mdw@git.distorted.org.uk Git - catacomb/blob - math/mp-gcd.c

   1 /* -*-c-*-
   2  *
   3  * Extended GCD calculation
   4  *
   5  * (c) 1999 Straylight/Edgeware
   6  */
   7
   8 /*----- Licensing notice --------------------------------------------------*
   9  *
  10  * This file is part of Catacomb.
  11  *
  12  * Catacomb is free software; you can redistribute it and/or modify
  13  * it under the terms of the GNU Library General Public License as
  14  * published by the Free Software Foundation; either version 2 of the
  15  * License, or (at your option) any later version.
  16  *
  17  * Catacomb is distributed in the hope that it will be useful,
  18  * but WITHOUT ANY WARRANTY; without even the implied warranty of
  19  * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  20  * GNU Library General Public License for more details.
  21  *
  22  * You should have received a copy of the GNU Library General Public
  23  * License along with Catacomb; if not, write to the Free
  24  * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
  25  * MA 02111-1307, USA.
  26  */
  27
  28 /*----- Header files ------------------------------------------------------*/
  29
  30 #include "mp.h"
  31
  32 /*----- Main code ---------------------------------------------------------*/
  33
  34 /* --- @mp_gcd@ --- *
  35  *
  36  * Arguments:   @mp **gcd, **xx, **yy@ = where to write the results
  37  *              @mp *a, *b@ = sources (must be nonzero)
  38  *
  39  * Returns:     ---
  40  *
  41  * Use:         Calculates @gcd(a, b)@, and two numbers @x@ and @y@ such that
  42  *              @ax + by = gcd(a, b)@.  This is useful for computing modular
  43  *              inverses.
  44  */
  45
  46 void mp_gcd(mp **gcd, mp **xx, mp **yy, mp *a, mp *b)
  47 {
  48   mp *x = MP_ONE, *X = MP_ZERO;
  49   mp *y = MP_ZERO, *Y = MP_ONE;
  50   mp *u, *v;
  51   mp *q = MP_NEW, *t, *spare = MP_NEW;
  52   unsigned f = 0;
  53
  54 #define f_swap 1u
  55 #define f_aneg 2u
  56 #define f_bneg 4u
  57 #define f_ext 8u
  58
  59   /* --- Sort out some initial flags --- */
  60
  61   if (xx || yy) f |= f_ext;
  62
  63   if (MP_NEGP(a)) f |= f_aneg;
  64   if (MP_NEGP(b)) f |= f_bneg;
  65
  66   /* --- Ensure that @a@ is larger than @b@ --- *
  67    *
  68    * Use absolute values here!
  69    */
  70
  71   if (MPX_UCMP(a->v, a->vl, <, b->v, b->vl)) {
  72     t = a; a = b; b = t;
  73     f |= f_swap;
  74   }
  75
  76   /* --- Check for zeroness --- */
  77
  78   if (MP_ZEROP(b)) {
  79
  80     /* --- Store %$|a|$% as the GCD --- */
  81
  82     if (gcd) {
  83       if (*gcd) MP_DROP(*gcd);
  84       a = MP_COPY(a);
  85       if (MP_NEGP(a)) {
  86         MP_SPLIT(a);
  87         a->f &= ~MP_NEG;
  88         f |= f_aneg;
  89       }
  90       *gcd = a;
  91     }
  92
  93     /* --- Store %$1$% and %$0$% in the appropriate bins --- */
  94
  95     if (f & f_ext) {
  96       if (f & f_swap) {
  97         mp **tt = xx; xx = yy; yy = tt;
  98       }
  99       if (xx) {
 100         if (*xx) MP_DROP(*xx);
 101         if (MP_EQ(a, MP_ZERO)) *xx = MP_ZERO;
 102         else if (f & f_aneg) *xx = MP_MONE;
 103         else *xx = MP_ONE;
 104       }
 105       if (yy) {
 106         if (*yy) MP_DROP(*yy);
 107         *yy = MP_ZERO;
 108       }
 109     }
 110     return;
 111   }
 112
 113   /* --- Force the signs on the arguments and take copies --- */
 114
 115   a = MP_COPY(a);
 116   b = MP_COPY(b);
 117
 118   MP_SPLIT(a); a->f &= ~MP_NEG;
 119   MP_SPLIT(b); b->f &= ~MP_NEG;
 120
 121   u = MP_COPY(a);
 122   v = MP_COPY(b);
 123
 124   /* --- Main extended Euclidean algorithm --- */
 125
 126   while (!MP_ZEROP(v)) {
 127     mp_div(&q, &u, u, v);
 128     if (f & f_ext) {
 129       t = mp_mul(spare, X, q);
 130       t = mp_sub(t, x, t);
 131       spare = x; x = X; X = t;
 132       t = mp_mul(spare, Y, q);
 133       t = mp_sub(t, y, t);
 134       spare = y; y = Y; Y = t;
 135     }
 136     t = u; u = v; v = t;
 137   }
 138
 139   MP_DROP(q); if (spare) MP_DROP(spare);
 140   if (!gcd)
 141     MP_DROP(u);
 142   else {
 143     if (*gcd) MP_DROP(*gcd);
 144     u->f &= ~MP_NEG;
 145     *gcd = u;
 146   }
 147
 148   /* --- Perform a little normalization --- *
 149    *
 150    * Ensure that the coefficient returned is positive, if there is only one.
 151    * If there are two, favour @y@.  Of course, if the original arguments were
 152    * negative then I'll need to twiddle their signs as well.
 153    */
 154
 155   if (f & f_ext) {
 156
 157     /* --- If @a@ and @b@ got swapped, swap the coefficients back --- */
 158
 159     if (f & f_swap) {
 160       t = x; x = y; y = t;
 161       t = a; a = b; b = t;
 162     }
 163
 164     /* --- Sort out the signs --- *
 165      *
 166      * Note that %$ax + by = a(x - b) + b(y + a)$%.
 167      *
 168      * This is currently bodgy.  It needs sorting out at some time.
 169      */
 170
 171     if (yy) {
 172       if (MP_NEGP(y)) {
 173         do {
 174           y = mp_add(y, y, a);
 175           x = mp_sub(x, x, b);
 176         } while (MP_NEGP(y));
 177       } else {
 178         while (MP_CMP(y, >=, a)) {
 179           y = mp_sub(y, y, a);
 180           x = mp_add(x, x, b);
 181         }
 182       }
 183     } else {
 184       if (MP_NEGP(x))
 185         do x = mp_add(x, x, b); while (MP_NEGP(x));
 186       else
 187         while (MP_CMP(x, >=, b)) x = mp_sub(x, x, b);
 188     }
 189
 190     /* --- Twiddle the signs --- */
 191
 192     if (f & f_aneg) { MP_SPLIT(x); x->f ^= MP_NEG; }
 193     if (f & f_bneg) { MP_SPLIT(y); y->f ^= MP_NEG; }
 194
 195     /* --- Store the results --- */
 196
 197     if (!xx)
 198       MP_DROP(x);
 199     else {
 200       if (*xx) MP_DROP(*xx);
 201       *xx = x;
 202     }
 203
 204     if (!yy)
 205       MP_DROP(y);
 206     else {
 207       if (*yy) MP_DROP(*yy);
 208       *yy = y;
 209     }
 210   }
 211
 212   MP_DROP(v);
 213   MP_DROP(X); MP_DROP(Y);
 214   MP_DROP(a); MP_DROP(b);
 215 }
 216
 217 /* -- @mp_modinv@ --- *
 218  *
 219  * Arguments:   @mp *d@ = destination
 220  *              @mp *x@ = argument
 221  *              @mp *p@ = modulus
 222  *
 223  * Returns:     The inverse %$x^{-1} \bmod p$%.
 224  *
 225  * Use:         Computes a modular inverse.    An assertion fails if %$p$%
 226  *              has no inverse.
 227  */
 228
 229 mp *mp_modinv(mp *d, mp *x, mp *p)
 230 {
 231   mp *g = MP_NEW;
 232   mp_gcd(&g, 0, &d, p, x);
 233   assert(MP_EQ(g, MP_ONE));
 234   mp_drop(g);
 235   return (d);
 236 }
 237
 238 /*----- Test rig ----------------------------------------------------------*/
 239
 240 #ifdef TEST_RIG
 241
 242 static int modinv(dstr *v)
 243 {
 244   int ok = 1;
 245   mp *x = *(mp **)v[0].buf;
 246   mp *m = *(mp **)v[1].buf;
 247   mp *r = *(mp **)v[2].buf;
 248
 249   mp *y = mp_modinv(MP_NEW, x, m);
 250   if (!MP_EQ(y, r)) {
 251     fputs("\n*** mp_modinv failed", stderr);
 252     fputs("\nx      = ", stderr); mp_writefile(x, stderr, 10);
 253     fputs("\nm      = ", stderr); mp_writefile(m, stderr, 10);
 254     fputs("\nexpect = ", stderr); mp_writefile(r, stderr, 10);
 255     fputs("\nresult = ", stderr); mp_writefile(y, stderr, 10);
 256     ok = 0;
 257   }
 258   MP_DROP(x); MP_DROP(m); MP_DROP(r); MP_DROP(y);
 259   assert(mparena_count(MPARENA_GLOBAL) == 0);
 260   return (ok);
 261 }
 262
 263 static int gcd(dstr *v)
 264 {
 265   int ok = 1;
 266   mp *a = *(mp **)v[0].buf;
 267   mp *b = *(mp **)v[1].buf;
 268   mp *g = *(mp **)v[2].buf;
 269   mp *x = *(mp **)v[3].buf;
 270   mp *y = *(mp **)v[4].buf;
 271
 272   mp *gg = MP_NEW, *xx = MP_NEW, *yy = MP_NEW;
 273   mp_gcd(&gg, &xx, &yy, a, b);
 274   if (!MP_EQ(x, xx)) {
 275     fputs("\n*** mp_gcd(x) failed", stderr);
 276     fputs("\na      = ", stderr); mp_writefile(a, stderr, 10);
 277     fputs("\nb      = ", stderr); mp_writefile(b, stderr, 10);
 278     fputs("\nexpect = ", stderr); mp_writefile(x, stderr, 10);
 279     fputs("\nresult = ", stderr); mp_writefile(xx, stderr, 10);
 280     fputc('\n', stderr);
 281     ok = 0;
 282   }
 283   if (!MP_EQ(y, yy)) {
 284     fputs("\n*** mp_gcd(y) failed", stderr);
 285     fputs("\na      = ", stderr); mp_writefile(a, stderr, 10);
 286     fputs("\nb      = ", stderr); mp_writefile(b, stderr, 10);
 287     fputs("\nexpect = ", stderr); mp_writefile(y, stderr, 10);
 288     fputs("\nresult = ", stderr); mp_writefile(yy, stderr, 10);
 289     fputc('\n', stderr);
 290     ok = 0;
 291   }
 292
 293   if (!ok) {
 294     mp *ax = mp_mul(MP_NEW, a, xx);
 295     mp *by = mp_mul(MP_NEW, b, yy);
 296     ax = mp_add(ax, ax, by);
 297     if (MP_EQ(ax, gg))
 298       fputs("\n*** (Alternative result found.)\n", stderr);
 299     MP_DROP(ax);
 300     MP_DROP(by);
 301   }
 302
 303   if (!MP_EQ(g, gg)) {
 304     fputs("\n*** mp_gcd(gcd) failed", stderr);
 305     fputs("\na      = ", stderr); mp_writefile(a, stderr, 10);
 306     fputs("\nb      = ", stderr); mp_writefile(b, stderr, 10);
 307     fputs("\nexpect = ", stderr); mp_writefile(g, stderr, 10);
 308     fputs("\nresult = ", stderr); mp_writefile(gg, stderr, 10);
 309     fputc('\n', stderr);
 310     ok = 0;
 311   }
 312   MP_DROP(a); MP_DROP(b); MP_DROP(g); MP_DROP(x); MP_DROP(y);
 313   MP_DROP(gg); MP_DROP(xx); MP_DROP(yy);
 314   assert(mparena_count(MPARENA_GLOBAL) == 0);
 315   return (ok);
 316 }
 317
 318 static test_chunk tests[] = {
 319   { "gcd", gcd, { &type_mp, &type_mp, &type_mp, &type_mp, &type_mp, 0 } },
 320   { "modinv", modinv, { &type_mp, &type_mp, &type_mp, 0 } },
 321   { 0, 0, { 0 } }
 322 };
 323
 324 int main(int argc, char *argv[])
 325 {
 326   sub_init();
 327   test_run(argc, argv, tests, SRCDIR "/t/mp");
 328   return (0);
 329 }
 330
 331 #endif
 332
 333 /*----- That's all, folks -------------------------------------------------*/