1 /*
2  * Copyright 2001-2020 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 #include <openssl/symhacks.h>
13 
14 #include "ec_local.h"
15 
EC_GFp_simple_method(void)16 const EC_METHOD *EC_GFp_simple_method(void)
17 {
18     static const EC_METHOD ret = {
19         EC_FLAGS_DEFAULT_OCT,
20         NID_X9_62_prime_field,
21         ec_GFp_simple_group_init,
22         ec_GFp_simple_group_finish,
23         ec_GFp_simple_group_clear_finish,
24         ec_GFp_simple_group_copy,
25         ec_GFp_simple_group_set_curve,
26         ec_GFp_simple_group_get_curve,
27         ec_GFp_simple_group_get_degree,
28         ec_group_simple_order_bits,
29         ec_GFp_simple_group_check_discriminant,
30         ec_GFp_simple_point_init,
31         ec_GFp_simple_point_finish,
32         ec_GFp_simple_point_clear_finish,
33         ec_GFp_simple_point_copy,
34         ec_GFp_simple_point_set_to_infinity,
35         ec_GFp_simple_set_Jprojective_coordinates_GFp,
36         ec_GFp_simple_get_Jprojective_coordinates_GFp,
37         ec_GFp_simple_point_set_affine_coordinates,
38         ec_GFp_simple_point_get_affine_coordinates,
39         0, 0, 0,
40         ec_GFp_simple_add,
41         ec_GFp_simple_dbl,
42         ec_GFp_simple_invert,
43         ec_GFp_simple_is_at_infinity,
44         ec_GFp_simple_is_on_curve,
45         ec_GFp_simple_cmp,
46         ec_GFp_simple_make_affine,
47         ec_GFp_simple_points_make_affine,
48         0 /* mul */ ,
49         0 /* precompute_mult */ ,
50         0 /* have_precompute_mult */ ,
51         ec_GFp_simple_field_mul,
52         ec_GFp_simple_field_sqr,
53         0 /* field_div */ ,
54         ec_GFp_simple_field_inv,
55         0 /* field_encode */ ,
56         0 /* field_decode */ ,
57         0,                      /* field_set_to_one */
58         ec_key_simple_priv2oct,
59         ec_key_simple_oct2priv,
60         0, /* set private */
61         ec_key_simple_generate_key,
62         ec_key_simple_check_key,
63         ec_key_simple_generate_public_key,
64         0, /* keycopy */
65         0, /* keyfinish */
66         ecdh_simple_compute_key,
67         0, /* field_inverse_mod_ord */
68         ec_GFp_simple_blind_coordinates,
69         ec_GFp_simple_ladder_pre,
70         ec_GFp_simple_ladder_step,
71         ec_GFp_simple_ladder_post
72     };
73 
74     return &ret;
75 }
76 
77 /*
78  * Most method functions in this file are designed to work with
79  * non-trivial representations of field elements if necessary
80  * (see ecp_mont.c): while standard modular addition and subtraction
81  * are used, the field_mul and field_sqr methods will be used for
82  * multiplication, and field_encode and field_decode (if defined)
83  * will be used for converting between representations.
84  *
85  * Functions ec_GFp_simple_points_make_affine() and
86  * ec_GFp_simple_point_get_affine_coordinates() specifically assume
87  * that if a non-trivial representation is used, it is a Montgomery
88  * representation (i.e. 'encoding' means multiplying by some factor R).
89  */
90 
ec_GFp_simple_group_init(EC_GROUP * group)91 int ec_GFp_simple_group_init(EC_GROUP *group)
92 {
93     group->field = BN_new();
94     group->a = BN_new();
95     group->b = BN_new();
96     if (group->field == NULL || group->a == NULL || group->b == NULL) {
97         BN_free(group->field);
98         BN_free(group->a);
99         BN_free(group->b);
100         return 0;
101     }
102     group->a_is_minus3 = 0;
103     return 1;
104 }
105 
ec_GFp_simple_group_finish(EC_GROUP * group)106 void ec_GFp_simple_group_finish(EC_GROUP *group)
107 {
108     BN_free(group->field);
109     BN_free(group->a);
110     BN_free(group->b);
111 }
112 
ec_GFp_simple_group_clear_finish(EC_GROUP * group)113 void ec_GFp_simple_group_clear_finish(EC_GROUP *group)
114 {
115     BN_clear_free(group->field);
116     BN_clear_free(group->a);
117     BN_clear_free(group->b);
118 }
119 
ec_GFp_simple_group_copy(EC_GROUP * dest,const EC_GROUP * src)120 int ec_GFp_simple_group_copy(EC_GROUP *dest, const EC_GROUP *src)
121 {
122     if (!BN_copy(dest->field, src->field))
123         return 0;
124     if (!BN_copy(dest->a, src->a))
125         return 0;
126     if (!BN_copy(dest->b, src->b))
127         return 0;
128 
129     dest->a_is_minus3 = src->a_is_minus3;
130 
131     return 1;
132 }
133 
ec_GFp_simple_group_set_curve(EC_GROUP * group,const BIGNUM * p,const BIGNUM * a,const BIGNUM * b,BN_CTX * ctx)134 int ec_GFp_simple_group_set_curve(EC_GROUP *group,
135                                   const BIGNUM *p, const BIGNUM *a,
136                                   const BIGNUM *b, BN_CTX *ctx)
137 {
138     int ret = 0;
139     BN_CTX *new_ctx = NULL;
140     BIGNUM *tmp_a;
141 
142     /* p must be a prime > 3 */
143     if (BN_num_bits(p) <= 2 || !BN_is_odd(p)) {
144         ECerr(EC_F_EC_GFP_SIMPLE_GROUP_SET_CURVE, EC_R_INVALID_FIELD);
145         return 0;
146     }
147 
148     if (ctx == NULL) {
149         ctx = new_ctx = BN_CTX_new();
150         if (ctx == NULL)
151             return 0;
152     }
153 
154     BN_CTX_start(ctx);
155     tmp_a = BN_CTX_get(ctx);
156     if (tmp_a == NULL)
157         goto err;
158 
159     /* group->field */
160     if (!BN_copy(group->field, p))
161         goto err;
162     BN_set_negative(group->field, 0);
163 
164     /* group->a */
165     if (!BN_nnmod(tmp_a, a, p, ctx))
166         goto err;
167     if (group->meth->field_encode) {
168         if (!group->meth->field_encode(group, group->a, tmp_a, ctx))
169             goto err;
170     } else if (!BN_copy(group->a, tmp_a))
171         goto err;
172 
173     /* group->b */
174     if (!BN_nnmod(group->b, b, p, ctx))
175         goto err;
176     if (group->meth->field_encode)
177         if (!group->meth->field_encode(group, group->b, group->b, ctx))
178             goto err;
179 
180     /* group->a_is_minus3 */
181     if (!BN_add_word(tmp_a, 3))
182         goto err;
183     group->a_is_minus3 = (0 == BN_cmp(tmp_a, group->field));
184 
185     ret = 1;
186 
187  err:
188     BN_CTX_end(ctx);
189     BN_CTX_free(new_ctx);
190     return ret;
191 }
192 
ec_GFp_simple_group_get_curve(const EC_GROUP * group,BIGNUM * p,BIGNUM * a,BIGNUM * b,BN_CTX * ctx)193 int ec_GFp_simple_group_get_curve(const EC_GROUP *group, BIGNUM *p, BIGNUM *a,
194                                   BIGNUM *b, BN_CTX *ctx)
195 {
196     int ret = 0;
197     BN_CTX *new_ctx = NULL;
198 
199     if (p != NULL) {
200         if (!BN_copy(p, group->field))
201             return 0;
202     }
203 
204     if (a != NULL || b != NULL) {
205         if (group->meth->field_decode) {
206             if (ctx == NULL) {
207                 ctx = new_ctx = BN_CTX_new();
208                 if (ctx == NULL)
209                     return 0;
210             }
211             if (a != NULL) {
212                 if (!group->meth->field_decode(group, a, group->a, ctx))
213                     goto err;
214             }
215             if (b != NULL) {
216                 if (!group->meth->field_decode(group, b, group->b, ctx))
217                     goto err;
218             }
219         } else {
220             if (a != NULL) {
221                 if (!BN_copy(a, group->a))
222                     goto err;
223             }
224             if (b != NULL) {
225                 if (!BN_copy(b, group->b))
226                     goto err;
227             }
228         }
229     }
230 
231     ret = 1;
232 
233  err:
234     BN_CTX_free(new_ctx);
235     return ret;
236 }
237 
ec_GFp_simple_group_get_degree(const EC_GROUP * group)238 int ec_GFp_simple_group_get_degree(const EC_GROUP *group)
239 {
240     return BN_num_bits(group->field);
241 }
242 
ec_GFp_simple_group_check_discriminant(const EC_GROUP * group,BN_CTX * ctx)243 int ec_GFp_simple_group_check_discriminant(const EC_GROUP *group, BN_CTX *ctx)
244 {
245     int ret = 0;
246     BIGNUM *a, *b, *order, *tmp_1, *tmp_2;
247     const BIGNUM *p = group->field;
248     BN_CTX *new_ctx = NULL;
249 
250     if (ctx == NULL) {
251         ctx = new_ctx = BN_CTX_new();
252         if (ctx == NULL) {
253             ECerr(EC_F_EC_GFP_SIMPLE_GROUP_CHECK_DISCRIMINANT,
254                   ERR_R_MALLOC_FAILURE);
255             goto err;
256         }
257     }
258     BN_CTX_start(ctx);
259     a = BN_CTX_get(ctx);
260     b = BN_CTX_get(ctx);
261     tmp_1 = BN_CTX_get(ctx);
262     tmp_2 = BN_CTX_get(ctx);
263     order = BN_CTX_get(ctx);
264     if (order == NULL)
265         goto err;
266 
267     if (group->meth->field_decode) {
268         if (!group->meth->field_decode(group, a, group->a, ctx))
269             goto err;
270         if (!group->meth->field_decode(group, b, group->b, ctx))
271             goto err;
272     } else {
273         if (!BN_copy(a, group->a))
274             goto err;
275         if (!BN_copy(b, group->b))
276             goto err;
277     }
278 
279     /*-
280      * check the discriminant:
281      * y^2 = x^3 + a*x + b is an elliptic curve <=> 4*a^3 + 27*b^2 != 0 (mod p)
282      * 0 =< a, b < p
283      */
284     if (BN_is_zero(a)) {
285         if (BN_is_zero(b))
286             goto err;
287     } else if (!BN_is_zero(b)) {
288         if (!BN_mod_sqr(tmp_1, a, p, ctx))
289             goto err;
290         if (!BN_mod_mul(tmp_2, tmp_1, a, p, ctx))
291             goto err;
292         if (!BN_lshift(tmp_1, tmp_2, 2))
293             goto err;
294         /* tmp_1 = 4*a^3 */
295 
296         if (!BN_mod_sqr(tmp_2, b, p, ctx))
297             goto err;
298         if (!BN_mul_word(tmp_2, 27))
299             goto err;
300         /* tmp_2 = 27*b^2 */
301 
302         if (!BN_mod_add(a, tmp_1, tmp_2, p, ctx))
303             goto err;
304         if (BN_is_zero(a))
305             goto err;
306     }
307     ret = 1;
308 
309  err:
310     BN_CTX_end(ctx);
311     BN_CTX_free(new_ctx);
312     return ret;
313 }
314 
ec_GFp_simple_point_init(EC_POINT * point)315 int ec_GFp_simple_point_init(EC_POINT *point)
316 {
317     point->X = BN_new();
318     point->Y = BN_new();
319     point->Z = BN_new();
320     point->Z_is_one = 0;
321 
322     if (point->X == NULL || point->Y == NULL || point->Z == NULL) {
323         BN_free(point->X);
324         BN_free(point->Y);
325         BN_free(point->Z);
326         return 0;
327     }
328     return 1;
329 }
330 
ec_GFp_simple_point_finish(EC_POINT * point)331 void ec_GFp_simple_point_finish(EC_POINT *point)
332 {
333     BN_free(point->X);
334     BN_free(point->Y);
335     BN_free(point->Z);
336 }
337 
ec_GFp_simple_point_clear_finish(EC_POINT * point)338 void ec_GFp_simple_point_clear_finish(EC_POINT *point)
339 {
340     BN_clear_free(point->X);
341     BN_clear_free(point->Y);
342     BN_clear_free(point->Z);
343     point->Z_is_one = 0;
344 }
345 
ec_GFp_simple_point_copy(EC_POINT * dest,const EC_POINT * src)346 int ec_GFp_simple_point_copy(EC_POINT *dest, const EC_POINT *src)
347 {
348     if (!BN_copy(dest->X, src->X))
349         return 0;
350     if (!BN_copy(dest->Y, src->Y))
351         return 0;
352     if (!BN_copy(dest->Z, src->Z))
353         return 0;
354     dest->Z_is_one = src->Z_is_one;
355     dest->curve_name = src->curve_name;
356 
357     return 1;
358 }
359 
ec_GFp_simple_point_set_to_infinity(const EC_GROUP * group,EC_POINT * point)360 int ec_GFp_simple_point_set_to_infinity(const EC_GROUP *group,
361                                         EC_POINT *point)
362 {
363     point->Z_is_one = 0;
364     BN_zero(point->Z);
365     return 1;
366 }
367 
ec_GFp_simple_set_Jprojective_coordinates_GFp(const EC_GROUP * group,EC_POINT * point,const BIGNUM * x,const BIGNUM * y,const BIGNUM * z,BN_CTX * ctx)368 int ec_GFp_simple_set_Jprojective_coordinates_GFp(const EC_GROUP *group,
369                                                   EC_POINT *point,
370                                                   const BIGNUM *x,
371                                                   const BIGNUM *y,
372                                                   const BIGNUM *z,
373                                                   BN_CTX *ctx)
374 {
375     BN_CTX *new_ctx = NULL;
376     int ret = 0;
377 
378     if (ctx == NULL) {
379         ctx = new_ctx = BN_CTX_new();
380         if (ctx == NULL)
381             return 0;
382     }
383 
384     if (x != NULL) {
385         if (!BN_nnmod(point->X, x, group->field, ctx))
386             goto err;
387         if (group->meth->field_encode) {
388             if (!group->meth->field_encode(group, point->X, point->X, ctx))
389                 goto err;
390         }
391     }
392 
393     if (y != NULL) {
394         if (!BN_nnmod(point->Y, y, group->field, ctx))
395             goto err;
396         if (group->meth->field_encode) {
397             if (!group->meth->field_encode(group, point->Y, point->Y, ctx))
398                 goto err;
399         }
400     }
401 
402     if (z != NULL) {
403         int Z_is_one;
404 
405         if (!BN_nnmod(point->Z, z, group->field, ctx))
406             goto err;
407         Z_is_one = BN_is_one(point->Z);
408         if (group->meth->field_encode) {
409             if (Z_is_one && (group->meth->field_set_to_one != 0)) {
410                 if (!group->meth->field_set_to_one(group, point->Z, ctx))
411                     goto err;
412             } else {
413                 if (!group->
414                     meth->field_encode(group, point->Z, point->Z, ctx))
415                     goto err;
416             }
417         }
418         point->Z_is_one = Z_is_one;
419     }
420 
421     ret = 1;
422 
423  err:
424     BN_CTX_free(new_ctx);
425     return ret;
426 }
427 
ec_GFp_simple_get_Jprojective_coordinates_GFp(const EC_GROUP * group,const EC_POINT * point,BIGNUM * x,BIGNUM * y,BIGNUM * z,BN_CTX * ctx)428 int ec_GFp_simple_get_Jprojective_coordinates_GFp(const EC_GROUP *group,
429                                                   const EC_POINT *point,
430                                                   BIGNUM *x, BIGNUM *y,
431                                                   BIGNUM *z, BN_CTX *ctx)
432 {
433     BN_CTX *new_ctx = NULL;
434     int ret = 0;
435 
436     if (group->meth->field_decode != 0) {
437         if (ctx == NULL) {
438             ctx = new_ctx = BN_CTX_new();
439             if (ctx == NULL)
440                 return 0;
441         }
442 
443         if (x != NULL) {
444             if (!group->meth->field_decode(group, x, point->X, ctx))
445                 goto err;
446         }
447         if (y != NULL) {
448             if (!group->meth->field_decode(group, y, point->Y, ctx))
449                 goto err;
450         }
451         if (z != NULL) {
452             if (!group->meth->field_decode(group, z, point->Z, ctx))
453                 goto err;
454         }
455     } else {
456         if (x != NULL) {
457             if (!BN_copy(x, point->X))
458                 goto err;
459         }
460         if (y != NULL) {
461             if (!BN_copy(y, point->Y))
462                 goto err;
463         }
464         if (z != NULL) {
465             if (!BN_copy(z, point->Z))
466                 goto err;
467         }
468     }
469 
470     ret = 1;
471 
472  err:
473     BN_CTX_free(new_ctx);
474     return ret;
475 }
476 
ec_GFp_simple_point_set_affine_coordinates(const EC_GROUP * group,EC_POINT * point,const BIGNUM * x,const BIGNUM * y,BN_CTX * ctx)477 int ec_GFp_simple_point_set_affine_coordinates(const EC_GROUP *group,
478                                                EC_POINT *point,
479                                                const BIGNUM *x,
480                                                const BIGNUM *y, BN_CTX *ctx)
481 {
482     if (x == NULL || y == NULL) {
483         /*
484          * unlike for projective coordinates, we do not tolerate this
485          */
486         ECerr(EC_F_EC_GFP_SIMPLE_POINT_SET_AFFINE_COORDINATES,
487               ERR_R_PASSED_NULL_PARAMETER);
488         return 0;
489     }
490 
491     return EC_POINT_set_Jprojective_coordinates_GFp(group, point, x, y,
492                                                     BN_value_one(), ctx);
493 }
494 
ec_GFp_simple_point_get_affine_coordinates(const EC_GROUP * group,const EC_POINT * point,BIGNUM * x,BIGNUM * y,BN_CTX * ctx)495 int ec_GFp_simple_point_get_affine_coordinates(const EC_GROUP *group,
496                                                const EC_POINT *point,
497                                                BIGNUM *x, BIGNUM *y,
498                                                BN_CTX *ctx)
499 {
500     BN_CTX *new_ctx = NULL;
501     BIGNUM *Z, *Z_1, *Z_2, *Z_3;
502     const BIGNUM *Z_;
503     int ret = 0;
504 
505     if (EC_POINT_is_at_infinity(group, point)) {
506         ECerr(EC_F_EC_GFP_SIMPLE_POINT_GET_AFFINE_COORDINATES,
507               EC_R_POINT_AT_INFINITY);
508         return 0;
509     }
510 
511     if (ctx == NULL) {
512         ctx = new_ctx = BN_CTX_new();
513         if (ctx == NULL)
514             return 0;
515     }
516 
517     BN_CTX_start(ctx);
518     Z = BN_CTX_get(ctx);
519     Z_1 = BN_CTX_get(ctx);
520     Z_2 = BN_CTX_get(ctx);
521     Z_3 = BN_CTX_get(ctx);
522     if (Z_3 == NULL)
523         goto err;
524 
525     /* transform  (X, Y, Z)  into  (x, y) := (X/Z^2, Y/Z^3) */
526 
527     if (group->meth->field_decode) {
528         if (!group->meth->field_decode(group, Z, point->Z, ctx))
529             goto err;
530         Z_ = Z;
531     } else {
532         Z_ = point->Z;
533     }
534 
535     if (BN_is_one(Z_)) {
536         if (group->meth->field_decode) {
537             if (x != NULL) {
538                 if (!group->meth->field_decode(group, x, point->X, ctx))
539                     goto err;
540             }
541             if (y != NULL) {
542                 if (!group->meth->field_decode(group, y, point->Y, ctx))
543                     goto err;
544             }
545         } else {
546             if (x != NULL) {
547                 if (!BN_copy(x, point->X))
548                     goto err;
549             }
550             if (y != NULL) {
551                 if (!BN_copy(y, point->Y))
552                     goto err;
553             }
554         }
555     } else {
556         if (!group->meth->field_inv(group, Z_1, Z_, ctx)) {
557             ECerr(EC_F_EC_GFP_SIMPLE_POINT_GET_AFFINE_COORDINATES,
558                   ERR_R_BN_LIB);
559             goto err;
560         }
561 
562         if (group->meth->field_encode == 0) {
563             /* field_sqr works on standard representation */
564             if (!group->meth->field_sqr(group, Z_2, Z_1, ctx))
565                 goto err;
566         } else {
567             if (!BN_mod_sqr(Z_2, Z_1, group->field, ctx))
568                 goto err;
569         }
570 
571         if (x != NULL) {
572             /*
573              * in the Montgomery case, field_mul will cancel out Montgomery
574              * factor in X:
575              */
576             if (!group->meth->field_mul(group, x, point->X, Z_2, ctx))
577                 goto err;
578         }
579 
580         if (y != NULL) {
581             if (group->meth->field_encode == 0) {
582                 /*
583                  * field_mul works on standard representation
584                  */
585                 if (!group->meth->field_mul(group, Z_3, Z_2, Z_1, ctx))
586                     goto err;
587             } else {
588                 if (!BN_mod_mul(Z_3, Z_2, Z_1, group->field, ctx))
589                     goto err;
590             }
591 
592             /*
593              * in the Montgomery case, field_mul will cancel out Montgomery
594              * factor in Y:
595              */
596             if (!group->meth->field_mul(group, y, point->Y, Z_3, ctx))
597                 goto err;
598         }
599     }
600 
601     ret = 1;
602 
603  err:
604     BN_CTX_end(ctx);
605     BN_CTX_free(new_ctx);
606     return ret;
607 }
608 
ec_GFp_simple_add(const EC_GROUP * group,EC_POINT * r,const EC_POINT * a,const EC_POINT * b,BN_CTX * ctx)609 int ec_GFp_simple_add(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a,
610                       const EC_POINT *b, BN_CTX *ctx)
611 {
612     int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *,
613                       const BIGNUM *, BN_CTX *);
614     int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
615     const BIGNUM *p;
616     BN_CTX *new_ctx = NULL;
617     BIGNUM *n0, *n1, *n2, *n3, *n4, *n5, *n6;
618     int ret = 0;
619 
620     if (a == b)
621         return EC_POINT_dbl(group, r, a, ctx);
622     if (EC_POINT_is_at_infinity(group, a))
623         return EC_POINT_copy(r, b);
624     if (EC_POINT_is_at_infinity(group, b))
625         return EC_POINT_copy(r, a);
626 
627     field_mul = group->meth->field_mul;
628     field_sqr = group->meth->field_sqr;
629     p = group->field;
630 
631     if (ctx == NULL) {
632         ctx = new_ctx = BN_CTX_new();
633         if (ctx == NULL)
634             return 0;
635     }
636 
637     BN_CTX_start(ctx);
638     n0 = BN_CTX_get(ctx);
639     n1 = BN_CTX_get(ctx);
640     n2 = BN_CTX_get(ctx);
641     n3 = BN_CTX_get(ctx);
642     n4 = BN_CTX_get(ctx);
643     n5 = BN_CTX_get(ctx);
644     n6 = BN_CTX_get(ctx);
645     if (n6 == NULL)
646         goto end;
647 
648     /*
649      * Note that in this function we must not read components of 'a' or 'b'
650      * once we have written the corresponding components of 'r'. ('r' might
651      * be one of 'a' or 'b'.)
652      */
653 
654     /* n1, n2 */
655     if (b->Z_is_one) {
656         if (!BN_copy(n1, a->X))
657             goto end;
658         if (!BN_copy(n2, a->Y))
659             goto end;
660         /* n1 = X_a */
661         /* n2 = Y_a */
662     } else {
663         if (!field_sqr(group, n0, b->Z, ctx))
664             goto end;
665         if (!field_mul(group, n1, a->X, n0, ctx))
666             goto end;
667         /* n1 = X_a * Z_b^2 */
668 
669         if (!field_mul(group, n0, n0, b->Z, ctx))
670             goto end;
671         if (!field_mul(group, n2, a->Y, n0, ctx))
672             goto end;
673         /* n2 = Y_a * Z_b^3 */
674     }
675 
676     /* n3, n4 */
677     if (a->Z_is_one) {
678         if (!BN_copy(n3, b->X))
679             goto end;
680         if (!BN_copy(n4, b->Y))
681             goto end;
682         /* n3 = X_b */
683         /* n4 = Y_b */
684     } else {
685         if (!field_sqr(group, n0, a->Z, ctx))
686             goto end;
687         if (!field_mul(group, n3, b->X, n0, ctx))
688             goto end;
689         /* n3 = X_b * Z_a^2 */
690 
691         if (!field_mul(group, n0, n0, a->Z, ctx))
692             goto end;
693         if (!field_mul(group, n4, b->Y, n0, ctx))
694             goto end;
695         /* n4 = Y_b * Z_a^3 */
696     }
697 
698     /* n5, n6 */
699     if (!BN_mod_sub_quick(n5, n1, n3, p))
700         goto end;
701     if (!BN_mod_sub_quick(n6, n2, n4, p))
702         goto end;
703     /* n5 = n1 - n3 */
704     /* n6 = n2 - n4 */
705 
706     if (BN_is_zero(n5)) {
707         if (BN_is_zero(n6)) {
708             /* a is the same point as b */
709             BN_CTX_end(ctx);
710             ret = EC_POINT_dbl(group, r, a, ctx);
711             ctx = NULL;
712             goto end;
713         } else {
714             /* a is the inverse of b */
715             BN_zero(r->Z);
716             r->Z_is_one = 0;
717             ret = 1;
718             goto end;
719         }
720     }
721 
722     /* 'n7', 'n8' */
723     if (!BN_mod_add_quick(n1, n1, n3, p))
724         goto end;
725     if (!BN_mod_add_quick(n2, n2, n4, p))
726         goto end;
727     /* 'n7' = n1 + n3 */
728     /* 'n8' = n2 + n4 */
729 
730     /* Z_r */
731     if (a->Z_is_one && b->Z_is_one) {
732         if (!BN_copy(r->Z, n5))
733             goto end;
734     } else {
735         if (a->Z_is_one) {
736             if (!BN_copy(n0, b->Z))
737                 goto end;
738         } else if (b->Z_is_one) {
739             if (!BN_copy(n0, a->Z))
740                 goto end;
741         } else {
742             if (!field_mul(group, n0, a->Z, b->Z, ctx))
743                 goto end;
744         }
745         if (!field_mul(group, r->Z, n0, n5, ctx))
746             goto end;
747     }
748     r->Z_is_one = 0;
749     /* Z_r = Z_a * Z_b * n5 */
750 
751     /* X_r */
752     if (!field_sqr(group, n0, n6, ctx))
753         goto end;
754     if (!field_sqr(group, n4, n5, ctx))
755         goto end;
756     if (!field_mul(group, n3, n1, n4, ctx))
757         goto end;
758     if (!BN_mod_sub_quick(r->X, n0, n3, p))
759         goto end;
760     /* X_r = n6^2 - n5^2 * 'n7' */
761 
762     /* 'n9' */
763     if (!BN_mod_lshift1_quick(n0, r->X, p))
764         goto end;
765     if (!BN_mod_sub_quick(n0, n3, n0, p))
766         goto end;
767     /* n9 = n5^2 * 'n7' - 2 * X_r */
768 
769     /* Y_r */
770     if (!field_mul(group, n0, n0, n6, ctx))
771         goto end;
772     if (!field_mul(group, n5, n4, n5, ctx))
773         goto end;               /* now n5 is n5^3 */
774     if (!field_mul(group, n1, n2, n5, ctx))
775         goto end;
776     if (!BN_mod_sub_quick(n0, n0, n1, p))
777         goto end;
778     if (BN_is_odd(n0))
779         if (!BN_add(n0, n0, p))
780             goto end;
781     /* now  0 <= n0 < 2*p,  and n0 is even */
782     if (!BN_rshift1(r->Y, n0))
783         goto end;
784     /* Y_r = (n6 * 'n9' - 'n8' * 'n5^3') / 2 */
785 
786     ret = 1;
787 
788  end:
789     BN_CTX_end(ctx);
790     BN_CTX_free(new_ctx);
791     return ret;
792 }
793 
ec_GFp_simple_dbl(const EC_GROUP * group,EC_POINT * r,const EC_POINT * a,BN_CTX * ctx)794 int ec_GFp_simple_dbl(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a,
795                       BN_CTX *ctx)
796 {
797     int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *,
798                       const BIGNUM *, BN_CTX *);
799     int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
800     const BIGNUM *p;
801     BN_CTX *new_ctx = NULL;
802     BIGNUM *n0, *n1, *n2, *n3;
803     int ret = 0;
804 
805     if (EC_POINT_is_at_infinity(group, a)) {
806         BN_zero(r->Z);
807         r->Z_is_one = 0;
808         return 1;
809     }
810 
811     field_mul = group->meth->field_mul;
812     field_sqr = group->meth->field_sqr;
813     p = group->field;
814 
815     if (ctx == NULL) {
816         ctx = new_ctx = BN_CTX_new();
817         if (ctx == NULL)
818             return 0;
819     }
820 
821     BN_CTX_start(ctx);
822     n0 = BN_CTX_get(ctx);
823     n1 = BN_CTX_get(ctx);
824     n2 = BN_CTX_get(ctx);
825     n3 = BN_CTX_get(ctx);
826     if (n3 == NULL)
827         goto err;
828 
829     /*
830      * Note that in this function we must not read components of 'a' once we
831      * have written the corresponding components of 'r'. ('r' might the same
832      * as 'a'.)
833      */
834 
835     /* n1 */
836     if (a->Z_is_one) {
837         if (!field_sqr(group, n0, a->X, ctx))
838             goto err;
839         if (!BN_mod_lshift1_quick(n1, n0, p))
840             goto err;
841         if (!BN_mod_add_quick(n0, n0, n1, p))
842             goto err;
843         if (!BN_mod_add_quick(n1, n0, group->a, p))
844             goto err;
845         /* n1 = 3 * X_a^2 + a_curve */
846     } else if (group->a_is_minus3) {
847         if (!field_sqr(group, n1, a->Z, ctx))
848             goto err;
849         if (!BN_mod_add_quick(n0, a->X, n1, p))
850             goto err;
851         if (!BN_mod_sub_quick(n2, a->X, n1, p))
852             goto err;
853         if (!field_mul(group, n1, n0, n2, ctx))
854             goto err;
855         if (!BN_mod_lshift1_quick(n0, n1, p))
856             goto err;
857         if (!BN_mod_add_quick(n1, n0, n1, p))
858             goto err;
859         /*-
860          * n1 = 3 * (X_a + Z_a^2) * (X_a - Z_a^2)
861          *    = 3 * X_a^2 - 3 * Z_a^4
862          */
863     } else {
864         if (!field_sqr(group, n0, a->X, ctx))
865             goto err;
866         if (!BN_mod_lshift1_quick(n1, n0, p))
867             goto err;
868         if (!BN_mod_add_quick(n0, n0, n1, p))
869             goto err;
870         if (!field_sqr(group, n1, a->Z, ctx))
871             goto err;
872         if (!field_sqr(group, n1, n1, ctx))
873             goto err;
874         if (!field_mul(group, n1, n1, group->a, ctx))
875             goto err;
876         if (!BN_mod_add_quick(n1, n1, n0, p))
877             goto err;
878         /* n1 = 3 * X_a^2 + a_curve * Z_a^4 */
879     }
880 
881     /* Z_r */
882     if (a->Z_is_one) {
883         if (!BN_copy(n0, a->Y))
884             goto err;
885     } else {
886         if (!field_mul(group, n0, a->Y, a->Z, ctx))
887             goto err;
888     }
889     if (!BN_mod_lshift1_quick(r->Z, n0, p))
890         goto err;
891     r->Z_is_one = 0;
892     /* Z_r = 2 * Y_a * Z_a */
893 
894     /* n2 */
895     if (!field_sqr(group, n3, a->Y, ctx))
896         goto err;
897     if (!field_mul(group, n2, a->X, n3, ctx))
898         goto err;
899     if (!BN_mod_lshift_quick(n2, n2, 2, p))
900         goto err;
901     /* n2 = 4 * X_a * Y_a^2 */
902 
903     /* X_r */
904     if (!BN_mod_lshift1_quick(n0, n2, p))
905         goto err;
906     if (!field_sqr(group, r->X, n1, ctx))
907         goto err;
908     if (!BN_mod_sub_quick(r->X, r->X, n0, p))
909         goto err;
910     /* X_r = n1^2 - 2 * n2 */
911 
912     /* n3 */
913     if (!field_sqr(group, n0, n3, ctx))
914         goto err;
915     if (!BN_mod_lshift_quick(n3, n0, 3, p))
916         goto err;
917     /* n3 = 8 * Y_a^4 */
918 
919     /* Y_r */
920     if (!BN_mod_sub_quick(n0, n2, r->X, p))
921         goto err;
922     if (!field_mul(group, n0, n1, n0, ctx))
923         goto err;
924     if (!BN_mod_sub_quick(r->Y, n0, n3, p))
925         goto err;
926     /* Y_r = n1 * (n2 - X_r) - n3 */
927 
928     ret = 1;
929 
930  err:
931     BN_CTX_end(ctx);
932     BN_CTX_free(new_ctx);
933     return ret;
934 }
935 
ec_GFp_simple_invert(const EC_GROUP * group,EC_POINT * point,BN_CTX * ctx)936 int ec_GFp_simple_invert(const EC_GROUP *group, EC_POINT *point, BN_CTX *ctx)
937 {
938     if (EC_POINT_is_at_infinity(group, point) || BN_is_zero(point->Y))
939         /* point is its own inverse */
940         return 1;
941 
942     return BN_usub(point->Y, group->field, point->Y);
943 }
944 
ec_GFp_simple_is_at_infinity(const EC_GROUP * group,const EC_POINT * point)945 int ec_GFp_simple_is_at_infinity(const EC_GROUP *group, const EC_POINT *point)
946 {
947     return BN_is_zero(point->Z);
948 }
949 
ec_GFp_simple_is_on_curve(const EC_GROUP * group,const EC_POINT * point,BN_CTX * ctx)950 int ec_GFp_simple_is_on_curve(const EC_GROUP *group, const EC_POINT *point,
951                               BN_CTX *ctx)
952 {
953     int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *,
954                       const BIGNUM *, BN_CTX *);
955     int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
956     const BIGNUM *p;
957     BN_CTX *new_ctx = NULL;
958     BIGNUM *rh, *tmp, *Z4, *Z6;
959     int ret = -1;
960 
961     if (EC_POINT_is_at_infinity(group, point))
962         return 1;
963 
964     field_mul = group->meth->field_mul;
965     field_sqr = group->meth->field_sqr;
966     p = group->field;
967 
968     if (ctx == NULL) {
969         ctx = new_ctx = BN_CTX_new();
970         if (ctx == NULL)
971             return -1;
972     }
973 
974     BN_CTX_start(ctx);
975     rh = BN_CTX_get(ctx);
976     tmp = BN_CTX_get(ctx);
977     Z4 = BN_CTX_get(ctx);
978     Z6 = BN_CTX_get(ctx);
979     if (Z6 == NULL)
980         goto err;
981 
982     /*-
983      * We have a curve defined by a Weierstrass equation
984      *      y^2 = x^3 + a*x + b.
985      * The point to consider is given in Jacobian projective coordinates
986      * where  (X, Y, Z)  represents  (x, y) = (X/Z^2, Y/Z^3).
987      * Substituting this and multiplying by  Z^6  transforms the above equation into
988      *      Y^2 = X^3 + a*X*Z^4 + b*Z^6.
989      * To test this, we add up the right-hand side in 'rh'.
990      */
991 
992     /* rh := X^2 */
993     if (!field_sqr(group, rh, point->X, ctx))
994         goto err;
995 
996     if (!point->Z_is_one) {
997         if (!field_sqr(group, tmp, point->Z, ctx))
998             goto err;
999         if (!field_sqr(group, Z4, tmp, ctx))
1000             goto err;
1001         if (!field_mul(group, Z6, Z4, tmp, ctx))
1002             goto err;
1003 
1004         /* rh := (rh + a*Z^4)*X */
1005         if (group->a_is_minus3) {
1006             if (!BN_mod_lshift1_quick(tmp, Z4, p))
1007                 goto err;
1008             if (!BN_mod_add_quick(tmp, tmp, Z4, p))
1009                 goto err;
1010             if (!BN_mod_sub_quick(rh, rh, tmp, p))
1011                 goto err;
1012             if (!field_mul(group, rh, rh, point->X, ctx))
1013                 goto err;
1014         } else {
1015             if (!field_mul(group, tmp, Z4, group->a, ctx))
1016                 goto err;
1017             if (!BN_mod_add_quick(rh, rh, tmp, p))
1018                 goto err;
1019             if (!field_mul(group, rh, rh, point->X, ctx))
1020                 goto err;
1021         }
1022 
1023         /* rh := rh + b*Z^6 */
1024         if (!field_mul(group, tmp, group->b, Z6, ctx))
1025             goto err;
1026         if (!BN_mod_add_quick(rh, rh, tmp, p))
1027             goto err;
1028     } else {
1029         /* point->Z_is_one */
1030 
1031         /* rh := (rh + a)*X */
1032         if (!BN_mod_add_quick(rh, rh, group->a, p))
1033             goto err;
1034         if (!field_mul(group, rh, rh, point->X, ctx))
1035             goto err;
1036         /* rh := rh + b */
1037         if (!BN_mod_add_quick(rh, rh, group->b, p))
1038             goto err;
1039     }
1040 
1041     /* 'lh' := Y^2 */
1042     if (!field_sqr(group, tmp, point->Y, ctx))
1043         goto err;
1044 
1045     ret = (0 == BN_ucmp(tmp, rh));
1046 
1047  err:
1048     BN_CTX_end(ctx);
1049     BN_CTX_free(new_ctx);
1050     return ret;
1051 }
1052 
ec_GFp_simple_cmp(const EC_GROUP * group,const EC_POINT * a,const EC_POINT * b,BN_CTX * ctx)1053 int ec_GFp_simple_cmp(const EC_GROUP *group, const EC_POINT *a,
1054                       const EC_POINT *b, BN_CTX *ctx)
1055 {
1056     /*-
1057      * return values:
1058      *  -1   error
1059      *   0   equal (in affine coordinates)
1060      *   1   not equal
1061      */
1062 
1063     int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *,
1064                       const BIGNUM *, BN_CTX *);
1065     int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
1066     BN_CTX *new_ctx = NULL;
1067     BIGNUM *tmp1, *tmp2, *Za23, *Zb23;
1068     const BIGNUM *tmp1_, *tmp2_;
1069     int ret = -1;
1070 
1071     if (EC_POINT_is_at_infinity(group, a)) {
1072         return EC_POINT_is_at_infinity(group, b) ? 0 : 1;
1073     }
1074 
1075     if (EC_POINT_is_at_infinity(group, b))
1076         return 1;
1077 
1078     if (a->Z_is_one && b->Z_is_one) {
1079         return ((BN_cmp(a->X, b->X) == 0) && BN_cmp(a->Y, b->Y) == 0) ? 0 : 1;
1080     }
1081 
1082     field_mul = group->meth->field_mul;
1083     field_sqr = group->meth->field_sqr;
1084 
1085     if (ctx == NULL) {
1086         ctx = new_ctx = BN_CTX_new();
1087         if (ctx == NULL)
1088             return -1;
1089     }
1090 
1091     BN_CTX_start(ctx);
1092     tmp1 = BN_CTX_get(ctx);
1093     tmp2 = BN_CTX_get(ctx);
1094     Za23 = BN_CTX_get(ctx);
1095     Zb23 = BN_CTX_get(ctx);
1096     if (Zb23 == NULL)
1097         goto end;
1098 
1099     /*-
1100      * We have to decide whether
1101      *     (X_a/Z_a^2, Y_a/Z_a^3) = (X_b/Z_b^2, Y_b/Z_b^3),
1102      * or equivalently, whether
1103      *     (X_a*Z_b^2, Y_a*Z_b^3) = (X_b*Z_a^2, Y_b*Z_a^3).
1104      */
1105 
1106     if (!b->Z_is_one) {
1107         if (!field_sqr(group, Zb23, b->Z, ctx))
1108             goto end;
1109         if (!field_mul(group, tmp1, a->X, Zb23, ctx))
1110             goto end;
1111         tmp1_ = tmp1;
1112     } else
1113         tmp1_ = a->X;
1114     if (!a->Z_is_one) {
1115         if (!field_sqr(group, Za23, a->Z, ctx))
1116             goto end;
1117         if (!field_mul(group, tmp2, b->X, Za23, ctx))
1118             goto end;
1119         tmp2_ = tmp2;
1120     } else
1121         tmp2_ = b->X;
1122 
1123     /* compare  X_a*Z_b^2  with  X_b*Z_a^2 */
1124     if (BN_cmp(tmp1_, tmp2_) != 0) {
1125         ret = 1;                /* points differ */
1126         goto end;
1127     }
1128 
1129     if (!b->Z_is_one) {
1130         if (!field_mul(group, Zb23, Zb23, b->Z, ctx))
1131             goto end;
1132         if (!field_mul(group, tmp1, a->Y, Zb23, ctx))
1133             goto end;
1134         /* tmp1_ = tmp1 */
1135     } else
1136         tmp1_ = a->Y;
1137     if (!a->Z_is_one) {
1138         if (!field_mul(group, Za23, Za23, a->Z, ctx))
1139             goto end;
1140         if (!field_mul(group, tmp2, b->Y, Za23, ctx))
1141             goto end;
1142         /* tmp2_ = tmp2 */
1143     } else
1144         tmp2_ = b->Y;
1145 
1146     /* compare  Y_a*Z_b^3  with  Y_b*Z_a^3 */
1147     if (BN_cmp(tmp1_, tmp2_) != 0) {
1148         ret = 1;                /* points differ */
1149         goto end;
1150     }
1151 
1152     /* points are equal */
1153     ret = 0;
1154 
1155  end:
1156     BN_CTX_end(ctx);
1157     BN_CTX_free(new_ctx);
1158     return ret;
1159 }
1160 
ec_GFp_simple_make_affine(const EC_GROUP * group,EC_POINT * point,BN_CTX * ctx)1161 int ec_GFp_simple_make_affine(const EC_GROUP *group, EC_POINT *point,
1162                               BN_CTX *ctx)
1163 {
1164     BN_CTX *new_ctx = NULL;
1165     BIGNUM *x, *y;
1166     int ret = 0;
1167 
1168     if (point->Z_is_one || EC_POINT_is_at_infinity(group, point))
1169         return 1;
1170 
1171     if (ctx == NULL) {
1172         ctx = new_ctx = BN_CTX_new();
1173         if (ctx == NULL)
1174             return 0;
1175     }
1176 
1177     BN_CTX_start(ctx);
1178     x = BN_CTX_get(ctx);
1179     y = BN_CTX_get(ctx);
1180     if (y == NULL)
1181         goto err;
1182 
1183     if (!EC_POINT_get_affine_coordinates(group, point, x, y, ctx))
1184         goto err;
1185     if (!EC_POINT_set_affine_coordinates(group, point, x, y, ctx))
1186         goto err;
1187     if (!point->Z_is_one) {
1188         ECerr(EC_F_EC_GFP_SIMPLE_MAKE_AFFINE, ERR_R_INTERNAL_ERROR);
1189         goto err;
1190     }
1191 
1192     ret = 1;
1193 
1194  err:
1195     BN_CTX_end(ctx);
1196     BN_CTX_free(new_ctx);
1197     return ret;
1198 }
1199 
ec_GFp_simple_points_make_affine(const EC_GROUP * group,size_t num,EC_POINT * points[],BN_CTX * ctx)1200 int ec_GFp_simple_points_make_affine(const EC_GROUP *group, size_t num,
1201                                      EC_POINT *points[], BN_CTX *ctx)
1202 {
1203     BN_CTX *new_ctx = NULL;
1204     BIGNUM *tmp, *tmp_Z;
1205     BIGNUM **prod_Z = NULL;
1206     size_t i;
1207     int ret = 0;
1208 
1209     if (num == 0)
1210         return 1;
1211 
1212     if (ctx == NULL) {
1213         ctx = new_ctx = BN_CTX_new();
1214         if (ctx == NULL)
1215             return 0;
1216     }
1217 
1218     BN_CTX_start(ctx);
1219     tmp = BN_CTX_get(ctx);
1220     tmp_Z = BN_CTX_get(ctx);
1221     if (tmp_Z == NULL)
1222         goto err;
1223 
1224     prod_Z = OPENSSL_malloc(num * sizeof(prod_Z[0]));
1225     if (prod_Z == NULL)
1226         goto err;
1227     for (i = 0; i < num; i++) {
1228         prod_Z[i] = BN_new();
1229         if (prod_Z[i] == NULL)
1230             goto err;
1231     }
1232 
1233     /*
1234      * Set each prod_Z[i] to the product of points[0]->Z .. points[i]->Z,
1235      * skipping any zero-valued inputs (pretend that they're 1).
1236      */
1237 
1238     if (!BN_is_zero(points[0]->Z)) {
1239         if (!BN_copy(prod_Z[0], points[0]->Z))
1240             goto err;
1241     } else {
1242         if (group->meth->field_set_to_one != 0) {
1243             if (!group->meth->field_set_to_one(group, prod_Z[0], ctx))
1244                 goto err;
1245         } else {
1246             if (!BN_one(prod_Z[0]))
1247                 goto err;
1248         }
1249     }
1250 
1251     for (i = 1; i < num; i++) {
1252         if (!BN_is_zero(points[i]->Z)) {
1253             if (!group->
1254                 meth->field_mul(group, prod_Z[i], prod_Z[i - 1], points[i]->Z,
1255                                 ctx))
1256                 goto err;
1257         } else {
1258             if (!BN_copy(prod_Z[i], prod_Z[i - 1]))
1259                 goto err;
1260         }
1261     }
1262 
1263     /*
1264      * Now use a single explicit inversion to replace every non-zero
1265      * points[i]->Z by its inverse.
1266      */
1267 
1268     if (!group->meth->field_inv(group, tmp, prod_Z[num - 1], ctx)) {
1269         ECerr(EC_F_EC_GFP_SIMPLE_POINTS_MAKE_AFFINE, ERR_R_BN_LIB);
1270         goto err;
1271     }
1272     if (group->meth->field_encode != 0) {
1273         /*
1274          * In the Montgomery case, we just turned R*H (representing H) into
1275          * 1/(R*H), but we need R*(1/H) (representing 1/H); i.e. we need to
1276          * multiply by the Montgomery factor twice.
1277          */
1278         if (!group->meth->field_encode(group, tmp, tmp, ctx))
1279             goto err;
1280         if (!group->meth->field_encode(group, tmp, tmp, ctx))
1281             goto err;
1282     }
1283 
1284     for (i = num - 1; i > 0; --i) {
1285         /*
1286          * Loop invariant: tmp is the product of the inverses of points[0]->Z
1287          * .. points[i]->Z (zero-valued inputs skipped).
1288          */
1289         if (!BN_is_zero(points[i]->Z)) {
1290             /*
1291              * Set tmp_Z to the inverse of points[i]->Z (as product of Z
1292              * inverses 0 .. i, Z values 0 .. i - 1).
1293              */
1294             if (!group->
1295                 meth->field_mul(group, tmp_Z, prod_Z[i - 1], tmp, ctx))
1296                 goto err;
1297             /*
1298              * Update tmp to satisfy the loop invariant for i - 1.
1299              */
1300             if (!group->meth->field_mul(group, tmp, tmp, points[i]->Z, ctx))
1301                 goto err;
1302             /* Replace points[i]->Z by its inverse. */
1303             if (!BN_copy(points[i]->Z, tmp_Z))
1304                 goto err;
1305         }
1306     }
1307 
1308     if (!BN_is_zero(points[0]->Z)) {
1309         /* Replace points[0]->Z by its inverse. */
1310         if (!BN_copy(points[0]->Z, tmp))
1311             goto err;
1312     }
1313 
1314     /* Finally, fix up the X and Y coordinates for all points. */
1315 
1316     for (i = 0; i < num; i++) {
1317         EC_POINT *p = points[i];
1318 
1319         if (!BN_is_zero(p->Z)) {
1320             /* turn  (X, Y, 1/Z)  into  (X/Z^2, Y/Z^3, 1) */
1321 
1322             if (!group->meth->field_sqr(group, tmp, p->Z, ctx))
1323                 goto err;
1324             if (!group->meth->field_mul(group, p->X, p->X, tmp, ctx))
1325                 goto err;
1326 
1327             if (!group->meth->field_mul(group, tmp, tmp, p->Z, ctx))
1328                 goto err;
1329             if (!group->meth->field_mul(group, p->Y, p->Y, tmp, ctx))
1330                 goto err;
1331 
1332             if (group->meth->field_set_to_one != 0) {
1333                 if (!group->meth->field_set_to_one(group, p->Z, ctx))
1334                     goto err;
1335             } else {
1336                 if (!BN_one(p->Z))
1337                     goto err;
1338             }
1339             p->Z_is_one = 1;
1340         }
1341     }
1342 
1343     ret = 1;
1344 
1345  err:
1346     BN_CTX_end(ctx);
1347     BN_CTX_free(new_ctx);
1348     if (prod_Z != NULL) {
1349         for (i = 0; i < num; i++) {
1350             if (prod_Z[i] == NULL)
1351                 break;
1352             BN_clear_free(prod_Z[i]);
1353         }
1354         OPENSSL_free(prod_Z);
1355     }
1356     return ret;
1357 }
1358 
ec_GFp_simple_field_mul(const EC_GROUP * group,BIGNUM * r,const BIGNUM * a,const BIGNUM * b,BN_CTX * ctx)1359 int ec_GFp_simple_field_mul(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a,
1360                             const BIGNUM *b, BN_CTX *ctx)
1361 {
1362     return BN_mod_mul(r, a, b, group->field, ctx);
1363 }
1364 
ec_GFp_simple_field_sqr(const EC_GROUP * group,BIGNUM * r,const BIGNUM * a,BN_CTX * ctx)1365 int ec_GFp_simple_field_sqr(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a,
1366                             BN_CTX *ctx)
1367 {
1368     return BN_mod_sqr(r, a, group->field, ctx);
1369 }
1370 
1371 /*-
1372  * Computes the multiplicative inverse of a in GF(p), storing the result in r.
1373  * If a is zero (or equivalent), you'll get a EC_R_CANNOT_INVERT error.
1374  * Since we don't have a Mont structure here, SCA hardening is with blinding.
1375  * NB: "a" must be in _decoded_ form. (i.e. field_decode must precede.)
1376  */
ec_GFp_simple_field_inv(const EC_GROUP * group,BIGNUM * r,const BIGNUM * a,BN_CTX * ctx)1377 int ec_GFp_simple_field_inv(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a,
1378                             BN_CTX *ctx)
1379 {
1380     BIGNUM *e = NULL;
1381     BN_CTX *new_ctx = NULL;
1382     int ret = 0;
1383 
1384     if (ctx == NULL && (ctx = new_ctx = BN_CTX_secure_new()) == NULL)
1385         return 0;
1386 
1387     BN_CTX_start(ctx);
1388     if ((e = BN_CTX_get(ctx)) == NULL)
1389         goto err;
1390 
1391     do {
1392         if (!BN_priv_rand_range(e, group->field))
1393         goto err;
1394     } while (BN_is_zero(e));
1395 
1396     /* r := a * e */
1397     if (!group->meth->field_mul(group, r, a, e, ctx))
1398         goto err;
1399     /* r := 1/(a * e) */
1400     if (!BN_mod_inverse(r, r, group->field, ctx)) {
1401         ECerr(EC_F_EC_GFP_SIMPLE_FIELD_INV, EC_R_CANNOT_INVERT);
1402         goto err;
1403     }
1404     /* r := e/(a * e) = 1/a */
1405     if (!group->meth->field_mul(group, r, r, e, ctx))
1406         goto err;
1407 
1408     ret = 1;
1409 
1410  err:
1411     BN_CTX_end(ctx);
1412     BN_CTX_free(new_ctx);
1413     return ret;
1414 }
1415 
1416 /*-
1417  * Apply randomization of EC point projective coordinates:
1418  *
1419  *   (X, Y ,Z ) = (lambda^2*X, lambda^3*Y, lambda*Z)
1420  *   lambda = [1,group->field)
1421  *
1422  */
ec_GFp_simple_blind_coordinates(const EC_GROUP * group,EC_POINT * p,BN_CTX * ctx)1423 int ec_GFp_simple_blind_coordinates(const EC_GROUP *group, EC_POINT *p,
1424                                     BN_CTX *ctx)
1425 {
1426     int ret = 0;
1427     BIGNUM *lambda = NULL;
1428     BIGNUM *temp = NULL;
1429 
1430     BN_CTX_start(ctx);
1431     lambda = BN_CTX_get(ctx);
1432     temp = BN_CTX_get(ctx);
1433     if (temp == NULL) {
1434         ECerr(EC_F_EC_GFP_SIMPLE_BLIND_COORDINATES, ERR_R_MALLOC_FAILURE);
1435         goto end;
1436     }
1437 
1438     /*-
1439      * Make sure lambda is not zero.
1440      * If the RNG fails, we cannot blind but nevertheless want
1441      * code to continue smoothly and not clobber the error stack.
1442      */
1443     do {
1444         ERR_set_mark();
1445         ret = BN_priv_rand_range(lambda, group->field);
1446         ERR_pop_to_mark();
1447         if (ret == 0) {
1448             ret = 1;
1449             goto end;
1450         }
1451     } while (BN_is_zero(lambda));
1452 
1453     /* if field_encode defined convert between representations */
1454     if ((group->meth->field_encode != NULL
1455          && !group->meth->field_encode(group, lambda, lambda, ctx))
1456         || !group->meth->field_mul(group, p->Z, p->Z, lambda, ctx)
1457         || !group->meth->field_sqr(group, temp, lambda, ctx)
1458         || !group->meth->field_mul(group, p->X, p->X, temp, ctx)
1459         || !group->meth->field_mul(group, temp, temp, lambda, ctx)
1460         || !group->meth->field_mul(group, p->Y, p->Y, temp, ctx))
1461         goto end;
1462 
1463     p->Z_is_one = 0;
1464     ret = 1;
1465 
1466  end:
1467     BN_CTX_end(ctx);
1468     return ret;
1469 }
1470 
1471 /*-
1472  * Input:
1473  * - p: affine coordinates
1474  *
1475  * Output:
1476  * - s := p, r := 2p: blinded projective (homogeneous) coordinates
1477  *
1478  * For doubling we use Formula 3 from Izu-Takagi "A fast parallel elliptic curve
1479  * multiplication resistant against side channel attacks" appendix, described at
1480  * https://hyperelliptic.org/EFD/g1p/auto-shortw-xz.html#doubling-dbl-2002-it-2
1481  * simplified for Z1=1.
1482  *
1483  * Blinding uses the equivalence relation (\lambda X, \lambda Y, \lambda Z)
1484  * for any non-zero \lambda that holds for projective (homogeneous) coords.
1485  */
ec_GFp_simple_ladder_pre(const EC_GROUP * group,EC_POINT * r,EC_POINT * s,EC_POINT * p,BN_CTX * ctx)1486 int ec_GFp_simple_ladder_pre(const EC_GROUP *group,
1487                              EC_POINT *r, EC_POINT *s,
1488                              EC_POINT *p, BN_CTX *ctx)
1489 {
1490     BIGNUM *t1, *t2, *t3, *t4, *t5 = NULL;
1491 
1492     t1 = s->Z;
1493     t2 = r->Z;
1494     t3 = s->X;
1495     t4 = r->X;
1496     t5 = s->Y;
1497 
1498     if (!p->Z_is_one /* r := 2p */
1499         || !group->meth->field_sqr(group, t3, p->X, ctx)
1500         || !BN_mod_sub_quick(t4, t3, group->a, group->field)
1501         || !group->meth->field_sqr(group, t4, t4, ctx)
1502         || !group->meth->field_mul(group, t5, p->X, group->b, ctx)
1503         || !BN_mod_lshift_quick(t5, t5, 3, group->field)
1504         /* r->X coord output */
1505         || !BN_mod_sub_quick(r->X, t4, t5, group->field)
1506         || !BN_mod_add_quick(t1, t3, group->a, group->field)
1507         || !group->meth->field_mul(group, t2, p->X, t1, ctx)
1508         || !BN_mod_add_quick(t2, group->b, t2, group->field)
1509         /* r->Z coord output */
1510         || !BN_mod_lshift_quick(r->Z, t2, 2, group->field))
1511         return 0;
1512 
1513     /* make sure lambda (r->Y here for storage) is not zero */
1514     do {
1515         if (!BN_priv_rand_range(r->Y, group->field))
1516             return 0;
1517     } while (BN_is_zero(r->Y));
1518 
1519     /* make sure lambda (s->Z here for storage) is not zero */
1520     do {
1521         if (!BN_priv_rand_range(s->Z, group->field))
1522             return 0;
1523     } while (BN_is_zero(s->Z));
1524 
1525     /* if field_encode defined convert between representations */
1526     if (group->meth->field_encode != NULL
1527         && (!group->meth->field_encode(group, r->Y, r->Y, ctx)
1528             || !group->meth->field_encode(group, s->Z, s->Z, ctx)))
1529         return 0;
1530 
1531     /* blind r and s independently */
1532     if (!group->meth->field_mul(group, r->Z, r->Z, r->Y, ctx)
1533         || !group->meth->field_mul(group, r->X, r->X, r->Y, ctx)
1534         || !group->meth->field_mul(group, s->X, p->X, s->Z, ctx)) /* s := p */
1535         return 0;
1536 
1537     r->Z_is_one = 0;
1538     s->Z_is_one = 0;
1539 
1540     return 1;
1541 }
1542 
1543 /*-
1544  * Input:
1545  * - s, r: projective (homogeneous) coordinates
1546  * - p: affine coordinates
1547  *
1548  * Output:
1549  * - s := r + s, r := 2r: projective (homogeneous) coordinates
1550  *
1551  * Differential addition-and-doubling using Eq. (9) and (10) from Izu-Takagi
1552  * "A fast parallel elliptic curve multiplication resistant against side channel
1553  * attacks", as described at
1554  * https://hyperelliptic.org/EFD/g1p/auto-shortw-xz.html#ladder-mladd-2002-it-4
1555  */
ec_GFp_simple_ladder_step(const EC_GROUP * group,EC_POINT * r,EC_POINT * s,EC_POINT * p,BN_CTX * ctx)1556 int ec_GFp_simple_ladder_step(const EC_GROUP *group,
1557                               EC_POINT *r, EC_POINT *s,
1558                               EC_POINT *p, BN_CTX *ctx)
1559 {
1560     int ret = 0;
1561     BIGNUM *t0, *t1, *t2, *t3, *t4, *t5, *t6 = NULL;
1562 
1563     BN_CTX_start(ctx);
1564     t0 = BN_CTX_get(ctx);
1565     t1 = BN_CTX_get(ctx);
1566     t2 = BN_CTX_get(ctx);
1567     t3 = BN_CTX_get(ctx);
1568     t4 = BN_CTX_get(ctx);
1569     t5 = BN_CTX_get(ctx);
1570     t6 = BN_CTX_get(ctx);
1571 
1572     if (t6 == NULL
1573         || !group->meth->field_mul(group, t6, r->X, s->X, ctx)
1574         || !group->meth->field_mul(group, t0, r->Z, s->Z, ctx)
1575         || !group->meth->field_mul(group, t4, r->X, s->Z, ctx)
1576         || !group->meth->field_mul(group, t3, r->Z, s->X, ctx)
1577         || !group->meth->field_mul(group, t5, group->a, t0, ctx)
1578         || !BN_mod_add_quick(t5, t6, t5, group->field)
1579         || !BN_mod_add_quick(t6, t3, t4, group->field)
1580         || !group->meth->field_mul(group, t5, t6, t5, ctx)
1581         || !group->meth->field_sqr(group, t0, t0, ctx)
1582         || !BN_mod_lshift_quick(t2, group->b, 2, group->field)
1583         || !group->meth->field_mul(group, t0, t2, t0, ctx)
1584         || !BN_mod_lshift1_quick(t5, t5, group->field)
1585         || !BN_mod_sub_quick(t3, t4, t3, group->field)
1586         /* s->Z coord output */
1587         || !group->meth->field_sqr(group, s->Z, t3, ctx)
1588         || !group->meth->field_mul(group, t4, s->Z, p->X, ctx)
1589         || !BN_mod_add_quick(t0, t0, t5, group->field)
1590         /* s->X coord output */
1591         || !BN_mod_sub_quick(s->X, t0, t4, group->field)
1592         || !group->meth->field_sqr(group, t4, r->X, ctx)
1593         || !group->meth->field_sqr(group, t5, r->Z, ctx)
1594         || !group->meth->field_mul(group, t6, t5, group->a, ctx)
1595         || !BN_mod_add_quick(t1, r->X, r->Z, group->field)
1596         || !group->meth->field_sqr(group, t1, t1, ctx)
1597         || !BN_mod_sub_quick(t1, t1, t4, group->field)
1598         || !BN_mod_sub_quick(t1, t1, t5, group->field)
1599         || !BN_mod_sub_quick(t3, t4, t6, group->field)
1600         || !group->meth->field_sqr(group, t3, t3, ctx)
1601         || !group->meth->field_mul(group, t0, t5, t1, ctx)
1602         || !group->meth->field_mul(group, t0, t2, t0, ctx)
1603         /* r->X coord output */
1604         || !BN_mod_sub_quick(r->X, t3, t0, group->field)
1605         || !BN_mod_add_quick(t3, t4, t6, group->field)
1606         || !group->meth->field_sqr(group, t4, t5, ctx)
1607         || !group->meth->field_mul(group, t4, t4, t2, ctx)
1608         || !group->meth->field_mul(group, t1, t1, t3, ctx)
1609         || !BN_mod_lshift1_quick(t1, t1, group->field)
1610         /* r->Z coord output */
1611         || !BN_mod_add_quick(r->Z, t4, t1, group->field))
1612         goto err;
1613 
1614     ret = 1;
1615 
1616  err:
1617     BN_CTX_end(ctx);
1618     return ret;
1619 }
1620 
1621 /*-
1622  * Input:
1623  * - s, r: projective (homogeneous) coordinates
1624  * - p: affine coordinates
1625  *
1626  * Output:
1627  * - r := (x,y): affine coordinates
1628  *
1629  * Recovers the y-coordinate of r using Eq. (8) from Brier-Joye, "Weierstrass
1630  * Elliptic Curves and Side-Channel Attacks", modified to work in mixed
1631  * projective coords, i.e. p is affine and (r,s) in projective (homogeneous)
1632  * coords, and return r in affine coordinates.
1633  *
1634  * X4 = two*Y1*X2*Z3*Z2;
1635  * Y4 = two*b*Z3*SQR(Z2) + Z3*(a*Z2+X1*X2)*(X1*Z2+X2) - X3*SQR(X1*Z2-X2);
1636  * Z4 = two*Y1*Z3*SQR(Z2);
1637  *
1638  * Z4 != 0 because:
1639  *  - Z2==0 implies r is at infinity (handled by the BN_is_zero(r->Z) branch);
1640  *  - Z3==0 implies s is at infinity (handled by the BN_is_zero(s->Z) branch);
1641  *  - Y1==0 implies p has order 2, so either r or s are infinity and handled by
1642  *    one of the BN_is_zero(...) branches.
1643  */
ec_GFp_simple_ladder_post(const EC_GROUP * group,EC_POINT * r,EC_POINT * s,EC_POINT * p,BN_CTX * ctx)1644 int ec_GFp_simple_ladder_post(const EC_GROUP *group,
1645                               EC_POINT *r, EC_POINT *s,
1646                               EC_POINT *p, BN_CTX *ctx)
1647 {
1648     int ret = 0;
1649     BIGNUM *t0, *t1, *t2, *t3, *t4, *t5, *t6 = NULL;
1650 
1651     if (BN_is_zero(r->Z))
1652         return EC_POINT_set_to_infinity(group, r);
1653 
1654     if (BN_is_zero(s->Z)) {
1655         if (!EC_POINT_copy(r, p)
1656             || !EC_POINT_invert(group, r, ctx))
1657             return 0;
1658         return 1;
1659     }
1660 
1661     BN_CTX_start(ctx);
1662     t0 = BN_CTX_get(ctx);
1663     t1 = BN_CTX_get(ctx);
1664     t2 = BN_CTX_get(ctx);
1665     t3 = BN_CTX_get(ctx);
1666     t4 = BN_CTX_get(ctx);
1667     t5 = BN_CTX_get(ctx);
1668     t6 = BN_CTX_get(ctx);
1669 
1670     if (t6 == NULL
1671         || !BN_mod_lshift1_quick(t4, p->Y, group->field)
1672         || !group->meth->field_mul(group, t6, r->X, t4, ctx)
1673         || !group->meth->field_mul(group, t6, s->Z, t6, ctx)
1674         || !group->meth->field_mul(group, t5, r->Z, t6, ctx)
1675         || !BN_mod_lshift1_quick(t1, group->b, group->field)
1676         || !group->meth->field_mul(group, t1, s->Z, t1, ctx)
1677         || !group->meth->field_sqr(group, t3, r->Z, ctx)
1678         || !group->meth->field_mul(group, t2, t3, t1, ctx)
1679         || !group->meth->field_mul(group, t6, r->Z, group->a, ctx)
1680         || !group->meth->field_mul(group, t1, p->X, r->X, ctx)
1681         || !BN_mod_add_quick(t1, t1, t6, group->field)
1682         || !group->meth->field_mul(group, t1, s->Z, t1, ctx)
1683         || !group->meth->field_mul(group, t0, p->X, r->Z, ctx)
1684         || !BN_mod_add_quick(t6, r->X, t0, group->field)
1685         || !group->meth->field_mul(group, t6, t6, t1, ctx)
1686         || !BN_mod_add_quick(t6, t6, t2, group->field)
1687         || !BN_mod_sub_quick(t0, t0, r->X, group->field)
1688         || !group->meth->field_sqr(group, t0, t0, ctx)
1689         || !group->meth->field_mul(group, t0, t0, s->X, ctx)
1690         || !BN_mod_sub_quick(t0, t6, t0, group->field)
1691         || !group->meth->field_mul(group, t1, s->Z, t4, ctx)
1692         || !group->meth->field_mul(group, t1, t3, t1, ctx)
1693         || (group->meth->field_decode != NULL
1694             && !group->meth->field_decode(group, t1, t1, ctx))
1695         || !group->meth->field_inv(group, t1, t1, ctx)
1696         || (group->meth->field_encode != NULL
1697             && !group->meth->field_encode(group, t1, t1, ctx))
1698         || !group->meth->field_mul(group, r->X, t5, t1, ctx)
1699         || !group->meth->field_mul(group, r->Y, t0, t1, ctx))
1700         goto err;
1701 
1702     if (group->meth->field_set_to_one != NULL) {
1703         if (!group->meth->field_set_to_one(group, r->Z, ctx))
1704             goto err;
1705     } else {
1706         if (!BN_one(r->Z))
1707             goto err;
1708     }
1709 
1710     r->Z_is_one = 1;
1711     ret = 1;
1712 
1713  err:
1714     BN_CTX_end(ctx);
1715     return ret;
1716 }
1717