/**
* return a sqrt root - the routine verifies that the calculation returns the right value - if
* none exists it returns null.
*/
public override ECFieldElement Sqrt()
{
/*
* Raise this element to the exponent 2^254 - 2^30 - 2^7 - 2^6 - 2^5 - 2^4 - 2^2
*
* Breaking up the exponent's binary representation into "repunits", we get:
* { 223 1s } { 1 0s } { 22 1s } { 4 0s } { 2 1s } { 2 0s}
*
* Therefore we need an addition chain containing 2, 22, 223 (the lengths of the repunits)
* We use: 1, [2], 3, 6, 9, 11, [22], 44, 88, 176, 220, [223]
*/
uint[] x1 = this.x;
if (Nat256.IsZero(x1) || Nat256.IsOne(x1))
{
return(this);
}
uint[] x2 = Nat256.Create();
SecP256K1Field.Square(x1, x2);
SecP256K1Field.Multiply(x2, x1, x2);
uint[] x3 = Nat256.Create();
SecP256K1Field.Square(x2, x3);
SecP256K1Field.Multiply(x3, x1, x3);
uint[] x6 = Nat256.Create();
SecP256K1Field.SquareN(x3, 3, x6);
SecP256K1Field.Multiply(x6, x3, x6);
uint[] x9 = x6;
SecP256K1Field.SquareN(x6, 3, x9);
SecP256K1Field.Multiply(x9, x3, x9);
uint[] x11 = x9;
SecP256K1Field.SquareN(x9, 2, x11);
SecP256K1Field.Multiply(x11, x2, x11);
uint[] x22 = Nat256.Create();
SecP256K1Field.SquareN(x11, 11, x22);
SecP256K1Field.Multiply(x22, x11, x22);
uint[] x44 = x11;
SecP256K1Field.SquareN(x22, 22, x44);
SecP256K1Field.Multiply(x44, x22, x44);
uint[] x88 = Nat256.Create();
SecP256K1Field.SquareN(x44, 44, x88);
SecP256K1Field.Multiply(x88, x44, x88);
uint[] x176 = Nat256.Create();
SecP256K1Field.SquareN(x88, 88, x176);
SecP256K1Field.Multiply(x176, x88, x176);
uint[] x220 = x88;
SecP256K1Field.SquareN(x176, 44, x220);
SecP256K1Field.Multiply(x220, x44, x220);
uint[] x223 = x44;
SecP256K1Field.SquareN(x220, 3, x223);
SecP256K1Field.Multiply(x223, x3, x223);
uint[] t1 = x223;
SecP256K1Field.SquareN(t1, 23, t1);
SecP256K1Field.Multiply(t1, x22, t1);
SecP256K1Field.SquareN(t1, 6, t1);
SecP256K1Field.Multiply(t1, x2, t1);
SecP256K1Field.SquareN(t1, 2, t1);
uint[] t2 = x2;
SecP256K1Field.Square(t1, t2);
return(Nat256.Eq(x1, t2) ? new SecP256K1FieldElement(t1) : null);
}
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;
SecP256K1FieldElement X1 = (SecP256K1FieldElement)this.RawXCoord, Y1 = (SecP256K1FieldElement)this.RawYCoord;
SecP256K1FieldElement X2 = (SecP256K1FieldElement)b.RawXCoord, Y2 = (SecP256K1FieldElement)b.RawYCoord;
SecP256K1FieldElement Z1 = (SecP256K1FieldElement)this.RawZCoords[0];
SecP256K1FieldElement Z2 = (SecP256K1FieldElement)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;
SecP256K1Field.Square(Z1.x, S2);
U2 = t2;
SecP256K1Field.Multiply(S2, X2.x, U2);
SecP256K1Field.Multiply(S2, Z1.x, S2);
SecP256K1Field.Multiply(S2, Y2.x, S2);
}
bool Z2IsOne = Z2.IsOne;
uint[] U1, S1;
if (Z2IsOne)
{
U1 = X1.x;
S1 = Y1.x;
}
else
{
S1 = t4;
SecP256K1Field.Square(Z2.x, S1);
U1 = tt1;
SecP256K1Field.Multiply(S1, X1.x, U1);
SecP256K1Field.Multiply(S1, Z2.x, S1);
SecP256K1Field.Multiply(S1, Y1.x, S1);
}
uint[] H = Nat256.Create();
SecP256K1Field.Subtract(U1, U2, H);
uint[] R = t2;
SecP256K1Field.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 = t3;
SecP256K1Field.Square(H, HSquared);
uint[] G = Nat256.Create();
SecP256K1Field.Multiply(HSquared, H, G);
uint[] V = t3;
SecP256K1Field.Multiply(HSquared, U1, V);
SecP256K1Field.Negate(G, G);
Nat256.Mul(S1, G, tt1);
c = Nat256.AddBothTo(V, V, G);
SecP256K1Field.Reduce32(c, G);
SecP256K1FieldElement X3 = new SecP256K1FieldElement(t4);
SecP256K1Field.Square(R, X3.x);
SecP256K1Field.Subtract(X3.x, G, X3.x);
SecP256K1FieldElement Y3 = new SecP256K1FieldElement(G);
SecP256K1Field.Subtract(V, X3.x, Y3.x);
SecP256K1Field.MultiplyAddToExt(Y3.x, R, tt1);
SecP256K1Field.Reduce(tt1, Y3.x);
SecP256K1FieldElement Z3 = new SecP256K1FieldElement(H);
if (!Z1IsOne)
{
SecP256K1Field.Multiply(Z3.x, Z1.x, Z3.x);
}
if (!Z2IsOne)
{
SecP256K1Field.Multiply(Z3.x, Z2.x, Z3.x);
}
ECFieldElement[] zs = new ECFieldElement[] { Z3 };
return(new SecP256K1Point(curve, X3, Y3, zs, IsCompressed));
}
public override ECFieldElement Square()
{
uint[] z = Nat256.Create();
SecP256K1Field.Square(x, z);
return(new SecP256K1FieldElement(z));
}
public override ECPoint Twice()
{
if (this.IsInfinity)
{
return(this);
}
ECCurve curve = this.Curve;
SecP256K1FieldElement Y1 = (SecP256K1FieldElement)this.RawYCoord;
if (Y1.IsZero)
{
return(curve.Infinity);
}
SecP256K1FieldElement X1 = (SecP256K1FieldElement)this.RawXCoord, Z1 = (SecP256K1FieldElement)this.RawZCoords[0];
uint c;
uint[] Y1Squared = Nat256.Create();
SecP256K1Field.Square(Y1.x, Y1Squared);
uint[] T = Nat256.Create();
SecP256K1Field.Square(Y1Squared, T);
uint[] M = Nat256.Create();
SecP256K1Field.Square(X1.x, M);
c = Nat256.AddBothTo(M, M, M);
SecP256K1Field.Reduce32(c, M);
uint[] S = Y1Squared;
SecP256K1Field.Multiply(Y1Squared, X1.x, S);
c = Nat.ShiftUpBits(8, S, 2, 0);
SecP256K1Field.Reduce32(c, S);
uint[] t1 = Nat256.Create();
c = Nat.ShiftUpBits(8, T, 3, 0, t1);
SecP256K1Field.Reduce32(c, t1);
SecP256K1FieldElement X3 = new SecP256K1FieldElement(T);
SecP256K1Field.Square(M, X3.x);
SecP256K1Field.Subtract(X3.x, S, X3.x);
SecP256K1Field.Subtract(X3.x, S, X3.x);
SecP256K1FieldElement Y3 = new SecP256K1FieldElement(S);
SecP256K1Field.Subtract(S, X3.x, Y3.x);
SecP256K1Field.Multiply(Y3.x, M, Y3.x);
SecP256K1Field.Subtract(Y3.x, t1, Y3.x);
SecP256K1FieldElement Z3 = new SecP256K1FieldElement(M);
SecP256K1Field.Twice(Y1.x, Z3.x);
if (!Z1.IsOne)
{
SecP256K1Field.Multiply(Z3.x, Z1.x, Z3.x);
}
return(new SecP256K1Point(curve, X3, Y3, new ECFieldElement[] { Z3 }, IsCompressed));
}