summaryrefslogtreecommitdiff
path: root/drivers/builtin_openssl2/crypto/ec/ec2_smpl.c
blob: e0e59c7d8299b0ccebb7e2027a7e48b6c02c8dc7 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
/* crypto/ec/ec2_smpl.c */
/* ====================================================================
 * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED.
 *
 * The Elliptic Curve Public-Key Crypto Library (ECC Code) included
 * herein is developed by SUN MICROSYSTEMS, INC., and is contributed
 * to the OpenSSL project.
 *
 * The ECC Code is licensed pursuant to the OpenSSL open source
 * license provided below.
 *
 * The software is originally written by Sheueling Chang Shantz and
 * Douglas Stebila of Sun Microsystems Laboratories.
 *
 */
/* ====================================================================
 * Copyright (c) 1998-2005 The OpenSSL Project.  All rights reserved.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 *
 * 1. Redistributions of source code must retain the above copyright
 *    notice, this list of conditions and the following disclaimer. 
 *
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in
 *    the documentation and/or other materials provided with the
 *    distribution.
 *
 * 3. All advertising materials mentioning features or use of this
 *    software must display the following acknowledgment:
 *    "This product includes software developed by the OpenSSL Project
 *    for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
 *
 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
 *    endorse or promote products derived from this software without
 *    prior written permission. For written permission, please contact
 *    openssl-core@openssl.org.
 *
 * 5. Products derived from this software may not be called "OpenSSL"
 *    nor may "OpenSSL" appear in their names without prior written
 *    permission of the OpenSSL Project.
 *
 * 6. Redistributions of any form whatsoever must retain the following
 *    acknowledgment:
 *    "This product includes software developed by the OpenSSL Project
 *    for use in the OpenSSL Toolkit (http://www.openssl.org/)"
 *
 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
 * PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL THE OpenSSL PROJECT OR
 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
 * OF THE POSSIBILITY OF SUCH DAMAGE.
 * ====================================================================
 *
 * This product includes cryptographic software written by Eric Young
 * (eay@cryptsoft.com).  This product includes software written by Tim
 * Hudson (tjh@cryptsoft.com).
 *
 */

#include <openssl/err.h>

#include "ec_lcl.h"

#ifndef OPENSSL_NO_EC2M

#ifdef OPENSSL_FIPS
#include <openssl/fips.h>
#endif


const EC_METHOD *EC_GF2m_simple_method(void)
	{
#ifdef OPENSSL_FIPS
	return fips_ec_gf2m_simple_method();
#else
	static const EC_METHOD ret = {
		EC_FLAGS_DEFAULT_OCT,
		NID_X9_62_characteristic_two_field,
		ec_GF2m_simple_group_init,
		ec_GF2m_simple_group_finish,
		ec_GF2m_simple_group_clear_finish,
		ec_GF2m_simple_group_copy,
		ec_GF2m_simple_group_set_curve,
		ec_GF2m_simple_group_get_curve,
		ec_GF2m_simple_group_get_degree,
		ec_GF2m_simple_group_check_discriminant,
		ec_GF2m_simple_point_init,
		ec_GF2m_simple_point_finish,
		ec_GF2m_simple_point_clear_finish,
		ec_GF2m_simple_point_copy,
		ec_GF2m_simple_point_set_to_infinity,
		0 /* set_Jprojective_coordinates_GFp */,
		0 /* get_Jprojective_coordinates_GFp */,
		ec_GF2m_simple_point_set_affine_coordinates,
		ec_GF2m_simple_point_get_affine_coordinates,
		0,0,0,
		ec_GF2m_simple_add,
		ec_GF2m_simple_dbl,
		ec_GF2m_simple_invert,
		ec_GF2m_simple_is_at_infinity,
		ec_GF2m_simple_is_on_curve,
		ec_GF2m_simple_cmp,
		ec_GF2m_simple_make_affine,
		ec_GF2m_simple_points_make_affine,

		/* the following three method functions are defined in ec2_mult.c */
		ec_GF2m_simple_mul,
		ec_GF2m_precompute_mult,
		ec_GF2m_have_precompute_mult,

		ec_GF2m_simple_field_mul,
		ec_GF2m_simple_field_sqr,
		ec_GF2m_simple_field_div,
		0 /* field_encode */,
		0 /* field_decode */,
		0 /* field_set_to_one */ };

	return &ret;
#endif
	}


/* Initialize a GF(2^m)-based EC_GROUP structure.
 * Note that all other members are handled by EC_GROUP_new.
 */
int ec_GF2m_simple_group_init(EC_GROUP *group)
	{
	BN_init(&group->field);
	BN_init(&group->a);
	BN_init(&group->b);
	return 1;
	}


/* Free a GF(2^m)-based EC_GROUP structure.
 * Note that all other members are handled by EC_GROUP_free.
 */
void ec_GF2m_simple_group_finish(EC_GROUP *group)
	{
	BN_free(&group->field);
	BN_free(&group->a);
	BN_free(&group->b);
	}


/* Clear and free a GF(2^m)-based EC_GROUP structure.
 * Note that all other members are handled by EC_GROUP_clear_free.
 */
void ec_GF2m_simple_group_clear_finish(EC_GROUP *group)
	{
	BN_clear_free(&group->field);
	BN_clear_free(&group->a);
	BN_clear_free(&group->b);
	group->poly[0] = 0;
	group->poly[1] = 0;
	group->poly[2] = 0;
	group->poly[3] = 0;
	group->poly[4] = 0;
	group->poly[5] = -1;
	}


/* Copy a GF(2^m)-based EC_GROUP structure.
 * Note that all other members are handled by EC_GROUP_copy.
 */
int ec_GF2m_simple_group_copy(EC_GROUP *dest, const EC_GROUP *src)
	{
	int i;
	if (!BN_copy(&dest->field, &src->field)) return 0;
	if (!BN_copy(&dest->a, &src->a)) return 0;
	if (!BN_copy(&dest->b, &src->b)) return 0;
	dest->poly[0] = src->poly[0];
	dest->poly[1] = src->poly[1];
	dest->poly[2] = src->poly[2];
	dest->poly[3] = src->poly[3];
	dest->poly[4] = src->poly[4];
	dest->poly[5] = src->poly[5];
	if (bn_wexpand(&dest->a, (int)(dest->poly[0] + BN_BITS2 - 1) / BN_BITS2) == NULL) return 0;
	if (bn_wexpand(&dest->b, (int)(dest->poly[0] + BN_BITS2 - 1) / BN_BITS2) == NULL) return 0;
	for (i = dest->a.top; i < dest->a.dmax; i++) dest->a.d[i] = 0;
	for (i = dest->b.top; i < dest->b.dmax; i++) dest->b.d[i] = 0;
	return 1;
	}


/* Set the curve parameters of an EC_GROUP structure. */
int ec_GF2m_simple_group_set_curve(EC_GROUP *group,
	const BIGNUM *p, const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
	{
	int ret = 0, i;

	/* group->field */
	if (!BN_copy(&group->field, p)) goto err;
	i = BN_GF2m_poly2arr(&group->field, group->poly, 6) - 1;
	if ((i != 5) && (i != 3))
		{
		ECerr(EC_F_EC_GF2M_SIMPLE_GROUP_SET_CURVE, EC_R_UNSUPPORTED_FIELD);
		goto err;
		}

	/* group->a */
	if (!BN_GF2m_mod_arr(&group->a, a, group->poly)) goto err;
	if(bn_wexpand(&group->a, (int)(group->poly[0] + BN_BITS2 - 1) / BN_BITS2) == NULL) goto err;
	for (i = group->a.top; i < group->a.dmax; i++) group->a.d[i] = 0;
	
	/* group->b */
	if (!BN_GF2m_mod_arr(&group->b, b, group->poly)) goto err;
	if(bn_wexpand(&group->b, (int)(group->poly[0] + BN_BITS2 - 1) / BN_BITS2) == NULL) goto err;
	for (i = group->b.top; i < group->b.dmax; i++) group->b.d[i] = 0;
		
	ret = 1;
  err:
	return ret;
	}


/* Get the curve parameters of an EC_GROUP structure.
 * If p, a, or b are NULL then there values will not be set but the method will return with success.
 */
int ec_GF2m_simple_group_get_curve(const EC_GROUP *group, BIGNUM *p, BIGNUM *a, BIGNUM *b, BN_CTX *ctx)
	{
	int ret = 0;
	
	if (p != NULL)
		{
		if (!BN_copy(p, &group->field)) return 0;
		}

	if (a != NULL)
		{
		if (!BN_copy(a, &group->a)) goto err;
		}

	if (b != NULL)
		{
		if (!BN_copy(b, &group->b)) goto err;
		}
	
	ret = 1;
	
  err:
	return ret;
	}


/* Gets the degree of the field.  For a curve over GF(2^m) this is the value m. */
int ec_GF2m_simple_group_get_degree(const EC_GROUP *group)
	{
	return BN_num_bits(&group->field)-1;
	}


/* Checks the discriminant of the curve.
 * y^2 + x*y = x^3 + a*x^2 + b is an elliptic curve <=> b != 0 (mod p) 
 */
int ec_GF2m_simple_group_check_discriminant(const EC_GROUP *group, BN_CTX *ctx)
	{
	int ret = 0;
	BIGNUM *b;
	BN_CTX *new_ctx = NULL;

	if (ctx == NULL)
		{
		ctx = new_ctx = BN_CTX_new();
		if (ctx == NULL)
			{
			ECerr(EC_F_EC_GF2M_SIMPLE_GROUP_CHECK_DISCRIMINANT, ERR_R_MALLOC_FAILURE);
			goto err;
			}
		}
	BN_CTX_start(ctx);
	b = BN_CTX_get(ctx);
	if (b == NULL) goto err;

	if (!BN_GF2m_mod_arr(b, &group->b, group->poly)) goto err;
	
	/* check the discriminant:
	 * y^2 + x*y = x^3 + a*x^2 + b is an elliptic curve <=> b != 0 (mod p) 
	 */
	if (BN_is_zero(b)) goto err;

	ret = 1;

err:
	if (ctx != NULL)
		BN_CTX_end(ctx);
	if (new_ctx != NULL)
		BN_CTX_free(new_ctx);
	return ret;
	}


/* Initializes an EC_POINT. */
int ec_GF2m_simple_point_init(EC_POINT *point)
	{
	BN_init(&point->X);
	BN_init(&point->Y);
	BN_init(&point->Z);
	return 1;
	}


/* Frees an EC_POINT. */
void ec_GF2m_simple_point_finish(EC_POINT *point)
	{
	BN_free(&point->X);
	BN_free(&point->Y);
	BN_free(&point->Z);
	}


/* Clears and frees an EC_POINT. */
void ec_GF2m_simple_point_clear_finish(EC_POINT *point)
	{
	BN_clear_free(&point->X);
	BN_clear_free(&point->Y);
	BN_clear_free(&point->Z);
	point->Z_is_one = 0;
	}


/* Copy the contents of one EC_POINT into another.  Assumes dest is initialized. */
int ec_GF2m_simple_point_copy(EC_POINT *dest, const EC_POINT *src)
	{
	if (!BN_copy(&dest->X, &src->X)) return 0;
	if (!BN_copy(&dest->Y, &src->Y)) return 0;
	if (!BN_copy(&dest->Z, &src->Z)) return 0;
	dest->Z_is_one = src->Z_is_one;

	return 1;
	}


/* Set an EC_POINT to the point at infinity.  
 * A point at infinity is represented by having Z=0.
 */
int ec_GF2m_simple_point_set_to_infinity(const EC_GROUP *group, EC_POINT *point)
	{
	point->Z_is_one = 0;
	BN_zero(&point->Z);
	return 1;
	}


/* Set the coordinates of an EC_POINT using affine coordinates. 
 * Note that the simple implementation only uses affine coordinates.
 */
int ec_GF2m_simple_point_set_affine_coordinates(const EC_GROUP *group, EC_POINT *point,
	const BIGNUM *x, const BIGNUM *y, BN_CTX *ctx)
	{
	int ret = 0;	
	if (x == NULL || y == NULL)
		{
		ECerr(EC_F_EC_GF2M_SIMPLE_POINT_SET_AFFINE_COORDINATES, ERR_R_PASSED_NULL_PARAMETER);
		return 0;
		}

	if (!BN_copy(&point->X, x)) goto err;
	BN_set_negative(&point->X, 0);
	if (!BN_copy(&point->Y, y)) goto err;
	BN_set_negative(&point->Y, 0);
	if (!BN_copy(&point->Z, BN_value_one())) goto err;
	BN_set_negative(&point->Z, 0);
	point->Z_is_one = 1;
	ret = 1;

  err:
	return ret;
	}


/* Gets the affine coordinates of an EC_POINT. 
 * Note that the simple implementation only uses affine coordinates.
 */
int ec_GF2m_simple_point_get_affine_coordinates(const EC_GROUP *group, const EC_POINT *point,
	BIGNUM *x, BIGNUM *y, BN_CTX *ctx)
	{
	int ret = 0;

	if (EC_POINT_is_at_infinity(group, point))
		{
		ECerr(EC_F_EC_GF2M_SIMPLE_POINT_GET_AFFINE_COORDINATES, EC_R_POINT_AT_INFINITY);
		return 0;
		}

	if (BN_cmp(&point->Z, BN_value_one())) 
		{
		ECerr(EC_F_EC_GF2M_SIMPLE_POINT_GET_AFFINE_COORDINATES, ERR_R_SHOULD_NOT_HAVE_BEEN_CALLED);
		return 0;
		}
	if (x != NULL)
		{
		if (!BN_copy(x, &point->X)) goto err;
		BN_set_negative(x, 0);
		}
	if (y != NULL)
		{
		if (!BN_copy(y, &point->Y)) goto err;
		BN_set_negative(y, 0);
		}
	ret = 1;
		
 err:
	return ret;
	}

/* Computes a + b and stores the result in r.  r could be a or b, a could be b.
 * Uses algorithm A.10.2 of IEEE P1363.
 */
int ec_GF2m_simple_add(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a, const EC_POINT *b, BN_CTX *ctx)
	{
	BN_CTX *new_ctx = NULL;
	BIGNUM *x0, *y0, *x1, *y1, *x2, *y2, *s, *t;
	int ret = 0;
	
	if (EC_POINT_is_at_infinity(group, a))
		{
		if (!EC_POINT_copy(r, b)) return 0;
		return 1;
		}

	if (EC_POINT_is_at_infinity(group, b))
		{
		if (!EC_POINT_copy(r, a)) return 0;
		return 1;
		}

	if (ctx == NULL)
		{
		ctx = new_ctx = BN_CTX_new();
		if (ctx == NULL)
			return 0;
		}

	BN_CTX_start(ctx);
	x0 = BN_CTX_get(ctx);
	y0 = BN_CTX_get(ctx);
	x1 = BN_CTX_get(ctx);
	y1 = BN_CTX_get(ctx);
	x2 = BN_CTX_get(ctx);
	y2 = BN_CTX_get(ctx);
	s = BN_CTX_get(ctx);
	t = BN_CTX_get(ctx);
	if (t == NULL) goto err;

	if (a->Z_is_one) 
		{
		if (!BN_copy(x0, &a->X)) goto err;
		if (!BN_copy(y0, &a->Y)) goto err;
		}
	else
		{
		if (!EC_POINT_get_affine_coordinates_GF2m(group, a, x0, y0, ctx)) goto err;
		}
	if (b->Z_is_one) 
		{
		if (!BN_copy(x1, &b->X)) goto err;
		if (!BN_copy(y1, &b->Y)) goto err;
		}
	else
		{
		if (!EC_POINT_get_affine_coordinates_GF2m(group, b, x1, y1, ctx)) goto err;
		}


	if (BN_GF2m_cmp(x0, x1))
		{
		if (!BN_GF2m_add(t, x0, x1)) goto err;
		if (!BN_GF2m_add(s, y0, y1)) goto err;
		if (!group->meth->field_div(group, s, s, t, ctx)) goto err;
		if (!group->meth->field_sqr(group, x2, s, ctx)) goto err;
		if (!BN_GF2m_add(x2, x2, &group->a)) goto err;
		if (!BN_GF2m_add(x2, x2, s)) goto err;
		if (!BN_GF2m_add(x2, x2, t)) goto err;
		}
	else
		{
		if (BN_GF2m_cmp(y0, y1) || BN_is_zero(x1))
			{
			if (!EC_POINT_set_to_infinity(group, r)) goto err;
			ret = 1;
			goto err;
			}
		if (!group->meth->field_div(group, s, y1, x1, ctx)) goto err;
		if (!BN_GF2m_add(s, s, x1)) goto err;
		
		if (!group->meth->field_sqr(group, x2, s, ctx)) goto err;
		if (!BN_GF2m_add(x2, x2, s)) goto err;
		if (!BN_GF2m_add(x2, x2, &group->a)) goto err;
		}

	if (!BN_GF2m_add(y2, x1, x2)) goto err;
	if (!group->meth->field_mul(group, y2, y2, s, ctx)) goto err;
	if (!BN_GF2m_add(y2, y2, x2)) goto err;
	if (!BN_GF2m_add(y2, y2, y1)) goto err;

	if (!EC_POINT_set_affine_coordinates_GF2m(group, r, x2, y2, ctx)) goto err;

	ret = 1;

 err:
	BN_CTX_end(ctx);
	if (new_ctx != NULL)
		BN_CTX_free(new_ctx);
	return ret;
	}


/* Computes 2 * a and stores the result in r.  r could be a.
 * Uses algorithm A.10.2 of IEEE P1363.
 */
int ec_GF2m_simple_dbl(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a, BN_CTX *ctx)
	{
	return ec_GF2m_simple_add(group, r, a, a, ctx);
	}


int ec_GF2m_simple_invert(const EC_GROUP *group, EC_POINT *point, BN_CTX *ctx)
	{
	if (EC_POINT_is_at_infinity(group, point) || BN_is_zero(&point->Y))
		/* point is its own inverse */
		return 1;
	
	if (!EC_POINT_make_affine(group, point, ctx)) return 0;
	return BN_GF2m_add(&point->Y, &point->X, &point->Y);
	}


/* Indicates whether the given point is the point at infinity. */
int ec_GF2m_simple_is_at_infinity(const EC_GROUP *group, const EC_POINT *point)
	{
	return BN_is_zero(&point->Z);
	}


/* Determines whether the given EC_POINT is an actual point on the curve defined
 * in the EC_GROUP.  A point is valid if it satisfies the Weierstrass equation:
 *      y^2 + x*y = x^3 + a*x^2 + b.
 */
int ec_GF2m_simple_is_on_curve(const EC_GROUP *group, const EC_POINT *point, BN_CTX *ctx)
	{
	int ret = -1;
	BN_CTX *new_ctx = NULL;
	BIGNUM *lh, *y2;
	int (*field_mul)(const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *, BN_CTX *);
	int (*field_sqr)(const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);

	if (EC_POINT_is_at_infinity(group, point))
		return 1;

	field_mul = group->meth->field_mul;
	field_sqr = group->meth->field_sqr;	

	/* only support affine coordinates */
	if (!point->Z_is_one) return -1;

	if (ctx == NULL)
		{
		ctx = new_ctx = BN_CTX_new();
		if (ctx == NULL)
			return -1;
		}

	BN_CTX_start(ctx);
	y2 = BN_CTX_get(ctx);
	lh = BN_CTX_get(ctx);
	if (lh == NULL) goto err;

	/* We have a curve defined by a Weierstrass equation
	 *      y^2 + x*y = x^3 + a*x^2 + b.
	 *  <=> x^3 + a*x^2 + x*y + b + y^2 = 0
	 *  <=> ((x + a) * x + y ) * x + b + y^2 = 0
	 */
	if (!BN_GF2m_add(lh, &point->X, &group->a)) goto err;
	if (!field_mul(group, lh, lh, &point->X, ctx)) goto err;
	if (!BN_GF2m_add(lh, lh, &point->Y)) goto err;
	if (!field_mul(group, lh, lh, &point->X, ctx)) goto err;
	if (!BN_GF2m_add(lh, lh, &group->b)) goto err;
	if (!field_sqr(group, y2, &point->Y, ctx)) goto err;
	if (!BN_GF2m_add(lh, lh, y2)) goto err;
	ret = BN_is_zero(lh);
 err:
	if (ctx) BN_CTX_end(ctx);
	if (new_ctx) BN_CTX_free(new_ctx);
	return ret;
	}


/* Indicates whether two points are equal.
 * Return values:
 *  -1   error
 *   0   equal (in affine coordinates)
 *   1   not equal
 */
int ec_GF2m_simple_cmp(const EC_GROUP *group, const EC_POINT *a, const EC_POINT *b, BN_CTX *ctx)
	{
	BIGNUM *aX, *aY, *bX, *bY;
	BN_CTX *new_ctx = NULL;
	int ret = -1;

	if (EC_POINT_is_at_infinity(group, a))
		{
		return EC_POINT_is_at_infinity(group, b) ? 0 : 1;
		}

	if (EC_POINT_is_at_infinity(group, b))
		return 1;
	
	if (a->Z_is_one && b->Z_is_one)
		{
		return ((BN_cmp(&a->X, &b->X) == 0) && BN_cmp(&a->Y, &b->Y) == 0) ? 0 : 1;
		}

	if (ctx == NULL)
		{
		ctx = new_ctx = BN_CTX_new();
		if (ctx == NULL)
			return -1;
		}

	BN_CTX_start(ctx);
	aX = BN_CTX_get(ctx);
	aY = BN_CTX_get(ctx);
	bX = BN_CTX_get(ctx);
	bY = BN_CTX_get(ctx);
	if (bY == NULL) goto err;

	if (!EC_POINT_get_affine_coordinates_GF2m(group, a, aX, aY, ctx)) goto err;
	if (!EC_POINT_get_affine_coordinates_GF2m(group, b, bX, bY, ctx)) goto err;
	ret = ((BN_cmp(aX, bX) == 0) && BN_cmp(aY, bY) == 0) ? 0 : 1;

  err:	
	if (ctx) BN_CTX_end(ctx);
	if (new_ctx) BN_CTX_free(new_ctx);
	return ret;
	}


/* Forces the given EC_POINT to internally use affine coordinates. */
int ec_GF2m_simple_make_affine(const EC_GROUP *group, EC_POINT *point, BN_CTX *ctx)
	{
	BN_CTX *new_ctx = NULL;
	BIGNUM *x, *y;
	int ret = 0;

	if (point->Z_is_one || EC_POINT_is_at_infinity(group, point))
		return 1;
	
	if (ctx == NULL)
		{
		ctx = new_ctx = BN_CTX_new();
		if (ctx == NULL)
			return 0;
		}

	BN_CTX_start(ctx);
	x = BN_CTX_get(ctx);
	y = BN_CTX_get(ctx);
	if (y == NULL) goto err;
	
	if (!EC_POINT_get_affine_coordinates_GF2m(group, point, x, y, ctx)) goto err;
	if (!BN_copy(&point->X, x)) goto err;
	if (!BN_copy(&point->Y, y)) goto err;
	if (!BN_one(&point->Z)) goto err;
	
	ret = 1;		

  err:
	if (ctx) BN_CTX_end(ctx);
	if (new_ctx) BN_CTX_free(new_ctx);
	return ret;
	}


/* Forces each of the EC_POINTs in the given array to use affine coordinates. */
int ec_GF2m_simple_points_make_affine(const EC_GROUP *group, size_t num, EC_POINT *points[], BN_CTX *ctx)
	{
	size_t i;

	for (i = 0; i < num; i++)
		{
		if (!group->meth->make_affine(group, points[i], ctx)) return 0;
		}

	return 1;
	}


/* Wrapper to simple binary polynomial field multiplication implementation. */
int ec_GF2m_simple_field_mul(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
	{
	return BN_GF2m_mod_mul_arr(r, a, b, group->poly, ctx);
	}


/* Wrapper to simple binary polynomial field squaring implementation. */
int ec_GF2m_simple_field_sqr(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a, BN_CTX *ctx)
	{
	return BN_GF2m_mod_sqr_arr(r, a, group->poly, ctx);
	}


/* Wrapper to simple binary polynomial field division implementation. */
int ec_GF2m_simple_field_div(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
	{
	return BN_GF2m_mod_div(r, a, b, &group->field, ctx);
	}

#endif