| package org.bouncycastle.math.ec; |
| |
| import java.math.BigInteger; |
| |
| import org.bouncycastle.math.ec.endo.ECEndomorphism; |
| import org.bouncycastle.math.ec.endo.GLVEndomorphism; |
| import org.bouncycastle.math.field.FiniteField; |
| import org.bouncycastle.math.field.PolynomialExtensionField; |
| |
| 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 validatePoint(implSumOfMultipliesGLV(imported, ks, (GLVEndomorphism)endomorphism)); |
| } |
| |
| return validatePoint(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 validatePoint(P.multiply(a).add(Q.multiply(b))); |
| } |
| } |
| |
| ECEndomorphism endomorphism = cp.getEndomorphism(); |
| if (endomorphism instanceof GLVEndomorphism) |
| { |
| return validatePoint( |
| implSumOfMultipliesGLV(new ECPoint[]{ P, Q }, new BigInteger[]{ a, b }, (GLVEndomorphism)endomorphism)); |
| } |
| |
| return validatePoint(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) |
| * <pre> |
| * 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 |
| * </pre> |
| */ |
| public static ECPoint shamirsTrick(ECPoint P, BigInteger k, |
| ECPoint Q, BigInteger l) |
| { |
| ECCurve cp = P.getCurve(); |
| Q = importPoint(cp, Q); |
| |
| return validatePoint(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. |
| * |
| * @param p |
| * The point to multiply. |
| * @param k |
| * The multiplier. |
| * @return The result of the point multiplication <code>kP</code>. |
| */ |
| 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 IllegalArgumentException("Invalid point"); |
| } |
| |
| 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; |
| |
| k = k.abs(); |
| l = l.abs(); |
| |
| int widthP = Math.max(2, Math.min(16, WNafUtil.getWindowSize(k.bitLength()))); |
| int widthQ = Math.max(2, Math.min(16, WNafUtil.getWindowSize(l.bitLength()))); |
| |
| WNafPreCompInfo infoP = WNafUtil.precompute(P, widthP, true); |
| WNafPreCompInfo infoQ = WNafUtil.precompute(Q, widthQ, true); |
| |
| 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); |
| } |
| |
| static ECPoint implShamirsTrickWNaf(ECPoint P, BigInteger k, ECPointMap pointMapQ, BigInteger l) |
| { |
| boolean negK = k.signum() < 0, negL = l.signum() < 0; |
| |
| k = k.abs(); |
| l = l.abs(); |
| |
| int width = Math.max(2, Math.min(16, WNafUtil.getWindowSize(Math.max(k.bitLength(), l.bitLength())))); |
| |
| ECPoint Q = WNafUtil.mapPointWithPrecomp(P, width, true, pointMapQ); |
| WNafPreCompInfo infoP = WNafUtil.getWNafPreCompInfo(P); |
| WNafPreCompInfo infoQ = WNafUtil.getWNafPreCompInfo(Q); |
| |
| 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(width, k); |
| byte[] wnafQ = WNafUtil.generateWindowNaf(width, 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 width = Math.max(2, Math.min(16, WNafUtil.getWindowSize(ki.bitLength()))); |
| infos[i] = WNafUtil.precompute(ps[i], width, true); |
| 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]; |
| } |
| |
| ECPointMap pointMap = glvEndomorphism.getPointMap(); |
| if (glvEndomorphism.hasEfficientPointMap()) |
| { |
| return ECAlgorithms.implSumOfMultiplies(ps, pointMap, abs); |
| } |
| |
| ECPoint[] pqs = new ECPoint[len << 1]; |
| for (int i = 0, j = 0; i < len; ++i) |
| { |
| ECPoint p = ps[i], q = pointMap.map(p); |
| pqs[j++] = p; |
| pqs[j++] = q; |
| } |
| |
| return ECAlgorithms.implSumOfMultiplies(pqs, abs); |
| |
| } |
| |
| static ECPoint implSumOfMultiplies(ECPoint[] ps, ECPointMap pointMap, BigInteger[] ks) |
| { |
| int halfCount = ps.length, fullCount = halfCount << 1; |
| |
| boolean[] negs = new boolean[fullCount]; |
| WNafPreCompInfo[] infos = new WNafPreCompInfo[fullCount]; |
| byte[][] wnafs = new byte[fullCount][]; |
| |
| 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 width = Math.max(2, Math.min(16, WNafUtil.getWindowSize(Math.max(kj0.bitLength(), kj1.bitLength())))); |
| |
| ECPoint P = ps[i], Q = WNafUtil.mapPointWithPrecomp(P, width, true, pointMap); |
| infos[j0] = WNafUtil.getWNafPreCompInfo(P); |
| infos[j1] = WNafUtil.getWNafPreCompInfo(Q); |
| wnafs[j0] = WNafUtil.generateWindowNaf(width, kj0); |
| wnafs[j1] = WNafUtil.generateWindowNaf(width, 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; |
| } |
| } |
