crypto/ec/ecp_nistp224.c

   1 /* crypto/ec/ecp_nistp224.c */
   2 /*
   3  * Written by Emilia Kasper (Google) for the OpenSSL project.
   4  */
   5 /* Copyright 2011 Google Inc.
   6  *
   7  * Licensed under the Apache License, Version 2.0 (the "License");
   8  *
   9  * you may not use this file except in compliance with the License.
  10  * You may obtain a copy of the License at
  11  *
  12  *     http://www.apache.org/licenses/LICENSE-2.0
  13  *
  14  *  Unless required by applicable law or agreed to in writing, software
  15  *  distributed under the License is distributed on an "AS IS" BASIS,
  16  *  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  17  *  See the License for the specific language governing permissions and
  18  *  limitations under the License.
  19  */
  20
  21 /*
  22  * A 64-bit implementation of the NIST P-224 elliptic curve point multiplication
  23  *
  24  * Inspired by Daniel J. Bernstein's public domain nistp224 implementation
  25  * and Adam Langley's public domain 64-bit C implementation of curve25519
  26  */
  27
  28 #include <openssl/opensslconf.h>
  29 #ifndef OPENSSL_NO_EC_NISTP_64_GCC_128
  30
  31 #ifndef OPENSSL_SYS_VMS
  32 #include <stdint.h>
  33 #else
  34 #include <inttypes.h>
  35 #endif
  36
  37 #include <string.h>
  38 #include <openssl/err.h>
  39 #include "ec_lcl.h"
  40
  41 #if defined(__GNUC__) && (__GNUC__ > 3 || (__GNUC__ == 3 && __GNUC_MINOR__ >= 1))
  42   /* even with gcc, the typedef won't work for 32-bit platforms */
  43   typedef __uint128_t uint128_t; /* nonstandard; implemented by gcc on 64-bit platforms */
  44 #else
  45   #error "Need GCC 3.1 or later to define type uint128_t"
  46 #endif
  47
  48 typedef uint8_t u8;
  49 typedef uint64_t u64;
  50 typedef int64_t s64;
  51
  52
  53 /******************************************************************************/
  54 /*-
  55  * INTERNAL REPRESENTATION OF FIELD ELEMENTS
  56  *
  57  * Field elements are represented as a_0 + 2^56*a_1 + 2^112*a_2 + 2^168*a_3
  58  * using 64-bit coefficients called 'limbs',
  59  * and sometimes (for multiplication results) as
  60  * b_0 + 2^56*b_1 + 2^112*b_2 + 2^168*b_3 + 2^224*b_4 + 2^280*b_5 + 2^336*b_6
  61  * using 128-bit coefficients called 'widelimbs'.
  62  * A 4-limb representation is an 'felem';
  63  * a 7-widelimb representation is a 'widefelem'.
  64  * Even within felems, bits of adjacent limbs overlap, and we don't always
  65  * reduce the representations: we ensure that inputs to each felem
  66  * multiplication satisfy a_i < 2^60, so outputs satisfy b_i < 4*2^60*2^60,
  67  * and fit into a 128-bit word without overflow. The coefficients are then
  68  * again partially reduced to obtain an felem satisfying a_i < 2^57.
  69  * We only reduce to the unique minimal representation at the end of the
  70  * computation.
  71  */
  72
  73 typedef uint64_t limb;
  74 typedef uint128_t widelimb;
  75
  76 typedef limb felem[4];
  77 typedef widelimb widefelem[7];
  78
  79 /* Field element represented as a byte arrary.
  80  * 28*8 = 224 bits is also the group order size for the elliptic curve,
  81  * and we also use this type for scalars for point multiplication.
  82   */
  83 typedef u8 felem_bytearray[28];
  84
  85 static const felem_bytearray nistp224_curve_params[5] = {
  86         {0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,    /* p */
  87          0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0x00,0x00,0x00,0x00,
  88          0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x01},
  89         {0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,    /* a */
  90          0xFF,0xFF,0xFF,0xFF,0xFF,0xFE,0xFF,0xFF,0xFF,0xFF,
  91          0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFE},
  92         {0xB4,0x05,0x0A,0x85,0x0C,0x04,0xB3,0xAB,0xF5,0x41,    /* b */
  93          0x32,0x56,0x50,0x44,0xB0,0xB7,0xD7,0xBF,0xD8,0xBA,
  94          0x27,0x0B,0x39,0x43,0x23,0x55,0xFF,0xB4},
  95         {0xB7,0x0E,0x0C,0xBD,0x6B,0xB4,0xBF,0x7F,0x32,0x13,    /* x */
  96          0x90,0xB9,0x4A,0x03,0xC1,0xD3,0x56,0xC2,0x11,0x22,
  97          0x34,0x32,0x80,0xD6,0x11,0x5C,0x1D,0x21},
  98         {0xbd,0x37,0x63,0x88,0xb5,0xf7,0x23,0xfb,0x4c,0x22,    /* y */
  99          0xdf,0xe6,0xcd,0x43,0x75,0xa0,0x5a,0x07,0x47,0x64,
 100          0x44,0xd5,0x81,0x99,0x85,0x00,0x7e,0x34}
 101 };
 102
 103 /*-
 104  * Precomputed multiples of the standard generator
 105  * Points are given in coordinates (X, Y, Z) where Z normally is 1
 106  * (0 for the point at infinity).
 107  * For each field element, slice a_0 is word 0, etc.
 108  *
 109  * The table has 2 * 16 elements, starting with the following:
 110  * index | bits    | point
 111  * ------+---------+------------------------------
 112  *     0 | 0 0 0 0 | 0G
 113  *     1 | 0 0 0 1 | 1G
 114  *     2 | 0 0 1 0 | 2^56G
 115  *     3 | 0 0 1 1 | (2^56 + 1)G
 116  *     4 | 0 1 0 0 | 2^112G
 117  *     5 | 0 1 0 1 | (2^112 + 1)G
 118  *     6 | 0 1 1 0 | (2^112 + 2^56)G
 119  *     7 | 0 1 1 1 | (2^112 + 2^56 + 1)G
 120  *     8 | 1 0 0 0 | 2^168G
 121  *     9 | 1 0 0 1 | (2^168 + 1)G
 122  *    10 | 1 0 1 0 | (2^168 + 2^56)G
 123  *    11 | 1 0 1 1 | (2^168 + 2^56 + 1)G
 124  *    12 | 1 1 0 0 | (2^168 + 2^112)G
 125  *    13 | 1 1 0 1 | (2^168 + 2^112 + 1)G
 126  *    14 | 1 1 1 0 | (2^168 + 2^112 + 2^56)G
 127  *    15 | 1 1 1 1 | (2^168 + 2^112 + 2^56 + 1)G
 128  * followed by a copy of this with each element multiplied by 2^28.
 129  *
 130  * The reason for this is so that we can clock bits into four different
 131  * locations when doing simple scalar multiplies against the base point,
 132  * and then another four locations using the second 16 elements.
 133  */
 134 static const felem gmul[2][16][3] =
 135 {{{{0, 0, 0, 0},
 136    {0, 0, 0, 0},
 137    {0, 0, 0, 0}},
 138   {{0x3280d6115c1d21, 0xc1d356c2112234, 0x7f321390b94a03, 0xb70e0cbd6bb4bf},
 139    {0xd5819985007e34, 0x75a05a07476444, 0xfb4c22dfe6cd43, 0xbd376388b5f723},
 140    {1, 0, 0, 0}},
 141   {{0xfd9675666ebbe9, 0xbca7664d40ce5e, 0x2242df8d8a2a43, 0x1f49bbb0f99bc5},
 142    {0x29e0b892dc9c43, 0xece8608436e662, 0xdc858f185310d0, 0x9812dd4eb8d321},
 143    {1, 0, 0, 0}},
 144   {{0x6d3e678d5d8eb8, 0x559eed1cb362f1, 0x16e9a3bbce8a3f, 0xeedcccd8c2a748},
 145    {0xf19f90ed50266d, 0xabf2b4bf65f9df, 0x313865468fafec, 0x5cb379ba910a17},
 146    {1, 0, 0, 0}},
 147   {{0x0641966cab26e3, 0x91fb2991fab0a0, 0xefec27a4e13a0b, 0x0499aa8a5f8ebe},
 148    {0x7510407766af5d, 0x84d929610d5450, 0x81d77aae82f706, 0x6916f6d4338c5b},
 149    {1, 0, 0, 0}},
 150   {{0xea95ac3b1f15c6, 0x086000905e82d4, 0xdd323ae4d1c8b1, 0x932b56be7685a3},
 151    {0x9ef93dea25dbbf, 0x41665960f390f0, 0xfdec76dbe2a8a7, 0x523e80f019062a},
 152    {1, 0, 0, 0}},
 153   {{0x822fdd26732c73, 0xa01c83531b5d0f, 0x363f37347c1ba4, 0xc391b45c84725c},
 154    {0xbbd5e1b2d6ad24, 0xddfbcde19dfaec, 0xc393da7e222a7f, 0x1efb7890ede244},
 155    {1, 0, 0, 0}},
 156   {{0x4c9e90ca217da1, 0xd11beca79159bb, 0xff8d33c2c98b7c, 0x2610b39409f849},
 157    {0x44d1352ac64da0, 0xcdbb7b2c46b4fb, 0x966c079b753c89, 0xfe67e4e820b112},
 158    {1, 0, 0, 0}},
 159   {{0xe28cae2df5312d, 0xc71b61d16f5c6e, 0x79b7619a3e7c4c, 0x05c73240899b47},
 160    {0x9f7f6382c73e3a, 0x18615165c56bda, 0x641fab2116fd56, 0x72855882b08394},
 161    {1, 0, 0, 0}},
 162   {{0x0469182f161c09, 0x74a98ca8d00fb5, 0xb89da93489a3e0, 0x41c98768fb0c1d},
 163    {0xe5ea05fb32da81, 0x3dce9ffbca6855, 0x1cfe2d3fbf59e6, 0x0e5e03408738a7},
 164    {1, 0, 0, 0}},
 165   {{0xdab22b2333e87f, 0x4430137a5dd2f6, 0xe03ab9f738beb8, 0xcb0c5d0dc34f24},
 166    {0x764a7df0c8fda5, 0x185ba5c3fa2044, 0x9281d688bcbe50, 0xc40331df893881},
 167    {1, 0, 0, 0}},
 168   {{0xb89530796f0f60, 0xade92bd26909a3, 0x1a0c83fb4884da, 0x1765bf22a5a984},
 169    {0x772a9ee75db09e, 0x23bc6c67cec16f, 0x4c1edba8b14e2f, 0xe2a215d9611369},
 170    {1, 0, 0, 0}},
 171   {{0x571e509fb5efb3, 0xade88696410552, 0xc8ae85fada74fe, 0x6c7e4be83bbde3},
 172    {0xff9f51160f4652, 0xb47ce2495a6539, 0xa2946c53b582f4, 0x286d2db3ee9a60},
 173    {1, 0, 0, 0}},
 174   {{0x40bbd5081a44af, 0x0995183b13926c, 0xbcefba6f47f6d0, 0x215619e9cc0057},
 175    {0x8bc94d3b0df45e, 0xf11c54a3694f6f, 0x8631b93cdfe8b5, 0xe7e3f4b0982db9},
 176    {1, 0, 0, 0}},
 177   {{0xb17048ab3e1c7b, 0xac38f36ff8a1d8, 0x1c29819435d2c6, 0xc813132f4c07e9},
 178    {0x2891425503b11f, 0x08781030579fea, 0xf5426ba5cc9674, 0x1e28ebf18562bc},
 179    {1, 0, 0, 0}},
 180   {{0x9f31997cc864eb, 0x06cd91d28b5e4c, 0xff17036691a973, 0xf1aef351497c58},
 181    {0xdd1f2d600564ff, 0xdead073b1402db, 0x74a684435bd693, 0xeea7471f962558},
 182    {1, 0, 0, 0}}},
 183  {{{0, 0, 0, 0},
 184    {0, 0, 0, 0},
 185    {0, 0, 0, 0}},
 186   {{0x9665266dddf554, 0x9613d78b60ef2d, 0xce27a34cdba417, 0xd35ab74d6afc31},
 187    {0x85ccdd22deb15e, 0x2137e5783a6aab, 0xa141cffd8c93c6, 0x355a1830e90f2d},
 188    {1, 0, 0, 0}},
 189   {{0x1a494eadaade65, 0xd6da4da77fe53c, 0xe7992996abec86, 0x65c3553c6090e3},
 190    {0xfa610b1fb09346, 0xf1c6540b8a4aaf, 0xc51a13ccd3cbab, 0x02995b1b18c28a},
 191    {1, 0, 0, 0}},
 192   {{0x7874568e7295ef, 0x86b419fbe38d04, 0xdc0690a7550d9a, 0xd3966a44beac33},
 193    {0x2b7280ec29132f, 0xbeaa3b6a032df3, 0xdc7dd88ae41200, 0xd25e2513e3a100},
 194    {1, 0, 0, 0}},
 195   {{0x924857eb2efafd, 0xac2bce41223190, 0x8edaa1445553fc, 0x825800fd3562d5},
 196    {0x8d79148ea96621, 0x23a01c3dd9ed8d, 0xaf8b219f9416b5, 0xd8db0cc277daea},
 197    {1, 0, 0, 0}},
 198   {{0x76a9c3b1a700f0, 0xe9acd29bc7e691, 0x69212d1a6b0327, 0x6322e97fe154be},
 199    {0x469fc5465d62aa, 0x8d41ed18883b05, 0x1f8eae66c52b88, 0xe4fcbe9325be51},
 200    {1, 0, 0, 0}},
 201   {{0x825fdf583cac16, 0x020b857c7b023a, 0x683c17744b0165, 0x14ffd0a2daf2f1},
 202    {0x323b36184218f9, 0x4944ec4e3b47d4, 0xc15b3080841acf, 0x0bced4b01a28bb},
 203    {1, 0, 0, 0}},
 204   {{0x92ac22230df5c4, 0x52f33b4063eda8, 0xcb3f19870c0c93, 0x40064f2ba65233},
 205    {0xfe16f0924f8992, 0x012da25af5b517, 0x1a57bb24f723a6, 0x06f8bc76760def},
 206    {1, 0, 0, 0}},
 207   {{0x4a7084f7817cb9, 0xbcab0738ee9a78, 0x3ec11e11d9c326, 0xdc0fe90e0f1aae},
 208    {0xcf639ea5f98390, 0x5c350aa22ffb74, 0x9afae98a4047b7, 0x956ec2d617fc45},
 209    {1, 0, 0, 0}},
 210   {{0x4306d648c1be6a, 0x9247cd8bc9a462, 0xf5595e377d2f2e, 0xbd1c3caff1a52e},
 211    {0x045e14472409d0, 0x29f3e17078f773, 0x745a602b2d4f7d, 0x191837685cdfbb},
 212    {1, 0, 0, 0}},
 213   {{0x5b6ee254a8cb79, 0x4953433f5e7026, 0xe21faeb1d1def4, 0xc4c225785c09de},
 214    {0x307ce7bba1e518, 0x31b125b1036db8, 0x47e91868839e8f, 0xc765866e33b9f3},
 215    {1, 0, 0, 0}},
 216   {{0x3bfece24f96906, 0x4794da641e5093, 0xde5df64f95db26, 0x297ecd89714b05},
 217    {0x701bd3ebb2c3aa, 0x7073b4f53cb1d5, 0x13c5665658af16, 0x9895089d66fe58},
 218    {1, 0, 0, 0}},
 219   {{0x0fef05f78c4790, 0x2d773633b05d2e, 0x94229c3a951c94, 0xbbbd70df4911bb},
 220    {0xb2c6963d2c1168, 0x105f47a72b0d73, 0x9fdf6111614080, 0x7b7e94b39e67b0},
 221    {1, 0, 0, 0}},
 222   {{0xad1a7d6efbe2b3, 0xf012482c0da69d, 0x6b3bdf12438345, 0x40d7558d7aa4d9},
 223    {0x8a09fffb5c6d3d, 0x9a356e5d9ffd38, 0x5973f15f4f9b1c, 0xdcd5f59f63c3ea},
 224    {1, 0, 0, 0}},
 225   {{0xacf39f4c5ca7ab, 0x4c8071cc5fd737, 0xc64e3602cd1184, 0x0acd4644c9abba},
 226    {0x6c011a36d8bf6e, 0xfecd87ba24e32a, 0x19f6f56574fad8, 0x050b204ced9405},
 227    {1, 0, 0, 0}},
 228   {{0xed4f1cae7d9a96, 0x5ceef7ad94c40a, 0x778e4a3bf3ef9b, 0x7405783dc3b55e},
 229    {0x32477c61b6e8c6, 0xb46a97570f018b, 0x91176d0a7e95d1, 0x3df90fbc4c7d0e},
 230    {1, 0, 0, 0}}}};
 231
 232 /* Precomputation for the group generator. */
 233 typedef struct {
 234         felem g_pre_comp[2][16][3];
 235         int references;
 236 } NISTP224_PRE_COMP;
 237
 238 const EC_METHOD *EC_GFp_nistp224_method(void)
 239         {
 240         static const EC_METHOD ret = {
 241                 EC_FLAGS_DEFAULT_OCT,
 242                 NID_X9_62_prime_field,
 243                 ec_GFp_nistp224_group_init,
 244                 ec_GFp_simple_group_finish,
 245                 ec_GFp_simple_group_clear_finish,
 246                 ec_GFp_nist_group_copy,
 247                 ec_GFp_nistp224_group_set_curve,
 248                 ec_GFp_simple_group_get_curve,
 249                 ec_GFp_simple_group_get_degree,
 250                 ec_GFp_simple_group_check_discriminant,
 251                 ec_GFp_simple_point_init,
 252                 ec_GFp_simple_point_finish,
 253                 ec_GFp_simple_point_clear_finish,
 254                 ec_GFp_simple_point_copy,
 255                 ec_GFp_simple_point_set_to_infinity,
 256                 ec_GFp_simple_set_Jprojective_coordinates_GFp,
 257                 ec_GFp_simple_get_Jprojective_coordinates_GFp,
 258                 ec_GFp_simple_point_set_affine_coordinates,
 259                 ec_GFp_nistp224_point_get_affine_coordinates,
 260                 0 /* point_set_compressed_coordinates */,
 261                 0 /* point2oct */,
 262                 0 /* oct2point */,
 263                 ec_GFp_simple_add,
 264                 ec_GFp_simple_dbl,
 265                 ec_GFp_simple_invert,
 266                 ec_GFp_simple_is_at_infinity,
 267                 ec_GFp_simple_is_on_curve,
 268                 ec_GFp_simple_cmp,
 269                 ec_GFp_simple_make_affine,
 270                 ec_GFp_simple_points_make_affine,
 271                 ec_GFp_nistp224_points_mul,
 272                 ec_GFp_nistp224_precompute_mult,
 273                 ec_GFp_nistp224_have_precompute_mult,
 274                 ec_GFp_nist_field_mul,
 275                 ec_GFp_nist_field_sqr,
 276                 0 /* field_div */,
 277                 0 /* field_encode */,
 278                 0 /* field_decode */,
 279                 0 /* field_set_to_one */ };
 280
 281         return &ret;
 282         }
 283
 284 /* Helper functions to convert field elements to/from internal representation */
 285 static void bin28_to_felem(felem out, const u8 in[28])
 286         {
 287         out[0] = *((const uint64_t *)(in)) & 0x00ffffffffffffff;
 288         out[1] = (*((const uint64_t *)(in+7))) & 0x00ffffffffffffff;
 289         out[2] = (*((const uint64_t *)(in+14))) & 0x00ffffffffffffff;
 290         out[3] = (*((const uint64_t *)(in+21))) & 0x00ffffffffffffff;
 291         }
 292
 293 static void felem_to_bin28(u8 out[28], const felem in)
 294         {
 295         unsigned i;
 296         for (i = 0; i < 7; ++i)
 297                 {
 298                 out[i]    = in[0]>>(8*i);
 299                 out[i+7]  = in[1]>>(8*i);
 300                 out[i+14] = in[2]>>(8*i);
 301                 out[i+21] = in[3]>>(8*i);
 302                 }
 303         }
 304
 305 /* To preserve endianness when using BN_bn2bin and BN_bin2bn */
 306 static void flip_endian(u8 *out, const u8 *in, unsigned len)
 307         {
 308         unsigned i;
 309         for (i = 0; i < len; ++i)
 310                 out[i] = in[len-1-i];
 311         }
 312
 313 /* From OpenSSL BIGNUM to internal representation */
 314 static int BN_to_felem(felem out, const BIGNUM *bn)
 315         {
 316         felem_bytearray b_in;
 317         felem_bytearray b_out;
 318         unsigned num_bytes;
 319
 320         /* BN_bn2bin eats leading zeroes */
 321         memset(b_out, 0, sizeof b_out);
 322         num_bytes = BN_num_bytes(bn);
 323         if (num_bytes > sizeof b_out)
 324                 {
 325                 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
 326                 return 0;
 327                 }
 328         if (BN_is_negative(bn))
 329                 {
 330                 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
 331                 return 0;
 332                 }
 333         num_bytes = BN_bn2bin(bn, b_in);
 334         flip_endian(b_out, b_in, num_bytes);
 335         bin28_to_felem(out, b_out);
 336         return 1;
 337         }
 338
 339 /* From internal representation to OpenSSL BIGNUM */
 340 static BIGNUM *felem_to_BN(BIGNUM *out, const felem in)
 341         {
 342         felem_bytearray b_in, b_out;
 343         felem_to_bin28(b_in, in);
 344         flip_endian(b_out, b_in, sizeof b_out);
 345         return BN_bin2bn(b_out, sizeof b_out, out);
 346         }
 347
 348 /******************************************************************************/
 349 /*                              FIELD OPERATIONS
 350  *
 351  * Field operations, using the internal representation of field elements.
 352  * NB! These operations are specific to our point multiplication and cannot be
 353  * expected to be correct in general - e.g., multiplication with a large scalar
 354  * will cause an overflow.
 355  *
 356  */
 357
 358 static void felem_one(felem out)
 359         {
 360         out[0] = 1;
 361         out[1] = 0;
 362         out[2] = 0;
 363         out[3] = 0;
 364         }
 365
 366 static void felem_assign(felem out, const felem in)
 367         {
 368         out[0] = in[0];
 369         out[1] = in[1];
 370         out[2] = in[2];
 371         out[3] = in[3];
 372         }
 373
 374 /* Sum two field elements: out += in */
 375 static void felem_sum(felem out, const felem in)
 376         {
 377         out[0] += in[0];
 378         out[1] += in[1];
 379         out[2] += in[2];
 380         out[3] += in[3];
 381         }
 382
 383 /* Get negative value: out = -in */
 384 /* Assumes in[i] < 2^57 */
 385 static void felem_neg(felem out, const felem in)
 386         {
 387         static const limb two58p2 = (((limb) 1) << 58) + (((limb) 1) << 2);
 388         static const limb two58m2 = (((limb) 1) << 58) - (((limb) 1) << 2);
 389         static const limb two58m42m2 = (((limb) 1) << 58) -
 390             (((limb) 1) << 42) - (((limb) 1) << 2);
 391
 392         /* Set to 0 mod 2^224-2^96+1 to ensure out > in */
 393         out[0] = two58p2 - in[0];
 394         out[1] = two58m42m2 - in[1];
 395         out[2] = two58m2 - in[2];
 396         out[3] = two58m2 - in[3];
 397         }
 398
 399 /* Subtract field elements: out -= in */
 400 /* Assumes in[i] < 2^57 */
 401 static void felem_diff(felem out, const felem in)
 402         {
 403         static const limb two58p2 = (((limb) 1) << 58) + (((limb) 1) << 2);
 404         static const limb two58m2 = (((limb) 1) << 58) - (((limb) 1) << 2);
 405         static const limb two58m42m2 = (((limb) 1) << 58) -
 406             (((limb) 1) << 42) - (((limb) 1) << 2);
 407
 408         /* Add 0 mod 2^224-2^96+1 to ensure out > in */
 409         out[0] += two58p2;
 410         out[1] += two58m42m2;
 411         out[2] += two58m2;
 412         out[3] += two58m2;
 413
 414         out[0] -= in[0];
 415         out[1] -= in[1];
 416         out[2] -= in[2];
 417         out[3] -= in[3];
 418         }
 419
 420 /* Subtract in unreduced 128-bit mode: out -= in */
 421 /* Assumes in[i] < 2^119 */
 422 static void widefelem_diff(widefelem out, const widefelem in)
 423         {
 424         static const widelimb two120 = ((widelimb) 1) << 120;
 425         static const widelimb two120m64 = (((widelimb) 1) << 120) -
 426                 (((widelimb) 1) << 64);
 427         static const widelimb two120m104m64 = (((widelimb) 1) << 120) -
 428                 (((widelimb) 1) << 104) - (((widelimb) 1) << 64);
 429
 430         /* Add 0 mod 2^224-2^96+1 to ensure out > in */
 431         out[0] += two120;
 432         out[1] += two120m64;
 433         out[2] += two120m64;
 434         out[3] += two120;
 435         out[4] += two120m104m64;
 436         out[5] += two120m64;
 437         out[6] += two120m64;
 438
 439         out[0] -= in[0];
 440         out[1] -= in[1];
 441         out[2] -= in[2];
 442         out[3] -= in[3];
 443         out[4] -= in[4];
 444         out[5] -= in[5];
 445         out[6] -= in[6];
 446         }
 447
 448 /* Subtract in mixed mode: out128 -= in64 */
 449 /* in[i] < 2^63 */
 450 static void felem_diff_128_64(widefelem out, const felem in)
 451         {
 452         static const widelimb two64p8 = (((widelimb) 1) << 64) +
 453                 (((widelimb) 1) << 8);
 454         static const widelimb two64m8 = (((widelimb) 1) << 64) -
 455                 (((widelimb) 1) << 8);
 456         static const widelimb two64m48m8 = (((widelimb) 1) << 64) -
 457                 (((widelimb) 1) << 48) - (((widelimb) 1) << 8);
 458
 459         /* Add 0 mod 2^224-2^96+1 to ensure out > in */
 460         out[0] += two64p8;
 461         out[1] += two64m48m8;
 462         out[2] += two64m8;
 463         out[3] += two64m8;
 464
 465         out[0] -= in[0];
 466         out[1] -= in[1];
 467         out[2] -= in[2];
 468         out[3] -= in[3];
 469         }
 470
 471 /* Multiply a field element by a scalar: out = out * scalar
 472  * The scalars we actually use are small, so results fit without overflow */
 473 static void felem_scalar(felem out, const limb scalar)
 474         {
 475         out[0] *= scalar;
 476         out[1] *= scalar;
 477         out[2] *= scalar;
 478         out[3] *= scalar;
 479         }
 480
 481 /* Multiply an unreduced field element by a scalar: out = out * scalar
 482  * The scalars we actually use are small, so results fit without overflow */
 483 static void widefelem_scalar(widefelem out, const widelimb scalar)
 484         {
 485         out[0] *= scalar;
 486         out[1] *= scalar;
 487         out[2] *= scalar;
 488         out[3] *= scalar;
 489         out[4] *= scalar;
 490         out[5] *= scalar;
 491         out[6] *= scalar;
 492         }
 493
 494 /* Square a field element: out = in^2 */
 495 static void felem_square(widefelem out, const felem in)
 496         {
 497         limb tmp0, tmp1, tmp2;
 498         tmp0 = 2 * in[0]; tmp1 = 2 * in[1]; tmp2 = 2 * in[2];
 499         out[0] = ((widelimb) in[0]) * in[0];
 500         out[1] = ((widelimb) in[0]) * tmp1;
 501         out[2] = ((widelimb) in[0]) * tmp2 + ((widelimb) in[1]) * in[1];
 502         out[3] = ((widelimb) in[3]) * tmp0 +
 503                 ((widelimb) in[1]) * tmp2;
 504         out[4] = ((widelimb) in[3]) * tmp1 + ((widelimb) in[2]) * in[2];
 505         out[5] = ((widelimb) in[3]) * tmp2;
 506         out[6] = ((widelimb) in[3]) * in[3];
 507         }
 508
 509 /* Multiply two field elements: out = in1 * in2 */
 510 static void felem_mul(widefelem out, const felem in1, const felem in2)
 511         {
 512         out[0] = ((widelimb) in1[0]) * in2[0];
 513         out[1] = ((widelimb) in1[0]) * in2[1] + ((widelimb) in1[1]) * in2[0];
 514         out[2] = ((widelimb) in1[0]) * in2[2] + ((widelimb) in1[1]) * in2[1] +
 515                 ((widelimb) in1[2]) * in2[0];
 516         out[3] = ((widelimb) in1[0]) * in2[3] + ((widelimb) in1[1]) * in2[2] +
 517                 ((widelimb) in1[2]) * in2[1] + ((widelimb) in1[3]) * in2[0];
 518         out[4] = ((widelimb) in1[1]) * in2[3] + ((widelimb) in1[2]) * in2[2] +
 519                 ((widelimb) in1[3]) * in2[1];
 520         out[5] = ((widelimb) in1[2]) * in2[3] + ((widelimb) in1[3]) * in2[2];
 521         out[6] = ((widelimb) in1[3]) * in2[3];
 522         }
 523
 524 /* Reduce seven 128-bit coefficients to four 64-bit coefficients.
 525  * Requires in[i] < 2^126,
 526  * ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] <= 2^56 + 2^16 */
 527 static void felem_reduce(felem out, const widefelem in)
 528         {
 529         static const widelimb two127p15 = (((widelimb) 1) << 127) +
 530                 (((widelimb) 1) << 15);
 531         static const widelimb two127m71 = (((widelimb) 1) << 127) -
 532                 (((widelimb) 1) << 71);
 533         static const widelimb two127m71m55 = (((widelimb) 1) << 127) -
 534                 (((widelimb) 1) << 71) - (((widelimb) 1) << 55);
 535         widelimb output[5];
 536
 537         /* Add 0 mod 2^224-2^96+1 to ensure all differences are positive */
 538         output[0] = in[0] + two127p15;
 539         output[1] = in[1] + two127m71m55;
 540         output[2] = in[2] + two127m71;
 541         output[3] = in[3];
 542         output[4] = in[4];
 543
 544         /* Eliminate in[4], in[5], in[6] */
 545         output[4] += in[6] >> 16;
 546         output[3] += (in[6] & 0xffff) << 40;
 547         output[2] -= in[6];
 548
 549         output[3] += in[5] >> 16;
 550         output[2] += (in[5] & 0xffff) << 40;
 551         output[1] -= in[5];
 552
 553         output[2] += output[4] >> 16;
 554         output[1] += (output[4] & 0xffff) << 40;
 555         output[0] -= output[4];
 556
 557         /* Carry 2 -> 3 -> 4 */
 558         output[3] += output[2] >> 56;
 559         output[2] &= 0x00ffffffffffffff;
 560
 561         output[4] = output[3] >> 56;
 562         output[3] &= 0x00ffffffffffffff;
 563
 564         /* Now output[2] < 2^56, output[3] < 2^56, output[4] < 2^72 */
 565
 566         /* Eliminate output[4] */
 567         output[2] += output[4] >> 16;
 568         /* output[2] < 2^56 + 2^56 = 2^57 */
 569         output[1] += (output[4] & 0xffff) << 40;
 570         output[0] -= output[4];
 571
 572         /* Carry 0 -> 1 -> 2 -> 3 */
 573         output[1] += output[0] >> 56;
 574         out[0] = output[0] & 0x00ffffffffffffff;
 575
 576         output[2] += output[1] >> 56;
 577         /* output[2] < 2^57 + 2^72 */
 578         out[1] = output[1] & 0x00ffffffffffffff;
 579         output[3] += output[2] >> 56;
 580         /* output[3] <= 2^56 + 2^16 */
 581         out[2] = output[2] & 0x00ffffffffffffff;
 582
 583         /*-
 584          * out[0] < 2^56, out[1] < 2^56, out[2] < 2^56,
 585          * out[3] <= 2^56 + 2^16 (due to final carry),
 586          * so out < 2*p
 587          */
 588         out[3] = output[3];
 589         }
 590
 591 static void felem_square_reduce(felem out, const felem in)
 592         {
 593         widefelem tmp;
 594         felem_square(tmp, in);
 595         felem_reduce(out, tmp);
 596         }
 597
 598 static void felem_mul_reduce(felem out, const felem in1, const felem in2)
 599         {
 600         widefelem tmp;
 601         felem_mul(tmp, in1, in2);
 602         felem_reduce(out, tmp);
 603         }
 604
 605 /* Reduce to unique minimal representation.
 606  * Requires 0 <= in < 2*p (always call felem_reduce first) */
 607 static void felem_contract(felem out, const felem in)
 608         {
 609         static const int64_t two56 = ((limb) 1) << 56;
 610         /* 0 <= in < 2*p, p = 2^224 - 2^96 + 1 */
 611         /* if in > p , reduce in = in - 2^224 + 2^96 - 1 */
 612         int64_t tmp[4], a;
 613         tmp[0] = in[0];
 614         tmp[1] = in[1];
 615         tmp[2] = in[2];
 616         tmp[3] = in[3];
 617         /* Case 1: a = 1 iff in >= 2^224 */
 618         a = (in[3] >> 56);
 619         tmp[0] -= a;
 620         tmp[1] += a << 40;
 621         tmp[3] &= 0x00ffffffffffffff;
 622         /* Case 2: a = 0 iff p <= in < 2^224, i.e.,
 623          * the high 128 bits are all 1 and the lower part is non-zero */
 624         a = ((in[3] & in[2] & (in[1] | 0x000000ffffffffff)) + 1) |
 625                 (((int64_t)(in[0] + (in[1] & 0x000000ffffffffff)) - 1) >> 63);
 626         a &= 0x00ffffffffffffff;
 627         /* turn a into an all-one mask (if a = 0) or an all-zero mask */
 628         a = (a - 1) >> 63;
 629         /* subtract 2^224 - 2^96 + 1 if a is all-one*/
 630         tmp[3] &= a ^ 0xffffffffffffffff;
 631         tmp[2] &= a ^ 0xffffffffffffffff;
 632         tmp[1] &= (a ^ 0xffffffffffffffff) | 0x000000ffffffffff;
 633         tmp[0] -= 1 & a;
 634
 635         /* eliminate negative coefficients: if tmp[0] is negative, tmp[1] must
 636          * be non-zero, so we only need one step */
 637         a = tmp[0] >> 63;
 638         tmp[0] += two56 & a;
 639         tmp[1] -= 1 & a;
 640
 641         /* carry 1 -> 2 -> 3 */
 642         tmp[2] += tmp[1] >> 56;
 643         tmp[1] &= 0x00ffffffffffffff;
 644
 645         tmp[3] += tmp[2] >> 56;
 646         tmp[2] &= 0x00ffffffffffffff;
 647
 648         /* Now 0 <= out < p */
 649         out[0] = tmp[0];
 650         out[1] = tmp[1];
 651         out[2] = tmp[2];
 652         out[3] = tmp[3];
 653         }
 654
 655 /* Zero-check: returns 1 if input is 0, and 0 otherwise.
 656  * We know that field elements are reduced to in < 2^225,
 657  * so we only need to check three cases: 0, 2^224 - 2^96 + 1,
 658  * and 2^225 - 2^97 + 2 */
 659 static limb felem_is_zero(const felem in)
 660         {
 661         limb zero, two224m96p1, two225m97p2;
 662
 663         zero = in[0] | in[1] | in[2] | in[3];
 664         zero = (((int64_t)(zero) - 1) >> 63) & 1;
 665         two224m96p1 = (in[0] ^ 1) | (in[1] ^ 0x00ffff0000000000)
 666                 | (in[2] ^ 0x00ffffffffffffff) | (in[3] ^ 0x00ffffffffffffff);
 667         two224m96p1 = (((int64_t)(two224m96p1) - 1) >> 63) & 1;
 668         two225m97p2 = (in[0] ^ 2) | (in[1] ^ 0x00fffe0000000000)
 669                 | (in[2] ^ 0x00ffffffffffffff) | (in[3] ^ 0x01ffffffffffffff);
 670         two225m97p2 = (((int64_t)(two225m97p2) - 1) >> 63) & 1;
 671         return (zero | two224m96p1 | two225m97p2);
 672         }
 673
 674 static limb felem_is_zero_int(const felem in)
 675         {
 676         return (int) (felem_is_zero(in) & ((limb)1));
 677         }
 678
 679 /* Invert a field element */
 680 /* Computation chain copied from djb's code */
 681 static void felem_inv(felem out, const felem in)
 682         {
 683         felem ftmp, ftmp2, ftmp3, ftmp4;
 684         widefelem tmp;
 685         unsigned i;
 686
 687         felem_square(tmp, in); felem_reduce(ftmp, tmp);         /* 2 */
 688         felem_mul(tmp, in, ftmp); felem_reduce(ftmp, tmp);      /* 2^2 - 1 */
 689         felem_square(tmp, ftmp); felem_reduce(ftmp, tmp);       /* 2^3 - 2 */
 690         felem_mul(tmp, in, ftmp); felem_reduce(ftmp, tmp);      /* 2^3 - 1 */
 691         felem_square(tmp, ftmp); felem_reduce(ftmp2, tmp);      /* 2^4 - 2 */
 692         felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp);     /* 2^5 - 4 */
 693         felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp);     /* 2^6 - 8 */
 694         felem_mul(tmp, ftmp2, ftmp); felem_reduce(ftmp, tmp);   /* 2^6 - 1 */
 695         felem_square(tmp, ftmp); felem_reduce(ftmp2, tmp);      /* 2^7 - 2 */
 696         for (i = 0; i < 5; ++i)                                 /* 2^12 - 2^6 */
 697                 {
 698                 felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp);
 699                 }
 700         felem_mul(tmp, ftmp2, ftmp); felem_reduce(ftmp2, tmp);  /* 2^12 - 1 */
 701         felem_square(tmp, ftmp2); felem_reduce(ftmp3, tmp);     /* 2^13 - 2 */
 702         for (i = 0; i < 11; ++i)                                /* 2^24 - 2^12 */
 703                 {
 704                 felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp);
 705                 }
 706         felem_mul(tmp, ftmp3, ftmp2); felem_reduce(ftmp2, tmp); /* 2^24 - 1 */
 707         felem_square(tmp, ftmp2); felem_reduce(ftmp3, tmp);     /* 2^25 - 2 */
 708         for (i = 0; i < 23; ++i)                                /* 2^48 - 2^24 */
 709                 {
 710                 felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp);
 711                 }
 712         felem_mul(tmp, ftmp3, ftmp2); felem_reduce(ftmp3, tmp); /* 2^48 - 1 */
 713         felem_square(tmp, ftmp3); felem_reduce(ftmp4, tmp);     /* 2^49 - 2 */
 714         for (i = 0; i < 47; ++i)                                /* 2^96 - 2^48 */
 715                 {
 716                 felem_square(tmp, ftmp4); felem_reduce(ftmp4, tmp);
 717                 }
 718         felem_mul(tmp, ftmp3, ftmp4); felem_reduce(ftmp3, tmp); /* 2^96 - 1 */
 719         felem_square(tmp, ftmp3); felem_reduce(ftmp4, tmp);     /* 2^97 - 2 */
 720         for (i = 0; i < 23; ++i)                                /* 2^120 - 2^24 */
 721                 {
 722                 felem_square(tmp, ftmp4); felem_reduce(ftmp4, tmp);
 723                 }
 724         felem_mul(tmp, ftmp2, ftmp4); felem_reduce(ftmp2, tmp); /* 2^120 - 1 */
 725         for (i = 0; i < 6; ++i)                                 /* 2^126 - 2^6 */
 726                 {
 727                 felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp);
 728                 }
 729         felem_mul(tmp, ftmp2, ftmp); felem_reduce(ftmp, tmp);   /* 2^126 - 1 */
 730         felem_square(tmp, ftmp); felem_reduce(ftmp, tmp);       /* 2^127 - 2 */
 731         felem_mul(tmp, ftmp, in); felem_reduce(ftmp, tmp);      /* 2^127 - 1 */
 732         for (i = 0; i < 97; ++i)                                /* 2^224 - 2^97 */
 733                 {
 734                 felem_square(tmp, ftmp); felem_reduce(ftmp, tmp);
 735                 }
 736         felem_mul(tmp, ftmp, ftmp3); felem_reduce(out, tmp);    /* 2^224 - 2^96 - 1 */
 737         }
 738
 739 /* Copy in constant time:
 740  * if icopy == 1, copy in to out,
 741  * if icopy == 0, copy out to itself. */
 742 static void
 743 copy_conditional(felem out, const felem in, limb icopy)
 744         {
 745         unsigned i;
 746         /* icopy is a (64-bit) 0 or 1, so copy is either all-zero or all-one */
 747         const limb copy = -icopy;
 748         for (i = 0; i < 4; ++i)
 749                 {
 750                 const limb tmp = copy & (in[i] ^ out[i]);
 751                 out[i] ^= tmp;
 752                 }
 753         }
 754
 755 /******************************************************************************/
 756 /*                       ELLIPTIC CURVE POINT OPERATIONS
 757  *
 758  * Points are represented in Jacobian projective coordinates:
 759  * (X, Y, Z) corresponds to the affine point (X/Z^2, Y/Z^3),
 760  * or to the point at infinity if Z == 0.
 761  *
 762  */
 763
 764 /*-
 765  * Double an elliptic curve point:
 766  * (X', Y', Z') = 2 * (X, Y, Z), where
 767  * X' = (3 * (X - Z^2) * (X + Z^2))^2 - 8 * X * Y^2
 768  * Y' = 3 * (X - Z^2) * (X + Z^2) * (4 * X * Y^2 - X') - 8 * Y^2
 769  * Z' = (Y + Z)^2 - Y^2 - Z^2 = 2 * Y * Z
 770  * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed,
 771  * while x_out == y_in is not (maybe this works, but it's not tested).
 772  */
 773 static void
 774 point_double(felem x_out, felem y_out, felem z_out,
 775              const felem x_in, const felem y_in, const felem z_in)
 776         {
 777         widefelem tmp, tmp2;
 778         felem delta, gamma, beta, alpha, ftmp, ftmp2;
 779
 780         felem_assign(ftmp, x_in);
 781         felem_assign(ftmp2, x_in);
 782
 783         /* delta = z^2 */
 784         felem_square(tmp, z_in);
 785         felem_reduce(delta, tmp);
 786
 787         /* gamma = y^2 */
 788         felem_square(tmp, y_in);
 789         felem_reduce(gamma, tmp);
 790
 791         /* beta = x*gamma */
 792         felem_mul(tmp, x_in, gamma);
 793         felem_reduce(beta, tmp);
 794
 795         /* alpha = 3*(x-delta)*(x+delta) */
 796         felem_diff(ftmp, delta);
 797         /* ftmp[i] < 2^57 + 2^58 + 2 < 2^59 */
 798         felem_sum(ftmp2, delta);
 799         /* ftmp2[i] < 2^57 + 2^57 = 2^58 */
 800         felem_scalar(ftmp2, 3);
 801         /* ftmp2[i] < 3 * 2^58 < 2^60 */
 802         felem_mul(tmp, ftmp, ftmp2);
 803         /* tmp[i] < 2^60 * 2^59 * 4 = 2^121 */
 804         felem_reduce(alpha, tmp);
 805
 806         /* x' = alpha^2 - 8*beta */
 807         felem_square(tmp, alpha);
 808         /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
 809         felem_assign(ftmp, beta);
 810         felem_scalar(ftmp, 8);
 811         /* ftmp[i] < 8 * 2^57 = 2^60 */
 812         felem_diff_128_64(tmp, ftmp);
 813         /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
 814         felem_reduce(x_out, tmp);
 815
 816         /* z' = (y + z)^2 - gamma - delta */
 817         felem_sum(delta, gamma);
 818         /* delta[i] < 2^57 + 2^57 = 2^58 */
 819         felem_assign(ftmp, y_in);
 820         felem_sum(ftmp, z_in);
 821         /* ftmp[i] < 2^57 + 2^57 = 2^58 */
 822         felem_square(tmp, ftmp);
 823         /* tmp[i] < 4 * 2^58 * 2^58 = 2^118 */
 824         felem_diff_128_64(tmp, delta);
 825         /* tmp[i] < 2^118 + 2^64 + 8 < 2^119 */
 826         felem_reduce(z_out, tmp);
 827
 828         /* y' = alpha*(4*beta - x') - 8*gamma^2 */
 829         felem_scalar(beta, 4);
 830         /* beta[i] < 4 * 2^57 = 2^59 */
 831         felem_diff(beta, x_out);
 832         /* beta[i] < 2^59 + 2^58 + 2 < 2^60 */
 833         felem_mul(tmp, alpha, beta);
 834         /* tmp[i] < 4 * 2^57 * 2^60 = 2^119 */
 835         felem_square(tmp2, gamma);
 836         /* tmp2[i] < 4 * 2^57 * 2^57 = 2^116 */
 837         widefelem_scalar(tmp2, 8);
 838         /* tmp2[i] < 8 * 2^116 = 2^119 */
 839         widefelem_diff(tmp, tmp2);
 840         /* tmp[i] < 2^119 + 2^120 < 2^121 */
 841         felem_reduce(y_out, tmp);
 842         }
 843
 844 /*-
 845  * Add two elliptic curve points:
 846  * (X_1, Y_1, Z_1) + (X_2, Y_2, Z_2) = (X_3, Y_3, Z_3), where
 847  * X_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1)^2 - (Z_1^2 * X_2 - Z_2^2 * X_1)^3 -
 848  * 2 * Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^2
 849  * Y_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1) * (Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^2 - X_3) -
 850  *        Z_2^3 * Y_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^3
 851  * Z_3 = (Z_1^2 * X_2 - Z_2^2 * X_1) * (Z_1 * Z_2)
 852  *
 853  * This runs faster if 'mixed' is set, which requires Z_2 = 1 or Z_2 = 0.
 854  */
 855
 856 /* This function is not entirely constant-time:
 857  * it includes a branch for checking whether the two input points are equal,
 858  * (while not equal to the point at infinity).
 859  * This case never happens during single point multiplication,
 860  * so there is no timing leak for ECDH or ECDSA signing. */
 861 static void point_add(felem x3, felem y3, felem z3,
 862         const felem x1, const felem y1, const felem z1,
 863         const int mixed, const felem x2, const felem y2, const felem z2)
 864         {
 865         felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, x_out, y_out, z_out;
 866         widefelem tmp, tmp2;
 867         limb z1_is_zero, z2_is_zero, x_equal, y_equal;
 868
 869         if (!mixed)
 870                 {
 871                 /* ftmp2 = z2^2 */
 872                 felem_square(tmp, z2);
 873                 felem_reduce(ftmp2, tmp);
 874
 875                 /* ftmp4 = z2^3 */
 876                 felem_mul(tmp, ftmp2, z2);
 877                 felem_reduce(ftmp4, tmp);
 878
 879                 /* ftmp4 = z2^3*y1 */
 880                 felem_mul(tmp2, ftmp4, y1);
 881                 felem_reduce(ftmp4, tmp2);
 882
 883                 /* ftmp2 = z2^2*x1 */
 884                 felem_mul(tmp2, ftmp2, x1);
 885                 felem_reduce(ftmp2, tmp2);
 886                 }
 887         else
 888                 {
 889                 /* We'll assume z2 = 1 (special case z2 = 0 is handled later) */
 890
 891                 /* ftmp4 = z2^3*y1 */
 892                 felem_assign(ftmp4, y1);
 893
 894                 /* ftmp2 = z2^2*x1 */
 895                 felem_assign(ftmp2, x1);
 896                 }
 897
 898         /* ftmp = z1^2 */
 899         felem_square(tmp, z1);
 900         felem_reduce(ftmp, tmp);
 901
 902         /* ftmp3 = z1^3 */
 903         felem_mul(tmp, ftmp, z1);
 904         felem_reduce(ftmp3, tmp);
 905
 906         /* tmp = z1^3*y2 */
 907         felem_mul(tmp, ftmp3, y2);
 908         /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
 909
 910         /* ftmp3 = z1^3*y2 - z2^3*y1 */
 911         felem_diff_128_64(tmp, ftmp4);
 912         /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
 913         felem_reduce(ftmp3, tmp);
 914
 915         /* tmp = z1^2*x2 */
 916         felem_mul(tmp, ftmp, x2);
 917         /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
 918
 919         /* ftmp = z1^2*x2 - z2^2*x1 */
 920         felem_diff_128_64(tmp, ftmp2);
 921         /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
 922         felem_reduce(ftmp, tmp);
 923
 924         /* the formulae are incorrect if the points are equal
 925          * so we check for this and do doubling if this happens */
 926         x_equal = felem_is_zero(ftmp);
 927         y_equal = felem_is_zero(ftmp3);
 928         z1_is_zero = felem_is_zero(z1);
 929         z2_is_zero = felem_is_zero(z2);
 930         /* In affine coordinates, (X_1, Y_1) == (X_2, Y_2) */
 931         if (x_equal && y_equal && !z1_is_zero && !z2_is_zero)
 932                 {
 933                 point_double(x3, y3, z3, x1, y1, z1);
 934                 return;
 935                 }
 936
 937         /* ftmp5 = z1*z2 */
 938         if (!mixed)
 939                 {
 940                 felem_mul(tmp, z1, z2);
 941                 felem_reduce(ftmp5, tmp);
 942                 }
 943         else
 944                 {
 945                 /* special case z2 = 0 is handled later */
 946                 felem_assign(ftmp5, z1);
 947                 }
 948
 949         /* z_out = (z1^2*x2 - z2^2*x1)*(z1*z2) */
 950         felem_mul(tmp, ftmp, ftmp5);
 951         felem_reduce(z_out, tmp);
 952
 953         /* ftmp = (z1^2*x2 - z2^2*x1)^2 */
 954         felem_assign(ftmp5, ftmp);
 955         felem_square(tmp, ftmp);
 956         felem_reduce(ftmp, tmp);
 957
 958         /* ftmp5 = (z1^2*x2 - z2^2*x1)^3 */
 959         felem_mul(tmp, ftmp, ftmp5);
 960         felem_reduce(ftmp5, tmp);
 961
 962         /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
 963         felem_mul(tmp, ftmp2, ftmp);
 964         felem_reduce(ftmp2, tmp);
 965
 966         /* tmp = z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */
 967         felem_mul(tmp, ftmp4, ftmp5);
 968         /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
 969
 970         /* tmp2 = (z1^3*y2 - z2^3*y1)^2 */
 971         felem_square(tmp2, ftmp3);
 972         /* tmp2[i] < 4 * 2^57 * 2^57 < 2^116 */
 973
 974         /* tmp2 = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 */
 975         felem_diff_128_64(tmp2, ftmp5);
 976         /* tmp2[i] < 2^116 + 2^64 + 8 < 2^117 */
 977
 978         /* ftmp5 = 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
 979         felem_assign(ftmp5, ftmp2);
 980         felem_scalar(ftmp5, 2);
 981         /* ftmp5[i] < 2 * 2^57 = 2^58 */
 982
 983         /*-
 984          * x_out = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 -
 985          *  2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2
 986          */
 987         felem_diff_128_64(tmp2, ftmp5);
 988         /* tmp2[i] < 2^117 + 2^64 + 8 < 2^118 */
 989         felem_reduce(x_out, tmp2);
 990
 991         /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out */
 992         felem_diff(ftmp2, x_out);
 993         /* ftmp2[i] < 2^57 + 2^58 + 2 < 2^59 */
 994
 995         /* tmp2 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) */
 996         felem_mul(tmp2, ftmp3, ftmp2);
 997         /* tmp2[i] < 4 * 2^57 * 2^59 = 2^118 */
 998
 999         /*-
1000          * y_out = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) -
1001          *  z2^3*y1*(z1^2*x2 - z2^2*x1)^3
1002          */
1003         widefelem_diff(tmp2, tmp);
1004         /* tmp2[i] < 2^118 + 2^120 < 2^121 */
1005         felem_reduce(y_out, tmp2);
1006
1007         /* the result (x_out, y_out, z_out) is incorrect if one of the inputs is
1008          * the point at infinity, so we need to check for this separately */
1009
1010         /* if point 1 is at infinity, copy point 2 to output, and vice versa */
1011         copy_conditional(x_out, x2, z1_is_zero);
1012         copy_conditional(x_out, x1, z2_is_zero);
1013         copy_conditional(y_out, y2, z1_is_zero);
1014         copy_conditional(y_out, y1, z2_is_zero);
1015         copy_conditional(z_out, z2, z1_is_zero);
1016         copy_conditional(z_out, z1, z2_is_zero);
1017         felem_assign(x3, x_out);
1018         felem_assign(y3, y_out);
1019         felem_assign(z3, z_out);
1020         }
1021
1022 /* select_point selects the |idx|th point from a precomputation table and
1023  * copies it to out. */
1024 static void select_point(const u64 idx, unsigned int size, const felem pre_comp[/*size*/][3], felem out[3])
1025         {
1026         unsigned i, j;
1027         limb *outlimbs = &out[0][0];
1028         memset(outlimbs, 0, 3 * sizeof(felem));
1029
1030         for (i = 0; i < size; i++)
1031                 {
1032                 const limb *inlimbs = &pre_comp[i][0][0];
1033                 u64 mask = i ^ idx;
1034                 mask |= mask >> 4;
1035                 mask |= mask >> 2;
1036                 mask |= mask >> 1;
1037                 mask &= 1;
1038                 mask--;
1039                 for (j = 0; j < 4 * 3; j++)
1040                         outlimbs[j] |= inlimbs[j] & mask;
1041                 }
1042         }
1043
1044 /* get_bit returns the |i|th bit in |in| */
1045 static char get_bit(const felem_bytearray in, unsigned i)
1046         {
1047         if (i >= 224)
1048                 return 0;
1049         return (in[i >> 3] >> (i & 7)) & 1;
1050         }
1051
1052 /* Interleaved point multiplication using precomputed point multiples:
1053  * The small point multiples 0*P, 1*P, ..., 16*P are in pre_comp[],
1054  * the scalars in scalars[]. If g_scalar is non-NULL, we also add this multiple
1055  * of the generator, using certain (large) precomputed multiples in g_pre_comp.
1056  * Output point (X, Y, Z) is stored in x_out, y_out, z_out */
1057 static void batch_mul(felem x_out, felem y_out, felem z_out,
1058         const felem_bytearray scalars[], const unsigned num_points, const u8 *g_scalar,
1059         const int mixed, const felem pre_comp[][17][3], const felem g_pre_comp[2][16][3])
1060         {
1061         int i, skip;
1062         unsigned num;
1063         unsigned gen_mul = (g_scalar != NULL);
1064         felem nq[3], tmp[4];
1065         u64 bits;
1066         u8 sign, digit;
1067
1068         /* set nq to the point at infinity */
1069         memset(nq, 0, 3 * sizeof(felem));
1070
1071         /* Loop over all scalars msb-to-lsb, interleaving additions
1072          * of multiples of the generator (two in each of the last 28 rounds)
1073          * and additions of other points multiples (every 5th round).
1074          */
1075         skip = 1; /* save two point operations in the first round */
1076         for (i = (num_points ? 220 : 27); i >= 0; --i)
1077                 {
1078                 /* double */
1079                 if (!skip)
1080                         point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
1081
1082                 /* add multiples of the generator */
1083                 if (gen_mul && (i <= 27))
1084                         {
1085                         /* first, look 28 bits upwards */
1086                         bits = get_bit(g_scalar, i + 196) << 3;
1087                         bits |= get_bit(g_scalar, i + 140) << 2;
1088                         bits |= get_bit(g_scalar, i + 84) << 1;
1089                         bits |= get_bit(g_scalar, i + 28);
1090                         /* select the point to add, in constant time */
1091                         select_point(bits, 16, g_pre_comp[1], tmp);
1092
1093                         if (!skip)
1094                                 {
1095                                 point_add(nq[0], nq[1], nq[2],
1096                                         nq[0], nq[1], nq[2],
1097                                         1 /* mixed */, tmp[0], tmp[1], tmp[2]);
1098                                 }
1099                         else
1100                                 {
1101                                 memcpy(nq, tmp, 3 * sizeof(felem));
1102                                 skip = 0;
1103                                 }
1104
1105                         /* second, look at the current position */
1106                         bits = get_bit(g_scalar, i + 168) << 3;
1107                         bits |= get_bit(g_scalar, i + 112) << 2;
1108                         bits |= get_bit(g_scalar, i + 56) << 1;
1109                         bits |= get_bit(g_scalar, i);
1110                         /* select the point to add, in constant time */
1111                         select_point(bits, 16, g_pre_comp[0], tmp);
1112                         point_add(nq[0], nq[1], nq[2],
1113                                 nq[0], nq[1], nq[2],
1114                                 1 /* mixed */, tmp[0], tmp[1], tmp[2]);
1115                         }
1116
1117                 /* do other additions every 5 doublings */
1118                 if (num_points && (i % 5 == 0))
1119                         {
1120                         /* loop over all scalars */
1121                         for (num = 0; num < num_points; ++num)
1122                                 {
1123                                 bits = get_bit(scalars[num], i + 4) << 5;
1124                                 bits |= get_bit(scalars[num], i + 3) << 4;
1125                                 bits |= get_bit(scalars[num], i + 2) << 3;
1126                                 bits |= get_bit(scalars[num], i + 1) << 2;
1127                                 bits |= get_bit(scalars[num], i) << 1;
1128                                 bits |= get_bit(scalars[num], i - 1);
1129                                 ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits);
1130
1131                                 /* select the point to add or subtract */
1132                                 select_point(digit, 17, pre_comp[num], tmp);
1133                                 felem_neg(tmp[3], tmp[1]); /* (X, -Y, Z) is the negative point */
1134                                 copy_conditional(tmp[1], tmp[3], sign);
1135
1136                                 if (!skip)
1137                                         {
1138                                         point_add(nq[0], nq[1], nq[2],
1139                                                 nq[0], nq[1], nq[2],
1140                                                 mixed, tmp[0], tmp[1], tmp[2]);
1141                                         }
1142                                 else
1143                                         {
1144                                         memcpy(nq, tmp, 3 * sizeof(felem));
1145                                         skip = 0;
1146                                         }
1147                                 }
1148                         }
1149                 }
1150         felem_assign(x_out, nq[0]);
1151         felem_assign(y_out, nq[1]);
1152         felem_assign(z_out, nq[2]);
1153         }
1154
1155 /******************************************************************************/
1156 /*                     FUNCTIONS TO MANAGE PRECOMPUTATION
1157  */
1158
1159 static NISTP224_PRE_COMP *nistp224_pre_comp_new()
1160         {
1161         NISTP224_PRE_COMP *ret = NULL;
1162         ret = (NISTP224_PRE_COMP *) OPENSSL_malloc(sizeof *ret);
1163         if (!ret)
1164                 {
1165                 ECerr(EC_F_NISTP224_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1166                 return ret;
1167                 }
1168         memset(ret->g_pre_comp, 0, sizeof(ret->g_pre_comp));
1169         ret->references = 1;
1170         return ret;
1171         }
1172
1173 static void *nistp224_pre_comp_dup(void *src_)
1174         {
1175         NISTP224_PRE_COMP *src = src_;
1176
1177         /* no need to actually copy, these objects never change! */
1178         CRYPTO_add(&src->references, 1, CRYPTO_LOCK_EC_PRE_COMP);
1179
1180         return src_;
1181         }
1182
1183 static void nistp224_pre_comp_free(void *pre_)
1184         {
1185         int i;
1186         NISTP224_PRE_COMP *pre = pre_;
1187
1188         if (!pre)
1189                 return;
1190
1191         i = CRYPTO_add(&pre->references, -1, CRYPTO_LOCK_EC_PRE_COMP);
1192         if (i > 0)
1193                 return;
1194
1195         OPENSSL_free(pre);
1196         }
1197
1198 static void nistp224_pre_comp_clear_free(void *pre_)
1199         {
1200         int i;
1201         NISTP224_PRE_COMP *pre = pre_;
1202
1203         if (!pre)
1204                 return;
1205
1206         i = CRYPTO_add(&pre->references, -1, CRYPTO_LOCK_EC_PRE_COMP);
1207         if (i > 0)
1208                 return;
1209
1210         OPENSSL_cleanse(pre, sizeof *pre);
1211         OPENSSL_free(pre);
1212         }
1213
1214 /******************************************************************************/
1215 /*                         OPENSSL EC_METHOD FUNCTIONS
1216  */
1217
1218 int ec_GFp_nistp224_group_init(EC_GROUP *group)
1219         {
1220         int ret;
1221         ret = ec_GFp_simple_group_init(group);
1222         group->a_is_minus3 = 1;
1223         return ret;
1224         }
1225
1226 int ec_GFp_nistp224_group_set_curve(EC_GROUP *group, const BIGNUM *p,
1227         const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
1228         {
1229         int ret = 0;
1230         BN_CTX *new_ctx = NULL;
1231         BIGNUM *curve_p, *curve_a, *curve_b;
1232
1233         if (ctx == NULL)
1234                 if ((ctx = new_ctx = BN_CTX_new()) == NULL) return 0;
1235         BN_CTX_start(ctx);
1236         if (((curve_p = BN_CTX_get(ctx)) == NULL) ||
1237                 ((curve_a = BN_CTX_get(ctx)) == NULL) ||
1238                 ((curve_b = BN_CTX_get(ctx)) == NULL)) goto err;
1239         BN_bin2bn(nistp224_curve_params[0], sizeof(felem_bytearray), curve_p);
1240         BN_bin2bn(nistp224_curve_params[1], sizeof(felem_bytearray), curve_a);
1241         BN_bin2bn(nistp224_curve_params[2], sizeof(felem_bytearray), curve_b);
1242         if ((BN_cmp(curve_p, p)) || (BN_cmp(curve_a, a)) ||
1243                 (BN_cmp(curve_b, b)))
1244                 {
1245                 ECerr(EC_F_EC_GFP_NISTP224_GROUP_SET_CURVE,
1246                         EC_R_WRONG_CURVE_PARAMETERS);
1247                 goto err;
1248                 }
1249         group->field_mod_func = BN_nist_mod_224;
1250         ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx);
1251 err:
1252         BN_CTX_end(ctx);
1253         if (new_ctx != NULL)
1254                 BN_CTX_free(new_ctx);
1255         return ret;
1256         }
1257
1258 /* Takes the Jacobian coordinates (X, Y, Z) of a point and returns
1259  * (X', Y') = (X/Z^2, Y/Z^3) */
1260 int ec_GFp_nistp224_point_get_affine_coordinates(const EC_GROUP *group,
1261         const EC_POINT *point, BIGNUM *x, BIGNUM *y, BN_CTX *ctx)
1262         {
1263         felem z1, z2, x_in, y_in, x_out, y_out;
1264         widefelem tmp;
1265
1266         if (EC_POINT_is_at_infinity(group, point))
1267                 {
1268                 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1269                         EC_R_POINT_AT_INFINITY);
1270                 return 0;
1271                 }
1272         if ((!BN_to_felem(x_in, &point->X)) || (!BN_to_felem(y_in, &point->Y)) ||
1273                 (!BN_to_felem(z1, &point->Z))) return 0;
1274         felem_inv(z2, z1);
1275         felem_square(tmp, z2); felem_reduce(z1, tmp);
1276         felem_mul(tmp, x_in, z1); felem_reduce(x_in, tmp);
1277         felem_contract(x_out, x_in);
1278         if (x != NULL)
1279                 {
1280                 if (!felem_to_BN(x, x_out)) {
1281                 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1282                         ERR_R_BN_LIB);
1283                 return 0;
1284                 }
1285                 }
1286         felem_mul(tmp, z1, z2); felem_reduce(z1, tmp);
1287         felem_mul(tmp, y_in, z1); felem_reduce(y_in, tmp);
1288         felem_contract(y_out, y_in);
1289         if (y != NULL)
1290                 {
1291                 if (!felem_to_BN(y, y_out)) {
1292                 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1293                         ERR_R_BN_LIB);
1294                 return 0;
1295                 }
1296                 }
1297         return 1;
1298         }
1299
1300 static void make_points_affine(size_t num, felem points[/*num*/][3], felem tmp_felems[/*num+1*/])
1301         {
1302         /* Runs in constant time, unless an input is the point at infinity
1303          * (which normally shouldn't happen). */
1304         ec_GFp_nistp_points_make_affine_internal(
1305                 num,
1306                 points,
1307                 sizeof(felem),
1308                 tmp_felems,
1309                 (void (*)(void *)) felem_one,
1310                 (int (*)(const void *)) felem_is_zero_int,
1311                 (void (*)(void *, const void *)) felem_assign,
1312                 (void (*)(void *, const void *)) felem_square_reduce,
1313                 (void (*)(void *, const void *, const void *)) felem_mul_reduce,
1314                 (void (*)(void *, const void *)) felem_inv,
1315                 (void (*)(void *, const void *)) felem_contract);
1316         }
1317
1318 /* Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL values
1319  * Result is stored in r (r can equal one of the inputs). */
1320 int ec_GFp_nistp224_points_mul(const EC_GROUP *group, EC_POINT *r,
1321         const BIGNUM *scalar, size_t num, const EC_POINT *points[],
1322         const BIGNUM *scalars[], BN_CTX *ctx)
1323         {
1324         int ret = 0;
1325         int j;
1326         unsigned i;
1327         int mixed = 0;
1328         BN_CTX *new_ctx = NULL;
1329         BIGNUM *x, *y, *z, *tmp_scalar;
1330         felem_bytearray g_secret;
1331         felem_bytearray *secrets = NULL;
1332         felem (*pre_comp)[17][3] = NULL;
1333         felem *tmp_felems = NULL;
1334         felem_bytearray tmp;
1335         unsigned num_bytes;
1336         int have_pre_comp = 0;
1337         size_t num_points = num;
1338         felem x_in, y_in, z_in, x_out, y_out, z_out;
1339         NISTP224_PRE_COMP *pre = NULL;
1340         const felem (*g_pre_comp)[16][3] = NULL;
1341         EC_POINT *generator = NULL;
1342         const EC_POINT *p = NULL;
1343         const BIGNUM *p_scalar = NULL;
1344
1345         if (ctx == NULL)
1346                 if ((ctx = new_ctx = BN_CTX_new()) == NULL) return 0;
1347         BN_CTX_start(ctx);
1348         if (((x = BN_CTX_get(ctx)) == NULL) ||
1349                 ((y = BN_CTX_get(ctx)) == NULL) ||
1350                 ((z = BN_CTX_get(ctx)) == NULL) ||
1351                 ((tmp_scalar = BN_CTX_get(ctx)) == NULL))
1352                 goto err;
1353
1354         if (scalar != NULL)
1355                 {
1356                 pre = EC_EX_DATA_get_data(group->extra_data,
1357                         nistp224_pre_comp_dup, nistp224_pre_comp_free,
1358                         nistp224_pre_comp_clear_free);
1359                 if (pre)
1360                         /* we have precomputation, try to use it */
1361                         g_pre_comp = (const felem (*)[16][3]) pre->g_pre_comp;
1362                 else
1363                         /* try to use the standard precomputation */
1364                         g_pre_comp = &gmul[0];
1365                 generator = EC_POINT_new(group);
1366                 if (generator == NULL)
1367                         goto err;
1368                 /* get the generator from precomputation */
1369                 if (!felem_to_BN(x, g_pre_comp[0][1][0]) ||
1370                         !felem_to_BN(y, g_pre_comp[0][1][1]) ||
1371                         !felem_to_BN(z, g_pre_comp[0][1][2]))
1372                         {
1373                         ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1374                         goto err;
1375                         }
1376                 if (!EC_POINT_set_Jprojective_coordinates_GFp(group,
1377                                 generator, x, y, z, ctx))
1378                         goto err;
1379                 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
1380                         /* precomputation matches generator */
1381                         have_pre_comp = 1;
1382                 else
1383                         /* we don't have valid precomputation:
1384                          * treat the generator as a random point */
1385                         num_points = num_points + 1;
1386                 }
1387
1388         if (num_points > 0)
1389                 {
1390                 if (num_points >= 3)
1391                         {
1392                         /* unless we precompute multiples for just one or two points,
1393                          * converting those into affine form is time well spent  */
1394                         mixed = 1;
1395                         }
1396                 secrets = OPENSSL_malloc(num_points * sizeof(felem_bytearray));
1397                 pre_comp = OPENSSL_malloc(num_points * 17 * 3 * sizeof(felem));
1398                 if (mixed)
1399                         tmp_felems = OPENSSL_malloc((num_points * 17 + 1) * sizeof(felem));
1400                 if ((secrets == NULL) || (pre_comp == NULL) || (mixed && (tmp_felems == NULL)))
1401                         {
1402                         ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_MALLOC_FAILURE);
1403                         goto err;
1404                         }
1405
1406                 /* we treat NULL scalars as 0, and NULL points as points at infinity,
1407                  * i.e., they contribute nothing to the linear combination */
1408                 memset(secrets, 0, num_points * sizeof(felem_bytearray));
1409                 memset(pre_comp, 0, num_points * 17 * 3 * sizeof(felem));
1410                 for (i = 0; i < num_points; ++i)
1411                         {
1412                         if (i == num)
1413                                 /* the generator */
1414                                 {
1415                                 p = EC_GROUP_get0_generator(group);
1416                                 p_scalar = scalar;
1417                                 }
1418                         else
1419                                 /* the i^th point */
1420                                 {
1421                                 p = points[i];
1422                                 p_scalar = scalars[i];
1423                                 }
1424                         if ((p_scalar != NULL) && (p != NULL))
1425                                 {
1426                                 /* reduce scalar to 0 <= scalar < 2^224 */
1427                                 if ((BN_num_bits(p_scalar) > 224) || (BN_is_negative(p_scalar)))
1428                                         {
1429                                         /* this is an unusual input, and we don't guarantee
1430                                          * constant-timeness */
1431                                         if (!BN_nnmod(tmp_scalar, p_scalar, &group->order, ctx))
1432                                                 {
1433                                                 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1434                                                 goto err;
1435                                                 }
1436                                         num_bytes = BN_bn2bin(tmp_scalar, tmp);
1437                                         }
1438                                 else
1439                                         num_bytes = BN_bn2bin(p_scalar, tmp);
1440                                 flip_endian(secrets[i], tmp, num_bytes);
1441                                 /* precompute multiples */
1442                                 if ((!BN_to_felem(x_out, &p->X)) ||
1443                                         (!BN_to_felem(y_out, &p->Y)) ||
1444                                         (!BN_to_felem(z_out, &p->Z))) goto err;
1445                                 felem_assign(pre_comp[i][1][0], x_out);
1446                                 felem_assign(pre_comp[i][1][1], y_out);
1447                                 felem_assign(pre_comp[i][1][2], z_out);
1448                                 for (j = 2; j <= 16; ++j)
1449                                         {
1450                                         if (j & 1)
1451                                                 {
1452                                                 point_add(
1453                                                         pre_comp[i][j][0], pre_comp[i][j][1], pre_comp[i][j][2],
1454                                                         pre_comp[i][1][0], pre_comp[i][1][1], pre_comp[i][1][2],
1455                                                         0, pre_comp[i][j-1][0], pre_comp[i][j-1][1], pre_comp[i][j-1][2]);
1456                                                 }
1457                                         else
1458                                                 {
1459                                                 point_double(
1460                                                         pre_comp[i][j][0], pre_comp[i][j][1], pre_comp[i][j][2],
1461                                                         pre_comp[i][j/2][0], pre_comp[i][j/2][1], pre_comp[i][j/2][2]);
1462                                                 }
1463                                         }
1464                                 }
1465                         }
1466                 if (mixed)
1467                         make_points_affine(num_points * 17, pre_comp[0], tmp_felems);
1468                 }
1469
1470         /* the scalar for the generator */
1471         if ((scalar != NULL) && (have_pre_comp))
1472                 {
1473                 memset(g_secret, 0, sizeof g_secret);
1474                 /* reduce scalar to 0 <= scalar < 2^224 */
1475                 if ((BN_num_bits(scalar) > 224) || (BN_is_negative(scalar)))
1476                         {
1477                         /* this is an unusual input, and we don't guarantee
1478                          * constant-timeness */
1479                         if (!BN_nnmod(tmp_scalar, scalar, &group->order, ctx))
1480                                 {
1481                                 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1482                                 goto err;
1483                                 }
1484                         num_bytes = BN_bn2bin(tmp_scalar, tmp);
1485                         }
1486                 else
1487                         num_bytes = BN_bn2bin(scalar, tmp);
1488                 flip_endian(g_secret, tmp, num_bytes);
1489                 /* do the multiplication with generator precomputation*/
1490                 batch_mul(x_out, y_out, z_out,
1491                         (const felem_bytearray (*)) secrets, num_points,
1492                         g_secret,
1493                         mixed, (const felem (*)[17][3]) pre_comp,
1494                         g_pre_comp);
1495                 }
1496         else
1497                 /* do the multiplication without generator precomputation */
1498                 batch_mul(x_out, y_out, z_out,
1499                         (const felem_bytearray (*)) secrets, num_points,
1500                         NULL, mixed, (const felem (*)[17][3]) pre_comp, NULL);
1501         /* reduce the output to its unique minimal representation */
1502         felem_contract(x_in, x_out);
1503         felem_contract(y_in, y_out);
1504         felem_contract(z_in, z_out);
1505         if ((!felem_to_BN(x, x_in)) || (!felem_to_BN(y, y_in)) ||
1506                 (!felem_to_BN(z, z_in)))
1507                 {
1508                 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1509                 goto err;
1510                 }
1511         ret = EC_POINT_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx);
1512
1513 err:
1514         BN_CTX_end(ctx);
1515         if (generator != NULL)
1516                 EC_POINT_free(generator);
1517         if (new_ctx != NULL)
1518                 BN_CTX_free(new_ctx);
1519         if (secrets != NULL)
1520                 OPENSSL_free(secrets);
1521         if (pre_comp != NULL)
1522                 OPENSSL_free(pre_comp);
1523         if (tmp_felems != NULL)
1524                 OPENSSL_free(tmp_felems);
1525         return ret;
1526         }
1527
1528 int ec_GFp_nistp224_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
1529         {
1530         int ret = 0;
1531         NISTP224_PRE_COMP *pre = NULL;
1532         int i, j;
1533         BN_CTX *new_ctx = NULL;
1534         BIGNUM *x, *y;
1535         EC_POINT *generator = NULL;
1536         felem tmp_felems[32];
1537
1538         /* throw away old precomputation */
1539         EC_EX_DATA_free_data(&group->extra_data, nistp224_pre_comp_dup,
1540                 nistp224_pre_comp_free, nistp224_pre_comp_clear_free);
1541         if (ctx == NULL)
1542                 if ((ctx = new_ctx = BN_CTX_new()) == NULL) return 0;
1543         BN_CTX_start(ctx);
1544         if (((x = BN_CTX_get(ctx)) == NULL) ||
1545                 ((y = BN_CTX_get(ctx)) == NULL))
1546                 goto err;
1547         /* get the generator */
1548         if (group->generator == NULL) goto err;
1549         generator = EC_POINT_new(group);
1550         if (generator == NULL)
1551                 goto err;
1552         BN_bin2bn(nistp224_curve_params[3], sizeof (felem_bytearray), x);
1553         BN_bin2bn(nistp224_curve_params[4], sizeof (felem_bytearray), y);
1554         if (!EC_POINT_set_affine_coordinates_GFp(group, generator, x, y, ctx))
1555                 goto err;
1556         if ((pre = nistp224_pre_comp_new()) == NULL)
1557                 goto err;
1558         /* if the generator is the standard one, use built-in precomputation */
1559         if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
1560                 {
1561                 memcpy(pre->g_pre_comp, gmul, sizeof(pre->g_pre_comp));
1562                 ret = 1;
1563                 goto err;
1564                 }
1565         if ((!BN_to_felem(pre->g_pre_comp[0][1][0], &group->generator->X)) ||
1566                 (!BN_to_felem(pre->g_pre_comp[0][1][1], &group->generator->Y)) ||
1567                 (!BN_to_felem(pre->g_pre_comp[0][1][2], &group->generator->Z)))
1568                 goto err;
1569         /* compute 2^56*G, 2^112*G, 2^168*G for the first table,
1570          * 2^28*G, 2^84*G, 2^140*G, 2^196*G for the second one
1571          */
1572         for (i = 1; i <= 8; i <<= 1)
1573                 {
1574                 point_double(
1575                         pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2],
1576                         pre->g_pre_comp[0][i][0], pre->g_pre_comp[0][i][1], pre->g_pre_comp[0][i][2]);
1577                 for (j = 0; j < 27; ++j)
1578                         {
1579                         point_double(
1580                                 pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2],
1581                                 pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2]);
1582                         }
1583                 if (i == 8)
1584                         break;
1585                 point_double(
1586                         pre->g_pre_comp[0][2*i][0], pre->g_pre_comp[0][2*i][1], pre->g_pre_comp[0][2*i][2],
1587                         pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2]);
1588                 for (j = 0; j < 27; ++j)
1589                         {
1590                         point_double(
1591                                 pre->g_pre_comp[0][2*i][0], pre->g_pre_comp[0][2*i][1], pre->g_pre_comp[0][2*i][2],
1592                                 pre->g_pre_comp[0][2*i][0], pre->g_pre_comp[0][2*i][1], pre->g_pre_comp[0][2*i][2]);
1593                         }
1594                 }
1595         for (i = 0; i < 2; i++)
1596                 {
1597                 /* g_pre_comp[i][0] is the point at infinity */
1598                 memset(pre->g_pre_comp[i][0], 0, sizeof(pre->g_pre_comp[i][0]));
1599                 /* the remaining multiples */
1600                 /* 2^56*G + 2^112*G resp. 2^84*G + 2^140*G */
1601                 point_add(
1602                         pre->g_pre_comp[i][6][0], pre->g_pre_comp[i][6][1],
1603                         pre->g_pre_comp[i][6][2], pre->g_pre_comp[i][4][0],
1604                         pre->g_pre_comp[i][4][1], pre->g_pre_comp[i][4][2],
1605                         0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
1606                         pre->g_pre_comp[i][2][2]);
1607                 /* 2^56*G + 2^168*G resp. 2^84*G + 2^196*G */
1608                 point_add(
1609                         pre->g_pre_comp[i][10][0], pre->g_pre_comp[i][10][1],
1610                         pre->g_pre_comp[i][10][2], pre->g_pre_comp[i][8][0],
1611                         pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2],
1612                         0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
1613                         pre->g_pre_comp[i][2][2]);
1614                 /* 2^112*G + 2^168*G resp. 2^140*G + 2^196*G */
1615                 point_add(
1616                         pre->g_pre_comp[i][12][0], pre->g_pre_comp[i][12][1],
1617                         pre->g_pre_comp[i][12][2], pre->g_pre_comp[i][8][0],
1618                         pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2],
1619                         0, pre->g_pre_comp[i][4][0], pre->g_pre_comp[i][4][1],
1620                         pre->g_pre_comp[i][4][2]);
1621                 /* 2^56*G + 2^112*G + 2^168*G resp. 2^84*G + 2^140*G + 2^196*G */
1622                 point_add(
1623                         pre->g_pre_comp[i][14][0], pre->g_pre_comp[i][14][1],
1624                         pre->g_pre_comp[i][14][2], pre->g_pre_comp[i][12][0],
1625                         pre->g_pre_comp[i][12][1], pre->g_pre_comp[i][12][2],
1626                         0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
1627                         pre->g_pre_comp[i][2][2]);
1628                 for (j = 1; j < 8; ++j)
1629                         {
1630                         /* odd multiples: add G resp. 2^28*G */
1631                         point_add(
1632                                 pre->g_pre_comp[i][2*j+1][0], pre->g_pre_comp[i][2*j+1][1],
1633                                 pre->g_pre_comp[i][2*j+1][2], pre->g_pre_comp[i][2*j][0],
1634                                 pre->g_pre_comp[i][2*j][1], pre->g_pre_comp[i][2*j][2],
1635                                 0, pre->g_pre_comp[i][1][0], pre->g_pre_comp[i][1][1],
1636                                 pre->g_pre_comp[i][1][2]);
1637                         }
1638                 }
1639         make_points_affine(31, &(pre->g_pre_comp[0][1]), tmp_felems);
1640
1641         if (!EC_EX_DATA_set_data(&group->extra_data, pre, nistp224_pre_comp_dup,
1642                         nistp224_pre_comp_free, nistp224_pre_comp_clear_free))
1643                 goto err;
1644         ret = 1;
1645         pre = NULL;
1646  err:
1647         BN_CTX_end(ctx);
1648         if (generator != NULL)
1649                 EC_POINT_free(generator);
1650         if (new_ctx != NULL)
1651                 BN_CTX_free(new_ctx);
1652         if (pre)
1653                 nistp224_pre_comp_free(pre);
1654         return ret;
1655         }
1656
1657 int ec_GFp_nistp224_have_precompute_mult(const EC_GROUP *group)
1658         {
1659         if (EC_EX_DATA_get_data(group->extra_data, nistp224_pre_comp_dup,
1660                         nistp224_pre_comp_free, nistp224_pre_comp_clear_free)
1661                 != NULL)
1662                 return 1;
1663         else
1664                 return 0;
1665         }
1666
1667 #else
1668 static void *dummy=&dummy;
1669 #endif