5 * Elliptic curve information management
7 * (c) 2004 Straylight/Edgeware
10 /*----- Licensing notice --------------------------------------------------*
12 * This file is part of Catacomb.
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.
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.
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,
30 /*----- Header files ------------------------------------------------------*/
40 /*----- Main code ---------------------------------------------------------*/
42 /* --- @ec_curveparse@ --- *
44 * Arguments: @qd_parse *qd@ = parser context
46 * Returns: Elliptic curve pointer if OK, or null.
48 * Use: Parses an elliptic curve description, which has the form
50 * * a field description
52 * * `prime', `primeproj', `bin', or `binproj'
54 * * the %$a$% parameter
56 * * the %$b$% parameter
59 ec_curve *ec_curveparse(qd_parse *qd)
61 mp *a = MP_NEW, *b = MP_NEW;
65 if ((f = field_parse(qd)) == 0) goto fail;
67 switch (qd_enum(qd, "prime,primeproj,bin,binproj")) {
69 if (F_TYPE(f) != FTY_PRIME) {
70 qd->e = "field not prime";
74 if ((a = qd_getmp(qd)) == 0) goto fail;
76 if ((b = qd_getmp(qd)) == 0) goto fail;
77 c = ec_prime(f, a, b);
80 if (F_TYPE(f) != FTY_PRIME) {
81 qd->e = "field not prime";
85 if ((a = qd_getmp(qd)) == 0) goto fail;
87 if ((b = qd_getmp(qd)) == 0) goto fail;
88 c = ec_primeproj(f, a, b);
91 if (F_TYPE(f) != FTY_BINARY) {
92 qd->e = "field not binary";
96 if ((a = qd_getmp(qd)) == 0) goto fail;
98 if ((b = qd_getmp(qd)) == 0) goto fail;
102 if (F_TYPE(f) != FTY_BINARY) {
103 qd->e = "field not binary";
107 if ((a = qd_getmp(qd)) == 0) goto fail;
109 if ((b = qd_getmp(qd)) == 0) goto fail;
110 c = ec_binproj(f, a, b);
116 qd->e = "bad curve parameters";
130 /* --- @ec_ptparse@ --- *
132 * Arguments: @qd_parse *qd@ = parser context
133 * @ec *p@ = where to put the point
135 * Returns: The point address, or null.
137 * Use: Parses an elliptic curve point. This has the form
144 ec *ec_ptparse(qd_parse *qd, ec *p)
146 mp *x = MP_NEW, *y = MP_NEW;
148 if (qd_enum(qd, "inf") >= 0) {
152 if ((x = qd_getmp(qd)) == 0) goto fail;
154 if ((y = qd_getmp(qd)) == 0) goto fail;
167 /* --- @ec_infofromdata@ --- *
169 * Arguments: @ec_info *ei@ = where to write the information
170 * @ecdata *ed@ = raw data
174 * Use: Loads elliptic curve information about one of the standard
178 void ec_infofromdata(ec_info *ei, ecdata *ed)
184 f = field_prime(&ed->p);
185 ei->c = ec_primeproj(f, &ed->a, &ed->b);
188 f = field_niceprime(&ed->p);
189 ei->c = ec_primeproj(f, &ed->a, &ed->b);
192 f = field_binpoly(&ed->p);
193 ei->c = ec_binproj(f, &ed->a, &ed->b);
196 f = field_binnorm(&ed->p, &ed->beta);
197 ei->c = ec_binproj(f, &ed->a, &ed->b);
203 assert(f); assert(ei->c);
204 EC_CREATE(&ei->g); ei->g.x = &ed->gx; ei->g.y = &ed->gy; ei->g.z = 0;
205 ei->r = &ed->r; ei->h = &ed->h;
208 /* --- @ec_infoparse@ --- *
210 * Arguments: @qd_parse *qd@ = parser context
211 * @ec_info *ei@ = curve information block, currently
214 * Returns: Zero on success, nonzero on failure.
216 * Use: Parses an elliptic curve information string, and stores the
217 * information in @ei@. This is either the name of a standard
218 * curve, or it has the form
220 * * elliptic curve description
229 int ec_infoparse(qd_parse *qd, ec_info *ei)
235 mp *r = MP_NEW, *h = MP_NEW;
237 for (ee = ectab; ee->name; ee++) {
238 if (qd_enum(qd, ee->name) >= 0) {
239 ec_infofromdata(ei, ee->data);
244 if ((c = ec_curveparse(qd)) == 0) goto fail;
245 qd_delim(qd, ';'); if (!ec_ptparse(qd, &g)) goto fail;
246 qd_delim(qd, ':'); if ((r = qd_getmp(qd)) == 0) goto fail;
247 qd_delim(qd, '*'); if ((h = qd_getmp(qd)) == 0) goto fail;
248 ei->c = c; ei->g = g; ei->r = r; ei->h = h;
257 if (c) { f = c->f; ec_destroycurve(c); F_DESTROY(f); }
261 /* --- @ec_getinfo@ --- *
263 * Arguments: @ec_info *ei@ = where to write the information
264 * @const char *p@ = string describing a curve
266 * Returns: Null on success, or a pointer to an error message.
268 * Use: Parses out information about a curve. The string is either a
269 * standard curve name, or a curve info string.
272 const char *ec_getinfo(ec_info *ei, const char *p)
278 if (ec_infoparse(&qd, ei))
282 return ("junk found at end of string");
287 /* --- @ec_sameinfop@ --- *
289 * Arguments: @ec_info *ei, *ej@ = two elliptic curve parameter sets
291 * Returns: Nonzero if the curves are identical (not just isomorphic).
293 * Use: Checks for sameness of curve parameters.
296 int ec_sameinfop(ec_info *ei, ec_info *ej)
298 return (ec_samep(ei->c, ej->c) &&
299 MP_EQ(ei->r, ej->r) && MP_EQ(ei->h, ej->h) &&
300 EC_EQ(&ei->g, &ej->g));
303 /* --- @ec_freeinfo@ --- *
305 * Arguments: @ec_info *ei@ = elliptic curve information block to free
309 * Use: Frees the information block.
312 void ec_freeinfo(ec_info *ei)
319 f = ei->c->f; ec_destroycurve(ei->c); F_DESTROY(f);
322 /* --- @ec_checkinfo@ --- *
324 * Arguments: @const ec_info *ei@ = elliptic curve information block
326 * Returns: Null if OK, or pointer to error message.
328 * Use: Checks an elliptic curve according to the rules in SEC1.
331 static const char *gencheck(const ec_info *ei, grand *gr, mp *q)
342 /* --- Check %$G \in E$% --- */
344 if (EC_ATINF(&ei->g)) return ("generator at infinity");
345 if (ec_check(c, &ei->g)) return ("generator not on curve");
347 /* --- Check %$r$% is prime --- */
349 if (!pgen_primep(ei->r, gr)) return ("generator order not prime");
351 /* --- Check that the cofactor is correct --- *
353 * Let %$q$% be the size of the field, and let %$n = h r = \#E(\gf{q})$% be
354 * the number of %$\gf{q}$%-rational points on our curve. Hasse's theorem
357 * %$|q + 1 - n| \le 2\sqrt{q}$%
359 * or, if we square both sides,
361 * %$(q + 1 - n)^2 \le 4 q$%.
363 * We'd like the cofactor to be uniquely determined by this equation, which
364 * is possible as long as it's not too big. (If it is, we have to mess
365 * about with Weil pairings, which is no fun.) For this, we need the
366 * following inequalities:
368 * * %$A = (q + 1 - n)^2 \le 4 q$% (both lower and upper bounds from
371 * * %$B = (q + 1 - n - r)^2 > 4 q$% (check %$h - 1$% isn't possible);
374 * * %$C = (q + 1 - n + r)^2 > 4 q$% (check %$h + 1$% isn't possible).
378 qq = mp_add(MP_NEW, q, MP_ONE);
379 nn = mp_mul(MP_NEW, ei->r, ei->h);
380 nn = mp_sub(nn, qq, nn);
381 qq = mp_lsl(qq, q, 2);
383 y = mp_sqr(MP_NEW, nn);
384 if (MP_CMP(y, >, qq)) rc = 0;
386 x = mp_sub(MP_NEW, nn, ei->r);
388 if (MP_CMP(y, <=, qq)) rc = 0;
390 x = mp_add(x, nn, ei->r);
392 if (MP_CMP(y, <=, qq)) rc = 0;
398 if (!rc) return ("incorrect or ambiguous cofactor");
400 /* --- Check %$n G = O$% --- */
403 ec_mul(c, &p, &ei->g, ei->r);
406 if (!rc) return ("incorrect group order");
408 /* --- Check %$q^B \not\equiv 1 \pmod{r}$% for %$1 \le B < 20$% --- *
410 * Actually, give up if %$q^B \ge 2^{2000}$% because that's probably
411 * good enough for jazz.
415 mp_div(0, &x, q, ei->r);
416 n = mp_bits(ei->r) - 1;
417 for (i = 0, j = n; i < 20; i++, j += n) {
420 if (MP_EQ(x, MP_ONE)) {
422 return("curve embedding degree too low");
424 x = mp_mul(x, x, f->m);
425 mp_div(0, &x, x, ei->r);
434 static int primeeltp(mp *x, field *f)
435 { return (!MP_NEGP(x) && MP_CMP(x, <, f->m)); }
437 static const char *primecheck(const ec_info *ei, grand *gr)
445 /* --- Check %$p$% is an odd prime --- */
447 if (!pgen_primep(f->m, gr)) return ("p not prime");
449 /* --- Check %$a$%, %$b$%, %$G_x$% and %$G_y$% are in %$[0, p)$% --- */
451 if (!primeeltp(c->a, f)) return ("a out of range");
452 if (!primeeltp(c->b, f)) return ("b out of range");
453 if (!primeeltp(ei->g.x, f)) return ("G_x out of range");
454 if (!primeeltp(ei->g.x, f)) return ("G_y out of range");
456 /* --- Check %$4 a^3 + 27 b^2 \not\equiv 0 \pmod{p}$% --- */
458 x = F_SQR(f, MP_NEW, c->a);
459 x = F_MUL(f, x, x, c->a);
461 y = F_SQR(f, MP_NEW, c->b);
465 x = F_ADD(f, x, x, y);
469 if (rc) return ("not an elliptic curve");
471 /* --- Now do the general checks --- */
473 err = gencheck(ei, gr, f->m);
477 static const char *bincheck(const ec_info *ei, grand *gr)
485 /* --- Check that %$m$% is prime --- */
487 x = mp_fromuint(MP_NEW, f->nbits);
488 rc = pfilt_smallfactor(x);
490 if (rc != PGEN_DONE) return ("degree not prime");
492 /* --- Check that %$p$% is irreducible --- */
494 if (!gf_irreduciblep(f->m)) return ("p not irreducible");
496 /* --- Check that %$a, b, G_x, G_y$% have degree less than %$p$% --- */
498 if (mp_bits(c->a) > f->nbits) return ("a out of range");
499 if (mp_bits(c->b) > f->nbits) return ("a out of range");
500 if (mp_bits(ei->g.x) > f->nbits) return ("G_x out of range");
501 if (mp_bits(ei->g.y) > f->nbits) return ("G_y out of range");
503 /* --- Check that %$b \ne 0$% --- */
505 if (F_ZEROP(f, c->b)) return ("b is zero");
507 /* --- Now do the general checks --- */
509 x = mp_lsl(MP_NEW, MP_ONE, f->nbits);
510 err = gencheck(ei, gr, x);
515 const char *ec_checkinfo(const ec_info *ei, grand *gr)
517 switch (F_TYPE(ei->c->f)) {
518 case FTY_PRIME: return (primecheck(ei, gr)); break;
519 case FTY_BINARY: return (bincheck(ei, gr)); break;
521 return ("unknown curve type");
524 /*----- Test rig ----------------------------------------------------------*/
530 int main(int argc, char *argv[])
538 gr = fibrand_create(0);
540 for (i = 1; i < argc; i++) {
542 if ((e = ec_getinfo(&ei, argv[i])) != 0)
543 fprintf(stderr, "bad curve spec `%s': %s\n", argv[i], e);
545 e = ec_checkinfo(&ei, gr);
548 printf("OK %s\n", argv[i]);
550 printf("BAD %s: %s\n", argv[i], e);
554 assert(mparena_count(MPARENA_GLOBAL) == 0);
557 fputs("checking standard curves:", stdout);
559 for (ee = ectab; ee->name; ee++) {
561 ec_infofromdata(&ei, ee->data);
562 e = ec_checkinfo(&ei, gr);
565 printf(" [%s fails: %s]", ee->name, e);
568 printf(" %s", ee->name);
570 assert(mparena_count(MPARENA_GLOBAL) == 0);
572 fputs(ok ? " ok\n" : " failed\n", stdout);
574 gr->ops->destroy(gr);
580 /*----- That's all, folks -------------------------------------------------*/