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 #ifdef EC_NISTP_64_GCC_128
  28 #include <stdint.h>
  29 #include <string.h>
  30 #include <openssl/err.h>
  31 #include "ec_lcl.h"
  32
  33 #if defined(__GNUC__) && (__GNUC__ > 3 || (__GNUC__ == 3 && __GNUC_MINOR__ >= 1))
  34   /* even with gcc, the typedef won't work for 32-bit platforms */
  35   typedef __uint128_t uint128_t; /* nonstandard; implemented by gcc on 64-bit platforms */
  36 #else
  37   #error "Need GCC 3.1 or later to define type uint128_t"
  38 #endif
  39
  40 typedef uint8_t u8;
  41 typedef uint64_t u64;
  42 typedef int64_t s64;
  43
  44
  45 /******************************************************************************/
  46 /*                  INTERNAL REPRESENTATION OF FIELD ELEMENTS
  47  *
  48  * Field elements are represented as a_0 + 2^56*a_1 + 2^112*a_2 + 2^168*a_3
  49  * using 64-bit coefficients called 'limbs',
  50  * and sometimes (for multiplication results) as
  51  * 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
  52  * using 128-bit coefficients called 'widelimbs'.
  53  * A 4-limb representation is an 'felem';
  54  * a 7-widelimb representation is a 'widefelem'.
  55  * Even within felems, bits of adjacent limbs overlap, and we don't always
  56  * reduce the representations: we ensure that inputs to each felem
  57  * multiplication satisfy a_i < 2^60, so outputs satisfy b_i < 4*2^60*2^60,
  58  * and fit into a 128-bit word without overflow. The coefficients are then
  59  * again partially reduced to obtain an felem satisfying a_i < 2^57.
  60  * We only reduce to the unique minimal representation at the end of the
  61  * computation.
  62  */
  63
  64 typedef uint64_t limb;
  65 typedef uint128_t widelimb;
  66
  67 typedef limb felem[4];
  68 typedef widelimb widefelem[7];
  69
  70 /* Field element represented as a byte arrary.
  71  * 28*8 = 224 bits is also the group order size for the elliptic curve,
  72  * and we also use this type for scalars for point multiplication.
  73   */
  74 typedef u8 felem_bytearray[28];
  75
  76 static const felem_bytearray nistp224_curve_params[5] = {
  77         {0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,    /* p */
  78          0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0x00,0x00,0x00,0x00,
  79          0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x01},
  80         {0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,    /* a */
  81          0xFF,0xFF,0xFF,0xFF,0xFF,0xFE,0xFF,0xFF,0xFF,0xFF,
  82          0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFE},
  83         {0xB4,0x05,0x0A,0x85,0x0C,0x04,0xB3,0xAB,0xF5,0x41,    /* b */
  84          0x32,0x56,0x50,0x44,0xB0,0xB7,0xD7,0xBF,0xD8,0xBA,
  85          0x27,0x0B,0x39,0x43,0x23,0x55,0xFF,0xB4},
  86         {0xB7,0x0E,0x0C,0xBD,0x6B,0xB4,0xBF,0x7F,0x32,0x13,    /* x */
  87          0x90,0xB9,0x4A,0x03,0xC1,0xD3,0x56,0xC2,0x11,0x22,
  88          0x34,0x32,0x80,0xD6,0x11,0x5C,0x1D,0x21},
  89         {0xbd,0x37,0x63,0x88,0xb5,0xf7,0x23,0xfb,0x4c,0x22,    /* y */
  90          0xdf,0xe6,0xcd,0x43,0x75,0xa0,0x5a,0x07,0x47,0x64,
  91          0x44,0xd5,0x81,0x99,0x85,0x00,0x7e,0x34}
  92 };
  93
  94 /* Precomputed multiples of the standard generator
  95  * Points are given in coordinates (X, Y, Z) where Z normally is 1
  96  * (0 for the point at infinity).
  97  * For each field element, slice a_0 is word 0, etc.
  98  *
  99  * The table has 2 * 16 elements, starting with the following:
 100  * index | bits    | point
 101  * ------+---------+------------------------------
 102  *     0 | 0 0 0 0 | 0G
 103  *     1 | 0 0 0 1 | 1G
 104  *     2 | 0 0 1 0 | 2^56G
 105  *     3 | 0 0 1 1 | (2^56 + 1)G
 106  *     4 | 0 1 0 0 | 2^112G
 107  *     5 | 0 1 0 1 | (2^112 + 1)G
 108  *     6 | 0 1 1 0 | (2^112 + 2^56)G
 109  *     7 | 0 1 1 1 | (2^112 + 2^56 + 1)G
 110  *     8 | 1 0 0 0 | 2^168G
 111  *     9 | 1 0 0 1 | (2^168 + 1)G
 112  *    10 | 1 0 1 0 | (2^168 + 2^56)G
 113  *    11 | 1 0 1 1 | (2^168 + 2^56 + 1)G
 114  *    12 | 1 1 0 0 | (2^168 + 2^112)G
 115  *    13 | 1 1 0 1 | (2^168 + 2^112 + 1)G
 116  *    14 | 1 1 1 0 | (2^168 + 2^112 + 2^56)G
 117  *    15 | 1 1 1 1 | (2^168 + 2^112 + 2^56 + 1)G
 118  * followed by a copy of this with each element multiplied by 2^28.
 119  *
 120  * The reason for this is so that we can clock bits into four different
 121  * locations when doing simple scalar multiplies against the base point,
 122  * and then another four locations using the second 16 elements.
 123  */
 124 static const felem gmul[2][16][3] =
 125 {{{{0, 0, 0, 0},
 126    {0, 0, 0, 0},
 127    {0, 0, 0, 0}},
 128   {{0x3280d6115c1d21, 0xc1d356c2112234, 0x7f321390b94a03, 0xb70e0cbd6bb4bf},
 129    {0xd5819985007e34, 0x75a05a07476444, 0xfb4c22dfe6cd43, 0xbd376388b5f723},
 130    {1, 0, 0, 0}},
 131   {{0xfd9675666ebbe9, 0xbca7664d40ce5e, 0x2242df8d8a2a43, 0x1f49bbb0f99bc5},
 132    {0x29e0b892dc9c43, 0xece8608436e662, 0xdc858f185310d0, 0x9812dd4eb8d321},
 133    {1, 0, 0, 0}},
 134   {{0x6d3e678d5d8eb8, 0x559eed1cb362f1, 0x16e9a3bbce8a3f, 0xeedcccd8c2a748},
 135    {0xf19f90ed50266d, 0xabf2b4bf65f9df, 0x313865468fafec, 0x5cb379ba910a17},
 136    {1, 0, 0, 0}},
 137   {{0x0641966cab26e3, 0x91fb2991fab0a0, 0xefec27a4e13a0b, 0x0499aa8a5f8ebe},
 138    {0x7510407766af5d, 0x84d929610d5450, 0x81d77aae82f706, 0x6916f6d4338c5b},
 139    {1, 0, 0, 0}},
 140   {{0xea95ac3b1f15c6, 0x086000905e82d4, 0xdd323ae4d1c8b1, 0x932b56be7685a3},
 141    {0x9ef93dea25dbbf, 0x41665960f390f0, 0xfdec76dbe2a8a7, 0x523e80f019062a},
 142    {1, 0, 0, 0}},
 143   {{0x822fdd26732c73, 0xa01c83531b5d0f, 0x363f37347c1ba4, 0xc391b45c84725c},
 144    {0xbbd5e1b2d6ad24, 0xddfbcde19dfaec, 0xc393da7e222a7f, 0x1efb7890ede244},
 145    {1, 0, 0, 0}},
 146   {{0x4c9e90ca217da1, 0xd11beca79159bb, 0xff8d33c2c98b7c, 0x2610b39409f849},
 147    {0x44d1352ac64da0, 0xcdbb7b2c46b4fb, 0x966c079b753c89, 0xfe67e4e820b112},
 148    {1, 0, 0, 0}},
 149   {{0xe28cae2df5312d, 0xc71b61d16f5c6e, 0x79b7619a3e7c4c, 0x05c73240899b47},
 150    {0x9f7f6382c73e3a, 0x18615165c56bda, 0x641fab2116fd56, 0x72855882b08394},
 151    {1, 0, 0, 0}},
 152   {{0x0469182f161c09, 0x74a98ca8d00fb5, 0xb89da93489a3e0, 0x41c98768fb0c1d},
 153    {0xe5ea05fb32da81, 0x3dce9ffbca6855, 0x1cfe2d3fbf59e6, 0x0e5e03408738a7},
 154    {1, 0, 0, 0}},
 155   {{0xdab22b2333e87f, 0x4430137a5dd2f6, 0xe03ab9f738beb8, 0xcb0c5d0dc34f24},
 156    {0x764a7df0c8fda5, 0x185ba5c3fa2044, 0x9281d688bcbe50, 0xc40331df893881},
 157    {1, 0, 0, 0}},
 158   {{0xb89530796f0f60, 0xade92bd26909a3, 0x1a0c83fb4884da, 0x1765bf22a5a984},
 159    {0x772a9ee75db09e, 0x23bc6c67cec16f, 0x4c1edba8b14e2f, 0xe2a215d9611369},
 160    {1, 0, 0, 0}},
 161   {{0x571e509fb5efb3, 0xade88696410552, 0xc8ae85fada74fe, 0x6c7e4be83bbde3},
 162    {0xff9f51160f4652, 0xb47ce2495a6539, 0xa2946c53b582f4, 0x286d2db3ee9a60},
 163    {1, 0, 0, 0}},
 164   {{0x40bbd5081a44af, 0x0995183b13926c, 0xbcefba6f47f6d0, 0x215619e9cc0057},
 165    {0x8bc94d3b0df45e, 0xf11c54a3694f6f, 0x8631b93cdfe8b5, 0xe7e3f4b0982db9},
 166    {1, 0, 0, 0}},
 167   {{0xb17048ab3e1c7b, 0xac38f36ff8a1d8, 0x1c29819435d2c6, 0xc813132f4c07e9},
 168    {0x2891425503b11f, 0x08781030579fea, 0xf5426ba5cc9674, 0x1e28ebf18562bc},
 169    {1, 0, 0, 0}},
 170   {{0x9f31997cc864eb, 0x06cd91d28b5e4c, 0xff17036691a973, 0xf1aef351497c58},
 171    {0xdd1f2d600564ff, 0xdead073b1402db, 0x74a684435bd693, 0xeea7471f962558},
 172    {1, 0, 0, 0}}},
 173  {{{0, 0, 0, 0},
 174    {0, 0, 0, 0},
 175    {0, 0, 0, 0}},
 176   {{0x9665266dddf554, 0x9613d78b60ef2d, 0xce27a34cdba417, 0xd35ab74d6afc31},
 177    {0x85ccdd22deb15e, 0x2137e5783a6aab, 0xa141cffd8c93c6, 0x355a1830e90f2d},
 178    {1, 0, 0, 0}},
 179   {{0x1a494eadaade65, 0xd6da4da77fe53c, 0xe7992996abec86, 0x65c3553c6090e3},
 180    {0xfa610b1fb09346, 0xf1c6540b8a4aaf, 0xc51a13ccd3cbab, 0x02995b1b18c28a},
 181    {1, 0, 0, 0}},
 182   {{0x7874568e7295ef, 0x86b419fbe38d04, 0xdc0690a7550d9a, 0xd3966a44beac33},
 183    {0x2b7280ec29132f, 0xbeaa3b6a032df3, 0xdc7dd88ae41200, 0xd25e2513e3a100},
 184    {1, 0, 0, 0}},
 185   {{0x924857eb2efafd, 0xac2bce41223190, 0x8edaa1445553fc, 0x825800fd3562d5},
 186    {0x8d79148ea96621, 0x23a01c3dd9ed8d, 0xaf8b219f9416b5, 0xd8db0cc277daea},
 187    {1, 0, 0, 0}},
 188   {{0x76a9c3b1a700f0, 0xe9acd29bc7e691, 0x69212d1a6b0327, 0x6322e97fe154be},
 189    {0x469fc5465d62aa, 0x8d41ed18883b05, 0x1f8eae66c52b88, 0xe4fcbe9325be51},
 190    {1, 0, 0, 0}},
 191   {{0x825fdf583cac16, 0x020b857c7b023a, 0x683c17744b0165, 0x14ffd0a2daf2f1},
 192    {0x323b36184218f9, 0x4944ec4e3b47d4, 0xc15b3080841acf, 0x0bced4b01a28bb},
 193    {1, 0, 0, 0}},
 194   {{0x92ac22230df5c4, 0x52f33b4063eda8, 0xcb3f19870c0c93, 0x40064f2ba65233},
 195    {0xfe16f0924f8992, 0x012da25af5b517, 0x1a57bb24f723a6, 0x06f8bc76760def},
 196    {1, 0, 0, 0}},
 197   {{0x4a7084f7817cb9, 0xbcab0738ee9a78, 0x3ec11e11d9c326, 0xdc0fe90e0f1aae},
 198    {0xcf639ea5f98390, 0x5c350aa22ffb74, 0x9afae98a4047b7, 0x956ec2d617fc45},
 199    {1, 0, 0, 0}},
 200   {{0x4306d648c1be6a, 0x9247cd8bc9a462, 0xf5595e377d2f2e, 0xbd1c3caff1a52e},
 201    {0x045e14472409d0, 0x29f3e17078f773, 0x745a602b2d4f7d, 0x191837685cdfbb},
 202    {1, 0, 0, 0}},
 203   {{0x5b6ee254a8cb79, 0x4953433f5e7026, 0xe21faeb1d1def4, 0xc4c225785c09de},
 204    {0x307ce7bba1e518, 0x31b125b1036db8, 0x47e91868839e8f, 0xc765866e33b9f3},
 205    {1, 0, 0, 0}},
 206   {{0x3bfece24f96906, 0x4794da641e5093, 0xde5df64f95db26, 0x297ecd89714b05},
 207    {0x701bd3ebb2c3aa, 0x7073b4f53cb1d5, 0x13c5665658af16, 0x9895089d66fe58},
 208    {1, 0, 0, 0}},
 209   {{0x0fef05f78c4790, 0x2d773633b05d2e, 0x94229c3a951c94, 0xbbbd70df4911bb},
 210    {0xb2c6963d2c1168, 0x105f47a72b0d73, 0x9fdf6111614080, 0x7b7e94b39e67b0},
 211    {1, 0, 0, 0}},
 212   {{0xad1a7d6efbe2b3, 0xf012482c0da69d, 0x6b3bdf12438345, 0x40d7558d7aa4d9},
 213    {0x8a09fffb5c6d3d, 0x9a356e5d9ffd38, 0x5973f15f4f9b1c, 0xdcd5f59f63c3ea},
 214    {1, 0, 0, 0}},
 215   {{0xacf39f4c5ca7ab, 0x4c8071cc5fd737, 0xc64e3602cd1184, 0x0acd4644c9abba},
 216    {0x6c011a36d8bf6e, 0xfecd87ba24e32a, 0x19f6f56574fad8, 0x050b204ced9405},
 217    {1, 0, 0, 0}},
 218   {{0xed4f1cae7d9a96, 0x5ceef7ad94c40a, 0x778e4a3bf3ef9b, 0x7405783dc3b55e},
 219    {0x32477c61b6e8c6, 0xb46a97570f018b, 0x91176d0a7e95d1, 0x3df90fbc4c7d0e},
 220    {1, 0, 0, 0}}}};
 221
 222 /* Precomputation for the group generator. */
 223 typedef struct {
 224         felem g_pre_comp[2][16][3];
 225         int references;
 226 } NISTP224_PRE_COMP;
 227
 228 const EC_METHOD *EC_GFp_nistp224_method(void)
 229         {
 230         static const EC_METHOD ret = {
 231                 EC_FLAGS_DEFAULT_OCT,
 232                 NID_X9_62_prime_field,
 233                 ec_GFp_nistp224_group_init,
 234                 ec_GFp_simple_group_finish,
 235                 ec_GFp_simple_group_clear_finish,
 236                 ec_GFp_nist_group_copy,
 237                 ec_GFp_nistp224_group_set_curve,
 238                 ec_GFp_simple_group_get_curve,
 239                 ec_GFp_simple_group_get_degree,
 240                 ec_GFp_simple_group_check_discriminant,
 241                 ec_GFp_simple_point_init,
 242                 ec_GFp_simple_point_finish,
 243                 ec_GFp_simple_point_clear_finish,
 244                 ec_GFp_simple_point_copy,
 245                 ec_GFp_simple_point_set_to_infinity,
 246                 ec_GFp_simple_set_Jprojective_coordinates_GFp,
 247                 ec_GFp_simple_get_Jprojective_coordinates_GFp,
 248                 ec_GFp_simple_point_set_affine_coordinates,
 249                 ec_GFp_nistp224_point_get_affine_coordinates,
 250                 0 /* point_set_compressed_coordinates */,
 251                 0 /* point2oct */,
 252                 0 /* oct2point */,
 253                 ec_GFp_simple_add,
 254                 ec_GFp_simple_dbl,
 255                 ec_GFp_simple_invert,
 256                 ec_GFp_simple_is_at_infinity,
 257                 ec_GFp_simple_is_on_curve,
 258                 ec_GFp_simple_cmp,
 259                 ec_GFp_simple_make_affine,
 260                 ec_GFp_simple_points_make_affine,
 261                 ec_GFp_nistp224_points_mul,
 262                 ec_GFp_nistp224_precompute_mult,
 263                 ec_GFp_nistp224_have_precompute_mult,
 264                 ec_GFp_nist_field_mul,
 265                 ec_GFp_nist_field_sqr,
 266                 0 /* field_div */,
 267                 0 /* field_encode */,
 268                 0 /* field_decode */,
 269                 0 /* field_set_to_one */ };
 270
 271         return &ret;
 272         }
 273
 274 /* Helper functions to convert field elements to/from internal representation */
 275 static void bin28_to_felem(felem out, const u8 in[28])
 276         {
 277         out[0] = *((const uint64_t *)(in)) & 0x00ffffffffffffff;
 278         out[1] = (*((const uint64_t *)(in+7))) & 0x00ffffffffffffff;
 279         out[2] = (*((const uint64_t *)(in+14))) & 0x00ffffffffffffff;
 280         out[3] = (*((const uint64_t *)(in+21))) & 0x00ffffffffffffff;
 281         }
 282
 283 static void felem_to_bin28(u8 out[28], const felem in)
 284         {
 285         unsigned i;
 286         for (i = 0; i < 7; ++i)
 287                 {
 288                 out[i]    = in[0]>>(8*i);
 289                 out[i+7]  = in[1]>>(8*i);
 290                 out[i+14] = in[2]>>(8*i);
 291                 out[i+21] = in[3]>>(8*i);
 292                 }
 293         }
 294
 295 /* To preserve endianness when using BN_bn2bin and BN_bin2bn */
 296 static void flip_endian(u8 *out, const u8 *in, unsigned len)
 297         {
 298         unsigned i;
 299         for (i = 0; i < len; ++i)
 300                 out[i] = in[len-1-i];
 301         }
 302
 303 /* From OpenSSL BIGNUM to internal representation */
 304 static int BN_to_felem(felem out, const BIGNUM *bn)
 305         {
 306         felem_bytearray b_in;
 307         felem_bytearray b_out;
 308         unsigned num_bytes;
 309
 310         /* BN_bn2bin eats leading zeroes */
 311         memset(b_out, 0, sizeof b_out);
 312         num_bytes = BN_num_bytes(bn);
 313         if (num_bytes > sizeof b_out)
 314                 {
 315                 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
 316                 return 0;
 317                 }
 318         if (BN_is_negative(bn))
 319                 {
 320                 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
 321                 return 0;
 322                 }
 323         num_bytes = BN_bn2bin(bn, b_in);
 324         flip_endian(b_out, b_in, num_bytes);
 325         bin28_to_felem(out, b_out);
 326         return 1;
 327         }
 328
 329 /* From internal representation to OpenSSL BIGNUM */
 330 static BIGNUM *felem_to_BN(BIGNUM *out, const felem in)
 331         {
 332         felem_bytearray b_in, b_out;
 333         felem_to_bin28(b_in, in);
 334         flip_endian(b_out, b_in, sizeof b_out);
 335         return BN_bin2bn(b_out, sizeof b_out, out);
 336         }
 337
 338 /******************************************************************************/
 339 /*                              FIELD OPERATIONS
 340  *
 341  * Field operations, using the internal representation of field elements.
 342  * NB! These operations are specific to our point multiplication and cannot be
 343  * expected to be correct in general - e.g., multiplication with a large scalar
 344  * will cause an overflow.
 345  *
 346  */
 347
 348 static void felem_one(felem out)
 349         {
 350         out[0] = 1;
 351         out[1] = 0;
 352         out[2] = 0;
 353         out[3] = 0;
 354         }
 355
 356 static void felem_assign(felem out, const felem in)
 357         {
 358         out[0] = in[0];
 359         out[1] = in[1];
 360         out[2] = in[2];
 361         out[3] = in[3];
 362         }
 363
 364 /* Sum two field elements: out += in */
 365 static void felem_sum(felem out, const felem in)
 366         {
 367         out[0] += in[0];
 368         out[1] += in[1];
 369         out[2] += in[2];
 370         out[3] += in[3];
 371         }
 372
 373 /* Get negative value: out = -in */
 374 /* Assumes in[i] < 2^57 */
 375 static void felem_neg(felem out, const felem in)
 376         {
 377         static const limb two58p2 = (((limb) 1) << 58) + (((limb) 1) << 2);
 378         static const limb two58m2 = (((limb) 1) << 58) - (((limb) 1) << 2);
 379         static const limb two58m42m2 = (((limb) 1) << 58) -
 380             (((limb) 1) << 42) - (((limb) 1) << 2);
 381
 382         /* Set to 0 mod 2^224-2^96+1 to ensure out > in */
 383         out[0] = two58p2 - in[0];
 384         out[1] = two58m42m2 - in[1];
 385         out[2] = two58m2 - in[2];
 386         out[3] = two58m2 - in[3];
 387         }
 388
 389 /* Subtract field elements: out -= in */
 390 /* Assumes in[i] < 2^57 */
 391 static void felem_diff(felem out, const felem in)
 392         {
 393         static const limb two58p2 = (((limb) 1) << 58) + (((limb) 1) << 2);
 394         static const limb two58m2 = (((limb) 1) << 58) - (((limb) 1) << 2);
 395         static const limb two58m42m2 = (((limb) 1) << 58) -
 396             (((limb) 1) << 42) - (((limb) 1) << 2);
 397
 398         /* Add 0 mod 2^224-2^96+1 to ensure out > in */
 399         out[0] += two58p2;
 400         out[1] += two58m42m2;
 401         out[2] += two58m2;
 402         out[3] += two58m2;
 403
 404         out[0] -= in[0];
 405         out[1] -= in[1];
 406         out[2] -= in[2];
 407         out[3] -= in[3];
 408         }
 409
 410 /* Subtract in unreduced 128-bit mode: out -= in */
 411 /* Assumes in[i] < 2^119 */
 412 static void widefelem_diff(widefelem out, const widefelem in)
 413         {
 414         static const widelimb two120 = ((widelimb) 1) << 120;
 415         static const widelimb two120m64 = (((widelimb) 1) << 120) -
 416                 (((widelimb) 1) << 64);
 417         static const widelimb two120m104m64 = (((widelimb) 1) << 120) -
 418                 (((widelimb) 1) << 104) - (((widelimb) 1) << 64);
 419
 420         /* Add 0 mod 2^224-2^96+1 to ensure out > in */
 421         out[0] += two120;
 422         out[1] += two120m64;
 423         out[2] += two120m64;
 424         out[3] += two120;
 425         out[4] += two120m104m64;
 426         out[5] += two120m64;
 427         out[6] += two120m64;
 428
 429         out[0] -= in[0];
 430         out[1] -= in[1];
 431         out[2] -= in[2];
 432         out[3] -= in[3];
 433         out[4] -= in[4];
 434         out[5] -= in[5];
 435         out[6] -= in[6];
 436         }
 437
 438 /* Subtract in mixed mode: out128 -= in64 */
 439 /* in[i] < 2^63 */
 440 static void felem_diff_128_64(widefelem out, const felem in)
 441         {
 442         static const widelimb two64p8 = (((widelimb) 1) << 64) +
 443                 (((widelimb) 1) << 8);
 444         static const widelimb two64m8 = (((widelimb) 1) << 64) -
 445                 (((widelimb) 1) << 8);
 446         static const widelimb two64m48m8 = (((widelimb) 1) << 64) -
 447                 (((widelimb) 1) << 48) - (((widelimb) 1) << 8);
 448
 449         /* Add 0 mod 2^224-2^96+1 to ensure out > in */
 450         out[0] += two64p8;
 451         out[1] += two64m48m8;
 452         out[2] += two64m8;
 453         out[3] += two64m8;
 454
 455         out[0] -= in[0];
 456         out[1] -= in[1];
 457         out[2] -= in[2];
 458         out[3] -= in[3];
 459         }
 460
 461 /* Multiply a field element by a scalar: out = out * scalar
 462  * The scalars we actually use are small, so results fit without overflow */
 463 static void felem_scalar(felem out, const limb scalar)
 464         {
 465         out[0] *= scalar;
 466         out[1] *= scalar;
 467         out[2] *= scalar;
 468         out[3] *= scalar;
 469         }
 470
 471 /* Multiply an unreduced field element by a scalar: out = out * scalar
 472  * The scalars we actually use are small, so results fit without overflow */
 473 static void widefelem_scalar(widefelem out, const widelimb scalar)
 474         {
 475         out[0] *= scalar;
 476         out[1] *= scalar;
 477         out[2] *= scalar;
 478         out[3] *= scalar;
 479         out[4] *= scalar;
 480         out[5] *= scalar;
 481         out[6] *= scalar;
 482         }
 483
 484 /* Square a field element: out = in^2 */
 485 static void felem_square(widefelem out, const felem in)
 486         {
 487         limb tmp0, tmp1, tmp2;
 488         tmp0 = 2 * in[0]; tmp1 = 2 * in[1]; tmp2 = 2 * in[2];
 489         out[0] = ((widelimb) in[0]) * in[0];
 490         out[1] = ((widelimb) in[0]) * tmp1;
 491         out[2] = ((widelimb) in[0]) * tmp2 + ((widelimb) in[1]) * in[1];
 492         out[3] = ((widelimb) in[3]) * tmp0 +
 493                 ((widelimb) in[1]) * tmp2;
 494         out[4] = ((widelimb) in[3]) * tmp1 + ((widelimb) in[2]) * in[2];
 495         out[5] = ((widelimb) in[3]) * tmp2;
 496         out[6] = ((widelimb) in[3]) * in[3];
 497         }
 498
 499 /* Multiply two field elements: out = in1 * in2 */
 500 static void felem_mul(widefelem out, const felem in1, const felem in2)
 501         {
 502         out[0] = ((widelimb) in1[0]) * in2[0];
 503         out[1] = ((widelimb) in1[0]) * in2[1] + ((widelimb) in1[1]) * in2[0];
 504         out[2] = ((widelimb) in1[0]) * in2[2] + ((widelimb) in1[1]) * in2[1] +
 505                 ((widelimb) in1[2]) * in2[0];
 506         out[3] = ((widelimb) in1[0]) * in2[3] + ((widelimb) in1[1]) * in2[2] +
 507                 ((widelimb) in1[2]) * in2[1] + ((widelimb) in1[3]) * in2[0];
 508         out[4] = ((widelimb) in1[1]) * in2[3] + ((widelimb) in1[2]) * in2[2] +
 509                 ((widelimb) in1[3]) * in2[1];
 510         out[5] = ((widelimb) in1[2]) * in2[3] + ((widelimb) in1[3]) * in2[2];
 511         out[6] = ((widelimb) in1[3]) * in2[3];
 512         }
 513
 514 /* Reduce seven 128-bit coefficients to four 64-bit coefficients.
 515  * Requires in[i] < 2^126,
 516  * ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] <= 2^56 + 2^16 */
 517 static void felem_reduce(felem out, const widefelem in)
 518         {
 519         static const widelimb two127p15 = (((widelimb) 1) << 127) +
 520                 (((widelimb) 1) << 15);
 521         static const widelimb two127m71 = (((widelimb) 1) << 127) -
 522                 (((widelimb) 1) << 71);
 523         static const widelimb two127m71m55 = (((widelimb) 1) << 127) -
 524                 (((widelimb) 1) << 71) - (((widelimb) 1) << 55);
 525         widelimb output[5];
 526
 527         /* Add 0 mod 2^224-2^96+1 to ensure all differences are positive */
 528         output[0] = in[0] + two127p15;
 529         output[1] = in[1] + two127m71m55;
 530         output[2] = in[2] + two127m71;
 531         output[3] = in[3];
 532         output[4] = in[4];
 533
 534         /* Eliminate in[4], in[5], in[6] */
 535         output[4] += in[6] >> 16;
 536         output[3] += (in[6] & 0xffff) << 40;
 537         output[2] -= in[6];
 538
 539         output[3] += in[5] >> 16;
 540         output[2] += (in[5] & 0xffff) << 40;
 541         output[1] -= in[5];
 542
 543         output[2] += output[4] >> 16;
 544         output[1] += (output[4] & 0xffff) << 40;
 545         output[0] -= output[4];
 546
 547         /* Carry 2 -> 3 -> 4 */
 548         output[3] += output[2] >> 56;
 549         output[2] &= 0x00ffffffffffffff;
 550
 551         output[4] = output[3] >> 56;
 552         output[3] &= 0x00ffffffffffffff;
 553
 554         /* Now output[2] < 2^56, output[3] < 2^56, output[4] < 2^72 */
 555
 556         /* Eliminate output[4] */
 557         output[2] += output[4] >> 16;
 558         /* output[2] < 2^56 + 2^56 = 2^57 */
 559         output[1] += (output[4] & 0xffff) << 40;
 560         output[0] -= output[4];
 561
 562         /* Carry 0 -> 1 -> 2 -> 3 */
 563         output[1] += output[0] >> 56;
 564         out[0] = output[0] & 0x00ffffffffffffff;
 565
 566         output[2] += output[1] >> 56;
 567         /* output[2] < 2^57 + 2^72 */
 568         out[1] = output[1] & 0x00ffffffffffffff;
 569         output[3] += output[2] >> 56;
 570         /* output[3] <= 2^56 + 2^16 */
 571         out[2] = output[2] & 0x00ffffffffffffff;
 572
 573         /* out[0] < 2^56, out[1] < 2^56, out[2] < 2^56,
 574          * out[3] <= 2^56 + 2^16 (due to final carry),
 575          * so out < 2*p */
 576         out[3] = output[3];
 577         }
 578
 579 static void felem_square_reduce(felem out, const felem in)
 580         {
 581         widefelem tmp;
 582         felem_square(tmp, in);
 583         felem_reduce(out, tmp);
 584         }
 585
 586 static void felem_mul_reduce(felem out, const felem in1, const felem in2)
 587         {
 588         widefelem tmp;
 589         felem_mul(tmp, in1, in2);
 590         felem_reduce(out, tmp);
 591         }
 592
 593 /* Reduce to unique minimal representation.
 594  * Requires 0 <= in < 2*p (always call felem_reduce first) */
 595 static void felem_contract(felem out, const felem in)
 596         {
 597         static const int64_t two56 = ((limb) 1) << 56;
 598         /* 0 <= in < 2*p, p = 2^224 - 2^96 + 1 */
 599         /* if in > p , reduce in = in - 2^224 + 2^96 - 1 */
 600         int64_t tmp[4], a;
 601         tmp[0] = in[0];
 602         tmp[1] = in[1];
 603         tmp[2] = in[2];
 604         tmp[3] = in[3];
 605         /* Case 1: a = 1 iff in >= 2^224 */
 606         a = (in[3] >> 56);
 607         tmp[0] -= a;
 608         tmp[1] += a << 40;
 609         tmp[3] &= 0x00ffffffffffffff;
 610         /* Case 2: a = 0 iff p <= in < 2^224, i.e.,
 611          * the high 128 bits are all 1 and the lower part is non-zero */
 612         a = ((in[3] & in[2] & (in[1] | 0x000000ffffffffff)) + 1) |
 613                 (((int64_t)(in[0] + (in[1] & 0x000000ffffffffff)) - 1) >> 63);
 614         a &= 0x00ffffffffffffff;
 615         /* turn a into an all-one mask (if a = 0) or an all-zero mask */
 616         a = (a - 1) >> 63;
 617         /* subtract 2^224 - 2^96 + 1 if a is all-one*/
 618         tmp[3] &= a ^ 0xffffffffffffffff;
 619         tmp[2] &= a ^ 0xffffffffffffffff;
 620         tmp[1] &= (a ^ 0xffffffffffffffff) | 0x000000ffffffffff;
 621         tmp[0] -= 1 & a;
 622
 623         /* eliminate negative coefficients: if tmp[0] is negative, tmp[1] must
 624          * be non-zero, so we only need one step */
 625         a = tmp[0] >> 63;
 626         tmp[0] += two56 & a;
 627         tmp[1] -= 1 & a;
 628
 629         /* carry 1 -> 2 -> 3 */
 630         tmp[2] += tmp[1] >> 56;
 631         tmp[1] &= 0x00ffffffffffffff;
 632
 633         tmp[3] += tmp[2] >> 56;
 634         tmp[2] &= 0x00ffffffffffffff;
 635
 636         /* Now 0 <= out < p */
 637         out[0] = tmp[0];
 638         out[1] = tmp[1];
 639         out[2] = tmp[2];
 640         out[3] = tmp[3];
 641         }
 642
 643 /* Zero-check: returns 1 if input is 0, and 0 otherwise.
 644  * We know that field elements are reduced to in < 2^225,
 645  * so we only need to check three cases: 0, 2^224 - 2^96 + 1,
 646  * and 2^225 - 2^97 + 2 */
 647 static limb felem_is_zero(const felem in)
 648         {
 649         limb zero, two224m96p1, two225m97p2;
 650
 651         zero = in[0] | in[1] | in[2] | in[3];
 652         zero = (((int64_t)(zero) - 1) >> 63) & 1;
 653         two224m96p1 = (in[0] ^ 1) | (in[1] ^ 0x00ffff0000000000)
 654                 | (in[2] ^ 0x00ffffffffffffff) | (in[3] ^ 0x00ffffffffffffff);
 655         two224m96p1 = (((int64_t)(two224m96p1) - 1) >> 63) & 1;
 656         two225m97p2 = (in[0] ^ 2) | (in[1] ^ 0x00fffe0000000000)
 657                 | (in[2] ^ 0x00ffffffffffffff) | (in[3] ^ 0x01ffffffffffffff);
 658         two225m97p2 = (((int64_t)(two225m97p2) - 1) >> 63) & 1;
 659         return (zero | two224m96p1 | two225m97p2);
 660         }
 661
 662 static limb felem_is_zero_int(const felem in)
 663         {
 664         return (int) (felem_is_zero(in) & ((limb)1));
 665         }
 666
 667 /* Invert a field element */
 668 /* Computation chain copied from djb's code */
 669 static void felem_inv(felem out, const felem in)
 670         {
 671         felem ftmp, ftmp2, ftmp3, ftmp4;
 672         widefelem tmp;
 673         unsigned i;
 674
 675         felem_square(tmp, in); felem_reduce(ftmp, tmp);         /* 2 */
 676         felem_mul(tmp, in, ftmp); felem_reduce(ftmp, tmp);      /* 2^2 - 1 */
 677         felem_square(tmp, ftmp); felem_reduce(ftmp, tmp);       /* 2^3 - 2 */
 678         felem_mul(tmp, in, ftmp); felem_reduce(ftmp, tmp);      /* 2^3 - 1 */
 679         felem_square(tmp, ftmp); felem_reduce(ftmp2, tmp);      /* 2^4 - 2 */
 680         felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp);     /* 2^5 - 4 */
 681         felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp);     /* 2^6 - 8 */
 682         felem_mul(tmp, ftmp2, ftmp); felem_reduce(ftmp, tmp);   /* 2^6 - 1 */
 683         felem_square(tmp, ftmp); felem_reduce(ftmp2, tmp);      /* 2^7 - 2 */
 684         for (i = 0; i < 5; ++i)                                 /* 2^12 - 2^6 */
 685                 {
 686                 felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp);
 687                 }
 688         felem_mul(tmp, ftmp2, ftmp); felem_reduce(ftmp2, tmp);  /* 2^12 - 1 */
 689         felem_square(tmp, ftmp2); felem_reduce(ftmp3, tmp);     /* 2^13 - 2 */
 690         for (i = 0; i < 11; ++i)                                /* 2^24 - 2^12 */
 691                 {
 692                 felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp);
 693                 }
 694         felem_mul(tmp, ftmp3, ftmp2); felem_reduce(ftmp2, tmp); /* 2^24 - 1 */
 695         felem_square(tmp, ftmp2); felem_reduce(ftmp3, tmp);     /* 2^25 - 2 */
 696         for (i = 0; i < 23; ++i)                                /* 2^48 - 2^24 */
 697                 {
 698                 felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp);
 699                 }
 700         felem_mul(tmp, ftmp3, ftmp2); felem_reduce(ftmp3, tmp); /* 2^48 - 1 */
 701         felem_square(tmp, ftmp3); felem_reduce(ftmp4, tmp);     /* 2^49 - 2 */
 702         for (i = 0; i < 47; ++i)                                /* 2^96 - 2^48 */
 703                 {
 704                 felem_square(tmp, ftmp4); felem_reduce(ftmp4, tmp);
 705                 }
 706         felem_mul(tmp, ftmp3, ftmp4); felem_reduce(ftmp3, tmp); /* 2^96 - 1 */
 707         felem_square(tmp, ftmp3); felem_reduce(ftmp4, tmp);     /* 2^97 - 2 */
 708         for (i = 0; i < 23; ++i)                                /* 2^120 - 2^24 */
 709                 {
 710                 felem_square(tmp, ftmp4); felem_reduce(ftmp4, tmp);
 711                 }
 712         felem_mul(tmp, ftmp2, ftmp4); felem_reduce(ftmp2, tmp); /* 2^120 - 1 */
 713         for (i = 0; i < 6; ++i)                                 /* 2^126 - 2^6 */
 714                 {
 715                 felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp);
 716                 }
 717         felem_mul(tmp, ftmp2, ftmp); felem_reduce(ftmp, tmp);   /* 2^126 - 1 */
 718         felem_square(tmp, ftmp); felem_reduce(ftmp, tmp);       /* 2^127 - 2 */
 719         felem_mul(tmp, ftmp, in); felem_reduce(ftmp, tmp);      /* 2^127 - 1 */
 720         for (i = 0; i < 97; ++i)                                /* 2^224 - 2^97 */
 721                 {
 722                 felem_square(tmp, ftmp); felem_reduce(ftmp, tmp);
 723                 }
 724         felem_mul(tmp, ftmp, ftmp3); felem_reduce(out, tmp);    /* 2^224 - 2^96 - 1 */
 725         }
 726
 727 /* Copy in constant time:
 728  * if icopy == 1, copy in to out,
 729  * if icopy == 0, copy out to itself. */
 730 static void
 731 copy_conditional(felem out, const felem in, limb icopy)
 732         {
 733         unsigned i;
 734         /* icopy is a (64-bit) 0 or 1, so copy is either all-zero or all-one */
 735         const limb copy = -icopy;
 736         for (i = 0; i < 4; ++i)
 737                 {
 738                 const limb tmp = copy & (in[i] ^ out[i]);
 739                 out[i] ^= tmp;
 740                 }
 741         }
 742
 743 /******************************************************************************/
 744 /*                       ELLIPTIC CURVE POINT OPERATIONS
 745  *
 746  * Points are represented in Jacobian projective coordinates:
 747  * (X, Y, Z) corresponds to the affine point (X/Z^2, Y/Z^3),
 748  * or to the point at infinity if Z == 0.
 749  *
 750  */
 751
 752 /* Double an elliptic curve point:
 753  * (X', Y', Z') = 2 * (X, Y, Z), where
 754  * X' = (3 * (X - Z^2) * (X + Z^2))^2 - 8 * X * Y^2
 755  * Y' = 3 * (X - Z^2) * (X + Z^2) * (4 * X * Y^2 - X') - 8 * Y^2
 756  * Z' = (Y + Z)^2 - Y^2 - Z^2 = 2 * Y * Z
 757  * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed,
 758  * while x_out == y_in is not (maybe this works, but it's not tested). */
 759 static void
 760 point_double(felem x_out, felem y_out, felem z_out,
 761              const felem x_in, const felem y_in, const felem z_in)
 762         {
 763         widefelem tmp, tmp2;
 764         felem delta, gamma, beta, alpha, ftmp, ftmp2;
 765
 766         felem_assign(ftmp, x_in);
 767         felem_assign(ftmp2, x_in);
 768
 769         /* delta = z^2 */
 770         felem_square(tmp, z_in);
 771         felem_reduce(delta, tmp);
 772
 773         /* gamma = y^2 */
 774         felem_square(tmp, y_in);
 775         felem_reduce(gamma, tmp);
 776
 777         /* beta = x*gamma */
 778         felem_mul(tmp, x_in, gamma);
 779         felem_reduce(beta, tmp);
 780
 781         /* alpha = 3*(x-delta)*(x+delta) */
 782         felem_diff(ftmp, delta);
 783         /* ftmp[i] < 2^57 + 2^58 + 2 < 2^59 */
 784         felem_sum(ftmp2, delta);
 785         /* ftmp2[i] < 2^57 + 2^57 = 2^58 */
 786         felem_scalar(ftmp2, 3);
 787         /* ftmp2[i] < 3 * 2^58 < 2^60 */
 788         felem_mul(tmp, ftmp, ftmp2);
 789         /* tmp[i] < 2^60 * 2^59 * 4 = 2^121 */
 790         felem_reduce(alpha, tmp);
 791
 792         /* x' = alpha^2 - 8*beta */
 793         felem_square(tmp, alpha);
 794         /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
 795         felem_assign(ftmp, beta);
 796         felem_scalar(ftmp, 8);
 797         /* ftmp[i] < 8 * 2^57 = 2^60 */
 798         felem_diff_128_64(tmp, ftmp);
 799         /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
 800         felem_reduce(x_out, tmp);
 801
 802         /* z' = (y + z)^2 - gamma - delta */
 803         felem_sum(delta, gamma);
 804         /* delta[i] < 2^57 + 2^57 = 2^58 */
 805         felem_assign(ftmp, y_in);
 806         felem_sum(ftmp, z_in);
 807         /* ftmp[i] < 2^57 + 2^57 = 2^58 */
 808         felem_square(tmp, ftmp);
 809         /* tmp[i] < 4 * 2^58 * 2^58 = 2^118 */
 810         felem_diff_128_64(tmp, delta);
 811         /* tmp[i] < 2^118 + 2^64 + 8 < 2^119 */
 812         felem_reduce(z_out, tmp);
 813
 814         /* y' = alpha*(4*beta - x') - 8*gamma^2 */
 815         felem_scalar(beta, 4);
 816         /* beta[i] < 4 * 2^57 = 2^59 */
 817         felem_diff(beta, x_out);
 818         /* beta[i] < 2^59 + 2^58 + 2 < 2^60 */
 819         felem_mul(tmp, alpha, beta);
 820         /* tmp[i] < 4 * 2^57 * 2^60 = 2^119 */
 821         felem_square(tmp2, gamma);
 822         /* tmp2[i] < 4 * 2^57 * 2^57 = 2^116 */
 823         widefelem_scalar(tmp2, 8);
 824         /* tmp2[i] < 8 * 2^116 = 2^119 */
 825         widefelem_diff(tmp, tmp2);
 826         /* tmp[i] < 2^119 + 2^120 < 2^121 */
 827         felem_reduce(y_out, tmp);
 828         }
 829
 830 /* Add two elliptic curve points:
 831  * (X_1, Y_1, Z_1) + (X_2, Y_2, Z_2) = (X_3, Y_3, Z_3), where
 832  * X_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1)^2 - (Z_1^2 * X_2 - Z_2^2 * X_1)^3 -
 833  * 2 * Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^2
 834  * 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) -
 835  *        Z_2^3 * Y_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^3
 836  * Z_3 = (Z_1^2 * X_2 - Z_2^2 * X_1) * (Z_1 * Z_2)
 837  *
 838  * This runs faster if 'mixed' is set, which requires Z_2 = 1 or Z_2 = 0.
 839  */
 840
 841 /* This function is not entirely constant-time:
 842  * it includes a branch for checking whether the two input points are equal,
 843  * (while not equal to the point at infinity).
 844  * This case never happens during single point multiplication,
 845  * so there is no timing leak for ECDH or ECDSA signing. */
 846 static void point_add(felem x3, felem y3, felem z3,
 847         const felem x1, const felem y1, const felem z1,
 848         const int mixed, const felem x2, const felem y2, const felem z2)
 849         {
 850         felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, x_out, y_out, z_out;
 851         widefelem tmp, tmp2;
 852         limb z1_is_zero, z2_is_zero, x_equal, y_equal;
 853
 854         if (!mixed)
 855                 {
 856                 /* ftmp2 = z2^2 */
 857                 felem_square(tmp, z2);
 858                 felem_reduce(ftmp2, tmp);
 859
 860                 /* ftmp4 = z2^3 */
 861                 felem_mul(tmp, ftmp2, z2);
 862                 felem_reduce(ftmp4, tmp);
 863
 864                 /* ftmp4 = z2^3*y1 */
 865                 felem_mul(tmp2, ftmp4, y1);
 866                 felem_reduce(ftmp4, tmp2);
 867
 868                 /* ftmp2 = z2^2*x1 */
 869                 felem_mul(tmp2, ftmp2, x1);
 870                 felem_reduce(ftmp2, tmp2);
 871                 }
 872         else
 873                 {
 874                 /* We'll assume z2 = 1 (special case z2 = 0 is handled later) */
 875
 876                 /* ftmp4 = z2^3*y1 */
 877                 felem_assign(ftmp4, y1);
 878
 879                 /* ftmp2 = z2^2*x1 */
 880                 felem_assign(ftmp2, x1);
 881                 }
 882
 883         /* ftmp = z1^2 */
 884         felem_square(tmp, z1);
 885         felem_reduce(ftmp, tmp);
 886
 887         /* ftmp3 = z1^3 */
 888         felem_mul(tmp, ftmp, z1);
 889         felem_reduce(ftmp3, tmp);
 890
 891         /* tmp = z1^3*y2 */
 892         felem_mul(tmp, ftmp3, y2);
 893         /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
 894
 895         /* ftmp3 = z1^3*y2 - z2^3*y1 */
 896         felem_diff_128_64(tmp, ftmp4);
 897         /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
 898         felem_reduce(ftmp3, tmp);
 899
 900         /* tmp = z1^2*x2 */
 901         felem_mul(tmp, ftmp, x2);
 902         /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
 903
 904         /* ftmp = z1^2*x2 - z2^2*x1 */
 905         felem_diff_128_64(tmp, ftmp2);
 906         /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
 907         felem_reduce(ftmp, tmp);
 908
 909         /* the formulae are incorrect if the points are equal
 910          * so we check for this and do doubling if this happens */
 911         x_equal = felem_is_zero(ftmp);
 912         y_equal = felem_is_zero(ftmp3);
 913         z1_is_zero = felem_is_zero(z1);
 914         z2_is_zero = felem_is_zero(z2);
 915         /* In affine coordinates, (X_1, Y_1) == (X_2, Y_2) */
 916         if (x_equal && y_equal && !z1_is_zero && !z2_is_zero)
 917                 {
 918                 point_double(x3, y3, z3, x1, y1, z1);
 919                 return;
 920                 }
 921
 922         /* ftmp5 = z1*z2 */
 923         if (!mixed)
 924                 {
 925                 felem_mul(tmp, z1, z2);
 926                 felem_reduce(ftmp5, tmp);
 927                 }
 928         else
 929                 {
 930                 /* special case z2 = 0 is handled later */
 931                 felem_assign(ftmp5, z1);
 932                 }
 933
 934         /* z_out = (z1^2*x2 - z2^2*x1)*(z1*z2) */
 935         felem_mul(tmp, ftmp, ftmp5);
 936         felem_reduce(z_out, tmp);
 937
 938         /* ftmp = (z1^2*x2 - z2^2*x1)^2 */
 939         felem_assign(ftmp5, ftmp);
 940         felem_square(tmp, ftmp);
 941         felem_reduce(ftmp, tmp);
 942
 943         /* ftmp5 = (z1^2*x2 - z2^2*x1)^3 */
 944         felem_mul(tmp, ftmp, ftmp5);
 945         felem_reduce(ftmp5, tmp);
 946
 947         /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
 948         felem_mul(tmp, ftmp2, ftmp);
 949         felem_reduce(ftmp2, tmp);
 950
 951         /* tmp = z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */
 952         felem_mul(tmp, ftmp4, ftmp5);
 953         /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
 954
 955         /* tmp2 = (z1^3*y2 - z2^3*y1)^2 */
 956         felem_square(tmp2, ftmp3);
 957         /* tmp2[i] < 4 * 2^57 * 2^57 < 2^116 */
 958
 959         /* tmp2 = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 */
 960         felem_diff_128_64(tmp2, ftmp5);
 961         /* tmp2[i] < 2^116 + 2^64 + 8 < 2^117 */
 962
 963         /* ftmp5 = 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
 964         felem_assign(ftmp5, ftmp2);
 965         felem_scalar(ftmp5, 2);
 966         /* ftmp5[i] < 2 * 2^57 = 2^58 */
 967
 968         /* x_out = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 -
 969            2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
 970         felem_diff_128_64(tmp2, ftmp5);
 971         /* tmp2[i] < 2^117 + 2^64 + 8 < 2^118 */
 972         felem_reduce(x_out, tmp2);
 973
 974         /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out */
 975         felem_diff(ftmp2, x_out);
 976         /* ftmp2[i] < 2^57 + 2^58 + 2 < 2^59 */
 977
 978         /* tmp2 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) */
 979         felem_mul(tmp2, ftmp3, ftmp2);
 980         /* tmp2[i] < 4 * 2^57 * 2^59 = 2^118 */
 981
 982         /* y_out = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) -
 983            z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */
 984         widefelem_diff(tmp2, tmp);
 985         /* tmp2[i] < 2^118 + 2^120 < 2^121 */
 986         felem_reduce(y_out, tmp2);
 987
 988         /* the result (x_out, y_out, z_out) is incorrect if one of the inputs is
 989          * the point at infinity, so we need to check for this separately */
 990
 991         /* if point 1 is at infinity, copy point 2 to output, and vice versa */
 992         copy_conditional(x_out, x2, z1_is_zero);
 993         copy_conditional(x_out, x1, z2_is_zero);
 994         copy_conditional(y_out, y2, z1_is_zero);
 995         copy_conditional(y_out, y1, z2_is_zero);
 996         copy_conditional(z_out, z2, z1_is_zero);
 997         copy_conditional(z_out, z1, z2_is_zero);
 998         felem_assign(x3, x_out);
 999         felem_assign(y3, y_out);
1000         felem_assign(z3, z_out);
1001         }
1002
1003 /* select_point selects the |index|th point from a precomputation table and
1004  * copies it to out. */
1005 static void select_point(const u64 index, unsigned int size, const felem pre_comp[/*size*/][3], felem out[3])
1006         {
1007         unsigned i, j;
1008         limb *outlimbs = &out[0][0];
1009         memset(outlimbs, 0, 3 * sizeof(felem));
1010
1011         for (i = 0; i < size; i++)
1012                 {
1013                 const limb *inlimbs = &pre_comp[i][0][0];
1014                 u64 mask = i ^ index;
1015                 mask |= mask >> 4;
1016                 mask |= mask >> 2;
1017                 mask |= mask >> 1;
1018                 mask &= 1;
1019                 mask--;
1020                 for (j = 0; j < 4 * 3; j++)
1021                         outlimbs[j] |= inlimbs[j] & mask;
1022                 }
1023         }
1024
1025 /* get_bit returns the |i|th bit in |in| */
1026 static char get_bit(const felem_bytearray in, unsigned i)
1027         {
1028         if (i >= 224)
1029                 return 0;
1030         return (in[i >> 3] >> (i & 7)) & 1;
1031         }
1032
1033 /* Interleaved point multiplication using precomputed point multiples:
1034  * The small point multiples 0*P, 1*P, ..., 16*P are in pre_comp[],
1035  * the scalars in scalars[]. If g_scalar is non-NULL, we also add this multiple
1036  * of the generator, using certain (large) precomputed multiples in g_pre_comp.
1037  * Output point (X, Y, Z) is stored in x_out, y_out, z_out */
1038 static void batch_mul(felem x_out, felem y_out, felem z_out,
1039         const felem_bytearray scalars[], const unsigned num_points, const u8 *g_scalar,
1040         const int mixed, const felem pre_comp[][17][3], const felem g_pre_comp[2][16][3])
1041         {
1042         int i, skip;
1043         unsigned num;
1044         unsigned gen_mul = (g_scalar != NULL);
1045         felem nq[3], tmp[4];
1046         u64 bits;
1047         u8 sign, digit;
1048
1049         /* set nq to the point at infinity */
1050         memset(nq, 0, 3 * sizeof(felem));
1051
1052         /* Loop over all scalars msb-to-lsb, interleaving additions
1053          * of multiples of the generator (two in each of the last 28 rounds)
1054          * and additions of other points multiples (every 5th round).
1055          */
1056         skip = 1; /* save two point operations in the first round */
1057         for (i = (num_points ? 220 : 27); i >= 0; --i)
1058                 {
1059                 /* double */
1060                 if (!skip)
1061                         point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
1062
1063                 /* add multiples of the generator */
1064                 if (gen_mul && (i <= 27))
1065                         {
1066                         /* first, look 28 bits upwards */
1067                         bits = get_bit(g_scalar, i + 196) << 3;
1068                         bits |= get_bit(g_scalar, i + 140) << 2;
1069                         bits |= get_bit(g_scalar, i + 84) << 1;
1070                         bits |= get_bit(g_scalar, i + 28);
1071                         /* select the point to add, in constant time */
1072                         select_point(bits, 16, g_pre_comp[1], tmp);
1073
1074                         if (!skip)
1075                                 {
1076                                 point_add(nq[0], nq[1], nq[2],
1077                                         nq[0], nq[1], nq[2],
1078                                         1 /* mixed */, tmp[0], tmp[1], tmp[2]);
1079                                 }
1080                         else
1081                                 {
1082                                 memcpy(nq, tmp, 3 * sizeof(felem));
1083                                 skip = 0;
1084                                 }
1085
1086                         /* second, look at the current position */
1087                         bits = get_bit(g_scalar, i + 168) << 3;
1088                         bits |= get_bit(g_scalar, i + 112) << 2;
1089                         bits |= get_bit(g_scalar, i + 56) << 1;
1090                         bits |= get_bit(g_scalar, i);
1091                         /* select the point to add, in constant time */
1092                         select_point(bits, 16, g_pre_comp[0], tmp);
1093                         point_add(nq[0], nq[1], nq[2],
1094                                 nq[0], nq[1], nq[2],
1095                                 1 /* mixed */, tmp[0], tmp[1], tmp[2]);
1096                         }
1097
1098                 /* do other additions every 5 doublings */
1099                 if (num_points && (i % 5 == 0))
1100                         {
1101                         /* loop over all scalars */
1102                         for (num = 0; num < num_points; ++num)
1103                                 {
1104                                 bits = get_bit(scalars[num], i + 4) << 5;
1105                                 bits |= get_bit(scalars[num], i + 3) << 4;
1106                                 bits |= get_bit(scalars[num], i + 2) << 3;
1107                                 bits |= get_bit(scalars[num], i + 1) << 2;
1108                                 bits |= get_bit(scalars[num], i) << 1;
1109                                 bits |= get_bit(scalars[num], i - 1);
1110                                 ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits);
1111
1112                                 /* select the point to add or subtract */
1113                                 select_point(digit, 17, pre_comp[num], tmp);
1114                                 felem_neg(tmp[3], tmp[1]); /* (X, -Y, Z) is the negative point */
1115                                 copy_conditional(tmp[1], tmp[3], sign);
1116
1117                                 if (!skip)
1118                                         {
1119                                         point_add(nq[0], nq[1], nq[2],
1120                                                 nq[0], nq[1], nq[2],
1121                                                 mixed, tmp[0], tmp[1], tmp[2]);
1122                                         }
1123                                 else
1124                                         {
1125                                         memcpy(nq, tmp, 3 * sizeof(felem));
1126                                         skip = 0;
1127                                         }
1128                                 }
1129                         }
1130                 }
1131         felem_assign(x_out, nq[0]);
1132         felem_assign(y_out, nq[1]);
1133         felem_assign(z_out, nq[2]);
1134         }
1135
1136 /******************************************************************************/
1137 /*                     FUNCTIONS TO MANAGE PRECOMPUTATION
1138  */
1139
1140 static NISTP224_PRE_COMP *nistp224_pre_comp_new()
1141         {
1142         NISTP224_PRE_COMP *ret = NULL;
1143         ret = (NISTP224_PRE_COMP *) OPENSSL_malloc(sizeof *ret);
1144         if (!ret)
1145                 {
1146                 ECerr(EC_F_NISTP224_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1147                 return ret;
1148                 }
1149         memset(ret->g_pre_comp, 0, sizeof(ret->g_pre_comp));
1150         ret->references = 1;
1151         return ret;
1152         }
1153
1154 static void *nistp224_pre_comp_dup(void *src_)
1155         {
1156         NISTP224_PRE_COMP *src = src_;
1157
1158         /* no need to actually copy, these objects never change! */
1159         CRYPTO_add(&src->references, 1, CRYPTO_LOCK_EC_PRE_COMP);
1160
1161         return src_;
1162         }
1163
1164 static void nistp224_pre_comp_free(void *pre_)
1165         {
1166         int i;
1167         NISTP224_PRE_COMP *pre = pre_;
1168
1169         if (!pre)
1170                 return;
1171
1172         i = CRYPTO_add(&pre->references, -1, CRYPTO_LOCK_EC_PRE_COMP);
1173         if (i > 0)
1174                 return;
1175
1176         OPENSSL_free(pre);
1177         }
1178
1179 static void nistp224_pre_comp_clear_free(void *pre_)
1180         {
1181         int i;
1182         NISTP224_PRE_COMP *pre = pre_;
1183
1184         if (!pre)
1185                 return;
1186
1187         i = CRYPTO_add(&pre->references, -1, CRYPTO_LOCK_EC_PRE_COMP);
1188         if (i > 0)
1189                 return;
1190
1191         OPENSSL_cleanse(pre, sizeof *pre);
1192         OPENSSL_free(pre);
1193         }
1194
1195 /******************************************************************************/
1196 /*                         OPENSSL EC_METHOD FUNCTIONS
1197  */
1198
1199 int ec_GFp_nistp224_group_init(EC_GROUP *group)
1200         {
1201         int ret;
1202         ret = ec_GFp_simple_group_init(group);
1203         group->a_is_minus3 = 1;
1204         return ret;
1205         }
1206
1207 int ec_GFp_nistp224_group_set_curve(EC_GROUP *group, const BIGNUM *p,
1208         const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
1209         {
1210         int ret = 0;
1211         BN_CTX *new_ctx = NULL;
1212         BIGNUM *curve_p, *curve_a, *curve_b;
1213
1214         if (ctx == NULL)
1215                 if ((ctx = new_ctx = BN_CTX_new()) == NULL) return 0;
1216         BN_CTX_start(ctx);
1217         if (((curve_p = BN_CTX_get(ctx)) == NULL) ||
1218                 ((curve_a = BN_CTX_get(ctx)) == NULL) ||
1219                 ((curve_b = BN_CTX_get(ctx)) == NULL)) goto err;
1220         BN_bin2bn(nistp224_curve_params[0], sizeof(felem_bytearray), curve_p);
1221         BN_bin2bn(nistp224_curve_params[1], sizeof(felem_bytearray), curve_a);
1222         BN_bin2bn(nistp224_curve_params[2], sizeof(felem_bytearray), curve_b);
1223         if ((BN_cmp(curve_p, p)) || (BN_cmp(curve_a, a)) ||
1224                 (BN_cmp(curve_b, b)))
1225                 {
1226                 ECerr(EC_F_EC_GFP_NISTP224_GROUP_SET_CURVE,
1227                         EC_R_WRONG_CURVE_PARAMETERS);
1228                 goto err;
1229                 }
1230         group->field_mod_func = BN_nist_mod_224;
1231         ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx);
1232 err:
1233         BN_CTX_end(ctx);
1234         if (new_ctx != NULL)
1235                 BN_CTX_free(new_ctx);
1236         return ret;
1237         }
1238
1239 /* Takes the Jacobian coordinates (X, Y, Z) of a point and returns
1240  * (X', Y') = (X/Z^2, Y/Z^3) */
1241 int ec_GFp_nistp224_point_get_affine_coordinates(const EC_GROUP *group,
1242         const EC_POINT *point, BIGNUM *x, BIGNUM *y, BN_CTX *ctx)
1243         {
1244         felem z1, z2, x_in, y_in, x_out, y_out;
1245         widefelem tmp;
1246
1247         if (EC_POINT_is_at_infinity(group, point))
1248                 {
1249                 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1250                         EC_R_POINT_AT_INFINITY);
1251                 return 0;
1252                 }
1253         if ((!BN_to_felem(x_in, &point->X)) || (!BN_to_felem(y_in, &point->Y)) ||
1254                 (!BN_to_felem(z1, &point->Z))) return 0;
1255         felem_inv(z2, z1);
1256         felem_square(tmp, z2); felem_reduce(z1, tmp);
1257         felem_mul(tmp, x_in, z1); felem_reduce(x_in, tmp);
1258         felem_contract(x_out, x_in);
1259         if (x != NULL)
1260                 {
1261                 if (!felem_to_BN(x, x_out)) {
1262                 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1263                         ERR_R_BN_LIB);
1264                 return 0;
1265                 }
1266                 }
1267         felem_mul(tmp, z1, z2); felem_reduce(z1, tmp);
1268         felem_mul(tmp, y_in, z1); felem_reduce(y_in, tmp);
1269         felem_contract(y_out, y_in);
1270         if (y != NULL)
1271                 {
1272                 if (!felem_to_BN(y, y_out)) {
1273                 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1274                         ERR_R_BN_LIB);
1275                 return 0;
1276                 }
1277                 }
1278         return 1;
1279         }
1280
1281 static void make_points_affine(size_t num, felem points[/*num*/][3], felem tmp_felems[/*num+1*/])
1282         {
1283         /* Runs in constant time, unless an input is the point at infinity
1284          * (which normally shouldn't happen). */
1285         ec_GFp_nistp_points_make_affine_internal(
1286                 num,
1287                 points,
1288                 sizeof(felem),
1289                 tmp_felems,
1290                 (void (*)(void *)) felem_one,
1291                 (int (*)(const void *)) felem_is_zero_int,
1292                 (void (*)(void *, const void *)) felem_assign,
1293                 (void (*)(void *, const void *)) felem_square_reduce,
1294                 (void (*)(void *, const void *, const void *)) felem_mul_reduce,
1295                 (void (*)(void *, const void *)) felem_inv,
1296                 (void (*)(void *, const void *)) felem_contract);
1297         }
1298
1299 /* Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL values
1300  * Result is stored in r (r can equal one of the inputs). */
1301 int ec_GFp_nistp224_points_mul(const EC_GROUP *group, EC_POINT *r,
1302         const BIGNUM *scalar, size_t num, const EC_POINT *points[],
1303         const BIGNUM *scalars[], BN_CTX *ctx)
1304         {
1305         int ret = 0;
1306         int j;
1307         unsigned i;
1308         int mixed = 0;
1309         BN_CTX *new_ctx = NULL;
1310         BIGNUM *x, *y, *z, *tmp_scalar;
1311         felem_bytearray g_secret;
1312         felem_bytearray *secrets = NULL;
1313         felem (*pre_comp)[17][3] = NULL;
1314         felem *tmp_felems = NULL;
1315         felem_bytearray tmp;
1316         unsigned num_bytes;
1317         int have_pre_comp = 0;
1318         size_t num_points = num;
1319         felem x_in, y_in, z_in, x_out, y_out, z_out;
1320         NISTP224_PRE_COMP *pre = NULL;
1321         const felem (*g_pre_comp)[16][3] = NULL;
1322         EC_POINT *generator = NULL;
1323         const EC_POINT *p = NULL;
1324         const BIGNUM *p_scalar = NULL;
1325
1326         if (ctx == NULL)
1327                 if ((ctx = new_ctx = BN_CTX_new()) == NULL) return 0;
1328         BN_CTX_start(ctx);
1329         if (((x = BN_CTX_get(ctx)) == NULL) ||
1330                 ((y = BN_CTX_get(ctx)) == NULL) ||
1331                 ((z = BN_CTX_get(ctx)) == NULL) ||
1332                 ((tmp_scalar = BN_CTX_get(ctx)) == NULL))
1333                 goto err;
1334
1335         if (scalar != NULL)
1336                 {
1337                 pre = EC_EX_DATA_get_data(group->extra_data,
1338                         nistp224_pre_comp_dup, nistp224_pre_comp_free,
1339                         nistp224_pre_comp_clear_free);
1340                 if (pre)
1341                         /* we have precomputation, try to use it */
1342                         g_pre_comp = (const felem (*)[16][3]) pre->g_pre_comp;
1343                 else
1344                         /* try to use the standard precomputation */
1345                         g_pre_comp = &gmul[0];
1346                 generator = EC_POINT_new(group);
1347                 if (generator == NULL)
1348                         goto err;
1349                 /* get the generator from precomputation */
1350                 if (!felem_to_BN(x, g_pre_comp[0][1][0]) ||
1351                         !felem_to_BN(y, g_pre_comp[0][1][1]) ||
1352                         !felem_to_BN(z, g_pre_comp[0][1][2]))
1353                         {
1354                         ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1355                         goto err;
1356                         }
1357                 if (!EC_POINT_set_Jprojective_coordinates_GFp(group,
1358                                 generator, x, y, z, ctx))
1359                         goto err;
1360                 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
1361                         /* precomputation matches generator */
1362                         have_pre_comp = 1;
1363                 else
1364                         /* we don't have valid precomputation:
1365                          * treat the generator as a random point */
1366                         num_points = num_points + 1;
1367                 }
1368
1369         if (num_points > 0)
1370                 {
1371                 if (num_points >= 3)
1372                         {
1373                         /* unless we precompute multiples for just one or two points,
1374                          * converting those into affine form is time well spent  */
1375                         mixed = 1;
1376                         }
1377                 secrets = OPENSSL_malloc(num_points * sizeof(felem_bytearray));
1378                 pre_comp = OPENSSL_malloc(num_points * 17 * 3 * sizeof(felem));
1379                 if (mixed)
1380                         tmp_felems = OPENSSL_malloc((num_points * 17 + 1) * sizeof(felem));
1381                 if ((secrets == NULL) || (pre_comp == NULL) || (mixed && (tmp_felems == NULL)))
1382                         {
1383                         ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_MALLOC_FAILURE);
1384                         goto err;
1385                         }
1386
1387                 /* we treat NULL scalars as 0, and NULL points as points at infinity,
1388                  * i.e., they contribute nothing to the linear combination */
1389                 memset(secrets, 0, num_points * sizeof(felem_bytearray));
1390                 memset(pre_comp, 0, num_points * 17 * 3 * sizeof(felem));
1391                 for (i = 0; i < num_points; ++i)
1392                         {
1393                         if (i == num)
1394                                 /* the generator */
1395                                 {
1396                                 p = EC_GROUP_get0_generator(group);
1397                                 p_scalar = scalar;
1398                                 }
1399                         else
1400                                 /* the i^th point */
1401                                 {
1402                                 p = points[i];
1403                                 p_scalar = scalars[i];
1404                                 }
1405                         if ((p_scalar != NULL) && (p != NULL))
1406                                 {
1407                                 /* reduce scalar to 0 <= scalar < 2^224 */
1408                                 if ((BN_num_bits(p_scalar) > 224) || (BN_is_negative(p_scalar)))
1409                                         {
1410                                         /* this is an unusual input, and we don't guarantee
1411                                          * constant-timeness */
1412                                         if (!BN_nnmod(tmp_scalar, p_scalar, &group->order, ctx))
1413                                                 {
1414                                                 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1415                                                 goto err;
1416                                                 }
1417                                         num_bytes = BN_bn2bin(tmp_scalar, tmp);
1418                                         }
1419                                 else
1420                                         num_bytes = BN_bn2bin(p_scalar, tmp);
1421                                 flip_endian(secrets[i], tmp, num_bytes);
1422                                 /* precompute multiples */
1423                                 if ((!BN_to_felem(x_out, &p->X)) ||
1424                                         (!BN_to_felem(y_out, &p->Y)) ||
1425                                         (!BN_to_felem(z_out, &p->Z))) goto err;
1426                                 felem_assign(pre_comp[i][1][0], x_out);
1427                                 felem_assign(pre_comp[i][1][1], y_out);
1428                                 felem_assign(pre_comp[i][1][2], z_out);
1429                                 for (j = 2; j <= 16; ++j)
1430                                         {
1431                                         if (j & 1)
1432                                                 {
1433                                                 point_add(
1434                                                         pre_comp[i][j][0], pre_comp[i][j][1], pre_comp[i][j][2],
1435                                                         pre_comp[i][1][0], pre_comp[i][1][1], pre_comp[i][1][2],
1436                                                         0, pre_comp[i][j-1][0], pre_comp[i][j-1][1], pre_comp[i][j-1][2]);
1437                                                 }
1438                                         else
1439                                                 {
1440                                                 point_double(
1441                                                         pre_comp[i][j][0], pre_comp[i][j][1], pre_comp[i][j][2],
1442                                                         pre_comp[i][j/2][0], pre_comp[i][j/2][1], pre_comp[i][j/2][2]);
1443                                                 }
1444                                         }
1445                                 }
1446                         }
1447                 if (mixed)
1448                         make_points_affine(num_points * 17, pre_comp[0], tmp_felems);
1449                 }
1450
1451         /* the scalar for the generator */
1452         if ((scalar != NULL) && (have_pre_comp))
1453                 {
1454                 memset(g_secret, 0, sizeof g_secret);
1455                 /* reduce scalar to 0 <= scalar < 2^224 */
1456                 if ((BN_num_bits(scalar) > 224) || (BN_is_negative(scalar)))
1457                         {
1458                         /* this is an unusual input, and we don't guarantee
1459                          * constant-timeness */
1460                         if (!BN_nnmod(tmp_scalar, scalar, &group->order, ctx))
1461                                 {
1462                                 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1463                                 goto err;
1464                                 }
1465                         num_bytes = BN_bn2bin(tmp_scalar, tmp);
1466                         }
1467                 else
1468                         num_bytes = BN_bn2bin(scalar, tmp);
1469                 flip_endian(g_secret, tmp, num_bytes);
1470                 /* do the multiplication with generator precomputation*/
1471                 batch_mul(x_out, y_out, z_out,
1472                         (const felem_bytearray (*)) secrets, num_points,
1473                         g_secret,
1474                         mixed, (const felem (*)[17][3]) pre_comp,
1475                         g_pre_comp);
1476                 }
1477         else
1478                 /* do the multiplication without generator precomputation */
1479                 batch_mul(x_out, y_out, z_out,
1480                         (const felem_bytearray (*)) secrets, num_points,
1481                         NULL, mixed, (const felem (*)[17][3]) pre_comp, NULL);
1482         /* reduce the output to its unique minimal representation */
1483         felem_contract(x_in, x_out);
1484         felem_contract(y_in, y_out);
1485         felem_contract(z_in, z_out);
1486         if ((!felem_to_BN(x, x_in)) || (!felem_to_BN(y, y_in)) ||
1487                 (!felem_to_BN(z, z_in)))
1488                 {
1489                 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1490                 goto err;
1491                 }
1492         ret = EC_POINT_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx);
1493
1494 err:
1495         BN_CTX_end(ctx);
1496         if (generator != NULL)
1497                 EC_POINT_free(generator);
1498         if (new_ctx != NULL)
1499                 BN_CTX_free(new_ctx);
1500         if (secrets != NULL)
1501                 OPENSSL_free(secrets);
1502         if (pre_comp != NULL)
1503                 OPENSSL_free(pre_comp);
1504         if (tmp_felems != NULL)
1505                 OPENSSL_free(tmp_felems);
1506         return ret;
1507         }
1508
1509 int ec_GFp_nistp224_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
1510         {
1511         int ret = 0;
1512         NISTP224_PRE_COMP *pre = NULL;
1513         int i, j;
1514         BN_CTX *new_ctx = NULL;
1515         BIGNUM *x, *y;
1516         EC_POINT *generator = NULL;
1517         felem tmp_felems[32];
1518
1519         /* throw away old precomputation */
1520         EC_EX_DATA_free_data(&group->extra_data, nistp224_pre_comp_dup,
1521                 nistp224_pre_comp_free, nistp224_pre_comp_clear_free);
1522         if (ctx == NULL)
1523                 if ((ctx = new_ctx = BN_CTX_new()) == NULL) return 0;
1524         BN_CTX_start(ctx);
1525         if (((x = BN_CTX_get(ctx)) == NULL) ||
1526                 ((y = BN_CTX_get(ctx)) == NULL))
1527                 goto err;
1528         /* get the generator */
1529         if (group->generator == NULL) goto err;
1530         generator = EC_POINT_new(group);
1531         if (generator == NULL)
1532                 goto err;
1533         BN_bin2bn(nistp224_curve_params[3], sizeof (felem_bytearray), x);
1534         BN_bin2bn(nistp224_curve_params[4], sizeof (felem_bytearray), y);
1535         if (!EC_POINT_set_affine_coordinates_GFp(group, generator, x, y, ctx))
1536                 goto err;
1537         if ((pre = nistp224_pre_comp_new()) == NULL)
1538                 goto err;
1539         /* if the generator is the standard one, use built-in precomputation */
1540         if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
1541                 {
1542                 memcpy(pre->g_pre_comp, gmul, sizeof(pre->g_pre_comp));
1543                 ret = 1;
1544                 goto err;
1545                 }
1546         if ((!BN_to_felem(pre->g_pre_comp[0][1][0], &group->generator->X)) ||
1547                 (!BN_to_felem(pre->g_pre_comp[0][1][1], &group->generator->Y)) ||
1548                 (!BN_to_felem(pre->g_pre_comp[0][1][2], &group->generator->Z)))
1549                 goto err;
1550         /* compute 2^56*G, 2^112*G, 2^168*G for the first table,
1551          * 2^28*G, 2^84*G, 2^140*G, 2^196*G for the second one
1552          */
1553         for (i = 1; i <= 8; i <<= 1)
1554                 {
1555                 point_double(
1556                         pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2],
1557                         pre->g_pre_comp[0][i][0], pre->g_pre_comp[0][i][1], pre->g_pre_comp[0][i][2]);
1558                 for (j = 0; j < 27; ++j)
1559                         {
1560                         point_double(
1561                                 pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2],
1562                                 pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2]);
1563                         }
1564                 if (i == 8)
1565                         break;
1566                 point_double(
1567                         pre->g_pre_comp[0][2*i][0], pre->g_pre_comp[0][2*i][1], pre->g_pre_comp[0][2*i][2],
1568                         pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2]);
1569                 for (j = 0; j < 27; ++j)
1570                         {
1571                         point_double(
1572                                 pre->g_pre_comp[0][2*i][0], pre->g_pre_comp[0][2*i][1], pre->g_pre_comp[0][2*i][2],
1573                                 pre->g_pre_comp[0][2*i][0], pre->g_pre_comp[0][2*i][1], pre->g_pre_comp[0][2*i][2]);
1574                         }
1575                 }
1576         for (i = 0; i < 2; i++)
1577                 {
1578                 /* g_pre_comp[i][0] is the point at infinity */
1579                 memset(pre->g_pre_comp[i][0], 0, sizeof(pre->g_pre_comp[i][0]));
1580                 /* the remaining multiples */
1581                 /* 2^56*G + 2^112*G resp. 2^84*G + 2^140*G */
1582                 point_add(
1583                         pre->g_pre_comp[i][6][0], pre->g_pre_comp[i][6][1],
1584                         pre->g_pre_comp[i][6][2], pre->g_pre_comp[i][4][0],
1585                         pre->g_pre_comp[i][4][1], pre->g_pre_comp[i][4][2],
1586                         0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
1587                         pre->g_pre_comp[i][2][2]);
1588                 /* 2^56*G + 2^168*G resp. 2^84*G + 2^196*G */
1589                 point_add(
1590                         pre->g_pre_comp[i][10][0], pre->g_pre_comp[i][10][1],
1591                         pre->g_pre_comp[i][10][2], pre->g_pre_comp[i][8][0],
1592                         pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2],
1593                         0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
1594                         pre->g_pre_comp[i][2][2]);
1595                 /* 2^112*G + 2^168*G resp. 2^140*G + 2^196*G */
1596                 point_add(
1597                         pre->g_pre_comp[i][12][0], pre->g_pre_comp[i][12][1],
1598                         pre->g_pre_comp[i][12][2], pre->g_pre_comp[i][8][0],
1599                         pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2],
1600                         0, pre->g_pre_comp[i][4][0], pre->g_pre_comp[i][4][1],
1601                         pre->g_pre_comp[i][4][2]);
1602                 /* 2^56*G + 2^112*G + 2^168*G resp. 2^84*G + 2^140*G + 2^196*G */
1603                 point_add(
1604                         pre->g_pre_comp[i][14][0], pre->g_pre_comp[i][14][1],
1605                         pre->g_pre_comp[i][14][2], pre->g_pre_comp[i][12][0],
1606                         pre->g_pre_comp[i][12][1], pre->g_pre_comp[i][12][2],
1607                         0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
1608                         pre->g_pre_comp[i][2][2]);
1609                 for (j = 1; j < 8; ++j)
1610                         {
1611                         /* odd multiples: add G resp. 2^28*G */
1612                         point_add(
1613                                 pre->g_pre_comp[i][2*j+1][0], pre->g_pre_comp[i][2*j+1][1],
1614                                 pre->g_pre_comp[i][2*j+1][2], pre->g_pre_comp[i][2*j][0],
1615                                 pre->g_pre_comp[i][2*j][1], pre->g_pre_comp[i][2*j][2],
1616                                 0, pre->g_pre_comp[i][1][0], pre->g_pre_comp[i][1][1],
1617                                 pre->g_pre_comp[i][1][2]);
1618                         }
1619                 }
1620         make_points_affine(31, &(pre->g_pre_comp[0][1]), tmp_felems);
1621
1622         if (!EC_EX_DATA_set_data(&group->extra_data, pre, nistp224_pre_comp_dup,
1623                         nistp224_pre_comp_free, nistp224_pre_comp_clear_free))
1624                 goto err;
1625         ret = 1;
1626         pre = NULL;
1627  err:
1628         BN_CTX_end(ctx);
1629         if (generator != NULL)
1630                 EC_POINT_free(generator);
1631         if (new_ctx != NULL)
1632                 BN_CTX_free(new_ctx);
1633         if (pre)
1634                 nistp224_pre_comp_free(pre);
1635         return ret;
1636         }
1637
1638 int ec_GFp_nistp224_have_precompute_mult(const EC_GROUP *group)
1639         {
1640         if (EC_EX_DATA_get_data(group->extra_data, nistp224_pre_comp_dup,
1641                         nistp224_pre_comp_free, nistp224_pre_comp_clear_free)
1642                 != NULL)
1643                 return 1;
1644         else
1645                 return 0;
1646         }
1647
1648 #else
1649 static void *dummy=&dummy;
1650 #endif