path: root/sbin/isakmpd/math_2n.c

/*	$OpenBSD: math_2n.c,v 1.8 2001/04/09 22:09:52 ho Exp $	*/
/*	$EOM: math_2n.c,v 1.15 1999/04/20 09:23:30 niklas Exp $	*/

/*
 * Copyright (c) 1998 Niels Provos.  All rights reserved.
 * Copyright (c) 1999 Niklas Hallqvist.  All rights reserved.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 * 1. Redistributions of source code must retain the above copyright
 *    notice, this list of conditions and the following disclaimer.
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in the
 *    documentation and/or other materials provided with the distribution.
 * 3. All advertising materials mentioning features or use of this software
 *    must display the following acknowledgement:
 *	This product includes software developed by Ericsson Radio Systems.
 * 4. The name of the author may not be used to endorse or promote products
 *    derived from this software without specific prior written permission.
 *
 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
 */

/*
 * This code was written under funding by Ericsson Radio Systems.
 */

/*
 * B2N is a module for doing arithmetic on the Field GF(2**n) which is
 * isomorph to ring of polynomials GF(2)[x]/p(x) where p(x) is an
 * irreduciable polynomial over GF(2)[x] with grade n.
 * 
 * First we need functions which operate on GF(2)[x], operation
 * on GF(2)[x]/p(x) can be done as for Z_p then.
 */

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

#include "sysdep.h"

#include "math_2n.h"
#include "util.h"

static u_int8_t hex2int (char);

static char int2hex[] = "0123456789abcdef";
CHUNK_TYPE b2n_mask[CHUNK_BITS] = {
  0x01,0x02,0x04,0x08,0x10,0x20,0x40,0x80,
#if CHUNK_BITS > 8
  0x0100,0x0200,0x0400,0x0800,0x1000,0x2000,0x4000,0x8000,
#if CHUNK_BITS > 16
  0x00010000,0x00020000,0x00040000,0x00080000,
  0x00100000,0x00200000,0x00400000,0x00800000,
  0x01000000,0x02000000,0x04000000,0x08000000,
  0x10000000,0x20000000,0x40000000,0x80000000,
#endif
#endif
};

/* Convert a hex character to its integer value.  */
static u_int8_t
hex2int (char c)
{
  if (c <= '9')
    return c - '0';
  if (c <= 'f')
    return 10 + c - 'a';

  return 0;
}

int
b2n_random (b2n_ptr n, u_int32_t bits)
{
  if (b2n_resize (n, (CHUNK_MASK + bits) >> CHUNK_SHIFTS))
    return -1;

  getrandom ((u_int8_t *)n->limp, CHUNK_BYTES * n->chunks);

  /* Get the number of significant bits right */
  if (bits & CHUNK_MASK)
    {
      CHUNK_TYPE m = (((1 << ((bits & CHUNK_MASK)-1)) - 1) << 1) | 1;
      n->limp[n->chunks-1] &= m;
    }

  n->dirty = 1;
  return 0;
}

/* b2n management functions */

void
b2n_init (b2n_ptr n)
{
  n->chunks = 0;
  n->limp = 0;
}

void
b2n_clear (b2n_ptr n)
{
  if (n->limp)
    free (n->limp);
}

int
b2n_resize (b2n_ptr n, unsigned int chunks)
{
  int old = n->chunks;
  int size;
  CHUNK_TYPE *new;

  if (chunks == 0)
    chunks = 1;

  if (chunks == old)
    return 0;
  
  size = CHUNK_BYTES * chunks;

  new = realloc (n->limp, size);
  if (!new)
    return -1;

  n->limp = new;
  n->chunks = chunks;
  n->bits = chunks << CHUNK_SHIFTS;
  n->dirty = 1;

  if (chunks > old)
    memset (n->limp + old, 0, size - CHUNK_BYTES * old);

  return 0;
}

/* Simple assignment functions.  */

int
b2n_set (b2n_ptr d, b2n_ptr s)
{
  if (d == s)
    return 0;

  b2n_sigbit (s);
  if (b2n_resize (d, (CHUNK_MASK + s->bits) >> CHUNK_SHIFTS))
    return -1;
  memcpy (d->limp, s->limp, CHUNK_BYTES * d->chunks);
  d->bits = s->bits;
  d->dirty = s->dirty;
  return 0;
}

int
b2n_set_null (b2n_ptr n)
{
  if (b2n_resize (n, 1))
    return -1;
  n->limp[0] = n->bits = n->dirty = 0;
  return 0;
}

int
b2n_set_ui (b2n_ptr n, unsigned int val)
{
#if CHUNK_BITS < 32
  int i, chunks;

  chunks = (CHUNK_BYTES - 1 + sizeof (val)) / CHUNK_BYTES;

  if (b2n_resize (n, chunks))
      return -1;

  for (i = 0; i < chunks; i++)
    {
      n->limp[i] = val & CHUNK_BMASK;
      val >>= CHUNK_BITS;
    }
#else
  if (b2n_resize (n, 1))
    return -1;
  n->limp[0] = val;
#endif
  n->dirty = 1;
  return 0;
}

/* XXX This one only takes hex at the moment.  */
int
b2n_set_str (b2n_ptr n, char *str)
{
  int i, j, w, len, chunks;
  CHUNK_TYPE tmp;

  if (strncasecmp (str, "0x", 2))
    return -1;

  /* Make the hex string even lengthed */
  len = strlen (str) - 2;
  if (len & 1)
    {
      len ++;
      str ++;
    }
  else
    str += 2;

  len /= 2;

  chunks = (CHUNK_BYTES - 1 + len) / CHUNK_BYTES;
  if (b2n_resize (n, chunks))
    return -1;
  memset (n->limp, 0, CHUNK_BYTES * n->chunks);

  for (w = 0, i = 0; i < chunks; i++)
    {
      tmp = 0;
      for (j = (i == 0 ? ((len - 1) % CHUNK_BYTES) + 1 : CHUNK_BYTES); j > 0;
	   j--)
	{
	  tmp <<= 8;
	  tmp |= (hex2int (str[w]) << 4) | hex2int (str[w + 1]);
	  w += 2;
	}
      n->limp[chunks - 1 - i] = tmp;
    }

  n->dirty = 1;
  return 0;
}

/* Output function, mainly for debugging purposes.  */
void
b2n_print (b2n_ptr n)
{
  int i, j, w, flag = 0;
  int left;
  char buffer[2 * CHUNK_BYTES];
  CHUNK_TYPE tmp;

  left = ((((7 + b2n_sigbit (n)) >> 3) - 1) % CHUNK_BYTES) + 1;
  printf ("0x");
  for (i = 0; i < n->chunks; i++)
    {
      tmp = n->limp[n->chunks - 1 - i];
      memset (buffer, '0', sizeof (buffer));
      for (w = 0, j = (i == 0 ? left : CHUNK_BYTES); j > 0; j--)
	{
	  buffer[w++] = int2hex[(tmp >> 4) & 0xf];
	  buffer[w++] = int2hex[tmp & 0xf];
	  tmp >>= 8;
	}

      for (j = (i == 0 ? left - 1: CHUNK_BYTES - 1); j >= 0; j--)
	if (flag || (i == n->chunks - 1 && j == 0) ||
	    buffer[2 * j] != '0' || buffer[2 * j + 1] != '0')
	  {
	    putchar (buffer[2 * j]);
	    putchar (buffer[2 * j + 1]);
	    flag = 1;
	  }
    }
  printf ("\n");
}

int
b2n_sprint (char *buf, b2n_ptr n)
{
  int i, k, j, w, flag = 0;
  int left;
  char buffer[2 * CHUNK_BYTES];
  CHUNK_TYPE tmp;

  left = ((((7 + b2n_sigbit (n)) >> 3) - 1) % CHUNK_BYTES) + 1;

  strcpy (buf, "0x"); k = 2;
  for (i = 0; i < n->chunks; i++)
    {
      tmp = n->limp[n->chunks - 1 - i];
      memset (buffer, '0', sizeof (buffer));
      for (w = 0, j = (i == 0 ? left : CHUNK_BYTES); j > 0; j--)
	{
	  buffer[w++] = int2hex[(tmp >> 4) & 0xf];
	  buffer[w++] = int2hex[tmp & 0xf];
	  tmp >>= 8;
	}

      for (j = (i == 0 ? left - 1: CHUNK_BYTES - 1); j >= 0; j--)
	if (flag || (i == n->chunks - 1 && j == 0) ||
	    buffer[2 * j] != '0' || buffer[2 * j + 1] != '0')
	  {
	    buf[k++] = buffer[2 * j];
	    buf[k++] = buffer[2 * j + 1];
	    flag = 1;
	  }
    }

  buf[k++] = 0;
  return k;
}

/* Arithmetic functions.  */

u_int32_t
b2n_sigbit (b2n_ptr n)
{
  int i, j;
  
  if (!n->dirty)
    return n->bits;

  for (i = n->chunks - 1; i > 0; i--)
    if (n->limp[i])
      break;

  if (!n->limp[i])
    return 0;

  for (j = CHUNK_MASK; j > 0; j--)
    if (n->limp[i] & b2n_mask[j])
      break;
  
  n->bits = (i << CHUNK_SHIFTS) + j + 1;
  n->dirty = 0;
  return n->bits;
}

/* Addition on GF(2)[x] is nice, its just an XOR.  */
int
b2n_add (b2n_ptr d, b2n_ptr a, b2n_ptr b)
{
  int i;
  b2n_ptr bmin, bmax;

  if (!b2n_cmp_null (a))
    return b2n_set (d, b);

  if (!b2n_cmp_null (b))
    return b2n_set (d, a);

  bmin = B2N_MIN (a,b);
  bmax = B2N_MAX (a,b);
    
  if (b2n_resize (d, bmax->chunks))
    return -1;

  for (i = 0; i < bmin->chunks; i++)
      d->limp[i] = bmax->limp[i] ^ bmin->limp[i];

  /* 
   * If d is not bmax, we have to copy the rest of the bytes, and also
   * need to adjust to number of relevant bits.
   */
  if (d != bmax)
    {
      for ( ; i < bmax->chunks; i++) 
	  d->limp[i] = bmax->limp[i];
      
      d->bits = bmax->bits;
    }

  /*
   * Help to converse memory. When the result of the addition is zero
   * truncate the used amount of memory.
   */
  if (d != bmax && !b2n_cmp_null (d)) 
    return b2n_set_null (d);
  else
    d->dirty = 1;
  return 0;
}

/* Compare two polynomials.  */
int
b2n_cmp (b2n_ptr n, b2n_ptr m)
{
  int sn, sm;
  int i;

  sn = b2n_sigbit (n);
  sm = b2n_sigbit (m);

  if (sn > sm)
    return 1;
  if (sn < sm)
    return -1;

  for (i = n->chunks-1; i >= 0; i--)
    if (n->limp[i] > m->limp[i])
      return 1;
    else if (n->limp[i] < m->limp[i])
      return -1;
  
  return 0;
}

int
b2n_cmp_null (b2n_ptr a)
{
  int i = 0;

  do
    {
      if (a->limp[i])
	return 1;
    }
  while (++i < a->chunks);

  return 0;
}

/* Left shift, needed for polynomial multiplication.  */
int
b2n_lshift (b2n_ptr d, b2n_ptr n, unsigned int s)
{
  int i, maj, min, chunks;
  u_int16_t bits = b2n_sigbit (n), add;
  CHUNK_TYPE *p, *op;

  if (!s)
    return b2n_set (d, n);

  maj = s >> CHUNK_SHIFTS;
  min = s & CHUNK_MASK;

  add = (!(bits & CHUNK_MASK) || ((bits & CHUNK_MASK) + min) > CHUNK_MASK)
    ? 1 : 0;
  chunks = n->chunks;
  if (b2n_resize (d, chunks + maj + add))
    return -1;
  memmove (d->limp + maj, n->limp, CHUNK_BYTES * chunks);

  if (maj)
    memset (d->limp, 0, CHUNK_BYTES * maj);
  if (add) 
    d->limp[d->chunks - 1] = 0;

  /* If !min there are no bit shifts, we are done */
  if (!min)
    return 0;

  op = p = &d->limp[d->chunks - 1];
  for (i = d->chunks - 2; i >= maj; i--)
    {
      op--;
      *p-- = (*p << min) | (*op >> (CHUNK_BITS - min));
    }
  *p <<= min;

  d->dirty = 0;
  d->bits = bits + (maj << CHUNK_SHIFTS) + min;
  return 0;
}

/* Right shift, needed for polynomial division.  */
int
b2n_rshift (b2n_ptr d, b2n_ptr n, unsigned int s)
{
  int maj, min, size = n->chunks, newsize;
  b2n_ptr tmp;

  if (!s)
    return b2n_set (d, n);

  maj = s >> CHUNK_SHIFTS;

  newsize = size - maj;

  if (size < maj)
    return b2n_set_null (d);

  min = (CHUNK_BITS - (s & CHUNK_MASK)) & CHUNK_MASK;
  if (min)
    {
      if ((b2n_sigbit (n) & CHUNK_MASK) > min)
	newsize++;

      if (b2n_lshift (d, n, min))
	return -1;
      tmp = d;
    }
  else
    tmp = n;

  memmove (d->limp, tmp->limp + maj + (min ? 1 : 0), CHUNK_BYTES * newsize);
  if (b2n_resize (d, newsize))
    return -1;

  d->bits = tmp->bits - ((maj + (min ? 1 : 0)) << CHUNK_SHIFTS);
  return 0;
}

/* Normal polynomial multiplication.  */
int
b2n_mul (b2n_ptr d, b2n_ptr n, b2n_ptr m)
{
  int i, j;
  b2n_t tmp, tmp2;

  if (!b2n_cmp_null (m) || !b2n_cmp_null (n))
    return b2n_set_null (d);

  if (b2n_sigbit (m) == 1)
    return b2n_set (d, n);

  if (b2n_sigbit (n) == 1)
    return b2n_set (d, m);
  
  b2n_init (tmp);
  b2n_init (tmp2);

  if (b2n_set (tmp, B2N_MAX (n, m)))
    goto fail;
  if (b2n_set (tmp2, B2N_MIN (n, m)))
    goto fail;

  if (b2n_set_null (d))
    goto fail;

  for (i = 0; i < tmp2->chunks; i++)
    if (tmp2->limp[i])
      for (j = 0; j < CHUNK_BITS; j++)
	{
	  if (tmp2->limp[i] & b2n_mask[j]) 
	    if (b2n_add (d, d, tmp))
	      goto fail;
	  
	  if (b2n_lshift (tmp, tmp, 1))
	    goto fail;
	}
    else
      if (b2n_lshift (tmp, tmp, CHUNK_BITS))
	goto fail;

  b2n_clear (tmp);
  b2n_clear (tmp2);
  return 0;

 fail:
  b2n_clear (tmp);
  b2n_clear (tmp2);
  return -1;
}

/*
 * Squaring in this polynomial ring is more efficient than normal
 * multiplication.
 */
int
b2n_square (b2n_ptr d, b2n_ptr n)
{
  int i, j, maj, min, bits, chunk;
  b2n_t t;

  maj = b2n_sigbit (n);
  min = maj & CHUNK_MASK;
  maj = (maj + CHUNK_MASK) >> CHUNK_SHIFTS;

  b2n_init (t);
  if (b2n_resize (t, 2 * maj + ((CHUNK_MASK + 2 * min) >> CHUNK_SHIFTS)))
    {
      b2n_clear (t);
      return -1;
    }

  chunk = 0;
  bits = 0;

  for (i = 0; i < maj; i++)
    if (n->limp[i])
      for (j = 0; j < CHUNK_BITS; j++)
	{
	  if (n->limp[i] & b2n_mask[j])
	    t->limp[chunk] ^= b2n_mask[bits];

	  bits += 2;
	  if (bits >= CHUNK_BITS)
	    {
	      chunk++;
	      bits &= CHUNK_MASK;
	    }
	}
    else
      chunk += 2;
 
  t->dirty = 1;
  B2N_SWAP (d, t);
  b2n_clear (t);
  return 0;
}

/*
 * Normal polynomial division.
 * These functions are far from optimal in speed.
 */
int
b2n_div_q (b2n_ptr d, b2n_ptr n, b2n_ptr m)
{
  b2n_t r;
  int rv;

  b2n_init (r);
  rv = b2n_div (d, r, n, m);
  b2n_clear (r);
  return rv;
}

int
b2n_div_r (b2n_ptr r, b2n_ptr n, b2n_ptr m)
{
  b2n_t q;
  int rv;

  b2n_init (q);
  rv = b2n_div (q, r, n, m);
  b2n_clear (q);
  return rv;
}

int
b2n_div (b2n_ptr q, b2n_ptr r, b2n_ptr n, b2n_ptr m)
{
  int sn, sm, i, j, len, bits;
  b2n_t nenn, div, shift, mask;

  /* If Teiler > Zaehler, the result is 0 */
  if ((sm = b2n_sigbit (m)) > (sn = b2n_sigbit (n)))
    {
      if (b2n_set_null (q))
	return -1;
      return b2n_set (r, n);
    }

  if (sm == 0)
    /* Division by Zero */
    return -1;
  else if (sm == 1)
    {
      /* Division by the One-Element */
      if (b2n_set (q, n))
	return -1;
      return b2n_set_null (r);
    }

  b2n_init (nenn);
  b2n_init (div);
  b2n_init (shift);
  b2n_init (mask);

  if (b2n_set (nenn, n))
    goto fail;
  if (b2n_set (div, m))
    goto fail;
  if (b2n_set (shift, m))
    goto fail;
  if (b2n_set_ui (mask, 1))
    goto fail;

  if (b2n_resize (q, (sn - sm + CHUNK_MASK) >> CHUNK_SHIFTS))
    goto fail;
  memset (q->limp, 0, CHUNK_BYTES * q->chunks);

  if (b2n_lshift (shift, shift, sn - sm))
    goto fail;
  if (b2n_lshift (mask, mask, sn - sm))
    goto fail;
  
  /* Number of significant octets */
  len = (sn - 1) >> CHUNK_SHIFTS;
  /* The first iteration is done over the relevant bits */
  bits = (CHUNK_MASK + sn) & CHUNK_MASK;
  for (i = len; i >= 0 && b2n_sigbit (nenn) >= sm; i--)
    for (j = (i == len ? bits : CHUNK_MASK); j >= 0 && b2n_sigbit (nenn) >= sm;
	 j--)
      {
	if (nenn->limp[i] & b2n_mask[j])
	  {
	    if (b2n_sub (nenn, nenn, shift))
	      goto fail;
	    if (b2n_add (q, q, mask))
	      goto fail;
	  }
	if (b2n_rshift (shift, shift, 1))
	  goto fail;
	if (b2n_rshift (mask, mask, 1))
	  goto fail;
      }

  B2N_SWAP (r, nenn);

  b2n_clear (nenn);
  b2n_clear (div);
  b2n_clear (shift);
  b2n_clear (mask);
  return 0;

fail:
  b2n_clear (nenn);
  b2n_clear (div);
  b2n_clear (shift);
  b2n_clear (mask);
  return -1;
}

/* Functions for Operation on GF(2**n) ~= GF(2)[x]/p(x).  */
int
b2n_mod (b2n_ptr m, b2n_ptr n, b2n_ptr p)
{
  int bits, size;

  if (b2n_div_r (m, n, p))
    return -1;

  bits = b2n_sigbit (m);
  size = ((CHUNK_MASK + bits) >> CHUNK_SHIFTS);
  if (size == 0)
    size = 1;
  if (m->chunks > size)
    if (b2n_resize (m, size))
      return -1;

  m->bits = bits;
  m->dirty = 0;
  return 0;
}

int
b2n_gcd (b2n_ptr e, b2n_ptr go, b2n_ptr ho)
{
  b2n_t g, h;

  b2n_init (g);
  b2n_init (h);
  if (b2n_set (g, go))
    goto fail;
  if (b2n_set (h, ho))
    goto fail;
  
  while (b2n_cmp_null (h))
    {
      if (b2n_mod (g, g, h))
	goto fail;
      B2N_SWAP (g, h);
    }

  B2N_SWAP (e, g);

  b2n_clear (g);
  b2n_clear (h);
  return 0;

fail:
  b2n_clear (g);
  b2n_clear (h);
  return -1;
}

int
b2n_mul_inv (b2n_ptr ga, b2n_ptr be, b2n_ptr p)
{
  b2n_t a;

  b2n_init (a);
  if (b2n_set_ui (a, 1))
    goto fail;

  if (b2n_div_mod (ga, a, be, p))
    goto fail;

  b2n_clear (a);
  return 0;

 fail:
  b2n_clear (a);
  return -1;
}

int
b2n_div_mod (b2n_ptr ga, b2n_ptr a, b2n_ptr be, b2n_ptr p)
{
  b2n_t s0, s1, s2, q, r0, r1;

  /* There is no multiplicative inverse to Null.  */
  if (!b2n_cmp_null (be))
    return b2n_set_null (ga);

  b2n_init (s0);
  b2n_init (s1);
  b2n_init (s2);
  b2n_init (r0);
  b2n_init (r1);
  b2n_init (q);

  if (b2n_set (r0, p))
    goto fail;
  if (b2n_set (r1, be))
    goto fail;

  if (b2n_set_null (s0))
    goto fail;
  if (b2n_set (s1, a))
    goto fail;

  while (b2n_cmp_null (r1))
    {
      if (b2n_div (q, r0, r0, r1))
	goto fail;
      B2N_SWAP (r0, r1);

      if (b2n_mul (s2, q, s1))
	goto fail;
      if (b2n_mod (s2, s2, p))
	goto fail;
      if (b2n_sub (s2, s0, s2))
	goto fail;

      B2N_SWAP (s0, s1);
      B2N_SWAP (s1, s2);
    }
  B2N_SWAP (ga, s0);

  b2n_clear (s0);
  b2n_clear (s1);
  b2n_clear (s2);
  b2n_clear (r0);
  b2n_clear (r1);
  b2n_clear (q);
  return 0;

fail:
  b2n_clear (s0);
  b2n_clear (s1);
  b2n_clear (s2);
  b2n_clear (r0);
  b2n_clear (r1);
  b2n_clear (q);
  return -1;
}

/*
 * The trace tells us if there do exist any square roots 
 * for 'a' in GF(2)[x]/p(x). The number of square roots is
 * 2 - 2*Trace. 
 * If z is a square root, z + 1 is the other.
 */
int
b2n_trace (b2n_ptr ho, b2n_ptr a, b2n_ptr p)
{
  int i, m = b2n_sigbit (p) - 1;
  b2n_t h;

  b2n_init (h);
  if (b2n_set (h, a))
    goto fail;
  
  for (i = 0; i < m - 1; i++)
    {
      if (b2n_square (h, h))
	goto fail;
      if (b2n_mod (h, h, p))
	goto fail;

      if (b2n_add (h, h, a))
	goto fail;
    }
  B2N_SWAP (ho, h);

  b2n_clear (h);
  return 0;

 fail:
  b2n_clear (h);
  return -1;
}

/*
 * The halftrace yields the square root if the degree of the
 * irreduceable polynomial is odd.
 */
int
b2n_halftrace (b2n_ptr ho, b2n_ptr a, b2n_ptr p)
{
  int i, m = b2n_sigbit (p) - 1;
  b2n_t h;

  b2n_init (h);
  if (b2n_set (h, a))
    goto fail;
  
  for (i = 0; i < (m - 1) / 2; i++)
    {
      if (b2n_square (h, h))
	goto fail;
      if (b2n_mod (h, h, p))
	goto fail;
      if (b2n_square (h, h))
	goto fail;
      if (b2n_mod (h, h, p))
	goto fail;

      if (b2n_add (h, h, a))
	goto fail;
    }

  B2N_SWAP (ho, h);

  b2n_clear (h);
  return 0;

 fail:
  b2n_clear (h);
  return -1;
}

/*
 * Solving the equation: y**2 + y = b in GF(2**m) where ip is the 
 * irreduceable polynomial. If m is odd, use the half trace.
 */
int
b2n_sqrt (b2n_ptr zo, b2n_ptr b, b2n_ptr ip)
{
  int i, m = b2n_sigbit (ip) - 1;
  b2n_t w, p, temp, z;

  if (!b2n_cmp_null (b))
    return b2n_set_null (z);

  if (m & 1)
    return b2n_halftrace (zo, b, ip);

  b2n_init (z);
  b2n_init (w);
  b2n_init (p);
  b2n_init (temp);

  do
    {
      if (b2n_random (p, m))
	goto fail;
      if (b2n_set_null (z))
	goto fail;
      if (b2n_set (w, p))
	goto fail;

      for (i = 1; i < m; i++)
	{
	  if (b2n_square (z, z))	/* z**2 */
	    goto fail;
	  if (b2n_mod (z, z, ip))
	    goto fail;

	  if (b2n_square (w, w))	/* w**2 */
	    goto fail;
	  if (b2n_mod (w, w, ip))
	    goto fail;

	  if (b2n_mul (temp, w, b))	/* w**2 * b */
	    goto fail;
	  if (b2n_mod (temp, temp, ip))
	    goto fail;
	  if (b2n_add (z, z, temp))	/* z**2 + w**2 + b */
	    goto fail;

	  if (b2n_add (w, w, p))	/* w**2 + p */
	    goto fail;
	}
    }
  while (!b2n_cmp_null (w));

  B2N_SWAP (zo, z);

  b2n_clear (w);
  b2n_clear (p);
  b2n_clear (temp);
  b2n_clear (z);
  return 0;

 fail:
  b2n_clear (w);
  b2n_clear (p);
  b2n_clear (temp);
  b2n_clear (z);
  return -1;
}

/* Exponentiation modulo a polynomial.  */
int
b2n_exp_mod (b2n_ptr d, b2n_ptr b0, u_int32_t e, b2n_ptr p)
{
  b2n_t u, b;

  b2n_init (u);
  b2n_init (b);
  if (b2n_set_ui (u, 1))
    goto fail;
  if (b2n_mod (b, b0, p))
    goto fail;
  
  while (e)
    {
      if (e & 1)
	{
	  if (b2n_mul (u, u, b))
	    goto fail;
	  if (b2n_mod (u, u, p))
	    goto fail;
	}
      if (b2n_square (b, b))
	goto fail;
      if (b2n_mod (b, b, p))
	goto fail;
      e >>= 1;
    }

  B2N_SWAP (d, u);

  b2n_clear (u);
  b2n_clear (b);
  return 0;

 fail:
  b2n_clear (u);
  b2n_clear (b);
  return -1;
}

/*
 * Low-level function to speed up scalar multiplication with
 * elliptic curves.
 * Multiplies a normal number by 3.
 */ 

/* Normal addition behaves as Z_{2**n} and not F_{2**n}.  */
int
b2n_nadd (b2n_ptr d0, b2n_ptr a0, b2n_ptr b0)
{
  int i, carry;
  b2n_ptr a, b;
  b2n_t d;

  if (!b2n_cmp_null (a0))
    return b2n_set (d0, b0);

  if (!b2n_cmp_null (b0))
    return b2n_set (d0, a0);

  b2n_init (d);
  a = B2N_MAX (a0, b0);
  b = B2N_MIN (a0, b0);

  if (b2n_resize (d, a->chunks + 1))
    {
      b2n_clear (d);
      return -1;
    }

  for (carry = i = 0; i < b->chunks; i++)
    {
      d->limp[i] = a->limp[i] + b->limp[i] + carry;
      carry = (d->limp[i] < a->limp[i] ? 1 : 0);
    }
  
  for (; i < a->chunks && carry; i++)
    {
      d->limp[i] = a->limp[i] + carry;
      carry = (d->limp[i] < a->limp[i] ? 1 : 0);
    }

  if (i < a->chunks)
      memcpy (d->limp + i, a->limp + i, CHUNK_BYTES * (a->chunks - i));

  d->dirty = 1;
  B2N_SWAP (d0, d);

  b2n_clear (d);
  return 0;
}

/* Very special sub, a > b.  */
int
b2n_nsub (b2n_ptr d0, b2n_ptr a, b2n_ptr b)
{
  int i, carry;
  b2n_t d;

  if (b2n_cmp (a, b) <= 0)
    return b2n_set_null (d0);
  
  b2n_init (d);
  if (b2n_resize (d, a->chunks))
    {
      b2n_clear (d);
      return -1;
    }

  for (carry = i = 0; i < b->chunks; i++)
    {
      d->limp[i] = a->limp[i] - b->limp[i] - carry;
      carry = (d->limp[i] > a->limp[i] ? 1 : 0);
    }
  
  for (; i < a->chunks && carry; i++)
    {
      d->limp[i] = a->limp[i] - carry;
      carry = (d->limp[i] > a->limp[i] ? 1 : 0);
    }

  if (i < a->chunks)
      memcpy (d->limp + i, a->limp + i, CHUNK_BYTES*(a->chunks - i));

  d->dirty = 1;

  B2N_SWAP (d0, d);

  b2n_clear (d);
  return 0;
}

int
b2n_3mul (b2n_ptr d0, b2n_ptr e)
{
  b2n_t d;

  b2n_init (d);
  if (b2n_lshift (d, e, 1))
    goto fail;

  if (b2n_nadd (d0, d, e))
    goto fail;

  b2n_clear (d);
  return 0;

 fail:
  b2n_clear (d);
  return -1;
}