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

   1 /* -*-c-*-
   2  *
   3  * $Id: ec-prime.c,v 1.9 2004/04/01 12:50:09 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 1.9  2004/04/01 12:50:09  mdw
  34  * Add cyclic group abstraction, with test code.  Separate off exponentation
  35  * functions for better static linking.  Fix a buttload of bugs on the way.
  36  * Generally ensure that negative exponents do inversion correctly.  Add
  37  * table of standard prime-field subgroups.  (Binary field subgroups are
  38  * currently unimplemented but easy to add if anyone ever finds a good one.)
  39  *
  40  * Revision 1.8  2004/03/27 17:54:11  mdw
  41  * Standard curves and curve checking.
  42  *
  43  * Revision 1.7  2004/03/27 00:04:46  mdw
  44  * Implement efficient reduction for pleasant-looking primes.
  45  *
  46  * Revision 1.6  2004/03/23 15:19:32  mdw
  47  * Test elliptic curves more thoroughly.
  48  *
  49  * Revision 1.5  2004/03/22 02:19:10  mdw
  50  * Rationalise the sliding-window threshold.  Drop guarantee that right
  51  * arguments to EC @add@ are canonical, and fix up projective implementations
  52  * to cope.
  53  *
  54  * Revision 1.4  2004/03/21 22:52:06  mdw
  55  * Merge and close elliptic curve branch.
  56  *
  57  * Revision 1.3.4.3  2004/03/21 22:39:46  mdw
  58  * Elliptic curves on binary fields work.
  59  *
  60  * Revision 1.3.4.2  2004/03/20 00:13:31  mdw
  61  * Projective coordinates for prime curves
  62  *
  63  * Revision 1.3.4.1  2003/06/10 13:43:53  mdw
  64  * Simple (non-projective) curves over prime fields now seem to work.
  65  *
  66  * Revision 1.3  2003/05/15 23:25:59  mdw
  67  * Make elliptic curve stuff build.
  68  *
  69  * Revision 1.2  2002/01/13 13:48:44  mdw
  70  * Further progress.
  71  *
  72  * Revision 1.1  2001/04/29 18:12:33  mdw
  73  * Prototype version.
  74  *
  75  */
  76
  77 /*----- Header files ------------------------------------------------------*/
  78
  79 #include <mLib/sub.h>
  80
  81 #include "ec.h"
  82
  83 /*----- Simple prime curves -----------------------------------------------*/
  84
  85 static const ec_ops ec_primeops, ec_primeprojops, ec_primeprojxops;
  86
  87 static ec *ecneg(ec_curve *c, ec *d, const ec *p)
  88 {
  89   EC_COPY(d, p);
  90   if (d->y)
  91     d->y = F_NEG(c->f, d->y, d->y);
  92   return (d);
  93 }
  94
  95 static ec *ecfind(ec_curve *c, ec *d, mp *x)
  96 {
  97   mp *p, *q;
  98   field *f = c->f;
  99
 100   q = F_SQR(f, MP_NEW, x);
 101   p = F_MUL(f, MP_NEW, x, q);
 102   q = F_MUL(f, q, x, c->a);
 103   p = F_ADD(f, p, p, q);
 104   p = F_ADD(f, p, p, c->b);
 105   MP_DROP(q);
 106   p = F_SQRT(f, p, p);
 107   if (!p)
 108     return (0);
 109   EC_DESTROY(d);
 110   d->x = MP_COPY(x);
 111   d->y = p;
 112   d->z = MP_COPY(f->one);
 113   return (d);
 114 }
 115
 116 static ec *ecdbl(ec_curve *c, ec *d, const ec *a)
 117 {
 118   if (EC_ATINF(a))
 119     EC_SETINF(d);
 120   else if (F_ZEROP(c->f, a->y))
 121     EC_COPY(d, a);
 122   else {
 123     field *f = c->f;
 124     mp *lambda;
 125     mp *dy, *dx;
 126
 127     dx = F_SQR(f, MP_NEW, a->x);        /* %$x^2$% */
 128     dy = F_DBL(f, MP_NEW, a->y);        /* %$2 y$% */
 129     dx = F_TPL(f, dx, dx);              /* %$3 x^2$% */
 130     dx = F_ADD(f, dx, dx, c->a);        /* %$3 x^2 + A$% */
 131     dy = F_INV(f, dy, dy);              /* %$(2 y)^{-1}$% */
 132     lambda = F_MUL(f, MP_NEW, dx, dy);  /* %$\lambda = (3 x^2 + A)/(2 y)$% */
 133
 134     dx = F_SQR(f, dx, lambda);          /* %$\lambda^2$% */
 135     dy = F_DBL(f, dy, a->x);            /* %$2 x$% */
 136     dx = F_SUB(f, dx, dx, dy);          /* %$x' = \lambda^2 - 2 x */
 137     dy = F_SUB(f, dy, a->x, dx);        /* %$x - x'$% */
 138     dy = F_MUL(f, dy, lambda, dy);      /* %$\lambda (x - x')$% */
 139     dy = F_SUB(f, dy, dy, a->y);        /* %$y' = \lambda (x - x') - y$% */
 140
 141     EC_DESTROY(d);
 142     d->x = dx;
 143     d->y = dy;
 144     d->z = 0;
 145     MP_DROP(lambda);
 146   }
 147   return (d);
 148 }
 149
 150 static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a)
 151 {
 152   if (EC_ATINF(a))
 153     EC_SETINF(d);
 154   else if (F_ZEROP(c->f, a->y))
 155     EC_COPY(d, a);
 156   else {
 157     field *f = c->f;
 158     mp *p, *q, *m, *s, *dx, *dy, *dz;
 159
 160     p = F_SQR(f, MP_NEW, a->z);         /* %$z^2$% */
 161     q = F_SQR(f, MP_NEW, p);            /* %$z^4$% */
 162     p = F_MUL(f, p, q, c->a);           /* %$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       dx = F_SQR(f, MP_NEW, a->x);      /* %$x_0^2$% */
 261       dx = F_TPL(f, dx, dx);            /* %$3 x_0^2$% */
 262       dx = F_ADD(f, dx, dx, c->a);      /* %$3 x_0^2 + A$% */
 263       dy = F_DBL(f, MP_NEW, a->y);      /* %$2 y_0$% */
 264       dy = F_INV(f, dy, dy);            /* %$(2 y_0)^{-1}$% */
 265       lambda = F_MUL(f, MP_NEW, dx, dy);
 266                                     /* %$\lambda = (3 x_0^2 + A)/(2 y_0)$% */
 267     }
 268
 269     dx = F_SQR(f, dx, lambda);          /* %$\lambda^2$% */
 270     dx = F_SUB(f, dx, dx, a->x);        /* %$\lambda^2 - x_0$% */
 271     dx = F_SUB(f, dx, dx, b->x);        /* %$x' = \lambda^2 - x_0 - x_1$% */
 272     dy = F_SUB(f, dy, b->x, dx);        /* %$x_1 - x'$% */
 273     dy = F_MUL(f, dy, lambda, dy);      /* %$\lambda (x_1 - x')$% */
 274     dy = F_SUB(f, dy, dy, b->y);      /* %$y' = \lambda (x_1 - x') - y_1$% */
 275
 276     EC_DESTROY(d);
 277     d->x = dx;
 278     d->y = dy;
 279     d->z = 0;
 280     MP_DROP(lambda);
 281   }
 282   return (d);
 283 }
 284
 285 static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b)
 286 {
 287   if (a == b)
 288     c->ops->dbl(c, d, a);
 289   else if (EC_ATINF(a))
 290     EC_COPY(d, b);
 291   else if (EC_ATINF(b))
 292     EC_COPY(d, a);
 293   else {
 294     field *f = c->f;
 295     mp *p, *q, *r, *w, *u, *uu, *s, *ss, *dx, *dy, *dz;
 296
 297     q = F_SQR(f, MP_NEW, a->z);         /* %$z_0^2$% */
 298     u = F_MUL(f, MP_NEW, q, b->x);      /* %$u = x_1 z_0^2$% */
 299     p = F_MUL(f, MP_NEW, q, b->y);      /* %$y_1 z_0^2$% */
 300     s = F_MUL(f, q, p, a->z);           /* %$s = y_1 z_0^3$% */
 301
 302     q = F_SQR(f, MP_NEW, b->z);         /* %$z_1^2$% */
 303     uu = F_MUL(f, MP_NEW, q, a->x);     /* %$uu = x_0 z_1^2$%*/
 304     p = F_MUL(f, p, q, a->y);           /* %$y_0 z_1^2$% */
 305     ss = F_MUL(f, q, p, b->z);          /* %$ss = y_0 z_1^3$% */
 306
 307     w = F_SUB(f, p, uu, u);             /* %$w = uu - u$% */
 308     r = F_SUB(f, MP_NEW, ss, s);        /* %$r = ss - s$% */
 309     if (F_ZEROP(f, w)) {
 310       MP_DROP(w);
 311       MP_DROP(u);
 312       MP_DROP(s);
 313       MP_DROP(uu);
 314       MP_DROP(ss);
 315       if (F_ZEROP(f, r)) {
 316         MP_DROP(r);
 317         return (c->ops->dbl(c, d, a));
 318       } else {
 319         MP_DROP(r);
 320         EC_SETINF(d);
 321         return (d);
 322       }
 323     }
 324     u = F_ADD(f, u, u, uu);             /* %$t = uu + u$% */
 325     s = F_ADD(f, s, s, ss);             /* %$m = ss + r$% */
 326
 327     uu = F_MUL(f, uu, a->z, w);         /* %$z_0 w$% */
 328     dz = F_MUL(f, ss, uu, b->z);        /* %$z' = z_0 z_1 w$% */
 329
 330     p = F_SQR(f, uu, w);                /* %$w^2$% */
 331     q = F_MUL(f, MP_NEW, p, u);         /* %$t w^2$% */
 332     u = F_MUL(f, u, p, w);              /* %$w^3$% */
 333     p = F_MUL(f, p, u, s);              /* %$m w^3$% */
 334
 335     dx = F_SQR(f, u, r);                /* %$r^2$% */
 336     dx = F_SUB(f, dx, dx, q);           /* %$x' = r^2 - t w^2$% */
 337
 338     s = F_DBL(f, s, dx);                /* %$2 x'$% */
 339     q = F_SUB(f, q, q, s);              /* %$v = t w^2 - 2 x'$% */
 340     dy = F_MUL(f, s, q, r);             /* %$v r$% */
 341     dy = F_SUB(f, dy, dy, p);           /* %$v r - m w^3$% */
 342     dy = F_HLV(f, dy, dy);              /* %$y' = (v r - m w^3)/2$% */
 343
 344     EC_DESTROY(d);
 345     d->x = dx;
 346     d->y = dy;
 347     d->z = dz;
 348     MP_DROP(p);
 349     MP_DROP(q);
 350     MP_DROP(r);
 351     MP_DROP(w);
 352   }
 353   return (d);
 354 }
 355
 356 static int eccheck(ec_curve *c, const ec *p)
 357 {
 358   field *f = c->f;
 359   mp *l, *x, *r;
 360   int rc;
 361   if (EC_ATINF(p)) return (0);
 362   l = F_SQR(f, MP_NEW, p->y);
 363   x = F_SQR(f, MP_NEW, p->x);
 364   r = F_MUL(f, MP_NEW, x, p->x);
 365   x = F_MUL(f, x, c->a, p->x);
 366   r = F_ADD(f, r, r, x);
 367   r = F_ADD(f, r, r, c->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   MP_DROP(c->a);
 389   MP_DROP(c->b);
 390   DESTROY(c);
 391 }
 392
 393 /* --- @ec_prime@, @ec_primeproj@ --- *
 394  *
 395  * Arguments:   @field *f@ = the underlying field for this elliptic curve
 396  *              @mp *a, *b@ = the coefficients for this curve
 397  *
 398  * Returns:     A pointer to the curve.
 399  *
 400  * Use:         Creates a curve structure for an elliptic curve defined over
 401  *              a prime field.  The @primeproj@ variant uses projective
 402  *              coordinates, which can be a win.
 403  */
 404
 405 extern ec_curve *ec_prime(field *f, mp *a, mp *b)
 406 {
 407   ec_curve *c = CREATE(ec_curve);
 408   c->ops = &ec_primeops;
 409   c->f = f;
 410   c->a = F_IN(f, MP_NEW, a);
 411   c->b = F_IN(f, MP_NEW, b);
 412   return (c);
 413 }
 414
 415 extern ec_curve *ec_primeproj(field *f, mp *a, mp *b)
 416 {
 417   ec_curve *c = CREATE(ec_curve);
 418   mp *ax;
 419
 420   ax = mp_add(MP_NEW, a, MP_THREE);
 421   ax = F_IN(f, ax, ax);
 422   if (F_ZEROP(f, ax))
 423     c->ops = &ec_primeprojxops;
 424   else
 425     c->ops = &ec_primeprojops;
 426   MP_DROP(ax);
 427   c->f = f;
 428   c->a = F_IN(f, MP_NEW, a);
 429   c->b = F_IN(f, MP_NEW, b);
 430   return (c);
 431 }
 432
 433 static const ec_ops ec_primeops = {
 434   ecdestroy, ec_stdsamep, ec_idin, ec_idout, ec_idfix,
 435   ecfind, ecneg, ecadd, ec_stdsub, ecdbl, eccheck
 436 };
 437
 438 static const ec_ops ec_primeprojops = {
 439   ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix,
 440   ecfind, ecneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck
 441 };
 442
 443 static const ec_ops ec_primeprojxops = {
 444   ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix,
 445   ecfind, ecneg, ecprojadd, ec_stdsub, ecprojxdbl, ecprojcheck
 446 };
 447
 448 /*----- Test rig ----------------------------------------------------------*/
 449
 450 #ifdef TEST_RIG
 451
 452 #define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
 453
 454 int main(int argc, char *argv[])
 455 {
 456   field *f;
 457   ec_curve *c;
 458   ec g = EC_INIT, d = EC_INIT;
 459   mp *p, *a, *b, *r;
 460   int i, n = argc == 1 ? 1 : atoi(argv[1]);
 461
 462   printf("ec-prime: ");
 463   fflush(stdout);
 464   a = MP(-3);
 465   b = MP(0xb3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875ac656398d8a2ed19d2a85c8edd3ec2aef);
 466   p = MP(39402006196394479212279040100143613805079739270465446667948293404245721771496870329047266088258938001861606973112319);
 467   r = MP(39402006196394479212279040100143613805079739270465446667946905279627659399113263569398956308152294913554433653942642);
 468
 469   f = field_niceprime(p);
 470   c = ec_primeproj(f, a, b);
 471
 472   g.x = MP(0xaa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a385502f25dbf55296c3a545e3872760ab7);
 473   g.y = MP(0x3617de4a96262c6f5d9e98bf9292dc29f8f41dbd289a147ce9da3113b5f0b8c00a60b1ce1d7e819d7a431d7c90ea0e5f);
 474
 475   for (i = 0; i < n; i++) {
 476     ec_mul(c, &d, &g, r);
 477     if (EC_ATINF(&d)) {
 478       fprintf(stderr, "zero too early\n");
 479       return (1);
 480     }
 481     ec_add(c, &d, &d, &g);
 482     if (!EC_ATINF(&d)) {
 483       fprintf(stderr, "didn't reach zero\n");
 484       MP_EPRINT("d.x", d.x);
 485       MP_EPRINT("d.y", d.y);
 486       return (1);
 487     }
 488     ec_destroy(&d);
 489   }
 490   ec_destroy(&g);
 491   ec_destroycurve(c);
 492   F_DESTROY(f);
 493   MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r);
 494   assert(!mparena_count(&mparena_global));
 495   printf("ok\n");
 496   return (0);
 497 }
 498
 499 #endif
 500
 501 /*----- That's all, folks -------------------------------------------------*/