package com.google.bitcoin.bouncycastle.math.ec; import java.math.BigInteger; /** * Class holding methods for point multiplication based on the window * τ-adic nonadjacent form (WTNAF). The algorithms are based on the * paper "Improved Algorithms for Arithmetic on Anomalous Binary Curves" * by Jerome A. Solinas. The paper first appeared in the Proceedings of * Crypto 1997. */ class Tnaf { private static final BigInteger MINUS_ONE = ECConstants.ONE.negate(); private static final BigInteger MINUS_TWO = ECConstants.TWO.negate(); private static final BigInteger MINUS_THREE = ECConstants.THREE.negate(); /** * The window width of WTNAF. The standard value of 4 is slightly less * than optimal for running time, but keeps space requirements for * precomputation low. For typical curves, a value of 5 or 6 results in * a better running time. When changing this value, the * <code>α<sub>u</sub></code>'s must be computed differently, see * e.g. "Guide to Elliptic Curve Cryptography", Darrel Hankerson, * Alfred Menezes, Scott Vanstone, Springer-Verlag New York Inc., 2004, * p. 121-122 */ public static final byte WIDTH = 4; /** * 2<sup>4</sup> */ public static final byte POW_2_WIDTH = 16; /** * The <code>α<sub>u</sub></code>'s for <code>a=0</code> as an array * of <code>ZTauElement</code>s. */ public static final ZTauElement[] alpha0 = { null, new ZTauElement(ECConstants.ONE, ECConstants.ZERO), null, new ZTauElement(MINUS_THREE, MINUS_ONE), null, new ZTauElement(MINUS_ONE, MINUS_ONE), null, new ZTauElement(ECConstants.ONE, MINUS_ONE), null }; /** * The <code>α<sub>u</sub></code>'s for <code>a=0</code> as an array * of TNAFs. */ public static final byte[][] alpha0Tnaf = { null, {1}, null, {-1, 0, 1}, null, {1, 0, 1}, null, {-1, 0, 0, 1} }; /** * The <code>α<sub>u</sub></code>'s for <code>a=1</code> as an array * of <code>ZTauElement</code>s. */ public static final ZTauElement[] alpha1 = {null, new ZTauElement(ECConstants.ONE, ECConstants.ZERO), null, new ZTauElement(MINUS_THREE, ECConstants.ONE), null, new ZTauElement(MINUS_ONE, ECConstants.ONE), null, new ZTauElement(ECConstants.ONE, ECConstants.ONE), null }; /** * The <code>α<sub>u</sub></code>'s for <code>a=1</code> as an array * of TNAFs. */ public static final byte[][] alpha1Tnaf = { null, {1}, null, {-1, 0, 1}, null, {1, 0, 1}, null, {-1, 0, 0, -1} }; /** * Computes the norm of an element <code>λ</code> of * <code><b>Z</b>[τ]</code>. * @param mu The parameter <code>μ</code> of the elliptic curve. * @param lambda The element <code>λ</code> of * <code><b>Z</b>[τ]</code>. * @return The norm of <code>λ</code>. */ public static BigInteger norm(final byte mu, ZTauElement lambda) { BigInteger norm; // s1 = u^2 BigInteger s1 = lambda.u.multiply(lambda.u); // s2 = u * v BigInteger s2 = lambda.u.multiply(lambda.v); // s3 = 2 * v^2 BigInteger s3 = lambda.v.multiply(lambda.v).shiftLeft(1); if (mu == 1) { norm = s1.add(s2).add(s3); } else if (mu == -1) { norm = s1.subtract(s2).add(s3); } else { throw new IllegalArgumentException("mu must be 1 or -1"); } return norm; } /** * Computes the norm of an element <code>λ</code> of * <code><b>R</b>[τ]</code>, where <code>λ = u + vτ</code> * and <code>u</code> and <code>u</code> are real numbers (elements of * <code><b>R</b></code>). * @param mu The parameter <code>μ</code> of the elliptic curve. * @param u The real part of the element <code>λ</code> of * <code><b>R</b>[τ]</code>. * @param v The <code>τ</code>-adic part of the element * <code>λ</code> of <code><b>R</b>[τ]</code>. * @return The norm of <code>λ</code>. */ public static SimpleBigDecimal norm(final byte mu, SimpleBigDecimal u, SimpleBigDecimal v) { SimpleBigDecimal norm; // s1 = u^2 SimpleBigDecimal s1 = u.multiply(u); // s2 = u * v SimpleBigDecimal s2 = u.multiply(v); // s3 = 2 * v^2 SimpleBigDecimal s3 = v.multiply(v).shiftLeft(1); if (mu == 1) { norm = s1.add(s2).add(s3); } else if (mu == -1) { norm = s1.subtract(s2).add(s3); } else { throw new IllegalArgumentException("mu must be 1 or -1"); } return norm; } /** * Rounds an element <code>λ</code> of <code><b>R</b>[τ]</code> * to an element of <code><b>Z</b>[τ]</code>, such that their difference * has minimal norm. <code>λ</code> is given as * <code>λ = λ<sub>0</sub> + λ<sub>1</sub>τ</code>. * @param lambda0 The component <code>λ<sub>0</sub></code>. * @param lambda1 The component <code>λ<sub>1</sub></code>. * @param mu The parameter <code>μ</code> of the elliptic curve. Must * equal 1 or -1. * @return The rounded element of <code><b>Z</b>[τ]</code>. * @throws IllegalArgumentException if <code>lambda0</code> and * <code>lambda1</code> do not have same scale. */ public static ZTauElement round(SimpleBigDecimal lambda0, SimpleBigDecimal lambda1, byte mu) { int scale = lambda0.getScale(); if (lambda1.getScale() != scale) { throw new IllegalArgumentException("lambda0 and lambda1 do not " + "have same scale"); } if (!((mu == 1) || (mu == -1))) { throw new IllegalArgumentException("mu must be 1 or -1"); } BigInteger f0 = lambda0.round(); BigInteger f1 = lambda1.round(); SimpleBigDecimal eta0 = lambda0.subtract(f0); SimpleBigDecimal eta1 = lambda1.subtract(f1); // eta = 2*eta0 + mu*eta1 SimpleBigDecimal eta = eta0.add(eta0); if (mu == 1) { eta = eta.add(eta1); } else { // mu == -1 eta = eta.subtract(eta1); } // check1 = eta0 - 3*mu*eta1 // check2 = eta0 + 4*mu*eta1 SimpleBigDecimal threeEta1 = eta1.add(eta1).add(eta1); SimpleBigDecimal fourEta1 = threeEta1.add(eta1); SimpleBigDecimal check1; SimpleBigDecimal check2; if (mu == 1) { check1 = eta0.subtract(threeEta1); check2 = eta0.add(fourEta1); } else { // mu == -1 check1 = eta0.add(threeEta1); check2 = eta0.subtract(fourEta1); } byte h0 = 0; byte h1 = 0; // if eta >= 1 if (eta.compareTo(ECConstants.ONE) >= 0) { if (check1.compareTo(MINUS_ONE) < 0) { h1 = mu; } else { h0 = 1; } } else { // eta < 1 if (check2.compareTo(ECConstants.TWO) >= 0) { h1 = mu; } } // if eta < -1 if (eta.compareTo(MINUS_ONE) < 0) { if (check1.compareTo(ECConstants.ONE) >= 0) { h1 = (byte)-mu; } else { h0 = -1; } } else { // eta >= -1 if (check2.compareTo(MINUS_TWO) < 0) { h1 = (byte)-mu; } } BigInteger q0 = f0.add(BigInteger.valueOf(h0)); BigInteger q1 = f1.add(BigInteger.valueOf(h1)); return new ZTauElement(q0, q1); } /** * Approximate division by <code>n</code>. For an integer * <code>k</code>, the value <code>λ = s k / n</code> is * computed to <code>c</code> bits of accuracy. * @param k The parameter <code>k</code>. * @param s The curve parameter <code>s<sub>0</sub></code> or * <code>s<sub>1</sub></code>. * @param vm The Lucas Sequence element <code>V<sub>m</sub></code>. * @param a The parameter <code>a</code> of the elliptic curve. * @param m The bit length of the finite field * <code><b>F</b><sub>m</sub></code>. * @param c The number of bits of accuracy, i.e. the scale of the returned * <code>SimpleBigDecimal</code>. * @return The value <code>λ = s k / n</code> computed to * <code>c</code> bits of accuracy. */ public static SimpleBigDecimal approximateDivisionByN(BigInteger k, BigInteger s, BigInteger vm, byte a, int m, int c) { int _k = (m + 5)/2 + c; BigInteger ns = k.shiftRight(m - _k - 2 + a); BigInteger gs = s.multiply(ns); BigInteger hs = gs.shiftRight(m); BigInteger js = vm.multiply(hs); BigInteger gsPlusJs = gs.add(js); BigInteger ls = gsPlusJs.shiftRight(_k-c); if (gsPlusJs.testBit(_k-c-1)) { // round up ls = ls.add(ECConstants.ONE); } return new SimpleBigDecimal(ls, c); } /** * Computes the <code>τ</code>-adic NAF (non-adjacent form) of an * element <code>λ</code> of <code><b>Z</b>[τ]</code>. * @param mu The parameter <code>μ</code> of the elliptic curve. * @param lambda The element <code>λ</code> of * <code><b>Z</b>[τ]</code>. * @return The <code>τ</code>-adic NAF of <code>λ</code>. */ public static byte[] tauAdicNaf(byte mu, ZTauElement lambda) { if (!((mu == 1) || (mu == -1))) { throw new IllegalArgumentException("mu must be 1 or -1"); } BigInteger norm = norm(mu, lambda); // Ceiling of log2 of the norm int log2Norm = norm.bitLength(); // If length(TNAF) > 30, then length(TNAF) < log2Norm + 3.52 int maxLength = log2Norm > 30 ? log2Norm + 4 : 34; // The array holding the TNAF byte[] u = new byte[maxLength]; int i = 0; // The actual length of the TNAF int length = 0; BigInteger r0 = lambda.u; BigInteger r1 = lambda.v; while(!((r0.equals(ECConstants.ZERO)) && (r1.equals(ECConstants.ZERO)))) { // If r0 is odd if (r0.testBit(0)) { u[i] = (byte) ECConstants.TWO.subtract((r0.subtract(r1.shiftLeft(1))).mod(ECConstants.FOUR)).intValue(); // r0 = r0 - u[i] if (u[i] == 1) { r0 = r0.clearBit(0); } else { // u[i] == -1 r0 = r0.add(ECConstants.ONE); } length = i; } else { u[i] = 0; } BigInteger t = r0; BigInteger s = r0.shiftRight(1); if (mu == 1) { r0 = r1.add(s); } else { // mu == -1 r0 = r1.subtract(s); } r1 = t.shiftRight(1).negate(); i++; } length++; // Reduce the TNAF array to its actual length byte[] tnaf = new byte[length]; System.arraycopy(u, 0, tnaf, 0, length); return tnaf; } /** * Applies the operation <code>τ()</code> to an * <code>ECPoint.F2m</code>. * @param p The ECPoint.F2m to which <code>τ()</code> is applied. * @return <code>τ(p)</code> */ public static ECPoint.F2m tau(ECPoint.F2m p) { if (p.isInfinity()) { return p; } ECFieldElement x = p.getX(); ECFieldElement y = p.getY(); return new ECPoint.F2m(p.getCurve(), x.square(), y.square(), p.isCompressed()); } /** * Returns the parameter <code>μ</code> of the elliptic curve. * @param curve The elliptic curve from which to obtain <code>μ</code>. * The curve must be a Koblitz curve, i.e. <code>a</code> equals * <code>0</code> or <code>1</code> and <code>b</code> equals * <code>1</code>. * @return <code>μ</code> of the elliptic curve. * @throws IllegalArgumentException if the given ECCurve is not a Koblitz * curve. */ public static byte getMu(ECCurve.F2m curve) { BigInteger a = curve.getA().toBigInteger(); byte mu; if (a.equals(ECConstants.ZERO)) { mu = -1; } else if (a.equals(ECConstants.ONE)) { mu = 1; } else { throw new IllegalArgumentException("No Koblitz curve (ABC), " + "TNAF multiplication not possible"); } return mu; } /** * Calculates the Lucas Sequence elements <code>U<sub>k-1</sub></code> and * <code>U<sub>k</sub></code> or <code>V<sub>k-1</sub></code> and * <code>V<sub>k</sub></code>. * @param mu The parameter <code>μ</code> of the elliptic curve. * @param k The index of the second element of the Lucas Sequence to be * returned. * @param doV If set to true, computes <code>V<sub>k-1</sub></code> and * <code>V<sub>k</sub></code>, otherwise <code>U<sub>k-1</sub></code> and * <code>U<sub>k</sub></code>. * @return An array with 2 elements, containing <code>U<sub>k-1</sub></code> * and <code>U<sub>k</sub></code> or <code>V<sub>k-1</sub></code> * and <code>V<sub>k</sub></code>. */ public static BigInteger[] getLucas(byte mu, int k, boolean doV) { if (!((mu == 1) || (mu == -1))) { throw new IllegalArgumentException("mu must be 1 or -1"); } BigInteger u0; BigInteger u1; BigInteger u2; if (doV) { u0 = ECConstants.TWO; u1 = BigInteger.valueOf(mu); } else { u0 = ECConstants.ZERO; u1 = ECConstants.ONE; } for (int i = 1; i < k; i++) { // u2 = mu*u1 - 2*u0; BigInteger s = null; if (mu == 1) { s = u1; } else { // mu == -1 s = u1.negate(); } u2 = s.subtract(u0.shiftLeft(1)); u0 = u1; u1 = u2; // System.out.println(i + ": " + u2); // System.out.println(); } BigInteger[] retVal = {u0, u1}; return retVal; } /** * Computes the auxiliary value <code>t<sub>w</sub></code>. If the width is * 4, then for <code>mu = 1</code>, <code>t<sub>w</sub> = 6</code> and for * <code>mu = -1</code>, <code>t<sub>w</sub> = 10</code> * @param mu The parameter <code>μ</code> of the elliptic curve. * @param w The window width of the WTNAF. * @return the auxiliary value <code>t<sub>w</sub></code> */ public static BigInteger getTw(byte mu, int w) { if (w == 4) { if (mu == 1) { return BigInteger.valueOf(6); } else { // mu == -1 return BigInteger.valueOf(10); } } else { // For w <> 4, the values must be computed BigInteger[] us = getLucas(mu, w, false); BigInteger twoToW = ECConstants.ZERO.setBit(w); BigInteger u1invert = us[1].modInverse(twoToW); BigInteger tw; tw = ECConstants.TWO.multiply(us[0]).multiply(u1invert).mod(twoToW); // System.out.println("mu = " + mu); // System.out.println("tw = " + tw); return tw; } } /** * Computes the auxiliary values <code>s<sub>0</sub></code> and * <code>s<sub>1</sub></code> used for partial modular reduction. * @param curve The elliptic curve for which to compute * <code>s<sub>0</sub></code> and <code>s<sub>1</sub></code>. * @throws IllegalArgumentException if <code>curve</code> is not a * Koblitz curve (Anomalous Binary Curve, ABC). */ public static BigInteger[] getSi(ECCurve.F2m curve) { if (!curve.isKoblitz()) { throw new IllegalArgumentException("si is defined for Koblitz curves only"); } int m = curve.getM(); int a = curve.getA().toBigInteger().intValue(); byte mu = curve.getMu(); int h = curve.getH().intValue(); int index = m + 3 - a; BigInteger[] ui = getLucas(mu, index, false); BigInteger dividend0; BigInteger dividend1; if (mu == 1) { dividend0 = ECConstants.ONE.subtract(ui[1]); dividend1 = ECConstants.ONE.subtract(ui[0]); } else if (mu == -1) { dividend0 = ECConstants.ONE.add(ui[1]); dividend1 = ECConstants.ONE.add(ui[0]); } else { throw new IllegalArgumentException("mu must be 1 or -1"); } BigInteger[] si = new BigInteger[2]; if (h == 2) { si[0] = dividend0.shiftRight(1); si[1] = dividend1.shiftRight(1).negate(); } else if (h == 4) { si[0] = dividend0.shiftRight(2); si[1] = dividend1.shiftRight(2).negate(); } else { throw new IllegalArgumentException("h (Cofactor) must be 2 or 4"); } return si; } /** * Partial modular reduction modulo * <code>(τ<sup>m</sup> - 1)/(τ - 1)</code>. * @param k The integer to be reduced. * @param m The bitlength of the underlying finite field. * @param a The parameter <code>a</code> of the elliptic curve. * @param s The auxiliary values <code>s<sub>0</sub></code> and * <code>s<sub>1</sub></code>. * @param mu The parameter μ of the elliptic curve. * @param c The precision (number of bits of accuracy) of the partial * modular reduction. * @return <code>ρ := k partmod (τ<sup>m</sup> - 1)/(τ - 1)</code> */ public static ZTauElement partModReduction(BigInteger k, int m, byte a, BigInteger[] s, byte mu, byte c) { // d0 = s[0] + mu*s[1]; mu is either 1 or -1 BigInteger d0; if (mu == 1) { d0 = s[0].add(s[1]); } else { d0 = s[0].subtract(s[1]); } BigInteger[] v = getLucas(mu, m, true); BigInteger vm = v[1]; SimpleBigDecimal lambda0 = approximateDivisionByN( k, s[0], vm, a, m, c); SimpleBigDecimal lambda1 = approximateDivisionByN( k, s[1], vm, a, m, c); ZTauElement q = round(lambda0, lambda1, mu); // r0 = n - d0*q0 - 2*s1*q1 BigInteger r0 = k.subtract(d0.multiply(q.u)).subtract( BigInteger.valueOf(2).multiply(s[1]).multiply(q.v)); // r1 = s1*q0 - s0*q1 BigInteger r1 = s[1].multiply(q.u).subtract(s[0].multiply(q.v)); return new ZTauElement(r0, r1); } /** * Multiplies a {@link com.google.bitcoin.bouncycastle.math.ec.ECPoint.F2m ECPoint.F2m} * by a <code>BigInteger</code> using the reduced <code>τ</code>-adic * NAF (RTNAF) method. * @param p The ECPoint.F2m to multiply. * @param k The <code>BigInteger</code> by which to multiply <code>p</code>. * @return <code>k * p</code> */ public static ECPoint.F2m multiplyRTnaf(ECPoint.F2m p, BigInteger k) { ECCurve.F2m curve = (ECCurve.F2m) p.getCurve(); int m = curve.getM(); byte a = (byte) curve.getA().toBigInteger().intValue(); byte mu = curve.getMu(); BigInteger[] s = curve.getSi(); ZTauElement rho = partModReduction(k, m, a, s, mu, (byte)10); return multiplyTnaf(p, rho); } /** * Multiplies a {@link com.google.bitcoin.bouncycastle.math.ec.ECPoint.F2m ECPoint.F2m} * by an element <code>λ</code> of <code><b>Z</b>[τ]</code> * using the <code>τ</code>-adic NAF (TNAF) method. * @param p The ECPoint.F2m to multiply. * @param lambda The element <code>λ</code> of * <code><b>Z</b>[τ]</code>. * @return <code>λ * p</code> */ public static ECPoint.F2m multiplyTnaf(ECPoint.F2m p, ZTauElement lambda) { ECCurve.F2m curve = (ECCurve.F2m)p.getCurve(); byte mu = curve.getMu(); byte[] u = tauAdicNaf(mu, lambda); ECPoint.F2m q = multiplyFromTnaf(p, u); return q; } /** * Multiplies a {@link com.google.bitcoin.bouncycastle.math.ec.ECPoint.F2m ECPoint.F2m} * by an element <code>λ</code> of <code><b>Z</b>[τ]</code> * using the <code>τ</code>-adic NAF (TNAF) method, given the TNAF * of <code>λ</code>. * @param p The ECPoint.F2m to multiply. * @param u The the TNAF of <code>λ</code>.. * @return <code>λ * p</code> */ public static ECPoint.F2m multiplyFromTnaf(ECPoint.F2m p, byte[] u) { ECCurve.F2m curve = (ECCurve.F2m)p.getCurve(); ECPoint.F2m q = (ECPoint.F2m) curve.getInfinity(); for (int i = u.length - 1; i >= 0; i--) { q = tau(q); if (u[i] == 1) { q = (ECPoint.F2m)q.addSimple(p); } else if (u[i] == -1) { q = (ECPoint.F2m)q.subtractSimple(p); } } return q; } /** * Computes the <code>[τ]</code>-adic window NAF of an element * <code>λ</code> of <code><b>Z</b>[τ]</code>. * @param mu The parameter μ of the elliptic curve. * @param lambda The element <code>λ</code> of * <code><b>Z</b>[τ]</code> of which to compute the * <code>[τ]</code>-adic NAF. * @param width The window width of the resulting WNAF. * @param pow2w 2<sup>width</sup>. * @param tw The auxiliary value <code>t<sub>w</sub></code>. * @param alpha The <code>α<sub>u</sub></code>'s for the window width. * @return The <code>[τ]</code>-adic window NAF of * <code>λ</code>. */ public static byte[] tauAdicWNaf(byte mu, ZTauElement lambda, byte width, BigInteger pow2w, BigInteger tw, ZTauElement[] alpha) { if (!((mu == 1) || (mu == -1))) { throw new IllegalArgumentException("mu must be 1 or -1"); } BigInteger norm = norm(mu, lambda); // Ceiling of log2 of the norm int log2Norm = norm.bitLength(); // If length(TNAF) > 30, then length(TNAF) < log2Norm + 3.52 int maxLength = log2Norm > 30 ? log2Norm + 4 + width : 34 + width; // The array holding the TNAF byte[] u = new byte[maxLength]; // 2^(width - 1) BigInteger pow2wMin1 = pow2w.shiftRight(1); // Split lambda into two BigIntegers to simplify calculations BigInteger r0 = lambda.u; BigInteger r1 = lambda.v; int i = 0; // while lambda <> (0, 0) while (!((r0.equals(ECConstants.ZERO))&&(r1.equals(ECConstants.ZERO)))) { // if r0 is odd if (r0.testBit(0)) { // uUnMod = r0 + r1*tw mod 2^width BigInteger uUnMod = r0.add(r1.multiply(tw)).mod(pow2w); byte uLocal; // if uUnMod >= 2^(width - 1) if (uUnMod.compareTo(pow2wMin1) >= 0) { uLocal = (byte) uUnMod.subtract(pow2w).intValue(); } else { uLocal = (byte) uUnMod.intValue(); } // uLocal is now in [-2^(width-1), 2^(width-1)-1] u[i] = uLocal; boolean s = true; if (uLocal < 0) { s = false; uLocal = (byte)-uLocal; } // uLocal is now >= 0 if (s) { r0 = r0.subtract(alpha[uLocal].u); r1 = r1.subtract(alpha[uLocal].v); } else { r0 = r0.add(alpha[uLocal].u); r1 = r1.add(alpha[uLocal].v); } } else { u[i] = 0; } BigInteger t = r0; if (mu == 1) { r0 = r1.add(r0.shiftRight(1)); } else { // mu == -1 r0 = r1.subtract(r0.shiftRight(1)); } r1 = t.shiftRight(1).negate(); i++; } return u; } /** * Does the precomputation for WTNAF multiplication. * @param p The <code>ECPoint</code> for which to do the precomputation. * @param a The parameter <code>a</code> of the elliptic curve. * @return The precomputation array for <code>p</code>. */ public static ECPoint.F2m[] getPreComp(ECPoint.F2m p, byte a) { ECPoint.F2m[] pu; pu = new ECPoint.F2m[16]; pu[1] = p; byte[][] alphaTnaf; if (a == 0) { alphaTnaf = Tnaf.alpha0Tnaf; } else { // a == 1 alphaTnaf = Tnaf.alpha1Tnaf; } int precompLen = alphaTnaf.length; for (int i = 3; i < precompLen; i = i + 2) { pu[i] = Tnaf.multiplyFromTnaf(p, alphaTnaf[i]); } return pu; } }