git.distorted.org.uk Git - u/mdw/catacomb/blob - ec-prime.c

   1 /* -*-c-*-
   2  *
   3  * $Id: ec-prime.c,v 1.7 2004/03/27 00:04:46 mdw Exp $
   4  *
   5  * Elliptic curves over prime fields
   6  *
   7  * (c) 2001 Straylight/Edgeware
   8  */
   9
  10 /*----- Licensing notice --------------------------------------------------*
  11  *
  12  * This file is part of Catacomb.
  13  *
  14  * Catacomb is free software; you can redistribute it and/or modify
  15  * it under the terms of the GNU Library General Public License as
  16  * published by the Free Software Foundation; either version 2 of the
  17  * License, or (at your option) any later version.
  18  *
  19  * Catacomb is distributed in the hope that it will be useful,
  20  * but WITHOUT ANY WARRANTY; without even the implied warranty of
  21  * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  22  * GNU Library General Public License for more details.
  23  *
  24  * You should have received a copy of the GNU Library General Public
  25  * License along with Catacomb; if not, write to the Free
  26  * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
  27  * MA 02111-1307, USA.
  28  */
  29
  30 /*----- Revision history --------------------------------------------------*
  31  *
  32  * $Log: ec-prime.c,v $
  33  * Revision 1.7  2004/03/27 00:04:46  mdw
  34  * Implement efficient reduction for pleasant-looking primes.
  35  *
  36  * Revision 1.6  2004/03/23 15:19:32  mdw
  37  * Test elliptic curves more thoroughly.
  38  *
  39  * Revision 1.5  2004/03/22 02:19:10  mdw
  40  * Rationalise the sliding-window threshold.  Drop guarantee that right
  41  * arguments to EC @add@ are canonical, and fix up projective implementations
  42  * to cope.
  43  *
  44  * Revision 1.4  2004/03/21 22:52:06  mdw
  45  * Merge and close elliptic curve branch.
  46  *
  47  * Revision 1.3.4.3  2004/03/21 22:39:46  mdw
  48  * Elliptic curves on binary fields work.
  49  *
  50  * Revision 1.3.4.2  2004/03/20 00:13:31  mdw
  51  * Projective coordinates for prime curves
  52  *
  53  * Revision 1.3.4.1  2003/06/10 13:43:53  mdw
  54  * Simple (non-projective) curves over prime fields now seem to work.
  55  *
  56  * Revision 1.3  2003/05/15 23:25:59  mdw
  57  * Make elliptic curve stuff build.
  58  *
  59  * Revision 1.2  2002/01/13 13:48:44  mdw
  60  * Further progress.
  61  *
  62  * Revision 1.1  2001/04/29 18:12:33  mdw
  63  * Prototype version.
  64  *
  65  */
  66
  67 /*----- Header files ------------------------------------------------------*/
  68
  69 #include <mLib/sub.h>
  70
  71 #include "ec.h"
  72
  73 /*----- Data structures ---------------------------------------------------*/
  74
  75 typedef struct ecctx {
  76   ec_curve c;
  77   mp *a, *b;
  78 } ecctx;
  79
  80 /*----- Simple prime curves -----------------------------------------------*/
  81
  82 static const ec_ops ec_primeops, ec_primeprojops, ec_primeprojxops;
  83
  84 static ec *ecneg(ec_curve *c, ec *d, const ec *p)
  85 {
  86   EC_COPY(d, p);
  87   if (d->y)
  88     d->y = F_NEG(c->f, d->y, d->y);
  89   return (d);
  90 }
  91
  92 static ec *ecfind(ec_curve *c, ec *d, mp *x)
  93 {
  94   mp *p, *q;
  95   ecctx *cc = (ecctx *)c;
  96   field *f = c->f;
  97
  98   q = F_SQR(f, MP_NEW, x);
  99   p = F_MUL(f, MP_NEW, x, q);
 100   q = F_MUL(f, q, x, cc->a);
 101   p = F_ADD(f, p, p, q);
 102   p = F_ADD(f, p, p, cc->b);
 103   MP_DROP(q);
 104   p = F_SQRT(f, p, p);
 105   if (!p)
 106     return (0);
 107   EC_DESTROY(d);
 108   d->x = MP_COPY(x);
 109   d->y = p;
 110   d->z = MP_COPY(f->one);
 111   return (d);
 112 }
 113
 114 static ec *ecdbl(ec_curve *c, ec *d, const ec *a)
 115 {
 116   if (EC_ATINF(a))
 117     EC_SETINF(d);
 118   else if (F_ZEROP(c->f, a->y))
 119     EC_COPY(d, a);
 120   else {
 121     field *f = c->f;
 122     ecctx *cc = (ecctx *)c;
 123     mp *lambda;
 124     mp *dy, *dx;
 125
 126     dx = F_SQR(f, MP_NEW, a->x);        /* %$x^2$% */
 127     dy = F_DBL(f, MP_NEW, a->y);        /* %$2 y$% */
 128     dx = F_TPL(f, dx, dx);              /* %$3 x^2$% */
 129     dx = F_ADD(f, dx, dx, cc->a);       /* %$3 x^2 + A$% */
 130     dy = F_INV(f, dy, dy);              /* %$(2 y)^{-1}$% */
 131     lambda = F_MUL(f, MP_NEW, dx, dy);  /* %$\lambda = (3 x^2 + A)/(2 y)$% */
 132
 133     dx = F_SQR(f, dx, lambda);          /* %$\lambda^2$% */
 134     dy = F_DBL(f, dy, a->x);            /* %$2 x$% */
 135     dx = F_SUB(f, dx, dx, dy);          /* %$x' = \lambda^2 - 2 x */
 136     dy = F_SUB(f, dy, a->x, dx);        /* %$x - x'$% */
 137     dy = F_MUL(f, dy, lambda, dy);      /* %$\lambda (x - x')$% */
 138     dy = F_SUB(f, dy, dy, a->y);        /* %$y' = \lambda (x - x') - y$% */
 139
 140     EC_DESTROY(d);
 141     d->x = dx;
 142     d->y = dy;
 143     d->z = 0;
 144     MP_DROP(lambda);
 145   }
 146   return (d);
 147 }
 148
 149 static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a)
 150 {
 151   if (EC_ATINF(a))
 152     EC_SETINF(d);
 153   else if (F_ZEROP(c->f, a->y))
 154     EC_COPY(d, a);
 155   else {
 156     field *f = c->f;
 157     ecctx *cc = (ecctx *)c;
 158     mp *p, *q, *m, *s, *dx, *dy, *dz;
 159
 160     p = F_SQR(f, MP_NEW, a->z);         /* %$z^2$% */
 161     q = F_SQR(f, MP_NEW, p);            /* %$z^4$% */
 162     p = F_MUL(f, p, q, cc->a);          /* %$A z^4$% */
 163     m = F_SQR(f, MP_NEW, a->x);         /* %$x^2$% */
 164     m = F_TPL(f, m, m);                 /* %$3 x^2$% */
 165     m = F_ADD(f, m, m, p);              /* %$m = 3 x^2 + A z^4$% */
 166
 167     q = F_DBL(f, q, a->y);              /* %$2 y$% */
 168     dz = F_MUL(f, MP_NEW, q, a->z);     /* %$z' = 2 y z$% */
 169
 170     p = F_SQR(f, p, q);                 /* %$4 y^2$% */
 171     s = F_MUL(f, MP_NEW, p, a->x);      /* %$s = 4 x y^2$% */
 172     q = F_SQR(f, q, p);                 /* %$16 y^4$% */
 173     q = F_HLV(f, q, q);                 /* %$t = 8 y^4$% */
 174
 175     p = F_DBL(f, p, s);                 /* %$2 s$% */
 176     dx = F_SQR(f, MP_NEW, m);           /* %$m^2$% */
 177     dx = F_SUB(f, dx, dx, p);           /* %$x' = m^2 - 2 s$% */
 178
 179     s = F_SUB(f, s, s, dx);             /* %$s - x'$% */
 180     dy = F_MUL(f, p, m, s);             /* %$m (s - x')$% */
 181     dy = F_SUB(f, dy, dy, q);           /* %$y' = m (s - x') - t$% */
 182
 183     EC_DESTROY(d);
 184     d->x = dx;
 185     d->y = dy;
 186     d->z = dz;
 187     MP_DROP(m);
 188     MP_DROP(q);
 189     MP_DROP(s);
 190   }
 191   return (d);
 192 }
 193
 194 static ec *ecprojxdbl(ec_curve *c, ec *d, const ec *a)
 195 {
 196   if (EC_ATINF(a))
 197     EC_SETINF(d);
 198   else if (F_ZEROP(c->f, a->y))
 199     EC_COPY(d, a);
 200   else {
 201     field *f = c->f;
 202     mp *p, *q, *m, *s, *dx, *dy, *dz;
 203
 204     m = F_SQR(f, MP_NEW, a->z);         /* %$z^2$% */
 205     p = F_SUB(f, MP_NEW, a->x, m);      /* %$x - z^2$% */
 206     q = F_ADD(f, MP_NEW, a->x, m);      /* %$x + z^2$% */
 207     m = F_MUL(f, m, p, q);              /* %$x^2 - z^4$% */
 208     m = F_TPL(f, m, m);                 /* %$m = 3 x^2 - 3 z^4$% */
 209
 210     q = F_DBL(f, q, a->y);              /* %$2 y$% */
 211     dz = F_MUL(f, MP_NEW, q, a->z);     /* %$z' = 2 y z$% */
 212
 213     p = F_SQR(f, p, q);                 /* %$4 y^2$% */
 214     s = F_MUL(f, MP_NEW, p, a->x);      /* %$s = 4 x y^2$% */
 215     q = F_SQR(f, q, p);                 /* %$16 y^4$% */
 216     q = F_HLV(f, q, q);                 /* %$t = 8 y^4$% */
 217
 218     p = F_DBL(f, p, s);                 /* %$2 s$% */
 219     dx = F_SQR(f, MP_NEW, m);           /* %$m^2$% */
 220     dx = F_SUB(f, dx, dx, p);           /* %$x' = m^2 - 2 s$% */
 221
 222     s = F_SUB(f, s, s, dx);             /* %$s - x'$% */
 223     dy = F_MUL(f, p, m, s);             /* %$m (s - x')$% */
 224     dy = F_SUB(f, dy, dy, q);           /* %$y' = m (s - x') - t$% */
 225
 226     EC_DESTROY(d);
 227     d->x = dx;
 228     d->y = dy;
 229     d->z = dz;
 230     MP_DROP(m);
 231     MP_DROP(q);
 232     MP_DROP(s);
 233   }
 234   return (d);
 235 }
 236
 237 static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b)
 238 {
 239   if (a == b)
 240     ecdbl(c, d, a);
 241   else if (EC_ATINF(a))
 242     EC_COPY(d, b);
 243   else if (EC_ATINF(b))
 244     EC_COPY(d, a);
 245   else {
 246     field *f = c->f;
 247     mp *lambda;
 248     mp *dy, *dx;
 249
 250     if (!MP_EQ(a->x, b->x)) {
 251       dy = F_SUB(f, MP_NEW, a->y, b->y); /* %$y_0 - y_1$% */
 252       dx = F_SUB(f, MP_NEW, a->x, b->x); /* %$x_0 - x_1$% */
 253       dx = F_INV(f, dx, dx);            /* %$(x_0 - x_1)^{-1}$% */
 254       lambda = F_MUL(f, MP_NEW, dy, dx);
 255                                    /* %$\lambda = (y_0 - y1)/(x_0 - x_1)$% */
 256     } else if (F_ZEROP(c->f, a->y) || !MP_EQ(a->y, b->y)) {
 257       EC_SETINF(d);
 258       return (d);
 259     } else {
 260       ecctx *cc = (ecctx *)c;
 261       dx = F_SQR(f, MP_NEW, a->x);      /* %$x_0^2$% */
 262       dx = F_TPL(f, dx, dx);            /* %$3 x_0^2$% */
 263       dx = F_ADD(f, dx, dx, cc->a);     /* %$3 x_0^2 + A$% */
 264       dy = F_DBL(f, MP_NEW, a->y);      /* %$2 y_0$% */
 265       dy = F_INV(f, dy, dy);            /* %$(2 y_0)^{-1}$% */
 266       lambda = F_MUL(f, MP_NEW, dx, dy);
 267                                     /* %$\lambda = (3 x_0^2 + A)/(2 y_0)$% */
 268     }
 269
 270     dx = F_SQR(f, dx, lambda);          /* %$\lambda^2$% */
 271     dx = F_SUB(f, dx, dx, a->x);        /* %$\lambda^2 - x_0$% */
 272     dx = F_SUB(f, dx, dx, b->x);        /* %$x' = \lambda^2 - x_0 - x_1$% */
 273     dy = F_SUB(f, dy, b->x, dx);        /* %$x_1 - x'$% */
 274     dy = F_MUL(f, dy, lambda, dy);      /* %$\lambda (x_1 - x')$% */
 275     dy = F_SUB(f, dy, dy, b->y);      /* %$y' = \lambda (x_1 - x') - y_1$% */
 276
 277     EC_DESTROY(d);
 278     d->x = dx;
 279     d->y = dy;
 280     d->z = 0;
 281     MP_DROP(lambda);
 282   }
 283   return (d);
 284 }
 285
 286 static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b)
 287 {
 288   if (a == b)
 289     c->ops->dbl(c, d, a);
 290   else if (EC_ATINF(a))
 291     EC_COPY(d, b);
 292   else if (EC_ATINF(b))
 293     EC_COPY(d, a);
 294   else {
 295     field *f = c->f;
 296     mp *p, *q, *r, *w, *u, *uu, *s, *ss, *dx, *dy, *dz;
 297
 298     q = F_SQR(f, MP_NEW, a->z);         /* %$z_0^2$% */
 299     u = F_MUL(f, MP_NEW, q, b->x);      /* %$u = x_1 z_0^2$% */
 300     p = F_MUL(f, MP_NEW, q, b->y);      /* %$y_1 z_0^2$% */
 301     s = F_MUL(f, q, p, a->z);           /* %$s = y_1 z_0^3$% */
 302
 303     q = F_SQR(f, MP_NEW, b->z);         /* %$z_1^2$% */
 304     uu = F_MUL(f, MP_NEW, q, a->x);     /* %$uu = x_0 z_1^2$%*/
 305     p = F_MUL(f, p, q, a->y);           /* %$y_0 z_1^2$% */
 306     ss = F_MUL(f, q, p, b->z);          /* %$ss = y_0 z_1^3$% */
 307
 308     w = F_SUB(f, p, uu, u);             /* %$w = uu - u$% */
 309     r = F_SUB(f, MP_NEW, ss, s);        /* %$r = ss - s$% */
 310     if (F_ZEROP(f, w)) {
 311       MP_DROP(w);
 312       MP_DROP(u);
 313       MP_DROP(s);
 314       MP_DROP(uu);
 315       MP_DROP(ss);
 316       if (F_ZEROP(f, r)) {
 317         MP_DROP(r);
 318         return (c->ops->dbl(c, d, a));
 319       } else {
 320         MP_DROP(r);
 321         EC_SETINF(d);
 322         return (d);
 323       }
 324     }
 325     u = F_ADD(f, u, u, uu);             /* %$t = uu + u$% */
 326     s = F_ADD(f, s, s, ss);             /* %$m = ss + r$% */
 327
 328     uu = F_MUL(f, uu, a->z, w);         /* %$z_0 w$% */
 329     dz = F_MUL(f, ss, uu, b->z);        /* %$z' = z_0 z_1 w$% */
 330
 331     p = F_SQR(f, uu, w);                /* %$w^2$% */
 332     q = F_MUL(f, MP_NEW, p, u);         /* %$t w^2$% */
 333     u = F_MUL(f, u, p, w);              /* %$w^3$% */
 334     p = F_MUL(f, p, u, s);              /* %$m w^3$% */
 335
 336     dx = F_SQR(f, u, r);                /* %$r^2$% */
 337     dx = F_SUB(f, dx, dx, q);           /* %$x' = r^2 - t w^2$% */
 338
 339     s = F_DBL(f, s, dx);                /* %$2 x'$% */
 340     q = F_SUB(f, q, q, s);              /* %$v = t w^2 - 2 x'$% */
 341     dy = F_MUL(f, s, q, r);             /* %$v r$% */
 342     dy = F_SUB(f, dy, dy, p);           /* %$v r - m w^3$% */
 343     dy = F_HLV(f, dy, dy);              /* %$y' = (v r - m w^3)/2$% */
 344
 345     EC_DESTROY(d);
 346     d->x = dx;
 347     d->y = dy;
 348     d->z = dz;
 349     MP_DROP(p);
 350     MP_DROP(q);
 351     MP_DROP(r);
 352     MP_DROP(w);
 353   }
 354   return (d);
 355 }
 356
 357 static int eccheck(ec_curve *c, const ec *p)
 358 {
 359   ecctx *cc = (ecctx *)c;
 360   field *f = c->f;
 361   int rc;
 362   mp *l = F_SQR(f, MP_NEW, p->y);
 363   mp *x = F_SQR(f, MP_NEW, p->x);
 364   mp *r = F_MUL(f, MP_NEW, x, p->x);
 365   x = F_MUL(f, x, cc->a, p->x);
 366   r = F_ADD(f, r, r, x);
 367   r = F_ADD(f, r, r, cc->b);
 368   rc = MP_EQ(l, r) ? 0 : -1;
 369   mp_drop(l);
 370   mp_drop(x);
 371   mp_drop(r);
 372   return (rc);
 373 }
 374
 375 static int ecprojcheck(ec_curve *c, const ec *p)
 376 {
 377   ec t = EC_INIT;
 378   int rc;
 379
 380   c->ops->fix(c, &t, p);
 381   rc = eccheck(c, &t);
 382   EC_DESTROY(&t);
 383   return (rc);
 384 }
 385
 386 static void ecdestroy(ec_curve *c)
 387 {
 388   ecctx *cc = (ecctx *)c;
 389   MP_DROP(cc->a);
 390   MP_DROP(cc->b);
 391   DESTROY(cc);
 392 }
 393
 394 /* --- @ec_prime@, @ec_primeproj@ --- *
 395  *
 396  * Arguments:   @field *f@ = the underlying field for this elliptic curve
 397  *              @mp *a, *b@ = the coefficients for this curve
 398  *
 399  * Returns:     A pointer to the curve.
 400  *
 401  * Use:         Creates a curve structure for an elliptic curve defined over
 402  *              a prime field.  The @primeproj@ variant uses projective
 403  *              coordinates, which can be a win.
 404  */
 405
 406 extern ec_curve *ec_prime(field *f, mp *a, mp *b)
 407 {
 408   ecctx *cc = CREATE(ecctx);
 409   cc->c.ops = &ec_primeops;
 410   cc->c.f = f;
 411   cc->a = F_IN(f, MP_NEW, a);
 412   cc->b = F_IN(f, MP_NEW, b);
 413   return (&cc->c);
 414 }
 415
 416 extern ec_curve *ec_primeproj(field *f, mp *a, mp *b)
 417 {
 418   ecctx *cc = CREATE(ecctx);
 419   mp *ax;
 420
 421   ax = mp_add(MP_NEW, a, MP_THREE);
 422   ax = F_IN(f, ax, ax);
 423   if (F_ZEROP(f, ax))
 424     cc->c.ops = &ec_primeprojxops;
 425   else
 426     cc->c.ops = &ec_primeprojops;
 427   MP_DROP(ax);
 428   cc->c.f = f;
 429   cc->a = F_IN(f, MP_NEW, a);
 430   cc->b = F_IN(f, MP_NEW, b);
 431   return (&cc->c);
 432 }
 433
 434 static const ec_ops ec_primeops = {
 435   ecdestroy, ec_idin, ec_idout, ec_idfix,
 436   ecfind, ecneg, ecadd, ec_stdsub, ecdbl, eccheck
 437 };
 438
 439 static const ec_ops ec_primeprojops = {
 440   ecdestroy, ec_projin, ec_projout, ec_projfix,
 441   ecfind, ecneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck
 442 };
 443
 444 static const ec_ops ec_primeprojxops = {
 445   ecdestroy, ec_projin, ec_projout, ec_projfix,
 446   ecfind, ecneg, ecprojadd, ec_stdsub, ecprojxdbl, ecprojcheck
 447 };
 448
 449 /*----- Test rig ----------------------------------------------------------*/
 450
 451 #ifdef TEST_RIG
 452
 453 #define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
 454
 455 int main(int argc, char *argv[])
 456 {
 457   field *f;
 458   ec_curve *c;
 459   ec g = EC_INIT, d = EC_INIT;
 460   mp *p, *a, *b, *r;
 461   int i, n = argc == 1 ? 1 : atoi(argv[1]);
 462
 463   printf("ec-prime: ");
 464   fflush(stdout);
 465   a = MP(-3);
 466   b = MP(0x64210519e59c80e70fa7e9ab72243049feb8deecc146b9b1);
 467   p = MP(6277101735386680763835789423207666416083908700390324961279);
 468   r = MP(6277101735386680763835789423176059013767194773182842284080);
 469
 470   f = field_niceprime(p);
 471   c = ec_primeproj(f, a, b);
 472
 473   g.x = MP(0x188da80eb03090f67cbf20eb43a18800f4ff0afd82ff1012);
 474   g.y = MP(0x07192b95ffc8da78631011ed6b24cdd573f977a11e794811);
 475
 476   for (i = 0; i < n; i++) {
 477     ec_mul(c, &d, &g, r);
 478     if (EC_ATINF(&d)) {
 479       fprintf(stderr, "zero too early\n");
 480       return (1);
 481     }
 482     ec_add(c, &d, &d, &g);
 483     if (!EC_ATINF(&d)) {
 484       fprintf(stderr, "didn't reach zero\n");
 485       MP_EPRINT("d.x", d.x);
 486       MP_EPRINT("d.y", d.y);
 487       return (1);
 488     }
 489     ec_destroy(&d);
 490   }
 491   ec_destroy(&g);
 492   ec_destroycurve(c);
 493   F_DESTROY(f);
 494   MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r);
 495   assert(!mparena_count(&mparena_global));
 496   printf("ok\n");
 497   return (0);
 498 }
 499
 500 #endif
 501
 502 /*----- That's all, folks -------------------------------------------------*/