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package org.bouncycastle.math.ec;
import java.math.BigInteger;
import java.security.SecureRandom;
import java.util.Hashtable;
import org.bouncycastle.crypto.CryptoServicesRegistrar;
/**
* base class for points on elliptic curves.
*/
public abstract class ECPoint
{
protected final static ECFieldElement[] EMPTY_ZS = new ECFieldElement[0];
protected static ECFieldElement[] getInitialZCoords(ECCurve curve)
{
// Cope with null curve, most commonly used by implicitlyCa
int coord = null == curve ? ECCurve.COORD_AFFINE : curve.getCoordinateSystem();
switch (coord)
{
case ECCurve.COORD_AFFINE:
case ECCurve.COORD_LAMBDA_AFFINE:
return EMPTY_ZS;
default:
break;
}
ECFieldElement one = curve.fromBigInteger(ECConstants.ONE);
switch (coord)
{
case ECCurve.COORD_HOMOGENEOUS:
case ECCurve.COORD_JACOBIAN:
case ECCurve.COORD_LAMBDA_PROJECTIVE:
return new ECFieldElement[]{ one };
case ECCurve.COORD_JACOBIAN_CHUDNOVSKY:
return new ECFieldElement[]{ one, one, one };
case ECCurve.COORD_JACOBIAN_MODIFIED:
return new ECFieldElement[]{ one, curve.getA() };
default:
throw new IllegalArgumentException("unknown coordinate system");
}
}
protected ECCurve curve;
protected ECFieldElement x;
protected ECFieldElement y;
protected ECFieldElement[] zs;
// Hashtable is (String -> PreCompInfo)
protected Hashtable preCompTable = null;
protected ECPoint(ECCurve curve, ECFieldElement x, ECFieldElement y)
{
this(curve, x, y, getInitialZCoords(curve));
}
protected ECPoint(ECCurve curve, ECFieldElement x, ECFieldElement y, ECFieldElement[] zs)
{
this.curve = curve;
this.x = x;
this.y = y;
this.zs = zs;
}
protected abstract boolean satisfiesCurveEquation();
protected boolean satisfiesOrder()
{
if (ECConstants.ONE.equals(curve.getCofactor()))
{
return true;
}
BigInteger n = curve.getOrder();
// TODO Require order to be available for all curves
return n == null || ECAlgorithms.referenceMultiply(this, n).isInfinity();
}
public final ECPoint getDetachedPoint()
{
return normalize().detach();
}
public ECCurve getCurve()
{
return curve;
}
protected abstract ECPoint detach();
protected int getCurveCoordinateSystem()
{
// Cope with null curve, most commonly used by implicitlyCa
return null == curve ? ECCurve.COORD_AFFINE : curve.getCoordinateSystem();
}
/**
* Returns the affine x-coordinate after checking that this point is normalized.
*
* @return The affine x-coordinate of this point
* @throws IllegalStateException if the point is not normalized
*/
public ECFieldElement getAffineXCoord()
{
checkNormalized();
return getXCoord();
}
/**
* Returns the affine y-coordinate after checking that this point is normalized
*
* @return The affine y-coordinate of this point
* @throws IllegalStateException if the point is not normalized
*/
public ECFieldElement getAffineYCoord()
{
checkNormalized();
return getYCoord();
}
/**
* Returns the x-coordinate.
*
* Caution: depending on the curve's coordinate system, this may not be the same value as in an
* affine coordinate system; use normalize() to get a point where the coordinates have their
* affine values, or use getAffineXCoord() if you expect the point to already have been
* normalized.
*
* @return the x-coordinate of this point
*/
public ECFieldElement getXCoord()
{
return x;
}
/**
* Returns the y-coordinate.
*
* Caution: depending on the curve's coordinate system, this may not be the same value as in an
* affine coordinate system; use normalize() to get a point where the coordinates have their
* affine values, or use getAffineYCoord() if you expect the point to already have been
* normalized.
*
* @return the y-coordinate of this point
*/
public ECFieldElement getYCoord()
{
return y;
}
public ECFieldElement getZCoord(int index)
{
return (index < 0 || index >= zs.length) ? null : zs[index];
}
public ECFieldElement[] getZCoords()
{
int zsLen = zs.length;
if (zsLen == 0)
{
return EMPTY_ZS;
}
ECFieldElement[] copy = new ECFieldElement[zsLen];
System.arraycopy(zs, 0, copy, 0, zsLen);
return copy;
}
public final ECFieldElement getRawXCoord()
{
return x;
}
public final ECFieldElement getRawYCoord()
{
return y;
}
protected final ECFieldElement[] getRawZCoords()
{
return zs;
}
protected void checkNormalized()
{
if (!isNormalized())
{
throw new IllegalStateException("point not in normal form");
}
}
public boolean isNormalized()
{
int coord = this.getCurveCoordinateSystem();
return coord == ECCurve.COORD_AFFINE
|| coord == ECCurve.COORD_LAMBDA_AFFINE
|| isInfinity()
|| zs[0].isOne();
}
/**
* Normalization ensures that any projective coordinate is 1, and therefore that the x, y
* coordinates reflect those of the equivalent point in an affine coordinate system.
*
* @return a new ECPoint instance representing the same point, but with normalized coordinates
*/
public ECPoint normalize()
{
if (this.isInfinity())
{
return this;
}
switch (this.getCurveCoordinateSystem())
{
case ECCurve.COORD_AFFINE:
case ECCurve.COORD_LAMBDA_AFFINE:
{
return this;
}
default:
{
ECFieldElement z = getZCoord(0);
if (z.isOne())
{
return this;
}
if (null == curve)
{
throw new IllegalStateException("Detached points must be in affine coordinates");
}
/*
* Use blinding to avoid the side-channel leak identified and analyzed in the paper
* "Yet another GCD based inversion side-channel affecting ECC implementations" by Nir
* Drucker and Shay Gueron.
*
* To blind the calculation of z^-1, choose a multiplicative (i.e. non-zero) field
* element 'b' uniformly at random, then calculate the result instead as (z * b)^-1 * b.
* Any side-channel in the implementation of 'inverse' now only leaks information about
* the value (z * b), and no longer reveals information about 'z' itself.
*/
SecureRandom r = CryptoServicesRegistrar.getSecureRandom();
ECFieldElement b = curve.randomFieldElementMult(r);
ECFieldElement zInv = z.multiply(b).invert().multiply(b);
return normalize(zInv);
}
}
}
ECPoint normalize(ECFieldElement zInv)
{
switch (this.getCurveCoordinateSystem())
{
case ECCurve.COORD_HOMOGENEOUS:
case ECCurve.COORD_LAMBDA_PROJECTIVE:
{
return createScaledPoint(zInv, zInv);
}
case ECCurve.COORD_JACOBIAN:
case ECCurve.COORD_JACOBIAN_CHUDNOVSKY:
case ECCurve.COORD_JACOBIAN_MODIFIED:
{
ECFieldElement zInv2 = zInv.square(), zInv3 = zInv2.multiply(zInv);
return createScaledPoint(zInv2, zInv3);
}
default:
{
throw new IllegalStateException("not a projective coordinate system");
}
}
}
protected ECPoint createScaledPoint(ECFieldElement sx, ECFieldElement sy)
{
return this.getCurve().createRawPoint(getRawXCoord().multiply(sx), getRawYCoord().multiply(sy));
}
public boolean isInfinity()
{
return x == null || y == null || (zs.length > 0 && zs[0].isZero());
}
public boolean isValid()
{
return implIsValid(false, true);
}
boolean isValidPartial()
{
return implIsValid(false, false);
}
boolean implIsValid(final boolean decompressed, final boolean checkOrder)
{
if (isInfinity())
{
return true;
}
ValidityPrecompInfo validity = (ValidityPrecompInfo)getCurve().precompute(this, ValidityPrecompInfo.PRECOMP_NAME, new PreCompCallback()
{
public PreCompInfo precompute(PreCompInfo existing)
{
ValidityPrecompInfo info = (existing instanceof ValidityPrecompInfo) ? (ValidityPrecompInfo)existing : null;
if (info == null)
{
info = new ValidityPrecompInfo();
}
if (info.hasFailed())
{
return info;
}
if (!info.hasCurveEquationPassed())
{
if (!decompressed && !satisfiesCurveEquation())
{
info.reportFailed();
return info;
}
info.reportCurveEquationPassed();
}
if (checkOrder && !info.hasOrderPassed())
{
if (!satisfiesOrder())
{
info.reportFailed();
return info;
}
info.reportOrderPassed();
}
return info;
}
});
return !validity.hasFailed();
}
public ECPoint scaleX(ECFieldElement scale)
{
return isInfinity()
? this
: getCurve().createRawPoint(getRawXCoord().multiply(scale), getRawYCoord(), getRawZCoords());
}
public ECPoint scaleXNegateY(ECFieldElement scale)
{
return isInfinity()
? this
: getCurve().createRawPoint(getRawXCoord().multiply(scale), getRawYCoord().negate(), getRawZCoords());
}
public ECPoint scaleY(ECFieldElement scale)
{
return isInfinity()
? this
: getCurve().createRawPoint(getRawXCoord(), getRawYCoord().multiply(scale), getRawZCoords());
}
public ECPoint scaleYNegateX(ECFieldElement scale)
{
return isInfinity()
? this
: getCurve().createRawPoint(getRawXCoord().negate(), getRawYCoord().multiply(scale), getRawZCoords());
}
public boolean equals(ECPoint other)
{
if (null == other)
{
return false;
}
ECCurve c1 = this.getCurve(), c2 = other.getCurve();
boolean n1 = (null == c1), n2 = (null == c2);
boolean i1 = isInfinity(), i2 = other.isInfinity();
if (i1 || i2)
{
return (i1 && i2) && (n1 || n2 || c1.equals(c2));
}
ECPoint p1 = this, p2 = other;
if (n1 && n2)
{
// Points with null curve are in affine form, so already normalized
}
else if (n1)
{
p2 = p2.normalize();
}
else if (n2)
{
p1 = p1.normalize();
}
else if (!c1.equals(c2))
{
return false;
}
else
{
// TODO Consider just requiring already normalized, to avoid silent performance degradation
ECPoint[] points = new ECPoint[]{ this, c1.importPoint(p2) };
// TODO This is a little strong, really only requires coZNormalizeAll to get Zs equal
c1.normalizeAll(points);
p1 = points[0];
p2 = points[1];
}
return p1.getXCoord().equals(p2.getXCoord()) && p1.getYCoord().equals(p2.getYCoord());
}
public boolean equals(Object other)
{
if (other == this)
{
return true;
}
if (!(other instanceof ECPoint))
{
return false;
}
return equals((ECPoint)other);
}
public int hashCode()
{
ECCurve c = this.getCurve();
int hc = (null == c) ? 0 : ~c.hashCode();
if (!this.isInfinity())
{
// TODO Consider just requiring already normalized, to avoid silent performance degradation
ECPoint p = normalize();
hc ^= p.getXCoord().hashCode() * 17;
hc ^= p.getYCoord().hashCode() * 257;
}
return hc;
}
public String toString()
{
if (this.isInfinity())
{
return "INF";
}
StringBuffer sb = new StringBuffer();
sb.append('(');
sb.append(getRawXCoord());
sb.append(',');
sb.append(getRawYCoord());
for (int i = 0; i < zs.length; ++i)
{
sb.append(',');
sb.append(zs[i]);
}
sb.append(')');
return sb.toString();
}
/**
* Get an encoding of the point value, optionally in compressed format.
*
* @param compressed whether to generate a compressed point encoding.
* @return the point encoding
*/
public byte[] getEncoded(boolean compressed)
{
if (this.isInfinity())
{
return new byte[1];
}
ECPoint normed = normalize();
byte[] X = normed.getXCoord().getEncoded();
if (compressed)
{
byte[] PO = new byte[X.length + 1];
PO[0] = (byte)(normed.getCompressionYTilde() ? 0x03 : 0x02);
System.arraycopy(X, 0, PO, 1, X.length);
return PO;
}
byte[] Y = normed.getYCoord().getEncoded();
byte[] PO = new byte[X.length + Y.length + 1];
PO[0] = 0x04;
System.arraycopy(X, 0, PO, 1, X.length);
System.arraycopy(Y, 0, PO, X.length + 1, Y.length);
return PO;
}
protected abstract boolean getCompressionYTilde();
public abstract ECPoint add(ECPoint b);
public abstract ECPoint negate();
public abstract ECPoint subtract(ECPoint b);
public ECPoint timesPow2(int e)
{
if (e < 0)
{
throw new IllegalArgumentException("'e' cannot be negative");
}
ECPoint p = this;
while (--e >= 0)
{
p = p.twice();
}
return p;
}
public abstract ECPoint twice();
public ECPoint twicePlus(ECPoint b)
{
return twice().add(b);
}
public ECPoint threeTimes()
{
return twicePlus(this);
}
/**
* Multiplies this ECPoint by the given number.
* @param k The multiplicator.
* @return k * this.
*/
public ECPoint multiply(BigInteger k)
{
return this.getCurve().getMultiplier().multiply(this, k);
}
public static abstract class AbstractFp extends ECPoint
{
protected AbstractFp(ECCurve curve, ECFieldElement x, ECFieldElement y)
{
super(curve, x, y);
}
protected AbstractFp(ECCurve curve, ECFieldElement x, ECFieldElement y, ECFieldElement[] zs)
{
super(curve, x, y, zs);
}
protected boolean getCompressionYTilde()
{
return this.getAffineYCoord().testBitZero();
}
protected boolean satisfiesCurveEquation()
{
ECFieldElement X = this.x, Y = this.y, A = curve.getA(), B = curve.getB();
ECFieldElement lhs = Y.square();
switch (this.getCurveCoordinateSystem())
{
case ECCurve.COORD_AFFINE:
break;
case ECCurve.COORD_HOMOGENEOUS:
{
ECFieldElement Z = this.zs[0];
if (!Z.isOne())
{
ECFieldElement Z2 = Z.square(), Z3 = Z.multiply(Z2);
lhs = lhs.multiply(Z);
A = A.multiply(Z2);
B = B.multiply(Z3);
}
break;
}
case ECCurve.COORD_JACOBIAN:
case ECCurve.COORD_JACOBIAN_CHUDNOVSKY:
case ECCurve.COORD_JACOBIAN_MODIFIED:
{
ECFieldElement Z = this.zs[0];
if (!Z.isOne())
{
ECFieldElement Z2 = Z.square(), Z4 = Z2.square(), Z6 = Z2.multiply(Z4);
A = A.multiply(Z4);
B = B.multiply(Z6);
}
break;
}
default:
throw new IllegalStateException("unsupported coordinate system");
}
ECFieldElement rhs = X.square().add(A).multiply(X).add(B);
return lhs.equals(rhs);
}
public ECPoint subtract(ECPoint b)
{
if (b.isInfinity())
{
return this;
}
// Add -b
return this.add(b.negate());
}
}
/**
* Elliptic curve points over Fp
*/
public static class Fp extends AbstractFp
{
Fp(ECCurve curve, ECFieldElement x, ECFieldElement y)
{
super(curve, x, y);
}
Fp(ECCurve curve, ECFieldElement x, ECFieldElement y, ECFieldElement[] zs)
{
super(curve, x, y, zs);
}
protected ECPoint detach()
{
return new ECPoint.Fp(null, this.getAffineXCoord(), this.getAffineYCoord());
}
public ECFieldElement getZCoord(int index)
{
if (index == 1 && ECCurve.COORD_JACOBIAN_MODIFIED == this.getCurveCoordinateSystem())
{
return getJacobianModifiedW();
}
return super.getZCoord(index);
}
// B.3 pg 62
public ECPoint add(ECPoint b)
{
if (this.isInfinity())
{
return b;
}
if (b.isInfinity())
{
return this;
}
if (this == b)
{
return twice();
}
ECCurve curve = this.getCurve();
int coord = curve.getCoordinateSystem();
ECFieldElement X1 = this.x, Y1 = this.y;
ECFieldElement X2 = b.x, Y2 = b.y;
switch (coord)
{
case ECCurve.COORD_AFFINE:
{
ECFieldElement dx = X2.subtract(X1), dy = Y2.subtract(Y1);
if (dx.isZero())
{
if (dy.isZero())
{
// this == b, i.e. this must be doubled
return twice();
}
// this == -b, i.e. the result is the point at infinity
return curve.getInfinity();
}
ECFieldElement gamma = dy.divide(dx);
ECFieldElement X3 = gamma.square().subtract(X1).subtract(X2);
ECFieldElement Y3 = gamma.multiply(X1.subtract(X3)).subtract(Y1);
return new ECPoint.Fp(curve, X3, Y3);
}
case ECCurve.COORD_HOMOGENEOUS:
{
ECFieldElement Z1 = this.zs[0];
ECFieldElement Z2 = b.zs[0];
boolean Z1IsOne = Z1.isOne();
boolean Z2IsOne = Z2.isOne();
ECFieldElement u1 = Z1IsOne ? Y2 : Y2.multiply(Z1);
ECFieldElement u2 = Z2IsOne ? Y1 : Y1.multiply(Z2);
ECFieldElement u = u1.subtract(u2);
ECFieldElement v1 = Z1IsOne ? X2 : X2.multiply(Z1);
ECFieldElement v2 = Z2IsOne ? X1 : X1.multiply(Z2);
ECFieldElement v = v1.subtract(v2);
// Check if b == this or b == -this
if (v.isZero())
{
if (u.isZero())
{
// this == b, i.e. this must be doubled
return this.twice();
}
// this == -b, i.e. the result is the point at infinity
return curve.getInfinity();
}
// TODO Optimize for when w == 1
ECFieldElement w = Z1IsOne ? Z2 : Z2IsOne ? Z1 : Z1.multiply(Z2);
ECFieldElement vSquared = v.square();
ECFieldElement vCubed = vSquared.multiply(v);
ECFieldElement vSquaredV2 = vSquared.multiply(v2);
ECFieldElement A = u.square().multiply(w).subtract(vCubed).subtract(two(vSquaredV2));
ECFieldElement X3 = v.multiply(A);
ECFieldElement Y3 = vSquaredV2.subtract(A).multiplyMinusProduct(u, u2, vCubed);
ECFieldElement Z3 = vCubed.multiply(w);
return new ECPoint.Fp(curve, X3, Y3, new ECFieldElement[]{ Z3 });
}
case ECCurve.COORD_JACOBIAN:
case ECCurve.COORD_JACOBIAN_MODIFIED:
{
ECFieldElement Z1 = this.zs[0];
ECFieldElement Z2 = b.zs[0];
boolean Z1IsOne = Z1.isOne();
ECFieldElement X3, Y3, Z3, Z3Squared = null;
if (!Z1IsOne && Z1.equals(Z2))
{
// TODO Make this available as public method coZAdd?
ECFieldElement dx = X1.subtract(X2), dy = Y1.subtract(Y2);
if (dx.isZero())
{
if (dy.isZero())
{
return twice();
}
return curve.getInfinity();
}
ECFieldElement C = dx.square();
ECFieldElement W1 = X1.multiply(C), W2 = X2.multiply(C);
ECFieldElement A1 = W1.subtract(W2).multiply(Y1);
X3 = dy.square().subtract(W1).subtract(W2);
Y3 = W1.subtract(X3).multiply(dy).subtract(A1);
Z3 = dx;
Z3 = Z3.multiply(Z1);
}
else
{
ECFieldElement Z1Squared, U2, S2;
if (Z1IsOne)
{
Z1Squared = Z1; U2 = X2; S2 = Y2;
}
else
{
Z1Squared = Z1.square();
U2 = Z1Squared.multiply(X2);
ECFieldElement Z1Cubed = Z1Squared.multiply(Z1);
S2 = Z1Cubed.multiply(Y2);
}
boolean Z2IsOne = Z2.isOne();
ECFieldElement Z2Squared, U1, S1;
if (Z2IsOne)
{
Z2Squared = Z2; U1 = X1; S1 = Y1;
}
else
{
Z2Squared = Z2.square();
U1 = Z2Squared.multiply(X1);
ECFieldElement Z2Cubed = Z2Squared.multiply(Z2);
S1 = Z2Cubed.multiply(Y1);
}
ECFieldElement H = U1.subtract(U2);
ECFieldElement R = S1.subtract(S2);
// Check if b == this or b == -this
if (H.isZero())
{
if (R.isZero())
{
// this == b, i.e. this must be doubled
return this.twice();
}
// this == -b, i.e. the result is the point at infinity
return curve.getInfinity();
}
ECFieldElement HSquared = H.square();
ECFieldElement G = HSquared.multiply(H);
ECFieldElement V = HSquared.multiply(U1);
X3 = R.square().add(G).subtract(two(V));
Y3 = V.subtract(X3).multiplyMinusProduct(R, G, S1);
Z3 = H;
if (!Z1IsOne)
{
Z3 = Z3.multiply(Z1);
}
if (!Z2IsOne)
{
Z3 = Z3.multiply(Z2);
}
// Alternative calculation of Z3 using fast square
// X3 = four(X3);
// Y3 = eight(Y3);
// Z3 = doubleProductFromSquares(Z1, Z2, Z1Squared, Z2Squared).multiply(H);
if (Z3 == H)
{
Z3Squared = HSquared;
}
}
ECFieldElement[] zs;
if (coord == ECCurve.COORD_JACOBIAN_MODIFIED)
{
// TODO If the result will only be used in a subsequent addition, we don't need W3
ECFieldElement W3 = calculateJacobianModifiedW(Z3, Z3Squared);
zs = new ECFieldElement[]{ Z3, W3 };
}
else
{
zs = new ECFieldElement[]{ Z3 };
}
return new ECPoint.Fp(curve, X3, Y3, zs);
}
default:
{
throw new IllegalStateException("unsupported coordinate system");
}
}
}
// B.3 pg 62
public ECPoint twice()
{
if (this.isInfinity())
{
return this;
}
ECCurve curve = this.getCurve();
ECFieldElement Y1 = this.y;
if (Y1.isZero())
{
return curve.getInfinity();
}
int coord = curve.getCoordinateSystem();
ECFieldElement X1 = this.x;
switch (coord)
{
case ECCurve.COORD_AFFINE:
{
ECFieldElement X1Squared = X1.square();
ECFieldElement gamma = three(X1Squared).add(this.getCurve().getA()).divide(two(Y1));
ECFieldElement X3 = gamma.square().subtract(two(X1));
ECFieldElement Y3 = gamma.multiply(X1.subtract(X3)).subtract(Y1);
return new ECPoint.Fp(curve, X3, Y3);
}
case ECCurve.COORD_HOMOGENEOUS:
{
ECFieldElement Z1 = this.zs[0];
boolean Z1IsOne = Z1.isOne();
// TODO Optimize for small negative a4 and -3
ECFieldElement w = curve.getA();
if (!w.isZero() && !Z1IsOne)
{
w = w.multiply(Z1.square());
}
w = w.add(three(X1.square()));
ECFieldElement s = Z1IsOne ? Y1 : Y1.multiply(Z1);
ECFieldElement t = Z1IsOne ? Y1.square() : s.multiply(Y1);
ECFieldElement B = X1.multiply(t);
ECFieldElement _4B = four(B);
ECFieldElement h = w.square().subtract(two(_4B));
ECFieldElement _2s = two(s);
ECFieldElement X3 = h.multiply(_2s);
ECFieldElement _2t = two(t);
ECFieldElement Y3 = _4B.subtract(h).multiply(w).subtract(two(_2t.square()));
ECFieldElement _4sSquared = Z1IsOne ? two(_2t) : _2s.square();
ECFieldElement Z3 = two(_4sSquared).multiply(s);
return new ECPoint.Fp(curve, X3, Y3, new ECFieldElement[]{ Z3 });
}
case ECCurve.COORD_JACOBIAN:
{
ECFieldElement Z1 = this.zs[0];
boolean Z1IsOne = Z1.isOne();
ECFieldElement Y1Squared = Y1.square();
ECFieldElement T = Y1Squared.square();
ECFieldElement a4 = curve.getA();
ECFieldElement a4Neg = a4.negate();
ECFieldElement M, S;
if (a4Neg.toBigInteger().equals(BigInteger.valueOf(3)))
{
ECFieldElement Z1Squared = Z1IsOne ? Z1 : Z1.square();
M = three(X1.add(Z1Squared).multiply(X1.subtract(Z1Squared)));
S = four(Y1Squared.multiply(X1));
}
else
{
ECFieldElement X1Squared = X1.square();
M = three(X1Squared);
if (Z1IsOne)
{
M = M.add(a4);
}
else if (!a4.isZero())
{
ECFieldElement Z1Squared = Z1.square();
ECFieldElement Z1Pow4 = Z1Squared.square();
if (a4Neg.bitLength() < a4.bitLength())
{
M = M.subtract(Z1Pow4.multiply(a4Neg));
}
else
{
M = M.add(Z1Pow4.multiply(a4));
}
}
// S = two(doubleProductFromSquares(X1, Y1Squared, X1Squared, T));
S = four(X1.multiply(Y1Squared));
}
ECFieldElement X3 = M.square().subtract(two(S));
ECFieldElement Y3 = S.subtract(X3).multiply(M).subtract(eight(T));
ECFieldElement Z3 = two(Y1);
if (!Z1IsOne)
{
Z3 = Z3.multiply(Z1);
}
// Alternative calculation of Z3 using fast square
// ECFieldElement Z3 = doubleProductFromSquares(Y1, Z1, Y1Squared, Z1Squared);
return new ECPoint.Fp(curve, X3, Y3, new ECFieldElement[]{ Z3 });
}
case ECCurve.COORD_JACOBIAN_MODIFIED:
{
return twiceJacobianModified(true);
}
default:
{
throw new IllegalStateException("unsupported coordinate system");
}
}
}
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;
}
ECCurve curve = this.getCurve();
int coord = curve.getCoordinateSystem();
switch (coord)
{
case ECCurve.COORD_AFFINE:
{
ECFieldElement X1 = this.x;
ECFieldElement X2 = b.x, Y2 = b.y;
ECFieldElement dx = X2.subtract(X1), dy = Y2.subtract(Y1);
if (dx.isZero())
{
if (dy.isZero())
{
// this == b i.e. the result is 3P
return threeTimes();
}
// this == -b, i.e. the result is P
return this;
}
/*
* Optimized calculation of 2P + Q, as described in "Trading Inversions for
* Multiplications in Elliptic Curve Cryptography", by Ciet, Joye, Lauter, Montgomery.
*/
ECFieldElement X = dx.square(), Y = dy.square();
ECFieldElement d = X.multiply(two(X1).add(X2)).subtract(Y);
if (d.isZero())
{
return curve.getInfinity();
}
ECFieldElement D = d.multiply(dx);
ECFieldElement I = D.invert();
ECFieldElement L1 = d.multiply(I).multiply(dy);
ECFieldElement L2 = two(Y1).multiply(X).multiply(dx).multiply(I).subtract(L1);
ECFieldElement X4 = (L2.subtract(L1)).multiply(L1.add(L2)).add(X2);
ECFieldElement Y4 = (X1.subtract(X4)).multiply(L2).subtract(Y1);
return new ECPoint.Fp(curve, X4, Y4);
}
case ECCurve.COORD_JACOBIAN_MODIFIED:
{
return twiceJacobianModified(false).add(b);
}
default:
{
return twice().add(b);
}
}
}
public ECPoint threeTimes()
{
if (this.isInfinity())
{
return this;
}
ECFieldElement Y1 = this.y;
if (Y1.isZero())
{
return this;
}
ECCurve curve = this.getCurve();
int coord = curve.getCoordinateSystem();
switch (coord)
{
case ECCurve.COORD_AFFINE:
{
ECFieldElement X1 = this.x;
ECFieldElement _2Y1 = two(Y1);
ECFieldElement X = _2Y1.square();
ECFieldElement Z = three(X1.square()).add(this.getCurve().getA());
ECFieldElement Y = Z.square();
ECFieldElement d = three(X1).multiply(X).subtract(Y);
if (d.isZero())
{
return this.getCurve().getInfinity();
}
ECFieldElement D = d.multiply(_2Y1);
ECFieldElement I = D.invert();
ECFieldElement L1 = d.multiply(I).multiply(Z);
ECFieldElement L2 = X.square().multiply(I).subtract(L1);
ECFieldElement X4 = (L2.subtract(L1)).multiply(L1.add(L2)).add(X1);
ECFieldElement Y4 = (X1.subtract(X4)).multiply(L2).subtract(Y1);
return new ECPoint.Fp(curve, X4, Y4);
}
case ECCurve.COORD_JACOBIAN_MODIFIED:
{
return twiceJacobianModified(false).add(this);
}
default:
{
// NOTE: Be careful about recursions between twicePlus and threeTimes
return twice().add(this);
}
}
}
public ECPoint timesPow2(int e)
{
if (e < 0)
{
throw new IllegalArgumentException("'e' cannot be negative");
}
if (e == 0 || this.isInfinity())
{
return this;
}
if (e == 1)
{
return twice();
}
ECCurve curve = this.getCurve();
ECFieldElement Y1 = this.y;
if (Y1.isZero())
{
return curve.getInfinity();
}
int coord = curve.getCoordinateSystem();
ECFieldElement W1 = curve.getA();
ECFieldElement X1 = this.x;
ECFieldElement Z1 = this.zs.length < 1 ? curve.fromBigInteger(ECConstants.ONE) : this.zs[0];
if (!Z1.isOne())
{
switch (coord)
{
case ECCurve.COORD_AFFINE:
break;
case ECCurve.COORD_HOMOGENEOUS:
ECFieldElement Z1Sq = Z1.square();
X1 = X1.multiply(Z1);
Y1 = Y1.multiply(Z1Sq);
W1 = calculateJacobianModifiedW(Z1, Z1Sq);
break;
case ECCurve.COORD_JACOBIAN:
W1 = calculateJacobianModifiedW(Z1, null);
break;
case ECCurve.COORD_JACOBIAN_MODIFIED:
W1 = getJacobianModifiedW();
break;
default:
throw new IllegalStateException("unsupported coordinate system");
}
}
for (int i = 0; i < e; ++i)
{
if (Y1.isZero())
{
return curve.getInfinity();
}
ECFieldElement X1Squared = X1.square();
ECFieldElement M = three(X1Squared);
ECFieldElement _2Y1 = two(Y1);
ECFieldElement _2Y1Squared = _2Y1.multiply(Y1);
ECFieldElement S = two(X1.multiply(_2Y1Squared));
ECFieldElement _4T = _2Y1Squared.square();
ECFieldElement _8T = two(_4T);
if (!W1.isZero())
{
M = M.add(W1);
W1 = two(_8T.multiply(W1));
}
X1 = M.square().subtract(two(S));
Y1 = M.multiply(S.subtract(X1)).subtract(_8T);
Z1 = Z1.isOne() ? _2Y1 : _2Y1.multiply(Z1);
}
switch (coord)
{
case ECCurve.COORD_AFFINE:
ECFieldElement zInv = Z1.invert(), zInv2 = zInv.square(), zInv3 = zInv2.multiply(zInv);
return new Fp(curve, X1.multiply(zInv2), Y1.multiply(zInv3));
case ECCurve.COORD_HOMOGENEOUS:
X1 = X1.multiply(Z1);
Z1 = Z1.multiply(Z1.square());
return new Fp(curve, X1, Y1, new ECFieldElement[]{ Z1 });
case ECCurve.COORD_JACOBIAN:
return new Fp(curve, X1, Y1, new ECFieldElement[]{ Z1 });
case ECCurve.COORD_JACOBIAN_MODIFIED:
return new Fp(curve, X1, Y1, new ECFieldElement[]{ Z1, W1 });
default:
throw new IllegalStateException("unsupported coordinate system");
}
}
protected ECFieldElement two(ECFieldElement x)
{
return x.add(x);
}
protected ECFieldElement three(ECFieldElement x)
{
return two(x).add(x);
}
protected ECFieldElement four(ECFieldElement x)
{
return two(two(x));
}
protected ECFieldElement eight(ECFieldElement x)
{
return four(two(x));
}
protected ECFieldElement doubleProductFromSquares(ECFieldElement a, ECFieldElement b,
ECFieldElement aSquared, ECFieldElement bSquared)
{
/*
* NOTE: If squaring in the field is faster than multiplication, then this is a quicker
* way to calculate 2.A.B, if A^2 and B^2 are already known.
*/
return a.add(b).square().subtract(aSquared).subtract(bSquared);
}
public ECPoint negate()
{
if (this.isInfinity())
{
return this;
}
ECCurve curve = this.getCurve();
int coord = curve.getCoordinateSystem();
if (ECCurve.COORD_AFFINE != coord)
{
return new ECPoint.Fp(curve, this.x, this.y.negate(), this.zs);
}
return new ECPoint.Fp(curve, this.x, this.y.negate());
}
protected ECFieldElement calculateJacobianModifiedW(ECFieldElement Z, ECFieldElement ZSquared)
{
ECFieldElement a4 = this.getCurve().getA();
if (a4.isZero() || Z.isOne())
{
return a4;
}
if (ZSquared == null)
{
ZSquared = Z.square();
}
ECFieldElement W = ZSquared.square();
ECFieldElement a4Neg = a4.negate();
if (a4Neg.bitLength() < a4.bitLength())
{
W = W.multiply(a4Neg).negate();
}
else
{
W = W.multiply(a4);
}
return W;
}
protected ECFieldElement getJacobianModifiedW()
{
ECFieldElement W = this.zs[1];
if (W == null)
{
// NOTE: Rarely, twicePlus will result in the need for a lazy W1 calculation here
this.zs[1] = W = calculateJacobianModifiedW(this.zs[0], null);
}
return W;
}
protected ECPoint.Fp twiceJacobianModified(boolean calculateW)
{
ECFieldElement X1 = this.x, Y1 = this.y, Z1 = this.zs[0], W1 = getJacobianModifiedW();
ECFieldElement X1Squared = X1.square();
ECFieldElement M = three(X1Squared).add(W1);
ECFieldElement _2Y1 = two(Y1);
ECFieldElement _2Y1Squared = _2Y1.multiply(Y1);
ECFieldElement S = two(X1.multiply(_2Y1Squared));
ECFieldElement X3 = M.square().subtract(two(S));
ECFieldElement _4T = _2Y1Squared.square();
ECFieldElement _8T = two(_4T);
ECFieldElement Y3 = M.multiply(S.subtract(X3)).subtract(_8T);
ECFieldElement W3 = calculateW ? two(_8T.multiply(W1)) : null;
ECFieldElement Z3 = Z1.isOne() ? _2Y1 : _2Y1.multiply(Z1);
return new ECPoint.Fp(this.getCurve(), X3, Y3, new ECFieldElement[]{ Z3, W3 });
}
}
public static abstract class AbstractF2m extends ECPoint
{
protected AbstractF2m(ECCurve curve, ECFieldElement x, ECFieldElement y)
{
super(curve, x, y);
}
protected AbstractF2m(ECCurve curve, ECFieldElement x, ECFieldElement y, ECFieldElement[] zs)
{
super(curve, x, y, zs);
}
protected boolean satisfiesCurveEquation()
{
ECCurve curve = this.getCurve();
ECFieldElement X = this.x, A = curve.getA(), B = curve.getB();
int coord = curve.getCoordinateSystem();
if (coord == ECCurve.COORD_LAMBDA_PROJECTIVE)
{
ECFieldElement Z = this.zs[0];
boolean ZIsOne = Z.isOne();
if (X.isZero())
{
// NOTE: For x == 0, we expect the affine-y instead of the lambda-y
ECFieldElement Y = this.y;
ECFieldElement lhs = Y.square(), rhs = B;
if (!ZIsOne)
{
rhs = rhs.multiply(Z.square());
}
return lhs.equals(rhs);
}
ECFieldElement L = this.y, X2 = X.square();
ECFieldElement lhs, rhs;
if (ZIsOne)
{
lhs = L.square().add(L).add(A);
rhs = X2.square().add(B);
}
else
{
ECFieldElement Z2 = Z.square(), Z4 = Z2.square();
lhs = L.add(Z).multiplyPlusProduct(L, A, Z2);
// TODO If sqrt(b) is precomputed this can be simplified to a single square
rhs = X2.squarePlusProduct(B, Z4);
}
lhs = lhs.multiply(X2);
return lhs.equals(rhs);
}
ECFieldElement Y = this.y;
ECFieldElement lhs = Y.add(X).multiply(Y);
switch (coord)
{
case ECCurve.COORD_AFFINE:
break;
case ECCurve.COORD_HOMOGENEOUS:
{
ECFieldElement Z = this.zs[0];
if (!Z.isOne())
{
ECFieldElement Z2 = Z.square(), Z3 = Z.multiply(Z2);
lhs = lhs.multiply(Z);
A = A.multiply(Z);
B = B.multiply(Z3);
}
break;
}
default:
throw new IllegalStateException("unsupported coordinate system");
}
ECFieldElement rhs = X.add(A).multiply(X.square()).add(B);
return lhs.equals(rhs);
}
protected boolean satisfiesOrder()
{
BigInteger cofactor = curve.getCofactor();
if (ECConstants.TWO.equals(cofactor))
{
/*
* Check that 0 == Tr(X + A); then there exists a solution to L^2 + L = X + A, and
* so a halving is possible, so this point is the double of another.
*
* Note: Tr(A) == 1 for cofactor 2 curves.
*/
ECPoint N = this.normalize();
ECFieldElement X = N.getAffineXCoord();
return 0 != ((ECFieldElement.AbstractF2m)X).trace();
}
if (ECConstants.FOUR.equals(cofactor))
{
/*
* Solve L^2 + L = X + A to find the half of this point, if it exists (fail if not).
*
* Note: Tr(A) == 0 for cofactor 4 curves.
*/
ECPoint N = this.normalize();
ECFieldElement X = N.getAffineXCoord();
ECFieldElement L = ((ECCurve.AbstractF2m)curve).solveQuadraticEquation(X.add(curve.getA()));
if (null == L)
{
return false;
}
/*
* A solution exists, therefore 0 == Tr(X + A) == Tr(X).
*/
ECFieldElement Y = N.getAffineYCoord();
ECFieldElement T = X.multiply(L).add(Y);
/*
* Either T or (T + X) is the square of a half-point's x coordinate (hx). In either
* case, the half-point can be halved again when 0 == Tr(hx + A).
*
* Note: Tr(hx + A) == Tr(hx) == Tr(hx^2) == Tr(T) == Tr(T + X)
*
* Check that 0 == Tr(T); then there exists a solution to L^2 + L = hx + A, and so a
* second halving is possible and this point is four times some other.
*/
return 0 == ((ECFieldElement.AbstractF2m)T).trace();
}
return super.satisfiesOrder();
}
public ECPoint scaleX(ECFieldElement scale)
{
if (this.isInfinity())
{
return this;
}
int coord = this.getCurveCoordinateSystem();
switch (coord)
{
case ECCurve.COORD_LAMBDA_AFFINE:
{
// Y is actually Lambda (X + Y/X) here
ECFieldElement X = this.getRawXCoord(), L = this.getRawYCoord(); // earlier JDK
ECFieldElement X2 = X.multiply(scale);
ECFieldElement L2 = L.add(X).divide(scale).add(X2);
return this.getCurve().createRawPoint(X, L2, this.getRawZCoords()); // earlier JDK
}
case ECCurve.COORD_LAMBDA_PROJECTIVE:
{
// Y is actually Lambda (X + Y/X) here
ECFieldElement X = this.getRawXCoord(), L = this.getRawYCoord(), Z = this.getRawZCoords()[0]; // earlier JDK
// We scale the Z coordinate also, to avoid an inversion
ECFieldElement X2 = X.multiply(scale.square());
ECFieldElement L2 = L.add(X).add(X2);
ECFieldElement Z2 = Z.multiply(scale);
return this.getCurve().createRawPoint(X2, L2, new ECFieldElement[]{ Z2 }); // earlier JDK
}
default:
{
return super.scaleX(scale);
}
}
}
public ECPoint scaleXNegateY(ECFieldElement scale)
{
return scaleX(scale);
}
public ECPoint scaleY(ECFieldElement scale)
{
if (this.isInfinity())
{
return this;
}
int coord = this.getCurveCoordinateSystem();
switch (coord)
{
case ECCurve.COORD_LAMBDA_AFFINE:
case ECCurve.COORD_LAMBDA_PROJECTIVE:
{
ECFieldElement X = this.getRawXCoord(), L = this.getRawYCoord(); // earlier JDK
// Y is actually Lambda (X + Y/X) here
ECFieldElement L2 = L.add(X).multiply(scale).add(X);
return this.getCurve().createRawPoint(X, L2, this.getRawZCoords()); // earlier JDK
}
default:
{
return super.scaleY(scale);
}
}
}
public ECPoint scaleYNegateX(ECFieldElement scale)
{
return scaleY(scale);
}
public ECPoint subtract(ECPoint b)
{
if (b.isInfinity())
{
return this;
}
// Add -b
return this.add(b.negate());
}
public ECPoint.AbstractF2m tau()
{
if (this.isInfinity())
{
return this;
}
ECCurve curve = this.getCurve();
int coord = curve.getCoordinateSystem();
ECFieldElement X1 = this.x;
switch (coord)
{
case ECCurve.COORD_AFFINE:
case ECCurve.COORD_LAMBDA_AFFINE:
{
ECFieldElement Y1 = this.y;
return (ECPoint.AbstractF2m)curve.createRawPoint(X1.square(), Y1.square());
}
case ECCurve.COORD_HOMOGENEOUS:
case ECCurve.COORD_LAMBDA_PROJECTIVE:
{
ECFieldElement Y1 = this.y, Z1 = this.zs[0];
return (ECPoint.AbstractF2m)curve.createRawPoint(X1.square(), Y1.square(),
new ECFieldElement[]{ Z1.square() });
}
default:
{
throw new IllegalStateException("unsupported coordinate system");
}
}
}
public ECPoint.AbstractF2m tauPow(int pow)
{
if (this.isInfinity())
{
return this;
}
ECCurve curve = this.getCurve();
int coord = curve.getCoordinateSystem();
ECFieldElement X1 = this.x;
switch (coord)
{
case ECCurve.COORD_AFFINE:
case ECCurve.COORD_LAMBDA_AFFINE:
{
ECFieldElement Y1 = this.y;
return (ECPoint.AbstractF2m)curve.createRawPoint(X1.squarePow(pow), Y1.squarePow(pow));
}
case ECCurve.COORD_HOMOGENEOUS:
case ECCurve.COORD_LAMBDA_PROJECTIVE:
{
ECFieldElement Y1 = this.y, Z1 = this.zs[0];
return (ECPoint.AbstractF2m)curve.createRawPoint(X1.squarePow(pow), Y1.squarePow(pow),
new ECFieldElement[]{ Z1.squarePow(pow) });
}
default:
{
throw new IllegalStateException("unsupported coordinate system");
}
}
}
}
/**
* Elliptic curve points over F2m
*/
public static class F2m extends AbstractF2m
{
F2m(ECCurve curve, ECFieldElement x, ECFieldElement y)
{
super(curve, x, y);
// checkCurveEquation();
}
F2m(ECCurve curve, ECFieldElement x, ECFieldElement y, ECFieldElement[] zs)
{
super(curve, x, y, zs);
// checkCurveEquation();
}
protected ECPoint detach()
{
return new ECPoint.F2m(null, this.getAffineXCoord(), this.getAffineYCoord()); // earlier JDK
}
public ECFieldElement getYCoord()
{
int coord = this.getCurveCoordinateSystem();
switch (coord)
{
case ECCurve.COORD_LAMBDA_AFFINE:
case ECCurve.COORD_LAMBDA_PROJECTIVE:
{
ECFieldElement X = x, L = y;
if (this.isInfinity() || X.isZero())
{
return L;
}
// Y is actually Lambda (X + Y/X) here; convert to affine value on the fly
ECFieldElement Y = L.add(X).multiply(X);
if (ECCurve.COORD_LAMBDA_PROJECTIVE == coord)
{
ECFieldElement Z = zs[0];
if (!Z.isOne())
{
Y = Y.divide(Z);
}
}
return Y;
}
default:
{
return y;
}
}
}
protected boolean getCompressionYTilde()
{
ECFieldElement X = this.getRawXCoord();
if (X.isZero())
{
return false;
}
ECFieldElement Y = this.getRawYCoord();
switch (this.getCurveCoordinateSystem())
{
case ECCurve.COORD_LAMBDA_AFFINE:
case ECCurve.COORD_LAMBDA_PROJECTIVE:
{
// Y is actually Lambda (X + Y/X) here
return Y.testBitZero() != X.testBitZero();
}
default:
{
return Y.divide(X).testBitZero();
}
}
}
public ECPoint add(ECPoint b)
{
if (this.isInfinity())
{
return b;
}
if (b.isInfinity())
{
return this;
}
ECCurve curve = this.getCurve();
int coord = curve.getCoordinateSystem();
ECFieldElement X1 = this.x;
ECFieldElement X2 = b.x;
switch (coord)
{
case ECCurve.COORD_AFFINE:
{
ECFieldElement Y1 = this.y;
ECFieldElement Y2 = b.y;
ECFieldElement dx = X1.add(X2), dy = Y1.add(Y2);
if (dx.isZero())
{
if (dy.isZero())
{
return twice();
}
return curve.getInfinity();
}
ECFieldElement L = dy.divide(dx);
ECFieldElement X3 = L.square().add(L).add(dx).add(curve.getA());
ECFieldElement Y3 = L.multiply(X1.add(X3)).add(X3).add(Y1);
return new ECPoint.F2m(curve, X3, Y3);
}
case ECCurve.COORD_HOMOGENEOUS:
{
ECFieldElement Y1 = this.y, Z1 = this.zs[0];
ECFieldElement Y2 = b.y, Z2 = b.zs[0];
boolean Z2IsOne = Z2.isOne();
ECFieldElement U1 = Z1.multiply(Y2);
ECFieldElement U2 = Z2IsOne ? Y1 : Y1.multiply(Z2);
ECFieldElement U = U1.add(U2);
ECFieldElement V1 = Z1.multiply(X2);
ECFieldElement V2 = Z2IsOne ? X1 : X1.multiply(Z2);
ECFieldElement V = V1.add(V2);
if (V.isZero())
{
if (U.isZero())
{
return twice();
}
return curve.getInfinity();
}
ECFieldElement VSq = V.square();
ECFieldElement VCu = VSq.multiply(V);
ECFieldElement W = Z2IsOne ? Z1 : Z1.multiply(Z2);
ECFieldElement uv = U.add(V);
ECFieldElement A = uv.multiplyPlusProduct(U, VSq, curve.getA()).multiply(W).add(VCu);
ECFieldElement X3 = V.multiply(A);
ECFieldElement VSqZ2 = Z2IsOne ? VSq : VSq.multiply(Z2);
ECFieldElement Y3 = U.multiplyPlusProduct(X1, V, Y1).multiplyPlusProduct(VSqZ2, uv, A);
ECFieldElement Z3 = VCu.multiply(W);
return new ECPoint.F2m(curve, X3, Y3, new ECFieldElement[]{ Z3 });
}
case ECCurve.COORD_LAMBDA_PROJECTIVE:
{
if (X1.isZero())
{
if (X2.isZero())
{
return curve.getInfinity();
}
return b.add(this);
}
ECFieldElement L1 = this.y, Z1 = this.zs[0];
ECFieldElement L2 = b.y, Z2 = b.zs[0];
boolean Z1IsOne = Z1.isOne();
ECFieldElement U2 = X2, S2 = L2;
if (!Z1IsOne)
{
U2 = U2.multiply(Z1);
S2 = S2.multiply(Z1);
}
boolean Z2IsOne = Z2.isOne();
ECFieldElement U1 = X1, S1 = L1;
if (!Z2IsOne)
{
U1 = U1.multiply(Z2);
S1 = S1.multiply(Z2);
}
ECFieldElement A = S1.add(S2);
ECFieldElement B = U1.add(U2);
if (B.isZero())
{
if (A.isZero())
{
return twice();
}
return curve.getInfinity();
}
ECFieldElement X3, L3, Z3;
if (X2.isZero())
{
// TODO This can probably be optimized quite a bit
ECPoint p = this.normalize();
X1 = p.getXCoord();
ECFieldElement Y1 = p.getYCoord();
ECFieldElement Y2 = L2;
ECFieldElement L = Y1.add(Y2).divide(X1);
X3 = L.square().add(L).add(X1).add(curve.getA());
if (X3.isZero())
{
return new ECPoint.F2m(curve, X3, curve.getB().sqrt());
}
ECFieldElement Y3 = L.multiply(X1.add(X3)).add(X3).add(Y1);
L3 = Y3.divide(X3).add(X3);
Z3 = curve.fromBigInteger(ECConstants.ONE);
}
else
{
B = B.square();
ECFieldElement AU1 = A.multiply(U1);
ECFieldElement AU2 = A.multiply(U2);
X3 = AU1.multiply(AU2);
if (X3.isZero())
{
return new ECPoint.F2m(curve, X3, curve.getB().sqrt());
}
ECFieldElement ABZ2 = A.multiply(B);
if (!Z2IsOne)
{
ABZ2 = ABZ2.multiply(Z2);
}
L3 = AU2.add(B).squarePlusProduct(ABZ2, L1.add(Z1));
Z3 = ABZ2;
if (!Z1IsOne)
{
Z3 = Z3.multiply(Z1);
}
}
return new ECPoint.F2m(curve, X3, L3, new ECFieldElement[]{ Z3 });
}
default:
{
throw new IllegalStateException("unsupported coordinate system");
}
}
}
public ECPoint twice()
{
if (this.isInfinity())
{
return this;
}
ECCurve curve = this.getCurve();
ECFieldElement X1 = this.x;
if (X1.isZero())
{
// A point with X == 0 is it's own additive inverse
return curve.getInfinity();
}
int coord = curve.getCoordinateSystem();
switch (coord)
{
case ECCurve.COORD_AFFINE:
{
ECFieldElement Y1 = this.y;
ECFieldElement L1 = Y1.divide(X1).add(X1);
ECFieldElement X3 = L1.square().add(L1).add(curve.getA());
ECFieldElement Y3 = X1.squarePlusProduct(X3, L1.addOne());
return new ECPoint.F2m(curve, X3, Y3);
}
case ECCurve.COORD_HOMOGENEOUS:
{
ECFieldElement Y1 = this.y, Z1 = this.zs[0];
boolean Z1IsOne = Z1.isOne();
ECFieldElement X1Z1 = Z1IsOne ? X1 : X1.multiply(Z1);
ECFieldElement Y1Z1 = Z1IsOne ? Y1 : Y1.multiply(Z1);
ECFieldElement X1Sq = X1.square();
ECFieldElement S = X1Sq.add(Y1Z1);
ECFieldElement V = X1Z1;
ECFieldElement vSquared = V.square();
ECFieldElement sv = S.add(V);
ECFieldElement h = sv.multiplyPlusProduct(S, vSquared, curve.getA());
ECFieldElement X3 = V.multiply(h);
ECFieldElement Y3 = X1Sq.square().multiplyPlusProduct(V, h, sv);
ECFieldElement Z3 = V.multiply(vSquared);
return new ECPoint.F2m(curve, X3, Y3, new ECFieldElement[]{ Z3 });
}
case ECCurve.COORD_LAMBDA_PROJECTIVE:
{
ECFieldElement L1 = this.y, Z1 = this.zs[0];
boolean Z1IsOne = Z1.isOne();
ECFieldElement L1Z1 = Z1IsOne ? L1 : L1.multiply(Z1);
ECFieldElement Z1Sq = Z1IsOne ? Z1 : Z1.square();
ECFieldElement a = curve.getA();
ECFieldElement aZ1Sq = Z1IsOne ? a : a.multiply(Z1Sq);
ECFieldElement T = L1.square().add(L1Z1).add(aZ1Sq);
if (T.isZero())
{
return new ECPoint.F2m(curve, T, curve.getB().sqrt());
}
ECFieldElement X3 = T.square();
ECFieldElement Z3 = Z1IsOne ? T : T.multiply(Z1Sq);
ECFieldElement b = curve.getB();
ECFieldElement L3;
if (b.bitLength() < (curve.getFieldSize() >> 1))
{
ECFieldElement t1 = L1.add(X1).square();
ECFieldElement t2;
if (b.isOne())
{
t2 = aZ1Sq.add(Z1Sq).square();
}
else
{
// TODO Can be calculated with one square if we pre-compute sqrt(b)
t2 = aZ1Sq.squarePlusProduct(b, Z1Sq.square());
}
L3 = t1.add(T).add(Z1Sq).multiply(t1).add(t2).add(X3);
if (a.isZero())
{
L3 = L3.add(Z3);
}
else if (!a.isOne())
{
L3 = L3.add(a.addOne().multiply(Z3));
}
}
else
{
ECFieldElement X1Z1 = Z1IsOne ? X1 : X1.multiply(Z1);
L3 = X1Z1.squarePlusProduct(T, L1Z1).add(X3).add(Z3);
}
return new ECPoint.F2m(curve, X3, L3, new ECFieldElement[]{ Z3 });
}
default:
{
throw new IllegalStateException("unsupported coordinate system");
}
}
}
public ECPoint twicePlus(ECPoint b)
{
if (this.isInfinity())
{
return b;
}
if (b.isInfinity())
{
return twice();
}
ECCurve curve = this.getCurve();
ECFieldElement X1 = this.x;
if (X1.isZero())
{
// A point with X == 0 is it's own additive inverse
return b;
}
int coord = curve.getCoordinateSystem();
switch (coord)
{
case ECCurve.COORD_LAMBDA_PROJECTIVE:
{
// NOTE: twicePlus() only optimized for lambda-affine argument
ECFieldElement X2 = b.x, Z2 = b.zs[0];
if (X2.isZero() || !Z2.isOne())
{
return twice().add(b);
}
ECFieldElement L1 = this.y, Z1 = this.zs[0];
ECFieldElement L2 = b.y;
ECFieldElement X1Sq = X1.square();
ECFieldElement L1Sq = L1.square();
ECFieldElement Z1Sq = Z1.square();
ECFieldElement L1Z1 = L1.multiply(Z1);
ECFieldElement T = curve.getA().multiply(Z1Sq).add(L1Sq).add(L1Z1);
ECFieldElement L2plus1 = L2.addOne();
ECFieldElement A = curve.getA().add(L2plus1).multiply(Z1Sq).add(L1Sq).multiplyPlusProduct(T, X1Sq, Z1Sq);
ECFieldElement X2Z1Sq = X2.multiply(Z1Sq);
ECFieldElement B = X2Z1Sq.add(T).square();
if (B.isZero())
{
if (A.isZero())
{
return b.twice();
}
return curve.getInfinity();
}
if (A.isZero())
{
return new ECPoint.F2m(curve, A, curve.getB().sqrt());
}
ECFieldElement X3 = A.square().multiply(X2Z1Sq);
ECFieldElement Z3 = A.multiply(B).multiply(Z1Sq);
ECFieldElement L3 = A.add(B).square().multiplyPlusProduct(T, L2plus1, Z3);
return new ECPoint.F2m(curve, X3, L3, new ECFieldElement[]{ Z3 });
}
default:
{
return twice().add(b);
}
}
}
public ECPoint negate()
{
if (this.isInfinity())
{
return this;
}
ECFieldElement X = this.x;
if (X.isZero())
{
return this;
}
switch (this.getCurveCoordinateSystem())
{
case ECCurve.COORD_AFFINE:
{
ECFieldElement Y = this.y;
return new ECPoint.F2m(curve, X, Y.add(X));
}
case ECCurve.COORD_HOMOGENEOUS:
{
ECFieldElement Y = this.y, Z = this.zs[0];
return new ECPoint.F2m(curve, X, Y.add(X), new ECFieldElement[]{ Z });
}
case ECCurve.COORD_LAMBDA_AFFINE:
{
ECFieldElement L = this.y;
return new ECPoint.F2m(curve, X, L.addOne());
}
case ECCurve.COORD_LAMBDA_PROJECTIVE:
{
// L is actually Lambda (X + Y/X) here
ECFieldElement L = this.y, Z = this.zs[0];
return new ECPoint.F2m(curve, X, L.add(Z), new ECFieldElement[]{ Z });
}
default:
{
throw new IllegalStateException("unsupported coordinate system");
}
}
}
}
}