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

   1 /* -*-c-*-
   2  *
   3  * Elliptic curves over prime fields
   4  *
   5  * (c) 2001 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 <mLib/sub.h>
  31
  32 #include "ec.h"
  33
  34 /*----- Simple prime curves -----------------------------------------------*/
  35
  36 static const ec_ops ec_primeops, ec_primeprojops, ec_primeprojxops;
  37
  38 static ec *ecneg(ec_curve *c, ec *d, const ec *p)
  39 {
  40   EC_COPY(d, p);
  41   if (d->y)
  42     d->y = F_NEG(c->f, d->y, d->y);
  43   return (d);
  44 }
  45
  46 static ec *ecfind(ec_curve *c, ec *d, mp *x)
  47 {
  48   mp *p, *q;
  49   field *f = c->f;
  50
  51   q = F_SQR(f, MP_NEW, x);
  52   p = F_MUL(f, MP_NEW, x, q);
  53   q = F_MUL(f, q, x, c->a);
  54   p = F_ADD(f, p, p, q);
  55   p = F_ADD(f, p, p, c->b);
  56   MP_DROP(q);
  57   p = F_SQRT(f, p, p);
  58   if (!p)
  59     return (0);
  60   EC_DESTROY(d);
  61   d->x = MP_COPY(x);
  62   d->y = p;
  63   d->z = MP_COPY(f->one);
  64   return (d);
  65 }
  66
  67 static ec *ecdbl(ec_curve *c, ec *d, const ec *a)
  68 {
  69   if (EC_ATINF(a) || F_ZEROP(c->f, a->y))
  70     EC_SETINF(d);
  71   else {
  72     field *f = c->f;
  73     mp *lambda;
  74     mp *dy, *dx;
  75
  76     dx = F_SQR(f, MP_NEW, a->x);        /* %$x^2$% */
  77     dy = F_DBL(f, MP_NEW, a->y);        /* %$2 y$% */
  78     dx = F_TPL(f, dx, dx);              /* %$3 x^2$% */
  79     dx = F_ADD(f, dx, dx, c->a);        /* %$3 x^2 + A$% */
  80     dy = F_INV(f, dy, dy);              /* %$(2 y)^{-1}$% */
  81     lambda = F_MUL(f, MP_NEW, dx, dy);  /* %$\lambda = (3 x^2 + A)/(2 y)$% */
  82
  83     dx = F_SQR(f, dx, lambda);          /* %$\lambda^2$% */
  84     dy = F_DBL(f, dy, a->x);            /* %$2 x$% */
  85     dx = F_SUB(f, dx, dx, dy);          /* %$x' = \lambda^2 - 2 x */
  86     dy = F_SUB(f, dy, a->x, dx);        /* %$x - x'$% */
  87     dy = F_MUL(f, dy, lambda, dy);      /* %$\lambda (x - x')$% */
  88     dy = F_SUB(f, dy, dy, a->y);        /* %$y' = \lambda (x - x') - y$% */
  89
  90     EC_DESTROY(d);
  91     d->x = dx;
  92     d->y = dy;
  93     d->z = 0;
  94     MP_DROP(lambda);
  95   }
  96   return (d);
  97 }
  98
  99 static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a)
 100 {
 101   if (EC_ATINF(a) || F_ZEROP(c->f, a->y))
 102     EC_SETINF(d);
 103   else {
 104     field *f = c->f;
 105     mp *p, *q, *m, *s, *dx, *dy, *dz;
 106
 107     p = F_SQR(f, MP_NEW, a->z);         /* %$z^2$% */
 108     q = F_SQR(f, MP_NEW, p);            /* %$z^4$% */
 109     p = F_MUL(f, p, q, c->a);           /* %$A z^4$% */
 110     m = F_SQR(f, MP_NEW, a->x);         /* %$x^2$% */
 111     m = F_TPL(f, m, m);                 /* %$3 x^2$% */
 112     m = F_ADD(f, m, m, p);              /* %$m = 3 x^2 + A z^4$% */
 113
 114     q = F_DBL(f, q, a->y);              /* %$2 y$% */
 115     dz = F_MUL(f, MP_NEW, q, a->z);     /* %$z' = 2 y z$% */
 116
 117     p = F_SQR(f, p, q);                 /* %$4 y^2$% */
 118     s = F_MUL(f, MP_NEW, p, a->x);      /* %$s = 4 x y^2$% */
 119     q = F_SQR(f, q, p);                 /* %$16 y^4$% */
 120     q = F_HLV(f, q, q);                 /* %$t = 8 y^4$% */
 121
 122     p = F_DBL(f, p, s);                 /* %$2 s$% */
 123     dx = F_SQR(f, MP_NEW, m);           /* %$m^2$% */
 124     dx = F_SUB(f, dx, dx, p);           /* %$x' = m^2 - 2 s$% */
 125
 126     s = F_SUB(f, s, s, dx);             /* %$s - x'$% */
 127     dy = F_MUL(f, p, m, s);             /* %$m (s - x')$% */
 128     dy = F_SUB(f, dy, dy, q);           /* %$y' = m (s - x') - t$% */
 129
 130     EC_DESTROY(d);
 131     d->x = dx;
 132     d->y = dy;
 133     d->z = dz;
 134     MP_DROP(m);
 135     MP_DROP(q);
 136     MP_DROP(s);
 137   }
 138   return (d);
 139 }
 140
 141 static ec *ecprojxdbl(ec_curve *c, ec *d, const ec *a)
 142 {
 143   if (EC_ATINF(a) || F_ZEROP(c->f, a->y))
 144     EC_SETINF(d);
 145   else {
 146     field *f = c->f;
 147     mp *p, *q, *m, *s, *dx, *dy, *dz;
 148
 149     m = F_SQR(f, MP_NEW, a->z);         /* %$z^2$% */
 150     p = F_SUB(f, MP_NEW, a->x, m);      /* %$x - z^2$% */
 151     q = F_ADD(f, MP_NEW, a->x, m);      /* %$x + z^2$% */
 152     m = F_MUL(f, m, p, q);              /* %$x^2 - z^4$% */
 153     m = F_TPL(f, m, m);                 /* %$m = 3 x^2 - 3 z^4$% */
 154
 155     q = F_DBL(f, q, a->y);              /* %$2 y$% */
 156     dz = F_MUL(f, MP_NEW, q, a->z);     /* %$z' = 2 y z$% */
 157
 158     p = F_SQR(f, p, q);                 /* %$4 y^2$% */
 159     s = F_MUL(f, MP_NEW, p, a->x);      /* %$s = 4 x y^2$% */
 160     q = F_SQR(f, q, p);                 /* %$16 y^4$% */
 161     q = F_HLV(f, q, q);                 /* %$t = 8 y^4$% */
 162
 163     p = F_DBL(f, p, s);                 /* %$2 s$% */
 164     dx = F_SQR(f, MP_NEW, m);           /* %$m^2$% */
 165     dx = F_SUB(f, dx, dx, p);           /* %$x' = m^2 - 2 s$% */
 166
 167     s = F_SUB(f, s, s, dx);             /* %$s - x'$% */
 168     dy = F_MUL(f, p, m, s);             /* %$m (s - x')$% */
 169     dy = F_SUB(f, dy, dy, q);           /* %$y' = m (s - x') - t$% */
 170
 171     EC_DESTROY(d);
 172     d->x = dx;
 173     d->y = dy;
 174     d->z = dz;
 175     MP_DROP(m);
 176     MP_DROP(q);
 177     MP_DROP(s);
 178   }
 179   return (d);
 180 }
 181
 182 static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b)
 183 {
 184   if (a == b)
 185     ecdbl(c, d, a);
 186   else if (EC_ATINF(a))
 187     EC_COPY(d, b);
 188   else if (EC_ATINF(b))
 189     EC_COPY(d, a);
 190   else {
 191     field *f = c->f;
 192     mp *lambda;
 193     mp *dy, *dx;
 194
 195     if (!MP_EQ(a->x, b->x)) {
 196       dy = F_SUB(f, MP_NEW, a->y, b->y); /* %$y_0 - y_1$% */
 197       dx = F_SUB(f, MP_NEW, a->x, b->x); /* %$x_0 - x_1$% */
 198       dx = F_INV(f, dx, dx);            /* %$(x_0 - x_1)^{-1}$% */
 199       lambda = F_MUL(f, MP_NEW, dy, dx);
 200                                    /* %$\lambda = (y_0 - y1)/(x_0 - x_1)$% */
 201     } else if (F_ZEROP(c->f, a->y) || !MP_EQ(a->y, b->y)) {
 202       EC_SETINF(d);
 203       return (d);
 204     } else {
 205       dx = F_SQR(f, MP_NEW, a->x);      /* %$x_0^2$% */
 206       dx = F_TPL(f, dx, dx);            /* %$3 x_0^2$% */
 207       dx = F_ADD(f, dx, dx, c->a);      /* %$3 x_0^2 + A$% */
 208       dy = F_DBL(f, MP_NEW, a->y);      /* %$2 y_0$% */
 209       dy = F_INV(f, dy, dy);            /* %$(2 y_0)^{-1}$% */
 210       lambda = F_MUL(f, MP_NEW, dx, dy);
 211                                     /* %$\lambda = (3 x_0^2 + A)/(2 y_0)$% */
 212     }
 213
 214     dx = F_SQR(f, dx, lambda);          /* %$\lambda^2$% */
 215     dx = F_SUB(f, dx, dx, a->x);        /* %$\lambda^2 - x_0$% */
 216     dx = F_SUB(f, dx, dx, b->x);        /* %$x' = \lambda^2 - x_0 - x_1$% */
 217     dy = F_SUB(f, dy, b->x, dx);        /* %$x_1 - x'$% */
 218     dy = F_MUL(f, dy, lambda, dy);      /* %$\lambda (x_1 - x')$% */
 219     dy = F_SUB(f, dy, dy, b->y);      /* %$y' = \lambda (x_1 - x') - y_1$% */
 220
 221     EC_DESTROY(d);
 222     d->x = dx;
 223     d->y = dy;
 224     d->z = 0;
 225     MP_DROP(lambda);
 226   }
 227   return (d);
 228 }
 229
 230 static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b)
 231 {
 232   if (a == b)
 233     c->ops->dbl(c, d, a);
 234   else if (EC_ATINF(a))
 235     EC_COPY(d, b);
 236   else if (EC_ATINF(b))
 237     EC_COPY(d, a);
 238   else {
 239     field *f = c->f;
 240     mp *p, *q, *r, *w, *u, *uu, *s, *ss, *dx, *dy, *dz;
 241
 242     q = F_SQR(f, MP_NEW, a->z);         /* %$z_0^2$% */
 243     u = F_MUL(f, MP_NEW, q, b->x);      /* %$u = x_1 z_0^2$% */
 244     p = F_MUL(f, MP_NEW, q, b->y);      /* %$y_1 z_0^2$% */
 245     s = F_MUL(f, q, p, a->z);           /* %$s = y_1 z_0^3$% */
 246
 247     q = F_SQR(f, MP_NEW, b->z);         /* %$z_1^2$% */
 248     uu = F_MUL(f, MP_NEW, q, a->x);     /* %$uu = x_0 z_1^2$%*/
 249     p = F_MUL(f, p, q, a->y);           /* %$y_0 z_1^2$% */
 250     ss = F_MUL(f, q, p, b->z);          /* %$ss = y_0 z_1^3$% */
 251
 252     w = F_SUB(f, p, uu, u);             /* %$w = uu - u$% */
 253     r = F_SUB(f, MP_NEW, ss, s);        /* %$r = ss - s$% */
 254     if (F_ZEROP(f, w)) {
 255       MP_DROP(w);
 256       MP_DROP(u);
 257       MP_DROP(s);
 258       MP_DROP(uu);
 259       MP_DROP(ss);
 260       if (F_ZEROP(f, r)) {
 261         MP_DROP(r);
 262         return (c->ops->dbl(c, d, a));
 263       } else {
 264         MP_DROP(r);
 265         EC_SETINF(d);
 266         return (d);
 267       }
 268     }
 269     u = F_ADD(f, u, u, uu);             /* %$t = uu + u$% */
 270     s = F_ADD(f, s, s, ss);             /* %$m = ss + r$% */
 271
 272     uu = F_MUL(f, uu, a->z, w);         /* %$z_0 w$% */
 273     dz = F_MUL(f, ss, uu, b->z);        /* %$z' = z_0 z_1 w$% */
 274
 275     p = F_SQR(f, uu, w);                /* %$w^2$% */
 276     q = F_MUL(f, MP_NEW, p, u);         /* %$t w^2$% */
 277     u = F_MUL(f, u, p, w);              /* %$w^3$% */
 278     p = F_MUL(f, p, u, s);              /* %$m w^3$% */
 279
 280     dx = F_SQR(f, u, r);                /* %$r^2$% */
 281     dx = F_SUB(f, dx, dx, q);           /* %$x' = r^2 - t w^2$% */
 282
 283     s = F_DBL(f, s, dx);                /* %$2 x'$% */
 284     q = F_SUB(f, q, q, s);              /* %$v = t w^2 - 2 x'$% */
 285     dy = F_MUL(f, s, q, r);             /* %$v r$% */
 286     dy = F_SUB(f, dy, dy, p);           /* %$v r - m w^3$% */
 287     dy = F_HLV(f, dy, dy);              /* %$y' = (v r - m w^3)/2$% */
 288
 289     EC_DESTROY(d);
 290     d->x = dx;
 291     d->y = dy;
 292     d->z = dz;
 293     MP_DROP(p);
 294     MP_DROP(q);
 295     MP_DROP(r);
 296     MP_DROP(w);
 297   }
 298   return (d);
 299 }
 300
 301 static int eccheck(ec_curve *c, const ec *p)
 302 {
 303   field *f = c->f;
 304   mp *l, *x, *r;
 305   int rc;
 306   if (EC_ATINF(p)) return (0);
 307   l = F_SQR(f, MP_NEW, p->y);
 308   x = F_SQR(f, MP_NEW, p->x);
 309   r = F_MUL(f, MP_NEW, x, p->x);
 310   x = F_MUL(f, x, c->a, p->x);
 311   r = F_ADD(f, r, r, x);
 312   r = F_ADD(f, r, r, c->b);
 313   rc = MP_EQ(l, r) ? 0 : -1;
 314   mp_drop(l);
 315   mp_drop(x);
 316   mp_drop(r);
 317   return (rc);
 318 }
 319
 320 static int ecprojcheck(ec_curve *c, const ec *p)
 321 {
 322   ec t = EC_INIT;
 323   int rc;
 324
 325   c->ops->fix(c, &t, p);
 326   rc = eccheck(c, &t);
 327   EC_DESTROY(&t);
 328   return (rc);
 329 }
 330
 331 static void ecdestroy(ec_curve *c)
 332 {
 333   MP_DROP(c->a);
 334   MP_DROP(c->b);
 335   DESTROY(c);
 336 }
 337
 338 /* --- @ec_prime@, @ec_primeproj@ --- *
 339  *
 340  * Arguments:   @field *f@ = the underlying field for this elliptic curve
 341  *              @mp *a, *b@ = the coefficients for this curve
 342  *
 343  * Returns:     A pointer to the curve, or null.
 344  *
 345  * Use:         Creates a curve structure for an elliptic curve defined over
 346  *              a prime field.  The @primeproj@ variant uses projective
 347  *              coordinates, which can be a win.
 348  */
 349
 350 extern ec_curve *ec_prime(field *f, mp *a, mp *b)
 351 {
 352   ec_curve *c = CREATE(ec_curve);
 353   c->ops = &ec_primeops;
 354   c->f = f;
 355   c->a = F_IN(f, MP_NEW, a);
 356   c->b = F_IN(f, MP_NEW, b);
 357   return (c);
 358 }
 359
 360 extern ec_curve *ec_primeproj(field *f, mp *a, mp *b)
 361 {
 362   ec_curve *c = CREATE(ec_curve);
 363   mp *ax;
 364
 365   ax = mp_add(MP_NEW, a, MP_THREE);
 366   ax = F_IN(f, ax, ax);
 367   if (F_ZEROP(f, ax))
 368     c->ops = &ec_primeprojxops;
 369   else
 370     c->ops = &ec_primeprojops;
 371   MP_DROP(ax);
 372   c->f = f;
 373   c->a = F_IN(f, MP_NEW, a);
 374   c->b = F_IN(f, MP_NEW, b);
 375   return (c);
 376 }
 377
 378 static const ec_ops ec_primeops = {
 379   "prime",
 380   ecdestroy, ec_stdsamep, ec_idin, ec_idout, ec_idfix,
 381   ecfind, ecneg, ecadd, ec_stdsub, ecdbl, eccheck
 382 };
 383
 384 static const ec_ops ec_primeprojops = {
 385   "primeproj",
 386   ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix,
 387   ecfind, ecneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck
 388 };
 389
 390 static const ec_ops ec_primeprojxops = {
 391   "primeproj",
 392   ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix,
 393   ecfind, ecneg, ecprojadd, ec_stdsub, ecprojxdbl, ecprojcheck
 394 };
 395
 396 /*----- Test rig ----------------------------------------------------------*/
 397
 398 #ifdef TEST_RIG
 399
 400 #define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
 401
 402 int main(int argc, char *argv[])
 403 {
 404   field *f;
 405   ec_curve *c;
 406   ec g = EC_INIT, d = EC_INIT;
 407   mp *p, *a, *b, *r;
 408   int i, n = argc == 1 ? 1 : atoi(argv[1]);
 409
 410   printf("ec-prime: ");
 411   fflush(stdout);
 412   a = MP(-3);
 413   b = MP(0xb3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875ac656398d8a2ed19d2a85c8edd3ec2aef);
 414   p = MP(39402006196394479212279040100143613805079739270465446667948293404245721771496870329047266088258938001861606973112319);
 415   r = MP(39402006196394479212279040100143613805079739270465446667946905279627659399113263569398956308152294913554433653942642);
 416
 417   f = field_niceprime(p);
 418   c = ec_primeproj(f, a, b);
 419
 420   g.x = MP(0xaa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a385502f25dbf55296c3a545e3872760ab7);
 421   g.y = MP(0x3617de4a96262c6f5d9e98bf9292dc29f8f41dbd289a147ce9da3113b5f0b8c00a60b1ce1d7e819d7a431d7c90ea0e5f);
 422
 423   for (i = 0; i < n; i++) {
 424     ec_mul(c, &d, &g, r);
 425     if (EC_ATINF(&d)) {
 426       fprintf(stderr, "zero too early\n");
 427       return (1);
 428     }
 429     ec_add(c, &d, &d, &g);
 430     if (!EC_ATINF(&d)) {
 431       fprintf(stderr, "didn't reach zero\n");
 432       MP_EPRINT("d.x", d.x);
 433       MP_EPRINT("d.y", d.y);
 434       return (1);
 435     }
 436     ec_destroy(&d);
 437   }
 438   ec_destroy(&g);
 439   ec_destroycurve(c);
 440   F_DESTROY(f);
 441   MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r);
 442   assert(!mparena_count(&mparena_global));
 443   printf("ok\n");
 444   return (0);
 445 }
 446
 447 #endif
 448
 449 /*----- That's all, folks -------------------------------------------------*/