b0ab12e6 |
1 | /* -*-c-*- |
2 | * |
34e4f738 |
3 | * $Id: ec.c,v 1.8 2004/04/01 12:50:09 mdw Exp $ |
b0ab12e6 |
4 | * |
5 | * Elliptic curve definitions |
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.c,v $ |
34e4f738 |
33 | * Revision 1.8 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 | * |
432c4e18 |
40 | * Revision 1.7 2004/03/27 17:54:11 mdw |
41 | * Standard curves and curve checking. |
42 | * |
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43 | * Revision 1.6 2004/03/23 15:19:32 mdw |
44 | * Test elliptic curves more thoroughly. |
45 | * |
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46 | * Revision 1.5 2004/03/21 22:52:06 mdw |
47 | * Merge and close elliptic curve branch. |
48 | * |
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49 | * Revision 1.4.4.2 2004/03/20 00:13:31 mdw |
50 | * Projective coordinates for prime curves |
51 | * |
dbfee00a |
52 | * Revision 1.4.4.1 2003/06/10 13:43:53 mdw |
53 | * Simple (non-projective) curves over prime fields now seem to work. |
54 | * |
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55 | * Revision 1.4 2003/05/15 23:25:59 mdw |
56 | * Make elliptic curve stuff build. |
57 | * |
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58 | * Revision 1.3 2002/01/13 13:48:44 mdw |
59 | * Further progress. |
60 | * |
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61 | * Revision 1.2 2001/05/07 17:29:44 mdw |
62 | * Treat projective coordinates as an internal representation. Various |
63 | * minor interface changes. |
64 | * |
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65 | * Revision 1.1 2001/04/29 18:12:33 mdw |
66 | * Prototype version. |
67 | * |
68 | */ |
69 | |
70 | /*----- Header files ------------------------------------------------------*/ |
71 | |
72 | #include "ec.h" |
73 | |
74 | /*----- Trivial wrappers --------------------------------------------------*/ |
75 | |
34e4f738 |
76 | /* --- @ec_samep@ --- * |
77 | * |
78 | * Arguments: @ec_curve *c, *d@ = two elliptic curves |
79 | * |
80 | * Returns: Nonzero if the curves are identical (not just isomorphic). |
81 | * |
82 | * Use: Checks for sameness of curves. This function does the full |
83 | * check, not just the curve-type-specific check done by the |
84 | * @sampep@ field operation. |
85 | */ |
86 | |
87 | int ec_samep(ec_curve *c, ec_curve *d) |
88 | { |
89 | return (field_samep(c->f, d->f) && c->ops == d->ops && EC_SAMEP(c, d)); |
90 | } |
91 | |
b0ab12e6 |
92 | /* --- @ec_create@ --- * |
93 | * |
94 | * Arguments: @ec *p@ = pointer to an elliptic-curve point |
95 | * |
41cb1beb |
96 | * Returns: The argument @p@. |
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97 | * |
98 | * Use: Initializes a new point. The initial value is the additive |
99 | * identity (which is universal for all curves). |
100 | */ |
101 | |
41cb1beb |
102 | ec *ec_create(ec *p) { EC_CREATE(p); return (p); } |
b0ab12e6 |
103 | |
104 | /* --- @ec_destroy@ --- * |
105 | * |
106 | * Arguments: @ec *p@ = pointer to an elliptic-curve point |
107 | * |
108 | * Returns: --- |
109 | * |
110 | * Use: Destroys a point, making it invalid. |
111 | */ |
112 | |
113 | void ec_destroy(ec *p) { EC_DESTROY(p); } |
114 | |
115 | /* --- @ec_atinf@ --- * |
116 | * |
117 | * Arguments: @const ec *p@ = pointer to a point |
118 | * |
119 | * Returns: Nonzero if %$p = O$% is the point at infinity, zero |
120 | * otherwise. |
121 | */ |
122 | |
123 | int ec_atinf(const ec *p) { return (EC_ATINF(p)); } |
124 | |
125 | /* --- @ec_setinf@ --- * |
126 | * |
127 | * Arguments: @ec *p@ = pointer to a point |
128 | * |
41cb1beb |
129 | * Returns: The argument @p@. |
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130 | * |
131 | * Use: Sets the given point to be the point %$O$% at infinity. |
132 | */ |
133 | |
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134 | ec *ec_setinf(ec *p) { EC_SETINF(p); return (p); } |
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135 | |
136 | /* --- @ec_copy@ --- * |
137 | * |
138 | * Arguments: @ec *d@ = pointer to destination point |
139 | * @const ec *p@ = pointer to source point |
140 | * |
41cb1beb |
141 | * Returns: The destination @d@. |
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142 | * |
143 | * Use: Creates a copy of an elliptic curve point. |
144 | */ |
145 | |
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146 | ec *ec_copy(ec *d, const ec *p) { EC_COPY(d, p); return (d); } |
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147 | |
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148 | /* --- @ec_eq@ --- * |
149 | * |
150 | * Arguments: @const ec *p, *q@ = two points |
151 | * |
152 | * Returns: Nonzero if the points are equal. Compares external-format |
153 | * points. |
154 | */ |
155 | |
156 | int ec_eq(const ec *p, const ec *q) { return (EC_EQ(p, q)); } |
157 | |
41a324a7 |
158 | /*----- Standard curve operations -----------------------------------------*/ |
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159 | |
34e4f738 |
160 | /* --- @ec_stdsamep@ --- * |
161 | * |
162 | * Arguments: @ec_curve *c, *d@ = two elliptic curves |
163 | * |
164 | * Returns: Nonzero if the curves are identical (not just isomorphic). |
165 | * |
166 | * Use: Simple sameness check on @a@ and @b@ curve members. |
167 | */ |
168 | |
169 | int ec_stdsamep(ec_curve *c, ec_curve *d) |
170 | { |
171 | return (MP_EQ(c->a, d->a) && MP_EQ(c->b, d->b)); |
172 | } |
173 | |
8823192f |
174 | /* --- @ec_idin@, @ec_idout@, @ec_idfix@ --- * |
b0ab12e6 |
175 | * |
176 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
41a324a7 |
177 | * @ec *d@ = pointer to the destination |
178 | * @const ec *p@ = pointer to a source point |
b0ab12e6 |
179 | * |
41a324a7 |
180 | * Returns: The destination @d@. |
b0ab12e6 |
181 | * |
41a324a7 |
182 | * Use: An identity operation if your curve has no internal |
183 | * representation. (The field internal representation is still |
184 | * used.) |
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185 | */ |
186 | |
41a324a7 |
187 | ec *ec_idin(ec_curve *c, ec *d, const ec *p) |
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188 | { |
189 | if (EC_ATINF(p)) |
190 | EC_SETINF(d); |
191 | else { |
192 | field *f = c->f; |
193 | d->x = F_IN(f, d->x, p->x); |
194 | d->y = F_IN(f, d->y, p->y); |
41a324a7 |
195 | mp_drop(d->z); d->z = 0; |
196 | } |
197 | return (d); |
198 | } |
199 | |
200 | ec *ec_idout(ec_curve *c, ec *d, const ec *p) |
201 | { |
202 | if (EC_ATINF(p)) |
203 | EC_SETINF(d); |
204 | else { |
205 | field *f = c->f; |
206 | d->x = F_OUT(f, d->x, p->x); |
207 | d->y = F_OUT(f, d->y, p->y); |
208 | mp_drop(d->z); d->z = 0; |
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209 | } |
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210 | return (d); |
b0ab12e6 |
211 | } |
212 | |
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213 | ec *ec_idfix(ec_curve *c, ec *d, const ec *p) |
214 | { |
215 | EC_COPY(d, p); |
216 | return (d); |
217 | } |
218 | |
41a324a7 |
219 | /* --- @ec_projin@, @ec_projout@ --- * |
b0ab12e6 |
220 | * |
221 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
41a324a7 |
222 | * @ec *d@ = pointer to the destination |
223 | * @const ec *p@ = pointer to a source point |
b0ab12e6 |
224 | * |
41a324a7 |
225 | * Returns: The destination @d@. |
b0ab12e6 |
226 | * |
41a324a7 |
227 | * Use: Conversion functions if your curve operations use a |
228 | * projective representation. |
b0ab12e6 |
229 | */ |
230 | |
41a324a7 |
231 | ec *ec_projin(ec_curve *c, ec *d, const ec *p) |
232 | { |
233 | if (EC_ATINF(p)) |
234 | EC_SETINF(d); |
235 | else { |
236 | field *f = c->f; |
237 | d->x = F_IN(f, d->x, p->x); |
238 | d->y = F_IN(f, d->y, p->y); |
239 | mp_drop(d->z); d->z = MP_COPY(f->one); |
240 | } |
241 | return (d); |
242 | } |
243 | |
244 | ec *ec_projout(ec_curve *c, ec *d, const ec *p) |
b0ab12e6 |
245 | { |
246 | if (EC_ATINF(p)) |
247 | EC_SETINF(d); |
248 | else { |
8823192f |
249 | mp *x, *y, *z, *zz; |
b0ab12e6 |
250 | field *f = c->f; |
251 | z = F_INV(f, MP_NEW, p->z); |
8823192f |
252 | zz = F_SQR(f, MP_NEW, z); |
253 | z = F_MUL(f, z, zz, z); |
254 | x = F_MUL(f, d->x, p->x, zz); |
b0ab12e6 |
255 | y = F_MUL(f, d->y, p->y, z); |
256 | mp_drop(z); |
8823192f |
257 | mp_drop(zz); |
b0ab12e6 |
258 | mp_drop(d->z); |
259 | d->x = F_OUT(f, x, x); |
260 | d->y = F_OUT(f, y, y); |
261 | d->z = 0; |
262 | } |
41a324a7 |
263 | return (d); |
b0ab12e6 |
264 | } |
265 | |
8823192f |
266 | ec *ec_projfix(ec_curve *c, ec *d, const ec *p) |
267 | { |
268 | if (EC_ATINF(p)) |
269 | EC_SETINF(d); |
270 | else if (d->z == c->f->one) |
271 | EC_COPY(d, p); |
272 | else { |
273 | mp *z, *zz; |
274 | field *f = c->f; |
275 | z = F_INV(f, MP_NEW, p->z); |
276 | zz = F_SQR(f, MP_NEW, z); |
277 | z = F_MUL(f, z, zz, z); |
278 | d->x = F_MUL(f, d->x, p->x, zz); |
279 | d->y = F_MUL(f, d->y, p->y, z); |
280 | mp_drop(z); |
281 | mp_drop(zz); |
282 | mp_drop(d->z); |
283 | d->z = MP_COPY(f->one); |
284 | } |
285 | return (d); |
286 | } |
287 | |
b085fd91 |
288 | /* --- @ec_stdsub@ --- * |
289 | * |
290 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
291 | * @ec *d@ = pointer to the destination |
41cb1beb |
292 | * @const ec *p, *q@ = the operand points |
b085fd91 |
293 | * |
294 | * Returns: The destination @d@. |
295 | * |
296 | * Use: Standard point subtraction operation, in terms of negation |
297 | * and addition. This isn't as efficient as a ready-made |
298 | * subtraction operator. |
299 | */ |
300 | |
41cb1beb |
301 | ec *ec_stdsub(ec_curve *c, ec *d, const ec *p, const ec *q) |
b085fd91 |
302 | { |
303 | ec t = EC_INIT; |
41cb1beb |
304 | EC_NEG(c, &t, q); |
8823192f |
305 | EC_FIX(c, &t, &t); |
41cb1beb |
306 | EC_ADD(c, d, p, &t); |
b085fd91 |
307 | EC_DESTROY(&t); |
308 | return (d); |
309 | } |
310 | |
41cb1beb |
311 | /*----- Creating curves ---------------------------------------------------*/ |
312 | |
313 | /* --- @ec_destroycurve@ --- * |
314 | * |
315 | * Arguments: @ec_curve *c@ = pointer to an ellptic curve |
316 | * |
317 | * Returns: --- |
318 | * |
319 | * Use: Destroys a description of an elliptic curve. |
320 | */ |
321 | |
322 | void ec_destroycurve(ec_curve *c) { c->ops->destroy(c); } |
323 | |
41a324a7 |
324 | /*----- Real arithmetic ---------------------------------------------------*/ |
325 | |
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326 | /* --- @ec_find@ --- * |
327 | * |
328 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
329 | * @ec *d@ = pointer to the destination point |
330 | * @mp *x@ = a possible x-coordinate |
331 | * |
332 | * Returns: Zero if OK, nonzero if there isn't a point there. |
333 | * |
334 | * Use: Finds a point on an elliptic curve with a given x-coordinate. |
335 | */ |
336 | |
41a324a7 |
337 | ec *ec_find(ec_curve *c, ec *d, mp *x) |
b0ab12e6 |
338 | { |
b0ab12e6 |
339 | x = F_IN(c->f, MP_NEW, x); |
41a324a7 |
340 | if ((d = EC_FIND(c, d, x)) != 0) |
341 | EC_OUT(c, d, d); |
8823192f |
342 | MP_DROP(x); |
41a324a7 |
343 | return (d); |
b0ab12e6 |
344 | } |
345 | |
dbfee00a |
346 | /* --- @ec_neg@ --- * |
347 | * |
348 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
349 | * @ec *d@ = pointer to the destination point |
350 | * @const ec *p@ = pointer to the operand point |
351 | * |
352 | * Returns: The destination point. |
353 | * |
354 | * Use: Computes the negation of the given point. |
355 | */ |
356 | |
357 | ec *ec_neg(ec_curve *c, ec *d, const ec *p) |
358 | { |
359 | EC_IN(c, d, p); |
360 | EC_NEG(c, d, d); |
361 | return (EC_OUT(c, d, d)); |
362 | } |
363 | |
b0ab12e6 |
364 | /* --- @ec_add@ --- * |
365 | * |
366 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
367 | * @ec *d@ = pointer to the destination point |
368 | * @const ec *p, *q@ = pointers to the operand points |
369 | * |
370 | * Returns: --- |
371 | * |
372 | * Use: Adds two points on an elliptic curve. |
373 | */ |
374 | |
41a324a7 |
375 | ec *ec_add(ec_curve *c, ec *d, const ec *p, const ec *q) |
b0ab12e6 |
376 | { |
377 | ec pp = EC_INIT, qq = EC_INIT; |
41a324a7 |
378 | EC_IN(c, &pp, p); |
379 | EC_IN(c, &qq, q); |
b0ab12e6 |
380 | EC_ADD(c, d, &pp, &qq); |
41a324a7 |
381 | EC_OUT(c, d, d); |
b0ab12e6 |
382 | EC_DESTROY(&pp); |
383 | EC_DESTROY(&qq); |
41a324a7 |
384 | return (d); |
b0ab12e6 |
385 | } |
386 | |
dbfee00a |
387 | /* --- @ec_sub@ --- * |
388 | * |
389 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
390 | * @ec *d@ = pointer to the destination point |
391 | * @const ec *p, *q@ = pointers to the operand points |
392 | * |
393 | * Returns: The destination @d@. |
394 | * |
395 | * Use: Subtracts one point from another on an elliptic curve. |
396 | */ |
397 | |
398 | ec *ec_sub(ec_curve *c, ec *d, const ec *p, const ec *q) |
399 | { |
432c4e18 |
400 | ec pp = EC_INIT, qq = EC_INIT; |
dbfee00a |
401 | EC_IN(c, &pp, p); |
402 | EC_IN(c, &qq, q); |
bc985cef |
403 | EC_SUB(c, d, &pp, &qq); |
dbfee00a |
404 | EC_OUT(c, d, d); |
405 | EC_DESTROY(&pp); |
406 | EC_DESTROY(&qq); |
407 | return (d); |
408 | } |
409 | |
b0ab12e6 |
410 | /* --- @ec_dbl@ --- * |
411 | * |
412 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
413 | * @ec *d@ = pointer to the destination point |
414 | * @const ec *p@ = pointer to the operand point |
415 | * |
416 | * Returns: --- |
417 | * |
418 | * Use: Doubles a point on an elliptic curve. |
419 | */ |
420 | |
41a324a7 |
421 | ec *ec_dbl(ec_curve *c, ec *d, const ec *p) |
b0ab12e6 |
422 | { |
41a324a7 |
423 | EC_IN(c, d, p); |
b0ab12e6 |
424 | EC_DBL(c, d, d); |
41a324a7 |
425 | return (EC_OUT(c, d, d)); |
b0ab12e6 |
426 | } |
427 | |
8823192f |
428 | /* --- @ec_check@ --- * |
429 | * |
430 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
431 | * @const ec *p@ = pointer to the point |
432 | * |
433 | * Returns: Zero if OK, nonzero if this is an invalid point. |
434 | * |
435 | * Use: Checks that a point is actually on an elliptic curve. |
436 | */ |
437 | |
438 | int ec_check(ec_curve *c, const ec *p) |
439 | { |
440 | ec t = EC_INIT; |
441 | int rc; |
442 | |
443 | if (EC_ATINF(p)) |
444 | return (0); |
445 | EC_IN(c, &t, p); |
446 | rc = EC_CHECK(c, &t); |
447 | EC_DESTROY(&t); |
448 | return (rc); |
449 | } |
450 | |
bc985cef |
451 | /* --- @ec_rand@ --- * |
452 | * |
453 | * Arguments: @ec_curve *c@ = pointer to an elliptic curve |
454 | * @ec *d@ = pointer to the destination point |
455 | * @grand *r@ = random number source |
456 | * |
457 | * Returns: The destination @d@. |
458 | * |
459 | * Use: Finds a random point on the given curve. |
460 | */ |
461 | |
462 | ec *ec_rand(ec_curve *c, ec *d, grand *r) |
463 | { |
464 | mp *x = MP_NEW; |
465 | do x = F_RAND(c->f, x, r); while (!EC_FIND(c, d, x)); |
466 | mp_drop(x); |
467 | if (grand_range(r, 2)) EC_NEG(c, d, d); |
468 | return (EC_OUT(c, d, d)); |
469 | } |
470 | |
b0ab12e6 |
471 | /*----- That's all, folks -------------------------------------------------*/ |