1 /*
2 * Copyright 2002-2019 The OpenSSL Project Authors. All Rights Reserved.
3 * Copyright (c) 2002, Oracle and/or its affiliates. All rights reserved
4 *
5 * Licensed under the OpenSSL license (the "License"). You may not use
6 * this file except in compliance with the License. You can obtain a copy
7 * in the file LICENSE in the source distribution or at
8 * https://www.openssl.org/source/license.html
9 */
10
11 #include <openssl/err.h>
12
13 #include "crypto/bn.h"
14 #include "ec_local.h"
15
16 #ifndef OPENSSL_NO_EC2M
17
18 /*
19 * Initialize a GF(2^m)-based EC_GROUP structure. Note that all other members
20 * are handled by EC_GROUP_new.
21 */
ec_GF2m_simple_group_init(EC_GROUP * group)22 int ec_GF2m_simple_group_init(EC_GROUP *group)
23 {
24 group->field = BN_new();
25 group->a = BN_new();
26 group->b = BN_new();
27
28 if (group->field == NULL || group->a == NULL || group->b == NULL) {
29 BN_free(group->field);
30 BN_free(group->a);
31 BN_free(group->b);
32 return 0;
33 }
34 return 1;
35 }
36
37 /*
38 * Free a GF(2^m)-based EC_GROUP structure. Note that all other members are
39 * handled by EC_GROUP_free.
40 */
ec_GF2m_simple_group_finish(EC_GROUP * group)41 void ec_GF2m_simple_group_finish(EC_GROUP *group)
42 {
43 BN_free(group->field);
44 BN_free(group->a);
45 BN_free(group->b);
46 }
47
48 /*
49 * Clear and free a GF(2^m)-based EC_GROUP structure. Note that all other
50 * members are handled by EC_GROUP_clear_free.
51 */
ec_GF2m_simple_group_clear_finish(EC_GROUP * group)52 void ec_GF2m_simple_group_clear_finish(EC_GROUP *group)
53 {
54 BN_clear_free(group->field);
55 BN_clear_free(group->a);
56 BN_clear_free(group->b);
57 group->poly[0] = 0;
58 group->poly[1] = 0;
59 group->poly[2] = 0;
60 group->poly[3] = 0;
61 group->poly[4] = 0;
62 group->poly[5] = -1;
63 }
64
65 /*
66 * Copy a GF(2^m)-based EC_GROUP structure. Note that all other members are
67 * handled by EC_GROUP_copy.
68 */
ec_GF2m_simple_group_copy(EC_GROUP * dest,const EC_GROUP * src)69 int ec_GF2m_simple_group_copy(EC_GROUP *dest, const EC_GROUP *src)
70 {
71 if (!BN_copy(dest->field, src->field))
72 return 0;
73 if (!BN_copy(dest->a, src->a))
74 return 0;
75 if (!BN_copy(dest->b, src->b))
76 return 0;
77 dest->poly[0] = src->poly[0];
78 dest->poly[1] = src->poly[1];
79 dest->poly[2] = src->poly[2];
80 dest->poly[3] = src->poly[3];
81 dest->poly[4] = src->poly[4];
82 dest->poly[5] = src->poly[5];
83 if (bn_wexpand(dest->a, (int)(dest->poly[0] + BN_BITS2 - 1) / BN_BITS2) ==
84 NULL)
85 return 0;
86 if (bn_wexpand(dest->b, (int)(dest->poly[0] + BN_BITS2 - 1) / BN_BITS2) ==
87 NULL)
88 return 0;
89 bn_set_all_zero(dest->a);
90 bn_set_all_zero(dest->b);
91 return 1;
92 }
93
94 /* Set the curve parameters of an EC_GROUP structure. */
ec_GF2m_simple_group_set_curve(EC_GROUP * group,const BIGNUM * p,const BIGNUM * a,const BIGNUM * b,BN_CTX * ctx)95 int ec_GF2m_simple_group_set_curve(EC_GROUP *group,
96 const BIGNUM *p, const BIGNUM *a,
97 const BIGNUM *b, BN_CTX *ctx)
98 {
99 int ret = 0, i;
100
101 /* group->field */
102 if (!BN_copy(group->field, p))
103 goto err;
104 i = BN_GF2m_poly2arr(group->field, group->poly, 6) - 1;
105 if ((i != 5) && (i != 3)) {
106 ECerr(EC_F_EC_GF2M_SIMPLE_GROUP_SET_CURVE, EC_R_UNSUPPORTED_FIELD);
107 goto err;
108 }
109
110 /* group->a */
111 if (!BN_GF2m_mod_arr(group->a, a, group->poly))
112 goto err;
113 if (bn_wexpand(group->a, (int)(group->poly[0] + BN_BITS2 - 1) / BN_BITS2)
114 == NULL)
115 goto err;
116 bn_set_all_zero(group->a);
117
118 /* group->b */
119 if (!BN_GF2m_mod_arr(group->b, b, group->poly))
120 goto err;
121 if (bn_wexpand(group->b, (int)(group->poly[0] + BN_BITS2 - 1) / BN_BITS2)
122 == NULL)
123 goto err;
124 bn_set_all_zero(group->b);
125
126 ret = 1;
127 err:
128 return ret;
129 }
130
131 /*
132 * Get the curve parameters of an EC_GROUP structure. If p, a, or b are NULL
133 * then there values will not be set but the method will return with success.
134 */
ec_GF2m_simple_group_get_curve(const EC_GROUP * group,BIGNUM * p,BIGNUM * a,BIGNUM * b,BN_CTX * ctx)135 int ec_GF2m_simple_group_get_curve(const EC_GROUP *group, BIGNUM *p,
136 BIGNUM *a, BIGNUM *b, BN_CTX *ctx)
137 {
138 int ret = 0;
139
140 if (p != NULL) {
141 if (!BN_copy(p, group->field))
142 return 0;
143 }
144
145 if (a != NULL) {
146 if (!BN_copy(a, group->a))
147 goto err;
148 }
149
150 if (b != NULL) {
151 if (!BN_copy(b, group->b))
152 goto err;
153 }
154
155 ret = 1;
156
157 err:
158 return ret;
159 }
160
161 /*
162 * Gets the degree of the field. For a curve over GF(2^m) this is the value
163 * m.
164 */
ec_GF2m_simple_group_get_degree(const EC_GROUP * group)165 int ec_GF2m_simple_group_get_degree(const EC_GROUP *group)
166 {
167 return BN_num_bits(group->field) - 1;
168 }
169
170 /*
171 * Checks the discriminant of the curve. y^2 + x*y = x^3 + a*x^2 + b is an
172 * elliptic curve <=> b != 0 (mod p)
173 */
ec_GF2m_simple_group_check_discriminant(const EC_GROUP * group,BN_CTX * ctx)174 int ec_GF2m_simple_group_check_discriminant(const EC_GROUP *group,
175 BN_CTX *ctx)
176 {
177 int ret = 0;
178 BIGNUM *b;
179 BN_CTX *new_ctx = NULL;
180
181 if (ctx == NULL) {
182 ctx = new_ctx = BN_CTX_new();
183 if (ctx == NULL) {
184 ECerr(EC_F_EC_GF2M_SIMPLE_GROUP_CHECK_DISCRIMINANT,
185 ERR_R_MALLOC_FAILURE);
186 goto err;
187 }
188 }
189 BN_CTX_start(ctx);
190 b = BN_CTX_get(ctx);
191 if (b == NULL)
192 goto err;
193
194 if (!BN_GF2m_mod_arr(b, group->b, group->poly))
195 goto err;
196
197 /*
198 * check the discriminant: y^2 + x*y = x^3 + a*x^2 + b is an elliptic
199 * curve <=> b != 0 (mod p)
200 */
201 if (BN_is_zero(b))
202 goto err;
203
204 ret = 1;
205
206 err:
207 BN_CTX_end(ctx);
208 BN_CTX_free(new_ctx);
209 return ret;
210 }
211
212 /* Initializes an EC_POINT. */
ec_GF2m_simple_point_init(EC_POINT * point)213 int ec_GF2m_simple_point_init(EC_POINT *point)
214 {
215 point->X = BN_new();
216 point->Y = BN_new();
217 point->Z = BN_new();
218
219 if (point->X == NULL || point->Y == NULL || point->Z == NULL) {
220 BN_free(point->X);
221 BN_free(point->Y);
222 BN_free(point->Z);
223 return 0;
224 }
225 return 1;
226 }
227
228 /* Frees an EC_POINT. */
ec_GF2m_simple_point_finish(EC_POINT * point)229 void ec_GF2m_simple_point_finish(EC_POINT *point)
230 {
231 BN_free(point->X);
232 BN_free(point->Y);
233 BN_free(point->Z);
234 }
235
236 /* Clears and frees an EC_POINT. */
ec_GF2m_simple_point_clear_finish(EC_POINT * point)237 void ec_GF2m_simple_point_clear_finish(EC_POINT *point)
238 {
239 BN_clear_free(point->X);
240 BN_clear_free(point->Y);
241 BN_clear_free(point->Z);
242 point->Z_is_one = 0;
243 }
244
245 /*
246 * Copy the contents of one EC_POINT into another. Assumes dest is
247 * initialized.
248 */
ec_GF2m_simple_point_copy(EC_POINT * dest,const EC_POINT * src)249 int ec_GF2m_simple_point_copy(EC_POINT *dest, const EC_POINT *src)
250 {
251 if (!BN_copy(dest->X, src->X))
252 return 0;
253 if (!BN_copy(dest->Y, src->Y))
254 return 0;
255 if (!BN_copy(dest->Z, src->Z))
256 return 0;
257 dest->Z_is_one = src->Z_is_one;
258 dest->curve_name = src->curve_name;
259
260 return 1;
261 }
262
263 /*
264 * Set an EC_POINT to the point at infinity. A point at infinity is
265 * represented by having Z=0.
266 */
ec_GF2m_simple_point_set_to_infinity(const EC_GROUP * group,EC_POINT * point)267 int ec_GF2m_simple_point_set_to_infinity(const EC_GROUP *group,
268 EC_POINT *point)
269 {
270 point->Z_is_one = 0;
271 BN_zero(point->Z);
272 return 1;
273 }
274
275 /*
276 * Set the coordinates of an EC_POINT using affine coordinates. Note that
277 * the simple implementation only uses affine coordinates.
278 */
ec_GF2m_simple_point_set_affine_coordinates(const EC_GROUP * group,EC_POINT * point,const BIGNUM * x,const BIGNUM * y,BN_CTX * ctx)279 int ec_GF2m_simple_point_set_affine_coordinates(const EC_GROUP *group,
280 EC_POINT *point,
281 const BIGNUM *x,
282 const BIGNUM *y, BN_CTX *ctx)
283 {
284 int ret = 0;
285 if (x == NULL || y == NULL) {
286 ECerr(EC_F_EC_GF2M_SIMPLE_POINT_SET_AFFINE_COORDINATES,
287 ERR_R_PASSED_NULL_PARAMETER);
288 return 0;
289 }
290
291 if (!BN_copy(point->X, x))
292 goto err;
293 BN_set_negative(point->X, 0);
294 if (!BN_copy(point->Y, y))
295 goto err;
296 BN_set_negative(point->Y, 0);
297 if (!BN_copy(point->Z, BN_value_one()))
298 goto err;
299 BN_set_negative(point->Z, 0);
300 point->Z_is_one = 1;
301 ret = 1;
302
303 err:
304 return ret;
305 }
306
307 /*
308 * Gets the affine coordinates of an EC_POINT. Note that the simple
309 * implementation only uses affine coordinates.
310 */
ec_GF2m_simple_point_get_affine_coordinates(const EC_GROUP * group,const EC_POINT * point,BIGNUM * x,BIGNUM * y,BN_CTX * ctx)311 int ec_GF2m_simple_point_get_affine_coordinates(const EC_GROUP *group,
312 const EC_POINT *point,
313 BIGNUM *x, BIGNUM *y,
314 BN_CTX *ctx)
315 {
316 int ret = 0;
317
318 if (EC_POINT_is_at_infinity(group, point)) {
319 ECerr(EC_F_EC_GF2M_SIMPLE_POINT_GET_AFFINE_COORDINATES,
320 EC_R_POINT_AT_INFINITY);
321 return 0;
322 }
323
324 if (BN_cmp(point->Z, BN_value_one())) {
325 ECerr(EC_F_EC_GF2M_SIMPLE_POINT_GET_AFFINE_COORDINATES,
326 ERR_R_SHOULD_NOT_HAVE_BEEN_CALLED);
327 return 0;
328 }
329 if (x != NULL) {
330 if (!BN_copy(x, point->X))
331 goto err;
332 BN_set_negative(x, 0);
333 }
334 if (y != NULL) {
335 if (!BN_copy(y, point->Y))
336 goto err;
337 BN_set_negative(y, 0);
338 }
339 ret = 1;
340
341 err:
342 return ret;
343 }
344
345 /*
346 * Computes a + b and stores the result in r. r could be a or b, a could be
347 * b. Uses algorithm A.10.2 of IEEE P1363.
348 */
ec_GF2m_simple_add(const EC_GROUP * group,EC_POINT * r,const EC_POINT * a,const EC_POINT * b,BN_CTX * ctx)349 int ec_GF2m_simple_add(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a,
350 const EC_POINT *b, BN_CTX *ctx)
351 {
352 BN_CTX *new_ctx = NULL;
353 BIGNUM *x0, *y0, *x1, *y1, *x2, *y2, *s, *t;
354 int ret = 0;
355
356 if (EC_POINT_is_at_infinity(group, a)) {
357 if (!EC_POINT_copy(r, b))
358 return 0;
359 return 1;
360 }
361
362 if (EC_POINT_is_at_infinity(group, b)) {
363 if (!EC_POINT_copy(r, a))
364 return 0;
365 return 1;
366 }
367
368 if (ctx == NULL) {
369 ctx = new_ctx = BN_CTX_new();
370 if (ctx == NULL)
371 return 0;
372 }
373
374 BN_CTX_start(ctx);
375 x0 = BN_CTX_get(ctx);
376 y0 = BN_CTX_get(ctx);
377 x1 = BN_CTX_get(ctx);
378 y1 = BN_CTX_get(ctx);
379 x2 = BN_CTX_get(ctx);
380 y2 = BN_CTX_get(ctx);
381 s = BN_CTX_get(ctx);
382 t = BN_CTX_get(ctx);
383 if (t == NULL)
384 goto err;
385
386 if (a->Z_is_one) {
387 if (!BN_copy(x0, a->X))
388 goto err;
389 if (!BN_copy(y0, a->Y))
390 goto err;
391 } else {
392 if (!EC_POINT_get_affine_coordinates(group, a, x0, y0, ctx))
393 goto err;
394 }
395 if (b->Z_is_one) {
396 if (!BN_copy(x1, b->X))
397 goto err;
398 if (!BN_copy(y1, b->Y))
399 goto err;
400 } else {
401 if (!EC_POINT_get_affine_coordinates(group, b, x1, y1, ctx))
402 goto err;
403 }
404
405 if (BN_GF2m_cmp(x0, x1)) {
406 if (!BN_GF2m_add(t, x0, x1))
407 goto err;
408 if (!BN_GF2m_add(s, y0, y1))
409 goto err;
410 if (!group->meth->field_div(group, s, s, t, ctx))
411 goto err;
412 if (!group->meth->field_sqr(group, x2, s, ctx))
413 goto err;
414 if (!BN_GF2m_add(x2, x2, group->a))
415 goto err;
416 if (!BN_GF2m_add(x2, x2, s))
417 goto err;
418 if (!BN_GF2m_add(x2, x2, t))
419 goto err;
420 } else {
421 if (BN_GF2m_cmp(y0, y1) || BN_is_zero(x1)) {
422 if (!EC_POINT_set_to_infinity(group, r))
423 goto err;
424 ret = 1;
425 goto err;
426 }
427 if (!group->meth->field_div(group, s, y1, x1, ctx))
428 goto err;
429 if (!BN_GF2m_add(s, s, x1))
430 goto err;
431
432 if (!group->meth->field_sqr(group, x2, s, ctx))
433 goto err;
434 if (!BN_GF2m_add(x2, x2, s))
435 goto err;
436 if (!BN_GF2m_add(x2, x2, group->a))
437 goto err;
438 }
439
440 if (!BN_GF2m_add(y2, x1, x2))
441 goto err;
442 if (!group->meth->field_mul(group, y2, y2, s, ctx))
443 goto err;
444 if (!BN_GF2m_add(y2, y2, x2))
445 goto err;
446 if (!BN_GF2m_add(y2, y2, y1))
447 goto err;
448
449 if (!EC_POINT_set_affine_coordinates(group, r, x2, y2, ctx))
450 goto err;
451
452 ret = 1;
453
454 err:
455 BN_CTX_end(ctx);
456 BN_CTX_free(new_ctx);
457 return ret;
458 }
459
460 /*
461 * Computes 2 * a and stores the result in r. r could be a. Uses algorithm
462 * A.10.2 of IEEE P1363.
463 */
ec_GF2m_simple_dbl(const EC_GROUP * group,EC_POINT * r,const EC_POINT * a,BN_CTX * ctx)464 int ec_GF2m_simple_dbl(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a,
465 BN_CTX *ctx)
466 {
467 return ec_GF2m_simple_add(group, r, a, a, ctx);
468 }
469
ec_GF2m_simple_invert(const EC_GROUP * group,EC_POINT * point,BN_CTX * ctx)470 int ec_GF2m_simple_invert(const EC_GROUP *group, EC_POINT *point, BN_CTX *ctx)
471 {
472 if (EC_POINT_is_at_infinity(group, point) || BN_is_zero(point->Y))
473 /* point is its own inverse */
474 return 1;
475
476 if (!EC_POINT_make_affine(group, point, ctx))
477 return 0;
478 return BN_GF2m_add(point->Y, point->X, point->Y);
479 }
480
481 /* Indicates whether the given point is the point at infinity. */
ec_GF2m_simple_is_at_infinity(const EC_GROUP * group,const EC_POINT * point)482 int ec_GF2m_simple_is_at_infinity(const EC_GROUP *group,
483 const EC_POINT *point)
484 {
485 return BN_is_zero(point->Z);
486 }
487
488 /*-
489 * Determines whether the given EC_POINT is an actual point on the curve defined
490 * in the EC_GROUP. A point is valid if it satisfies the Weierstrass equation:
491 * y^2 + x*y = x^3 + a*x^2 + b.
492 */
ec_GF2m_simple_is_on_curve(const EC_GROUP * group,const EC_POINT * point,BN_CTX * ctx)493 int ec_GF2m_simple_is_on_curve(const EC_GROUP *group, const EC_POINT *point,
494 BN_CTX *ctx)
495 {
496 int ret = -1;
497 BN_CTX *new_ctx = NULL;
498 BIGNUM *lh, *y2;
499 int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *,
500 const BIGNUM *, BN_CTX *);
501 int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
502
503 if (EC_POINT_is_at_infinity(group, point))
504 return 1;
505
506 field_mul = group->meth->field_mul;
507 field_sqr = group->meth->field_sqr;
508
509 /* only support affine coordinates */
510 if (!point->Z_is_one)
511 return -1;
512
513 if (ctx == NULL) {
514 ctx = new_ctx = BN_CTX_new();
515 if (ctx == NULL)
516 return -1;
517 }
518
519 BN_CTX_start(ctx);
520 y2 = BN_CTX_get(ctx);
521 lh = BN_CTX_get(ctx);
522 if (lh == NULL)
523 goto err;
524
525 /*-
526 * We have a curve defined by a Weierstrass equation
527 * y^2 + x*y = x^3 + a*x^2 + b.
528 * <=> x^3 + a*x^2 + x*y + b + y^2 = 0
529 * <=> ((x + a) * x + y ) * x + b + y^2 = 0
530 */
531 if (!BN_GF2m_add(lh, point->X, group->a))
532 goto err;
533 if (!field_mul(group, lh, lh, point->X, ctx))
534 goto err;
535 if (!BN_GF2m_add(lh, lh, point->Y))
536 goto err;
537 if (!field_mul(group, lh, lh, point->X, ctx))
538 goto err;
539 if (!BN_GF2m_add(lh, lh, group->b))
540 goto err;
541 if (!field_sqr(group, y2, point->Y, ctx))
542 goto err;
543 if (!BN_GF2m_add(lh, lh, y2))
544 goto err;
545 ret = BN_is_zero(lh);
546
547 err:
548 BN_CTX_end(ctx);
549 BN_CTX_free(new_ctx);
550 return ret;
551 }
552
553 /*-
554 * Indicates whether two points are equal.
555 * Return values:
556 * -1 error
557 * 0 equal (in affine coordinates)
558 * 1 not equal
559 */
ec_GF2m_simple_cmp(const EC_GROUP * group,const EC_POINT * a,const EC_POINT * b,BN_CTX * ctx)560 int ec_GF2m_simple_cmp(const EC_GROUP *group, const EC_POINT *a,
561 const EC_POINT *b, BN_CTX *ctx)
562 {
563 BIGNUM *aX, *aY, *bX, *bY;
564 BN_CTX *new_ctx = NULL;
565 int ret = -1;
566
567 if (EC_POINT_is_at_infinity(group, a)) {
568 return EC_POINT_is_at_infinity(group, b) ? 0 : 1;
569 }
570
571 if (EC_POINT_is_at_infinity(group, b))
572 return 1;
573
574 if (a->Z_is_one && b->Z_is_one) {
575 return ((BN_cmp(a->X, b->X) == 0) && BN_cmp(a->Y, b->Y) == 0) ? 0 : 1;
576 }
577
578 if (ctx == NULL) {
579 ctx = new_ctx = BN_CTX_new();
580 if (ctx == NULL)
581 return -1;
582 }
583
584 BN_CTX_start(ctx);
585 aX = BN_CTX_get(ctx);
586 aY = BN_CTX_get(ctx);
587 bX = BN_CTX_get(ctx);
588 bY = BN_CTX_get(ctx);
589 if (bY == NULL)
590 goto err;
591
592 if (!EC_POINT_get_affine_coordinates(group, a, aX, aY, ctx))
593 goto err;
594 if (!EC_POINT_get_affine_coordinates(group, b, bX, bY, ctx))
595 goto err;
596 ret = ((BN_cmp(aX, bX) == 0) && BN_cmp(aY, bY) == 0) ? 0 : 1;
597
598 err:
599 BN_CTX_end(ctx);
600 BN_CTX_free(new_ctx);
601 return ret;
602 }
603
604 /* Forces the given EC_POINT to internally use affine coordinates. */
ec_GF2m_simple_make_affine(const EC_GROUP * group,EC_POINT * point,BN_CTX * ctx)605 int ec_GF2m_simple_make_affine(const EC_GROUP *group, EC_POINT *point,
606 BN_CTX *ctx)
607 {
608 BN_CTX *new_ctx = NULL;
609 BIGNUM *x, *y;
610 int ret = 0;
611
612 if (point->Z_is_one || EC_POINT_is_at_infinity(group, point))
613 return 1;
614
615 if (ctx == NULL) {
616 ctx = new_ctx = BN_CTX_new();
617 if (ctx == NULL)
618 return 0;
619 }
620
621 BN_CTX_start(ctx);
622 x = BN_CTX_get(ctx);
623 y = BN_CTX_get(ctx);
624 if (y == NULL)
625 goto err;
626
627 if (!EC_POINT_get_affine_coordinates(group, point, x, y, ctx))
628 goto err;
629 if (!BN_copy(point->X, x))
630 goto err;
631 if (!BN_copy(point->Y, y))
632 goto err;
633 if (!BN_one(point->Z))
634 goto err;
635 point->Z_is_one = 1;
636
637 ret = 1;
638
639 err:
640 BN_CTX_end(ctx);
641 BN_CTX_free(new_ctx);
642 return ret;
643 }
644
645 /*
646 * Forces each of the EC_POINTs in the given array to use affine coordinates.
647 */
ec_GF2m_simple_points_make_affine(const EC_GROUP * group,size_t num,EC_POINT * points[],BN_CTX * ctx)648 int ec_GF2m_simple_points_make_affine(const EC_GROUP *group, size_t num,
649 EC_POINT *points[], BN_CTX *ctx)
650 {
651 size_t i;
652
653 for (i = 0; i < num; i++) {
654 if (!group->meth->make_affine(group, points[i], ctx))
655 return 0;
656 }
657
658 return 1;
659 }
660
661 /* Wrapper to simple binary polynomial field multiplication implementation. */
ec_GF2m_simple_field_mul(const EC_GROUP * group,BIGNUM * r,const BIGNUM * a,const BIGNUM * b,BN_CTX * ctx)662 int ec_GF2m_simple_field_mul(const EC_GROUP *group, BIGNUM *r,
663 const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
664 {
665 return BN_GF2m_mod_mul_arr(r, a, b, group->poly, ctx);
666 }
667
668 /* Wrapper to simple binary polynomial field squaring implementation. */
ec_GF2m_simple_field_sqr(const EC_GROUP * group,BIGNUM * r,const BIGNUM * a,BN_CTX * ctx)669 int ec_GF2m_simple_field_sqr(const EC_GROUP *group, BIGNUM *r,
670 const BIGNUM *a, BN_CTX *ctx)
671 {
672 return BN_GF2m_mod_sqr_arr(r, a, group->poly, ctx);
673 }
674
675 /* Wrapper to simple binary polynomial field division implementation. */
ec_GF2m_simple_field_div(const EC_GROUP * group,BIGNUM * r,const BIGNUM * a,const BIGNUM * b,BN_CTX * ctx)676 int ec_GF2m_simple_field_div(const EC_GROUP *group, BIGNUM *r,
677 const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
678 {
679 return BN_GF2m_mod_div(r, a, b, group->field, ctx);
680 }
681
682 /*-
683 * Lopez-Dahab ladder, pre step.
684 * See e.g. "Guide to ECC" Alg 3.40.
685 * Modified to blind s and r independently.
686 * s:= p, r := 2p
687 */
688 static
ec_GF2m_simple_ladder_pre(const EC_GROUP * group,EC_POINT * r,EC_POINT * s,EC_POINT * p,BN_CTX * ctx)689 int ec_GF2m_simple_ladder_pre(const EC_GROUP *group,
690 EC_POINT *r, EC_POINT *s,
691 EC_POINT *p, BN_CTX *ctx)
692 {
693 /* if p is not affine, something is wrong */
694 if (p->Z_is_one == 0)
695 return 0;
696
697 /* s blinding: make sure lambda (s->Z here) is not zero */
698 do {
699 if (!BN_priv_rand(s->Z, BN_num_bits(group->field) - 1,
700 BN_RAND_TOP_ANY, BN_RAND_BOTTOM_ANY)) {
701 ECerr(EC_F_EC_GF2M_SIMPLE_LADDER_PRE, ERR_R_BN_LIB);
702 return 0;
703 }
704 } while (BN_is_zero(s->Z));
705
706 /* if field_encode defined convert between representations */
707 if ((group->meth->field_encode != NULL
708 && !group->meth->field_encode(group, s->Z, s->Z, ctx))
709 || !group->meth->field_mul(group, s->X, p->X, s->Z, ctx))
710 return 0;
711
712 /* r blinding: make sure lambda (r->Y here for storage) is not zero */
713 do {
714 if (!BN_priv_rand(r->Y, BN_num_bits(group->field) - 1,
715 BN_RAND_TOP_ANY, BN_RAND_BOTTOM_ANY)) {
716 ECerr(EC_F_EC_GF2M_SIMPLE_LADDER_PRE, ERR_R_BN_LIB);
717 return 0;
718 }
719 } while (BN_is_zero(r->Y));
720
721 if ((group->meth->field_encode != NULL
722 && !group->meth->field_encode(group, r->Y, r->Y, ctx))
723 || !group->meth->field_sqr(group, r->Z, p->X, ctx)
724 || !group->meth->field_sqr(group, r->X, r->Z, ctx)
725 || !BN_GF2m_add(r->X, r->X, group->b)
726 || !group->meth->field_mul(group, r->Z, r->Z, r->Y, ctx)
727 || !group->meth->field_mul(group, r->X, r->X, r->Y, ctx))
728 return 0;
729
730 s->Z_is_one = 0;
731 r->Z_is_one = 0;
732
733 return 1;
734 }
735
736 /*-
737 * Ladder step: differential addition-and-doubling, mixed Lopez-Dahab coords.
738 * http://www.hyperelliptic.org/EFD/g12o/auto-code/shortw/xz/ladder/mladd-2003-s.op3
739 * s := r + s, r := 2r
740 */
741 static
ec_GF2m_simple_ladder_step(const EC_GROUP * group,EC_POINT * r,EC_POINT * s,EC_POINT * p,BN_CTX * ctx)742 int ec_GF2m_simple_ladder_step(const EC_GROUP *group,
743 EC_POINT *r, EC_POINT *s,
744 EC_POINT *p, BN_CTX *ctx)
745 {
746 if (!group->meth->field_mul(group, r->Y, r->Z, s->X, ctx)
747 || !group->meth->field_mul(group, s->X, r->X, s->Z, ctx)
748 || !group->meth->field_sqr(group, s->Y, r->Z, ctx)
749 || !group->meth->field_sqr(group, r->Z, r->X, ctx)
750 || !BN_GF2m_add(s->Z, r->Y, s->X)
751 || !group->meth->field_sqr(group, s->Z, s->Z, ctx)
752 || !group->meth->field_mul(group, s->X, r->Y, s->X, ctx)
753 || !group->meth->field_mul(group, r->Y, s->Z, p->X, ctx)
754 || !BN_GF2m_add(s->X, s->X, r->Y)
755 || !group->meth->field_sqr(group, r->Y, r->Z, ctx)
756 || !group->meth->field_mul(group, r->Z, r->Z, s->Y, ctx)
757 || !group->meth->field_sqr(group, s->Y, s->Y, ctx)
758 || !group->meth->field_mul(group, s->Y, s->Y, group->b, ctx)
759 || !BN_GF2m_add(r->X, r->Y, s->Y))
760 return 0;
761
762 return 1;
763 }
764
765 /*-
766 * Recover affine (x,y) result from Lopez-Dahab r and s, affine p.
767 * See e.g. "Fast Multiplication on Elliptic Curves over GF(2**m)
768 * without Precomputation" (Lopez and Dahab, CHES 1999),
769 * Appendix Alg Mxy.
770 */
771 static
ec_GF2m_simple_ladder_post(const EC_GROUP * group,EC_POINT * r,EC_POINT * s,EC_POINT * p,BN_CTX * ctx)772 int ec_GF2m_simple_ladder_post(const EC_GROUP *group,
773 EC_POINT *r, EC_POINT *s,
774 EC_POINT *p, BN_CTX *ctx)
775 {
776 int ret = 0;
777 BIGNUM *t0, *t1, *t2 = NULL;
778
779 if (BN_is_zero(r->Z))
780 return EC_POINT_set_to_infinity(group, r);
781
782 if (BN_is_zero(s->Z)) {
783 if (!EC_POINT_copy(r, p)
784 || !EC_POINT_invert(group, r, ctx)) {
785 ECerr(EC_F_EC_GF2M_SIMPLE_LADDER_POST, ERR_R_EC_LIB);
786 return 0;
787 }
788 return 1;
789 }
790
791 BN_CTX_start(ctx);
792 t0 = BN_CTX_get(ctx);
793 t1 = BN_CTX_get(ctx);
794 t2 = BN_CTX_get(ctx);
795 if (t2 == NULL) {
796 ECerr(EC_F_EC_GF2M_SIMPLE_LADDER_POST, ERR_R_MALLOC_FAILURE);
797 goto err;
798 }
799
800 if (!group->meth->field_mul(group, t0, r->Z, s->Z, ctx)
801 || !group->meth->field_mul(group, t1, p->X, r->Z, ctx)
802 || !BN_GF2m_add(t1, r->X, t1)
803 || !group->meth->field_mul(group, t2, p->X, s->Z, ctx)
804 || !group->meth->field_mul(group, r->Z, r->X, t2, ctx)
805 || !BN_GF2m_add(t2, t2, s->X)
806 || !group->meth->field_mul(group, t1, t1, t2, ctx)
807 || !group->meth->field_sqr(group, t2, p->X, ctx)
808 || !BN_GF2m_add(t2, p->Y, t2)
809 || !group->meth->field_mul(group, t2, t2, t0, ctx)
810 || !BN_GF2m_add(t1, t2, t1)
811 || !group->meth->field_mul(group, t2, p->X, t0, ctx)
812 || !group->meth->field_inv(group, t2, t2, ctx)
813 || !group->meth->field_mul(group, t1, t1, t2, ctx)
814 || !group->meth->field_mul(group, r->X, r->Z, t2, ctx)
815 || !BN_GF2m_add(t2, p->X, r->X)
816 || !group->meth->field_mul(group, t2, t2, t1, ctx)
817 || !BN_GF2m_add(r->Y, p->Y, t2)
818 || !BN_one(r->Z))
819 goto err;
820
821 r->Z_is_one = 1;
822
823 /* GF(2^m) field elements should always have BIGNUM::neg = 0 */
824 BN_set_negative(r->X, 0);
825 BN_set_negative(r->Y, 0);
826
827 ret = 1;
828
829 err:
830 BN_CTX_end(ctx);
831 return ret;
832 }
833
834 static
ec_GF2m_simple_points_mul(const EC_GROUP * group,EC_POINT * r,const BIGNUM * scalar,size_t num,const EC_POINT * points[],const BIGNUM * scalars[],BN_CTX * ctx)835 int ec_GF2m_simple_points_mul(const EC_GROUP *group, EC_POINT *r,
836 const BIGNUM *scalar, size_t num,
837 const EC_POINT *points[],
838 const BIGNUM *scalars[],
839 BN_CTX *ctx)
840 {
841 int ret = 0;
842 EC_POINT *t = NULL;
843
844 /*-
845 * We limit use of the ladder only to the following cases:
846 * - r := scalar * G
847 * Fixed point mul: scalar != NULL && num == 0;
848 * - r := scalars[0] * points[0]
849 * Variable point mul: scalar == NULL && num == 1;
850 * - r := scalar * G + scalars[0] * points[0]
851 * used, e.g., in ECDSA verification: scalar != NULL && num == 1
852 *
853 * In any other case (num > 1) we use the default wNAF implementation.
854 *
855 * We also let the default implementation handle degenerate cases like group
856 * order or cofactor set to 0.
857 */
858 if (num > 1 || BN_is_zero(group->order) || BN_is_zero(group->cofactor))
859 return ec_wNAF_mul(group, r, scalar, num, points, scalars, ctx);
860
861 if (scalar != NULL && num == 0)
862 /* Fixed point multiplication */
863 return ec_scalar_mul_ladder(group, r, scalar, NULL, ctx);
864
865 if (scalar == NULL && num == 1)
866 /* Variable point multiplication */
867 return ec_scalar_mul_ladder(group, r, scalars[0], points[0], ctx);
868
869 /*-
870 * Double point multiplication:
871 * r := scalar * G + scalars[0] * points[0]
872 */
873
874 if ((t = EC_POINT_new(group)) == NULL) {
875 ECerr(EC_F_EC_GF2M_SIMPLE_POINTS_MUL, ERR_R_MALLOC_FAILURE);
876 return 0;
877 }
878
879 if (!ec_scalar_mul_ladder(group, t, scalar, NULL, ctx)
880 || !ec_scalar_mul_ladder(group, r, scalars[0], points[0], ctx)
881 || !EC_POINT_add(group, r, t, r, ctx))
882 goto err;
883
884 ret = 1;
885
886 err:
887 EC_POINT_free(t);
888 return ret;
889 }
890
891 /*-
892 * Computes the multiplicative inverse of a in GF(2^m), storing the result in r.
893 * If a is zero (or equivalent), you'll get a EC_R_CANNOT_INVERT error.
894 * SCA hardening is with blinding: BN_GF2m_mod_inv does that.
895 */
ec_GF2m_simple_field_inv(const EC_GROUP * group,BIGNUM * r,const BIGNUM * a,BN_CTX * ctx)896 static int ec_GF2m_simple_field_inv(const EC_GROUP *group, BIGNUM *r,
897 const BIGNUM *a, BN_CTX *ctx)
898 {
899 int ret;
900
901 if (!(ret = BN_GF2m_mod_inv(r, a, group->field, ctx)))
902 ECerr(EC_F_EC_GF2M_SIMPLE_FIELD_INV, EC_R_CANNOT_INVERT);
903 return ret;
904 }
905
EC_GF2m_simple_method(void)906 const EC_METHOD *EC_GF2m_simple_method(void)
907 {
908 static const EC_METHOD ret = {
909 EC_FLAGS_DEFAULT_OCT,
910 NID_X9_62_characteristic_two_field,
911 ec_GF2m_simple_group_init,
912 ec_GF2m_simple_group_finish,
913 ec_GF2m_simple_group_clear_finish,
914 ec_GF2m_simple_group_copy,
915 ec_GF2m_simple_group_set_curve,
916 ec_GF2m_simple_group_get_curve,
917 ec_GF2m_simple_group_get_degree,
918 ec_group_simple_order_bits,
919 ec_GF2m_simple_group_check_discriminant,
920 ec_GF2m_simple_point_init,
921 ec_GF2m_simple_point_finish,
922 ec_GF2m_simple_point_clear_finish,
923 ec_GF2m_simple_point_copy,
924 ec_GF2m_simple_point_set_to_infinity,
925 0, /* set_Jprojective_coordinates_GFp */
926 0, /* get_Jprojective_coordinates_GFp */
927 ec_GF2m_simple_point_set_affine_coordinates,
928 ec_GF2m_simple_point_get_affine_coordinates,
929 0, /* point_set_compressed_coordinates */
930 0, /* point2oct */
931 0, /* oct2point */
932 ec_GF2m_simple_add,
933 ec_GF2m_simple_dbl,
934 ec_GF2m_simple_invert,
935 ec_GF2m_simple_is_at_infinity,
936 ec_GF2m_simple_is_on_curve,
937 ec_GF2m_simple_cmp,
938 ec_GF2m_simple_make_affine,
939 ec_GF2m_simple_points_make_affine,
940 ec_GF2m_simple_points_mul,
941 0, /* precompute_mult */
942 0, /* have_precompute_mult */
943 ec_GF2m_simple_field_mul,
944 ec_GF2m_simple_field_sqr,
945 ec_GF2m_simple_field_div,
946 ec_GF2m_simple_field_inv,
947 0, /* field_encode */
948 0, /* field_decode */
949 0, /* field_set_to_one */
950 ec_key_simple_priv2oct,
951 ec_key_simple_oct2priv,
952 0, /* set private */
953 ec_key_simple_generate_key,
954 ec_key_simple_check_key,
955 ec_key_simple_generate_public_key,
956 0, /* keycopy */
957 0, /* keyfinish */
958 ecdh_simple_compute_key,
959 0, /* field_inverse_mod_ord */
960 0, /* blind_coordinates */
961 ec_GF2m_simple_ladder_pre,
962 ec_GF2m_simple_ladder_step,
963 ec_GF2m_simple_ladder_post
964 };
965
966 return &ret;
967 }
968
969 #endif
970