org.bouncycastle/bcprov-jdk15/1.46 : org/bouncycastle/math/ec/Tnaf.java

Tnaf
http://www.bouncycastle.org/java.html: 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.
Bouncy Castle Licence
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 holding methods for point multiplication based on the window
 * &tau;-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
α_u'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
/**
     * 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>&alpha;<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;

    2⁴
/**
     * 2<sup>4</sup>
     */
    public static final byte POW_2_WIDTH = 16;

    The α_u's for a=0 as an array
of ZTauElements.
/**
     * The <code>&alpha;<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 α_u's for a=0 as an array
of TNAFs.
/**
     * The <code>&alpha;<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 α_u's for a=1 as an array
of ZTauElements.
/**
     * The <code>&alpha;<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 α_u's for a=1 as an array
of TNAFs.
/**
     * The <code>&alpha;<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 λ of
Z[τ].
Params: mu – The parameter μ of the elliptic curve.
lambda – The element λ of
Z[τ].
Returns: The norm of λ./**
     * Computes the norm of an element <code>&lambda;</code> of
     * <code><b>Z</b>[&tau;]</code>.
     * @param mu The parameter <code>&mu;</code> of the elliptic curve.
     * @param lambda The element <code>&lambda;</code> of
     * <code><b>Z</b>[&tau;]</code>.
     * @return The norm of <code>&lambda;</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 λ of
R[τ], where λ = u + vτ
and u and u are real numbers (elements of
R). 
Params: mu – The parameter μ of the elliptic curve.
u – The real part of the element λ of
R[τ].
v – The τ-adic part of the element
λ of R[τ].
Returns: The norm of λ./**
     * Computes the norm of an element <code>&lambda;</code> of
     * <code><b>R</b>[&tau;]</code>, where <code>&lambda; = u + v&tau;</code>
     * and <code>u</code> and <code>u</code> are real numbers (elements of
     * <code><b>R</b></code>). 
     * @param mu The parameter <code>&mu;</code> of the elliptic curve.
     * @param u The real part of the element <code>&lambda;</code> of
     * <code><b>R</b>[&tau;]</code>.
     * @param v The <code>&tau;</code>-adic part of the element
     * <code>&lambda;</code> of <code><b>R</b>[&tau;]</code>.
     * @return The norm of <code>&lambda;</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 λ of R[τ]
to an element of Z[τ], such that their difference
has minimal norm. λ is given as
λ = λ₀ + λ₁τ.
Params: lambda0 – The component λ₀.
lambda1 – The component λ₁.
mu – The parameter μ of the elliptic curve. Must
equal 1 or -1.
Throws: IllegalArgumentException – if lambda0 and
lambda1 do not have same scale.
Returns: The rounded element of Z[τ]./**
     * Rounds an element <code>&lambda;</code> of <code><b>R</b>[&tau;]</code>
     * to an element of <code><b>Z</b>[&tau;]</code>, such that their difference
     * has minimal norm. <code>&lambda;</code> is given as
     * <code>&lambda; = &lambda;<sub>0</sub> + &lambda;<sub>1</sub>&tau;</code>.
     * @param lambda0 The component <code>&lambda;<sub>0</sub></code>.
     * @param lambda1 The component <code>&lambda;<sub>1</sub></code>.
     * @param mu The parameter <code>&mu;</code> of the elliptic curve. Must
     * equal 1 or -1.
     * @return The rounded element of <code><b>Z</b>[&tau;]</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 n. For an integer
k, the value λ = s k / n is
computed to c bits of accuracy.
Params: k – The parameter k.
s – The curve parameter s₀ or
s₁.
vm – The Lucas Sequence element V_m.
a – The parameter a of the elliptic curve.
m – The bit length of the finite field
F_m.
c – The number of bits of accuracy, i.e. the scale of the returned
SimpleBigDecimal.
Returns: The value λ = s k / n computed to
c bits of accuracy./**
     * Approximate division by <code>n</code>. For an integer
     * <code>k</code>, the value <code>&lambda; = 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>&lambda; = 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 τ-adic NAF (non-adjacent form) of an
element λ of Z[τ].
Params: mu – The parameter μ of the elliptic curve.
lambda – The element λ of
Z[τ].
Returns: The τ-adic NAF of λ./**
     * Computes the <code>&tau;</code>-adic NAF (non-adjacent form) of an
     * element <code>&lambda;</code> of <code><b>Z</b>[&tau;]</code>.
     * @param mu The parameter <code>&mu;</code> of the elliptic curve.
     * @param lambda The element <code>&lambda;</code> of
     * <code><b>Z</b>[&tau;]</code>.
     * @return The <code>&tau;</code>-adic NAF of <code>&lambda;</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 τ() to an
ECPoint.F2m. 
Params: p – The ECPoint.F2m to which τ() is applied.
Returns: τ(p)/**
     * Applies the operation <code>&tau;()</code> to an
     * <code>ECPoint.F2m</code>. 
     * @param p The ECPoint.F2m to which <code>&tau;()</code> is applied.
     * @return <code>&tau;(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 μ of the elliptic curve.
Params: 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.
Throws: IllegalArgumentException – if the given ECCurve is not a Koblitz
curve.
Returns: μ of the elliptic curve./**
     * Returns the parameter <code>&mu;</code> of the elliptic curve.
     * @param curve The elliptic curve from which to obtain <code>&mu;</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>&mu;</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 U_k-1 and
U_k or V_k-1 and
V_k.
Params: mu – The parameter μ of the elliptic curve.
k – The index of the second element of the Lucas Sequence to be
returned.
doV – If set to true, computes V_k-1 and
V_k, otherwise U_k-1 and
U_k.
Returns: An array with 2 elements, containing U_k-1
and U_k or V_k-1
and V_k./**
     * 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>&mu;</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 t_w. If the width is
4, then for mu = 1, t_w = 6 and for
mu = -1, t_w = 10 
Params: mu – The parameter μ of the elliptic curve.
w – The window width of the WTNAF.
Returns: the auxiliary value t_w/**
     * 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>&mu;</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 s₀ and
s₁ used for partial modular reduction. 
Params: curve – The elliptic curve for which to compute
s₀ and s₁.
Throws: IllegalArgumentException – if curve is not a
Koblitz curve (Anomalous Binary Curve, ABC)./**
     * 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
(τ^m - 1)/(τ - 1).
Params: k – The integer to be reduced.
m – The bitlength of the underlying finite field.
a – The parameter a of the elliptic curve.
s – The auxiliary values s₀ and
s₁.
mu – The parameter μ of the elliptic curve.
c – The precision (number of bits of accuracy) of the partial
modular reduction.
Returns: ρ := k partmod (τ^m - 1)/(τ - 1)/**
     * Partial modular reduction modulo
     * <code>(&tau;<sup>m</sup> - 1)/(&tau; - 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 &mu; of the elliptic curve.
     * @param c The precision (number of bits of accuracy) of the partial
     * modular reduction.
     * @return <code>&rho; := k partmod (&tau;<sup>m</sup> - 1)/(&tau; - 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 ECPoint.F2m by a BigInteger using the reduced τ-adic
NAF (RTNAF) method.
Params: p – The ECPoint.F2m to multiply.
k – The BigInteger by which to multiply p.
Returns: k * p/**
     * Multiplies a {@link org.bouncycastle.math.ec.ECPoint.F2m ECPoint.F2m}
     * by a <code>BigInteger</code> using the reduced <code>&tau;</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 ECPoint.F2m by an element λ of Z[τ]
using the τ-adic NAF (TNAF) method.
Params: p – The ECPoint.F2m to multiply.
lambda – The element λ of
Z[τ].
Returns: λ * p/**
     * Multiplies a {@link org.bouncycastle.math.ec.ECPoint.F2m ECPoint.F2m}
     * by an element <code>&lambda;</code> of <code><b>Z</b>[&tau;]</code>
     * using the <code>&tau;</code>-adic NAF (TNAF) method.
     * @param p The ECPoint.F2m to multiply.
     * @param lambda The element <code>&lambda;</code> of
     * <code><b>Z</b>[&tau;]</code>.
     * @return <code>&lambda; * 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 ECPoint.F2m by an element λ of Z[τ]
using the τ-adic NAF (TNAF) method, given the TNAF
of λ.
Params: p – The ECPoint.F2m to multiply.
u – The the TNAF of λ..
Returns: λ * p/**
    * Multiplies a {@link org.bouncycastle.math.ec.ECPoint.F2m ECPoint.F2m}
    * by an element <code>&lambda;</code> of <code><b>Z</b>[&tau;]</code>
    * using the <code>&tau;</code>-adic NAF (TNAF) method, given the TNAF
    * of <code>&lambda;</code>.
    * @param p The ECPoint.F2m to multiply.
    * @param u The the TNAF of <code>&lambda;</code>..
    * @return <code>&lambda; * 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 [τ]-adic window NAF of an element
λ of Z[τ].
Params: mu – The parameter μ of the elliptic curve.
lambda – The element λ of
Z[τ] of which to compute the
[τ]-adic NAF.
width – The window width of the resulting WNAF.
pow2w – 2^width.
tw – The auxiliary value t_w.
alpha – The α_u's for the window width.
Returns: The [τ]-adic window NAF of
λ./**
     * Computes the <code>[&tau;]</code>-adic window NAF of an element
     * <code>&lambda;</code> of <code><b>Z</b>[&tau;]</code>.
     * @param mu The parameter &mu; of the elliptic curve.
     * @param lambda The element <code>&lambda;</code> of
     * <code><b>Z</b>[&tau;]</code> of which to compute the
     * <code>[&tau;]</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>&alpha;<sub>u</sub></code>'s for the window width.
     * @return The <code>[&tau;]</code>-adic window NAF of
     * <code>&lambda;</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.
Params: p – The ECPoint for which to do the precomputation.
a – The parameter a of the elliptic curve.
Returns: The precomputation array for p./**
     * 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;
    }
}
Params:	mu – The parameter `μ` of the elliptic curve. lambda – The element `λ` of `Z[τ]`.
Returns:	The norm of `λ`.
Params:	lambda0 – The component `λ₀`. lambda1 – The component `λ₁`. mu – The parameter `μ` of the elliptic curve. Must equal 1 or -1.
Throws:	IllegalArgumentException – if `lambda0` and `lambda1` do not have same scale.
Returns:	The rounded element of `Z[τ]`.
Params:	k – The parameter `k`. s – The curve parameter `s₀` or `s₁`. vm – The Lucas Sequence element `V_m`. a – The parameter `a` of the elliptic curve. m – The bit length of the finite field `F_m`. c – The number of bits of accuracy, i.e. the scale of the returned `SimpleBigDecimal`.
Returns:	The value `λ = s k / n` computed to `c` bits of accuracy.
Params:	p – The ECPoint.F2m to which `τ()` is applied.
Returns:	`τ(p)`
Params:	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`.
Throws:	IllegalArgumentException – if the given ECCurve is not a Koblitz curve.
Returns:	`μ` of the elliptic curve.
Params:	curve – The elliptic curve for which to compute `s₀` and `s₁`.
Throws:	IllegalArgumentException – if `curve` is not a Koblitz curve (Anomalous Binary Curve, ABC).
Params:	k – The integer to be reduced. m – The bitlength of the underlying finite field. a – The parameter `a` of the elliptic curve. s – The auxiliary values `s₀` and `s₁`. mu – The parameter μ of the elliptic curve. c – The precision (number of bits of accuracy) of the partial modular reduction.
Returns:	`ρ := k partmod (τ^m - 1)/(τ - 1)`
Params:	p – The ECPoint.F2m to multiply. k – The `BigInteger` by which to multiply `p`.
Returns:	`k * p`
Params:	p – The ECPoint.F2m to multiply. lambda – The element `λ` of `Z[τ]`.
Returns:	`λ * p`
Params:	p – The ECPoint.F2m to multiply. u – The the TNAF of `λ`..
Returns:	`λ * p`
Params:	mu – The parameter μ of the elliptic curve. lambda – The element `λ` of `Z[τ]` of which to compute the `[τ]`-adic NAF. width – The window width of the resulting WNAF. pow2w – 2^width. tw – The auxiliary value `t_w`. alpha – The `α_u`'s for the window width.
Returns:	The `[τ]`-adic window NAF of `λ`.
Params:	p – The `ECPoint` for which to do the precomputation. a – The parameter `a` of the elliptic curve.
Returns:	The precomputation array for `p`.
/

org.bouncycastle/ bcprov-jdk15/ 1.46/ org/bouncycastle/math/ec/Tnaf.java
