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