2 * Copyright 2010-2019 The OpenSSL Project Authors. All Rights Reserved.
4 * Licensed under the OpenSSL license (the "License"). You may not use
5 * this file except in compliance with the License. You can obtain a copy
6 * in the file LICENSE in the source distribution or at
7 * https://www.openssl.org/source/license.html
10 /* Copyright 2011 Google Inc.
12 * Licensed under the Apache License, Version 2.0 (the "License");
14 * you may not use this file except in compliance with the License.
15 * You may obtain a copy of the License at
17 * http://www.apache.org/licenses/LICENSE-2.0
19 * Unless required by applicable law or agreed to in writing, software
20 * distributed under the License is distributed on an "AS IS" BASIS,
21 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
22 * See the License for the specific language governing permissions and
23 * limitations under the License.
27 * A 64-bit implementation of the NIST P-224 elliptic curve point multiplication
29 * Inspired by Daniel J. Bernstein's public domain nistp224 implementation
30 * and Adam Langley's public domain 64-bit C implementation of curve25519
33 #include <openssl/opensslconf.h>
34 #ifdef OPENSSL_NO_EC_NISTP_64_GCC_128
35 NON_EMPTY_TRANSLATION_UNIT
40 # include <openssl/err.h>
43 # if defined(__SIZEOF_INT128__) && __SIZEOF_INT128__==16
44 /* even with gcc, the typedef won't work for 32-bit platforms */
45 typedef __uint128_t uint128_t; /* nonstandard; implemented by gcc on 64-bit
48 # error "Your compiler doesn't appear to support 128-bit integer types"
54 /******************************************************************************/
56 * INTERNAL REPRESENTATION OF FIELD ELEMENTS
58 * Field elements are represented as a_0 + 2^56*a_1 + 2^112*a_2 + 2^168*a_3
59 * using 64-bit coefficients called 'limbs',
60 * and sometimes (for multiplication results) as
61 * 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
62 * using 128-bit coefficients called 'widelimbs'.
63 * A 4-limb representation is an 'felem';
64 * a 7-widelimb representation is a 'widefelem'.
65 * Even within felems, bits of adjacent limbs overlap, and we don't always
66 * reduce the representations: we ensure that inputs to each felem
67 * multiplication satisfy a_i < 2^60, so outputs satisfy b_i < 4*2^60*2^60,
68 * and fit into a 128-bit word without overflow. The coefficients are then
69 * again partially reduced to obtain an felem satisfying a_i < 2^57.
70 * We only reduce to the unique minimal representation at the end of the
74 typedef uint64_t limb;
75 typedef uint128_t widelimb;
77 typedef limb felem[4];
78 typedef widelimb widefelem[7];
81 * Field element represented as a byte array. 28*8 = 224 bits is also the
82 * group order size for the elliptic curve, and we also use this type for
83 * scalars for point multiplication.
85 typedef u8 felem_bytearray[28];
87 static const felem_bytearray nistp224_curve_params[5] = {
88 {0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, /* p */
89 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0x00, 0x00, 0x00, 0x00,
90 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x01},
91 {0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, /* a */
92 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFE, 0xFF, 0xFF, 0xFF, 0xFF,
93 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFE},
94 {0xB4, 0x05, 0x0A, 0x85, 0x0C, 0x04, 0xB3, 0xAB, 0xF5, 0x41, /* b */
95 0x32, 0x56, 0x50, 0x44, 0xB0, 0xB7, 0xD7, 0xBF, 0xD8, 0xBA,
96 0x27, 0x0B, 0x39, 0x43, 0x23, 0x55, 0xFF, 0xB4},
97 {0xB7, 0x0E, 0x0C, 0xBD, 0x6B, 0xB4, 0xBF, 0x7F, 0x32, 0x13, /* x */
98 0x90, 0xB9, 0x4A, 0x03, 0xC1, 0xD3, 0x56, 0xC2, 0x11, 0x22,
99 0x34, 0x32, 0x80, 0xD6, 0x11, 0x5C, 0x1D, 0x21},
100 {0xbd, 0x37, 0x63, 0x88, 0xb5, 0xf7, 0x23, 0xfb, 0x4c, 0x22, /* y */
101 0xdf, 0xe6, 0xcd, 0x43, 0x75, 0xa0, 0x5a, 0x07, 0x47, 0x64,
102 0x44, 0xd5, 0x81, 0x99, 0x85, 0x00, 0x7e, 0x34}
106 * Precomputed multiples of the standard generator
107 * Points are given in coordinates (X, Y, Z) where Z normally is 1
108 * (0 for the point at infinity).
109 * For each field element, slice a_0 is word 0, etc.
111 * The table has 2 * 16 elements, starting with the following:
112 * index | bits | point
113 * ------+---------+------------------------------
116 * 2 | 0 0 1 0 | 2^56G
117 * 3 | 0 0 1 1 | (2^56 + 1)G
118 * 4 | 0 1 0 0 | 2^112G
119 * 5 | 0 1 0 1 | (2^112 + 1)G
120 * 6 | 0 1 1 0 | (2^112 + 2^56)G
121 * 7 | 0 1 1 1 | (2^112 + 2^56 + 1)G
122 * 8 | 1 0 0 0 | 2^168G
123 * 9 | 1 0 0 1 | (2^168 + 1)G
124 * 10 | 1 0 1 0 | (2^168 + 2^56)G
125 * 11 | 1 0 1 1 | (2^168 + 2^56 + 1)G
126 * 12 | 1 1 0 0 | (2^168 + 2^112)G
127 * 13 | 1 1 0 1 | (2^168 + 2^112 + 1)G
128 * 14 | 1 1 1 0 | (2^168 + 2^112 + 2^56)G
129 * 15 | 1 1 1 1 | (2^168 + 2^112 + 2^56 + 1)G
130 * followed by a copy of this with each element multiplied by 2^28.
132 * The reason for this is so that we can clock bits into four different
133 * locations when doing simple scalar multiplies against the base point,
134 * and then another four locations using the second 16 elements.
136 static const felem gmul[2][16][3] = {
140 {{0x3280d6115c1d21, 0xc1d356c2112234, 0x7f321390b94a03, 0xb70e0cbd6bb4bf},
141 {0xd5819985007e34, 0x75a05a07476444, 0xfb4c22dfe6cd43, 0xbd376388b5f723},
143 {{0xfd9675666ebbe9, 0xbca7664d40ce5e, 0x2242df8d8a2a43, 0x1f49bbb0f99bc5},
144 {0x29e0b892dc9c43, 0xece8608436e662, 0xdc858f185310d0, 0x9812dd4eb8d321},
146 {{0x6d3e678d5d8eb8, 0x559eed1cb362f1, 0x16e9a3bbce8a3f, 0xeedcccd8c2a748},
147 {0xf19f90ed50266d, 0xabf2b4bf65f9df, 0x313865468fafec, 0x5cb379ba910a17},
149 {{0x0641966cab26e3, 0x91fb2991fab0a0, 0xefec27a4e13a0b, 0x0499aa8a5f8ebe},
150 {0x7510407766af5d, 0x84d929610d5450, 0x81d77aae82f706, 0x6916f6d4338c5b},
152 {{0xea95ac3b1f15c6, 0x086000905e82d4, 0xdd323ae4d1c8b1, 0x932b56be7685a3},
153 {0x9ef93dea25dbbf, 0x41665960f390f0, 0xfdec76dbe2a8a7, 0x523e80f019062a},
155 {{0x822fdd26732c73, 0xa01c83531b5d0f, 0x363f37347c1ba4, 0xc391b45c84725c},
156 {0xbbd5e1b2d6ad24, 0xddfbcde19dfaec, 0xc393da7e222a7f, 0x1efb7890ede244},
158 {{0x4c9e90ca217da1, 0xd11beca79159bb, 0xff8d33c2c98b7c, 0x2610b39409f849},
159 {0x44d1352ac64da0, 0xcdbb7b2c46b4fb, 0x966c079b753c89, 0xfe67e4e820b112},
161 {{0xe28cae2df5312d, 0xc71b61d16f5c6e, 0x79b7619a3e7c4c, 0x05c73240899b47},
162 {0x9f7f6382c73e3a, 0x18615165c56bda, 0x641fab2116fd56, 0x72855882b08394},
164 {{0x0469182f161c09, 0x74a98ca8d00fb5, 0xb89da93489a3e0, 0x41c98768fb0c1d},
165 {0xe5ea05fb32da81, 0x3dce9ffbca6855, 0x1cfe2d3fbf59e6, 0x0e5e03408738a7},
167 {{0xdab22b2333e87f, 0x4430137a5dd2f6, 0xe03ab9f738beb8, 0xcb0c5d0dc34f24},
168 {0x764a7df0c8fda5, 0x185ba5c3fa2044, 0x9281d688bcbe50, 0xc40331df893881},
170 {{0xb89530796f0f60, 0xade92bd26909a3, 0x1a0c83fb4884da, 0x1765bf22a5a984},
171 {0x772a9ee75db09e, 0x23bc6c67cec16f, 0x4c1edba8b14e2f, 0xe2a215d9611369},
173 {{0x571e509fb5efb3, 0xade88696410552, 0xc8ae85fada74fe, 0x6c7e4be83bbde3},
174 {0xff9f51160f4652, 0xb47ce2495a6539, 0xa2946c53b582f4, 0x286d2db3ee9a60},
176 {{0x40bbd5081a44af, 0x0995183b13926c, 0xbcefba6f47f6d0, 0x215619e9cc0057},
177 {0x8bc94d3b0df45e, 0xf11c54a3694f6f, 0x8631b93cdfe8b5, 0xe7e3f4b0982db9},
179 {{0xb17048ab3e1c7b, 0xac38f36ff8a1d8, 0x1c29819435d2c6, 0xc813132f4c07e9},
180 {0x2891425503b11f, 0x08781030579fea, 0xf5426ba5cc9674, 0x1e28ebf18562bc},
182 {{0x9f31997cc864eb, 0x06cd91d28b5e4c, 0xff17036691a973, 0xf1aef351497c58},
183 {0xdd1f2d600564ff, 0xdead073b1402db, 0x74a684435bd693, 0xeea7471f962558},
188 {{0x9665266dddf554, 0x9613d78b60ef2d, 0xce27a34cdba417, 0xd35ab74d6afc31},
189 {0x85ccdd22deb15e, 0x2137e5783a6aab, 0xa141cffd8c93c6, 0x355a1830e90f2d},
191 {{0x1a494eadaade65, 0xd6da4da77fe53c, 0xe7992996abec86, 0x65c3553c6090e3},
192 {0xfa610b1fb09346, 0xf1c6540b8a4aaf, 0xc51a13ccd3cbab, 0x02995b1b18c28a},
194 {{0x7874568e7295ef, 0x86b419fbe38d04, 0xdc0690a7550d9a, 0xd3966a44beac33},
195 {0x2b7280ec29132f, 0xbeaa3b6a032df3, 0xdc7dd88ae41200, 0xd25e2513e3a100},
197 {{0x924857eb2efafd, 0xac2bce41223190, 0x8edaa1445553fc, 0x825800fd3562d5},
198 {0x8d79148ea96621, 0x23a01c3dd9ed8d, 0xaf8b219f9416b5, 0xd8db0cc277daea},
200 {{0x76a9c3b1a700f0, 0xe9acd29bc7e691, 0x69212d1a6b0327, 0x6322e97fe154be},
201 {0x469fc5465d62aa, 0x8d41ed18883b05, 0x1f8eae66c52b88, 0xe4fcbe9325be51},
203 {{0x825fdf583cac16, 0x020b857c7b023a, 0x683c17744b0165, 0x14ffd0a2daf2f1},
204 {0x323b36184218f9, 0x4944ec4e3b47d4, 0xc15b3080841acf, 0x0bced4b01a28bb},
206 {{0x92ac22230df5c4, 0x52f33b4063eda8, 0xcb3f19870c0c93, 0x40064f2ba65233},
207 {0xfe16f0924f8992, 0x012da25af5b517, 0x1a57bb24f723a6, 0x06f8bc76760def},
209 {{0x4a7084f7817cb9, 0xbcab0738ee9a78, 0x3ec11e11d9c326, 0xdc0fe90e0f1aae},
210 {0xcf639ea5f98390, 0x5c350aa22ffb74, 0x9afae98a4047b7, 0x956ec2d617fc45},
212 {{0x4306d648c1be6a, 0x9247cd8bc9a462, 0xf5595e377d2f2e, 0xbd1c3caff1a52e},
213 {0x045e14472409d0, 0x29f3e17078f773, 0x745a602b2d4f7d, 0x191837685cdfbb},
215 {{0x5b6ee254a8cb79, 0x4953433f5e7026, 0xe21faeb1d1def4, 0xc4c225785c09de},
216 {0x307ce7bba1e518, 0x31b125b1036db8, 0x47e91868839e8f, 0xc765866e33b9f3},
218 {{0x3bfece24f96906, 0x4794da641e5093, 0xde5df64f95db26, 0x297ecd89714b05},
219 {0x701bd3ebb2c3aa, 0x7073b4f53cb1d5, 0x13c5665658af16, 0x9895089d66fe58},
221 {{0x0fef05f78c4790, 0x2d773633b05d2e, 0x94229c3a951c94, 0xbbbd70df4911bb},
222 {0xb2c6963d2c1168, 0x105f47a72b0d73, 0x9fdf6111614080, 0x7b7e94b39e67b0},
224 {{0xad1a7d6efbe2b3, 0xf012482c0da69d, 0x6b3bdf12438345, 0x40d7558d7aa4d9},
225 {0x8a09fffb5c6d3d, 0x9a356e5d9ffd38, 0x5973f15f4f9b1c, 0xdcd5f59f63c3ea},
227 {{0xacf39f4c5ca7ab, 0x4c8071cc5fd737, 0xc64e3602cd1184, 0x0acd4644c9abba},
228 {0x6c011a36d8bf6e, 0xfecd87ba24e32a, 0x19f6f56574fad8, 0x050b204ced9405},
230 {{0xed4f1cae7d9a96, 0x5ceef7ad94c40a, 0x778e4a3bf3ef9b, 0x7405783dc3b55e},
231 {0x32477c61b6e8c6, 0xb46a97570f018b, 0x91176d0a7e95d1, 0x3df90fbc4c7d0e},
235 /* Precomputation for the group generator. */
236 struct nistp224_pre_comp_st {
237 felem g_pre_comp[2][16][3];
238 CRYPTO_REF_COUNT references;
242 const EC_METHOD *EC_GFp_nistp224_method(void)
244 static const EC_METHOD ret = {
245 EC_FLAGS_DEFAULT_OCT,
246 NID_X9_62_prime_field,
247 ec_GFp_nistp224_group_init,
248 ec_GFp_simple_group_finish,
249 ec_GFp_simple_group_clear_finish,
250 ec_GFp_nist_group_copy,
251 ec_GFp_nistp224_group_set_curve,
252 ec_GFp_simple_group_get_curve,
253 ec_GFp_simple_group_get_degree,
254 ec_group_simple_order_bits,
255 ec_GFp_simple_group_check_discriminant,
256 ec_GFp_simple_point_init,
257 ec_GFp_simple_point_finish,
258 ec_GFp_simple_point_clear_finish,
259 ec_GFp_simple_point_copy,
260 ec_GFp_simple_point_set_to_infinity,
261 ec_GFp_simple_set_Jprojective_coordinates_GFp,
262 ec_GFp_simple_get_Jprojective_coordinates_GFp,
263 ec_GFp_simple_point_set_affine_coordinates,
264 ec_GFp_nistp224_point_get_affine_coordinates,
265 0 /* point_set_compressed_coordinates */ ,
270 ec_GFp_simple_invert,
271 ec_GFp_simple_is_at_infinity,
272 ec_GFp_simple_is_on_curve,
274 ec_GFp_simple_make_affine,
275 ec_GFp_simple_points_make_affine,
276 ec_GFp_nistp224_points_mul,
277 ec_GFp_nistp224_precompute_mult,
278 ec_GFp_nistp224_have_precompute_mult,
279 ec_GFp_nist_field_mul,
280 ec_GFp_nist_field_sqr,
282 ec_GFp_simple_field_inv,
283 0 /* field_encode */ ,
284 0 /* field_decode */ ,
285 0, /* field_set_to_one */
286 ec_key_simple_priv2oct,
287 ec_key_simple_oct2priv,
289 ec_key_simple_generate_key,
290 ec_key_simple_check_key,
291 ec_key_simple_generate_public_key,
294 ecdh_simple_compute_key,
295 0, /* field_inverse_mod_ord */
296 0, /* blind_coordinates */
306 * Helper functions to convert field elements to/from internal representation
308 static void bin28_to_felem(felem out, const u8 in[28])
310 out[0] = *((const uint64_t *)(in)) & 0x00ffffffffffffff;
311 out[1] = (*((const uint64_t *)(in + 7))) & 0x00ffffffffffffff;
312 out[2] = (*((const uint64_t *)(in + 14))) & 0x00ffffffffffffff;
313 out[3] = (*((const uint64_t *)(in+20))) >> 8;
316 static void felem_to_bin28(u8 out[28], const felem in)
319 for (i = 0; i < 7; ++i) {
320 out[i] = in[0] >> (8 * i);
321 out[i + 7] = in[1] >> (8 * i);
322 out[i + 14] = in[2] >> (8 * i);
323 out[i + 21] = in[3] >> (8 * i);
327 /* From OpenSSL BIGNUM to internal representation */
328 static int BN_to_felem(felem out, const BIGNUM *bn)
330 felem_bytearray b_out;
333 if (BN_is_negative(bn)) {
334 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
337 num_bytes = BN_bn2lebinpad(bn, b_out, sizeof(b_out));
339 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
342 bin28_to_felem(out, b_out);
346 /* From internal representation to OpenSSL BIGNUM */
347 static BIGNUM *felem_to_BN(BIGNUM *out, const felem in)
349 felem_bytearray b_out;
350 felem_to_bin28(b_out, in);
351 return BN_lebin2bn(b_out, sizeof(b_out), out);
354 /******************************************************************************/
358 * Field operations, using the internal representation of field elements.
359 * NB! These operations are specific to our point multiplication and cannot be
360 * expected to be correct in general - e.g., multiplication with a large scalar
361 * will cause an overflow.
365 static void felem_one(felem out)
373 static void felem_assign(felem out, const felem in)
381 /* Sum two field elements: out += in */
382 static void felem_sum(felem out, const felem in)
390 /* Subtract field elements: out -= in */
391 /* Assumes in[i] < 2^57 */
392 static void felem_diff(felem out, const felem in)
394 static const limb two58p2 = (((limb) 1) << 58) + (((limb) 1) << 2);
395 static const limb two58m2 = (((limb) 1) << 58) - (((limb) 1) << 2);
396 static const limb two58m42m2 = (((limb) 1) << 58) -
397 (((limb) 1) << 42) - (((limb) 1) << 2);
399 /* Add 0 mod 2^224-2^96+1 to ensure out > in */
401 out[1] += two58m42m2;
411 /* Subtract in unreduced 128-bit mode: out -= in */
412 /* Assumes in[i] < 2^119 */
413 static void widefelem_diff(widefelem out, const widefelem in)
415 static const widelimb two120 = ((widelimb) 1) << 120;
416 static const widelimb two120m64 = (((widelimb) 1) << 120) -
417 (((widelimb) 1) << 64);
418 static const widelimb two120m104m64 = (((widelimb) 1) << 120) -
419 (((widelimb) 1) << 104) - (((widelimb) 1) << 64);
421 /* Add 0 mod 2^224-2^96+1 to ensure out > in */
426 out[4] += two120m104m64;
439 /* Subtract in mixed mode: out128 -= in64 */
441 static void felem_diff_128_64(widefelem out, const felem in)
443 static const widelimb two64p8 = (((widelimb) 1) << 64) +
444 (((widelimb) 1) << 8);
445 static const widelimb two64m8 = (((widelimb) 1) << 64) -
446 (((widelimb) 1) << 8);
447 static const widelimb two64m48m8 = (((widelimb) 1) << 64) -
448 (((widelimb) 1) << 48) - (((widelimb) 1) << 8);
450 /* Add 0 mod 2^224-2^96+1 to ensure out > in */
452 out[1] += two64m48m8;
463 * Multiply a field element by a scalar: out = out * scalar The scalars we
464 * actually use are small, so results fit without overflow
466 static void felem_scalar(felem out, const limb scalar)
475 * Multiply an unreduced field element by a scalar: out = out * scalar The
476 * scalars we actually use are small, so results fit without overflow
478 static void widefelem_scalar(widefelem out, const widelimb scalar)
489 /* Square a field element: out = in^2 */
490 static void felem_square(widefelem out, const felem in)
492 limb tmp0, tmp1, tmp2;
496 out[0] = ((widelimb) in[0]) * in[0];
497 out[1] = ((widelimb) in[0]) * tmp1;
498 out[2] = ((widelimb) in[0]) * tmp2 + ((widelimb) in[1]) * in[1];
499 out[3] = ((widelimb) in[3]) * tmp0 + ((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);
603 * Reduce to unique minimal representation. Requires 0 <= in < 2*p (always
604 * call felem_reduce first)
606 static void felem_contract(felem out, const felem in)
608 static const int64_t two56 = ((limb) 1) << 56;
609 /* 0 <= in < 2*p, p = 2^224 - 2^96 + 1 */
610 /* if in > p , reduce in = in - 2^224 + 2^96 - 1 */
616 /* Case 1: a = 1 iff in >= 2^224 */
620 tmp[3] &= 0x00ffffffffffffff;
622 * Case 2: a = 0 iff p <= in < 2^224, i.e., the high 128 bits are all 1
623 * and the lower part is non-zero
625 a = ((in[3] & in[2] & (in[1] | 0x000000ffffffffff)) + 1) |
626 (((int64_t) (in[0] + (in[1] & 0x000000ffffffffff)) - 1) >> 63);
627 a &= 0x00ffffffffffffff;
628 /* turn a into an all-one mask (if a = 0) or an all-zero mask */
630 /* subtract 2^224 - 2^96 + 1 if a is all-one */
631 tmp[3] &= a ^ 0xffffffffffffffff;
632 tmp[2] &= a ^ 0xffffffffffffffff;
633 tmp[1] &= (a ^ 0xffffffffffffffff) | 0x000000ffffffffff;
637 * eliminate negative coefficients: if tmp[0] is negative, tmp[1] must be
638 * non-zero, so we only need one step
644 /* carry 1 -> 2 -> 3 */
645 tmp[2] += tmp[1] >> 56;
646 tmp[1] &= 0x00ffffffffffffff;
648 tmp[3] += tmp[2] >> 56;
649 tmp[2] &= 0x00ffffffffffffff;
651 /* Now 0 <= out < p */
659 * Get negative value: out = -in
660 * Requires in[i] < 2^63,
661 * ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] <= 2^56 + 2^16
663 static void felem_neg(felem out, const felem in)
666 felem_diff_128_64(tmp, in);
667 felem_reduce(out, tmp);
671 * Zero-check: returns 1 if input is 0, and 0 otherwise. We know that field
672 * elements are reduced to in < 2^225, so we only need to check three cases:
673 * 0, 2^224 - 2^96 + 1, and 2^225 - 2^97 + 2
675 static limb felem_is_zero(const felem in)
677 limb zero, two224m96p1, two225m97p2;
679 zero = in[0] | in[1] | in[2] | in[3];
680 zero = (((int64_t) (zero) - 1) >> 63) & 1;
681 two224m96p1 = (in[0] ^ 1) | (in[1] ^ 0x00ffff0000000000)
682 | (in[2] ^ 0x00ffffffffffffff) | (in[3] ^ 0x00ffffffffffffff);
683 two224m96p1 = (((int64_t) (two224m96p1) - 1) >> 63) & 1;
684 two225m97p2 = (in[0] ^ 2) | (in[1] ^ 0x00fffe0000000000)
685 | (in[2] ^ 0x00ffffffffffffff) | (in[3] ^ 0x01ffffffffffffff);
686 two225m97p2 = (((int64_t) (two225m97p2) - 1) >> 63) & 1;
687 return (zero | two224m96p1 | two225m97p2);
690 static int felem_is_zero_int(const void *in)
692 return (int)(felem_is_zero(in) & ((limb) 1));
695 /* Invert a field element */
696 /* Computation chain copied from djb's code */
697 static void felem_inv(felem out, const felem in)
699 felem ftmp, ftmp2, ftmp3, ftmp4;
703 felem_square(tmp, in);
704 felem_reduce(ftmp, tmp); /* 2 */
705 felem_mul(tmp, in, ftmp);
706 felem_reduce(ftmp, tmp); /* 2^2 - 1 */
707 felem_square(tmp, ftmp);
708 felem_reduce(ftmp, tmp); /* 2^3 - 2 */
709 felem_mul(tmp, in, ftmp);
710 felem_reduce(ftmp, tmp); /* 2^3 - 1 */
711 felem_square(tmp, ftmp);
712 felem_reduce(ftmp2, tmp); /* 2^4 - 2 */
713 felem_square(tmp, ftmp2);
714 felem_reduce(ftmp2, tmp); /* 2^5 - 4 */
715 felem_square(tmp, ftmp2);
716 felem_reduce(ftmp2, tmp); /* 2^6 - 8 */
717 felem_mul(tmp, ftmp2, ftmp);
718 felem_reduce(ftmp, tmp); /* 2^6 - 1 */
719 felem_square(tmp, ftmp);
720 felem_reduce(ftmp2, tmp); /* 2^7 - 2 */
721 for (i = 0; i < 5; ++i) { /* 2^12 - 2^6 */
722 felem_square(tmp, ftmp2);
723 felem_reduce(ftmp2, tmp);
725 felem_mul(tmp, ftmp2, ftmp);
726 felem_reduce(ftmp2, tmp); /* 2^12 - 1 */
727 felem_square(tmp, ftmp2);
728 felem_reduce(ftmp3, tmp); /* 2^13 - 2 */
729 for (i = 0; i < 11; ++i) { /* 2^24 - 2^12 */
730 felem_square(tmp, ftmp3);
731 felem_reduce(ftmp3, tmp);
733 felem_mul(tmp, ftmp3, ftmp2);
734 felem_reduce(ftmp2, tmp); /* 2^24 - 1 */
735 felem_square(tmp, ftmp2);
736 felem_reduce(ftmp3, tmp); /* 2^25 - 2 */
737 for (i = 0; i < 23; ++i) { /* 2^48 - 2^24 */
738 felem_square(tmp, ftmp3);
739 felem_reduce(ftmp3, tmp);
741 felem_mul(tmp, ftmp3, ftmp2);
742 felem_reduce(ftmp3, tmp); /* 2^48 - 1 */
743 felem_square(tmp, ftmp3);
744 felem_reduce(ftmp4, tmp); /* 2^49 - 2 */
745 for (i = 0; i < 47; ++i) { /* 2^96 - 2^48 */
746 felem_square(tmp, ftmp4);
747 felem_reduce(ftmp4, tmp);
749 felem_mul(tmp, ftmp3, ftmp4);
750 felem_reduce(ftmp3, tmp); /* 2^96 - 1 */
751 felem_square(tmp, ftmp3);
752 felem_reduce(ftmp4, tmp); /* 2^97 - 2 */
753 for (i = 0; i < 23; ++i) { /* 2^120 - 2^24 */
754 felem_square(tmp, ftmp4);
755 felem_reduce(ftmp4, tmp);
757 felem_mul(tmp, ftmp2, ftmp4);
758 felem_reduce(ftmp2, tmp); /* 2^120 - 1 */
759 for (i = 0; i < 6; ++i) { /* 2^126 - 2^6 */
760 felem_square(tmp, ftmp2);
761 felem_reduce(ftmp2, tmp);
763 felem_mul(tmp, ftmp2, ftmp);
764 felem_reduce(ftmp, tmp); /* 2^126 - 1 */
765 felem_square(tmp, ftmp);
766 felem_reduce(ftmp, tmp); /* 2^127 - 2 */
767 felem_mul(tmp, ftmp, in);
768 felem_reduce(ftmp, tmp); /* 2^127 - 1 */
769 for (i = 0; i < 97; ++i) { /* 2^224 - 2^97 */
770 felem_square(tmp, ftmp);
771 felem_reduce(ftmp, tmp);
773 felem_mul(tmp, ftmp, ftmp3);
774 felem_reduce(out, tmp); /* 2^224 - 2^96 - 1 */
778 * Copy in constant time: if icopy == 1, copy in to out, if icopy == 0, copy
781 static void copy_conditional(felem out, const felem in, limb icopy)
785 * icopy is a (64-bit) 0 or 1, so copy is either all-zero or all-one
787 const limb copy = -icopy;
788 for (i = 0; i < 4; ++i) {
789 const limb tmp = copy & (in[i] ^ out[i]);
794 /******************************************************************************/
796 * ELLIPTIC CURVE POINT OPERATIONS
798 * Points are represented in Jacobian projective coordinates:
799 * (X, Y, Z) corresponds to the affine point (X/Z^2, Y/Z^3),
800 * or to the point at infinity if Z == 0.
805 * Double an elliptic curve point:
806 * (X', Y', Z') = 2 * (X, Y, Z), where
807 * X' = (3 * (X - Z^2) * (X + Z^2))^2 - 8 * X * Y^2
808 * Y' = 3 * (X - Z^2) * (X + Z^2) * (4 * X * Y^2 - X') - 8 * Y^4
809 * Z' = (Y + Z)^2 - Y^2 - Z^2 = 2 * Y * Z
810 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed,
811 * while x_out == y_in is not (maybe this works, but it's not tested).
814 point_double(felem x_out, felem y_out, felem z_out,
815 const felem x_in, const felem y_in, const felem z_in)
818 felem delta, gamma, beta, alpha, ftmp, ftmp2;
820 felem_assign(ftmp, x_in);
821 felem_assign(ftmp2, x_in);
824 felem_square(tmp, z_in);
825 felem_reduce(delta, tmp);
828 felem_square(tmp, y_in);
829 felem_reduce(gamma, tmp);
832 felem_mul(tmp, x_in, gamma);
833 felem_reduce(beta, tmp);
835 /* alpha = 3*(x-delta)*(x+delta) */
836 felem_diff(ftmp, delta);
837 /* ftmp[i] < 2^57 + 2^58 + 2 < 2^59 */
838 felem_sum(ftmp2, delta);
839 /* ftmp2[i] < 2^57 + 2^57 = 2^58 */
840 felem_scalar(ftmp2, 3);
841 /* ftmp2[i] < 3 * 2^58 < 2^60 */
842 felem_mul(tmp, ftmp, ftmp2);
843 /* tmp[i] < 2^60 * 2^59 * 4 = 2^121 */
844 felem_reduce(alpha, tmp);
846 /* x' = alpha^2 - 8*beta */
847 felem_square(tmp, alpha);
848 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
849 felem_assign(ftmp, beta);
850 felem_scalar(ftmp, 8);
851 /* ftmp[i] < 8 * 2^57 = 2^60 */
852 felem_diff_128_64(tmp, ftmp);
853 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
854 felem_reduce(x_out, tmp);
856 /* z' = (y + z)^2 - gamma - delta */
857 felem_sum(delta, gamma);
858 /* delta[i] < 2^57 + 2^57 = 2^58 */
859 felem_assign(ftmp, y_in);
860 felem_sum(ftmp, z_in);
861 /* ftmp[i] < 2^57 + 2^57 = 2^58 */
862 felem_square(tmp, ftmp);
863 /* tmp[i] < 4 * 2^58 * 2^58 = 2^118 */
864 felem_diff_128_64(tmp, delta);
865 /* tmp[i] < 2^118 + 2^64 + 8 < 2^119 */
866 felem_reduce(z_out, tmp);
868 /* y' = alpha*(4*beta - x') - 8*gamma^2 */
869 felem_scalar(beta, 4);
870 /* beta[i] < 4 * 2^57 = 2^59 */
871 felem_diff(beta, x_out);
872 /* beta[i] < 2^59 + 2^58 + 2 < 2^60 */
873 felem_mul(tmp, alpha, beta);
874 /* tmp[i] < 4 * 2^57 * 2^60 = 2^119 */
875 felem_square(tmp2, gamma);
876 /* tmp2[i] < 4 * 2^57 * 2^57 = 2^116 */
877 widefelem_scalar(tmp2, 8);
878 /* tmp2[i] < 8 * 2^116 = 2^119 */
879 widefelem_diff(tmp, tmp2);
880 /* tmp[i] < 2^119 + 2^120 < 2^121 */
881 felem_reduce(y_out, tmp);
885 * Add two elliptic curve points:
886 * (X_1, Y_1, Z_1) + (X_2, Y_2, Z_2) = (X_3, Y_3, Z_3), where
887 * X_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1)^2 - (Z_1^2 * X_2 - Z_2^2 * X_1)^3 -
888 * 2 * Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^2
889 * 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) -
890 * Z_2^3 * Y_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^3
891 * Z_3 = (Z_1^2 * X_2 - Z_2^2 * X_1) * (Z_1 * Z_2)
893 * This runs faster if 'mixed' is set, which requires Z_2 = 1 or Z_2 = 0.
897 * This function is not entirely constant-time: it includes a branch for
898 * checking whether the two input points are equal, (while not equal to the
899 * point at infinity). This case never happens during single point
900 * multiplication, so there is no timing leak for ECDH or ECDSA signing.
902 static void point_add(felem x3, felem y3, felem z3,
903 const felem x1, const felem y1, const felem z1,
904 const int mixed, const felem x2, const felem y2,
907 felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, x_out, y_out, z_out;
909 limb z1_is_zero, z2_is_zero, x_equal, y_equal;
913 felem_square(tmp, z2);
914 felem_reduce(ftmp2, tmp);
917 felem_mul(tmp, ftmp2, z2);
918 felem_reduce(ftmp4, tmp);
920 /* ftmp4 = z2^3*y1 */
921 felem_mul(tmp2, ftmp4, y1);
922 felem_reduce(ftmp4, tmp2);
924 /* ftmp2 = z2^2*x1 */
925 felem_mul(tmp2, ftmp2, x1);
926 felem_reduce(ftmp2, tmp2);
929 * We'll assume z2 = 1 (special case z2 = 0 is handled later)
932 /* ftmp4 = z2^3*y1 */
933 felem_assign(ftmp4, y1);
935 /* ftmp2 = z2^2*x1 */
936 felem_assign(ftmp2, x1);
940 felem_square(tmp, z1);
941 felem_reduce(ftmp, tmp);
944 felem_mul(tmp, ftmp, z1);
945 felem_reduce(ftmp3, tmp);
948 felem_mul(tmp, ftmp3, y2);
949 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
951 /* ftmp3 = z1^3*y2 - z2^3*y1 */
952 felem_diff_128_64(tmp, ftmp4);
953 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
954 felem_reduce(ftmp3, tmp);
957 felem_mul(tmp, ftmp, x2);
958 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
960 /* ftmp = z1^2*x2 - z2^2*x1 */
961 felem_diff_128_64(tmp, ftmp2);
962 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
963 felem_reduce(ftmp, tmp);
966 * the formulae are incorrect if the points are equal so we check for
967 * this and do doubling if this happens
969 x_equal = felem_is_zero(ftmp);
970 y_equal = felem_is_zero(ftmp3);
971 z1_is_zero = felem_is_zero(z1);
972 z2_is_zero = felem_is_zero(z2);
973 /* In affine coordinates, (X_1, Y_1) == (X_2, Y_2) */
974 if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) {
975 point_double(x3, y3, z3, x1, y1, z1);
981 felem_mul(tmp, z1, z2);
982 felem_reduce(ftmp5, tmp);
984 /* special case z2 = 0 is handled later */
985 felem_assign(ftmp5, z1);
988 /* z_out = (z1^2*x2 - z2^2*x1)*(z1*z2) */
989 felem_mul(tmp, ftmp, ftmp5);
990 felem_reduce(z_out, tmp);
992 /* ftmp = (z1^2*x2 - z2^2*x1)^2 */
993 felem_assign(ftmp5, ftmp);
994 felem_square(tmp, ftmp);
995 felem_reduce(ftmp, tmp);
997 /* ftmp5 = (z1^2*x2 - z2^2*x1)^3 */
998 felem_mul(tmp, ftmp, ftmp5);
999 felem_reduce(ftmp5, tmp);
1001 /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
1002 felem_mul(tmp, ftmp2, ftmp);
1003 felem_reduce(ftmp2, tmp);
1005 /* tmp = z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */
1006 felem_mul(tmp, ftmp4, ftmp5);
1007 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
1009 /* tmp2 = (z1^3*y2 - z2^3*y1)^2 */
1010 felem_square(tmp2, ftmp3);
1011 /* tmp2[i] < 4 * 2^57 * 2^57 < 2^116 */
1013 /* tmp2 = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 */
1014 felem_diff_128_64(tmp2, ftmp5);
1015 /* tmp2[i] < 2^116 + 2^64 + 8 < 2^117 */
1017 /* ftmp5 = 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
1018 felem_assign(ftmp5, ftmp2);
1019 felem_scalar(ftmp5, 2);
1020 /* ftmp5[i] < 2 * 2^57 = 2^58 */
1023 * x_out = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 -
1024 * 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2
1026 felem_diff_128_64(tmp2, ftmp5);
1027 /* tmp2[i] < 2^117 + 2^64 + 8 < 2^118 */
1028 felem_reduce(x_out, tmp2);
1030 /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out */
1031 felem_diff(ftmp2, x_out);
1032 /* ftmp2[i] < 2^57 + 2^58 + 2 < 2^59 */
1035 * tmp2 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out)
1037 felem_mul(tmp2, ftmp3, ftmp2);
1038 /* tmp2[i] < 4 * 2^57 * 2^59 = 2^118 */
1041 * y_out = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) -
1042 * z2^3*y1*(z1^2*x2 - z2^2*x1)^3
1044 widefelem_diff(tmp2, tmp);
1045 /* tmp2[i] < 2^118 + 2^120 < 2^121 */
1046 felem_reduce(y_out, tmp2);
1049 * the result (x_out, y_out, z_out) is incorrect if one of the inputs is
1050 * the point at infinity, so we need to check for this separately
1054 * if point 1 is at infinity, copy point 2 to output, and vice versa
1056 copy_conditional(x_out, x2, z1_is_zero);
1057 copy_conditional(x_out, x1, z2_is_zero);
1058 copy_conditional(y_out, y2, z1_is_zero);
1059 copy_conditional(y_out, y1, z2_is_zero);
1060 copy_conditional(z_out, z2, z1_is_zero);
1061 copy_conditional(z_out, z1, z2_is_zero);
1062 felem_assign(x3, x_out);
1063 felem_assign(y3, y_out);
1064 felem_assign(z3, z_out);
1068 * select_point selects the |idx|th point from a precomputation table and
1070 * The pre_comp array argument should be size of |size| argument
1072 static void select_point(const u64 idx, unsigned int size,
1073 const felem pre_comp[][3], felem out[3])
1076 limb *outlimbs = &out[0][0];
1078 memset(out, 0, sizeof(*out) * 3);
1079 for (i = 0; i < size; i++) {
1080 const limb *inlimbs = &pre_comp[i][0][0];
1087 for (j = 0; j < 4 * 3; j++)
1088 outlimbs[j] |= inlimbs[j] & mask;
1092 /* get_bit returns the |i|th bit in |in| */
1093 static char get_bit(const felem_bytearray in, unsigned i)
1097 return (in[i >> 3] >> (i & 7)) & 1;
1101 * Interleaved point multiplication using precomputed point multiples: The
1102 * small point multiples 0*P, 1*P, ..., 16*P are in pre_comp[], the scalars
1103 * in scalars[]. If g_scalar is non-NULL, we also add this multiple of the
1104 * generator, using certain (large) precomputed multiples in g_pre_comp.
1105 * Output point (X, Y, Z) is stored in x_out, y_out, z_out
1107 static void batch_mul(felem x_out, felem y_out, felem z_out,
1108 const felem_bytearray scalars[],
1109 const unsigned num_points, const u8 *g_scalar,
1110 const int mixed, const felem pre_comp[][17][3],
1111 const felem g_pre_comp[2][16][3])
1115 unsigned gen_mul = (g_scalar != NULL);
1116 felem nq[3], tmp[4];
1120 /* set nq to the point at infinity */
1121 memset(nq, 0, sizeof(nq));
1124 * Loop over all scalars msb-to-lsb, interleaving additions of multiples
1125 * of the generator (two in each of the last 28 rounds) and additions of
1126 * other points multiples (every 5th round).
1128 skip = 1; /* save two point operations in the first
1130 for (i = (num_points ? 220 : 27); i >= 0; --i) {
1133 point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
1135 /* add multiples of the generator */
1136 if (gen_mul && (i <= 27)) {
1137 /* first, look 28 bits upwards */
1138 bits = get_bit(g_scalar, i + 196) << 3;
1139 bits |= get_bit(g_scalar, i + 140) << 2;
1140 bits |= get_bit(g_scalar, i + 84) << 1;
1141 bits |= get_bit(g_scalar, i + 28);
1142 /* select the point to add, in constant time */
1143 select_point(bits, 16, g_pre_comp[1], tmp);
1146 /* value 1 below is argument for "mixed" */
1147 point_add(nq[0], nq[1], nq[2],
1148 nq[0], nq[1], nq[2], 1, tmp[0], tmp[1], tmp[2]);
1150 memcpy(nq, tmp, 3 * sizeof(felem));
1154 /* second, look at the current position */
1155 bits = get_bit(g_scalar, i + 168) << 3;
1156 bits |= get_bit(g_scalar, i + 112) << 2;
1157 bits |= get_bit(g_scalar, i + 56) << 1;
1158 bits |= get_bit(g_scalar, i);
1159 /* select the point to add, in constant time */
1160 select_point(bits, 16, g_pre_comp[0], tmp);
1161 point_add(nq[0], nq[1], nq[2],
1162 nq[0], nq[1], nq[2],
1163 1 /* mixed */ , tmp[0], tmp[1], tmp[2]);
1166 /* do other additions every 5 doublings */
1167 if (num_points && (i % 5 == 0)) {
1168 /* loop over all scalars */
1169 for (num = 0; num < num_points; ++num) {
1170 bits = get_bit(scalars[num], i + 4) << 5;
1171 bits |= get_bit(scalars[num], i + 3) << 4;
1172 bits |= get_bit(scalars[num], i + 2) << 3;
1173 bits |= get_bit(scalars[num], i + 1) << 2;
1174 bits |= get_bit(scalars[num], i) << 1;
1175 bits |= get_bit(scalars[num], i - 1);
1176 ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits);
1178 /* select the point to add or subtract */
1179 select_point(digit, 17, pre_comp[num], tmp);
1180 felem_neg(tmp[3], tmp[1]); /* (X, -Y, Z) is the negative
1182 copy_conditional(tmp[1], tmp[3], sign);
1185 point_add(nq[0], nq[1], nq[2],
1186 nq[0], nq[1], nq[2],
1187 mixed, tmp[0], tmp[1], tmp[2]);
1189 memcpy(nq, tmp, 3 * sizeof(felem));
1195 felem_assign(x_out, nq[0]);
1196 felem_assign(y_out, nq[1]);
1197 felem_assign(z_out, nq[2]);
1200 /******************************************************************************/
1202 * FUNCTIONS TO MANAGE PRECOMPUTATION
1205 static NISTP224_PRE_COMP *nistp224_pre_comp_new(void)
1207 NISTP224_PRE_COMP *ret = OPENSSL_zalloc(sizeof(*ret));
1210 ECerr(EC_F_NISTP224_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1214 ret->references = 1;
1216 ret->lock = CRYPTO_THREAD_lock_new();
1217 if (ret->lock == NULL) {
1218 ECerr(EC_F_NISTP224_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1225 NISTP224_PRE_COMP *EC_nistp224_pre_comp_dup(NISTP224_PRE_COMP *p)
1229 CRYPTO_UP_REF(&p->references, &i, p->lock);
1233 void EC_nistp224_pre_comp_free(NISTP224_PRE_COMP *p)
1240 CRYPTO_DOWN_REF(&p->references, &i, p->lock);
1241 REF_PRINT_COUNT("EC_nistp224", x);
1244 REF_ASSERT_ISNT(i < 0);
1246 CRYPTO_THREAD_lock_free(p->lock);
1250 /******************************************************************************/
1252 * OPENSSL EC_METHOD FUNCTIONS
1255 int ec_GFp_nistp224_group_init(EC_GROUP *group)
1258 ret = ec_GFp_simple_group_init(group);
1259 group->a_is_minus3 = 1;
1263 int ec_GFp_nistp224_group_set_curve(EC_GROUP *group, const BIGNUM *p,
1264 const BIGNUM *a, const BIGNUM *b,
1268 BN_CTX *new_ctx = NULL;
1269 BIGNUM *curve_p, *curve_a, *curve_b;
1272 if ((ctx = new_ctx = BN_CTX_new()) == NULL)
1275 curve_p = BN_CTX_get(ctx);
1276 curve_a = BN_CTX_get(ctx);
1277 curve_b = BN_CTX_get(ctx);
1278 if (curve_b == NULL)
1280 BN_bin2bn(nistp224_curve_params[0], sizeof(felem_bytearray), curve_p);
1281 BN_bin2bn(nistp224_curve_params[1], sizeof(felem_bytearray), curve_a);
1282 BN_bin2bn(nistp224_curve_params[2], sizeof(felem_bytearray), curve_b);
1283 if ((BN_cmp(curve_p, p)) || (BN_cmp(curve_a, a)) || (BN_cmp(curve_b, b))) {
1284 ECerr(EC_F_EC_GFP_NISTP224_GROUP_SET_CURVE,
1285 EC_R_WRONG_CURVE_PARAMETERS);
1288 group->field_mod_func = BN_nist_mod_224;
1289 ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx);
1292 BN_CTX_free(new_ctx);
1297 * Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') =
1300 int ec_GFp_nistp224_point_get_affine_coordinates(const EC_GROUP *group,
1301 const EC_POINT *point,
1302 BIGNUM *x, BIGNUM *y,
1305 felem z1, z2, x_in, y_in, x_out, y_out;
1308 if (EC_POINT_is_at_infinity(group, point)) {
1309 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1310 EC_R_POINT_AT_INFINITY);
1313 if ((!BN_to_felem(x_in, point->X)) || (!BN_to_felem(y_in, point->Y)) ||
1314 (!BN_to_felem(z1, point->Z)))
1317 felem_square(tmp, z2);
1318 felem_reduce(z1, tmp);
1319 felem_mul(tmp, x_in, z1);
1320 felem_reduce(x_in, tmp);
1321 felem_contract(x_out, x_in);
1323 if (!felem_to_BN(x, x_out)) {
1324 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1329 felem_mul(tmp, z1, z2);
1330 felem_reduce(z1, tmp);
1331 felem_mul(tmp, y_in, z1);
1332 felem_reduce(y_in, tmp);
1333 felem_contract(y_out, y_in);
1335 if (!felem_to_BN(y, y_out)) {
1336 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1344 static void make_points_affine(size_t num, felem points[ /* num */ ][3],
1345 felem tmp_felems[ /* num+1 */ ])
1348 * Runs in constant time, unless an input is the point at infinity (which
1349 * normally shouldn't happen).
1351 ec_GFp_nistp_points_make_affine_internal(num,
1355 (void (*)(void *))felem_one,
1357 (void (*)(void *, const void *))
1359 (void (*)(void *, const void *))
1360 felem_square_reduce, (void (*)
1367 (void (*)(void *, const void *))
1369 (void (*)(void *, const void *))
1374 * Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL
1375 * values Result is stored in r (r can equal one of the inputs).
1377 int ec_GFp_nistp224_points_mul(const EC_GROUP *group, EC_POINT *r,
1378 const BIGNUM *scalar, size_t num,
1379 const EC_POINT *points[],
1380 const BIGNUM *scalars[], BN_CTX *ctx)
1386 BIGNUM *x, *y, *z, *tmp_scalar;
1387 felem_bytearray g_secret;
1388 felem_bytearray *secrets = NULL;
1389 felem (*pre_comp)[17][3] = NULL;
1390 felem *tmp_felems = NULL;
1392 int have_pre_comp = 0;
1393 size_t num_points = num;
1394 felem x_in, y_in, z_in, x_out, y_out, z_out;
1395 NISTP224_PRE_COMP *pre = NULL;
1396 const felem(*g_pre_comp)[16][3] = NULL;
1397 EC_POINT *generator = NULL;
1398 const EC_POINT *p = NULL;
1399 const BIGNUM *p_scalar = NULL;
1402 x = BN_CTX_get(ctx);
1403 y = BN_CTX_get(ctx);
1404 z = BN_CTX_get(ctx);
1405 tmp_scalar = BN_CTX_get(ctx);
1406 if (tmp_scalar == NULL)
1409 if (scalar != NULL) {
1410 pre = group->pre_comp.nistp224;
1412 /* we have precomputation, try to use it */
1413 g_pre_comp = (const felem(*)[16][3])pre->g_pre_comp;
1415 /* try to use the standard precomputation */
1416 g_pre_comp = &gmul[0];
1417 generator = EC_POINT_new(group);
1418 if (generator == NULL)
1420 /* get the generator from precomputation */
1421 if (!felem_to_BN(x, g_pre_comp[0][1][0]) ||
1422 !felem_to_BN(y, g_pre_comp[0][1][1]) ||
1423 !felem_to_BN(z, g_pre_comp[0][1][2])) {
1424 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1427 if (!EC_POINT_set_Jprojective_coordinates_GFp(group,
1431 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
1432 /* precomputation matches generator */
1436 * we don't have valid precomputation: treat the generator as a
1439 num_points = num_points + 1;
1442 if (num_points > 0) {
1443 if (num_points >= 3) {
1445 * unless we precompute multiples for just one or two points,
1446 * converting those into affine form is time well spent
1450 secrets = OPENSSL_zalloc(sizeof(*secrets) * num_points);
1451 pre_comp = OPENSSL_zalloc(sizeof(*pre_comp) * num_points);
1454 OPENSSL_malloc(sizeof(felem) * (num_points * 17 + 1));
1455 if ((secrets == NULL) || (pre_comp == NULL)
1456 || (mixed && (tmp_felems == NULL))) {
1457 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_MALLOC_FAILURE);
1462 * we treat NULL scalars as 0, and NULL points as points at infinity,
1463 * i.e., they contribute nothing to the linear combination
1465 for (i = 0; i < num_points; ++i) {
1468 p = EC_GROUP_get0_generator(group);
1471 /* the i^th point */
1473 p_scalar = scalars[i];
1475 if ((p_scalar != NULL) && (p != NULL)) {
1476 /* reduce scalar to 0 <= scalar < 2^224 */
1477 if ((BN_num_bits(p_scalar) > 224)
1478 || (BN_is_negative(p_scalar))) {
1480 * this is an unusual input, and we don't guarantee
1483 if (!BN_nnmod(tmp_scalar, p_scalar, group->order, ctx)) {
1484 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1487 num_bytes = BN_bn2lebinpad(tmp_scalar,
1488 secrets[i], sizeof(secrets[i]));
1490 num_bytes = BN_bn2lebinpad(p_scalar,
1491 secrets[i], sizeof(secrets[i]));
1493 if (num_bytes < 0) {
1494 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1497 /* precompute multiples */
1498 if ((!BN_to_felem(x_out, p->X)) ||
1499 (!BN_to_felem(y_out, p->Y)) ||
1500 (!BN_to_felem(z_out, p->Z)))
1502 felem_assign(pre_comp[i][1][0], x_out);
1503 felem_assign(pre_comp[i][1][1], y_out);
1504 felem_assign(pre_comp[i][1][2], z_out);
1505 for (j = 2; j <= 16; ++j) {
1507 point_add(pre_comp[i][j][0], pre_comp[i][j][1],
1508 pre_comp[i][j][2], pre_comp[i][1][0],
1509 pre_comp[i][1][1], pre_comp[i][1][2], 0,
1510 pre_comp[i][j - 1][0],
1511 pre_comp[i][j - 1][1],
1512 pre_comp[i][j - 1][2]);
1514 point_double(pre_comp[i][j][0], pre_comp[i][j][1],
1515 pre_comp[i][j][2], pre_comp[i][j / 2][0],
1516 pre_comp[i][j / 2][1],
1517 pre_comp[i][j / 2][2]);
1523 make_points_affine(num_points * 17, pre_comp[0], tmp_felems);
1526 /* the scalar for the generator */
1527 if ((scalar != NULL) && (have_pre_comp)) {
1528 memset(g_secret, 0, sizeof(g_secret));
1529 /* reduce scalar to 0 <= scalar < 2^224 */
1530 if ((BN_num_bits(scalar) > 224) || (BN_is_negative(scalar))) {
1532 * this is an unusual input, and we don't guarantee
1535 if (!BN_nnmod(tmp_scalar, scalar, group->order, ctx)) {
1536 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1539 num_bytes = BN_bn2lebinpad(tmp_scalar, g_secret, sizeof(g_secret));
1541 num_bytes = BN_bn2lebinpad(scalar, g_secret, sizeof(g_secret));
1543 /* do the multiplication with generator precomputation */
1544 batch_mul(x_out, y_out, z_out,
1545 (const felem_bytearray(*))secrets, num_points,
1547 mixed, (const felem(*)[17][3])pre_comp, g_pre_comp);
1549 /* do the multiplication without generator precomputation */
1550 batch_mul(x_out, y_out, z_out,
1551 (const felem_bytearray(*))secrets, num_points,
1552 NULL, mixed, (const felem(*)[17][3])pre_comp, NULL);
1554 /* reduce the output to its unique minimal representation */
1555 felem_contract(x_in, x_out);
1556 felem_contract(y_in, y_out);
1557 felem_contract(z_in, z_out);
1558 if ((!felem_to_BN(x, x_in)) || (!felem_to_BN(y, y_in)) ||
1559 (!felem_to_BN(z, z_in))) {
1560 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1563 ret = EC_POINT_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx);
1567 EC_POINT_free(generator);
1568 OPENSSL_free(secrets);
1569 OPENSSL_free(pre_comp);
1570 OPENSSL_free(tmp_felems);
1574 int ec_GFp_nistp224_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
1577 NISTP224_PRE_COMP *pre = NULL;
1579 BN_CTX *new_ctx = NULL;
1581 EC_POINT *generator = NULL;
1582 felem tmp_felems[32];
1584 /* throw away old precomputation */
1585 EC_pre_comp_free(group);
1587 if ((ctx = new_ctx = BN_CTX_new()) == NULL)
1590 x = BN_CTX_get(ctx);
1591 y = BN_CTX_get(ctx);
1594 /* get the generator */
1595 if (group->generator == NULL)
1597 generator = EC_POINT_new(group);
1598 if (generator == NULL)
1600 BN_bin2bn(nistp224_curve_params[3], sizeof(felem_bytearray), x);
1601 BN_bin2bn(nistp224_curve_params[4], sizeof(felem_bytearray), y);
1602 if (!EC_POINT_set_affine_coordinates(group, generator, x, y, ctx))
1604 if ((pre = nistp224_pre_comp_new()) == NULL)
1607 * if the generator is the standard one, use built-in precomputation
1609 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) {
1610 memcpy(pre->g_pre_comp, gmul, sizeof(pre->g_pre_comp));
1613 if ((!BN_to_felem(pre->g_pre_comp[0][1][0], group->generator->X)) ||
1614 (!BN_to_felem(pre->g_pre_comp[0][1][1], group->generator->Y)) ||
1615 (!BN_to_felem(pre->g_pre_comp[0][1][2], group->generator->Z)))
1618 * compute 2^56*G, 2^112*G, 2^168*G for the first table, 2^28*G, 2^84*G,
1619 * 2^140*G, 2^196*G for the second one
1621 for (i = 1; i <= 8; i <<= 1) {
1622 point_double(pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1],
1623 pre->g_pre_comp[1][i][2], pre->g_pre_comp[0][i][0],
1624 pre->g_pre_comp[0][i][1], pre->g_pre_comp[0][i][2]);
1625 for (j = 0; j < 27; ++j) {
1626 point_double(pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1],
1627 pre->g_pre_comp[1][i][2], pre->g_pre_comp[1][i][0],
1628 pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2]);
1632 point_double(pre->g_pre_comp[0][2 * i][0],
1633 pre->g_pre_comp[0][2 * i][1],
1634 pre->g_pre_comp[0][2 * i][2], pre->g_pre_comp[1][i][0],
1635 pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2]);
1636 for (j = 0; j < 27; ++j) {
1637 point_double(pre->g_pre_comp[0][2 * i][0],
1638 pre->g_pre_comp[0][2 * i][1],
1639 pre->g_pre_comp[0][2 * i][2],
1640 pre->g_pre_comp[0][2 * i][0],
1641 pre->g_pre_comp[0][2 * i][1],
1642 pre->g_pre_comp[0][2 * i][2]);
1645 for (i = 0; i < 2; i++) {
1646 /* g_pre_comp[i][0] is the point at infinity */
1647 memset(pre->g_pre_comp[i][0], 0, sizeof(pre->g_pre_comp[i][0]));
1648 /* the remaining multiples */
1649 /* 2^56*G + 2^112*G resp. 2^84*G + 2^140*G */
1650 point_add(pre->g_pre_comp[i][6][0], pre->g_pre_comp[i][6][1],
1651 pre->g_pre_comp[i][6][2], pre->g_pre_comp[i][4][0],
1652 pre->g_pre_comp[i][4][1], pre->g_pre_comp[i][4][2],
1653 0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
1654 pre->g_pre_comp[i][2][2]);
1655 /* 2^56*G + 2^168*G resp. 2^84*G + 2^196*G */
1656 point_add(pre->g_pre_comp[i][10][0], pre->g_pre_comp[i][10][1],
1657 pre->g_pre_comp[i][10][2], pre->g_pre_comp[i][8][0],
1658 pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2],
1659 0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
1660 pre->g_pre_comp[i][2][2]);
1661 /* 2^112*G + 2^168*G resp. 2^140*G + 2^196*G */
1662 point_add(pre->g_pre_comp[i][12][0], pre->g_pre_comp[i][12][1],
1663 pre->g_pre_comp[i][12][2], pre->g_pre_comp[i][8][0],
1664 pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2],
1665 0, pre->g_pre_comp[i][4][0], pre->g_pre_comp[i][4][1],
1666 pre->g_pre_comp[i][4][2]);
1668 * 2^56*G + 2^112*G + 2^168*G resp. 2^84*G + 2^140*G + 2^196*G
1670 point_add(pre->g_pre_comp[i][14][0], pre->g_pre_comp[i][14][1],
1671 pre->g_pre_comp[i][14][2], pre->g_pre_comp[i][12][0],
1672 pre->g_pre_comp[i][12][1], pre->g_pre_comp[i][12][2],
1673 0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
1674 pre->g_pre_comp[i][2][2]);
1675 for (j = 1; j < 8; ++j) {
1676 /* odd multiples: add G resp. 2^28*G */
1677 point_add(pre->g_pre_comp[i][2 * j + 1][0],
1678 pre->g_pre_comp[i][2 * j + 1][1],
1679 pre->g_pre_comp[i][2 * j + 1][2],
1680 pre->g_pre_comp[i][2 * j][0],
1681 pre->g_pre_comp[i][2 * j][1],
1682 pre->g_pre_comp[i][2 * j][2], 0,
1683 pre->g_pre_comp[i][1][0], pre->g_pre_comp[i][1][1],
1684 pre->g_pre_comp[i][1][2]);
1687 make_points_affine(31, &(pre->g_pre_comp[0][1]), tmp_felems);
1690 SETPRECOMP(group, nistp224, pre);
1695 EC_POINT_free(generator);
1696 BN_CTX_free(new_ctx);
1697 EC_nistp224_pre_comp_free(pre);
1701 int ec_GFp_nistp224_have_precompute_mult(const EC_GROUP *group)
1703 return HAVEPRECOMP(group, nistp224);