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The Bouncy Castle Crypto package is a Java implementation of cryptographic algorithms. This jar contains JCE provider and lightweight API for the Bouncy Castle Cryptography APIs for JDK 1.5 to JDK 1.8.
package org.bouncycastle.math.ec;
import java.math.BigInteger;
import org.bouncycastle.math.ec.endo.ECEndomorphism;
import org.bouncycastle.math.ec.endo.EndoUtil;
import org.bouncycastle.math.ec.endo.GLVEndomorphism;
import org.bouncycastle.math.field.FiniteField;
import org.bouncycastle.math.field.PolynomialExtensionField;
import org.bouncycastle.math.raw.Nat;
public class ECAlgorithms
{
public static boolean isF2mCurve(ECCurve c)
{
return isF2mField(c.getField());
}
public static boolean isF2mField(FiniteField field)
{
return field.getDimension() > 1 && field.getCharacteristic().equals(ECConstants.TWO)
&& field instanceof PolynomialExtensionField;
}
public static boolean isFpCurve(ECCurve c)
{
return isFpField(c.getField());
}
public static boolean isFpField(FiniteField field)
{
return field.getDimension() == 1;
}
public static ECPoint sumOfMultiplies(ECPoint[] ps, BigInteger[] ks)
{
if (ps == null || ks == null || ps.length != ks.length || ps.length < 1)
{
throw new IllegalArgumentException("point and scalar arrays should be non-null, and of equal, non-zero, length");
}
int count = ps.length;
switch (count)
{
case 1:
return ps[0].multiply(ks[0]);
case 2:
return sumOfTwoMultiplies(ps[0], ks[0], ps[1], ks[1]);
default:
break;
}
ECPoint p = ps[0];
ECCurve c = p.getCurve();
ECPoint[] imported = new ECPoint[count];
imported[0] = p;
for (int i = 1; i < count; ++i)
{
imported[i] = importPoint(c, ps[i]);
}
ECEndomorphism endomorphism = c.getEndomorphism();
if (endomorphism instanceof GLVEndomorphism)
{
return implCheckResult(implSumOfMultipliesGLV(imported, ks, (GLVEndomorphism)endomorphism));
}
return implCheckResult(implSumOfMultiplies(imported, ks));
}
public static ECPoint sumOfTwoMultiplies(ECPoint P, BigInteger a,
ECPoint Q, BigInteger b)
{
ECCurve cp = P.getCurve();
Q = importPoint(cp, Q);
// Point multiplication for Koblitz curves (using WTNAF) beats Shamir's trick
if (cp instanceof ECCurve.AbstractF2m)
{
ECCurve.AbstractF2m f2mCurve = (ECCurve.AbstractF2m)cp;
if (f2mCurve.isKoblitz())
{
return implCheckResult(P.multiply(a).add(Q.multiply(b)));
}
}
ECEndomorphism endomorphism = cp.getEndomorphism();
if (endomorphism instanceof GLVEndomorphism)
{
return implCheckResult(
implSumOfMultipliesGLV(new ECPoint[]{ P, Q }, new BigInteger[]{ a, b }, (GLVEndomorphism)endomorphism));
}
return implCheckResult(implShamirsTrickWNaf(P, a, Q, b));
}
/*
* "Shamir's Trick", originally due to E. G. Straus
* (Addition chains of vectors. American Mathematical Monthly,
* 71(7):806-808, Aug./Sept. 1964)
*
* Input: The points P, Q, scalar k = (km?, ... , k1, k0)
* and scalar l = (lm?, ... , l1, l0).
* Output: R = k * P + l * Q.
* 1: Z <- P + Q
* 2: R <- O
* 3: for i from m-1 down to 0 do
* 4: R <- R + R {point doubling}
* 5: if (ki = 1) and (li = 0) then R <- R + P end if
* 6: if (ki = 0) and (li = 1) then R <- R + Q end if
* 7: if (ki = 1) and (li = 1) then R <- R + Z end if
* 8: end for
* 9: return R
*
*/
public static ECPoint shamirsTrick(ECPoint P, BigInteger k,
ECPoint Q, BigInteger l)
{
ECCurve cp = P.getCurve();
Q = importPoint(cp, Q);
return implCheckResult(implShamirsTrickJsf(P, k, Q, l));
}
public static ECPoint importPoint(ECCurve c, ECPoint p)
{
ECCurve cp = p.getCurve();
if (!c.equals(cp))
{
throw new IllegalArgumentException("Point must be on the same curve");
}
return c.importPoint(p);
}
public static void montgomeryTrick(ECFieldElement[] zs, int off, int len)
{
montgomeryTrick(zs, off, len, null);
}
public static void montgomeryTrick(ECFieldElement[] zs, int off, int len, ECFieldElement scale)
{
/*
* Uses the "Montgomery Trick" to invert many field elements, with only a single actual
* field inversion. See e.g. the paper:
* "Fast Multi-scalar Multiplication Methods on Elliptic Curves with Precomputation Strategy Using Montgomery Trick"
* by Katsuyuki Okeya, Kouichi Sakurai.
*/
ECFieldElement[] c = new ECFieldElement[len];
c[0] = zs[off];
int i = 0;
while (++i < len)
{
c[i] = c[i - 1].multiply(zs[off + i]);
}
--i;
if (scale != null)
{
c[i] = c[i].multiply(scale);
}
ECFieldElement u = c[i].invert();
while (i > 0)
{
int j = off + i--;
ECFieldElement tmp = zs[j];
zs[j] = c[i].multiply(u);
u = u.multiply(tmp);
}
zs[off] = u;
}
/**
* Simple shift-and-add multiplication. Serves as reference implementation to verify (possibly
* faster) implementations, and for very small scalars. CAUTION: This implementation is NOT
* constant-time in any way. It is only intended to be used for diagnostics.
*
* @param p
* The point to multiply.
* @param k
* The multiplier.
* @return The result of the point multiplication kP.
*/
public static ECPoint referenceMultiply(ECPoint p, BigInteger k)
{
BigInteger x = k.abs();
ECPoint q = p.getCurve().getInfinity();
int t = x.bitLength();
if (t > 0)
{
if (x.testBit(0))
{
q = p;
}
for (int i = 1; i < t; i++)
{
p = p.twice();
if (x.testBit(i))
{
q = q.add(p);
}
}
}
return k.signum() < 0 ? q.negate() : q;
}
public static ECPoint validatePoint(ECPoint p)
{
if (!p.isValid())
{
throw new IllegalStateException("Invalid point");
}
return p;
}
public static ECPoint cleanPoint(ECCurve c, ECPoint p)
{
ECCurve cp = p.getCurve();
if (!c.equals(cp))
{
throw new IllegalArgumentException("Point must be on the same curve");
}
return c.decodePoint(p.getEncoded(false));
}
static ECPoint implCheckResult(ECPoint p)
{
if (!p.isValidPartial())
{
throw new IllegalStateException("Invalid result");
}
return p;
}
static ECPoint implShamirsTrickJsf(ECPoint P, BigInteger k,
ECPoint Q, BigInteger l)
{
ECCurve curve = P.getCurve();
ECPoint infinity = curve.getInfinity();
// TODO conjugate co-Z addition (ZADDC) can return both of these
ECPoint PaddQ = P.add(Q);
ECPoint PsubQ = P.subtract(Q);
ECPoint[] points = new ECPoint[]{ Q, PsubQ, P, PaddQ };
curve.normalizeAll(points);
ECPoint[] table = new ECPoint[] {
points[3].negate(), points[2].negate(), points[1].negate(),
points[0].negate(), infinity, points[0],
points[1], points[2], points[3] };
byte[] jsf = WNafUtil.generateJSF(k, l);
ECPoint R = infinity;
int i = jsf.length;
while (--i >= 0)
{
int jsfi = jsf[i];
// NOTE: The shifting ensures the sign is extended correctly
int kDigit = ((jsfi << 24) >> 28), lDigit = ((jsfi << 28) >> 28);
int index = 4 + (kDigit * 3) + lDigit;
R = R.twicePlus(table[index]);
}
return R;
}
static ECPoint implShamirsTrickWNaf(ECPoint P, BigInteger k,
ECPoint Q, BigInteger l)
{
boolean negK = k.signum() < 0, negL = l.signum() < 0;
BigInteger kAbs = k.abs(), lAbs = l.abs();
int minWidthP = WNafUtil.getWindowSize(kAbs.bitLength(), 8);
int minWidthQ = WNafUtil.getWindowSize(lAbs.bitLength(), 8);
WNafPreCompInfo infoP = WNafUtil.precompute(P, minWidthP, true);
WNafPreCompInfo infoQ = WNafUtil.precompute(Q, minWidthQ, true);
// When P, Q are 'promoted' (i.e. reused several times), switch to fixed-point algorithm
{
ECCurve c = P.getCurve();
int combSize = FixedPointUtil.getCombSize(c);
if (!negK && !negL
&& k.bitLength() <= combSize && l.bitLength() <= combSize
&& infoP.isPromoted() && infoQ.isPromoted())
{
return implShamirsTrickFixedPoint(P, k, Q, l);
}
}
int widthP = Math.min(8, infoP.getWidth());
int widthQ = Math.min(8, infoQ.getWidth());
ECPoint[] preCompP = negK ? infoP.getPreCompNeg() : infoP.getPreComp();
ECPoint[] preCompQ = negL ? infoQ.getPreCompNeg() : infoQ.getPreComp();
ECPoint[] preCompNegP = negK ? infoP.getPreComp() : infoP.getPreCompNeg();
ECPoint[] preCompNegQ = negL ? infoQ.getPreComp() : infoQ.getPreCompNeg();
byte[] wnafP = WNafUtil.generateWindowNaf(widthP, kAbs);
byte[] wnafQ = WNafUtil.generateWindowNaf(widthQ, lAbs);
return implShamirsTrickWNaf(preCompP, preCompNegP, wnafP, preCompQ, preCompNegQ, wnafQ);
}
static ECPoint implShamirsTrickWNaf(ECEndomorphism endomorphism, ECPoint P, BigInteger k, BigInteger l)
{
boolean negK = k.signum() < 0, negL = l.signum() < 0;
k = k.abs();
l = l.abs();
int minWidth = WNafUtil.getWindowSize(Math.max(k.bitLength(), l.bitLength()), 8);
WNafPreCompInfo infoP = WNafUtil.precompute(P, minWidth, true);
ECPoint Q = EndoUtil.mapPoint(endomorphism, P);
WNafPreCompInfo infoQ = WNafUtil.precomputeWithPointMap(Q, endomorphism.getPointMap(), infoP, true);
int widthP = Math.min(8, infoP.getWidth());
int widthQ = Math.min(8, infoQ.getWidth());
ECPoint[] preCompP = negK ? infoP.getPreCompNeg() : infoP.getPreComp();
ECPoint[] preCompQ = negL ? infoQ.getPreCompNeg() : infoQ.getPreComp();
ECPoint[] preCompNegP = negK ? infoP.getPreComp() : infoP.getPreCompNeg();
ECPoint[] preCompNegQ = negL ? infoQ.getPreComp() : infoQ.getPreCompNeg();
byte[] wnafP = WNafUtil.generateWindowNaf(widthP, k);
byte[] wnafQ = WNafUtil.generateWindowNaf(widthQ, l);
return implShamirsTrickWNaf(preCompP, preCompNegP, wnafP, preCompQ, preCompNegQ, wnafQ);
}
private static ECPoint implShamirsTrickWNaf(ECPoint[] preCompP, ECPoint[] preCompNegP, byte[] wnafP,
ECPoint[] preCompQ, ECPoint[] preCompNegQ, byte[] wnafQ)
{
int len = Math.max(wnafP.length, wnafQ.length);
ECCurve curve = preCompP[0].getCurve();
ECPoint infinity = curve.getInfinity();
ECPoint R = infinity;
int zeroes = 0;
for (int i = len - 1; i >= 0; --i)
{
int wiP = i < wnafP.length ? wnafP[i] : 0;
int wiQ = i < wnafQ.length ? wnafQ[i] : 0;
if ((wiP | wiQ) == 0)
{
++zeroes;
continue;
}
ECPoint r = infinity;
if (wiP != 0)
{
int nP = Math.abs(wiP);
ECPoint[] tableP = wiP < 0 ? preCompNegP : preCompP;
r = r.add(tableP[nP >>> 1]);
}
if (wiQ != 0)
{
int nQ = Math.abs(wiQ);
ECPoint[] tableQ = wiQ < 0 ? preCompNegQ : preCompQ;
r = r.add(tableQ[nQ >>> 1]);
}
if (zeroes > 0)
{
R = R.timesPow2(zeroes);
zeroes = 0;
}
R = R.twicePlus(r);
}
if (zeroes > 0)
{
R = R.timesPow2(zeroes);
}
return R;
}
static ECPoint implSumOfMultiplies(ECPoint[] ps, BigInteger[] ks)
{
int count = ps.length;
boolean[] negs = new boolean[count];
WNafPreCompInfo[] infos = new WNafPreCompInfo[count];
byte[][] wnafs = new byte[count][];
for (int i = 0; i < count; ++i)
{
BigInteger ki = ks[i]; negs[i] = ki.signum() < 0; ki = ki.abs();
int minWidth = WNafUtil.getWindowSize(ki.bitLength(), 8);
WNafPreCompInfo info = WNafUtil.precompute(ps[i], minWidth, true);
int width = Math.min(8, info.getWidth());
infos[i] = info;
wnafs[i] = WNafUtil.generateWindowNaf(width, ki);
}
return implSumOfMultiplies(negs, infos, wnafs);
}
static ECPoint implSumOfMultipliesGLV(ECPoint[] ps, BigInteger[] ks, GLVEndomorphism glvEndomorphism)
{
BigInteger n = ps[0].getCurve().getOrder();
int len = ps.length;
BigInteger[] abs = new BigInteger[len << 1];
for (int i = 0, j = 0; i < len; ++i)
{
BigInteger[] ab = glvEndomorphism.decomposeScalar(ks[i].mod(n));
abs[j++] = ab[0];
abs[j++] = ab[1];
}
if (glvEndomorphism.hasEfficientPointMap())
{
return implSumOfMultiplies(glvEndomorphism, ps, abs);
}
ECPoint[] pqs = new ECPoint[len << 1];
for (int i = 0, j = 0; i < len; ++i)
{
ECPoint p = ps[i];
ECPoint q = EndoUtil.mapPoint(glvEndomorphism, p);
pqs[j++] = p;
pqs[j++] = q;
}
return implSumOfMultiplies(pqs, abs);
}
static ECPoint implSumOfMultiplies(ECEndomorphism endomorphism, ECPoint[] ps, BigInteger[] ks)
{
int halfCount = ps.length, fullCount = halfCount << 1;
boolean[] negs = new boolean[fullCount];
WNafPreCompInfo[] infos = new WNafPreCompInfo[fullCount];
byte[][] wnafs = new byte[fullCount][];
ECPointMap pointMap = endomorphism.getPointMap();
for (int i = 0; i < halfCount; ++i)
{
int j0 = i << 1, j1 = j0 + 1;
BigInteger kj0 = ks[j0]; negs[j0] = kj0.signum() < 0; kj0 = kj0.abs();
BigInteger kj1 = ks[j1]; negs[j1] = kj1.signum() < 0; kj1 = kj1.abs();
int minWidth = WNafUtil.getWindowSize(Math.max(kj0.bitLength(), kj1.bitLength()), 8);
ECPoint P = ps[i];
WNafPreCompInfo infoP = WNafUtil.precompute(P, minWidth, true);
ECPoint Q = EndoUtil.mapPoint(endomorphism, P);
WNafPreCompInfo infoQ = WNafUtil.precomputeWithPointMap(Q, pointMap, infoP, true);
int widthP = Math.min(8, infoP.getWidth());
int widthQ = Math.min(8, infoQ.getWidth());
infos[j0] = infoP;
infos[j1] = infoQ;
wnafs[j0] = WNafUtil.generateWindowNaf(widthP, kj0);
wnafs[j1] = WNafUtil.generateWindowNaf(widthQ, kj1);
}
return implSumOfMultiplies(negs, infos, wnafs);
}
private static ECPoint implSumOfMultiplies(boolean[] negs, WNafPreCompInfo[] infos, byte[][] wnafs)
{
int len = 0, count = wnafs.length;
for (int i = 0; i < count; ++i)
{
len = Math.max(len, wnafs[i].length);
}
ECCurve curve = infos[0].getPreComp()[0].getCurve();
ECPoint infinity = curve.getInfinity();
ECPoint R = infinity;
int zeroes = 0;
for (int i = len - 1; i >= 0; --i)
{
ECPoint r = infinity;
for (int j = 0; j < count; ++j)
{
byte[] wnaf = wnafs[j];
int wi = i < wnaf.length ? wnaf[i] : 0;
if (wi != 0)
{
int n = Math.abs(wi);
WNafPreCompInfo info = infos[j];
ECPoint[] table = (wi < 0 == negs[j]) ? info.getPreComp() : info.getPreCompNeg();
r = r.add(table[n >>> 1]);
}
}
if (r == infinity)
{
++zeroes;
continue;
}
if (zeroes > 0)
{
R = R.timesPow2(zeroes);
zeroes = 0;
}
R = R.twicePlus(r);
}
if (zeroes > 0)
{
R = R.timesPow2(zeroes);
}
return R;
}
private static ECPoint implShamirsTrickFixedPoint(ECPoint p, BigInteger k, ECPoint q, BigInteger l)
{
ECCurve c = p.getCurve();
int combSize = FixedPointUtil.getCombSize(c);
if (k.bitLength() > combSize || l.bitLength() > combSize)
{
/*
* TODO The comb works best when the scalars are less than the (possibly unknown) order.
* Still, if we want to handle larger scalars, we could allow customization of the comb
* size, or alternatively we could deal with the 'extra' bits either by running the comb
* multiple times as necessary, or by using an alternative multiplier as prelude.
*/
throw new IllegalStateException("fixed-point comb doesn't support scalars larger than the curve order");
}
FixedPointPreCompInfo infoP = FixedPointUtil.precompute(p);
FixedPointPreCompInfo infoQ = FixedPointUtil.precompute(q);
ECLookupTable lookupTableP = infoP.getLookupTable();
ECLookupTable lookupTableQ = infoQ.getLookupTable();
int widthP = infoP.getWidth();
int widthQ = infoQ.getWidth();
// TODO This shouldn't normally happen, but a better "solution" is desirable anyway
if (widthP != widthQ)
{
FixedPointCombMultiplier m = new FixedPointCombMultiplier();
ECPoint r1 = m.multiply(p, k);
ECPoint r2 = m.multiply(q, l);
return r1.add(r2);
}
int width = widthP;
int d = (combSize + width - 1) / width;
ECPoint R = c.getInfinity();
int fullComb = d * width;
int[] K = Nat.fromBigInteger(fullComb, k);
int[] L = Nat.fromBigInteger(fullComb, l);
int top = fullComb - 1;
for (int i = 0; i < d; ++i)
{
int secretIndexK = 0, secretIndexL = 0;
for (int j = top - i; j >= 0; j -= d)
{
int secretBitK = K[j >>> 5] >>> (j & 0x1F);
secretIndexK ^= secretBitK >>> 1;
secretIndexK <<= 1;
secretIndexK ^= secretBitK;
int secretBitL = L[j >>> 5] >>> (j & 0x1F);
secretIndexL ^= secretBitL >>> 1;
secretIndexL <<= 1;
secretIndexL ^= secretBitL;
}
ECPoint addP = lookupTableP.lookupVar(secretIndexK);
ECPoint addQ = lookupTableQ.lookupVar(secretIndexL);
ECPoint T = addP.add(addQ);
R = R.twicePlus(T);
}
return R.add(infoP.getOffset()).add(infoQ.getOffset());
}
}