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

   1 /* -*-c-*-
   2  *
   3  * $Id$
   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 /*----- Header files ------------------------------------------------------*/
  31
  32 #include <mLib/sub.h>
  33
  34 #include "ec.h"
  35
  36 /*----- Simple prime curves -----------------------------------------------*/
  37
  38 static const ec_ops ec_primeops, ec_primeprojops, ec_primeprojxops;
  39
  40 static ec *ecneg(ec_curve *c, ec *d, const ec *p)
  41 {
  42   EC_COPY(d, p);
  43   if (d->y)
  44     d->y = F_NEG(c->f, d->y, d->y);
  45   return (d);
  46 }
  47
  48 static ec *ecfind(ec_curve *c, ec *d, mp *x)
  49 {
  50   mp *p, *q;
  51   field *f = c->f;
  52
  53   q = F_SQR(f, MP_NEW, x);
  54   p = F_MUL(f, MP_NEW, x, q);
  55   q = F_MUL(f, q, x, c->a);
  56   p = F_ADD(f, p, p, q);
  57   p = F_ADD(f, p, p, c->b);
  58   MP_DROP(q);
  59   p = F_SQRT(f, p, p);
  60   if (!p)
  61     return (0);
  62   EC_DESTROY(d);
  63   d->x = MP_COPY(x);
  64   d->y = p;
  65   d->z = MP_COPY(f->one);
  66   return (d);
  67 }
  68
  69 static ec *ecdbl(ec_curve *c, ec *d, const ec *a)
  70 {
  71   if (EC_ATINF(a) || F_ZEROP(c->f, a->y))
  72     EC_SETINF(d);
  73   else {
  74     field *f = c->f;
  75     mp *lambda;
  76     mp *dy, *dx;
  77
  78     dx = F_SQR(f, MP_NEW, a->x);        /* %$x^2$% */
  79     dy = F_DBL(f, MP_NEW, a->y);        /* %$2 y$% */
  80     dx = F_TPL(f, dx, dx);              /* %$3 x^2$% */
  81     dx = F_ADD(f, dx, dx, c->a);        /* %$3 x^2 + A$% */
  82     dy = F_INV(f, dy, dy);              /* %$(2 y)^{-1}$% */
  83     lambda = F_MUL(f, MP_NEW, dx, dy);  /* %$\lambda = (3 x^2 + A)/(2 y)$% */
  84
  85     dx = F_SQR(f, dx, lambda);          /* %$\lambda^2$% */
  86     dy = F_DBL(f, dy, a->x);            /* %$2 x$% */
  87     dx = F_SUB(f, dx, dx, dy);          /* %$x' = \lambda^2 - 2 x */
  88     dy = F_SUB(f, dy, a->x, dx);        /* %$x - x'$% */
  89     dy = F_MUL(f, dy, lambda, dy);      /* %$\lambda (x - x')$% */
  90     dy = F_SUB(f, dy, dy, a->y);        /* %$y' = \lambda (x - x') - y$% */
  91
  92     EC_DESTROY(d);
  93     d->x = dx;
  94     d->y = dy;
  95     d->z = 0;
  96     MP_DROP(lambda);
  97   }
  98   return (d);
  99 }
 100
 101 static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a)
 102 {
 103   if (EC_ATINF(a) || F_ZEROP(c->f, a->y))
 104     EC_SETINF(d);
 105   else {
 106     field *f = c->f;
 107     mp *p, *q, *m, *s, *dx, *dy, *dz;
 108
 109     p = F_SQR(f, MP_NEW, a->z);         /* %$z^2$% */
 110     q = F_SQR(f, MP_NEW, p);            /* %$z^4$% */
 111     p = F_MUL(f, p, q, c->a);           /* %$A z^4$% */
 112     m = F_SQR(f, MP_NEW, a->x);         /* %$x^2$% */
 113     m = F_TPL(f, m, m);                 /* %$3 x^2$% */
 114     m = F_ADD(f, m, m, p);              /* %$m = 3 x^2 + A z^4$% */
 115
 116     q = F_DBL(f, q, a->y);              /* %$2 y$% */
 117     dz = F_MUL(f, MP_NEW, q, a->z);     /* %$z' = 2 y z$% */
 118
 119     p = F_SQR(f, p, q);                 /* %$4 y^2$% */
 120     s = F_MUL(f, MP_NEW, p, a->x);      /* %$s = 4 x y^2$% */
 121     q = F_SQR(f, q, p);                 /* %$16 y^4$% */
 122     q = F_HLV(f, q, q);                 /* %$t = 8 y^4$% */
 123
 124     p = F_DBL(f, p, s);                 /* %$2 s$% */
 125     dx = F_SQR(f, MP_NEW, m);           /* %$m^2$% */
 126     dx = F_SUB(f, dx, dx, p);           /* %$x' = m^2 - 2 s$% */
 127
 128     s = F_SUB(f, s, s, dx);             /* %$s - x'$% */
 129     dy = F_MUL(f, p, m, s);             /* %$m (s - x')$% */
 130     dy = F_SUB(f, dy, dy, q);           /* %$y' = m (s - x') - t$% */
 131
 132     EC_DESTROY(d);
 133     d->x = dx;
 134     d->y = dy;
 135     d->z = dz;
 136     MP_DROP(m);
 137     MP_DROP(q);
 138     MP_DROP(s);
 139   }
 140   return (d);
 141 }
 142
 143 static ec *ecprojxdbl(ec_curve *c, ec *d, const ec *a)
 144 {
 145   if (EC_ATINF(a) || F_ZEROP(c->f, a->y))
 146     EC_SETINF(d);
 147   else {
 148     field *f = c->f;
 149     mp *p, *q, *m, *s, *dx, *dy, *dz;
 150
 151     m = F_SQR(f, MP_NEW, a->z);         /* %$z^2$% */
 152     p = F_SUB(f, MP_NEW, a->x, m);      /* %$x - z^2$% */
 153     q = F_ADD(f, MP_NEW, a->x, m);      /* %$x + z^2$% */
 154     m = F_MUL(f, m, p, q);              /* %$x^2 - z^4$% */
 155     m = F_TPL(f, m, m);                 /* %$m = 3 x^2 - 3 z^4$% */
 156
 157     q = F_DBL(f, q, a->y);              /* %$2 y$% */
 158     dz = F_MUL(f, MP_NEW, q, a->z);     /* %$z' = 2 y z$% */
 159
 160     p = F_SQR(f, p, q);                 /* %$4 y^2$% */
 161     s = F_MUL(f, MP_NEW, p, a->x);      /* %$s = 4 x y^2$% */
 162     q = F_SQR(f, q, p);                 /* %$16 y^4$% */
 163     q = F_HLV(f, q, q);                 /* %$t = 8 y^4$% */
 164
 165     p = F_DBL(f, p, s);                 /* %$2 s$% */
 166     dx = F_SQR(f, MP_NEW, m);           /* %$m^2$% */
 167     dx = F_SUB(f, dx, dx, p);           /* %$x' = m^2 - 2 s$% */
 168
 169     s = F_SUB(f, s, s, dx);             /* %$s - x'$% */
 170     dy = F_MUL(f, p, m, s);             /* %$m (s - x')$% */
 171     dy = F_SUB(f, dy, dy, q);           /* %$y' = m (s - x') - t$% */
 172
 173     EC_DESTROY(d);
 174     d->x = dx;
 175     d->y = dy;
 176     d->z = dz;
 177     MP_DROP(m);
 178     MP_DROP(q);
 179     MP_DROP(s);
 180   }
 181   return (d);
 182 }
 183
 184 static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b)
 185 {
 186   if (a == b)
 187     ecdbl(c, d, a);
 188   else if (EC_ATINF(a))
 189     EC_COPY(d, b);
 190   else if (EC_ATINF(b))
 191     EC_COPY(d, a);
 192   else {
 193     field *f = c->f;
 194     mp *lambda;
 195     mp *dy, *dx;
 196
 197     if (!MP_EQ(a->x, b->x)) {
 198       dy = F_SUB(f, MP_NEW, a->y, b->y); /* %$y_0 - y_1$% */
 199       dx = F_SUB(f, MP_NEW, a->x, b->x); /* %$x_0 - x_1$% */
 200       dx = F_INV(f, dx, dx);            /* %$(x_0 - x_1)^{-1}$% */
 201       lambda = F_MUL(f, MP_NEW, dy, dx);
 202                                    /* %$\lambda = (y_0 - y1)/(x_0 - x_1)$% */
 203     } else if (F_ZEROP(c->f, a->y) || !MP_EQ(a->y, b->y)) {
 204       EC_SETINF(d);
 205       return (d);
 206     } else {
 207       dx = F_SQR(f, MP_NEW, a->x);      /* %$x_0^2$% */
 208       dx = F_TPL(f, dx, dx);            /* %$3 x_0^2$% */
 209       dx = F_ADD(f, dx, dx, c->a);      /* %$3 x_0^2 + A$% */
 210       dy = F_DBL(f, MP_NEW, a->y);      /* %$2 y_0$% */
 211       dy = F_INV(f, dy, dy);            /* %$(2 y_0)^{-1}$% */
 212       lambda = F_MUL(f, MP_NEW, dx, dy);
 213                                     /* %$\lambda = (3 x_0^2 + A)/(2 y_0)$% */
 214     }
 215
 216     dx = F_SQR(f, dx, lambda);          /* %$\lambda^2$% */
 217     dx = F_SUB(f, dx, dx, a->x);        /* %$\lambda^2 - x_0$% */
 218     dx = F_SUB(f, dx, dx, b->x);        /* %$x' = \lambda^2 - x_0 - x_1$% */
 219     dy = F_SUB(f, dy, b->x, dx);        /* %$x_1 - x'$% */
 220     dy = F_MUL(f, dy, lambda, dy);      /* %$\lambda (x_1 - x')$% */
 221     dy = F_SUB(f, dy, dy, b->y);      /* %$y' = \lambda (x_1 - x') - y_1$% */
 222
 223     EC_DESTROY(d);
 224     d->x = dx;
 225     d->y = dy;
 226     d->z = 0;
 227     MP_DROP(lambda);
 228   }
 229   return (d);
 230 }
 231
 232 static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b)
 233 {
 234   if (a == b)
 235     c->ops->dbl(c, d, a);
 236   else if (EC_ATINF(a))
 237     EC_COPY(d, b);
 238   else if (EC_ATINF(b))
 239     EC_COPY(d, a);
 240   else {
 241     field *f = c->f;
 242     mp *p, *q, *r, *w, *u, *uu, *s, *ss, *dx, *dy, *dz;
 243
 244     q = F_SQR(f, MP_NEW, a->z);         /* %$z_0^2$% */
 245     u = F_MUL(f, MP_NEW, q, b->x);      /* %$u = x_1 z_0^2$% */
 246     p = F_MUL(f, MP_NEW, q, b->y);      /* %$y_1 z_0^2$% */
 247     s = F_MUL(f, q, p, a->z);           /* %$s = y_1 z_0^3$% */
 248
 249     q = F_SQR(f, MP_NEW, b->z);         /* %$z_1^2$% */
 250     uu = F_MUL(f, MP_NEW, q, a->x);     /* %$uu = x_0 z_1^2$%*/
 251     p = F_MUL(f, p, q, a->y);           /* %$y_0 z_1^2$% */
 252     ss = F_MUL(f, q, p, b->z);          /* %$ss = y_0 z_1^3$% */
 253
 254     w = F_SUB(f, p, uu, u);             /* %$w = uu - u$% */
 255     r = F_SUB(f, MP_NEW, ss, s);        /* %$r = ss - s$% */
 256     if (F_ZEROP(f, w)) {
 257       MP_DROP(w);
 258       MP_DROP(u);
 259       MP_DROP(s);
 260       MP_DROP(uu);
 261       MP_DROP(ss);
 262       if (F_ZEROP(f, r)) {
 263         MP_DROP(r);
 264         return (c->ops->dbl(c, d, a));
 265       } else {
 266         MP_DROP(r);
 267         EC_SETINF(d);
 268         return (d);
 269       }
 270     }
 271     u = F_ADD(f, u, u, uu);             /* %$t = uu + u$% */
 272     s = F_ADD(f, s, s, ss);             /* %$m = ss + r$% */
 273
 274     uu = F_MUL(f, uu, a->z, w);         /* %$z_0 w$% */
 275     dz = F_MUL(f, ss, uu, b->z);        /* %$z' = z_0 z_1 w$% */
 276
 277     p = F_SQR(f, uu, w);                /* %$w^2$% */
 278     q = F_MUL(f, MP_NEW, p, u);         /* %$t w^2$% */
 279     u = F_MUL(f, u, p, w);              /* %$w^3$% */
 280     p = F_MUL(f, p, u, s);              /* %$m w^3$% */
 281
 282     dx = F_SQR(f, u, r);                /* %$r^2$% */
 283     dx = F_SUB(f, dx, dx, q);           /* %$x' = r^2 - t w^2$% */
 284
 285     s = F_DBL(f, s, dx);                /* %$2 x'$% */
 286     q = F_SUB(f, q, q, s);              /* %$v = t w^2 - 2 x'$% */
 287     dy = F_MUL(f, s, q, r);             /* %$v r$% */
 288     dy = F_SUB(f, dy, dy, p);           /* %$v r - m w^3$% */
 289     dy = F_HLV(f, dy, dy);              /* %$y' = (v r - m w^3)/2$% */
 290
 291     EC_DESTROY(d);
 292     d->x = dx;
 293     d->y = dy;
 294     d->z = dz;
 295     MP_DROP(p);
 296     MP_DROP(q);
 297     MP_DROP(r);
 298     MP_DROP(w);
 299   }
 300   return (d);
 301 }
 302
 303 static int eccheck(ec_curve *c, const ec *p)
 304 {
 305   field *f = c->f;
 306   mp *l, *x, *r;
 307   int rc;
 308   if (EC_ATINF(p)) return (0);
 309   l = F_SQR(f, MP_NEW, p->y);
 310   x = F_SQR(f, MP_NEW, p->x);
 311   r = F_MUL(f, MP_NEW, x, p->x);
 312   x = F_MUL(f, x, c->a, p->x);
 313   r = F_ADD(f, r, r, x);
 314   r = F_ADD(f, r, r, c->b);
 315   rc = MP_EQ(l, r) ? 0 : -1;
 316   mp_drop(l);
 317   mp_drop(x);
 318   mp_drop(r);
 319   return (rc);
 320 }
 321
 322 static int ecprojcheck(ec_curve *c, const ec *p)
 323 {
 324   ec t = EC_INIT;
 325   int rc;
 326
 327   c->ops->fix(c, &t, p);
 328   rc = eccheck(c, &t);
 329   EC_DESTROY(&t);
 330   return (rc);
 331 }
 332
 333 static void ecdestroy(ec_curve *c)
 334 {
 335   MP_DROP(c->a);
 336   MP_DROP(c->b);
 337   DESTROY(c);
 338 }
 339
 340 /* --- @ec_prime@, @ec_primeproj@ --- *
 341  *
 342  * Arguments:   @field *f@ = the underlying field for this elliptic curve
 343  *              @mp *a, *b@ = the coefficients for this curve
 344  *
 345  * Returns:     A pointer to the curve, or null.
 346  *
 347  * Use:         Creates a curve structure for an elliptic curve defined over
 348  *              a prime field.  The @primeproj@ variant uses projective
 349  *              coordinates, which can be a win.
 350  */
 351
 352 extern ec_curve *ec_prime(field *f, mp *a, mp *b)
 353 {
 354   ec_curve *c = CREATE(ec_curve);
 355   c->ops = &ec_primeops;
 356   c->f = f;
 357   c->a = F_IN(f, MP_NEW, a);
 358   c->b = F_IN(f, MP_NEW, b);
 359   return (c);
 360 }
 361
 362 extern ec_curve *ec_primeproj(field *f, mp *a, mp *b)
 363 {
 364   ec_curve *c = CREATE(ec_curve);
 365   mp *ax;
 366
 367   ax = mp_add(MP_NEW, a, MP_THREE);
 368   ax = F_IN(f, ax, ax);
 369   if (F_ZEROP(f, ax))
 370     c->ops = &ec_primeprojxops;
 371   else
 372     c->ops = &ec_primeprojops;
 373   MP_DROP(ax);
 374   c->f = f;
 375   c->a = F_IN(f, MP_NEW, a);
 376   c->b = F_IN(f, MP_NEW, b);
 377   return (c);
 378 }
 379
 380 static const ec_ops ec_primeops = {
 381   "prime",
 382   ecdestroy, ec_stdsamep, ec_idin, ec_idout, ec_idfix,
 383   ecfind, ecneg, ecadd, ec_stdsub, ecdbl, eccheck
 384 };
 385
 386 static const ec_ops ec_primeprojops = {
 387   "primeproj",
 388   ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix,
 389   ecfind, ecneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck
 390 };
 391
 392 static const ec_ops ec_primeprojxops = {
 393   "primeproj",
 394   ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix,
 395   ecfind, ecneg, ecprojadd, ec_stdsub, ecprojxdbl, ecprojcheck
 396 };
 397
 398 /*----- Test rig ----------------------------------------------------------*/
 399
 400 #ifdef TEST_RIG
 401
 402 #define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
 403
 404 int main(int argc, char *argv[])
 405 {
 406   field *f;
 407   ec_curve *c;
 408   ec g = EC_INIT, d = EC_INIT;
 409   mp *p, *a, *b, *r;
 410   int i, n = argc == 1 ? 1 : atoi(argv[1]);
 411
 412   printf("ec-prime: ");
 413   fflush(stdout);
 414   a = MP(-3);
 415   b = MP(0xb3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875ac656398d8a2ed19d2a85c8edd3ec2aef);
 416   p = MP(39402006196394479212279040100143613805079739270465446667948293404245721771496870329047266088258938001861606973112319);
 417   r = MP(39402006196394479212279040100143613805079739270465446667946905279627659399113263569398956308152294913554433653942642);
 418
 419   f = field_niceprime(p);
 420   c = ec_primeproj(f, a, b);
 421
 422   g.x = MP(0xaa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a385502f25dbf55296c3a545e3872760ab7);
 423   g.y = MP(0x3617de4a96262c6f5d9e98bf9292dc29f8f41dbd289a147ce9da3113b5f0b8c00a60b1ce1d7e819d7a431d7c90ea0e5f);
 424
 425   for (i = 0; i < n; i++) {
 426     ec_mul(c, &d, &g, r);
 427     if (EC_ATINF(&d)) {
 428       fprintf(stderr, "zero too early\n");
 429       return (1);
 430     }
 431     ec_add(c, &d, &d, &g);
 432     if (!EC_ATINF(&d)) {
 433       fprintf(stderr, "didn't reach zero\n");
 434       MP_EPRINT("d.x", d.x);
 435       MP_EPRINT("d.y", d.y);
 436       return (1);
 437     }
 438     ec_destroy(&d);
 439   }
 440   ec_destroy(&g);
 441   ec_destroycurve(c);
 442   F_DESTROY(f);
 443   MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r);
 444   assert(!mparena_count(&mparena_global));
 445   printf("ok\n");
 446   return (0);
 447 }
 448
 449 #endif
 450
 451 /*----- That's all, folks -------------------------------------------------*/