// B.3 pg 62
public override ECPoint Add(ECPoint b)
{
if (this.IsInfinity)
{
return(b);
}
if (b.IsInfinity)
{
return(this);
}
if (this == b)
{
return(Twice());
}
ECCurve curve = this.Curve;
int coord = curve.CoordinateSystem;
ECFieldElement X1 = this.RawXCoord, Y1 = this.RawYCoord;
ECFieldElement X2 = b.RawXCoord, Y2 = b.RawYCoord;
switch (coord)
{
case ECCurve.COORD_AFFINE:
{
ECFieldElement dx = X2.Subtract(X1), dy = Y2.Subtract(Y1);
if (dx.IsZero)
{
if (dy.IsZero)
{
// this == b, i.e. this must be doubled
return(Twice());
}
// this == -b, i.e. the result is the point at infinity
return(Curve.Infinity);
}
ECFieldElement gamma = dy.Divide(dx);
ECFieldElement X3 = gamma.Square().Subtract(X1).Subtract(X2);
ECFieldElement Y3 = gamma.Multiply(X1.Subtract(X3)).Subtract(Y1);
return(new FpPoint(Curve, X3, Y3, IsCompressed));
}
case ECCurve.COORD_HOMOGENEOUS:
{
ECFieldElement Z1 = this.RawZCoords[0];
ECFieldElement Z2 = b.RawZCoords[0];
bool Z1IsOne = Z1.IsOne;
bool Z2IsOne = Z2.IsOne;
ECFieldElement u1 = Z1IsOne ? Y2 : Y2.Multiply(Z1);
ECFieldElement u2 = Z2IsOne ? Y1 : Y1.Multiply(Z2);
ECFieldElement u = u1.Subtract(u2);
ECFieldElement v1 = Z1IsOne ? X2 : X2.Multiply(Z1);
ECFieldElement v2 = Z2IsOne ? X1 : X1.Multiply(Z2);
ECFieldElement v = v1.Subtract(v2);
// Check if b == this or b == -this
if (v.IsZero)
{
if (u.IsZero)
{
// this == b, i.e. this must be doubled
return(this.Twice());
}
// this == -b, i.e. the result is the point at infinity
return(curve.Infinity);
}
// TODO Optimize for when w == 1
ECFieldElement w = Z1IsOne ? Z2 : Z2IsOne ? Z1 : Z1.Multiply(Z2);
ECFieldElement vSquared = v.Square();
ECFieldElement vCubed = vSquared.Multiply(v);
ECFieldElement vSquaredV2 = vSquared.Multiply(v2);
ECFieldElement A = u.Square().Multiply(w).Subtract(vCubed).Subtract(Two(vSquaredV2));
ECFieldElement X3 = v.Multiply(A);
ECFieldElement Y3 = vSquaredV2.Subtract(A).Multiply(u).Subtract(vCubed.Multiply(u2));
ECFieldElement Z3 = vCubed.Multiply(w);
return(new FpPoint(curve, X3, Y3, new ECFieldElement[] { Z3 }, IsCompressed));
}
case ECCurve.COORD_JACOBIAN:
case ECCurve.COORD_JACOBIAN_MODIFIED:
{
ECFieldElement Z1 = this.RawZCoords[0];
ECFieldElement Z2 = b.RawZCoords[0];
bool Z1IsOne = Z1.IsOne;
ECFieldElement X3, Y3, Z3, Z3Squared = null;
if (!Z1IsOne && Z1.Equals(Z2))
{
// TODO Make this available as public method coZAdd?
ECFieldElement dx = X1.Subtract(X2), dy = Y1.Subtract(Y2);
if (dx.IsZero)
{
if (dy.IsZero)
{
return(Twice());
}
return(curve.Infinity);
}
ECFieldElement C = dx.Square();
ECFieldElement W1 = X1.Multiply(C), W2 = X2.Multiply(C);
ECFieldElement A1 = W1.Subtract(W2).Multiply(Y1);
X3 = dy.Square().Subtract(W1).Subtract(W2);
Y3 = W1.Subtract(X3).Multiply(dy).Subtract(A1);
Z3 = dx;
if (Z1IsOne)
{
Z3Squared = C;
}
else
{
Z3 = Z3.Multiply(Z1);
}
}
else
{
ECFieldElement Z1Squared, U2, S2;
if (Z1IsOne)
{
Z1Squared = Z1; U2 = X2; S2 = Y2;
}
else
{
Z1Squared = Z1.Square();
U2 = Z1Squared.Multiply(X2);
ECFieldElement Z1Cubed = Z1Squared.Multiply(Z1);
S2 = Z1Cubed.Multiply(Y2);
}
bool Z2IsOne = Z2.IsOne;
ECFieldElement Z2Squared, U1, S1;
if (Z2IsOne)
{
Z2Squared = Z2; U1 = X1; S1 = Y1;
}
else
{
Z2Squared = Z2.Square();
U1 = Z2Squared.Multiply(X1);
ECFieldElement Z2Cubed = Z2Squared.Multiply(Z2);
S1 = Z2Cubed.Multiply(Y1);
}
ECFieldElement H = U1.Subtract(U2);
ECFieldElement R = S1.Subtract(S2);
// Check if b == this or b == -this
if (H.IsZero)
{
if (R.IsZero)
{
// this == b, i.e. this must be doubled
return(this.Twice());
}
// this == -b, i.e. the result is the point at infinity
return(curve.Infinity);
}
ECFieldElement HSquared = H.Square();
ECFieldElement G = HSquared.Multiply(H);
ECFieldElement V = HSquared.Multiply(U1);
X3 = R.Square().Add(G).Subtract(Two(V));
Y3 = V.Subtract(X3).Multiply(R).Subtract(S1.Multiply(G));
Z3 = H;
if (!Z1IsOne)
{
Z3 = Z3.Multiply(Z1);
}
if (!Z2IsOne)
{
Z3 = Z3.Multiply(Z2);
}
// Alternative calculation of Z3 using fast square
//X3 = four(X3);
//Y3 = eight(Y3);
//Z3 = doubleProductFromSquares(Z1, Z2, Z1Squared, Z2Squared).multiply(H);
if (Z3 == H)
{
Z3Squared = HSquared;
}
}
ECFieldElement[] zs;
if (coord == ECCurve.COORD_JACOBIAN_MODIFIED)
{
// TODO If the result will only be used in a subsequent addition, we don't need W3
ECFieldElement W3 = CalculateJacobianModifiedW(Z3, Z3Squared);
zs = new ECFieldElement[] { Z3, W3 };
}
else
{
zs = new ECFieldElement[] { Z3 };
}
return(new FpPoint(curve, X3, Y3, zs, IsCompressed));
}
default:
{
throw new InvalidOperationException("unsupported coordinate system");
}
}
}