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
16const 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
91int 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
106void 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
113void 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
120int 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
134int 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
193int 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
238int ec_GFp_simple_group_get_degree(const EC_GROUP *group)
239{
240    return BN_num_bits(group->field);
241}
242
243int 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
315int 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
331void 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
338void 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
346int 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
360int 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
368int 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
428int 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
477int 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
495int 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
609int 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
794int 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
936int 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
945int ec_GFp_simple_is_at_infinity(const EC_GROUP *group, const EC_POINT *point)
946{
947    return BN_is_zero(point->Z);
948}
949
950int 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
1053int 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
1161int 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
1200int 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
1359int 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
1365int 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 */
1377int 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 */
1423int 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 */
1486int 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 */
1556int 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 */
1644int 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