Curve25519.java

package nxt.crypto;

/* Ported from C to Java by Dmitry Skiba [sahn0], 23/02/08.
 * Original: http://cds.xs4all.nl:8081/ecdh/
 */
/* Generic 64-bit integer implementation of Curve25519 ECDH
 * Written by Matthijs van Duin, 200608242056
 * Public domain.
 *
 * Based on work by Daniel J Bernstein, http://cr.yp.to/ecdh.html
 */
final class Curve25519 {

    /* key size */
    public static final int KEY_SIZE = 32;

    /* 0 */
    public static final byte[] ZERO = {
            0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
            0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
    };

    /* the prime 2^255-19 */
    public static final byte[] PRIME = {
            (byte)237, (byte)255, (byte)255, (byte)255,
            (byte)255, (byte)255, (byte)255, (byte)255,
            (byte)255, (byte)255, (byte)255, (byte)255,
            (byte)255, (byte)255, (byte)255, (byte)255,
            (byte)255, (byte)255, (byte)255, (byte)255,
            (byte)255, (byte)255, (byte)255, (byte)255,
            (byte)255, (byte)255, (byte)255, (byte)255,
            (byte)255, (byte)255, (byte)255, (byte)127
    };

    /* group order (a prime near 2^252+2^124) */
    public static final byte[] ORDER = {
            (byte)237, (byte)211, (byte)245, (byte)92,
            (byte)26,  (byte)99,  (byte)18,  (byte)88,
            (byte)214, (byte)156, (byte)247, (byte)162,
            (byte)222, (byte)249, (byte)222, (byte)20,
            (byte)0,   (byte)0,   (byte)0,   (byte)0,
            (byte)0,   (byte)0,   (byte)0,   (byte)0,
            (byte)0,   (byte)0,   (byte)0,   (byte)0,
            (byte)0,   (byte)0,   (byte)0,   (byte)16
    };

    /********* KEY AGREEMENT *********/

    /* Private key clamping
     *   k [out] your private key for key agreement
     *   k  [in]  32 random bytes
     */
    public static void clamp(byte[] k) {
        k[31] &= 0x7F;
        k[31] |= 0x40;
        k[ 0] &= 0xF8;
    }

    /* Key-pair generation
     *   P  [out] your public key
     *   s  [out] your private key for signing
     *   k  [out] your private key for key agreement
     *   k  [in]  32 random bytes
     * s may be NULL if you don't care
     *
     * WARNING: if s is not NULL, this function has data-dependent timing */
    public static void keygen(byte[] P, byte[] s, byte[] k) {
        clamp(k);
        core(P, s, k, null);
    }

    /* Key agreement
     *   Z  [out] shared secret (needs hashing before use)
     *   k  [in]  your private key for key agreement
     *   P  [in]  peer's public key
     */
    public static void curve(byte[] Z, byte[] k, byte[] P) {
        core(Z, null, k, P);
    }

    /********* DIGITAL SIGNATURES *********/

    /* deterministic EC-KCDSA
     *
     *    s is the private key for signing
     *    P is the corresponding public key
     *    Z is the context data (signer public key or certificate, etc)
     *
     * signing:
     *
     *    m = hash(Z, message)
     *    x = hash(m, s)
     *    keygen25519(Y, NULL, x);
     *    r = hash(Y);
     *    h = m XOR r
     *    sign25519(v, h, x, s);
     *
     *    output (v,r) as the signature
     *
     * verification:
     *
     *    m = hash(Z, message);
     *    h = m XOR r
     *    verify25519(Y, v, h, P)
     *
     *    confirm  r == hash(Y)
     *
     * It would seem to me that it would be simpler to have the signer directly do
     * h = hash(m, Y) and send that to the recipient instead of r, who can verify
     * the signature by checking h == hash(m, Y).  If there are any problems with
     * such a scheme, please let me know.
     *
     * Also, EC-KCDSA (like most DS algorithms) picks x random, which is a waste of
     * perfectly good entropy, but does allow Y to be calculated in advance of (or
     * parallel to) hashing the message.
     */

    /* Signature generation primitive, calculates (x-h)s mod q
     *   v  [out] signature value
     *   h  [in]  signature hash (of message, signature pub key, and context data)
     *   x  [in]  signature private key
     *   s  [in]  private key for signing
     * returns true on success, false on failure (use different x or h)
     */
    public static boolean sign(byte[] v, byte[] h, byte[] x, byte[] s) {
        // v = (x - h) s  mod q
        int w, i;
        byte[] h1 = new byte[32], x1 = new byte[32];
        byte[] tmp1 = new byte[64];
        byte[] tmp2 = new byte[64];

        // Don't clobber the arguments, be nice!
        cpy32(h1, h);
        cpy32(x1, x);

        // Reduce modulo group order
        byte[] tmp3=new byte[32];
        divmod(tmp3, h1, 32, ORDER, 32);
        divmod(tmp3, x1, 32, ORDER, 32);

        // v = x1 - h1
        // If v is negative, add the group order to it to become positive.
        // If v was already positive we don't have to worry about overflow
        // when adding the order because v < ORDER and 2*ORDER < 2^256
        mula_small(v, x1, 0, h1, 32, -1);
        mula_small(v, v , 0, ORDER, 32, 1);

        // tmp1 = (x-h)*s mod q
        mula32(tmp1, v, s, 32, 1);
        divmod(tmp2, tmp1, 64, ORDER, 32);

        for (w = 0, i = 0; i < 32; i++)
            w |= v[i] = tmp1[i];
        return w != 0;
    }

    /* Signature verification primitive, calculates Y = vP + hG
     *   Y  [out] signature public key
     *   v  [in]  signature value
     *   h  [in]  signature hash
     *   P  [in]  public key
     */
    public static void verify(byte[] Y, byte[] v, byte[] h, byte[] P) {
        /* Y = v abs(P) + h G  */
        byte[] d=new byte[32];
        long10[]
                p=new long10[]{new long10(),new long10()},
                s=new long10[]{new long10(),new long10()},
                yx=new long10[]{new long10(),new long10(),new long10()},
                yz=new long10[]{new long10(),new long10(),new long10()},
                t1=new long10[]{new long10(),new long10(),new long10()},
                t2=new long10[]{new long10(),new long10(),new long10()};

        int vi = 0, hi = 0, di = 0, nvh=0, i, j, k;

        /* set p[0] to G and p[1] to P  */

        set(p[0], 9);
        unpack(p[1], P);

        /* set s[0] to P+G and s[1] to P-G  */

        /* s[0] = (Py^2 + Gy^2 - 2 Py Gy)/(Px - Gx)^2 - Px - Gx - 486662  */
        /* s[1] = (Py^2 + Gy^2 + 2 Py Gy)/(Px - Gx)^2 - Px - Gx - 486662  */

        x_to_y2(t1[0], t2[0], p[1]);	/* t2[0] = Py^2  */
        sqrt(t1[0], t2[0]);	/* t1[0] = Py or -Py  */
        j = is_negative(t1[0]);		/*      ... check which  */
        t2[0]._0 += 39420360;		/* t2[0] = Py^2 + Gy^2  */
        mul(t2[1], BASE_2Y, t1[0]);/* t2[1] = 2 Py Gy or -2 Py Gy  */
        sub(t1[j], t2[0], t2[1]);	/* t1[0] = Py^2 + Gy^2 - 2 Py Gy  */
        add(t1[1-j], t2[0], t2[1]);/* t1[1] = Py^2 + Gy^2 + 2 Py Gy  */
        cpy(t2[0], p[1]);		/* t2[0] = Px  */
        t2[0]._0 -= 9;			/* t2[0] = Px - Gx  */
        sqr(t2[1], t2[0]);		/* t2[1] = (Px - Gx)^2  */
        recip(t2[0], t2[1], 0);	/* t2[0] = 1/(Px - Gx)^2  */
        mul(s[0], t1[0], t2[0]);	/* s[0] = t1[0]/(Px - Gx)^2  */
        sub(s[0], s[0], p[1]);	/* s[0] = t1[0]/(Px - Gx)^2 - Px  */
        s[0]._0 -= 9 + 486662;		/* s[0] = X(P+G)  */
        mul(s[1], t1[1], t2[0]);	/* s[1] = t1[1]/(Px - Gx)^2  */
        sub(s[1], s[1], p[1]);	/* s[1] = t1[1]/(Px - Gx)^2 - Px  */
        s[1]._0 -= 9 + 486662;		/* s[1] = X(P-G)  */
        mul_small(s[0], s[0], 1);	/* reduce s[0] */
        mul_small(s[1], s[1], 1);	/* reduce s[1] */


        /* prepare the chain  */
        for (i = 0; i < 32; i++) {
            vi = (vi >> 8) ^ (v[i] & 0xFF) ^ ((v[i] & 0xFF) << 1);
            hi = (hi >> 8) ^ (h[i] & 0xFF) ^ ((h[i] & 0xFF) << 1);
            nvh = ~(vi ^ hi);
            di = (nvh & (di & 0x80) >> 7) ^ vi;
            di ^= nvh & (di & 0x01) << 1;
            di ^= nvh & (di & 0x02) << 1;
            di ^= nvh & (di & 0x04) << 1;
            di ^= nvh & (di & 0x08) << 1;
            di ^= nvh & (di & 0x10) << 1;
            di ^= nvh & (di & 0x20) << 1;
            di ^= nvh & (di & 0x40) << 1;
            d[i] = (byte)di;
        }

        di = ((nvh & (di & 0x80) << 1) ^ vi) >> 8;

        /* initialize state */
        set(yx[0], 1);
        cpy(yx[1], p[di]);
        cpy(yx[2], s[0]);
        set(yz[0], 0);
        set(yz[1], 1);
        set(yz[2], 1);

        /* y[0] is (even)P + (even)G
         * y[1] is (even)P + (odd)G  if current d-bit is 0
         * y[1] is (odd)P + (even)G  if current d-bit is 1
         * y[2] is (odd)P + (odd)G
         */

        vi = 0;
        hi = 0;

        /* and go for it! */
        for (i = 32; i--!=0; ) {
            vi = (vi << 8) | (v[i] & 0xFF);
            hi = (hi << 8) | (h[i] & 0xFF);
            di = (di << 8) | (d[i] & 0xFF);

            for (j = 8; j--!=0; ) {
                mont_prep(t1[0], t2[0], yx[0], yz[0]);
                mont_prep(t1[1], t2[1], yx[1], yz[1]);
                mont_prep(t1[2], t2[2], yx[2], yz[2]);

                k = ((vi ^ vi >> 1) >> j & 1)
                        + ((hi ^ hi >> 1) >> j & 1);
                mont_dbl(yx[2], yz[2], t1[k], t2[k], yx[0], yz[0]);

                k = (di >> j & 2) ^ ((di >> j & 1) << 1);
                mont_add(t1[1], t2[1], t1[k], t2[k], yx[1], yz[1],
                        p[di >> j & 1]);

                mont_add(t1[2], t2[2], t1[0], t2[0], yx[2], yz[2],
                        s[((vi ^ hi) >> j & 2) >> 1]);
            }
        }

        k = (vi & 1) + (hi & 1);
        recip(t1[0], yz[k], 0);
        mul(t1[1], yx[k], t1[0]);

        pack(t1[1], Y);
    }

    public static boolean isCanonicalSignature(byte[] v) {
        byte[] vCopy = java.util.Arrays.copyOfRange(v, 0, 32);
        byte[] tmp = new byte[32];
        divmod(tmp, vCopy, 32, ORDER, 32);
        for (int i = 0; i < 32; i++){
            if (v[i] != vCopy[i])
                return false;
        }
        return true;
    }

    public static boolean isCanonicalPublicKey(byte[] publicKey) {
        if ( publicKey.length != 32 ) {
            return false;
        }
        long10 publicKeyUnpacked = new long10();
        unpack(publicKeyUnpacked, publicKey);
        byte[] publicKeyCopy = new byte[32];
        pack(publicKeyUnpacked, publicKeyCopy);
        for (int i = 0; i < 32; i++){
            if (publicKeyCopy[i] != publicKey[i]) {
                return false;
            }
        }
        return true;
    }


    ///////////////////////////////////////////////////////////////////////////

    /* sahn0:
     * Using this class instead of long[10] to avoid bounds checks. */
    private static final class long10 {
        public long10() {}
        public long10(
                long _0, long _1, long _2, long _3, long _4,
                long _5, long _6, long _7, long _8, long _9)
        {
            this._0=_0; this._1=_1; this._2=_2;
            this._3=_3; this._4=_4; this._5=_5;
            this._6=_6; this._7=_7; this._8=_8;
            this._9=_9;
        }
        public long _0,_1,_2,_3,_4,_5,_6,_7,_8,_9;
    }

    /********************* radix 2^8 math *********************/

    private static void cpy32(byte[] d, byte[] s) {
        int i;
        for (i = 0; i < 32; i++)
            d[i] = s[i];
    }

    /* p[m..n+m-1] = q[m..n+m-1] + z * x */
    /* n is the size of x */
    /* n+m is the size of p and q */
    private static int mula_small(byte[] p,byte[] q,int m,byte[] x,int n,int z) {
        int v=0;
        for (int i=0;i<n;++i) {
            v+=(q[i+m] & 0xFF)+z*(x[i] & 0xFF);
            p[i+m]=(byte)v;
            v>>=8;
        }
        return v;
    }

    /* p += x * y * z  where z is a small integer
     * x is size 32, y is size t, p is size 32+t
     * y is allowed to overlap with p+32 if you don't care about the upper half  */
    private static int mula32(byte[] p, byte[] x, byte[] y, int t, int z) {
        final int n = 31;
        int w = 0;
        int i = 0;
        for (; i < t; i++) {
            int zy = z * (y[i] & 0xFF);
            w += mula_small(p, p, i, x, n, zy) +
                    (p[i+n] & 0xFF) + zy * (x[n] & 0xFF);
            p[i+n] = (byte)w;
            w >>= 8;
        }
        p[i+n] = (byte)(w + (p[i+n] & 0xFF));
        return w >> 8;
    }

    /* divide r (size n) by d (size t), returning quotient q and remainder r
     * quotient is size n-t+1, remainder is size t
     * requires t > 0 && d[t-1] != 0
     * requires that r[-1] and d[-1] are valid memory locations
     * q may overlap with r+t */
    private static void divmod(byte[] q, byte[] r, int n, byte[] d, int t) {
        int rn = 0;
        int dt = ((d[t-1] & 0xFF) << 8);
        if (t>1) {
            dt |= (d[t-2] & 0xFF);
        }
        while (n-- >= t) {
            int z = (rn << 16) | ((r[n] & 0xFF) << 8);
            if (n>0) {
                z |= (r[n-1] & 0xFF);
            }
            z/=dt;
            rn += mula_small(r,r, n-t+1, d, t, -z);
            q[n-t+1] = (byte)((z + rn) & 0xFF); /* rn is 0 or -1 (underflow) */
            mula_small(r,r, n-t+1, d, t, -rn);
            rn = (r[n] & 0xFF);
            r[n] = 0;
        }
        r[t-1] = (byte)rn;
    }

    private static int numsize(byte[] x,int n) {
        while (n--!=0 && x[n]==0)
            ;
        return n+1;
    }

    /* Returns x if a contains the gcd, y if b.
     * Also, the returned buffer contains the inverse of a mod b,
     * as 32-byte signed.
     * x and y must have 64 bytes space for temporary use.
     * requires that a[-1] and b[-1] are valid memory locations  */
    private static byte[] egcd32(byte[] x,byte[] y,byte[] a,byte[] b) {
        int an, bn = 32, qn, i;
        for (i = 0; i < 32; i++)
            x[i] = y[i] = 0;
        x[0] = 1;
        an = numsize(a, 32);
        if (an==0)
            return y;	/* division by zero */
        byte[] temp=new byte[32];
        while (true) {
            qn = bn - an + 1;
            divmod(temp, b, bn, a, an);
            bn = numsize(b, bn);
            if (bn==0)
                return x;
            mula32(y, x, temp, qn, -1);

            qn = an - bn + 1;
            divmod(temp, a, an, b, bn);
            an = numsize(a, an);
            if (an==0)
                return y;
            mula32(x, y, temp, qn, -1);
        }
    }

    /********************* radix 2^25.5 GF(2^255-19) math *********************/

    private static final int P25=33554431;	/* (1 << 25) - 1 */
    private static final int P26=67108863;	/* (1 << 26) - 1 */

    /* Convert to internal format from little-endian byte format */
    private static void unpack(long10 x,byte[] m) {
        x._0 = ((m[0] & 0xFF))         | ((m[1] & 0xFF))<<8 |
                (m[2] & 0xFF)<<16      | ((m[3] & 0xFF)& 3)<<24;
        x._1 = ((m[3] & 0xFF)&~ 3)>>2  | (m[4] & 0xFF)<<6 |
                (m[5] & 0xFF)<<14 | ((m[6] & 0xFF)& 7)<<22;
        x._2 = ((m[6] & 0xFF)&~ 7)>>3  | (m[7] & 0xFF)<<5 |
                (m[8] & 0xFF)<<13 | ((m[9] & 0xFF)&31)<<21;
        x._3 = ((m[9] & 0xFF)&~31)>>5  | (m[10] & 0xFF)<<3 |
                (m[11] & 0xFF)<<11 | ((m[12] & 0xFF)&63)<<19;
        x._4 = ((m[12] & 0xFF)&~63)>>6 | (m[13] & 0xFF)<<2 |
                (m[14] & 0xFF)<<10 |  (m[15] & 0xFF)    <<18;
        x._5 =  (m[16] & 0xFF)         | (m[17] & 0xFF)<<8 |
                (m[18] & 0xFF)<<16 | ((m[19] & 0xFF)& 1)<<24;
        x._6 = ((m[19] & 0xFF)&~ 1)>>1 | (m[20] & 0xFF)<<7 |
                (m[21] & 0xFF)<<15 | ((m[22] & 0xFF)& 7)<<23;
        x._7 = ((m[22] & 0xFF)&~ 7)>>3 | (m[23] & 0xFF)<<5 |
                (m[24] & 0xFF)<<13 | ((m[25] & 0xFF)&15)<<21;
        x._8 = ((m[25] & 0xFF)&~15)>>4 | (m[26] & 0xFF)<<4 |
                (m[27] & 0xFF)<<12 | ((m[28] & 0xFF)&63)<<20;
        x._9 = ((m[28] & 0xFF)&~63)>>6 | (m[29] & 0xFF)<<2 |
                (m[30] & 0xFF)<<10 |  (m[31] & 0xFF)    <<18;
    }

    /* Check if reduced-form input >= 2^255-19 */
    private static boolean is_overflow(long10 x) {
        return (
                ((x._0 > P26-19)) &&
                        ((x._1 & x._3 & x._5 & x._7 & x._9) == P25) &&
                        ((x._2 & x._4 & x._6 & x._8) == P26)
        ) || (x._9 > P25);
    }

    /* Convert from internal format to little-endian byte format.  The
     * number must be in a reduced form which is output by the following ops:
     *     unpack, mul, sqr
     *     set --  if input in range 0 .. P25
     * If you're unsure if the number is reduced, first multiply it by 1.  */
    private static void pack(long10 x,byte[] m) {
        int ld = 0, ud = 0;
        long t;
        ld = (is_overflow(x)?1:0) - ((x._9 < 0)?1:0);
        ud = ld * -(P25+1);
        ld *= 19;
        t = ld + x._0 + (x._1 << 26);
        m[ 0] = (byte)t;
        m[ 1] = (byte)(t >> 8);
        m[ 2] = (byte)(t >> 16);
        m[ 3] = (byte)(t >> 24);
        t = (t >> 32) + (x._2 << 19);
        m[ 4] = (byte)t;
        m[ 5] = (byte)(t >> 8);
        m[ 6] = (byte)(t >> 16);
        m[ 7] = (byte)(t >> 24);
        t = (t >> 32) + (x._3 << 13);
        m[ 8] = (byte)t;
        m[ 9] = (byte)(t >> 8);
        m[10] = (byte)(t >> 16);
        m[11] = (byte)(t >> 24);
        t = (t >> 32) + (x._4 <<  6);
        m[12] = (byte)t;
        m[13] = (byte)(t >> 8);
        m[14] = (byte)(t >> 16);
        m[15] = (byte)(t >> 24);
        t = (t >> 32) + x._5 + (x._6 << 25);
        m[16] = (byte)t;
        m[17] = (byte)(t >> 8);
        m[18] = (byte)(t >> 16);
        m[19] = (byte)(t >> 24);
        t = (t >> 32) + (x._7 << 19);
        m[20] = (byte)t;
        m[21] = (byte)(t >> 8);
        m[22] = (byte)(t >> 16);
        m[23] = (byte)(t >> 24);
        t = (t >> 32) + (x._8 << 12);
        m[24] = (byte)t;
        m[25] = (byte)(t >> 8);
        m[26] = (byte)(t >> 16);
        m[27] = (byte)(t >> 24);
        t = (t >> 32) + ((x._9 + ud) << 6);
        m[28] = (byte)t;
        m[29] = (byte)(t >> 8);
        m[30] = (byte)(t >> 16);
        m[31] = (byte)(t >> 24);
    }

    /* Copy a number */
    private static void cpy(long10 out, long10 in) {
        out._0=in._0;	out._1=in._1;
        out._2=in._2;	out._3=in._3;
        out._4=in._4;	out._5=in._5;
        out._6=in._6;	out._7=in._7;
        out._8=in._8;	out._9=in._9;
    }

    /* Set a number to value, which must be in range -185861411 .. 185861411 */
    private static void set(long10 out, int in) {
        out._0=in;	out._1=0;
        out._2=0;	out._3=0;
        out._4=0;	out._5=0;
        out._6=0;	out._7=0;
        out._8=0;	out._9=0;
    }

    /* Add/subtract two numbers.  The inputs must be in reduced form, and the
     * output isn't, so to do another addition or subtraction on the output,
     * first multiply it by one to reduce it. */
    private static void add(long10 xy, long10 x, long10 y) {
        xy._0 = x._0 + y._0;	xy._1 = x._1 + y._1;
        xy._2 = x._2 + y._2;	xy._3 = x._3 + y._3;
        xy._4 = x._4 + y._4;	xy._5 = x._5 + y._5;
        xy._6 = x._6 + y._6;	xy._7 = x._7 + y._7;
        xy._8 = x._8 + y._8;	xy._9 = x._9 + y._9;
    }
    private static void sub(long10 xy, long10 x, long10 y) {
        xy._0 = x._0 - y._0;	xy._1 = x._1 - y._1;
        xy._2 = x._2 - y._2;	xy._3 = x._3 - y._3;
        xy._4 = x._4 - y._4;	xy._5 = x._5 - y._5;
        xy._6 = x._6 - y._6;	xy._7 = x._7 - y._7;
        xy._8 = x._8 - y._8;	xy._9 = x._9 - y._9;
    }

    /* Multiply a number by a small integer in range -185861411 .. 185861411.
     * The output is in reduced form, the input x need not be.  x and xy may point
     * to the same buffer. */
    private static long10 mul_small(long10 xy, long10 x, long y) {
        long t;
        t = (x._8*y);
        xy._8 = (t & ((1 << 26) - 1));
        t = (t >> 26) + (x._9*y);
        xy._9 = (t & ((1 << 25) - 1));
        t = 19 * (t >> 25) + (x._0*y);
        xy._0 = (t & ((1 << 26) - 1));
        t = (t >> 26) + (x._1*y);
        xy._1 = (t & ((1 << 25) - 1));
        t = (t >> 25) + (x._2*y);
        xy._2 = (t & ((1 << 26) - 1));
        t = (t >> 26) + (x._3*y);
        xy._3 = (t & ((1 << 25) - 1));
        t = (t >> 25) + (x._4*y);
        xy._4 = (t & ((1 << 26) - 1));
        t = (t >> 26) + (x._5*y);
        xy._5 = (t & ((1 << 25) - 1));
        t = (t >> 25) + (x._6*y);
        xy._6 = (t & ((1 << 26) - 1));
        t = (t >> 26) + (x._7*y);
        xy._7 = (t & ((1 << 25) - 1));
        t = (t >> 25) + xy._8;
        xy._8 = (t & ((1 << 26) - 1));
        xy._9 += (t >> 26);
        return xy;
    }

    /* Multiply two numbers.  The output is in reduced form, the inputs need not
     * be. */
    private static long10 mul(long10 xy, long10 x, long10 y) {
        /* sahn0:
         * Using local variables to avoid class access.
         * This seem to improve performance a bit...
         */
        long
                x_0=x._0,x_1=x._1,x_2=x._2,x_3=x._3,x_4=x._4,
                x_5=x._5,x_6=x._6,x_7=x._7,x_8=x._8,x_9=x._9;
        long
                y_0=y._0,y_1=y._1,y_2=y._2,y_3=y._3,y_4=y._4,
                y_5=y._5,y_6=y._6,y_7=y._7,y_8=y._8,y_9=y._9;
        long t;
        t = (x_0*y_8) + (x_2*y_6) + (x_4*y_4) + (x_6*y_2) +
                (x_8*y_0) + 2 * ((x_1*y_7) + (x_3*y_5) +
                (x_5*y_3) + (x_7*y_1)) + 38 *
                (x_9*y_9);
        xy._8 = (t & ((1 << 26) - 1));
        t = (t >> 26) + (x_0*y_9) + (x_1*y_8) + (x_2*y_7) +
                (x_3*y_6) + (x_4*y_5) + (x_5*y_4) +
                (x_6*y_3) + (x_7*y_2) + (x_8*y_1) +
                (x_9*y_0);
        xy._9 = (t & ((1 << 25) - 1));
        t = (x_0*y_0) + 19 * ((t >> 25) + (x_2*y_8) + (x_4*y_6)
                + (x_6*y_4) + (x_8*y_2)) + 38 *
                ((x_1*y_9) + (x_3*y_7) + (x_5*y_5) +
                        (x_7*y_3) + (x_9*y_1));
        xy._0 = (t & ((1 << 26) - 1));
        t = (t >> 26) + (x_0*y_1) + (x_1*y_0) + 19 * ((x_2*y_9)
                + (x_3*y_8) + (x_4*y_7) + (x_5*y_6) +
                (x_6*y_5) + (x_7*y_4) + (x_8*y_3) +
                (x_9*y_2));
        xy._1 = (t & ((1 << 25) - 1));
        t = (t >> 25) + (x_0*y_2) + (x_2*y_0) + 19 * ((x_4*y_8)
                + (x_6*y_6) + (x_8*y_4)) + 2 * (x_1*y_1)
                + 38 * ((x_3*y_9) + (x_5*y_7) +
                (x_7*y_5) + (x_9*y_3));
        xy._2 = (t & ((1 << 26) - 1));
        t = (t >> 26) + (x_0*y_3) + (x_1*y_2) + (x_2*y_1) +
                (x_3*y_0) + 19 * ((x_4*y_9) + (x_5*y_8) +
                (x_6*y_7) + (x_7*y_6) +
                (x_8*y_5) + (x_9*y_4));
        xy._3 = (t & ((1 << 25) - 1));
        t = (t >> 25) + (x_0*y_4) + (x_2*y_2) + (x_4*y_0) + 19 *
                ((x_6*y_8) + (x_8*y_6)) + 2 * ((x_1*y_3) +
                (x_3*y_1)) + 38 *
                ((x_5*y_9) + (x_7*y_7) + (x_9*y_5));
        xy._4 = (t & ((1 << 26) - 1));
        t = (t >> 26) + (x_0*y_5) + (x_1*y_4) + (x_2*y_3) +
                (x_3*y_2) + (x_4*y_1) + (x_5*y_0) + 19 *
                ((x_6*y_9) + (x_7*y_8) + (x_8*y_7) +
                        (x_9*y_6));
        xy._5 = (t & ((1 << 25) - 1));
        t = (t >> 25) + (x_0*y_6) + (x_2*y_4) + (x_4*y_2) +
                (x_6*y_0) + 19 * (x_8*y_8) + 2 * ((x_1*y_5) +
                (x_3*y_3) + (x_5*y_1)) + 38 *
                ((x_7*y_9) + (x_9*y_7));
        xy._6 = (t & ((1 << 26) - 1));
        t = (t >> 26) + (x_0*y_7) + (x_1*y_6) + (x_2*y_5) +
                (x_3*y_4) + (x_4*y_3) + (x_5*y_2) +
                (x_6*y_1) + (x_7*y_0) + 19 * ((x_8*y_9) +
                (x_9*y_8));
        xy._7 = (t & ((1 << 25) - 1));
        t = (t >> 25) + xy._8;
        xy._8 = (t & ((1 << 26) - 1));
        xy._9 += (t >> 26);
        return xy;
    }

    /* Square a number.  Optimization of  mul25519(x2, x, x)  */
    private static long10 sqr(long10 x2, long10 x) {
        long
                x_0=x._0,x_1=x._1,x_2=x._2,x_3=x._3,x_4=x._4,
                x_5=x._5,x_6=x._6,x_7=x._7,x_8=x._8,x_9=x._9;
        long t;
        t = (x_4*x_4) + 2 * ((x_0*x_8) + (x_2*x_6)) + 38 *
                (x_9*x_9) + 4 * ((x_1*x_7) + (x_3*x_5));
        x2._8 = (t & ((1 << 26) - 1));
        t = (t >> 26) + 2 * ((x_0*x_9) + (x_1*x_8) + (x_2*x_7) +
                (x_3*x_6) + (x_4*x_5));
        x2._9 = (t & ((1 << 25) - 1));
        t = 19 * (t >> 25) + (x_0*x_0) + 38 * ((x_2*x_8) +
                (x_4*x_6) + (x_5*x_5)) + 76 * ((x_1*x_9)
                + (x_3*x_7));
        x2._0 = (t & ((1 << 26) - 1));
        t = (t >> 26) + 2 * (x_0*x_1) + 38 * ((x_2*x_9) +
                (x_3*x_8) + (x_4*x_7) + (x_5*x_6));
        x2._1 = (t & ((1 << 25) - 1));
        t = (t >> 25) + 19 * (x_6*x_6) + 2 * ((x_0*x_2) +
                (x_1*x_1)) + 38 * (x_4*x_8) + 76 *
                ((x_3*x_9) + (x_5*x_7));
        x2._2 = (t & ((1 << 26) - 1));
        t = (t >> 26) + 2 * ((x_0*x_3) + (x_1*x_2)) + 38 *
                ((x_4*x_9) + (x_5*x_8) + (x_6*x_7));
        x2._3 = (t & ((1 << 25) - 1));
        t = (t >> 25) + (x_2*x_2) + 2 * (x_0*x_4) + 38 *
                ((x_6*x_8) + (x_7*x_7)) + 4 * (x_1*x_3) + 76 *
                (x_5*x_9);
        x2._4 = (t & ((1 << 26) - 1));
        t = (t >> 26) + 2 * ((x_0*x_5) + (x_1*x_4) + (x_2*x_3))
                + 38 * ((x_6*x_9) + (x_7*x_8));
        x2._5 = (t & ((1 << 25) - 1));
        t = (t >> 25) + 19 * (x_8*x_8) + 2 * ((x_0*x_6) +
                (x_2*x_4) + (x_3*x_3)) + 4 * (x_1*x_5) +
                76 * (x_7*x_9);
        x2._6 = (t & ((1 << 26) - 1));
        t = (t >> 26) + 2 * ((x_0*x_7) + (x_1*x_6) + (x_2*x_5) +
                (x_3*x_4)) + 38 * (x_8*x_9);
        x2._7 = (t & ((1 << 25) - 1));
        t = (t >> 25) + x2._8;
        x2._8 = (t & ((1 << 26) - 1));
        x2._9 += (t >> 26);
        return x2;
    }

    /* Calculates a reciprocal.  The output is in reduced form, the inputs need not
     * be.  Simply calculates  y = x^(p-2)  so it's not too fast. */
    /* When sqrtassist is true, it instead calculates y = x^((p-5)/8) */
    private static void recip(long10 y, long10 x, int sqrtassist) {
        long10
                t0=new long10(),
                t1=new long10(),
                t2=new long10(),
                t3=new long10(),
                t4=new long10();
        int i;
        /* the chain for x^(2^255-21) is straight from djb's implementation */
        sqr(t1, x);	/*  2 == 2 * 1	*/
        sqr(t2, t1);	/*  4 == 2 * 2	*/
        sqr(t0, t2);	/*  8 == 2 * 4	*/
        mul(t2, t0, x);	/*  9 == 8 + 1	*/
        mul(t0, t2, t1);	/* 11 == 9 + 2	*/
        sqr(t1, t0);	/* 22 == 2 * 11	*/
        mul(t3, t1, t2);	/* 31 == 22 + 9
                    == 2^5   - 2^0	*/
        sqr(t1, t3);	/* 2^6   - 2^1	*/
        sqr(t2, t1);	/* 2^7   - 2^2	*/
        sqr(t1, t2);	/* 2^8   - 2^3	*/
        sqr(t2, t1);	/* 2^9   - 2^4	*/
        sqr(t1, t2);	/* 2^10  - 2^5	*/
        mul(t2, t1, t3);	/* 2^10  - 2^0	*/
        sqr(t1, t2);	/* 2^11  - 2^1	*/
        sqr(t3, t1);	/* 2^12  - 2^2	*/
        for (i = 1; i < 5; i++) {
            sqr(t1, t3);
            sqr(t3, t1);
        } /* t3 */		/* 2^20  - 2^10	*/
        mul(t1, t3, t2);	/* 2^20  - 2^0	*/
        sqr(t3, t1);	/* 2^21  - 2^1	*/
        sqr(t4, t3);	/* 2^22  - 2^2	*/
        for (i = 1; i < 10; i++) {
            sqr(t3, t4);
            sqr(t4, t3);
        } /* t4 */		/* 2^40  - 2^20	*/
        mul(t3, t4, t1);	/* 2^40  - 2^0	*/
        for (i = 0; i < 5; i++) {
            sqr(t1, t3);
            sqr(t3, t1);
        } /* t3 */		/* 2^50  - 2^10	*/
        mul(t1, t3, t2);	/* 2^50  - 2^0	*/
        sqr(t2, t1);	/* 2^51  - 2^1	*/
        sqr(t3, t2);	/* 2^52  - 2^2	*/
        for (i = 1; i < 25; i++) {
            sqr(t2, t3);
            sqr(t3, t2);
        } /* t3 */		/* 2^100 - 2^50 */
        mul(t2, t3, t1);	/* 2^100 - 2^0	*/
        sqr(t3, t2);	/* 2^101 - 2^1	*/
        sqr(t4, t3);	/* 2^102 - 2^2	*/
        for (i = 1; i < 50; i++) {
            sqr(t3, t4);
            sqr(t4, t3);
        } /* t4 */		/* 2^200 - 2^100 */
        mul(t3, t4, t2);	/* 2^200 - 2^0	*/
        for (i = 0; i < 25; i++) {
            sqr(t4, t3);
            sqr(t3, t4);
        } /* t3 */		/* 2^250 - 2^50	*/
        mul(t2, t3, t1);	/* 2^250 - 2^0	*/
        sqr(t1, t2);	/* 2^251 - 2^1	*/
        sqr(t2, t1);	/* 2^252 - 2^2	*/
        if (sqrtassist!=0) {
            mul(y, x, t2);	/* 2^252 - 3 */
        } else {
            sqr(t1, t2);	/* 2^253 - 2^3	*/
            sqr(t2, t1);	/* 2^254 - 2^4	*/
            sqr(t1, t2);	/* 2^255 - 2^5	*/
            mul(y, t1, t0);	/* 2^255 - 21	*/
        }
    }

    /* checks if x is "negative", requires reduced input */
    private static int is_negative(long10 x) {
        return (int)(((is_overflow(x) || (x._9 < 0))?1:0) ^ (x._0 & 1));
    }

    /* a square root */
    private static void sqrt(long10 x, long10 u) {
        long10 v=new long10(), t1=new long10(), t2=new long10();
        add(t1, u, u);	/* t1 = 2u		*/
        recip(v, t1, 1);	/* v = (2u)^((p-5)/8)	*/
        sqr(x, v);		/* x = v^2		*/
        mul(t2, t1, x);	/* t2 = 2uv^2		*/
        t2._0--;		/* t2 = 2uv^2-1		*/
        mul(t1, v, t2);	/* t1 = v(2uv^2-1)	*/
        mul(x, u, t1);	/* x = uv(2uv^2-1)	*/
    }

    /********************* Elliptic curve *********************/

    /* y^2 = x^3 + 486662 x^2 + x  over GF(2^255-19) */

    /* t1 = ax + az
     * t2 = ax - az  */
    private static void mont_prep(long10 t1, long10 t2, long10 ax, long10 az) {
        add(t1, ax, az);
        sub(t2, ax, az);
    }

    /* A = P + Q   where
     *  X(A) = ax/az
     *  X(P) = (t1+t2)/(t1-t2)
     *  X(Q) = (t3+t4)/(t3-t4)
     *  X(P-Q) = dx
     * clobbers t1 and t2, preserves t3 and t4  */
    private static void mont_add(long10 t1, long10 t2, long10 t3, long10 t4,long10 ax, long10 az, long10 dx) {
        mul(ax, t2, t3);
        mul(az, t1, t4);
        add(t1, ax, az);
        sub(t2, ax, az);
        sqr(ax, t1);
        sqr(t1, t2);
        mul(az, t1, dx);
    }

    /* B = 2 * Q   where
     *  X(B) = bx/bz
     *  X(Q) = (t3+t4)/(t3-t4)
     * clobbers t1 and t2, preserves t3 and t4  */
    private static void mont_dbl(long10 t1, long10 t2, long10 t3, long10 t4,long10 bx, long10 bz) {
        sqr(t1, t3);
        sqr(t2, t4);
        mul(bx, t1, t2);
        sub(t2, t1, t2);
        mul_small(bz, t2, 121665);
        add(t1, t1, bz);
        mul(bz, t1, t2);
    }

    /* Y^2 = X^3 + 486662 X^2 + X
     * t is a temporary  */
    private static void x_to_y2(long10 t, long10 y2, long10 x) {
        sqr(t, x);
        mul_small(y2, x, 486662);
        add(t, t, y2);
        t._0++;
        mul(y2, t, x);
    }

    /* P = kG   and  s = sign(P)/k  */
    private static void core(byte[] Px, byte[] s, byte[] k, byte[] Gx) {
        long10
                dx=new long10(),
                t1=new long10(),
                t2=new long10(),
                t3=new long10(),
                t4=new long10();
        long10[]
                x=new long10[]{new long10(),new long10()},
                z=new long10[]{new long10(),new long10()};
        int i, j;

        /* unpack the base */
        if (Gx!=null)
            unpack(dx, Gx);
        else
            set(dx, 9);

        /* 0G = point-at-infinity */
        set(x[0], 1);
        set(z[0], 0);

        /* 1G = G */
        cpy(x[1], dx);
        set(z[1], 1);

        for (i = 32; i--!=0; ) {
            if (i==0) {
                i=0;
            }
            for (j = 8; j--!=0; ) {
                /* swap arguments depending on bit */
                int bit1 = (k[i] & 0xFF) >> j & 1;
                int bit0 = ~(k[i] & 0xFF) >> j & 1;
                long10 ax = x[bit0];
                long10 az = z[bit0];
                long10 bx = x[bit1];
                long10 bz = z[bit1];

                /* a' = a + b	*/
                /* b' = 2 b	*/
                mont_prep(t1, t2, ax, az);
                mont_prep(t3, t4, bx, bz);
                mont_add(t1, t2, t3, t4, ax, az, dx);
                mont_dbl(t1, t2, t3, t4, bx, bz);
            }
        }

        recip(t1, z[0], 0);
        mul(dx, x[0], t1);
        pack(dx, Px);

        /* calculate s such that s abs(P) = G  .. assumes G is std base point */
        if (s!=null) {
            x_to_y2(t2, t1, dx);	/* t1 = Py^2  */
            recip(t3, z[1], 0);	/* where Q=P+G ... */
            mul(t2, x[1], t3);	/* t2 = Qx  */
            add(t2, t2, dx);	/* t2 = Qx + Px  */
            t2._0 += 9 + 486662;	/* t2 = Qx + Px + Gx + 486662  */
            dx._0 -= 9;		/* dx = Px - Gx  */
            sqr(t3, dx);	/* t3 = (Px - Gx)^2  */
            mul(dx, t2, t3);	/* dx = t2 (Px - Gx)^2  */
            sub(dx, dx, t1);	/* dx = t2 (Px - Gx)^2 - Py^2  */
            dx._0 -= 39420360;	/* dx = t2 (Px - Gx)^2 - Py^2 - Gy^2  */
            mul(t1, dx, BASE_R2Y);	/* t1 = -Py  */
            if (is_negative(t1)!=0)	/* sign is 1, so just copy  */
                cpy32(s, k);
            else			/* sign is -1, so negate  */
                mula_small(s, ORDER_TIMES_8, 0, k, 32, -1);

            /* reduce s mod q
             * (is this needed?  do it just in case, it's fast anyway) */
            //divmod((dstptr) t1, s, 32, order25519, 32);

            /* take reciprocal of s mod q */
            byte[] temp1=new byte[32];
            byte[] temp2=new byte[64];
            byte[] temp3=new byte[64];
            cpy32(temp1, ORDER);
            cpy32(s, egcd32(temp2, temp3, s, temp1));
            if ((s[31] & 0x80)!=0)
                mula_small(s, s, 0, ORDER, 32, 1);
        }
    }

    /* smallest multiple of the order that's >= 2^255 */
    private static final byte[] ORDER_TIMES_8 = {
            (byte)104, (byte)159, (byte)174, (byte)231,
            (byte)210, (byte)24,  (byte)147, (byte)192,
            (byte)178, (byte)230, (byte)188, (byte)23,
            (byte)245, (byte)206, (byte)247, (byte)166,
            (byte)0,   (byte)0,   (byte)0,   (byte)0,
            (byte)0,   (byte)0,   (byte)0,   (byte)0,
            (byte)0,   (byte)0,   (byte)0,   (byte)0,
            (byte)0,   (byte)0,   (byte)0,   (byte)128
    };

    /* constants 2Gy and 1/(2Gy) */
    private static final long10 BASE_2Y = new long10(
            39999547, 18689728, 59995525, 1648697, 57546132,
            24010086, 19059592, 5425144, 63499247, 16420658
    );
    private static final long10 BASE_R2Y = new long10(
            5744, 8160848, 4790893, 13779497, 35730846,
            12541209, 49101323, 30047407, 40071253, 6226132
    );
}