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 /* To preserve endianness when using BN_bn2bin and BN_bin2bn */
328 static void flip_endian(u8 *out, const u8 *in, unsigned len)
331 for (i = 0; i < len; ++i)
332 out[i] = in[len - 1 - i];
335 /* From OpenSSL BIGNUM to internal representation */
336 static int BN_to_felem(felem out, const BIGNUM *bn)
338 felem_bytearray b_in;
339 felem_bytearray b_out;
342 /* BN_bn2bin eats leading zeroes */
343 memset(b_out, 0, sizeof(b_out));
344 num_bytes = BN_num_bytes(bn);
345 if (num_bytes > sizeof(b_out)) {
346 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
349 if (BN_is_negative(bn)) {
350 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
353 num_bytes = BN_bn2bin(bn, b_in);
354 flip_endian(b_out, b_in, num_bytes);
355 bin28_to_felem(out, b_out);
359 /* From internal representation to OpenSSL BIGNUM */
360 static BIGNUM *felem_to_BN(BIGNUM *out, const felem in)
362 felem_bytearray b_in, b_out;
363 felem_to_bin28(b_in, in);
364 flip_endian(b_out, b_in, sizeof(b_out));
365 return BN_bin2bn(b_out, sizeof(b_out), out);
368 /******************************************************************************/
372 * Field operations, using the internal representation of field elements.
373 * NB! These operations are specific to our point multiplication and cannot be
374 * expected to be correct in general - e.g., multiplication with a large scalar
375 * will cause an overflow.
379 static void felem_one(felem out)
387 static void felem_assign(felem out, const felem in)
395 /* Sum two field elements: out += in */
396 static void felem_sum(felem out, const felem in)
404 /* Subtract field elements: out -= in */
405 /* Assumes in[i] < 2^57 */
406 static void felem_diff(felem out, const felem in)
408 static const limb two58p2 = (((limb) 1) << 58) + (((limb) 1) << 2);
409 static const limb two58m2 = (((limb) 1) << 58) - (((limb) 1) << 2);
410 static const limb two58m42m2 = (((limb) 1) << 58) -
411 (((limb) 1) << 42) - (((limb) 1) << 2);
413 /* Add 0 mod 2^224-2^96+1 to ensure out > in */
415 out[1] += two58m42m2;
425 /* Subtract in unreduced 128-bit mode: out -= in */
426 /* Assumes in[i] < 2^119 */
427 static void widefelem_diff(widefelem out, const widefelem in)
429 static const widelimb two120 = ((widelimb) 1) << 120;
430 static const widelimb two120m64 = (((widelimb) 1) << 120) -
431 (((widelimb) 1) << 64);
432 static const widelimb two120m104m64 = (((widelimb) 1) << 120) -
433 (((widelimb) 1) << 104) - (((widelimb) 1) << 64);
435 /* Add 0 mod 2^224-2^96+1 to ensure out > in */
440 out[4] += two120m104m64;
453 /* Subtract in mixed mode: out128 -= in64 */
455 static void felem_diff_128_64(widefelem out, const felem in)
457 static const widelimb two64p8 = (((widelimb) 1) << 64) +
458 (((widelimb) 1) << 8);
459 static const widelimb two64m8 = (((widelimb) 1) << 64) -
460 (((widelimb) 1) << 8);
461 static const widelimb two64m48m8 = (((widelimb) 1) << 64) -
462 (((widelimb) 1) << 48) - (((widelimb) 1) << 8);
464 /* Add 0 mod 2^224-2^96+1 to ensure out > in */
466 out[1] += two64m48m8;
477 * Multiply a field element by a scalar: out = out * scalar The scalars we
478 * actually use are small, so results fit without overflow
480 static void felem_scalar(felem out, const limb scalar)
489 * Multiply an unreduced field element by a scalar: out = out * scalar The
490 * scalars we actually use are small, so results fit without overflow
492 static void widefelem_scalar(widefelem out, const widelimb scalar)
503 /* Square a field element: out = in^2 */
504 static void felem_square(widefelem out, const felem in)
506 limb tmp0, tmp1, tmp2;
510 out[0] = ((widelimb) in[0]) * in[0];
511 out[1] = ((widelimb) in[0]) * tmp1;
512 out[2] = ((widelimb) in[0]) * tmp2 + ((widelimb) in[1]) * in[1];
513 out[3] = ((widelimb) in[3]) * tmp0 + ((widelimb) in[1]) * tmp2;
514 out[4] = ((widelimb) in[3]) * tmp1 + ((widelimb) in[2]) * in[2];
515 out[5] = ((widelimb) in[3]) * tmp2;
516 out[6] = ((widelimb) in[3]) * in[3];
519 /* Multiply two field elements: out = in1 * in2 */
520 static void felem_mul(widefelem out, const felem in1, const felem in2)
522 out[0] = ((widelimb) in1[0]) * in2[0];
523 out[1] = ((widelimb) in1[0]) * in2[1] + ((widelimb) in1[1]) * in2[0];
524 out[2] = ((widelimb) in1[0]) * in2[2] + ((widelimb) in1[1]) * in2[1] +
525 ((widelimb) in1[2]) * in2[0];
526 out[3] = ((widelimb) in1[0]) * in2[3] + ((widelimb) in1[1]) * in2[2] +
527 ((widelimb) in1[2]) * in2[1] + ((widelimb) in1[3]) * in2[0];
528 out[4] = ((widelimb) in1[1]) * in2[3] + ((widelimb) in1[2]) * in2[2] +
529 ((widelimb) in1[3]) * in2[1];
530 out[5] = ((widelimb) in1[2]) * in2[3] + ((widelimb) in1[3]) * in2[2];
531 out[6] = ((widelimb) in1[3]) * in2[3];
535 * Reduce seven 128-bit coefficients to four 64-bit coefficients.
536 * Requires in[i] < 2^126,
537 * ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] <= 2^56 + 2^16 */
538 static void felem_reduce(felem out, const widefelem in)
540 static const widelimb two127p15 = (((widelimb) 1) << 127) +
541 (((widelimb) 1) << 15);
542 static const widelimb two127m71 = (((widelimb) 1) << 127) -
543 (((widelimb) 1) << 71);
544 static const widelimb two127m71m55 = (((widelimb) 1) << 127) -
545 (((widelimb) 1) << 71) - (((widelimb) 1) << 55);
548 /* Add 0 mod 2^224-2^96+1 to ensure all differences are positive */
549 output[0] = in[0] + two127p15;
550 output[1] = in[1] + two127m71m55;
551 output[2] = in[2] + two127m71;
555 /* Eliminate in[4], in[5], in[6] */
556 output[4] += in[6] >> 16;
557 output[3] += (in[6] & 0xffff) << 40;
560 output[3] += in[5] >> 16;
561 output[2] += (in[5] & 0xffff) << 40;
564 output[2] += output[4] >> 16;
565 output[1] += (output[4] & 0xffff) << 40;
566 output[0] -= output[4];
568 /* Carry 2 -> 3 -> 4 */
569 output[3] += output[2] >> 56;
570 output[2] &= 0x00ffffffffffffff;
572 output[4] = output[3] >> 56;
573 output[3] &= 0x00ffffffffffffff;
575 /* Now output[2] < 2^56, output[3] < 2^56, output[4] < 2^72 */
577 /* Eliminate output[4] */
578 output[2] += output[4] >> 16;
579 /* output[2] < 2^56 + 2^56 = 2^57 */
580 output[1] += (output[4] & 0xffff) << 40;
581 output[0] -= output[4];
583 /* Carry 0 -> 1 -> 2 -> 3 */
584 output[1] += output[0] >> 56;
585 out[0] = output[0] & 0x00ffffffffffffff;
587 output[2] += output[1] >> 56;
588 /* output[2] < 2^57 + 2^72 */
589 out[1] = output[1] & 0x00ffffffffffffff;
590 output[3] += output[2] >> 56;
591 /* output[3] <= 2^56 + 2^16 */
592 out[2] = output[2] & 0x00ffffffffffffff;
595 * out[0] < 2^56, out[1] < 2^56, out[2] < 2^56,
596 * out[3] <= 2^56 + 2^16 (due to final carry),
602 static void felem_square_reduce(felem out, const felem in)
605 felem_square(tmp, in);
606 felem_reduce(out, tmp);
609 static void felem_mul_reduce(felem out, const felem in1, const felem in2)
612 felem_mul(tmp, in1, in2);
613 felem_reduce(out, tmp);
617 * Reduce to unique minimal representation. Requires 0 <= in < 2*p (always
618 * call felem_reduce first)
620 static void felem_contract(felem out, const felem in)
622 static const int64_t two56 = ((limb) 1) << 56;
623 /* 0 <= in < 2*p, p = 2^224 - 2^96 + 1 */
624 /* if in > p , reduce in = in - 2^224 + 2^96 - 1 */
630 /* Case 1: a = 1 iff in >= 2^224 */
634 tmp[3] &= 0x00ffffffffffffff;
636 * Case 2: a = 0 iff p <= in < 2^224, i.e., the high 128 bits are all 1
637 * and the lower part is non-zero
639 a = ((in[3] & in[2] & (in[1] | 0x000000ffffffffff)) + 1) |
640 (((int64_t) (in[0] + (in[1] & 0x000000ffffffffff)) - 1) >> 63);
641 a &= 0x00ffffffffffffff;
642 /* turn a into an all-one mask (if a = 0) or an all-zero mask */
644 /* subtract 2^224 - 2^96 + 1 if a is all-one */
645 tmp[3] &= a ^ 0xffffffffffffffff;
646 tmp[2] &= a ^ 0xffffffffffffffff;
647 tmp[1] &= (a ^ 0xffffffffffffffff) | 0x000000ffffffffff;
651 * eliminate negative coefficients: if tmp[0] is negative, tmp[1] must be
652 * non-zero, so we only need one step
658 /* carry 1 -> 2 -> 3 */
659 tmp[2] += tmp[1] >> 56;
660 tmp[1] &= 0x00ffffffffffffff;
662 tmp[3] += tmp[2] >> 56;
663 tmp[2] &= 0x00ffffffffffffff;
665 /* Now 0 <= out < p */
673 * Get negative value: out = -in
674 * Requires in[i] < 2^63,
675 * ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] <= 2^56 + 2^16
677 static void felem_neg(felem out, const felem in)
680 felem_diff_128_64(tmp, in);
681 felem_reduce(out, tmp);
685 * Zero-check: returns 1 if input is 0, and 0 otherwise. We know that field
686 * elements are reduced to in < 2^225, so we only need to check three cases:
687 * 0, 2^224 - 2^96 + 1, and 2^225 - 2^97 + 2
689 static limb felem_is_zero(const felem in)
691 limb zero, two224m96p1, two225m97p2;
693 zero = in[0] | in[1] | in[2] | in[3];
694 zero = (((int64_t) (zero) - 1) >> 63) & 1;
695 two224m96p1 = (in[0] ^ 1) | (in[1] ^ 0x00ffff0000000000)
696 | (in[2] ^ 0x00ffffffffffffff) | (in[3] ^ 0x00ffffffffffffff);
697 two224m96p1 = (((int64_t) (two224m96p1) - 1) >> 63) & 1;
698 two225m97p2 = (in[0] ^ 2) | (in[1] ^ 0x00fffe0000000000)
699 | (in[2] ^ 0x00ffffffffffffff) | (in[3] ^ 0x01ffffffffffffff);
700 two225m97p2 = (((int64_t) (two225m97p2) - 1) >> 63) & 1;
701 return (zero | two224m96p1 | two225m97p2);
704 static int felem_is_zero_int(const void *in)
706 return (int)(felem_is_zero(in) & ((limb) 1));
709 /* Invert a field element */
710 /* Computation chain copied from djb's code */
711 static void felem_inv(felem out, const felem in)
713 felem ftmp, ftmp2, ftmp3, ftmp4;
717 felem_square(tmp, in);
718 felem_reduce(ftmp, tmp); /* 2 */
719 felem_mul(tmp, in, ftmp);
720 felem_reduce(ftmp, tmp); /* 2^2 - 1 */
721 felem_square(tmp, ftmp);
722 felem_reduce(ftmp, tmp); /* 2^3 - 2 */
723 felem_mul(tmp, in, ftmp);
724 felem_reduce(ftmp, tmp); /* 2^3 - 1 */
725 felem_square(tmp, ftmp);
726 felem_reduce(ftmp2, tmp); /* 2^4 - 2 */
727 felem_square(tmp, ftmp2);
728 felem_reduce(ftmp2, tmp); /* 2^5 - 4 */
729 felem_square(tmp, ftmp2);
730 felem_reduce(ftmp2, tmp); /* 2^6 - 8 */
731 felem_mul(tmp, ftmp2, ftmp);
732 felem_reduce(ftmp, tmp); /* 2^6 - 1 */
733 felem_square(tmp, ftmp);
734 felem_reduce(ftmp2, tmp); /* 2^7 - 2 */
735 for (i = 0; i < 5; ++i) { /* 2^12 - 2^6 */
736 felem_square(tmp, ftmp2);
737 felem_reduce(ftmp2, tmp);
739 felem_mul(tmp, ftmp2, ftmp);
740 felem_reduce(ftmp2, tmp); /* 2^12 - 1 */
741 felem_square(tmp, ftmp2);
742 felem_reduce(ftmp3, tmp); /* 2^13 - 2 */
743 for (i = 0; i < 11; ++i) { /* 2^24 - 2^12 */
744 felem_square(tmp, ftmp3);
745 felem_reduce(ftmp3, tmp);
747 felem_mul(tmp, ftmp3, ftmp2);
748 felem_reduce(ftmp2, tmp); /* 2^24 - 1 */
749 felem_square(tmp, ftmp2);
750 felem_reduce(ftmp3, tmp); /* 2^25 - 2 */
751 for (i = 0; i < 23; ++i) { /* 2^48 - 2^24 */
752 felem_square(tmp, ftmp3);
753 felem_reduce(ftmp3, tmp);
755 felem_mul(tmp, ftmp3, ftmp2);
756 felem_reduce(ftmp3, tmp); /* 2^48 - 1 */
757 felem_square(tmp, ftmp3);
758 felem_reduce(ftmp4, tmp); /* 2^49 - 2 */
759 for (i = 0; i < 47; ++i) { /* 2^96 - 2^48 */
760 felem_square(tmp, ftmp4);
761 felem_reduce(ftmp4, tmp);
763 felem_mul(tmp, ftmp3, ftmp4);
764 felem_reduce(ftmp3, tmp); /* 2^96 - 1 */
765 felem_square(tmp, ftmp3);
766 felem_reduce(ftmp4, tmp); /* 2^97 - 2 */
767 for (i = 0; i < 23; ++i) { /* 2^120 - 2^24 */
768 felem_square(tmp, ftmp4);
769 felem_reduce(ftmp4, tmp);
771 felem_mul(tmp, ftmp2, ftmp4);
772 felem_reduce(ftmp2, tmp); /* 2^120 - 1 */
773 for (i = 0; i < 6; ++i) { /* 2^126 - 2^6 */
774 felem_square(tmp, ftmp2);
775 felem_reduce(ftmp2, tmp);
777 felem_mul(tmp, ftmp2, ftmp);
778 felem_reduce(ftmp, tmp); /* 2^126 - 1 */
779 felem_square(tmp, ftmp);
780 felem_reduce(ftmp, tmp); /* 2^127 - 2 */
781 felem_mul(tmp, ftmp, in);
782 felem_reduce(ftmp, tmp); /* 2^127 - 1 */
783 for (i = 0; i < 97; ++i) { /* 2^224 - 2^97 */
784 felem_square(tmp, ftmp);
785 felem_reduce(ftmp, tmp);
787 felem_mul(tmp, ftmp, ftmp3);
788 felem_reduce(out, tmp); /* 2^224 - 2^96 - 1 */
792 * Copy in constant time: if icopy == 1, copy in to out, if icopy == 0, copy
795 static void copy_conditional(felem out, const felem in, limb icopy)
799 * icopy is a (64-bit) 0 or 1, so copy is either all-zero or all-one
801 const limb copy = -icopy;
802 for (i = 0; i < 4; ++i) {
803 const limb tmp = copy & (in[i] ^ out[i]);
808 /******************************************************************************/
810 * ELLIPTIC CURVE POINT OPERATIONS
812 * Points are represented in Jacobian projective coordinates:
813 * (X, Y, Z) corresponds to the affine point (X/Z^2, Y/Z^3),
814 * or to the point at infinity if Z == 0.
819 * Double an elliptic curve point:
820 * (X', Y', Z') = 2 * (X, Y, Z), where
821 * X' = (3 * (X - Z^2) * (X + Z^2))^2 - 8 * X * Y^2
822 * Y' = 3 * (X - Z^2) * (X + Z^2) * (4 * X * Y^2 - X') - 8 * Y^4
823 * Z' = (Y + Z)^2 - Y^2 - Z^2 = 2 * Y * Z
824 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed,
825 * while x_out == y_in is not (maybe this works, but it's not tested).
828 point_double(felem x_out, felem y_out, felem z_out,
829 const felem x_in, const felem y_in, const felem z_in)
832 felem delta, gamma, beta, alpha, ftmp, ftmp2;
834 felem_assign(ftmp, x_in);
835 felem_assign(ftmp2, x_in);
838 felem_square(tmp, z_in);
839 felem_reduce(delta, tmp);
842 felem_square(tmp, y_in);
843 felem_reduce(gamma, tmp);
846 felem_mul(tmp, x_in, gamma);
847 felem_reduce(beta, tmp);
849 /* alpha = 3*(x-delta)*(x+delta) */
850 felem_diff(ftmp, delta);
851 /* ftmp[i] < 2^57 + 2^58 + 2 < 2^59 */
852 felem_sum(ftmp2, delta);
853 /* ftmp2[i] < 2^57 + 2^57 = 2^58 */
854 felem_scalar(ftmp2, 3);
855 /* ftmp2[i] < 3 * 2^58 < 2^60 */
856 felem_mul(tmp, ftmp, ftmp2);
857 /* tmp[i] < 2^60 * 2^59 * 4 = 2^121 */
858 felem_reduce(alpha, tmp);
860 /* x' = alpha^2 - 8*beta */
861 felem_square(tmp, alpha);
862 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
863 felem_assign(ftmp, beta);
864 felem_scalar(ftmp, 8);
865 /* ftmp[i] < 8 * 2^57 = 2^60 */
866 felem_diff_128_64(tmp, ftmp);
867 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
868 felem_reduce(x_out, tmp);
870 /* z' = (y + z)^2 - gamma - delta */
871 felem_sum(delta, gamma);
872 /* delta[i] < 2^57 + 2^57 = 2^58 */
873 felem_assign(ftmp, y_in);
874 felem_sum(ftmp, z_in);
875 /* ftmp[i] < 2^57 + 2^57 = 2^58 */
876 felem_square(tmp, ftmp);
877 /* tmp[i] < 4 * 2^58 * 2^58 = 2^118 */
878 felem_diff_128_64(tmp, delta);
879 /* tmp[i] < 2^118 + 2^64 + 8 < 2^119 */
880 felem_reduce(z_out, tmp);
882 /* y' = alpha*(4*beta - x') - 8*gamma^2 */
883 felem_scalar(beta, 4);
884 /* beta[i] < 4 * 2^57 = 2^59 */
885 felem_diff(beta, x_out);
886 /* beta[i] < 2^59 + 2^58 + 2 < 2^60 */
887 felem_mul(tmp, alpha, beta);
888 /* tmp[i] < 4 * 2^57 * 2^60 = 2^119 */
889 felem_square(tmp2, gamma);
890 /* tmp2[i] < 4 * 2^57 * 2^57 = 2^116 */
891 widefelem_scalar(tmp2, 8);
892 /* tmp2[i] < 8 * 2^116 = 2^119 */
893 widefelem_diff(tmp, tmp2);
894 /* tmp[i] < 2^119 + 2^120 < 2^121 */
895 felem_reduce(y_out, tmp);
899 * Add two elliptic curve points:
900 * (X_1, Y_1, Z_1) + (X_2, Y_2, Z_2) = (X_3, Y_3, Z_3), where
901 * X_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1)^2 - (Z_1^2 * X_2 - Z_2^2 * X_1)^3 -
902 * 2 * Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^2
903 * 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) -
904 * Z_2^3 * Y_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^3
905 * Z_3 = (Z_1^2 * X_2 - Z_2^2 * X_1) * (Z_1 * Z_2)
907 * This runs faster if 'mixed' is set, which requires Z_2 = 1 or Z_2 = 0.
911 * This function is not entirely constant-time: it includes a branch for
912 * checking whether the two input points are equal, (while not equal to the
913 * point at infinity). This case never happens during single point
914 * multiplication, so there is no timing leak for ECDH or ECDSA signing.
916 static void point_add(felem x3, felem y3, felem z3,
917 const felem x1, const felem y1, const felem z1,
918 const int mixed, const felem x2, const felem y2,
921 felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, x_out, y_out, z_out;
923 limb z1_is_zero, z2_is_zero, x_equal, y_equal;
927 felem_square(tmp, z2);
928 felem_reduce(ftmp2, tmp);
931 felem_mul(tmp, ftmp2, z2);
932 felem_reduce(ftmp4, tmp);
934 /* ftmp4 = z2^3*y1 */
935 felem_mul(tmp2, ftmp4, y1);
936 felem_reduce(ftmp4, tmp2);
938 /* ftmp2 = z2^2*x1 */
939 felem_mul(tmp2, ftmp2, x1);
940 felem_reduce(ftmp2, tmp2);
943 * We'll assume z2 = 1 (special case z2 = 0 is handled later)
946 /* ftmp4 = z2^3*y1 */
947 felem_assign(ftmp4, y1);
949 /* ftmp2 = z2^2*x1 */
950 felem_assign(ftmp2, x1);
954 felem_square(tmp, z1);
955 felem_reduce(ftmp, tmp);
958 felem_mul(tmp, ftmp, z1);
959 felem_reduce(ftmp3, tmp);
962 felem_mul(tmp, ftmp3, y2);
963 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
965 /* ftmp3 = z1^3*y2 - z2^3*y1 */
966 felem_diff_128_64(tmp, ftmp4);
967 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
968 felem_reduce(ftmp3, tmp);
971 felem_mul(tmp, ftmp, x2);
972 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
974 /* ftmp = z1^2*x2 - z2^2*x1 */
975 felem_diff_128_64(tmp, ftmp2);
976 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
977 felem_reduce(ftmp, tmp);
980 * the formulae are incorrect if the points are equal so we check for
981 * this and do doubling if this happens
983 x_equal = felem_is_zero(ftmp);
984 y_equal = felem_is_zero(ftmp3);
985 z1_is_zero = felem_is_zero(z1);
986 z2_is_zero = felem_is_zero(z2);
987 /* In affine coordinates, (X_1, Y_1) == (X_2, Y_2) */
988 if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) {
989 point_double(x3, y3, z3, x1, y1, z1);
995 felem_mul(tmp, z1, z2);
996 felem_reduce(ftmp5, tmp);
998 /* special case z2 = 0 is handled later */
999 felem_assign(ftmp5, z1);
1002 /* z_out = (z1^2*x2 - z2^2*x1)*(z1*z2) */
1003 felem_mul(tmp, ftmp, ftmp5);
1004 felem_reduce(z_out, tmp);
1006 /* ftmp = (z1^2*x2 - z2^2*x1)^2 */
1007 felem_assign(ftmp5, ftmp);
1008 felem_square(tmp, ftmp);
1009 felem_reduce(ftmp, tmp);
1011 /* ftmp5 = (z1^2*x2 - z2^2*x1)^3 */
1012 felem_mul(tmp, ftmp, ftmp5);
1013 felem_reduce(ftmp5, tmp);
1015 /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
1016 felem_mul(tmp, ftmp2, ftmp);
1017 felem_reduce(ftmp2, tmp);
1019 /* tmp = z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */
1020 felem_mul(tmp, ftmp4, ftmp5);
1021 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
1023 /* tmp2 = (z1^3*y2 - z2^3*y1)^2 */
1024 felem_square(tmp2, ftmp3);
1025 /* tmp2[i] < 4 * 2^57 * 2^57 < 2^116 */
1027 /* tmp2 = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 */
1028 felem_diff_128_64(tmp2, ftmp5);
1029 /* tmp2[i] < 2^116 + 2^64 + 8 < 2^117 */
1031 /* ftmp5 = 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
1032 felem_assign(ftmp5, ftmp2);
1033 felem_scalar(ftmp5, 2);
1034 /* ftmp5[i] < 2 * 2^57 = 2^58 */
1037 * x_out = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 -
1038 * 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2
1040 felem_diff_128_64(tmp2, ftmp5);
1041 /* tmp2[i] < 2^117 + 2^64 + 8 < 2^118 */
1042 felem_reduce(x_out, tmp2);
1044 /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out */
1045 felem_diff(ftmp2, x_out);
1046 /* ftmp2[i] < 2^57 + 2^58 + 2 < 2^59 */
1049 * tmp2 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out)
1051 felem_mul(tmp2, ftmp3, ftmp2);
1052 /* tmp2[i] < 4 * 2^57 * 2^59 = 2^118 */
1055 * y_out = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) -
1056 * z2^3*y1*(z1^2*x2 - z2^2*x1)^3
1058 widefelem_diff(tmp2, tmp);
1059 /* tmp2[i] < 2^118 + 2^120 < 2^121 */
1060 felem_reduce(y_out, tmp2);
1063 * the result (x_out, y_out, z_out) is incorrect if one of the inputs is
1064 * the point at infinity, so we need to check for this separately
1068 * if point 1 is at infinity, copy point 2 to output, and vice versa
1070 copy_conditional(x_out, x2, z1_is_zero);
1071 copy_conditional(x_out, x1, z2_is_zero);
1072 copy_conditional(y_out, y2, z1_is_zero);
1073 copy_conditional(y_out, y1, z2_is_zero);
1074 copy_conditional(z_out, z2, z1_is_zero);
1075 copy_conditional(z_out, z1, z2_is_zero);
1076 felem_assign(x3, x_out);
1077 felem_assign(y3, y_out);
1078 felem_assign(z3, z_out);
1082 * select_point selects the |idx|th point from a precomputation table and
1084 * The pre_comp array argument should be size of |size| argument
1086 static void select_point(const u64 idx, unsigned int size,
1087 const felem pre_comp[][3], felem out[3])
1090 limb *outlimbs = &out[0][0];
1092 memset(out, 0, sizeof(*out) * 3);
1093 for (i = 0; i < size; i++) {
1094 const limb *inlimbs = &pre_comp[i][0][0];
1101 for (j = 0; j < 4 * 3; j++)
1102 outlimbs[j] |= inlimbs[j] & mask;
1106 /* get_bit returns the |i|th bit in |in| */
1107 static char get_bit(const felem_bytearray in, unsigned i)
1111 return (in[i >> 3] >> (i & 7)) & 1;
1115 * Interleaved point multiplication using precomputed point multiples: The
1116 * small point multiples 0*P, 1*P, ..., 16*P are in pre_comp[], the scalars
1117 * in scalars[]. If g_scalar is non-NULL, we also add this multiple of the
1118 * generator, using certain (large) precomputed multiples in g_pre_comp.
1119 * Output point (X, Y, Z) is stored in x_out, y_out, z_out
1121 static void batch_mul(felem x_out, felem y_out, felem z_out,
1122 const felem_bytearray scalars[],
1123 const unsigned num_points, const u8 *g_scalar,
1124 const int mixed, const felem pre_comp[][17][3],
1125 const felem g_pre_comp[2][16][3])
1129 unsigned gen_mul = (g_scalar != NULL);
1130 felem nq[3], tmp[4];
1134 /* set nq to the point at infinity */
1135 memset(nq, 0, sizeof(nq));
1138 * Loop over all scalars msb-to-lsb, interleaving additions of multiples
1139 * of the generator (two in each of the last 28 rounds) and additions of
1140 * other points multiples (every 5th round).
1142 skip = 1; /* save two point operations in the first
1144 for (i = (num_points ? 220 : 27); i >= 0; --i) {
1147 point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
1149 /* add multiples of the generator */
1150 if (gen_mul && (i <= 27)) {
1151 /* first, look 28 bits upwards */
1152 bits = get_bit(g_scalar, i + 196) << 3;
1153 bits |= get_bit(g_scalar, i + 140) << 2;
1154 bits |= get_bit(g_scalar, i + 84) << 1;
1155 bits |= get_bit(g_scalar, i + 28);
1156 /* select the point to add, in constant time */
1157 select_point(bits, 16, g_pre_comp[1], tmp);
1160 /* value 1 below is argument for "mixed" */
1161 point_add(nq[0], nq[1], nq[2],
1162 nq[0], nq[1], nq[2], 1, tmp[0], tmp[1], tmp[2]);
1164 memcpy(nq, tmp, 3 * sizeof(felem));
1168 /* second, look at the current position */
1169 bits = get_bit(g_scalar, i + 168) << 3;
1170 bits |= get_bit(g_scalar, i + 112) << 2;
1171 bits |= get_bit(g_scalar, i + 56) << 1;
1172 bits |= get_bit(g_scalar, i);
1173 /* select the point to add, in constant time */
1174 select_point(bits, 16, g_pre_comp[0], tmp);
1175 point_add(nq[0], nq[1], nq[2],
1176 nq[0], nq[1], nq[2],
1177 1 /* mixed */ , tmp[0], tmp[1], tmp[2]);
1180 /* do other additions every 5 doublings */
1181 if (num_points && (i % 5 == 0)) {
1182 /* loop over all scalars */
1183 for (num = 0; num < num_points; ++num) {
1184 bits = get_bit(scalars[num], i + 4) << 5;
1185 bits |= get_bit(scalars[num], i + 3) << 4;
1186 bits |= get_bit(scalars[num], i + 2) << 3;
1187 bits |= get_bit(scalars[num], i + 1) << 2;
1188 bits |= get_bit(scalars[num], i) << 1;
1189 bits |= get_bit(scalars[num], i - 1);
1190 ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits);
1192 /* select the point to add or subtract */
1193 select_point(digit, 17, pre_comp[num], tmp);
1194 felem_neg(tmp[3], tmp[1]); /* (X, -Y, Z) is the negative
1196 copy_conditional(tmp[1], tmp[3], sign);
1199 point_add(nq[0], nq[1], nq[2],
1200 nq[0], nq[1], nq[2],
1201 mixed, tmp[0], tmp[1], tmp[2]);
1203 memcpy(nq, tmp, 3 * sizeof(felem));
1209 felem_assign(x_out, nq[0]);
1210 felem_assign(y_out, nq[1]);
1211 felem_assign(z_out, nq[2]);
1214 /******************************************************************************/
1216 * FUNCTIONS TO MANAGE PRECOMPUTATION
1219 static NISTP224_PRE_COMP *nistp224_pre_comp_new(void)
1221 NISTP224_PRE_COMP *ret = OPENSSL_zalloc(sizeof(*ret));
1224 ECerr(EC_F_NISTP224_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1228 ret->references = 1;
1230 ret->lock = CRYPTO_THREAD_lock_new();
1231 if (ret->lock == NULL) {
1232 ECerr(EC_F_NISTP224_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1239 NISTP224_PRE_COMP *EC_nistp224_pre_comp_dup(NISTP224_PRE_COMP *p)
1243 CRYPTO_UP_REF(&p->references, &i, p->lock);
1247 void EC_nistp224_pre_comp_free(NISTP224_PRE_COMP *p)
1254 CRYPTO_DOWN_REF(&p->references, &i, p->lock);
1255 REF_PRINT_COUNT("EC_nistp224", x);
1258 REF_ASSERT_ISNT(i < 0);
1260 CRYPTO_THREAD_lock_free(p->lock);
1264 /******************************************************************************/
1266 * OPENSSL EC_METHOD FUNCTIONS
1269 int ec_GFp_nistp224_group_init(EC_GROUP *group)
1272 ret = ec_GFp_simple_group_init(group);
1273 group->a_is_minus3 = 1;
1277 int ec_GFp_nistp224_group_set_curve(EC_GROUP *group, const BIGNUM *p,
1278 const BIGNUM *a, const BIGNUM *b,
1282 BN_CTX *new_ctx = NULL;
1283 BIGNUM *curve_p, *curve_a, *curve_b;
1286 if ((ctx = new_ctx = BN_CTX_new()) == NULL)
1289 curve_p = BN_CTX_get(ctx);
1290 curve_a = BN_CTX_get(ctx);
1291 curve_b = BN_CTX_get(ctx);
1292 if (curve_b == NULL)
1294 BN_bin2bn(nistp224_curve_params[0], sizeof(felem_bytearray), curve_p);
1295 BN_bin2bn(nistp224_curve_params[1], sizeof(felem_bytearray), curve_a);
1296 BN_bin2bn(nistp224_curve_params[2], sizeof(felem_bytearray), curve_b);
1297 if ((BN_cmp(curve_p, p)) || (BN_cmp(curve_a, a)) || (BN_cmp(curve_b, b))) {
1298 ECerr(EC_F_EC_GFP_NISTP224_GROUP_SET_CURVE,
1299 EC_R_WRONG_CURVE_PARAMETERS);
1302 group->field_mod_func = BN_nist_mod_224;
1303 ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx);
1306 BN_CTX_free(new_ctx);
1311 * Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') =
1314 int ec_GFp_nistp224_point_get_affine_coordinates(const EC_GROUP *group,
1315 const EC_POINT *point,
1316 BIGNUM *x, BIGNUM *y,
1319 felem z1, z2, x_in, y_in, x_out, y_out;
1322 if (EC_POINT_is_at_infinity(group, point)) {
1323 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1324 EC_R_POINT_AT_INFINITY);
1327 if ((!BN_to_felem(x_in, point->X)) || (!BN_to_felem(y_in, point->Y)) ||
1328 (!BN_to_felem(z1, point->Z)))
1331 felem_square(tmp, z2);
1332 felem_reduce(z1, tmp);
1333 felem_mul(tmp, x_in, z1);
1334 felem_reduce(x_in, tmp);
1335 felem_contract(x_out, x_in);
1337 if (!felem_to_BN(x, x_out)) {
1338 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1343 felem_mul(tmp, z1, z2);
1344 felem_reduce(z1, tmp);
1345 felem_mul(tmp, y_in, z1);
1346 felem_reduce(y_in, tmp);
1347 felem_contract(y_out, y_in);
1349 if (!felem_to_BN(y, y_out)) {
1350 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1358 static void make_points_affine(size_t num, felem points[ /* num */ ][3],
1359 felem tmp_felems[ /* num+1 */ ])
1362 * Runs in constant time, unless an input is the point at infinity (which
1363 * normally shouldn't happen).
1365 ec_GFp_nistp_points_make_affine_internal(num,
1369 (void (*)(void *))felem_one,
1371 (void (*)(void *, const void *))
1373 (void (*)(void *, const void *))
1374 felem_square_reduce, (void (*)
1381 (void (*)(void *, const void *))
1383 (void (*)(void *, const void *))
1388 * Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL
1389 * values Result is stored in r (r can equal one of the inputs).
1391 int ec_GFp_nistp224_points_mul(const EC_GROUP *group, EC_POINT *r,
1392 const BIGNUM *scalar, size_t num,
1393 const EC_POINT *points[],
1394 const BIGNUM *scalars[], BN_CTX *ctx)
1400 BIGNUM *x, *y, *z, *tmp_scalar;
1401 felem_bytearray g_secret;
1402 felem_bytearray *secrets = NULL;
1403 felem (*pre_comp)[17][3] = NULL;
1404 felem *tmp_felems = NULL;
1405 felem_bytearray tmp;
1407 int have_pre_comp = 0;
1408 size_t num_points = num;
1409 felem x_in, y_in, z_in, x_out, y_out, z_out;
1410 NISTP224_PRE_COMP *pre = NULL;
1411 const felem(*g_pre_comp)[16][3] = NULL;
1412 EC_POINT *generator = NULL;
1413 const EC_POINT *p = NULL;
1414 const BIGNUM *p_scalar = NULL;
1417 x = BN_CTX_get(ctx);
1418 y = BN_CTX_get(ctx);
1419 z = BN_CTX_get(ctx);
1420 tmp_scalar = BN_CTX_get(ctx);
1421 if (tmp_scalar == NULL)
1424 if (scalar != NULL) {
1425 pre = group->pre_comp.nistp224;
1427 /* we have precomputation, try to use it */
1428 g_pre_comp = (const felem(*)[16][3])pre->g_pre_comp;
1430 /* try to use the standard precomputation */
1431 g_pre_comp = &gmul[0];
1432 generator = EC_POINT_new(group);
1433 if (generator == NULL)
1435 /* get the generator from precomputation */
1436 if (!felem_to_BN(x, g_pre_comp[0][1][0]) ||
1437 !felem_to_BN(y, g_pre_comp[0][1][1]) ||
1438 !felem_to_BN(z, g_pre_comp[0][1][2])) {
1439 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1442 if (!EC_POINT_set_Jprojective_coordinates_GFp(group,
1446 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
1447 /* precomputation matches generator */
1451 * we don't have valid precomputation: treat the generator as a
1454 num_points = num_points + 1;
1457 if (num_points > 0) {
1458 if (num_points >= 3) {
1460 * unless we precompute multiples for just one or two points,
1461 * converting those into affine form is time well spent
1465 secrets = OPENSSL_zalloc(sizeof(*secrets) * num_points);
1466 pre_comp = OPENSSL_zalloc(sizeof(*pre_comp) * num_points);
1469 OPENSSL_malloc(sizeof(felem) * (num_points * 17 + 1));
1470 if ((secrets == NULL) || (pre_comp == NULL)
1471 || (mixed && (tmp_felems == NULL))) {
1472 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_MALLOC_FAILURE);
1477 * we treat NULL scalars as 0, and NULL points as points at infinity,
1478 * i.e., they contribute nothing to the linear combination
1480 for (i = 0; i < num_points; ++i) {
1484 p = EC_GROUP_get0_generator(group);
1487 /* the i^th point */
1490 p_scalar = scalars[i];
1492 if ((p_scalar != NULL) && (p != NULL)) {
1493 /* reduce scalar to 0 <= scalar < 2^224 */
1494 if ((BN_num_bits(p_scalar) > 224)
1495 || (BN_is_negative(p_scalar))) {
1497 * this is an unusual input, and we don't guarantee
1500 if (!BN_nnmod(tmp_scalar, p_scalar, group->order, ctx)) {
1501 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1504 num_bytes = BN_bn2bin(tmp_scalar, tmp);
1506 num_bytes = BN_bn2bin(p_scalar, tmp);
1507 flip_endian(secrets[i], tmp, num_bytes);
1508 /* precompute multiples */
1509 if ((!BN_to_felem(x_out, p->X)) ||
1510 (!BN_to_felem(y_out, p->Y)) ||
1511 (!BN_to_felem(z_out, p->Z)))
1513 felem_assign(pre_comp[i][1][0], x_out);
1514 felem_assign(pre_comp[i][1][1], y_out);
1515 felem_assign(pre_comp[i][1][2], z_out);
1516 for (j = 2; j <= 16; ++j) {
1518 point_add(pre_comp[i][j][0], pre_comp[i][j][1],
1519 pre_comp[i][j][2], pre_comp[i][1][0],
1520 pre_comp[i][1][1], pre_comp[i][1][2], 0,
1521 pre_comp[i][j - 1][0],
1522 pre_comp[i][j - 1][1],
1523 pre_comp[i][j - 1][2]);
1525 point_double(pre_comp[i][j][0], pre_comp[i][j][1],
1526 pre_comp[i][j][2], pre_comp[i][j / 2][0],
1527 pre_comp[i][j / 2][1],
1528 pre_comp[i][j / 2][2]);
1534 make_points_affine(num_points * 17, pre_comp[0], tmp_felems);
1537 /* the scalar for the generator */
1538 if ((scalar != NULL) && (have_pre_comp)) {
1539 memset(g_secret, 0, sizeof(g_secret));
1540 /* reduce scalar to 0 <= scalar < 2^224 */
1541 if ((BN_num_bits(scalar) > 224) || (BN_is_negative(scalar))) {
1543 * this is an unusual input, and we don't guarantee
1546 if (!BN_nnmod(tmp_scalar, scalar, group->order, ctx)) {
1547 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1550 num_bytes = BN_bn2bin(tmp_scalar, tmp);
1552 num_bytes = BN_bn2bin(scalar, tmp);
1553 flip_endian(g_secret, tmp, num_bytes);
1554 /* do the multiplication with generator precomputation */
1555 batch_mul(x_out, y_out, z_out,
1556 (const felem_bytearray(*))secrets, num_points,
1558 mixed, (const felem(*)[17][3])pre_comp, g_pre_comp);
1560 /* do the multiplication without generator precomputation */
1561 batch_mul(x_out, y_out, z_out,
1562 (const felem_bytearray(*))secrets, num_points,
1563 NULL, mixed, (const felem(*)[17][3])pre_comp, NULL);
1564 /* reduce the output to its unique minimal representation */
1565 felem_contract(x_in, x_out);
1566 felem_contract(y_in, y_out);
1567 felem_contract(z_in, z_out);
1568 if ((!felem_to_BN(x, x_in)) || (!felem_to_BN(y, y_in)) ||
1569 (!felem_to_BN(z, z_in))) {
1570 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1573 ret = EC_POINT_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx);
1577 EC_POINT_free(generator);
1578 OPENSSL_free(secrets);
1579 OPENSSL_free(pre_comp);
1580 OPENSSL_free(tmp_felems);
1584 int ec_GFp_nistp224_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
1587 NISTP224_PRE_COMP *pre = NULL;
1589 BN_CTX *new_ctx = NULL;
1591 EC_POINT *generator = NULL;
1592 felem tmp_felems[32];
1594 /* throw away old precomputation */
1595 EC_pre_comp_free(group);
1597 if ((ctx = new_ctx = BN_CTX_new()) == NULL)
1600 x = BN_CTX_get(ctx);
1601 y = BN_CTX_get(ctx);
1604 /* get the generator */
1605 if (group->generator == NULL)
1607 generator = EC_POINT_new(group);
1608 if (generator == NULL)
1610 BN_bin2bn(nistp224_curve_params[3], sizeof(felem_bytearray), x);
1611 BN_bin2bn(nistp224_curve_params[4], sizeof(felem_bytearray), y);
1612 if (!EC_POINT_set_affine_coordinates(group, generator, x, y, ctx))
1614 if ((pre = nistp224_pre_comp_new()) == NULL)
1617 * if the generator is the standard one, use built-in precomputation
1619 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) {
1620 memcpy(pre->g_pre_comp, gmul, sizeof(pre->g_pre_comp));
1623 if ((!BN_to_felem(pre->g_pre_comp[0][1][0], group->generator->X)) ||
1624 (!BN_to_felem(pre->g_pre_comp[0][1][1], group->generator->Y)) ||
1625 (!BN_to_felem(pre->g_pre_comp[0][1][2], group->generator->Z)))
1628 * compute 2^56*G, 2^112*G, 2^168*G for the first table, 2^28*G, 2^84*G,
1629 * 2^140*G, 2^196*G for the second one
1631 for (i = 1; i <= 8; i <<= 1) {
1632 point_double(pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1],
1633 pre->g_pre_comp[1][i][2], pre->g_pre_comp[0][i][0],
1634 pre->g_pre_comp[0][i][1], pre->g_pre_comp[0][i][2]);
1635 for (j = 0; j < 27; ++j) {
1636 point_double(pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1],
1637 pre->g_pre_comp[1][i][2], pre->g_pre_comp[1][i][0],
1638 pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2]);
1642 point_double(pre->g_pre_comp[0][2 * i][0],
1643 pre->g_pre_comp[0][2 * i][1],
1644 pre->g_pre_comp[0][2 * i][2], pre->g_pre_comp[1][i][0],
1645 pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2]);
1646 for (j = 0; j < 27; ++j) {
1647 point_double(pre->g_pre_comp[0][2 * i][0],
1648 pre->g_pre_comp[0][2 * i][1],
1649 pre->g_pre_comp[0][2 * i][2],
1650 pre->g_pre_comp[0][2 * i][0],
1651 pre->g_pre_comp[0][2 * i][1],
1652 pre->g_pre_comp[0][2 * i][2]);
1655 for (i = 0; i < 2; i++) {
1656 /* g_pre_comp[i][0] is the point at infinity */
1657 memset(pre->g_pre_comp[i][0], 0, sizeof(pre->g_pre_comp[i][0]));
1658 /* the remaining multiples */
1659 /* 2^56*G + 2^112*G resp. 2^84*G + 2^140*G */
1660 point_add(pre->g_pre_comp[i][6][0], pre->g_pre_comp[i][6][1],
1661 pre->g_pre_comp[i][6][2], pre->g_pre_comp[i][4][0],
1662 pre->g_pre_comp[i][4][1], pre->g_pre_comp[i][4][2],
1663 0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
1664 pre->g_pre_comp[i][2][2]);
1665 /* 2^56*G + 2^168*G resp. 2^84*G + 2^196*G */
1666 point_add(pre->g_pre_comp[i][10][0], pre->g_pre_comp[i][10][1],
1667 pre->g_pre_comp[i][10][2], pre->g_pre_comp[i][8][0],
1668 pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2],
1669 0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
1670 pre->g_pre_comp[i][2][2]);
1671 /* 2^112*G + 2^168*G resp. 2^140*G + 2^196*G */
1672 point_add(pre->g_pre_comp[i][12][0], pre->g_pre_comp[i][12][1],
1673 pre->g_pre_comp[i][12][2], pre->g_pre_comp[i][8][0],
1674 pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2],
1675 0, pre->g_pre_comp[i][4][0], pre->g_pre_comp[i][4][1],
1676 pre->g_pre_comp[i][4][2]);
1678 * 2^56*G + 2^112*G + 2^168*G resp. 2^84*G + 2^140*G + 2^196*G
1680 point_add(pre->g_pre_comp[i][14][0], pre->g_pre_comp[i][14][1],
1681 pre->g_pre_comp[i][14][2], pre->g_pre_comp[i][12][0],
1682 pre->g_pre_comp[i][12][1], pre->g_pre_comp[i][12][2],
1683 0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
1684 pre->g_pre_comp[i][2][2]);
1685 for (j = 1; j < 8; ++j) {
1686 /* odd multiples: add G resp. 2^28*G */
1687 point_add(pre->g_pre_comp[i][2 * j + 1][0],
1688 pre->g_pre_comp[i][2 * j + 1][1],
1689 pre->g_pre_comp[i][2 * j + 1][2],
1690 pre->g_pre_comp[i][2 * j][0],
1691 pre->g_pre_comp[i][2 * j][1],
1692 pre->g_pre_comp[i][2 * j][2], 0,
1693 pre->g_pre_comp[i][1][0], pre->g_pre_comp[i][1][1],
1694 pre->g_pre_comp[i][1][2]);
1697 make_points_affine(31, &(pre->g_pre_comp[0][1]), tmp_felems);
1700 SETPRECOMP(group, nistp224, pre);
1705 EC_POINT_free(generator);
1706 BN_CTX_free(new_ctx);
1707 EC_nistp224_pre_comp_free(pre);
1711 int ec_GFp_nistp224_have_precompute_mult(const EC_GROUP *group)
1713 return HAVEPRECOMP(group, nistp224);