1 /* crypto/ec/ecp_nistp224.c */
3 * Written by Emilia Kasper (Google) for the OpenSSL project.
5 /* Copyright 2011 Google Inc.
7 * Licensed under the Apache License, Version 2.0 (the "License");
9 * you may not use this file except in compliance with the License.
10 * You may obtain a copy of the License at
12 * http://www.apache.org/licenses/LICENSE-2.0
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.
22 * A 64-bit implementation of the NIST P-224 elliptic curve point multiplication
24 * Inspired by Daniel J. Bernstein's public domain nistp224 implementation
25 * and Adam Langley's public domain 64-bit C implementation of curve25519
28 #include <openssl/opensslconf.h>
29 #ifndef OPENSSL_NO_EC_NISTP_64_GCC_128
33 #include <openssl/err.h>
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 */
40 #error "Need GCC 3.1 or later to define type uint128_t"
48 /******************************************************************************/
50 * INTERNAL REPRESENTATION OF FIELD ELEMENTS
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
68 typedef uint64_t limb;
69 typedef uint128_t widelimb;
71 typedef limb felem[4];
72 typedef widelimb widefelem[7];
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.
78 typedef u8 felem_bytearray[28];
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}
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.
104 * The table has 2 * 16 elements, starting with the following:
105 * index | bits | point
106 * ------+---------+------------------------------
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.
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.
129 static const felem gmul[2][16][3] =
133 {{0x3280d6115c1d21, 0xc1d356c2112234, 0x7f321390b94a03, 0xb70e0cbd6bb4bf},
134 {0xd5819985007e34, 0x75a05a07476444, 0xfb4c22dfe6cd43, 0xbd376388b5f723},
136 {{0xfd9675666ebbe9, 0xbca7664d40ce5e, 0x2242df8d8a2a43, 0x1f49bbb0f99bc5},
137 {0x29e0b892dc9c43, 0xece8608436e662, 0xdc858f185310d0, 0x9812dd4eb8d321},
139 {{0x6d3e678d5d8eb8, 0x559eed1cb362f1, 0x16e9a3bbce8a3f, 0xeedcccd8c2a748},
140 {0xf19f90ed50266d, 0xabf2b4bf65f9df, 0x313865468fafec, 0x5cb379ba910a17},
142 {{0x0641966cab26e3, 0x91fb2991fab0a0, 0xefec27a4e13a0b, 0x0499aa8a5f8ebe},
143 {0x7510407766af5d, 0x84d929610d5450, 0x81d77aae82f706, 0x6916f6d4338c5b},
145 {{0xea95ac3b1f15c6, 0x086000905e82d4, 0xdd323ae4d1c8b1, 0x932b56be7685a3},
146 {0x9ef93dea25dbbf, 0x41665960f390f0, 0xfdec76dbe2a8a7, 0x523e80f019062a},
148 {{0x822fdd26732c73, 0xa01c83531b5d0f, 0x363f37347c1ba4, 0xc391b45c84725c},
149 {0xbbd5e1b2d6ad24, 0xddfbcde19dfaec, 0xc393da7e222a7f, 0x1efb7890ede244},
151 {{0x4c9e90ca217da1, 0xd11beca79159bb, 0xff8d33c2c98b7c, 0x2610b39409f849},
152 {0x44d1352ac64da0, 0xcdbb7b2c46b4fb, 0x966c079b753c89, 0xfe67e4e820b112},
154 {{0xe28cae2df5312d, 0xc71b61d16f5c6e, 0x79b7619a3e7c4c, 0x05c73240899b47},
155 {0x9f7f6382c73e3a, 0x18615165c56bda, 0x641fab2116fd56, 0x72855882b08394},
157 {{0x0469182f161c09, 0x74a98ca8d00fb5, 0xb89da93489a3e0, 0x41c98768fb0c1d},
158 {0xe5ea05fb32da81, 0x3dce9ffbca6855, 0x1cfe2d3fbf59e6, 0x0e5e03408738a7},
160 {{0xdab22b2333e87f, 0x4430137a5dd2f6, 0xe03ab9f738beb8, 0xcb0c5d0dc34f24},
161 {0x764a7df0c8fda5, 0x185ba5c3fa2044, 0x9281d688bcbe50, 0xc40331df893881},
163 {{0xb89530796f0f60, 0xade92bd26909a3, 0x1a0c83fb4884da, 0x1765bf22a5a984},
164 {0x772a9ee75db09e, 0x23bc6c67cec16f, 0x4c1edba8b14e2f, 0xe2a215d9611369},
166 {{0x571e509fb5efb3, 0xade88696410552, 0xc8ae85fada74fe, 0x6c7e4be83bbde3},
167 {0xff9f51160f4652, 0xb47ce2495a6539, 0xa2946c53b582f4, 0x286d2db3ee9a60},
169 {{0x40bbd5081a44af, 0x0995183b13926c, 0xbcefba6f47f6d0, 0x215619e9cc0057},
170 {0x8bc94d3b0df45e, 0xf11c54a3694f6f, 0x8631b93cdfe8b5, 0xe7e3f4b0982db9},
172 {{0xb17048ab3e1c7b, 0xac38f36ff8a1d8, 0x1c29819435d2c6, 0xc813132f4c07e9},
173 {0x2891425503b11f, 0x08781030579fea, 0xf5426ba5cc9674, 0x1e28ebf18562bc},
175 {{0x9f31997cc864eb, 0x06cd91d28b5e4c, 0xff17036691a973, 0xf1aef351497c58},
176 {0xdd1f2d600564ff, 0xdead073b1402db, 0x74a684435bd693, 0xeea7471f962558},
181 {{0x9665266dddf554, 0x9613d78b60ef2d, 0xce27a34cdba417, 0xd35ab74d6afc31},
182 {0x85ccdd22deb15e, 0x2137e5783a6aab, 0xa141cffd8c93c6, 0x355a1830e90f2d},
184 {{0x1a494eadaade65, 0xd6da4da77fe53c, 0xe7992996abec86, 0x65c3553c6090e3},
185 {0xfa610b1fb09346, 0xf1c6540b8a4aaf, 0xc51a13ccd3cbab, 0x02995b1b18c28a},
187 {{0x7874568e7295ef, 0x86b419fbe38d04, 0xdc0690a7550d9a, 0xd3966a44beac33},
188 {0x2b7280ec29132f, 0xbeaa3b6a032df3, 0xdc7dd88ae41200, 0xd25e2513e3a100},
190 {{0x924857eb2efafd, 0xac2bce41223190, 0x8edaa1445553fc, 0x825800fd3562d5},
191 {0x8d79148ea96621, 0x23a01c3dd9ed8d, 0xaf8b219f9416b5, 0xd8db0cc277daea},
193 {{0x76a9c3b1a700f0, 0xe9acd29bc7e691, 0x69212d1a6b0327, 0x6322e97fe154be},
194 {0x469fc5465d62aa, 0x8d41ed18883b05, 0x1f8eae66c52b88, 0xe4fcbe9325be51},
196 {{0x825fdf583cac16, 0x020b857c7b023a, 0x683c17744b0165, 0x14ffd0a2daf2f1},
197 {0x323b36184218f9, 0x4944ec4e3b47d4, 0xc15b3080841acf, 0x0bced4b01a28bb},
199 {{0x92ac22230df5c4, 0x52f33b4063eda8, 0xcb3f19870c0c93, 0x40064f2ba65233},
200 {0xfe16f0924f8992, 0x012da25af5b517, 0x1a57bb24f723a6, 0x06f8bc76760def},
202 {{0x4a7084f7817cb9, 0xbcab0738ee9a78, 0x3ec11e11d9c326, 0xdc0fe90e0f1aae},
203 {0xcf639ea5f98390, 0x5c350aa22ffb74, 0x9afae98a4047b7, 0x956ec2d617fc45},
205 {{0x4306d648c1be6a, 0x9247cd8bc9a462, 0xf5595e377d2f2e, 0xbd1c3caff1a52e},
206 {0x045e14472409d0, 0x29f3e17078f773, 0x745a602b2d4f7d, 0x191837685cdfbb},
208 {{0x5b6ee254a8cb79, 0x4953433f5e7026, 0xe21faeb1d1def4, 0xc4c225785c09de},
209 {0x307ce7bba1e518, 0x31b125b1036db8, 0x47e91868839e8f, 0xc765866e33b9f3},
211 {{0x3bfece24f96906, 0x4794da641e5093, 0xde5df64f95db26, 0x297ecd89714b05},
212 {0x701bd3ebb2c3aa, 0x7073b4f53cb1d5, 0x13c5665658af16, 0x9895089d66fe58},
214 {{0x0fef05f78c4790, 0x2d773633b05d2e, 0x94229c3a951c94, 0xbbbd70df4911bb},
215 {0xb2c6963d2c1168, 0x105f47a72b0d73, 0x9fdf6111614080, 0x7b7e94b39e67b0},
217 {{0xad1a7d6efbe2b3, 0xf012482c0da69d, 0x6b3bdf12438345, 0x40d7558d7aa4d9},
218 {0x8a09fffb5c6d3d, 0x9a356e5d9ffd38, 0x5973f15f4f9b1c, 0xdcd5f59f63c3ea},
220 {{0xacf39f4c5ca7ab, 0x4c8071cc5fd737, 0xc64e3602cd1184, 0x0acd4644c9abba},
221 {0x6c011a36d8bf6e, 0xfecd87ba24e32a, 0x19f6f56574fad8, 0x050b204ced9405},
223 {{0xed4f1cae7d9a96, 0x5ceef7ad94c40a, 0x778e4a3bf3ef9b, 0x7405783dc3b55e},
224 {0x32477c61b6e8c6, 0xb46a97570f018b, 0x91176d0a7e95d1, 0x3df90fbc4c7d0e},
227 /* Precomputation for the group generator. */
229 felem g_pre_comp[2][16][3];
233 const EC_METHOD *EC_GFp_nistp224_method(void)
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 */,
260 ec_GFp_simple_invert,
261 ec_GFp_simple_is_at_infinity,
262 ec_GFp_simple_is_on_curve,
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,
272 0 /* field_encode */,
273 0 /* field_decode */,
274 0 /* field_set_to_one */ };
279 /* Helper functions to convert field elements to/from internal representation */
280 static void bin28_to_felem(felem out, const u8 in[28])
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;
288 static void felem_to_bin28(u8 out[28], const felem in)
291 for (i = 0; i < 7; ++i)
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);
300 /* To preserve endianness when using BN_bn2bin and BN_bin2bn */
301 static void flip_endian(u8 *out, const u8 *in, unsigned len)
304 for (i = 0; i < len; ++i)
305 out[i] = in[len-1-i];
308 /* From OpenSSL BIGNUM to internal representation */
309 static int BN_to_felem(felem out, const BIGNUM *bn)
311 felem_bytearray b_in;
312 felem_bytearray b_out;
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)
320 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
323 if (BN_is_negative(bn))
325 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
328 num_bytes = BN_bn2bin(bn, b_in);
329 flip_endian(b_out, b_in, num_bytes);
330 bin28_to_felem(out, b_out);
334 /* From internal representation to OpenSSL BIGNUM */
335 static BIGNUM *felem_to_BN(BIGNUM *out, const felem in)
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);
343 /******************************************************************************/
347 * Field operations, using the internal representation of field elements.
348 * NB! These operations are specific to our point multiplication and cannot be
349 * expected to be correct in general - e.g., multiplication with a large scalar
350 * will cause an overflow.
354 static void felem_one(felem out)
362 static void felem_assign(felem out, const felem in)
370 /* Sum two field elements: out += in */
371 static void felem_sum(felem out, const felem in)
379 /* Get negative value: out = -in */
380 /* Assumes in[i] < 2^57 */
381 static void felem_neg(felem out, const felem in)
383 static const limb two58p2 = (((limb) 1) << 58) + (((limb) 1) << 2);
384 static const limb two58m2 = (((limb) 1) << 58) - (((limb) 1) << 2);
385 static const limb two58m42m2 = (((limb) 1) << 58) -
386 (((limb) 1) << 42) - (((limb) 1) << 2);
388 /* Set to 0 mod 2^224-2^96+1 to ensure out > in */
389 out[0] = two58p2 - in[0];
390 out[1] = two58m42m2 - in[1];
391 out[2] = two58m2 - in[2];
392 out[3] = two58m2 - in[3];
395 /* Subtract field elements: out -= in */
396 /* Assumes in[i] < 2^57 */
397 static void felem_diff(felem out, const felem in)
399 static const limb two58p2 = (((limb) 1) << 58) + (((limb) 1) << 2);
400 static const limb two58m2 = (((limb) 1) << 58) - (((limb) 1) << 2);
401 static const limb two58m42m2 = (((limb) 1) << 58) -
402 (((limb) 1) << 42) - (((limb) 1) << 2);
404 /* Add 0 mod 2^224-2^96+1 to ensure out > in */
406 out[1] += two58m42m2;
416 /* Subtract in unreduced 128-bit mode: out -= in */
417 /* Assumes in[i] < 2^119 */
418 static void widefelem_diff(widefelem out, const widefelem in)
420 static const widelimb two120 = ((widelimb) 1) << 120;
421 static const widelimb two120m64 = (((widelimb) 1) << 120) -
422 (((widelimb) 1) << 64);
423 static const widelimb two120m104m64 = (((widelimb) 1) << 120) -
424 (((widelimb) 1) << 104) - (((widelimb) 1) << 64);
426 /* Add 0 mod 2^224-2^96+1 to ensure out > in */
431 out[4] += two120m104m64;
444 /* Subtract in mixed mode: out128 -= in64 */
446 static void felem_diff_128_64(widefelem out, const felem in)
448 static const widelimb two64p8 = (((widelimb) 1) << 64) +
449 (((widelimb) 1) << 8);
450 static const widelimb two64m8 = (((widelimb) 1) << 64) -
451 (((widelimb) 1) << 8);
452 static const widelimb two64m48m8 = (((widelimb) 1) << 64) -
453 (((widelimb) 1) << 48) - (((widelimb) 1) << 8);
455 /* Add 0 mod 2^224-2^96+1 to ensure out > in */
457 out[1] += two64m48m8;
467 /* Multiply a field element by a scalar: out = out * scalar
468 * The scalars we actually use are small, so results fit without overflow */
469 static void felem_scalar(felem out, const limb scalar)
477 /* Multiply an unreduced field element by a scalar: out = out * scalar
478 * The scalars we actually use are small, so results fit without overflow */
479 static void widefelem_scalar(widefelem out, const widelimb scalar)
490 /* Square a field element: out = in^2 */
491 static void felem_square(widefelem out, const felem in)
493 limb tmp0, tmp1, tmp2;
494 tmp0 = 2 * in[0]; tmp1 = 2 * in[1]; tmp2 = 2 * in[2];
495 out[0] = ((widelimb) in[0]) * in[0];
496 out[1] = ((widelimb) in[0]) * tmp1;
497 out[2] = ((widelimb) in[0]) * tmp2 + ((widelimb) in[1]) * in[1];
498 out[3] = ((widelimb) in[3]) * tmp0 +
499 ((widelimb) in[1]) * tmp2;
500 out[4] = ((widelimb) in[3]) * tmp1 + ((widelimb) in[2]) * in[2];
501 out[5] = ((widelimb) in[3]) * tmp2;
502 out[6] = ((widelimb) in[3]) * in[3];
505 /* Multiply two field elements: out = in1 * in2 */
506 static void felem_mul(widefelem out, const felem in1, const felem in2)
508 out[0] = ((widelimb) in1[0]) * in2[0];
509 out[1] = ((widelimb) in1[0]) * in2[1] + ((widelimb) in1[1]) * in2[0];
510 out[2] = ((widelimb) in1[0]) * in2[2] + ((widelimb) in1[1]) * in2[1] +
511 ((widelimb) in1[2]) * in2[0];
512 out[3] = ((widelimb) in1[0]) * in2[3] + ((widelimb) in1[1]) * in2[2] +
513 ((widelimb) in1[2]) * in2[1] + ((widelimb) in1[3]) * in2[0];
514 out[4] = ((widelimb) in1[1]) * in2[3] + ((widelimb) in1[2]) * in2[2] +
515 ((widelimb) in1[3]) * in2[1];
516 out[5] = ((widelimb) in1[2]) * in2[3] + ((widelimb) in1[3]) * in2[2];
517 out[6] = ((widelimb) in1[3]) * in2[3];
521 * Reduce seven 128-bit coefficients to four 64-bit coefficients.
522 * Requires in[i] < 2^126,
523 * ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] <= 2^56 + 2^16 */
524 static void felem_reduce(felem out, const widefelem in)
526 static const widelimb two127p15 = (((widelimb) 1) << 127) +
527 (((widelimb) 1) << 15);
528 static const widelimb two127m71 = (((widelimb) 1) << 127) -
529 (((widelimb) 1) << 71);
530 static const widelimb two127m71m55 = (((widelimb) 1) << 127) -
531 (((widelimb) 1) << 71) - (((widelimb) 1) << 55);
534 /* Add 0 mod 2^224-2^96+1 to ensure all differences are positive */
535 output[0] = in[0] + two127p15;
536 output[1] = in[1] + two127m71m55;
537 output[2] = in[2] + two127m71;
541 /* Eliminate in[4], in[5], in[6] */
542 output[4] += in[6] >> 16;
543 output[3] += (in[6] & 0xffff) << 40;
546 output[3] += in[5] >> 16;
547 output[2] += (in[5] & 0xffff) << 40;
550 output[2] += output[4] >> 16;
551 output[1] += (output[4] & 0xffff) << 40;
552 output[0] -= output[4];
554 /* Carry 2 -> 3 -> 4 */
555 output[3] += output[2] >> 56;
556 output[2] &= 0x00ffffffffffffff;
558 output[4] = output[3] >> 56;
559 output[3] &= 0x00ffffffffffffff;
561 /* Now output[2] < 2^56, output[3] < 2^56, output[4] < 2^72 */
563 /* Eliminate output[4] */
564 output[2] += output[4] >> 16;
565 /* output[2] < 2^56 + 2^56 = 2^57 */
566 output[1] += (output[4] & 0xffff) << 40;
567 output[0] -= output[4];
569 /* Carry 0 -> 1 -> 2 -> 3 */
570 output[1] += output[0] >> 56;
571 out[0] = output[0] & 0x00ffffffffffffff;
573 output[2] += output[1] >> 56;
574 /* output[2] < 2^57 + 2^72 */
575 out[1] = output[1] & 0x00ffffffffffffff;
576 output[3] += output[2] >> 56;
577 /* output[3] <= 2^56 + 2^16 */
578 out[2] = output[2] & 0x00ffffffffffffff;
581 * out[0] < 2^56, out[1] < 2^56, out[2] < 2^56,
582 * out[3] <= 2^56 + 2^16 (due to final carry),
588 static void felem_square_reduce(felem out, const felem in)
591 felem_square(tmp, in);
592 felem_reduce(out, tmp);
595 static void felem_mul_reduce(felem out, const felem in1, const felem in2)
598 felem_mul(tmp, in1, in2);
599 felem_reduce(out, tmp);
602 /* Reduce to unique minimal representation.
603 * Requires 0 <= in < 2*p (always call felem_reduce first) */
604 static void felem_contract(felem out, const felem in)
606 static const int64_t two56 = ((limb) 1) << 56;
607 /* 0 <= in < 2*p, p = 2^224 - 2^96 + 1 */
608 /* if in > p , reduce in = in - 2^224 + 2^96 - 1 */
614 /* Case 1: a = 1 iff in >= 2^224 */
618 tmp[3] &= 0x00ffffffffffffff;
619 /* Case 2: a = 0 iff p <= in < 2^224, i.e.,
620 * the high 128 bits are all 1 and the lower part is non-zero */
621 a = ((in[3] & in[2] & (in[1] | 0x000000ffffffffff)) + 1) |
622 (((int64_t)(in[0] + (in[1] & 0x000000ffffffffff)) - 1) >> 63);
623 a &= 0x00ffffffffffffff;
624 /* turn a into an all-one mask (if a = 0) or an all-zero mask */
626 /* subtract 2^224 - 2^96 + 1 if a is all-one*/
627 tmp[3] &= a ^ 0xffffffffffffffff;
628 tmp[2] &= a ^ 0xffffffffffffffff;
629 tmp[1] &= (a ^ 0xffffffffffffffff) | 0x000000ffffffffff;
632 /* eliminate negative coefficients: if tmp[0] is negative, tmp[1] must
633 * be non-zero, so we only need one step */
638 /* carry 1 -> 2 -> 3 */
639 tmp[2] += tmp[1] >> 56;
640 tmp[1] &= 0x00ffffffffffffff;
642 tmp[3] += tmp[2] >> 56;
643 tmp[2] &= 0x00ffffffffffffff;
645 /* Now 0 <= out < p */
652 /* Zero-check: returns 1 if input is 0, and 0 otherwise.
653 * We know that field elements are reduced to in < 2^225,
654 * so we only need to check three cases: 0, 2^224 - 2^96 + 1,
655 * and 2^225 - 2^97 + 2 */
656 static limb felem_is_zero(const felem in)
658 limb zero, two224m96p1, two225m97p2;
660 zero = in[0] | in[1] | in[2] | in[3];
661 zero = (((int64_t)(zero) - 1) >> 63) & 1;
662 two224m96p1 = (in[0] ^ 1) | (in[1] ^ 0x00ffff0000000000)
663 | (in[2] ^ 0x00ffffffffffffff) | (in[3] ^ 0x00ffffffffffffff);
664 two224m96p1 = (((int64_t)(two224m96p1) - 1) >> 63) & 1;
665 two225m97p2 = (in[0] ^ 2) | (in[1] ^ 0x00fffe0000000000)
666 | (in[2] ^ 0x00ffffffffffffff) | (in[3] ^ 0x01ffffffffffffff);
667 two225m97p2 = (((int64_t)(two225m97p2) - 1) >> 63) & 1;
668 return (zero | two224m96p1 | two225m97p2);
671 static limb felem_is_zero_int(const felem in)
673 return (int) (felem_is_zero(in) & ((limb)1));
676 /* Invert a field element */
677 /* Computation chain copied from djb's code */
678 static void felem_inv(felem out, const felem in)
680 felem ftmp, ftmp2, ftmp3, ftmp4;
684 felem_square(tmp, in); felem_reduce(ftmp, tmp); /* 2 */
685 felem_mul(tmp, in, ftmp); felem_reduce(ftmp, tmp); /* 2^2 - 1 */
686 felem_square(tmp, ftmp); felem_reduce(ftmp, tmp); /* 2^3 - 2 */
687 felem_mul(tmp, in, ftmp); felem_reduce(ftmp, tmp); /* 2^3 - 1 */
688 felem_square(tmp, ftmp); felem_reduce(ftmp2, tmp); /* 2^4 - 2 */
689 felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp); /* 2^5 - 4 */
690 felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp); /* 2^6 - 8 */
691 felem_mul(tmp, ftmp2, ftmp); felem_reduce(ftmp, tmp); /* 2^6 - 1 */
692 felem_square(tmp, ftmp); felem_reduce(ftmp2, tmp); /* 2^7 - 2 */
693 for (i = 0; i < 5; ++i) /* 2^12 - 2^6 */
695 felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp);
697 felem_mul(tmp, ftmp2, ftmp); felem_reduce(ftmp2, tmp); /* 2^12 - 1 */
698 felem_square(tmp, ftmp2); felem_reduce(ftmp3, tmp); /* 2^13 - 2 */
699 for (i = 0; i < 11; ++i) /* 2^24 - 2^12 */
701 felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp);
703 felem_mul(tmp, ftmp3, ftmp2); felem_reduce(ftmp2, tmp); /* 2^24 - 1 */
704 felem_square(tmp, ftmp2); felem_reduce(ftmp3, tmp); /* 2^25 - 2 */
705 for (i = 0; i < 23; ++i) /* 2^48 - 2^24 */
707 felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp);
709 felem_mul(tmp, ftmp3, ftmp2); felem_reduce(ftmp3, tmp); /* 2^48 - 1 */
710 felem_square(tmp, ftmp3); felem_reduce(ftmp4, tmp); /* 2^49 - 2 */
711 for (i = 0; i < 47; ++i) /* 2^96 - 2^48 */
713 felem_square(tmp, ftmp4); felem_reduce(ftmp4, tmp);
715 felem_mul(tmp, ftmp3, ftmp4); felem_reduce(ftmp3, tmp); /* 2^96 - 1 */
716 felem_square(tmp, ftmp3); felem_reduce(ftmp4, tmp); /* 2^97 - 2 */
717 for (i = 0; i < 23; ++i) /* 2^120 - 2^24 */
719 felem_square(tmp, ftmp4); felem_reduce(ftmp4, tmp);
721 felem_mul(tmp, ftmp2, ftmp4); felem_reduce(ftmp2, tmp); /* 2^120 - 1 */
722 for (i = 0; i < 6; ++i) /* 2^126 - 2^6 */
724 felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp);
726 felem_mul(tmp, ftmp2, ftmp); felem_reduce(ftmp, tmp); /* 2^126 - 1 */
727 felem_square(tmp, ftmp); felem_reduce(ftmp, tmp); /* 2^127 - 2 */
728 felem_mul(tmp, ftmp, in); felem_reduce(ftmp, tmp); /* 2^127 - 1 */
729 for (i = 0; i < 97; ++i) /* 2^224 - 2^97 */
731 felem_square(tmp, ftmp); felem_reduce(ftmp, tmp);
733 felem_mul(tmp, ftmp, ftmp3); felem_reduce(out, tmp); /* 2^224 - 2^96 - 1 */
736 /* Copy in constant time:
737 * if icopy == 1, copy in to out,
738 * if icopy == 0, copy out to itself. */
740 copy_conditional(felem out, const felem in, limb icopy)
743 /* icopy is a (64-bit) 0 or 1, so copy is either all-zero or all-one */
744 const limb copy = -icopy;
745 for (i = 0; i < 4; ++i)
747 const limb tmp = copy & (in[i] ^ out[i]);
752 /******************************************************************************/
754 * ELLIPTIC CURVE POINT OPERATIONS
756 * Points are represented in Jacobian projective coordinates:
757 * (X, Y, Z) corresponds to the affine point (X/Z^2, Y/Z^3),
758 * or to the point at infinity if Z == 0.
763 * Double an elliptic curve point:
764 * (X', Y', Z') = 2 * (X, Y, Z), where
765 * X' = (3 * (X - Z^2) * (X + Z^2))^2 - 8 * X * Y^2
766 * Y' = 3 * (X - Z^2) * (X + Z^2) * (4 * X * Y^2 - X') - 8 * Y^2
767 * Z' = (Y + Z)^2 - Y^2 - Z^2 = 2 * Y * Z
768 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed,
769 * while x_out == y_in is not (maybe this works, but it's not tested).
772 point_double(felem x_out, felem y_out, felem z_out,
773 const felem x_in, const felem y_in, const felem z_in)
776 felem delta, gamma, beta, alpha, ftmp, ftmp2;
778 felem_assign(ftmp, x_in);
779 felem_assign(ftmp2, x_in);
782 felem_square(tmp, z_in);
783 felem_reduce(delta, tmp);
786 felem_square(tmp, y_in);
787 felem_reduce(gamma, tmp);
790 felem_mul(tmp, x_in, gamma);
791 felem_reduce(beta, tmp);
793 /* alpha = 3*(x-delta)*(x+delta) */
794 felem_diff(ftmp, delta);
795 /* ftmp[i] < 2^57 + 2^58 + 2 < 2^59 */
796 felem_sum(ftmp2, delta);
797 /* ftmp2[i] < 2^57 + 2^57 = 2^58 */
798 felem_scalar(ftmp2, 3);
799 /* ftmp2[i] < 3 * 2^58 < 2^60 */
800 felem_mul(tmp, ftmp, ftmp2);
801 /* tmp[i] < 2^60 * 2^59 * 4 = 2^121 */
802 felem_reduce(alpha, tmp);
804 /* x' = alpha^2 - 8*beta */
805 felem_square(tmp, alpha);
806 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
807 felem_assign(ftmp, beta);
808 felem_scalar(ftmp, 8);
809 /* ftmp[i] < 8 * 2^57 = 2^60 */
810 felem_diff_128_64(tmp, ftmp);
811 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
812 felem_reduce(x_out, tmp);
814 /* z' = (y + z)^2 - gamma - delta */
815 felem_sum(delta, gamma);
816 /* delta[i] < 2^57 + 2^57 = 2^58 */
817 felem_assign(ftmp, y_in);
818 felem_sum(ftmp, z_in);
819 /* ftmp[i] < 2^57 + 2^57 = 2^58 */
820 felem_square(tmp, ftmp);
821 /* tmp[i] < 4 * 2^58 * 2^58 = 2^118 */
822 felem_diff_128_64(tmp, delta);
823 /* tmp[i] < 2^118 + 2^64 + 8 < 2^119 */
824 felem_reduce(z_out, tmp);
826 /* y' = alpha*(4*beta - x') - 8*gamma^2 */
827 felem_scalar(beta, 4);
828 /* beta[i] < 4 * 2^57 = 2^59 */
829 felem_diff(beta, x_out);
830 /* beta[i] < 2^59 + 2^58 + 2 < 2^60 */
831 felem_mul(tmp, alpha, beta);
832 /* tmp[i] < 4 * 2^57 * 2^60 = 2^119 */
833 felem_square(tmp2, gamma);
834 /* tmp2[i] < 4 * 2^57 * 2^57 = 2^116 */
835 widefelem_scalar(tmp2, 8);
836 /* tmp2[i] < 8 * 2^116 = 2^119 */
837 widefelem_diff(tmp, tmp2);
838 /* tmp[i] < 2^119 + 2^120 < 2^121 */
839 felem_reduce(y_out, tmp);
843 * Add two elliptic curve points:
844 * (X_1, Y_1, Z_1) + (X_2, Y_2, Z_2) = (X_3, Y_3, Z_3), where
845 * X_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1)^2 - (Z_1^2 * X_2 - Z_2^2 * X_1)^3 -
846 * 2 * Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^2
847 * 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) -
848 * Z_2^3 * Y_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^3
849 * Z_3 = (Z_1^2 * X_2 - Z_2^2 * X_1) * (Z_1 * Z_2)
851 * This runs faster if 'mixed' is set, which requires Z_2 = 1 or Z_2 = 0.
854 /* This function is not entirely constant-time:
855 * it includes a branch for checking whether the two input points are equal,
856 * (while not equal to the point at infinity).
857 * This case never happens during single point multiplication,
858 * so there is no timing leak for ECDH or ECDSA signing. */
859 static void point_add(felem x3, felem y3, felem z3,
860 const felem x1, const felem y1, const felem z1,
861 const int mixed, const felem x2, const felem y2, const felem z2)
863 felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, x_out, y_out, z_out;
865 limb z1_is_zero, z2_is_zero, x_equal, y_equal;
870 felem_square(tmp, z2);
871 felem_reduce(ftmp2, tmp);
874 felem_mul(tmp, ftmp2, z2);
875 felem_reduce(ftmp4, tmp);
877 /* ftmp4 = z2^3*y1 */
878 felem_mul(tmp2, ftmp4, y1);
879 felem_reduce(ftmp4, tmp2);
881 /* ftmp2 = z2^2*x1 */
882 felem_mul(tmp2, ftmp2, x1);
883 felem_reduce(ftmp2, tmp2);
887 /* We'll assume z2 = 1 (special case z2 = 0 is handled later) */
889 /* ftmp4 = z2^3*y1 */
890 felem_assign(ftmp4, y1);
892 /* ftmp2 = z2^2*x1 */
893 felem_assign(ftmp2, x1);
897 felem_square(tmp, z1);
898 felem_reduce(ftmp, tmp);
901 felem_mul(tmp, ftmp, z1);
902 felem_reduce(ftmp3, tmp);
905 felem_mul(tmp, ftmp3, y2);
906 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
908 /* ftmp3 = z1^3*y2 - z2^3*y1 */
909 felem_diff_128_64(tmp, ftmp4);
910 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
911 felem_reduce(ftmp3, tmp);
914 felem_mul(tmp, ftmp, x2);
915 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
917 /* ftmp = z1^2*x2 - z2^2*x1 */
918 felem_diff_128_64(tmp, ftmp2);
919 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
920 felem_reduce(ftmp, tmp);
922 /* the formulae are incorrect if the points are equal
923 * so we check for this and do doubling if this happens */
924 x_equal = felem_is_zero(ftmp);
925 y_equal = felem_is_zero(ftmp3);
926 z1_is_zero = felem_is_zero(z1);
927 z2_is_zero = felem_is_zero(z2);
928 /* In affine coordinates, (X_1, Y_1) == (X_2, Y_2) */
929 if (x_equal && y_equal && !z1_is_zero && !z2_is_zero)
931 point_double(x3, y3, z3, x1, y1, z1);
938 felem_mul(tmp, z1, z2);
939 felem_reduce(ftmp5, tmp);
943 /* special case z2 = 0 is handled later */
944 felem_assign(ftmp5, z1);
947 /* z_out = (z1^2*x2 - z2^2*x1)*(z1*z2) */
948 felem_mul(tmp, ftmp, ftmp5);
949 felem_reduce(z_out, tmp);
951 /* ftmp = (z1^2*x2 - z2^2*x1)^2 */
952 felem_assign(ftmp5, ftmp);
953 felem_square(tmp, ftmp);
954 felem_reduce(ftmp, tmp);
956 /* ftmp5 = (z1^2*x2 - z2^2*x1)^3 */
957 felem_mul(tmp, ftmp, ftmp5);
958 felem_reduce(ftmp5, tmp);
960 /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
961 felem_mul(tmp, ftmp2, ftmp);
962 felem_reduce(ftmp2, tmp);
964 /* tmp = z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */
965 felem_mul(tmp, ftmp4, ftmp5);
966 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
968 /* tmp2 = (z1^3*y2 - z2^3*y1)^2 */
969 felem_square(tmp2, ftmp3);
970 /* tmp2[i] < 4 * 2^57 * 2^57 < 2^116 */
972 /* tmp2 = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 */
973 felem_diff_128_64(tmp2, ftmp5);
974 /* tmp2[i] < 2^116 + 2^64 + 8 < 2^117 */
976 /* ftmp5 = 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
977 felem_assign(ftmp5, ftmp2);
978 felem_scalar(ftmp5, 2);
979 /* ftmp5[i] < 2 * 2^57 = 2^58 */
982 * x_out = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 -
983 * 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2
985 felem_diff_128_64(tmp2, ftmp5);
986 /* tmp2[i] < 2^117 + 2^64 + 8 < 2^118 */
987 felem_reduce(x_out, tmp2);
989 /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out */
990 felem_diff(ftmp2, x_out);
991 /* ftmp2[i] < 2^57 + 2^58 + 2 < 2^59 */
993 /* tmp2 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) */
994 felem_mul(tmp2, ftmp3, ftmp2);
995 /* tmp2[i] < 4 * 2^57 * 2^59 = 2^118 */
998 * y_out = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) -
999 * z2^3*y1*(z1^2*x2 - z2^2*x1)^3
1001 widefelem_diff(tmp2, tmp);
1002 /* tmp2[i] < 2^118 + 2^120 < 2^121 */
1003 felem_reduce(y_out, tmp2);
1005 /* the result (x_out, y_out, z_out) is incorrect if one of the inputs is
1006 * the point at infinity, so we need to check for this separately */
1008 /* if point 1 is at infinity, copy point 2 to output, and vice versa */
1009 copy_conditional(x_out, x2, z1_is_zero);
1010 copy_conditional(x_out, x1, z2_is_zero);
1011 copy_conditional(y_out, y2, z1_is_zero);
1012 copy_conditional(y_out, y1, z2_is_zero);
1013 copy_conditional(z_out, z2, z1_is_zero);
1014 copy_conditional(z_out, z1, z2_is_zero);
1015 felem_assign(x3, x_out);
1016 felem_assign(y3, y_out);
1017 felem_assign(z3, z_out);
1021 * select_point selects the |idx|th point from a precomputation table and
1023 * The pre_comp array argument should be size of |size| argument
1025 static void select_point(const u64 idx, unsigned int size, const felem pre_comp[][3], felem out[3])
1028 limb *outlimbs = &out[0][0];
1029 memset(outlimbs, 0, 3 * sizeof(felem));
1031 for (i = 0; i < size; i++)
1033 const limb *inlimbs = &pre_comp[i][0][0];
1040 for (j = 0; j < 4 * 3; j++)
1041 outlimbs[j] |= inlimbs[j] & mask;
1045 /* get_bit returns the |i|th bit in |in| */
1046 static char get_bit(const felem_bytearray in, unsigned i)
1050 return (in[i >> 3] >> (i & 7)) & 1;
1053 /* Interleaved point multiplication using precomputed point multiples:
1054 * The small point multiples 0*P, 1*P, ..., 16*P are in pre_comp[],
1055 * the scalars in scalars[]. If g_scalar is non-NULL, we also add this multiple
1056 * of the generator, using certain (large) precomputed multiples in g_pre_comp.
1057 * Output point (X, Y, Z) is stored in x_out, y_out, z_out */
1058 static void batch_mul(felem x_out, felem y_out, felem z_out,
1059 const felem_bytearray scalars[], const unsigned num_points, const u8 *g_scalar,
1060 const int mixed, const felem pre_comp[][17][3], const felem g_pre_comp[2][16][3])
1064 unsigned gen_mul = (g_scalar != NULL);
1065 felem nq[3], tmp[4];
1069 /* set nq to the point at infinity */
1070 memset(nq, 0, 3 * sizeof(felem));
1072 /* Loop over all scalars msb-to-lsb, interleaving additions
1073 * of multiples of the generator (two in each of the last 28 rounds)
1074 * and additions of other points multiples (every 5th round).
1076 skip = 1; /* save two point operations in the first round */
1077 for (i = (num_points ? 220 : 27); i >= 0; --i)
1081 point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
1083 /* add multiples of the generator */
1084 if (gen_mul && (i <= 27))
1086 /* first, look 28 bits upwards */
1087 bits = get_bit(g_scalar, i + 196) << 3;
1088 bits |= get_bit(g_scalar, i + 140) << 2;
1089 bits |= get_bit(g_scalar, i + 84) << 1;
1090 bits |= get_bit(g_scalar, i + 28);
1091 /* select the point to add, in constant time */
1092 select_point(bits, 16, g_pre_comp[1], tmp);
1096 point_add(nq[0], nq[1], nq[2],
1097 nq[0], nq[1], nq[2],
1098 1 /* mixed */, tmp[0], tmp[1], tmp[2]);
1102 memcpy(nq, tmp, 3 * sizeof(felem));
1106 /* second, look at the current position */
1107 bits = get_bit(g_scalar, i + 168) << 3;
1108 bits |= get_bit(g_scalar, i + 112) << 2;
1109 bits |= get_bit(g_scalar, i + 56) << 1;
1110 bits |= get_bit(g_scalar, i);
1111 /* select the point to add, in constant time */
1112 select_point(bits, 16, g_pre_comp[0], tmp);
1113 point_add(nq[0], nq[1], nq[2],
1114 nq[0], nq[1], nq[2],
1115 1 /* mixed */, tmp[0], tmp[1], tmp[2]);
1118 /* do other additions every 5 doublings */
1119 if (num_points && (i % 5 == 0))
1121 /* loop over all scalars */
1122 for (num = 0; num < num_points; ++num)
1124 bits = get_bit(scalars[num], i + 4) << 5;
1125 bits |= get_bit(scalars[num], i + 3) << 4;
1126 bits |= get_bit(scalars[num], i + 2) << 3;
1127 bits |= get_bit(scalars[num], i + 1) << 2;
1128 bits |= get_bit(scalars[num], i) << 1;
1129 bits |= get_bit(scalars[num], i - 1);
1130 ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits);
1132 /* select the point to add or subtract */
1133 select_point(digit, 17, pre_comp[num], tmp);
1134 felem_neg(tmp[3], tmp[1]); /* (X, -Y, Z) is the negative point */
1135 copy_conditional(tmp[1], tmp[3], sign);
1139 point_add(nq[0], nq[1], nq[2],
1140 nq[0], nq[1], nq[2],
1141 mixed, tmp[0], tmp[1], tmp[2]);
1145 memcpy(nq, tmp, 3 * sizeof(felem));
1151 felem_assign(x_out, nq[0]);
1152 felem_assign(y_out, nq[1]);
1153 felem_assign(z_out, nq[2]);
1156 /******************************************************************************/
1157 /* FUNCTIONS TO MANAGE PRECOMPUTATION
1160 static NISTP224_PRE_COMP *nistp224_pre_comp_new()
1162 NISTP224_PRE_COMP *ret = NULL;
1163 ret = (NISTP224_PRE_COMP *) OPENSSL_malloc(sizeof *ret);
1166 ECerr(EC_F_NISTP224_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1169 memset(ret->g_pre_comp, 0, sizeof(ret->g_pre_comp));
1170 ret->references = 1;
1174 static void *nistp224_pre_comp_dup(void *src_)
1176 NISTP224_PRE_COMP *src = src_;
1178 /* no need to actually copy, these objects never change! */
1179 CRYPTO_add(&src->references, 1, CRYPTO_LOCK_EC_PRE_COMP);
1184 static void nistp224_pre_comp_free(void *pre_)
1187 NISTP224_PRE_COMP *pre = pre_;
1192 i = CRYPTO_add(&pre->references, -1, CRYPTO_LOCK_EC_PRE_COMP);
1199 static void nistp224_pre_comp_clear_free(void *pre_)
1202 NISTP224_PRE_COMP *pre = pre_;
1207 i = CRYPTO_add(&pre->references, -1, CRYPTO_LOCK_EC_PRE_COMP);
1211 OPENSSL_cleanse(pre, sizeof *pre);
1215 /******************************************************************************/
1216 /* OPENSSL EC_METHOD FUNCTIONS
1219 int ec_GFp_nistp224_group_init(EC_GROUP *group)
1222 ret = ec_GFp_simple_group_init(group);
1223 group->a_is_minus3 = 1;
1227 int ec_GFp_nistp224_group_set_curve(EC_GROUP *group, const BIGNUM *p,
1228 const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
1231 BN_CTX *new_ctx = NULL;
1232 BIGNUM *curve_p, *curve_a, *curve_b;
1235 if ((ctx = new_ctx = BN_CTX_new()) == NULL) return 0;
1237 if (((curve_p = BN_CTX_get(ctx)) == NULL) ||
1238 ((curve_a = BN_CTX_get(ctx)) == NULL) ||
1239 ((curve_b = BN_CTX_get(ctx)) == NULL)) goto err;
1240 BN_bin2bn(nistp224_curve_params[0], sizeof(felem_bytearray), curve_p);
1241 BN_bin2bn(nistp224_curve_params[1], sizeof(felem_bytearray), curve_a);
1242 BN_bin2bn(nistp224_curve_params[2], sizeof(felem_bytearray), curve_b);
1243 if ((BN_cmp(curve_p, p)) || (BN_cmp(curve_a, a)) ||
1244 (BN_cmp(curve_b, b)))
1246 ECerr(EC_F_EC_GFP_NISTP224_GROUP_SET_CURVE,
1247 EC_R_WRONG_CURVE_PARAMETERS);
1250 group->field_mod_func = BN_nist_mod_224;
1251 ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx);
1254 if (new_ctx != NULL)
1255 BN_CTX_free(new_ctx);
1259 /* Takes the Jacobian coordinates (X, Y, Z) of a point and returns
1260 * (X', Y') = (X/Z^2, Y/Z^3) */
1261 int ec_GFp_nistp224_point_get_affine_coordinates(const EC_GROUP *group,
1262 const EC_POINT *point, BIGNUM *x, BIGNUM *y, BN_CTX *ctx)
1264 felem z1, z2, x_in, y_in, x_out, y_out;
1267 if (EC_POINT_is_at_infinity(group, point))
1269 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1270 EC_R_POINT_AT_INFINITY);
1273 if ((!BN_to_felem(x_in, &point->X)) || (!BN_to_felem(y_in, &point->Y)) ||
1274 (!BN_to_felem(z1, &point->Z))) return 0;
1276 felem_square(tmp, z2); felem_reduce(z1, tmp);
1277 felem_mul(tmp, x_in, z1); felem_reduce(x_in, tmp);
1278 felem_contract(x_out, x_in);
1281 if (!felem_to_BN(x, x_out)) {
1282 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1287 felem_mul(tmp, z1, z2); felem_reduce(z1, tmp);
1288 felem_mul(tmp, y_in, z1); felem_reduce(y_in, tmp);
1289 felem_contract(y_out, y_in);
1292 if (!felem_to_BN(y, y_out)) {
1293 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1301 static void make_points_affine(size_t num, felem points[/*num*/][3], felem tmp_felems[/*num+1*/])
1303 /* Runs in constant time, unless an input is the point at infinity
1304 * (which normally shouldn't happen). */
1305 ec_GFp_nistp_points_make_affine_internal(
1310 (void (*)(void *)) felem_one,
1311 (int (*)(const void *)) felem_is_zero_int,
1312 (void (*)(void *, const void *)) felem_assign,
1313 (void (*)(void *, const void *)) felem_square_reduce,
1314 (void (*)(void *, const void *, const void *)) felem_mul_reduce,
1315 (void (*)(void *, const void *)) felem_inv,
1316 (void (*)(void *, const void *)) felem_contract);
1319 /* Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL values
1320 * Result is stored in r (r can equal one of the inputs). */
1321 int ec_GFp_nistp224_points_mul(const EC_GROUP *group, EC_POINT *r,
1322 const BIGNUM *scalar, size_t num, const EC_POINT *points[],
1323 const BIGNUM *scalars[], BN_CTX *ctx)
1329 BN_CTX *new_ctx = NULL;
1330 BIGNUM *x, *y, *z, *tmp_scalar;
1331 felem_bytearray g_secret;
1332 felem_bytearray *secrets = NULL;
1333 felem (*pre_comp)[17][3] = NULL;
1334 felem *tmp_felems = NULL;
1335 felem_bytearray tmp;
1337 int have_pre_comp = 0;
1338 size_t num_points = num;
1339 felem x_in, y_in, z_in, x_out, y_out, z_out;
1340 NISTP224_PRE_COMP *pre = NULL;
1341 const felem (*g_pre_comp)[16][3] = NULL;
1342 EC_POINT *generator = NULL;
1343 const EC_POINT *p = NULL;
1344 const BIGNUM *p_scalar = NULL;
1347 if ((ctx = new_ctx = BN_CTX_new()) == NULL) return 0;
1349 if (((x = BN_CTX_get(ctx)) == NULL) ||
1350 ((y = BN_CTX_get(ctx)) == NULL) ||
1351 ((z = BN_CTX_get(ctx)) == NULL) ||
1352 ((tmp_scalar = BN_CTX_get(ctx)) == NULL))
1357 pre = EC_EX_DATA_get_data(group->extra_data,
1358 nistp224_pre_comp_dup, nistp224_pre_comp_free,
1359 nistp224_pre_comp_clear_free);
1361 /* we have precomputation, try to use it */
1362 g_pre_comp = (const felem (*)[16][3]) pre->g_pre_comp;
1364 /* try to use the standard precomputation */
1365 g_pre_comp = &gmul[0];
1366 generator = EC_POINT_new(group);
1367 if (generator == NULL)
1369 /* get the generator from precomputation */
1370 if (!felem_to_BN(x, g_pre_comp[0][1][0]) ||
1371 !felem_to_BN(y, g_pre_comp[0][1][1]) ||
1372 !felem_to_BN(z, g_pre_comp[0][1][2]))
1374 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1377 if (!EC_POINT_set_Jprojective_coordinates_GFp(group,
1378 generator, x, y, z, ctx))
1380 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
1381 /* precomputation matches generator */
1384 /* we don't have valid precomputation:
1385 * treat the generator as a random point */
1386 num_points = num_points + 1;
1391 if (num_points >= 3)
1393 /* unless we precompute multiples for just one or two points,
1394 * converting those into affine form is time well spent */
1397 secrets = OPENSSL_malloc(num_points * sizeof(felem_bytearray));
1398 pre_comp = OPENSSL_malloc(num_points * 17 * 3 * sizeof(felem));
1400 tmp_felems = OPENSSL_malloc((num_points * 17 + 1) * sizeof(felem));
1401 if ((secrets == NULL) || (pre_comp == NULL) || (mixed && (tmp_felems == NULL)))
1403 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_MALLOC_FAILURE);
1407 /* we treat NULL scalars as 0, and NULL points as points at infinity,
1408 * i.e., they contribute nothing to the linear combination */
1409 memset(secrets, 0, num_points * sizeof(felem_bytearray));
1410 memset(pre_comp, 0, num_points * 17 * 3 * sizeof(felem));
1411 for (i = 0; i < num_points; ++i)
1416 p = EC_GROUP_get0_generator(group);
1420 /* the i^th point */
1423 p_scalar = scalars[i];
1425 if ((p_scalar != NULL) && (p != NULL))
1427 /* reduce scalar to 0 <= scalar < 2^224 */
1428 if ((BN_num_bits(p_scalar) > 224) || (BN_is_negative(p_scalar)))
1430 /* this is an unusual input, and we don't guarantee
1431 * constant-timeness */
1432 if (!BN_nnmod(tmp_scalar, p_scalar, &group->order, ctx))
1434 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1437 num_bytes = BN_bn2bin(tmp_scalar, tmp);
1440 num_bytes = BN_bn2bin(p_scalar, tmp);
1441 flip_endian(secrets[i], tmp, num_bytes);
1442 /* precompute multiples */
1443 if ((!BN_to_felem(x_out, &p->X)) ||
1444 (!BN_to_felem(y_out, &p->Y)) ||
1445 (!BN_to_felem(z_out, &p->Z))) goto err;
1446 felem_assign(pre_comp[i][1][0], x_out);
1447 felem_assign(pre_comp[i][1][1], y_out);
1448 felem_assign(pre_comp[i][1][2], z_out);
1449 for (j = 2; j <= 16; ++j)
1454 pre_comp[i][j][0], pre_comp[i][j][1], pre_comp[i][j][2],
1455 pre_comp[i][1][0], pre_comp[i][1][1], pre_comp[i][1][2],
1456 0, pre_comp[i][j-1][0], pre_comp[i][j-1][1], pre_comp[i][j-1][2]);
1461 pre_comp[i][j][0], pre_comp[i][j][1], pre_comp[i][j][2],
1462 pre_comp[i][j/2][0], pre_comp[i][j/2][1], pre_comp[i][j/2][2]);
1468 make_points_affine(num_points * 17, pre_comp[0], tmp_felems);
1471 /* the scalar for the generator */
1472 if ((scalar != NULL) && (have_pre_comp))
1474 memset(g_secret, 0, sizeof g_secret);
1475 /* reduce scalar to 0 <= scalar < 2^224 */
1476 if ((BN_num_bits(scalar) > 224) || (BN_is_negative(scalar)))
1478 /* this is an unusual input, and we don't guarantee
1479 * constant-timeness */
1480 if (!BN_nnmod(tmp_scalar, scalar, &group->order, ctx))
1482 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1485 num_bytes = BN_bn2bin(tmp_scalar, tmp);
1488 num_bytes = BN_bn2bin(scalar, tmp);
1489 flip_endian(g_secret, tmp, num_bytes);
1490 /* do the multiplication with generator precomputation*/
1491 batch_mul(x_out, y_out, z_out,
1492 (const felem_bytearray (*)) secrets, num_points,
1494 mixed, (const felem (*)[17][3]) pre_comp,
1498 /* do the multiplication without generator precomputation */
1499 batch_mul(x_out, y_out, z_out,
1500 (const felem_bytearray (*)) secrets, num_points,
1501 NULL, mixed, (const felem (*)[17][3]) pre_comp, NULL);
1502 /* reduce the output to its unique minimal representation */
1503 felem_contract(x_in, x_out);
1504 felem_contract(y_in, y_out);
1505 felem_contract(z_in, z_out);
1506 if ((!felem_to_BN(x, x_in)) || (!felem_to_BN(y, y_in)) ||
1507 (!felem_to_BN(z, z_in)))
1509 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1512 ret = EC_POINT_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx);
1516 if (generator != NULL)
1517 EC_POINT_free(generator);
1518 if (new_ctx != NULL)
1519 BN_CTX_free(new_ctx);
1520 if (secrets != NULL)
1521 OPENSSL_free(secrets);
1522 if (pre_comp != NULL)
1523 OPENSSL_free(pre_comp);
1524 if (tmp_felems != NULL)
1525 OPENSSL_free(tmp_felems);
1529 int ec_GFp_nistp224_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
1532 NISTP224_PRE_COMP *pre = NULL;
1534 BN_CTX *new_ctx = NULL;
1536 EC_POINT *generator = NULL;
1537 felem tmp_felems[32];
1539 /* throw away old precomputation */
1540 EC_EX_DATA_free_data(&group->extra_data, nistp224_pre_comp_dup,
1541 nistp224_pre_comp_free, nistp224_pre_comp_clear_free);
1543 if ((ctx = new_ctx = BN_CTX_new()) == NULL) return 0;
1545 if (((x = BN_CTX_get(ctx)) == NULL) ||
1546 ((y = BN_CTX_get(ctx)) == NULL))
1548 /* get the generator */
1549 if (group->generator == NULL) goto err;
1550 generator = EC_POINT_new(group);
1551 if (generator == NULL)
1553 BN_bin2bn(nistp224_curve_params[3], sizeof (felem_bytearray), x);
1554 BN_bin2bn(nistp224_curve_params[4], sizeof (felem_bytearray), y);
1555 if (!EC_POINT_set_affine_coordinates_GFp(group, generator, x, y, ctx))
1557 if ((pre = nistp224_pre_comp_new()) == NULL)
1559 /* if the generator is the standard one, use built-in precomputation */
1560 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
1562 memcpy(pre->g_pre_comp, gmul, sizeof(pre->g_pre_comp));
1566 if ((!BN_to_felem(pre->g_pre_comp[0][1][0], &group->generator->X)) ||
1567 (!BN_to_felem(pre->g_pre_comp[0][1][1], &group->generator->Y)) ||
1568 (!BN_to_felem(pre->g_pre_comp[0][1][2], &group->generator->Z)))
1570 /* compute 2^56*G, 2^112*G, 2^168*G for the first table,
1571 * 2^28*G, 2^84*G, 2^140*G, 2^196*G for the second one
1573 for (i = 1; i <= 8; i <<= 1)
1576 pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2],
1577 pre->g_pre_comp[0][i][0], pre->g_pre_comp[0][i][1], pre->g_pre_comp[0][i][2]);
1578 for (j = 0; j < 27; ++j)
1581 pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2],
1582 pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][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 pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2]);
1589 for (j = 0; j < 27; ++j)
1592 pre->g_pre_comp[0][2*i][0], pre->g_pre_comp[0][2*i][1], pre->g_pre_comp[0][2*i][2],
1593 pre->g_pre_comp[0][2*i][0], pre->g_pre_comp[0][2*i][1], pre->g_pre_comp[0][2*i][2]);
1596 for (i = 0; i < 2; i++)
1598 /* g_pre_comp[i][0] is the point at infinity */
1599 memset(pre->g_pre_comp[i][0], 0, sizeof(pre->g_pre_comp[i][0]));
1600 /* the remaining multiples */
1601 /* 2^56*G + 2^112*G resp. 2^84*G + 2^140*G */
1603 pre->g_pre_comp[i][6][0], pre->g_pre_comp[i][6][1],
1604 pre->g_pre_comp[i][6][2], pre->g_pre_comp[i][4][0],
1605 pre->g_pre_comp[i][4][1], pre->g_pre_comp[i][4][2],
1606 0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
1607 pre->g_pre_comp[i][2][2]);
1608 /* 2^56*G + 2^168*G resp. 2^84*G + 2^196*G */
1610 pre->g_pre_comp[i][10][0], pre->g_pre_comp[i][10][1],
1611 pre->g_pre_comp[i][10][2], pre->g_pre_comp[i][8][0],
1612 pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2],
1613 0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
1614 pre->g_pre_comp[i][2][2]);
1615 /* 2^112*G + 2^168*G resp. 2^140*G + 2^196*G */
1617 pre->g_pre_comp[i][12][0], pre->g_pre_comp[i][12][1],
1618 pre->g_pre_comp[i][12][2], pre->g_pre_comp[i][8][0],
1619 pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2],
1620 0, pre->g_pre_comp[i][4][0], pre->g_pre_comp[i][4][1],
1621 pre->g_pre_comp[i][4][2]);
1622 /* 2^56*G + 2^112*G + 2^168*G resp. 2^84*G + 2^140*G + 2^196*G */
1624 pre->g_pre_comp[i][14][0], pre->g_pre_comp[i][14][1],
1625 pre->g_pre_comp[i][14][2], pre->g_pre_comp[i][12][0],
1626 pre->g_pre_comp[i][12][1], pre->g_pre_comp[i][12][2],
1627 0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
1628 pre->g_pre_comp[i][2][2]);
1629 for (j = 1; j < 8; ++j)
1631 /* odd multiples: add G resp. 2^28*G */
1633 pre->g_pre_comp[i][2*j+1][0], pre->g_pre_comp[i][2*j+1][1],
1634 pre->g_pre_comp[i][2*j+1][2], pre->g_pre_comp[i][2*j][0],
1635 pre->g_pre_comp[i][2*j][1], pre->g_pre_comp[i][2*j][2],
1636 0, pre->g_pre_comp[i][1][0], pre->g_pre_comp[i][1][1],
1637 pre->g_pre_comp[i][1][2]);
1640 make_points_affine(31, &(pre->g_pre_comp[0][1]), tmp_felems);
1642 if (!EC_EX_DATA_set_data(&group->extra_data, pre, nistp224_pre_comp_dup,
1643 nistp224_pre_comp_free, nistp224_pre_comp_clear_free))
1649 if (generator != NULL)
1650 EC_POINT_free(generator);
1651 if (new_ctx != NULL)
1652 BN_CTX_free(new_ctx);
1654 nistp224_pre_comp_free(pre);
1658 int ec_GFp_nistp224_have_precompute_mult(const EC_GROUP *group)
1660 if (EC_EX_DATA_get_data(group->extra_data, nistp224_pre_comp_dup,
1661 nistp224_pre_comp_free, nistp224_pre_comp_clear_free)
1669 static void *dummy=&dummy;