public ECC521Point Multiply(BigInteger b)
{
if (b.Sign == -1)
{
throw new FormatException("The multiplicator cannot be negative");
}
if (b >= OrderN)
{
b %= OrderN;
}
BigInteger exp = (b * 3) ^ b;
ECC521Point result = ECC521Point.Infinity;
ECC521Point affine = this.Normalize();
ECC521Point negative = affine.Negate();
int high = exp <= 0 ? 0 : (int)Math.Floor(BigInteger.Log(exp, 2));
BigInteger bit = BigInteger.One << high;
for (int i = high; --i >= 0; bit >>= 1)
{
result = result.Twice();
if (!(exp & bit).IsZero)
{
result = result.Add(!(b & bit).IsZero ? negative : affine);
}
}
result = result.Normalize();
return(result);
}
public ECC521Point Negate()
{
ECC521Point r = (ECC521Point)Clone();
r._y = P - r._y;
return(r);
}
public ECC521Point Normalize()
{
if (this.Z == BigInteger.One || this.IsInfinity)
{
return(this);
}
var inverse = this.Z.ModInverse(P);
var squareInverse = modP(inverse.Square());
var power3Inverse = modP(inverse * squareInverse);
var x = modP(this.X * squareInverse);
var y = modP(this.Y * power3Inverse);
remodP(ref x);
remodP(ref y);
var result = new ECC521Point(x, y, this.IsInfinity);
return(result);
}
//if (Y == 0)
// return POINT_AT_INFINITY
//S = 4*X*Y^2
//M = 3*X^2 + a*Z^4 = 3(X - Z^2) * (X + Z^2)
//X' = M^2 - 2*S
//Y' = M*(S - X') - 8*Y^4
//Z' = 2*Y*Z
//return (X', Y', Z')
public ECC521Point Twice()
{ // (4M + 4S)
if (this.Y.IsZero || this.IsInfinity)
{
return(ECC521Point.Infinity);
}
var y2 = modP(this.Y.Square());
var z2 = modP(this.Z.Square());
var S = modP(this.X * y2 << 2);
var M = (this.X - z2) * (this.X + z2);
M = modP(M + (M << 1));
var x = modP(M.Square() - (S << 1));
var y = modP(M * (S - x) - (y2.Square() << 3));
var z = modP(this.Z * (this.Y << 1));
var result = new ECC521Point(x, y, z, false);
return(result);
}
//U1 = X1*Z2^2
//U2 = X2*Z1^2
//S1 = Y1*Z2^3
//S2 = Y2*Z1^3
//if (U1 == U2)
// if (S1 != S2)
// return POINT_AT_INFINITY
// else
// return POINT_DOUBLE(X1, Y1, Z1)
//H = U2 - U1
//R = S2 - S1
//X3 = R^2 - H^3 - 2*U1*H^2
//Y3 = R*(U1*H^2 - X3) - S1*H^3
//Z3 = H*Z1*Z2
//return (X3, Y3, Z3)
public ECC521Point Add(ECC521Point q)
{ // (8M + 3S)
if (this.IsInfinity)
{
return(q);
}
if (q.IsInfinity)
{
return(this);
}
var z2 = modP(this.Z.Square());
var z3 = modP(z2 * this.Z);
var U2 = modP(q.X * z2);
var S2 = modP(q.Y * z3);
var H = U2 - this.X;
var R = S2 - this.Y;
if (H.IsZero)
{
if (!R.IsZero)
{
return(ECC521Point.Infinity);
}
return(this.Twice());
}
var H2 = modP(H.Square());
var H3 = modP(H2 * H);
var U1 = modP(this.X * H2);
var X3 = modP(R.Square() - (H3 + (U1 << 1)));
var Y3 = modP(R * (U1 - X3) - this.Y * H3);
var Z3 = modP(H * this.Z);
var result = new ECC521Point(X3, Y3, Z3, false);
return(result);
}
public ECC521Point Subtract(ECC521Point b)
{
return(Add(b.Negate()));
}
public override ECPoint ECMultiplication(BigInteger factor)
{
ECC521Point point = ECC521Point.G.Multiply(factor);
return(new ECPoint(point.X, point.Y, point.IsInfinity));
}