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sshbn.c

/*
 * 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)
#else
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;
}

/*
 * 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(BignumInt *a, BignumInt *b,
                   BignumInt *c, int len)
{
    int i, j;
    BignumDblInt t;

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

    for (i = len - 1; i >= 0; i--) {
      t = 0;
      for (j = len - 1; j >= 0; j--) {
          t += MUL_WORD(a[i], (BignumDblInt) b[j]);
          t += (BignumDblInt) c[i + j + 1];
          c[i + j + 1] = (BignumInt) t;
          t = t >> BIGNUM_INT_BITS;
      }
      c[i] = (BignumInt) t;
    }
}

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.
 */
Bignum modpow(Bignum base_in, Bignum exp, Bignum mod)
{
    BignumInt *a, *b, *n, *m;
    int mshift;
    int mlen, 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;

    /* 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);
          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 {
            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 < 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 (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;
    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 = 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);

    /* 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] >> (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 < 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;
    BignumInt *workspace;
    Bignum ret;

    /* mlen space for a, mlen space for b, 2*mlen for result */
    workspace = snewn(mlen * 4, 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);

    /* 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;

    sfree(workspace);
    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;
    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;
}

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