/**
* 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 qradratic equation for.
* @return the solution for <code>z<sup>2</sup> + z = beta</code> or <code>null</code> if no
* solution exists.
*/
private ECFieldElement solveQuadradicEquation(ECFieldElement beta) {
ECFieldElement zeroElement =
new ECFieldElement.F2m(this.m, this.k1, this.k2, this.k3, ECConstants.ZERO);
if (beta.toBigInteger().equals(ECConstants.ZERO)) {
return zeroElement;
}
ECFieldElement z = null;
ECFieldElement gamma = zeroElement;
Random rand = new Random();
do {
ECFieldElement t =
new ECFieldElement.F2m(this.m, this.k1, this.k2, this.k3, 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.toBigInteger().equals(ECConstants.ZERO)) {
return null;
}
gamma = z.square().add(z);
} while (gamma.toBigInteger().equals(ECConstants.ZERO));
return z;
}
public ECFieldElement multiplyPlusProduct(
ECFieldElement b, ECFieldElement x, ECFieldElement y) {
BigInteger ax = this.x, bx = b.toBigInteger(), xx = x.toBigInteger(), yx = y.toBigInteger();
BigInteger ab = ax.multiply(bx);
BigInteger xy = xx.multiply(yx);
return new Fp(q, r, modReduce(ab.add(xy)));
}
/**
* Decompresses a compressed point P = (xp, yp) (X9.62 s 4.2.2).
*
* @param xEnc The encoding of field element xp.
* @param ypBit ~yp, an indication bit for the decompression of yp.
* @return the decompressed point.
*/
private ECPoint decompressPoint(byte[] xEnc, int ypBit) {
ECFieldElement xp =
new ECFieldElement.F2m(this.m, this.k1, this.k2, this.k3, new BigInteger(1, xEnc));
ECFieldElement yp = null;
if (xp.toBigInteger().equals(ECConstants.ZERO)) {
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 = solveQuadradicEquation(beta);
if (z == null) {
throw new RuntimeException("Invalid point compression");
}
int zBit = 0;
if (z.toBigInteger().testBit(0)) {
zBit = 1;
}
if (zBit != ypBit) {
z = z.add(new ECFieldElement.F2m(this.m, this.k1, this.k2, this.k3, ECConstants.ONE));
}
yp = xp.multiply(z);
}
return new ECPoint.F2m(this, xp, yp);
}
/**
* Decode a point on this curve from its ASN.1 encoding. The different encodings are taken
* account of, including point compression for <code>F<sub>p</sub></code> (X9.62 s 4.2.1 pg 17).
*
* @return The decoded point.
*/
public ECPoint decodePoint(byte[] encoded) {
ECPoint p = null;
switch (encoded[0]) {
// infinity
case 0x00:
p = getInfinity();
break;
// compressed
case 0x02:
case 0x03:
int ytilde = encoded[0] & 1;
byte[] i = new byte[encoded.length - 1];
System.arraycopy(encoded, 1, i, 0, i.length);
ECFieldElement x = new ECFieldElement.Fp(this.q, new BigInteger(1, i));
ECFieldElement alpha = x.multiply(x.square().add(a)).add(b);
ECFieldElement beta = alpha.sqrt();
//
// if we can't find a sqrt we haven't got a point on the
// curve - run!
//
if (beta == null) {
throw new RuntimeException("Invalid point compression");
}
int bit0 = (beta.toBigInteger().testBit(0) ? 1 : 0);
if (bit0 == ytilde) {
p = new ECPoint.Fp(this, x, beta, true);
} else {
p =
new ECPoint.Fp(
this, x, new ECFieldElement.Fp(this.q, q.subtract(beta.toBigInteger())), true);
}
break;
// uncompressed
case 0x04:
// hybrid
case 0x06:
case 0x07:
byte[] xEnc = new byte[(encoded.length - 1) / 2];
byte[] yEnc = new byte[(encoded.length - 1) / 2];
System.arraycopy(encoded, 1, xEnc, 0, xEnc.length);
System.arraycopy(encoded, xEnc.length + 1, yEnc, 0, yEnc.length);
p =
new ECPoint.Fp(
this,
new ECFieldElement.Fp(this.q, new BigInteger(1, xEnc)),
new ECFieldElement.Fp(this.q, new BigInteger(1, yEnc)));
break;
default:
throw new RuntimeException(
"Invalid point encoding 0x" + Integer.toString(encoded[0], 16));
}
return p;
}
public ECFieldElement squarePlusProduct(ECFieldElement x, ECFieldElement y) {
BigInteger ax = this.x, xx = x.toBigInteger(), yx = y.toBigInteger();
BigInteger aa = ax.multiply(ax);
BigInteger xy = xx.multiply(yx);
return new Fp(q, r, modReduce(aa.add(xy)));
}
public ECFieldElement divide(ECFieldElement b) {
return new Fp(q, r, modMult(x, modInverse(b.toBigInteger())));
}
public ECFieldElement subtract(ECFieldElement b) {
return new Fp(q, r, modSubtract(x, b.toBigInteger()));
}
public ECFieldElement multiply(ECFieldElement b) {
return new Fp(q, r, modMult(x, b.toBigInteger()));
}
public ECFieldElement add(ECFieldElement b) {
return new Fp(q, r, modAdd(x, b.toBigInteger()));
}
public ECFieldElement divide(ECFieldElement b) {
return new Fp(q, x.multiply(b.toBigInteger().modInverse(q)).mod(q));
}
public ECFieldElement multiply(ECFieldElement b) {
return new Fp(q, x.multiply(b.toBigInteger()).mod(q));
}
public ECFieldElement subtract(ECFieldElement b) {
return new Fp(q, x.subtract(b.toBigInteger()).mod(q));
}
public ECFieldElement add(ECFieldElement b) {
return new Fp(q, x.add(b.toBigInteger()).mod(q));
}