plink: updated to revision 10170 of putty

1 files changed, 2001 insertions, 1918 deletions

diff --git a/tools/plink/sshbn.c b/tools/plink/sshbn.c
index 51cecdf2b..a5e0552ff 100644
--- a/tools/plink/sshbn.c
+++ b/tools/plink/sshbn.c
@@ -1,1918 +1,2001 @@
-/*

- * Bignum routines for RSA and DH and stuff.

- */

-

-#include <stdio.h>

-#include <assert.h>

-#include <stdlib.h>

-#include <string.h>

-

-#include "misc.h"

-

-/*

- * Usage notes:

- *  * Do not call the DIVMOD_WORD macro with expressions such as array

- *    subscripts, as some implementations object to this (see below).

- *  * Note that none of the division methods below will cope if the

- *    quotient won't fit into BIGNUM_INT_BITS. Callers should be careful

- *    to avoid this case.

- *    If this condition occurs, in the case of the x86 DIV instruction,

- *    an overflow exception will occur, which (according to a correspondent)

- *    will manifest on Windows as something like

- *      0xC0000095: Integer overflow

- *    The C variant won't give the right answer, either.

- */

-

-#if defined __GNUC__ && defined __i386__

-typedef unsigned long BignumInt;

-typedef unsigned long long BignumDblInt;

-#define BIGNUM_INT_MASK  0xFFFFFFFFUL

-#define BIGNUM_TOP_BIT   0x80000000UL

-#define BIGNUM_INT_BITS  32

-#define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)

-#define DIVMOD_WORD(q, r, hi, lo, w) \

-    __asm__("div %2" : \

-	    "=d" (r), "=a" (q) : \

-	    "r" (w), "d" (hi), "a" (lo))

-#elif defined _MSC_VER && defined _M_IX86

-typedef unsigned __int32 BignumInt;

-typedef unsigned __int64 BignumDblInt;

-#define BIGNUM_INT_MASK  0xFFFFFFFFUL

-#define BIGNUM_TOP_BIT   0x80000000UL

-#define BIGNUM_INT_BITS  32

-#define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)

-/* Note: MASM interprets array subscripts in the macro arguments as

- * assembler syntax, which gives the wrong answer. Don't supply them.

- * <http://msdn2.microsoft.com/en-us/library/bf1dw62z.aspx> */

-#define DIVMOD_WORD(q, r, hi, lo, w) do { \

-    __asm mov edx, hi \

-    __asm mov eax, lo \

-    __asm div w \

-    __asm mov r, edx \

-    __asm mov q, eax \

-} while(0)

-#elif defined _LP64

-/* 64-bit architectures can do 32x32->64 chunks at a time */

-typedef unsigned int BignumInt;

-typedef unsigned long BignumDblInt;

-#define BIGNUM_INT_MASK  0xFFFFFFFFU

-#define BIGNUM_TOP_BIT   0x80000000U

-#define BIGNUM_INT_BITS  32

-#define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)

-#define DIVMOD_WORD(q, r, hi, lo, w) do { \

-    BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \

-    q = n / w; \

-    r = n % w; \

-} while (0)

-#elif defined _LLP64

-/* 64-bit architectures in which unsigned long is 32 bits, not 64 */

-typedef unsigned long BignumInt;

-typedef unsigned long long BignumDblInt;

-#define BIGNUM_INT_MASK  0xFFFFFFFFUL

-#define BIGNUM_TOP_BIT   0x80000000UL

-#define BIGNUM_INT_BITS  32

-#define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)

-#define DIVMOD_WORD(q, r, hi, lo, w) do { \

-    BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \

-    q = n / w; \

-    r = n % w; \

-} while (0)

-#else

-/* Fallback for all other cases */

-typedef unsigned short BignumInt;

-typedef unsigned long BignumDblInt;

-#define BIGNUM_INT_MASK  0xFFFFU

-#define BIGNUM_TOP_BIT   0x8000U

-#define BIGNUM_INT_BITS  16

-#define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)

-#define DIVMOD_WORD(q, r, hi, lo, w) do { \

-    BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \

-    q = n / w; \

-    r = n % w; \

-} while (0)

-#endif

-

-#define BIGNUM_INT_BYTES (BIGNUM_INT_BITS / 8)

-

-#define BIGNUM_INTERNAL

-typedef BignumInt *Bignum;

-

-#include "ssh.h"

-

-BignumInt bnZero[1] = { 0 };

-BignumInt bnOne[2] = { 1, 1 };

-

-/*

- * The Bignum format is an array of `BignumInt'. The first

- * element of the array counts the remaining elements. The

- * remaining elements express the actual number, base 2^BIGNUM_INT_BITS, _least_

- * significant digit first. (So it's trivial to extract the bit

- * with value 2^n for any n.)

- *

- * All Bignums in this module are positive. Negative numbers must

- * be dealt with outside it.

- *

- * INVARIANT: the most significant word of any Bignum must be

- * nonzero.

- */

-

-Bignum Zero = bnZero, One = bnOne;

-

-static Bignum newbn(int length)

-{

-    Bignum b = snewn(length + 1, BignumInt);

-    if (!b)

-	abort();		       /* FIXME */

-    memset(b, 0, (length + 1) * sizeof(*b));

-    b[0] = length;

-    return b;

-}

-

-void bn_restore_invariant(Bignum b)

-{

-    while (b[0] > 1 && b[b[0]] == 0)

-	b[0]--;

-}

-

-Bignum copybn(Bignum orig)

-{

-    Bignum b = snewn(orig[0] + 1, BignumInt);

-    if (!b)

-	abort();		       /* FIXME */

-    memcpy(b, orig, (orig[0] + 1) * sizeof(*b));

-    return b;

-}

-

-void freebn(Bignum b)

-{

-    /*

-     * Burn the evidence, just in case.

-     */

-    memset(b, 0, sizeof(b[0]) * (b[0] + 1));

-    sfree(b);

-}

-

-Bignum bn_power_2(int n)

-{

-    Bignum ret = newbn(n / BIGNUM_INT_BITS + 1);

-    bignum_set_bit(ret, n, 1);

-    return ret;

-}

-

-/*

- * Internal addition. Sets c = a - b, where 'a', 'b' and 'c' are all

- * big-endian arrays of 'len' BignumInts. Returns a BignumInt carried

- * off the top.

- */

-static BignumInt internal_add(const BignumInt *a, const BignumInt *b,

-                              BignumInt *c, int len)

-{

-    int i;

-    BignumDblInt carry = 0;

-

-    for (i = len-1; i >= 0; i--) {

-        carry += (BignumDblInt)a[i] + b[i];

-        c[i] = (BignumInt)carry;

-        carry >>= BIGNUM_INT_BITS;

-    }

-

-    return (BignumInt)carry;

-}

-

-/*

- * Internal subtraction. Sets c = a - b, where 'a', 'b' and 'c' are

- * all big-endian arrays of 'len' BignumInts. Any borrow from the top

- * is ignored.

- */

-static void internal_sub(const BignumInt *a, const BignumInt *b,

-                         BignumInt *c, int len)

-{

-    int i;

-    BignumDblInt carry = 1;

-

-    for (i = len-1; i >= 0; i--) {

-        carry += (BignumDblInt)a[i] + (b[i] ^ BIGNUM_INT_MASK);

-        c[i] = (BignumInt)carry;

-        carry >>= BIGNUM_INT_BITS;

-    }

-}

-

-/*

- * Compute c = a * b.

- * Input is in the first len words of a and b.

- * Result is returned in the first 2*len words of c.

- *

- * 'scratch' must point to an array of BignumInt of size at least

- * mul_compute_scratch(len). (This covers the needs of internal_mul

- * and all its recursive calls to itself.)

- */

-#define KARATSUBA_THRESHOLD 50

-static int mul_compute_scratch(int len)

-{

-    int ret = 0;

-    while (len > KARATSUBA_THRESHOLD) {

-        int toplen = len/2, botlen = len - toplen; /* botlen is the bigger */

-        int midlen = botlen + 1;

-        ret += 4*midlen;

-        len = midlen;

-    }

-    return ret;

-}

-static void internal_mul(const BignumInt *a, const BignumInt *b,

-			 BignumInt *c, int len, BignumInt *scratch)

-{

-    if (len > KARATSUBA_THRESHOLD) {

-        int i;

-

-        /*

-         * Karatsuba divide-and-conquer algorithm. Cut each input in

-         * half, so that it's expressed as two big 'digits' in a giant

-         * base D:

-         *

-         *   a = a_1 D + a_0

-         *   b = b_1 D + b_0

-         *

-         * Then the product is of course

-         *

-         *  ab = a_1 b_1 D^2 + (a_1 b_0 + a_0 b_1) D + a_0 b_0

-         *

-         * and we compute the three coefficients by recursively

-         * calling ourself to do half-length multiplications.

-         *

-         * The clever bit that makes this worth doing is that we only

-         * need _one_ half-length multiplication for the central

-         * coefficient rather than the two that it obviouly looks

-         * like, because we can use a single multiplication to compute

-         *

-         *   (a_1 + a_0) (b_1 + b_0) = a_1 b_1 + a_1 b_0 + a_0 b_1 + a_0 b_0

-         *

-         * and then we subtract the other two coefficients (a_1 b_1

-         * and a_0 b_0) which we were computing anyway.

-         *

-         * Hence we get to multiply two numbers of length N in about

-         * three times as much work as it takes to multiply numbers of

-         * length N/2, which is obviously better than the four times

-         * as much work it would take if we just did a long

-         * conventional multiply.

-         */

-

-        int toplen = len/2, botlen = len - toplen; /* botlen is the bigger */

-        int midlen = botlen + 1;

-        BignumDblInt carry;

-#ifdef KARA_DEBUG

-        int i;

-#endif

-

-        /*

-         * The coefficients a_1 b_1 and a_0 b_0 just avoid overlapping

-         * in the output array, so we can compute them immediately in

-         * place.

-         */

-

-#ifdef KARA_DEBUG

-        printf("a1,a0 = 0x");

-        for (i = 0; i < len; i++) {

-            if (i == toplen) printf(", 0x");

-            printf("%0*x", BIGNUM_INT_BITS/4, a[i]);

-        }

-        printf("\n");

-        printf("b1,b0 = 0x");

-        for (i = 0; i < len; i++) {

-            if (i == toplen) printf(", 0x");

-            printf("%0*x", BIGNUM_INT_BITS/4, b[i]);

-        }

-        printf("\n");

-#endif

-

-        /* a_1 b_1 */

-        internal_mul(a, b, c, toplen, scratch);

-#ifdef KARA_DEBUG

-        printf("a1b1 = 0x");

-        for (i = 0; i < 2*toplen; i++) {

-            printf("%0*x", BIGNUM_INT_BITS/4, c[i]);

-        }

-        printf("\n");

-#endif

-

-        /* a_0 b_0 */

-        internal_mul(a + toplen, b + toplen, c + 2*toplen, botlen, scratch);

-#ifdef KARA_DEBUG

-        printf("a0b0 = 0x");

-        for (i = 0; i < 2*botlen; i++) {

-            printf("%0*x", BIGNUM_INT_BITS/4, c[2*toplen+i]);

-        }

-        printf("\n");

-#endif

-

-        /* Zero padding. midlen exceeds toplen by at most 2, so just

-         * zero the first two words of each input and the rest will be

-         * copied over. */

-        scratch[0] = scratch[1] = scratch[midlen] = scratch[midlen+1] = 0;

-

-        for (i = 0; i < toplen; i++) {

-            scratch[midlen - toplen + i] = a[i]; /* a_1 */

-            scratch[2*midlen - toplen + i] = b[i]; /* b_1 */

-        }

-

-        /* compute a_1 + a_0 */

-        scratch[0] = internal_add(scratch+1, a+toplen, scratch+1, botlen);

-#ifdef KARA_DEBUG

-        printf("a1plusa0 = 0x");

-        for (i = 0; i < midlen; i++) {

-            printf("%0*x", BIGNUM_INT_BITS/4, scratch[i]);

-        }

-        printf("\n");

-#endif

-        /* compute b_1 + b_0 */

-        scratch[midlen] = internal_add(scratch+midlen+1, b+toplen,

-                                       scratch+midlen+1, botlen);

-#ifdef KARA_DEBUG

-        printf("b1plusb0 = 0x");

-        for (i = 0; i < midlen; i++) {

-            printf("%0*x", BIGNUM_INT_BITS/4, scratch[midlen+i]);

-        }

-        printf("\n");

-#endif

-

-        /*

-         * Now we can do the third multiplication.

-         */

-        internal_mul(scratch, scratch + midlen, scratch + 2*midlen, midlen,

-                     scratch + 4*midlen);

-#ifdef KARA_DEBUG

-        printf("a1plusa0timesb1plusb0 = 0x");

-        for (i = 0; i < 2*midlen; i++) {

-            printf("%0*x", BIGNUM_INT_BITS/4, scratch[2*midlen+i]);

-        }

-        printf("\n");

-#endif

-

-        /*

-         * Now we can reuse the first half of 'scratch' to compute the

-         * sum of the outer two coefficients, to subtract from that

-         * product to obtain the middle one.

-         */

-        scratch[0] = scratch[1] = scratch[2] = scratch[3] = 0;

-        for (i = 0; i < 2*toplen; i++)

-            scratch[2*midlen - 2*toplen + i] = c[i];

-        scratch[1] = internal_add(scratch+2, c + 2*toplen,

-                                  scratch+2, 2*botlen);

-#ifdef KARA_DEBUG

-        printf("a1b1plusa0b0 = 0x");

-        for (i = 0; i < 2*midlen; i++) {

-            printf("%0*x", BIGNUM_INT_BITS/4, scratch[i]);

-        }

-        printf("\n");

-#endif

-

-        internal_sub(scratch + 2*midlen, scratch,

-                     scratch + 2*midlen, 2*midlen);

-#ifdef KARA_DEBUG

-        printf("a1b0plusa0b1 = 0x");

-        for (i = 0; i < 2*midlen; i++) {

-            printf("%0*x", BIGNUM_INT_BITS/4, scratch[2*midlen+i]);

-        }

-        printf("\n");

-#endif

-

-        /*

-         * And now all we need to do is to add that middle coefficient

-         * back into the output. We may have to propagate a carry

-         * further up the output, but we can be sure it won't

-         * propagate right the way off the top.

-         */

-        carry = internal_add(c + 2*len - botlen - 2*midlen,

-                             scratch + 2*midlen,

-                             c + 2*len - botlen - 2*midlen, 2*midlen);

-        i = 2*len - botlen - 2*midlen - 1;

-        while (carry) {

-            assert(i >= 0);

-            carry += c[i];

-            c[i] = (BignumInt)carry;

-            carry >>= BIGNUM_INT_BITS;

-            i--;

-        }

-#ifdef KARA_DEBUG

-        printf("ab = 0x");

-        for (i = 0; i < 2*len; i++) {

-            printf("%0*x", BIGNUM_INT_BITS/4, c[i]);

-        }

-        printf("\n");

-#endif

-

-    } else {

-        int i;

-        BignumInt carry;

-        BignumDblInt t;

-        const BignumInt *ap, *bp;

-        BignumInt *cp, *cps;

-

-        /*

-         * Multiply in the ordinary O(N^2) way.

-         */

-

-        for (i = 0; i < 2 * len; i++)

-            c[i] = 0;

-

-        for (cps = c + 2*len, ap = a + len; ap-- > a; cps--) {

-            carry = 0;

-            for (cp = cps, bp = b + len; cp--, bp-- > b ;) {

-                t = (MUL_WORD(*ap, *bp) + carry) + *cp;

-                *cp = (BignumInt) t;

-                carry = (BignumInt)(t >> BIGNUM_INT_BITS);

-            }

-            *cp = carry;

-        }

-    }

-}

-

-/*

- * Variant form of internal_mul used for the initial step of

- * Montgomery reduction. Only bothers outputting 'len' words

- * (everything above that is thrown away).

- */

-static void internal_mul_low(const BignumInt *a, const BignumInt *b,

-                             BignumInt *c, int len, BignumInt *scratch)

-{

-    if (len > KARATSUBA_THRESHOLD) {

-        int i;

-

-        /*

-         * Karatsuba-aware version of internal_mul_low. As before, we

-         * express each input value as a shifted combination of two

-         * halves:

-         *

-         *   a = a_1 D + a_0

-         *   b = b_1 D + b_0

-         *

-         * Then the full product is, as before,

-         *

-         *  ab = a_1 b_1 D^2 + (a_1 b_0 + a_0 b_1) D + a_0 b_0

-         *

-         * Provided we choose D on the large side (so that a_0 and b_0

-         * are _at least_ as long as a_1 and b_1), we don't need the

-         * topmost term at all, and we only need half of the middle

-         * term. So there's no point in doing the proper Karatsuba

-         * optimisation which computes the middle term using the top

-         * one, because we'd take as long computing the top one as

-         * just computing the middle one directly.

-         *

-         * So instead, we do a much more obvious thing: we call the

-         * fully optimised internal_mul to compute a_0 b_0, and we

-         * recursively call ourself to compute the _bottom halves_ of

-         * a_1 b_0 and a_0 b_1, each of which we add into the result

-         * in the obvious way.

-         *

-         * In other words, there's no actual Karatsuba _optimisation_

-         * in this function; the only benefit in doing it this way is

-         * that we call internal_mul proper for a large part of the

-         * work, and _that_ can optimise its operation.

-         */

-

-        int toplen = len/2, botlen = len - toplen; /* botlen is the bigger */

-

-        /*

-         * Scratch space for the various bits and pieces we're going

-         * to be adding together: we need botlen*2 words for a_0 b_0

-         * (though we may end up throwing away its topmost word), and

-         * toplen words for each of a_1 b_0 and a_0 b_1. That adds up

-         * to exactly 2*len.

-         */

-

-        /* a_0 b_0 */

-        internal_mul(a + toplen, b + toplen, scratch + 2*toplen, botlen,

-                     scratch + 2*len);

-

-        /* a_1 b_0 */

-        internal_mul_low(a, b + len - toplen, scratch + toplen, toplen,

-                         scratch + 2*len);

-

-        /* a_0 b_1 */

-        internal_mul_low(a + len - toplen, b, scratch, toplen,

-                         scratch + 2*len);

-

-        /* Copy the bottom half of the big coefficient into place */

-        for (i = 0; i < botlen; i++)

-            c[toplen + i] = scratch[2*toplen + botlen + i];

-

-        /* Add the two small coefficients, throwing away the returned carry */

-        internal_add(scratch, scratch + toplen, scratch, toplen);

-

-        /* And add that to the large coefficient, leaving the result in c. */

-        internal_add(scratch, scratch + 2*toplen + botlen - toplen,

-                     c, toplen);

-

-    } else {

-        int i;

-        BignumInt carry;

-        BignumDblInt t;

-        const BignumInt *ap, *bp;

-        BignumInt *cp, *cps;

-

-        /*

-         * Multiply in the ordinary O(N^2) way.

-         */

-

-        for (i = 0; i < len; i++)

-            c[i] = 0;

-

-        for (cps = c + len, ap = a + len; ap-- > a; cps--) {

-            carry = 0;

-            for (cp = cps, bp = b + len; bp--, cp-- > c ;) {

-                t = (MUL_WORD(*ap, *bp) + carry) + *cp;

-                *cp = (BignumInt) t;

-                carry = (BignumInt)(t >> BIGNUM_INT_BITS);

-            }

-        }

-    }

-}

-

-/*

- * Montgomery reduction. Expects x to be a big-endian array of 2*len

- * BignumInts whose value satisfies 0 <= x < rn (where r = 2^(len *

- * BIGNUM_INT_BITS) is the Montgomery base). Returns in the same array

- * a value x' which is congruent to xr^{-1} mod n, and satisfies 0 <=

- * x' < n.

- *

- * 'n' and 'mninv' should be big-endian arrays of 'len' BignumInts

- * each, containing respectively n and the multiplicative inverse of

- * -n mod r.

- *

- * 'tmp' is an array of BignumInt used as scratch space, of length at

- * least 3*len + mul_compute_scratch(len).

- */

-static void monty_reduce(BignumInt *x, const BignumInt *n,

-                         const BignumInt *mninv, BignumInt *tmp, int len)

-{

-    int i;

-    BignumInt carry;

-

-    /*

-     * Multiply x by (-n)^{-1} mod r. This gives us a value m such

-     * that mn is congruent to -x mod r. Hence, mn+x is an exact

-     * multiple of r, and is also (obviously) congruent to x mod n.

-     */

-    internal_mul_low(x + len, mninv, tmp, len, tmp + 3*len);

-

-    /*

-     * Compute t = (mn+x)/r in ordinary, non-modular, integer

-     * arithmetic. By construction this is exact, and is congruent mod

-     * n to x * r^{-1}, i.e. the answer we want.

-     *

-     * The following multiply leaves that answer in the _most_

-     * significant half of the 'x' array, so then we must shift it

-     * down.

-     */

-    internal_mul(tmp, n, tmp+len, len, tmp + 3*len);

-    carry = internal_add(x, tmp+len, x, 2*len);

-    for (i = 0; i < len; i++)

-        x[len + i] = x[i], x[i] = 0;

-

-    /*

-     * Reduce t mod n. This doesn't require a full-on division by n,

-     * but merely a test and single optional subtraction, since we can

-     * show that 0 <= t < 2n.

-     *

-     * Proof:

-     *  + we computed m mod r, so 0 <= m < r.

-     *  + so 0 <= mn < rn, obviously

-     *  + hence we only need 0 <= x < rn to guarantee that 0 <= mn+x < 2rn

-     *  + yielding 0 <= (mn+x)/r < 2n as required.

-     */

-    if (!carry) {

-        for (i = 0; i < len; i++)

-            if (x[len + i] != n[i])

-                break;

-    }

-    if (carry || i >= len || x[len + i] > n[i])

-        internal_sub(x+len, n, x+len, len);

-}

-

-static void internal_add_shifted(BignumInt *number,

-				 unsigned n, int shift)

-{

-    int word = 1 + (shift / BIGNUM_INT_BITS);

-    int bshift = shift % BIGNUM_INT_BITS;

-    BignumDblInt addend;

-

-    addend = (BignumDblInt)n << bshift;

-

-    while (addend) {

-	addend += number[word];

-	number[word] = (BignumInt) addend & BIGNUM_INT_MASK;

-	addend >>= BIGNUM_INT_BITS;

-	word++;

-    }

-}

-

-/*

- * Compute a = a % m.

- * Input in first alen words of a and first mlen words of m.

- * Output in first alen words of a

- * (of which first alen-mlen words will be zero).

- * The MSW of m MUST have its high bit set.

- * Quotient is accumulated in the `quotient' array, which is a Bignum

- * rather than the internal bigendian format. Quotient parts are shifted

- * left by `qshift' before adding into quot.

- */

-static void internal_mod(BignumInt *a, int alen,

-			 BignumInt *m, int mlen,

-			 BignumInt *quot, int qshift)

-{

-    BignumInt m0, m1;

-    unsigned int h;

-    int i, k;

-

-    m0 = m[0];

-    if (mlen > 1)

-	m1 = m[1];

-    else

-	m1 = 0;

-

-    for (i = 0; i <= alen - mlen; i++) {

-	BignumDblInt t;

-	unsigned int q, r, c, ai1;

-

-	if (i == 0) {

-	    h = 0;

-	} else {

-	    h = a[i - 1];

-	    a[i - 1] = 0;

-	}

-

-	if (i == alen - 1)

-	    ai1 = 0;

-	else

-	    ai1 = a[i + 1];

-

-	/* Find q = h:a[i] / m0 */

-	if (h >= m0) {

-	    /*

-	     * Special case.

-	     * 

-	     * To illustrate it, suppose a BignumInt is 8 bits, and

-	     * we are dividing (say) A1:23:45:67 by A1:B2:C3. Then

-	     * our initial division will be 0xA123 / 0xA1, which

-	     * will give a quotient of 0x100 and a divide overflow.

-	     * However, the invariants in this division algorithm

-	     * are not violated, since the full number A1:23:... is

-	     * _less_ than the quotient prefix A1:B2:... and so the

-	     * following correction loop would have sorted it out.

-	     * 

-	     * In this situation we set q to be the largest

-	     * quotient we _can_ stomach (0xFF, of course).

-	     */

-	    q = BIGNUM_INT_MASK;

-	} else {

-	    /* Macro doesn't want an array subscript expression passed

-	     * into it (see definition), so use a temporary. */

-	    BignumInt tmplo = a[i];

-	    DIVMOD_WORD(q, r, h, tmplo, m0);

-

-	    /* Refine our estimate of q by looking at

-	     h:a[i]:a[i+1] / m0:m1 */

-	    t = MUL_WORD(m1, q);

-	    if (t > ((BignumDblInt) r << BIGNUM_INT_BITS) + ai1) {

-		q--;

-		t -= m1;

-		r = (r + m0) & BIGNUM_INT_MASK;     /* overflow? */

-		if (r >= (BignumDblInt) m0 &&

-		    t > ((BignumDblInt) r << BIGNUM_INT_BITS) + ai1) q--;

-	    }

-	}

-

-	/* Subtract q * m from a[i...] */

-	c = 0;

-	for (k = mlen - 1; k >= 0; k--) {

-	    t = MUL_WORD(q, m[k]);

-	    t += c;

-	    c = (unsigned)(t >> BIGNUM_INT_BITS);

-	    if ((BignumInt) t > a[i + k])

-		c++;

-	    a[i + k] -= (BignumInt) t;

-	}

-

-	/* Add back m in case of borrow */

-	if (c != h) {

-	    t = 0;

-	    for (k = mlen - 1; k >= 0; k--) {

-		t += m[k];

-		t += a[i + k];

-		a[i + k] = (BignumInt) t;

-		t = t >> BIGNUM_INT_BITS;

-	    }

-	    q--;

-	}

-	if (quot)

-	    internal_add_shifted(quot, q, qshift + BIGNUM_INT_BITS * (alen - mlen - i));

-    }

-}

-

-/*

- * Compute (base ^ exp) % mod, the pedestrian way.

- */

-Bignum modpow_simple(Bignum base_in, Bignum exp, Bignum mod)

-{

-    BignumInt *a, *b, *n, *m, *scratch;

-    int mshift;

-    int mlen, scratchlen, i, j;

-    Bignum base, result;

-

-    /*

-     * The most significant word of mod needs to be non-zero. It

-     * should already be, but let's make sure.

-     */

-    assert(mod[mod[0]] != 0);

-

-    /*

-     * Make sure the base is smaller than the modulus, by reducing

-     * it modulo the modulus if not.

-     */

-    base = bigmod(base_in, mod);

-

-    /* Allocate m of size mlen, copy mod to m */

-    /* We use big endian internally */

-    mlen = mod[0];

-    m = snewn(mlen, BignumInt);

-    for (j = 0; j < mlen; j++)

-	m[j] = mod[mod[0] - j];

-

-    /* Shift m left to make msb bit set */

-    for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++)

-	if ((m[0] << mshift) & BIGNUM_TOP_BIT)

-	    break;

-    if (mshift) {

-	for (i = 0; i < mlen - 1; i++)

-	    m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift));

-	m[mlen - 1] = m[mlen - 1] << mshift;

-    }

-

-    /* Allocate n of size mlen, copy base to n */

-    n = snewn(mlen, BignumInt);

-    i = mlen - base[0];

-    for (j = 0; j < i; j++)

-	n[j] = 0;

-    for (j = 0; j < (int)base[0]; j++)

-	n[i + j] = base[base[0] - j];

-

-    /* Allocate a and b of size 2*mlen. Set a = 1 */

-    a = snewn(2 * mlen, BignumInt);

-    b = snewn(2 * mlen, BignumInt);

-    for (i = 0; i < 2 * mlen; i++)

-	a[i] = 0;

-    a[2 * mlen - 1] = 1;

-

-    /* Scratch space for multiplies */

-    scratchlen = mul_compute_scratch(mlen);

-    scratch = snewn(scratchlen, BignumInt);

-

-    /* Skip leading zero bits of exp. */

-    i = 0;

-    j = BIGNUM_INT_BITS-1;

-    while (i < (int)exp[0] && (exp[exp[0] - i] & (1 << j)) == 0) {

-	j--;

-	if (j < 0) {

-	    i++;

-	    j = BIGNUM_INT_BITS-1;

-	}

-    }

-

-    /* Main computation */

-    while (i < (int)exp[0]) {

-	while (j >= 0) {

-	    internal_mul(a + mlen, a + mlen, b, mlen, scratch);

-	    internal_mod(b, mlen * 2, m, mlen, NULL, 0);

-	    if ((exp[exp[0] - i] & (1 << j)) != 0) {

-		internal_mul(b + mlen, n, a, mlen, scratch);

-		internal_mod(a, mlen * 2, m, mlen, NULL, 0);

-	    } else {

-		BignumInt *t;

-		t = a;

-		a = b;

-		b = t;

-	    }

-	    j--;

-	}

-	i++;

-	j = BIGNUM_INT_BITS-1;

-    }

-

-    /* Fixup result in case the modulus was shifted */

-    if (mshift) {

-	for (i = mlen - 1; i < 2 * mlen - 1; i++)

-	    a[i] = (a[i] << mshift) | (a[i + 1] >> (BIGNUM_INT_BITS - mshift));

-	a[2 * mlen - 1] = a[2 * mlen - 1] << mshift;

-	internal_mod(a, mlen * 2, m, mlen, NULL, 0);

-	for (i = 2 * mlen - 1; i >= mlen; i--)

-	    a[i] = (a[i] >> mshift) | (a[i - 1] << (BIGNUM_INT_BITS - mshift));

-    }

-

-    /* Copy result to buffer */

-    result = newbn(mod[0]);

-    for (i = 0; i < mlen; i++)

-	result[result[0] - i] = a[i + mlen];

-    while (result[0] > 1 && result[result[0]] == 0)

-	result[0]--;

-

-    /* Free temporary arrays */

-    for (i = 0; i < 2 * mlen; i++)

-	a[i] = 0;

-    sfree(a);

-    for (i = 0; i < scratchlen; i++)

-	scratch[i] = 0;

-    sfree(scratch);

-    for (i = 0; i < 2 * mlen; i++)

-	b[i] = 0;

-    sfree(b);

-    for (i = 0; i < mlen; i++)

-	m[i] = 0;

-    sfree(m);

-    for (i = 0; i < mlen; i++)

-	n[i] = 0;

-    sfree(n);

-

-    freebn(base);

-

-    return result;

-}

-

-/*

- * Compute (base ^ exp) % mod. Uses the Montgomery multiplication

- * technique where possible, falling back to modpow_simple otherwise.

- */

-Bignum modpow(Bignum base_in, Bignum exp, Bignum mod)

-{

-    BignumInt *a, *b, *x, *n, *mninv, *scratch;

-    int len, scratchlen, i, j;

-    Bignum base, base2, r, rn, inv, result;

-

-    /*

-     * The most significant word of mod needs to be non-zero. It

-     * should already be, but let's make sure.

-     */

-    assert(mod[mod[0]] != 0);

-

-    /*

-     * mod had better be odd, or we can't do Montgomery multiplication

-     * using a power of two at all.

-     */

-    if (!(mod[1] & 1))

-        return modpow_simple(base_in, exp, mod);

-

-    /*

-     * Make sure the base is smaller than the modulus, by reducing

-     * it modulo the modulus if not.

-     */

-    base = bigmod(base_in, mod);

-

-    /*

-     * Compute the inverse of n mod r, for monty_reduce. (In fact we

-     * want the inverse of _minus_ n mod r, but we'll sort that out

-     * below.)

-     */

-    len = mod[0];

-    r = bn_power_2(BIGNUM_INT_BITS * len);

-    inv = modinv(mod, r);

-

-    /*

-     * Multiply the base by r mod n, to get it into Montgomery

-     * representation.

-     */

-    base2 = modmul(base, r, mod);

-    freebn(base);

-    base = base2;

-

-    rn = bigmod(r, mod);               /* r mod n, i.e. Montgomerified 1 */

-

-    freebn(r);                         /* won't need this any more */

-

-    /*

-     * Set up internal arrays of the right lengths, in big-endian

-     * format, containing the base, the modulus, and the modulus's

-     * inverse.

-     */

-    n = snewn(len, BignumInt);

-    for (j = 0; j < len; j++)

-	n[len - 1 - j] = mod[j + 1];

-

-    mninv = snewn(len, BignumInt);

-    for (j = 0; j < len; j++)

-	mninv[len - 1 - j] = (j < (int)inv[0] ? inv[j + 1] : 0);

-    freebn(inv);         /* we don't need this copy of it any more */

-    /* Now negate mninv mod r, so it's the inverse of -n rather than +n. */

-    x = snewn(len, BignumInt);

-    for (j = 0; j < len; j++)

-        x[j] = 0;

-    internal_sub(x, mninv, mninv, len);

-

-    /* x = snewn(len, BignumInt); */ /* already done above */

-    for (j = 0; j < len; j++)

-	x[len - 1 - j] = (j < (int)base[0] ? base[j + 1] : 0);

-    freebn(base);        /* we don't need this copy of it any more */

-

-    a = snewn(2*len, BignumInt);

-    b = snewn(2*len, BignumInt);

-    for (j = 0; j < len; j++)

-	a[2*len - 1 - j] = (j < (int)rn[0] ? rn[j + 1] : 0);

-    freebn(rn);

-

-    /* Scratch space for multiplies */

-    scratchlen = 3*len + mul_compute_scratch(len);

-    scratch = snewn(scratchlen, BignumInt);

-

-    /* Skip leading zero bits of exp. */

-    i = 0;

-    j = BIGNUM_INT_BITS-1;

-    while (i < (int)exp[0] && (exp[exp[0] - i] & (1 << j)) == 0) {

-	j--;

-	if (j < 0) {

-	    i++;

-	    j = BIGNUM_INT_BITS-1;

-	}

-    }

-

-    /* Main computation */

-    while (i < (int)exp[0]) {

-	while (j >= 0) {

-	    internal_mul(a + len, a + len, b, len, scratch);

-            monty_reduce(b, n, mninv, scratch, len);

-	    if ((exp[exp[0] - i] & (1 << j)) != 0) {

-                internal_mul(b + len, x, a, len,  scratch);

-                monty_reduce(a, n, mninv, scratch, len);

-	    } else {

-		BignumInt *t;

-		t = a;

-		a = b;

-		b = t;

-	    }

-	    j--;

-	}

-	i++;

-	j = BIGNUM_INT_BITS-1;

-    }

-

-    /*

-     * Final monty_reduce to get back from the adjusted Montgomery

-     * representation.

-     */

-    monty_reduce(a, n, mninv, scratch, len);

-

-    /* Copy result to buffer */

-    result = newbn(mod[0]);

-    for (i = 0; i < len; i++)

-	result[result[0] - i] = a[i + len];

-    while (result[0] > 1 && result[result[0]] == 0)

-	result[0]--;

-

-    /* Free temporary arrays */

-    for (i = 0; i < scratchlen; i++)

-	scratch[i] = 0;

-    sfree(scratch);

-    for (i = 0; i < 2 * len; i++)

-	a[i] = 0;

-    sfree(a);

-    for (i = 0; i < 2 * len; i++)

-	b[i] = 0;

-    sfree(b);

-    for (i = 0; i < len; i++)

-	mninv[i] = 0;

-    sfree(mninv);

-    for (i = 0; i < len; i++)

-	n[i] = 0;

-    sfree(n);

-    for (i = 0; i < len; i++)

-	x[i] = 0;

-    sfree(x);

-

-    return result;

-}

-

-/*

- * Compute (p * q) % mod.

- * The most significant word of mod MUST be non-zero.

- * We assume that the result array is the same size as the mod array.

- */

-Bignum modmul(Bignum p, Bignum q, Bignum mod)

-{

-    BignumInt *a, *n, *m, *o, *scratch;

-    int mshift, scratchlen;

-    int pqlen, mlen, rlen, i, j;

-    Bignum result;

-

-    /* Allocate m of size mlen, copy mod to m */

-    /* We use big endian internally */

-    mlen = mod[0];

-    m = snewn(mlen, BignumInt);

-    for (j = 0; j < mlen; j++)

-	m[j] = mod[mod[0] - j];

-

-    /* Shift m left to make msb bit set */

-    for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++)

-	if ((m[0] << mshift) & BIGNUM_TOP_BIT)

-	    break;

-    if (mshift) {

-	for (i = 0; i < mlen - 1; i++)

-	    m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift));

-	m[mlen - 1] = m[mlen - 1] << mshift;

-    }

-

-    pqlen = (p[0] > q[0] ? p[0] : q[0]);

-

-    /* Allocate n of size pqlen, copy p to n */

-    n = snewn(pqlen, BignumInt);

-    i = pqlen - p[0];

-    for (j = 0; j < i; j++)

-	n[j] = 0;

-    for (j = 0; j < (int)p[0]; j++)

-	n[i + j] = p[p[0] - j];

-

-    /* Allocate o of size pqlen, copy q to o */

-    o = snewn(pqlen, BignumInt);

-    i = pqlen - q[0];

-    for (j = 0; j < i; j++)

-	o[j] = 0;

-    for (j = 0; j < (int)q[0]; j++)

-	o[i + j] = q[q[0] - j];

-

-    /* Allocate a of size 2*pqlen for result */

-    a = snewn(2 * pqlen, BignumInt);

-

-    /* Scratch space for multiplies */

-    scratchlen = mul_compute_scratch(pqlen);

-    scratch = snewn(scratchlen, BignumInt);

-

-    /* Main computation */

-    internal_mul(n, o, a, pqlen, scratch);

-    internal_mod(a, pqlen * 2, m, mlen, NULL, 0);

-

-    /* Fixup result in case the modulus was shifted */

-    if (mshift) {

-	for (i = 2 * pqlen - mlen - 1; i < 2 * pqlen - 1; i++)

-	    a[i] = (a[i] << mshift) | (a[i + 1] >> (BIGNUM_INT_BITS - mshift));

-	a[2 * pqlen - 1] = a[2 * pqlen - 1] << mshift;

-	internal_mod(a, pqlen * 2, m, mlen, NULL, 0);

-	for (i = 2 * pqlen - 1; i >= 2 * pqlen - mlen; i--)

-	    a[i] = (a[i] >> mshift) | (a[i - 1] << (BIGNUM_INT_BITS - mshift));

-    }

-

-    /* Copy result to buffer */

-    rlen = (mlen < pqlen * 2 ? mlen : pqlen * 2);

-    result = newbn(rlen);

-    for (i = 0; i < rlen; i++)

-	result[result[0] - i] = a[i + 2 * pqlen - rlen];

-    while (result[0] > 1 && result[result[0]] == 0)

-	result[0]--;

-

-    /* Free temporary arrays */

-    for (i = 0; i < scratchlen; i++)

-	scratch[i] = 0;

-    sfree(scratch);

-    for (i = 0; i < 2 * pqlen; i++)

-	a[i] = 0;

-    sfree(a);

-    for (i = 0; i < mlen; i++)

-	m[i] = 0;

-    sfree(m);

-    for (i = 0; i < pqlen; i++)

-	n[i] = 0;

-    sfree(n);

-    for (i = 0; i < pqlen; i++)

-	o[i] = 0;

-    sfree(o);

-

-    return result;

-}

-

-/*

- * Compute p % mod.

- * The most significant word of mod MUST be non-zero.

- * We assume that the result array is the same size as the mod array.

- * We optionally write out a quotient if `quotient' is non-NULL.

- * We can avoid writing out the result if `result' is NULL.

- */

-static void bigdivmod(Bignum p, Bignum mod, Bignum result, Bignum quotient)

-{

-    BignumInt *n, *m;

-    int mshift;

-    int plen, mlen, i, j;

-

-    /* Allocate m of size mlen, copy mod to m */

-    /* We use big endian internally */

-    mlen = mod[0];

-    m = snewn(mlen, BignumInt);

-    for (j = 0; j < mlen; j++)

-	m[j] = mod[mod[0] - j];

-

-    /* Shift m left to make msb bit set */

-    for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++)

-	if ((m[0] << mshift) & BIGNUM_TOP_BIT)

-	    break;

-    if (mshift) {

-	for (i = 0; i < mlen - 1; i++)

-	    m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift));

-	m[mlen - 1] = m[mlen - 1] << mshift;

-    }

-

-    plen = p[0];

-    /* Ensure plen > mlen */

-    if (plen <= mlen)

-	plen = mlen + 1;

-

-    /* Allocate n of size plen, copy p to n */

-    n = snewn(plen, BignumInt);

-    for (j = 0; j < plen; j++)

-	n[j] = 0;

-    for (j = 1; j <= (int)p[0]; j++)

-	n[plen - j] = p[j];

-

-    /* Main computation */

-    internal_mod(n, plen, m, mlen, quotient, mshift);

-

-    /* Fixup result in case the modulus was shifted */

-    if (mshift) {

-	for (i = plen - mlen - 1; i < plen - 1; i++)

-	    n[i] = (n[i] << mshift) | (n[i + 1] >> (BIGNUM_INT_BITS - mshift));

-	n[plen - 1] = n[plen - 1] << mshift;

-	internal_mod(n, plen, m, mlen, quotient, 0);

-	for (i = plen - 1; i >= plen - mlen; i--)

-	    n[i] = (n[i] >> mshift) | (n[i - 1] << (BIGNUM_INT_BITS - mshift));

-    }

-

-    /* Copy result to buffer */

-    if (result) {

-	for (i = 1; i <= (int)result[0]; i++) {

-	    int j = plen - i;

-	    result[i] = j >= 0 ? n[j] : 0;

-	}

-    }

-

-    /* Free temporary arrays */

-    for (i = 0; i < mlen; i++)

-	m[i] = 0;

-    sfree(m);

-    for (i = 0; i < plen; i++)

-	n[i] = 0;

-    sfree(n);

-}

-

-/*

- * Decrement a number.

- */

-void decbn(Bignum bn)

-{

-    int i = 1;

-    while (i < (int)bn[0] && bn[i] == 0)

-	bn[i++] = BIGNUM_INT_MASK;

-    bn[i]--;

-}

-

-Bignum bignum_from_bytes(const unsigned char *data, int nbytes)

-{

-    Bignum result;

-    int w, i;

-

-    w = (nbytes + BIGNUM_INT_BYTES - 1) / BIGNUM_INT_BYTES; /* bytes->words */

-

-    result = newbn(w);

-    for (i = 1; i <= w; i++)

-	result[i] = 0;

-    for (i = nbytes; i--;) {

-	unsigned char byte = *data++;

-	result[1 + i / BIGNUM_INT_BYTES] |= byte << (8*i % BIGNUM_INT_BITS);

-    }

-

-    while (result[0] > 1 && result[result[0]] == 0)

-	result[0]--;

-    return result;

-}

-

-/*

- * Read an SSH-1-format bignum from a data buffer. Return the number

- * of bytes consumed, or -1 if there wasn't enough data.

- */

-int ssh1_read_bignum(const unsigned char *data, int len, Bignum * result)

-{

-    const unsigned char *p = data;

-    int i;

-    int w, b;

-

-    if (len < 2)

-	return -1;

-

-    w = 0;

-    for (i = 0; i < 2; i++)

-	w = (w << 8) + *p++;

-    b = (w + 7) / 8;		       /* bits -> bytes */

-

-    if (len < b+2)

-	return -1;

-

-    if (!result)		       /* just return length */

-	return b + 2;

-

-    *result = bignum_from_bytes(p, b);

-

-    return p + b - data;

-}

-

-/*

- * Return the bit count of a bignum, for SSH-1 encoding.

- */

-int bignum_bitcount(Bignum bn)

-{

-    int bitcount = bn[0] * BIGNUM_INT_BITS - 1;

-    while (bitcount >= 0

-	   && (bn[bitcount / BIGNUM_INT_BITS + 1] >> (bitcount % BIGNUM_INT_BITS)) == 0) bitcount--;

-    return bitcount + 1;

-}

-

-/*

- * Return the byte length of a bignum when SSH-1 encoded.

- */

-int ssh1_bignum_length(Bignum bn)

-{

-    return 2 + (bignum_bitcount(bn) + 7) / 8;

-}

-

-/*

- * Return the byte length of a bignum when SSH-2 encoded.

- */

-int ssh2_bignum_length(Bignum bn)

-{

-    return 4 + (bignum_bitcount(bn) + 8) / 8;

-}

-

-/*

- * Return a byte from a bignum; 0 is least significant, etc.

- */

-int bignum_byte(Bignum bn, int i)

-{

-    if (i >= (int)(BIGNUM_INT_BYTES * bn[0]))

-	return 0;		       /* beyond the end */

-    else

-	return (bn[i / BIGNUM_INT_BYTES + 1] >>

-		((i % BIGNUM_INT_BYTES)*8)) & 0xFF;

-}

-

-/*

- * Return a bit from a bignum; 0 is least significant, etc.

- */

-int bignum_bit(Bignum bn, int i)

-{

-    if (i >= (int)(BIGNUM_INT_BITS * bn[0]))

-	return 0;		       /* beyond the end */

-    else

-	return (bn[i / BIGNUM_INT_BITS + 1] >> (i % BIGNUM_INT_BITS)) & 1;

-}

-

-/*

- * Set a bit in a bignum; 0 is least significant, etc.

- */

-void bignum_set_bit(Bignum bn, int bitnum, int value)

-{

-    if (bitnum >= (int)(BIGNUM_INT_BITS * bn[0]))

-	abort();		       /* beyond the end */

-    else {

-	int v = bitnum / BIGNUM_INT_BITS + 1;

-	int mask = 1 << (bitnum % BIGNUM_INT_BITS);

-	if (value)

-	    bn[v] |= mask;

-	else

-	    bn[v] &= ~mask;

-    }

-}

-

-/*

- * Write a SSH-1-format bignum into a buffer. It is assumed the

- * buffer is big enough. Returns the number of bytes used.

- */

-int ssh1_write_bignum(void *data, Bignum bn)

-{

-    unsigned char *p = data;

-    int len = ssh1_bignum_length(bn);

-    int i;

-    int bitc = bignum_bitcount(bn);

-

-    *p++ = (bitc >> 8) & 0xFF;

-    *p++ = (bitc) & 0xFF;

-    for (i = len - 2; i--;)

-	*p++ = bignum_byte(bn, i);

-    return len;

-}

-

-/*

- * Compare two bignums. Returns like strcmp.

- */

-int bignum_cmp(Bignum a, Bignum b)

-{

-    int amax = a[0], bmax = b[0];

-    int i = (amax > bmax ? amax : bmax);

-    while (i) {

-	BignumInt aval = (i > amax ? 0 : a[i]);

-	BignumInt bval = (i > bmax ? 0 : b[i]);

-	if (aval < bval)

-	    return -1;

-	if (aval > bval)

-	    return +1;

-	i--;

-    }

-    return 0;

-}

-

-/*

- * Right-shift one bignum to form another.

- */

-Bignum bignum_rshift(Bignum a, int shift)

-{

-    Bignum ret;

-    int i, shiftw, shiftb, shiftbb, bits;

-    BignumInt ai, ai1;

-

-    bits = bignum_bitcount(a) - shift;

-    ret = newbn((bits + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS);

-

-    if (ret) {

-	shiftw = shift / BIGNUM_INT_BITS;

-	shiftb = shift % BIGNUM_INT_BITS;

-	shiftbb = BIGNUM_INT_BITS - shiftb;

-

-	ai1 = a[shiftw + 1];

-	for (i = 1; i <= (int)ret[0]; i++) {

-	    ai = ai1;

-	    ai1 = (i + shiftw + 1 <= (int)a[0] ? a[i + shiftw + 1] : 0);

-	    ret[i] = ((ai >> shiftb) | (ai1 << shiftbb)) & BIGNUM_INT_MASK;

-	}

-    }

-

-    return ret;

-}

-

-/*

- * Non-modular multiplication and addition.

- */

-Bignum bigmuladd(Bignum a, Bignum b, Bignum addend)

-{

-    int alen = a[0], blen = b[0];

-    int mlen = (alen > blen ? alen : blen);

-    int rlen, i, maxspot;

-    int wslen;

-    BignumInt *workspace;

-    Bignum ret;

-

-    /* mlen space for a, mlen space for b, 2*mlen for result,

-     * plus scratch space for multiplication */

-    wslen = mlen * 4 + mul_compute_scratch(mlen);

-    workspace = snewn(wslen, BignumInt);

-    for (i = 0; i < mlen; i++) {

-	workspace[0 * mlen + i] = (mlen - i <= (int)a[0] ? a[mlen - i] : 0);

-	workspace[1 * mlen + i] = (mlen - i <= (int)b[0] ? b[mlen - i] : 0);

-    }

-

-    internal_mul(workspace + 0 * mlen, workspace + 1 * mlen,

-		 workspace + 2 * mlen, mlen, workspace + 4 * mlen);

-

-    /* now just copy the result back */

-    rlen = alen + blen + 1;

-    if (addend && rlen <= (int)addend[0])

-	rlen = addend[0] + 1;

-    ret = newbn(rlen);

-    maxspot = 0;

-    for (i = 1; i <= (int)ret[0]; i++) {

-	ret[i] = (i <= 2 * mlen ? workspace[4 * mlen - i] : 0);

-	if (ret[i] != 0)

-	    maxspot = i;

-    }

-    ret[0] = maxspot;

-

-    /* now add in the addend, if any */

-    if (addend) {

-	BignumDblInt carry = 0;

-	for (i = 1; i <= rlen; i++) {

-	    carry += (i <= (int)ret[0] ? ret[i] : 0);

-	    carry += (i <= (int)addend[0] ? addend[i] : 0);

-	    ret[i] = (BignumInt) carry & BIGNUM_INT_MASK;

-	    carry >>= BIGNUM_INT_BITS;

-	    if (ret[i] != 0 && i > maxspot)

-		maxspot = i;

-	}

-    }

-    ret[0] = maxspot;

-

-    for (i = 0; i < wslen; i++)

-        workspace[i] = 0;

-    sfree(workspace);

-    return ret;

-}

-

-/*

- * Non-modular multiplication.

- */

-Bignum bigmul(Bignum a, Bignum b)

-{

-    return bigmuladd(a, b, NULL);

-}

-

-/*

- * Simple addition.

- */

-Bignum bigadd(Bignum a, Bignum b)

-{

-    int alen = a[0], blen = b[0];

-    int rlen = (alen > blen ? alen : blen) + 1;

-    int i, maxspot;

-    Bignum ret;

-    BignumDblInt carry;

-

-    ret = newbn(rlen);

-

-    carry = 0;

-    maxspot = 0;

-    for (i = 1; i <= rlen; i++) {

-        carry += (i <= (int)a[0] ? a[i] : 0);

-        carry += (i <= (int)b[0] ? b[i] : 0);

-        ret[i] = (BignumInt) carry & BIGNUM_INT_MASK;

-        carry >>= BIGNUM_INT_BITS;

-        if (ret[i] != 0 && i > maxspot)

-            maxspot = i;

-    }

-    ret[0] = maxspot;

-

-    return ret;

-}

-

-/*

- * Subtraction. Returns a-b, or NULL if the result would come out

- * negative (recall that this entire bignum module only handles

- * positive numbers).

- */

-Bignum bigsub(Bignum a, Bignum b)

-{

-    int alen = a[0], blen = b[0];

-    int rlen = (alen > blen ? alen : blen);

-    int i, maxspot;

-    Bignum ret;

-    BignumDblInt carry;

-

-    ret = newbn(rlen);

-

-    carry = 1;

-    maxspot = 0;

-    for (i = 1; i <= rlen; i++) {

-        carry += (i <= (int)a[0] ? a[i] : 0);

-        carry += (i <= (int)b[0] ? b[i] ^ BIGNUM_INT_MASK : BIGNUM_INT_MASK);

-        ret[i] = (BignumInt) carry & BIGNUM_INT_MASK;

-        carry >>= BIGNUM_INT_BITS;

-        if (ret[i] != 0 && i > maxspot)

-            maxspot = i;

-    }

-    ret[0] = maxspot;

-

-    if (!carry) {

-        freebn(ret);

-        return NULL;

-    }

-

-    return ret;

-}

-

-/*

- * Create a bignum which is the bitmask covering another one. That

- * is, the smallest integer which is >= N and is also one less than

- * a power of two.

- */

-Bignum bignum_bitmask(Bignum n)

-{

-    Bignum ret = copybn(n);

-    int i;

-    BignumInt j;

-

-    i = ret[0];

-    while (n[i] == 0 && i > 0)

-	i--;

-    if (i <= 0)

-	return ret;		       /* input was zero */

-    j = 1;

-    while (j < n[i])

-	j = 2 * j + 1;

-    ret[i] = j;

-    while (--i > 0)

-	ret[i] = BIGNUM_INT_MASK;

-    return ret;

-}

-

-/*

- * Convert a (max 32-bit) long into a bignum.

- */

-Bignum bignum_from_long(unsigned long nn)

-{

-    Bignum ret;

-    BignumDblInt n = nn;

-

-    ret = newbn(3);

-    ret[1] = (BignumInt)(n & BIGNUM_INT_MASK);

-    ret[2] = (BignumInt)((n >> BIGNUM_INT_BITS) & BIGNUM_INT_MASK);

-    ret[3] = 0;

-    ret[0] = (ret[2]  ? 2 : 1);

-    return ret;

-}

-

-/*

- * Add a long to a bignum.

- */

-Bignum bignum_add_long(Bignum number, unsigned long addendx)

-{

-    Bignum ret = newbn(number[0] + 1);

-    int i, maxspot = 0;

-    BignumDblInt carry = 0, addend = addendx;

-

-    for (i = 1; i <= (int)ret[0]; i++) {

-	carry += addend & BIGNUM_INT_MASK;

-	carry += (i <= (int)number[0] ? number[i] : 0);

-	addend >>= BIGNUM_INT_BITS;

-	ret[i] = (BignumInt) carry & BIGNUM_INT_MASK;

-	carry >>= BIGNUM_INT_BITS;

-	if (ret[i] != 0)

-	    maxspot = i;

-    }

-    ret[0] = maxspot;

-    return ret;

-}

-

-/*

- * Compute the residue of a bignum, modulo a (max 16-bit) short.

- */

-unsigned short bignum_mod_short(Bignum number, unsigned short modulus)

-{

-    BignumDblInt mod, r;

-    int i;

-

-    r = 0;

-    mod = modulus;

-    for (i = number[0]; i > 0; i--)

-	r = (r * (BIGNUM_TOP_BIT % mod) * 2 + number[i] % mod) % mod;

-    return (unsigned short) r;

-}

-

-#ifdef DEBUG

-void diagbn(char *prefix, Bignum md)

-{

-    int i, nibbles, morenibbles;

-    static const char hex[] = "0123456789ABCDEF";

-

-    debug(("%s0x", prefix ? prefix : ""));

-

-    nibbles = (3 + bignum_bitcount(md)) / 4;

-    if (nibbles < 1)

-	nibbles = 1;

-    morenibbles = 4 * md[0] - nibbles;

-    for (i = 0; i < morenibbles; i++)

-	debug(("-"));

-    for (i = nibbles; i--;)

-	debug(("%c",

-	       hex[(bignum_byte(md, i / 2) >> (4 * (i % 2))) & 0xF]));

-

-    if (prefix)

-	debug(("\n"));

-}

-#endif

-

-/*

- * Simple division.

- */

-Bignum bigdiv(Bignum a, Bignum b)

-{

-    Bignum q = newbn(a[0]);

-    bigdivmod(a, b, NULL, q);

-    return q;

-}

-

-/*

- * Simple remainder.

- */

-Bignum bigmod(Bignum a, Bignum b)

-{

-    Bignum r = newbn(b[0]);

-    bigdivmod(a, b, r, NULL);

-    return r;

-}

-

-/*

- * Greatest common divisor.

- */

-Bignum biggcd(Bignum av, Bignum bv)

-{

-    Bignum a = copybn(av);

-    Bignum b = copybn(bv);

-

-    while (bignum_cmp(b, Zero) != 0) {

-	Bignum t = newbn(b[0]);

-	bigdivmod(a, b, t, NULL);

-	while (t[0] > 1 && t[t[0]] == 0)

-	    t[0]--;

-	freebn(a);

-	a = b;

-	b = t;

-    }

-

-    freebn(b);

-    return a;

-}

-

-/*

- * Modular inverse, using Euclid's extended algorithm.

- */

-Bignum modinv(Bignum number, Bignum modulus)

-{

-    Bignum a = copybn(modulus);

-    Bignum b = copybn(number);

-    Bignum xp = copybn(Zero);

-    Bignum x = copybn(One);

-    int sign = +1;

-

-    while (bignum_cmp(b, One) != 0) {

-	Bignum t = newbn(b[0]);

-	Bignum q = newbn(a[0]);

-	bigdivmod(a, b, t, q);

-	while (t[0] > 1 && t[t[0]] == 0)

-	    t[0]--;

-	freebn(a);

-	a = b;

-	b = t;

-	t = xp;

-	xp = x;

-	x = bigmuladd(q, xp, t);

-	sign = -sign;

-	freebn(t);

-	freebn(q);

-    }

-

-    freebn(b);

-    freebn(a);

-    freebn(xp);

-

-    /* now we know that sign * x == 1, and that x < modulus */

-    if (sign < 0) {

-	/* set a new x to be modulus - x */

-	Bignum newx = newbn(modulus[0]);

-	BignumInt carry = 0;

-	int maxspot = 1;

-	int i;

-

-	for (i = 1; i <= (int)newx[0]; i++) {

-	    BignumInt aword = (i <= (int)modulus[0] ? modulus[i] : 0);

-	    BignumInt bword = (i <= (int)x[0] ? x[i] : 0);

-	    newx[i] = aword - bword - carry;

-	    bword = ~bword;

-	    carry = carry ? (newx[i] >= bword) : (newx[i] > bword);

-	    if (newx[i] != 0)

-		maxspot = i;

-	}

-	newx[0] = maxspot;

-	freebn(x);

-	x = newx;

-    }

-

-    /* and return. */

-    return x;

-}

-

-/*

- * Render a bignum into decimal. Return a malloced string holding

- * the decimal representation.

- */

-char *bignum_decimal(Bignum x)

-{

-    int ndigits, ndigit;

-    int i, iszero;

-    BignumDblInt carry;

-    char *ret;

-    BignumInt *workspace;

-

-    /*

-     * First, estimate the number of digits. Since log(10)/log(2)

-     * is just greater than 93/28 (the joys of continued fraction

-     * approximations...) we know that for every 93 bits, we need

-     * at most 28 digits. This will tell us how much to malloc.

-     *

-     * Formally: if x has i bits, that means x is strictly less

-     * than 2^i. Since 2 is less than 10^(28/93), this is less than

-     * 10^(28i/93). We need an integer power of ten, so we must

-     * round up (rounding down might make it less than x again).

-     * Therefore if we multiply the bit count by 28/93, rounding

-     * up, we will have enough digits.

-     *

-     * i=0 (i.e., x=0) is an irritating special case.

-     */

-    i = bignum_bitcount(x);

-    if (!i)

-	ndigits = 1;		       /* x = 0 */

-    else

-	ndigits = (28 * i + 92) / 93;  /* multiply by 28/93 and round up */

-    ndigits++;			       /* allow for trailing \0 */

-    ret = snewn(ndigits, char);

-

-    /*

-     * Now allocate some workspace to hold the binary form as we

-     * repeatedly divide it by ten. Initialise this to the

-     * big-endian form of the number.

-     */

-    workspace = snewn(x[0], BignumInt);

-    for (i = 0; i < (int)x[0]; i++)

-	workspace[i] = x[x[0] - i];

-

-    /*

-     * Next, write the decimal number starting with the last digit.

-     * We use ordinary short division, dividing 10 into the

-     * workspace.

-     */

-    ndigit = ndigits - 1;

-    ret[ndigit] = '\0';

-    do {

-	iszero = 1;

-	carry = 0;

-	for (i = 0; i < (int)x[0]; i++) {

-	    carry = (carry << BIGNUM_INT_BITS) + workspace[i];

-	    workspace[i] = (BignumInt) (carry / 10);

-	    if (workspace[i])

-		iszero = 0;

-	    carry %= 10;

-	}

-	ret[--ndigit] = (char) (carry + '0');

-    } while (!iszero);

-

-    /*

-     * There's a chance we've fallen short of the start of the

-     * string. Correct if so.

-     */

-    if (ndigit > 0)

-	memmove(ret, ret + ndigit, ndigits - ndigit);

-

-    /*

-     * Done.

-     */

-    sfree(workspace);

-    return ret;

-}

-

-#ifdef TESTBN

-

-#include <stdio.h>

-#include <stdlib.h>

-#include <ctype.h>

-

-/*

- * gcc -g -O0 -DTESTBN -o testbn sshbn.c misc.c -I unix -I charset

- *

- * Then feed to this program's standard input the output of

- * testdata/bignum.py .

- */

-

-void modalfatalbox(char *p, ...)

-{

-    va_list ap;

-    fprintf(stderr, "FATAL ERROR: ");

-    va_start(ap, p);

-    vfprintf(stderr, p, ap);

-    va_end(ap);

-    fputc('\n', stderr);

-    exit(1);

-}

-

-#define fromxdigit(c) ( (c)>'9' ? ((c)&0xDF) - 'A' + 10 : (c) - '0' )

-

-int main(int argc, char **argv)

-{

-    char *buf;

-    int line = 0;

-    int passes = 0, fails = 0;

-

-    while ((buf = fgetline(stdin)) != NULL) {

-        int maxlen = strlen(buf);

-        unsigned char *data = snewn(maxlen, unsigned char);

-        unsigned char *ptrs[5], *q;

-        int ptrnum;

-        char *bufp = buf;

-

-        line++;

-

-        q = data;

-        ptrnum = 0;

-

-        while (*bufp && !isspace((unsigned char)*bufp))

-            bufp++;

-        if (bufp)

-            *bufp++ = '\0';

-

-        while (*bufp) {

-            char *start, *end;

-            int i;

-

-            while (*bufp && !isxdigit((unsigned char)*bufp))

-                bufp++;

-            start = bufp;

-

-            if (!*bufp)

-                break;

-

-            while (*bufp && isxdigit((unsigned char)*bufp))

-                bufp++;

-            end = bufp;

-

-            if (ptrnum >= lenof(ptrs))

-                break;

-            ptrs[ptrnum++] = q;

-            

-            for (i = -((end - start) & 1); i < end-start; i += 2) {

-                unsigned char val = (i < 0 ? 0 : fromxdigit(start[i]));

-                val = val * 16 + fromxdigit(start[i+1]);

-                *q++ = val;

-            }

-

-            ptrs[ptrnum] = q;

-        }

-

-        if (!strcmp(buf, "mul")) {

-            Bignum a, b, c, p;

-

-            if (ptrnum != 3) {

-                printf("%d: mul with %d parameters, expected 3\n", line);

-                exit(1);

-            }

-            a = bignum_from_bytes(ptrs[0], ptrs[1]-ptrs[0]);

-            b = bignum_from_bytes(ptrs[1], ptrs[2]-ptrs[1]);

-            c = bignum_from_bytes(ptrs[2], ptrs[3]-ptrs[2]);

-            p = bigmul(a, b);

-

-            if (bignum_cmp(c, p) == 0) {

-                passes++;

-            } else {

-                char *as = bignum_decimal(a);

-                char *bs = bignum_decimal(b);

-                char *cs = bignum_decimal(c);

-                char *ps = bignum_decimal(p);

-                

-                printf("%d: fail: %s * %s gave %s expected %s\n",

-                       line, as, bs, ps, cs);

-                fails++;

-

-                sfree(as);

-                sfree(bs);

-                sfree(cs);

-                sfree(ps);

-            }

-            freebn(a);

-            freebn(b);

-            freebn(c);

-            freebn(p);

-        } else if (!strcmp(buf, "pow")) {

-            Bignum base, expt, modulus, expected, answer;

-

-            if (ptrnum != 4) {

-                printf("%d: mul with %d parameters, expected 3\n", line);

-                exit(1);

-            }

-

-            base = bignum_from_bytes(ptrs[0], ptrs[1]-ptrs[0]);

-            expt = bignum_from_bytes(ptrs[1], ptrs[2]-ptrs[1]);

-            modulus = bignum_from_bytes(ptrs[2], ptrs[3]-ptrs[2]);

-            expected = bignum_from_bytes(ptrs[3], ptrs[4]-ptrs[3]);

-            answer = modpow(base, expt, modulus);

-

-            if (bignum_cmp(expected, answer) == 0) {

-                passes++;

-            } else {

-                char *as = bignum_decimal(base);

-                char *bs = bignum_decimal(expt);

-                char *cs = bignum_decimal(modulus);

-                char *ds = bignum_decimal(answer);

-                char *ps = bignum_decimal(expected);

-                

-                printf("%d: fail: %s ^ %s mod %s gave %s expected %s\n",

-                       line, as, bs, cs, ds, ps);

-                fails++;

-

-                sfree(as);

-                sfree(bs);

-                sfree(cs);

-                sfree(ds);

-                sfree(ps);

-            }

-            freebn(base);

-            freebn(expt);

-            freebn(modulus);

-            freebn(expected);

-            freebn(answer);

-        } else {

-            printf("%d: unrecognised test keyword: '%s'\n", line, buf);

-            exit(1);

-        }

-

-        sfree(buf);

-        sfree(data);

-    }

-

-    printf("passed %d failed %d total %d\n", passes, fails, passes+fails);

-    return fails != 0;

-}

-

-#endif

+/*
+ * Bignum routines for RSA and DH and stuff.
+ */
+
+#include <stdio.h>
+#include <assert.h>
+#include <stdlib.h>
+#include <string.h>
+#include <limits.h>
+
+#include "misc.h"
+
+/*
+ * Usage notes:
+ *  * Do not call the DIVMOD_WORD macro with expressions such as array
+ *    subscripts, as some implementations object to this (see below).
+ *  * Note that none of the division methods below will cope if the
+ *    quotient won't fit into BIGNUM_INT_BITS. Callers should be careful
+ *    to avoid this case.
+ *    If this condition occurs, in the case of the x86 DIV instruction,
+ *    an overflow exception will occur, which (according to a correspondent)
+ *    will manifest on Windows as something like
+ *      0xC0000095: Integer overflow
+ *    The C variant won't give the right answer, either.
+ */
+
+#if defined __GNUC__ && defined __i386__
+typedef unsigned long BignumInt;
+typedef unsigned long long BignumDblInt;
+#define BIGNUM_INT_MASK  0xFFFFFFFFUL
+#define BIGNUM_TOP_BIT   0x80000000UL
+#define BIGNUM_INT_BITS  32
+#define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
+#define DIVMOD_WORD(q, r, hi, lo, w) \
+    __asm__("div %2" : \
+	    "=d" (r), "=a" (q) : \
+	    "r" (w), "d" (hi), "a" (lo))
+#elif defined _MSC_VER && defined _M_IX86
+typedef unsigned __int32 BignumInt;
+typedef unsigned __int64 BignumDblInt;
+#define BIGNUM_INT_MASK  0xFFFFFFFFUL
+#define BIGNUM_TOP_BIT   0x80000000UL
+#define BIGNUM_INT_BITS  32
+#define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
+/* Note: MASM interprets array subscripts in the macro arguments as
+ * assembler syntax, which gives the wrong answer. Don't supply them.
+ * <http://msdn2.microsoft.com/en-us/library/bf1dw62z.aspx> */
+#define DIVMOD_WORD(q, r, hi, lo, w) do { \
+    __asm mov edx, hi \
+    __asm mov eax, lo \
+    __asm div w \
+    __asm mov r, edx \
+    __asm mov q, eax \
+} while(0)
+#elif defined _LP64
+/* 64-bit architectures can do 32x32->64 chunks at a time */
+typedef unsigned int BignumInt;
+typedef unsigned long BignumDblInt;
+#define BIGNUM_INT_MASK  0xFFFFFFFFU
+#define BIGNUM_TOP_BIT   0x80000000U
+#define BIGNUM_INT_BITS  32
+#define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
+#define DIVMOD_WORD(q, r, hi, lo, w) do { \
+    BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \
+    q = n / w; \
+    r = n % w; \
+} while (0)
+#elif defined _LLP64
+/* 64-bit architectures in which unsigned long is 32 bits, not 64 */
+typedef unsigned long BignumInt;
+typedef unsigned long long BignumDblInt;
+#define BIGNUM_INT_MASK  0xFFFFFFFFUL
+#define BIGNUM_TOP_BIT   0x80000000UL
+#define BIGNUM_INT_BITS  32
+#define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
+#define DIVMOD_WORD(q, r, hi, lo, w) do { \
+    BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \
+    q = n / w; \
+    r = n % w; \
+} while (0)
+#else
+/* Fallback for all other cases */
+typedef unsigned short BignumInt;
+typedef unsigned long BignumDblInt;
+#define BIGNUM_INT_MASK  0xFFFFU
+#define BIGNUM_TOP_BIT   0x8000U
+#define BIGNUM_INT_BITS  16
+#define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
+#define DIVMOD_WORD(q, r, hi, lo, w) do { \
+    BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \
+    q = n / w; \
+    r = n % w; \
+} while (0)
+#endif
+
+#define BIGNUM_INT_BYTES (BIGNUM_INT_BITS / 8)
+
+#define BIGNUM_INTERNAL
+typedef BignumInt *Bignum;
+
+#include "ssh.h"
+
+BignumInt bnZero[1] = { 0 };
+BignumInt bnOne[2] = { 1, 1 };
+
+/*
+ * The Bignum format is an array of `BignumInt'. The first
+ * element of the array counts the remaining elements. The
+ * remaining elements express the actual number, base 2^BIGNUM_INT_BITS, _least_
+ * significant digit first. (So it's trivial to extract the bit
+ * with value 2^n for any n.)
+ *
+ * All Bignums in this module are positive. Negative numbers must
+ * be dealt with outside it.
+ *
+ * INVARIANT: the most significant word of any Bignum must be
+ * nonzero.
+ */
+
+Bignum Zero = bnZero, One = bnOne;
+
+static Bignum newbn(int length)
+{
+    Bignum b;
+
+    assert(length >= 0 && length < INT_MAX / BIGNUM_INT_BITS);
+
+    b = snewn(length + 1, BignumInt);
+    if (!b)
+	abort();		       /* FIXME */
+    memset(b, 0, (length + 1) * sizeof(*b));
+    b[0] = length;
+    return b;
+}
+
+void bn_restore_invariant(Bignum b)
+{
+    while (b[0] > 1 && b[b[0]] == 0)
+	b[0]--;
+}
+
+Bignum copybn(Bignum orig)
+{
+    Bignum b = snewn(orig[0] + 1, BignumInt);
+    if (!b)
+	abort();		       /* FIXME */
+    memcpy(b, orig, (orig[0] + 1) * sizeof(*b));
+    return b;
+}
+
+void freebn(Bignum b)
+{
+    /*
+     * Burn the evidence, just in case.
+     */
+    smemclr(b, sizeof(b[0]) * (b[0] + 1));
+    sfree(b);
+}
+
+Bignum bn_power_2(int n)
+{
+    Bignum ret;
+
+    assert(n >= 0);
+
+    ret = newbn(n / BIGNUM_INT_BITS + 1);
+    bignum_set_bit(ret, n, 1);
+    return ret;
+}
+
+/*
+ * Internal addition. Sets c = a - b, where 'a', 'b' and 'c' are all
+ * big-endian arrays of 'len' BignumInts. Returns a BignumInt carried
+ * off the top.
+ */
+static BignumInt internal_add(const BignumInt *a, const BignumInt *b,
+                              BignumInt *c, int len)
+{
+    int i;
+    BignumDblInt carry = 0;
+
+    for (i = len-1; i >= 0; i--) {
+        carry += (BignumDblInt)a[i] + b[i];
+        c[i] = (BignumInt)carry;
+        carry >>= BIGNUM_INT_BITS;
+    }
+
+    return (BignumInt)carry;
+}
+
+/*
+ * Internal subtraction. Sets c = a - b, where 'a', 'b' and 'c' are
+ * all big-endian arrays of 'len' BignumInts. Any borrow from the top
+ * is ignored.
+ */
+static void internal_sub(const BignumInt *a, const BignumInt *b,
+                         BignumInt *c, int len)
+{
+    int i;
+    BignumDblInt carry = 1;
+
+    for (i = len-1; i >= 0; i--) {
+        carry += (BignumDblInt)a[i] + (b[i] ^ BIGNUM_INT_MASK);
+        c[i] = (BignumInt)carry;
+        carry >>= BIGNUM_INT_BITS;
+    }
+}
+
+/*
+ * Compute c = a * b.
+ * Input is in the first len words of a and b.
+ * Result is returned in the first 2*len words of c.
+ *
+ * 'scratch' must point to an array of BignumInt of size at least
+ * mul_compute_scratch(len). (This covers the needs of internal_mul
+ * and all its recursive calls to itself.)
+ */
+#define KARATSUBA_THRESHOLD 50
+static int mul_compute_scratch(int len)
+{
+    int ret = 0;
+    while (len > KARATSUBA_THRESHOLD) {
+        int toplen = len/2, botlen = len - toplen; /* botlen is the bigger */
+        int midlen = botlen + 1;
+        ret += 4*midlen;
+        len = midlen;
+    }
+    return ret;
+}
+static void internal_mul(const BignumInt *a, const BignumInt *b,
+			 BignumInt *c, int len, BignumInt *scratch)
+{
+    if (len > KARATSUBA_THRESHOLD) {
+        int i;
+
+        /*
+         * Karatsuba divide-and-conquer algorithm. Cut each input in
+         * half, so that it's expressed as two big 'digits' in a giant
+         * base D:
+         *
+         *   a = a_1 D + a_0
+         *   b = b_1 D + b_0
+         *
+         * Then the product is of course
+         *
+         *  ab = a_1 b_1 D^2 + (a_1 b_0 + a_0 b_1) D + a_0 b_0
+         *
+         * and we compute the three coefficients by recursively
+         * calling ourself to do half-length multiplications.
+         *
+         * The clever bit that makes this worth doing is that we only
+         * need _one_ half-length multiplication for the central
+         * coefficient rather than the two that it obviouly looks
+         * like, because we can use a single multiplication to compute
+         *
+         *   (a_1 + a_0) (b_1 + b_0) = a_1 b_1 + a_1 b_0 + a_0 b_1 + a_0 b_0
+         *
+         * and then we subtract the other two coefficients (a_1 b_1
+         * and a_0 b_0) which we were computing anyway.
+         *
+         * Hence we get to multiply two numbers of length N in about
+         * three times as much work as it takes to multiply numbers of
+         * length N/2, which is obviously better than the four times
+         * as much work it would take if we just did a long
+         * conventional multiply.
+         */
+
+        int toplen = len/2, botlen = len - toplen; /* botlen is the bigger */
+        int midlen = botlen + 1;
+        BignumDblInt carry;
+#ifdef KARA_DEBUG
+        int i;
+#endif
+
+        /*
+         * The coefficients a_1 b_1 and a_0 b_0 just avoid overlapping
+         * in the output array, so we can compute them immediately in
+         * place.
+         */
+
+#ifdef KARA_DEBUG
+        printf("a1,a0 = 0x");
+        for (i = 0; i < len; i++) {
+            if (i == toplen) printf(", 0x");
+            printf("%0*x", BIGNUM_INT_BITS/4, a[i]);
+        }
+        printf("\n");
+        printf("b1,b0 = 0x");
+        for (i = 0; i < len; i++) {
+            if (i == toplen) printf(", 0x");
+            printf("%0*x", BIGNUM_INT_BITS/4, b[i]);
+        }
+        printf("\n");
+#endif
+
+        /* a_1 b_1 */
+        internal_mul(a, b, c, toplen, scratch);
+#ifdef KARA_DEBUG
+        printf("a1b1 = 0x");
+        for (i = 0; i < 2*toplen; i++) {
+            printf("%0*x", BIGNUM_INT_BITS/4, c[i]);
+        }
+        printf("\n");
+#endif
+
+        /* a_0 b_0 */
+        internal_mul(a + toplen, b + toplen, c + 2*toplen, botlen, scratch);
+#ifdef KARA_DEBUG
+        printf("a0b0 = 0x");
+        for (i = 0; i < 2*botlen; i++) {
+            printf("%0*x", BIGNUM_INT_BITS/4, c[2*toplen+i]);
+        }
+        printf("\n");
+#endif
+
+        /* Zero padding. midlen exceeds toplen by at most 2, so just
+         * zero the first two words of each input and the rest will be
+         * copied over. */
+        scratch[0] = scratch[1] = scratch[midlen] = scratch[midlen+1] = 0;
+
+        for (i = 0; i < toplen; i++) {
+            scratch[midlen - toplen + i] = a[i]; /* a_1 */
+            scratch[2*midlen - toplen + i] = b[i]; /* b_1 */
+        }
+
+        /* compute a_1 + a_0 */
+        scratch[0] = internal_add(scratch+1, a+toplen, scratch+1, botlen);
+#ifdef KARA_DEBUG
+        printf("a1plusa0 = 0x");
+        for (i = 0; i < midlen; i++) {
+            printf("%0*x", BIGNUM_INT_BITS/4, scratch[i]);
+        }
+        printf("\n");
+#endif
+        /* compute b_1 + b_0 */
+        scratch[midlen] = internal_add(scratch+midlen+1, b+toplen,
+                                       scratch+midlen+1, botlen);
+#ifdef KARA_DEBUG
+        printf("b1plusb0 = 0x");
+        for (i = 0; i < midlen; i++) {
+            printf("%0*x", BIGNUM_INT_BITS/4, scratch[midlen+i]);
+        }
+        printf("\n");
+#endif
+
+        /*
+         * Now we can do the third multiplication.
+         */
+        internal_mul(scratch, scratch + midlen, scratch + 2*midlen, midlen,
+                     scratch + 4*midlen);
+#ifdef KARA_DEBUG
+        printf("a1plusa0timesb1plusb0 = 0x");
+        for (i = 0; i < 2*midlen; i++) {
+            printf("%0*x", BIGNUM_INT_BITS/4, scratch[2*midlen+i]);
+        }
+        printf("\n");
+#endif
+
+        /*
+         * Now we can reuse the first half of 'scratch' to compute the
+         * sum of the outer two coefficients, to subtract from that
+         * product to obtain the middle one.
+         */
+        scratch[0] = scratch[1] = scratch[2] = scratch[3] = 0;
+        for (i = 0; i < 2*toplen; i++)
+            scratch[2*midlen - 2*toplen + i] = c[i];
+        scratch[1] = internal_add(scratch+2, c + 2*toplen,
+                                  scratch+2, 2*botlen);
+#ifdef KARA_DEBUG
+        printf("a1b1plusa0b0 = 0x");
+        for (i = 0; i < 2*midlen; i++) {
+            printf("%0*x", BIGNUM_INT_BITS/4, scratch[i]);
+        }
+        printf("\n");
+#endif
+
+        internal_sub(scratch + 2*midlen, scratch,
+                     scratch + 2*midlen, 2*midlen);
+#ifdef KARA_DEBUG
+        printf("a1b0plusa0b1 = 0x");
+        for (i = 0; i < 2*midlen; i++) {
+            printf("%0*x", BIGNUM_INT_BITS/4, scratch[2*midlen+i]);
+        }
+        printf("\n");
+#endif
+
+        /*
+         * And now all we need to do is to add that middle coefficient
+         * back into the output. We may have to propagate a carry
+         * further up the output, but we can be sure it won't
+         * propagate right the way off the top.
+         */
+        carry = internal_add(c + 2*len - botlen - 2*midlen,
+                             scratch + 2*midlen,
+                             c + 2*len - botlen - 2*midlen, 2*midlen);
+        i = 2*len - botlen - 2*midlen - 1;
+        while (carry) {
+            assert(i >= 0);
+            carry += c[i];
+            c[i] = (BignumInt)carry;
+            carry >>= BIGNUM_INT_BITS;
+            i--;
+        }
+#ifdef KARA_DEBUG
+        printf("ab = 0x");
+        for (i = 0; i < 2*len; i++) {
+            printf("%0*x", BIGNUM_INT_BITS/4, c[i]);
+        }
+        printf("\n");
+#endif
+
+    } else {
+        int i;
+        BignumInt carry;
+        BignumDblInt t;
+        const BignumInt *ap, *bp;
+        BignumInt *cp, *cps;
+
+        /*
+         * Multiply in the ordinary O(N^2) way.
+         */
+
+        for (i = 0; i < 2 * len; i++)
+            c[i] = 0;
+
+        for (cps = c + 2*len, ap = a + len; ap-- > a; cps--) {
+            carry = 0;
+            for (cp = cps, bp = b + len; cp--, bp-- > b ;) {
+                t = (MUL_WORD(*ap, *bp) + carry) + *cp;
+                *cp = (BignumInt) t;
+                carry = (BignumInt)(t >> BIGNUM_INT_BITS);
+            }
+            *cp = carry;
+        }
+    }
+}
+
+/*
+ * Variant form of internal_mul used for the initial step of
+ * Montgomery reduction. Only bothers outputting 'len' words
+ * (everything above that is thrown away).
+ */
+static void internal_mul_low(const BignumInt *a, const BignumInt *b,
+                             BignumInt *c, int len, BignumInt *scratch)
+{
+    if (len > KARATSUBA_THRESHOLD) {
+        int i;
+
+        /*
+         * Karatsuba-aware version of internal_mul_low. As before, we
+         * express each input value as a shifted combination of two
+         * halves:
+         *
+         *   a = a_1 D + a_0
+         *   b = b_1 D + b_0
+         *
+         * Then the full product is, as before,
+         *
+         *  ab = a_1 b_1 D^2 + (a_1 b_0 + a_0 b_1) D + a_0 b_0
+         *
+         * Provided we choose D on the large side (so that a_0 and b_0
+         * are _at least_ as long as a_1 and b_1), we don't need the
+         * topmost term at all, and we only need half of the middle
+         * term. So there's no point in doing the proper Karatsuba
+         * optimisation which computes the middle term using the top
+         * one, because we'd take as long computing the top one as
+         * just computing the middle one directly.
+         *
+         * So instead, we do a much more obvious thing: we call the
+         * fully optimised internal_mul to compute a_0 b_0, and we
+         * recursively call ourself to compute the _bottom halves_ of
+         * a_1 b_0 and a_0 b_1, each of which we add into the result
+         * in the obvious way.
+         *
+         * In other words, there's no actual Karatsuba _optimisation_
+         * in this function; the only benefit in doing it this way is
+         * that we call internal_mul proper for a large part of the
+         * work, and _that_ can optimise its operation.
+         */
+
+        int toplen = len/2, botlen = len - toplen; /* botlen is the bigger */
+
+        /*
+         * Scratch space for the various bits and pieces we're going
+         * to be adding together: we need botlen*2 words for a_0 b_0
+         * (though we may end up throwing away its topmost word), and
+         * toplen words for each of a_1 b_0 and a_0 b_1. That adds up
+         * to exactly 2*len.
+         */
+
+        /* a_0 b_0 */
+        internal_mul(a + toplen, b + toplen, scratch + 2*toplen, botlen,
+                     scratch + 2*len);
+
+        /* a_1 b_0 */
+        internal_mul_low(a, b + len - toplen, scratch + toplen, toplen,
+                         scratch + 2*len);
+
+        /* a_0 b_1 */
+        internal_mul_low(a + len - toplen, b, scratch, toplen,
+                         scratch + 2*len);
+
+        /* Copy the bottom half of the big coefficient into place */
+        for (i = 0; i < botlen; i++)
+            c[toplen + i] = scratch[2*toplen + botlen + i];
+
+        /* Add the two small coefficients, throwing away the returned carry */
+        internal_add(scratch, scratch + toplen, scratch, toplen);
+
+        /* And add that to the large coefficient, leaving the result in c. */
+        internal_add(scratch, scratch + 2*toplen + botlen - toplen,
+                     c, toplen);
+
+    } else {
+        int i;
+        BignumInt carry;
+        BignumDblInt t;
+        const BignumInt *ap, *bp;
+        BignumInt *cp, *cps;
+
+        /*
+         * Multiply in the ordinary O(N^2) way.
+         */
+
+        for (i = 0; i < len; i++)
+            c[i] = 0;
+
+        for (cps = c + len, ap = a + len; ap-- > a; cps--) {
+            carry = 0;
+            for (cp = cps, bp = b + len; bp--, cp-- > c ;) {
+                t = (MUL_WORD(*ap, *bp) + carry) + *cp;
+                *cp = (BignumInt) t;
+                carry = (BignumInt)(t >> BIGNUM_INT_BITS);
+            }
+        }
+    }
+}
+
+/*
+ * Montgomery reduction. Expects x to be a big-endian array of 2*len
+ * BignumInts whose value satisfies 0 <= x < rn (where r = 2^(len *
+ * BIGNUM_INT_BITS) is the Montgomery base). Returns in the same array
+ * a value x' which is congruent to xr^{-1} mod n, and satisfies 0 <=
+ * x' < n.
+ *
+ * 'n' and 'mninv' should be big-endian arrays of 'len' BignumInts
+ * each, containing respectively n and the multiplicative inverse of
+ * -n mod r.
+ *
+ * 'tmp' is an array of BignumInt used as scratch space, of length at
+ * least 3*len + mul_compute_scratch(len).
+ */
+static void monty_reduce(BignumInt *x, const BignumInt *n,
+                         const BignumInt *mninv, BignumInt *tmp, int len)
+{
+    int i;
+    BignumInt carry;
+
+    /*
+     * Multiply x by (-n)^{-1} mod r. This gives us a value m such
+     * that mn is congruent to -x mod r. Hence, mn+x is an exact
+     * multiple of r, and is also (obviously) congruent to x mod n.
+     */
+    internal_mul_low(x + len, mninv, tmp, len, tmp + 3*len);
+
+    /*
+     * Compute t = (mn+x)/r in ordinary, non-modular, integer
+     * arithmetic. By construction this is exact, and is congruent mod
+     * n to x * r^{-1}, i.e. the answer we want.
+     *
+     * The following multiply leaves that answer in the _most_
+     * significant half of the 'x' array, so then we must shift it
+     * down.
+     */
+    internal_mul(tmp, n, tmp+len, len, tmp + 3*len);
+    carry = internal_add(x, tmp+len, x, 2*len);
+    for (i = 0; i < len; i++)
+        x[len + i] = x[i], x[i] = 0;
+
+    /*
+     * Reduce t mod n. This doesn't require a full-on division by n,
+     * but merely a test and single optional subtraction, since we can
+     * show that 0 <= t < 2n.
+     *
+     * Proof:
+     *  + we computed m mod r, so 0 <= m < r.
+     *  + so 0 <= mn < rn, obviously
+     *  + hence we only need 0 <= x < rn to guarantee that 0 <= mn+x < 2rn
+     *  + yielding 0 <= (mn+x)/r < 2n as required.
+     */
+    if (!carry) {
+        for (i = 0; i < len; i++)
+            if (x[len + i] != n[i])
+                break;
+    }
+    if (carry || i >= len || x[len + i] > n[i])
+        internal_sub(x+len, n, x+len, len);
+}
+
+static void internal_add_shifted(BignumInt *number,
+				 unsigned n, int shift)
+{
+    int word = 1 + (shift / BIGNUM_INT_BITS);
+    int bshift = shift % BIGNUM_INT_BITS;
+    BignumDblInt addend;
+
+    addend = (BignumDblInt)n << bshift;
+
+    while (addend) {
+        assert(word <= number[0]);
+	addend += number[word];
+	number[word] = (BignumInt) addend & BIGNUM_INT_MASK;
+	addend >>= BIGNUM_INT_BITS;
+	word++;
+    }
+}
+
+/*
+ * Compute a = a % m.
+ * Input in first alen words of a and first mlen words of m.
+ * Output in first alen words of a
+ * (of which first alen-mlen words will be zero).
+ * The MSW of m MUST have its high bit set.
+ * Quotient is accumulated in the `quotient' array, which is a Bignum
+ * rather than the internal bigendian format. Quotient parts are shifted
+ * left by `qshift' before adding into quot.
+ */
+static void internal_mod(BignumInt *a, int alen,
+			 BignumInt *m, int mlen,
+			 BignumInt *quot, int qshift)
+{
+    BignumInt m0, m1;
+    unsigned int h;
+    int i, k;
+
+    m0 = m[0];
+    assert(m0 >> (BIGNUM_INT_BITS-1) == 1);
+    if (mlen > 1)
+	m1 = m[1];
+    else
+	m1 = 0;
+
+    for (i = 0; i <= alen - mlen; i++) {
+	BignumDblInt t;
+	unsigned int q, r, c, ai1;
+
+	if (i == 0) {
+	    h = 0;
+	} else {
+	    h = a[i - 1];
+	    a[i - 1] = 0;
+	}
+
+	if (i == alen - 1)
+	    ai1 = 0;
+	else
+	    ai1 = a[i + 1];
+
+	/* Find q = h:a[i] / m0 */
+	if (h >= m0) {
+	    /*
+	     * Special case.
+	     * 
+	     * To illustrate it, suppose a BignumInt is 8 bits, and
+	     * we are dividing (say) A1:23:45:67 by A1:B2:C3. Then
+	     * our initial division will be 0xA123 / 0xA1, which
+	     * will give a quotient of 0x100 and a divide overflow.
+	     * However, the invariants in this division algorithm
+	     * are not violated, since the full number A1:23:... is
+	     * _less_ than the quotient prefix A1:B2:... and so the
+	     * following correction loop would have sorted it out.
+	     * 
+	     * In this situation we set q to be the largest
+	     * quotient we _can_ stomach (0xFF, of course).
+	     */
+	    q = BIGNUM_INT_MASK;
+	} else {
+	    /* Macro doesn't want an array subscript expression passed
+	     * into it (see definition), so use a temporary. */
+	    BignumInt tmplo = a[i];
+	    DIVMOD_WORD(q, r, h, tmplo, m0);
+
+	    /* Refine our estimate of q by looking at
+	     h:a[i]:a[i+1] / m0:m1 */
+	    t = MUL_WORD(m1, q);
+	    if (t > ((BignumDblInt) r << BIGNUM_INT_BITS) + ai1) {
+		q--;
+		t -= m1;
+		r = (r + m0) & BIGNUM_INT_MASK;     /* overflow? */
+		if (r >= (BignumDblInt) m0 &&
+		    t > ((BignumDblInt) r << BIGNUM_INT_BITS) + ai1) q--;
+	    }
+	}
+
+	/* Subtract q * m from a[i...] */
+	c = 0;
+	for (k = mlen - 1; k >= 0; k--) {
+	    t = MUL_WORD(q, m[k]);
+	    t += c;
+	    c = (unsigned)(t >> BIGNUM_INT_BITS);
+	    if ((BignumInt) t > a[i + k])
+		c++;
+	    a[i + k] -= (BignumInt) t;
+	}
+
+	/* Add back m in case of borrow */
+	if (c != h) {
+	    t = 0;
+	    for (k = mlen - 1; k >= 0; k--) {
+		t += m[k];
+		t += a[i + k];
+		a[i + k] = (BignumInt) t;
+		t = t >> BIGNUM_INT_BITS;
+	    }
+	    q--;
+	}
+	if (quot)
+	    internal_add_shifted(quot, q, qshift + BIGNUM_INT_BITS * (alen - mlen - i));
+    }
+}
+
+/*
+ * Compute (base ^ exp) % mod, the pedestrian way.
+ */
+Bignum modpow_simple(Bignum base_in, Bignum exp, Bignum mod)
+{
+    BignumInt *a, *b, *n, *m, *scratch;
+    int mshift;
+    int mlen, scratchlen, i, j;
+    Bignum base, result;
+
+    /*
+     * The most significant word of mod needs to be non-zero. It
+     * should already be, but let's make sure.
+     */
+    assert(mod[mod[0]] != 0);
+
+    /*
+     * Make sure the base is smaller than the modulus, by reducing
+     * it modulo the modulus if not.
+     */
+    base = bigmod(base_in, mod);
+
+    /* Allocate m of size mlen, copy mod to m */
+    /* We use big endian internally */
+    mlen = mod[0];
+    m = snewn(mlen, BignumInt);
+    for (j = 0; j < mlen; j++)
+	m[j] = mod[mod[0] - j];
+
+    /* Shift m left to make msb bit set */
+    for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++)
+	if ((m[0] << mshift) & BIGNUM_TOP_BIT)
+	    break;
+    if (mshift) {
+	for (i = 0; i < mlen - 1; i++)
+	    m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift));
+	m[mlen - 1] = m[mlen - 1] << mshift;
+    }
+
+    /* Allocate n of size mlen, copy base to n */
+    n = snewn(mlen, BignumInt);
+    i = mlen - base[0];
+    for (j = 0; j < i; j++)
+	n[j] = 0;
+    for (j = 0; j < (int)base[0]; j++)
+	n[i + j] = base[base[0] - j];
+
+    /* Allocate a and b of size 2*mlen. Set a = 1 */
+    a = snewn(2 * mlen, BignumInt);
+    b = snewn(2 * mlen, BignumInt);
+    for (i = 0; i < 2 * mlen; i++)
+	a[i] = 0;
+    a[2 * mlen - 1] = 1;
+
+    /* Scratch space for multiplies */
+    scratchlen = mul_compute_scratch(mlen);
+    scratch = snewn(scratchlen, BignumInt);
+
+    /* Skip leading zero bits of exp. */
+    i = 0;
+    j = BIGNUM_INT_BITS-1;
+    while (i < (int)exp[0] && (exp[exp[0] - i] & (1 << j)) == 0) {
+	j--;
+	if (j < 0) {
+	    i++;
+	    j = BIGNUM_INT_BITS-1;
+	}
+    }
+
+    /* Main computation */
+    while (i < (int)exp[0]) {
+	while (j >= 0) {
+	    internal_mul(a + mlen, a + mlen, b, mlen, scratch);
+	    internal_mod(b, mlen * 2, m, mlen, NULL, 0);
+	    if ((exp[exp[0] - i] & (1 << j)) != 0) {
+		internal_mul(b + mlen, n, a, mlen, scratch);
+		internal_mod(a, mlen * 2, m, mlen, NULL, 0);
+	    } else {
+		BignumInt *t;
+		t = a;
+		a = b;
+		b = t;
+	    }
+	    j--;
+	}
+	i++;
+	j = BIGNUM_INT_BITS-1;
+    }
+
+    /* Fixup result in case the modulus was shifted */
+    if (mshift) {
+	for (i = mlen - 1; i < 2 * mlen - 1; i++)
+	    a[i] = (a[i] << mshift) | (a[i + 1] >> (BIGNUM_INT_BITS - mshift));
+	a[2 * mlen - 1] = a[2 * mlen - 1] << mshift;
+	internal_mod(a, mlen * 2, m, mlen, NULL, 0);
+	for (i = 2 * mlen - 1; i >= mlen; i--)
+	    a[i] = (a[i] >> mshift) | (a[i - 1] << (BIGNUM_INT_BITS - mshift));
+    }
+
+    /* Copy result to buffer */
+    result = newbn(mod[0]);
+    for (i = 0; i < mlen; i++)
+	result[result[0] - i] = a[i + mlen];
+    while (result[0] > 1 && result[result[0]] == 0)
+	result[0]--;
+
+    /* Free temporary arrays */
+    smemclr(a, 2 * mlen * sizeof(*a));
+    sfree(a);
+    smemclr(scratch, scratchlen * sizeof(*scratch));
+    sfree(scratch);
+    smemclr(b, 2 * mlen * sizeof(*b));
+    sfree(b);
+    smemclr(m, mlen * sizeof(*m));
+    sfree(m);
+    smemclr(n, mlen * sizeof(*n));
+    sfree(n);
+
+    freebn(base);
+
+    return result;
+}
+
+/*
+ * Compute (base ^ exp) % mod. Uses the Montgomery multiplication
+ * technique where possible, falling back to modpow_simple otherwise.
+ */
+Bignum modpow(Bignum base_in, Bignum exp, Bignum mod)
+{
+    BignumInt *a, *b, *x, *n, *mninv, *scratch;
+    int len, scratchlen, i, j;
+    Bignum base, base2, r, rn, inv, result;
+
+    /*
+     * The most significant word of mod needs to be non-zero. It
+     * should already be, but let's make sure.
+     */
+    assert(mod[mod[0]] != 0);
+
+    /*
+     * mod had better be odd, or we can't do Montgomery multiplication
+     * using a power of two at all.
+     */
+    if (!(mod[1] & 1))
+        return modpow_simple(base_in, exp, mod);
+
+    /*
+     * Make sure the base is smaller than the modulus, by reducing
+     * it modulo the modulus if not.
+     */
+    base = bigmod(base_in, mod);
+
+    /*
+     * Compute the inverse of n mod r, for monty_reduce. (In fact we
+     * want the inverse of _minus_ n mod r, but we'll sort that out
+     * below.)
+     */
+    len = mod[0];
+    r = bn_power_2(BIGNUM_INT_BITS * len);
+    inv = modinv(mod, r);
+    assert(inv); /* cannot fail, since mod is odd and r is a power of 2 */
+
+    /*
+     * Multiply the base by r mod n, to get it into Montgomery
+     * representation.
+     */
+    base2 = modmul(base, r, mod);
+    freebn(base);
+    base = base2;
+
+    rn = bigmod(r, mod);               /* r mod n, i.e. Montgomerified 1 */
+
+    freebn(r);                         /* won't need this any more */
+
+    /*
+     * Set up internal arrays of the right lengths, in big-endian
+     * format, containing the base, the modulus, and the modulus's
+     * inverse.
+     */
+    n = snewn(len, BignumInt);
+    for (j = 0; j < len; j++)
+	n[len - 1 - j] = mod[j + 1];
+
+    mninv = snewn(len, BignumInt);
+    for (j = 0; j < len; j++)
+	mninv[len - 1 - j] = (j < (int)inv[0] ? inv[j + 1] : 0);
+    freebn(inv);         /* we don't need this copy of it any more */
+    /* Now negate mninv mod r, so it's the inverse of -n rather than +n. */
+    x = snewn(len, BignumInt);
+    for (j = 0; j < len; j++)
+        x[j] = 0;
+    internal_sub(x, mninv, mninv, len);
+
+    /* x = snewn(len, BignumInt); */ /* already done above */
+    for (j = 0; j < len; j++)
+	x[len - 1 - j] = (j < (int)base[0] ? base[j + 1] : 0);
+    freebn(base);        /* we don't need this copy of it any more */
+
+    a = snewn(2*len, BignumInt);
+    b = snewn(2*len, BignumInt);
+    for (j = 0; j < len; j++)
+	a[2*len - 1 - j] = (j < (int)rn[0] ? rn[j + 1] : 0);
+    freebn(rn);
+
+    /* Scratch space for multiplies */
+    scratchlen = 3*len + mul_compute_scratch(len);
+    scratch = snewn(scratchlen, BignumInt);
+
+    /* Skip leading zero bits of exp. */
+    i = 0;
+    j = BIGNUM_INT_BITS-1;
+    while (i < (int)exp[0] && (exp[exp[0] - i] & (1 << j)) == 0) {
+	j--;
+	if (j < 0) {
+	    i++;
+	    j = BIGNUM_INT_BITS-1;
+	}
+    }
+
+    /* Main computation */
+    while (i < (int)exp[0]) {
+	while (j >= 0) {
+	    internal_mul(a + len, a + len, b, len, scratch);
+            monty_reduce(b, n, mninv, scratch, len);
+	    if ((exp[exp[0] - i] & (1 << j)) != 0) {
+                internal_mul(b + len, x, a, len,  scratch);
+                monty_reduce(a, n, mninv, scratch, len);
+	    } else {
+		BignumInt *t;
+		t = a;
+		a = b;
+		b = t;
+	    }
+	    j--;
+	}
+	i++;
+	j = BIGNUM_INT_BITS-1;
+    }
+
+    /*
+     * Final monty_reduce to get back from the adjusted Montgomery
+     * representation.
+     */
+    monty_reduce(a, n, mninv, scratch, len);
+
+    /* Copy result to buffer */
+    result = newbn(mod[0]);
+    for (i = 0; i < len; i++)
+	result[result[0] - i] = a[i + len];
+    while (result[0] > 1 && result[result[0]] == 0)
+	result[0]--;
+
+    /* Free temporary arrays */
+    smemclr(scratch, scratchlen * sizeof(*scratch));
+    sfree(scratch);
+    smemclr(a, 2 * len * sizeof(*a));
+    sfree(a);
+    smemclr(b, 2 * len * sizeof(*b));
+    sfree(b);
+    smemclr(mninv, len * sizeof(*mninv));
+    sfree(mninv);
+    smemclr(n, len * sizeof(*n));
+    sfree(n);
+    smemclr(x, len * sizeof(*x));
+    sfree(x);
+
+    return result;
+}
+
+/*
+ * Compute (p * q) % mod.
+ * The most significant word of mod MUST be non-zero.
+ * We assume that the result array is the same size as the mod array.
+ */
+Bignum modmul(Bignum p, Bignum q, Bignum mod)
+{
+    BignumInt *a, *n, *m, *o, *scratch;
+    int mshift, scratchlen;
+    int pqlen, mlen, rlen, i, j;
+    Bignum result;
+
+    /*
+     * The most significant word of mod needs to be non-zero. It
+     * should already be, but let's make sure.
+     */
+    assert(mod[mod[0]] != 0);
+
+    /* Allocate m of size mlen, copy mod to m */
+    /* We use big endian internally */
+    mlen = mod[0];
+    m = snewn(mlen, BignumInt);
+    for (j = 0; j < mlen; j++)
+	m[j] = mod[mod[0] - j];
+
+    /* Shift m left to make msb bit set */
+    for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++)
+	if ((m[0] << mshift) & BIGNUM_TOP_BIT)
+	    break;
+    if (mshift) {
+	for (i = 0; i < mlen - 1; i++)
+	    m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift));
+	m[mlen - 1] = m[mlen - 1] << mshift;
+    }
+
+    pqlen = (p[0] > q[0] ? p[0] : q[0]);
+
+    /*
+     * Make sure that we're allowing enough space. The shifting below
+     * will underflow the vectors we allocate if pqlen is too small.
+     */
+    if (2*pqlen <= mlen)
+        pqlen = mlen/2 + 1;
+
+    /* Allocate n of size pqlen, copy p to n */
+    n = snewn(pqlen, BignumInt);
+    i = pqlen - p[0];
+    for (j = 0; j < i; j++)
+	n[j] = 0;
+    for (j = 0; j < (int)p[0]; j++)
+	n[i + j] = p[p[0] - j];
+
+    /* Allocate o of size pqlen, copy q to o */
+    o = snewn(pqlen, BignumInt);
+    i = pqlen - q[0];
+    for (j = 0; j < i; j++)
+	o[j] = 0;
+    for (j = 0; j < (int)q[0]; j++)
+	o[i + j] = q[q[0] - j];
+
+    /* Allocate a of size 2*pqlen for result */
+    a = snewn(2 * pqlen, BignumInt);
+
+    /* Scratch space for multiplies */
+    scratchlen = mul_compute_scratch(pqlen);
+    scratch = snewn(scratchlen, BignumInt);
+
+    /* Main computation */
+    internal_mul(n, o, a, pqlen, scratch);
+    internal_mod(a, pqlen * 2, m, mlen, NULL, 0);
+
+    /* Fixup result in case the modulus was shifted */
+    if (mshift) {
+	for (i = 2 * pqlen - mlen - 1; i < 2 * pqlen - 1; i++)
+	    a[i] = (a[i] << mshift) | (a[i + 1] >> (BIGNUM_INT_BITS - mshift));
+	a[2 * pqlen - 1] = a[2 * pqlen - 1] << mshift;
+	internal_mod(a, pqlen * 2, m, mlen, NULL, 0);
+	for (i = 2 * pqlen - 1; i >= 2 * pqlen - mlen; i--)
+	    a[i] = (a[i] >> mshift) | (a[i - 1] << (BIGNUM_INT_BITS - mshift));
+    }
+
+    /* Copy result to buffer */
+    rlen = (mlen < pqlen * 2 ? mlen : pqlen * 2);
+    result = newbn(rlen);
+    for (i = 0; i < rlen; i++)
+	result[result[0] - i] = a[i + 2 * pqlen - rlen];
+    while (result[0] > 1 && result[result[0]] == 0)
+	result[0]--;
+
+    /* Free temporary arrays */
+    smemclr(scratch, scratchlen * sizeof(*scratch));
+    sfree(scratch);
+    smemclr(a, 2 * pqlen * sizeof(*a));
+    sfree(a);
+    smemclr(m, mlen * sizeof(*m));
+    sfree(m);
+    smemclr(n, pqlen * sizeof(*n));
+    sfree(n);
+    smemclr(o, pqlen * sizeof(*o));
+    sfree(o);
+
+    return result;
+}
+
+/*
+ * Compute p % mod.
+ * The most significant word of mod MUST be non-zero.
+ * We assume that the result array is the same size as the mod array.
+ * We optionally write out a quotient if `quotient' is non-NULL.
+ * We can avoid writing out the result if `result' is NULL.
+ */
+static void bigdivmod(Bignum p, Bignum mod, Bignum result, Bignum quotient)
+{
+    BignumInt *n, *m;
+    int mshift;
+    int plen, mlen, i, j;
+
+    /*
+     * The most significant word of mod needs to be non-zero. It
+     * should already be, but let's make sure.
+     */
+    assert(mod[mod[0]] != 0);
+
+    /* Allocate m of size mlen, copy mod to m */
+    /* We use big endian internally */
+    mlen = mod[0];
+    m = snewn(mlen, BignumInt);
+    for (j = 0; j < mlen; j++)
+	m[j] = mod[mod[0] - j];
+
+    /* Shift m left to make msb bit set */
+    for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++)
+	if ((m[0] << mshift) & BIGNUM_TOP_BIT)
+	    break;
+    if (mshift) {
+	for (i = 0; i < mlen - 1; i++)
+	    m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift));
+	m[mlen - 1] = m[mlen - 1] << mshift;
+    }
+
+    plen = p[0];
+    /* Ensure plen > mlen */
+    if (plen <= mlen)
+	plen = mlen + 1;
+
+    /* Allocate n of size plen, copy p to n */
+    n = snewn(plen, BignumInt);
+    for (j = 0; j < plen; j++)
+	n[j] = 0;
+    for (j = 1; j <= (int)p[0]; j++)
+	n[plen - j] = p[j];
+
+    /* Main computation */
+    internal_mod(n, plen, m, mlen, quotient, mshift);
+
+    /* Fixup result in case the modulus was shifted */
+    if (mshift) {
+	for (i = plen - mlen - 1; i < plen - 1; i++)
+	    n[i] = (n[i] << mshift) | (n[i + 1] >> (BIGNUM_INT_BITS - mshift));
+	n[plen - 1] = n[plen - 1] << mshift;
+	internal_mod(n, plen, m, mlen, quotient, 0);
+	for (i = plen - 1; i >= plen - mlen; i--)
+	    n[i] = (n[i] >> mshift) | (n[i - 1] << (BIGNUM_INT_BITS - mshift));
+    }
+
+    /* Copy result to buffer */
+    if (result) {
+	for (i = 1; i <= (int)result[0]; i++) {
+	    int j = plen - i;
+	    result[i] = j >= 0 ? n[j] : 0;
+	}
+    }
+
+    /* Free temporary arrays */
+    smemclr(m, mlen * sizeof(*m));
+    sfree(m);
+    smemclr(n, plen * sizeof(*n));
+    sfree(n);
+}
+
+/*
+ * Decrement a number.
+ */
+void decbn(Bignum bn)
+{
+    int i = 1;
+    while (i < (int)bn[0] && bn[i] == 0)
+	bn[i++] = BIGNUM_INT_MASK;
+    bn[i]--;
+}
+
+Bignum bignum_from_bytes(const unsigned char *data, int nbytes)
+{
+    Bignum result;
+    int w, i;
+
+    assert(nbytes >= 0 && nbytes < INT_MAX/8);
+
+    w = (nbytes + BIGNUM_INT_BYTES - 1) / BIGNUM_INT_BYTES; /* bytes->words */
+
+    result = newbn(w);
+    for (i = 1; i <= w; i++)
+	result[i] = 0;
+    for (i = nbytes; i--;) {
+	unsigned char byte = *data++;
+	result[1 + i / BIGNUM_INT_BYTES] |= byte << (8*i % BIGNUM_INT_BITS);
+    }
+
+    while (result[0] > 1 && result[result[0]] == 0)
+	result[0]--;
+    return result;
+}
+
+/*
+ * Read an SSH-1-format bignum from a data buffer. Return the number
+ * of bytes consumed, or -1 if there wasn't enough data.
+ */
+int ssh1_read_bignum(const unsigned char *data, int len, Bignum * result)
+{
+    const unsigned char *p = data;
+    int i;
+    int w, b;
+
+    if (len < 2)
+	return -1;
+
+    w = 0;
+    for (i = 0; i < 2; i++)
+	w = (w << 8) + *p++;
+    b = (w + 7) / 8;		       /* bits -> bytes */
+
+    if (len < b+2)
+	return -1;
+
+    if (!result)		       /* just return length */
+	return b + 2;
+
+    *result = bignum_from_bytes(p, b);
+
+    return p + b - data;
+}
+
+/*
+ * Return the bit count of a bignum, for SSH-1 encoding.
+ */
+int bignum_bitcount(Bignum bn)
+{
+    int bitcount = bn[0] * BIGNUM_INT_BITS - 1;
+    while (bitcount >= 0
+	   && (bn[bitcount / BIGNUM_INT_BITS + 1] >> (bitcount % BIGNUM_INT_BITS)) == 0) bitcount--;
+    return bitcount + 1;
+}
+
+/*
+ * Return the byte length of a bignum when SSH-1 encoded.
+ */
+int ssh1_bignum_length(Bignum bn)
+{
+    return 2 + (bignum_bitcount(bn) + 7) / 8;
+}
+
+/*
+ * Return the byte length of a bignum when SSH-2 encoded.
+ */
+int ssh2_bignum_length(Bignum bn)
+{
+    return 4 + (bignum_bitcount(bn) + 8) / 8;
+}
+
+/*
+ * Return a byte from a bignum; 0 is least significant, etc.
+ */
+int bignum_byte(Bignum bn, int i)
+{
+    if (i < 0 || i >= (int)(BIGNUM_INT_BYTES * bn[0]))
+	return 0;		       /* beyond the end */
+    else
+	return (bn[i / BIGNUM_INT_BYTES + 1] >>
+		((i % BIGNUM_INT_BYTES)*8)) & 0xFF;
+}
+
+/*
+ * Return a bit from a bignum; 0 is least significant, etc.
+ */
+int bignum_bit(Bignum bn, int i)
+{
+    if (i < 0 || i >= (int)(BIGNUM_INT_BITS * bn[0]))
+	return 0;		       /* beyond the end */
+    else
+	return (bn[i / BIGNUM_INT_BITS + 1] >> (i % BIGNUM_INT_BITS)) & 1;
+}
+
+/*
+ * Set a bit in a bignum; 0 is least significant, etc.
+ */
+void bignum_set_bit(Bignum bn, int bitnum, int value)
+{
+    if (bitnum < 0 || bitnum >= (int)(BIGNUM_INT_BITS * bn[0]))
+	abort();		       /* beyond the end */
+    else {
+	int v = bitnum / BIGNUM_INT_BITS + 1;
+	int mask = 1 << (bitnum % BIGNUM_INT_BITS);
+	if (value)
+	    bn[v] |= mask;
+	else
+	    bn[v] &= ~mask;
+    }
+}
+
+/*
+ * Write a SSH-1-format bignum into a buffer. It is assumed the
+ * buffer is big enough. Returns the number of bytes used.
+ */
+int ssh1_write_bignum(void *data, Bignum bn)
+{
+    unsigned char *p = data;
+    int len = ssh1_bignum_length(bn);
+    int i;
+    int bitc = bignum_bitcount(bn);
+
+    *p++ = (bitc >> 8) & 0xFF;
+    *p++ = (bitc) & 0xFF;
+    for (i = len - 2; i--;)
+	*p++ = bignum_byte(bn, i);
+    return len;
+}
+
+/*
+ * Compare two bignums. Returns like strcmp.
+ */
+int bignum_cmp(Bignum a, Bignum b)
+{
+    int amax = a[0], bmax = b[0];
+    int i;
+
+    /* Annoyingly we have two representations of zero */
+    if (amax == 1 && a[amax] == 0)
+        amax = 0;
+    if (bmax == 1 && b[bmax] == 0)
+        bmax = 0;
+
+    assert(amax == 0 || a[amax] != 0);
+    assert(bmax == 0 || b[bmax] != 0);
+
+    i = (amax > bmax ? amax : bmax);
+    while (i) {
+	BignumInt aval = (i > amax ? 0 : a[i]);
+	BignumInt bval = (i > bmax ? 0 : b[i]);
+	if (aval < bval)
+	    return -1;
+	if (aval > bval)
+	    return +1;
+	i--;
+    }
+    return 0;
+}
+
+/*
+ * Right-shift one bignum to form another.
+ */
+Bignum bignum_rshift(Bignum a, int shift)
+{
+    Bignum ret;
+    int i, shiftw, shiftb, shiftbb, bits;
+    BignumInt ai, ai1;
+
+    assert(shift >= 0);
+
+    bits = bignum_bitcount(a) - shift;
+    ret = newbn((bits + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS);
+
+    if (ret) {
+	shiftw = shift / BIGNUM_INT_BITS;
+	shiftb = shift % BIGNUM_INT_BITS;
+	shiftbb = BIGNUM_INT_BITS - shiftb;
+
+	ai1 = a[shiftw + 1];
+	for (i = 1; i <= (int)ret[0]; i++) {
+	    ai = ai1;
+	    ai1 = (i + shiftw + 1 <= (int)a[0] ? a[i + shiftw + 1] : 0);
+	    ret[i] = ((ai >> shiftb) | (ai1 << shiftbb)) & BIGNUM_INT_MASK;
+	}
+    }
+
+    return ret;
+}
+
+/*
+ * Non-modular multiplication and addition.
+ */
+Bignum bigmuladd(Bignum a, Bignum b, Bignum addend)
+{
+    int alen = a[0], blen = b[0];
+    int mlen = (alen > blen ? alen : blen);
+    int rlen, i, maxspot;
+    int wslen;
+    BignumInt *workspace;
+    Bignum ret;
+
+    /* mlen space for a, mlen space for b, 2*mlen for result,
+     * plus scratch space for multiplication */
+    wslen = mlen * 4 + mul_compute_scratch(mlen);
+    workspace = snewn(wslen, BignumInt);
+    for (i = 0; i < mlen; i++) {
+	workspace[0 * mlen + i] = (mlen - i <= (int)a[0] ? a[mlen - i] : 0);
+	workspace[1 * mlen + i] = (mlen - i <= (int)b[0] ? b[mlen - i] : 0);
+    }
+
+    internal_mul(workspace + 0 * mlen, workspace + 1 * mlen,
+		 workspace + 2 * mlen, mlen, workspace + 4 * mlen);
+
+    /* now just copy the result back */
+    rlen = alen + blen + 1;
+    if (addend && rlen <= (int)addend[0])
+	rlen = addend[0] + 1;
+    ret = newbn(rlen);
+    maxspot = 0;
+    for (i = 1; i <= (int)ret[0]; i++) {
+	ret[i] = (i <= 2 * mlen ? workspace[4 * mlen - i] : 0);
+	if (ret[i] != 0)
+	    maxspot = i;
+    }
+    ret[0] = maxspot;
+
+    /* now add in the addend, if any */
+    if (addend) {
+	BignumDblInt carry = 0;
+	for (i = 1; i <= rlen; i++) {
+	    carry += (i <= (int)ret[0] ? ret[i] : 0);
+	    carry += (i <= (int)addend[0] ? addend[i] : 0);
+	    ret[i] = (BignumInt) carry & BIGNUM_INT_MASK;
+	    carry >>= BIGNUM_INT_BITS;
+	    if (ret[i] != 0 && i > maxspot)
+		maxspot = i;
+	}
+    }
+    ret[0] = maxspot;
+
+    smemclr(workspace, wslen * sizeof(*workspace));
+    sfree(workspace);
+    return ret;
+}
+
+/*
+ * Non-modular multiplication.
+ */
+Bignum bigmul(Bignum a, Bignum b)
+{
+    return bigmuladd(a, b, NULL);
+}
+
+/*
+ * Simple addition.
+ */
+Bignum bigadd(Bignum a, Bignum b)
+{
+    int alen = a[0], blen = b[0];
+    int rlen = (alen > blen ? alen : blen) + 1;
+    int i, maxspot;
+    Bignum ret;
+    BignumDblInt carry;
+
+    ret = newbn(rlen);
+
+    carry = 0;
+    maxspot = 0;
+    for (i = 1; i <= rlen; i++) {
+        carry += (i <= (int)a[0] ? a[i] : 0);
+        carry += (i <= (int)b[0] ? b[i] : 0);
+        ret[i] = (BignumInt) carry & BIGNUM_INT_MASK;
+        carry >>= BIGNUM_INT_BITS;
+        if (ret[i] != 0 && i > maxspot)
+            maxspot = i;
+    }
+    ret[0] = maxspot;
+
+    return ret;
+}
+
+/*
+ * Subtraction. Returns a-b, or NULL if the result would come out
+ * negative (recall that this entire bignum module only handles
+ * positive numbers).
+ */
+Bignum bigsub(Bignum a, Bignum b)
+{
+    int alen = a[0], blen = b[0];
+    int rlen = (alen > blen ? alen : blen);
+    int i, maxspot;
+    Bignum ret;
+    BignumDblInt carry;
+
+    ret = newbn(rlen);
+
+    carry = 1;
+    maxspot = 0;
+    for (i = 1; i <= rlen; i++) {
+        carry += (i <= (int)a[0] ? a[i] : 0);
+        carry += (i <= (int)b[0] ? b[i] ^ BIGNUM_INT_MASK : BIGNUM_INT_MASK);
+        ret[i] = (BignumInt) carry & BIGNUM_INT_MASK;
+        carry >>= BIGNUM_INT_BITS;
+        if (ret[i] != 0 && i > maxspot)
+            maxspot = i;
+    }
+    ret[0] = maxspot;
+
+    if (!carry) {
+        freebn(ret);
+        return NULL;
+    }
+
+    return ret;
+}
+
+/*
+ * Create a bignum which is the bitmask covering another one. That
+ * is, the smallest integer which is >= N and is also one less than
+ * a power of two.
+ */
+Bignum bignum_bitmask(Bignum n)
+{
+    Bignum ret = copybn(n);
+    int i;
+    BignumInt j;
+
+    i = ret[0];
+    while (n[i] == 0 && i > 0)
+	i--;
+    if (i <= 0)
+	return ret;		       /* input was zero */
+    j = 1;
+    while (j < n[i])
+	j = 2 * j + 1;
+    ret[i] = j;
+    while (--i > 0)
+	ret[i] = BIGNUM_INT_MASK;
+    return ret;
+}
+
+/*
+ * Convert a (max 32-bit) long into a bignum.
+ */
+Bignum bignum_from_long(unsigned long nn)
+{
+    Bignum ret;
+    BignumDblInt n = nn;
+
+    ret = newbn(3);
+    ret[1] = (BignumInt)(n & BIGNUM_INT_MASK);
+    ret[2] = (BignumInt)((n >> BIGNUM_INT_BITS) & BIGNUM_INT_MASK);
+    ret[3] = 0;
+    ret[0] = (ret[2]  ? 2 : 1);
+    return ret;
+}
+
+/*
+ * Add a long to a bignum.
+ */
+Bignum bignum_add_long(Bignum number, unsigned long addendx)
+{
+    Bignum ret = newbn(number[0] + 1);
+    int i, maxspot = 0;
+    BignumDblInt carry = 0, addend = addendx;
+
+    for (i = 1; i <= (int)ret[0]; i++) {
+	carry += addend & BIGNUM_INT_MASK;
+	carry += (i <= (int)number[0] ? number[i] : 0);
+	addend >>= BIGNUM_INT_BITS;
+	ret[i] = (BignumInt) carry & BIGNUM_INT_MASK;
+	carry >>= BIGNUM_INT_BITS;
+	if (ret[i] != 0)
+	    maxspot = i;
+    }
+    ret[0] = maxspot;
+    return ret;
+}
+
+/*
+ * Compute the residue of a bignum, modulo a (max 16-bit) short.
+ */
+unsigned short bignum_mod_short(Bignum number, unsigned short modulus)
+{
+    BignumDblInt mod, r;
+    int i;
+
+    r = 0;
+    mod = modulus;
+    for (i = number[0]; i > 0; i--)
+	r = (r * (BIGNUM_TOP_BIT % mod) * 2 + number[i] % mod) % mod;
+    return (unsigned short) r;
+}
+
+#ifdef DEBUG
+void diagbn(char *prefix, Bignum md)
+{
+    int i, nibbles, morenibbles;
+    static const char hex[] = "0123456789ABCDEF";
+
+    debug(("%s0x", prefix ? prefix : ""));
+
+    nibbles = (3 + bignum_bitcount(md)) / 4;
+    if (nibbles < 1)
+	nibbles = 1;
+    morenibbles = 4 * md[0] - nibbles;
+    for (i = 0; i < morenibbles; i++)
+	debug(("-"));
+    for (i = nibbles; i--;)
+	debug(("%c",
+	       hex[(bignum_byte(md, i / 2) >> (4 * (i % 2))) & 0xF]));
+
+    if (prefix)
+	debug(("\n"));
+}
+#endif
+
+/*
+ * Simple division.
+ */
+Bignum bigdiv(Bignum a, Bignum b)
+{
+    Bignum q = newbn(a[0]);
+    bigdivmod(a, b, NULL, q);
+    return q;
+}
+
+/*
+ * Simple remainder.
+ */
+Bignum bigmod(Bignum a, Bignum b)
+{
+    Bignum r = newbn(b[0]);
+    bigdivmod(a, b, r, NULL);
+    return r;
+}
+
+/*
+ * Greatest common divisor.
+ */
+Bignum biggcd(Bignum av, Bignum bv)
+{
+    Bignum a = copybn(av);
+    Bignum b = copybn(bv);
+
+    while (bignum_cmp(b, Zero) != 0) {
+	Bignum t = newbn(b[0]);
+	bigdivmod(a, b, t, NULL);
+	while (t[0] > 1 && t[t[0]] == 0)
+	    t[0]--;
+	freebn(a);
+	a = b;
+	b = t;
+    }
+
+    freebn(b);
+    return a;
+}
+
+/*
+ * Modular inverse, using Euclid's extended algorithm.
+ */
+Bignum modinv(Bignum number, Bignum modulus)
+{
+    Bignum a = copybn(modulus);
+    Bignum b = copybn(number);
+    Bignum xp = copybn(Zero);
+    Bignum x = copybn(One);
+    int sign = +1;
+
+    assert(number[number[0]] != 0);
+    assert(modulus[modulus[0]] != 0);
+
+    while (bignum_cmp(b, One) != 0) {
+	Bignum t, q;
+
+        if (bignum_cmp(b, Zero) == 0) {
+            /*
+             * Found a common factor between the inputs, so we cannot
+             * return a modular inverse at all.
+             */
+            freebn(b);
+            freebn(a);
+            freebn(xp);
+            freebn(x);
+            return NULL;
+        }
+
+        t = newbn(b[0]);
+	q = newbn(a[0]);
+	bigdivmod(a, b, t, q);
+	while (t[0] > 1 && t[t[0]] == 0)
+	    t[0]--;
+	freebn(a);
+	a = b;
+	b = t;
+	t = xp;
+	xp = x;
+	x = bigmuladd(q, xp, t);
+	sign = -sign;
+	freebn(t);
+	freebn(q);
+    }
+
+    freebn(b);
+    freebn(a);
+    freebn(xp);
+
+    /* now we know that sign * x == 1, and that x < modulus */
+    if (sign < 0) {
+	/* set a new x to be modulus - x */
+	Bignum newx = newbn(modulus[0]);
+	BignumInt carry = 0;
+	int maxspot = 1;
+	int i;
+
+	for (i = 1; i <= (int)newx[0]; i++) {
+	    BignumInt aword = (i <= (int)modulus[0] ? modulus[i] : 0);
+	    BignumInt bword = (i <= (int)x[0] ? x[i] : 0);
+	    newx[i] = aword - bword - carry;
+	    bword = ~bword;
+	    carry = carry ? (newx[i] >= bword) : (newx[i] > bword);
+	    if (newx[i] != 0)
+		maxspot = i;
+	}
+	newx[0] = maxspot;
+	freebn(x);
+	x = newx;
+    }
+
+    /* and return. */
+    return x;
+}
+
+/*
+ * Render a bignum into decimal. Return a malloced string holding
+ * the decimal representation.
+ */
+char *bignum_decimal(Bignum x)
+{
+    int ndigits, ndigit;
+    int i, iszero;
+    BignumDblInt carry;
+    char *ret;
+    BignumInt *workspace;
+
+    /*
+     * First, estimate the number of digits. Since log(10)/log(2)
+     * is just greater than 93/28 (the joys of continued fraction
+     * approximations...) we know that for every 93 bits, we need
+     * at most 28 digits. This will tell us how much to malloc.
+     *
+     * Formally: if x has i bits, that means x is strictly less
+     * than 2^i. Since 2 is less than 10^(28/93), this is less than
+     * 10^(28i/93). We need an integer power of ten, so we must
+     * round up (rounding down might make it less than x again).
+     * Therefore if we multiply the bit count by 28/93, rounding
+     * up, we will have enough digits.
+     *
+     * i=0 (i.e., x=0) is an irritating special case.
+     */
+    i = bignum_bitcount(x);
+    if (!i)
+	ndigits = 1;		       /* x = 0 */
+    else
+	ndigits = (28 * i + 92) / 93;  /* multiply by 28/93 and round up */
+    ndigits++;			       /* allow for trailing \0 */
+    ret = snewn(ndigits, char);
+
+    /*
+     * Now allocate some workspace to hold the binary form as we
+     * repeatedly divide it by ten. Initialise this to the
+     * big-endian form of the number.
+     */
+    workspace = snewn(x[0], BignumInt);
+    for (i = 0; i < (int)x[0]; i++)
+	workspace[i] = x[x[0] - i];
+
+    /*
+     * Next, write the decimal number starting with the last digit.
+     * We use ordinary short division, dividing 10 into the
+     * workspace.
+     */
+    ndigit = ndigits - 1;
+    ret[ndigit] = '\0';
+    do {
+	iszero = 1;
+	carry = 0;
+	for (i = 0; i < (int)x[0]; i++) {
+	    carry = (carry << BIGNUM_INT_BITS) + workspace[i];
+	    workspace[i] = (BignumInt) (carry / 10);
+	    if (workspace[i])
+		iszero = 0;
+	    carry %= 10;
+	}
+	ret[--ndigit] = (char) (carry + '0');
+    } while (!iszero);
+
+    /*
+     * There's a chance we've fallen short of the start of the
+     * string. Correct if so.
+     */
+    if (ndigit > 0)
+	memmove(ret, ret + ndigit, ndigits - ndigit);
+
+    /*
+     * Done.
+     */
+    smemclr(workspace, x[0] * sizeof(*workspace));
+    sfree(workspace);
+    return ret;
+}
+
+#ifdef TESTBN
+
+#include <stdio.h>
+#include <stdlib.h>
+#include <ctype.h>
+
+/*
+ * gcc -Wall -g -O0 -DTESTBN -o testbn sshbn.c misc.c conf.c tree234.c unix/uxmisc.c -I. -I unix -I charset
+ *
+ * Then feed to this program's standard input the output of
+ * testdata/bignum.py .
+ */
+
+void modalfatalbox(char *p, ...)
+{
+    va_list ap;
+    fprintf(stderr, "FATAL ERROR: ");
+    va_start(ap, p);
+    vfprintf(stderr, p, ap);
+    va_end(ap);
+    fputc('\n', stderr);
+    exit(1);
+}
+
+#define fromxdigit(c) ( (c)>'9' ? ((c)&0xDF) - 'A' + 10 : (c) - '0' )
+
+int main(int argc, char **argv)
+{
+    char *buf;
+    int line = 0;
+    int passes = 0, fails = 0;
+
+    while ((buf = fgetline(stdin)) != NULL) {
+        int maxlen = strlen(buf);
+        unsigned char *data = snewn(maxlen, unsigned char);
+        unsigned char *ptrs[5], *q;
+        int ptrnum;
+        char *bufp = buf;
+
+        line++;
+
+        q = data;
+        ptrnum = 0;
+
+        while (*bufp && !isspace((unsigned char)*bufp))
+            bufp++;
+        if (bufp)
+            *bufp++ = '\0';
+
+        while (*bufp) {
+            char *start, *end;
+            int i;
+
+            while (*bufp && !isxdigit((unsigned char)*bufp))
+                bufp++;
+            start = bufp;
+
+            if (!*bufp)
+                break;
+
+            while (*bufp && isxdigit((unsigned char)*bufp))
+                bufp++;
+            end = bufp;
+
+            if (ptrnum >= lenof(ptrs))
+                break;
+            ptrs[ptrnum++] = q;
+            
+            for (i = -((end - start) & 1); i < end-start; i += 2) {
+                unsigned char val = (i < 0 ? 0 : fromxdigit(start[i]));
+                val = val * 16 + fromxdigit(start[i+1]);
+                *q++ = val;
+            }
+
+            ptrs[ptrnum] = q;
+        }
+
+        if (!strcmp(buf, "mul")) {
+            Bignum a, b, c, p;
+
+            if (ptrnum != 3) {
+                printf("%d: mul with %d parameters, expected 3\n", line, ptrnum);
+                exit(1);
+            }
+            a = bignum_from_bytes(ptrs[0], ptrs[1]-ptrs[0]);
+            b = bignum_from_bytes(ptrs[1], ptrs[2]-ptrs[1]);
+            c = bignum_from_bytes(ptrs[2], ptrs[3]-ptrs[2]);
+            p = bigmul(a, b);
+
+            if (bignum_cmp(c, p) == 0) {
+                passes++;
+            } else {
+                char *as = bignum_decimal(a);
+                char *bs = bignum_decimal(b);
+                char *cs = bignum_decimal(c);
+                char *ps = bignum_decimal(p);
+                
+                printf("%d: fail: %s * %s gave %s expected %s\n",
+                       line, as, bs, ps, cs);
+                fails++;
+
+                sfree(as);
+                sfree(bs);
+                sfree(cs);
+                sfree(ps);
+            }
+            freebn(a);
+            freebn(b);
+            freebn(c);
+            freebn(p);
+        } else if (!strcmp(buf, "modmul")) {
+            Bignum a, b, m, c, p;
+
+            if (ptrnum != 4) {
+                printf("%d: modmul with %d parameters, expected 4\n",
+                       line, ptrnum);
+                exit(1);
+            }
+            a = bignum_from_bytes(ptrs[0], ptrs[1]-ptrs[0]);
+            b = bignum_from_bytes(ptrs[1], ptrs[2]-ptrs[1]);
+            m = bignum_from_bytes(ptrs[2], ptrs[3]-ptrs[2]);
+            c = bignum_from_bytes(ptrs[3], ptrs[4]-ptrs[3]);
+            p = modmul(a, b, m);
+
+            if (bignum_cmp(c, p) == 0) {
+                passes++;
+            } else {
+                char *as = bignum_decimal(a);
+                char *bs = bignum_decimal(b);
+                char *ms = bignum_decimal(m);
+                char *cs = bignum_decimal(c);
+                char *ps = bignum_decimal(p);
+                
+                printf("%d: fail: %s * %s mod %s gave %s expected %s\n",
+                       line, as, bs, ms, ps, cs);
+                fails++;
+
+                sfree(as);
+                sfree(bs);
+                sfree(ms);
+                sfree(cs);
+                sfree(ps);
+            }
+            freebn(a);
+            freebn(b);
+            freebn(m);
+            freebn(c);
+            freebn(p);
+        } else if (!strcmp(buf, "pow")) {
+            Bignum base, expt, modulus, expected, answer;
+
+            if (ptrnum != 4) {
+                printf("%d: mul with %d parameters, expected 4\n", line, ptrnum);
+                exit(1);
+            }
+
+            base = bignum_from_bytes(ptrs[0], ptrs[1]-ptrs[0]);
+            expt = bignum_from_bytes(ptrs[1], ptrs[2]-ptrs[1]);
+            modulus = bignum_from_bytes(ptrs[2], ptrs[3]-ptrs[2]);
+            expected = bignum_from_bytes(ptrs[3], ptrs[4]-ptrs[3]);
+            answer = modpow(base, expt, modulus);
+
+            if (bignum_cmp(expected, answer) == 0) {
+                passes++;
+            } else {
+                char *as = bignum_decimal(base);
+                char *bs = bignum_decimal(expt);
+                char *cs = bignum_decimal(modulus);
+                char *ds = bignum_decimal(answer);
+                char *ps = bignum_decimal(expected);
+                
+                printf("%d: fail: %s ^ %s mod %s gave %s expected %s\n",
+                       line, as, bs, cs, ds, ps);
+                fails++;
+
+                sfree(as);
+                sfree(bs);
+                sfree(cs);
+                sfree(ds);
+                sfree(ps);
+            }
+            freebn(base);
+            freebn(expt);
+            freebn(modulus);
+            freebn(expected);
+            freebn(answer);
+        } else {
+            printf("%d: unrecognised test keyword: '%s'\n", line, buf);
+            exit(1);
+        }
+
+        sfree(buf);
+        sfree(data);
+    }
+
+    printf("passed %d failed %d total %d\n", passes, fails, passes+fails);
+    return fails != 0;
+}
+
+#endif