OpenXM_contrib/gmp/mpz/powm.c - diff

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Diff for /OpenXM_contrib/gmp/mpz/Attic/powm.c between version 1.1.1.1 and 1.1.1.2

-version 1.1.1.1, 2000/01/10 15:35:27
+version 1.1.1.2, 2000/09/09 14:12:56
 Line 1
 Line 1
 Line 1
  /* mpz_powm(res,base,exp,mod) -- Set RES to (base**exp) mod MOD.
- Copyright (C) 1991, 1993, 1994, 1996 Free Software Foundation, Inc.
+ Copyright (C) 1991, 1993, 1994, 1996, 1997, 2000 Free Software Foundation, Inc.
+ Contributed by Paul Zimmermann.
  This file is part of the GNU MP Library.
  The GNU MP Library is free software; you can redistribute it and/or modify
- it under the terms of the GNU Library General Public License as published by
+ it under the terms of the GNU Lesser General Public License as published by
- the Free Software Foundation; either version 2 of the License, or (at your
+ the Free Software Foundation; either version 2.1 of the License, or (at your
  option) any later version.
  The GNU MP Library is distributed in the hope that it will be useful, but
  WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
- or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU Library General Public
+ or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU Lesser General Public
  License for more details.
- You should have received a copy of the GNU Library General Public License
+ You should have received a copy of the GNU Lesser General Public License
  along with the GNU MP Library; see the file COPYING.LIB.  If not, write to
  the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
  MA 02111-1307, USA. */
-Line 22  MA 02111-1307, USA. */
+Line 23  MA 02111-1307, USA. */
 Line 22  MA 02111-1307, USA. */
 Line 23  MA 02111-1307, USA. */
  #include "gmp.h"
  #include "gmp-impl.h"
  #include "longlong.h"
+ #ifdef BERKELEY_MP
+ #include "mp.h"
+ #endif
+ /* set c <- (a*b)/R^n mod m c has to have at least (2n) allocated limbs */
+ static void
+ #if __STDC__
+ mpz_redc (mpz_ptr c, mpz_srcptr a, mpz_srcptr b, mpz_srcptr m, mp_limb_t Nprim)
+ #else
+ mpz_redc (c, a, b, m, Nprim)
+      mpz_ptr c;
+      mpz_srcptr a;
+      mpz_srcptr b;
+      mpz_srcptr m;
+      mp_limb_t Nprim;
+ #endif
+ {
+   mp_ptr cp, mp = PTR (m);
+   mp_limb_t cy, cout = 0;
+   mp_limb_t q;
+   size_t j, n = ABSIZ (m);
+   ASSERT (ALLOC (c) >= 2 * n);
+   mpz_mul (c, a, b);
+   cp = PTR (c);
+   j = ABSIZ (c);
+   MPN_ZERO (cp + j, 2 * n - j);
+   for (j = 0; j < n; j++)
+     {
+       q = cp[0] * Nprim;
+       cy = mpn_addmul_1 (cp, mp, n, q);
+       cout += mpn_add_1 (cp + n, cp + n, n - j, cy);
+       cp++;
+     }
+   cp -= n;
+   if (cout)
+     {
+       cy = cout - mpn_sub_n (cp, cp + n, mp, n);
+       while (cy)
+         cy -= mpn_sub_n (cp, cp, mp, n);
+     }
+   else
+     MPN_COPY (cp, cp + n, n);
+   MPN_NORMALIZE (cp, n);
+   SIZ (c) = SIZ (c) < 0 ? -n : n;
+ }
+ /* average number of calls to redc for an exponent of n bits
+    with the sliding window algorithm of base 2^k: the optimal is
+    obtained for the value of k which minimizes 2^(k-1)+n/(k+1):
+    n\k    4     5     6     7     8
+   156*  159   171   200   261
+   309   307*  316   343   403
+   617   607*  610   632   688
+  1231  1204  1195* 1207  1256
+  2461  2399  2366  2360* 2396
+  4918  4787  4707  4665* 4670
+ */
  #ifndef BERKELEY_MP
  void
  #if __STDC__
- mpz_powm (mpz_ptr res, mpz_srcptr base, mpz_srcptr exp, mpz_srcptr mod)
+ mpz_powm (mpz_ptr res, mpz_srcptr base, mpz_srcptr e, mpz_srcptr mod)
  #else
- mpz_powm (res, base, exp, mod)
+ mpz_powm (res, base, e, mod)
       mpz_ptr res;
       mpz_srcptr base;
-      mpz_srcptr exp;
+      mpz_srcptr e;
       mpz_srcptr mod;
  #endif
  #else /* BERKELEY_MP */
  void
  #if __STDC__
- pow (mpz_srcptr base, mpz_srcptr exp, mpz_srcptr mod, mpz_ptr res)
+ pow (mpz_srcptr base, mpz_srcptr e, mpz_srcptr mod, mpz_ptr res)
  #else
- pow (base, exp, mod, res)
+ pow (base, e, mod, res)
       mpz_srcptr base;
-      mpz_srcptr exp;
+      mpz_srcptr e;
       mpz_srcptr mod;
       mpz_ptr res;
  #endif
  #endif /* BERKELEY_MP */
  {
-   mp_ptr rp, ep, mp, bp;
+   mp_limb_t invm, *ep, c, mask;
-   mp_size_t esize, msize, bsize, rsize;
+   mpz_t xx, *g;
-   mp_size_t size;
+   mp_size_t n, i, K, j, l, k;
-   int mod_shift_cnt;
+   int sh;
-   int negative_result;
+   int use_redc;
-   mp_limb_t *free_me = NULL;
-   size_t free_me_size;
-   TMP_DECL (marker);
-   esize = ABS (exp->_mp_size);
+ #ifdef POWM_DEBUG
-   msize = ABS (mod->_mp_size);
+   mpz_t exp;
-   size = 2 * msize;
+   mpz_init (exp);
+ #endif
-   rp = res->_mp_d;
+   n = ABSIZ (mod);
-   ep = exp->_mp_d;
-   if (msize == 0)
+   if (n == 0)
-     msize = 1 / msize;          /* provoke a signal */
+     DIVIDE_BY_ZERO;
-   if (esize == 0)
+   if (SIZ (e) == 0)
      {
        /* Exponent is zero, result is 1 mod MOD, i.e., 1 or 0
           depending on if MOD equals 1.  */
-       rp[0] = 1;
+       SIZ(res) = (ABSIZ (mod) == 1 && (PTR(mod))[0] == 1) ? 0 : 1;
-       res->_mp_size = (msize == 1 && (mod->_mp_d)[0] == 1) ? 0 : 1;
+       PTR(res)[0] = 1;
        return;
      }
-   TMP_MARK (marker);
+   /* Use REDC instead of usual reduction for sizes < POWM_THRESHOLD.
+      In REDC each modular multiplication costs about 2*n^2 limbs operations,
+      whereas using usual reduction it costs 3*K(n), where K(n) is the cost of a
+      multiplication using Karatsuba, and a division is assumed to cost 2*K(n),
+      for example using Burnikel-Ziegler's algorithm. This gives a theoretical
+      threshold of a*KARATSUBA_SQR_THRESHOLD, with a=(3/2)^(1/(2-ln(3)/ln(2))) ~
+.66.  */
+   /* For now, also disable REDC when MOD is even, as the inverse can't
+      handle that.  */
-   /* Normalize MOD (i.e. make its most significant bit set) as required by
+ #ifndef POWM_THRESHOLD
-      mpn_divmod.  This will make the intermediate values in the calculation
+ #define POWM_THRESHOLD  ((8 * KARATSUBA_SQR_THRESHOLD) / 3)
-      slightly larger, but the correct result is obtained after a final
+ #endif
-      reduction using the original MOD value.  */
-   mp = (mp_ptr) TMP_ALLOC (msize * BYTES_PER_MP_LIMB);
+   use_redc = (n < POWM_THRESHOLD && PTR(mod)[0] % 2 != 0);
-   count_leading_zeros (mod_shift_cnt, mod->_mp_d[msize - 1]);
+   if (use_redc)
-   if (mod_shift_cnt != 0)
+     {
-     mpn_lshift (mp, mod->_mp_d, msize, mod_shift_cnt);
+       /* invm = -1/m mod 2^BITS_PER_MP_LIMB, must have m odd */
-   else
+       modlimb_invert (invm, PTR(mod)[0]);
-     MPN_COPY (mp, mod->_mp_d, msize);
+       invm = -invm;
+     }
-   bsize = ABS (base->_mp_size);
+   /* determines optimal value of k */
-   if (bsize > msize)
+   l = ABSIZ (e) * BITS_PER_MP_LIMB; /* number of bits of exponent */
+   k = 1;
+   K = 2;
+   while (2 * l > K * (2 + k * (3 + k)))
      {
-       /* The base is larger than the module.  Reduce it.  */
+       k++;
+       K *= 2;
+     }
-       /* Allocate (BSIZE + 1) with space for remainder and quotient.
+   g = (mpz_t *) (*_mp_allocate_func) (K / 2 * sizeof (mpz_t));
-          (The quotient is (bsize - msize + 1) limbs.)  */
+   /* compute x*R^n where R=2^BITS_PER_MP_LIMB */
-       bp = (mp_ptr) TMP_ALLOC ((bsize + 1) * BYTES_PER_MP_LIMB);
+   mpz_init (g[0]);
-       MPN_COPY (bp, base->_mp_d, bsize);
+   if (use_redc)
-       /* We don't care about the quotient, store it above the remainder,
+     {
-          at BP + MSIZE.  */
+       mpz_mul_2exp (g[0], base, n * BITS_PER_MP_LIMB);
-       mpn_divmod (bp + msize, bp, bsize, mp, msize);
+       mpz_mod (g[0], g[0], mod);
-       bsize = msize;
-       /* Canonicalize the base, since we are going to multiply with it
-          quite a few times.  */
-       MPN_NORMALIZE (bp, bsize);
      }
    else
-     bp = base->_mp_d;
+     mpz_mod (g[0], base, mod);
-   if (bsize == 0)
+   /* compute xx^g for odd g < 2^k */
+   mpz_init (xx);
+   if (use_redc)
      {
-       res->_mp_size = 0;
+       _mpz_realloc (xx, 2 * n);
-       TMP_FREE (marker);
+       mpz_redc (xx, g[0], g[0], mod, invm); /* xx = x^2*R^n */
-       return;
      }
+   else
-   if (res->_mp_alloc < size)
      {
-       /* We have to allocate more space for RES.  If any of the input
+       mpz_mul (xx, g[0], g[0]);
-          parameters are identical to RES, defer deallocation of the old
+       mpz_mod (xx, xx, mod);
-          space.  */
+     }
+   for (i = 1; i < K / 2; i++)
-       if (rp == ep || rp == mp || rp == bp)
+     {
+       mpz_init (g[i]);
+       if (use_redc)
          {
-           free_me = rp;
+           _mpz_realloc (g[i], 2 * n);
-           free_me_size = res->_mp_alloc;
+           mpz_redc (g[i], g[i - 1], xx, mod, invm); /* g[i] = x^(2i+1)*R^n */
          }
        else
-         (*_mp_free_func) (rp, res->_mp_alloc * BYTES_PER_MP_LIMB);
+         {
+           mpz_mul (g[i], g[i - 1], xx);
+           mpz_mod (g[i], g[i], mod);
+         }
+     }
-       rp = (mp_ptr) (*_mp_allocate_func) (size * BYTES_PER_MP_LIMB);
+   /* now starts the real stuff */
-       res->_mp_alloc = size;
+   mask = (mp_limb_t) ((1<<k) - 1);
-       res->_mp_d = rp;
+   ep = PTR (e);
+   i = ABSIZ (e) - 1;                    /* current index */
+   c = ep[i];                            /* current limb */
+   count_leading_zeros (sh, c);
+   sh = BITS_PER_MP_LIMB - sh;           /* significant bits in ep[i] */
+   sh -= k;                              /* index of lower bit of ep[i] to take into account */
+   if (sh < 0)
+     {                                   /* k-sh extra bits are needed */
+       if (i > 0)
+         {
+           i--;
+           c = (c << (-sh)) | (ep[i] >> (BITS_PER_MP_LIMB + sh));
+           sh += BITS_PER_MP_LIMB;
+         }
      }
    else
+     c = c >> sh;
+ #ifdef POWM_DEBUG
+   printf ("-1/m mod 2^%u = %lu\n", BITS_PER_MP_LIMB, invm);
+   mpz_set_ui (exp, c);
+ #endif
+   j=0;
+   while (c % 2 == 0)
      {
-       /* Make BASE, EXP and MOD not overlap with RES.  */
+       j++;
-       if (rp == bp)
+       c = (c >> 1);
+     }
+   mpz_set (xx, g[c >> 1]);
+   while (j--)
+     {
+       if (use_redc)
+         mpz_redc (xx, xx, xx, mod, invm);
+       else
          {
-           /* RES and BASE are identical.  Allocate temp. space for BASE.  */
+           mpz_mul (xx, xx, xx);
-           bp = (mp_ptr) TMP_ALLOC (bsize * BYTES_PER_MP_LIMB);
+           mpz_mod (xx, xx, mod);
-           MPN_COPY (bp, rp, bsize);
          }
-       if (rp == ep)
+     }
+ #ifdef POWM_DEBUG
+   printf ("x^"); mpz_out_str (0, 10, exp);
+   printf ("*2^%u mod m = ", n * BITS_PER_MP_LIMB); mpz_out_str (0, 10, xx);
+   putchar ('\n');
+ #endif
+   while (i > 0 || sh > 0)
+     {
+       c = ep[i];
+       sh -= k;
+       l = k;                            /* number of bits treated */
+       if (sh < 0)
          {
-           /* RES and EXP are identical.  Allocate temp. space for EXP.  */
+           if (i > 0)
-           ep = (mp_ptr) TMP_ALLOC (esize * BYTES_PER_MP_LIMB);
+             {
-           MPN_COPY (ep, rp, esize);
+               i--;
+               c = (c << (-sh)) | (ep[i] >> (BITS_PER_MP_LIMB + sh));
+               sh += BITS_PER_MP_LIMB;
+             }
+           else
+             {
+               l += sh;                  /* may be less bits than k here */
+               c = c & ((1<<l) - 1);
+             }
          }
-       if (rp == mp)
+       else
+         c = c >> sh;
+       c = c & mask;
+       /* this while loop implements the sliding window improvement */
+       while ((c & (1 << (k - 1))) == 0 && (i > 0 || sh > 0))
          {
-           /* RES and MOD are identical.  Allocate temporary space for MOD.  */
+           if (use_redc) mpz_redc (xx, xx, xx, mod, invm);
-           mp = (mp_ptr) TMP_ALLOC (msize * BYTES_PER_MP_LIMB);
+           else
-           MPN_COPY (mp, rp, msize);
+             {
+               mpz_mul (xx, xx, xx);
+               mpz_mod (xx, xx, mod);
+             }
+           if (sh)
+             {
+               sh--;
+               c = (c<<1) + ((ep[i]>>sh) & 1);
+             }
+           else
+             {
+               i--;
+               sh = BITS_PER_MP_LIMB - 1;
+               c = (c<<1) + (ep[i]>>sh);
+             }
          }
-     }
-   MPN_COPY (rp, bp, bsize);
+ #ifdef POWM_DEBUG
-   rsize = bsize;
+       printf ("l=%u c=%lu\n", l, c);
+       mpz_mul_2exp (exp, exp, k);
+       mpz_add_ui (exp, exp, c);
+ #endif
-   {
+       /* now replace xx by xx^(2^k)*x^c */
-     mp_size_t i;
+       if (c != 0)
-     mp_ptr xp = (mp_ptr) TMP_ALLOC (2 * (msize + 1) * BYTES_PER_MP_LIMB);
+         {
-     int c;
+           j = 0;
-     mp_limb_t e;
+           while (c % 2 == 0)
-     mp_limb_t carry_limb;
+             {
+               j++;
-     negative_result = (ep[0] & 1) && base->_mp_size < 0;
+               c = c >> 1;
+             }
-     i = esize - 1;
+           /* c0 = c * 2^j, i.e. xx^(2^k)*x^c = (A^(2^(k - j))*c)^(2^j) */
-     e = ep[i];
+           l -= j;
-     count_leading_zeros (c, e);
+           while (l--)
-     e = (e << c) << 1;          /* shift the exp bits to the left, lose msb */
+             if (use_redc) mpz_redc (xx, xx, xx, mod, invm);
-     c = BITS_PER_MP_LIMB - 1 - c;
+             else
-     /* Main loop.
-        Make the result be pointed to alternately by XP and RP.  This
-        helps us avoid block copying, which would otherwise be necessary
-        with the overlap restrictions of mpn_divmod.  With 50% probability
-        the result after this loop will be in the area originally pointed
-        by RP (==RES->_mp_d), and with 50% probability in the area originally
-        pointed to by XP.  */
-     for (;;)
-       {
-         while (c != 0)
-           {
-             mp_ptr tp;
-             mp_size_t xsize;
-             mpn_mul_n (xp, rp, rp, rsize);
-             xsize = 2 * rsize;
-             if (xsize > msize)
                {
-                 mpn_divmod (xp + msize, xp, xsize, mp, msize);
+                 mpz_mul (xx, xx, xx);
-                 xsize = msize;
+                 mpz_mod (xx, xx, mod);
                }
+           if (use_redc)
+             mpz_redc (xx, xx, g[c >> 1], mod, invm);
+           else
+             {
+               mpz_mul (xx, xx, g[c >> 1]);
+               mpz_mod (xx, xx, mod);
+             }
+         }
+       else
+         j = l;                          /* case c=0 */
+       while (j--)
+         {
+           if (use_redc)
+             mpz_redc (xx, xx, xx, mod, invm);
+           else
+             {
+               mpz_mul (xx, xx, xx);
+               mpz_mod (xx, xx, mod);
+             }
+         }
+ #ifdef POWM_DEBUG
+       printf ("x^"); mpz_out_str (0, 10, exp);
+       printf ("*2^%u mod m = ", n * BITS_PER_MP_LIMB); mpz_out_str (0, 10, xx);
+       putchar ('\n');
+ #endif
+     }
-             tp = rp; rp = xp; xp = tp;
+   /* now convert back xx to xx/R^n */
-             rsize = xsize;
+   if (use_redc)
-             if ((mp_limb_signed_t) e < 0)
-               {
-                 mpn_mul (xp, rp, rsize, bp, bsize);
-                 xsize = rsize + bsize;
-                 if (xsize > msize)
-                   {
-                     mpn_divmod (xp + msize, xp, xsize, mp, msize);
-                     xsize = msize;
-                   }
-                 tp = rp; rp = xp; xp = tp;
-                 rsize = xsize;
-               }
-             e <<= 1;
-             c--;
-           }
-         i--;
-         if (i < 0)
-           break;
-         e = ep[i];
-         c = BITS_PER_MP_LIMB;
-       }
-     /* We shifted MOD, the modulo reduction argument, left MOD_SHIFT_CNT
-        steps.  Adjust the result by reducing it with the original MOD.
-        Also make sure the result is put in RES->_mp_d (where it already
-        might be, see above).  */
-     if (mod_shift_cnt != 0)
-       {
-         carry_limb = mpn_lshift (res->_mp_d, rp, rsize, mod_shift_cnt);
-         rp = res->_mp_d;
-         if (carry_limb != 0)
-           {
-             rp[rsize] = carry_limb;
-             rsize++;
-           }
-       }
-     else
-       {
-         MPN_COPY (res->_mp_d, rp, rsize);
-         rp = res->_mp_d;
-       }
-     if (rsize >= msize)
-       {
-         mpn_divmod (rp + msize, rp, rsize, mp, msize);
-         rsize = msize;
-       }
-     /* Remove any leading zero words from the result.  */
-     if (mod_shift_cnt != 0)
-       mpn_rshift (rp, rp, rsize, mod_shift_cnt);
-     MPN_NORMALIZE (rp, rsize);
-   }
-   if (negative_result && rsize != 0)
      {
-       if (mod_shift_cnt != 0)
+       mpz_set_ui (g[0], 1);
-         mpn_rshift (mp, mp, msize, mod_shift_cnt);
+       mpz_redc (xx, xx, g[0], mod, invm);
-       mpn_sub (rp, mp, msize, rp, rsize);
+       if (mpz_cmp (xx, mod) >= 0)
-       rsize = msize;
+         mpz_sub (xx, xx, mod);
-       MPN_NORMALIZE (rp, rsize);
      }
-   res->_mp_size = rsize;
+   mpz_set (res, xx);
-   if (free_me != NULL)
+   mpz_clear (xx);
-     (*_mp_free_func) (free_me, free_me_size * BYTES_PER_MP_LIMB);
+   for (i = 0; i < K / 2; i++)
-   TMP_FREE (marker);
+     mpz_clear (g[i]);
+   (*_mp_free_func) (g, K / 2 * sizeof (mpz_t));
  }

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