mpi.c

/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * SPDX-License-Identifier: Unlicense
 */

#include <stdarg.h>
#include "tommath_private.h"

/* Start: bn_fast_mp_invmod.c */

/* computes the modular inverse via binary extended euclidean algorithm,
 * that is c = 1/a mod b
 *
 * Based on slow invmod except this is optimized for the case where b is
 * odd as per HAC Note 14.64 on pp. 610
 */
int fast_mp_invmod(const mp_int *a, const mp_int *b, mp_int *c)
{
   mp_int  x, y, u, v, B, D;
   int     res, neg;

   /* 2. [modified] b must be odd   */
   if (mp_iseven(b) == MP_YES) {
      return MP_VAL;
   }

   /* init all our temps */
   if ((res = mp_init_multi(&x, &y, &u, &v, &B, &D, NULL)) != MP_OKAY) {
      return res;
   }

   /* x == modulus, y == value to invert */
   if ((res = mp_copy(b, &x)) != MP_OKAY) {
      goto LBL_ERR;
   }

   /* we need y = |a| */
   if ((res = mp_mod(a, b, &y)) != MP_OKAY) {
      goto LBL_ERR;
   }

   /* if one of x,y is zero return an error! */
   if ((mp_iszero(&x) == MP_YES) || (mp_iszero(&y) == MP_YES)) {
      res = MP_VAL;
      goto LBL_ERR;
   }

   /* 3. u=x, v=y, A=1, B=0, C=0,D=1 */
   if ((res = mp_copy(&x, &u)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_copy(&y, &v)) != MP_OKAY) {
      goto LBL_ERR;
   }
   mp_set(&D, 1uL);

top:
   /* 4.  while u is even do */
   while (mp_iseven(&u) == MP_YES) {
      /* 4.1 u = u/2 */
      if ((res = mp_div_2(&u, &u)) != MP_OKAY) {
         goto LBL_ERR;
      }
      /* 4.2 if B is odd then */
      if (mp_isodd(&B) == MP_YES) {
         if ((res = mp_sub(&B, &x, &B)) != MP_OKAY) {
            goto LBL_ERR;
         }
      }
      /* B = B/2 */
      if ((res = mp_div_2(&B, &B)) != MP_OKAY) {
         goto LBL_ERR;
      }
   }

   /* 5.  while v is even do */
   while (mp_iseven(&v) == MP_YES) {
      /* 5.1 v = v/2 */
      if ((res = mp_div_2(&v, &v)) != MP_OKAY) {
         goto LBL_ERR;
      }
      /* 5.2 if D is odd then */
      if (mp_isodd(&D) == MP_YES) {
         /* D = (D-x)/2 */
         if ((res = mp_sub(&D, &x, &D)) != MP_OKAY) {
            goto LBL_ERR;
         }
      }
      /* D = D/2 */
      if ((res = mp_div_2(&D, &D)) != MP_OKAY) {
         goto LBL_ERR;
      }
   }

   /* 6.  if u >= v then */
   if (mp_cmp(&u, &v) != MP_LT) {
      /* u = u - v, B = B - D */
      if ((res = mp_sub(&u, &v, &u)) != MP_OKAY) {
         goto LBL_ERR;
      }

      if ((res = mp_sub(&B, &D, &B)) != MP_OKAY) {
         goto LBL_ERR;
      }
   } else {
      /* v - v - u, D = D - B */
      if ((res = mp_sub(&v, &u, &v)) != MP_OKAY) {
         goto LBL_ERR;
      }

      if ((res = mp_sub(&D, &B, &D)) != MP_OKAY) {
         goto LBL_ERR;
      }
   }

   /* if not zero goto step 4 */
   if (mp_iszero(&u) == MP_NO) {
      goto top;
   }

   /* now a = C, b = D, gcd == g*v */

   /* if v != 1 then there is no inverse */
   if (mp_cmp_d(&v, 1uL) != MP_EQ) {
      res = MP_VAL;
      goto LBL_ERR;
   }

   /* b is now the inverse */
   neg = a->sign;
   while (D.sign == MP_NEG) {
      if ((res = mp_add(&D, b, &D)) != MP_OKAY) {
         goto LBL_ERR;
      }
   }

   /* too big */
   while (mp_cmp_mag(&D, b) != MP_LT) {
      if ((res = mp_sub(&D, b, &D)) != MP_OKAY) {
         goto LBL_ERR;
      }
   }

   mp_exch(&D, c);
   c->sign = neg;
   res = MP_OKAY;

LBL_ERR:
   mp_clear_multi(&x, &y, &u, &v, &B, &D, NULL);
   return res;
}

/* End: bn_fast_mp_invmod.c */

/* Start: bn_fast_mp_montgomery_reduce.c */

/* computes xR**-1 == x (mod N) via Montgomery Reduction
 *
 * This is an optimized implementation of montgomery_reduce
 * which uses the comba method to quickly calculate the columns of the
 * reduction.
 *
 * Based on Algorithm 14.32 on pp.601 of HAC.
*/
int fast_mp_montgomery_reduce(mp_int *x, const mp_int *n, mp_digit rho)
{
   int     ix, res, olduse;
   mp_word W[MP_WARRAY];

   if (x->used > (int)MP_WARRAY) {
      return MP_VAL;
   }

   /* get old used count */
   olduse = x->used;

   /* grow a as required */
   if (x->alloc < (n->used + 1)) {
      if ((res = mp_grow(x, n->used + 1)) != MP_OKAY) {
         return res;
      }
   }

   /* first we have to get the digits of the input into
    * an array of double precision words W[...]
    */
   {
      mp_word *_W;
      mp_digit *tmpx;

      /* alias for the W[] array */
      _W   = W;

      /* alias for the digits of  x*/
      tmpx = x->dp;

      /* copy the digits of a into W[0..a->used-1] */
      for (ix = 0; ix < x->used; ix++) {
         *_W++ = *tmpx++;
      }

      /* zero the high words of W[a->used..m->used*2] */
      for (; ix < ((n->used * 2) + 1); ix++) {
         *_W++ = 0;
      }
   }

   /* now we proceed to zero successive digits
    * from the least significant upwards
    */
   for (ix = 0; ix < n->used; ix++) {
      /* mu = ai * m' mod b
       *
       * We avoid a double precision multiplication (which isn't required)
       * by casting the value down to a mp_digit.  Note this requires
       * that W[ix-1] have  the carry cleared (see after the inner loop)
       */
      mp_digit mu;
      mu = ((W[ix] & MP_MASK) * rho) & MP_MASK;

      /* a = a + mu * m * b**i
       *
       * This is computed in place and on the fly.  The multiplication
       * by b**i is handled by offseting which columns the results
       * are added to.
       *
       * Note the comba method normally doesn't handle carries in the
       * inner loop In this case we fix the carry from the previous
       * column since the Montgomery reduction requires digits of the
       * result (so far) [see above] to work.  This is
       * handled by fixing up one carry after the inner loop.  The
       * carry fixups are done in order so after these loops the
       * first m->used words of W[] have the carries fixed
       */
      {
         int iy;
         mp_digit *tmpn;
         mp_word *_W;

         /* alias for the digits of the modulus */
         tmpn = n->dp;

         /* Alias for the columns set by an offset of ix */
         _W = W + ix;

         /* inner loop */
         for (iy = 0; iy < n->used; iy++) {
            *_W++ += (mp_word)mu * (mp_word)*tmpn++;
         }
      }

      /* now fix carry for next digit, W[ix+1] */
      W[ix + 1] += W[ix] >> (mp_word)DIGIT_BIT;
   }

   /* now we have to propagate the carries and
    * shift the words downward [all those least
    * significant digits we zeroed].
    */
   {
      mp_digit *tmpx;
      mp_word *_W, *_W1;

      /* nox fix rest of carries */

      /* alias for current word */
      _W1 = W + ix;

      /* alias for next word, where the carry goes */
      _W = W + ++ix;

      for (; ix <= ((n->used * 2) + 1); ix++) {
         *_W++ += *_W1++ >> (mp_word)DIGIT_BIT;
      }

      /* copy out, A = A/b**n
       *
       * The result is A/b**n but instead of converting from an
       * array of mp_word to mp_digit than calling mp_rshd
       * we just copy them in the right order
       */

      /* alias for destination word */
      tmpx = x->dp;

      /* alias for shifted double precision result */
      _W = W + n->used;

      for (ix = 0; ix < (n->used + 1); ix++) {
         *tmpx++ = *_W++ & (mp_word)MP_MASK;
      }

      /* zero oldused digits, if the input a was larger than
       * m->used+1 we'll have to clear the digits
       */
      for (; ix < olduse; ix++) {
         *tmpx++ = 0;
      }
   }

   /* set the max used and clamp */
   x->used = n->used + 1;
   mp_clamp(x);

   /* if A >= m then A = A - m */
   if (mp_cmp_mag(x, n) != MP_LT) {
      return s_mp_sub(x, n, x);
   }
   return MP_OKAY;
}

/* End: bn_fast_mp_montgomery_reduce.c */

/* Start: bn_fast_s_mp_mul_digs.c */

/* Fast (comba) multiplier
 *
 * This is the fast column-array [comba] multiplier.  It is
 * designed to compute the columns of the product first
 * then handle the carries afterwards.  This has the effect
 * of making the nested loops that compute the columns very
 * simple and schedulable on super-scalar processors.
 *
 * This has been modified to produce a variable number of
 * digits of output so if say only a half-product is required
 * you don't have to compute the upper half (a feature
 * required for fast Barrett reduction).
 *
 * Based on Algorithm 14.12 on pp.595 of HAC.
 *
 */
int fast_s_mp_mul_digs(const mp_int *a, const mp_int *b, mp_int *c, int digs)
{
   int     olduse, res, pa, ix, iz;
   mp_digit W[MP_WARRAY];
   mp_word  _W;

   /* grow the destination as required */
   if (c->alloc < digs) {
      if ((res = mp_grow(c, digs)) != MP_OKAY) {
         return res;
      }
   }

   /* number of output digits to produce */
   pa = MIN(digs, a->used + b->used);

   /* clear the carry */
   _W = 0;
   for (ix = 0; ix < pa; ix++) {
      int      tx, ty;
      int      iy;
      mp_digit *tmpx, *tmpy;

      /* get offsets into the two bignums */
      ty = MIN(b->used-1, ix);
      tx = ix - ty;

      /* setup temp aliases */
      tmpx = a->dp + tx;
      tmpy = b->dp + ty;

      /* this is the number of times the loop will iterrate, essentially
         while (tx++ < a->used && ty-- >= 0) { ... }
       */
      iy = MIN(a->used-tx, ty+1);

      /* execute loop */
      for (iz = 0; iz < iy; ++iz) {
         _W += (mp_word)*tmpx++ * (mp_word)*tmpy--;

      }

      /* store term */
      W[ix] = (mp_digit)_W & MP_MASK;

      /* make next carry */
      _W = _W >> (mp_word)DIGIT_BIT;
   }

   /* setup dest */
   olduse  = c->used;
   c->used = pa;

   {
      mp_digit *tmpc;
      tmpc = c->dp;
      for (ix = 0; ix < pa; ix++) {
         /* now extract the previous digit [below the carry] */
         *tmpc++ = W[ix];
      }

      /* clear unused digits [that existed in the old copy of c] */
      for (; ix < olduse; ix++) {
         *tmpc++ = 0;
      }
   }
   mp_clamp(c);
   return MP_OKAY;
}

/* End: bn_fast_s_mp_mul_digs.c */

/* Start: bn_fast_s_mp_mul_high_digs.c */

/* this is a modified version of fast_s_mul_digs that only produces
 * output digits *above* digs.  See the comments for fast_s_mul_digs
 * to see how it works.
 *
 * This is used in the Barrett reduction since for one of the multiplications
 * only the higher digits were needed.  This essentially halves the work.
 *
 * Based on Algorithm 14.12 on pp.595 of HAC.
 */
int fast_s_mp_mul_high_digs(const mp_int *a, const mp_int *b, mp_int *c, int digs)
{
   int     olduse, res, pa, ix, iz;
   mp_digit W[MP_WARRAY];
   mp_word  _W;

   /* grow the destination as required */
   pa = a->used + b->used;
   if (c->alloc < pa) {
      if ((res = mp_grow(c, pa)) != MP_OKAY) {
         return res;
      }
   }

   /* number of output digits to produce */
   pa = a->used + b->used;
   _W = 0;
   for (ix = digs; ix < pa; ix++) {
      int      tx, ty, iy;
      mp_digit *tmpx, *tmpy;

      /* get offsets into the two bignums */
      ty = MIN(b->used-1, ix);
      tx = ix - ty;

      /* setup temp aliases */
      tmpx = a->dp + tx;
      tmpy = b->dp + ty;

      /* this is the number of times the loop will iterrate, essentially its
         while (tx++ < a->used && ty-- >= 0) { ... }
       */
      iy = MIN(a->used-tx, ty+1);

      /* execute loop */
      for (iz = 0; iz < iy; iz++) {
         _W += (mp_word)*tmpx++ * (mp_word)*tmpy--;
      }

      /* store term */
      W[ix] = (mp_digit)_W & MP_MASK;

      /* make next carry */
      _W = _W >> (mp_word)DIGIT_BIT;
   }

   /* setup dest */
   olduse  = c->used;
   c->used = pa;

   {
      mp_digit *tmpc;

      tmpc = c->dp + digs;
      for (ix = digs; ix < pa; ix++) {
         /* now extract the previous digit [below the carry] */
         *tmpc++ = W[ix];
      }

      /* clear unused digits [that existed in the old copy of c] */
      for (; ix < olduse; ix++) {
         *tmpc++ = 0;
      }
   }
   mp_clamp(c);
   return MP_OKAY;
}

/* End: bn_fast_s_mp_mul_high_digs.c */

/* Start: bn_fast_s_mp_sqr.c */

/* the jist of squaring...
 * you do like mult except the offset of the tmpx [one that
 * starts closer to zero] can't equal the offset of tmpy.
 * So basically you set up iy like before then you min it with
 * (ty-tx) so that it never happens.  You double all those
 * you add in the inner loop

After that loop you do the squares and add them in.
*/

int fast_s_mp_sqr(const mp_int *a, mp_int *b)
{
   int       olduse, res, pa, ix, iz;
   mp_digit   W[MP_WARRAY], *tmpx;
   mp_word   W1;

   /* grow the destination as required */
   pa = a->used + a->used;
   if (b->alloc < pa) {
      if ((res = mp_grow(b, pa)) != MP_OKAY) {
         return res;
      }
   }

   /* number of output digits to produce */
   W1 = 0;
   for (ix = 0; ix < pa; ix++) {
      int      tx, ty, iy;
      mp_word  _W;
      mp_digit *tmpy;

      /* clear counter */
      _W = 0;

      /* get offsets into the two bignums */
      ty = MIN(a->used-1, ix);
      tx = ix - ty;

      /* setup temp aliases */
      tmpx = a->dp + tx;
      tmpy = a->dp + ty;

      /* this is the number of times the loop will iterrate, essentially
         while (tx++ < a->used && ty-- >= 0) { ... }
       */
      iy = MIN(a->used-tx, ty+1);

      /* now for squaring tx can never equal ty
       * we halve the distance since they approach at a rate of 2x
       * and we have to round because odd cases need to be executed
       */
      iy = MIN(iy, ((ty-tx)+1)>>1);

      /* execute loop */
      for (iz = 0; iz < iy; iz++) {
         _W += (mp_word)*tmpx++ * (mp_word)*tmpy--;
      }

      /* double the inner product and add carry */
      _W = _W + _W + W1;

      /* even columns have the square term in them */
      if (((unsigned)ix & 1u) == 0u) {
         _W += (mp_word)a->dp[ix>>1] * (mp_word)a->dp[ix>>1];
      }

      /* store it */
      W[ix] = _W & MP_MASK;

      /* make next carry */
      W1 = _W >> (mp_word)DIGIT_BIT;
   }

   /* setup dest */
   olduse  = b->used;
   b->used = a->used+a->used;

   {
      mp_digit *tmpb;
      tmpb = b->dp;
      for (ix = 0; ix < pa; ix++) {
         *tmpb++ = W[ix] & MP_MASK;
      }

      /* clear unused digits [that existed in the old copy of c] */
      for (; ix < olduse; ix++) {
         *tmpb++ = 0;
      }
   }
   mp_clamp(b);
   return MP_OKAY;
}

/* End: bn_fast_s_mp_sqr.c */

/* Start: bn_mp_2expt.c */

/* computes a = 2**b
 *
 * Simple algorithm which zeroes the int, grows it then just sets one bit
 * as required.
 */
int mp_2expt(mp_int *a, int b)
{
   int     res;

   /* zero a as per default */
   mp_zero(a);

   /* grow a to accomodate the single bit */
   if ((res = mp_grow(a, (b / DIGIT_BIT) + 1)) != MP_OKAY) {
      return res;
   }

   /* set the used count of where the bit will go */
   a->used = (b / DIGIT_BIT) + 1;

   /* put the single bit in its place */
   a->dp[b / DIGIT_BIT] = (mp_digit)1 << (mp_digit)(b % DIGIT_BIT);

   return MP_OKAY;
}

/* End: bn_mp_2expt.c */

/* Start: bn_mp_abs.c */

/* b = |a|
 *
 * Simple function copies the input and fixes the sign to positive
 */
int mp_abs(const mp_int *a, mp_int *b)
{
   int     res;

   /* copy a to b */
   if (a != b) {
      if ((res = mp_copy(a, b)) != MP_OKAY) {
         return res;
      }
   }

   /* force the sign of b to positive */
   b->sign = MP_ZPOS;

   return MP_OKAY;
}

/* End: bn_mp_abs.c */

/* Start: bn_mp_add.c */

/* high level addition (handles signs) */
int mp_add(const mp_int *a, const mp_int *b, mp_int *c)
{
   int     sa, sb, res;

   /* get sign of both inputs */
   sa = a->sign;
   sb = b->sign;

   /* handle two cases, not four */
   if (sa == sb) {
      /* both positive or both negative */
      /* add their magnitudes, copy the sign */
      c->sign = sa;
      res = s_mp_add(a, b, c);
   } else {
      /* one positive, the other negative */
      /* subtract the one with the greater magnitude from */
      /* the one of the lesser magnitude.  The result gets */
      /* the sign of the one with the greater magnitude. */
      if (mp_cmp_mag(a, b) == MP_LT) {
         c->sign = sb;
         res = s_mp_sub(b, a, c);
      } else {
         c->sign = sa;
         res = s_mp_sub(a, b, c);
      }
   }
   return res;
}

/* End: bn_mp_add.c */

/* Start: bn_mp_add_d.c */

/* single digit addition */
int mp_add_d(const mp_int *a, mp_digit b, mp_int *c)
{
   int     res, ix, oldused;
   mp_digit *tmpa, *tmpc, mu;

   /* grow c as required */
   if (c->alloc < (a->used + 1)) {
      if ((res = mp_grow(c, a->used + 1)) != MP_OKAY) {
         return res;
      }
   }

   /* if a is negative and |a| >= b, call c = |a| - b */
   if ((a->sign == MP_NEG) && ((a->used > 1) || (a->dp[0] >= b))) {
      mp_int a_ = *a;
      /* temporarily fix sign of a */
      a_.sign = MP_ZPOS;

      /* c = |a| - b */
      res = mp_sub_d(&a_, b, c);

      /* fix sign  */
      c->sign = MP_NEG;

      /* clamp */
      mp_clamp(c);

      return res;
   }

   /* old number of used digits in c */
   oldused = c->used;

   /* source alias */
   tmpa    = a->dp;

   /* destination alias */
   tmpc    = c->dp;

   /* if a is positive */
   if (a->sign == MP_ZPOS) {
      /* add digit, after this we're propagating
       * the carry.
       */
      *tmpc   = *tmpa++ + b;
      mu      = *tmpc >> DIGIT_BIT;
      *tmpc++ &= MP_MASK;

      /* now handle rest of the digits */
      for (ix = 1; ix < a->used; ix++) {
         *tmpc   = *tmpa++ + mu;
         mu      = *tmpc >> DIGIT_BIT;
         *tmpc++ &= MP_MASK;
      }
      /* set final carry */
      ix++;
      *tmpc++  = mu;

      /* setup size */
      c->used = a->used + 1;
   } else {
      /* a was negative and |a| < b */
      c->used  = 1;

      /* the result is a single digit */
      if (a->used == 1) {
         *tmpc++  =  b - a->dp[0];
      } else {
         *tmpc++  =  b;
      }

      /* setup count so the clearing of oldused
       * can fall through correctly
       */
      ix       = 1;
   }

   /* sign always positive */
   c->sign = MP_ZPOS;

   /* now zero to oldused */
   while (ix++ < oldused) {
      *tmpc++ = 0;
   }
   mp_clamp(c);

   return MP_OKAY;
}

/* End: bn_mp_add_d.c */

/* Start: bn_mp_addmod.c */

/* d = a + b (mod c) */
int mp_addmod(const mp_int *a, const mp_int *b, const mp_int *c, mp_int *d)
{
   int     res;
   mp_int  t;

   if ((res = mp_init(&t)) != MP_OKAY) {
      return res;
   }

   if ((res = mp_add(a, b, &t)) != MP_OKAY) {
      mp_clear(&t);
      return res;
   }
   res = mp_mod(&t, c, d);
   mp_clear(&t);
   return res;
}

/* End: bn_mp_addmod.c */

/* Start: bn_mp_and.c */

/* AND two ints together */
int mp_and(const mp_int *a, const mp_int *b, mp_int *c)
{
   int     res, ix, px;
   mp_int  t;
   const mp_int *x;

   if (a->used > b->used) {
      if ((res = mp_init_copy(&t, a)) != MP_OKAY) {
         return res;
      }
      px = b->used;
      x = b;
   } else {
      if ((res = mp_init_copy(&t, b)) != MP_OKAY) {
         return res;
      }
      px = a->used;
      x = a;
   }

   for (ix = 0; ix < px; ix++) {
      t.dp[ix] &= x->dp[ix];
   }

   /* zero digits above the last from the smallest mp_int */
   for (; ix < t.used; ix++) {
      t.dp[ix] = 0;
   }

   mp_clamp(&t);
   mp_exch(c, &t);
   mp_clear(&t);
   return MP_OKAY;
}

/* End: bn_mp_and.c */

/* Start: bn_mp_clamp.c */

/* trim unused digits
 *
 * This is used to ensure that leading zero digits are
 * trimed and the leading "used" digit will be non-zero
 * Typically very fast.  Also fixes the sign if there
 * are no more leading digits
 */
void mp_clamp(mp_int *a)
{
   /* decrease used while the most significant digit is
    * zero.
    */
   while ((a->used > 0) && (a->dp[a->used - 1] == 0u)) {
      --(a->used);
   }

   /* reset the sign flag if used == 0 */
   if (a->used == 0) {
      a->sign = MP_ZPOS;
   }
}

/* End: bn_mp_clamp.c */

/* Start: bn_mp_clear.c */

/* clear one (frees)  */
void mp_clear(mp_int *a)
{
   int i;

   /* only do anything if a hasn't been freed previously */
   if (a->dp != NULL) {
      /* first zero the digits */
      for (i = 0; i < a->used; i++) {
         a->dp[i] = 0;
      }

      /* free ram */
      XFREE(a->dp);

      /* reset members to make debugging easier */
      a->dp    = NULL;
      a->alloc = a->used = 0;
      a->sign  = MP_ZPOS;
   }
}

/* End: bn_mp_clear.c */

/* Start: bn_mp_clear_multi.c */

void mp_clear_multi(mp_int *mp, ...)
{
   mp_int *next_mp = mp;
   va_list args;
   va_start(args, mp);
   while (next_mp != NULL) {
      mp_clear(next_mp);
      next_mp = va_arg(args, mp_int *);
   }
   va_end(args);
}

/* End: bn_mp_clear_multi.c */

/* Start: bn_mp_cmp.c */

/* compare two ints (signed)*/
int mp_cmp(const mp_int *a, const mp_int *b)
{
   /* compare based on sign */
   if (a->sign != b->sign) {
      if (a->sign == MP_NEG) {
         return MP_LT;
      } else {
         return MP_GT;
      }
   }

   /* compare digits */
   if (a->sign == MP_NEG) {
      /* if negative compare opposite direction */
      return mp_cmp_mag(b, a);
   } else {
      return mp_cmp_mag(a, b);
   }
}

/* End: bn_mp_cmp.c */

/* Start: bn_mp_cmp_d.c */

/* compare a digit */
int mp_cmp_d(const mp_int *a, mp_digit b)
{
   /* compare based on sign */
   if (a->sign == MP_NEG) {
      return MP_LT;
   }

   /* compare based on magnitude */
   if (a->used > 1) {
      return MP_GT;
   }

   /* compare the only digit of a to b */
   if (a->dp[0] > b) {
      return MP_GT;
   } else if (a->dp[0] < b) {
      return MP_LT;
   } else {
      return MP_EQ;
   }
}

/* End: bn_mp_cmp_d.c */

/* Start: bn_mp_cmp_mag.c */

/* compare maginitude of two ints (unsigned) */
int mp_cmp_mag(const mp_int *a, const mp_int *b)
{
   int     n;
   mp_digit *tmpa, *tmpb;

   /* compare based on # of non-zero digits */
   if (a->used > b->used) {
      return MP_GT;
   }

   if (a->used < b->used) {
      return MP_LT;
   }

   /* alias for a */
   tmpa = a->dp + (a->used - 1);

   /* alias for b */
   tmpb = b->dp + (a->used - 1);

   /* compare based on digits  */
   for (n = 0; n < a->used; ++n, --tmpa, --tmpb) {
      if (*tmpa > *tmpb) {
         return MP_GT;
      }

      if (*tmpa < *tmpb) {
         return MP_LT;
      }
   }
   return MP_EQ;
}

/* End: bn_mp_cmp_mag.c */

/* Start: bn_mp_cnt_lsb.c */

static const int lnz[16] = {
   4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0
};

/* Counts the number of lsbs which are zero before the first zero bit */
int mp_cnt_lsb(const mp_int *a)
{
   int x;
   mp_digit q, qq;

   /* easy out */
   if (mp_iszero(a) == MP_YES) {
      return 0;
   }

   /* scan lower digits until non-zero */
   for (x = 0; (x < a->used) && (a->dp[x] == 0u); x++) {}
   q = a->dp[x];
   x *= DIGIT_BIT;

   /* now scan this digit until a 1 is found */
   if ((q & 1u) == 0u) {
      do {
         qq  = q & 15u;
         x  += lnz[qq];
         q >>= 4;
      } while (qq == 0u);
   }
   return x;
}

/* End: bn_mp_cnt_lsb.c */

/* Start: bn_mp_complement.c */

/* b = ~a */
int mp_complement(const mp_int *a, mp_int *b)
{
   int res = mp_neg(a, b);
   return (res == MP_OKAY) ? mp_sub_d(b, 1uL, b) : res;
}

/* End: bn_mp_complement.c */

/* Start: bn_mp_copy.c */

/* copy, b = a */
int mp_copy(const mp_int *a, mp_int *b)
{
   int     res, n;

   /* if dst == src do nothing */
   if (a == b) {
      return MP_OKAY;
   }

   /* grow dest */
   if (b->alloc < a->used) {
      if ((res = mp_grow(b, a->used)) != MP_OKAY) {
         return res;
      }
   }

   /* zero b and copy the parameters over */
   {
      mp_digit *tmpa, *tmpb;

      /* pointer aliases */

      /* source */
      tmpa = a->dp;

      /* destination */
      tmpb = b->dp;

      /* copy all the digits */
      for (n = 0; n < a->used; n++) {
         *tmpb++ = *tmpa++;
      }

      /* clear high digits */
      for (; n < b->used; n++) {
         *tmpb++ = 0;
      }
   }

   /* copy used count and sign */
   b->used = a->used;
   b->sign = a->sign;
   return MP_OKAY;
}

/* End: bn_mp_copy.c */

/* Start: bn_mp_count_bits.c */

/* returns the number of bits in an int */
int mp_count_bits(const mp_int *a)
{
   int     r;
   mp_digit q;

   /* shortcut */
   if (a->used == 0) {
      return 0;
   }

   /* get number of digits and add that */
   r = (a->used - 1) * DIGIT_BIT;

   /* take the last digit and count the bits in it */
   q = a->dp[a->used - 1];
   while (q > (mp_digit)0) {
      ++r;
      q >>= (mp_digit)1;
   }
   return r;
}

/* End: bn_mp_count_bits.c */

/* Start: bn_mp_div.c */

/* integer signed division.
 * c*b + d == a [e.g. a/b, c=quotient, d=remainder]
 * HAC pp.598 Algorithm 14.20
 *
 * Note that the description in HAC is horribly
 * incomplete.  For example, it doesn't consider
 * the case where digits are removed from 'x' in
 * the inner loop.  It also doesn't consider the
 * case that y has fewer than three digits, etc..
 *
 * The overall algorithm is as described as
 * 14.20 from HAC but fixed to treat these cases.
*/
int mp_div(const mp_int *a, const mp_int *b, mp_int *c, mp_int *d)
{
   mp_int  q, x, y, t1, t2;
   int     res, n, t, i, norm, neg;

   /* is divisor zero ? */
   if (mp_iszero(b) == MP_YES) {
      return MP_VAL;
   }

   /* if a < b then q=0, r = a */
   if (mp_cmp_mag(a, b) == MP_LT) {
      if (d != NULL) {
         res = mp_copy(a, d);
      } else {
         res = MP_OKAY;
      }
      if (c != NULL) {
         mp_zero(c);
      }
      return res;
   }

   if ((res = mp_init_size(&q, a->used + 2)) != MP_OKAY) {
      return res;
   }
   q.used = a->used + 2;

   if ((res = mp_init(&t1)) != MP_OKAY) {
      goto LBL_Q;
   }

   if ((res = mp_init(&t2)) != MP_OKAY) {
      goto LBL_T1;
   }

   if ((res = mp_init_copy(&x, a)) != MP_OKAY) {
      goto LBL_T2;
   }

   if ((res = mp_init_copy(&y, b)) != MP_OKAY) {
      goto LBL_X;
   }

   /* fix the sign */
   neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG;
   x.sign = y.sign = MP_ZPOS;

   /* normalize both x and y, ensure that y >= b/2, [b == 2**DIGIT_BIT] */
   norm = mp_count_bits(&y) % DIGIT_BIT;
   if (norm < (DIGIT_BIT - 1)) {
      norm = (DIGIT_BIT - 1) - norm;
      if ((res = mp_mul_2d(&x, norm, &x)) != MP_OKAY) {
         goto LBL_Y;
      }
      if ((res = mp_mul_2d(&y, norm, &y)) != MP_OKAY) {
         goto LBL_Y;
      }
   } else {
      norm = 0;
   }

   /* note hac does 0 based, so if used==5 then its 0,1,2,3,4, e.g. use 4 */
   n = x.used - 1;
   t = y.used - 1;

   /* while (x >= y*b**n-t) do { q[n-t] += 1; x -= y*b**{n-t} } */
   if ((res = mp_lshd(&y, n - t)) != MP_OKAY) { /* y = y*b**{n-t} */
      goto LBL_Y;
   }

   while (mp_cmp(&x, &y) != MP_LT) {
      ++(q.dp[n - t]);
      if ((res = mp_sub(&x, &y, &x)) != MP_OKAY) {
         goto LBL_Y;
      }
   }

   /* reset y by shifting it back down */
   mp_rshd(&y, n - t);

   /* step 3. for i from n down to (t + 1) */
   for (i = n; i >= (t + 1); i--) {
      if (i > x.used) {
         continue;
      }

      /* step 3.1 if xi == yt then set q{i-t-1} to b-1,
       * otherwise set q{i-t-1} to (xi*b + x{i-1})/yt */
      if (x.dp[i] == y.dp[t]) {
         q.dp[(i - t) - 1] = ((mp_digit)1 << (mp_digit)DIGIT_BIT) - (mp_digit)1;
      } else {
         mp_word tmp;
         tmp = (mp_word)x.dp[i] << (mp_word)DIGIT_BIT;
         tmp |= (mp_word)x.dp[i - 1];
         tmp /= (mp_word)y.dp[t];
         if (tmp > (mp_word)MP_MASK) {
            tmp = MP_MASK;
         }
         q.dp[(i - t) - 1] = (mp_digit)(tmp & (mp_word)MP_MASK);
      }

      /* while (q{i-t-1} * (yt * b + y{t-1})) >
               xi * b**2 + xi-1 * b + xi-2

         do q{i-t-1} -= 1;
      */
      q.dp[(i - t) - 1] = (q.dp[(i - t) - 1] + 1uL) & (mp_digit)MP_MASK;
      do {
         q.dp[(i - t) - 1] = (q.dp[(i - t) - 1] - 1uL) & (mp_digit)MP_MASK;

         /* find left hand */
         mp_zero(&t1);
         t1.dp[0] = ((t - 1) < 0) ? 0u : y.dp[t - 1];
         t1.dp[1] = y.dp[t];
         t1.used = 2;
         if ((res = mp_mul_d(&t1, q.dp[(i - t) - 1], &t1)) != MP_OKAY) {
            goto LBL_Y;
         }

         /* find right hand */
         t2.dp[0] = ((i - 2) < 0) ? 0u : x.dp[i - 2];
         t2.dp[1] = ((i - 1) < 0) ? 0u : x.dp[i - 1];
         t2.dp[2] = x.dp[i];
         t2.used = 3;
      } while (mp_cmp_mag(&t1, &t2) == MP_GT);

      /* step 3.3 x = x - q{i-t-1} * y * b**{i-t-1} */
      if ((res = mp_mul_d(&y, q.dp[(i - t) - 1], &t1)) != MP_OKAY) {
         goto LBL_Y;
      }

      if ((res = mp_lshd(&t1, (i - t) - 1)) != MP_OKAY) {
         goto LBL_Y;
      }

      if ((res = mp_sub(&x, &t1, &x)) != MP_OKAY) {
         goto LBL_Y;
      }

      /* if x < 0 then { x = x + y*b**{i-t-1}; q{i-t-1} -= 1; } */
      if (x.sign == MP_NEG) {
         if ((res = mp_copy(&y, &t1)) != MP_OKAY) {
            goto LBL_Y;
         }
         if ((res = mp_lshd(&t1, (i - t) - 1)) != MP_OKAY) {
            goto LBL_Y;
         }
         if ((res = mp_add(&x, &t1, &x)) != MP_OKAY) {
            goto LBL_Y;
         }

         q.dp[(i - t) - 1] = (q.dp[(i - t) - 1] - 1uL) & MP_MASK;
      }
   }

   /* now q is the quotient and x is the remainder
    * [which we have to normalize]
    */

   /* get sign before writing to c */
   x.sign = (x.used == 0) ? MP_ZPOS : a->sign;

   if (c != NULL) {
      mp_clamp(&q);
      mp_exch(&q, c);
      c->sign = neg;
   }

   if (d != NULL) {
      if ((res = mp_div_2d(&x, norm, &x, NULL)) != MP_OKAY) {
         goto LBL_Y;
      }
      mp_exch(&x, d);
   }

   res = MP_OKAY;

LBL_Y:
   mp_clear(&y);
LBL_X:
   mp_clear(&x);
LBL_T2:
   mp_clear(&t2);
LBL_T1:
   mp_clear(&t1);
LBL_Q:
   mp_clear(&q);
   return res;
}

/* End: bn_mp_div.c */

/* Start: bn_mp_div_2.c */

/* b = a/2 */
int mp_div_2(const mp_int *a, mp_int *b)
{
   int     x, res, oldused;

   /* copy */
   if (b->alloc < a->used) {
      if ((res = mp_grow(b, a->used)) != MP_OKAY) {
         return res;
      }
   }

   oldused = b->used;
   b->used = a->used;
   {
      mp_digit r, rr, *tmpa, *tmpb;

      /* source alias */
      tmpa = a->dp + b->used - 1;

      /* dest alias */
      tmpb = b->dp + b->used - 1;

      /* carry */
      r = 0;
      for (x = b->used - 1; x >= 0; x--) {
         /* get the carry for the next iteration */
         rr = *tmpa & 1u;

         /* shift the current digit, add in carry and store */
         *tmpb-- = (*tmpa-- >> 1) | (r << (DIGIT_BIT - 1));

         /* forward carry to next iteration */
         r = rr;
      }

      /* zero excess digits */
      tmpb = b->dp + b->used;
      for (x = b->used; x < oldused; x++) {
         *tmpb++ = 0;
      }
   }
   b->sign = a->sign;
   mp_clamp(b);
   return MP_OKAY;
}

/* End: bn_mp_div_2.c */

/* Start: bn_mp_div_2d.c */

/* shift right by a certain bit count (store quotient in c, optional remainder in d) */
int mp_div_2d(const mp_int *a, int b, mp_int *c, mp_int *d)
{
   mp_digit D, r, rr;
   int     x, res;

   /* if the shift count is <= 0 then we do no work */
   if (b <= 0) {
      res = mp_copy(a, c);
      if (d != NULL) {
         mp_zero(d);
      }
      return res;
   }

   /* copy */
   if ((res = mp_copy(a, c)) != MP_OKAY) {
      return res;
   }
   /* 'a' should not be used after here - it might be the same as d */

   /* get the remainder */
   if (d != NULL) {
      if ((res = mp_mod_2d(a, b, d)) != MP_OKAY) {
         return res;
      }
   }

   /* shift by as many digits in the bit count */
   if (b >= DIGIT_BIT) {
      mp_rshd(c, b / DIGIT_BIT);
   }

   /* shift any bit count < DIGIT_BIT */
   D = (mp_digit)(b % DIGIT_BIT);
   if (D != 0u) {
      mp_digit *tmpc, mask, shift;

      /* mask */
      mask = ((mp_digit)1 << D) - 1uL;

      /* shift for lsb */
      shift = (mp_digit)DIGIT_BIT - D;

      /* alias */
      tmpc = c->dp + (c->used - 1);

      /* carry */
      r = 0;
      for (x = c->used - 1; x >= 0; x--) {
         /* get the lower  bits of this word in a temp */
         rr = *tmpc & mask;

         /* shift the current word and mix in the carry bits from the previous word */
         *tmpc = (*tmpc >> D) | (r << shift);
         --tmpc;

         /* set the carry to the carry bits of the current word found above */
         r = rr;
      }
   }
   mp_clamp(c);
   return MP_OKAY;
}

/* End: bn_mp_div_2d.c */

/* Start: bn_mp_div_3.c */

/* divide by three (based on routine from MPI and the GMP manual) */
int mp_div_3(const mp_int *a, mp_int *c, mp_digit *d)
{
   mp_int   q;
   mp_word  w, t;
   mp_digit b;
   int      res, ix;

   /* b = 2**DIGIT_BIT / 3 */
   b = ((mp_word)1 << (mp_word)DIGIT_BIT) / (mp_word)3;

   if ((res = mp_init_size(&q, a->used)) != MP_OKAY) {
      return res;
   }

   q.used = a->used;
   q.sign = a->sign;
   w = 0;
   for (ix = a->used - 1; ix >= 0; ix--) {
      w = (w << (mp_word)DIGIT_BIT) | (mp_word)a->dp[ix];

      if (w >= 3u) {
         /* multiply w by [1/3] */
         t = (w * (mp_word)b) >> (mp_word)DIGIT_BIT;

         /* now subtract 3 * [w/3] from w, to get the remainder */
         w -= t+t+t;

         /* fixup the remainder as required since
          * the optimization is not exact.
          */
         while (w >= 3u) {
            t += 1u;
            w -= 3u;
         }
      } else {
         t = 0;
      }
      q.dp[ix] = (mp_digit)t;
   }

   /* [optional] store the remainder */
   if (d != NULL) {
      *d = (mp_digit)w;
   }

   /* [optional] store the quotient */
   if (c != NULL) {
      mp_clamp(&q);
      mp_exch(&q, c);
   }
   mp_clear(&q);

   return res;
}

/* End: bn_mp_div_3.c */

/* Start: bn_mp_div_d.c */

static int s_is_power_of_two(mp_digit b, int *p)
{
   int x;

   /* fast return if no power of two */
   if ((b == 0u) || ((b & (b-1u)) != 0u)) {
      return 0;
   }

   for (x = 0; x < DIGIT_BIT; x++) {
      if (b == ((mp_digit)1<<(mp_digit)x)) {
         *p = x;
         return 1;
      }
   }
   return 0;
}

/* single digit division (based on routine from MPI) */
int mp_div_d(const mp_int *a, mp_digit b, mp_int *c, mp_digit *d)
{
   mp_int  q;
   mp_word w;
   mp_digit t;
   int     res, ix;

   /* cannot divide by zero */
   if (b == 0u) {
      return MP_VAL;
   }

   /* quick outs */
   if ((b == 1u) || (mp_iszero(a) == MP_YES)) {
      if (d != NULL) {
         *d = 0;
      }
      if (c != NULL) {
         return mp_copy(a, c);
      }
      return MP_OKAY;
   }

   /* power of two ? */
   if (s_is_power_of_two(b, &ix) == 1) {
      if (d != NULL) {
         *d = a->dp[0] & (((mp_digit)1<<(mp_digit)ix) - 1uL);
      }
      if (c != NULL) {
         return mp_div_2d(a, ix, c, NULL);
      }
      return MP_OKAY;
   }

   /* three? */
   if (b == 3u) {
      return mp_div_3(a, c, d);
   }

   /* no easy answer [c'est la vie].  Just division */
   if ((res = mp_init_size(&q, a->used)) != MP_OKAY) {
      return res;
   }

   q.used = a->used;
   q.sign = a->sign;
   w = 0;
   for (ix = a->used - 1; ix >= 0; ix--) {
      w = (w << (mp_word)DIGIT_BIT) | (mp_word)a->dp[ix];

      if (w >= b) {
         t = (mp_digit)(w / b);
         w -= (mp_word)t * (mp_word)b;
      } else {
         t = 0;
      }
      q.dp[ix] = t;
   }

   if (d != NULL) {
      *d = (mp_digit)w;
   }

   if (c != NULL) {
      mp_clamp(&q);
      mp_exch(&q, c);
   }
   mp_clear(&q);

   return res;
}

/* End: bn_mp_div_d.c */

/* Start: bn_mp_dr_is_modulus.c */

/* determines if a number is a valid DR modulus */
int mp_dr_is_modulus(const mp_int *a)
{
   int ix;

   /* must be at least two digits */
   if (a->used < 2) {
      return 0;
   }

   /* must be of the form b**k - a [a <= b] so all
    * but the first digit must be equal to -1 (mod b).
    */
   for (ix = 1; ix < a->used; ix++) {
      if (a->dp[ix] != MP_MASK) {
         return 0;
      }
   }
   return 1;
}

/* End: bn_mp_dr_is_modulus.c */

/* Start: bn_mp_dr_reduce.c */

/* reduce "x" in place modulo "n" using the Diminished Radix algorithm.
 *
 * Based on algorithm from the paper
 *
 * "Generating Efficient Primes for Discrete Log Cryptosystems"
 *                 Chae Hoon Lim, Pil Joong Lee,
 *          POSTECH Information Research Laboratories
 *
 * The modulus must be of a special format [see manual]
 *
 * Has been modified to use algorithm 7.10 from the LTM book instead
 *
 * Input x must be in the range 0 <= x <= (n-1)**2
 */
int mp_dr_reduce(mp_int *x, const mp_int *n, mp_digit k)
{
   int      err, i, m;
   mp_word  r;
   mp_digit mu, *tmpx1, *tmpx2;

   /* m = digits in modulus */
   m = n->used;

   /* ensure that "x" has at least 2m digits */
   if (x->alloc < (m + m)) {
      if ((err = mp_grow(x, m + m)) != MP_OKAY) {
         return err;
      }
   }

   /* top of loop, this is where the code resumes if
    * another reduction pass is required.
    */
top:
   /* aliases for digits */
   /* alias for lower half of x */
   tmpx1 = x->dp;

   /* alias for upper half of x, or x/B**m */
   tmpx2 = x->dp + m;

   /* set carry to zero */
   mu = 0;

   /* compute (x mod B**m) + k * [x/B**m] inline and inplace */
   for (i = 0; i < m; i++) {
      r         = ((mp_word)*tmpx2++ * (mp_word)k) + *tmpx1 + mu;
      *tmpx1++  = (mp_digit)(r & MP_MASK);
      mu        = (mp_digit)(r >> ((mp_word)DIGIT_BIT));
   }

   /* set final carry */
   *tmpx1++ = mu;

   /* zero words above m */
   for (i = m + 1; i < x->used; i++) {
      *tmpx1++ = 0;
   }

   /* clamp, sub and return */
   mp_clamp(x);

   /* if x >= n then subtract and reduce again
    * Each successive "recursion" makes the input smaller and smaller.
    */
   if (mp_cmp_mag(x, n) != MP_LT) {
      if ((err = s_mp_sub(x, n, x)) != MP_OKAY) {
         return err;
      }
      goto top;
   }
   return MP_OKAY;
}

/* End: bn_mp_dr_reduce.c */

/* Start: bn_mp_dr_setup.c */
#include "tommath_private.h"

/* determines the setup value */
void mp_dr_setup(const mp_int *a, mp_digit *d)
{
   /* the casts are required if DIGIT_BIT is one less than
    * the number of bits in a mp_digit [e.g. DIGIT_BIT==31]
    */
   *d = (mp_digit)(((mp_word)1 << (mp_word)DIGIT_BIT) - (mp_word)a->dp[0]);
}

/* End: bn_mp_dr_setup.c */

/* Start: bn_mp_exch.c */

/* swap the elements of two integers, for cases where you can't simply swap the
 * mp_int pointers around
 */
void mp_exch(mp_int *a, mp_int *b)
{
   mp_int  t;

   t  = *a;
   *a = *b;
   *b = t;
}

/* End: bn_mp_exch.c */

/* Start: bn_mp_export.c */

/* based on gmp's mpz_export.
 * see http://gmplib.org/manual/Integer-Import-and-Export.html
 */
int mp_export(void *rop, size_t *countp, int order, size_t size,
              int endian, size_t nails, const mp_int *op)
{
   int result;
   size_t odd_nails, nail_bytes, i, j, bits, count;
   unsigned char odd_nail_mask;

   mp_int t;

   if ((result = mp_init_copy(&t, op)) != MP_OKAY) {
      return result;
   }

   if (endian == 0) {
      union {
         unsigned int i;
         char c[4];
      } lint;
      lint.i = 0x01020304;

      endian = (lint.c[0] == '\x04') ? -1 : 1;
   }

   odd_nails = (nails % 8u);
   odd_nail_mask = 0xff;
   for (i = 0; i < odd_nails; ++i) {
      odd_nail_mask ^= (unsigned char)(1u << (7u - i));
   }
   nail_bytes = nails / 8u;

   bits = (size_t)mp_count_bits(&t);
   count = (bits / ((size * 8u) - nails)) + (((bits % ((size * 8u) - nails)) != 0u) ? 1u : 0u);

   for (i = 0; i < count; ++i) {
      for (j = 0; j < size; ++j) {
         unsigned char *byte = (unsigned char *)rop +
                               (((order == -1) ? i : ((count - 1u) - i)) * size) +
                               ((endian == -1) ? j : ((size - 1u) - j));

         if (j >= (size - nail_bytes)) {
            *byte = 0;
            continue;
         }

         *byte = (unsigned char)((j == ((size - nail_bytes) - 1u)) ? (t.dp[0] & odd_nail_mask) : (t.dp[0] & 0xFFuL));

         if ((result = mp_div_2d(&t, (j == ((size - nail_bytes) - 1u)) ? (int)(8u - odd_nails) : 8, &t, NULL)) != MP_OKAY) {
            mp_clear(&t);
            return result;
         }
      }
   }

   mp_clear(&t);

   if (countp != NULL) {
      *countp = count;
   }

   return MP_OKAY;
}

/* End: bn_mp_export.c */

/* Start: bn_mp_expt_d.c */

/* wrapper function for mp_expt_d_ex() */
int mp_expt_d(const mp_int *a, mp_digit b, mp_int *c)
{
   return mp_expt_d_ex(a, b, c, 0);
}

/* End: bn_mp_expt_d.c */

/* Start: bn_mp_expt_d_ex.c */

/* calculate c = a**b  using a square-multiply algorithm */
int mp_expt_d_ex(const mp_int *a, mp_digit b, mp_int *c, int fast)
{
   int     res;
   unsigned int x;

   mp_int  g;

   if ((res = mp_init_copy(&g, a)) != MP_OKAY) {
      return res;
   }

   /* set initial result */
   mp_set(c, 1uL);

   if (fast != 0) {
      while (b > 0u) {
         /* if the bit is set multiply */
         if ((b & 1u) != 0u) {
            if ((res = mp_mul(c, &g, c)) != MP_OKAY) {
               mp_clear(&g);
               return res;
            }
         }

         /* square */
         if (b > 1u) {
            if ((res = mp_sqr(&g, &g)) != MP_OKAY) {
               mp_clear(&g);
               return res;
            }
         }

         /* shift to next bit */
         b >>= 1;
      }
   } else {
      for (x = 0; x < (unsigned)DIGIT_BIT; x++) {
         /* square */
         if ((res = mp_sqr(c, c)) != MP_OKAY) {
            mp_clear(&g);
            return res;
         }

         /* if the bit is set multiply */
         if ((b & ((mp_digit)1 << (DIGIT_BIT - 1))) != 0u) {
            if ((res = mp_mul(c, &g, c)) != MP_OKAY) {
               mp_clear(&g);
               return res;
            }
         }

         /* shift to next bit */
         b <<= 1;
      }
   } /* if ... else */

   mp_clear(&g);
   return MP_OKAY;
}

/* End: bn_mp_expt_d_ex.c */

/* Start: bn_mp_exptmod.c */

/* this is a shell function that calls either the normal or Montgomery
 * exptmod functions.  Originally the call to the montgomery code was
 * embedded in the normal function but that wasted alot of stack space
 * for nothing (since 99% of the time the Montgomery code would be called)
 */
int mp_exptmod(const mp_int *G, const mp_int *X, const mp_int *P, mp_int *Y)
{
   int dr;

   /* modulus P must be positive */
   if (P->sign == MP_NEG) {
      return MP_VAL;
   }

   /* if exponent X is negative we have to recurse */
   if (X->sign == MP_NEG) {
      mp_int tmpG, tmpX;
      int err;

      /* first compute 1/G mod P */
      if ((err = mp_init(&tmpG)) != MP_OKAY) {
         return err;
      }
      if ((err = mp_invmod(G, P, &tmpG)) != MP_OKAY) {
         mp_clear(&tmpG);
         return err;
      }

      /* now get |X| */
      if ((err = mp_init(&tmpX)) != MP_OKAY) {
         mp_clear(&tmpG);
         return err;
      }
      if ((err = mp_abs(X, &tmpX)) != MP_OKAY) {
         mp_clear_multi(&tmpG, &tmpX, NULL);
         return err;
      }

      /* and now compute (1/G)**|X| instead of G**X [X < 0] */
      err = mp_exptmod(&tmpG, &tmpX, P, Y);
      mp_clear_multi(&tmpG, &tmpX, NULL);
      return err;
   }

   /* modified diminished radix reduction */
   if (mp_reduce_is_2k_l(P) == MP_YES) {
      return s_mp_exptmod(G, X, P, Y, 1);
   }

   /* is it a DR modulus? */
   dr = mp_dr_is_modulus(P);

   /* if not, is it a unrestricted DR modulus? */
   if (dr == 0) {
      dr = mp_reduce_is_2k(P) << 1;
   }

   /* if the modulus is odd or dr != 0 use the montgomery method */
   if ((mp_isodd(P) == MP_YES) || (dr !=  0)) {
      return mp_exptmod_fast(G, X, P, Y, dr);
   } else {
      /* otherwise use the generic Barrett reduction technique */
      return s_mp_exptmod(G, X, P, Y, 0);
   }
}

/* End: bn_mp_exptmod.c */

/* Start: bn_mp_exptmod_fast.c */

/* computes Y == G**X mod P, HAC pp.616, Algorithm 14.85
 *
 * Uses a left-to-right k-ary sliding window to compute the modular exponentiation.
 * The value of k changes based on the size of the exponent.
 *
 * Uses Montgomery or Diminished Radix reduction [whichever appropriate]
 */

#ifdef MP_LOW_MEM
#   define TAB_SIZE 32
#else
#   define TAB_SIZE 256
#endif

int mp_exptmod_fast(const mp_int *G, const mp_int *X, const mp_int *P, mp_int *Y, int redmode)
{
   mp_int  M[TAB_SIZE], res;
   mp_digit buf, mp;
   int     err, bitbuf, bitcpy, bitcnt, mode, digidx, x, y, winsize;

   /* use a pointer to the reduction algorithm.  This allows us to use
    * one of many reduction algorithms without modding the guts of
    * the code with if statements everywhere.
    */
   int (*redux)(mp_int *x, const mp_int *n, mp_digit rho);

   /* find window size */
   x = mp_count_bits(X);
   if (x <= 7) {
      winsize = 2;
   } else if (x <= 36) {
      winsize = 3;
   } else if (x <= 140) {
      winsize = 4;
   } else if (x <= 450) {
      winsize = 5;
   } else if (x <= 1303) {
      winsize = 6;
   } else if (x <= 3529) {
      winsize = 7;
   } else {
      winsize = 8;
   }

#ifdef MP_LOW_MEM
   if (winsize > 5) {
      winsize = 5;
   }
#endif

   /* init M array */
   /* init first cell */
   if ((err = mp_init_size(&M[1], P->alloc)) != MP_OKAY) {
      return err;
   }

   /* now init the second half of the array */
   for (x = 1<<(winsize-1); x < (1 << winsize); x++) {
      if ((err = mp_init_size(&M[x], P->alloc)) != MP_OKAY) {
         for (y = 1<<(winsize-1); y < x; y++) {
            mp_clear(&M[y]);
         }
         mp_clear(&M[1]);
         return err;
      }
   }

   /* determine and setup reduction code */
   if (redmode == 0) {
      /* now setup montgomery  */
      if ((err = mp_montgomery_setup(P, &mp)) != MP_OKAY) {
         goto LBL_M;
      }

      /* automatically pick the comba one if available (saves quite a few calls/ifs) */
      if ((((P->used * 2) + 1) < (int)MP_WARRAY) &&
          (P->used < (1 << ((CHAR_BIT * sizeof(mp_word)) - (2 * DIGIT_BIT))))) {
         redux = fast_mp_montgomery_reduce;
      } else
      {
         /* use slower baseline Montgomery method */
         redux = mp_montgomery_reduce;
      }
   } else if (redmode == 1) {
      /* setup DR reduction for moduli of the form B**k - b */
      mp_dr_setup(P, &mp);
      redux = mp_dr_reduce;
   } else {
      /* setup DR reduction for moduli of the form 2**k - b */
      if ((err = mp_reduce_2k_setup(P, &mp)) != MP_OKAY) {
         goto LBL_M;
      }
      redux = mp_reduce_2k;
   }

   /* setup result */
   if ((err = mp_init_size(&res, P->alloc)) != MP_OKAY) {
      goto LBL_M;
   }

   /* create M table
    *

    *
    * The first half of the table is not computed though accept for M[0] and M[1]
    */

   if (redmode == 0) {
      /* now we need R mod m */
      if ((err = mp_montgomery_calc_normalization(&res, P)) != MP_OKAY) {
         goto LBL_RES;
      }

      /* now set M[1] to G * R mod m */
      if ((err = mp_mulmod(G, &res, P, &M[1])) != MP_OKAY) {
         goto LBL_RES;
      }
   } else {
      mp_set(&res, 1uL);
      if ((err = mp_mod(G, P, &M[1])) != MP_OKAY) {
         goto LBL_RES;
      }
   }

   /* compute the value at M[1<<(winsize-1)] by squaring M[1] (winsize-1) times */
   if ((err = mp_copy(&M[1], &M[(size_t)1 << (winsize - 1)])) != MP_OKAY) {
      goto LBL_RES;
   }

   for (x = 0; x < (winsize - 1); x++) {
      if ((err = mp_sqr(&M[(size_t)1 << (winsize - 1)], &M[(size_t)1 << (winsize - 1)])) != MP_OKAY) {
         goto LBL_RES;
      }
      if ((err = redux(&M[(size_t)1 << (winsize - 1)], P, mp)) != MP_OKAY) {
         goto LBL_RES;
      }
   }

   /* create upper table */
   for (x = (1 << (winsize - 1)) + 1; x < (1 << winsize); x++) {
      if ((err = mp_mul(&M[x - 1], &M[1], &M[x])) != MP_OKAY) {
         goto LBL_RES;
      }
      if ((err = redux(&M[x], P, mp)) != MP_OKAY) {
         goto LBL_RES;
      }
   }

   /* set initial mode and bit cnt */
   mode   = 0;
   bitcnt = 1;
   buf    = 0;
   digidx = X->used - 1;
   bitcpy = 0;
   bitbuf = 0;

   for (;;) {
      /* grab next digit as required */
      if (--bitcnt == 0) {
         /* if digidx == -1 we are out of digits so break */
         if (digidx == -1) {
            break;
         }
         /* read next digit and reset bitcnt */
         buf    = X->dp[digidx--];
         bitcnt = (int)DIGIT_BIT;
      }

      /* grab the next msb from the exponent */
      y     = (mp_digit)(buf >> (DIGIT_BIT - 1)) & 1;
      buf <<= (mp_digit)1;

      /* if the bit is zero and mode == 0 then we ignore it
       * These represent the leading zero bits before the first 1 bit
       * in the exponent.  Technically this opt is not required but it
       * does lower the # of trivial squaring/reductions used
       */
      if ((mode == 0) && (y == 0)) {
         continue;
      }

      /* if the bit is zero and mode == 1 then we square */
      if ((mode == 1) && (y == 0)) {
         if ((err = mp_sqr(&res, &res)) != MP_OKAY) {
            goto LBL_RES;
         }
         if ((err = redux(&res, P, mp)) != MP_OKAY) {
            goto LBL_RES;
         }
         continue;
      }

      /* else we add it to the window */
      bitbuf |= (y << (winsize - ++bitcpy));
      mode    = 2;

      if (bitcpy == winsize) {
         /* ok window is filled so square as required and multiply  */
         /* square first */
         for (x = 0; x < winsize; x++) {
            if ((err = mp_sqr(&res, &res)) != MP_OKAY) {
               goto LBL_RES;
            }
            if ((err = redux(&res, P, mp)) != MP_OKAY) {
               goto LBL_RES;
            }
         }

         /* then multiply */
         if ((err = mp_mul(&res, &M[bitbuf], &res)) != MP_OKAY) {
            goto LBL_RES;
         }
         if ((err = redux(&res, P, mp)) != MP_OKAY) {
            goto LBL_RES;
         }

         /* empty window and reset */
         bitcpy = 0;
         bitbuf = 0;
         mode   = 1;
      }
   }

   /* if bits remain then square/multiply */
   if ((mode == 2) && (bitcpy > 0)) {
      /* square then multiply if the bit is set */
      for (x = 0; x < bitcpy; x++) {
         if ((err = mp_sqr(&res, &res)) != MP_OKAY) {
            goto LBL_RES;
         }
         if ((err = redux(&res, P, mp)) != MP_OKAY) {
            goto LBL_RES;
         }

         /* get next bit of the window */
         bitbuf <<= 1;
         if ((bitbuf & (1 << winsize)) != 0) {
            /* then multiply */
            if ((err = mp_mul(&res, &M[1], &res)) != MP_OKAY) {
               goto LBL_RES;
            }
            if ((err = redux(&res, P, mp)) != MP_OKAY) {
               goto LBL_RES;
            }
         }
      }
   }

   if (redmode == 0) {
      /* fixup result if Montgomery reduction is used
       * recall that any value in a Montgomery system is
       * actually multiplied by R mod n.  So we have
       * to reduce one more time to cancel out the factor
       * of R.
       */
      if ((err = redux(&res, P, mp)) != MP_OKAY) {
         goto LBL_RES;
      }
   }

   /* swap res with Y */
   mp_exch(&res, Y);
   err = MP_OKAY;
LBL_RES:
   mp_clear(&res);
LBL_M:
   mp_clear(&M[1]);
   for (x = 1<<(winsize-1); x < (1 << winsize); x++) {
      mp_clear(&M[x]);
   }
   return err;
}

/* End: bn_mp_exptmod_fast.c */

/* Start: bn_mp_exteuclid.c */

/* Extended euclidean algorithm of (a, b) produces
   a*u1 + b*u2 = u3
 */
int mp_exteuclid(const mp_int *a, const mp_int *b, mp_int *U1, mp_int *U2, mp_int *U3)
{
   mp_int u1, u2, u3, v1, v2, v3, t1, t2, t3, q, tmp;
   int err;

   if ((err = mp_init_multi(&u1, &u2, &u3, &v1, &v2, &v3, &t1, &t2, &t3, &q, &tmp, NULL)) != MP_OKAY) {
      return err;
   }

   /* initialize, (u1,u2,u3) = (1,0,a) */
   mp_set(&u1, 1uL);
   if ((err = mp_copy(a, &u3)) != MP_OKAY) {
      goto LBL_ERR;
   }

   /* initialize, (v1,v2,v3) = (0,1,b) */
   mp_set(&v2, 1uL);
   if ((err = mp_copy(b, &v3)) != MP_OKAY) {
      goto LBL_ERR;
   }

   /* loop while v3 != 0 */
   while (mp_iszero(&v3) == MP_NO) {
      /* q = u3/v3 */
      if ((err = mp_div(&u3, &v3, &q, NULL)) != MP_OKAY) {
         goto LBL_ERR;
      }

      /* (t1,t2,t3) = (u1,u2,u3) - (v1,v2,v3)q */
      if ((err = mp_mul(&v1, &q, &tmp)) != MP_OKAY) {
         goto LBL_ERR;
      }
      if ((err = mp_sub(&u1, &tmp, &t1)) != MP_OKAY) {
         goto LBL_ERR;
      }
      if ((err = mp_mul(&v2, &q, &tmp)) != MP_OKAY) {
         goto LBL_ERR;
      }
      if ((err = mp_sub(&u2, &tmp, &t2)) != MP_OKAY) {
         goto LBL_ERR;
      }
      if ((err = mp_mul(&v3, &q, &tmp)) != MP_OKAY) {
         goto LBL_ERR;
      }
      if ((err = mp_sub(&u3, &tmp, &t3)) != MP_OKAY) {
         goto LBL_ERR;
      }

      /* (u1,u2,u3) = (v1,v2,v3) */
      if ((err = mp_copy(&v1, &u1)) != MP_OKAY) {
         goto LBL_ERR;
      }
      if ((err = mp_copy(&v2, &u2)) != MP_OKAY) {
         goto LBL_ERR;
      }
      if ((err = mp_copy(&v3, &u3)) != MP_OKAY) {
         goto LBL_ERR;
      }

      /* (v1,v2,v3) = (t1,t2,t3) */
      if ((err = mp_copy(&t1, &v1)) != MP_OKAY) {
         goto LBL_ERR;
      }
      if ((err = mp_copy(&t2, &v2)) != MP_OKAY) {
         goto LBL_ERR;
      }
      if ((err = mp_copy(&t3, &v3)) != MP_OKAY) {
         goto LBL_ERR;
      }
   }

   /* make sure U3 >= 0 */
   if (u3.sign == MP_NEG) {
      if ((err = mp_neg(&u1, &u1)) != MP_OKAY) {
         goto LBL_ERR;
      }
      if ((err = mp_neg(&u2, &u2)) != MP_OKAY) {
         goto LBL_ERR;
      }
      if ((err = mp_neg(&u3, &u3)) != MP_OKAY) {
         goto LBL_ERR;
      }
   }

   /* copy result out */
   if (U1 != NULL) {
      mp_exch(U1, &u1);
   }
   if (U2 != NULL) {
      mp_exch(U2, &u2);
   }
   if (U3 != NULL) {
      mp_exch(U3, &u3);
   }

   err = MP_OKAY;
LBL_ERR:
   mp_clear_multi(&u1, &u2, &u3, &v1, &v2, &v3, &t1, &t2, &t3, &q, &tmp, NULL);
   return err;
}

/* End: bn_mp_exteuclid.c */

/* Start: bn_mp_gcd.c */

/* Greatest Common Divisor using the binary method */
int mp_gcd(const mp_int *a, const mp_int *b, mp_int *c)
{
   mp_int  u, v;
   int     k, u_lsb, v_lsb, res;

   /* either zero than gcd is the largest */
   if (mp_iszero(a) == MP_YES) {
      return mp_abs(b, c);
   }
   if (mp_iszero(b) == MP_YES) {
      return mp_abs(a, c);
   }

   /* get copies of a and b we can modify */
   if ((res = mp_init_copy(&u, a)) != MP_OKAY) {
      return res;
   }

   if ((res = mp_init_copy(&v, b)) != MP_OKAY) {
      goto LBL_U;
   }

   /* must be positive for the remainder of the algorithm */
   u.sign = v.sign = MP_ZPOS;

   /* B1.  Find the common power of two for u and v */
   u_lsb = mp_cnt_lsb(&u);
   v_lsb = mp_cnt_lsb(&v);
   k     = MIN(u_lsb, v_lsb);

   if (k > 0) {
      /* divide the power of two out */
      if ((res = mp_div_2d(&u, k, &u, NULL)) != MP_OKAY) {
         goto LBL_V;
      }

      if ((res = mp_div_2d(&v, k, &v, NULL)) != MP_OKAY) {
         goto LBL_V;
      }
   }

   /* divide any remaining factors of two out */
   if (u_lsb != k) {
      if ((res = mp_div_2d(&u, u_lsb - k, &u, NULL)) != MP_OKAY) {
         goto LBL_V;
      }
   }

   if (v_lsb != k) {
      if ((res = mp_div_2d(&v, v_lsb - k, &v, NULL)) != MP_OKAY) {
         goto LBL_V;
      }
   }

   while (mp_iszero(&v) == MP_NO) {
      /* make sure v is the largest */
      if (mp_cmp_mag(&u, &v) == MP_GT) {
         /* swap u and v to make sure v is >= u */
         mp_exch(&u, &v);
      }

      /* subtract smallest from largest */
      if ((res = s_mp_sub(&v, &u, &v)) != MP_OKAY) {
         goto LBL_V;
      }

      /* Divide out all factors of two */
      if ((res = mp_div_2d(&v, mp_cnt_lsb(&v), &v, NULL)) != MP_OKAY) {
         goto LBL_V;
      }
   }

   /* multiply by 2**k which we divided out at the beginning */
   if ((res = mp_mul_2d(&u, k, c)) != MP_OKAY) {
      goto LBL_V;
   }
   c->sign = MP_ZPOS;
   res = MP_OKAY;
LBL_V:
   mp_clear(&u);
LBL_U:
   mp_clear(&v);
   return res;
}

/* End: bn_mp_gcd.c */

/* Start: bn_mp_get_bit.c */

/* Checks the bit at position b and returns MP_YES
   if the bit is 1, MP_NO if it is 0 and MP_VAL
   in case of error */
int mp_get_bit(const mp_int *a, int b)
{
   int limb;
   mp_digit bit, isset;

   if (b < 0) {
      return MP_VAL;
   }

   limb = b / DIGIT_BIT;

   /*
    * Zero is a special value with the member "used" set to zero.
    * Needs to be tested before the check for the upper boundary
    * otherwise (limb >= a->used) would be true for a = 0
    */

   if (mp_iszero(a) != MP_NO) {
      return MP_NO;
   }

   if (limb >= a->used) {
      return MP_VAL;
   }

   bit = (mp_digit)(1) << (b % DIGIT_BIT);

   isset = a->dp[limb] & bit;
   return (isset != 0u) ? MP_YES : MP_NO;
}

/* End: bn_mp_get_bit.c */

/* Start: bn_mp_get_double.c */

double mp_get_double(const mp_int *a)
{
   int i;
   double d = 0.0, fac = 1.0;
   for (i = 0; i < DIGIT_BIT; ++i) {
      fac *= 2.0;
   }
   for (i = USED(a); i --> 0;) {
      d = (d * fac) + (double)DIGIT(a, i);
   }
   return (mp_isneg(a) != MP_NO) ? -d : d;
}

/* End: bn_mp_get_double.c */

/* Start: bn_mp_get_int.c */

/* get the lower 32-bits of an mp_int */
unsigned long mp_get_int(const mp_int *a)
{
   int i;
   mp_min_u32 res;

   if (a->used == 0) {
      return 0;
   }

   /* get number of digits of the lsb we have to read */
   i = MIN(a->used, ((((int)sizeof(unsigned long) * CHAR_BIT) + DIGIT_BIT - 1) / DIGIT_BIT)) - 1;

   /* get most significant digit of result */
   res = DIGIT(a, i);

   while (--i >= 0) {
      res = (res << DIGIT_BIT) | DIGIT(a, i);
   }

   /* force result to 32-bits always so it is consistent on non 32-bit platforms */
   return res & 0xFFFFFFFFUL;
}

/* End: bn_mp_get_int.c */

/* Start: bn_mp_get_long.c */

/* get the lower unsigned long of an mp_int, platform dependent */
unsigned long mp_get_long(const mp_int *a)
{
   int i;
   unsigned long res;

   if (a->used == 0) {
      return 0;
   }

   /* get number of digits of the lsb we have to read */
   i = MIN(a->used, ((((int)sizeof(unsigned long) * CHAR_BIT) + DIGIT_BIT - 1) / DIGIT_BIT)) - 1;

   /* get most significant digit of result */
   res = DIGIT(a, i);

#if (ULONG_MAX != 0xffffffffuL) || (DIGIT_BIT < 32)
   while (--i >= 0) {
      res = (res << DIGIT_BIT) | DIGIT(a, i);
   }
#endif
   return res;
}

/* End: bn_mp_get_long.c */

/* Start: bn_mp_get_long_long.c */

/* get the lower unsigned long long of an mp_int, platform dependent */
unsigned long long mp_get_long_long(const mp_int *a)
{
   int i;
   unsigned long long res;

   if (a->used == 0) {
      return 0;
   }

   /* get number of digits of the lsb we have to read */
   i = MIN(a->used, ((((int)sizeof(unsigned long long) * CHAR_BIT) + DIGIT_BIT - 1) / DIGIT_BIT)) - 1;

   /* get most significant digit of result */
   res = DIGIT(a, i);

#if DIGIT_BIT < 64
   while (--i >= 0) {
      res = (res << DIGIT_BIT) | DIGIT(a, i);
   }
#endif
   return res;
}

/* End: bn_mp_get_long_long.c */

/* Start: bn_mp_grow.c */

/* grow as required */
int mp_grow(mp_int *a, int size)
{
   int     i;
   mp_digit *tmp;

   /* if the alloc size is smaller alloc more ram */
   if (a->alloc < size) {
      /* ensure there are always at least MP_PREC digits extra on top */
      size += (MP_PREC * 2) - (size % MP_PREC);

      /* reallocate the array a->dp
       *
       * We store the return in a temporary variable
       * in case the operation failed we don't want
       * to overwrite the dp member of a.
       */
      tmp = OPT_CAST(mp_digit) XREALLOC(a->dp, sizeof(mp_digit) * (size_t)size);
      if (tmp == NULL) {
         /* reallocation failed but "a" is still valid [can be freed] */
         return MP_MEM;
      }

      /* reallocation succeeded so set a->dp */
      a->dp = tmp;

      /* zero excess digits */
      i        = a->alloc;
      a->alloc = size;
      for (; i < a->alloc; i++) {
         a->dp[i] = 0;
      }
   }
   return MP_OKAY;
}

/* End: bn_mp_grow.c */

/* Start: bn_mp_import.c */

/* based on gmp's mpz_import.
 * see http://gmplib.org/manual/Integer-Import-and-Export.html
 */
int mp_import(mp_int *rop, size_t count, int order, size_t size,
              int endian, size_t nails, const void *op)
{
   int result;
   size_t odd_nails, nail_bytes, i, j;
   unsigned char odd_nail_mask;

   mp_zero(rop);

   if (endian == 0) {
      union {
         unsigned int i;
         char c[4];
      } lint;
      lint.i = 0x01020304;

      endian = (lint.c[0] == '\x04') ? -1 : 1;
   }

   odd_nails = (nails % 8u);
   odd_nail_mask = 0xff;
   for (i = 0; i < odd_nails; ++i) {
      odd_nail_mask ^= (unsigned char)(1u << (7u - i));
   }
   nail_bytes = nails / 8u;

   for (i = 0; i < count; ++i) {
      for (j = 0; j < (size - nail_bytes); ++j) {
         unsigned char byte = *((unsigned char *)op +
                                (((order == 1) ? i : ((count - 1u) - i)) * size) +
                                ((endian == 1) ? (j + nail_bytes) : (((size - 1u) - j) - nail_bytes)));

         if ((result = mp_mul_2d(rop, (j == 0u) ? (int)(8u - odd_nails) : 8, rop)) != MP_OKAY) {
            return result;
         }

         rop->dp[0] |= (j == 0u) ? (mp_digit)(byte & odd_nail_mask) : (mp_digit)byte;
         rop->used  += 1;
      }
   }

   mp_clamp(rop);

   return MP_OKAY;
}

/* End: bn_mp_import.c */

/* Start: bn_mp_init.c */

/* init a new mp_int */
int mp_init(mp_int *a)
{
   int i;

   /* allocate memory required and clear it */
   a->dp = OPT_CAST(mp_digit) XMALLOC(sizeof(mp_digit) * (size_t)MP_PREC);
   if (a->dp == NULL) {
      return MP_MEM;
   }

   /* set the digits to zero */
   for (i = 0; i < MP_PREC; i++) {
      a->dp[i] = 0;
   }

   /* set the used to zero, allocated digits to the default precision
    * and sign to positive */
   a->used  = 0;
   a->alloc = MP_PREC;
   a->sign  = MP_ZPOS;

   return MP_OKAY;
}

/* End: bn_mp_init.c */

/* Start: bn_mp_init_copy.c */

/* creates "a" then copies b into it */
int mp_init_copy(mp_int *a, const mp_int *b)
{
   int     res;

   if ((res = mp_init_size(a, b->used)) != MP_OKAY) {
      return res;
   }

   if ((res = mp_copy(b, a)) != MP_OKAY) {
      mp_clear(a);
   }

   return res;
}

/* End: bn_mp_init_copy.c */

/* Start: bn_mp_init_multi.c */

int mp_init_multi(mp_int *mp, ...)
{
   mp_err res = MP_OKAY;      /* Assume ok until proven otherwise */
   int n = 0;                 /* Number of ok inits */
   mp_int *cur_arg = mp;
   va_list args;

   va_start(args, mp);        /* init args to next argument from caller */
   while (cur_arg != NULL) {
      if (mp_init(cur_arg) != MP_OKAY) {
         /* Oops - error! Back-track and mp_clear what we already
            succeeded in init-ing, then return error.
         */
         va_list clean_args;

         /* now start cleaning up */
         cur_arg = mp;
         va_start(clean_args, mp);
         while (n-- != 0) {
            mp_clear(cur_arg);
            cur_arg = va_arg(clean_args, mp_int *);
         }
         va_end(clean_args);
         res = MP_MEM;
         break;
      }
      n++;
      cur_arg = va_arg(args, mp_int *);
   }
   va_end(args);
   return res;                /* Assumed ok, if error flagged above. */
}

/* End: bn_mp_init_multi.c */

/* Start: bn_mp_init_set.c */

/* initialize and set a digit */
int mp_init_set(mp_int *a, mp_digit b)
{
   int err;
   if ((err = mp_init(a)) != MP_OKAY) {
      return err;
   }
   mp_set(a, b);
   return err;
}

/* End: bn_mp_init_set.c */

/* Start: bn_mp_init_set_int.c */

/* initialize and set a digit */
int mp_init_set_int(mp_int *a, unsigned long b)
{
   int err;
   if ((err = mp_init(a)) != MP_OKAY) {
      return err;
   }
   return mp_set_int(a, b);
}

/* End: bn_mp_init_set_int.c */

/* Start: bn_mp_init_size.c */

/* init an mp_init for a given size */
int mp_init_size(mp_int *a, int size)
{
   int x;

   /* pad size so there are always extra digits */
   size += (MP_PREC * 2) - (size % MP_PREC);

   /* alloc mem */
   a->dp = OPT_CAST(mp_digit) XMALLOC(sizeof(mp_digit) * (size_t)size);
   if (a->dp == NULL) {
      return MP_MEM;
   }

   /* set the members */
   a->used  = 0;
   a->alloc = size;
   a->sign  = MP_ZPOS;

   /* zero the digits */
   for (x = 0; x < size; x++) {
      a->dp[x] = 0;
   }

   return MP_OKAY;
}

/* End: bn_mp_init_size.c */

/* Start: bn_mp_invmod.c */

/* hac 14.61, pp608 */
int mp_invmod(const mp_int *a, const mp_int *b, mp_int *c)
{
   /* b cannot be negative and has to be >1 */
   if ((b->sign == MP_NEG) || (mp_cmp_d(b, 1uL) != MP_GT)) {
      return MP_VAL;
   }

   /* if the modulus is odd we can use a faster routine instead */
   if ((mp_isodd(b) == MP_YES)) {
      return fast_mp_invmod(a, b, c);
   }

   return mp_invmod_slow(a, b, c);
}

/* End: bn_mp_invmod.c */

/* Start: bn_mp_invmod_slow.c */

/* hac 14.61, pp608 */
int mp_invmod_slow(const mp_int *a, const mp_int *b, mp_int *c)
{
   mp_int  x, y, u, v, A, B, C, D;
   int     res;

   /* b cannot be negative */
   if ((b->sign == MP_NEG) || (mp_iszero(b) == MP_YES)) {
      return MP_VAL;
   }

   /* init temps */
   if ((res = mp_init_multi(&x, &y, &u, &v,
                            &A, &B, &C, &D, NULL)) != MP_OKAY) {
      return res;
   }

   /* x = a, y = b */
   if ((res = mp_mod(a, b, &x)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_copy(b, &y)) != MP_OKAY) {
      goto LBL_ERR;
   }

   /* 2. [modified] if x,y are both even then return an error! */
   if ((mp_iseven(&x) == MP_YES) && (mp_iseven(&y) == MP_YES)) {
      res = MP_VAL;
      goto LBL_ERR;
   }

   /* 3. u=x, v=y, A=1, B=0, C=0,D=1 */
   if ((res = mp_copy(&x, &u)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_copy(&y, &v)) != MP_OKAY) {
      goto LBL_ERR;
   }
   mp_set(&A, 1uL);
   mp_set(&D, 1uL);

top:
   /* 4.  while u is even do */
   while (mp_iseven(&u) == MP_YES) {
      /* 4.1 u = u/2 */
      if ((res = mp_div_2(&u, &u)) != MP_OKAY) {
         goto LBL_ERR;
      }
      /* 4.2 if A or B is odd then */
      if ((mp_isodd(&A) == MP_YES) || (mp_isodd(&B) == MP_YES)) {
         /* A = (A+y)/2, B = (B-x)/2 */
         if ((res = mp_add(&A, &y, &A)) != MP_OKAY) {
            goto LBL_ERR;
         }
         if ((res = mp_sub(&B, &x, &B)) != MP_OKAY) {
            goto LBL_ERR;
         }
      }
      /* A = A/2, B = B/2 */
      if ((res = mp_div_2(&A, &A)) != MP_OKAY) {
         goto LBL_ERR;
      }
      if ((res = mp_div_2(&B, &B)) != MP_OKAY) {
         goto LBL_ERR;
      }
   }

   /* 5.  while v is even do */
   while (mp_iseven(&v) == MP_YES) {
      /* 5.1 v = v/2 */
      if ((res = mp_div_2(&v, &v)) != MP_OKAY) {
         goto LBL_ERR;
      }
      /* 5.2 if C or D is odd then */
      if ((mp_isodd(&C) == MP_YES) || (mp_isodd(&D) == MP_YES)) {
         /* C = (C+y)/2, D = (D-x)/2 */
         if ((res = mp_add(&C, &y, &C)) != MP_OKAY) {
            goto LBL_ERR;
         }
         if ((res = mp_sub(&D, &x, &D)) != MP_OKAY) {
            goto LBL_ERR;
         }
      }
      /* C = C/2, D = D/2 */
      if ((res = mp_div_2(&C, &C)) != MP_OKAY) {
         goto LBL_ERR;
      }
      if ((res = mp_div_2(&D, &D)) != MP_OKAY) {
         goto LBL_ERR;
      }
   }

   /* 6.  if u >= v then */
   if (mp_cmp(&u, &v) != MP_LT) {
      /* u = u - v, A = A - C, B = B - D */
      if ((res = mp_sub(&u, &v, &u)) != MP_OKAY) {
         goto LBL_ERR;
      }

      if ((res = mp_sub(&A, &C, &A)) != MP_OKAY) {
         goto LBL_ERR;
      }

      if ((res = mp_sub(&B, &D, &B)) != MP_OKAY) {
         goto LBL_ERR;
      }
   } else {
      /* v - v - u, C = C - A, D = D - B */
      if ((res = mp_sub(&v, &u, &v)) != MP_OKAY) {
         goto LBL_ERR;
      }

      if ((res = mp_sub(&C, &A, &C)) != MP_OKAY) {
         goto LBL_ERR;
      }

      if ((res = mp_sub(&D, &B, &D)) != MP_OKAY) {
         goto LBL_ERR;
      }
   }

   /* if not zero goto step 4 */
   if (mp_iszero(&u) == MP_NO)
      goto top;

   /* now a = C, b = D, gcd == g*v */

   /* if v != 1 then there is no inverse */
   if (mp_cmp_d(&v, 1uL) != MP_EQ) {
      res = MP_VAL;
      goto LBL_ERR;
   }

   /* if its too low */
   while (mp_cmp_d(&C, 0uL) == MP_LT) {
      if ((res = mp_add(&C, b, &C)) != MP_OKAY) {
         goto LBL_ERR;
      }
   }

   /* too big */
   while (mp_cmp_mag(&C, b) != MP_LT) {
      if ((res = mp_sub(&C, b, &C)) != MP_OKAY) {
         goto LBL_ERR;
      }
   }

   /* C is now the inverse */
   mp_exch(&C, c);
   res = MP_OKAY;
LBL_ERR:
   mp_clear_multi(&x, &y, &u, &v, &A, &B, &C, &D, NULL);
   return res;
}

/* End: bn_mp_invmod_slow.c */

/* Start: bn_mp_is_square.c */

/* Check if remainders are possible squares - fast exclude non-squares */
static const char rem_128[128] = {
   0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1,
   0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1,
   1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1,
   1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1,
   0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1,
   1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1,
   1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1,
   1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1
};

static const char rem_105[105] = {
   0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1,
   0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1,
   0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1,
   1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1,
   0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1,
   1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1,
   1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1
};

/* Store non-zero to ret if arg is square, and zero if not */
int mp_is_square(const mp_int *arg, int *ret)
{
   int           res;
   mp_digit      c;
   mp_int        t;
   unsigned long r;

   /* Default to Non-square :) */
   *ret = MP_NO;

   if (arg->sign == MP_NEG) {
      return MP_VAL;
   }

   /* digits used?  (TSD) */
   if (arg->used == 0) {
      return MP_OKAY;
   }

   /* First check mod 128 (suppose that DIGIT_BIT is at least 7) */
   if (rem_128[127u & DIGIT(arg, 0)] == (char)1) {
      return MP_OKAY;
   }

   /* Next check mod 105 (3*5*7) */
   if ((res = mp_mod_d(arg, 105uL, &c)) != MP_OKAY) {
      return res;
   }
   if (rem_105[c] == (char)1) {
      return MP_OKAY;
   }


   if ((res = mp_init_set_int(&t, 11L*13L*17L*19L*23L*29L*31L)) != MP_OKAY) {
      return res;
   }
   if ((res = mp_mod(arg, &t, &t)) != MP_OKAY) {
      goto LBL_ERR;
   }
   r = mp_get_int(&t);
   /* Check for other prime modules, note it's not an ERROR but we must
    * free "t" so the easiest way is to goto LBL_ERR.  We know that res
    * is already equal to MP_OKAY from the mp_mod call
    */
   if (((1uL<<(r%11uL)) & 0x5C4uL) != 0uL)         goto LBL_ERR;
   if (((1uL<<(r%13uL)) & 0x9E4uL) != 0uL)         goto LBL_ERR;
   if (((1uL<<(r%17uL)) & 0x5CE8uL) != 0uL)        goto LBL_ERR;
   if (((1uL<<(r%19uL)) & 0x4F50CuL) != 0uL)       goto LBL_ERR;
   if (((1uL<<(r%23uL)) & 0x7ACCA0uL) != 0uL)      goto LBL_ERR;
   if (((1uL<<(r%29uL)) & 0xC2EDD0CuL) != 0uL)     goto LBL_ERR;
   if (((1uL<<(r%31uL)) & 0x6DE2B848uL) != 0uL)    goto LBL_ERR;

   /* Final check - is sqr(sqrt(arg)) == arg ? */
   if ((res = mp_sqrt(arg, &t)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_sqr(&t, &t)) != MP_OKAY) {
      goto LBL_ERR;
   }

   *ret = (mp_cmp_mag(&t, arg) == MP_EQ) ? MP_YES : MP_NO;
LBL_ERR:
   mp_clear(&t);
   return res;
}

/* End: bn_mp_is_square.c */

/* Start: bn_mp_jacobi.c */

/* computes the jacobi c = (a | n) (or Legendre if n is prime)
 * Kept for legacy reasons, please use mp_kronecker() instead
 */
int mp_jacobi(const mp_int *a, const mp_int *n, int *c)
{
   /* if a < 0 return MP_VAL */
   if (mp_isneg(a) == MP_YES) {
      return MP_VAL;
   }

   /* if n <= 0 return MP_VAL */
   if (mp_cmp_d(n, 0uL) != MP_GT) {
      return MP_VAL;
   }

   return mp_kronecker(a, n, c);
}

/* End: bn_mp_jacobi.c */

/* Start: bn_mp_karatsuba_mul.c */

/* c = |a| * |b| using Karatsuba Multiplication using
 * three half size multiplications
 *
 * Let B represent the radix [e.g. 2**DIGIT_BIT] and
 * let n represent half of the number of digits in
 * the min(a,b)
 *
 * a = a1 * B**n + a0
 * b = b1 * B**n + b0
 *
 * Then, a * b =>
   a1b1 * B**2n + ((a1 + a0)(b1 + b0) - (a0b0 + a1b1)) * B + a0b0
 *
 * Note that a1b1 and a0b0 are used twice and only need to be
 * computed once.  So in total three half size (half # of
 * digit) multiplications are performed, a0b0, a1b1 and
 * (a1+b1)(a0+b0)
 *
 * Note that a multiplication of half the digits requires
 * 1/4th the number of single precision multiplications so in
 * total after one call 25% of the single precision multiplications
 * are saved.  Note also that the call to mp_mul can end up back
 * in this function if the a0, a1, b0, or b1 are above the threshold.
 * This is known as divide-and-conquer and leads to the famous
 * O(N**lg(3)) or O(N**1.584) work which is asymptopically lower than
 * the standard O(N**2) that the baseline/comba methods use.
 * Generally though the overhead of this method doesn't pay off
 * until a certain size (N ~ 80) is reached.
 */
int mp_karatsuba_mul(const mp_int *a, const mp_int *b, mp_int *c)
{
   mp_int  x0, x1, y0, y1, t1, x0y0, x1y1;
   int     B, err;

   /* default the return code to an error */
   err = MP_MEM;

   /* min # of digits */
   B = MIN(a->used, b->used);

   /* now divide in two */
   B = B >> 1;

   /* init copy all the temps */
   if (mp_init_size(&x0, B) != MP_OKAY)
      goto LBL_ERR;
   if (mp_init_size(&x1, a->used - B) != MP_OKAY)
      goto X0;
   if (mp_init_size(&y0, B) != MP_OKAY)
      goto X1;
   if (mp_init_size(&y1, b->used - B) != MP_OKAY)
      goto Y0;

   /* init temps */
   if (mp_init_size(&t1, B * 2) != MP_OKAY)
      goto Y1;
   if (mp_init_size(&x0y0, B * 2) != MP_OKAY)
      goto T1;
   if (mp_init_size(&x1y1, B * 2) != MP_OKAY)
      goto X0Y0;

   /* now shift the digits */
   x0.used = y0.used = B;
   x1.used = a->used - B;
   y1.used = b->used - B;

   {
      int x;
      mp_digit *tmpa, *tmpb, *tmpx, *tmpy;

      /* we copy the digits directly instead of using higher level functions
       * since we also need to shift the digits
       */
      tmpa = a->dp;
      tmpb = b->dp;

      tmpx = x0.dp;
      tmpy = y0.dp;
      for (x = 0; x < B; x++) {
         *tmpx++ = *tmpa++;
         *tmpy++ = *tmpb++;
      }

      tmpx = x1.dp;
      for (x = B; x < a->used; x++) {
         *tmpx++ = *tmpa++;
      }

      tmpy = y1.dp;
      for (x = B; x < b->used; x++) {
         *tmpy++ = *tmpb++;
      }
   }

   /* only need to clamp the lower words since by definition the
    * upper words x1/y1 must have a known number of digits
    */
   mp_clamp(&x0);
   mp_clamp(&y0);

   /* now calc the products x0y0 and x1y1 */
   /* after this x0 is no longer required, free temp [x0==t2]! */
   if (mp_mul(&x0, &y0, &x0y0) != MP_OKAY)
      goto X1Y1;          /* x0y0 = x0*y0 */
   if (mp_mul(&x1, &y1, &x1y1) != MP_OKAY)
      goto X1Y1;          /* x1y1 = x1*y1 */

   /* now calc x1+x0 and y1+y0 */
   if (s_mp_add(&x1, &x0, &t1) != MP_OKAY)
      goto X1Y1;          /* t1 = x1 - x0 */
   if (s_mp_add(&y1, &y0, &x0) != MP_OKAY)
      goto X1Y1;          /* t2 = y1 - y0 */
   if (mp_mul(&t1, &x0, &t1) != MP_OKAY)
      goto X1Y1;          /* t1 = (x1 + x0) * (y1 + y0) */

   /* add x0y0 */
   if (mp_add(&x0y0, &x1y1, &x0) != MP_OKAY)
      goto X1Y1;          /* t2 = x0y0 + x1y1 */
   if (s_mp_sub(&t1, &x0, &t1) != MP_OKAY)
      goto X1Y1;          /* t1 = (x1+x0)*(y1+y0) - (x1y1 + x0y0) */

   /* shift by B */
   if (mp_lshd(&t1, B) != MP_OKAY)
      goto X1Y1;          /* t1 = (x0y0 + x1y1 - (x1-x0)*(y1-y0))<<B */
   if (mp_lshd(&x1y1, B * 2) != MP_OKAY)
      goto X1Y1;          /* x1y1 = x1y1 << 2*B */

   if (mp_add(&x0y0, &t1, &t1) != MP_OKAY)
      goto X1Y1;          /* t1 = x0y0 + t1 */
   if (mp_add(&t1, &x1y1, c) != MP_OKAY)
      goto X1Y1;          /* t1 = x0y0 + t1 + x1y1 */

   /* Algorithm succeeded set the return code to MP_OKAY */
   err = MP_OKAY;

X1Y1:
   mp_clear(&x1y1);
X0Y0:
   mp_clear(&x0y0);
T1:
   mp_clear(&t1);
Y1:
   mp_clear(&y1);
Y0:
   mp_clear(&y0);
X1:
   mp_clear(&x1);
X0:
   mp_clear(&x0);
LBL_ERR:
   return err;
}

/* End: bn_mp_karatsuba_mul.c */

/* Start: bn_mp_karatsuba_sqr.c */

/* Karatsuba squaring, computes b = a*a using three
 * half size squarings
 *
 * See comments of karatsuba_mul for details.  It
 * is essentially the same algorithm but merely
 * tuned to perform recursive squarings.
 */
int mp_karatsuba_sqr(const mp_int *a, mp_int *b)
{
   mp_int  x0, x1, t1, t2, x0x0, x1x1;
   int     B, err;

   err = MP_MEM;

   /* min # of digits */
   B = a->used;

   /* now divide in two */
   B = B >> 1;

   /* init copy all the temps */
   if (mp_init_size(&x0, B) != MP_OKAY)
      goto LBL_ERR;
   if (mp_init_size(&x1, a->used - B) != MP_OKAY)
      goto X0;

   /* init temps */
   if (mp_init_size(&t1, a->used * 2) != MP_OKAY)
      goto X1;
   if (mp_init_size(&t2, a->used * 2) != MP_OKAY)
      goto T1;
   if (mp_init_size(&x0x0, B * 2) != MP_OKAY)
      goto T2;
   if (mp_init_size(&x1x1, (a->used - B) * 2) != MP_OKAY)
      goto X0X0;

   {
      int x;
      mp_digit *dst, *src;

      src = a->dp;

      /* now shift the digits */
      dst = x0.dp;
      for (x = 0; x < B; x++) {
         *dst++ = *src++;
      }

      dst = x1.dp;
      for (x = B; x < a->used; x++) {
         *dst++ = *src++;
      }
   }

   x0.used = B;
   x1.used = a->used - B;

   mp_clamp(&x0);

   /* now calc the products x0*x0 and x1*x1 */
   if (mp_sqr(&x0, &x0x0) != MP_OKAY)
      goto X1X1;           /* x0x0 = x0*x0 */
   if (mp_sqr(&x1, &x1x1) != MP_OKAY)
      goto X1X1;           /* x1x1 = x1*x1 */

   /* now calc (x1+x0)**2 */
   if (s_mp_add(&x1, &x0, &t1) != MP_OKAY)
      goto X1X1;           /* t1 = x1 - x0 */
   if (mp_sqr(&t1, &t1) != MP_OKAY)
      goto X1X1;           /* t1 = (x1 - x0) * (x1 - x0) */

   /* add x0y0 */
   if (s_mp_add(&x0x0, &x1x1, &t2) != MP_OKAY)
      goto X1X1;           /* t2 = x0x0 + x1x1 */
   if (s_mp_sub(&t1, &t2, &t1) != MP_OKAY)
      goto X1X1;           /* t1 = (x1+x0)**2 - (x0x0 + x1x1) */

   /* shift by B */
   if (mp_lshd(&t1, B) != MP_OKAY)
      goto X1X1;           /* t1 = (x0x0 + x1x1 - (x1-x0)*(x1-x0))<<B */
   if (mp_lshd(&x1x1, B * 2) != MP_OKAY)
      goto X1X1;           /* x1x1 = x1x1 << 2*B */

   if (mp_add(&x0x0, &t1, &t1) != MP_OKAY)
      goto X1X1;           /* t1 = x0x0 + t1 */
   if (mp_add(&t1, &x1x1, b) != MP_OKAY)
      goto X1X1;           /* t1 = x0x0 + t1 + x1x1 */

   err = MP_OKAY;

X1X1:
   mp_clear(&x1x1);
X0X0:
   mp_clear(&x0x0);
T2:
   mp_clear(&t2);
T1:
   mp_clear(&t1);
X1:
   mp_clear(&x1);
X0:
   mp_clear(&x0);
LBL_ERR:
   return err;
}

/* End: bn_mp_karatsuba_sqr.c */

/* Start: bn_mp_kronecker.c */

/*
   Kronecker symbol (a|p)
   Straightforward implementation of algorithm 1.4.10 in
   Henri Cohen: "A Course in Computational Algebraic Number Theory"

   @book{cohen2013course,
     title={A course in computational algebraic number theory},
     author={Cohen, Henri},
     volume={138},
     year={2013},
     publisher={Springer Science \& Business Media}
    }
 */
int mp_kronecker(const mp_int *a, const mp_int *p, int *c)
{
   mp_int a1, p1, r;

   int e = MP_OKAY;
   int v, k;

   static const int table[8] = {0, 1, 0, -1, 0, -1, 0, 1};

   if (mp_iszero(p) != MP_NO) {
      if ((a->used == 1) && (a->dp[0] == 1u)) {
         *c = 1;
         return e;
      } else {
         *c = 0;
         return e;
      }
   }

   if ((mp_iseven(a) != MP_NO) && (mp_iseven(p) != MP_NO)) {
      *c = 0;
      return e;
   }

   if ((e = mp_init_copy(&a1, a)) != MP_OKAY) {
      return e;
   }
   if ((e = mp_init_copy(&p1, p)) != MP_OKAY) {
      goto LBL_KRON_0;
   }

   v = mp_cnt_lsb(&p1);
   if ((e = mp_div_2d(&p1, v, &p1, NULL)) != MP_OKAY) {
      goto LBL_KRON_1;
   }

   if ((v & 0x1) == 0) {
      k = 1;
   } else {
      k = table[a->dp[0] & 7u];
   }

   if (p1.sign == MP_NEG) {
      p1.sign = MP_ZPOS;
      if (a1.sign == MP_NEG) {
         k = -k;
      }
   }

   if ((e = mp_init(&r)) != MP_OKAY) {
      goto LBL_KRON_1;
   }

   for (;;) {
      if (mp_iszero(&a1) != MP_NO) {
         if (mp_cmp_d(&p1, 1uL) == MP_EQ) {
            *c = k;
            goto LBL_KRON;
         } else {
            *c = 0;
            goto LBL_KRON;
         }
      }

      v = mp_cnt_lsb(&a1);
      if ((e = mp_div_2d(&a1, v, &a1, NULL)) != MP_OKAY) {
         goto LBL_KRON;
      }

      if ((v & 0x1) == 1) {
         k = k * table[p1.dp[0] & 7u];
      }

      if (a1.sign == MP_NEG) {
         /*
          * Compute k = (-1)^((a1)*(p1-1)/4) * k
          * a1.dp[0] + 1 cannot overflow because the MSB
          * of the type mp_digit is not set by definition
          */
         if (((a1.dp[0] + 1u) & p1.dp[0] & 2u) != 0u) {
            k = -k;
         }
      } else {
         /* compute k = (-1)^((a1-1)*(p1-1)/4) * k */
         if ((a1.dp[0] & p1.dp[0] & 2u) != 0u) {
            k = -k;
         }
      }

      if ((e = mp_copy(&a1, &r)) != MP_OKAY) {
         goto LBL_KRON;
      }
      r.sign = MP_ZPOS;
      if ((e = mp_mod(&p1, &r, &a1)) != MP_OKAY) {
         goto LBL_KRON;
      }
      if ((e = mp_copy(&r, &p1)) != MP_OKAY) {
         goto LBL_KRON;
      }
   }

LBL_KRON:
   mp_clear(&r);
LBL_KRON_1:
   mp_clear(&p1);
LBL_KRON_0:
   mp_clear(&a1);

   return e;
}

/* End: bn_mp_kronecker.c */

/* Start: bn_mp_lcm.c */

/* computes least common multiple as |a*b|/(a, b) */
int mp_lcm(const mp_int *a, const mp_int *b, mp_int *c)
{
   int     res;
   mp_int  t1, t2;


   if ((res = mp_init_multi(&t1, &t2, NULL)) != MP_OKAY) {
      return res;
   }

   /* t1 = get the GCD of the two inputs */
   if ((res = mp_gcd(a, b, &t1)) != MP_OKAY) {
      goto LBL_T;
   }

   /* divide the smallest by the GCD */
   if (mp_cmp_mag(a, b) == MP_LT) {
      /* store quotient in t2 such that t2 * b is the LCM */
      if ((res = mp_div(a, &t1, &t2, NULL)) != MP_OKAY) {
         goto LBL_T;
      }
      res = mp_mul(b, &t2, c);
   } else {
      /* store quotient in t2 such that t2 * a is the LCM */
      if ((res = mp_div(b, &t1, &t2, NULL)) != MP_OKAY) {
         goto LBL_T;
      }
      res = mp_mul(a, &t2, c);
   }

   /* fix the sign to positive */
   c->sign = MP_ZPOS;

LBL_T:
   mp_clear_multi(&t1, &t2, NULL);
   return res;
}

/* End: bn_mp_lcm.c */

/* Start: bn_mp_lshd.c */

/* shift left a certain amount of digits */
int mp_lshd(mp_int *a, int b)
{
   int     x, res;

   /* if its less than zero return */
   if (b <= 0) {
      return MP_OKAY;
   }
   /* no need to shift 0 around */
   if (mp_iszero(a) == MP_YES) {
      return MP_OKAY;
   }

   /* grow to fit the new digits */
   if (a->alloc < (a->used + b)) {
      if ((res = mp_grow(a, a->used + b)) != MP_OKAY) {
         return res;
      }
   }

   {
      mp_digit *top, *bottom;

      /* increment the used by the shift amount then copy upwards */
      a->used += b;

      /* top */
      top = a->dp + a->used - 1;

      /* base */
      bottom = (a->dp + a->used - 1) - b;

      /* much like mp_rshd this is implemented using a sliding window
       * except the window goes the otherway around.  Copying from
       * the bottom to the top.  see bn_mp_rshd.c for more info.
       */
      for (x = a->used - 1; x >= b; x--) {
         *top-- = *bottom--;
      }

      /* zero the lower digits */
      top = a->dp;
      for (x = 0; x < b; x++) {
         *top++ = 0;
      }
   }
   return MP_OKAY;
}

/* End: bn_mp_lshd.c */

/* Start: bn_mp_mod.c */

/* c = a mod b, 0 <= c < b if b > 0, b < c <= 0 if b < 0 */
int mp_mod(const mp_int *a, const mp_int *b, mp_int *c)
{
   mp_int  t;
   int     res;

   if ((res = mp_init_size(&t, b->used)) != MP_OKAY) {
      return res;
   }

   if ((res = mp_div(a, b, NULL, &t)) != MP_OKAY) {
      mp_clear(&t);
      return res;
   }

   if ((mp_iszero(&t) != MP_NO) || (t.sign == b->sign)) {
      res = MP_OKAY;
      mp_exch(&t, c);
   } else {
      res = mp_add(b, &t, c);
   }

   mp_clear(&t);
   return res;
}

/* End: bn_mp_mod.c */

/* Start: bn_mp_mod_2d.c */

/* calc a value mod 2**b */
int mp_mod_2d(const mp_int *a, int b, mp_int *c)
{
   int     x, res;

   /* if b is <= 0 then zero the int */
   if (b <= 0) {
      mp_zero(c);
      return MP_OKAY;
   }

   /* if the modulus is larger than the value than return */
   if (b >= (a->used * DIGIT_BIT)) {
      res = mp_copy(a, c);
      return res;
   }

   /* copy */
   if ((res = mp_copy(a, c)) != MP_OKAY) {
      return res;
   }

   /* zero digits above the last digit of the modulus */
   for (x = (b / DIGIT_BIT) + (((b % DIGIT_BIT) == 0) ? 0 : 1); x < c->used; x++) {
      c->dp[x] = 0;
   }
   /* clear the digit that is not completely outside/inside the modulus */
   c->dp[b / DIGIT_BIT] &=
      ((mp_digit)1 << (mp_digit)(b % DIGIT_BIT)) - (mp_digit)1;
   mp_clamp(c);
   return MP_OKAY;
}

/* End: bn_mp_mod_2d.c */

/* Start: bn_mp_mod_d.c */

int mp_mod_d(const mp_int *a, mp_digit b, mp_digit *c)
{
   return mp_div_d(a, b, NULL, c);
}

/* End: bn_mp_mod_d.c */

/* Start: bn_mp_montgomery_calc_normalization.c */

/*
 * shifts with subtractions when the result is greater than b.
 *
 * The method is slightly modified to shift B unconditionally upto just under
 * the leading bit of b.  This saves alot of multiple precision shifting.
 */
int mp_montgomery_calc_normalization(mp_int *a, const mp_int *b)
{
   int     x, bits, res;

   /* how many bits of last digit does b use */
   bits = mp_count_bits(b) % DIGIT_BIT;

   if (b->used > 1) {
      if ((res = mp_2expt(a, ((b->used - 1) * DIGIT_BIT) + bits - 1)) != MP_OKAY) {
         return res;
      }
   } else {
      mp_set(a, 1uL);
      bits = 1;
   }


   /* now compute C = A * B mod b */
   for (x = bits - 1; x < (int)DIGIT_BIT; x++) {
      if ((res = mp_mul_2(a, a)) != MP_OKAY) {
         return res;
      }
      if (mp_cmp_mag(a, b) != MP_LT) {
         if ((res = s_mp_sub(a, b, a)) != MP_OKAY) {
            return res;
         }
      }
   }

   return MP_OKAY;
}

/* End: bn_mp_montgomery_calc_normalization.c */

/* Start: bn_mp_montgomery_reduce.c */

/* computes xR**-1 == x (mod N) via Montgomery Reduction */
int mp_montgomery_reduce(mp_int *x, const mp_int *n, mp_digit rho)
{
   int     ix, res, digs;
   mp_digit mu;

   /* can the fast reduction [comba] method be used?
    *
    * Note that unlike in mul you're safely allowed *less*
    * than the available columns [255 per default] since carries
    * are fixed up in the inner loop.
    */
   digs = (n->used * 2) + 1;
   if ((digs < (int)MP_WARRAY) &&
       (x->used <= (int)MP_WARRAY) &&
       (n->used <
        (int)(1u << (((size_t)CHAR_BIT * sizeof(mp_word)) - (2u * (size_t)DIGIT_BIT))))) {
      return fast_mp_montgomery_reduce(x, n, rho);
   }

   /* grow the input as required */
   if (x->alloc < digs) {
      if ((res = mp_grow(x, digs)) != MP_OKAY) {
         return res;
      }
   }
   x->used = digs;

   for (ix = 0; ix < n->used; ix++) {
      /* mu = ai * rho mod b
       *
       * The value of rho must be precalculated via
       * montgomery_setup() such that
       * it equals -1/n0 mod b this allows the
       * following inner loop to reduce the
       * input one digit at a time
       */
      mu = (mp_digit)(((mp_word)x->dp[ix] * (mp_word)rho) & MP_MASK);

      /* a = a + mu * m * b**i */
      {
         int iy;
         mp_digit *tmpn, *tmpx, u;
         mp_word r;

         /* alias for digits of the modulus */
         tmpn = n->dp;

         /* alias for the digits of x [the input] */
         tmpx = x->dp + ix;

         /* set the carry to zero */
         u = 0;

         /* Multiply and add in place */
         for (iy = 0; iy < n->used; iy++) {
            /* compute product and sum */
            r       = ((mp_word)mu * (mp_word)*tmpn++) +
                      (mp_word)u + (mp_word)*tmpx;

            /* get carry */
            u       = (mp_digit)(r >> (mp_word)DIGIT_BIT);

            /* fix digit */
            *tmpx++ = (mp_digit)(r & (mp_word)MP_MASK);
         }
         /* At this point the ix'th digit of x should be zero */


         /* propagate carries upwards as required*/
         while (u != 0u) {
            *tmpx   += u;
            u        = *tmpx >> DIGIT_BIT;
            *tmpx++ &= MP_MASK;
         }
      }
   }

   /* at this point the n.used'th least
    * significant digits of x are all zero
    * which means we can shift x to the
    * right by n.used digits and the
    * residue is unchanged.
    */

   /* x = x/b**n.used */
   mp_clamp(x);
   mp_rshd(x, n->used);

   /* if x >= n then x = x - n */
   if (mp_cmp_mag(x, n) != MP_LT) {
      return s_mp_sub(x, n, x);
   }

   return MP_OKAY;
}

/* End: bn_mp_montgomery_reduce.c */

/* Start: bn_mp_montgomery_setup.c */

/* setups the montgomery reduction stuff */
int mp_montgomery_setup(const mp_int *n, mp_digit *rho)
{
   mp_digit x, b;

   /* fast inversion mod 2**k
    *
    * Based on the fact that
    *
    * XA = 1 (mod 2**n)  =>  (X(2-XA)) A = 1 (mod 2**2n)
    *                    =>  2*X*A - X*X*A*A = 1
    *                    =>  2*(1) - (1)     = 1
    */
   b = n->dp[0];

   if ((b & 1u) == 0u) {
      return MP_VAL;
   }

   x = (((b + 2u) & 4u) << 1) + b; /* here x*a==1 mod 2**4 */
   x *= 2u - (b * x);              /* here x*a==1 mod 2**8 */
#if !defined(MP_8BIT)
   x *= 2u - (b * x);              /* here x*a==1 mod 2**16 */
#endif
#if defined(MP_64BIT) || !(defined(MP_8BIT) || defined(MP_16BIT))
   x *= 2u - (b * x);              /* here x*a==1 mod 2**32 */
#endif
#ifdef MP_64BIT
   x *= 2u - (b * x);              /* here x*a==1 mod 2**64 */
#endif

   /* rho = -1/m mod b */
   *rho = (mp_digit)(((mp_word)1 << (mp_word)DIGIT_BIT) - x) & MP_MASK;

   return MP_OKAY;
}

/* End: bn_mp_montgomery_setup.c */

/* Start: bn_mp_mul.c */

/* high level multiplication (handles sign) */
int mp_mul(const mp_int *a, const mp_int *b, mp_int *c)
{
   int     res, neg;
   neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG;

   /* use Toom-Cook? */
   if (MIN(a->used, b->used) >= TOOM_MUL_CUTOFF) {
      res = mp_toom_mul(a, b, c);
   } else
      /* use Karatsuba? */
      if (MIN(a->used, b->used) >= KARATSUBA_MUL_CUTOFF) {
         res = mp_karatsuba_mul(a, b, c);
      } else
      {
         /* can we use the fast multiplier?
          *
          * The fast multiplier can be used if the output will
          * have less than MP_WARRAY digits and the number of
          * digits won't affect carry propagation
          */
         int     digs = a->used + b->used + 1;

         if ((digs < (int)MP_WARRAY) &&
             (MIN(a->used, b->used) <=
              (int)(1u << (((size_t)CHAR_BIT * sizeof(mp_word)) - (2u * (size_t)DIGIT_BIT))))) {
            res = fast_s_mp_mul_digs(a, b, c, digs);
         } else
         {
            res = s_mp_mul(a, b, c); /* uses s_mp_mul_digs */
         }
      }
   c->sign = (c->used > 0) ? neg : MP_ZPOS;
   return res;
}

/* End: bn_mp_mul.c */

/* Start: bn_mp_mul_2.c */

/* b = a*2 */
int mp_mul_2(const mp_int *a, mp_int *b)
{
   int     x, res, oldused;

   /* grow to accomodate result */
   if (b->alloc < (a->used + 1)) {
      if ((res = mp_grow(b, a->used + 1)) != MP_OKAY) {
         return res;
      }
   }

   oldused = b->used;
   b->used = a->used;

   {
      mp_digit r, rr, *tmpa, *tmpb;

      /* alias for source */
      tmpa = a->dp;

      /* alias for dest */
      tmpb = b->dp;

      /* carry */
      r = 0;
      for (x = 0; x < a->used; x++) {

         /* get what will be the *next* carry bit from the
          * MSB of the current digit
          */
         rr = *tmpa >> (mp_digit)(DIGIT_BIT - 1);

         /* now shift up this digit, add in the carry [from the previous] */
         *tmpb++ = ((*tmpa++ << 1uL) | r) & MP_MASK;

         /* copy the carry that would be from the source
          * digit into the next iteration
          */
         r = rr;
      }

      /* new leading digit? */
      if (r != 0u) {
         /* add a MSB which is always 1 at this point */
         *tmpb = 1;
         ++(b->used);
      }

      /* now zero any excess digits on the destination
       * that we didn't write to
       */
      tmpb = b->dp + b->used;
      for (x = b->used; x < oldused; x++) {
         *tmpb++ = 0;
      }
   }
   b->sign = a->sign;
   return MP_OKAY;
}

/* End: bn_mp_mul_2.c */

/* Start: bn_mp_mul_2d.c */

/* shift left by a certain bit count */
int mp_mul_2d(const mp_int *a, int b, mp_int *c)
{
   mp_digit d;
   int      res;

   /* copy */
   if (a != c) {
      if ((res = mp_copy(a, c)) != MP_OKAY) {
         return res;
      }
   }

   if (c->alloc < (c->used + (b / DIGIT_BIT) + 1)) {
      if ((res = mp_grow(c, c->used + (b / DIGIT_BIT) + 1)) != MP_OKAY) {
         return res;
      }
   }

   /* shift by as many digits in the bit count */
   if (b >= DIGIT_BIT) {
      if ((res = mp_lshd(c, b / DIGIT_BIT)) != MP_OKAY) {
         return res;
      }
   }

   /* shift any bit count < DIGIT_BIT */
   d = (mp_digit)(b % DIGIT_BIT);
   if (d != 0u) {
      mp_digit *tmpc, shift, mask, r, rr;
      int x;

      /* bitmask for carries */
      mask = ((mp_digit)1 << d) - (mp_digit)1;

      /* shift for msbs */
      shift = (mp_digit)DIGIT_BIT - d;

      /* alias */
      tmpc = c->dp;

      /* carry */
      r    = 0;
      for (x = 0; x < c->used; x++) {
         /* get the higher bits of the current word */
         rr = (*tmpc >> shift) & mask;

         /* shift the current word and OR in the carry */
         *tmpc = ((*tmpc << d) | r) & MP_MASK;
         ++tmpc;

         /* set the carry to the carry bits of the current word */
         r = rr;
      }

      /* set final carry */
      if (r != 0u) {
         c->dp[(c->used)++] = r;
      }
   }
   mp_clamp(c);
   return MP_OKAY;
}

/* End: bn_mp_mul_2d.c */

/* Start: bn_mp_mul_d.c */

/* multiply by a digit */
int mp_mul_d(const mp_int *a, mp_digit b, mp_int *c)
{
   mp_digit u, *tmpa, *tmpc;
   mp_word  r;
   int      ix, res, olduse;

   /* make sure c is big enough to hold a*b */
   if (c->alloc < (a->used + 1)) {
      if ((res = mp_grow(c, a->used + 1)) != MP_OKAY) {
         return res;
      }
   }

   /* get the original destinations used count */
   olduse = c->used;

   /* set the sign */
   c->sign = a->sign;

   /* alias for a->dp [source] */
   tmpa = a->dp;

   /* alias for c->dp [dest] */
   tmpc = c->dp;

   /* zero carry */
   u = 0;

   /* compute columns */
   for (ix = 0; ix < a->used; ix++) {
      /* compute product and carry sum for this term */
      r       = (mp_word)u + ((mp_word)*tmpa++ * (mp_word)b);

      /* mask off higher bits to get a single digit */
      *tmpc++ = (mp_digit)(r & (mp_word)MP_MASK);

      /* send carry into next iteration */
      u       = (mp_digit)(r >> (mp_word)DIGIT_BIT);
   }

   /* store final carry [if any] and increment ix offset  */
   *tmpc++ = u;
   ++ix;

   /* now zero digits above the top */
   while (ix++ < olduse) {
      *tmpc++ = 0;
   }

   /* set used count */
   c->used = a->used + 1;
   mp_clamp(c);

   return MP_OKAY;
}

/* End: bn_mp_mul_d.c */

/* Start: bn_mp_mulmod.c */

/* d = a * b (mod c) */
int mp_mulmod(const mp_int *a, const mp_int *b, const mp_int *c, mp_int *d)
{
   int     res;
   mp_int  t;

   if ((res = mp_init_size(&t, c->used)) != MP_OKAY) {
      return res;
   }

   if ((res = mp_mul(a, b, &t)) != MP_OKAY) {
      mp_clear(&t);
      return res;
   }
   res = mp_mod(&t, c, d);
   mp_clear(&t);
   return res;
}

/* End: bn_mp_mulmod.c */

/* Start: bn_mp_n_root.c */

/* wrapper function for mp_n_root_ex()
 * computes c = (a)**(1/b) such that (c)**b <= a and (c+1)**b > a
 */
int mp_n_root(const mp_int *a, mp_digit b, mp_int *c)
{
   return mp_n_root_ex(a, b, c, 0);
}

/* End: bn_mp_n_root.c */

/* Start: bn_mp_n_root_ex.c */

/* find the n'th root of an integer
 *
 * Result found such that (c)**b <= a and (c+1)**b > a
 *
 * This algorithm uses Newton's approximation
 * x[i+1] = x[i] - f(x[i])/f'(x[i])
 * which will find the root in log(N) time where
 * each step involves a fair bit.  This is not meant to
 * find huge roots [square and cube, etc].
 */
int mp_n_root_ex(const mp_int *a, mp_digit b, mp_int *c, int fast)
{
   mp_int  t1, t2, t3, a_;
   int     res;

   /* input must be positive if b is even */
   if (((b & 1u) == 0u) && (a->sign == MP_NEG)) {
      return MP_VAL;
   }

   if ((res = mp_init(&t1)) != MP_OKAY) {
      return res;
   }

   if ((res = mp_init(&t2)) != MP_OKAY) {
      goto LBL_T1;
   }

   if ((res = mp_init(&t3)) != MP_OKAY) {
      goto LBL_T2;
   }

   /* if a is negative fudge the sign but keep track */
   a_ = *a;
   a_.sign = MP_ZPOS;

   /* t2 = 2 */
   mp_set(&t2, 2uL);

   do {
      /* t1 = t2 */
      if ((res = mp_copy(&t2, &t1)) != MP_OKAY) {
         goto LBL_T3;
      }

      /* t2 = t1 - ((t1**b - a) / (b * t1**(b-1))) */

      /* t3 = t1**(b-1) */
      if ((res = mp_expt_d_ex(&t1, b - 1u, &t3, fast)) != MP_OKAY) {
         goto LBL_T3;
      }

      /* numerator */
      /* t2 = t1**b */
      if ((res = mp_mul(&t3, &t1, &t2)) != MP_OKAY) {
         goto LBL_T3;
      }

      /* t2 = t1**b - a */
      if ((res = mp_sub(&t2, &a_, &t2)) != MP_OKAY) {
         goto LBL_T3;
      }

      /* denominator */
      /* t3 = t1**(b-1) * b  */
      if ((res = mp_mul_d(&t3, b, &t3)) != MP_OKAY) {
         goto LBL_T3;
      }

      /* t3 = (t1**b - a)/(b * t1**(b-1)) */
      if ((res = mp_div(&t2, &t3, &t3, NULL)) != MP_OKAY) {
         goto LBL_T3;
      }

      if ((res = mp_sub(&t1, &t3, &t2)) != MP_OKAY) {
         goto LBL_T3;
      }
   }  while (mp_cmp(&t1, &t2) != MP_EQ);

   /* result can be off by a few so check */
   for (;;) {
      if ((res = mp_expt_d_ex(&t1, b, &t2, fast)) != MP_OKAY) {
         goto LBL_T3;
      }

      if (mp_cmp(&t2, &a_) == MP_GT) {
         if ((res = mp_sub_d(&t1, 1uL, &t1)) != MP_OKAY) {
            goto LBL_T3;
         }
      } else {
         break;
      }
   }

   /* set the result */
   mp_exch(&t1, c);

   /* set the sign of the result */
   c->sign = a->sign;

   res = MP_OKAY;

LBL_T3:
   mp_clear(&t3);
LBL_T2:
   mp_clear(&t2);
LBL_T1:
   mp_clear(&t1);
   return res;
}

/* End: bn_mp_n_root_ex.c */

/* Start: bn_mp_neg.c */

/* b = -a */
int mp_neg(const mp_int *a, mp_int *b)
{
   int     res;
   if (a != b) {
      if ((res = mp_copy(a, b)) != MP_OKAY) {
         return res;
      }
   }

   if (mp_iszero(b) != MP_YES) {
      b->sign = (a->sign == MP_ZPOS) ? MP_NEG : MP_ZPOS;
   } else {
      b->sign = MP_ZPOS;
   }

   return MP_OKAY;
}

/* End: bn_mp_neg.c */

/* Start: bn_mp_or.c */

/* OR two ints together */
int mp_or(const mp_int *a, const mp_int *b, mp_int *c)
{
   int     res, ix, px;
   mp_int  t;
   const mp_int *x;

   if (a->used > b->used) {
      if ((res = mp_init_copy(&t, a)) != MP_OKAY) {
         return res;
      }
      px = b->used;
      x = b;
   } else {
      if ((res = mp_init_copy(&t, b)) != MP_OKAY) {
         return res;
      }
      px = a->used;
      x = a;
   }

   for (ix = 0; ix < px; ix++) {
      t.dp[ix] |= x->dp[ix];
   }
   mp_clamp(&t);
   mp_exch(c, &t);
   mp_clear(&t);
   return MP_OKAY;
}

/* End: bn_mp_or.c */

/* Start: bn_mp_prime_fermat.c */

/* performs one Fermat test.
 *
 * If "a" were prime then b**a == b (mod a) since the order of
 * the multiplicative sub-group would be phi(a) = a-1.  That means
 * it would be the same as b**(a mod (a-1)) == b**1 == b (mod a).
 *
 * Sets result to 1 if the congruence holds, or zero otherwise.
 */
int mp_prime_fermat(const mp_int *a, const mp_int *b, int *result)
{
   mp_int  t;
   int     err;

   /* default to composite  */
   *result = MP_NO;

   /* ensure b > 1 */
   if (mp_cmp_d(b, 1uL) != MP_GT) {
      return MP_VAL;
   }

   /* init t */
   if ((err = mp_init(&t)) != MP_OKAY) {
      return err;
   }

   /* compute t = b**a mod a */
   if ((err = mp_exptmod(b, a, a, &t)) != MP_OKAY) {
      goto LBL_T;
   }

   /* is it equal to b? */
   if (mp_cmp(&t, b) == MP_EQ) {
      *result = MP_YES;
   }

   err = MP_OKAY;
LBL_T:
   mp_clear(&t);
   return err;
}

/* End: bn_mp_prime_fermat.c */

/* Start: bn_mp_prime_frobenius_underwood.c */

/*
 *  See file bn_mp_prime_is_prime.c or the documentation in doc/bn.tex for the details
 */
#ifndef LTM_USE_FIPS_ONLY

#ifdef MP_8BIT
/*
 * floor of positive solution of
 * (2^16)-1 = (a+4)*(2*a+5)
 * TODO: Both values are smaller than N^(1/4), would have to use a bigint
 *       for a instead but any a biger than about 120 are already so rare that
 *       it is possible to ignore them and still get enough pseudoprimes.
 *       But it is still a restriction of the set of available pseudoprimes
 *       which makes this implementation less secure if used stand-alone.
 */
#define LTM_FROBENIUS_UNDERWOOD_A 177
#else
#define LTM_FROBENIUS_UNDERWOOD_A 32764
#endif
int mp_prime_frobenius_underwood(const mp_int *N, int *result)
{
   mp_int T1z, T2z, Np1z, sz, tz;

   int a, ap2, length, i, j, isset;
   int e;

   *result = MP_NO;

   if ((e = mp_init_multi(&T1z, &T2z, &Np1z, &sz, &tz, NULL)) != MP_OKAY) {
      return e;
   }

   for (a = 0; a < LTM_FROBENIUS_UNDERWOOD_A; a++) {
      /* TODO: That's ugly! No, really, it is! */
      if ((a==2) || (a==4) || (a==7) || (a==8) || (a==10) ||
          (a==14) || (a==18) || (a==23) || (a==26) || (a==28)) {
         continue;
      }
      /* (32764^2 - 4) < 2^31, no bigint for >MP_8BIT needed) */
      if ((e = mp_set_long(&T1z, (unsigned long)a)) != MP_OKAY) {
         goto LBL_FU_ERR;
      }

      if ((e = mp_sqr(&T1z, &T1z)) != MP_OKAY) {
         goto LBL_FU_ERR;
      }

      if ((e = mp_sub_d(&T1z, 4uL, &T1z)) != MP_OKAY) {
         goto LBL_FU_ERR;
      }

      if ((e = mp_kronecker(&T1z, N, &j)) != MP_OKAY) {
         goto LBL_FU_ERR;
      }

      if (j == -1) {
         break;
      }

      if (j == 0) {
         /* composite */
         goto LBL_FU_ERR;
      }
   }
   /* Tell it a composite and set return value accordingly */
   if (a >= LTM_FROBENIUS_UNDERWOOD_A) {
      e = MP_ITER;
      goto LBL_FU_ERR;
   }
   /* Composite if N and (a+4)*(2*a+5) are not coprime */
   if ((e = mp_set_long(&T1z, (unsigned long)((a+4)*((2*a)+5)))) != MP_OKAY) {
      goto LBL_FU_ERR;
   }

   if ((e = mp_gcd(N, &T1z, &T1z)) != MP_OKAY) {
      goto LBL_FU_ERR;
   }

   if (!((T1z.used == 1) && (T1z.dp[0] == 1u))) {
      goto LBL_FU_ERR;
   }

   ap2 = a + 2;
   if ((e = mp_add_d(N, 1uL, &Np1z)) != MP_OKAY) {
      goto LBL_FU_ERR;
   }

   mp_set(&sz, 1uL);
   mp_set(&tz, 2uL);
   length = mp_count_bits(&Np1z);

   for (i = length - 2; i >= 0; i--) {
      /*
       * temp = (sz*(a*sz+2*tz))%N;
       * tz   = ((tz-sz)*(tz+sz))%N;
       * sz   = temp;
       */
      if ((e = mp_mul_2(&tz, &T2z)) != MP_OKAY) {
         goto LBL_FU_ERR;
      }

      /* a = 0 at about 50% of the cases (non-square and odd input) */
      if (a != 0) {
         if ((e = mp_mul_d(&sz, (mp_digit)a, &T1z)) != MP_OKAY) {
            goto LBL_FU_ERR;
         }
         if ((e = mp_add(&T1z, &T2z, &T2z)) != MP_OKAY) {
            goto LBL_FU_ERR;
         }
      }

      if ((e = mp_mul(&T2z, &sz, &T1z)) != MP_OKAY) {
         goto LBL_FU_ERR;
      }
      if ((e = mp_sub(&tz, &sz, &T2z)) != MP_OKAY) {
         goto LBL_FU_ERR;
      }
      if ((e = mp_add(&sz, &tz, &sz)) != MP_OKAY) {
         goto LBL_FU_ERR;
      }
      if ((e = mp_mul(&sz, &T2z, &tz)) != MP_OKAY) {
         goto LBL_FU_ERR;
      }
      if ((e = mp_mod(&tz, N, &tz)) != MP_OKAY) {
         goto LBL_FU_ERR;
      }
      if ((e = mp_mod(&T1z, N, &sz)) != MP_OKAY) {
         goto LBL_FU_ERR;
      }
      if ((isset = mp_get_bit(&Np1z, i)) == MP_VAL) {
         e = isset;
         goto LBL_FU_ERR;
      }
      if (isset == MP_YES) {
         /*
          *  temp = (a+2) * sz + tz
          *  tz   = 2 * tz - sz
          *  sz   = temp
          */
         if (a == 0) {
            if ((e = mp_mul_2(&sz, &T1z)) != MP_OKAY) {
               goto LBL_FU_ERR;
            }
         } else {
            if ((e = mp_mul_d(&sz, (mp_digit)ap2, &T1z)) != MP_OKAY) {
               goto LBL_FU_ERR;
            }
         }
         if ((e = mp_add(&T1z, &tz, &T1z)) != MP_OKAY) {
            goto LBL_FU_ERR;
         }
         if ((e = mp_mul_2(&tz, &T2z)) != MP_OKAY) {
            goto LBL_FU_ERR;
         }
         if ((e = mp_sub(&T2z, &sz, &tz)) != MP_OKAY) {
            goto LBL_FU_ERR;
         }
         mp_exch(&sz, &T1z);
      }
   }

   if ((e = mp_set_long(&T1z, (unsigned long)((2 * a) + 5))) != MP_OKAY) {
      goto LBL_FU_ERR;
   }
   if ((e = mp_mod(&T1z, N, &T1z)) != MP_OKAY) {
      goto LBL_FU_ERR;
   }
   if ((mp_iszero(&sz) != MP_NO) && (mp_cmp(&tz, &T1z) == MP_EQ)) {
      *result = MP_YES;
      goto LBL_FU_ERR;
   }

LBL_FU_ERR:
   mp_clear_multi(&tz, &sz, &Np1z, &T2z, &T1z, NULL);
   return e;
}

#endif

/* End: bn_mp_prime_frobenius_underwood.c */

/* Start: bn_mp_prime_is_divisible.c */

/* determines if an integers is divisible by one
 * of the first PRIME_SIZE primes or not
 *
 * sets result to 0 if not, 1 if yes
 */
int mp_prime_is_divisible(const mp_int *a, int *result)
{
   int     err, ix;
   mp_digit res;

   /* default to not */
   *result = MP_NO;

   for (ix = 0; ix < PRIME_SIZE; ix++) {
      /* what is a mod LBL_prime_tab[ix] */
      if ((err = mp_mod_d(a, ltm_prime_tab[ix], &res)) != MP_OKAY) {
         return err;
      }

      /* is the residue zero? */
      if (res == 0u) {
         *result = MP_YES;
         return MP_OKAY;
      }
   }

   return MP_OKAY;
}

/* End: bn_mp_prime_is_divisible.c */

/* Start: bn_mp_prime_is_prime.c */

/* portable integer log of two with small footprint */
static unsigned int s_floor_ilog2(int value)
{
   unsigned int r = 0;
   while ((value >>= 1) != 0) {
      r++;
   }
   return r;
}


int mp_prime_is_prime(const mp_int *a, int t, int *result)
{
   mp_int  b;
   int     ix, err, res, p_max = 0, size_a, len;
   unsigned int fips_rand, mask;

   /* default to no */
   *result = MP_NO;

   /* valid value of t? */
   if (t > PRIME_SIZE) {
      return MP_VAL;
   }

   /* Some shortcuts */
   /* N > 3 */
   if (a->used == 1) {
      if ((a->dp[0] == 0u) || (a->dp[0] == 1u)) {
         *result = 0;
         return MP_OKAY;
      }
      if (a->dp[0] == 2u) {
         *result = 1;
         return MP_OKAY;
      }
   }

   /* N must be odd */
   if (mp_iseven(a) == MP_YES) {
      return MP_OKAY;
   }
   /* N is not a perfect square: floor(sqrt(N))^2 != N */
   if ((err = mp_is_square(a, &res)) != MP_OKAY) {
      return err;
   }
   if (res != 0) {
      return MP_OKAY;
   }

   /* is the input equal to one of the primes in the table? */
   for (ix = 0; ix < PRIME_SIZE; ix++) {
      if (mp_cmp_d(a, ltm_prime_tab[ix]) == MP_EQ) {
         *result = MP_YES;
         return MP_OKAY;
      }
   }
#ifdef MP_8BIT
   /* The search in the loop above was exhaustive in this case */
   if ((a->used == 1) && (PRIME_SIZE >= 31)) {
      return MP_OKAY;
   }
#endif

   /* first perform trial division */
   if ((err = mp_prime_is_divisible(a, &res)) != MP_OKAY) {
      return err;
   }

   /* return if it was trivially divisible */
   if (res == MP_YES) {
      return MP_OKAY;
   }

   /*
       Run the Miller-Rabin test with base 2 for the BPSW test.
    */
   if ((err = mp_init_set(&b, 2uL)) != MP_OKAY) {
      return err;
   }

   if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) {
      goto LBL_B;
   }
   if (res == MP_NO) {
      goto LBL_B;
   }
   /*
      Rumours have it that Mathematica does a second M-R test with base 3.
      Other rumours have it that their strong L-S test is slightly different.
      It does not hurt, though, beside a bit of extra runtime.
   */
   b.dp[0]++;
   if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) {
      goto LBL_B;
   }
   if (res == MP_NO) {
      goto LBL_B;
   }

   /*
    * Both, the Frobenius-Underwood test and the the Lucas-Selfridge test are quite
    * slow so if speed is an issue, define LTM_USE_FIPS_ONLY to use M-R tests with
    * bases 2, 3 and t random bases.
    */
#ifndef LTM_USE_FIPS_ONLY
   if (t >= 0) {
      /*
       * Use a Frobenius-Underwood test instead of the Lucas-Selfridge test for
       * MP_8BIT (It is unknown if the Lucas-Selfridge test works with 16-bit
       * integers but the necesssary analysis is on the todo-list).
       */
#if defined (MP_8BIT) || defined (LTM_USE_FROBENIUS_TEST)
      err = mp_prime_frobenius_underwood(a, &res);
      if ((err != MP_OKAY) && (err != MP_ITER)) {
         goto LBL_B;
      }
      if (res == MP_NO) {
         goto LBL_B;
      }
#else
      if ((err = mp_prime_strong_lucas_selfridge(a, &res)) != MP_OKAY) {
         goto LBL_B;
      }
      if (res == MP_NO) {
         goto LBL_B;
      }
#endif
   }
#endif

   /* run at least one Miller-Rabin test with a random base */
   if (t == 0) {
      t = 1;
   }

   /*
      abs(t) extra rounds of M-R to extend the range of primes it can find if t < 0.
      Only recommended if the input range is known to be < 3317044064679887385961981

      It uses the bases for a deterministic M-R test if input < 3317044064679887385961981
      The caller has to check the size.

      Not for cryptographic use because with known bases strong M-R pseudoprimes can
      be constructed. Use at least one M-R test with a random base (t >= 1).

      The 1119 bit large number

      80383745745363949125707961434194210813883768828755814583748891752229742737653\
      33652186502336163960045457915042023603208766569966760987284043965408232928738\
      79185086916685732826776177102938969773947016708230428687109997439976544144845\
      34115587245063340927902227529622941498423068816854043264575340183297861112989\
      60644845216191652872597534901

      has been constructed by F. Arnault (F. Arnault, "Rabin-Miller primality test:
      composite numbers which pass it.",  Mathematics of Computation, 1995, 64. Jg.,
      Nr. 209, S. 355-361), is a semiprime with the two factors

      40095821663949960541830645208454685300518816604113250877450620473800321707011\
      96242716223191597219733582163165085358166969145233813917169287527980445796800\
      452592031836601

      20047910831974980270915322604227342650259408302056625438725310236900160853505\
      98121358111595798609866791081582542679083484572616906958584643763990222898400\
      226296015918301

      and it is a strong pseudoprime to all forty-six prime M-R bases up to 200

      It does not fail the strong Bailley-PSP test as implemented here, it is just
      given as an example, if not the reason to use the BPSW-test instead of M-R-tests
      with a sequence of primes 2...n.

   */
   if (t < 0) {
      t = -t;
      /*
          Sorenson, Jonathan; Webster, Jonathan (2015).
           "Strong Pseudoprimes to Twelve Prime Bases".
       */
      /* 0x437ae92817f9fc85b7e5 = 318665857834031151167461 */
      if ((err =   mp_read_radix(&b, "437ae92817f9fc85b7e5", 16)) != MP_OKAY) {
         goto LBL_B;
      }

      if (mp_cmp(a, &b) == MP_LT) {
         p_max = 12;
      } else {
         /* 0x2be6951adc5b22410a5fd = 3317044064679887385961981 */
         if ((err = mp_read_radix(&b, "2be6951adc5b22410a5fd", 16)) != MP_OKAY) {
            goto LBL_B;
         }

         if (mp_cmp(a, &b) == MP_LT) {
            p_max = 13;
         } else {
            err = MP_VAL;
            goto LBL_B;
         }
      }

      /* for compatibility with the current API (well, compatible within a sign's width) */
      if (p_max < t) {
         p_max = t;
      }

      if (p_max > PRIME_SIZE) {
         err = MP_VAL;
         goto LBL_B;
      }
      /* we did bases 2 and 3  already, skip them */
      for (ix = 2; ix < p_max; ix++) {
         mp_set(&b, ltm_prime_tab[ix]);
         if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) {
            goto LBL_B;
         }
         if (res == MP_NO) {
            goto LBL_B;
         }
      }
   }
   /*
       Do "t" M-R tests with random bases between 3 and "a".
       See Fips 186.4 p. 126ff
   */
   else if (t > 0) {
      /*
       * The mp_digit's have a defined bit-size but the size of the
       * array a.dp is a simple 'int' and this library can not assume full
       * compliance to the current C-standard (ISO/IEC 9899:2011) because
       * it gets used for small embeded processors, too. Some of those MCUs
       * have compilers that one cannot call standard compliant by any means.
       * Hence the ugly type-fiddling in the following code.
       */
      size_a = mp_count_bits(a);
      mask = (1u << s_floor_ilog2(size_a)) - 1u;
      /*
         Assuming the General Rieman hypothesis (never thought to write that in a
         comment) the upper bound can be lowered to  2*(log a)^2.
         E. Bach, "Explicit bounds for primality testing and related problems,"
         Math. Comp. 55 (1990), 355-380.

            size_a = (size_a/10) * 7;
            len = 2 * (size_a * size_a);

         E.g.: a number of size 2^2048 would be reduced to the upper limit

            floor(2048/10)*7 = 1428
            2 * 1428^2       = 4078368

         (would have been ~4030331.9962 with floats and natural log instead)
         That number is smaller than 2^28, the default bit-size of mp_digit.
      */

      /*
        How many tests, you might ask? Dana Jacobsen of Math::Prime::Util fame
        does exactly 1. In words: one. Look at the end of _GMP_is_prime() in
        Math-Prime-Util-GMP-0.50/primality.c if you do not believe it.

        The function mp_rand() goes to some length to use a cryptographically
        good PRNG. That also means that the chance to always get the same base
        in the loop is non-zero, although very low.
        If the BPSW test and/or the addtional Frobenious test have been
        performed instead of just the Miller-Rabin test with the bases 2 and 3,
        a single extra test should suffice, so such a very unlikely event
        will not do much harm.

        To preemptivly answer the dangling question: no, a witness does not
        need to be prime.
      */
      for (ix = 0; ix < t; ix++) {
         /* mp_rand() guarantees the first digit to be non-zero */
         if ((err = mp_rand(&b, 1)) != MP_OKAY) {
            goto LBL_B;
         }
         /*
          * Reduce digit before casting because mp_digit might be bigger than
          * an unsigned int and "mask" on the other side is most probably not.
          */
         fips_rand = (unsigned int)(b.dp[0] & (mp_digit) mask);
#ifdef MP_8BIT
         /*
          * One 8-bit digit is too small, so concatenate two if the size of
          * unsigned int allows for it.
          */
         if (((sizeof(unsigned int) * CHAR_BIT)/2) >= (sizeof(mp_digit) * CHAR_BIT)) {
            if ((err = mp_rand(&b, 1)) != MP_OKAY) {
               goto LBL_B;
            }
            fips_rand <<= sizeof(mp_digit) * CHAR_BIT;
            fips_rand |= (unsigned int) b.dp[0];
            fips_rand &= mask;
         }
#endif
         if (fips_rand > (unsigned int)(INT_MAX - DIGIT_BIT)) {
            len = INT_MAX / DIGIT_BIT;
         } else {
            len = (((int)fips_rand + DIGIT_BIT) / DIGIT_BIT);
         }
         /*  Unlikely. */
         if (len < 0) {
            ix--;
            continue;
         }
         /*
          * As mentioned above, one 8-bit digit is too small and
          * although it can only happen in the unlikely case that
          * an "unsigned int" is smaller than 16 bit a simple test
          * is cheap and the correction even cheaper.
          */
#ifdef MP_8BIT
         /* All "a" < 2^8 have been caught before */
         if (len == 1) {
            len++;
         }
#endif
         if ((err = mp_rand(&b, len)) != MP_OKAY) {
            goto LBL_B;
         }
         /*
          * That number might got too big and the witness has to be
          * smaller than or equal to "a"
          */
         len = mp_count_bits(&b);
         if (len > size_a) {
            len = len - size_a;
            if ((err = mp_div_2d(&b, len, &b, NULL)) != MP_OKAY) {
               goto LBL_B;
            }
         }

         /* Although the chance for b <= 3 is miniscule, try again. */
         if (mp_cmp_d(&b, 3uL) != MP_GT) {
            ix--;
            continue;
         }
         if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) {
            goto LBL_B;
         }
         if (res == MP_NO) {
            goto LBL_B;
         }
      }
   }

   /* passed the test */
   *result = MP_YES;
LBL_B:
   mp_clear(&b);
   return err;
}

/* End: bn_mp_prime_is_prime.c */

/* Start: bn_mp_prime_miller_rabin.c */

/* Miller-Rabin test of "a" to the base of "b" as described in
 * HAC pp. 139 Algorithm 4.24
 *
 * Sets result to 0 if definitely composite or 1 if probably prime.
 * Randomly the chance of error is no more than 1/4 and often
 * very much lower.
 */
int mp_prime_miller_rabin(const mp_int *a, const mp_int *b, int *result)
{
   mp_int  n1, y, r;
   int     s, j, err;

   /* default */
   *result = MP_NO;

   /* ensure b > 1 */
   if (mp_cmp_d(b, 1uL) != MP_GT) {
      return MP_VAL;
   }

   /* get n1 = a - 1 */
   if ((err = mp_init_copy(&n1, a)) != MP_OKAY) {
      return err;
   }
   if ((err = mp_sub_d(&n1, 1uL, &n1)) != MP_OKAY) {
      goto LBL_N1;
   }

   /* set 2**s * r = n1 */
   if ((err = mp_init_copy(&r, &n1)) != MP_OKAY) {
      goto LBL_N1;
   }

   /* count the number of least significant bits
    * which are zero
    */
   s = mp_cnt_lsb(&r);

   /* now divide n - 1 by 2**s */
   if ((err = mp_div_2d(&r, s, &r, NULL)) != MP_OKAY) {
      goto LBL_R;
   }

   /* compute y = b**r mod a */
   if ((err = mp_init(&y)) != MP_OKAY) {
      goto LBL_R;
   }
   if ((err = mp_exptmod(b, &r, a, &y)) != MP_OKAY) {
      goto LBL_Y;
   }

   /* if y != 1 and y != n1 do */
   if ((mp_cmp_d(&y, 1uL) != MP_EQ) && (mp_cmp(&y, &n1) != MP_EQ)) {
      j = 1;
      /* while j <= s-1 and y != n1 */
      while ((j <= (s - 1)) && (mp_cmp(&y, &n1) != MP_EQ)) {
         if ((err = mp_sqrmod(&y, a, &y)) != MP_OKAY) {
            goto LBL_Y;
         }

         /* if y == 1 then composite */
         if (mp_cmp_d(&y, 1uL) == MP_EQ) {
            goto LBL_Y;
         }

         ++j;
      }

      /* if y != n1 then composite */
      if (mp_cmp(&y, &n1) != MP_EQ) {
         goto LBL_Y;
      }
   }

   /* probably prime now */
   *result = MP_YES;
LBL_Y:
   mp_clear(&y);
LBL_R:
   mp_clear(&r);
LBL_N1:
   mp_clear(&n1);
   return err;
}

/* End: bn_mp_prime_miller_rabin.c */

/* Start: bn_mp_prime_next_prime.c */

/* finds the next prime after the number "a" using "t" trials
 * of Miller-Rabin.
 *
 * bbs_style = 1 means the prime must be congruent to 3 mod 4
 */
int mp_prime_next_prime(mp_int *a, int t, int bbs_style)
{
   int      err, res = MP_NO, x, y;
   mp_digit res_tab[PRIME_SIZE], step, kstep;
   mp_int   b;

   /* force positive */
   a->sign = MP_ZPOS;

   /* simple algo if a is less than the largest prime in the table */
   if (mp_cmp_d(a, ltm_prime_tab[PRIME_SIZE-1]) == MP_LT) {
      /* find which prime it is bigger than */
      for (x = PRIME_SIZE - 2; x >= 0; x--) {
         if (mp_cmp_d(a, ltm_prime_tab[x]) != MP_LT) {
            if (bbs_style == 1) {
               /* ok we found a prime smaller or
                * equal [so the next is larger]
                *
                * however, the prime must be
                * congruent to 3 mod 4
                */
               if ((ltm_prime_tab[x + 1] & 3u) != 3u) {
                  /* scan upwards for a prime congruent to 3 mod 4 */
                  for (y = x + 1; y < PRIME_SIZE; y++) {
                     if ((ltm_prime_tab[y] & 3u) == 3u) {
                        mp_set(a, ltm_prime_tab[y]);
                        return MP_OKAY;
                     }
                  }
               }
            } else {
               mp_set(a, ltm_prime_tab[x + 1]);
               return MP_OKAY;
            }
         }
      }
      /* at this point a maybe 1 */
      if (mp_cmp_d(a, 1uL) == MP_EQ) {
         mp_set(a, 2uL);
         return MP_OKAY;
      }
      /* fall through to the sieve */
   }

   /* generate a prime congruent to 3 mod 4 or 1/3 mod 4? */
   if (bbs_style == 1) {
      kstep   = 4;
   } else {
      kstep   = 2;
   }

   /* at this point we will use a combination of a sieve and Miller-Rabin */

   if (bbs_style == 1) {
      /* if a mod 4 != 3 subtract the correct value to make it so */
      if ((a->dp[0] & 3u) != 3u) {
         if ((err = mp_sub_d(a, (a->dp[0] & 3u) + 1u, a)) != MP_OKAY) {
            return err;
         };
      }
   } else {
      if (mp_iseven(a) == MP_YES) {
         /* force odd */
         if ((err = mp_sub_d(a, 1uL, a)) != MP_OKAY) {
            return err;
         }
      }
   }

   /* generate the restable */
   for (x = 1; x < PRIME_SIZE; x++) {
      if ((err = mp_mod_d(a, ltm_prime_tab[x], res_tab + x)) != MP_OKAY) {
         return err;
      }
   }

   /* init temp used for Miller-Rabin Testing */
   if ((err = mp_init(&b)) != MP_OKAY) {
      return err;
   }

   for (;;) {
      /* skip to the next non-trivially divisible candidate */
      step = 0;
      do {
         /* y == 1 if any residue was zero [e.g. cannot be prime] */
         y     =  0;

         /* increase step to next candidate */
         step += kstep;

         /* compute the new residue without using division */
         for (x = 1; x < PRIME_SIZE; x++) {
            /* add the step to each residue */
            res_tab[x] += kstep;

            /* subtract the modulus [instead of using division] */
            if (res_tab[x] >= ltm_prime_tab[x]) {
               res_tab[x]  -= ltm_prime_tab[x];
            }

            /* set flag if zero */
            if (res_tab[x] == 0u) {
               y = 1;
            }
         }
      } while ((y == 1) && (step < (((mp_digit)1 << DIGIT_BIT) - kstep)));

      /* add the step */
      if ((err = mp_add_d(a, step, a)) != MP_OKAY) {
         goto LBL_ERR;
      }

      /* if didn't pass sieve and step == MAX then skip test */
      if ((y == 1) && (step >= (((mp_digit)1 << DIGIT_BIT) - kstep))) {
         continue;
      }

      if ((err = mp_prime_is_prime(a, t, &res)) != MP_OKAY) {
         goto LBL_ERR;
      }
      if (res == MP_YES) {
         break;
      }
   }

   err = MP_OKAY;
LBL_ERR:
   mp_clear(&b);
   return err;
}

/* End: bn_mp_prime_next_prime.c */

/* Start: bn_mp_prime_rabin_miller_trials.c */

static const struct {
   int k, t;
} sizes[] = {
   {    80,    -1 }, /* Use deterministic algorithm for size <= 80 bits */
   {    81,    39 },
   {    96,    37 },
   {   128,    32 },
   {   160,    27 },
   {   192,    21 },
   {   256,    16 },
   {   384,    10 },
   {   512,     7 },
   {   640,     6 },
   {   768,     5 },
   {   896,     4 },
   {  1024,     4 },
   {  2048,     2 },
   {  4096,     1 },
};

/* returns # of RM trials required for a given bit size and max. error of 2^(-96)*/
int mp_prime_rabin_miller_trials(int size)
{
   int x;

   for (x = 0; x < (int)(sizeof(sizes)/(sizeof(sizes[0]))); x++) {
      if (sizes[x].k == size) {
         return sizes[x].t;
      } else if (sizes[x].k > size) {
         return (x == 0) ? sizes[0].t : sizes[x - 1].t;
      }
   }
   return sizes[x-1].t + 1;
}

/* End: bn_mp_prime_rabin_miller_trials.c */

/* Start: bn_mp_prime_random_ex.c */

/* makes a truly random prime of a given size (bits),
 *
 * Flags are as follows:
 *
 *   LTM_PRIME_BBS      - make prime congruent to 3 mod 4
 *   LTM_PRIME_SAFE     - make sure (p-1)/2 is prime as well (implies LTM_PRIME_BBS)
 *   LTM_PRIME_2MSB_ON  - make the 2nd highest bit one
 *
 * You have to supply a callback which fills in a buffer with random bytes.  "dat" is a parameter you can
 * have passed to the callback (e.g. a state or something).  This function doesn't use "dat" itself
 * so it can be NULL
 *
 */

/* This is possibly the mother of all prime generation functions, muahahahahaha! */
int mp_prime_random_ex(mp_int *a, int t, int size, int flags, ltm_prime_callback cb, void *dat)
{
   unsigned char *tmp, maskAND, maskOR_msb, maskOR_lsb;
   int res, err, bsize, maskOR_msb_offset;

   /* sanity check the input */
   if ((size <= 1) || (t <= 0)) {
      return MP_VAL;
   }

   /* LTM_PRIME_SAFE implies LTM_PRIME_BBS */
   if ((flags & LTM_PRIME_SAFE) != 0) {
      flags |= LTM_PRIME_BBS;
   }

   /* calc the byte size */
   bsize = (size>>3) + ((size&7)?1:0);

   /* we need a buffer of bsize bytes */
   tmp = OPT_CAST(unsigned char) XMALLOC((size_t)bsize);
   if (tmp == NULL) {
      return MP_MEM;
   }

   /* calc the maskAND value for the MSbyte*/
   maskAND = ((size&7) == 0) ? 0xFF : (0xFF >> (8 - (size & 7)));

   /* calc the maskOR_msb */
   maskOR_msb        = 0;
   maskOR_msb_offset = ((size & 7) == 1) ? 1 : 0;
   if ((flags & LTM_PRIME_2MSB_ON) != 0) {
      maskOR_msb       |= 0x80 >> ((9 - size) & 7);
   }

   /* get the maskOR_lsb */
   maskOR_lsb         = 1;
   if ((flags & LTM_PRIME_BBS) != 0) {
      maskOR_lsb     |= 3;
   }

   do {
      /* read the bytes */
      if (cb(tmp, bsize, dat) != bsize) {
         err = MP_VAL;
         goto error;
      }

      /* work over the MSbyte */
      tmp[0]    &= maskAND;
      tmp[0]    |= 1 << ((size - 1) & 7);

      /* mix in the maskORs */
      tmp[maskOR_msb_offset]   |= maskOR_msb;
      tmp[bsize-1]             |= maskOR_lsb;

      /* read it in */
      if ((err = mp_read_unsigned_bin(a, tmp, bsize)) != MP_OKAY) {
         goto error;
      }

      /* is it prime? */
      if ((err = mp_prime_is_prime(a, t, &res)) != MP_OKAY) {
         goto error;
      }
      if (res == MP_NO) {
         continue;
      }

      if ((flags & LTM_PRIME_SAFE) != 0) {
         /* see if (a-1)/2 is prime */
         if ((err = mp_sub_d(a, 1uL, a)) != MP_OKAY) {
            goto error;
         }
         if ((err = mp_div_2(a, a)) != MP_OKAY) {
            goto error;
         }

         /* is it prime? */
         if ((err = mp_prime_is_prime(a, t, &res)) != MP_OKAY) {
            goto error;
         }
      }
   } while (res == MP_NO);

   if ((flags & LTM_PRIME_SAFE) != 0) {
      /* restore a to the original value */
      if ((err = mp_mul_2(a, a)) != MP_OKAY) {
         goto error;
      }
      if ((err = mp_add_d(a, 1uL, a)) != MP_OKAY) {
         goto error;
      }
   }

   err = MP_OKAY;
error:
   XFREE(tmp);
   return err;
}

/* End: bn_mp_prime_random_ex.c */

/* Start: bn_mp_prime_strong_lucas_selfridge.c */

/*
 *  See file bn_mp_prime_is_prime.c or the documentation in doc/bn.tex for the details
 */
#ifndef LTM_USE_FIPS_ONLY

/*
 *  8-bit is just too small. You can try the Frobenius test
 *  but that frobenius test can fail, too, for the same reason.
 */
#ifndef MP_8BIT

/*
 * multiply bigint a with int d and put the result in c
 * Like mp_mul_d() but with a signed long as the small input
 */
static int s_mp_mul_si(const mp_int *a, long d, mp_int *c)
{
   mp_int t;
   int err, neg = 0;

   if ((err = mp_init(&t)) != MP_OKAY) {
      return err;
   }
   if (d < 0) {
      neg = 1;
      d = -d;
   }

   /*
    * mp_digit might be smaller than a long, which excludes
    * the use of mp_mul_d() here.
    */
   if ((err = mp_set_long(&t, (unsigned long) d)) != MP_OKAY) {
      goto LBL_MPMULSI_ERR;
   }
   if ((err = mp_mul(a, &t, c)) != MP_OKAY) {
      goto LBL_MPMULSI_ERR;
   }
   if (neg ==  1) {
      c->sign = (a->sign == MP_NEG) ? MP_ZPOS: MP_NEG;
   }
LBL_MPMULSI_ERR:
   mp_clear(&t);
   return err;
}
/*
    Strong Lucas-Selfridge test.
    returns MP_YES if it is a strong L-S prime, MP_NO if it is composite

    Code ported from  Thomas Ray Nicely's implementation of the BPSW test
    at http://www.trnicely.net/misc/bpsw.html

    Freeware copyright (C) 2016 Thomas R. Nicely <http://www.trnicely.net>.
    Released into the public domain by the author, who disclaims any legal
    liability arising from its use

    The multi-line comments are made by Thomas R. Nicely and are copied verbatim.
    Additional comments marked "CZ" (without the quotes) are by the code-portist.

    (If that name sounds familiar, he is the guy who found the fdiv bug in the
     Pentium (P5x, I think) Intel processor)
*/
int mp_prime_strong_lucas_selfridge(const mp_int *a, int *result)
{
   /* CZ TODO: choose better variable names! */
   mp_int Dz, gcd, Np1, Uz, Vz, U2mz, V2mz, Qmz, Q2mz, Qkdz, T1z, T2z, T3z, T4z, Q2kdz;
   /* CZ TODO: Some of them need the full 32 bit, hence the (temporary) exclusion of MP_8BIT */
   int32_t D, Ds, J, sign, P, Q, r, s, u, Nbits;
   int e;
   int isset, oddness;

   *result = MP_NO;
   /*
   Find the first element D in the sequence {5, -7, 9, -11, 13, ...}
   such that Jacobi(D,N) = -1 (Selfridge's algorithm). Theory
   indicates that, if N is not a perfect square, D will "nearly
   always" be "small." Just in case, an overflow trap for D is
   included.
   */

   if ((e = mp_init_multi(&Dz, &gcd, &Np1, &Uz, &Vz, &U2mz, &V2mz, &Qmz, &Q2mz, &Qkdz, &T1z, &T2z, &T3z, &T4z, &Q2kdz,
                          NULL)) != MP_OKAY) {
      return e;
   }

   D = 5;
   sign = 1;

   for (;;) {
      Ds   = sign * D;
      sign = -sign;
      if ((e = mp_set_long(&Dz, (unsigned long)D)) != MP_OKAY) {
         goto LBL_LS_ERR;
      }
      if ((e = mp_gcd(a, &Dz, &gcd)) != MP_OKAY) {
         goto LBL_LS_ERR;
      }
      /* if 1 < GCD < N then N is composite with factor "D", and
         Jacobi(D,N) is technically undefined (but often returned
         as zero). */
      if ((mp_cmp_d(&gcd, 1uL) == MP_GT) && (mp_cmp(&gcd, a) == MP_LT)) {
         goto LBL_LS_ERR;
      }
      if (Ds < 0) {
         Dz.sign = MP_NEG;
      }
      if ((e = mp_kronecker(&Dz, a, &J)) != MP_OKAY) {
         goto LBL_LS_ERR;
      }

      if (J == -1) {
         break;
      }
      D += 2;

      if (D > (INT_MAX - 2)) {
         e = MP_VAL;
         goto LBL_LS_ERR;
      }
   }



   P = 1;              /* Selfridge's choice */
   Q = (1 - Ds) / 4;   /* Required so D = P*P - 4*Q */

   /* NOTE: The conditions (a) N does not divide Q, and
      (b) D is square-free or not a perfect square, are included by
      some authors; e.g., "Prime numbers and computer methods for
      factorization," Hans Riesel (2nd ed., 1994, Birkhauser, Boston),
      p. 130. For this particular application of Lucas sequences,
      these conditions were found to be immaterial. */

   /* Now calculate N - Jacobi(D,N) = N + 1 (even), and calculate the
      odd positive integer d and positive integer s for which
      N + 1 = 2^s*d (similar to the step for N - 1 in Miller's test).
      The strong Lucas-Selfridge test then returns N as a strong
      Lucas probable prime (slprp) if any of the following
      conditions is met: U_d=0, V_d=0, V_2d=0, V_4d=0, V_8d=0,
      V_16d=0, ..., etc., ending with V_{2^(s-1)*d}=V_{(N+1)/2}=0
      (all equalities mod N). Thus d is the highest index of U that
      must be computed (since V_2m is independent of U), compared
      to U_{N+1} for the standard Lucas-Selfridge test; and no
      index of V beyond (N+1)/2 is required, just as in the
      standard Lucas-Selfridge test. However, the quantity Q^d must
      be computed for use (if necessary) in the latter stages of
      the test. The result is that the strong Lucas-Selfridge test
      has a running time only slightly greater (order of 10 %) than
      that of the standard Lucas-Selfridge test, while producing
      only (roughly) 30 % as many pseudoprimes (and every strong
      Lucas pseudoprime is also a standard Lucas pseudoprime). Thus
      the evidence indicates that the strong Lucas-Selfridge test is
      more effective than the standard Lucas-Selfridge test, and a
      Baillie-PSW test based on the strong Lucas-Selfridge test
      should be more reliable. */

   if ((e = mp_add_d(a, 1uL, &Np1)) != MP_OKAY) {
      goto LBL_LS_ERR;
   }
   s = mp_cnt_lsb(&Np1);

   /* CZ
    * This should round towards zero because
    * Thomas R. Nicely used GMP's mpz_tdiv_q_2exp()
    * and mp_div_2d() is equivalent. Additionally:
    * dividing an even number by two does not produce
    * any leftovers.
    */
   if ((e = mp_div_2d(&Np1, s, &Dz, NULL)) != MP_OKAY) {
      goto LBL_LS_ERR;
   }
   /* We must now compute U_d and V_d. Since d is odd, the accumulated
      values U and V are initialized to U_1 and V_1 (if the target
      index were even, U and V would be initialized instead to U_0=0
      and V_0=2). The values of U_2m and V_2m are also initialized to
      U_1 and V_1; the FOR loop calculates in succession U_2 and V_2,
      U_4 and V_4, U_8 and V_8, etc. If the corresponding bits
      (1, 2, 3, ...) of t are on (the zero bit having been accounted
      for in the initialization of U and V), these values are then
      combined with the previous totals for U and V, using the
      composition formulas for addition of indices. */

   mp_set(&Uz, 1uL);    /* U=U_1 */
   mp_set(&Vz, (mp_digit)P);    /* V=V_1 */
   mp_set(&U2mz, 1uL);  /* U_1 */
   mp_set(&V2mz, (mp_digit)P);  /* V_1 */

   if (Q < 0) {
      Q = -Q;
      if ((e = mp_set_long(&Qmz, (unsigned long)Q)) != MP_OKAY) {
         goto LBL_LS_ERR;
      }
      if ((e = mp_mul_2(&Qmz, &Q2mz)) != MP_OKAY) {
         goto LBL_LS_ERR;
      }
      /* Initializes calculation of Q^d */
      if ((e = mp_set_long(&Qkdz, (unsigned long)Q)) != MP_OKAY) {
         goto LBL_LS_ERR;
      }
      Qmz.sign = MP_NEG;
      Q2mz.sign = MP_NEG;
      Qkdz.sign = MP_NEG;
      Q = -Q;
   } else {
      if ((e = mp_set_long(&Qmz, (unsigned long)Q)) != MP_OKAY) {
         goto LBL_LS_ERR;
      }
      if ((e = mp_mul_2(&Qmz, &Q2mz)) != MP_OKAY) {
         goto LBL_LS_ERR;
      }
      /* Initializes calculation of Q^d */
      if ((e = mp_set_long(&Qkdz, (unsigned long)Q)) != MP_OKAY) {
         goto LBL_LS_ERR;
      }
   }

   Nbits = mp_count_bits(&Dz);

   for (u = 1; u < Nbits; u++) { /* zero bit off, already accounted for */
      /* Formulas for doubling of indices (carried out mod N). Note that
       * the indices denoted as "2m" are actually powers of 2, specifically
       * 2^(ul-1) beginning each loop and 2^ul ending each loop.
       *
       * U_2m = U_m*V_m
       * V_2m = V_m*V_m - 2*Q^m
       */

      if ((e = mp_mul(&U2mz, &V2mz, &U2mz)) != MP_OKAY) {
         goto LBL_LS_ERR;
      }
      if ((e = mp_mod(&U2mz, a, &U2mz)) != MP_OKAY) {
         goto LBL_LS_ERR;
      }
      if ((e = mp_sqr(&V2mz, &V2mz)) != MP_OKAY) {
         goto LBL_LS_ERR;
      }
      if ((e = mp_sub(&V2mz, &Q2mz, &V2mz)) != MP_OKAY) {
         goto LBL_LS_ERR;
      }
      if ((e = mp_mod(&V2mz, a, &V2mz)) != MP_OKAY) {
         goto LBL_LS_ERR;
      }
      /* Must calculate powers of Q for use in V_2m, also for Q^d later */
      if ((e = mp_sqr(&Qmz, &Qmz)) != MP_OKAY) {
         goto LBL_LS_ERR;
      }
      /* prevents overflow */ /* CZ  still necessary without a fixed prealloc'd mem.? */
      if ((e = mp_mod(&Qmz, a, &Qmz)) != MP_OKAY) {
         goto LBL_LS_ERR;
      }
      if ((e = mp_mul_2(&Qmz, &Q2mz)) != MP_OKAY) {
         goto LBL_LS_ERR;
      }
      if ((isset = mp_get_bit(&Dz, u)) == MP_VAL) {
         e = isset;
         goto LBL_LS_ERR;
      }
      if (isset == MP_YES) {
         /* Formulas for addition of indices (carried out mod N);
          *
          * U_(m+n) = (U_m*V_n + U_n*V_m)/2
          * V_(m+n) = (V_m*V_n + D*U_m*U_n)/2
          *
          * Be careful with division by 2 (mod N)!
          */
         if ((e = mp_mul(&U2mz, &Vz, &T1z)) != MP_OKAY) {
            goto LBL_LS_ERR;
         }
         if ((e = mp_mul(&Uz, &V2mz, &T2z)) != MP_OKAY) {
            goto LBL_LS_ERR;
         }
         if ((e = mp_mul(&V2mz, &Vz, &T3z)) != MP_OKAY) {
            goto LBL_LS_ERR;
         }
         if ((e = mp_mul(&U2mz, &Uz, &T4z)) != MP_OKAY) {
            goto LBL_LS_ERR;
         }
         if ((e = s_mp_mul_si(&T4z, (long)Ds, &T4z)) != MP_OKAY) {
            goto LBL_LS_ERR;
         }
         if ((e = mp_add(&T1z, &T2z, &Uz)) != MP_OKAY) {
            goto LBL_LS_ERR;
         }
         if (mp_isodd(&Uz) != MP_NO) {
            if ((e = mp_add(&Uz, a, &Uz)) != MP_OKAY) {
               goto LBL_LS_ERR;
            }
         }
         /* CZ
          * This should round towards negative infinity because
          * Thomas R. Nicely used GMP's mpz_fdiv_q_2exp().
          * But mp_div_2() does not do so, it is truncating instead.
          */
         oddness = mp_isodd(&Uz);
         if ((e = mp_div_2(&Uz, &Uz)) != MP_OKAY) {
            goto LBL_LS_ERR;
         }
         if ((Uz.sign == MP_NEG) && (oddness != MP_NO)) {
            if ((e = mp_sub_d(&Uz, 1uL, &Uz)) != MP_OKAY) {
               goto LBL_LS_ERR;
            }
         }
         if ((e = mp_add(&T3z, &T4z, &Vz)) != MP_OKAY) {
            goto LBL_LS_ERR;
         }
         if (mp_isodd(&Vz) != MP_NO) {
            if ((e = mp_add(&Vz, a, &Vz)) != MP_OKAY) {
               goto LBL_LS_ERR;
            }
         }
         oddness = mp_isodd(&Vz);
         if ((e = mp_div_2(&Vz, &Vz)) != MP_OKAY) {
            goto LBL_LS_ERR;
         }
         if ((Vz.sign == MP_NEG) && (oddness != MP_NO)) {
            if ((e = mp_sub_d(&Vz, 1uL, &Vz)) != MP_OKAY) {
               goto LBL_LS_ERR;
            }
         }
         if ((e = mp_mod(&Uz, a, &Uz)) != MP_OKAY) {
            goto LBL_LS_ERR;
         }
         if ((e = mp_mod(&Vz, a, &Vz)) != MP_OKAY) {
            goto LBL_LS_ERR;
         }
         /* Calculating Q^d for later use */
         if ((e = mp_mul(&Qkdz, &Qmz, &Qkdz)) != MP_OKAY) {
            goto LBL_LS_ERR;
         }
         if ((e = mp_mod(&Qkdz, a, &Qkdz)) != MP_OKAY) {
            goto LBL_LS_ERR;
         }
      }
   }

   /* If U_d or V_d is congruent to 0 mod N, then N is a prime or a
      strong Lucas pseudoprime. */
   if ((mp_iszero(&Uz) != MP_NO) || (mp_iszero(&Vz) != MP_NO)) {
      *result = MP_YES;
      goto LBL_LS_ERR;
   }

   /* NOTE: Ribenboim ("The new book of prime number records," 3rd ed.,
      1995/6) omits the condition V0 on p.142, but includes it on
      p. 130. The condition is NECESSARY; otherwise the test will
      return false negatives---e.g., the primes 29 and 2000029 will be
      returned as composite. */

   /* Otherwise, we must compute V_2d, V_4d, V_8d, ..., V_{2^(s-1)*d}
      by repeated use of the formula V_2m = V_m*V_m - 2*Q^m. If any of
      these are congruent to 0 mod N, then N is a prime or a strong
      Lucas pseudoprime. */

   /* Initialize 2*Q^(d*2^r) for V_2m */
   if ((e = mp_mul_2(&Qkdz, &Q2kdz)) != MP_OKAY) {
      goto LBL_LS_ERR;
   }

   for (r = 1; r < s; r++) {
      if ((e = mp_sqr(&Vz, &Vz)) != MP_OKAY) {
         goto LBL_LS_ERR;
      }
      if ((e = mp_sub(&Vz, &Q2kdz, &Vz)) != MP_OKAY) {
         goto LBL_LS_ERR;
      }
      if ((e = mp_mod(&Vz, a, &Vz)) != MP_OKAY) {
         goto LBL_LS_ERR;
      }
      if (mp_iszero(&Vz) != MP_NO) {
         *result = MP_YES;
         goto LBL_LS_ERR;
      }
      /* Calculate Q^{d*2^r} for next r (final iteration irrelevant). */
      if (r < (s - 1)) {
         if ((e = mp_sqr(&Qkdz, &Qkdz)) != MP_OKAY) {
            goto LBL_LS_ERR;
         }
         if ((e = mp_mod(&Qkdz, a, &Qkdz)) != MP_OKAY) {
            goto LBL_LS_ERR;
         }
         if ((e = mp_mul_2(&Qkdz, &Q2kdz)) != MP_OKAY) {
            goto LBL_LS_ERR;
         }
      }
   }
LBL_LS_ERR:
   mp_clear_multi(&Q2kdz, &T4z, &T3z, &T2z, &T1z, &Qkdz, &Q2mz, &Qmz, &V2mz, &U2mz, &Vz, &Uz, &Np1, &gcd, &Dz, NULL);
   return e;
}
#endif
#endif

/* End: bn_mp_prime_strong_lucas_selfridge.c */

/* Start: bn_mp_radix_size.c */

/* returns size of ASCII reprensentation */
int mp_radix_size(const mp_int *a, int radix, int *size)
{
   int     res, digs;
   mp_int  t;
   mp_digit d;

   *size = 0;

   /* make sure the radix is in range */
   if ((radix < 2) || (radix > 64)) {
      return MP_VAL;
   }

   if (mp_iszero(a) == MP_YES) {
      *size = 2;
      return MP_OKAY;
   }

   /* special case for binary */
   if (radix == 2) {
      *size = mp_count_bits(a) + ((a->sign == MP_NEG) ? 1 : 0) + 1;
      return MP_OKAY;
   }

   /* digs is the digit count */
   digs = 0;

   /* if it's negative add one for the sign */
   if (a->sign == MP_NEG) {
      ++digs;
   }

   /* init a copy of the input */
   if ((res = mp_init_copy(&t, a)) != MP_OKAY) {
      return res;
   }

   /* force temp to positive */
   t.sign = MP_ZPOS;

   /* fetch out all of the digits */
   while (mp_iszero(&t) == MP_NO) {
      if ((res = mp_div_d(&t, (mp_digit)radix, &t, &d)) != MP_OKAY) {
         mp_clear(&t);
         return res;
      }
      ++digs;
   }
   mp_clear(&t);

   /* return digs + 1, the 1 is for the NULL byte that would be required. */
   *size = digs + 1;
   return MP_OKAY;
}

/* End: bn_mp_radix_size.c */

/* Start: bn_mp_radix_smap.c */

/* chars used in radix conversions */
const char *const mp_s_rmap = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz+/";
const uint8_t mp_s_rmap_reverse[] = {
   0xff, 0xff, 0xff, 0x3e, 0xff, 0xff, 0xff, 0x3f, /* ()*+,-./ */
   0x00, 0x01, 0x02, 0x03, 0x04, 0x05, 0x06, 0x07, /* 01234567 */
   0x08, 0x09, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, /* 89:;<=>? */
   0xff, 0x0a, 0x0b, 0x0c, 0x0d, 0x0e, 0x0f, 0x10, /* @ABCDEFG */
   0x11, 0x12, 0x13, 0x14, 0x15, 0x16, 0x17, 0x18, /* HIJKLMNO */
   0x19, 0x1a, 0x1b, 0x1c, 0x1d, 0x1e, 0x1f, 0x20, /* PQRSTUVW */
   0x21, 0x22, 0x23, 0xff, 0xff, 0xff, 0xff, 0xff, /* XYZ[\]^_ */
   0xff, 0x24, 0x25, 0x26, 0x27, 0x28, 0x29, 0x2a, /* `abcdefg */
   0x2b, 0x2c, 0x2d, 0x2e, 0x2f, 0x30, 0x31, 0x32, /* hijklmno */
   0x33, 0x34, 0x35, 0x36, 0x37, 0x38, 0x39, 0x3a, /* pqrstuvw */
   0x3b, 0x3c, 0x3d, 0xff, 0xff, 0xff, 0xff, 0xff, /* xyz{|}~. */
};
const size_t mp_s_rmap_reverse_sz = sizeof(mp_s_rmap_reverse);

/* End: bn_mp_radix_smap.c */

/* Start: bn_mp_rand.c */

/* First the OS-specific special cases
 * - *BSD
 * - Windows
 */
#if defined(__FreeBSD__) || defined(__OpenBSD__) || defined(__NetBSD__) || defined(__DragonFly__)
#define MP_ARC4RANDOM
#define MP_GEN_RANDOM_MAX     0xffffffffu
#define MP_GEN_RANDOM_SHIFT   32

static int s_read_arc4random(mp_digit *p)
{
   mp_digit d = 0, msk = 0;
   do {
      d <<= MP_GEN_RANDOM_SHIFT;
      d |= ((mp_digit) arc4random());
      msk <<= MP_GEN_RANDOM_SHIFT;
      msk |= (MP_MASK & MP_GEN_RANDOM_MAX);
   } while ((MP_MASK & msk) != MP_MASK);
   *p = d;
   return MP_OKAY;
}
#endif

#if defined(_WIN32) || defined(_WIN32_WCE)
#define MP_WIN_CSP

#ifndef _WIN32_WINNT
#define _WIN32_WINNT 0x0400
#endif
#ifdef _WIN32_WCE
#define UNDER_CE
#define ARM
#endif

#define WIN32_LEAN_AND_MEAN
#include <windows.h>
#include <ntsecapi.h>

static int s_read_win_csp(mp_digit *p)
{
   int ret = -1;
   if (RtlGenRandom(p, sizeof(*p)) == TRUE) {
      ret = MP_OKAY;
   }
   return ret;
}
#endif /* WIN32 */

#if !defined(MP_WIN_CSP) && defined(__linux__) && defined(__GLIBC_PREREQ)
#if __GLIBC_PREREQ(2, 25)
#define MP_GETRANDOM
#include <sys/random.h>
#include <errno.h>

static int s_read_getrandom(mp_digit *p)
{
   int ret;
   do {
      ret = getrandom(p, sizeof(*p), 0);
   } while ((ret == -1) && (errno == EINTR));
   if (ret == sizeof(*p)) return MP_OKAY;
   return -1;
}
#endif
#endif

/* We assume all platforms besides windows provide "/dev/urandom".
 * In case yours doesn't, define MP_NO_DEV_URANDOM at compile-time.
 */
#if !defined(MP_WIN_CSP) && !defined(MP_NO_DEV_URANDOM)
#ifndef MP_DEV_URANDOM
#define MP_DEV_URANDOM "/dev/urandom"
#endif
#include <fcntl.h>
#include <errno.h>
#include <unistd.h>

static int s_read_dev_urandom(mp_digit *p)
{
   ssize_t r;
   int fd;
   do {
      fd = open(MP_DEV_URANDOM, O_RDONLY);
   } while ((fd == -1) && (errno == EINTR));
   if (fd == -1) return -1;
   do {
      r = read(fd, p, sizeof(*p));
   } while ((r == -1) && (errno == EINTR));
   close(fd);
   if (r != sizeof(*p)) return -1;
   return MP_OKAY;
}
#endif

#if defined(MP_PRNG_ENABLE_LTM_RNG)
unsigned long (*ltm_rng)(unsigned char *out, unsigned long outlen, void (*callback)(void));
void (*ltm_rng_callback)(void);

static int s_read_ltm_rng(mp_digit *p)
{
   unsigned long ret;
   if (ltm_rng == NULL) return -1;
   ret = ltm_rng((void *)p, sizeof(*p), ltm_rng_callback);
   if (ret != sizeof(*p)) return -1;
   return MP_OKAY;
}
#endif

static int s_rand_digit(mp_digit *p)
{
   int ret = -1;

#if defined(MP_ARC4RANDOM)
   ret = s_read_arc4random(p);
   if (ret == MP_OKAY) return ret;
#endif

#if defined(MP_WIN_CSP)
   ret = s_read_win_csp(p);
   if (ret == MP_OKAY) return ret;
#else

#if defined(MP_GETRANDOM)
   ret = s_read_getrandom(p);
   if (ret == MP_OKAY) return ret;
#endif
#if defined(MP_DEV_URANDOM)
   ret = s_read_dev_urandom(p);
   if (ret == MP_OKAY) return ret;
#endif

#endif /* MP_WIN_CSP */

#if defined(MP_PRNG_ENABLE_LTM_RNG)
   ret = s_read_ltm_rng(p);
   if (ret == MP_OKAY) return ret;
#endif

   return ret;
}

/* makes a pseudo-random int of a given size */
int mp_rand_digit(mp_digit *r)
{
   int ret = s_rand_digit(r);
   *r &= MP_MASK;
   return ret;
}

int mp_rand(mp_int *a, int digits)
{
   int     res;
   mp_digit d;

   mp_zero(a);
   if (digits <= 0) {
      return MP_OKAY;
   }

   /* first place a random non-zero digit */
   do {
      if (mp_rand_digit(&d) != MP_OKAY) {
         return MP_VAL;
      }
   } while (d == 0u);

   if ((res = mp_add_d(a, d, a)) != MP_OKAY) {
      return res;
   }

   while (--digits > 0) {
      if ((res = mp_lshd(a, 1)) != MP_OKAY) {
         return res;
      }

      if (mp_rand_digit(&d) != MP_OKAY) {
         return MP_VAL;
      }
      if ((res = mp_add_d(a, d, a)) != MP_OKAY) {
         return res;
      }
   }

   return MP_OKAY;
}

/* Start: bn_mp_read_radix.c */

/* read a string [ASCII] in a given radix */
int mp_read_radix(mp_int *a, const char *str, int radix)
{
   int     y, res, neg;
   unsigned pos;
   char    ch;

   /* zero the digit bignum */
   mp_zero(a);

   /* make sure the radix is ok */
   if ((radix < 2) || (radix > 64)) {
      return MP_VAL;
   }

   /* if the leading digit is a
    * minus set the sign to negative.
    */
   if (*str == '-') {
      ++str;
      neg = MP_NEG;
   } else {
      neg = MP_ZPOS;
   }

   /* set the integer to the default of zero */
   mp_zero(a);

   /* process each digit of the string */
   while (*str != '\0') {
      /* if the radix <= 36 the conversion is case insensitive
       * this allows numbers like 1AB and 1ab to represent the same  value
       * [e.g. in hex]
       */
      ch = (radix <= 36) ? (char)toupper((int)*str) : *str;
      pos = (unsigned)(ch - '(');
      if (mp_s_rmap_reverse_sz < pos) {
         break;
      }
      y = (int)mp_s_rmap_reverse[pos];

      /* if the char was found in the map
       * and is less than the given radix add it
       * to the number, otherwise exit the loop.
       */
      if ((y == 0xff) || (y >= radix)) {
         break;
      }
      if ((res = mp_mul_d(a, (mp_digit)radix, a)) != MP_OKAY) {
         return res;
      }
      if ((res = mp_add_d(a, (mp_digit)y, a)) != MP_OKAY) {
         return res;
      }
      ++str;
   }

   /* if an illegal character was found, fail. */
   if (!((*str == '\0') || (*str == '\r') || (*str == '\n'))) {
      mp_zero(a);
      return MP_VAL;
   }

   /* set the sign only if a != 0 */
   if (mp_iszero(a) != MP_YES) {
      a->sign = neg;
   }
   return MP_OKAY;
}

/* End: bn_mp_read_radix.c */

/* Start: bn_mp_read_signed_bin.c */

/* read signed bin, big endian, first byte is 0==positive or 1==negative */
int mp_read_signed_bin(mp_int *a, const unsigned char *b, int c)
{
   int     res;

   /* read magnitude */
   if ((res = mp_read_unsigned_bin(a, b + 1, c - 1)) != MP_OKAY) {
      return res;
   }

   /* first byte is 0 for positive, non-zero for negative */
   if (b[0] == (unsigned char)0) {
      a->sign = MP_ZPOS;
   } else {
      a->sign = MP_NEG;
   }

   return MP_OKAY;
}

/* End: bn_mp_read_signed_bin.c */

/* Start: bn_mp_read_unsigned_bin.c */

/* reads a unsigned char array, assumes the msb is stored first [big endian] */
int mp_read_unsigned_bin(mp_int *a, const unsigned char *b, int c)
{
   int     res;

   /* make sure there are at least two digits */
   if (a->alloc < 2) {
      if ((res = mp_grow(a, 2)) != MP_OKAY) {
         return res;
      }
   }

   /* zero the int */
   mp_zero(a);

   /* read the bytes in */
   while (c-- > 0) {
      if ((res = mp_mul_2d(a, 8, a)) != MP_OKAY) {
         return res;
      }

#ifndef MP_8BIT
      a->dp[0] |= *b++;
      a->used += 1;
#else
      a->dp[0] = (*b & MP_MASK);
      a->dp[1] |= ((*b++ >> 7) & 1u);
      a->used += 2;
#endif
   }
   mp_clamp(a);
   return MP_OKAY;
}

/* End: bn_mp_read_unsigned_bin.c */

/* Start: bn_mp_reduce.c */

/* reduces x mod m, assumes 0 < x < m**2, mu is
 * precomputed via mp_reduce_setup.
 * From HAC pp.604 Algorithm 14.42
 */
int mp_reduce(mp_int *x, const mp_int *m, const mp_int *mu)
{
   mp_int  q;
   int     res, um = m->used;

   /* q = x */
   if ((res = mp_init_copy(&q, x)) != MP_OKAY) {
      return res;
   }

   /* q1 = x / b**(k-1)  */
   mp_rshd(&q, um - 1);

   /* according to HAC this optimization is ok */
   if ((mp_digit)um > ((mp_digit)1 << (DIGIT_BIT - 1))) {
      if ((res = mp_mul(&q, mu, &q)) != MP_OKAY) {
         goto CLEANUP;
      }
   } else {
      if ((res = s_mp_mul_high_digs(&q, mu, &q, um)) != MP_OKAY) {
         goto CLEANUP;
      }
   }

   /* q3 = q2 / b**(k+1) */
   mp_rshd(&q, um + 1);

   /* x = x mod b**(k+1), quick (no division) */
   if ((res = mp_mod_2d(x, DIGIT_BIT * (um + 1), x)) != MP_OKAY) {
      goto CLEANUP;
   }

   /* q = q * m mod b**(k+1), quick (no division) */
   if ((res = s_mp_mul_digs(&q, m, &q, um + 1)) != MP_OKAY) {
      goto CLEANUP;
   }

   /* x = x - q */
   if ((res = mp_sub(x, &q, x)) != MP_OKAY) {
      goto CLEANUP;
   }

   /* If x < 0, add b**(k+1) to it */
   if (mp_cmp_d(x, 0uL) == MP_LT) {
      mp_set(&q, 1uL);
      if ((res = mp_lshd(&q, um + 1)) != MP_OKAY)
         goto CLEANUP;
      if ((res = mp_add(x, &q, x)) != MP_OKAY)
         goto CLEANUP;
   }

   /* Back off if it's too big */
   while (mp_cmp(x, m) != MP_LT) {
      if ((res = s_mp_sub(x, m, x)) != MP_OKAY) {
         goto CLEANUP;
      }
   }

CLEANUP:
   mp_clear(&q);

   return res;
}

/* End: bn_mp_reduce.c */

/* Start: bn_mp_reduce_2k.c */

/* reduces a modulo n where n is of the form 2**p - d */
int mp_reduce_2k(mp_int *a, const mp_int *n, mp_digit d)
{
   mp_int q;
   int    p, res;

   if ((res = mp_init(&q)) != MP_OKAY) {
      return res;
   }

   p = mp_count_bits(n);
top:
   /* q = a/2**p, a = a mod 2**p */
   if ((res = mp_div_2d(a, p, &q, a)) != MP_OKAY) {
      goto LBL_ERR;
   }

   if (d != 1u) {
      /* q = q * d */
      if ((res = mp_mul_d(&q, d, &q)) != MP_OKAY) {
         goto LBL_ERR;
      }
   }

   /* a = a + q */
   if ((res = s_mp_add(a, &q, a)) != MP_OKAY) {
      goto LBL_ERR;
   }

   if (mp_cmp_mag(a, n) != MP_LT) {
      if ((res = s_mp_sub(a, n, a)) != MP_OKAY) {
         goto LBL_ERR;
      }
      goto top;
   }

LBL_ERR:
   mp_clear(&q);
   return res;
}

/* End: bn_mp_reduce_2k.c */

/* Start: bn_mp_reduce_2k_l.c */

/* reduces a modulo n where n is of the form 2**p - d
   This differs from reduce_2k since "d" can be larger
   than a single digit.
*/
int mp_reduce_2k_l(mp_int *a, const mp_int *n, const mp_int *d)
{
   mp_int q;
   int    p, res;

   if ((res = mp_init(&q)) != MP_OKAY) {
      return res;
   }

   p = mp_count_bits(n);
top:
   /* q = a/2**p, a = a mod 2**p */
   if ((res = mp_div_2d(a, p, &q, a)) != MP_OKAY) {
      goto LBL_ERR;
   }

   /* q = q * d */
   if ((res = mp_mul(&q, d, &q)) != MP_OKAY) {
      goto LBL_ERR;
   }

   /* a = a + q */
   if ((res = s_mp_add(a, &q, a)) != MP_OKAY) {
      goto LBL_ERR;
   }

   if (mp_cmp_mag(a, n) != MP_LT) {
      if ((res = s_mp_sub(a, n, a)) != MP_OKAY) {
         goto LBL_ERR;
      }
      goto top;
   }

LBL_ERR:
   mp_clear(&q);
   return res;
}

/* End: bn_mp_reduce_2k_l.c */

/* Start: bn_mp_reduce_2k_setup.c */

/* determines the setup value */
int mp_reduce_2k_setup(const mp_int *a, mp_digit *d)
{
   int res, p;
   mp_int tmp;

   if ((res = mp_init(&tmp)) != MP_OKAY) {
      return res;
   }

   p = mp_count_bits(a);
   if ((res = mp_2expt(&tmp, p)) != MP_OKAY) {
      mp_clear(&tmp);
      return res;
   }

   if ((res = s_mp_sub(&tmp, a, &tmp)) != MP_OKAY) {
      mp_clear(&tmp);
      return res;
   }

   *d = tmp.dp[0];
   mp_clear(&tmp);
   return MP_OKAY;
}

/* End: bn_mp_reduce_2k_setup.c */

/* Start: bn_mp_reduce_2k_setup_l.c */

/* determines the setup value */
int mp_reduce_2k_setup_l(const mp_int *a, mp_int *d)
{
   int    res;
   mp_int tmp;

   if ((res = mp_init(&tmp)) != MP_OKAY) {
      return res;
   }

   if ((res = mp_2expt(&tmp, mp_count_bits(a))) != MP_OKAY) {
      goto LBL_ERR;
   }

   if ((res = s_mp_sub(&tmp, a, d)) != MP_OKAY) {
      goto LBL_ERR;
   }

LBL_ERR:
   mp_clear(&tmp);
   return res;
}

/* End: bn_mp_reduce_2k_setup_l.c */

/* Start: bn_mp_reduce_is_2k.c */

/* determines if mp_reduce_2k can be used */
int mp_reduce_is_2k(const mp_int *a)
{
   int ix, iy, iw;
   mp_digit iz;

   if (a->used == 0) {
      return MP_NO;
   } else if (a->used == 1) {
      return MP_YES;
   } else if (a->used > 1) {
      iy = mp_count_bits(a);
      iz = 1;
      iw = 1;

      /* Test every bit from the second digit up, must be 1 */
      for (ix = DIGIT_BIT; ix < iy; ix++) {
         if ((a->dp[iw] & iz) == 0u) {
            return MP_NO;
         }
         iz <<= 1;
         if (iz > (mp_digit)MP_MASK) {
            ++iw;
            iz = 1;
         }
      }
   }
   return MP_YES;
}

/* End: bn_mp_reduce_is_2k.c */

/* Start: bn_mp_reduce_is_2k_l.c */

/* determines if reduce_2k_l can be used */
int mp_reduce_is_2k_l(const mp_int *a)
{
   int ix, iy;

   if (a->used == 0) {
      return MP_NO;
   } else if (a->used == 1) {
      return MP_YES;
   } else if (a->used > 1) {
      /* if more than half of the digits are -1 we're sold */
      for (iy = ix = 0; ix < a->used; ix++) {
         if (a->dp[ix] == MP_MASK) {
            ++iy;
         }
      }
      return (iy >= (a->used/2)) ? MP_YES : MP_NO;

   }
   return MP_NO;
}

/* End: bn_mp_reduce_is_2k_l.c */

/* Start: bn_mp_reduce_setup.c */

/* pre-calculate the value required for Barrett reduction
 * For a given modulus "b" it calulates the value required in "a"
 */
int mp_reduce_setup(mp_int *a, const mp_int *b)
{
   int     res;

   if ((res = mp_2expt(a, b->used * 2 * DIGIT_BIT)) != MP_OKAY) {
      return res;
   }
   return mp_div(a, b, a, NULL);
}

/* End: bn_mp_reduce_setup.c */

/* Start: bn_mp_rshd.c */

/* shift right a certain amount of digits */
void mp_rshd(mp_int *a, int b)
{
   int     x;

   /* if b <= 0 then ignore it */
   if (b <= 0) {
      return;
   }

   /* if b > used then simply zero it and return */
   if (a->used <= b) {
      mp_zero(a);
      return;
   }

   {
      mp_digit *bottom, *top;

      /* shift the digits down */

      /* bottom */
      bottom = a->dp;

      /* top [offset into digits] */
      top = a->dp + b;

      /* this is implemented as a sliding window where
       * the window is b-digits long and digits from
       * the top of the window are copied to the bottom
       *
       * e.g.

       b-2 | b-1 | b0 | b1 | b2 | ... | bb |   ---->
                   /\                   |      ---->
                    \-------------------/      ---->
       */
      for (x = 0; x < (a->used - b); x++) {
         *bottom++ = *top++;
      }

      /* zero the top digits */
      for (; x < a->used; x++) {
         *bottom++ = 0;
      }
   }

   /* remove excess digits */
   a->used -= b;
}

/* End: bn_mp_rshd.c */

/* Start: bn_mp_set.c */

/* set to a digit */
void mp_set(mp_int *a, mp_digit b)
{
   mp_zero(a);
   a->dp[0] = b & MP_MASK;
   a->used  = (a->dp[0] != 0u) ? 1 : 0;
}

/* End: bn_mp_set.c */

/* Start: bn_mp_set_int.c */

/* set a 32-bit const */
int mp_set_int(mp_int *a, unsigned long b)
{
   int     x, res;

   mp_zero(a);

   /* set four bits at a time */
   for (x = 0; x < 8; x++) {
      /* shift the number up four bits */
      if ((res = mp_mul_2d(a, 4, a)) != MP_OKAY) {
         return res;
      }

      /* OR in the top four bits of the source */
      a->dp[0] |= (mp_digit)(b >> 28) & 15uL;

      /* shift the source up to the next four bits */
      b <<= 4;

      /* ensure that digits are not clamped off */
      a->used += 1;
   }
   mp_clamp(a);
   return MP_OKAY;
}

/* End: bn_mp_set_int.c */

/* Start: bn_mp_set_long.c */

/* set a platform dependent unsigned long int */
MP_SET_XLONG(mp_set_long, unsigned long)

/* End: bn_mp_set_long.c */

/* Start: bn_mp_set_long_long.c */

/* set a platform dependent unsigned long long int */
MP_SET_XLONG(mp_set_long_long, unsigned long long)

/* End: bn_mp_set_long_long.c */

/* Start: bn_mp_shrink.c */

/* shrink a bignum */
int mp_shrink(mp_int *a)
{
   mp_digit *tmp;
   int used = 1;

   if (a->used > 0) {
      used = a->used;
   }

   if (a->alloc != used) {
      if ((tmp = OPT_CAST(mp_digit) XREALLOC(a->dp, sizeof(mp_digit) * (size_t)used)) == NULL) {
         return MP_MEM;
      }
      a->dp    = tmp;
      a->alloc = used;
   }
   return MP_OKAY;
}

/* End: bn_mp_shrink.c */

/* Start: bn_mp_signed_bin_size.c */

/* get the size for an signed equivalent */
int mp_signed_bin_size(const mp_int *a)
{
   return 1 + mp_unsigned_bin_size(a);
}

/* End: bn_mp_signed_bin_size.c */

/* Start: bn_mp_sqr.c */

/* computes b = a*a */
int mp_sqr(const mp_int *a, mp_int *b)
{
   int     res;

   /* use Toom-Cook? */
   if (a->used >= TOOM_SQR_CUTOFF) {
      res = mp_toom_sqr(a, b);
      /* Karatsuba? */
   } else
      if (a->used >= KARATSUBA_SQR_CUTOFF) {
         res = mp_karatsuba_sqr(a, b);
      } else
      {
         /* can we use the fast comba multiplier? */
         if ((((a->used * 2) + 1) < (int)MP_WARRAY) &&
             (a->used <
              (int)(1u << (((sizeof(mp_word) * (size_t)CHAR_BIT) - (2u * (size_t)DIGIT_BIT)) - 1u)))) {
            res = fast_s_mp_sqr(a, b);
         } else
         {
            res = s_mp_sqr(a, b);
         }
      }
   b->sign = MP_ZPOS;
   return res;
}

/* End: bn_mp_sqr.c */

/* Start: bn_mp_sqrmod.c */

/* c = a * a (mod b) */
int mp_sqrmod(const mp_int *a, const mp_int *b, mp_int *c)
{
   int     res;
   mp_int  t;

   if ((res = mp_init(&t)) != MP_OKAY) {
      return res;
   }

   if ((res = mp_sqr(a, &t)) != MP_OKAY) {
      mp_clear(&t);
      return res;
   }
   res = mp_mod(&t, b, c);
   mp_clear(&t);
   return res;
}

/* End: bn_mp_sqrmod.c */

/* Start: bn_mp_sqrt.c */

/* this function is less generic than mp_n_root, simpler and faster */
int mp_sqrt(const mp_int *arg, mp_int *ret)
{
   int res;
   mp_int t1, t2;

   /* must be positive */
   if (arg->sign == MP_NEG) {
      return MP_VAL;
   }

   /* easy out */
   if (mp_iszero(arg) == MP_YES) {
      mp_zero(ret);
      return MP_OKAY;
   }

   if ((res = mp_init_copy(&t1, arg)) != MP_OKAY) {
      return res;
   }

   if ((res = mp_init(&t2)) != MP_OKAY) {
      goto E2;
   }

   /* First approx. (not very bad for large arg) */
   mp_rshd(&t1, t1.used/2);

   /* t1 > 0  */
   if ((res = mp_div(arg, &t1, &t2, NULL)) != MP_OKAY) {
      goto E1;
   }
   if ((res = mp_add(&t1, &t2, &t1)) != MP_OKAY) {
      goto E1;
   }
   if ((res = mp_div_2(&t1, &t1)) != MP_OKAY) {
      goto E1;
   }
   /* And now t1 > sqrt(arg) */
   do {
      if ((res = mp_div(arg, &t1, &t2, NULL)) != MP_OKAY) {
         goto E1;
      }
      if ((res = mp_add(&t1, &t2, &t1)) != MP_OKAY) {
         goto E1;
      }
      if ((res = mp_div_2(&t1, &t1)) != MP_OKAY) {
         goto E1;
      }
      /* t1 >= sqrt(arg) >= t2 at this point */
   } while (mp_cmp_mag(&t1, &t2) == MP_GT);

   mp_exch(&t1, ret);

E1:
   mp_clear(&t2);
E2:
   mp_clear(&t1);
   return res;
}

/* End: bn_mp_sqrt.c */

/* Start: bn_mp_sqrtmod_prime.c */

/* Tonelli-Shanks algorithm
 * https://en.wikipedia.org/wiki/Tonelli%E2%80%93Shanks_algorithm
 * https://gmplib.org/list-archives/gmp-discuss/2013-April/005300.html
 *
 */

int mp_sqrtmod_prime(const mp_int *n, const mp_int *prime, mp_int *ret)
{
   int res, legendre;
   mp_int t1, C, Q, S, Z, M, T, R, two;
   mp_digit i;

   /* first handle the simple cases */
   if (mp_cmp_d(n, 0uL) == MP_EQ) {
      mp_zero(ret);
      return MP_OKAY;
   }
   if (mp_cmp_d(prime, 2uL) == MP_EQ)                            return MP_VAL; /* prime must be odd */
   if ((res = mp_jacobi(n, prime, &legendre)) != MP_OKAY)        return res;
   if (legendre == -1)                                           return MP_VAL; /* quadratic non-residue mod prime */

   if ((res = mp_init_multi(&t1, &C, &Q, &S, &Z, &M, &T, &R, &two, NULL)) != MP_OKAY) {
      return res;
   }

   /* SPECIAL CASE: if prime mod 4 == 3
    * compute directly: res = n^(prime+1)/4 mod prime
    * Handbook of Applied Cryptography algorithm 3.36
    */
   if ((res = mp_mod_d(prime, 4uL, &i)) != MP_OKAY)               goto cleanup;
   if (i == 3u) {
      if ((res = mp_add_d(prime, 1uL, &t1)) != MP_OKAY)           goto cleanup;
      if ((res = mp_div_2(&t1, &t1)) != MP_OKAY)                  goto cleanup;
      if ((res = mp_div_2(&t1, &t1)) != MP_OKAY)                  goto cleanup;
      if ((res = mp_exptmod(n, &t1, prime, ret)) != MP_OKAY)      goto cleanup;
      res = MP_OKAY;
      goto cleanup;
   }

   /* NOW: Tonelli-Shanks algorithm */

   /* factor out powers of 2 from prime-1, defining Q and S as: prime-1 = Q*2^S */
   if ((res = mp_copy(prime, &Q)) != MP_OKAY)                    goto cleanup;
   if ((res = mp_sub_d(&Q, 1uL, &Q)) != MP_OKAY)                 goto cleanup;
   /* Q = prime - 1 */
   mp_zero(&S);
   /* S = 0 */
   while (mp_iseven(&Q) != MP_NO) {
      if ((res = mp_div_2(&Q, &Q)) != MP_OKAY)                    goto cleanup;
      /* Q = Q / 2 */
      if ((res = mp_add_d(&S, 1uL, &S)) != MP_OKAY)               goto cleanup;
      /* S = S + 1 */
   }

   /* find a Z such that the Legendre symbol (Z|prime) == -1 */
   if ((res = mp_set_int(&Z, 2uL)) != MP_OKAY)                    goto cleanup;
   /* Z = 2 */
   while (1) {
      if ((res = mp_jacobi(&Z, prime, &legendre)) != MP_OKAY)     goto cleanup;
      if (legendre == -1) break;
      if ((res = mp_add_d(&Z, 1uL, &Z)) != MP_OKAY)               goto cleanup;
      /* Z = Z + 1 */
   }

   if ((res = mp_exptmod(&Z, &Q, prime, &C)) != MP_OKAY)         goto cleanup;
   /* C = Z ^ Q mod prime */
   if ((res = mp_add_d(&Q, 1uL, &t1)) != MP_OKAY)                goto cleanup;
   if ((res = mp_div_2(&t1, &t1)) != MP_OKAY)                    goto cleanup;
   /* t1 = (Q + 1) / 2 */
   if ((res = mp_exptmod(n, &t1, prime, &R)) != MP_OKAY)         goto cleanup;
   /* R = n ^ ((Q + 1) / 2) mod prime */
   if ((res = mp_exptmod(n, &Q, prime, &T)) != MP_OKAY)          goto cleanup;
   /* T = n ^ Q mod prime */
   if ((res = mp_copy(&S, &M)) != MP_OKAY)                       goto cleanup;
   /* M = S */
   if ((res = mp_set_int(&two, 2uL)) != MP_OKAY)                 goto cleanup;

   res = MP_VAL;
   while (1) {
      if ((res = mp_copy(&T, &t1)) != MP_OKAY)                    goto cleanup;
      i = 0;
      while (1) {
         if (mp_cmp_d(&t1, 1uL) == MP_EQ) break;
         if ((res = mp_exptmod(&t1, &two, prime, &t1)) != MP_OKAY) goto cleanup;
         i++;
      }
      if (i == 0u) {
         if ((res = mp_copy(&R, ret)) != MP_OKAY)                  goto cleanup;
         res = MP_OKAY;
         goto cleanup;
      }
      if ((res = mp_sub_d(&M, i, &t1)) != MP_OKAY)                goto cleanup;
      if ((res = mp_sub_d(&t1, 1uL, &t1)) != MP_OKAY)             goto cleanup;
      if ((res = mp_exptmod(&two, &t1, prime, &t1)) != MP_OKAY)   goto cleanup;
      /* t1 = 2 ^ (M - i - 1) */
      if ((res = mp_exptmod(&C, &t1, prime, &t1)) != MP_OKAY)     goto cleanup;
      /* t1 = C ^ (2 ^ (M - i - 1)) mod prime */
      if ((res = mp_sqrmod(&t1, prime, &C)) != MP_OKAY)           goto cleanup;
      /* C = (t1 * t1) mod prime */
      if ((res = mp_mulmod(&R, &t1, prime, &R)) != MP_OKAY)       goto cleanup;
      /* R = (R * t1) mod prime */
      if ((res = mp_mulmod(&T, &C, prime, &T)) != MP_OKAY)        goto cleanup;
      /* T = (T * C) mod prime */
      mp_set(&M, i);
      /* M = i */
   }

cleanup:
   mp_clear_multi(&t1, &C, &Q, &S, &Z, &M, &T, &R, &two, NULL);
   return res;
}

/* End: bn_mp_sqrtmod_prime.c */

/* Start: bn_mp_sub.c */

/* high level subtraction (handles signs) */
int mp_sub(const mp_int *a, const mp_int *b, mp_int *c)
{
   int     sa, sb, res;

   sa = a->sign;
   sb = b->sign;

   if (sa != sb) {
      /* subtract a negative from a positive, OR */
      /* subtract a positive from a negative. */
      /* In either case, ADD their magnitudes, */
      /* and use the sign of the first number. */
      c->sign = sa;
      res = s_mp_add(a, b, c);
   } else {
      /* subtract a positive from a positive, OR */
      /* subtract a negative from a negative. */
      /* First, take the difference between their */
      /* magnitudes, then... */
      if (mp_cmp_mag(a, b) != MP_LT) {
         /* Copy the sign from the first */
         c->sign = sa;
         /* The first has a larger or equal magnitude */
         res = s_mp_sub(a, b, c);
      } else {
         /* The result has the *opposite* sign from */
         /* the first number. */
         c->sign = (sa == MP_ZPOS) ? MP_NEG : MP_ZPOS;
         /* The second has a larger magnitude */
         res = s_mp_sub(b, a, c);
      }
   }
   return res;
}

/* End: bn_mp_sub.c */

/* Start: bn_mp_sub_d.c */

/* single digit subtraction */
int mp_sub_d(const mp_int *a, mp_digit b, mp_int *c)
{
   mp_digit *tmpa, *tmpc, mu;
   int       res, ix, oldused;

   /* grow c as required */
   if (c->alloc < (a->used + 1)) {
      if ((res = mp_grow(c, a->used + 1)) != MP_OKAY) {
         return res;
      }
   }

   /* if a is negative just do an unsigned
    * addition [with fudged signs]
    */
   if (a->sign == MP_NEG) {
      mp_int a_ = *a;
      a_.sign = MP_ZPOS;
      res     = mp_add_d(&a_, b, c);
      c->sign = MP_NEG;

      /* clamp */
      mp_clamp(c);

      return res;
   }

   /* setup regs */
   oldused = c->used;
   tmpa    = a->dp;
   tmpc    = c->dp;

   /* if a <= b simply fix the single digit */
   if (((a->used == 1) && (a->dp[0] <= b)) || (a->used == 0)) {
      if (a->used == 1) {
         *tmpc++ = b - *tmpa;
      } else {
         *tmpc++ = b;
      }
      ix      = 1;

      /* negative/1digit */
      c->sign = MP_NEG;
      c->used = 1;
   } else {
      /* positive/size */
      c->sign = MP_ZPOS;
      c->used = a->used;

      /* subtract first digit */
      *tmpc    = *tmpa++ - b;
      mu       = *tmpc >> ((sizeof(mp_digit) * (size_t)CHAR_BIT) - 1u);
      *tmpc++ &= MP_MASK;

      /* handle rest of the digits */
      for (ix = 1; ix < a->used; ix++) {
         *tmpc    = *tmpa++ - mu;
         mu       = *tmpc >> ((sizeof(mp_digit) * (size_t)CHAR_BIT) - 1u);
         *tmpc++ &= MP_MASK;
      }
   }

   /* zero excess digits */
   while (ix++ < oldused) {
      *tmpc++ = 0;
   }
   mp_clamp(c);
   return MP_OKAY;
}

/* End: bn_mp_sub_d.c */

/* Start: bn_mp_submod.c */

/* d = a - b (mod c) */
int mp_submod(const mp_int *a, const mp_int *b, const mp_int *c, mp_int *d)
{
   int     res;
   mp_int  t;


   if ((res = mp_init(&t)) != MP_OKAY) {
      return res;
   }

   if ((res = mp_sub(a, b, &t)) != MP_OKAY) {
      mp_clear(&t);
      return res;
   }
   res = mp_mod(&t, c, d);
   mp_clear(&t);
   return res;
}

/* End: bn_mp_submod.c */

/* Start: bn_mp_tc_and.c */

/* two complement and */
int mp_tc_and(const mp_int *a, const mp_int *b, mp_int *c)
{
   int res = MP_OKAY, bits, abits, bbits;
   int as = mp_isneg(a), bs = mp_isneg(b);
   mp_int *mx = NULL, _mx, acpy, bcpy;

   if ((as != MP_NO) || (bs != MP_NO)) {
      abits = mp_count_bits(a);
      bbits = mp_count_bits(b);
      bits = MAX(abits, bbits);
      res = mp_init_set_int(&_mx, 1uL);
      if (res != MP_OKAY) {
         goto end;
      }

      mx = &_mx;
      res = mp_mul_2d(mx, bits + 1, mx);
      if (res != MP_OKAY) {
         goto end;
      }

      if (as != MP_NO) {
         res = mp_init(&acpy);
         if (res != MP_OKAY) {
            goto end;
         }

         res = mp_add(mx, a, &acpy);
         if (res != MP_OKAY) {
            mp_clear(&acpy);
            goto end;
         }
         a = &acpy;
      }
      if (bs != MP_NO) {
         res = mp_init(&bcpy);
         if (res != MP_OKAY) {
            goto end;
         }

         res = mp_add(mx, b, &bcpy);
         if (res != MP_OKAY) {
            mp_clear(&bcpy);
            goto end;
         }
         b = &bcpy;
      }
   }

   res = mp_and(a, b, c);

   if ((as != MP_NO) && (bs != MP_NO) && (res == MP_OKAY)) {
      res = mp_sub(c, mx, c);
   }

end:
   if (a == &acpy) {
      mp_clear(&acpy);
   }

   if (b == &bcpy) {
      mp_clear(&bcpy);
   }

   if (mx == &_mx) {
      mp_clear(mx);
   }

   return res;
}

/* End: bn_mp_tc_and.c */

/* Start: bn_mp_tc_div_2d.c */

/* two complement right shift */
int mp_tc_div_2d(const mp_int *a, int b, mp_int *c)
{
   int res;
   if (mp_isneg(a) == MP_NO) {
      return mp_div_2d(a, b, c, NULL);
   }

   res = mp_add_d(a, 1uL, c);
   if (res != MP_OKAY) {
      return res;
   }

   res = mp_div_2d(c, b, c, NULL);
   return (res == MP_OKAY) ? mp_sub_d(c, 1uL, c) : res;
}

/* End: bn_mp_tc_div_2d.c */

/* Start: bn_mp_tc_or.c */

/* two complement or */
int mp_tc_or(const mp_int *a, const mp_int *b, mp_int *c)
{
   int res = MP_OKAY, bits, abits, bbits;
   int as = mp_isneg(a), bs = mp_isneg(b);
   mp_int *mx = NULL, _mx, acpy, bcpy;

   if ((as != MP_NO) || (bs != MP_NO)) {
      abits = mp_count_bits(a);
      bbits = mp_count_bits(b);
      bits = MAX(abits, bbits);
      res = mp_init_set_int(&_mx, 1uL);
      if (res != MP_OKAY) {
         goto end;
      }

      mx = &_mx;
      res = mp_mul_2d(mx, bits + 1, mx);
      if (res != MP_OKAY) {
         goto end;
      }

      if (as != MP_NO) {
         res = mp_init(&acpy);
         if (res != MP_OKAY) {
            goto end;
         }

         res = mp_add(mx, a, &acpy);
         if (res != MP_OKAY) {
            mp_clear(&acpy);
            goto end;
         }
         a = &acpy;
      }
      if (bs != MP_NO) {
         res = mp_init(&bcpy);
         if (res != MP_OKAY) {
            goto end;
         }

         res = mp_add(mx, b, &bcpy);
         if (res != MP_OKAY) {
            mp_clear(&bcpy);
            goto end;
         }
         b = &bcpy;
      }
   }

   res = mp_or(a, b, c);

   if (((as != MP_NO) || (bs != MP_NO)) && (res == MP_OKAY)) {
      res = mp_sub(c, mx, c);
   }

end:
   if (a == &acpy) {
      mp_clear(&acpy);
   }

   if (b == &bcpy) {
      mp_clear(&bcpy);
   }

   if (mx == &_mx) {
      mp_clear(mx);
   }

   return res;
}

/* End: bn_mp_tc_or.c */

/* Start: bn_mp_tc_xor.c */

/* two complement xor */
int mp_tc_xor(const mp_int *a, const mp_int *b, mp_int *c)
{
   int res = MP_OKAY, bits, abits, bbits;
   int as = mp_isneg(a), bs = mp_isneg(b);
   mp_int *mx = NULL, _mx, acpy, bcpy;

   if ((as != MP_NO) || (bs != MP_NO)) {
      abits = mp_count_bits(a);
      bbits = mp_count_bits(b);
      bits = MAX(abits, bbits);
      res = mp_init_set_int(&_mx, 1uL);
      if (res != MP_OKAY) {
         goto end;
      }

      mx = &_mx;
      res = mp_mul_2d(mx, bits + 1, mx);
      if (res != MP_OKAY) {
         goto end;
      }

      if (as != MP_NO) {
         res = mp_init(&acpy);
         if (res != MP_OKAY) {
            goto end;
         }

         res = mp_add(mx, a, &acpy);
         if (res != MP_OKAY) {
            mp_clear(&acpy);
            goto end;
         }
         a = &acpy;
      }
      if (bs != MP_NO) {
         res = mp_init(&bcpy);
         if (res != MP_OKAY) {
            goto end;
         }

         res = mp_add(mx, b, &bcpy);
         if (res != MP_OKAY) {
            mp_clear(&bcpy);
            goto end;
         }
         b = &bcpy;
      }
   }

   res = mp_xor(a, b, c);

   if ((as != bs) && (res == MP_OKAY)) {
      res = mp_sub(c, mx, c);
   }

end:
   if (a == &acpy) {
      mp_clear(&acpy);
   }

   if (b == &bcpy) {
      mp_clear(&bcpy);
   }

   if (mx == &_mx) {
      mp_clear(mx);
   }

   return res;
}

/* End: bn_mp_tc_xor.c */

/* Start: bn_mp_to_signed_bin.c */

/* store in signed [big endian] format */
int mp_to_signed_bin(const mp_int *a, unsigned char *b)
{
   int     res;

   if ((res = mp_to_unsigned_bin(a, b + 1)) != MP_OKAY) {
      return res;
   }
   b[0] = (a->sign == MP_ZPOS) ? (unsigned char)0 : (unsigned char)1;
   return MP_OKAY;
}

/* End: bn_mp_to_signed_bin.c */

/* Start: bn_mp_to_signed_bin_n.c */

/* store in signed [big endian] format */
int mp_to_signed_bin_n(const mp_int *a, unsigned char *b, unsigned long *outlen)
{
   if (*outlen < (unsigned long)mp_signed_bin_size(a)) {
      return MP_VAL;
   }
   *outlen = (unsigned long)mp_signed_bin_size(a);
   return mp_to_signed_bin(a, b);
}

/* End: bn_mp_to_signed_bin_n.c */

/* Start: bn_mp_to_unsigned_bin.c */

/* store in unsigned [big endian] format */
int mp_to_unsigned_bin(const mp_int *a, unsigned char *b)
{
   int     x, res;
   mp_int  t;

   if ((res = mp_init_copy(&t, a)) != MP_OKAY) {
      return res;
   }

   x = 0;
   while (mp_iszero(&t) == MP_NO) {
#ifndef MP_8BIT
      b[x++] = (unsigned char)(t.dp[0] & 255u);
#else
      b[x++] = (unsigned char)(t.dp[0] | ((t.dp[1] & 1u) << 7));
#endif
      if ((res = mp_div_2d(&t, 8, &t, NULL)) != MP_OKAY) {
         mp_clear(&t);
         return res;
      }
   }
   bn_reverse(b, x);
   mp_clear(&t);
   return MP_OKAY;
}

/* End: bn_mp_to_unsigned_bin.c */

/* Start: bn_mp_to_unsigned_bin_n.c */

/* store in unsigned [big endian] format */
int mp_to_unsigned_bin_n(const mp_int *a, unsigned char *b, unsigned long *outlen)
{
   if (*outlen < (unsigned long)mp_unsigned_bin_size(a)) {
      return MP_VAL;
   }
   *outlen = (unsigned long)mp_unsigned_bin_size(a);
   return mp_to_unsigned_bin(a, b);
}

/* End: bn_mp_to_unsigned_bin_n.c */

/* Start: bn_mp_toom_mul.c */

/* multiplication using the Toom-Cook 3-way algorithm
 *
 * Much more complicated than Karatsuba but has a lower
 * asymptotic running time of O(N**1.464).  This algorithm is
 * only particularly useful on VERY large inputs
 * (we're talking 1000s of digits here...).
*/
int mp_toom_mul(const mp_int *a, const mp_int *b, mp_int *c)
{
   mp_int w0, w1, w2, w3, w4, tmp1, tmp2, a0, a1, a2, b0, b1, b2;
   int res, B;

   /* init temps */
   if ((res = mp_init_multi(&w0, &w1, &w2, &w3, &w4,
                            &a0, &a1, &a2, &b0, &b1,
                            &b2, &tmp1, &tmp2, NULL)) != MP_OKAY) {
      return res;
   }

   /* B */
   B = MIN(a->used, b->used) / 3;

   /* a = a2 * B**2 + a1 * B + a0 */
   if ((res = mp_mod_2d(a, DIGIT_BIT * B, &a0)) != MP_OKAY) {
      goto LBL_ERR;
   }

   if ((res = mp_copy(a, &a1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   mp_rshd(&a1, B);
   if ((res = mp_mod_2d(&a1, DIGIT_BIT * B, &a1)) != MP_OKAY) {
      goto LBL_ERR;
   }

   if ((res = mp_copy(a, &a2)) != MP_OKAY) {
      goto LBL_ERR;
   }
   mp_rshd(&a2, B*2);

   /* b = b2 * B**2 + b1 * B + b0 */
   if ((res = mp_mod_2d(b, DIGIT_BIT * B, &b0)) != MP_OKAY) {
      goto LBL_ERR;
   }

   if ((res = mp_copy(b, &b1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   mp_rshd(&b1, B);
   (void)mp_mod_2d(&b1, DIGIT_BIT * B, &b1);

   if ((res = mp_copy(b, &b2)) != MP_OKAY) {
      goto LBL_ERR;
   }
   mp_rshd(&b2, B*2);

   /* w0 = a0*b0 */
   if ((res = mp_mul(&a0, &b0, &w0)) != MP_OKAY) {
      goto LBL_ERR;
   }

   /* w4 = a2 * b2 */
   if ((res = mp_mul(&a2, &b2, &w4)) != MP_OKAY) {
      goto LBL_ERR;
   }

   /* w1 = (a2 + 2(a1 + 2a0))(b2 + 2(b1 + 2b0)) */
   if ((res = mp_mul_2(&a0, &tmp1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_add(&tmp1, &a2, &tmp1)) != MP_OKAY) {
      goto LBL_ERR;
   }

   if ((res = mp_mul_2(&b0, &tmp2)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_add(&tmp2, &b2, &tmp2)) != MP_OKAY) {
      goto LBL_ERR;
   }

   if ((res = mp_mul(&tmp1, &tmp2, &w1)) != MP_OKAY) {
      goto LBL_ERR;
   }

   /* w3 = (a0 + 2(a1 + 2a2))(b0 + 2(b1 + 2b2)) */
   if ((res = mp_mul_2(&a2, &tmp1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) {
      goto LBL_ERR;
   }

   if ((res = mp_mul_2(&b2, &tmp2)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) {
      goto LBL_ERR;
   }

   if ((res = mp_mul(&tmp1, &tmp2, &w3)) != MP_OKAY) {
      goto LBL_ERR;
   }


   /* w2 = (a2 + a1 + a0)(b2 + b1 + b0) */
   if ((res = mp_add(&a2, &a1, &tmp1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_add(&b2, &b1, &tmp2)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_mul(&tmp1, &tmp2, &w2)) != MP_OKAY) {
      goto LBL_ERR;
   }

   /* now solve the matrix

      0  0  0  0  1
      1  2  4  8  16
      1  1  1  1  1
      16 8  4  2  1
      1  0  0  0  0

      using 12 subtractions, 4 shifts,
             2 small divisions and 1 small multiplication
    */

   /* r1 - r4 */
   if ((res = mp_sub(&w1, &w4, &w1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   /* r3 - r0 */
   if ((res = mp_sub(&w3, &w0, &w3)) != MP_OKAY) {
      goto LBL_ERR;
   }
   /* r1/2 */
   if ((res = mp_div_2(&w1, &w1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   /* r3/2 */
   if ((res = mp_div_2(&w3, &w3)) != MP_OKAY) {
      goto LBL_ERR;
   }
   /* r2 - r0 - r4 */
   if ((res = mp_sub(&w2, &w0, &w2)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_sub(&w2, &w4, &w2)) != MP_OKAY) {
      goto LBL_ERR;
   }
   /* r1 - r2 */
   if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   /* r3 - r2 */
   if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) {
      goto LBL_ERR;
   }
   /* r1 - 8r0 */
   if ((res = mp_mul_2d(&w0, 3, &tmp1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_sub(&w1, &tmp1, &w1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   /* r3 - 8r4 */
   if ((res = mp_mul_2d(&w4, 3, &tmp1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_sub(&w3, &tmp1, &w3)) != MP_OKAY) {
      goto LBL_ERR;
   }
   /* 3r2 - r1 - r3 */
   if ((res = mp_mul_d(&w2, 3uL, &w2)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_sub(&w2, &w1, &w2)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_sub(&w2, &w3, &w2)) != MP_OKAY) {
      goto LBL_ERR;
   }
   /* r1 - r2 */
   if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   /* r3 - r2 */
   if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) {
      goto LBL_ERR;
   }
   /* r1/3 */
   if ((res = mp_div_3(&w1, &w1, NULL)) != MP_OKAY) {
      goto LBL_ERR;
   }
   /* r3/3 */
   if ((res = mp_div_3(&w3, &w3, NULL)) != MP_OKAY) {
      goto LBL_ERR;
   }

   /* at this point shift W[n] by B*n */
   if ((res = mp_lshd(&w1, 1*B)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_lshd(&w2, 2*B)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_lshd(&w3, 3*B)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_lshd(&w4, 4*B)) != MP_OKAY) {
      goto LBL_ERR;
   }

   if ((res = mp_add(&w0, &w1, c)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_add(&w2, &w3, &tmp1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_add(&w4, &tmp1, &tmp1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_add(&tmp1, c, c)) != MP_OKAY) {
      goto LBL_ERR;
   }

LBL_ERR:
   mp_clear_multi(&w0, &w1, &w2, &w3, &w4,
                  &a0, &a1, &a2, &b0, &b1,
                  &b2, &tmp1, &tmp2, NULL);
   return res;
}

/* End: bn_mp_toom_mul.c */

/* Start: bn_mp_toom_sqr.c */

/* squaring using Toom-Cook 3-way algorithm */
int mp_toom_sqr(const mp_int *a, mp_int *b)
{
   mp_int w0, w1, w2, w3, w4, tmp1, a0, a1, a2;
   int res, B;

   /* init temps */
   if ((res = mp_init_multi(&w0, &w1, &w2, &w3, &w4, &a0, &a1, &a2, &tmp1, NULL)) != MP_OKAY) {
      return res;
   }

   /* B */
   B = a->used / 3;

   /* a = a2 * B**2 + a1 * B + a0 */
   if ((res = mp_mod_2d(a, DIGIT_BIT * B, &a0)) != MP_OKAY) {
      goto LBL_ERR;
   }

   if ((res = mp_copy(a, &a1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   mp_rshd(&a1, B);
   if ((res = mp_mod_2d(&a1, DIGIT_BIT * B, &a1)) != MP_OKAY) {
      goto LBL_ERR;
   }

   if ((res = mp_copy(a, &a2)) != MP_OKAY) {
      goto LBL_ERR;
   }
   mp_rshd(&a2, B*2);

   /* w0 = a0*a0 */
   if ((res = mp_sqr(&a0, &w0)) != MP_OKAY) {
      goto LBL_ERR;
   }

   /* w4 = a2 * a2 */
   if ((res = mp_sqr(&a2, &w4)) != MP_OKAY) {
      goto LBL_ERR;
   }

   /* w1 = (a2 + 2(a1 + 2a0))**2 */
   if ((res = mp_mul_2(&a0, &tmp1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_add(&tmp1, &a2, &tmp1)) != MP_OKAY) {
      goto LBL_ERR;
   }

   if ((res = mp_sqr(&tmp1, &w1)) != MP_OKAY) {
      goto LBL_ERR;
   }

   /* w3 = (a0 + 2(a1 + 2a2))**2 */
   if ((res = mp_mul_2(&a2, &tmp1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) {
      goto LBL_ERR;
   }

   if ((res = mp_sqr(&tmp1, &w3)) != MP_OKAY) {
      goto LBL_ERR;
   }


   /* w2 = (a2 + a1 + a0)**2 */
   if ((res = mp_add(&a2, &a1, &tmp1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_sqr(&tmp1, &w2)) != MP_OKAY) {
      goto LBL_ERR;
   }

   /* now solve the matrix

      0  0  0  0  1
      1  2  4  8  16
      1  1  1  1  1
      16 8  4  2  1
      1  0  0  0  0

      using 12 subtractions, 4 shifts, 2 small divisions and 1 small multiplication.
    */

   /* r1 - r4 */
   if ((res = mp_sub(&w1, &w4, &w1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   /* r3 - r0 */
   if ((res = mp_sub(&w3, &w0, &w3)) != MP_OKAY) {
      goto LBL_ERR;
   }
   /* r1/2 */
   if ((res = mp_div_2(&w1, &w1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   /* r3/2 */
   if ((res = mp_div_2(&w3, &w3)) != MP_OKAY) {
      goto LBL_ERR;
   }
   /* r2 - r0 - r4 */
   if ((res = mp_sub(&w2, &w0, &w2)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_sub(&w2, &w4, &w2)) != MP_OKAY) {
      goto LBL_ERR;
   }
   /* r1 - r2 */
   if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   /* r3 - r2 */
   if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) {
      goto LBL_ERR;
   }
   /* r1 - 8r0 */
   if ((res = mp_mul_2d(&w0, 3, &tmp1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_sub(&w1, &tmp1, &w1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   /* r3 - 8r4 */
   if ((res = mp_mul_2d(&w4, 3, &tmp1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_sub(&w3, &tmp1, &w3)) != MP_OKAY) {
      goto LBL_ERR;
   }
   /* 3r2 - r1 - r3 */
   if ((res = mp_mul_d(&w2, 3uL, &w2)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_sub(&w2, &w1, &w2)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_sub(&w2, &w3, &w2)) != MP_OKAY) {
      goto LBL_ERR;
   }
   /* r1 - r2 */
   if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   /* r3 - r2 */
   if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) {
      goto LBL_ERR;
   }
   /* r1/3 */
   if ((res = mp_div_3(&w1, &w1, NULL)) != MP_OKAY) {
      goto LBL_ERR;
   }
   /* r3/3 */
   if ((res = mp_div_3(&w3, &w3, NULL)) != MP_OKAY) {
      goto LBL_ERR;
   }

   /* at this point shift W[n] by B*n */
   if ((res = mp_lshd(&w1, 1*B)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_lshd(&w2, 2*B)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_lshd(&w3, 3*B)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_lshd(&w4, 4*B)) != MP_OKAY) {
      goto LBL_ERR;
   }

   if ((res = mp_add(&w0, &w1, b)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_add(&w2, &w3, &tmp1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_add(&w4, &tmp1, &tmp1)) != MP_OKAY) {
      goto LBL_ERR;
   }
   if ((res = mp_add(&tmp1, b, b)) != MP_OKAY) {
      goto LBL_ERR;
   }

LBL_ERR:
   mp_clear_multi(&w0, &w1, &w2, &w3, &w4, &a0, &a1, &a2, &tmp1, NULL);
   return res;
}

/* End: bn_mp_toom_sqr.c */

/* Start: bn_mp_toradix.c */

/* stores a bignum as a ASCII string in a given radix (2..64) */
int mp_toradix(const mp_int *a, char *str, int radix)
{
   int     res, digs;
   mp_int  t;
   mp_digit d;
   char   *_s = str;

   /* check range of the radix */
   if ((radix < 2) || (radix > 64)) {
      return MP_VAL;
   }

   /* quick out if its zero */
   if (mp_iszero(a) == MP_YES) {
      *str++ = '0';
      *str = '\0';
      return MP_OKAY;
   }

   if ((res = mp_init_copy(&t, a)) != MP_OKAY) {
      return res;
   }

   /* if it is negative output a - */
   if (t.sign == MP_NEG) {
      ++_s;
      *str++ = '-';
      t.sign = MP_ZPOS;
   }

   digs = 0;
   while (mp_iszero(&t) == MP_NO) {
      if ((res = mp_div_d(&t, (mp_digit)radix, &t, &d)) != MP_OKAY) {
         mp_clear(&t);
         return res;
      }
      *str++ = mp_s_rmap[d];
      ++digs;
   }

   /* reverse the digits of the string.  In this case _s points
    * to the first digit [exluding the sign] of the number]
    */
   bn_reverse((unsigned char *)_s, digs);

   /* append a NULL so the string is properly terminated */
   *str = '\0';

   mp_clear(&t);
   return MP_OKAY;
}

/* End: bn_mp_toradix.c */

/* Start: bn_mp_toradix_n.c */

/* stores a bignum as a ASCII string in a given radix (2..64)
 *
 * Stores upto maxlen-1 chars and always a NULL byte
 */
int mp_toradix_n(const mp_int *a, char *str, int radix, int maxlen)
{
   int     res, digs;
   mp_int  t;
   mp_digit d;
   char   *_s = str;

   /* check range of the maxlen, radix */
   if ((maxlen < 2) || (radix < 2) || (radix > 64)) {
      return MP_VAL;
   }

   /* quick out if its zero */
   if (mp_iszero(a) == MP_YES) {
      *str++ = '0';
      *str = '\0';
      return MP_OKAY;
   }

   if ((res = mp_init_copy(&t, a)) != MP_OKAY) {
      return res;
   }

   /* if it is negative output a - */
   if (t.sign == MP_NEG) {
      /* we have to reverse our digits later... but not the - sign!! */
      ++_s;

      /* store the flag and mark the number as positive */
      *str++ = '-';
      t.sign = MP_ZPOS;

      /* subtract a char */
      --maxlen;
   }

   digs = 0;
   while (mp_iszero(&t) == MP_NO) {
      if (--maxlen < 1) {
         /* no more room */
         break;
      }
      if ((res = mp_div_d(&t, (mp_digit)radix, &t, &d)) != MP_OKAY) {
         mp_clear(&t);
         return res;
      }
      *str++ = mp_s_rmap[d];
      ++digs;
   }

   /* reverse the digits of the string.  In this case _s points
    * to the first digit [exluding the sign] of the number
    */
   bn_reverse((unsigned char *)_s, digs);

   /* append a NULL so the string is properly terminated */
   *str = '\0';

   mp_clear(&t);
   return MP_OKAY;
}

/* End: bn_mp_toradix_n.c */

/* Start: bn_mp_unsigned_bin_size.c */

/* get the size for an unsigned equivalent */
int mp_unsigned_bin_size(const mp_int *a)
{
   int     size = mp_count_bits(a);
   return (size / 8) + ((((unsigned)size & 7u) != 0u) ? 1 : 0);
}

/* End: bn_mp_unsigned_bin_size.c */

/* Start: bn_mp_xor.c */

/* XOR two ints together */
int mp_xor(const mp_int *a, const mp_int *b, mp_int *c)
{
   int     res, ix, px;
   mp_int  t;
   const mp_int *x;

   if (a->used > b->used) {
      if ((res = mp_init_copy(&t, a)) != MP_OKAY) {
         return res;
      }
      px = b->used;
      x = b;
   } else {
      if ((res = mp_init_copy(&t, b)) != MP_OKAY) {
         return res;
      }
      px = a->used;
      x = a;
   }

   for (ix = 0; ix < px; ix++) {
      t.dp[ix] ^= x->dp[ix];
   }
   mp_clamp(&t);
   mp_exch(c, &t);
   mp_clear(&t);
   return MP_OKAY;
}

/* End: bn_mp_xor.c */

/* Start: bn_mp_zero.c */

/* set to zero */
void mp_zero(mp_int *a)
{
   int       n;
   mp_digit *tmp;

   a->sign = MP_ZPOS;
   a->used = 0;

   tmp = a->dp;
   for (n = 0; n < a->alloc; n++) {
      *tmp++ = 0;
   }
}

/* End: bn_mp_zero.c */

/* Start: bn_prime_tab.c */

const mp_digit ltm_prime_tab[] = {
   0x0002, 0x0003, 0x0005, 0x0007, 0x000B, 0x000D, 0x0011, 0x0013,
   0x0017, 0x001D, 0x001F, 0x0025, 0x0029, 0x002B, 0x002F, 0x0035,
   0x003B, 0x003D, 0x0043, 0x0047, 0x0049, 0x004F, 0x0053, 0x0059,
   0x0061, 0x0065, 0x0067, 0x006B, 0x006D, 0x0071, 0x007F,
#ifndef MP_8BIT
   0x0083,
   0x0089, 0x008B, 0x0095, 0x0097, 0x009D, 0x00A3, 0x00A7, 0x00AD,
   0x00B3, 0x00B5, 0x00BF, 0x00C1, 0x00C5, 0x00C7, 0x00D3, 0x00DF,
   0x00E3, 0x00E5, 0x00E9, 0x00EF, 0x00F1, 0x00FB, 0x0101, 0x0107,
   0x010D, 0x010F, 0x0115, 0x0119, 0x011B, 0x0125, 0x0133, 0x0137,

   0x0139, 0x013D, 0x014B, 0x0151, 0x015B, 0x015D, 0x0161, 0x0167,
   0x016F, 0x0175, 0x017B, 0x017F, 0x0185, 0x018D, 0x0191, 0x0199,
   0x01A3, 0x01A5, 0x01AF, 0x01B1, 0x01B7, 0x01BB, 0x01C1, 0x01C9,
   0x01CD, 0x01CF, 0x01D3, 0x01DF, 0x01E7, 0x01EB, 0x01F3, 0x01F7,
   0x01FD, 0x0209, 0x020B, 0x021D, 0x0223, 0x022D, 0x0233, 0x0239,
   0x023B, 0x0241, 0x024B, 0x0251, 0x0257, 0x0259, 0x025F, 0x0265,
   0x0269, 0x026B, 0x0277, 0x0281, 0x0283, 0x0287, 0x028D, 0x0293,
   0x0295, 0x02A1, 0x02A5, 0x02AB, 0x02B3, 0x02BD, 0x02C5, 0x02CF,

   0x02D7, 0x02DD, 0x02E3, 0x02E7, 0x02EF, 0x02F5, 0x02F9, 0x0301,
   0x0305, 0x0313, 0x031D, 0x0329, 0x032B, 0x0335, 0x0337, 0x033B,
   0x033D, 0x0347, 0x0355, 0x0359, 0x035B, 0x035F, 0x036D, 0x0371,
   0x0373, 0x0377, 0x038B, 0x038F, 0x0397, 0x03A1, 0x03A9, 0x03AD,
   0x03B3, 0x03B9, 0x03C7, 0x03CB, 0x03D1, 0x03D7, 0x03DF, 0x03E5,
   0x03F1, 0x03F5, 0x03FB, 0x03FD, 0x0407, 0x0409, 0x040F, 0x0419,
   0x041B, 0x0425, 0x0427, 0x042D, 0x043F, 0x0443, 0x0445, 0x0449,
   0x044F, 0x0455, 0x045D, 0x0463, 0x0469, 0x047F, 0x0481, 0x048B,

   0x0493, 0x049D, 0x04A3, 0x04A9, 0x04B1, 0x04BD, 0x04C1, 0x04C7,
   0x04CD, 0x04CF, 0x04D5, 0x04E1, 0x04EB, 0x04FD, 0x04FF, 0x0503,
   0x0509, 0x050B, 0x0511, 0x0515, 0x0517, 0x051B, 0x0527, 0x0529,
   0x052F, 0x0551, 0x0557, 0x055D, 0x0565, 0x0577, 0x0581, 0x058F,
   0x0593, 0x0595, 0x0599, 0x059F, 0x05A7, 0x05AB, 0x05AD, 0x05B3,
   0x05BF, 0x05C9, 0x05CB, 0x05CF, 0x05D1, 0x05D5, 0x05DB, 0x05E7,
   0x05F3, 0x05FB, 0x0607, 0x060D, 0x0611, 0x0617, 0x061F, 0x0623,
   0x062B, 0x062F, 0x063D, 0x0641, 0x0647, 0x0649, 0x064D, 0x0653
#endif
};

/* End: bn_prime_tab.c */

/* Start: bn_reverse.c */

/* reverse an array, used for radix code */
void bn_reverse(unsigned char *s, int len)
{
   int     ix, iy;
   unsigned char t;

   ix = 0;
   iy = len - 1;
   while (ix < iy) {
      t     = s[ix];
      s[ix] = s[iy];
      s[iy] = t;
      ++ix;
      --iy;
   }
}

/* End: bn_reverse.c */

/* Start: bn_s_mp_add.c */

/* low level addition, based on HAC pp.594, Algorithm 14.7 */
int s_mp_add(const mp_int *a, const mp_int *b, mp_int *c)
{
   const mp_int *x;
   int     olduse, res, min, max;

   /* find sizes, we let |a| <= |b| which means we have to sort
    * them.  "x" will point to the input with the most digits
    */
   if (a->used > b->used) {
      min = b->used;
      max = a->used;
      x = a;
   } else {
      min = a->used;
      max = b->used;
      x = b;
   }

   /* init result */
   if (c->alloc < (max + 1)) {
      if ((res = mp_grow(c, max + 1)) != MP_OKAY) {
         return res;
      }
   }

   /* get old used digit count and set new one */
   olduse = c->used;
   c->used = max + 1;

   {
      mp_digit u, *tmpa, *tmpb, *tmpc;
      int i;

      /* alias for digit pointers */

      /* first input */
      tmpa = a->dp;

      /* second input */
      tmpb = b->dp;

      /* destination */
      tmpc = c->dp;

      /* zero the carry */
      u = 0;
      for (i = 0; i < min; i++) {
         /* Compute the sum at one digit, T[i] = A[i] + B[i] + U */
         *tmpc = *tmpa++ + *tmpb++ + u;

         /* U = carry bit of T[i] */
         u = *tmpc >> (mp_digit)DIGIT_BIT;

         /* take away carry bit from T[i] */
         *tmpc++ &= MP_MASK;
      }

      /* now copy higher words if any, that is in A+B
       * if A or B has more digits add those in
       */
      if (min != max) {
         for (; i < max; i++) {
            /* T[i] = X[i] + U */
            *tmpc = x->dp[i] + u;

            /* U = carry bit of T[i] */
            u = *tmpc >> (mp_digit)DIGIT_BIT;

            /* take away carry bit from T[i] */
            *tmpc++ &= MP_MASK;
         }
      }

      /* add carry */
      *tmpc++ = u;

      /* clear digits above oldused */
      for (i = c->used; i < olduse; i++) {
         *tmpc++ = 0;
      }
   }

   mp_clamp(c);
   return MP_OKAY;
}

/* End: bn_s_mp_add.c */

/* Start: bn_s_mp_exptmod.c */

#ifdef MP_LOW_MEM
#   define TAB_SIZE 32
#else
#   define TAB_SIZE 256
#endif

int s_mp_exptmod(const mp_int *G, const mp_int *X, const mp_int *P, mp_int *Y, int redmode)
{
   mp_int  M[TAB_SIZE], res, mu;
   mp_digit buf;
   int     err, bitbuf, bitcpy, bitcnt, mode, digidx, x, y, winsize;
   int (*redux)(mp_int *x, const mp_int *m, const mp_int *mu);

   /* find window size */
   x = mp_count_bits(X);
   if (x <= 7) {
      winsize = 2;
   } else if (x <= 36) {
      winsize = 3;
   } else if (x <= 140) {
      winsize = 4;
   } else if (x <= 450) {
      winsize = 5;
   } else if (x <= 1303) {
      winsize = 6;
   } else if (x <= 3529) {
      winsize = 7;
   } else {
      winsize = 8;
   }

#ifdef MP_LOW_MEM
   if (winsize > 5) {
      winsize = 5;
   }
#endif

   /* init M array */
   /* init first cell */
   if ((err = mp_init(&M[1])) != MP_OKAY) {
      return err;
   }

   /* now init the second half of the array */
   for (x = 1<<(winsize-1); x < (1 << winsize); x++) {
      if ((err = mp_init(&M[x])) != MP_OKAY) {
         for (y = 1<<(winsize-1); y < x; y++) {
            mp_clear(&M[y]);
         }
         mp_clear(&M[1]);
         return err;
      }
   }

   /* create mu, used for Barrett reduction */
   if ((err = mp_init(&mu)) != MP_OKAY) {
      goto LBL_M;
   }

   if (redmode == 0) {
      if ((err = mp_reduce_setup(&mu, P)) != MP_OKAY) {
         goto LBL_MU;
      }
      redux = mp_reduce;
   } else {
      if ((err = mp_reduce_2k_setup_l(P, &mu)) != MP_OKAY) {
         goto LBL_MU;
      }
      redux = mp_reduce_2k_l;
   }

   /* create M table
    *
    * The M table contains powers of the base,
    * e.g. M[x] = G**x mod P
    *
    * The first half of the table is not
    * computed though accept for M[0] and M[1]
    */
   if ((err = mp_mod(G, P, &M[1])) != MP_OKAY) {
      goto LBL_MU;
   }

   /* compute the value at M[1<<(winsize-1)] by squaring
    * M[1] (winsize-1) times
    */
   if ((err = mp_copy(&M[1], &M[(size_t)1 << (winsize - 1)])) != MP_OKAY) {
      goto LBL_MU;
   }

   for (x = 0; x < (winsize - 1); x++) {
      /* square it */
      if ((err = mp_sqr(&M[(size_t)1 << (winsize - 1)],
                        &M[(size_t)1 << (winsize - 1)])) != MP_OKAY) {
         goto LBL_MU;
      }

      /* reduce modulo P */
      if ((err = redux(&M[(size_t)1 << (winsize - 1)], P, &mu)) != MP_OKAY) {
         goto LBL_MU;
      }
   }

   /* create upper table, that is M[x] = M[x-1] * M[1] (mod P)
    * for x = (2**(winsize - 1) + 1) to (2**winsize - 1)
    */
   for (x = (1 << (winsize - 1)) + 1; x < (1 << winsize); x++) {
      if ((err = mp_mul(&M[x - 1], &M[1], &M[x])) != MP_OKAY) {
         goto LBL_MU;
      }
      if ((err = redux(&M[x], P, &mu)) != MP_OKAY) {
         goto LBL_MU;
      }
   }

   /* setup result */
   if ((err = mp_init(&res)) != MP_OKAY) {
      goto LBL_MU;
   }
   mp_set(&res, 1uL);

   /* set initial mode and bit cnt */
   mode   = 0;
   bitcnt = 1;
   buf    = 0;
   digidx = X->used - 1;
   bitcpy = 0;
   bitbuf = 0;

   for (;;) {
      /* grab next digit as required */
      if (--bitcnt == 0) {
         /* if digidx == -1 we are out of digits */
         if (digidx == -1) {
            break;
         }
         /* read next digit and reset the bitcnt */
         buf    = X->dp[digidx--];
         bitcnt = (int)DIGIT_BIT;
      }

      /* grab the next msb from the exponent */
      y     = (buf >> (mp_digit)(DIGIT_BIT - 1)) & 1;
      buf <<= (mp_digit)1;

      /* if the bit is zero and mode == 0 then we ignore it
       * These represent the leading zero bits before the first 1 bit
       * in the exponent.  Technically this opt is not required but it
       * does lower the # of trivial squaring/reductions used
       */
      if ((mode == 0) && (y == 0)) {
         continue;
      }

      /* if the bit is zero and mode == 1 then we square */
      if ((mode == 1) && (y == 0)) {
         if ((err = mp_sqr(&res, &res)) != MP_OKAY) {
            goto LBL_RES;
         }
         if ((err = redux(&res, P, &mu)) != MP_OKAY) {
            goto LBL_RES;
         }
         continue;
      }

      /* else we add it to the window */
      bitbuf |= (y << (winsize - ++bitcpy));
      mode    = 2;

      if (bitcpy == winsize) {
         /* ok window is filled so square as required and multiply  */
         /* square first */
         for (x = 0; x < winsize; x++) {
            if ((err = mp_sqr(&res, &res)) != MP_OKAY) {
               goto LBL_RES;
            }
            if ((err = redux(&res, P, &mu)) != MP_OKAY) {
               goto LBL_RES;
            }
         }

         /* then multiply */
         if ((err = mp_mul(&res, &M[bitbuf], &res)) != MP_OKAY) {
            goto LBL_RES;
         }
         if ((err = redux(&res, P, &mu)) != MP_OKAY) {
            goto LBL_RES;
         }

         /* empty window and reset */
         bitcpy = 0;
         bitbuf = 0;
         mode   = 1;
      }
   }

   /* if bits remain then square/multiply */
   if ((mode == 2) && (bitcpy > 0)) {
      /* square then multiply if the bit is set */
      for (x = 0; x < bitcpy; x++) {
         if ((err = mp_sqr(&res, &res)) != MP_OKAY) {
            goto LBL_RES;
         }
         if ((err = redux(&res, P, &mu)) != MP_OKAY) {
            goto LBL_RES;
         }

         bitbuf <<= 1;
         if ((bitbuf & (1 << winsize)) != 0) {
            /* then multiply */
            if ((err = mp_mul(&res, &M[1], &res)) != MP_OKAY) {
               goto LBL_RES;
            }
            if ((err = redux(&res, P, &mu)) != MP_OKAY) {
               goto LBL_RES;
            }
         }
      }
   }

   mp_exch(&res, Y);
   err = MP_OKAY;
LBL_RES:
   mp_clear(&res);
LBL_MU:
   mp_clear(&mu);
LBL_M:
   mp_clear(&M[1]);
   for (x = 1<<(winsize-1); x < (1 << winsize); x++) {
      mp_clear(&M[x]);
   }
   return err;
}

/* End: bn_s_mp_exptmod.c */

/* Start: bn_s_mp_mul_digs.c */

/* multiplies |a| * |b| and only computes upto digs digits of result
 * HAC pp. 595, Algorithm 14.12  Modified so you can control how
 * many digits of output are created.
 */
int s_mp_mul_digs(const mp_int *a, const mp_int *b, mp_int *c, int digs)
{
   mp_int  t;
   int     res, pa, pb, ix, iy;
   mp_digit u;
   mp_word r;
   mp_digit tmpx, *tmpt, *tmpy;

   /* can we use the fast multiplier? */
   if ((digs < (int)MP_WARRAY) &&
       (MIN(a->used, b->used) <
        (int)(1u << (((size_t)CHAR_BIT * sizeof(mp_word)) - (2u * (size_t)DIGIT_BIT))))) {
      return fast_s_mp_mul_digs(a, b, c, digs);
   }

   if ((res = mp_init_size(&t, digs)) != MP_OKAY) {
      return res;
   }
   t.used = digs;

   /* compute the digits of the product directly */
   pa = a->used;
   for (ix = 0; ix < pa; ix++) {
      /* set the carry to zero */
      u = 0;

      /* limit ourselves to making digs digits of output */
      pb = MIN(b->used, digs - ix);

      /* setup some aliases */
      /* copy of the digit from a used within the nested loop */
      tmpx = a->dp[ix];

      /* an alias for the destination shifted ix places */
      tmpt = t.dp + ix;

      /* an alias for the digits of b */
      tmpy = b->dp;

      /* compute the columns of the output and propagate the carry */
      for (iy = 0; iy < pb; iy++) {
         /* compute the column as a mp_word */
         r       = (mp_word)*tmpt +
                   ((mp_word)tmpx * (mp_word)*tmpy++) +
                   (mp_word)u;

         /* the new column is the lower part of the result */
         *tmpt++ = (mp_digit)(r & (mp_word)MP_MASK);

         /* get the carry word from the result */
         u       = (mp_digit)(r >> (mp_word)DIGIT_BIT);
      }
      /* set carry if it is placed below digs */
      if ((ix + iy) < digs) {
         *tmpt = u;
      }
   }

   mp_clamp(&t);
   mp_exch(&t, c);

   mp_clear(&t);
   return MP_OKAY;
}

/* End: bn_s_mp_mul_digs.c */

/* Start: bn_s_mp_mul_high_digs.c */

/* multiplies |a| * |b| and does not compute the lower digs digits
 * [meant to get the higher part of the product]
 */
int s_mp_mul_high_digs(const mp_int *a, const mp_int *b, mp_int *c, int digs)
{
   mp_int  t;
   int     res, pa, pb, ix, iy;
   mp_digit u;
   mp_word r;
   mp_digit tmpx, *tmpt, *tmpy;

   /* can we use the fast multiplier? */
   if (((a->used + b->used + 1) < (int)MP_WARRAY)
       && (MIN(a->used, b->used) < (int)(1u << (((size_t)CHAR_BIT * sizeof(mp_word)) - (2u * (size_t)DIGIT_BIT))))) {
      return fast_s_mp_mul_high_digs(a, b, c, digs);
   }

   if ((res = mp_init_size(&t, a->used + b->used + 1)) != MP_OKAY) {
      return res;
   }
   t.used = a->used + b->used + 1;

   pa = a->used;
   pb = b->used;
   for (ix = 0; ix < pa; ix++) {
      /* clear the carry */
      u = 0;

      /* left hand side of A[ix] * B[iy] */
      tmpx = a->dp[ix];

      /* alias to the address of where the digits will be stored */
      tmpt = &(t.dp[digs]);

      /* alias for where to read the right hand side from */
      tmpy = b->dp + (digs - ix);

      for (iy = digs - ix; iy < pb; iy++) {
         /* calculate the double precision result */
         r       = (mp_word)*tmpt +
                   ((mp_word)tmpx * (mp_word)*tmpy++) +
                   (mp_word)u;

         /* get the lower part */
         *tmpt++ = (mp_digit)(r & (mp_word)MP_MASK);

         /* carry the carry */
         u       = (mp_digit)(r >> (mp_word)DIGIT_BIT);
      }
      *tmpt = u;
   }
   mp_clamp(&t);
   mp_exch(&t, c);
   mp_clear(&t);
   return MP_OKAY;
}

/* End: bn_s_mp_mul_high_digs.c */

/* Start: bn_s_mp_sqr.c */

/* low level squaring, b = a*a, HAC pp.596-597, Algorithm 14.16 */
int s_mp_sqr(const mp_int *a, mp_int *b)
{
   mp_int  t;
   int     res, ix, iy, pa;
   mp_word r;
   mp_digit u, tmpx, *tmpt;

   pa = a->used;
   if ((res = mp_init_size(&t, (2 * pa) + 1)) != MP_OKAY) {
      return res;
   }

   /* default used is maximum possible size */
   t.used = (2 * pa) + 1;

   for (ix = 0; ix < pa; ix++) {
      /* first calculate the digit at 2*ix */
      /* calculate double precision result */
      r = (mp_word)t.dp[2*ix] +
          ((mp_word)a->dp[ix] * (mp_word)a->dp[ix]);

      /* store lower part in result */
      t.dp[ix+ix] = (mp_digit)(r & (mp_word)MP_MASK);

      /* get the carry */
      u           = (mp_digit)(r >> (mp_word)DIGIT_BIT);

      /* left hand side of A[ix] * A[iy] */
      tmpx        = a->dp[ix];

      /* alias for where to store the results */
      tmpt        = t.dp + ((2 * ix) + 1);

      for (iy = ix + 1; iy < pa; iy++) {
         /* first calculate the product */
         r       = (mp_word)tmpx * (mp_word)a->dp[iy];

         /* now calculate the double precision result, note we use
          * addition instead of *2 since it's easier to optimize
          */
         r       = (mp_word)*tmpt + r + r + (mp_word)u;

         /* store lower part */
         *tmpt++ = (mp_digit)(r & (mp_word)MP_MASK);

         /* get carry */
         u       = (mp_digit)(r >> (mp_word)DIGIT_BIT);
      }
      /* propagate upwards */
      while (u != 0uL) {
         r       = (mp_word)*tmpt + (mp_word)u;
         *tmpt++ = (mp_digit)(r & (mp_word)MP_MASK);
         u       = (mp_digit)(r >> (mp_word)DIGIT_BIT);
      }
   }

   mp_clamp(&t);
   mp_exch(&t, b);
   mp_clear(&t);
   return MP_OKAY;
}

/* End: bn_s_mp_sqr.c */

/* Start: bn_s_mp_sub.c */

/* low level subtraction (assumes |a| > |b|), HAC pp.595 Algorithm 14.9 */
int s_mp_sub(const mp_int *a, const mp_int *b, mp_int *c)
{
   int     olduse, res, min, max;

   /* find sizes */
   min = b->used;
   max = a->used;

   /* init result */
   if (c->alloc < max) {
      if ((res = mp_grow(c, max)) != MP_OKAY) {
         return res;
      }
   }
   olduse = c->used;
   c->used = max;

   {
      mp_digit u, *tmpa, *tmpb, *tmpc;
      int i;

      /* alias for digit pointers */
      tmpa = a->dp;
      tmpb = b->dp;
      tmpc = c->dp;

      /* set carry to zero */
      u = 0;
      for (i = 0; i < min; i++) {
         /* T[i] = A[i] - B[i] - U */
         *tmpc = (*tmpa++ - *tmpb++) - u;

         /* U = carry bit of T[i]
          * Note this saves performing an AND operation since
          * if a carry does occur it will propagate all the way to the
          * MSB.  As a result a single shift is enough to get the carry
          */
         u = *tmpc >> (((size_t)CHAR_BIT * sizeof(mp_digit)) - 1u);

         /* Clear carry from T[i] */
         *tmpc++ &= MP_MASK;
      }

      /* now copy higher words if any, e.g. if A has more digits than B  */
      for (; i < max; i++) {
         /* T[i] = A[i] - U */
         *tmpc = *tmpa++ - u;

         /* U = carry bit of T[i] */
         u = *tmpc >> (((size_t)CHAR_BIT * sizeof(mp_digit)) - 1u);

         /* Clear carry from T[i] */
         *tmpc++ &= MP_MASK;
      }

      /* clear digits above used (since we may not have grown result above) */
      for (i = c->used; i < olduse; i++) {
         *tmpc++ = 0;
      }
   }

   mp_clamp(c);
   return MP_OKAY;
}

/* End: bn_s_mp_sub.c */

/* Start: bncore.c */

/* Known optimal configurations

 CPU                    /Compiler     /MUL CUTOFF/SQR CUTOFF
-------------------------------------------------------------
 Intel P4 Northwood     /GCC v3.4.1   /        88/       128/LTM 0.32 ;-)
 AMD Athlon64           /GCC v3.4.4   /        80/       120/LTM 0.35

*/

int     KARATSUBA_MUL_CUTOFF = 80,      /* Min. number of digits before Karatsuba multiplication is used. */
        KARATSUBA_SQR_CUTOFF = 120,     /* Min. number of digits before Karatsuba squaring is used. */

        TOOM_MUL_CUTOFF      = 350,      /* no optimal values of these are known yet so set em high */
        TOOM_SQR_CUTOFF      = 400;

/* End: bncore.c */