }
}

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is_special()

bool libff::edwards_G2::is_special

(

)

const

Definition at line 131 of file edwards_g2.cpp.

{
    return (this->is_zero() || this->Z == edwards_Fq3::one());
}

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is_well_formed()

bool libff::edwards_G2::is_well_formed

(

)

const

Definition at line 297 of file edwards_g2.cpp.

{
    /* Note that point at infinity is the only special case we must check as
       inverted representation does no cover points (0, +-c) and (+-c, 0). */
    if (this->is_zero())
    {
        return true;
    }
    else
    {
        /*
          a x^2 + y^2 = 1 + d x^2 y^2
 
          We are using inverted, so equation we need to check is actually
 
          a (z/x)^2 + (z/y)^2 = 1 + d z^4 / (x^2 * y^2)
          z^2 (a y^2 + x^2 - dz^2) = x^2 y^2
        */
        edwards_Fq3 X2 = this->X.squared();
        edwards_Fq3 Y2 = this->Y.squared();
        edwards_Fq3 Z2 = this->Z.squared();
        edwards_Fq3 aY2 = edwards_G2::mul_by_a(Y2);
        edwards_Fq3 dZ2 = edwards_G2::mul_by_d(Z2);
        return (Z2 * (aY2 + X2 - dZ2) == X2 * Y2);
    }
}

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is_zero()

bool libff::edwards_G2::is_zero

(

)

const

Definition at line 136 of file edwards_g2.cpp.

{
    return (this->Y.is_zero() && this->Z.is_zero());
}

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mixed_add()

edwards_G2 libff::edwards_G2::mixed_add

(

const edwards_G2 &

other

)

const

Definition at line 224 of file edwards_g2.cpp.

{
#ifdef PROFILE_OP_COUNTS
    this->add_cnt++;
#endif
    // handle special cases having to do with O
    if (this->is_zero())
    {
        return other;
    }
 
    if (other.is_zero())
    {
        return *this;
    }
 
#ifdef DEBUG
    assert(other.is_special());
#endif
 
    // NOTE: does not handle O and pts of order 2,4
    // http://www.hyperelliptic.org/EFD/g1p/auto-edwards-inverted.html#addition-madd-2007-lb
 
    const edwards_Fq3 A = this->Z;                                     // A = Z1*Z2
    const edwards_Fq3 B = edwards_G2::mul_by_d(A.squared());           // B = d*A^2
    const edwards_Fq3 C = (this->X) * (other.X);                       // C = X1*X2
    const edwards_Fq3 D = (this->Y) * (other.Y);                       // D = Y1*Y2
    const edwards_Fq3 E = C*D;                                         // E = C*D
    const edwards_Fq3 H = C - edwards_G2::mul_by_a(D);                 // H = C-a*D
    const edwards_Fq3 I = (this->X+this->Y)*(other.X+other.Y)-C-D;     // I = (X1+Y1)*(X2+Y2)-C-D
    const edwards_Fq3 X3 = (E+B)*H;                                    // X3 = (E+B)*H
    const edwards_Fq3 Y3 = (E-B)*I;                                    // Y3 = (E-B)*I
    const edwards_Fq3 Z3 = A*H*I;                                      // Z3 = A*H*I
 
    return edwards_G2(X3, Y3, Z3);
}

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mul_by_a()

edwards_Fq3 libff::edwards_G2::mul_by_a ( const edwards_Fq3 & elt )

static

Definition at line 34 of file edwards_g2.cpp.

{
    // should be
    //  edwards_Fq3(edwards_twist_mul_by_a_c0 * elt.c2, edwards_twist_mul_by_a_c1 * elt.c0, edwards_twist_mul_by_a_c2 * elt.c1)
    // but optimizing the fact that edwards_twist_mul_by_a_c1 = edwards_twist_mul_by_a_c2 = 1
    return edwards_Fq3(edwards_twist_mul_by_a_c0 * elt.c2, elt.c0, elt.c1);
}

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mul_by_d()

edwards_Fq3 libff::edwards_G2::mul_by_d ( const edwards_Fq3 & elt )

static

Definition at line 42 of file edwards_g2.cpp.

{
    return edwards_Fq3(edwards_twist_mul_by_d_c0 * elt.c2, edwards_twist_mul_by_d_c1 * elt.c0, edwards_twist_mul_by_d_c2 * elt.c1);
}

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mul_by_q()

edwards_G2 libff::edwards_G2::mul_by_q

(

)

const

Definition at line 290 of file edwards_g2.cpp.

{
    return edwards_G2((this->X).Frobenius_map(1),
                      edwards_twist_mul_by_q_Y * (this->Y).Frobenius_map(1),
                      edwards_twist_mul_by_q_Z * (this->Z).Frobenius_map(1));
}

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one()

edwards_G2 libff::edwards_G2::one ( )

static

Definition at line 329 of file edwards_g2.cpp.

{
    return G2_one;
}

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operator!=()

bool libff::edwards_G2::operator!=

(

const edwards_G2 &

other

)

const

Definition at line 170 of file edwards_g2.cpp.

{
    return !(operator==(other));
}

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operator+()

edwards_G2 libff::edwards_G2::operator+

(

const edwards_G2 &

other

)

const

Definition at line 175 of file edwards_g2.cpp.

{
    // handle special cases having to do with O
    if (this->is_zero())
    {
        return other;
    }
 
    if (other.is_zero())
    {
        return (*this);
    }
 
    return this->add(other);
}

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operator-() [1/2]

edwards_G2 libff::edwards_G2::operator-

(

)

const

Definition at line 191 of file edwards_g2.cpp.

{
    return edwards_G2(-(this->X), this->Y, this->Z);
}

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operator-() [2/2]

edwards_G2 libff::edwards_G2::operator-

(

const edwards_G2 &

other

)

const

Definition at line 197 of file edwards_g2.cpp.

{
    return (*this) + (-other);
}

operator==()

bool libff::edwards_G2::operator==

(

const edwards_G2 &

other

)

const

Definition at line 141 of file edwards_g2.cpp.

{
    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;
}

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order()

static bigint< scalar_field::num_limbs > libff::edwards_G2::order ( )

inlinestatic

Definition at line 78 of file edwards_g2.hpp.

{ return scalar_field::field_char(); }

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print()

void libff::edwards_G2::print

(

)

const

Definition at line 47 of file edwards_g2.cpp.

{
    if (this->is_zero())
    {
        printf("O\n");
    }
    else
    {
        edwards_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, edwards_Fq::num_limbs,
                   copy.X.c1.as_bigint().data, edwards_Fq::num_limbs,
                   copy.X.c0.as_bigint().data, edwards_Fq::num_limbs,
                   copy.Y.c2.as_bigint().data, edwards_Fq::num_limbs,
                   copy.Y.c1.as_bigint().data, edwards_Fq::num_limbs,
                   copy.Y.c0.as_bigint().data, edwards_Fq::num_limbs);
    }
}

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print_coordinates()

void libff::edwards_G2::print_coordinates

(

)

const

Definition at line 67 of file edwards_g2.cpp.

{
    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, edwards_Fq::num_limbs,
                   this->X.c1.as_bigint().data, edwards_Fq::num_limbs,
                   this->X.c0.as_bigint().data, edwards_Fq::num_limbs,
                   this->Y.c2.as_bigint().data, edwards_Fq::num_limbs,
                   this->Y.c1.as_bigint().data, edwards_Fq::num_limbs,
                   this->Y.c0.as_bigint().data, edwards_Fq::num_limbs,
                   this->Z.c2.as_bigint().data, edwards_Fq::num_limbs,
                   this->Z.c1.as_bigint().data, edwards_Fq::num_limbs,
                   this->Z.c0.as_bigint().data, edwards_Fq::num_limbs);
    }
}

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random_element()

edwards_G2 libff::edwards_G2::random_element ( )

static

Definition at line 334 of file edwards_g2.cpp.

{
    return edwards_Fr::random_element().as_bigint() * G2_one;
}

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size_in_bits()

static size_t libff::edwards_G2::size_in_bits ( )

inlinestatic

Definition at line 76 of file edwards_g2.hpp.

{ return twist_field::size_in_bits() + 1; }

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to_affine_coordinates()

void libff::edwards_G2::to_affine_coordinates

(

)

Definition at line 88 of file edwards_g2.cpp.

{
    if (this->is_zero())
    {
        this->X = edwards_Fq3::zero();
        this->Y = edwards_Fq3::one();
        this->Z = edwards_Fq3::one();
    }
    else
    {
        // go from inverted coordinates to projective coordinates
        edwards_Fq3 tX = this->Y * this->Z;
        edwards_Fq3 tY = this->X * this->Z;
        edwards_Fq3 tZ = this->X * this->Y;
        // go from projective coordinates to affine coordinates
        edwards_Fq3 tZ_inv = tZ.inverse();
        this->X = tX * tZ_inv;
        this->Y = tY * tZ_inv;
        this->Z = edwards_Fq3::one();
    }
}

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to_special()

void libff::edwards_G2::to_special

(

)

Definition at line 110 of file edwards_g2.cpp.

{
    if (this->Z.is_zero())
    {
        return;
    }
 
#ifdef DEBUG
    const edwards_G2 copy(*this);
#endif
 
    edwards_Fq3 Z_inv = this->Z.inverse();
    this->X = this->X * Z_inv;
    this->Y = this->Y * Z_inv;
    this->Z = edwards_Fq3::one();
 
#ifdef DEBUG
    assert((*this) == copy);
#endif
}

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zero()

edwards_G2 libff::edwards_G2::zero ( )

static

Definition at line 324 of file edwards_g2.cpp.

{
    return G2_zero;
}

Friends And Related Symbol Documentation

operator<<

std::ostream & operator<< ( std::ostream & out, const edwards_G2 & g )

friend

Definition at line 339 of file edwards_g2.cpp.

{
    edwards_G2 copy(g);
    copy.to_affine_coordinates();
#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;
}

operator>>

std::istream & operator>> ( std::istream & in, edwards_G2 & g )

friend

Definition at line 352 of file edwards_g2.cpp.

{
    edwards_Fq3 tX, tY;
 
#ifdef NO_PT_COMPRESSION
    in >> tX;
    consume_OUTPUT_SEPARATOR(in);
    in >> tY;
#else
    /*
      a x^2 + y^2 = 1 + d x^2 y^2
      y = sqrt((1-ax^2)/(1-dx^2))
    */
    unsigned char Y_lsb;
    in >> tX;
    consume_OUTPUT_SEPARATOR(in);
 
    in.read((char*)&Y_lsb, 1);
    Y_lsb -= '0';
 
    edwards_Fq3 tX2 = tX.squared();
    const edwards_Fq3 tY2 =
        (edwards_Fq3::one() - edwards_G2::mul_by_a(tX2)) *
        (edwards_Fq3::one() - edwards_G2::mul_by_d(tX2)).inverse();
    tY = tY2.sqrt();
 
    if ((tY.c0.as_bigint().data[0] & 1) != Y_lsb)
    {
        tY = -tY;
    }
#endif
 
    // using inverted coordinates
    g.X = tY;
    g.Y = tX;
    g.Z = tX * tY;
 
#ifdef USE_MIXED_ADDITION
    g.to_special();
#endif
 
    return in;
}

Member Data Documentation

fixed_base_exp_window_table

std::vector< size_t > libff::edwards_G2::fixed_base_exp_window_table

static

Definition at line 29 of file edwards_g2.hpp.

G2_one

edwards_G2 libff::edwards_G2::G2_one = {}

static

Definition at line 32 of file edwards_g2.hpp.

G2_zero

edwards_G2 libff::edwards_G2::G2_zero = {}

static

Definition at line 31 of file edwards_g2.hpp.

initialized

bool libff::edwards_G2::initialized = false

static

Definition at line 33 of file edwards_g2.hpp.

wnaf_window_table

std::vector< size_t > libff::edwards_G2::wnaf_window_table

static

Definition at line 28 of file edwards_g2.hpp.

X

edwards_Fq3 libff::edwards_G2::X

Definition at line 35 of file edwards_g2.hpp.

Y

edwards_Fq3 libff::edwards_G2::Y

Definition at line 35 of file edwards_g2.hpp.

Z

edwards_Fq3 libff::edwards_G2::Z

Definition at line 35 of file edwards_g2.hpp.

---

The documentation for this class was generated from the following files:

• libraries/fc/libraries/ff/libff/algebra/curves/edwards/edwards_g2.hpp
• libraries/fc/libraries/ff/libff/algebra/curves/edwards/edwards_g2.cpp