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• ECAlgorithms.java

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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 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. * * @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; 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; } }