source/libs/gmp/gmp-src/mpn/generic/brootinv.c

   1 /* mpn_brootinv, compute r such that r^k * y = 1 (mod 2^b).
   2
   3    Contributed to the GNU project by Martin Boij (as part of perfpow.c).
   4
   5 Copyright 2009, 2010, 2012, 2013 Free Software Foundation, Inc.
   6
   7 This file is part of the GNU MP Library.
   8
   9 The GNU MP Library is free software; you can redistribute it and/or modify
  10 it under the terms of either:
  11
  12   * the GNU Lesser General Public License as published by the Free
  13     Software Foundation; either version 3 of the License, or (at your
  14     option) any later version.
  15
  16 or
  17
  18   * the GNU General Public License as published by the Free Software
  19     Foundation; either version 2 of the License, or (at your option) any
  20     later version.
  21
  22 or both in parallel, as here.
  23
  24 The GNU MP Library is distributed in the hope that it will be useful, but
  25 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
  26 or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
  27 for more details.
  28
  29 You should have received copies of the GNU General Public License and the
  30 GNU Lesser General Public License along with the GNU MP Library.  If not,
  31 see https://www.gnu.org/licenses/.  */
  32
  33 #include "gmp.h"
  34 #include "gmp-impl.h"
  35
  36 /* Computes a^e (mod B). Uses right-to-left binary algorithm, since
  37    typical use will have e small. */
  38 static mp_limb_t
  39 powlimb (mp_limb_t a, mp_limb_t e)
  40 {
  41   mp_limb_t r;
  42
  43   for (r = 1; e > 0; e >>= 1, a *= a)
  44     if (e & 1)
  45       r *= a;
  46
  47   return r;
  48 }
  49
  50 /* Compute r such that r^k * y = 1 (mod B^n).
  51
  52    Iterates
  53      r' <-- k^{-1} ((k+1) r - r^{k+1} y) (mod 2^b)
  54    using Hensel lifting, each time doubling the number of known bits in r.
  55
  56    Works just for odd k.  Else the Hensel lifting degenerates.
  57
  58    FIXME:
  59
  60      (1) Make it work for k == GMP_LIMB_MAX (k+1 below overflows).
  61
  62      (2) Rewrite iteration as
  63            r' <-- r - k^{-1} r (r^k y - 1)
  64          and take advantage of the zero low part of r^k y - 1.
  65
  66      (3) Use wrap-around trick.
  67
  68      (4) Use a small table to get starting value.
  69
  70    Scratch need: 5*bn, where bn = ceil (bnb / GMP_NUMB_BITS).
  71 */
  72
  73 void
  74 mpn_brootinv (mp_ptr rp, mp_srcptr yp, mp_size_t bn, mp_limb_t k, mp_ptr tp)
  75 {
  76   mp_ptr tp2, tp3;
  77   mp_limb_t kinv, k2, r0, y0;
  78   mp_size_t order[GMP_LIMB_BITS + 1];
  79   int i, d;
  80
  81   ASSERT (bn > 0);
  82   ASSERT ((k & 1) != 0);
  83
  84   tp2 = tp + bn;
  85   tp3 = tp + 2 * bn;
  86   k2 = k + 1;
  87
  88   binvert_limb (kinv, k);
  89
  90   /* 4-bit initial approximation:
  91
  92    y%16 | 1  3  5  7  9 11 13 15,
  93     k%4 +-------------------------+k2%4
  94      1  | 1 11 13  7  9  3  5 15  |  2
  95      3  | 1  3  5  7  9 11 13 15  |  0
  96
  97   */
  98   y0 = yp[0];
  99
 100   r0 = y0 ^ (((y0 << 1) ^ (y0 << 2)) & (k2 << 2) & 8);          /* 4 bits */
 101   r0 = kinv * (k2 * r0 - y0 * powlimb(r0, k2 & 0x7f));          /* 8 bits */
 102   r0 = kinv * (k2 * r0 - y0 * powlimb(r0, k2 & 0x7fff));        /* 16 bits */
 103 #if GMP_NUMB_BITS > 16
 104   {
 105     unsigned prec = 16;
 106     do
 107       {
 108         r0 = kinv * (k2 * r0 - y0 * powlimb(r0, k2));
 109         prec *= 2;
 110       }
 111     while (prec < GMP_NUMB_BITS);
 112   }
 113 #endif
 114
 115   rp[0] = r0;
 116   if (bn == 1)
 117     return;
 118
 119   /* This initialization doesn't matter for the result (any garbage is
 120      cancelled in the iteration), but proper initialization makes
 121      valgrind happier. */
 122   MPN_ZERO (rp+1, bn-1);
 123
 124   d = 0;
 125   for (; bn > 1; bn = (bn + 1) >> 1)
 126     order[d++] = bn;
 127
 128   for (i = d - 1; i >= 0; i--)
 129     {
 130       bn = order[i];
 131
 132       mpn_mul_1 (tp, rp, bn, k2);
 133
 134       mpn_powlo (tp2, rp, &k2, 1, bn, tp3);
 135       mpn_mullo_n (rp, yp, tp2, bn);
 136
 137       mpn_sub_n (tp2, tp, rp, bn);
 138       mpn_pi1_bdiv_q_1 (rp, tp2, bn, k, kinv, 0);
 139     }
 140 }