/**
* Solves a quadratic equation <code>z<sup>2</sup> + z = beta</code>(X9.62 D.1.6) The other
* solution is <code>z + 1</code>.
*
* @param beta The value to solve the quadratic equation for.
* @return the solution for <code>z<sup>2</sup> + z = beta</code> or <code>null</code> if no
* solution exists.
*/
private ECFieldElement solveQuadraticEquation(ECFieldElement beta) {
if (beta.isZero()) {
return beta;
}
ECFieldElement zeroElement = fromBigInteger(ECConstants.ZERO);
ECFieldElement z = null;
ECFieldElement gamma = null;
Random rand = new Random();
do {
ECFieldElement t = fromBigInteger(new BigInteger(m, rand));
z = zeroElement;
ECFieldElement w = beta;
for (int i = 1; i <= m - 1; i++) {
ECFieldElement w2 = w.square();
z = z.square().add(w2.multiply(t));
w = w2.add(beta);
}
if (!w.isZero()) {
return null;
}
gamma = z.square().add(z);
} while (gamma.isZero());
return z;
}
public ECPoint twicePlus(ECPoint b) {
if (this == b) {
return threeTimes();
}
if (this.isInfinity()) {
return b;
}
if (b.isInfinity()) {
return twice();
}
ECFieldElement Y1 = this.y;
if (Y1.isZero()) {
return b;
}
return twice().add(b);
}
public ECPoint createPoint(BigInteger x, BigInteger y, boolean withCompression) {
ECFieldElement X = fromBigInteger(x), Y = fromBigInteger(y);
switch (this.getCoordinateSystem()) {
case COORD_LAMBDA_AFFINE:
case COORD_LAMBDA_PROJECTIVE:
{
if (!X.isZero()) {
// Y becomes Lambda (X + Y/X) here
Y = Y.divide(X).add(X);
}
break;
}
default:
{
break;
}
}
return createRawPoint(X, Y, withCompression);
}
/**
* Decompresses a compressed point P = (xp, yp) (X9.62 s 4.2.2).
*
* @param yTilde ~yp, an indication bit for the decompression of yp.
* @param X1 The field element xp.
* @return the decompressed point.
*/
protected ECPoint decompressPoint(int yTilde, BigInteger X1) {
ECFieldElement xp = fromBigInteger(X1);
ECFieldElement yp = null;
if (xp.isZero()) {
yp = (ECFieldElement.F2m) b;
for (int i = 0; i < m - 1; i++) {
yp = yp.square();
}
} else {
ECFieldElement beta = xp.add(a).add(b.multiply(xp.square().invert()));
ECFieldElement z = solveQuadraticEquation(beta);
if (z == null) {
throw new IllegalArgumentException("Invalid point compression");
}
if (z.testBitZero() != (yTilde == 1)) {
z = z.addOne();
}
yp = xp.multiply(z);
switch (this.getCoordinateSystem()) {
case COORD_LAMBDA_AFFINE:
case COORD_LAMBDA_PROJECTIVE:
{
yp = yp.divide(xp).add(xp);
break;
}
default:
{
break;
}
}
}
return new ECPoint.F2m(this, xp, yp, true);
}