/**
* 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 2^221 - 2^29 - 2^9 - 2^8 - 2^6 - 2^4 - 2^1 (i.e. m + 1)
*
* Breaking up the exponent's binary representation into "repunits", we get:
* { 191 1s } { 1 0s } { 19 1s } { 2 0s } { 1 1s } { 1 0s} { 1 1s } { 1 0s} { 3 1s } { 1 0s}
*
* Therefore we need an addition chain containing 1, 3, 19, 191 (the lengths of the repunits)
* We use: [1], 2, [3], 4, 8, 11, [19], 23, 42, 84, 107, [191]
*/
uint[] x1 = this.x;
if (Nat224.IsZero(x1) || Nat224.IsOne(x1))
{
return(this);
}
uint[] x2 = Nat224.Create();
SecP224K1Field.Square(x1, x2);
SecP224K1Field.Multiply(x2, x1, x2);
uint[] x3 = x2;
SecP224K1Field.Square(x2, x3);
SecP224K1Field.Multiply(x3, x1, x3);
uint[] x4 = Nat224.Create();
SecP224K1Field.Square(x3, x4);
SecP224K1Field.Multiply(x4, x1, x4);
uint[] x8 = Nat224.Create();
SecP224K1Field.SquareN(x4, 4, x8);
SecP224K1Field.Multiply(x8, x4, x8);
uint[] x11 = Nat224.Create();
SecP224K1Field.SquareN(x8, 3, x11);
SecP224K1Field.Multiply(x11, x3, x11);
uint[] x19 = x11;
SecP224K1Field.SquareN(x11, 8, x19);
SecP224K1Field.Multiply(x19, x8, x19);
uint[] x23 = x8;
SecP224K1Field.SquareN(x19, 4, x23);
SecP224K1Field.Multiply(x23, x4, x23);
uint[] x42 = x4;
SecP224K1Field.SquareN(x23, 19, x42);
SecP224K1Field.Multiply(x42, x19, x42);
uint[] x84 = Nat224.Create();
SecP224K1Field.SquareN(x42, 42, x84);
SecP224K1Field.Multiply(x84, x42, x84);
uint[] x107 = x42;
SecP224K1Field.SquareN(x84, 23, x107);
SecP224K1Field.Multiply(x107, x23, x107);
uint[] x191 = x23;
SecP224K1Field.SquareN(x107, 84, x191);
SecP224K1Field.Multiply(x191, x84, x191);
uint[] t1 = x191;
SecP224K1Field.SquareN(t1, 20, t1);
SecP224K1Field.Multiply(t1, x19, t1);
SecP224K1Field.SquareN(t1, 3, t1);
SecP224K1Field.Multiply(t1, x1, t1);
SecP224K1Field.SquareN(t1, 2, t1);
SecP224K1Field.Multiply(t1, x1, t1);
SecP224K1Field.SquareN(t1, 4, t1);
SecP224K1Field.Multiply(t1, x3, t1);
SecP224K1Field.Square(t1, t1);
uint[] t2 = x84;
SecP224K1Field.Square(t1, t2);
if (Nat224.Eq(x1, t2))
{
return(new SecP224K1FieldElement(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
*/
SecP224K1Field.Multiply(t1, PRECOMP_POW2, t1);
SecP224K1Field.Square(t1, t2);
if (Nat224.Eq(x1, t2))
{
return(new SecP224K1FieldElement(t1));
}
return(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;
SecP224K1FieldElement X1 = (SecP224K1FieldElement)this.RawXCoord, Y1 = (SecP224K1FieldElement)this.RawYCoord;
SecP224K1FieldElement X2 = (SecP224K1FieldElement)b.RawXCoord, Y2 = (SecP224K1FieldElement)b.RawYCoord;
SecP224K1FieldElement Z1 = (SecP224K1FieldElement)this.RawZCoords[0];
SecP224K1FieldElement Z2 = (SecP224K1FieldElement)b.RawZCoords[0];
uint c;
uint[] tt1 = Nat224.CreateExt();
uint[] t2 = Nat224.Create();
uint[] t3 = Nat224.Create();
uint[] t4 = Nat224.Create();
bool Z1IsOne = Z1.IsOne;
uint[] U2, S2;
if (Z1IsOne)
{
U2 = X2.x;
S2 = Y2.x;
}
else
{
S2 = t3;
SecP224K1Field.Square(Z1.x, S2);
U2 = t2;
SecP224K1Field.Multiply(S2, X2.x, U2);
SecP224K1Field.Multiply(S2, Z1.x, S2);
SecP224K1Field.Multiply(S2, Y2.x, S2);
}
bool Z2IsOne = Z2.IsOne;
uint[] U1, S1;
if (Z2IsOne)
{
U1 = X1.x;
S1 = Y1.x;
}
else
{
S1 = t4;
SecP224K1Field.Square(Z2.x, S1);
U1 = tt1;
SecP224K1Field.Multiply(S1, X1.x, U1);
SecP224K1Field.Multiply(S1, Z2.x, S1);
SecP224K1Field.Multiply(S1, Y1.x, S1);
}
uint[] H = Nat224.Create();
SecP224K1Field.Subtract(U1, U2, H);
uint[] R = t2;
SecP224K1Field.Subtract(S1, S2, R);
// Check if b == this or b == -this
if (Nat224.IsZero(H))
{
if (Nat224.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;
SecP224K1Field.Square(H, HSquared);
uint[] G = Nat224.Create();
SecP224K1Field.Multiply(HSquared, H, G);
uint[] V = t3;
SecP224K1Field.Multiply(HSquared, U1, V);
SecP224K1Field.Negate(G, G);
Nat224.Mul(S1, G, tt1);
c = Nat224.AddBothTo(V, V, G);
SecP224K1Field.Reduce32(c, G);
SecP224K1FieldElement X3 = new SecP224K1FieldElement(t4);
SecP224K1Field.Square(R, X3.x);
SecP224K1Field.Subtract(X3.x, G, X3.x);
SecP224K1FieldElement Y3 = new SecP224K1FieldElement(G);
SecP224K1Field.Subtract(V, X3.x, Y3.x);
SecP224K1Field.MultiplyAddToExt(Y3.x, R, tt1);
SecP224K1Field.Reduce(tt1, Y3.x);
SecP224K1FieldElement Z3 = new SecP224K1FieldElement(H);
if (!Z1IsOne)
{
SecP224K1Field.Multiply(Z3.x, Z1.x, Z3.x);
}
if (!Z2IsOne)
{
SecP224K1Field.Multiply(Z3.x, Z2.x, Z3.x);
}
ECFieldElement[] zs = new ECFieldElement[] { Z3 };
return(new SecP224K1Point(curve, X3, Y3, zs, IsCompressed));
}
public override ECFieldElement Square()
{
uint[] z = Nat224.Create();
SecP224K1Field.Square(x, z);
return(new SecP224K1FieldElement(z));
}
public override ECPoint Twice()
{
if (this.IsInfinity)
{
return(this);
}
ECCurve curve = this.Curve;
SecP224K1FieldElement Y1 = (SecP224K1FieldElement)this.RawYCoord;
if (Y1.IsZero)
{
return(curve.Infinity);
}
SecP224K1FieldElement X1 = (SecP224K1FieldElement)this.RawXCoord, Z1 = (SecP224K1FieldElement)this.RawZCoords[0];
uint c;
uint[] Y1Squared = Nat224.Create();
SecP224K1Field.Square(Y1.x, Y1Squared);
uint[] T = Nat224.Create();
SecP224K1Field.Square(Y1Squared, T);
uint[] M = Nat224.Create();
SecP224K1Field.Square(X1.x, M);
c = Nat224.AddBothTo(M, M, M);
SecP224K1Field.Reduce32(c, M);
uint[] S = Y1Squared;
SecP224K1Field.Multiply(Y1Squared, X1.x, S);
c = Nat.ShiftUpBits(7, S, 2, 0);
SecP224K1Field.Reduce32(c, S);
uint[] t1 = Nat224.Create();
c = Nat.ShiftUpBits(7, T, 3, 0, t1);
SecP224K1Field.Reduce32(c, t1);
SecP224K1FieldElement X3 = new SecP224K1FieldElement(T);
SecP224K1Field.Square(M, X3.x);
SecP224K1Field.Subtract(X3.x, S, X3.x);
SecP224K1Field.Subtract(X3.x, S, X3.x);
SecP224K1FieldElement Y3 = new SecP224K1FieldElement(S);
SecP224K1Field.Subtract(S, X3.x, Y3.x);
SecP224K1Field.Multiply(Y3.x, M, Y3.x);
SecP224K1Field.Subtract(Y3.x, t1, Y3.x);
SecP224K1FieldElement Z3 = new SecP224K1FieldElement(M);
SecP224K1Field.Twice(Y1.x, Z3.x);
if (!Z1.IsOne)
{
SecP224K1Field.Multiply(Z3.x, Z1.x, Z3.x);
}
return(new SecP224K1Point(curve, X3, Y3, new ECFieldElement[] { Z3 }, IsCompressed));
}