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
41 # 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];
75 * Field element represented as a byte arrary. 28*8 = 224 bits is also the
76 * group order size for the elliptic curve, and we also use this type for
77 * scalars for point multiplication.
79 typedef u8 felem_bytearray[28];
81 static const felem_bytearray nistp224_curve_params[5] = {
82 {0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, /* p */
83 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0x00, 0x00, 0x00, 0x00,
84 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x01},
85 {0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, /* a */
86 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFE, 0xFF, 0xFF, 0xFF, 0xFF,
87 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFE},
88 {0xB4, 0x05, 0x0A, 0x85, 0x0C, 0x04, 0xB3, 0xAB, 0xF5, 0x41, /* b */
89 0x32, 0x56, 0x50, 0x44, 0xB0, 0xB7, 0xD7, 0xBF, 0xD8, 0xBA,
90 0x27, 0x0B, 0x39, 0x43, 0x23, 0x55, 0xFF, 0xB4},
91 {0xB7, 0x0E, 0x0C, 0xBD, 0x6B, 0xB4, 0xBF, 0x7F, 0x32, 0x13, /* x */
92 0x90, 0xB9, 0x4A, 0x03, 0xC1, 0xD3, 0x56, 0xC2, 0x11, 0x22,
93 0x34, 0x32, 0x80, 0xD6, 0x11, 0x5C, 0x1D, 0x21},
94 {0xbd, 0x37, 0x63, 0x88, 0xb5, 0xf7, 0x23, 0xfb, 0x4c, 0x22, /* y */
95 0xdf, 0xe6, 0xcd, 0x43, 0x75, 0xa0, 0x5a, 0x07, 0x47, 0x64,
96 0x44, 0xd5, 0x81, 0x99, 0x85, 0x00, 0x7e, 0x34}
100 * Precomputed multiples of the standard generator
101 * Points are given in coordinates (X, Y, Z) where Z normally is 1
102 * (0 for the point at infinity).
103 * For each field element, slice a_0 is word 0, etc.
105 * The table has 2 * 16 elements, starting with the following:
106 * index | bits | point
107 * ------+---------+------------------------------
110 * 2 | 0 0 1 0 | 2^56G
111 * 3 | 0 0 1 1 | (2^56 + 1)G
112 * 4 | 0 1 0 0 | 2^112G
113 * 5 | 0 1 0 1 | (2^112 + 1)G
114 * 6 | 0 1 1 0 | (2^112 + 2^56)G
115 * 7 | 0 1 1 1 | (2^112 + 2^56 + 1)G
116 * 8 | 1 0 0 0 | 2^168G
117 * 9 | 1 0 0 1 | (2^168 + 1)G
118 * 10 | 1 0 1 0 | (2^168 + 2^56)G
119 * 11 | 1 0 1 1 | (2^168 + 2^56 + 1)G
120 * 12 | 1 1 0 0 | (2^168 + 2^112)G
121 * 13 | 1 1 0 1 | (2^168 + 2^112 + 1)G
122 * 14 | 1 1 1 0 | (2^168 + 2^112 + 2^56)G
123 * 15 | 1 1 1 1 | (2^168 + 2^112 + 2^56 + 1)G
124 * followed by a copy of this with each element multiplied by 2^28.
126 * The reason for this is so that we can clock bits into four different
127 * locations when doing simple scalar multiplies against the base point,
128 * and then another four locations using the second 16 elements.
130 static const felem gmul[2][16][3] = {
134 {{0x3280d6115c1d21, 0xc1d356c2112234, 0x7f321390b94a03, 0xb70e0cbd6bb4bf},
135 {0xd5819985007e34, 0x75a05a07476444, 0xfb4c22dfe6cd43, 0xbd376388b5f723},
137 {{0xfd9675666ebbe9, 0xbca7664d40ce5e, 0x2242df8d8a2a43, 0x1f49bbb0f99bc5},
138 {0x29e0b892dc9c43, 0xece8608436e662, 0xdc858f185310d0, 0x9812dd4eb8d321},
140 {{0x6d3e678d5d8eb8, 0x559eed1cb362f1, 0x16e9a3bbce8a3f, 0xeedcccd8c2a748},
141 {0xf19f90ed50266d, 0xabf2b4bf65f9df, 0x313865468fafec, 0x5cb379ba910a17},
143 {{0x0641966cab26e3, 0x91fb2991fab0a0, 0xefec27a4e13a0b, 0x0499aa8a5f8ebe},
144 {0x7510407766af5d, 0x84d929610d5450, 0x81d77aae82f706, 0x6916f6d4338c5b},
146 {{0xea95ac3b1f15c6, 0x086000905e82d4, 0xdd323ae4d1c8b1, 0x932b56be7685a3},
147 {0x9ef93dea25dbbf, 0x41665960f390f0, 0xfdec76dbe2a8a7, 0x523e80f019062a},
149 {{0x822fdd26732c73, 0xa01c83531b5d0f, 0x363f37347c1ba4, 0xc391b45c84725c},
150 {0xbbd5e1b2d6ad24, 0xddfbcde19dfaec, 0xc393da7e222a7f, 0x1efb7890ede244},
152 {{0x4c9e90ca217da1, 0xd11beca79159bb, 0xff8d33c2c98b7c, 0x2610b39409f849},
153 {0x44d1352ac64da0, 0xcdbb7b2c46b4fb, 0x966c079b753c89, 0xfe67e4e820b112},
155 {{0xe28cae2df5312d, 0xc71b61d16f5c6e, 0x79b7619a3e7c4c, 0x05c73240899b47},
156 {0x9f7f6382c73e3a, 0x18615165c56bda, 0x641fab2116fd56, 0x72855882b08394},
158 {{0x0469182f161c09, 0x74a98ca8d00fb5, 0xb89da93489a3e0, 0x41c98768fb0c1d},
159 {0xe5ea05fb32da81, 0x3dce9ffbca6855, 0x1cfe2d3fbf59e6, 0x0e5e03408738a7},
161 {{0xdab22b2333e87f, 0x4430137a5dd2f6, 0xe03ab9f738beb8, 0xcb0c5d0dc34f24},
162 {0x764a7df0c8fda5, 0x185ba5c3fa2044, 0x9281d688bcbe50, 0xc40331df893881},
164 {{0xb89530796f0f60, 0xade92bd26909a3, 0x1a0c83fb4884da, 0x1765bf22a5a984},
165 {0x772a9ee75db09e, 0x23bc6c67cec16f, 0x4c1edba8b14e2f, 0xe2a215d9611369},
167 {{0x571e509fb5efb3, 0xade88696410552, 0xc8ae85fada74fe, 0x6c7e4be83bbde3},
168 {0xff9f51160f4652, 0xb47ce2495a6539, 0xa2946c53b582f4, 0x286d2db3ee9a60},
170 {{0x40bbd5081a44af, 0x0995183b13926c, 0xbcefba6f47f6d0, 0x215619e9cc0057},
171 {0x8bc94d3b0df45e, 0xf11c54a3694f6f, 0x8631b93cdfe8b5, 0xe7e3f4b0982db9},
173 {{0xb17048ab3e1c7b, 0xac38f36ff8a1d8, 0x1c29819435d2c6, 0xc813132f4c07e9},
174 {0x2891425503b11f, 0x08781030579fea, 0xf5426ba5cc9674, 0x1e28ebf18562bc},
176 {{0x9f31997cc864eb, 0x06cd91d28b5e4c, 0xff17036691a973, 0xf1aef351497c58},
177 {0xdd1f2d600564ff, 0xdead073b1402db, 0x74a684435bd693, 0xeea7471f962558},
182 {{0x9665266dddf554, 0x9613d78b60ef2d, 0xce27a34cdba417, 0xd35ab74d6afc31},
183 {0x85ccdd22deb15e, 0x2137e5783a6aab, 0xa141cffd8c93c6, 0x355a1830e90f2d},
185 {{0x1a494eadaade65, 0xd6da4da77fe53c, 0xe7992996abec86, 0x65c3553c6090e3},
186 {0xfa610b1fb09346, 0xf1c6540b8a4aaf, 0xc51a13ccd3cbab, 0x02995b1b18c28a},
188 {{0x7874568e7295ef, 0x86b419fbe38d04, 0xdc0690a7550d9a, 0xd3966a44beac33},
189 {0x2b7280ec29132f, 0xbeaa3b6a032df3, 0xdc7dd88ae41200, 0xd25e2513e3a100},
191 {{0x924857eb2efafd, 0xac2bce41223190, 0x8edaa1445553fc, 0x825800fd3562d5},
192 {0x8d79148ea96621, 0x23a01c3dd9ed8d, 0xaf8b219f9416b5, 0xd8db0cc277daea},
194 {{0x76a9c3b1a700f0, 0xe9acd29bc7e691, 0x69212d1a6b0327, 0x6322e97fe154be},
195 {0x469fc5465d62aa, 0x8d41ed18883b05, 0x1f8eae66c52b88, 0xe4fcbe9325be51},
197 {{0x825fdf583cac16, 0x020b857c7b023a, 0x683c17744b0165, 0x14ffd0a2daf2f1},
198 {0x323b36184218f9, 0x4944ec4e3b47d4, 0xc15b3080841acf, 0x0bced4b01a28bb},
200 {{0x92ac22230df5c4, 0x52f33b4063eda8, 0xcb3f19870c0c93, 0x40064f2ba65233},
201 {0xfe16f0924f8992, 0x012da25af5b517, 0x1a57bb24f723a6, 0x06f8bc76760def},
203 {{0x4a7084f7817cb9, 0xbcab0738ee9a78, 0x3ec11e11d9c326, 0xdc0fe90e0f1aae},
204 {0xcf639ea5f98390, 0x5c350aa22ffb74, 0x9afae98a4047b7, 0x956ec2d617fc45},
206 {{0x4306d648c1be6a, 0x9247cd8bc9a462, 0xf5595e377d2f2e, 0xbd1c3caff1a52e},
207 {0x045e14472409d0, 0x29f3e17078f773, 0x745a602b2d4f7d, 0x191837685cdfbb},
209 {{0x5b6ee254a8cb79, 0x4953433f5e7026, 0xe21faeb1d1def4, 0xc4c225785c09de},
210 {0x307ce7bba1e518, 0x31b125b1036db8, 0x47e91868839e8f, 0xc765866e33b9f3},
212 {{0x3bfece24f96906, 0x4794da641e5093, 0xde5df64f95db26, 0x297ecd89714b05},
213 {0x701bd3ebb2c3aa, 0x7073b4f53cb1d5, 0x13c5665658af16, 0x9895089d66fe58},
215 {{0x0fef05f78c4790, 0x2d773633b05d2e, 0x94229c3a951c94, 0xbbbd70df4911bb},
216 {0xb2c6963d2c1168, 0x105f47a72b0d73, 0x9fdf6111614080, 0x7b7e94b39e67b0},
218 {{0xad1a7d6efbe2b3, 0xf012482c0da69d, 0x6b3bdf12438345, 0x40d7558d7aa4d9},
219 {0x8a09fffb5c6d3d, 0x9a356e5d9ffd38, 0x5973f15f4f9b1c, 0xdcd5f59f63c3ea},
221 {{0xacf39f4c5ca7ab, 0x4c8071cc5fd737, 0xc64e3602cd1184, 0x0acd4644c9abba},
222 {0x6c011a36d8bf6e, 0xfecd87ba24e32a, 0x19f6f56574fad8, 0x050b204ced9405},
224 {{0xed4f1cae7d9a96, 0x5ceef7ad94c40a, 0x778e4a3bf3ef9b, 0x7405783dc3b55e},
225 {0x32477c61b6e8c6, 0xb46a97570f018b, 0x91176d0a7e95d1, 0x3df90fbc4c7d0e},
229 /* Precomputation for the group generator. */
230 struct nistp224_pre_comp_st {
231 felem g_pre_comp[2][16][3];
235 const EC_METHOD *EC_GFp_nistp224_method(void)
237 static const EC_METHOD ret = {
238 EC_FLAGS_DEFAULT_OCT,
239 NID_X9_62_prime_field,
240 ec_GFp_nistp224_group_init,
241 ec_GFp_simple_group_finish,
242 ec_GFp_simple_group_clear_finish,
243 ec_GFp_nist_group_copy,
244 ec_GFp_nistp224_group_set_curve,
245 ec_GFp_simple_group_get_curve,
246 ec_GFp_simple_group_get_degree,
247 ec_GFp_simple_group_check_discriminant,
248 ec_GFp_simple_point_init,
249 ec_GFp_simple_point_finish,
250 ec_GFp_simple_point_clear_finish,
251 ec_GFp_simple_point_copy,
252 ec_GFp_simple_point_set_to_infinity,
253 ec_GFp_simple_set_Jprojective_coordinates_GFp,
254 ec_GFp_simple_get_Jprojective_coordinates_GFp,
255 ec_GFp_simple_point_set_affine_coordinates,
256 ec_GFp_nistp224_point_get_affine_coordinates,
257 0 /* point_set_compressed_coordinates */ ,
262 ec_GFp_simple_invert,
263 ec_GFp_simple_is_at_infinity,
264 ec_GFp_simple_is_on_curve,
266 ec_GFp_simple_make_affine,
267 ec_GFp_simple_points_make_affine,
268 ec_GFp_nistp224_points_mul,
269 ec_GFp_nistp224_precompute_mult,
270 ec_GFp_nistp224_have_precompute_mult,
271 ec_GFp_nist_field_mul,
272 ec_GFp_nist_field_sqr,
274 0 /* field_encode */ ,
275 0 /* field_decode */ ,
276 0 /* field_set_to_one */
283 * Helper functions to convert field elements to/from internal representation
285 static void bin28_to_felem(felem out, const u8 in[28])
287 out[0] = *((const uint64_t *)(in)) & 0x00ffffffffffffff;
288 out[1] = (*((const uint64_t *)(in + 7))) & 0x00ffffffffffffff;
289 out[2] = (*((const uint64_t *)(in + 14))) & 0x00ffffffffffffff;
290 out[3] = (*((const uint64_t *)(in+20))) >> 8;
293 static void felem_to_bin28(u8 out[28], const felem in)
296 for (i = 0; i < 7; ++i) {
297 out[i] = in[0] >> (8 * i);
298 out[i + 7] = in[1] >> (8 * i);
299 out[i + 14] = in[2] >> (8 * i);
300 out[i + 21] = in[3] >> (8 * i);
304 /* To preserve endianness when using BN_bn2bin and BN_bin2bn */
305 static void flip_endian(u8 *out, const u8 *in, unsigned len)
308 for (i = 0; i < len; ++i)
309 out[i] = in[len - 1 - i];
312 /* From OpenSSL BIGNUM to internal representation */
313 static int BN_to_felem(felem out, const BIGNUM *bn)
315 felem_bytearray b_in;
316 felem_bytearray b_out;
319 /* BN_bn2bin eats leading zeroes */
320 memset(b_out, 0, sizeof(b_out));
321 num_bytes = BN_num_bytes(bn);
322 if (num_bytes > sizeof b_out) {
323 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
326 if (BN_is_negative(bn)) {
327 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
330 num_bytes = BN_bn2bin(bn, b_in);
331 flip_endian(b_out, b_in, num_bytes);
332 bin28_to_felem(out, b_out);
336 /* From internal representation to OpenSSL BIGNUM */
337 static BIGNUM *felem_to_BN(BIGNUM *out, const felem in)
339 felem_bytearray b_in, b_out;
340 felem_to_bin28(b_in, in);
341 flip_endian(b_out, b_in, sizeof b_out);
342 return BN_bin2bn(b_out, sizeof b_out, out);
345 /******************************************************************************/
349 * Field operations, using the internal representation of field elements.
350 * NB! These operations are specific to our point multiplication and cannot be
351 * expected to be correct in general - e.g., multiplication with a large scalar
352 * will cause an overflow.
356 static void felem_one(felem out)
364 static void felem_assign(felem out, const felem in)
372 /* Sum two field elements: out += in */
373 static void felem_sum(felem out, const felem in)
381 /* Get negative value: out = -in */
382 /* Assumes in[i] < 2^57 */
383 static void felem_neg(felem out, const felem in)
385 static const limb two58p2 = (((limb) 1) << 58) + (((limb) 1) << 2);
386 static const limb two58m2 = (((limb) 1) << 58) - (((limb) 1) << 2);
387 static const limb two58m42m2 = (((limb) 1) << 58) -
388 (((limb) 1) << 42) - (((limb) 1) << 2);
390 /* Set to 0 mod 2^224-2^96+1 to ensure out > in */
391 out[0] = two58p2 - in[0];
392 out[1] = two58m42m2 - in[1];
393 out[2] = two58m2 - in[2];
394 out[3] = two58m2 - in[3];
397 /* Subtract field elements: out -= in */
398 /* Assumes in[i] < 2^57 */
399 static void felem_diff(felem out, const felem in)
401 static const limb two58p2 = (((limb) 1) << 58) + (((limb) 1) << 2);
402 static const limb two58m2 = (((limb) 1) << 58) - (((limb) 1) << 2);
403 static const limb two58m42m2 = (((limb) 1) << 58) -
404 (((limb) 1) << 42) - (((limb) 1) << 2);
406 /* Add 0 mod 2^224-2^96+1 to ensure out > in */
408 out[1] += two58m42m2;
418 /* Subtract in unreduced 128-bit mode: out -= in */
419 /* Assumes in[i] < 2^119 */
420 static void widefelem_diff(widefelem out, const widefelem in)
422 static const widelimb two120 = ((widelimb) 1) << 120;
423 static const widelimb two120m64 = (((widelimb) 1) << 120) -
424 (((widelimb) 1) << 64);
425 static const widelimb two120m104m64 = (((widelimb) 1) << 120) -
426 (((widelimb) 1) << 104) - (((widelimb) 1) << 64);
428 /* Add 0 mod 2^224-2^96+1 to ensure out > in */
433 out[4] += two120m104m64;
446 /* Subtract in mixed mode: out128 -= in64 */
448 static void felem_diff_128_64(widefelem out, const felem in)
450 static const widelimb two64p8 = (((widelimb) 1) << 64) +
451 (((widelimb) 1) << 8);
452 static const widelimb two64m8 = (((widelimb) 1) << 64) -
453 (((widelimb) 1) << 8);
454 static const widelimb two64m48m8 = (((widelimb) 1) << 64) -
455 (((widelimb) 1) << 48) - (((widelimb) 1) << 8);
457 /* Add 0 mod 2^224-2^96+1 to ensure out > in */
459 out[1] += two64m48m8;
470 * Multiply a field element by a scalar: out = out * scalar The scalars we
471 * actually use are small, so results fit without overflow
473 static void felem_scalar(felem out, const limb scalar)
482 * Multiply an unreduced field element by a scalar: out = out * scalar The
483 * scalars we actually use are small, so results fit without overflow
485 static void widefelem_scalar(widefelem out, const widelimb scalar)
496 /* Square a field element: out = in^2 */
497 static void felem_square(widefelem out, const felem in)
499 limb tmp0, tmp1, tmp2;
503 out[0] = ((widelimb) in[0]) * in[0];
504 out[1] = ((widelimb) in[0]) * tmp1;
505 out[2] = ((widelimb) in[0]) * tmp2 + ((widelimb) in[1]) * in[1];
506 out[3] = ((widelimb) in[3]) * tmp0 + ((widelimb) in[1]) * tmp2;
507 out[4] = ((widelimb) in[3]) * tmp1 + ((widelimb) in[2]) * in[2];
508 out[5] = ((widelimb) in[3]) * tmp2;
509 out[6] = ((widelimb) in[3]) * in[3];
512 /* Multiply two field elements: out = in1 * in2 */
513 static void felem_mul(widefelem out, const felem in1, const felem in2)
515 out[0] = ((widelimb) in1[0]) * in2[0];
516 out[1] = ((widelimb) in1[0]) * in2[1] + ((widelimb) in1[1]) * in2[0];
517 out[2] = ((widelimb) in1[0]) * in2[2] + ((widelimb) in1[1]) * in2[1] +
518 ((widelimb) in1[2]) * in2[0];
519 out[3] = ((widelimb) in1[0]) * in2[3] + ((widelimb) in1[1]) * in2[2] +
520 ((widelimb) in1[2]) * in2[1] + ((widelimb) in1[3]) * in2[0];
521 out[4] = ((widelimb) in1[1]) * in2[3] + ((widelimb) in1[2]) * in2[2] +
522 ((widelimb) in1[3]) * in2[1];
523 out[5] = ((widelimb) in1[2]) * in2[3] + ((widelimb) in1[3]) * in2[2];
524 out[6] = ((widelimb) in1[3]) * in2[3];
528 * Reduce seven 128-bit coefficients to four 64-bit coefficients.
529 * Requires in[i] < 2^126,
530 * ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] <= 2^56 + 2^16 */
531 static void felem_reduce(felem out, const widefelem in)
533 static const widelimb two127p15 = (((widelimb) 1) << 127) +
534 (((widelimb) 1) << 15);
535 static const widelimb two127m71 = (((widelimb) 1) << 127) -
536 (((widelimb) 1) << 71);
537 static const widelimb two127m71m55 = (((widelimb) 1) << 127) -
538 (((widelimb) 1) << 71) - (((widelimb) 1) << 55);
541 /* Add 0 mod 2^224-2^96+1 to ensure all differences are positive */
542 output[0] = in[0] + two127p15;
543 output[1] = in[1] + two127m71m55;
544 output[2] = in[2] + two127m71;
548 /* Eliminate in[4], in[5], in[6] */
549 output[4] += in[6] >> 16;
550 output[3] += (in[6] & 0xffff) << 40;
553 output[3] += in[5] >> 16;
554 output[2] += (in[5] & 0xffff) << 40;
557 output[2] += output[4] >> 16;
558 output[1] += (output[4] & 0xffff) << 40;
559 output[0] -= output[4];
561 /* Carry 2 -> 3 -> 4 */
562 output[3] += output[2] >> 56;
563 output[2] &= 0x00ffffffffffffff;
565 output[4] = output[3] >> 56;
566 output[3] &= 0x00ffffffffffffff;
568 /* Now output[2] < 2^56, output[3] < 2^56, output[4] < 2^72 */
570 /* Eliminate output[4] */
571 output[2] += output[4] >> 16;
572 /* output[2] < 2^56 + 2^56 = 2^57 */
573 output[1] += (output[4] & 0xffff) << 40;
574 output[0] -= output[4];
576 /* Carry 0 -> 1 -> 2 -> 3 */
577 output[1] += output[0] >> 56;
578 out[0] = output[0] & 0x00ffffffffffffff;
580 output[2] += output[1] >> 56;
581 /* output[2] < 2^57 + 2^72 */
582 out[1] = output[1] & 0x00ffffffffffffff;
583 output[3] += output[2] >> 56;
584 /* output[3] <= 2^56 + 2^16 */
585 out[2] = output[2] & 0x00ffffffffffffff;
588 * out[0] < 2^56, out[1] < 2^56, out[2] < 2^56,
589 * out[3] <= 2^56 + 2^16 (due to final carry),
595 static void felem_square_reduce(felem out, const felem in)
598 felem_square(tmp, in);
599 felem_reduce(out, tmp);
602 static void felem_mul_reduce(felem out, const felem in1, const felem in2)
605 felem_mul(tmp, in1, in2);
606 felem_reduce(out, tmp);
610 * Reduce to unique minimal representation. Requires 0 <= in < 2*p (always
611 * call felem_reduce first)
613 static void felem_contract(felem out, const felem in)
615 static const int64_t two56 = ((limb) 1) << 56;
616 /* 0 <= in < 2*p, p = 2^224 - 2^96 + 1 */
617 /* if in > p , reduce in = in - 2^224 + 2^96 - 1 */
623 /* Case 1: a = 1 iff in >= 2^224 */
627 tmp[3] &= 0x00ffffffffffffff;
629 * Case 2: a = 0 iff p <= in < 2^224, i.e., the high 128 bits are all 1
630 * and the lower part is non-zero
632 a = ((in[3] & in[2] & (in[1] | 0x000000ffffffffff)) + 1) |
633 (((int64_t) (in[0] + (in[1] & 0x000000ffffffffff)) - 1) >> 63);
634 a &= 0x00ffffffffffffff;
635 /* turn a into an all-one mask (if a = 0) or an all-zero mask */
637 /* subtract 2^224 - 2^96 + 1 if a is all-one */
638 tmp[3] &= a ^ 0xffffffffffffffff;
639 tmp[2] &= a ^ 0xffffffffffffffff;
640 tmp[1] &= (a ^ 0xffffffffffffffff) | 0x000000ffffffffff;
644 * eliminate negative coefficients: if tmp[0] is negative, tmp[1] must be
645 * non-zero, so we only need one step
651 /* carry 1 -> 2 -> 3 */
652 tmp[2] += tmp[1] >> 56;
653 tmp[1] &= 0x00ffffffffffffff;
655 tmp[3] += tmp[2] >> 56;
656 tmp[2] &= 0x00ffffffffffffff;
658 /* Now 0 <= out < p */
666 * Zero-check: returns 1 if input is 0, and 0 otherwise. We know that field
667 * elements are reduced to in < 2^225, so we only need to check three cases:
668 * 0, 2^224 - 2^96 + 1, and 2^225 - 2^97 + 2
670 static limb felem_is_zero(const felem in)
672 limb zero, two224m96p1, two225m97p2;
674 zero = in[0] | in[1] | in[2] | in[3];
675 zero = (((int64_t) (zero) - 1) >> 63) & 1;
676 two224m96p1 = (in[0] ^ 1) | (in[1] ^ 0x00ffff0000000000)
677 | (in[2] ^ 0x00ffffffffffffff) | (in[3] ^ 0x00ffffffffffffff);
678 two224m96p1 = (((int64_t) (two224m96p1) - 1) >> 63) & 1;
679 two225m97p2 = (in[0] ^ 2) | (in[1] ^ 0x00fffe0000000000)
680 | (in[2] ^ 0x00ffffffffffffff) | (in[3] ^ 0x01ffffffffffffff);
681 two225m97p2 = (((int64_t) (two225m97p2) - 1) >> 63) & 1;
682 return (zero | two224m96p1 | two225m97p2);
685 static limb felem_is_zero_int(const felem in)
687 return (int)(felem_is_zero(in) & ((limb) 1));
690 /* Invert a field element */
691 /* Computation chain copied from djb's code */
692 static void felem_inv(felem out, const felem in)
694 felem ftmp, ftmp2, ftmp3, ftmp4;
698 felem_square(tmp, in);
699 felem_reduce(ftmp, tmp); /* 2 */
700 felem_mul(tmp, in, ftmp);
701 felem_reduce(ftmp, tmp); /* 2^2 - 1 */
702 felem_square(tmp, ftmp);
703 felem_reduce(ftmp, tmp); /* 2^3 - 2 */
704 felem_mul(tmp, in, ftmp);
705 felem_reduce(ftmp, tmp); /* 2^3 - 1 */
706 felem_square(tmp, ftmp);
707 felem_reduce(ftmp2, tmp); /* 2^4 - 2 */
708 felem_square(tmp, ftmp2);
709 felem_reduce(ftmp2, tmp); /* 2^5 - 4 */
710 felem_square(tmp, ftmp2);
711 felem_reduce(ftmp2, tmp); /* 2^6 - 8 */
712 felem_mul(tmp, ftmp2, ftmp);
713 felem_reduce(ftmp, tmp); /* 2^6 - 1 */
714 felem_square(tmp, ftmp);
715 felem_reduce(ftmp2, tmp); /* 2^7 - 2 */
716 for (i = 0; i < 5; ++i) { /* 2^12 - 2^6 */
717 felem_square(tmp, ftmp2);
718 felem_reduce(ftmp2, tmp);
720 felem_mul(tmp, ftmp2, ftmp);
721 felem_reduce(ftmp2, tmp); /* 2^12 - 1 */
722 felem_square(tmp, ftmp2);
723 felem_reduce(ftmp3, tmp); /* 2^13 - 2 */
724 for (i = 0; i < 11; ++i) { /* 2^24 - 2^12 */
725 felem_square(tmp, ftmp3);
726 felem_reduce(ftmp3, tmp);
728 felem_mul(tmp, ftmp3, ftmp2);
729 felem_reduce(ftmp2, tmp); /* 2^24 - 1 */
730 felem_square(tmp, ftmp2);
731 felem_reduce(ftmp3, tmp); /* 2^25 - 2 */
732 for (i = 0; i < 23; ++i) { /* 2^48 - 2^24 */
733 felem_square(tmp, ftmp3);
734 felem_reduce(ftmp3, tmp);
736 felem_mul(tmp, ftmp3, ftmp2);
737 felem_reduce(ftmp3, tmp); /* 2^48 - 1 */
738 felem_square(tmp, ftmp3);
739 felem_reduce(ftmp4, tmp); /* 2^49 - 2 */
740 for (i = 0; i < 47; ++i) { /* 2^96 - 2^48 */
741 felem_square(tmp, ftmp4);
742 felem_reduce(ftmp4, tmp);
744 felem_mul(tmp, ftmp3, ftmp4);
745 felem_reduce(ftmp3, tmp); /* 2^96 - 1 */
746 felem_square(tmp, ftmp3);
747 felem_reduce(ftmp4, tmp); /* 2^97 - 2 */
748 for (i = 0; i < 23; ++i) { /* 2^120 - 2^24 */
749 felem_square(tmp, ftmp4);
750 felem_reduce(ftmp4, tmp);
752 felem_mul(tmp, ftmp2, ftmp4);
753 felem_reduce(ftmp2, tmp); /* 2^120 - 1 */
754 for (i = 0; i < 6; ++i) { /* 2^126 - 2^6 */
755 felem_square(tmp, ftmp2);
756 felem_reduce(ftmp2, tmp);
758 felem_mul(tmp, ftmp2, ftmp);
759 felem_reduce(ftmp, tmp); /* 2^126 - 1 */
760 felem_square(tmp, ftmp);
761 felem_reduce(ftmp, tmp); /* 2^127 - 2 */
762 felem_mul(tmp, ftmp, in);
763 felem_reduce(ftmp, tmp); /* 2^127 - 1 */
764 for (i = 0; i < 97; ++i) { /* 2^224 - 2^97 */
765 felem_square(tmp, ftmp);
766 felem_reduce(ftmp, tmp);
768 felem_mul(tmp, ftmp, ftmp3);
769 felem_reduce(out, tmp); /* 2^224 - 2^96 - 1 */
773 * Copy in constant time: if icopy == 1, copy in to out, if icopy == 0, copy
776 static void copy_conditional(felem out, const felem in, limb icopy)
780 * icopy is a (64-bit) 0 or 1, so copy is either all-zero or all-one
782 const limb copy = -icopy;
783 for (i = 0; i < 4; ++i) {
784 const limb tmp = copy & (in[i] ^ out[i]);
789 /******************************************************************************/
791 * ELLIPTIC CURVE POINT OPERATIONS
793 * Points are represented in Jacobian projective coordinates:
794 * (X, Y, Z) corresponds to the affine point (X/Z^2, Y/Z^3),
795 * or to the point at infinity if Z == 0.
800 * Double an elliptic curve point:
801 * (X', Y', Z') = 2 * (X, Y, Z), where
802 * X' = (3 * (X - Z^2) * (X + Z^2))^2 - 8 * X * Y^2
803 * Y' = 3 * (X - Z^2) * (X + Z^2) * (4 * X * Y^2 - X') - 8 * Y^2
804 * Z' = (Y + Z)^2 - Y^2 - Z^2 = 2 * Y * Z
805 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed,
806 * while x_out == y_in is not (maybe this works, but it's not tested).
809 point_double(felem x_out, felem y_out, felem z_out,
810 const felem x_in, const felem y_in, const felem z_in)
813 felem delta, gamma, beta, alpha, ftmp, ftmp2;
815 felem_assign(ftmp, x_in);
816 felem_assign(ftmp2, x_in);
819 felem_square(tmp, z_in);
820 felem_reduce(delta, tmp);
823 felem_square(tmp, y_in);
824 felem_reduce(gamma, tmp);
827 felem_mul(tmp, x_in, gamma);
828 felem_reduce(beta, tmp);
830 /* alpha = 3*(x-delta)*(x+delta) */
831 felem_diff(ftmp, delta);
832 /* ftmp[i] < 2^57 + 2^58 + 2 < 2^59 */
833 felem_sum(ftmp2, delta);
834 /* ftmp2[i] < 2^57 + 2^57 = 2^58 */
835 felem_scalar(ftmp2, 3);
836 /* ftmp2[i] < 3 * 2^58 < 2^60 */
837 felem_mul(tmp, ftmp, ftmp2);
838 /* tmp[i] < 2^60 * 2^59 * 4 = 2^121 */
839 felem_reduce(alpha, tmp);
841 /* x' = alpha^2 - 8*beta */
842 felem_square(tmp, alpha);
843 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
844 felem_assign(ftmp, beta);
845 felem_scalar(ftmp, 8);
846 /* ftmp[i] < 8 * 2^57 = 2^60 */
847 felem_diff_128_64(tmp, ftmp);
848 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
849 felem_reduce(x_out, tmp);
851 /* z' = (y + z)^2 - gamma - delta */
852 felem_sum(delta, gamma);
853 /* delta[i] < 2^57 + 2^57 = 2^58 */
854 felem_assign(ftmp, y_in);
855 felem_sum(ftmp, z_in);
856 /* ftmp[i] < 2^57 + 2^57 = 2^58 */
857 felem_square(tmp, ftmp);
858 /* tmp[i] < 4 * 2^58 * 2^58 = 2^118 */
859 felem_diff_128_64(tmp, delta);
860 /* tmp[i] < 2^118 + 2^64 + 8 < 2^119 */
861 felem_reduce(z_out, tmp);
863 /* y' = alpha*(4*beta - x') - 8*gamma^2 */
864 felem_scalar(beta, 4);
865 /* beta[i] < 4 * 2^57 = 2^59 */
866 felem_diff(beta, x_out);
867 /* beta[i] < 2^59 + 2^58 + 2 < 2^60 */
868 felem_mul(tmp, alpha, beta);
869 /* tmp[i] < 4 * 2^57 * 2^60 = 2^119 */
870 felem_square(tmp2, gamma);
871 /* tmp2[i] < 4 * 2^57 * 2^57 = 2^116 */
872 widefelem_scalar(tmp2, 8);
873 /* tmp2[i] < 8 * 2^116 = 2^119 */
874 widefelem_diff(tmp, tmp2);
875 /* tmp[i] < 2^119 + 2^120 < 2^121 */
876 felem_reduce(y_out, tmp);
880 * Add two elliptic curve points:
881 * (X_1, Y_1, Z_1) + (X_2, Y_2, Z_2) = (X_3, Y_3, Z_3), where
882 * X_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1)^2 - (Z_1^2 * X_2 - Z_2^2 * X_1)^3 -
883 * 2 * Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^2
884 * 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) -
885 * Z_2^3 * Y_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^3
886 * Z_3 = (Z_1^2 * X_2 - Z_2^2 * X_1) * (Z_1 * Z_2)
888 * This runs faster if 'mixed' is set, which requires Z_2 = 1 or Z_2 = 0.
892 * This function is not entirely constant-time: it includes a branch for
893 * checking whether the two input points are equal, (while not equal to the
894 * point at infinity). This case never happens during single point
895 * multiplication, so there is no timing leak for ECDH or ECDSA signing.
897 static void point_add(felem x3, felem y3, felem z3,
898 const felem x1, const felem y1, const felem z1,
899 const int mixed, const felem x2, const felem y2,
902 felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, x_out, y_out, z_out;
904 limb z1_is_zero, z2_is_zero, x_equal, y_equal;
908 felem_square(tmp, z2);
909 felem_reduce(ftmp2, tmp);
912 felem_mul(tmp, ftmp2, z2);
913 felem_reduce(ftmp4, tmp);
915 /* ftmp4 = z2^3*y1 */
916 felem_mul(tmp2, ftmp4, y1);
917 felem_reduce(ftmp4, tmp2);
919 /* ftmp2 = z2^2*x1 */
920 felem_mul(tmp2, ftmp2, x1);
921 felem_reduce(ftmp2, tmp2);
924 * We'll assume z2 = 1 (special case z2 = 0 is handled later)
927 /* ftmp4 = z2^3*y1 */
928 felem_assign(ftmp4, y1);
930 /* ftmp2 = z2^2*x1 */
931 felem_assign(ftmp2, x1);
935 felem_square(tmp, z1);
936 felem_reduce(ftmp, tmp);
939 felem_mul(tmp, ftmp, z1);
940 felem_reduce(ftmp3, tmp);
943 felem_mul(tmp, ftmp3, y2);
944 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
946 /* ftmp3 = z1^3*y2 - z2^3*y1 */
947 felem_diff_128_64(tmp, ftmp4);
948 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
949 felem_reduce(ftmp3, tmp);
952 felem_mul(tmp, ftmp, x2);
953 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
955 /* ftmp = z1^2*x2 - z2^2*x1 */
956 felem_diff_128_64(tmp, ftmp2);
957 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
958 felem_reduce(ftmp, tmp);
961 * the formulae are incorrect if the points are equal so we check for
962 * this and do doubling if this happens
964 x_equal = felem_is_zero(ftmp);
965 y_equal = felem_is_zero(ftmp3);
966 z1_is_zero = felem_is_zero(z1);
967 z2_is_zero = felem_is_zero(z2);
968 /* In affine coordinates, (X_1, Y_1) == (X_2, Y_2) */
969 if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) {
970 point_double(x3, y3, z3, x1, y1, z1);
976 felem_mul(tmp, z1, z2);
977 felem_reduce(ftmp5, tmp);
979 /* special case z2 = 0 is handled later */
980 felem_assign(ftmp5, z1);
983 /* z_out = (z1^2*x2 - z2^2*x1)*(z1*z2) */
984 felem_mul(tmp, ftmp, ftmp5);
985 felem_reduce(z_out, tmp);
987 /* ftmp = (z1^2*x2 - z2^2*x1)^2 */
988 felem_assign(ftmp5, ftmp);
989 felem_square(tmp, ftmp);
990 felem_reduce(ftmp, tmp);
992 /* ftmp5 = (z1^2*x2 - z2^2*x1)^3 */
993 felem_mul(tmp, ftmp, ftmp5);
994 felem_reduce(ftmp5, tmp);
996 /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
997 felem_mul(tmp, ftmp2, ftmp);
998 felem_reduce(ftmp2, tmp);
1000 /* tmp = z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */
1001 felem_mul(tmp, ftmp4, ftmp5);
1002 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
1004 /* tmp2 = (z1^3*y2 - z2^3*y1)^2 */
1005 felem_square(tmp2, ftmp3);
1006 /* tmp2[i] < 4 * 2^57 * 2^57 < 2^116 */
1008 /* tmp2 = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 */
1009 felem_diff_128_64(tmp2, ftmp5);
1010 /* tmp2[i] < 2^116 + 2^64 + 8 < 2^117 */
1012 /* ftmp5 = 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
1013 felem_assign(ftmp5, ftmp2);
1014 felem_scalar(ftmp5, 2);
1015 /* ftmp5[i] < 2 * 2^57 = 2^58 */
1018 * x_out = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 -
1019 * 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2
1021 felem_diff_128_64(tmp2, ftmp5);
1022 /* tmp2[i] < 2^117 + 2^64 + 8 < 2^118 */
1023 felem_reduce(x_out, tmp2);
1025 /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out */
1026 felem_diff(ftmp2, x_out);
1027 /* ftmp2[i] < 2^57 + 2^58 + 2 < 2^59 */
1030 * tmp2 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out)
1032 felem_mul(tmp2, ftmp3, ftmp2);
1033 /* tmp2[i] < 4 * 2^57 * 2^59 = 2^118 */
1036 * y_out = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) -
1037 * z2^3*y1*(z1^2*x2 - z2^2*x1)^3
1039 widefelem_diff(tmp2, tmp);
1040 /* tmp2[i] < 2^118 + 2^120 < 2^121 */
1041 felem_reduce(y_out, tmp2);
1044 * the result (x_out, y_out, z_out) is incorrect if one of the inputs is
1045 * the point at infinity, so we need to check for this separately
1049 * if point 1 is at infinity, copy point 2 to output, and vice versa
1051 copy_conditional(x_out, x2, z1_is_zero);
1052 copy_conditional(x_out, x1, z2_is_zero);
1053 copy_conditional(y_out, y2, z1_is_zero);
1054 copy_conditional(y_out, y1, z2_is_zero);
1055 copy_conditional(z_out, z2, z1_is_zero);
1056 copy_conditional(z_out, z1, z2_is_zero);
1057 felem_assign(x3, x_out);
1058 felem_assign(y3, y_out);
1059 felem_assign(z3, z_out);
1063 * select_point selects the |idx|th point from a precomputation table and
1065 * The pre_comp array argument should be size of |size| argument
1067 static void select_point(const u64 idx, unsigned int size,
1068 const felem pre_comp[][3], felem out[3])
1071 limb *outlimbs = &out[0][0];
1073 memset(out, 0, sizeof(*out) * 3);
1074 for (i = 0; i < size; i++) {
1075 const limb *inlimbs = &pre_comp[i][0][0];
1082 for (j = 0; j < 4 * 3; j++)
1083 outlimbs[j] |= inlimbs[j] & mask;
1087 /* get_bit returns the |i|th bit in |in| */
1088 static char get_bit(const felem_bytearray in, unsigned i)
1092 return (in[i >> 3] >> (i & 7)) & 1;
1096 * Interleaved point multiplication using precomputed point multiples: The
1097 * small point multiples 0*P, 1*P, ..., 16*P are in pre_comp[], the scalars
1098 * in scalars[]. If g_scalar is non-NULL, we also add this multiple of the
1099 * generator, using certain (large) precomputed multiples in g_pre_comp.
1100 * Output point (X, Y, Z) is stored in x_out, y_out, z_out
1102 static void batch_mul(felem x_out, felem y_out, felem z_out,
1103 const felem_bytearray scalars[],
1104 const unsigned num_points, const u8 *g_scalar,
1105 const int mixed, const felem pre_comp[][17][3],
1106 const felem g_pre_comp[2][16][3])
1110 unsigned gen_mul = (g_scalar != NULL);
1111 felem nq[3], tmp[4];
1115 /* set nq to the point at infinity */
1116 memset(nq, 0, sizeof(nq));
1119 * Loop over all scalars msb-to-lsb, interleaving additions of multiples
1120 * of the generator (two in each of the last 28 rounds) and additions of
1121 * other points multiples (every 5th round).
1123 skip = 1; /* save two point operations in the first
1125 for (i = (num_points ? 220 : 27); i >= 0; --i) {
1128 point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
1130 /* add multiples of the generator */
1131 if (gen_mul && (i <= 27)) {
1132 /* first, look 28 bits upwards */
1133 bits = get_bit(g_scalar, i + 196) << 3;
1134 bits |= get_bit(g_scalar, i + 140) << 2;
1135 bits |= get_bit(g_scalar, i + 84) << 1;
1136 bits |= get_bit(g_scalar, i + 28);
1137 /* select the point to add, in constant time */
1138 select_point(bits, 16, g_pre_comp[1], tmp);
1141 /* value 1 below is argument for "mixed" */
1142 point_add(nq[0], nq[1], nq[2],
1143 nq[0], nq[1], nq[2], 1, tmp[0], tmp[1], tmp[2]);
1145 memcpy(nq, tmp, 3 * sizeof(felem));
1149 /* second, look at the current position */
1150 bits = get_bit(g_scalar, i + 168) << 3;
1151 bits |= get_bit(g_scalar, i + 112) << 2;
1152 bits |= get_bit(g_scalar, i + 56) << 1;
1153 bits |= get_bit(g_scalar, i);
1154 /* select the point to add, in constant time */
1155 select_point(bits, 16, g_pre_comp[0], tmp);
1156 point_add(nq[0], nq[1], nq[2],
1157 nq[0], nq[1], nq[2],
1158 1 /* mixed */ , tmp[0], tmp[1], tmp[2]);
1161 /* do other additions every 5 doublings */
1162 if (num_points && (i % 5 == 0)) {
1163 /* loop over all scalars */
1164 for (num = 0; num < num_points; ++num) {
1165 bits = get_bit(scalars[num], i + 4) << 5;
1166 bits |= get_bit(scalars[num], i + 3) << 4;
1167 bits |= get_bit(scalars[num], i + 2) << 3;
1168 bits |= get_bit(scalars[num], i + 1) << 2;
1169 bits |= get_bit(scalars[num], i) << 1;
1170 bits |= get_bit(scalars[num], i - 1);
1171 ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits);
1173 /* select the point to add or subtract */
1174 select_point(digit, 17, pre_comp[num], tmp);
1175 felem_neg(tmp[3], tmp[1]); /* (X, -Y, Z) is the negative
1177 copy_conditional(tmp[1], tmp[3], sign);
1180 point_add(nq[0], nq[1], nq[2],
1181 nq[0], nq[1], nq[2],
1182 mixed, tmp[0], tmp[1], tmp[2]);
1184 memcpy(nq, tmp, 3 * sizeof(felem));
1190 felem_assign(x_out, nq[0]);
1191 felem_assign(y_out, nq[1]);
1192 felem_assign(z_out, nq[2]);
1195 /******************************************************************************/
1197 * FUNCTIONS TO MANAGE PRECOMPUTATION
1200 static NISTP224_PRE_COMP *nistp224_pre_comp_new()
1202 NISTP224_PRE_COMP *ret = OPENSSL_zalloc(sizeof(*ret));
1205 ECerr(EC_F_NISTP224_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1208 ret->references = 1;
1212 NISTP224_PRE_COMP *EC_nistp224_pre_comp_dup(NISTP224_PRE_COMP *p)
1215 CRYPTO_add(&p->references, 1, CRYPTO_LOCK_EC_PRE_COMP);
1219 void EC_nistp224_pre_comp_free(NISTP224_PRE_COMP *p)
1222 || CRYPTO_add(&p->references, -1, CRYPTO_LOCK_EC_PRE_COMP) > 0)
1227 /******************************************************************************/
1229 * OPENSSL EC_METHOD FUNCTIONS
1232 int ec_GFp_nistp224_group_init(EC_GROUP *group)
1235 ret = ec_GFp_simple_group_init(group);
1236 group->a_is_minus3 = 1;
1240 int ec_GFp_nistp224_group_set_curve(EC_GROUP *group, const BIGNUM *p,
1241 const BIGNUM *a, const BIGNUM *b,
1245 BN_CTX *new_ctx = NULL;
1246 BIGNUM *curve_p, *curve_a, *curve_b;
1249 if ((ctx = new_ctx = BN_CTX_new()) == NULL)
1252 if (((curve_p = BN_CTX_get(ctx)) == NULL) ||
1253 ((curve_a = BN_CTX_get(ctx)) == NULL) ||
1254 ((curve_b = BN_CTX_get(ctx)) == NULL))
1256 BN_bin2bn(nistp224_curve_params[0], sizeof(felem_bytearray), curve_p);
1257 BN_bin2bn(nistp224_curve_params[1], sizeof(felem_bytearray), curve_a);
1258 BN_bin2bn(nistp224_curve_params[2], sizeof(felem_bytearray), curve_b);
1259 if ((BN_cmp(curve_p, p)) || (BN_cmp(curve_a, a)) || (BN_cmp(curve_b, b))) {
1260 ECerr(EC_F_EC_GFP_NISTP224_GROUP_SET_CURVE,
1261 EC_R_WRONG_CURVE_PARAMETERS);
1264 group->field_mod_func = BN_nist_mod_224;
1265 ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx);
1268 BN_CTX_free(new_ctx);
1273 * Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') =
1276 int ec_GFp_nistp224_point_get_affine_coordinates(const EC_GROUP *group,
1277 const EC_POINT *point,
1278 BIGNUM *x, BIGNUM *y,
1281 felem z1, z2, x_in, y_in, x_out, y_out;
1284 if (EC_POINT_is_at_infinity(group, point)) {
1285 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1286 EC_R_POINT_AT_INFINITY);
1289 if ((!BN_to_felem(x_in, point->X)) || (!BN_to_felem(y_in, point->Y)) ||
1290 (!BN_to_felem(z1, point->Z)))
1293 felem_square(tmp, z2);
1294 felem_reduce(z1, tmp);
1295 felem_mul(tmp, x_in, z1);
1296 felem_reduce(x_in, tmp);
1297 felem_contract(x_out, x_in);
1299 if (!felem_to_BN(x, x_out)) {
1300 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1305 felem_mul(tmp, z1, z2);
1306 felem_reduce(z1, tmp);
1307 felem_mul(tmp, y_in, z1);
1308 felem_reduce(y_in, tmp);
1309 felem_contract(y_out, y_in);
1311 if (!felem_to_BN(y, y_out)) {
1312 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1320 static void make_points_affine(size_t num, felem points[ /* num */ ][3],
1321 felem tmp_felems[ /* num+1 */ ])
1324 * Runs in constant time, unless an input is the point at infinity (which
1325 * normally shouldn't happen).
1327 ec_GFp_nistp_points_make_affine_internal(num,
1331 (void (*)(void *))felem_one,
1332 (int (*)(const void *))
1334 (void (*)(void *, const void *))
1336 (void (*)(void *, const void *))
1337 felem_square_reduce, (void (*)
1344 (void (*)(void *, const void *))
1346 (void (*)(void *, const void *))
1351 * Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL
1352 * values Result is stored in r (r can equal one of the inputs).
1354 int ec_GFp_nistp224_points_mul(const EC_GROUP *group, EC_POINT *r,
1355 const BIGNUM *scalar, size_t num,
1356 const EC_POINT *points[],
1357 const BIGNUM *scalars[], BN_CTX *ctx)
1363 BN_CTX *new_ctx = NULL;
1364 BIGNUM *x, *y, *z, *tmp_scalar;
1365 felem_bytearray g_secret;
1366 felem_bytearray *secrets = NULL;
1367 felem (*pre_comp)[17][3] = NULL;
1368 felem *tmp_felems = NULL;
1369 felem_bytearray tmp;
1371 int have_pre_comp = 0;
1372 size_t num_points = num;
1373 felem x_in, y_in, z_in, x_out, y_out, z_out;
1374 NISTP224_PRE_COMP *pre = NULL;
1375 const felem(*g_pre_comp)[16][3] = NULL;
1376 EC_POINT *generator = NULL;
1377 const EC_POINT *p = NULL;
1378 const BIGNUM *p_scalar = NULL;
1381 if ((ctx = new_ctx = BN_CTX_new()) == NULL)
1384 if (((x = BN_CTX_get(ctx)) == NULL) ||
1385 ((y = BN_CTX_get(ctx)) == NULL) ||
1386 ((z = BN_CTX_get(ctx)) == NULL) ||
1387 ((tmp_scalar = BN_CTX_get(ctx)) == NULL))
1390 if (scalar != NULL) {
1391 pre = group->pre_comp.nistp224;
1393 /* we have precomputation, try to use it */
1394 g_pre_comp = (const felem(*)[16][3])pre->g_pre_comp;
1396 /* try to use the standard precomputation */
1397 g_pre_comp = &gmul[0];
1398 generator = EC_POINT_new(group);
1399 if (generator == NULL)
1401 /* get the generator from precomputation */
1402 if (!felem_to_BN(x, g_pre_comp[0][1][0]) ||
1403 !felem_to_BN(y, g_pre_comp[0][1][1]) ||
1404 !felem_to_BN(z, g_pre_comp[0][1][2])) {
1405 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1408 if (!EC_POINT_set_Jprojective_coordinates_GFp(group,
1412 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
1413 /* precomputation matches generator */
1417 * we don't have valid precomputation: treat the generator as a
1420 num_points = num_points + 1;
1423 if (num_points > 0) {
1424 if (num_points >= 3) {
1426 * unless we precompute multiples for just one or two points,
1427 * converting those into affine form is time well spent
1431 secrets = OPENSSL_zalloc(sizeof(*secrets) * num_points);
1432 pre_comp = OPENSSL_zalloc(sizeof(*pre_comp) * num_points);
1435 OPENSSL_malloc(sizeof(felem) * (num_points * 17 + 1));
1436 if ((secrets == NULL) || (pre_comp == NULL)
1437 || (mixed && (tmp_felems == NULL))) {
1438 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_MALLOC_FAILURE);
1443 * we treat NULL scalars as 0, and NULL points as points at infinity,
1444 * i.e., they contribute nothing to the linear combination
1446 for (i = 0; i < num_points; ++i) {
1450 p = EC_GROUP_get0_generator(group);
1453 /* the i^th point */
1456 p_scalar = scalars[i];
1458 if ((p_scalar != NULL) && (p != NULL)) {
1459 /* reduce scalar to 0 <= scalar < 2^224 */
1460 if ((BN_num_bits(p_scalar) > 224)
1461 || (BN_is_negative(p_scalar))) {
1463 * this is an unusual input, and we don't guarantee
1466 if (!BN_nnmod(tmp_scalar, p_scalar, group->order, ctx)) {
1467 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1470 num_bytes = BN_bn2bin(tmp_scalar, tmp);
1472 num_bytes = BN_bn2bin(p_scalar, tmp);
1473 flip_endian(secrets[i], tmp, num_bytes);
1474 /* precompute multiples */
1475 if ((!BN_to_felem(x_out, p->X)) ||
1476 (!BN_to_felem(y_out, p->Y)) ||
1477 (!BN_to_felem(z_out, p->Z)))
1479 felem_assign(pre_comp[i][1][0], x_out);
1480 felem_assign(pre_comp[i][1][1], y_out);
1481 felem_assign(pre_comp[i][1][2], z_out);
1482 for (j = 2; j <= 16; ++j) {
1484 point_add(pre_comp[i][j][0], pre_comp[i][j][1],
1485 pre_comp[i][j][2], pre_comp[i][1][0],
1486 pre_comp[i][1][1], pre_comp[i][1][2], 0,
1487 pre_comp[i][j - 1][0],
1488 pre_comp[i][j - 1][1],
1489 pre_comp[i][j - 1][2]);
1491 point_double(pre_comp[i][j][0], pre_comp[i][j][1],
1492 pre_comp[i][j][2], pre_comp[i][j / 2][0],
1493 pre_comp[i][j / 2][1],
1494 pre_comp[i][j / 2][2]);
1500 make_points_affine(num_points * 17, pre_comp[0], tmp_felems);
1503 /* the scalar for the generator */
1504 if ((scalar != NULL) && (have_pre_comp)) {
1505 memset(g_secret, 0, sizeof(g_secret));
1506 /* reduce scalar to 0 <= scalar < 2^224 */
1507 if ((BN_num_bits(scalar) > 224) || (BN_is_negative(scalar))) {
1509 * this is an unusual input, and we don't guarantee
1512 if (!BN_nnmod(tmp_scalar, scalar, group->order, ctx)) {
1513 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1516 num_bytes = BN_bn2bin(tmp_scalar, tmp);
1518 num_bytes = BN_bn2bin(scalar, tmp);
1519 flip_endian(g_secret, tmp, num_bytes);
1520 /* do the multiplication with generator precomputation */
1521 batch_mul(x_out, y_out, z_out,
1522 (const felem_bytearray(*))secrets, num_points,
1524 mixed, (const felem(*)[17][3])pre_comp, g_pre_comp);
1526 /* do the multiplication without generator precomputation */
1527 batch_mul(x_out, y_out, z_out,
1528 (const felem_bytearray(*))secrets, num_points,
1529 NULL, mixed, (const felem(*)[17][3])pre_comp, NULL);
1530 /* reduce the output to its unique minimal representation */
1531 felem_contract(x_in, x_out);
1532 felem_contract(y_in, y_out);
1533 felem_contract(z_in, z_out);
1534 if ((!felem_to_BN(x, x_in)) || (!felem_to_BN(y, y_in)) ||
1535 (!felem_to_BN(z, z_in))) {
1536 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1539 ret = EC_POINT_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx);
1543 EC_POINT_free(generator);
1544 BN_CTX_free(new_ctx);
1545 OPENSSL_free(secrets);
1546 OPENSSL_free(pre_comp);
1547 OPENSSL_free(tmp_felems);
1551 int ec_GFp_nistp224_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
1554 NISTP224_PRE_COMP *pre = NULL;
1556 BN_CTX *new_ctx = NULL;
1558 EC_POINT *generator = NULL;
1559 felem tmp_felems[32];
1561 /* throw away old precomputation */
1562 EC_nistp224_pre_comp_free(group->pre_comp.nistp224);
1564 if ((ctx = new_ctx = BN_CTX_new()) == NULL)
1567 if (((x = BN_CTX_get(ctx)) == NULL) || ((y = BN_CTX_get(ctx)) == NULL))
1569 /* get the generator */
1570 if (group->generator == NULL)
1572 generator = EC_POINT_new(group);
1573 if (generator == NULL)
1575 BN_bin2bn(nistp224_curve_params[3], sizeof(felem_bytearray), x);
1576 BN_bin2bn(nistp224_curve_params[4], sizeof(felem_bytearray), y);
1577 if (!EC_POINT_set_affine_coordinates_GFp(group, generator, x, y, ctx))
1579 if ((pre = nistp224_pre_comp_new()) == NULL)
1582 * if the generator is the standard one, use built-in precomputation
1584 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) {
1585 memcpy(pre->g_pre_comp, gmul, sizeof(pre->g_pre_comp));
1589 if ((!BN_to_felem(pre->g_pre_comp[0][1][0], group->generator->X)) ||
1590 (!BN_to_felem(pre->g_pre_comp[0][1][1], group->generator->Y)) ||
1591 (!BN_to_felem(pre->g_pre_comp[0][1][2], group->generator->Z)))
1594 * compute 2^56*G, 2^112*G, 2^168*G for the first table, 2^28*G, 2^84*G,
1595 * 2^140*G, 2^196*G for the second one
1597 for (i = 1; i <= 8; i <<= 1) {
1598 point_double(pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1],
1599 pre->g_pre_comp[1][i][2], pre->g_pre_comp[0][i][0],
1600 pre->g_pre_comp[0][i][1], pre->g_pre_comp[0][i][2]);
1601 for (j = 0; j < 27; ++j) {
1602 point_double(pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1],
1603 pre->g_pre_comp[1][i][2], pre->g_pre_comp[1][i][0],
1604 pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2]);
1608 point_double(pre->g_pre_comp[0][2 * i][0],
1609 pre->g_pre_comp[0][2 * i][1],
1610 pre->g_pre_comp[0][2 * i][2], pre->g_pre_comp[1][i][0],
1611 pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2]);
1612 for (j = 0; j < 27; ++j) {
1613 point_double(pre->g_pre_comp[0][2 * i][0],
1614 pre->g_pre_comp[0][2 * i][1],
1615 pre->g_pre_comp[0][2 * i][2],
1616 pre->g_pre_comp[0][2 * i][0],
1617 pre->g_pre_comp[0][2 * i][1],
1618 pre->g_pre_comp[0][2 * i][2]);
1621 for (i = 0; i < 2; i++) {
1622 /* g_pre_comp[i][0] is the point at infinity */
1623 memset(pre->g_pre_comp[i][0], 0, sizeof(pre->g_pre_comp[i][0]));
1624 /* the remaining multiples */
1625 /* 2^56*G + 2^112*G resp. 2^84*G + 2^140*G */
1626 point_add(pre->g_pre_comp[i][6][0], pre->g_pre_comp[i][6][1],
1627 pre->g_pre_comp[i][6][2], pre->g_pre_comp[i][4][0],
1628 pre->g_pre_comp[i][4][1], pre->g_pre_comp[i][4][2],
1629 0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
1630 pre->g_pre_comp[i][2][2]);
1631 /* 2^56*G + 2^168*G resp. 2^84*G + 2^196*G */
1632 point_add(pre->g_pre_comp[i][10][0], pre->g_pre_comp[i][10][1],
1633 pre->g_pre_comp[i][10][2], pre->g_pre_comp[i][8][0],
1634 pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2],
1635 0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
1636 pre->g_pre_comp[i][2][2]);
1637 /* 2^112*G + 2^168*G resp. 2^140*G + 2^196*G */
1638 point_add(pre->g_pre_comp[i][12][0], pre->g_pre_comp[i][12][1],
1639 pre->g_pre_comp[i][12][2], pre->g_pre_comp[i][8][0],
1640 pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2],
1641 0, pre->g_pre_comp[i][4][0], pre->g_pre_comp[i][4][1],
1642 pre->g_pre_comp[i][4][2]);
1644 * 2^56*G + 2^112*G + 2^168*G resp. 2^84*G + 2^140*G + 2^196*G
1646 point_add(pre->g_pre_comp[i][14][0], pre->g_pre_comp[i][14][1],
1647 pre->g_pre_comp[i][14][2], pre->g_pre_comp[i][12][0],
1648 pre->g_pre_comp[i][12][1], pre->g_pre_comp[i][12][2],
1649 0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
1650 pre->g_pre_comp[i][2][2]);
1651 for (j = 1; j < 8; ++j) {
1652 /* odd multiples: add G resp. 2^28*G */
1653 point_add(pre->g_pre_comp[i][2 * j + 1][0],
1654 pre->g_pre_comp[i][2 * j + 1][1],
1655 pre->g_pre_comp[i][2 * j + 1][2],
1656 pre->g_pre_comp[i][2 * j][0],
1657 pre->g_pre_comp[i][2 * j][1],
1658 pre->g_pre_comp[i][2 * j][2], 0,
1659 pre->g_pre_comp[i][1][0], pre->g_pre_comp[i][1][1],
1660 pre->g_pre_comp[i][1][2]);
1663 make_points_affine(31, &(pre->g_pre_comp[0][1]), tmp_felems);
1665 SETPRECOMP(group, nistp224, pre);
1670 EC_POINT_free(generator);
1671 BN_CTX_free(new_ctx);
1672 EC_nistp224_pre_comp_free(pre);
1676 int ec_GFp_nistp224_have_precompute_mult(const EC_GROUP *group)
1678 return HAVEPRECOMP(group, nistp224);
1682 static void *dummy = &dummy;