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      1 /* Originally written by Bodo Moeller for the OpenSSL project.
      2  * ====================================================================
      3  * Copyright (c) 1998-2005 The OpenSSL Project.  All rights reserved.
      4  *
      5  * Redistribution and use in source and binary forms, with or without
      6  * modification, are permitted provided that the following conditions
      7  * are met:
      8  *
      9  * 1. Redistributions of source code must retain the above copyright
     10  *    notice, this list of conditions and the following disclaimer.
     11  *
     12  * 2. Redistributions in binary form must reproduce the above copyright
     13  *    notice, this list of conditions and the following disclaimer in
     14  *    the documentation and/or other materials provided with the
     15  *    distribution.
     16  *
     17  * 3. All advertising materials mentioning features or use of this
     18  *    software must display the following acknowledgment:
     19  *    "This product includes software developed by the OpenSSL Project
     20  *    for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
     21  *
     22  * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
     23  *    endorse or promote products derived from this software without
     24  *    prior written permission. For written permission, please contact
     25  *    openssl-core (at) openssl.org.
     26  *
     27  * 5. Products derived from this software may not be called "OpenSSL"
     28  *    nor may "OpenSSL" appear in their names without prior written
     29  *    permission of the OpenSSL Project.
     30  *
     31  * 6. Redistributions of any form whatsoever must retain the following
     32  *    acknowledgment:
     33  *    "This product includes software developed by the OpenSSL Project
     34  *    for use in the OpenSSL Toolkit (http://www.openssl.org/)"
     35  *
     36  * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
     37  * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
     38  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
     39  * PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL THE OpenSSL PROJECT OR
     40  * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
     41  * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
     42  * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
     43  * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
     44  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
     45  * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
     46  * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
     47  * OF THE POSSIBILITY OF SUCH DAMAGE.
     48  * ====================================================================
     49  *
     50  * This product includes cryptographic software written by Eric Young
     51  * (eay (at) cryptsoft.com).  This product includes software written by Tim
     52  * Hudson (tjh (at) cryptsoft.com).
     53  *
     54  */
     55 /* ====================================================================
     56  * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED.
     57  *
     58  * Portions of the attached software ("Contribution") are developed by
     59  * SUN MICROSYSTEMS, INC., and are contributed to the OpenSSL project.
     60  *
     61  * The Contribution is licensed pursuant to the OpenSSL open source
     62  * license provided above.
     63  *
     64  * The elliptic curve binary polynomial software is originally written by
     65  * Sheueling Chang Shantz and Douglas Stebila of Sun Microsystems
     66  * Laboratories. */
     67 
     68 #include <openssl/ec.h>
     69 
     70 #include <string.h>
     71 
     72 #include <openssl/bn.h>
     73 #include <openssl/err.h>
     74 #include <openssl/mem.h>
     75 
     76 #include "internal.h"
     77 #include "../internal.h"
     78 
     79 
     80 /* Most method functions in this file are designed to work with non-trivial
     81  * representations of field elements if necessary (see ecp_mont.c): while
     82  * standard modular addition and subtraction are used, the field_mul and
     83  * field_sqr methods will be used for multiplication, and field_encode and
     84  * field_decode (if defined) will be used for converting between
     85  * representations.
     86  *
     87  * Functions here specifically assume that if a non-trivial representation is
     88  * used, it is a Montgomery representation (i.e. 'encoding' means multiplying
     89  * by some factor R). */
     90 
     91 int ec_GFp_simple_group_init(EC_GROUP *group) {
     92   BN_init(&group->field);
     93   BN_init(&group->a);
     94   BN_init(&group->b);
     95   BN_init(&group->one);
     96   group->a_is_minus3 = 0;
     97   return 1;
     98 }
     99 
    100 void ec_GFp_simple_group_finish(EC_GROUP *group) {
    101   BN_free(&group->field);
    102   BN_free(&group->a);
    103   BN_free(&group->b);
    104   BN_free(&group->one);
    105 }
    106 
    107 int ec_GFp_simple_group_copy(EC_GROUP *dest, const EC_GROUP *src) {
    108   if (!BN_copy(&dest->field, &src->field) ||
    109       !BN_copy(&dest->a, &src->a) ||
    110       !BN_copy(&dest->b, &src->b) ||
    111       !BN_copy(&dest->one, &src->one)) {
    112     return 0;
    113   }
    114 
    115   dest->a_is_minus3 = src->a_is_minus3;
    116   return 1;
    117 }
    118 
    119 int ec_GFp_simple_group_set_curve(EC_GROUP *group, const BIGNUM *p,
    120                                   const BIGNUM *a, const BIGNUM *b,
    121                                   BN_CTX *ctx) {
    122   int ret = 0;
    123   BN_CTX *new_ctx = NULL;
    124   BIGNUM *tmp_a;
    125 
    126   /* p must be a prime > 3 */
    127   if (BN_num_bits(p) <= 2 || !BN_is_odd(p)) {
    128     OPENSSL_PUT_ERROR(EC, EC_R_INVALID_FIELD);
    129     return 0;
    130   }
    131 
    132   if (ctx == NULL) {
    133     ctx = new_ctx = BN_CTX_new();
    134     if (ctx == NULL) {
    135       return 0;
    136     }
    137   }
    138 
    139   BN_CTX_start(ctx);
    140   tmp_a = BN_CTX_get(ctx);
    141   if (tmp_a == NULL) {
    142     goto err;
    143   }
    144 
    145   /* group->field */
    146   if (!BN_copy(&group->field, p)) {
    147     goto err;
    148   }
    149   BN_set_negative(&group->field, 0);
    150 
    151   /* group->a */
    152   if (!BN_nnmod(tmp_a, a, p, ctx)) {
    153     goto err;
    154   }
    155   if (group->meth->field_encode) {
    156     if (!group->meth->field_encode(group, &group->a, tmp_a, ctx)) {
    157       goto err;
    158     }
    159   } else if (!BN_copy(&group->a, tmp_a)) {
    160     goto err;
    161   }
    162 
    163   /* group->b */
    164   if (!BN_nnmod(&group->b, b, p, ctx)) {
    165     goto err;
    166   }
    167   if (group->meth->field_encode &&
    168       !group->meth->field_encode(group, &group->b, &group->b, ctx)) {
    169     goto err;
    170   }
    171 
    172   /* group->a_is_minus3 */
    173   if (!BN_add_word(tmp_a, 3)) {
    174     goto err;
    175   }
    176   group->a_is_minus3 = (0 == BN_cmp(tmp_a, &group->field));
    177 
    178   if (group->meth->field_encode != NULL) {
    179     if (!group->meth->field_encode(group, &group->one, BN_value_one(), ctx)) {
    180       goto err;
    181     }
    182   } else if (!BN_copy(&group->one, BN_value_one())) {
    183     goto err;
    184   }
    185 
    186   ret = 1;
    187 
    188 err:
    189   BN_CTX_end(ctx);
    190   BN_CTX_free(new_ctx);
    191   return ret;
    192 }
    193 
    194 int ec_GFp_simple_group_get_curve(const EC_GROUP *group, BIGNUM *p, BIGNUM *a,
    195                                   BIGNUM *b, BN_CTX *ctx) {
    196   int ret = 0;
    197   BN_CTX *new_ctx = NULL;
    198 
    199   if (p != NULL && !BN_copy(p, &group->field)) {
    200     return 0;
    201   }
    202 
    203   if (a != NULL || b != NULL) {
    204     if (group->meth->field_decode) {
    205       if (ctx == NULL) {
    206         ctx = new_ctx = BN_CTX_new();
    207         if (ctx == NULL) {
    208           return 0;
    209         }
    210       }
    211       if (a != NULL && !group->meth->field_decode(group, a, &group->a, ctx)) {
    212         goto err;
    213       }
    214       if (b != NULL && !group->meth->field_decode(group, b, &group->b, ctx)) {
    215         goto err;
    216       }
    217     } else {
    218       if (a != NULL && !BN_copy(a, &group->a)) {
    219         goto err;
    220       }
    221       if (b != NULL && !BN_copy(b, &group->b)) {
    222         goto err;
    223       }
    224     }
    225   }
    226 
    227   ret = 1;
    228 
    229 err:
    230   BN_CTX_free(new_ctx);
    231   return ret;
    232 }
    233 
    234 unsigned ec_GFp_simple_group_get_degree(const EC_GROUP *group) {
    235   return BN_num_bits(&group->field);
    236 }
    237 
    238 int ec_GFp_simple_point_init(EC_POINT *point) {
    239   BN_init(&point->X);
    240   BN_init(&point->Y);
    241   BN_init(&point->Z);
    242 
    243   return 1;
    244 }
    245 
    246 void ec_GFp_simple_point_finish(EC_POINT *point) {
    247   BN_free(&point->X);
    248   BN_free(&point->Y);
    249   BN_free(&point->Z);
    250 }
    251 
    252 void ec_GFp_simple_point_clear_finish(EC_POINT *point) {
    253   BN_clear_free(&point->X);
    254   BN_clear_free(&point->Y);
    255   BN_clear_free(&point->Z);
    256 }
    257 
    258 int ec_GFp_simple_point_copy(EC_POINT *dest, const EC_POINT *src) {
    259   if (!BN_copy(&dest->X, &src->X) ||
    260       !BN_copy(&dest->Y, &src->Y) ||
    261       !BN_copy(&dest->Z, &src->Z)) {
    262     return 0;
    263   }
    264 
    265   return 1;
    266 }
    267 
    268 int ec_GFp_simple_point_set_to_infinity(const EC_GROUP *group,
    269                                         EC_POINT *point) {
    270   BN_zero(&point->Z);
    271   return 1;
    272 }
    273 
    274 static int set_Jprojective_coordinate_GFp(const EC_GROUP *group, BIGNUM *out,
    275                                           const BIGNUM *in, BN_CTX *ctx) {
    276   if (in == NULL) {
    277     return 1;
    278   }
    279   if (BN_is_negative(in) ||
    280       BN_cmp(in, &group->field) >= 0) {
    281     OPENSSL_PUT_ERROR(EC, EC_R_COORDINATES_OUT_OF_RANGE);
    282     return 0;
    283   }
    284   if (group->meth->field_encode) {
    285     return group->meth->field_encode(group, out, in, ctx);
    286   }
    287   return BN_copy(out, in) != NULL;
    288 }
    289 
    290 int ec_GFp_simple_set_Jprojective_coordinates_GFp(
    291     const EC_GROUP *group, EC_POINT *point, const BIGNUM *x, const BIGNUM *y,
    292     const BIGNUM *z, BN_CTX *ctx) {
    293   BN_CTX *new_ctx = NULL;
    294   int ret = 0;
    295 
    296   if (ctx == NULL) {
    297     ctx = new_ctx = BN_CTX_new();
    298     if (ctx == NULL) {
    299       return 0;
    300     }
    301   }
    302 
    303   if (!set_Jprojective_coordinate_GFp(group, &point->X, x, ctx) ||
    304       !set_Jprojective_coordinate_GFp(group, &point->Y, y, ctx) ||
    305       !set_Jprojective_coordinate_GFp(group, &point->Z, z, ctx)) {
    306     goto err;
    307   }
    308 
    309   ret = 1;
    310 
    311 err:
    312   BN_CTX_free(new_ctx);
    313   return ret;
    314 }
    315 
    316 int ec_GFp_simple_get_Jprojective_coordinates_GFp(const EC_GROUP *group,
    317                                                   const EC_POINT *point,
    318                                                   BIGNUM *x, BIGNUM *y,
    319                                                   BIGNUM *z, BN_CTX *ctx) {
    320   BN_CTX *new_ctx = NULL;
    321   int ret = 0;
    322 
    323   if (group->meth->field_decode != 0) {
    324     if (ctx == NULL) {
    325       ctx = new_ctx = BN_CTX_new();
    326       if (ctx == NULL) {
    327         return 0;
    328       }
    329     }
    330 
    331     if (x != NULL && !group->meth->field_decode(group, x, &point->X, ctx)) {
    332       goto err;
    333     }
    334     if (y != NULL && !group->meth->field_decode(group, y, &point->Y, ctx)) {
    335       goto err;
    336     }
    337     if (z != NULL && !group->meth->field_decode(group, z, &point->Z, ctx)) {
    338       goto err;
    339     }
    340   } else {
    341     if (x != NULL && !BN_copy(x, &point->X)) {
    342       goto err;
    343     }
    344     if (y != NULL && !BN_copy(y, &point->Y)) {
    345       goto err;
    346     }
    347     if (z != NULL && !BN_copy(z, &point->Z)) {
    348       goto err;
    349     }
    350   }
    351 
    352   ret = 1;
    353 
    354 err:
    355   BN_CTX_free(new_ctx);
    356   return ret;
    357 }
    358 
    359 int ec_GFp_simple_point_set_affine_coordinates(const EC_GROUP *group,
    360                                                EC_POINT *point, const BIGNUM *x,
    361                                                const BIGNUM *y, BN_CTX *ctx) {
    362   if (x == NULL || y == NULL) {
    363     /* unlike for projective coordinates, we do not tolerate this */
    364     OPENSSL_PUT_ERROR(EC, ERR_R_PASSED_NULL_PARAMETER);
    365     return 0;
    366   }
    367 
    368   return ec_point_set_Jprojective_coordinates_GFp(group, point, x, y,
    369                                                   BN_value_one(), ctx);
    370 }
    371 
    372 int ec_GFp_simple_add(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a,
    373                       const EC_POINT *b, BN_CTX *ctx) {
    374   int (*field_mul)(const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *,
    375                    BN_CTX *);
    376   int (*field_sqr)(const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
    377   const BIGNUM *p;
    378   BN_CTX *new_ctx = NULL;
    379   BIGNUM *n0, *n1, *n2, *n3, *n4, *n5, *n6;
    380   int ret = 0;
    381 
    382   if (a == b) {
    383     return EC_POINT_dbl(group, r, a, ctx);
    384   }
    385   if (EC_POINT_is_at_infinity(group, a)) {
    386     return EC_POINT_copy(r, b);
    387   }
    388   if (EC_POINT_is_at_infinity(group, b)) {
    389     return EC_POINT_copy(r, a);
    390   }
    391 
    392   field_mul = group->meth->field_mul;
    393   field_sqr = group->meth->field_sqr;
    394   p = &group->field;
    395 
    396   if (ctx == NULL) {
    397     ctx = new_ctx = BN_CTX_new();
    398     if (ctx == NULL) {
    399       return 0;
    400     }
    401   }
    402 
    403   BN_CTX_start(ctx);
    404   n0 = BN_CTX_get(ctx);
    405   n1 = BN_CTX_get(ctx);
    406   n2 = BN_CTX_get(ctx);
    407   n3 = BN_CTX_get(ctx);
    408   n4 = BN_CTX_get(ctx);
    409   n5 = BN_CTX_get(ctx);
    410   n6 = BN_CTX_get(ctx);
    411   if (n6 == NULL) {
    412     goto end;
    413   }
    414 
    415   /* Note that in this function we must not read components of 'a' or 'b'
    416    * once we have written the corresponding components of 'r'.
    417    * ('r' might be one of 'a' or 'b'.)
    418    */
    419 
    420   /* n1, n2 */
    421   int b_Z_is_one = BN_cmp(&b->Z, &group->one) == 0;
    422 
    423   if (b_Z_is_one) {
    424     if (!BN_copy(n1, &a->X) || !BN_copy(n2, &a->Y)) {
    425       goto end;
    426     }
    427     /* n1 = X_a */
    428     /* n2 = Y_a */
    429   } else {
    430     if (!field_sqr(group, n0, &b->Z, ctx) ||
    431         !field_mul(group, n1, &a->X, n0, ctx)) {
    432       goto end;
    433     }
    434     /* n1 = X_a * Z_b^2 */
    435 
    436     if (!field_mul(group, n0, n0, &b->Z, ctx) ||
    437         !field_mul(group, n2, &a->Y, n0, ctx)) {
    438       goto end;
    439     }
    440     /* n2 = Y_a * Z_b^3 */
    441   }
    442 
    443   /* n3, n4 */
    444   int a_Z_is_one = BN_cmp(&a->Z, &group->one) == 0;
    445   if (a_Z_is_one) {
    446     if (!BN_copy(n3, &b->X) || !BN_copy(n4, &b->Y)) {
    447       goto end;
    448     }
    449     /* n3 = X_b */
    450     /* n4 = Y_b */
    451   } else {
    452     if (!field_sqr(group, n0, &a->Z, ctx) ||
    453         !field_mul(group, n3, &b->X, n0, ctx)) {
    454       goto end;
    455     }
    456     /* n3 = X_b * Z_a^2 */
    457 
    458     if (!field_mul(group, n0, n0, &a->Z, ctx) ||
    459         !field_mul(group, n4, &b->Y, n0, ctx)) {
    460       goto end;
    461     }
    462     /* n4 = Y_b * Z_a^3 */
    463   }
    464 
    465   /* n5, n6 */
    466   if (!BN_mod_sub_quick(n5, n1, n3, p) ||
    467       !BN_mod_sub_quick(n6, n2, n4, p)) {
    468     goto end;
    469   }
    470   /* n5 = n1 - n3 */
    471   /* n6 = n2 - n4 */
    472 
    473   if (BN_is_zero(n5)) {
    474     if (BN_is_zero(n6)) {
    475       /* a is the same point as b */
    476       BN_CTX_end(ctx);
    477       ret = EC_POINT_dbl(group, r, a, ctx);
    478       ctx = NULL;
    479       goto end;
    480     } else {
    481       /* a is the inverse of b */
    482       BN_zero(&r->Z);
    483       ret = 1;
    484       goto end;
    485     }
    486   }
    487 
    488   /* 'n7', 'n8' */
    489   if (!BN_mod_add_quick(n1, n1, n3, p) ||
    490       !BN_mod_add_quick(n2, n2, n4, p)) {
    491     goto end;
    492   }
    493   /* 'n7' = n1 + n3 */
    494   /* 'n8' = n2 + n4 */
    495 
    496   /* Z_r */
    497   if (a_Z_is_one && b_Z_is_one) {
    498     if (!BN_copy(&r->Z, n5)) {
    499       goto end;
    500     }
    501   } else {
    502     if (a_Z_is_one) {
    503       if (!BN_copy(n0, &b->Z)) {
    504         goto end;
    505       }
    506     } else if (b_Z_is_one) {
    507       if (!BN_copy(n0, &a->Z)) {
    508         goto end;
    509       }
    510     } else if (!field_mul(group, n0, &a->Z, &b->Z, ctx)) {
    511       goto end;
    512     }
    513     if (!field_mul(group, &r->Z, n0, n5, ctx)) {
    514       goto end;
    515     }
    516   }
    517 
    518   /* Z_r = Z_a * Z_b * n5 */
    519 
    520   /* X_r */
    521   if (!field_sqr(group, n0, n6, ctx) ||
    522       !field_sqr(group, n4, n5, ctx) ||
    523       !field_mul(group, n3, n1, n4, ctx) ||
    524       !BN_mod_sub_quick(&r->X, n0, n3, p)) {
    525     goto end;
    526   }
    527   /* X_r = n6^2 - n5^2 * 'n7' */
    528 
    529   /* 'n9' */
    530   if (!BN_mod_lshift1_quick(n0, &r->X, p) ||
    531       !BN_mod_sub_quick(n0, n3, n0, p)) {
    532     goto end;
    533   }
    534   /* n9 = n5^2 * 'n7' - 2 * X_r */
    535 
    536   /* Y_r */
    537   if (!field_mul(group, n0, n0, n6, ctx) ||
    538       !field_mul(group, n5, n4, n5, ctx)) {
    539     goto end; /* now n5 is n5^3 */
    540   }
    541   if (!field_mul(group, n1, n2, n5, ctx) ||
    542       !BN_mod_sub_quick(n0, n0, n1, p)) {
    543     goto end;
    544   }
    545   if (BN_is_odd(n0) && !BN_add(n0, n0, p)) {
    546     goto end;
    547   }
    548   /* now  0 <= n0 < 2*p,  and n0 is even */
    549   if (!BN_rshift1(&r->Y, n0)) {
    550     goto end;
    551   }
    552   /* Y_r = (n6 * 'n9' - 'n8' * 'n5^3') / 2 */
    553 
    554   ret = 1;
    555 
    556 end:
    557   if (ctx) {
    558     /* otherwise we already called BN_CTX_end */
    559     BN_CTX_end(ctx);
    560   }
    561   BN_CTX_free(new_ctx);
    562   return ret;
    563 }
    564 
    565 int ec_GFp_simple_dbl(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a,
    566                       BN_CTX *ctx) {
    567   int (*field_mul)(const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *,
    568                    BN_CTX *);
    569   int (*field_sqr)(const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
    570   const BIGNUM *p;
    571   BN_CTX *new_ctx = NULL;
    572   BIGNUM *n0, *n1, *n2, *n3;
    573   int ret = 0;
    574 
    575   if (EC_POINT_is_at_infinity(group, a)) {
    576     BN_zero(&r->Z);
    577     return 1;
    578   }
    579 
    580   field_mul = group->meth->field_mul;
    581   field_sqr = group->meth->field_sqr;
    582   p = &group->field;
    583 
    584   if (ctx == NULL) {
    585     ctx = new_ctx = BN_CTX_new();
    586     if (ctx == NULL) {
    587       return 0;
    588     }
    589   }
    590 
    591   BN_CTX_start(ctx);
    592   n0 = BN_CTX_get(ctx);
    593   n1 = BN_CTX_get(ctx);
    594   n2 = BN_CTX_get(ctx);
    595   n3 = BN_CTX_get(ctx);
    596   if (n3 == NULL) {
    597     goto err;
    598   }
    599 
    600   /* Note that in this function we must not read components of 'a'
    601    * once we have written the corresponding components of 'r'.
    602    * ('r' might the same as 'a'.)
    603    */
    604 
    605   /* n1 */
    606   if (BN_cmp(&a->Z, &group->one) == 0) {
    607     if (!field_sqr(group, n0, &a->X, ctx) ||
    608         !BN_mod_lshift1_quick(n1, n0, p) ||
    609         !BN_mod_add_quick(n0, n0, n1, p) ||
    610         !BN_mod_add_quick(n1, n0, &group->a, p)) {
    611       goto err;
    612     }
    613     /* n1 = 3 * X_a^2 + a_curve */
    614   } else if (group->a_is_minus3) {
    615     if (!field_sqr(group, n1, &a->Z, ctx) ||
    616         !BN_mod_add_quick(n0, &a->X, n1, p) ||
    617         !BN_mod_sub_quick(n2, &a->X, n1, p) ||
    618         !field_mul(group, n1, n0, n2, ctx) ||
    619         !BN_mod_lshift1_quick(n0, n1, p) ||
    620         !BN_mod_add_quick(n1, n0, n1, p)) {
    621       goto err;
    622     }
    623     /* n1 = 3 * (X_a + Z_a^2) * (X_a - Z_a^2)
    624      *    = 3 * X_a^2 - 3 * Z_a^4 */
    625   } else {
    626     if (!field_sqr(group, n0, &a->X, ctx) ||
    627         !BN_mod_lshift1_quick(n1, n0, p) ||
    628         !BN_mod_add_quick(n0, n0, n1, p) ||
    629         !field_sqr(group, n1, &a->Z, ctx) ||
    630         !field_sqr(group, n1, n1, ctx) ||
    631         !field_mul(group, n1, n1, &group->a, ctx) ||
    632         !BN_mod_add_quick(n1, n1, n0, p)) {
    633       goto err;
    634     }
    635     /* n1 = 3 * X_a^2 + a_curve * Z_a^4 */
    636   }
    637 
    638   /* Z_r */
    639   if (BN_cmp(&a->Z, &group->one) == 0) {
    640     if (!BN_copy(n0, &a->Y)) {
    641       goto err;
    642     }
    643   } else if (!field_mul(group, n0, &a->Y, &a->Z, ctx)) {
    644     goto err;
    645   }
    646   if (!BN_mod_lshift1_quick(&r->Z, n0, p)) {
    647     goto err;
    648   }
    649   /* Z_r = 2 * Y_a * Z_a */
    650 
    651   /* n2 */
    652   if (!field_sqr(group, n3, &a->Y, ctx) ||
    653       !field_mul(group, n2, &a->X, n3, ctx) ||
    654       !BN_mod_lshift_quick(n2, n2, 2, p)) {
    655     goto err;
    656   }
    657   /* n2 = 4 * X_a * Y_a^2 */
    658 
    659   /* X_r */
    660   if (!BN_mod_lshift1_quick(n0, n2, p) ||
    661       !field_sqr(group, &r->X, n1, ctx) ||
    662       !BN_mod_sub_quick(&r->X, &r->X, n0, p)) {
    663     goto err;
    664   }
    665   /* X_r = n1^2 - 2 * n2 */
    666 
    667   /* n3 */
    668   if (!field_sqr(group, n0, n3, ctx) ||
    669       !BN_mod_lshift_quick(n3, n0, 3, p)) {
    670     goto err;
    671   }
    672   /* n3 = 8 * Y_a^4 */
    673 
    674   /* Y_r */
    675   if (!BN_mod_sub_quick(n0, n2, &r->X, p) ||
    676       !field_mul(group, n0, n1, n0, ctx) ||
    677       !BN_mod_sub_quick(&r->Y, n0, n3, p)) {
    678     goto err;
    679   }
    680   /* Y_r = n1 * (n2 - X_r) - n3 */
    681 
    682   ret = 1;
    683 
    684 err:
    685   BN_CTX_end(ctx);
    686   BN_CTX_free(new_ctx);
    687   return ret;
    688 }
    689 
    690 int ec_GFp_simple_invert(const EC_GROUP *group, EC_POINT *point, BN_CTX *ctx) {
    691   if (EC_POINT_is_at_infinity(group, point) || BN_is_zero(&point->Y)) {
    692     /* point is its own inverse */
    693     return 1;
    694   }
    695 
    696   return BN_usub(&point->Y, &group->field, &point->Y);
    697 }
    698 
    699 int ec_GFp_simple_is_at_infinity(const EC_GROUP *group, const EC_POINT *point) {
    700   return BN_is_zero(&point->Z);
    701 }
    702 
    703 int ec_GFp_simple_is_on_curve(const EC_GROUP *group, const EC_POINT *point,
    704                               BN_CTX *ctx) {
    705   int (*field_mul)(const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *,
    706                    BN_CTX *);
    707   int (*field_sqr)(const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
    708   const BIGNUM *p;
    709   BN_CTX *new_ctx = NULL;
    710   BIGNUM *rh, *tmp, *Z4, *Z6;
    711   int ret = 0;
    712 
    713   if (EC_POINT_is_at_infinity(group, point)) {
    714     return 1;
    715   }
    716 
    717   field_mul = group->meth->field_mul;
    718   field_sqr = group->meth->field_sqr;
    719   p = &group->field;
    720 
    721   if (ctx == NULL) {
    722     ctx = new_ctx = BN_CTX_new();
    723     if (ctx == NULL) {
    724       return 0;
    725     }
    726   }
    727 
    728   BN_CTX_start(ctx);
    729   rh = BN_CTX_get(ctx);
    730   tmp = BN_CTX_get(ctx);
    731   Z4 = BN_CTX_get(ctx);
    732   Z6 = BN_CTX_get(ctx);
    733   if (Z6 == NULL) {
    734     goto err;
    735   }
    736 
    737   /* We have a curve defined by a Weierstrass equation
    738    *      y^2 = x^3 + a*x + b.
    739    * The point to consider is given in Jacobian projective coordinates
    740    * where  (X, Y, Z)  represents  (x, y) = (X/Z^2, Y/Z^3).
    741    * Substituting this and multiplying by  Z^6  transforms the above equation
    742    * into
    743    *      Y^2 = X^3 + a*X*Z^4 + b*Z^6.
    744    * To test this, we add up the right-hand side in 'rh'.
    745    */
    746 
    747   /* rh := X^2 */
    748   if (!field_sqr(group, rh, &point->X, ctx)) {
    749     goto err;
    750   }
    751 
    752   if (BN_cmp(&point->Z, &group->one) != 0) {
    753     if (!field_sqr(group, tmp, &point->Z, ctx) ||
    754         !field_sqr(group, Z4, tmp, ctx) ||
    755         !field_mul(group, Z6, Z4, tmp, ctx)) {
    756       goto err;
    757     }
    758 
    759     /* rh := (rh + a*Z^4)*X */
    760     if (group->a_is_minus3) {
    761       if (!BN_mod_lshift1_quick(tmp, Z4, p) ||
    762           !BN_mod_add_quick(tmp, tmp, Z4, p) ||
    763           !BN_mod_sub_quick(rh, rh, tmp, p) ||
    764           !field_mul(group, rh, rh, &point->X, ctx)) {
    765         goto err;
    766       }
    767     } else {
    768       if (!field_mul(group, tmp, Z4, &group->a, ctx) ||
    769           !BN_mod_add_quick(rh, rh, tmp, p) ||
    770           !field_mul(group, rh, rh, &point->X, ctx)) {
    771         goto err;
    772       }
    773     }
    774 
    775     /* rh := rh + b*Z^6 */
    776     if (!field_mul(group, tmp, &group->b, Z6, ctx) ||
    777         !BN_mod_add_quick(rh, rh, tmp, p)) {
    778       goto err;
    779     }
    780   } else {
    781     /* rh := (rh + a)*X */
    782     if (!BN_mod_add_quick(rh, rh, &group->a, p) ||
    783         !field_mul(group, rh, rh, &point->X, ctx)) {
    784       goto err;
    785     }
    786     /* rh := rh + b */
    787     if (!BN_mod_add_quick(rh, rh, &group->b, p)) {
    788       goto err;
    789     }
    790   }
    791 
    792   /* 'lh' := Y^2 */
    793   if (!field_sqr(group, tmp, &point->Y, ctx)) {
    794     goto err;
    795   }
    796 
    797   ret = (0 == BN_ucmp(tmp, rh));
    798 
    799 err:
    800   BN_CTX_end(ctx);
    801   BN_CTX_free(new_ctx);
    802   return ret;
    803 }
    804 
    805 int ec_GFp_simple_cmp(const EC_GROUP *group, const EC_POINT *a,
    806                       const EC_POINT *b, BN_CTX *ctx) {
    807   /* return values:
    808    *  -1   error
    809    *   0   equal (in affine coordinates)
    810    *   1   not equal
    811    */
    812 
    813   int (*field_mul)(const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *,
    814                    BN_CTX *);
    815   int (*field_sqr)(const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
    816   BN_CTX *new_ctx = NULL;
    817   BIGNUM *tmp1, *tmp2, *Za23, *Zb23;
    818   const BIGNUM *tmp1_, *tmp2_;
    819   int ret = -1;
    820 
    821   if (EC_POINT_is_at_infinity(group, a)) {
    822     return EC_POINT_is_at_infinity(group, b) ? 0 : 1;
    823   }
    824 
    825   if (EC_POINT_is_at_infinity(group, b)) {
    826     return 1;
    827   }
    828 
    829   int a_Z_is_one = BN_cmp(&a->Z, &group->one) == 0;
    830   int b_Z_is_one = BN_cmp(&b->Z, &group->one) == 0;
    831 
    832   if (a_Z_is_one && b_Z_is_one) {
    833     return ((BN_cmp(&a->X, &b->X) == 0) && BN_cmp(&a->Y, &b->Y) == 0) ? 0 : 1;
    834   }
    835 
    836   field_mul = group->meth->field_mul;
    837   field_sqr = group->meth->field_sqr;
    838 
    839   if (ctx == NULL) {
    840     ctx = new_ctx = BN_CTX_new();
    841     if (ctx == NULL) {
    842       return -1;
    843     }
    844   }
    845 
    846   BN_CTX_start(ctx);
    847   tmp1 = BN_CTX_get(ctx);
    848   tmp2 = BN_CTX_get(ctx);
    849   Za23 = BN_CTX_get(ctx);
    850   Zb23 = BN_CTX_get(ctx);
    851   if (Zb23 == NULL) {
    852     goto end;
    853   }
    854 
    855   /* We have to decide whether
    856    *     (X_a/Z_a^2, Y_a/Z_a^3) = (X_b/Z_b^2, Y_b/Z_b^3),
    857    * or equivalently, whether
    858    *     (X_a*Z_b^2, Y_a*Z_b^3) = (X_b*Z_a^2, Y_b*Z_a^3).
    859    */
    860 
    861   if (!b_Z_is_one) {
    862     if (!field_sqr(group, Zb23, &b->Z, ctx) ||
    863         !field_mul(group, tmp1, &a->X, Zb23, ctx)) {
    864       goto end;
    865     }
    866     tmp1_ = tmp1;
    867   } else {
    868     tmp1_ = &a->X;
    869   }
    870   if (!a_Z_is_one) {
    871     if (!field_sqr(group, Za23, &a->Z, ctx) ||
    872         !field_mul(group, tmp2, &b->X, Za23, ctx)) {
    873       goto end;
    874     }
    875     tmp2_ = tmp2;
    876   } else {
    877     tmp2_ = &b->X;
    878   }
    879 
    880   /* compare  X_a*Z_b^2  with  X_b*Z_a^2 */
    881   if (BN_cmp(tmp1_, tmp2_) != 0) {
    882     ret = 1; /* points differ */
    883     goto end;
    884   }
    885 
    886 
    887   if (!b_Z_is_one) {
    888     if (!field_mul(group, Zb23, Zb23, &b->Z, ctx) ||
    889         !field_mul(group, tmp1, &a->Y, Zb23, ctx)) {
    890       goto end;
    891     }
    892     /* tmp1_ = tmp1 */
    893   } else {
    894     tmp1_ = &a->Y;
    895   }
    896   if (!a_Z_is_one) {
    897     if (!field_mul(group, Za23, Za23, &a->Z, ctx) ||
    898         !field_mul(group, tmp2, &b->Y, Za23, ctx)) {
    899       goto end;
    900     }
    901     /* tmp2_ = tmp2 */
    902   } else {
    903     tmp2_ = &b->Y;
    904   }
    905 
    906   /* compare  Y_a*Z_b^3  with  Y_b*Z_a^3 */
    907   if (BN_cmp(tmp1_, tmp2_) != 0) {
    908     ret = 1; /* points differ */
    909     goto end;
    910   }
    911 
    912   /* points are equal */
    913   ret = 0;
    914 
    915 end:
    916   BN_CTX_end(ctx);
    917   BN_CTX_free(new_ctx);
    918   return ret;
    919 }
    920 
    921 int ec_GFp_simple_make_affine(const EC_GROUP *group, EC_POINT *point,
    922                               BN_CTX *ctx) {
    923   BN_CTX *new_ctx = NULL;
    924   BIGNUM *x, *y;
    925   int ret = 0;
    926 
    927   if (BN_cmp(&point->Z, &group->one) == 0 ||
    928       EC_POINT_is_at_infinity(group, point)) {
    929     return 1;
    930   }
    931 
    932   if (ctx == NULL) {
    933     ctx = new_ctx = BN_CTX_new();
    934     if (ctx == NULL) {
    935       return 0;
    936     }
    937   }
    938 
    939   BN_CTX_start(ctx);
    940   x = BN_CTX_get(ctx);
    941   y = BN_CTX_get(ctx);
    942   if (y == NULL) {
    943     goto err;
    944   }
    945 
    946   if (!EC_POINT_get_affine_coordinates_GFp(group, point, x, y, ctx) ||
    947       !EC_POINT_set_affine_coordinates_GFp(group, point, x, y, ctx)) {
    948     goto err;
    949   }
    950   if (BN_cmp(&point->Z, &group->one) != 0) {
    951     OPENSSL_PUT_ERROR(EC, ERR_R_INTERNAL_ERROR);
    952     goto err;
    953   }
    954 
    955   ret = 1;
    956 
    957 err:
    958   BN_CTX_end(ctx);
    959   BN_CTX_free(new_ctx);
    960   return ret;
    961 }
    962 
    963 int ec_GFp_simple_points_make_affine(const EC_GROUP *group, size_t num,
    964                                      EC_POINT *points[], BN_CTX *ctx) {
    965   BN_CTX *new_ctx = NULL;
    966   BIGNUM *tmp, *tmp_Z;
    967   BIGNUM **prod_Z = NULL;
    968   int ret = 0;
    969 
    970   if (num == 0) {
    971     return 1;
    972   }
    973 
    974   if (ctx == NULL) {
    975     ctx = new_ctx = BN_CTX_new();
    976     if (ctx == NULL) {
    977       return 0;
    978     }
    979   }
    980 
    981   BN_CTX_start(ctx);
    982   tmp = BN_CTX_get(ctx);
    983   tmp_Z = BN_CTX_get(ctx);
    984   if (tmp == NULL || tmp_Z == NULL) {
    985     goto err;
    986   }
    987 
    988   prod_Z = OPENSSL_malloc(num * sizeof(prod_Z[0]));
    989   if (prod_Z == NULL) {
    990     goto err;
    991   }
    992   OPENSSL_memset(prod_Z, 0, num * sizeof(prod_Z[0]));
    993   for (size_t i = 0; i < num; i++) {
    994     prod_Z[i] = BN_new();
    995     if (prod_Z[i] == NULL) {
    996       goto err;
    997     }
    998   }
    999 
   1000   /* Set each prod_Z[i] to the product of points[0]->Z .. points[i]->Z,
   1001    * skipping any zero-valued inputs (pretend that they're 1). */
   1002 
   1003   if (!BN_is_zero(&points[0]->Z)) {
   1004     if (!BN_copy(prod_Z[0], &points[0]->Z)) {
   1005       goto err;
   1006     }
   1007   } else {
   1008     if (BN_copy(prod_Z[0], &group->one) == NULL) {
   1009       goto err;
   1010     }
   1011   }
   1012 
   1013   for (size_t i = 1; i < num; i++) {
   1014     if (!BN_is_zero(&points[i]->Z)) {
   1015       if (!group->meth->field_mul(group, prod_Z[i], prod_Z[i - 1],
   1016                                   &points[i]->Z, ctx)) {
   1017         goto err;
   1018       }
   1019     } else {
   1020       if (!BN_copy(prod_Z[i], prod_Z[i - 1])) {
   1021         goto err;
   1022       }
   1023     }
   1024   }
   1025 
   1026   /* Now use a single explicit inversion to replace every non-zero points[i]->Z
   1027    * by its inverse. We use |BN_mod_inverse_odd| instead of doing a constant-
   1028    * time inversion using Fermat's Little Theorem because this function is
   1029    * usually only used for converting multiples of a public key point to
   1030    * affine, and a public key point isn't secret. If we were to use Fermat's
   1031    * Little Theorem then the cost of the inversion would usually be so high
   1032    * that converting the multiples to affine would be counterproductive. */
   1033   int no_inverse;
   1034   if (!BN_mod_inverse_odd(tmp, &no_inverse, prod_Z[num - 1], &group->field,
   1035                           ctx)) {
   1036     OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB);
   1037     goto err;
   1038   }
   1039 
   1040   if (group->meth->field_encode != NULL) {
   1041     /* In the Montgomery case, we just turned R*H (representing H)
   1042      * into 1/(R*H), but we need R*(1/H) (representing 1/H);
   1043      * i.e. we need to multiply by the Montgomery factor twice. */
   1044     if (!group->meth->field_encode(group, tmp, tmp, ctx) ||
   1045         !group->meth->field_encode(group, tmp, tmp, ctx)) {
   1046       goto err;
   1047     }
   1048   }
   1049 
   1050   for (size_t i = num - 1; i > 0; --i) {
   1051     /* Loop invariant: tmp is the product of the inverses of
   1052      * points[0]->Z .. points[i]->Z (zero-valued inputs skipped). */
   1053     if (BN_is_zero(&points[i]->Z)) {
   1054       continue;
   1055     }
   1056 
   1057     /* Set tmp_Z to the inverse of points[i]->Z (as product
   1058      * of Z inverses 0 .. i, Z values 0 .. i - 1). */
   1059     if (!group->meth->field_mul(group, tmp_Z, prod_Z[i - 1], tmp, ctx) ||
   1060         /* Update tmp to satisfy the loop invariant for i - 1. */
   1061         !group->meth->field_mul(group, tmp, tmp, &points[i]->Z, ctx) ||
   1062         /* Replace points[i]->Z by its inverse. */
   1063         !BN_copy(&points[i]->Z, tmp_Z)) {
   1064       goto err;
   1065     }
   1066   }
   1067 
   1068   /* Replace points[0]->Z by its inverse. */
   1069   if (!BN_is_zero(&points[0]->Z) && !BN_copy(&points[0]->Z, tmp)) {
   1070     goto err;
   1071   }
   1072 
   1073   /* Finally, fix up the X and Y coordinates for all points. */
   1074   for (size_t i = 0; i < num; i++) {
   1075     EC_POINT *p = points[i];
   1076 
   1077     if (!BN_is_zero(&p->Z)) {
   1078       /* turn (X, Y, 1/Z) into (X/Z^2, Y/Z^3, 1). */
   1079       if (!group->meth->field_sqr(group, tmp, &p->Z, ctx) ||
   1080           !group->meth->field_mul(group, &p->X, &p->X, tmp, ctx) ||
   1081           !group->meth->field_mul(group, tmp, tmp, &p->Z, ctx) ||
   1082           !group->meth->field_mul(group, &p->Y, &p->Y, tmp, ctx)) {
   1083         goto err;
   1084       }
   1085 
   1086       if (BN_copy(&p->Z, &group->one) == NULL) {
   1087         goto err;
   1088       }
   1089     }
   1090   }
   1091 
   1092   ret = 1;
   1093 
   1094 err:
   1095   BN_CTX_end(ctx);
   1096   BN_CTX_free(new_ctx);
   1097   if (prod_Z != NULL) {
   1098     for (size_t i = 0; i < num; i++) {
   1099       if (prod_Z[i] == NULL) {
   1100         break;
   1101       }
   1102       BN_clear_free(prod_Z[i]);
   1103     }
   1104     OPENSSL_free(prod_Z);
   1105   }
   1106 
   1107   return ret;
   1108 }
   1109 
   1110 int ec_GFp_simple_field_mul(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a,
   1111                             const BIGNUM *b, BN_CTX *ctx) {
   1112   return BN_mod_mul(r, a, b, &group->field, ctx);
   1113 }
   1114 
   1115 int ec_GFp_simple_field_sqr(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a,
   1116                             BN_CTX *ctx) {
   1117   return BN_mod_sqr(r, a, &group->field, ctx);
   1118 }
   1119