Run entire source base through GNU indent to tidy up the varying

[u/mdw/putty] / sshbn.c

/*
 * Bignum routines for RSA and DH and stuff.
 */

#include <stdio.h>
#include <stdlib.h>
#include <string.h>

#if 0                                  // use PuTTY main debugging for diagbn()
#include <windows.h>
#include "putty.h"
#define debugprint debug
#else
#define debugprint(x) printf x
#endif

#define BIGNUM_INTERNAL
typedef unsigned short *Bignum;

#include "ssh.h"

unsigned short bnZero[1] = { 0 };
unsigned short bnOne[2] = { 1, 1 };

/*
 * The Bignum format is an array of `unsigned short'. The first
 * element of the array counts the remaining elements. The
 * remaining elements express the actual number, base 2^16, _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 = smalloc((length + 1) * sizeof(unsigned short));
    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 = smalloc((orig[0] + 1) * sizeof(unsigned short));
    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 / 16 + 1);
    bignum_set_bit(ret, n, 1);
    return ret;
}

/*
 * 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.
 */
static void internal_mul(unsigned short *a, unsigned short *b,
                         unsigned short *c, int len)
{
    int i, j;
    unsigned long ai, t;

    for (j = 0; j < 2 * len; j++)
        c[j] = 0;

    for (i = len - 1; i >= 0; i--) {
        ai = a[i];
        t = 0;
        for (j = len - 1; j >= 0; j--) {
            t += ai * (unsigned long) b[j];
            t += (unsigned long) c[i + j + 1];
            c[i + j + 1] = (unsigned short) t;
            t = t >> 16;
        }
        c[i] = (unsigned short) t;
    }
}

static void internal_add_shifted(unsigned short *number,
                                 unsigned n, int shift)
{
    int word = 1 + (shift / 16);
    int bshift = shift % 16;
    unsigned long addend;

    addend = n << bshift;

    while (addend) {
        addend += number[word];
        number[word] = (unsigned short) addend & 0xFFFF;
        addend >>= 16;
        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(unsigned short *a, int alen,
                         unsigned short *m, int mlen,
                         unsigned short *quot, int qshift)
{
    unsigned short 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++) {
        unsigned long 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 */
        t = ((unsigned long) h << 16) + a[i];
        q = t / m0;
        r = t % m0;

        /* Refine our estimate of q by looking at
           h:a[i]:a[i+1] / m0:m1 */
        t = (long) m1 *(long) q;
        if (t > ((unsigned long) r << 16) + ai1) {
            q--;
            t -= m1;
            r = (r + m0) & 0xffff;     /* overflow? */
            if (r >= (unsigned long) m0 &&
                t > ((unsigned long) r << 16) + ai1) q--;
        }

        /* Subtract q * m from a[i...] */
        c = 0;
        for (k = mlen - 1; k >= 0; k--) {
            t = (long) q *(long) m[k];
            t += c;
            c = t >> 16;
            if ((unsigned short) t > a[i + k])
                c++;
            a[i + k] -= (unsigned short) 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] = (unsigned short) t;
                t = t >> 16;
            }
            q--;
        }
        if (quot)
            internal_add_shifted(quot, q, qshift + 16 * (alen - mlen - i));
    }
}

/*
 * Compute (base ^ exp) % mod.
 * The base MUST be smaller than the modulus.
 * 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 modpow(Bignum base, Bignum exp, Bignum mod)
{
    unsigned short *a, *b, *n, *m;
    int mshift;
    int mlen, i, j;
    Bignum result;

    /* Allocate m of size mlen, copy mod to m */
    /* We use big endian internally */
    mlen = mod[0];
    m = smalloc(mlen * sizeof(unsigned short));
    for (j = 0; j < mlen; j++)
        m[j] = mod[mod[0] - j];

    /* Shift m left to make msb bit set */
    for (mshift = 0; mshift < 15; mshift++)
        if ((m[0] << mshift) & 0x8000)
            break;
    if (mshift) {
        for (i = 0; i < mlen - 1; i++)
            m[i] = (m[i] << mshift) | (m[i + 1] >> (16 - mshift));
        m[mlen - 1] = m[mlen - 1] << mshift;
    }

    /* Allocate n of size mlen, copy base to n */
    n = smalloc(mlen * sizeof(unsigned short));
    i = mlen - base[0];
    for (j = 0; j < i; j++)
        n[j] = 0;
    for (j = 0; j < base[0]; j++)
        n[i + j] = base[base[0] - j];

    /* Allocate a and b of size 2*mlen. Set a = 1 */
    a = smalloc(2 * mlen * sizeof(unsigned short));
    b = smalloc(2 * mlen * sizeof(unsigned short));
    for (i = 0; i < 2 * mlen; i++)
        a[i] = 0;
    a[2 * mlen - 1] = 1;

    /* Skip leading zero bits of exp. */
    i = 0;
    j = 15;
    while (i < exp[0] && (exp[exp[0] - i] & (1 << j)) == 0) {
        j--;
        if (j < 0) {
            i++;
            j = 15;
        }
    }

    /* Main computation */
    while (i < exp[0]) {
        while (j >= 0) {
            internal_mul(a + mlen, a + mlen, b, mlen);
            internal_mod(b, mlen * 2, m, mlen, NULL, 0);
            if ((exp[exp[0] - i] & (1 << j)) != 0) {
                internal_mul(b + mlen, n, a, mlen);
                internal_mod(a, mlen * 2, m, mlen, NULL, 0);
            } else {
                unsigned short *t;
                t = a;
                a = b;
                b = t;
            }
            j--;
        }
        i++;
        j = 15;
    }

    /* 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] >> (16 - 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] << (16 - 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 < 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);

    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)
{
    unsigned short *a, *n, *m, *o;
    int mshift;
    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 = smalloc(mlen * sizeof(unsigned short));
    for (j = 0; j < mlen; j++)
        m[j] = mod[mod[0] - j];

    /* Shift m left to make msb bit set */
    for (mshift = 0; mshift < 15; mshift++)
        if ((m[0] << mshift) & 0x8000)
            break;
    if (mshift) {
        for (i = 0; i < mlen - 1; i++)
            m[i] = (m[i] << mshift) | (m[i + 1] >> (16 - 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 = smalloc(pqlen * sizeof(unsigned short));
    i = pqlen - p[0];
    for (j = 0; j < i; j++)
        n[j] = 0;
    for (j = 0; j < p[0]; j++)
        n[i + j] = p[p[0] - j];

    /* Allocate o of size pqlen, copy q to o */
    o = smalloc(pqlen * sizeof(unsigned short));
    i = pqlen - q[0];
    for (j = 0; j < i; j++)
        o[j] = 0;
    for (j = 0; j < q[0]; j++)
        o[i + j] = q[q[0] - j];

    /* Allocate a of size 2*pqlen for result */
    a = smalloc(2 * pqlen * sizeof(unsigned short));

    /* Main computation */
    internal_mul(n, o, a, pqlen);
    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] >> (16 - 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] << (16 - 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 < 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.
 */
void bigmod(Bignum p, Bignum mod, Bignum result, Bignum quotient)
{
    unsigned short *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 = smalloc(mlen * sizeof(unsigned short));
    for (j = 0; j < mlen; j++)
        m[j] = mod[mod[0] - j];

    /* Shift m left to make msb bit set */
    for (mshift = 0; mshift < 15; mshift++)
        if ((m[0] << mshift) & 0x8000)
            break;
    if (mshift) {
        for (i = 0; i < mlen - 1; i++)
            m[i] = (m[i] << mshift) | (m[i + 1] >> (16 - 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 = smalloc(plen * sizeof(unsigned short));
    for (j = 0; j < plen; j++)
        n[j] = 0;
    for (j = 1; j <= 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] >> (16 - 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] << (16 - mshift));
    }

    /* Copy result to buffer */
    for (i = 1; i <= 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 < bn[0] && bn[i] == 0)
        bn[i++] = 0xFFFF;
    bn[i]--;
}

Bignum bignum_from_bytes(unsigned char *data, int nbytes)
{
    Bignum result;
    int w, i;

    w = (nbytes + 1) / 2;              /* bytes -> words */

    result = newbn(w);
    for (i = 1; i <= w; i++)
        result[i] = 0;
    for (i = nbytes; i--;) {
        unsigned char byte = *data++;
        if (i & 1)
            result[1 + i / 2] |= byte << 8;
        else
            result[1 + i / 2] |= byte;
    }

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

/*
 * Read an ssh1-format bignum from a data buffer. Return the number
 * of bytes consumed.
 */
int ssh1_read_bignum(unsigned char *data, Bignum * result)
{
    unsigned char *p = data;
    int i;
    int w, b;

    w = 0;
    for (i = 0; i < 2; i++)
        w = (w << 8) + *p++;
    b = (w + 7) / 8;                   /* bits -> bytes */

    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 ssh1 encoding.
 */
int bignum_bitcount(Bignum bn)
{
    int bitcount = bn[0] * 16 - 1;
    while (bitcount >= 0
           && (bn[bitcount / 16 + 1] >> (bitcount % 16)) == 0) bitcount--;
    return bitcount + 1;
}

/*
 * Return the byte length of a bignum when ssh1 encoded.
 */
int ssh1_bignum_length(Bignum bn)
{
    return 2 + (bignum_bitcount(bn) + 7) / 8;
}

/*
 * Return the byte length of a bignum when ssh2 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 >= 2 * bn[0])
        return 0;                      /* beyond the end */
    else if (i & 1)
        return (bn[i / 2 + 1] >> 8) & 0xFF;
    else
        return (bn[i / 2 + 1]) & 0xFF;
}

/*
 * Return a bit from a bignum; 0 is least significant, etc.
 */
int bignum_bit(Bignum bn, int i)
{
    if (i >= 16 * bn[0])
        return 0;                      /* beyond the end */
    else
        return (bn[i / 16 + 1] >> (i % 16)) & 1;
}

/*
 * Set a bit in a bignum; 0 is least significant, etc.
 */
void bignum_set_bit(Bignum bn, int bitnum, int value)
{
    if (bitnum >= 16 * bn[0])
        abort();                       /* beyond the end */
    else {
        int v = bitnum / 16 + 1;
        int mask = 1 << (bitnum % 16);
        if (value)
            bn[v] |= mask;
        else
            bn[v] &= ~mask;
    }
}

/*
 * Write a ssh1-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) {
        unsigned short aval = (i > amax ? 0 : a[i]);
        unsigned short 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;
    unsigned short ai, ai1;

    bits = bignum_bitcount(a) - shift;
    ret = newbn((bits + 15) / 16);

    if (ret) {
        shiftw = shift / 16;
        shiftb = shift % 16;
        shiftbb = 16 - shiftb;

        ai1 = a[shiftw + 1];
        for (i = 1; i <= ret[0]; i++) {
            ai = ai1;
            ai1 = (i + shiftw + 1 <= a[0] ? a[i + shiftw + 1] : 0);
            ret[i] = ((ai >> shiftb) | (ai1 << shiftbb)) & 0xFFFF;
        }
    }

    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;
    unsigned short *workspace;
    Bignum ret;

    /* mlen space for a, mlen space for b, 2*mlen for result */
    workspace = smalloc(mlen * 4 * sizeof(unsigned short));
    for (i = 0; i < mlen; i++) {
        workspace[0 * mlen + i] = (mlen - i <= a[0] ? a[mlen - i] : 0);
        workspace[1 * mlen + i] = (mlen - i <= b[0] ? b[mlen - i] : 0);
    }

    internal_mul(workspace + 0 * mlen, workspace + 1 * mlen,
                 workspace + 2 * mlen, mlen);

    /* now just copy the result back */
    rlen = alen + blen + 1;
    if (addend && rlen <= addend[0])
        rlen = addend[0] + 1;
    ret = newbn(rlen);
    maxspot = 0;
    for (i = 1; i <= 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) {
        unsigned long carry = 0;
        for (i = 1; i <= rlen; i++) {
            carry += (i <= ret[0] ? ret[i] : 0);
            carry += (i <= addend[0] ? addend[i] : 0);
            ret[i] = (unsigned short) carry & 0xFFFF;
            carry >>= 16;
            if (ret[i] != 0 && i > maxspot)
                maxspot = i;
        }
    }
    ret[0] = maxspot;

    return ret;
}

/*
 * Non-modular multiplication.
 */
Bignum bigmul(Bignum a, Bignum b)
{
    return bigmuladd(a, b, NULL);
}

/*
 * 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;
    unsigned short 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] = 0xFFFF;
    return ret;
}

/*
 * Convert a (max 16-bit) short into a bignum.
 */
Bignum bignum_from_short(unsigned short n)
{
    Bignum ret;

    ret = newbn(2);
    ret[1] = n & 0xFFFF;
    ret[2] = (n >> 16) & 0xFFFF;
    ret[0] = (ret[2] ? 2 : 1);
    return ret;
}

/*
 * Add a long to a bignum.
 */
Bignum bignum_add_long(Bignum number, unsigned long addend)
{
    Bignum ret = newbn(number[0] + 1);
    int i, maxspot = 0;
    unsigned long carry = 0;

    for (i = 1; i <= ret[0]; i++) {
        carry += addend & 0xFFFF;
        carry += (i <= number[0] ? number[i] : 0);
        addend >>= 16;
        ret[i] = (unsigned short) carry & 0xFFFF;
        carry >>= 16;
        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)
{
    unsigned long mod, r;
    int i;

    r = 0;
    mod = modulus;
    for (i = number[0]; i > 0; i--)
        r = (r * 65536 + number[i]) % mod;
    return (unsigned short) r;
}

void diagbn(char *prefix, Bignum md)
{
    int i, nibbles, morenibbles;
    static const char hex[] = "0123456789ABCDEF";

    debugprint(("%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++)
        debugprint(("-"));
    for (i = nibbles; i--;)
        debugprint(
                   ("%c",
                    hex[(bignum_byte(md, i / 2) >> (4 * (i % 2))) & 0xF]));

    if (prefix)
        debugprint(("\n"));
}

/*
 * Greatest common divisor.
 */
Bignum biggcd(Bignum av, Bignum bv)
{
    Bignum a = copybn(av);
    Bignum b = copybn(bv);

    diagbn("a = ", a);
    diagbn("b = ", b);
    while (bignum_cmp(b, Zero) != 0) {
        Bignum t = newbn(b[0]);
        bigmod(a, b, t, NULL);
        diagbn("t = ", t);
        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]);
        bigmod(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(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]);
        unsigned short carry = 0;
        int maxspot = 1;
        int i;

        for (i = 1; i <= newx[0]; i++) {
            unsigned short aword = (i <= modulus[0] ? modulus[i] : 0);
            unsigned short bword = (i <= 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;
    unsigned long carry;
    char *ret;
    unsigned short *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 = bignum_bitcount(x);
    ndigits = (28 * i + 92) / 93;      /* multiply by 28/93 and round up */
    ndigits++;                         /* allow for trailing \0 */
    ret = smalloc(ndigits);

    /*
     * 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 = smalloc(sizeof(unsigned short) * x[0]);
    for (i = 0; i < 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 < x[0]; i++) {
            carry = (carry << 16) + workspace[i];
            workspace[i] = (unsigned short) (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.
     */
    return ret;
}