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

   1 /* -*-c-*-
   2  *
   3  * $Id: ec-bin.c,v 1.7 2004/04/01 21:28:41 mdw Exp $
   4  *
   5  * Arithmetic for elliptic curves over binary fields
   6  *
   7  * (c) 2004 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-bin.c,v $
  33  * Revision 1.7  2004/04/01 21:28:41  mdw
  34  * Normal basis support (translates to poly basis internally).  Rewrite
  35  * EC and prime group table generators in awk, so that they can reuse data
  36  * for repeated constants.
  37  *
  38  * Revision 1.6  2004/04/01 12:50:09  mdw
  39  * Add cyclic group abstraction, with test code.  Separate off exponentation
  40  * functions for better static linking.  Fix a buttload of bugs on the way.
  41  * Generally ensure that negative exponents do inversion correctly.  Add
  42  * table of standard prime-field subgroups.  (Binary field subgroups are
  43  * currently unimplemented but easy to add if anyone ever finds a good one.)
  44  *
  45  * Revision 1.5  2004/03/27 17:54:11  mdw
  46  * Standard curves and curve checking.
  47  *
  48  * Revision 1.4  2004/03/23 15:19:32  mdw
  49  * Test elliptic curves more thoroughly.
  50  *
  51  * Revision 1.3  2004/03/22 02:19:09  mdw
  52  * Rationalise the sliding-window threshold.  Drop guarantee that right
  53  * arguments to EC @add@ are canonical, and fix up projective implementations
  54  * to cope.
  55  *
  56  * Revision 1.2  2004/03/21 22:52:06  mdw
  57  * Merge and close elliptic curve branch.
  58  *
  59  * Revision 1.1.2.1  2004/03/21 22:39:46  mdw
  60  * Elliptic curves on binary fields work.
  61  *
  62  */
  63
  64 /*----- Header files ------------------------------------------------------*/
  65
  66 #include <mLib/sub.h>
  67
  68 #include "ec.h"
  69
  70 /*----- Data structures ---------------------------------------------------*/
  71
  72 typedef struct ecctx {
  73   ec_curve c;
  74   mp *bb;
  75 } ecctx;
  76
  77 /*----- Main code ---------------------------------------------------------*/
  78
  79 static const ec_ops ec_binops, ec_binprojops;
  80
  81 static ec *ecneg(ec_curve *c, ec *d, const ec *p)
  82 {
  83   EC_COPY(d, p);
  84   if (d->x)
  85     d->y = F_ADD(c->f, d->y, d->y, d->x);
  86   return (d);
  87 }
  88
  89 static ec *ecprojneg(ec_curve *c, ec *d, const ec *p)
  90 {
  91   EC_COPY(d, p);
  92   if (d->x) {
  93     mp *t = F_MUL(c->f, MP_NEW, d->x, d->z);
  94     d->y = F_ADD(c->f, d->y, d->y, t);
  95     MP_DROP(t);
  96   }
  97   return (d);
  98 }
  99
 100 static ec *ecfind(ec_curve *c, ec *d, mp *x)
 101 {
 102   field *f = c->f;
 103   mp *y, *u, *v;
 104
 105   if (F_ZEROP(f, x))
 106     y = F_SQRT(f, MP_NEW, c->b);
 107   else {
 108     u = F_SQR(f, MP_NEW, x);            /* %$x^2$% */
 109     y = F_MUL(f, MP_NEW, u, c->a);      /* %$a x^2$% */
 110     y = F_ADD(f, y, y, c->b);           /* %$a x^2 + b$% */
 111     v = F_MUL(f, MP_NEW, u, x);         /* %$x^3$% */
 112     y = F_ADD(f, y, y, v);              /* %$A = x^3 + a x^2 + b$% */
 113     if (!F_ZEROP(f, y)) {
 114       u = F_INV(f, u, u);               /* %$x^{-2}$% */
 115       v = F_MUL(f, v, u, y);        /* %$B = A x^{-2} = x + a + b x^{-2}$% */
 116       y = F_QUADSOLVE(f, y, v);         /* %$z^2 + z = B$% */
 117       if (y) y = F_MUL(f, y, y, x);     /* %$y = z x$% */
 118     }
 119     MP_DROP(u);
 120     MP_DROP(v);
 121   }
 122   if (!y) return (0);
 123   EC_DESTROY(d);
 124   d->x = MP_COPY(x);
 125   d->y = y;
 126   d->z = MP_COPY(f->one);
 127   return (d);
 128 }
 129
 130 static ec *ecdbl(ec_curve *c, ec *d, const ec *a)
 131 {
 132   if (EC_ATINF(a) || F_ZEROP(c->f, a->x))
 133     EC_SETINF(d);
 134   else {
 135     field *f = c->f;
 136     mp *lambda;
 137     mp *dx, *dy;
 138
 139     dx = F_INV(f, MP_NEW, a->x);        /* %$x^{-1}$% */
 140     dy = F_MUL(f, MP_NEW, dx, a->y);    /* %$y/x$% */
 141     lambda = F_ADD(f, dy, dy, a->x);    /* %$\lambda = x + y/x$% */
 142
 143     dx = F_SQR(f, dx, lambda);          /* %$\lambda^2$% */
 144     dx = F_ADD(f, dx, dx, lambda);      /* %$\lambda^2 + \lambda$% */
 145     dx = F_ADD(f, dx, dx, c->a);       /* %$x' = a + \lambda^2 + \lambda$% */
 146
 147     dy = F_ADD(f, MP_NEW, a->x, dx);    /* %$ x + x' $% */
 148     dy = F_MUL(f, dy, dy, lambda);      /* %$ (x + x') \lambda$% */
 149     dy = F_ADD(f, dy, dy, a->y);        /* %$ (x + x') \lambda + y$% */
 150     dy = F_ADD(f, dy, dy, dx);      /* %$ y' = (x + x') \lambda + y + x'$% */
 151
 152     EC_DESTROY(d);
 153     d->x = dx;
 154     d->y = dy;
 155     d->z = 0;
 156     MP_DROP(lambda);
 157   }
 158   return (d);
 159 }
 160
 161 static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a)
 162 {
 163   if (EC_ATINF(a) || F_ZEROP(c->f, a->x))
 164     EC_SETINF(d);
 165   else {
 166     field *f = c->f;
 167     ecctx *cc = (ecctx *)c;
 168     mp *dx, *dy, *dz, *u, *v;
 169
 170     dy = F_SQR(f, MP_NEW, a->z);        /* %$z^2$% */
 171     dx = F_MUL(f, MP_NEW, dy, cc->bb);  /* %$c z^2$% */
 172     dx = F_ADD(f, dx, dx, a->x);        /* %$x + c z^2$% */
 173     dz = F_SQR(f, MP_NEW, dx);          /* %$(x + c z^2)^2$% */
 174     dx = F_SQR(f, dx, dz);              /* %$x' = (x + c z^2)^4$% */
 175
 176     dz = F_MUL(f, dz, dy, a->x);        /* %$z' = x z^2$% */
 177
 178     dy = F_SQR(f, dy, a->x);            /* %$x^2$% */
 179     u = F_MUL(f, MP_NEW, a->y, a->z);   /* %$y z$% */
 180     u = F_ADD(f, u, u, dz);             /* %$z' + y z$% */
 181     u = F_ADD(f, u, u, dy);             /* %$u = z' + x^2 + y z$% */
 182
 183     v = F_SQR(f, MP_NEW, dy);           /* %$x^4$% */
 184     dy = F_MUL(f, dy, v, dz);           /* %$x^4 z'$% */
 185     v = F_MUL(f, v, u, dx);             /* %$u x'$% */
 186     dy = F_ADD(f, dy, dy, v);           /* %$y' = x^4 z' + u x'$% */
 187
 188     EC_DESTROY(d);
 189     d->x = dx;
 190     d->y = dy;
 191     d->z = dz;
 192     MP_DROP(u);
 193     MP_DROP(v);
 194     assert(!(d->x->f & MP_DESTROYED));
 195     assert(!(d->y->f & MP_DESTROYED));
 196     assert(!(d->z->f & MP_DESTROYED));
 197   }
 198   return (d);
 199 }
 200
 201 static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b)
 202 {
 203   if (a == b)
 204     ecdbl(c, d, a);
 205   else if (EC_ATINF(a))
 206     EC_COPY(d, b);
 207   else if (EC_ATINF(b))
 208     EC_COPY(d, a);
 209   else {
 210     field *f = c->f;
 211     mp *lambda;
 212     mp *dx, *dy;
 213
 214     if (!MP_EQ(a->x, b->x)) {
 215       dx = F_ADD(f, MP_NEW, a->x, b->x); /* %$x_0 + x_1$% */
 216       dy = F_INV(f, MP_NEW, dx);        /* %$(x_0 + x_1)^{-1}$% */
 217       dx = F_ADD(f, dx, a->y, b->y);    /* %$y_0 + y_1$% */
 218       lambda = F_MUL(f, MP_NEW, dy, dx);
 219                                   /* %$\lambda = (y_0 + y_1)/(x_0 + x_1)$% */
 220
 221       dx = F_SQR(f, dx, lambda);        /* %$\lambda^2$% */
 222       dx = F_ADD(f, dx, dx, lambda);    /* %$\lambda^2 + \lambda$% */
 223       dx = F_ADD(f, dx, dx, c->a);      /* %$a + \lambda^2 + \lambda$% */
 224       dx = F_ADD(f, dx, dx, a->x);    /* %$a + \lambda^2 + \lambda + x_0$% */
 225       dx = F_ADD(f, dx, dx, b->x);
 226                            /* %$x' = a + \lambda^2 + \lambda + x_0 + x_1$% */
 227     } else if (!MP_EQ(a->y, b->y) || F_ZEROP(f, a->x)) {
 228       EC_SETINF(d);
 229       return (d);
 230     } else {
 231       dx = F_INV(f, MP_NEW, a->x);      /* %$x^{-1}$% */
 232       dy = F_MUL(f, MP_NEW, dx, a->y);  /* %$y/x$% */
 233       lambda = F_ADD(f, dy, dy, a->x);  /* %$\lambda = x + y/x$% */
 234
 235       dx = F_SQR(f, dx, lambda);        /* %$\lambda^2$% */
 236       dx = F_ADD(f, dx, dx, lambda);    /* %$\lambda^2 + \lambda$% */
 237       dx = F_ADD(f, dx, dx, c->a);    /* %$x' = a + \lambda^2 + \lambda$% */
 238       dy = MP_NEW;
 239     }
 240
 241     dy = F_ADD(f, dy, a->x, dx);        /* %$ x + x' $% */
 242     dy = F_MUL(f, dy, dy, lambda);      /* %$ (x + x') \lambda$% */
 243     dy = F_ADD(f, dy, dy, a->y);        /* %$ (x + x') \lambda + y$% */
 244     dy = F_ADD(f, dy, dy, dx);      /* %$ y' = (x + x') \lambda + y + x'$% */
 245
 246     EC_DESTROY(d);
 247     d->x = dx;
 248     d->y = dy;
 249     d->z = 0;
 250     MP_DROP(lambda);
 251   }
 252   return (d);
 253 }
 254
 255 static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b)
 256 {
 257   if (a == b)
 258     c->ops->dbl(c, d, a);
 259   else if (EC_ATINF(a))
 260     EC_COPY(d, b);
 261   else if (EC_ATINF(b))
 262     EC_COPY(d, a);
 263   else {
 264     field *f = c->f;
 265     mp *dx, *dy, *dz, *u, *uu, *v, *t, *s, *ss, *r, *w, *l;
 266
 267     dz = F_SQR(f, MP_NEW, b->z);        /* %$z_1^2$% */
 268     u = F_MUL(f, MP_NEW, dz, a->x);     /* %$u_0 = x_0 z_1^2$% */
 269     t = F_MUL(f, MP_NEW, dz, b->z);     /* %$z_1^3$% */
 270     s = F_MUL(f, MP_NEW, t, a->y);      /* %$s_0 = y_0 z_1^3$% */
 271
 272     dz = F_SQR(f, dz, a->z);            /* %$z_0^2$% */
 273     uu = F_MUL(f, MP_NEW, dz, b->x);    /* %$u_1 = x_1 z_0^2$% */
 274     t = F_MUL(f, t, dz, a->z);          /* %$z_0^3$% */
 275     ss = F_MUL(f, MP_NEW, t, b->y);     /* %$s_1 = y_1 z_0^3$% */
 276
 277     w = F_ADD(f, u, u, uu);             /* %$r = u_0 + u_1$% */
 278     r = F_ADD(f, s, s, ss);             /* %$w = s_0 + s_1$% */
 279     if (F_ZEROP(f, w)) {
 280       MP_DROP(w);
 281       MP_DROP(uu);
 282       MP_DROP(ss);
 283       MP_DROP(t);
 284       MP_DROP(dz);
 285       if (F_ZEROP(f, r)) {
 286         MP_DROP(r);
 287         return (c->ops->dbl(c, d, a));
 288       } else {
 289         MP_DROP(r);
 290         EC_SETINF(d);
 291         return (d);
 292       }
 293     }
 294
 295     l = F_MUL(f, t, a->z, w);           /* %$l = z_0 w$% */
 296
 297     dz = F_MUL(f, dz, l, b->z);         /* %$z' = l z_1$% */
 298
 299     ss = F_MUL(f, ss, r, b->x);         /* %$r x_1$% */
 300     t = F_MUL(f, uu, l, b->y);          /* %$l y_1$% */
 301     v = F_ADD(f, ss, ss, t);            /* %$v = r x_1 + l y_1$% */
 302
 303     t = F_ADD(f, t, r, dz);             /* %$t = r + z'$% */
 304
 305     uu = F_SQR(f, MP_NEW, dz);          /* %$z'^2$% */
 306     dx = F_MUL(f, MP_NEW, uu, c->a);    /* %$a z'^2$% */
 307     uu = F_MUL(f, uu, t, r);            /* %$t r$% */
 308     dx = F_ADD(f, dx, dx, uu);          /* %$a z'^2 + t r$% */
 309     r = F_SQR(f, r, w);                 /* %$w^2$% */
 310     uu = F_MUL(f, uu, r, w);            /* %$w^3$% */
 311     dx = F_ADD(f, dx, dx, uu);          /* %$x' = a z'^2 + t r + w^3$% */
 312
 313     r = F_SQR(f, r, l);                 /* %$l^2$% */
 314     dy = F_MUL(f, uu, v, r);            /* %$v l^2$% */
 315     l = F_MUL(f, l, t, dx);             /* %$t x'$% */
 316     dy = F_ADD(f, dy, dy, l);           /* %$y' = t x' + v l^2$% */
 317
 318     EC_DESTROY(d);
 319     d->x = dx;
 320     d->y = dy;
 321     d->z = dz;
 322     MP_DROP(l);
 323     MP_DROP(r);
 324     MP_DROP(w);
 325     MP_DROP(t);
 326     MP_DROP(v);
 327   }
 328   return (d);
 329 }
 330
 331 static int eccheck(ec_curve *c, const ec *p)
 332 {
 333   field *f = c->f;
 334   int rc;
 335   mp *u, *v;
 336
 337   if (EC_ATINF(p)) return (0);
 338   v = F_SQR(f, MP_NEW, p->x);
 339   u = F_MUL(f, MP_NEW, v, p->x);
 340   v = F_MUL(f, v, v, c->a);
 341   u = F_ADD(f, u, u, v);
 342   u = F_ADD(f, u, u, c->b);
 343   v = F_MUL(f, v, p->x, p->y);
 344   u = F_ADD(f, u, u, v);
 345   v = F_SQR(f, v, p->y);
 346   u = F_ADD(f, u, u, v);
 347   rc = F_ZEROP(f, u) ? 0 : -1;
 348   mp_drop(u);
 349   mp_drop(v);
 350   return (rc);
 351 }
 352
 353 static int ecprojcheck(ec_curve *c, const ec *p)
 354 {
 355   ec t = EC_INIT;
 356   int rc;
 357
 358   c->ops->fix(c, &t, p);
 359   rc = eccheck(c, &t);
 360   EC_DESTROY(&t);
 361   return (rc);
 362 }
 363
 364 static void ecdestroy(ec_curve *c)
 365 {
 366   ecctx *cc = (ecctx *)c;
 367   MP_DROP(cc->c.a);
 368   MP_DROP(cc->c.b);
 369   if (cc->bb) MP_DROP(cc->bb);
 370   DESTROY(cc);
 371 }
 372
 373 /* --- @ec_bin@, @ec_binproj@ --- *
 374  *
 375  * Arguments:   @field *f@ = the underlying field for this elliptic curve
 376  *              @mp *a, *b@ = the coefficients for this curve
 377  *
 378  * Returns:     A pointer to the curve.
 379  *
 380  * Use:         Creates a curve structure for an elliptic curve defined over
 381  *              a binary field.  The @binproj@ variant uses projective
 382  *              coordinates, which can be a win.
 383  */
 384
 385 ec_curve *ec_bin(field *f, mp *a, mp *b)
 386 {
 387   ecctx *cc = CREATE(ecctx);
 388   cc->c.ops = &ec_binops;
 389   cc->c.f = f;
 390   cc->c.a = F_IN(f, MP_NEW, a);
 391   cc->c.b = F_IN(f, MP_NEW, b);
 392   cc->bb = 0;
 393   return (&cc->c);
 394 }
 395
 396 ec_curve *ec_binproj(field *f, mp *a, mp *b)
 397 {
 398   ecctx *cc = CREATE(ecctx);
 399   cc->c.ops = &ec_binprojops;
 400   cc->c.f = f;
 401   cc->c.a = F_IN(f, MP_NEW, a);
 402   cc->c.b = F_IN(f, MP_NEW, b);
 403   cc->bb = F_SQRT(f, MP_NEW, cc->c.b);
 404   cc->bb = F_SQRT(f, cc->bb, cc->bb);
 405   return (&cc->c);
 406 }
 407
 408 static const ec_ops ec_binops = {
 409   ecdestroy, ec_stdsamep, ec_idin, ec_idout, ec_idfix,
 410   ecfind, ecneg, ecadd, ec_stdsub, ecdbl, eccheck
 411 };
 412
 413 static const ec_ops ec_binprojops = {
 414   ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix,
 415   ecfind, ecprojneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck
 416 };
 417
 418 /*----- Test rig ----------------------------------------------------------*/
 419
 420 #ifdef TEST_RIG
 421
 422 #define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
 423
 424 int main(int argc, char *argv[])
 425 {
 426   field *f;
 427   ec_curve *c;
 428   ec g = EC_INIT, d = EC_INIT;
 429   mp *p, *a, *b, *r, *beta;
 430   int i, n = argc == 1 ? 1 : atoi(argv[1]);
 431
 432   printf("ec-bin: ");
 433   fflush(stdout);
 434   a = MP(0x7ffffffffffffffffffffffffffffffffffffffff);
 435   b = MP(0x6645f3cacf1638e139c6cd13ef61734fbc9e3d9fb);
 436   p = MP(0x800000000000000000000000000000000000000c9);
 437   beta = MP(0x715169c109c612e390d347c748342bcd3b02a0bef);
 438   r = MP(0x040000000000000000000292fe77e70c12a4234c32);
 439
 440   f = field_binnorm(p, beta);
 441   c = ec_binproj(f, a, b);
 442   g.x = MP(0x0311103c17167564ace77ccb09c681f886ba54ee8);
 443   g.y = MP(0x333ac13c6447f2e67613bf7009daf98c87bb50c7f);
 444
 445   for (i = 0; i < n; i++) {
 446     ec_mul(c, &d, &g, r);
 447     if (EC_ATINF(&d)) {
 448       fprintf(stderr, "zero too early\n");
 449       return (1);
 450     }
 451     ec_add(c, &d, &d, &g);
 452     if (!EC_ATINF(&d)) {
 453       fprintf(stderr, "didn't reach zero\n");
 454       MP_EPRINTX("d.x", d.x);
 455       MP_EPRINTX("d.y", d.y);
 456       return (1);
 457     }
 458     ec_destroy(&d);
 459   }
 460
 461   ec_destroy(&g);
 462   ec_destroycurve(c);
 463   F_DESTROY(f);
 464   MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r); MP_DROP(beta);
 465   assert(!mparena_count(&mparena_global));
 466   printf("ok\n");
 467   return (0);
 468 }
 469
 470 #endif
 471
 472 /*----- That's all, folks -------------------------------------------------*/