// D.1.4 91
/**
* 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 2^158 - 2^29
*
* Breaking up the exponent's binary representation into "repunits", we get:
* { 129 1s } { 29 0s }
*
* Therefore we need an addition chain containing 129 (the length of the repunit) We use:
* 1, 2, 4, 8, 16, 32, 64, 128, [129]
*/
uint[] x1 = this.x;
if (Nat160.IsZero(x1) || Nat160.IsOne(x1))
{
return(this);
}
uint[] x2 = Nat160.Create();
SecP160R1Field.Square(x1, x2);
SecP160R1Field.Multiply(x2, x1, x2);
uint[] x4 = Nat160.Create();
SecP160R1Field.SquareN(x2, 2, x4);
SecP160R1Field.Multiply(x4, x2, x4);
uint[] x8 = x2;
SecP160R1Field.SquareN(x4, 4, x8);
SecP160R1Field.Multiply(x8, x4, x8);
uint[] x16 = x4;
SecP160R1Field.SquareN(x8, 8, x16);
SecP160R1Field.Multiply(x16, x8, x16);
uint[] x32 = x8;
SecP160R1Field.SquareN(x16, 16, x32);
SecP160R1Field.Multiply(x32, x16, x32);
uint[] x64 = x16;
SecP160R1Field.SquareN(x32, 32, x64);
SecP160R1Field.Multiply(x64, x32, x64);
uint[] x128 = x32;
SecP160R1Field.SquareN(x64, 64, x128);
SecP160R1Field.Multiply(x128, x64, x128);
uint[] x129 = x64;
SecP160R1Field.Square(x128, x129);
SecP160R1Field.Multiply(x129, x1, x129);
uint[] t1 = x129;
SecP160R1Field.SquareN(t1, 29, t1);
uint[] t2 = x128;
SecP160R1Field.Square(t1, t2);
return(Nat160.Eq(x1, t2) ? new SecP160R1FieldElement(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;
SecP160R1FieldElement X1 = (SecP160R1FieldElement)this.RawXCoord, Y1 = (SecP160R1FieldElement)this.RawYCoord;
SecP160R1FieldElement X2 = (SecP160R1FieldElement)b.RawXCoord, Y2 = (SecP160R1FieldElement)b.RawYCoord;
SecP160R1FieldElement Z1 = (SecP160R1FieldElement)this.RawZCoords[0];
SecP160R1FieldElement Z2 = (SecP160R1FieldElement)b.RawZCoords[0];
uint c;
uint[] tt1 = Nat160.CreateExt();
uint[] t2 = Nat160.Create();
uint[] t3 = Nat160.Create();
uint[] t4 = Nat160.Create();
bool Z1IsOne = Z1.IsOne;
uint[] U2, S2;
if (Z1IsOne)
{
U2 = X2.x;
S2 = Y2.x;
}
else
{
S2 = t3;
SecP160R1Field.Square(Z1.x, S2);
U2 = t2;
SecP160R1Field.Multiply(S2, X2.x, U2);
SecP160R1Field.Multiply(S2, Z1.x, S2);
SecP160R1Field.Multiply(S2, Y2.x, S2);
}
bool Z2IsOne = Z2.IsOne;
uint[] U1, S1;
if (Z2IsOne)
{
U1 = X1.x;
S1 = Y1.x;
}
else
{
S1 = t4;
SecP160R1Field.Square(Z2.x, S1);
U1 = tt1;
SecP160R1Field.Multiply(S1, X1.x, U1);
SecP160R1Field.Multiply(S1, Z2.x, S1);
SecP160R1Field.Multiply(S1, Y1.x, S1);
}
uint[] H = Nat160.Create();
SecP160R1Field.Subtract(U1, U2, H);
uint[] R = t2;
SecP160R1Field.Subtract(S1, S2, R);
// Check if b == this or b == -this
if (Nat160.IsZero(H))
{
if (Nat160.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;
SecP160R1Field.Square(H, HSquared);
uint[] G = Nat160.Create();
SecP160R1Field.Multiply(HSquared, H, G);
uint[] V = t3;
SecP160R1Field.Multiply(HSquared, U1, V);
SecP160R1Field.Negate(G, G);
Nat160.Mul(S1, G, tt1);
c = Nat160.AddBothTo(V, V, G);
SecP160R1Field.Reduce32(c, G);
SecP160R1FieldElement X3 = new SecP160R1FieldElement(t4);
SecP160R1Field.Square(R, X3.x);
SecP160R1Field.Subtract(X3.x, G, X3.x);
SecP160R1FieldElement Y3 = new SecP160R1FieldElement(G);
SecP160R1Field.Subtract(V, X3.x, Y3.x);
SecP160R1Field.MultiplyAddToExt(Y3.x, R, tt1);
SecP160R1Field.Reduce(tt1, Y3.x);
SecP160R1FieldElement Z3 = new SecP160R1FieldElement(H);
if (!Z1IsOne)
{
SecP160R1Field.Multiply(Z3.x, Z1.x, Z3.x);
}
if (!Z2IsOne)
{
SecP160R1Field.Multiply(Z3.x, Z2.x, Z3.x);
}
ECFieldElement[] zs = new ECFieldElement[] { Z3 };
return(new SecP160R1Point(curve, X3, Y3, zs, IsCompressed));
}
public override ECFieldElement Square()
{
uint[] z = Nat160.Create();
SecP160R1Field.Square(x, z);
return(new SecP160R1FieldElement(z));
}
public override ECPoint Twice()
{
if (this.IsInfinity)
{
return(this);
}
ECCurve curve = this.Curve;
SecP160R1FieldElement Y1 = (SecP160R1FieldElement)this.RawYCoord;
if (Y1.IsZero)
{
return(curve.Infinity);
}
SecP160R1FieldElement X1 = (SecP160R1FieldElement)this.RawXCoord, Z1 = (SecP160R1FieldElement)this.RawZCoords[0];
uint c;
uint[] t1 = Nat160.Create();
uint[] t2 = Nat160.Create();
uint[] Y1Squared = Nat160.Create();
SecP160R1Field.Square(Y1.x, Y1Squared);
uint[] T = Nat160.Create();
SecP160R1Field.Square(Y1Squared, T);
bool Z1IsOne = Z1.IsOne;
uint[] Z1Squared = Z1.x;
if (!Z1IsOne)
{
Z1Squared = t2;
SecP160R1Field.Square(Z1.x, Z1Squared);
}
SecP160R1Field.Subtract(X1.x, Z1Squared, t1);
uint[] M = t2;
SecP160R1Field.Add(X1.x, Z1Squared, M);
SecP160R1Field.Multiply(M, t1, M);
c = Nat160.AddBothTo(M, M, M);
SecP160R1Field.Reduce32(c, M);
uint[] S = Y1Squared;
SecP160R1Field.Multiply(Y1Squared, X1.x, S);
c = Nat.ShiftUpBits(5, S, 2, 0);
SecP160R1Field.Reduce32(c, S);
c = Nat.ShiftUpBits(5, T, 3, 0, t1);
SecP160R1Field.Reduce32(c, t1);
SecP160R1FieldElement X3 = new SecP160R1FieldElement(T);
SecP160R1Field.Square(M, X3.x);
SecP160R1Field.Subtract(X3.x, S, X3.x);
SecP160R1Field.Subtract(X3.x, S, X3.x);
SecP160R1FieldElement Y3 = new SecP160R1FieldElement(S);
SecP160R1Field.Subtract(S, X3.x, Y3.x);
SecP160R1Field.Multiply(Y3.x, M, Y3.x);
SecP160R1Field.Subtract(Y3.x, t1, Y3.x);
SecP160R1FieldElement Z3 = new SecP160R1FieldElement(M);
SecP160R1Field.Twice(Y1.x, Z3.x);
if (!Z1IsOne)
{
SecP160R1Field.Multiply(Z3.x, Z1.x, Z3.x);
}
return(new SecP160R1Point(curve, X3, Y3, new ECFieldElement[] { Z3 }, IsCompressed));
}