path: root/bcprov/src/main/java/org/bouncycastle/math/ec/ECCurve.java

package org.bouncycastle.math.ec;

import java.math.BigInteger;
import java.security.SecureRandom;
import java.util.Hashtable;
import java.util.Random;

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.FiniteFields;
import org.bouncycastle.math.raw.Nat;
import org.bouncycastle.util.BigIntegers;
import org.bouncycastle.util.Integers;

/**
 * base class for an elliptic curve
 */
public abstract class ECCurve
{
    public static final int COORD_AFFINE = 0;
    public static final int COORD_HOMOGENEOUS = 1;
    public static final int COORD_JACOBIAN = 2;
    public static final int COORD_JACOBIAN_CHUDNOVSKY = 3;
    public static final int COORD_JACOBIAN_MODIFIED = 4;
    public static final int COORD_LAMBDA_AFFINE = 5;
    public static final int COORD_LAMBDA_PROJECTIVE = 6;
    public static final int COORD_SKEWED = 7;

    public static int[] getAllCoordinateSystems()
    {
        return new int[]{ COORD_AFFINE, COORD_HOMOGENEOUS, COORD_JACOBIAN, COORD_JACOBIAN_CHUDNOVSKY,
            COORD_JACOBIAN_MODIFIED, COORD_LAMBDA_AFFINE, COORD_LAMBDA_PROJECTIVE, COORD_SKEWED };
    }

    public class Config
    {
        protected int coord;
        protected ECEndomorphism endomorphism;
        protected ECMultiplier multiplier;

        Config(int coord, ECEndomorphism endomorphism, ECMultiplier multiplier)
        {
            this.coord = coord;
            this.endomorphism = endomorphism;
            this.multiplier = multiplier;
        }

        public Config setCoordinateSystem(int coord)
        {
            this.coord = coord;
            return this;
        }

        public Config setEndomorphism(ECEndomorphism endomorphism)
        {
            this.endomorphism = endomorphism;
            return this;
        }

        public Config setMultiplier(ECMultiplier multiplier)
        {
            this.multiplier = multiplier;
            return this;
        }

        public ECCurve create()
        {
            if (!supportsCoordinateSystem(coord))
            {
                throw new IllegalStateException("unsupported coordinate system");
            }

            ECCurve c = cloneCurve();
            if (c == ECCurve.this)
            {
                throw new IllegalStateException("implementation returned current curve");
            }

            // NOTE: Synchronization added to keep FindBugs™ happy
            synchronized (c)
            {
                c.coord = coord;
                c.endomorphism = endomorphism;
                c.multiplier = multiplier;
            }

            return c;
        }
    }

    protected FiniteField field;
    protected ECFieldElement a, b;
    protected BigInteger order, cofactor;

    protected int coord = COORD_AFFINE;
    protected ECEndomorphism endomorphism = null;
    protected ECMultiplier multiplier = null;

    protected ECCurve(FiniteField field)
    {
        this.field = field;
    }

    public abstract int getFieldSize();

    public abstract ECFieldElement fromBigInteger(BigInteger x);

    public abstract boolean isValidFieldElement(BigInteger x);

    public abstract ECFieldElement randomFieldElement(SecureRandom r);

    public abstract ECFieldElement randomFieldElementMult(SecureRandom r);

    public synchronized Config configure()
    {
        return new Config(this.coord, this.endomorphism, this.multiplier);
    }

    public ECPoint validatePoint(BigInteger x, BigInteger y)
    {
        ECPoint p = createPoint(x, y);
        if (!p.isValid())
        {
            throw new IllegalArgumentException("Invalid point coordinates");
        }
        return p;
    }

    public ECPoint createPoint(BigInteger x, BigInteger y)
    {
        return createRawPoint(fromBigInteger(x), fromBigInteger(y));
    }

    protected abstract ECCurve cloneCurve();

    protected abstract ECPoint createRawPoint(ECFieldElement x, ECFieldElement y);

    protected abstract ECPoint createRawPoint(ECFieldElement x, ECFieldElement y, ECFieldElement[] zs);

    protected ECMultiplier createDefaultMultiplier()
    {
        if (endomorphism instanceof GLVEndomorphism)
        {
            return new GLVMultiplier(this, (GLVEndomorphism)endomorphism);
        }

        return new WNafL2RMultiplier();
    }

    public boolean supportsCoordinateSystem(int coord)
    {
        return coord == COORD_AFFINE;
    }

    public PreCompInfo getPreCompInfo(ECPoint point, String name)
    {
        checkPoint(point);

        Hashtable table;
        synchronized (point)
        {
            table = point.preCompTable;
        }

        if (null == table)
        {
            return null;
        }

        synchronized (table)
        {
            return (PreCompInfo)table.get(name);
        }
    }

    /**
     * Compute a <code>PreCompInfo</code> for a point on this curve, under a given name. Used by
     * <code>ECMultiplier</code>s to save the precomputation for this <code>ECPoint</code> for use
     * by subsequent multiplication.
     * 
     * @param point
     *            The <code>ECPoint</code> to store precomputations for.
     * @param name
     *            A <code>String</code> used to index precomputations of different types.
     * @param callback
     *            Called to calculate the <code>PreCompInfo</code>.
     */
    public PreCompInfo precompute(ECPoint point, String name, PreCompCallback callback)
    {
        checkPoint(point);

        Hashtable table;
        synchronized (point)
        {
            table = point.preCompTable;
            if (null == table)
            {
                point.preCompTable = table = new Hashtable(4);
            }
        }

        synchronized (table)
        {
            PreCompInfo existing = (PreCompInfo)table.get(name);
            PreCompInfo result = callback.precompute(existing);

            if (result != existing)
            {
                table.put(name, result);
            }

            return result;
        }
    }

    public ECPoint importPoint(ECPoint p)
    {
        if (this == p.getCurve())
        {
            return p;
        }
        if (p.isInfinity())
        {
            return getInfinity();
        }

        // TODO Default behaviour could be improved if the two curves have the same coordinate system by copying any Z coordinates.
        p = p.normalize();

        return createPoint(p.getXCoord().toBigInteger(), p.getYCoord().toBigInteger());
    }

    /**
     * Normalization ensures that any projective coordinate is 1, and therefore that the x, y
     * coordinates reflect those of the equivalent point in an affine coordinate system. Where more
     * than one point is to be normalized, this method will generally be more efficient than
     * normalizing each point separately.
     * 
     * @param points
     *            An array of points that will be updated in place with their normalized versions,
     *            where necessary
     */
    public void normalizeAll(ECPoint[] points)
    {
        normalizeAll(points, 0, points.length, null);
    }

    /**
     * Normalization ensures that any projective coordinate is 1, and therefore that the x, y
     * coordinates reflect those of the equivalent point in an affine coordinate system. Where more
     * than one point is to be normalized, this method will generally be more efficient than
     * normalizing each point separately. An (optional) z-scaling factor can be applied; effectively
     * each z coordinate is scaled by this value prior to normalization (but only one
     * actual multiplication is needed).
     * 
     * @param points
     *            An array of points that will be updated in place with their normalized versions,
     *            where necessary
     * @param off
     *            The start of the range of points to normalize
     * @param len
     *            The length of the range of points to normalize
     * @param iso
     *            The (optional) z-scaling factor - can be null
     */
    public void normalizeAll(ECPoint[] points, int off, int len, ECFieldElement iso)
    {
        checkPoints(points, off, len);

        switch (this.getCoordinateSystem())
        {
        case ECCurve.COORD_AFFINE:
        case ECCurve.COORD_LAMBDA_AFFINE:
        {
            if (iso != null)
            {
                throw new IllegalArgumentException("'iso' not valid for affine coordinates");
            }
            return;
        }
        }

        /*
         * Figure out which of the points actually need to be normalized
         */
        ECFieldElement[] zs = new ECFieldElement[len];
        int[] indices = new int[len];
        int count = 0;
        for (int i = 0; i < len; ++i)
        {
            ECPoint p = points[off + i];
            if (null != p && (iso != null || !p.isNormalized()))
            {
                zs[count] = p.getZCoord(0);
                indices[count++] = off + i;
            }
        }

        if (count == 0)
        {
            return;
        }

        ECAlgorithms.montgomeryTrick(zs, 0, count, iso);

        for (int j = 0; j < count; ++j)
        {
            int index = indices[j];
            points[index] = points[index].normalize(zs[j]);
        }
    }

    public abstract ECPoint getInfinity();

    public FiniteField getField()
    {
        return field;
    }

    public ECFieldElement getA()
    {
        return a;
    }

    public ECFieldElement getB()
    {
        return b;
    }

    public BigInteger getOrder()
    {
        return order;
    }

    public BigInteger getCofactor()
    {
        return cofactor;
    }

    public int getCoordinateSystem()
    {
        return coord;
    }

    protected abstract ECPoint decompressPoint(int yTilde, BigInteger X1);

    public ECEndomorphism getEndomorphism()
    {
        return endomorphism;
    }

    /**
     * Sets the default <code>ECMultiplier</code>, unless already set.
     * 
     * We avoid synchronizing for performance reasons, so there is no uniqueness guarantee.
     */
    public ECMultiplier getMultiplier()
    {
        if (this.multiplier == null)
        {
            this.multiplier = createDefaultMultiplier();
        }
        return this.multiplier;
    }

    /**
     * Decode a point on this curve from its ASN.1 encoding. The different
     * encodings are taken account of, including point compression for
     * <code>F<sub>p</sub></code> (X9.62 s 4.2.1 pg 17).
     * @return The decoded point.
     */
    public ECPoint decodePoint(byte[] encoded)
    {
        ECPoint p = null;
        int expectedLength = (getFieldSize() + 7) / 8;

        byte type = encoded[0];
        switch (type)
        {
        case 0x00: // infinity
        {
            if (encoded.length != 1)
            {
                throw new IllegalArgumentException("Incorrect length for infinity encoding");
            }

            p = getInfinity();
            break;
        }
        case 0x02: // compressed
        case 0x03: // compressed
        {
            if (encoded.length != (expectedLength + 1))
            {
                throw new IllegalArgumentException("Incorrect length for compressed encoding");
            }

            int yTilde = type & 1;
            BigInteger X = BigIntegers.fromUnsignedByteArray(encoded, 1, expectedLength);

            p = decompressPoint(yTilde, X);
            if (!p.implIsValid(true, true))
            {
                throw new IllegalArgumentException("Invalid point");
            }

            break;
        }
        case 0x04: // uncompressed
        {
            if (encoded.length != (2 * expectedLength + 1))
            {
                throw new IllegalArgumentException("Incorrect length for uncompressed encoding");
            }

            BigInteger X = BigIntegers.fromUnsignedByteArray(encoded, 1, expectedLength);
            BigInteger Y = BigIntegers.fromUnsignedByteArray(encoded, 1 + expectedLength, expectedLength);

            p = validatePoint(X, Y);
            break;
        }
        case 0x06: // hybrid
        case 0x07: // hybrid
        {
            if (encoded.length != (2 * expectedLength + 1))
            {
                throw new IllegalArgumentException("Incorrect length for hybrid encoding");
            }

            BigInteger X = BigIntegers.fromUnsignedByteArray(encoded, 1, expectedLength);
            BigInteger Y = BigIntegers.fromUnsignedByteArray(encoded, 1 + expectedLength, expectedLength);

            if (Y.testBit(0) != (type == 0x07))
            {
                throw new IllegalArgumentException("Inconsistent Y coordinate in hybrid encoding");
            }

            p = validatePoint(X, Y);
            break;
        }
        default:
            throw new IllegalArgumentException("Invalid point encoding 0x" + Integer.toString(type, 16));
        }

        if (type != 0x00 && p.isInfinity())
        {
            throw new IllegalArgumentException("Invalid infinity encoding");
        }

        return p;
    }

    /**
     * Create a cache-safe lookup table for the specified sequence of points. All the points MUST
     * belong to this {@link ECCurve} instance, and MUST already be normalized.
     */
    public ECLookupTable createCacheSafeLookupTable(final ECPoint[] points, int off, final int len)
    {
        final int FE_BYTES = (getFieldSize() + 7) >>> 3;

        final byte[] table = new byte[len * FE_BYTES * 2];
        {
            int pos = 0;
            for (int i = 0; i < len; ++i)
            {
                ECPoint p = points[off + i];
                byte[] px = p.getRawXCoord().toBigInteger().toByteArray();
                byte[] py = p.getRawYCoord().toBigInteger().toByteArray();

                int pxStart = px.length > FE_BYTES ? 1 : 0, pxLen = px.length - pxStart;
                int pyStart = py.length > FE_BYTES ? 1 : 0, pyLen = py.length - pyStart;

                System.arraycopy(px, pxStart, table, pos + FE_BYTES - pxLen, pxLen); pos += FE_BYTES;
                System.arraycopy(py, pyStart, table, pos + FE_BYTES - pyLen, pyLen); pos += FE_BYTES;
            }
        }

        return new AbstractECLookupTable()
        {
            public int getSize()
            {
                return len;
            }

            public ECPoint lookup(int index)
            {
                byte[] x = new byte[FE_BYTES], y = new byte[FE_BYTES];
                int pos = 0;

                for (int i = 0; i < len; ++i)
                {
                    int MASK = ((i ^ index) - 1) >> 31;

                    for (int j = 0; j < FE_BYTES; ++j)
                    {
                        x[j] ^= table[pos + j] & MASK;
                        y[j] ^= table[pos + FE_BYTES + j] & MASK;
                    }

                    pos += (FE_BYTES * 2);
                }

                return createPoint(x, y);
            }

            public ECPoint lookupVar(int index)
            {
                byte[] x = new byte[FE_BYTES], y = new byte[FE_BYTES];
                int pos = index * FE_BYTES * 2;

                for (int j = 0; j < FE_BYTES; ++j)
                {
                    x[j] = table[pos + j];
                    y[j] = table[pos + FE_BYTES + j];
                }

                return createPoint(x, y);
            }

            private ECPoint createPoint(byte[] x, byte[] y)
            {
                return createRawPoint(fromBigInteger(new BigInteger(1, x)), fromBigInteger(new BigInteger(1, y)));
            }
        };
    }

    protected void checkPoint(ECPoint point)
    {
        if (null == point || (this != point.getCurve()))
        {
            throw new IllegalArgumentException("'point' must be non-null and on this curve");
        }
    }

    protected void checkPoints(ECPoint[] points)
    {
        checkPoints(points, 0, points.length);
    }

    protected void checkPoints(ECPoint[] points, int off, int len)
    {
        if (points == null)
        {
            throw new IllegalArgumentException("'points' cannot be null");
        }
        if (off < 0 || len < 0 || (off > (points.length - len)))
        {
            throw new IllegalArgumentException("invalid range specified for 'points'");
        }

        for (int i = 0; i < len; ++i)
        {
            ECPoint point = points[off + i];
            if (null != point && this != point.getCurve())
            {
                throw new IllegalArgumentException("'points' entries must be null or on this curve");
            }
        }
    }

    public boolean equals(ECCurve other)
    {
        return this == other
            || (null != other
                && getField().equals(other.getField())
                && getA().toBigInteger().equals(other.getA().toBigInteger())
                && getB().toBigInteger().equals(other.getB().toBigInteger()));
    }

    public boolean equals(Object obj) 
    {
        return this == obj || (obj instanceof ECCurve && equals((ECCurve)obj));
    }

    public int hashCode() 
    {
        return getField().hashCode()
            ^ Integers.rotateLeft(getA().toBigInteger().hashCode(), 8)
            ^ Integers.rotateLeft(getB().toBigInteger().hashCode(), 16);
    }

    public static abstract class AbstractFp extends ECCurve
    {
        protected AbstractFp(BigInteger q)
        {
            super(FiniteFields.getPrimeField(q));
        }

        public boolean isValidFieldElement(BigInteger x)
        {
            return x != null && x.signum() >= 0 && x.compareTo(this.getField().getCharacteristic()) < 0;
        }

        public ECFieldElement randomFieldElement(SecureRandom r)
        {
            /*
             * NOTE: BigInteger comparisons in the rejection sampling are not constant-time, so we
             * use the product of two independent elements to mitigate side-channels.
             */
            BigInteger p = getField().getCharacteristic();
            ECFieldElement fe1 = fromBigInteger(implRandomFieldElement(r, p));
            ECFieldElement fe2 = fromBigInteger(implRandomFieldElement(r, p));
            return fe1.multiply(fe2);
        }

        public ECFieldElement randomFieldElementMult(SecureRandom r)
        {
            /*
             * NOTE: BigInteger comparisons in the rejection sampling are not constant-time, so we
             * use the product of two independent elements to mitigate side-channels.
             */
            BigInteger p = getField().getCharacteristic();
            ECFieldElement fe1 = fromBigInteger(implRandomFieldElementMult(r, p));
            ECFieldElement fe2 = fromBigInteger(implRandomFieldElementMult(r, p));
            return fe1.multiply(fe2);
        }

        protected ECPoint decompressPoint(int yTilde, BigInteger X1)
        {
            ECFieldElement x = this.fromBigInteger(X1);
            ECFieldElement rhs = x.square().add(this.a).multiply(x).add(this.b);
            ECFieldElement y = rhs.sqrt();

            /*
             * If y is not a square, then we haven't got a point on the curve
             */
            if (y == null)
            {
                throw new IllegalArgumentException("Invalid point compression");
            }

            if (y.testBitZero() != (yTilde == 1))
            {
                // Use the other root
                y = y.negate();
            }

            return this.createRawPoint(x, y);
        }

        private static BigInteger implRandomFieldElement(SecureRandom r, BigInteger p)
        {
            BigInteger x;
            do
            {
                x = BigIntegers.createRandomBigInteger(p.bitLength(), r);
            }
            while (x.compareTo(p) >= 0);
            return x;
        }

        private static BigInteger implRandomFieldElementMult(SecureRandom r, BigInteger p)
        {
            BigInteger x;
            do
            {
                x = BigIntegers.createRandomBigInteger(p.bitLength(), r);
            }
            while (x.signum() <= 0 || x.compareTo(p) >= 0);
            return x;
        }
    }

    /**
     * Elliptic curve over Fp
     */
    public static class Fp extends AbstractFp
    {
        private static final int FP_DEFAULT_COORDS = ECCurve.COORD_JACOBIAN_MODIFIED;

        BigInteger q, r;
        ECPoint.Fp infinity;

        /**
         * @deprecated use constructor taking order/cofactor
         */
        public Fp(BigInteger q, BigInteger a, BigInteger b)
        {
            this(q, a, b, null, null);
        }

        public Fp(BigInteger q, BigInteger a, BigInteger b, BigInteger order, BigInteger cofactor)
        {
            super(q);

            this.q = q;
            this.r = ECFieldElement.Fp.calculateResidue(q);
            this.infinity = new ECPoint.Fp(this, null, null);

            this.a = fromBigInteger(a);
            this.b = fromBigInteger(b);
            this.order = order;
            this.cofactor = cofactor;
            this.coord = FP_DEFAULT_COORDS;
        }

        protected Fp(BigInteger q, BigInteger r, ECFieldElement a, ECFieldElement b, BigInteger order, BigInteger cofactor)
        {
            super(q);

            this.q = q;
            this.r = r;
            this.infinity = new ECPoint.Fp(this, null, null);

            this.a = a;
            this.b = b;
            this.order = order;
            this.cofactor = cofactor;
            this.coord = FP_DEFAULT_COORDS;
        }

        protected ECCurve cloneCurve()
        {
            return new Fp(this.q, this.r, this.a, this.b, this.order, this.cofactor);
        }

        public boolean supportsCoordinateSystem(int coord)
        {
            switch (coord)
            {
            case ECCurve.COORD_AFFINE:
            case ECCurve.COORD_HOMOGENEOUS:
            case ECCurve.COORD_JACOBIAN:
            case ECCurve.COORD_JACOBIAN_MODIFIED:
                return true;
            default:
                return false;
            }
        }

        public BigInteger getQ()
        {
            return q;
        }

        public int getFieldSize()
        {
            return q.bitLength();
        }

        public ECFieldElement fromBigInteger(BigInteger x)
        {
            return new ECFieldElement.Fp(this.q, this.r, x);
        }

        protected ECPoint createRawPoint(ECFieldElement x, ECFieldElement y)
        {
            return new ECPoint.Fp(this, x, y);
        }

        protected ECPoint createRawPoint(ECFieldElement x, ECFieldElement y, ECFieldElement[] zs)
        {
            return new ECPoint.Fp(this, x, y, zs);
        }

        public ECPoint importPoint(ECPoint p)
        {
            if (this != p.getCurve() && this.getCoordinateSystem() == ECCurve.COORD_JACOBIAN && !p.isInfinity())
            {
                switch (p.getCurve().getCoordinateSystem())
                {
                case ECCurve.COORD_JACOBIAN:
                case ECCurve.COORD_JACOBIAN_CHUDNOVSKY:
                case ECCurve.COORD_JACOBIAN_MODIFIED:
                    return new ECPoint.Fp(this,
                        fromBigInteger(p.x.toBigInteger()),
                        fromBigInteger(p.y.toBigInteger()),
                        new ECFieldElement[]{ fromBigInteger(p.zs[0].toBigInteger()) });
                default:
                    break;
                }
            }

            return super.importPoint(p);
        }

        public ECPoint getInfinity()
        {
            return infinity;
        }
    }

    public static abstract class AbstractF2m extends ECCurve
    {
        public static BigInteger inverse(int m, int[] ks, BigInteger x)
        {
            return new LongArray(x).modInverse(m, ks).toBigInteger();
        }

        /**
         * The auxiliary values <code>s<sub>0</sub></code> and
         * <code>s<sub>1</sub></code> used for partial modular reduction for
         * Koblitz curves.
         */
        private BigInteger[] si = null;

        private static FiniteField buildField(int m, int k1, int k2, int k3)
        {
            if (k1 == 0)
            {
                throw new IllegalArgumentException("k1 must be > 0");
            }

            if (k2 == 0)
            {
                if (k3 != 0)
                {
                    throw new IllegalArgumentException("k3 must be 0 if k2 == 0");
                }

                return FiniteFields.getBinaryExtensionField(new int[]{ 0, k1, m });
            }

            if (k2 <= k1)
            {
                throw new IllegalArgumentException("k2 must be > k1");
            }

            if (k3 <= k2)
            {
                throw new IllegalArgumentException("k3 must be > k2");
            }

            return FiniteFields.getBinaryExtensionField(new int[]{ 0, k1, k2, k3, m });
        }

        protected AbstractF2m(int m, int k1, int k2, int k3)
        {
            super(buildField(m, k1, k2, k3));
        }

        public ECPoint createPoint(BigInteger x, BigInteger y)
        {
            ECFieldElement X = this.fromBigInteger(x), Y = this.fromBigInteger(y);

            int coord = this.getCoordinateSystem();

            switch (coord)
            {
            case ECCurve.COORD_LAMBDA_AFFINE:
            case ECCurve.COORD_LAMBDA_PROJECTIVE:
            {
                if (X.isZero())
                {
                    if (!Y.square().equals(this.getB()))
                    {
                        throw new IllegalArgumentException();
                    }
                }
                /*
                 * NOTE: A division could be avoided using a projective result, except at present
                 * callers will expect that the result is already normalized.
                 */
//                else if (coord == COORD_LAMBDA_PROJECTIVE)
//                {
//                    ECFieldElement Z = X;
//                    X = X.square();
//                    Y = Y.add(X);
//                    return createRawPoint(X, Y, new ECFieldElement[]{ Z });
//                }
                else
                {
                    // Y becomes Lambda (X + Y/X) here
                    Y = Y.divide(X).add(X);
                }
                break;
            }
            default:
            {
                break;
            }
            }

            return this.createRawPoint(X, Y);
        }

        public boolean isValidFieldElement(BigInteger x)
        {
            return x != null && x.signum() >= 0 && x.bitLength() <= this.getFieldSize();
        }

        public ECFieldElement randomFieldElement(SecureRandom r)
        {
            int m = getFieldSize();
            return fromBigInteger(BigIntegers.createRandomBigInteger(m, r));
        }

        public ECFieldElement randomFieldElementMult(SecureRandom r)
        {
            /*
             * NOTE: BigInteger comparisons in the rejection sampling are not constant-time, so we
             * use the product of two independent elements to mitigate side-channels.
             */
            int m = getFieldSize();
            ECFieldElement fe1 = fromBigInteger(implRandomFieldElementMult(r, m));
            ECFieldElement fe2 = fromBigInteger(implRandomFieldElementMult(r, m));
            return fe1.multiply(fe2);
        }

        /**
         * Decompresses a compressed point P = (xp, yp) (X9.62 s 4.2.2).
         * 
         * @param yTilde
         *            ~yp, an indication bit for the decompression of yp.
         * @param X1
         *            The field element xp.
         * @return the decompressed point.
         */
        protected ECPoint decompressPoint(int yTilde, BigInteger X1)
        {
            ECFieldElement x = this.fromBigInteger(X1), y = null;
            if (x.isZero())
            {
                y = this.getB().sqrt();
            }
            else
            {
                ECFieldElement beta = x.square().invert().multiply(this.getB()).add(this.getA()).add(x);
                ECFieldElement z = solveQuadraticEquation(beta);
                if (z != null)
                {
                    if (z.testBitZero() != (yTilde == 1))
                    {
                        z = z.addOne();
                    }

                    switch (this.getCoordinateSystem())
                    {
                    case ECCurve.COORD_LAMBDA_AFFINE:
                    case ECCurve.COORD_LAMBDA_PROJECTIVE:
                    {
                        y = z.add(x);
                        break;
                    }
                    default:
                    {
                        y = z.multiply(x);
                        break;
                    }
                    }
                }
            }

            if (y == null)
            {
                throw new IllegalArgumentException("Invalid point compression");
            }

            return this.createRawPoint(x, y);
        }

        /**
         * Solves a quadratic equation <code>z<sup>2</sup> + z = beta</code>(X9.62
         * D.1.6) The other solution is <code>z + 1</code>.
         * 
         * @param beta
         *            The value to solve the quadratic equation for.
         * @return the solution for <code>z<sup>2</sup> + z = beta</code> or
         *         <code>null</code> if no solution exists.
         */
        protected ECFieldElement solveQuadraticEquation(ECFieldElement beta)
        {
            ECFieldElement.AbstractF2m betaF2m = (ECFieldElement.AbstractF2m)beta;

            boolean fastTrace = betaF2m.hasFastTrace();
            if (fastTrace && 0 != betaF2m.trace())
            {
                return null;
            }

            int m = this.getFieldSize();

            // For odd m, use the half-trace 
            if (0 != (m & 1))
            {
                ECFieldElement r = betaF2m.halfTrace();
                if (fastTrace || r.square().add(r).add(beta).isZero())
                {
                    return r;
                }
                return null;
            }

            if (beta.isZero())
            {
                return beta;
            }

            ECFieldElement gamma, z, zeroElement = this.fromBigInteger(ECConstants.ZERO);

            Random rand = new Random();
            do
            {
                ECFieldElement t = this.fromBigInteger(new BigInteger(m, rand));
                z = zeroElement;
                ECFieldElement w = beta;
                for (int i = 1; i < m; i++)
                {
                    ECFieldElement w2 = w.square();
                    z = z.square().add(w2.multiply(t));
                    w = w2.add(beta);
                }
                if (!w.isZero())
                {
                    return null;
                }
                gamma = z.square().add(z);
            }
            while (gamma.isZero());

            return z;
        }

        /**
         * @return the auxiliary values <code>s<sub>0</sub></code> and
         * <code>s<sub>1</sub></code> used for partial modular reduction for
         * Koblitz curves.
         */
        synchronized BigInteger[] getSi()
        {
            if (si == null)
            {
                si = Tnaf.getSi(this);
            }
            return si;
        }

        /**
         * Returns true if this is a Koblitz curve (ABC curve).
         * @return true if this is a Koblitz curve (ABC curve), false otherwise
         */
        public boolean isKoblitz()
        {
            return this.order != null && this.cofactor != null && this.b.isOne() && (this.a.isZero() || this.a.isOne());
        }

        private static BigInteger implRandomFieldElementMult(SecureRandom r, int m)
        {
            BigInteger x;
            do
            {
                x = BigIntegers.createRandomBigInteger(m, r);
            }
            while (x.signum() <= 0);
            return x;
        }
    }

    /**
     * Elliptic curves over F2m. The Weierstrass equation is given by
     * <code>y<sup>2</sup> + xy = x<sup>3</sup> + ax<sup>2</sup> + b</code>.
     */
    public static class F2m extends AbstractF2m
    {
        private static final int F2M_DEFAULT_COORDS = ECCurve.COORD_LAMBDA_PROJECTIVE;

        /**
         * The exponent <code>m</code> of <code>F<sub>2<sup>m</sup></sub></code>.
         */
        private int m;  // can't be final - JDK 1.1

        /**
         * TPB: The integer <code>k</code> where <code>x<sup>m</sup> +
         * x<sup>k</sup> + 1</code> represents the reduction polynomial
         * <code>f(z)</code>.<br>
         * PPB: The integer <code>k1</code> where <code>x<sup>m</sup> +
         * x<sup>k3</sup> + x<sup>k2</sup> + x<sup>k1</sup> + 1</code>
         * represents the reduction polynomial <code>f(z)</code>.<br>
         */
        private int k1;  // can't be final - JDK 1.1

        /**
         * TPB: Always set to <code>0</code><br>
         * PPB: The integer <code>k2</code> where <code>x<sup>m</sup> +
         * x<sup>k3</sup> + x<sup>k2</sup> + x<sup>k1</sup> + 1</code>
         * represents the reduction polynomial <code>f(z)</code>.<br>
         */
        private int k2;  // can't be final - JDK 1.1

        /**
         * TPB: Always set to <code>0</code><br>
         * PPB: The integer <code>k3</code> where <code>x<sup>m</sup> +
         * x<sup>k3</sup> + x<sup>k2</sup> + x<sup>k1</sup> + 1</code>
         * represents the reduction polynomial <code>f(z)</code>.<br>
         */
        private int k3;  // can't be final - JDK 1.1
        
         /**
         * The point at infinity on this curve.
         */
        private ECPoint.F2m infinity;  // can't be final - JDK 1.1

        /**
         * Constructor for Trinomial Polynomial Basis (TPB).
         * @param m  The exponent <code>m</code> of
         * <code>F<sub>2<sup>m</sup></sub></code>.
         * @param k The integer <code>k</code> where <code>x<sup>m</sup> +
         * x<sup>k</sup> + 1</code> represents the reduction
         * polynomial <code>f(z)</code>.
         * @param a The coefficient <code>a</code> in the Weierstrass equation
         * for non-supersingular elliptic curves over
         * <code>F<sub>2<sup>m</sup></sub></code>.
         * @param b The coefficient <code>b</code> in the Weierstrass equation
         * for non-supersingular elliptic curves over
         * <code>F<sub>2<sup>m</sup></sub></code>.
         * @deprecated use constructor taking order/cofactor
         */
        public F2m(
            int m,
            int k,
            BigInteger a,
            BigInteger b)
        {
            this(m, k, 0, 0, a, b, null, null);
        }

        /**
         * Constructor for Trinomial Polynomial Basis (TPB).
         * @param m  The exponent <code>m</code> of
         * <code>F<sub>2<sup>m</sup></sub></code>.
         * @param k The integer <code>k</code> where <code>x<sup>m</sup> +
         * x<sup>k</sup> + 1</code> represents the reduction
         * polynomial <code>f(z)</code>.
         * @param a The coefficient <code>a</code> in the Weierstrass equation
         * for non-supersingular elliptic curves over
         * <code>F<sub>2<sup>m</sup></sub></code>.
         * @param b The coefficient <code>b</code> in the Weierstrass equation
         * for non-supersingular elliptic curves over
         * <code>F<sub>2<sup>m</sup></sub></code>.
         * @param order The order of the main subgroup of the elliptic curve.
         * @param cofactor The cofactor of the elliptic curve, i.e.
         * <code>#E<sub>a</sub>(F<sub>2<sup>m</sup></sub>) = h * n</code>.
         */
        public F2m(
            int m, 
            int k, 
            BigInteger a, 
            BigInteger b,
            BigInteger order,
            BigInteger cofactor)
        {
            this(m, k, 0, 0, a, b, order, cofactor);
        }

        /**
         * Constructor for Pentanomial Polynomial Basis (PPB).
         * @param m  The exponent <code>m</code> of
         * <code>F<sub>2<sup>m</sup></sub></code>.
         * @param k1 The integer <code>k1</code> where <code>x<sup>m</sup> +
         * x<sup>k3</sup> + x<sup>k2</sup> + x<sup>k1</sup> + 1</code>
         * represents the reduction polynomial <code>f(z)</code>.
         * @param k2 The integer <code>k2</code> where <code>x<sup>m</sup> +
         * x<sup>k3</sup> + x<sup>k2</sup> + x<sup>k1</sup> + 1</code>
         * represents the reduction polynomial <code>f(z)</code>.
         * @param k3 The integer <code>k3</code> where <code>x<sup>m</sup> +
         * x<sup>k3</sup> + x<sup>k2</sup> + x<sup>k1</sup> + 1</code>
         * represents the reduction polynomial <code>f(z)</code>.
         * @param a The coefficient <code>a</code> in the Weierstrass equation
         * for non-supersingular elliptic curves over
         * <code>F<sub>2<sup>m</sup></sub></code>.
         * @param b The coefficient <code>b</code> in the Weierstrass equation
         * for non-supersingular elliptic curves over
         * <code>F<sub>2<sup>m</sup></sub></code>.
         * @deprecated use constructor taking order/cofactor
         */
        public F2m(
            int m,
            int k1,
            int k2,
            int k3,
            BigInteger a,
            BigInteger b)
        {
            this(m, k1, k2, k3, a, b, null, null);
        }

        /**
         * Constructor for Pentanomial Polynomial Basis (PPB).
         * @param m  The exponent <code>m</code> of
         * <code>F<sub>2<sup>m</sup></sub></code>.
         * @param k1 The integer <code>k1</code> where <code>x<sup>m</sup> +
         * x<sup>k3</sup> + x<sup>k2</sup> + x<sup>k1</sup> + 1</code>
         * represents the reduction polynomial <code>f(z)</code>.
         * @param k2 The integer <code>k2</code> where <code>x<sup>m</sup> +
         * x<sup>k3</sup> + x<sup>k2</sup> + x<sup>k1</sup> + 1</code>
         * represents the reduction polynomial <code>f(z)</code>.
         * @param k3 The integer <code>k3</code> where <code>x<sup>m</sup> +
         * x<sup>k3</sup> + x<sup>k2</sup> + x<sup>k1</sup> + 1</code>
         * represents the reduction polynomial <code>f(z)</code>.
         * @param a The coefficient <code>a</code> in the Weierstrass equation
         * for non-supersingular elliptic curves over
         * <code>F<sub>2<sup>m</sup></sub></code>.
         * @param b The coefficient <code>b</code> in the Weierstrass equation
         * for non-supersingular elliptic curves over
         * <code>F<sub>2<sup>m</sup></sub></code>.
         * @param order The order of the main subgroup of the elliptic curve.
         * @param cofactor The cofactor of the elliptic curve, i.e.
         * <code>#E<sub>a</sub>(F<sub>2<sup>m</sup></sub>) = h * n</code>.
         */
        public F2m(
            int m, 
            int k1, 
            int k2, 
            int k3,
            BigInteger a, 
            BigInteger b,
            BigInteger order,
            BigInteger cofactor)
        {
            super(m, k1, k2, k3);

            this.m = m;
            this.k1 = k1;
            this.k2 = k2;
            this.k3 = k3;
            this.order = order;
            this.cofactor = cofactor;

            this.infinity = new ECPoint.F2m(this, null, null);
            this.a = fromBigInteger(a);
            this.b = fromBigInteger(b);
            this.coord = F2M_DEFAULT_COORDS;
        }

        protected F2m(int m, int k1, int k2, int k3, ECFieldElement a, ECFieldElement b, BigInteger order, BigInteger cofactor)
        {
            super(m, k1, k2, k3);

            this.m = m;
            this.k1 = k1;
            this.k2 = k2;
            this.k3 = k3;
            this.order = order;
            this.cofactor = cofactor;

            this.infinity = new ECPoint.F2m(this, null, null);
            this.a = a;
            this.b = b;
            this.coord = F2M_DEFAULT_COORDS;
        }

        protected ECCurve cloneCurve()
        {
            return new F2m(this.m, this.k1, this.k2, this.k3, this.a, this.b, this.order, this.cofactor);
        }

        public boolean supportsCoordinateSystem(int coord)
        {
            switch (coord)
            {
            case ECCurve.COORD_AFFINE:
            case ECCurve.COORD_HOMOGENEOUS:
            case ECCurve.COORD_LAMBDA_PROJECTIVE:
                return true;
            default:
                return false;
            }
        }

        protected ECMultiplier createDefaultMultiplier()
        {
            if (isKoblitz())
            {
                return new WTauNafMultiplier();
            }

            return super.createDefaultMultiplier();
        }

        public int getFieldSize()
        {
            return m;
        }

        public ECFieldElement fromBigInteger(BigInteger x)
        {
            return new ECFieldElement.F2m(this.m, this.k1, this.k2, this.k3, x);
        }

        protected ECPoint createRawPoint(ECFieldElement x, ECFieldElement y)
        {
            return new ECPoint.F2m(this, x, y);
        }

        protected ECPoint createRawPoint(ECFieldElement x, ECFieldElement y, ECFieldElement[] zs)
        {
            return new ECPoint.F2m(this, x, y, zs);
        }

        public ECPoint getInfinity()
        {
            return infinity;
        }

        public int getM()
        {
            return m;
        }

        /**
         * Return true if curve uses a Trinomial basis.
         * 
         * @return true if curve Trinomial, false otherwise.
         */
        public boolean isTrinomial()
        {
            return k2 == 0 && k3 == 0;
        }
        
        public int getK1()
        {
            return k1;
        }

        public int getK2()
        {
            return k2;
        }

        public int getK3()
        {
            return k3;
        }

        public ECLookupTable createCacheSafeLookupTable(ECPoint[] points, int off, final int len)
        {
            final int FE_LONGS = (m + 63) >>> 6;
            final int[] ks = isTrinomial() ? new int[]{ k1 } : new int[]{ k1, k2, k3 }; 

            final long[] table = new long[len * FE_LONGS * 2];
            {
                int pos = 0;
                for (int i = 0; i < len; ++i)
                {
                    ECPoint p = points[off + i];
                    ((ECFieldElement.F2m)p.getRawXCoord()).x.copyTo(table, pos); pos += FE_LONGS;
                    ((ECFieldElement.F2m)p.getRawYCoord()).x.copyTo(table, pos); pos += FE_LONGS;
                }
            }

            return new AbstractECLookupTable()
            {
                public int getSize()
                {
                    return len;
                }

                public ECPoint lookup(int index)
                {
                    long[] x = Nat.create64(FE_LONGS), y = Nat.create64(FE_LONGS);
                    int pos = 0;

                    for (int i = 0; i < len; ++i)
                    {
                        long MASK = ((i ^ index) - 1) >> 31;

                        for (int j = 0; j < FE_LONGS; ++j)
                        {
                            x[j] ^= table[pos + j] & MASK;
                            y[j] ^= table[pos + FE_LONGS + j] & MASK;
                        }

                        pos += (FE_LONGS * 2);
                    }

                    return createPoint(x, y);
                }

                public ECPoint lookupVar(int index)
                {
                    long[] x = Nat.create64(FE_LONGS), y = Nat.create64(FE_LONGS);
                    int pos = index * FE_LONGS * 2;

                    for (int j = 0; j < FE_LONGS; ++j)
                    {
                        x[j] = table[pos + j];
                        y[j] = table[pos + FE_LONGS + j];
                    }

                    return createPoint(x, y);
                }

                private ECPoint createPoint(long[] x, long[] y)
                {
                    ECFieldElement.F2m X = new ECFieldElement.F2m(m, ks, new LongArray(x));
                    ECFieldElement.F2m Y = new ECFieldElement.F2m(m, ks, new LongArray(y));
                    return createRawPoint(X, Y);
                }
            };
        }
    }
}