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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().get_value() * x;
g += xpow3;
g += curve.get_b().get_value();
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;
}
}
// construct the point at infinity or a random point
PointGFp::PointGFp(const CurveGFp& curve) :
mC(curve),
mX(curve.get_p(), 0),
mY(curve.get_p(), 1),
mZ(curve.get_p(), 0)
{
}
// construct a point given its jacobian projective coordinates
PointGFp::PointGFp(const CurveGFp& curve,
const BigInt& x,
const BigInt& y,
const BigInt& z) :
mC(curve),
mX(curve.get_p(), x),
mY(curve.get_p(), y),
mZ(curve.get_p(), z)
{
}
PointGFp::PointGFp(const CurveGFp& curve,
const BigInt& x,
const BigInt& y) :
mC(curve),
mX(curve.get_p(), x),
mY(curve.get_p(), y),
mZ(curve.get_p(), 1)
{
}
// arithmetic operators
PointGFp& PointGFp::operator+=(const PointGFp& rhs)
{
if(rhs.is_zero())
return *this;
if(is_zero())
{
*this = rhs;
return *this;
}
GFpElement U1 = mX;
GFpElement S1 = mY;
GFpElement rhs_z2 = rhs.mZ * rhs.mZ;
U1 *= rhs_z2;
S1 *= rhs_z2 * rhs.mZ;
GFpElement U2 = rhs.mX;
GFpElement S2 = rhs.mY;
GFpElement lhs_z2 = mZ * mZ;
U2 *= lhs_z2;
S2 *= lhs_z2 * mZ;
GFpElement H(U2 - U1);
GFpElement r(S2 - S1);
if(H.is_zero())
{
if(r.is_zero())
{
mult2_in_place();
return *this;
}
*this = PointGFp(mC); // setting myself to zero
return *this;
}
U2 = H * H;
S2 = U2 * H;
U2 *= U1;
GFpElement x(r*r - S2 - (U2+U2));
GFpElement z(S1 * S2);
GFpElement y(r * (U2-x) - z);
z = (mZ * rhs.mZ) * H;
mX = x;
mY = y;
mZ = 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)
{
PointGFp H(this->mC); // create as zero
PointGFp P(*this);
BigInt m(scalar);
if(m < BigInt(0))
{
m.flip_sign();
P.negate();
}
// Move upwards
if(P.is_zero() || (m == BigInt(0)))
{
*this = H;
return *this;
}
// FIXME: *this != P if m was -1 !
if(m == BigInt(1)) //*this == P already
return *this;
const int l = m.bits() - 1;
for(int i = l; i >= 0; --i)
{
H.mult2_in_place();
if(m.get_bit(i))
H += P;
}
if(!H.is_zero()) // cannot convert if H == O
*this = H.get_z_to_one();
else
*this = H;
return *this;
}
PointGFp& PointGFp::negate()
{
if(!is_zero())
mY.negate();
return *this;
}
// *this *= 2
PointGFp& PointGFp::mult2_in_place()
{
if(is_zero())
return *this;
else if(mY.is_zero())
{
*this = PointGFp(mC); // setting myself to zero
return *this;
}
GFpElement Y_squared = mY*mY;
GFpElement S = mX * Y_squared;
GFpElement x = S + S;
S = x + x;
GFpElement a_z4 = mC.get_a();
GFpElement z2 = mZ * mZ;
a_z4 *= z2;
a_z4 *= z2;
GFpElement y(mX * mX);
GFpElement M(y + y + y + a_z4);
x = M * M - (S+S);
y = Y_squared * Y_squared;
GFpElement U(y + y);
GFpElement z = U + U;
U = z + z;
y = M * (S - x) - U;
z = mY * mZ;
z = z + z;
mX = x;
mY = y;
mZ = z;
return *this;
}
/**
* returns a point equivalent to *this but were
* Z has value one, i.e. x and y correspond to
* their values in affine coordinates
*/
PointGFp PointGFp::get_z_to_one()
{
return PointGFp(*this).set_z_to_one();
}
/**
* changes the representation of *this so that
* Z has value one, i.e. x and y correspond to
* their values in affine coordinates.
* returns *this.
*/
const PointGFp& PointGFp::set_z_to_one()
{
if(mZ.is_zero())
throw Illegal_Transformation("cannot convert Z to one");
if(mZ.get_value() != 1)
{
// Converts to affine coordinates
GFpElement z = inverse(mZ);
GFpElement z2 = z * z;
z *= z2;
GFpElement x = mX * z2;
GFpElement y = mY * z;
mZ = GFpElement(mC.get_p(), BigInt(1));
mX = x;
mY = y;
}
return *this;
}
BigInt PointGFp::get_affine_x() const
{
if(is_zero())
throw Illegal_Transformation("cannot convert to affine");
GFpElement z2 = mZ * mZ;
z2.inverse_in_place();
z2 *= mX;
return z2.get_value();
}
BigInt PointGFp::get_affine_y() const
{
if(is_zero())
throw Illegal_Transformation("cannot convert to affine");
GFpElement z3 = mZ * mZ * mZ;
z3.inverse_in_place();
z3 *= mY;
return z3.get_value();
}
// Is this the point at infinity?
bool PointGFp::is_zero() const
{
return(mX.is_zero() && mZ.is_zero());
}
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 GFpElement y2 = mY * mY;
const GFpElement x3 = mX * mX * mX;
if(mZ.get_value() == BigInt(1))
{
GFpElement ax = mC.get_a() * mX;
if(y2 != (x3 + ax + mC.get_b()))
throw Illegal_Point();
}
GFpElement Zpow2 = mZ * mZ;
GFpElement Zpow3 = Zpow2 * mZ;
GFpElement AZpow4 = Zpow3 * mZ * mC.get_a();
const GFpElement aXZ4 = AZpow4 * mX;
const GFpElement bZ6 = mC.get_b() * Zpow3 * Zpow3;
if(y2 != (x3 + aXZ4 + bZ6))
throw Illegal_Point();
}
// swaps the states of *this and other, does not throw!
void PointGFp::swap(PointGFp& other)
{
mC.swap(other.mC);
mX.swap(other.mX);
mY.swap(other.mY);
mZ.swap(other.mZ);
}
bool PointGFp::operator==(const PointGFp& other) const
{
if(get_curve() != other.get_curve())
return false;
return (mX == other.mX && mY == other.mY && mZ == other.mZ);
}
// arithmetic operators
PointGFp operator+(const PointGFp& lhs, PointGFp const& rhs)
{
PointGFp tmp(lhs);
return tmp += rhs;
}
PointGFp operator-(const PointGFp& lhs, PointGFp const& rhs)
{
PointGFp tmp(lhs);
return tmp -= rhs;
}
PointGFp operator-(const PointGFp& lhs)
{
return PointGFp(lhs).negate();
}
PointGFp operator*(const BigInt& scalar, const PointGFp& point)
{
PointGFp result(point);
return result *= scalar;
}
PointGFp operator*(const PointGFp& point, const BigInt& scalar)
{
PointGFp result(point);
return result *= scalar;
}
// 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_bits = point.get_curve().get_p().bits();
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 MemoryRegion<byte>& os, const CurveGFp& curve)
{
if(os.size() == 1 && os[0] == 0)
return PointGFp(curve); // return zero
const byte pc = os[0];
BigInt x, y;
if(pc == 2 || pc == 3)
{
//compressed form
x = BigInt::decode(&os[1], os.size() - 1);
bool yMod2 = ((pc & 0x01) == 1);
y = decompress_point(yMod2, x, curve);
}
else if(pc == 4)
{
// uncompressed form
u32bit l = (os.size() - 1) / 2;
x = BigInt::decode(&os[1], l);
y = BigInt::decode(&os[l+1], l);
}
else if(pc == 6 || pc == 7)
{
// hybrid form
u32bit l = (os.size() - 1) / 2;
x = BigInt::decode(&os[1], l);
y = BigInt::decode(&os[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;
}
PointGFp create_random_point(RandomNumberGenerator& rng,
const CurveGFp& curve)
{
const BigInt& p = curve.get_p();
while(true)
{
BigInt r(rng, p.bits());
GFpElement x = GFpElement(p, r);
GFpElement x3 = x * x * x;
GFpElement y = (curve.get_a() * x) + (x3 * curve.get_b());
if(ressol(y.get_value(), p) > 0)
return PointGFp(curve, x.get_value(), y.get_value());
}
}
} // namespace Botan
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