/// <summary>
/// Point dooubling over the curve
/// </summary>
private bool PointDouble(
BigInteger.ModulusRing ring,
ECPoint p1,
out ECPoint p3) {
if (p1 is ECPointAtInfinity) {
p3 = p1;
return true;
}
if (p1.Y == 0) {
p3 = new ECPointAtInfinity();
return true;
}
// x3 = {(3 * x1^2 + a)/(2 * y1)}^2 - (2 * x1)
// y3 = {(3 * x1^2 + a)/(2 * y1)} * (x1 - x3) - y1
try {
BigInteger x1 = p1.X;
BigInteger y1 = p1.Y;
BigInteger x1_2 = ring.Multiply(x1, x1);
BigInteger lambda = ring.Multiply(x1_2 + x1_2 + x1_2 + a, (y1 + y1).ModInverse(p));
BigInteger x3 = ring.Difference(ring.Multiply(lambda, lambda), x1 + x1);
BigInteger y3 = ring.Difference(ring.Multiply(lambda, ring.Difference(x1, x3)), y1);
p3 = new ECPoint(x3, y3);
return true;
}
catch (Exception) {
p3 = null;
return false;
}
}
/// <summary>
/// Point addition over the curve
/// </summary>
private bool PointAdd(
BigInteger.ModulusRing ring,
ECPoint p1,
ECPoint p2,
out ECPoint p3) {
if (p1 is ECPointAtInfinity) {
p3 = p2;
return true;
}
if (p2 is ECPointAtInfinity) {
p3 = p1;
return true;
}
if (p1.X == p2.X) {
if (p1.Y == p2.Y) {
return PointDouble(ring, p1, out p3);
}
else {
p3 = new ECPointAtInfinity();
return true;
}
}
// x3 = {(y2 - y1)/(x2 - x1)}^2 - (x1 + x2)
// y3 = {(y2 - y1)/(x2 - x1)} * (x1 - x3) - y1
try {
BigInteger x1 = p1.X;
BigInteger y1 = p1.Y;
BigInteger x2 = p2.X;
BigInteger y2 = p2.Y;
BigInteger lambda = ring.Multiply(ring.Difference(y2, y1), ring.Difference(x2, x1).ModInverse(p));
BigInteger x3 = ring.Difference(ring.Multiply(lambda, lambda), x1 + x2);
BigInteger y3 = ring.Difference(ring.Multiply(lambda, ring.Difference(x1, x3)), y1);
p3 = new ECPoint(x3, y3);
return true;
}
catch (Exception) {
p3 = null;
return false;
}
}