public override ECFieldElement Multiply(ECFieldElement b)
{
uint[] z = Nat256.Create();
Curve25519Field.Multiply(x, ((Curve25519FieldElement)b).x, z);
return(new Curve25519FieldElement(z));
}
public override ECFieldElement Subtract(ECFieldElement b)
{
uint[] z = Nat256.Create();
Curve25519Field.Subtract(x, ((Curve25519FieldElement)b).x, z);
return(new Curve25519FieldElement(z));
}
/**
* return a sqrt root - the routine verifies that the calculation returns the right value - if
* none exists it returns null.
*/
public override ECFieldElement Sqrt()
{
/*
* Q == 8m + 5, so we use Pocklington's method for this case.
*
* First, raise this element to the exponent 2^252 - 2^1 (i.e. m + 1)
*
* Breaking up the exponent's binary representation into "repunits", we get:
* { 251 1s } { 1 0s }
*
* Therefore we need an addition chain containing 251 (the lengths of the repunits)
* We use: 1, 2, 3, 4, 7, 11, 15, 30, 60, 120, 131, [251]
*/
uint[] x1 = this.x;
if (Nat256.IsZero(x1) || Nat256.IsOne(x1))
{
return(this);
}
uint[] x2 = Nat256.Create();
Curve25519Field.Square(x1, x2);
Curve25519Field.Multiply(x2, x1, x2);
uint[] x3 = x2;
Curve25519Field.Square(x2, x3);
Curve25519Field.Multiply(x3, x1, x3);
uint[] x4 = Nat256.Create();
Curve25519Field.Square(x3, x4);
Curve25519Field.Multiply(x4, x1, x4);
uint[] x7 = Nat256.Create();
Curve25519Field.SquareN(x4, 3, x7);
Curve25519Field.Multiply(x7, x3, x7);
uint[] x11 = x3;
Curve25519Field.SquareN(x7, 4, x11);
Curve25519Field.Multiply(x11, x4, x11);
uint[] x15 = x7;
Curve25519Field.SquareN(x11, 4, x15);
Curve25519Field.Multiply(x15, x4, x15);
uint[] x30 = x4;
Curve25519Field.SquareN(x15, 15, x30);
Curve25519Field.Multiply(x30, x15, x30);
uint[] x60 = x15;
Curve25519Field.SquareN(x30, 30, x60);
Curve25519Field.Multiply(x60, x30, x60);
uint[] x120 = x30;
Curve25519Field.SquareN(x60, 60, x120);
Curve25519Field.Multiply(x120, x60, x120);
uint[] x131 = x60;
Curve25519Field.SquareN(x120, 11, x131);
Curve25519Field.Multiply(x131, x11, x131);
uint[] x251 = x11;
Curve25519Field.SquareN(x131, 120, x251);
Curve25519Field.Multiply(x251, x120, x251);
uint[] t1 = x251;
Curve25519Field.Square(t1, t1);
uint[] t2 = x120;
Curve25519Field.Square(t1, t2);
if (Nat256.Eq(x1, t2))
{
return(new Curve25519FieldElement(t1));
}
/*
* If the first guess is incorrect, we multiply by a precomputed power of 2 to get the second guess,
* which is ((4x)^(m + 1))/2 mod Q
*/
Curve25519Field.Multiply(t1, PRECOMP_POW2, t1);
Curve25519Field.Square(t1, t2);
if (Nat256.Eq(x1, t2))
{
return(new Curve25519FieldElement(t1));
}
return(null);
}
public override ECFieldElement AddOne()
{
uint[] z = Nat256.Create();
Curve25519Field.AddOne(x, z);
return(new Curve25519FieldElement(z));
}
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;
Curve25519FieldElement X1 = (Curve25519FieldElement)this.RawXCoord, Y1 = (Curve25519FieldElement)this.RawYCoord,
Z1 = (Curve25519FieldElement)this.RawZCoords[0];
Curve25519FieldElement X2 = (Curve25519FieldElement)b.RawXCoord, Y2 = (Curve25519FieldElement)b.RawYCoord,
Z2 = (Curve25519FieldElement)b.RawZCoords[0];
uint c;
uint[] tt1 = Nat256.CreateExt();
uint[] t2 = Nat256.Create();
uint[] t3 = Nat256.Create();
uint[] t4 = Nat256.Create();
bool Z1IsOne = Z1.IsOne;
uint[] U2, S2;
if (Z1IsOne)
{
U2 = X2.x;
S2 = Y2.x;
}
else
{
S2 = t3;
Curve25519Field.Square(Z1.x, S2);
U2 = t2;
Curve25519Field.Multiply(S2, X2.x, U2);
Curve25519Field.Multiply(S2, Z1.x, S2);
Curve25519Field.Multiply(S2, Y2.x, S2);
}
bool Z2IsOne = Z2.IsOne;
uint[] U1, S1;
if (Z2IsOne)
{
U1 = X1.x;
S1 = Y1.x;
}
else
{
S1 = t4;
Curve25519Field.Square(Z2.x, S1);
U1 = tt1;
Curve25519Field.Multiply(S1, X1.x, U1);
Curve25519Field.Multiply(S1, Z2.x, S1);
Curve25519Field.Multiply(S1, Y1.x, S1);
}
uint[] H = Nat256.Create();
Curve25519Field.Subtract(U1, U2, H);
uint[] R = t2;
Curve25519Field.Subtract(S1, S2, R);
// Check if b == this or b == -this
if (Nat256.IsZero(H))
{
if (Nat256.IsZero(R))
{
// 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);
}
uint[] HSquared = Nat256.Create();
Curve25519Field.Square(H, HSquared);
uint[] G = Nat256.Create();
Curve25519Field.Multiply(HSquared, H, G);
uint[] V = t3;
Curve25519Field.Multiply(HSquared, U1, V);
Curve25519Field.Negate(G, G);
Nat256.Mul(S1, G, tt1);
c = Nat256.AddBothTo(V, V, G);
Curve25519Field.Reduce27(c, G);
Curve25519FieldElement X3 = new Curve25519FieldElement(t4);
Curve25519Field.Square(R, X3.x);
Curve25519Field.Subtract(X3.x, G, X3.x);
Curve25519FieldElement Y3 = new Curve25519FieldElement(G);
Curve25519Field.Subtract(V, X3.x, Y3.x);
Curve25519Field.MultiplyAddToExt(Y3.x, R, tt1);
Curve25519Field.Reduce(tt1, Y3.x);
Curve25519FieldElement Z3 = new Curve25519FieldElement(H);
if (!Z1IsOne)
{
Curve25519Field.Multiply(Z3.x, Z1.x, Z3.x);
}
if (!Z2IsOne)
{
Curve25519Field.Multiply(Z3.x, Z2.x, Z3.x);
}
uint[] Z3Squared = (Z1IsOne && Z2IsOne) ? HSquared : null;
// TODO If the result will only be used in a subsequent addition, we don't need W3
Curve25519FieldElement W3 = CalculateJacobianModifiedW((Curve25519FieldElement)Z3, Z3Squared);
ECFieldElement[] zs = new ECFieldElement[] { Z3, W3 };
return(new Curve25519Point(curve, X3, Y3, zs, IsCompressed));
}