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/*
* Arithmetic for point groups of elliptic curves over GF(p)
*
* (C) 2007 Martin Doering, Christoph Ludwig, Falko Strenzke
* 2008-2010 Jack Lloyd
*
* Distributed under the terms of the Botan license
*/
#include <botan/point_gfp.h>
#include <botan/numthry.h>
namespace Botan {
namespace {
BigInt decompress_point(bool yMod2,
const BigInt& x,
const CurveGFp& curve)
{
BigInt xpow3 = x * x * x;
BigInt g = curve.get_a() * x;
g += xpow3;
g += curve.get_b();
g = g % curve.get_p();
BigInt z = ressol(g, curve.get_p());
if(z < 0)
throw Illegal_Point("error during decompression");
if(z.get_bit(0) != yMod2)
z = curve.get_p() - z;
return z;
}
}
PointGFp::PointGFp(const CurveGFp& curve) :
curve(curve), coord_x(0), coord_y(1), coord_z(0)
{
}
PointGFp::PointGFp(const CurveGFp& curve, const BigInt& x, const BigInt& y) :
curve(curve), coord_x(x), coord_y(y), coord_z(1)
{
}
// arithmetic operators
PointGFp& PointGFp::operator+=(const PointGFp& rhs)
{
if(rhs.is_zero())
return *this;
if(is_zero())
{
*this = rhs;
return *this;
}
const Modular_Reducer& mod_p = curve.mod_p();
BigInt rhs_z2 = mod_p.square(rhs.coord_z);
BigInt U1 = mod_p.multiply(coord_x, rhs_z2);
BigInt S1 = mod_p.multiply(coord_y, mod_p.multiply(rhs.coord_z, rhs_z2));
BigInt lhs_z2 = mod_p.square(coord_z);
BigInt U2 = mod_p.multiply(rhs.coord_x, lhs_z2);
BigInt S2 = mod_p.multiply(rhs.coord_y, mod_p.multiply(coord_z, lhs_z2));
BigInt H = mod_p.reduce(U2 - U1);
BigInt r = mod_p.reduce(S2 - S1);
if(H.is_zero())
{
if(r.is_zero())
{
mult2();
return *this;
}
*this = PointGFp(curve); // setting myself to zero
return *this;
}
U2 = mod_p.square(H);
S2 = mod_p.multiply(U2, H);
U2 = mod_p.multiply(U1, U2);
BigInt x = mod_p.reduce(mod_p.square(r) - S2 - mod_p.multiply(2, U2));
BigInt y = mod_p.reduce(mod_p.multiply(r, (U2-x)) - mod_p.multiply(S1, S2));
BigInt z = mod_p.multiply(mod_p.multiply(coord_z, rhs.coord_z), H);
coord_x = x;
coord_y = y;
coord_z = z;
return *this;
}
PointGFp& PointGFp::operator-=(const PointGFp& rhs)
{
PointGFp minus_rhs = PointGFp(rhs).negate();
if(is_zero())
*this = minus_rhs;
else
*this += minus_rhs;
return *this;
}
PointGFp& PointGFp::operator*=(const BigInt& scalar)
{
if(scalar.abs() <= 2) // special cases for small values
{
u32bit value = scalar.abs().to_u32bit();
if(value == 0)
*this = PointGFp(curve); // set to zero point
else if(value == 1)
{
if(scalar.is_negative())
this->negate();
}
else if(value == 2)
{
this->mult2();
if(scalar.is_negative())
this->negate();
}
return *this;
}
PointGFp H(this->curve); // create as zero
PointGFp P(*this);
if(scalar.is_negative())
P.negate();
for(int i = scalar.bits() - 1; i >= 0; --i)
{
H.mult2();
if(scalar.get_bit(i))
H += P;
}
if(!H.is_zero()) // cannot convert if H == O
{
/**
* Convert H to an equivalent point with z == 1, thus x and y
* correspond to their affine coordinates
*/
if(H.coord_z != 1)
{
const Modular_Reducer& mod_p = curve.mod_p();
BigInt z_inv = inverse_mod(H.coord_z, curve.get_p());
BigInt z_inv_2 = mod_p.square(z_inv);
H.coord_x = mod_p.multiply(H.coord_x, z_inv_2);
H.coord_y = mod_p.multiply(H.coord_y, mod_p.multiply(z_inv, z_inv_2));
H.coord_z = 1;
}
}
*this = H;
return *this;
}
// *this *= 2
void PointGFp::mult2()
{
if(is_zero())
return;
else if(coord_y.is_zero())
{
*this = PointGFp(curve); // setting myself to zero
return;
}
const Modular_Reducer& mod_p = curve.mod_p();
BigInt y_2 = mod_p.square(coord_y);
BigInt S = mod_p.multiply(4, mod_p.multiply(coord_x, y_2));
BigInt a_z4 = mod_p.multiply(curve.get_a(),
mod_p.square(mod_p.square(coord_z)));
BigInt M = mod_p.reduce(a_z4 + 3 * mod_p.square(coord_x));
BigInt x = mod_p.reduce(mod_p.square(M) - mod_p.multiply(2, S));
BigInt U = mod_p.multiply(8, mod_p.square(y_2));
BigInt y = mod_p.reduce(mod_p.multiply(M, S - x) - U);
BigInt z = mod_p.multiply(2, mod_p.multiply(coord_y, coord_z));
coord_x = x;
coord_y = y;
coord_z = z;
}
BigInt PointGFp::get_affine_x() const
{
if(is_zero())
throw Illegal_Transformation("Cannot convert zero point to affine");
const Modular_Reducer& mod_p = curve.mod_p();
BigInt z2 = mod_p.square(coord_z);
return mod_p.multiply(coord_x, inverse_mod(z2, curve.get_p()));
}
BigInt PointGFp::get_affine_y() const
{
if(is_zero())
throw Illegal_Transformation("Cannot convert zero point to affine");
const Modular_Reducer& mod_p = curve.mod_p();
BigInt z3 = mod_p.cube(coord_z);
return mod_p.multiply(coord_y, inverse_mod(z3, curve.get_p()));
}
void PointGFp::check_invariants() const
{
/*
Is the point still on the curve?? (If everything is correct, the
point is always on its curve; then the function will return
silently. If Oskar managed to corrupt this object's state, then it
will throw an exception.)
*/
if(is_zero())
return;
const Modular_Reducer& mod_p = curve.mod_p();
BigInt y2 = mod_p.square(coord_y);
BigInt x3 = mod_p.cube(coord_x);
BigInt ax = mod_p.multiply(coord_x, curve.get_a());
if(coord_z == 1)
{
if(mod_p.reduce(x3 + ax + curve.get_b()) != y2)
throw Illegal_Point("Invalid ECP point: y^2 != x^3 + a*x + b");
}
BigInt z2 = mod_p.square(coord_z);
BigInt z3 = mod_p.multiply(coord_z, z2);
BigInt ax_z4 = mod_p.multiply(mod_p.multiply(z3, coord_z), ax);
BigInt b_z6 = mod_p.multiply(curve.get_b(), mod_p.square(z3));
if(y2 != mod_p.reduce(x3 + ax_z4 + b_z6))
throw Illegal_Point("Invalid ECP point: y^2 != x^3 + a*x*z^4 + b*z^6");
}
// swaps the states of *this and other, does not throw!
void PointGFp::swap(PointGFp& other)
{
curve.swap(other.curve);
coord_x.swap(other.coord_x);
coord_y.swap(other.coord_y);
coord_z.swap(other.coord_z);
}
bool PointGFp::operator==(const PointGFp& other) const
{
if(get_curve() != other.get_curve())
return false;
// If this is zero, only equal if other is also zero
if(is_zero())
return other.is_zero();
return (get_affine_x() == other.get_affine_x() &&
get_affine_y() == other.get_affine_y());
}
// encoding and decoding
SecureVector<byte> EC2OSP(const PointGFp& point, byte format)
{
if(point.is_zero())
return SecureVector<byte>(1); // single 0 byte
const u32bit p_bytes = point.get_curve().get_p().bytes();
BigInt x = point.get_affine_x();
BigInt y = point.get_affine_y();
SecureVector<byte> bX = BigInt::encode_1363(x, p_bytes);
SecureVector<byte> bY = BigInt::encode_1363(y, p_bytes);
if(format == PointGFp::UNCOMPRESSED)
{
SecureVector<byte> result(2*p_bytes+1);
result[0] = 4;
result.copy(1, bX.begin(), p_bytes);
result.copy(p_bytes+1, bY.begin(), p_bytes);
return result;
}
else if(format == PointGFp::COMPRESSED)
{
SecureVector<byte> result(p_bytes+1);
result[0] = 2;
result.copy(1, bX.begin(), bX.size());
if(y.get_bit(0))
result[0] |= 1;
return result;
}
else if(format == PointGFp::HYBRID)
{
SecureVector<byte> result(2*p_bytes+1);
result[0] = 6;
result.copy(1, bX.begin(), bX.size());
result.copy(p_bytes+1, bY.begin(), bY.size());
if(y.get_bit(0))
result[0] |= 1;
return result;
}
else
throw Invalid_Argument("illegal point encoding format specification");
}
PointGFp OS2ECP(const byte data[], u32bit data_len,
const CurveGFp& curve)
{
if(data_len <= 1)
return PointGFp(curve); // return zero
const byte pc = data[0];
BigInt x, y;
if(pc == 2 || pc == 3)
{
//compressed form
x = BigInt::decode(&data[1], data_len - 1);
bool yMod2 = ((pc & 0x01) == 1);
y = decompress_point(yMod2, x, curve);
}
else if(pc == 4)
{
const u32bit l = (data_len - 1) / 2;
// uncompressed form
x = BigInt::decode(&data[1], l);
y = BigInt::decode(&data[l+1], l);
}
else if(pc == 6 || pc == 7)
{
const u32bit l = (data_len - 1) / 2;
// hybrid form
x = BigInt::decode(&data[1], l);
y = BigInt::decode(&data[l+1], l);
bool yMod2 = ((pc & 0x01) == 1);
if(decompress_point(yMod2, x, curve) != y)
throw Illegal_Point("OS2ECP: Decoding error in hybrid format");
}
else
throw Invalid_Argument("OS2ECP: Unknown format type");
PointGFp result(curve, x, y);
result.check_invariants();
return result;
}
}
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