/* 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
-the Free Software Foundation; either version 2 of the License, or (at your
+it under the terms of the GNU Lesser General Public License as published by
+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. */
#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
+ 128 156* 159 171 200 261
+ 256 309 307* 316 343 403
+ 512 617 607* 610 632 688
+ 1024 1231 1204 1195* 1207 1256
+ 2048 2461 2399 2366 2360* 2396
+ 4096 4918 4787 4707 4665* 4670
+*/
+\f
#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_size_t esize, msize, bsize, rsize;
- mp_size_t size;
- int mod_shift_cnt;
- int negative_result;
- mp_limb_t *free_me = NULL;
- size_t free_me_size;
- TMP_DECL (marker);
-
- esize = ABS (exp->_mp_size);
- msize = ABS (mod->_mp_size);
- size = 2 * msize;
+ mp_limb_t invm, *ep, c, mask;
+ mpz_t xx, *g;
+ mp_size_t n, i, K, j, l, k;
+ int sh;
+ int use_redc;
+
+#ifdef POWM_DEBUG
+ mpz_t exp;
+ mpz_init (exp);
+#endif
- rp = res->_mp_d;
- ep = exp->_mp_d;
+ n = ABSIZ (mod);
- if (msize == 0)
- msize = 1 / msize; /* provoke a signal */
+ if (n == 0)
+ 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;
- res->_mp_size = (msize == 1 && (mod->_mp_d)[0] == 1) ? 0 : 1;
+ depending on if MOD equals 1. */
+ SIZ(res) = (ABSIZ (mod) == 1 && (PTR(mod))[0] == 1) ? 0 : 1;
+ PTR(res)[0] = 1;
return;
}
- TMP_MARK (marker);
-
- /* Normalize MOD (i.e. make its most significant bit set) as required by
- mpn_divmod. This will make the intermediate values in the calculation
- slightly larger, but the correct result is obtained after a final
- reduction using the original MOD value. */
+ /* 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))) ~
+ 2.66. */
+ /* For now, also disable REDC when MOD is even, as the inverse can't
+ handle that. */
+
+#ifndef POWM_THRESHOLD
+#define POWM_THRESHOLD ((8 * KARATSUBA_SQR_THRESHOLD) / 3)
+#endif
- mp = (mp_ptr) TMP_ALLOC (msize * BYTES_PER_MP_LIMB);
- count_leading_zeros (mod_shift_cnt, mod->_mp_d[msize - 1]);
- if (mod_shift_cnt != 0)
- mpn_lshift (mp, mod->_mp_d, msize, mod_shift_cnt);
- else
- MPN_COPY (mp, mod->_mp_d, msize);
+ use_redc = (n < POWM_THRESHOLD && PTR(mod)[0] % 2 != 0);
+ if (use_redc)
+ {
+ /* invm = -1/m mod 2^BITS_PER_MP_LIMB, must have m odd */
+ modlimb_invert (invm, PTR(mod)[0]);
+ invm = -invm;
+ }
- bsize = ABS (base->_mp_size);
- if (bsize > msize)
+ /* determines optimal value of k */
+ 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.
- (The quotient is (bsize - msize + 1) limbs.) */
- bp = (mp_ptr) TMP_ALLOC ((bsize + 1) * BYTES_PER_MP_LIMB);
- MPN_COPY (bp, base->_mp_d, bsize);
- /* We don't care about the quotient, store it above the remainder,
- at BP + MSIZE. */
- mpn_divmod (bp + msize, bp, bsize, mp, msize);
- bsize = msize;
- /* Canonicalize the base, since we are going to multiply with it
- quite a few times. */
- MPN_NORMALIZE (bp, bsize);
+ g = (mpz_t *) (*_mp_allocate_func) (K / 2 * sizeof (mpz_t));
+ /* compute x*R^n where R=2^BITS_PER_MP_LIMB */
+ mpz_init (g[0]);
+ if (use_redc)
+ {
+ mpz_mul_2exp (g[0], base, n * BITS_PER_MP_LIMB);
+ mpz_mod (g[0], g[0], mod);
}
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;
- TMP_FREE (marker);
- return;
+ _mpz_realloc (xx, 2 * n);
+ mpz_redc (xx, g[0], g[0], mod, invm); /* xx = x^2*R^n */
}
-
- if (res->_mp_alloc < size)
+ else
{
- /* We have to allocate more space for RES. If any of the input
- parameters are identical to RES, defer deallocation of the old
- space. */
-
- if (rp == ep || rp == mp || rp == bp)
+ mpz_mul (xx, g[0], g[0]);
+ mpz_mod (xx, xx, mod);
+ }
+ for (i = 1; i < K / 2; i++)
+ {
+ mpz_init (g[i]);
+ if (use_redc)
{
- free_me = rp;
- free_me_size = res->_mp_alloc;
+ _mpz_realloc (g[i], 2 * n);
+ 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);
-
- rp = (mp_ptr) (*_mp_allocate_func) (size * BYTES_PER_MP_LIMB);
- res->_mp_alloc = size;
- res->_mp_d = rp;
- }
- else
- {
- /* Make BASE, EXP and MOD not overlap with RES. */
- if (rp == bp)
{
- /* RES and BASE are identical. Allocate temp. space for BASE. */
- bp = (mp_ptr) TMP_ALLOC (bsize * BYTES_PER_MP_LIMB);
- MPN_COPY (bp, rp, bsize);
+ mpz_mul (g[i], g[i - 1], xx);
+ mpz_mod (g[i], g[i], mod);
}
- if (rp == ep)
+ }
+
+ /* now starts the real stuff */
+ mask = (mp_limb_t) ((1<<k) - 1);
+ 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)
{
- /* RES and EXP are identical. Allocate temp. space for EXP. */
- 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;
}
- if (rp == mp)
+ }
+ 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)
+ {
+ j++;
+ c = (c >> 1);
+ }
+ mpz_set (xx, g[c >> 1]);
+ while (j--)
+ {
+ if (use_redc)
+ mpz_redc (xx, xx, xx, mod, invm);
+ else
{
- /* RES and MOD are identical. Allocate temporary space for MOD. */
- mp = (mp_ptr) TMP_ALLOC (msize * BYTES_PER_MP_LIMB);
- MPN_COPY (mp, rp, msize);
+ mpz_mul (xx, xx, xx);
+ mpz_mod (xx, xx, mod);
}
}
- MPN_COPY (rp, bp, bsize);
- rsize = bsize;
-
- {
- mp_size_t i;
- mp_ptr xp = (mp_ptr) TMP_ALLOC (2 * (msize + 1) * BYTES_PER_MP_LIMB);
- int c;
- mp_limb_t e;
- mp_limb_t carry_limb;
-
- negative_result = (ep[0] & 1) && base->_mp_size < 0;
-
- i = esize - 1;
- e = ep[i];
- count_leading_zeros (c, e);
- e = (e << c) << 1; /* shift the exp bits to the left, lose msb */
- c = BITS_PER_MP_LIMB - 1 - c;
-
- /* 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. */
+#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
- for (;;)
- {
- while (c != 0)
- {
- mp_ptr tp;
- mp_size_t xsize;
+ while (i > 0 || sh > 0)
+ {
+ c = ep[i];
+ sh -= k;
+ l = k; /* number of bits treated */
+ if (sh < 0)
+ {
+ if (i > 0)
+ {
+ 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);
+ }
+ }
+ else
+ c = c >> sh;
+ c = c & mask;
- mpn_mul_n (xp, rp, rp, rsize);
- xsize = 2 * rsize;
- if (xsize > msize)
- {
- mpn_divmod (xp + msize, xp, xsize, mp, msize);
- xsize = msize;
- }
+ /* this while loop implements the sliding window improvement */
+ while ((c & (1 << (k - 1))) == 0 && (i > 0 || sh > 0))
+ {
+ if (use_redc) mpz_redc (xx, xx, xx, mod, invm);
+ else
+ {
+ 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);
+ }
+ }
- tp = rp; rp = xp; xp = tp;
- rsize = xsize;
+#ifdef POWM_DEBUG
+ printf ("l=%u c=%lu\n", l, c);
+ mpz_mul_2exp (exp, exp, k);
+ mpz_add_ui (exp, exp, c);
+#endif
- if ((mp_limb_signed_t) e < 0)
+ /* now replace xx by xx^(2^k)*x^c */
+ if (c != 0)
+ {
+ j = 0;
+ while (c % 2 == 0)
+ {
+ j++;
+ c = c >> 1;
+ }
+ /* c0 = c * 2^j, i.e. xx^(2^k)*x^c = (A^(2^(k - j))*c)^(2^j) */
+ l -= j;
+ while (l--)
+ if (use_redc) mpz_redc (xx, xx, xx, mod, invm);
+ else
{
- 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;
+ mpz_mul (xx, xx, xx);
+ mpz_mod (xx, xx, mod);
}
- 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 (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
+ }
- if (negative_result && rsize != 0)
+ /* now convert back xx to xx/R^n */
+ if (use_redc)
{
- if (mod_shift_cnt != 0)
- mpn_rshift (mp, mp, msize, mod_shift_cnt);
- mpn_sub (rp, mp, msize, rp, rsize);
- rsize = msize;
- MPN_NORMALIZE (rp, rsize);
+ mpz_set_ui (g[0], 1);
+ mpz_redc (xx, xx, g[0], mod, invm);
+ if (mpz_cmp (xx, mod) >= 0)
+ mpz_sub (xx, xx, mod);
}
- res->_mp_size = rsize;
+ mpz_set (res, xx);
- if (free_me != NULL)
- (*_mp_free_func) (free_me, free_me_size * BYTES_PER_MP_LIMB);
- TMP_FREE (marker);
+ mpz_clear (xx);
+ for (i = 0; i < K / 2; i++)
+ mpz_clear (g[i]);
+ (*_mp_free_func) (g, K / 2 * sizeof (mpz_t));
}
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