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package org.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
* αu'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;
/**
* 24
*/
public static final byte POW_2_WIDTH = 16;
/**
* The αu's for a=0 as an array
* of ZTauElements.
*/
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 αu's for a=0 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 αu's for a=1 as an array
* of ZTauElements.
*/
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 αu's for a=1 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 λ of
* Z[τ].
* @param mu The parameter μ of the elliptic curve.
* @param lambda The element λ of
* Z[τ].
* @return The norm of λ.
*/
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 λ of
* R[τ], where λ = u + vτ
* and u and u are real numbers (elements of
* R).
* @param mu The parameter μ of the elliptic curve.
* @param u The real part of the element λ of
* R[τ].
* @param v The τ-adic part of the element
* λ of R[τ].
* @return The norm of λ.
*/
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 λ of R[τ]
* to an element of Z[τ], such that their difference
* has minimal norm. λ is given as
* λ = λ0 + λ1τ.
* @param lambda0 The component λ0.
* @param lambda1 The component λ1.
* @param mu The parameter μ of the elliptic curve. Must
* equal 1 or -1.
* @return The rounded element of Z[τ].
* @throws IllegalArgumentException if lambda0 and
* lambda1 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 n. For an integer
* k, the value λ = s k / n is
* computed to c bits of accuracy.
* @param k The parameter k.
* @param s The curve parameter s0 or
* s1.
* @param vm The Lucas Sequence element Vm.
* @param a The parameter a of the elliptic curve.
* @param m The bit length of the finite field
* Fm.
* @param c The number of bits of accuracy, i.e. the scale of the returned
* SimpleBigDecimal.
* @return The value λ = s k / n computed to
* c 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 τ-adic NAF (non-adjacent form) of an
* element λ of Z[τ].
* @param mu The parameter μ of the elliptic curve.
* @param lambda The element λ of
* Z[τ].
* @return The τ-adic NAF of λ.
*/
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 τ() to an
* ECPoint.F2m.
* @param p The ECPoint.F2m to which τ() is applied.
* @return τ(p)
*/
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 μ of the elliptic curve.
* @param curve The elliptic curve from which to obtain μ.
* The curve must be a Koblitz curve, i.e. a equals
* 0 or 1 and b equals
* 1.
* @return μ 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 Uk-1 and
* Uk or Vk-1 and
* Vk.
* @param mu The parameter μ 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 Vk-1 and
* Vk, otherwise Uk-1 and
* Uk.
* @return An array with 2 elements, containing Uk-1
* and Uk or Vk-1
* and Vk.
*/
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 tw. If the width is
* 4, then for mu = 1, tw = 6 and for
* mu = -1, tw = 10
* @param mu The parameter μ of the elliptic curve.
* @param w The window width of the WTNAF.
* @return the auxiliary value tw
*/
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 s0 and
* s1 used for partial modular reduction.
* @param curve The elliptic curve for which to compute
* s0 and s1.
* @throws IllegalArgumentException if curve 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
* (τm - 1)/(τ - 1).
* @param k The integer to be reduced.
* @param m The bitlength of the underlying finite field.
* @param a The parameter a of the elliptic curve.
* @param s The auxiliary values s0 and
* s1.
* @param mu The parameter μ of the elliptic curve.
* @param c The precision (number of bits of accuracy) of the partial
* modular reduction.
* @return ρ := k partmod (τm - 1)/(τ - 1)
*/
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 org.bouncycastle.math.ec.ECPoint.F2m ECPoint.F2m}
* by a BigInteger using the reduced τ-adic
* NAF (RTNAF) method.
* @param p The ECPoint.F2m to multiply.
* @param k The BigInteger by which to multiply p.
* @return k * p
*/
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 org.bouncycastle.math.ec.ECPoint.F2m ECPoint.F2m}
* by an element λ of Z[τ]
* using the τ-adic NAF (TNAF) method.
* @param p The ECPoint.F2m to multiply.
* @param lambda The element λ of
* Z[τ].
* @return λ * p
*/
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 org.bouncycastle.math.ec.ECPoint.F2m ECPoint.F2m}
* by an element λ of Z[τ]
* using the τ-adic NAF (TNAF) method, given the TNAF
* of λ.
* @param p The ECPoint.F2m to multiply.
* @param u The the TNAF of λ..
* @return λ * p
*/
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 [τ]-adic window NAF of an element
* λ of Z[τ].
* @param mu The parameter μ of the elliptic curve.
* @param lambda The element λ of
* Z[τ] of which to compute the
* [τ]-adic NAF.
* @param width The window width of the resulting WNAF.
* @param pow2w 2width.
* @param tw The auxiliary value tw.
* @param alpha The αu's for the window width.
* @return The [τ]-adic window NAF of
* λ.
*/
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 ECPoint for which to do the precomputation.
* @param a The parameter a of the elliptic curve.
* @return The precomputation array for p.
*/
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;
}
}