2 * Copyright 2010-2021 The OpenSSL Project Authors. All Rights Reserved.
4 * Licensed under the Apache License 2.0 (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 * ECDSA low level APIs are deprecated for public use, but still ok for
30 #include "internal/deprecated.h"
33 * A 64-bit implementation of the NIST P-224 elliptic curve point multiplication
35 * Inspired by Daniel J. Bernstein's public domain nistp224 implementation
36 * and Adam Langley's public domain 64-bit C implementation of curve25519
39 #include <openssl/opensslconf.h>
43 #include <openssl/err.h>
46 #include "internal/numbers.h"
49 # error "Your compiler doesn't appear to support 128-bit integer types"
55 /******************************************************************************/
57 * INTERNAL REPRESENTATION OF FIELD ELEMENTS
59 * Field elements are represented as a_0 + 2^56*a_1 + 2^112*a_2 + 2^168*a_3
60 * using 64-bit coefficients called 'limbs',
61 * and sometimes (for multiplication results) as
62 * 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
63 * using 128-bit coefficients called 'widelimbs'.
64 * A 4-limb representation is an 'felem';
65 * a 7-widelimb representation is a 'widefelem'.
66 * Even within felems, bits of adjacent limbs overlap, and we don't always
67 * reduce the representations: we ensure that inputs to each felem
68 * multiplication satisfy a_i < 2^60, so outputs satisfy b_i < 4*2^60*2^60,
69 * and fit into a 128-bit word without overflow. The coefficients are then
70 * again partially reduced to obtain an felem satisfying a_i < 2^57.
71 * We only reduce to the unique minimal representation at the end of the
75 typedef uint64_t limb;
76 typedef uint64_t limb_aX __attribute((__aligned__(1)));
77 typedef uint128_t widelimb;
79 typedef limb felem[4];
80 typedef widelimb widefelem[7];
83 * Field element represented as a byte array. 28*8 = 224 bits is also the
84 * group order size for the elliptic curve, and we also use this type for
85 * scalars for point multiplication.
87 typedef u8 felem_bytearray[28];
89 static const felem_bytearray nistp224_curve_params[5] = {
90 {0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, /* p */
91 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0x00, 0x00, 0x00, 0x00,
92 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x01},
93 {0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, /* a */
94 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFE, 0xFF, 0xFF, 0xFF, 0xFF,
95 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFE},
96 {0xB4, 0x05, 0x0A, 0x85, 0x0C, 0x04, 0xB3, 0xAB, 0xF5, 0x41, /* b */
97 0x32, 0x56, 0x50, 0x44, 0xB0, 0xB7, 0xD7, 0xBF, 0xD8, 0xBA,
98 0x27, 0x0B, 0x39, 0x43, 0x23, 0x55, 0xFF, 0xB4},
99 {0xB7, 0x0E, 0x0C, 0xBD, 0x6B, 0xB4, 0xBF, 0x7F, 0x32, 0x13, /* x */
100 0x90, 0xB9, 0x4A, 0x03, 0xC1, 0xD3, 0x56, 0xC2, 0x11, 0x22,
101 0x34, 0x32, 0x80, 0xD6, 0x11, 0x5C, 0x1D, 0x21},
102 {0xbd, 0x37, 0x63, 0x88, 0xb5, 0xf7, 0x23, 0xfb, 0x4c, 0x22, /* y */
103 0xdf, 0xe6, 0xcd, 0x43, 0x75, 0xa0, 0x5a, 0x07, 0x47, 0x64,
104 0x44, 0xd5, 0x81, 0x99, 0x85, 0x00, 0x7e, 0x34}
108 * Precomputed multiples of the standard generator
109 * Points are given in coordinates (X, Y, Z) where Z normally is 1
110 * (0 for the point at infinity).
111 * For each field element, slice a_0 is word 0, etc.
113 * The table has 2 * 16 elements, starting with the following:
114 * index | bits | point
115 * ------+---------+------------------------------
118 * 2 | 0 0 1 0 | 2^56G
119 * 3 | 0 0 1 1 | (2^56 + 1)G
120 * 4 | 0 1 0 0 | 2^112G
121 * 5 | 0 1 0 1 | (2^112 + 1)G
122 * 6 | 0 1 1 0 | (2^112 + 2^56)G
123 * 7 | 0 1 1 1 | (2^112 + 2^56 + 1)G
124 * 8 | 1 0 0 0 | 2^168G
125 * 9 | 1 0 0 1 | (2^168 + 1)G
126 * 10 | 1 0 1 0 | (2^168 + 2^56)G
127 * 11 | 1 0 1 1 | (2^168 + 2^56 + 1)G
128 * 12 | 1 1 0 0 | (2^168 + 2^112)G
129 * 13 | 1 1 0 1 | (2^168 + 2^112 + 1)G
130 * 14 | 1 1 1 0 | (2^168 + 2^112 + 2^56)G
131 * 15 | 1 1 1 1 | (2^168 + 2^112 + 2^56 + 1)G
132 * followed by a copy of this with each element multiplied by 2^28.
134 * The reason for this is so that we can clock bits into four different
135 * locations when doing simple scalar multiplies against the base point,
136 * and then another four locations using the second 16 elements.
138 static const felem gmul[2][16][3] = {
142 {{0x3280d6115c1d21, 0xc1d356c2112234, 0x7f321390b94a03, 0xb70e0cbd6bb4bf},
143 {0xd5819985007e34, 0x75a05a07476444, 0xfb4c22dfe6cd43, 0xbd376388b5f723},
145 {{0xfd9675666ebbe9, 0xbca7664d40ce5e, 0x2242df8d8a2a43, 0x1f49bbb0f99bc5},
146 {0x29e0b892dc9c43, 0xece8608436e662, 0xdc858f185310d0, 0x9812dd4eb8d321},
148 {{0x6d3e678d5d8eb8, 0x559eed1cb362f1, 0x16e9a3bbce8a3f, 0xeedcccd8c2a748},
149 {0xf19f90ed50266d, 0xabf2b4bf65f9df, 0x313865468fafec, 0x5cb379ba910a17},
151 {{0x0641966cab26e3, 0x91fb2991fab0a0, 0xefec27a4e13a0b, 0x0499aa8a5f8ebe},
152 {0x7510407766af5d, 0x84d929610d5450, 0x81d77aae82f706, 0x6916f6d4338c5b},
154 {{0xea95ac3b1f15c6, 0x086000905e82d4, 0xdd323ae4d1c8b1, 0x932b56be7685a3},
155 {0x9ef93dea25dbbf, 0x41665960f390f0, 0xfdec76dbe2a8a7, 0x523e80f019062a},
157 {{0x822fdd26732c73, 0xa01c83531b5d0f, 0x363f37347c1ba4, 0xc391b45c84725c},
158 {0xbbd5e1b2d6ad24, 0xddfbcde19dfaec, 0xc393da7e222a7f, 0x1efb7890ede244},
160 {{0x4c9e90ca217da1, 0xd11beca79159bb, 0xff8d33c2c98b7c, 0x2610b39409f849},
161 {0x44d1352ac64da0, 0xcdbb7b2c46b4fb, 0x966c079b753c89, 0xfe67e4e820b112},
163 {{0xe28cae2df5312d, 0xc71b61d16f5c6e, 0x79b7619a3e7c4c, 0x05c73240899b47},
164 {0x9f7f6382c73e3a, 0x18615165c56bda, 0x641fab2116fd56, 0x72855882b08394},
166 {{0x0469182f161c09, 0x74a98ca8d00fb5, 0xb89da93489a3e0, 0x41c98768fb0c1d},
167 {0xe5ea05fb32da81, 0x3dce9ffbca6855, 0x1cfe2d3fbf59e6, 0x0e5e03408738a7},
169 {{0xdab22b2333e87f, 0x4430137a5dd2f6, 0xe03ab9f738beb8, 0xcb0c5d0dc34f24},
170 {0x764a7df0c8fda5, 0x185ba5c3fa2044, 0x9281d688bcbe50, 0xc40331df893881},
172 {{0xb89530796f0f60, 0xade92bd26909a3, 0x1a0c83fb4884da, 0x1765bf22a5a984},
173 {0x772a9ee75db09e, 0x23bc6c67cec16f, 0x4c1edba8b14e2f, 0xe2a215d9611369},
175 {{0x571e509fb5efb3, 0xade88696410552, 0xc8ae85fada74fe, 0x6c7e4be83bbde3},
176 {0xff9f51160f4652, 0xb47ce2495a6539, 0xa2946c53b582f4, 0x286d2db3ee9a60},
178 {{0x40bbd5081a44af, 0x0995183b13926c, 0xbcefba6f47f6d0, 0x215619e9cc0057},
179 {0x8bc94d3b0df45e, 0xf11c54a3694f6f, 0x8631b93cdfe8b5, 0xe7e3f4b0982db9},
181 {{0xb17048ab3e1c7b, 0xac38f36ff8a1d8, 0x1c29819435d2c6, 0xc813132f4c07e9},
182 {0x2891425503b11f, 0x08781030579fea, 0xf5426ba5cc9674, 0x1e28ebf18562bc},
184 {{0x9f31997cc864eb, 0x06cd91d28b5e4c, 0xff17036691a973, 0xf1aef351497c58},
185 {0xdd1f2d600564ff, 0xdead073b1402db, 0x74a684435bd693, 0xeea7471f962558},
190 {{0x9665266dddf554, 0x9613d78b60ef2d, 0xce27a34cdba417, 0xd35ab74d6afc31},
191 {0x85ccdd22deb15e, 0x2137e5783a6aab, 0xa141cffd8c93c6, 0x355a1830e90f2d},
193 {{0x1a494eadaade65, 0xd6da4da77fe53c, 0xe7992996abec86, 0x65c3553c6090e3},
194 {0xfa610b1fb09346, 0xf1c6540b8a4aaf, 0xc51a13ccd3cbab, 0x02995b1b18c28a},
196 {{0x7874568e7295ef, 0x86b419fbe38d04, 0xdc0690a7550d9a, 0xd3966a44beac33},
197 {0x2b7280ec29132f, 0xbeaa3b6a032df3, 0xdc7dd88ae41200, 0xd25e2513e3a100},
199 {{0x924857eb2efafd, 0xac2bce41223190, 0x8edaa1445553fc, 0x825800fd3562d5},
200 {0x8d79148ea96621, 0x23a01c3dd9ed8d, 0xaf8b219f9416b5, 0xd8db0cc277daea},
202 {{0x76a9c3b1a700f0, 0xe9acd29bc7e691, 0x69212d1a6b0327, 0x6322e97fe154be},
203 {0x469fc5465d62aa, 0x8d41ed18883b05, 0x1f8eae66c52b88, 0xe4fcbe9325be51},
205 {{0x825fdf583cac16, 0x020b857c7b023a, 0x683c17744b0165, 0x14ffd0a2daf2f1},
206 {0x323b36184218f9, 0x4944ec4e3b47d4, 0xc15b3080841acf, 0x0bced4b01a28bb},
208 {{0x92ac22230df5c4, 0x52f33b4063eda8, 0xcb3f19870c0c93, 0x40064f2ba65233},
209 {0xfe16f0924f8992, 0x012da25af5b517, 0x1a57bb24f723a6, 0x06f8bc76760def},
211 {{0x4a7084f7817cb9, 0xbcab0738ee9a78, 0x3ec11e11d9c326, 0xdc0fe90e0f1aae},
212 {0xcf639ea5f98390, 0x5c350aa22ffb74, 0x9afae98a4047b7, 0x956ec2d617fc45},
214 {{0x4306d648c1be6a, 0x9247cd8bc9a462, 0xf5595e377d2f2e, 0xbd1c3caff1a52e},
215 {0x045e14472409d0, 0x29f3e17078f773, 0x745a602b2d4f7d, 0x191837685cdfbb},
217 {{0x5b6ee254a8cb79, 0x4953433f5e7026, 0xe21faeb1d1def4, 0xc4c225785c09de},
218 {0x307ce7bba1e518, 0x31b125b1036db8, 0x47e91868839e8f, 0xc765866e33b9f3},
220 {{0x3bfece24f96906, 0x4794da641e5093, 0xde5df64f95db26, 0x297ecd89714b05},
221 {0x701bd3ebb2c3aa, 0x7073b4f53cb1d5, 0x13c5665658af16, 0x9895089d66fe58},
223 {{0x0fef05f78c4790, 0x2d773633b05d2e, 0x94229c3a951c94, 0xbbbd70df4911bb},
224 {0xb2c6963d2c1168, 0x105f47a72b0d73, 0x9fdf6111614080, 0x7b7e94b39e67b0},
226 {{0xad1a7d6efbe2b3, 0xf012482c0da69d, 0x6b3bdf12438345, 0x40d7558d7aa4d9},
227 {0x8a09fffb5c6d3d, 0x9a356e5d9ffd38, 0x5973f15f4f9b1c, 0xdcd5f59f63c3ea},
229 {{0xacf39f4c5ca7ab, 0x4c8071cc5fd737, 0xc64e3602cd1184, 0x0acd4644c9abba},
230 {0x6c011a36d8bf6e, 0xfecd87ba24e32a, 0x19f6f56574fad8, 0x050b204ced9405},
232 {{0xed4f1cae7d9a96, 0x5ceef7ad94c40a, 0x778e4a3bf3ef9b, 0x7405783dc3b55e},
233 {0x32477c61b6e8c6, 0xb46a97570f018b, 0x91176d0a7e95d1, 0x3df90fbc4c7d0e},
237 /* Precomputation for the group generator. */
238 struct nistp224_pre_comp_st {
239 felem g_pre_comp[2][16][3];
240 CRYPTO_REF_COUNT references;
244 const EC_METHOD *EC_GFp_nistp224_method(void)
246 static const EC_METHOD ret = {
247 EC_FLAGS_DEFAULT_OCT,
248 NID_X9_62_prime_field,
249 ossl_ec_GFp_nistp224_group_init,
250 ossl_ec_GFp_simple_group_finish,
251 ossl_ec_GFp_simple_group_clear_finish,
252 ossl_ec_GFp_nist_group_copy,
253 ossl_ec_GFp_nistp224_group_set_curve,
254 ossl_ec_GFp_simple_group_get_curve,
255 ossl_ec_GFp_simple_group_get_degree,
256 ossl_ec_group_simple_order_bits,
257 ossl_ec_GFp_simple_group_check_discriminant,
258 ossl_ec_GFp_simple_point_init,
259 ossl_ec_GFp_simple_point_finish,
260 ossl_ec_GFp_simple_point_clear_finish,
261 ossl_ec_GFp_simple_point_copy,
262 ossl_ec_GFp_simple_point_set_to_infinity,
263 ossl_ec_GFp_simple_point_set_affine_coordinates,
264 ossl_ec_GFp_nistp224_point_get_affine_coordinates,
265 0 /* point_set_compressed_coordinates */ ,
268 ossl_ec_GFp_simple_add,
269 ossl_ec_GFp_simple_dbl,
270 ossl_ec_GFp_simple_invert,
271 ossl_ec_GFp_simple_is_at_infinity,
272 ossl_ec_GFp_simple_is_on_curve,
273 ossl_ec_GFp_simple_cmp,
274 ossl_ec_GFp_simple_make_affine,
275 ossl_ec_GFp_simple_points_make_affine,
276 ossl_ec_GFp_nistp224_points_mul,
277 ossl_ec_GFp_nistp224_precompute_mult,
278 ossl_ec_GFp_nistp224_have_precompute_mult,
279 ossl_ec_GFp_nist_field_mul,
280 ossl_ec_GFp_nist_field_sqr,
282 ossl_ec_GFp_simple_field_inv,
283 0 /* field_encode */ ,
284 0 /* field_decode */ ,
285 0, /* field_set_to_one */
286 ossl_ec_key_simple_priv2oct,
287 ossl_ec_key_simple_oct2priv,
289 ossl_ec_key_simple_generate_key,
290 ossl_ec_key_simple_check_key,
291 ossl_ec_key_simple_generate_public_key,
294 ossl_ecdh_simple_compute_key,
295 ossl_ecdsa_simple_sign_setup,
296 ossl_ecdsa_simple_sign_sig,
297 ossl_ecdsa_simple_verify_sig,
298 0, /* field_inverse_mod_ord */
299 0, /* blind_coordinates */
309 * Helper functions to convert field elements to/from internal representation
311 static void bin28_to_felem(felem out, const u8 in[28])
313 out[0] = *((const limb *)(in)) & 0x00ffffffffffffff;
314 out[1] = (*((const limb_aX *)(in + 7))) & 0x00ffffffffffffff;
315 out[2] = (*((const limb_aX *)(in + 14))) & 0x00ffffffffffffff;
316 out[3] = (*((const limb_aX *)(in + 20))) >> 8;
319 static void felem_to_bin28(u8 out[28], const felem in)
322 for (i = 0; i < 7; ++i) {
323 out[i] = in[0] >> (8 * i);
324 out[i + 7] = in[1] >> (8 * i);
325 out[i + 14] = in[2] >> (8 * i);
326 out[i + 21] = in[3] >> (8 * i);
330 /* From OpenSSL BIGNUM to internal representation */
331 static int BN_to_felem(felem out, const BIGNUM *bn)
333 felem_bytearray b_out;
336 if (BN_is_negative(bn)) {
337 ERR_raise(ERR_LIB_EC, EC_R_BIGNUM_OUT_OF_RANGE);
340 num_bytes = BN_bn2lebinpad(bn, b_out, sizeof(b_out));
342 ERR_raise(ERR_LIB_EC, EC_R_BIGNUM_OUT_OF_RANGE);
345 bin28_to_felem(out, b_out);
349 /* From internal representation to OpenSSL BIGNUM */
350 static BIGNUM *felem_to_BN(BIGNUM *out, const felem in)
352 felem_bytearray b_out;
353 felem_to_bin28(b_out, in);
354 return BN_lebin2bn(b_out, sizeof(b_out), out);
357 /******************************************************************************/
361 * Field operations, using the internal representation of field elements.
362 * NB! These operations are specific to our point multiplication and cannot be
363 * expected to be correct in general - e.g., multiplication with a large scalar
364 * will cause an overflow.
368 static void felem_one(felem out)
376 static void felem_assign(felem out, const felem in)
384 /* Sum two field elements: out += in */
385 static void felem_sum(felem out, const felem in)
393 /* Subtract field elements: out -= in */
394 /* Assumes in[i] < 2^57 */
395 static void felem_diff(felem out, const felem in)
397 static const limb two58p2 = (((limb) 1) << 58) + (((limb) 1) << 2);
398 static const limb two58m2 = (((limb) 1) << 58) - (((limb) 1) << 2);
399 static const limb two58m42m2 = (((limb) 1) << 58) -
400 (((limb) 1) << 42) - (((limb) 1) << 2);
402 /* Add 0 mod 2^224-2^96+1 to ensure out > in */
404 out[1] += two58m42m2;
414 /* Subtract in unreduced 128-bit mode: out -= in */
415 /* Assumes in[i] < 2^119 */
416 static void widefelem_diff(widefelem out, const widefelem in)
418 static const widelimb two120 = ((widelimb) 1) << 120;
419 static const widelimb two120m64 = (((widelimb) 1) << 120) -
420 (((widelimb) 1) << 64);
421 static const widelimb two120m104m64 = (((widelimb) 1) << 120) -
422 (((widelimb) 1) << 104) - (((widelimb) 1) << 64);
424 /* Add 0 mod 2^224-2^96+1 to ensure out > in */
429 out[4] += two120m104m64;
442 /* Subtract in mixed mode: out128 -= in64 */
444 static void felem_diff_128_64(widefelem out, const felem in)
446 static const widelimb two64p8 = (((widelimb) 1) << 64) +
447 (((widelimb) 1) << 8);
448 static const widelimb two64m8 = (((widelimb) 1) << 64) -
449 (((widelimb) 1) << 8);
450 static const widelimb two64m48m8 = (((widelimb) 1) << 64) -
451 (((widelimb) 1) << 48) - (((widelimb) 1) << 8);
453 /* Add 0 mod 2^224-2^96+1 to ensure out > in */
455 out[1] += two64m48m8;
466 * Multiply a field element by a scalar: out = out * scalar The scalars we
467 * actually use are small, so results fit without overflow
469 static void felem_scalar(felem out, const limb scalar)
478 * Multiply an unreduced field element by a scalar: out = out * scalar The
479 * scalars we actually use are small, so results fit without overflow
481 static void widefelem_scalar(widefelem out, const widelimb scalar)
492 /* Square a field element: out = in^2 */
493 static void felem_square(widefelem out, const felem in)
495 limb tmp0, tmp1, tmp2;
499 out[0] = ((widelimb) in[0]) * in[0];
500 out[1] = ((widelimb) in[0]) * tmp1;
501 out[2] = ((widelimb) in[0]) * tmp2 + ((widelimb) in[1]) * in[1];
502 out[3] = ((widelimb) in[3]) * tmp0 + ((widelimb) in[1]) * tmp2;
503 out[4] = ((widelimb) in[3]) * tmp1 + ((widelimb) in[2]) * in[2];
504 out[5] = ((widelimb) in[3]) * tmp2;
505 out[6] = ((widelimb) in[3]) * in[3];
508 /* Multiply two field elements: out = in1 * in2 */
509 static void felem_mul(widefelem out, const felem in1, const felem in2)
511 out[0] = ((widelimb) in1[0]) * in2[0];
512 out[1] = ((widelimb) in1[0]) * in2[1] + ((widelimb) in1[1]) * in2[0];
513 out[2] = ((widelimb) in1[0]) * in2[2] + ((widelimb) in1[1]) * in2[1] +
514 ((widelimb) in1[2]) * in2[0];
515 out[3] = ((widelimb) in1[0]) * in2[3] + ((widelimb) in1[1]) * in2[2] +
516 ((widelimb) in1[2]) * in2[1] + ((widelimb) in1[3]) * in2[0];
517 out[4] = ((widelimb) in1[1]) * in2[3] + ((widelimb) in1[2]) * in2[2] +
518 ((widelimb) in1[3]) * in2[1];
519 out[5] = ((widelimb) in1[2]) * in2[3] + ((widelimb) in1[3]) * in2[2];
520 out[6] = ((widelimb) in1[3]) * in2[3];
524 * Reduce seven 128-bit coefficients to four 64-bit coefficients.
525 * Requires in[i] < 2^126,
526 * ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] <= 2^56 + 2^16 */
527 static void felem_reduce(felem out, const widefelem in)
529 static const widelimb two127p15 = (((widelimb) 1) << 127) +
530 (((widelimb) 1) << 15);
531 static const widelimb two127m71 = (((widelimb) 1) << 127) -
532 (((widelimb) 1) << 71);
533 static const widelimb two127m71m55 = (((widelimb) 1) << 127) -
534 (((widelimb) 1) << 71) - (((widelimb) 1) << 55);
537 /* Add 0 mod 2^224-2^96+1 to ensure all differences are positive */
538 output[0] = in[0] + two127p15;
539 output[1] = in[1] + two127m71m55;
540 output[2] = in[2] + two127m71;
544 /* Eliminate in[4], in[5], in[6] */
545 output[4] += in[6] >> 16;
546 output[3] += (in[6] & 0xffff) << 40;
549 output[3] += in[5] >> 16;
550 output[2] += (in[5] & 0xffff) << 40;
553 output[2] += output[4] >> 16;
554 output[1] += (output[4] & 0xffff) << 40;
555 output[0] -= output[4];
557 /* Carry 2 -> 3 -> 4 */
558 output[3] += output[2] >> 56;
559 output[2] &= 0x00ffffffffffffff;
561 output[4] = output[3] >> 56;
562 output[3] &= 0x00ffffffffffffff;
564 /* Now output[2] < 2^56, output[3] < 2^56, output[4] < 2^72 */
566 /* Eliminate output[4] */
567 output[2] += output[4] >> 16;
568 /* output[2] < 2^56 + 2^56 = 2^57 */
569 output[1] += (output[4] & 0xffff) << 40;
570 output[0] -= output[4];
572 /* Carry 0 -> 1 -> 2 -> 3 */
573 output[1] += output[0] >> 56;
574 out[0] = output[0] & 0x00ffffffffffffff;
576 output[2] += output[1] >> 56;
577 /* output[2] < 2^57 + 2^72 */
578 out[1] = output[1] & 0x00ffffffffffffff;
579 output[3] += output[2] >> 56;
580 /* output[3] <= 2^56 + 2^16 */
581 out[2] = output[2] & 0x00ffffffffffffff;
584 * out[0] < 2^56, out[1] < 2^56, out[2] < 2^56,
585 * out[3] <= 2^56 + 2^16 (due to final carry),
591 static void felem_square_reduce(felem out, const felem in)
594 felem_square(tmp, in);
595 felem_reduce(out, tmp);
598 static void felem_mul_reduce(felem out, const felem in1, const felem in2)
601 felem_mul(tmp, in1, in2);
602 felem_reduce(out, tmp);
606 * Reduce to unique minimal representation. Requires 0 <= in < 2*p (always
607 * call felem_reduce first)
609 static void felem_contract(felem out, const felem in)
611 static const int64_t two56 = ((limb) 1) << 56;
612 /* 0 <= in < 2*p, p = 2^224 - 2^96 + 1 */
613 /* if in > p , reduce in = in - 2^224 + 2^96 - 1 */
619 /* Case 1: a = 1 iff in >= 2^224 */
623 tmp[3] &= 0x00ffffffffffffff;
625 * Case 2: a = 0 iff p <= in < 2^224, i.e., the high 128 bits are all 1
626 * and the lower part is non-zero
628 a = ((in[3] & in[2] & (in[1] | 0x000000ffffffffff)) + 1) |
629 (((int64_t) (in[0] + (in[1] & 0x000000ffffffffff)) - 1) >> 63);
630 a &= 0x00ffffffffffffff;
631 /* turn a into an all-one mask (if a = 0) or an all-zero mask */
633 /* subtract 2^224 - 2^96 + 1 if a is all-one */
634 tmp[3] &= a ^ 0xffffffffffffffff;
635 tmp[2] &= a ^ 0xffffffffffffffff;
636 tmp[1] &= (a ^ 0xffffffffffffffff) | 0x000000ffffffffff;
640 * eliminate negative coefficients: if tmp[0] is negative, tmp[1] must be
641 * non-zero, so we only need one step
647 /* carry 1 -> 2 -> 3 */
648 tmp[2] += tmp[1] >> 56;
649 tmp[1] &= 0x00ffffffffffffff;
651 tmp[3] += tmp[2] >> 56;
652 tmp[2] &= 0x00ffffffffffffff;
654 /* Now 0 <= out < p */
662 * Get negative value: out = -in
663 * Requires in[i] < 2^63,
664 * ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] <= 2^56 + 2^16
666 static void felem_neg(felem out, const felem in)
670 memset(tmp, 0, sizeof(tmp));
671 felem_diff_128_64(tmp, in);
672 felem_reduce(out, tmp);
676 * Zero-check: returns 1 if input is 0, and 0 otherwise. We know that field
677 * elements are reduced to in < 2^225, so we only need to check three cases:
678 * 0, 2^224 - 2^96 + 1, and 2^225 - 2^97 + 2
680 static limb felem_is_zero(const felem in)
682 limb zero, two224m96p1, two225m97p2;
684 zero = in[0] | in[1] | in[2] | in[3];
685 zero = (((int64_t) (zero) - 1) >> 63) & 1;
686 two224m96p1 = (in[0] ^ 1) | (in[1] ^ 0x00ffff0000000000)
687 | (in[2] ^ 0x00ffffffffffffff) | (in[3] ^ 0x00ffffffffffffff);
688 two224m96p1 = (((int64_t) (two224m96p1) - 1) >> 63) & 1;
689 two225m97p2 = (in[0] ^ 2) | (in[1] ^ 0x00fffe0000000000)
690 | (in[2] ^ 0x00ffffffffffffff) | (in[3] ^ 0x01ffffffffffffff);
691 two225m97p2 = (((int64_t) (two225m97p2) - 1) >> 63) & 1;
692 return (zero | two224m96p1 | two225m97p2);
695 static int felem_is_zero_int(const void *in)
697 return (int)(felem_is_zero(in) & ((limb) 1));
700 /* Invert a field element */
701 /* Computation chain copied from djb's code */
702 static void felem_inv(felem out, const felem in)
704 felem ftmp, ftmp2, ftmp3, ftmp4;
708 felem_square(tmp, in);
709 felem_reduce(ftmp, tmp); /* 2 */
710 felem_mul(tmp, in, ftmp);
711 felem_reduce(ftmp, tmp); /* 2^2 - 1 */
712 felem_square(tmp, ftmp);
713 felem_reduce(ftmp, tmp); /* 2^3 - 2 */
714 felem_mul(tmp, in, ftmp);
715 felem_reduce(ftmp, tmp); /* 2^3 - 1 */
716 felem_square(tmp, ftmp);
717 felem_reduce(ftmp2, tmp); /* 2^4 - 2 */
718 felem_square(tmp, ftmp2);
719 felem_reduce(ftmp2, tmp); /* 2^5 - 4 */
720 felem_square(tmp, ftmp2);
721 felem_reduce(ftmp2, tmp); /* 2^6 - 8 */
722 felem_mul(tmp, ftmp2, ftmp);
723 felem_reduce(ftmp, tmp); /* 2^6 - 1 */
724 felem_square(tmp, ftmp);
725 felem_reduce(ftmp2, tmp); /* 2^7 - 2 */
726 for (i = 0; i < 5; ++i) { /* 2^12 - 2^6 */
727 felem_square(tmp, ftmp2);
728 felem_reduce(ftmp2, tmp);
730 felem_mul(tmp, ftmp2, ftmp);
731 felem_reduce(ftmp2, tmp); /* 2^12 - 1 */
732 felem_square(tmp, ftmp2);
733 felem_reduce(ftmp3, tmp); /* 2^13 - 2 */
734 for (i = 0; i < 11; ++i) { /* 2^24 - 2^12 */
735 felem_square(tmp, ftmp3);
736 felem_reduce(ftmp3, tmp);
738 felem_mul(tmp, ftmp3, ftmp2);
739 felem_reduce(ftmp2, tmp); /* 2^24 - 1 */
740 felem_square(tmp, ftmp2);
741 felem_reduce(ftmp3, tmp); /* 2^25 - 2 */
742 for (i = 0; i < 23; ++i) { /* 2^48 - 2^24 */
743 felem_square(tmp, ftmp3);
744 felem_reduce(ftmp3, tmp);
746 felem_mul(tmp, ftmp3, ftmp2);
747 felem_reduce(ftmp3, tmp); /* 2^48 - 1 */
748 felem_square(tmp, ftmp3);
749 felem_reduce(ftmp4, tmp); /* 2^49 - 2 */
750 for (i = 0; i < 47; ++i) { /* 2^96 - 2^48 */
751 felem_square(tmp, ftmp4);
752 felem_reduce(ftmp4, tmp);
754 felem_mul(tmp, ftmp3, ftmp4);
755 felem_reduce(ftmp3, tmp); /* 2^96 - 1 */
756 felem_square(tmp, ftmp3);
757 felem_reduce(ftmp4, tmp); /* 2^97 - 2 */
758 for (i = 0; i < 23; ++i) { /* 2^120 - 2^24 */
759 felem_square(tmp, ftmp4);
760 felem_reduce(ftmp4, tmp);
762 felem_mul(tmp, ftmp2, ftmp4);
763 felem_reduce(ftmp2, tmp); /* 2^120 - 1 */
764 for (i = 0; i < 6; ++i) { /* 2^126 - 2^6 */
765 felem_square(tmp, ftmp2);
766 felem_reduce(ftmp2, tmp);
768 felem_mul(tmp, ftmp2, ftmp);
769 felem_reduce(ftmp, tmp); /* 2^126 - 1 */
770 felem_square(tmp, ftmp);
771 felem_reduce(ftmp, tmp); /* 2^127 - 2 */
772 felem_mul(tmp, ftmp, in);
773 felem_reduce(ftmp, tmp); /* 2^127 - 1 */
774 for (i = 0; i < 97; ++i) { /* 2^224 - 2^97 */
775 felem_square(tmp, ftmp);
776 felem_reduce(ftmp, tmp);
778 felem_mul(tmp, ftmp, ftmp3);
779 felem_reduce(out, tmp); /* 2^224 - 2^96 - 1 */
783 * Copy in constant time: if icopy == 1, copy in to out, if icopy == 0, copy
786 static void copy_conditional(felem out, const felem in, limb icopy)
790 * icopy is a (64-bit) 0 or 1, so copy is either all-zero or all-one
792 const limb copy = -icopy;
793 for (i = 0; i < 4; ++i) {
794 const limb tmp = copy & (in[i] ^ out[i]);
799 /******************************************************************************/
801 * ELLIPTIC CURVE POINT OPERATIONS
803 * Points are represented in Jacobian projective coordinates:
804 * (X, Y, Z) corresponds to the affine point (X/Z^2, Y/Z^3),
805 * or to the point at infinity if Z == 0.
810 * Double an elliptic curve point:
811 * (X', Y', Z') = 2 * (X, Y, Z), where
812 * X' = (3 * (X - Z^2) * (X + Z^2))^2 - 8 * X * Y^2
813 * Y' = 3 * (X - Z^2) * (X + Z^2) * (4 * X * Y^2 - X') - 8 * Y^4
814 * Z' = (Y + Z)^2 - Y^2 - Z^2 = 2 * Y * Z
815 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed,
816 * while x_out == y_in is not (maybe this works, but it's not tested).
819 point_double(felem x_out, felem y_out, felem z_out,
820 const felem x_in, const felem y_in, const felem z_in)
823 felem delta, gamma, beta, alpha, ftmp, ftmp2;
825 felem_assign(ftmp, x_in);
826 felem_assign(ftmp2, x_in);
829 felem_square(tmp, z_in);
830 felem_reduce(delta, tmp);
833 felem_square(tmp, y_in);
834 felem_reduce(gamma, tmp);
837 felem_mul(tmp, x_in, gamma);
838 felem_reduce(beta, tmp);
840 /* alpha = 3*(x-delta)*(x+delta) */
841 felem_diff(ftmp, delta);
842 /* ftmp[i] < 2^57 + 2^58 + 2 < 2^59 */
843 felem_sum(ftmp2, delta);
844 /* ftmp2[i] < 2^57 + 2^57 = 2^58 */
845 felem_scalar(ftmp2, 3);
846 /* ftmp2[i] < 3 * 2^58 < 2^60 */
847 felem_mul(tmp, ftmp, ftmp2);
848 /* tmp[i] < 2^60 * 2^59 * 4 = 2^121 */
849 felem_reduce(alpha, tmp);
851 /* x' = alpha^2 - 8*beta */
852 felem_square(tmp, alpha);
853 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
854 felem_assign(ftmp, beta);
855 felem_scalar(ftmp, 8);
856 /* ftmp[i] < 8 * 2^57 = 2^60 */
857 felem_diff_128_64(tmp, ftmp);
858 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
859 felem_reduce(x_out, tmp);
861 /* z' = (y + z)^2 - gamma - delta */
862 felem_sum(delta, gamma);
863 /* delta[i] < 2^57 + 2^57 = 2^58 */
864 felem_assign(ftmp, y_in);
865 felem_sum(ftmp, z_in);
866 /* ftmp[i] < 2^57 + 2^57 = 2^58 */
867 felem_square(tmp, ftmp);
868 /* tmp[i] < 4 * 2^58 * 2^58 = 2^118 */
869 felem_diff_128_64(tmp, delta);
870 /* tmp[i] < 2^118 + 2^64 + 8 < 2^119 */
871 felem_reduce(z_out, tmp);
873 /* y' = alpha*(4*beta - x') - 8*gamma^2 */
874 felem_scalar(beta, 4);
875 /* beta[i] < 4 * 2^57 = 2^59 */
876 felem_diff(beta, x_out);
877 /* beta[i] < 2^59 + 2^58 + 2 < 2^60 */
878 felem_mul(tmp, alpha, beta);
879 /* tmp[i] < 4 * 2^57 * 2^60 = 2^119 */
880 felem_square(tmp2, gamma);
881 /* tmp2[i] < 4 * 2^57 * 2^57 = 2^116 */
882 widefelem_scalar(tmp2, 8);
883 /* tmp2[i] < 8 * 2^116 = 2^119 */
884 widefelem_diff(tmp, tmp2);
885 /* tmp[i] < 2^119 + 2^120 < 2^121 */
886 felem_reduce(y_out, tmp);
890 * Add two elliptic curve points:
891 * (X_1, Y_1, Z_1) + (X_2, Y_2, Z_2) = (X_3, Y_3, Z_3), where
892 * X_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1)^2 - (Z_1^2 * X_2 - Z_2^2 * X_1)^3 -
893 * 2 * Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^2
894 * 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) -
895 * Z_2^3 * Y_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^3
896 * Z_3 = (Z_1^2 * X_2 - Z_2^2 * X_1) * (Z_1 * Z_2)
898 * This runs faster if 'mixed' is set, which requires Z_2 = 1 or Z_2 = 0.
902 * This function is not entirely constant-time: it includes a branch for
903 * checking whether the two input points are equal, (while not equal to the
904 * point at infinity). This case never happens during single point
905 * multiplication, so there is no timing leak for ECDH or ECDSA signing.
907 static void point_add(felem x3, felem y3, felem z3,
908 const felem x1, const felem y1, const felem z1,
909 const int mixed, const felem x2, const felem y2,
912 felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, x_out, y_out, z_out;
914 limb z1_is_zero, z2_is_zero, x_equal, y_equal;
919 felem_square(tmp, z2);
920 felem_reduce(ftmp2, tmp);
923 felem_mul(tmp, ftmp2, z2);
924 felem_reduce(ftmp4, tmp);
926 /* ftmp4 = z2^3*y1 */
927 felem_mul(tmp2, ftmp4, y1);
928 felem_reduce(ftmp4, tmp2);
930 /* ftmp2 = z2^2*x1 */
931 felem_mul(tmp2, ftmp2, x1);
932 felem_reduce(ftmp2, tmp2);
935 * We'll assume z2 = 1 (special case z2 = 0 is handled later)
938 /* ftmp4 = z2^3*y1 */
939 felem_assign(ftmp4, y1);
941 /* ftmp2 = z2^2*x1 */
942 felem_assign(ftmp2, x1);
946 felem_square(tmp, z1);
947 felem_reduce(ftmp, tmp);
950 felem_mul(tmp, ftmp, z1);
951 felem_reduce(ftmp3, tmp);
954 felem_mul(tmp, ftmp3, y2);
955 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
957 /* ftmp3 = z1^3*y2 - z2^3*y1 */
958 felem_diff_128_64(tmp, ftmp4);
959 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
960 felem_reduce(ftmp3, tmp);
963 felem_mul(tmp, ftmp, x2);
964 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
966 /* ftmp = z1^2*x2 - z2^2*x1 */
967 felem_diff_128_64(tmp, ftmp2);
968 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
969 felem_reduce(ftmp, tmp);
972 * The formulae are incorrect if the points are equal, in affine coordinates
973 * (X_1, Y_1) == (X_2, Y_2), so we check for this and do doubling if this
976 * We use bitwise operations to avoid potential side-channels introduced by
977 * the short-circuiting behaviour of boolean operators.
979 x_equal = felem_is_zero(ftmp);
980 y_equal = felem_is_zero(ftmp3);
982 * The special case of either point being the point at infinity (z1 and/or
983 * z2 are zero), is handled separately later on in this function, so we
984 * avoid jumping to point_double here in those special cases.
986 z1_is_zero = felem_is_zero(z1);
987 z2_is_zero = felem_is_zero(z2);
990 * Compared to `ecp_nistp256.c` and `ecp_nistp521.c`, in this
991 * specific implementation `felem_is_zero()` returns truth as `0x1`
992 * (rather than `0xff..ff`).
994 * This implies that `~true` in this implementation becomes
995 * `0xff..fe` (rather than `0x0`): for this reason, to be used in
996 * the if expression, we mask out only the last bit in the next
999 points_equal = (x_equal & y_equal & (~z1_is_zero) & (~z2_is_zero)) & 1;
1003 * This is obviously not constant-time but, as mentioned before, this
1004 * case never happens during single point multiplication, so there is no
1005 * timing leak for ECDH or ECDSA signing.
1007 point_double(x3, y3, z3, x1, y1, z1);
1013 felem_mul(tmp, z1, z2);
1014 felem_reduce(ftmp5, tmp);
1016 /* special case z2 = 0 is handled later */
1017 felem_assign(ftmp5, z1);
1020 /* z_out = (z1^2*x2 - z2^2*x1)*(z1*z2) */
1021 felem_mul(tmp, ftmp, ftmp5);
1022 felem_reduce(z_out, tmp);
1024 /* ftmp = (z1^2*x2 - z2^2*x1)^2 */
1025 felem_assign(ftmp5, ftmp);
1026 felem_square(tmp, ftmp);
1027 felem_reduce(ftmp, tmp);
1029 /* ftmp5 = (z1^2*x2 - z2^2*x1)^3 */
1030 felem_mul(tmp, ftmp, ftmp5);
1031 felem_reduce(ftmp5, tmp);
1033 /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
1034 felem_mul(tmp, ftmp2, ftmp);
1035 felem_reduce(ftmp2, tmp);
1037 /* tmp = z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */
1038 felem_mul(tmp, ftmp4, ftmp5);
1039 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
1041 /* tmp2 = (z1^3*y2 - z2^3*y1)^2 */
1042 felem_square(tmp2, ftmp3);
1043 /* tmp2[i] < 4 * 2^57 * 2^57 < 2^116 */
1045 /* tmp2 = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 */
1046 felem_diff_128_64(tmp2, ftmp5);
1047 /* tmp2[i] < 2^116 + 2^64 + 8 < 2^117 */
1049 /* ftmp5 = 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
1050 felem_assign(ftmp5, ftmp2);
1051 felem_scalar(ftmp5, 2);
1052 /* ftmp5[i] < 2 * 2^57 = 2^58 */
1055 * x_out = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 -
1056 * 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2
1058 felem_diff_128_64(tmp2, ftmp5);
1059 /* tmp2[i] < 2^117 + 2^64 + 8 < 2^118 */
1060 felem_reduce(x_out, tmp2);
1062 /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out */
1063 felem_diff(ftmp2, x_out);
1064 /* ftmp2[i] < 2^57 + 2^58 + 2 < 2^59 */
1067 * tmp2 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out)
1069 felem_mul(tmp2, ftmp3, ftmp2);
1070 /* tmp2[i] < 4 * 2^57 * 2^59 = 2^118 */
1073 * y_out = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) -
1074 * z2^3*y1*(z1^2*x2 - z2^2*x1)^3
1076 widefelem_diff(tmp2, tmp);
1077 /* tmp2[i] < 2^118 + 2^120 < 2^121 */
1078 felem_reduce(y_out, tmp2);
1081 * the result (x_out, y_out, z_out) is incorrect if one of the inputs is
1082 * the point at infinity, so we need to check for this separately
1086 * if point 1 is at infinity, copy point 2 to output, and vice versa
1088 copy_conditional(x_out, x2, z1_is_zero);
1089 copy_conditional(x_out, x1, z2_is_zero);
1090 copy_conditional(y_out, y2, z1_is_zero);
1091 copy_conditional(y_out, y1, z2_is_zero);
1092 copy_conditional(z_out, z2, z1_is_zero);
1093 copy_conditional(z_out, z1, z2_is_zero);
1094 felem_assign(x3, x_out);
1095 felem_assign(y3, y_out);
1096 felem_assign(z3, z_out);
1100 * select_point selects the |idx|th point from a precomputation table and
1102 * The pre_comp array argument should be size of |size| argument
1104 static void select_point(const u64 idx, unsigned int size,
1105 const felem pre_comp[][3], felem out[3])
1108 limb *outlimbs = &out[0][0];
1110 memset(out, 0, sizeof(*out) * 3);
1111 for (i = 0; i < size; i++) {
1112 const limb *inlimbs = &pre_comp[i][0][0];
1119 for (j = 0; j < 4 * 3; j++)
1120 outlimbs[j] |= inlimbs[j] & mask;
1124 /* get_bit returns the |i|th bit in |in| */
1125 static char get_bit(const felem_bytearray in, unsigned i)
1129 return (in[i >> 3] >> (i & 7)) & 1;
1133 * Interleaved point multiplication using precomputed point multiples: The
1134 * small point multiples 0*P, 1*P, ..., 16*P are in pre_comp[], the scalars
1135 * in scalars[]. If g_scalar is non-NULL, we also add this multiple of the
1136 * generator, using certain (large) precomputed multiples in g_pre_comp.
1137 * Output point (X, Y, Z) is stored in x_out, y_out, z_out
1139 static void batch_mul(felem x_out, felem y_out, felem z_out,
1140 const felem_bytearray scalars[],
1141 const unsigned num_points, const u8 *g_scalar,
1142 const int mixed, const felem pre_comp[][17][3],
1143 const felem g_pre_comp[2][16][3])
1147 unsigned gen_mul = (g_scalar != NULL);
1148 felem nq[3], tmp[4];
1152 /* set nq to the point at infinity */
1153 memset(nq, 0, sizeof(nq));
1156 * Loop over all scalars msb-to-lsb, interleaving additions of multiples
1157 * of the generator (two in each of the last 28 rounds) and additions of
1158 * other points multiples (every 5th round).
1160 skip = 1; /* save two point operations in the first
1162 for (i = (num_points ? 220 : 27); i >= 0; --i) {
1165 point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
1167 /* add multiples of the generator */
1168 if (gen_mul && (i <= 27)) {
1169 /* first, look 28 bits upwards */
1170 bits = get_bit(g_scalar, i + 196) << 3;
1171 bits |= get_bit(g_scalar, i + 140) << 2;
1172 bits |= get_bit(g_scalar, i + 84) << 1;
1173 bits |= get_bit(g_scalar, i + 28);
1174 /* select the point to add, in constant time */
1175 select_point(bits, 16, g_pre_comp[1], tmp);
1178 /* value 1 below is argument for "mixed" */
1179 point_add(nq[0], nq[1], nq[2],
1180 nq[0], nq[1], nq[2], 1, tmp[0], tmp[1], tmp[2]);
1182 memcpy(nq, tmp, 3 * sizeof(felem));
1186 /* second, look at the current position */
1187 bits = get_bit(g_scalar, i + 168) << 3;
1188 bits |= get_bit(g_scalar, i + 112) << 2;
1189 bits |= get_bit(g_scalar, i + 56) << 1;
1190 bits |= get_bit(g_scalar, i);
1191 /* select the point to add, in constant time */
1192 select_point(bits, 16, g_pre_comp[0], tmp);
1193 point_add(nq[0], nq[1], nq[2],
1194 nq[0], nq[1], nq[2],
1195 1 /* mixed */ , tmp[0], tmp[1], tmp[2]);
1198 /* do other additions every 5 doublings */
1199 if (num_points && (i % 5 == 0)) {
1200 /* loop over all scalars */
1201 for (num = 0; num < num_points; ++num) {
1202 bits = get_bit(scalars[num], i + 4) << 5;
1203 bits |= get_bit(scalars[num], i + 3) << 4;
1204 bits |= get_bit(scalars[num], i + 2) << 3;
1205 bits |= get_bit(scalars[num], i + 1) << 2;
1206 bits |= get_bit(scalars[num], i) << 1;
1207 bits |= get_bit(scalars[num], i - 1);
1208 ossl_ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits);
1210 /* select the point to add or subtract */
1211 select_point(digit, 17, pre_comp[num], tmp);
1212 felem_neg(tmp[3], tmp[1]); /* (X, -Y, Z) is the negative
1214 copy_conditional(tmp[1], tmp[3], sign);
1217 point_add(nq[0], nq[1], nq[2],
1218 nq[0], nq[1], nq[2],
1219 mixed, tmp[0], tmp[1], tmp[2]);
1221 memcpy(nq, tmp, 3 * sizeof(felem));
1227 felem_assign(x_out, nq[0]);
1228 felem_assign(y_out, nq[1]);
1229 felem_assign(z_out, nq[2]);
1232 /******************************************************************************/
1234 * FUNCTIONS TO MANAGE PRECOMPUTATION
1237 static NISTP224_PRE_COMP *nistp224_pre_comp_new(void)
1239 NISTP224_PRE_COMP *ret = OPENSSL_zalloc(sizeof(*ret));
1242 ERR_raise(ERR_LIB_EC, ERR_R_MALLOC_FAILURE);
1246 ret->references = 1;
1248 ret->lock = CRYPTO_THREAD_lock_new();
1249 if (ret->lock == NULL) {
1250 ERR_raise(ERR_LIB_EC, ERR_R_MALLOC_FAILURE);
1257 NISTP224_PRE_COMP *EC_nistp224_pre_comp_dup(NISTP224_PRE_COMP *p)
1261 CRYPTO_UP_REF(&p->references, &i, p->lock);
1265 void EC_nistp224_pre_comp_free(NISTP224_PRE_COMP *p)
1272 CRYPTO_DOWN_REF(&p->references, &i, p->lock);
1273 REF_PRINT_COUNT("EC_nistp224", p);
1276 REF_ASSERT_ISNT(i < 0);
1278 CRYPTO_THREAD_lock_free(p->lock);
1282 /******************************************************************************/
1284 * OPENSSL EC_METHOD FUNCTIONS
1287 int ossl_ec_GFp_nistp224_group_init(EC_GROUP *group)
1290 ret = ossl_ec_GFp_simple_group_init(group);
1291 group->a_is_minus3 = 1;
1295 int ossl_ec_GFp_nistp224_group_set_curve(EC_GROUP *group, const BIGNUM *p,
1296 const BIGNUM *a, const BIGNUM *b,
1300 BIGNUM *curve_p, *curve_a, *curve_b;
1302 BN_CTX *new_ctx = NULL;
1305 ctx = new_ctx = BN_CTX_new();
1311 curve_p = BN_CTX_get(ctx);
1312 curve_a = BN_CTX_get(ctx);
1313 curve_b = BN_CTX_get(ctx);
1314 if (curve_b == NULL)
1316 BN_bin2bn(nistp224_curve_params[0], sizeof(felem_bytearray), curve_p);
1317 BN_bin2bn(nistp224_curve_params[1], sizeof(felem_bytearray), curve_a);
1318 BN_bin2bn(nistp224_curve_params[2], sizeof(felem_bytearray), curve_b);
1319 if ((BN_cmp(curve_p, p)) || (BN_cmp(curve_a, a)) || (BN_cmp(curve_b, b))) {
1320 ERR_raise(ERR_LIB_EC, EC_R_WRONG_CURVE_PARAMETERS);
1323 group->field_mod_func = BN_nist_mod_224;
1324 ret = ossl_ec_GFp_simple_group_set_curve(group, p, a, b, ctx);
1328 BN_CTX_free(new_ctx);
1334 * Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') =
1337 int ossl_ec_GFp_nistp224_point_get_affine_coordinates(const EC_GROUP *group,
1338 const EC_POINT *point,
1339 BIGNUM *x, BIGNUM *y,
1342 felem z1, z2, x_in, y_in, x_out, y_out;
1345 if (EC_POINT_is_at_infinity(group, point)) {
1346 ERR_raise(ERR_LIB_EC, EC_R_POINT_AT_INFINITY);
1349 if ((!BN_to_felem(x_in, point->X)) || (!BN_to_felem(y_in, point->Y)) ||
1350 (!BN_to_felem(z1, point->Z)))
1353 felem_square(tmp, z2);
1354 felem_reduce(z1, tmp);
1355 felem_mul(tmp, x_in, z1);
1356 felem_reduce(x_in, tmp);
1357 felem_contract(x_out, x_in);
1359 if (!felem_to_BN(x, x_out)) {
1360 ERR_raise(ERR_LIB_EC, ERR_R_BN_LIB);
1364 felem_mul(tmp, z1, z2);
1365 felem_reduce(z1, tmp);
1366 felem_mul(tmp, y_in, z1);
1367 felem_reduce(y_in, tmp);
1368 felem_contract(y_out, y_in);
1370 if (!felem_to_BN(y, y_out)) {
1371 ERR_raise(ERR_LIB_EC, ERR_R_BN_LIB);
1378 static void make_points_affine(size_t num, felem points[ /* num */ ][3],
1379 felem tmp_felems[ /* num+1 */ ])
1382 * Runs in constant time, unless an input is the point at infinity (which
1383 * normally shouldn't happen).
1385 ossl_ec_GFp_nistp_points_make_affine_internal(num,
1389 (void (*)(void *))felem_one,
1391 (void (*)(void *, const void *))
1393 (void (*)(void *, const void *))
1394 felem_square_reduce, (void (*)
1401 (void (*)(void *, const void *))
1403 (void (*)(void *, const void *))
1408 * Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL
1409 * values Result is stored in r (r can equal one of the inputs).
1411 int ossl_ec_GFp_nistp224_points_mul(const EC_GROUP *group, EC_POINT *r,
1412 const BIGNUM *scalar, size_t num,
1413 const EC_POINT *points[],
1414 const BIGNUM *scalars[], BN_CTX *ctx)
1420 BIGNUM *x, *y, *z, *tmp_scalar;
1421 felem_bytearray g_secret;
1422 felem_bytearray *secrets = NULL;
1423 felem (*pre_comp)[17][3] = NULL;
1424 felem *tmp_felems = NULL;
1426 int have_pre_comp = 0;
1427 size_t num_points = num;
1428 felem x_in, y_in, z_in, x_out, y_out, z_out;
1429 NISTP224_PRE_COMP *pre = NULL;
1430 const felem(*g_pre_comp)[16][3] = NULL;
1431 EC_POINT *generator = NULL;
1432 const EC_POINT *p = NULL;
1433 const BIGNUM *p_scalar = NULL;
1436 x = BN_CTX_get(ctx);
1437 y = BN_CTX_get(ctx);
1438 z = BN_CTX_get(ctx);
1439 tmp_scalar = BN_CTX_get(ctx);
1440 if (tmp_scalar == NULL)
1443 if (scalar != NULL) {
1444 pre = group->pre_comp.nistp224;
1446 /* we have precomputation, try to use it */
1447 g_pre_comp = (const felem(*)[16][3])pre->g_pre_comp;
1449 /* try to use the standard precomputation */
1450 g_pre_comp = &gmul[0];
1451 generator = EC_POINT_new(group);
1452 if (generator == NULL)
1454 /* get the generator from precomputation */
1455 if (!felem_to_BN(x, g_pre_comp[0][1][0]) ||
1456 !felem_to_BN(y, g_pre_comp[0][1][1]) ||
1457 !felem_to_BN(z, g_pre_comp[0][1][2])) {
1458 ERR_raise(ERR_LIB_EC, ERR_R_BN_LIB);
1461 if (!ossl_ec_GFp_simple_set_Jprojective_coordinates_GFp(group,
1465 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
1466 /* precomputation matches generator */
1470 * we don't have valid precomputation: treat the generator as a
1473 num_points = num_points + 1;
1476 if (num_points > 0) {
1477 if (num_points >= 3) {
1479 * unless we precompute multiples for just one or two points,
1480 * converting those into affine form is time well spent
1484 secrets = OPENSSL_zalloc(sizeof(*secrets) * num_points);
1485 pre_comp = OPENSSL_zalloc(sizeof(*pre_comp) * num_points);
1488 OPENSSL_malloc(sizeof(felem) * (num_points * 17 + 1));
1489 if ((secrets == NULL) || (pre_comp == NULL)
1490 || (mixed && (tmp_felems == NULL))) {
1491 ERR_raise(ERR_LIB_EC, ERR_R_MALLOC_FAILURE);
1496 * we treat NULL scalars as 0, and NULL points as points at infinity,
1497 * i.e., they contribute nothing to the linear combination
1499 for (i = 0; i < num_points; ++i) {
1502 p = EC_GROUP_get0_generator(group);
1505 /* the i^th point */
1507 p_scalar = scalars[i];
1509 if ((p_scalar != NULL) && (p != NULL)) {
1510 /* reduce scalar to 0 <= scalar < 2^224 */
1511 if ((BN_num_bits(p_scalar) > 224)
1512 || (BN_is_negative(p_scalar))) {
1514 * this is an unusual input, and we don't guarantee
1517 if (!BN_nnmod(tmp_scalar, p_scalar, group->order, ctx)) {
1518 ERR_raise(ERR_LIB_EC, ERR_R_BN_LIB);
1521 num_bytes = BN_bn2lebinpad(tmp_scalar,
1522 secrets[i], sizeof(secrets[i]));
1524 num_bytes = BN_bn2lebinpad(p_scalar,
1525 secrets[i], sizeof(secrets[i]));
1527 if (num_bytes < 0) {
1528 ERR_raise(ERR_LIB_EC, ERR_R_BN_LIB);
1531 /* precompute multiples */
1532 if ((!BN_to_felem(x_out, p->X)) ||
1533 (!BN_to_felem(y_out, p->Y)) ||
1534 (!BN_to_felem(z_out, p->Z)))
1536 felem_assign(pre_comp[i][1][0], x_out);
1537 felem_assign(pre_comp[i][1][1], y_out);
1538 felem_assign(pre_comp[i][1][2], z_out);
1539 for (j = 2; j <= 16; ++j) {
1541 point_add(pre_comp[i][j][0], pre_comp[i][j][1],
1542 pre_comp[i][j][2], pre_comp[i][1][0],
1543 pre_comp[i][1][1], pre_comp[i][1][2], 0,
1544 pre_comp[i][j - 1][0],
1545 pre_comp[i][j - 1][1],
1546 pre_comp[i][j - 1][2]);
1548 point_double(pre_comp[i][j][0], pre_comp[i][j][1],
1549 pre_comp[i][j][2], pre_comp[i][j / 2][0],
1550 pre_comp[i][j / 2][1],
1551 pre_comp[i][j / 2][2]);
1557 make_points_affine(num_points * 17, pre_comp[0], tmp_felems);
1560 /* the scalar for the generator */
1561 if ((scalar != NULL) && (have_pre_comp)) {
1562 memset(g_secret, 0, sizeof(g_secret));
1563 /* reduce scalar to 0 <= scalar < 2^224 */
1564 if ((BN_num_bits(scalar) > 224) || (BN_is_negative(scalar))) {
1566 * this is an unusual input, and we don't guarantee
1569 if (!BN_nnmod(tmp_scalar, scalar, group->order, ctx)) {
1570 ERR_raise(ERR_LIB_EC, ERR_R_BN_LIB);
1573 num_bytes = BN_bn2lebinpad(tmp_scalar, g_secret, sizeof(g_secret));
1575 num_bytes = BN_bn2lebinpad(scalar, g_secret, sizeof(g_secret));
1577 /* do the multiplication with generator precomputation */
1578 batch_mul(x_out, y_out, z_out,
1579 (const felem_bytearray(*))secrets, num_points,
1581 mixed, (const felem(*)[17][3])pre_comp, g_pre_comp);
1583 /* do the multiplication without generator precomputation */
1584 batch_mul(x_out, y_out, z_out,
1585 (const felem_bytearray(*))secrets, num_points,
1586 NULL, mixed, (const felem(*)[17][3])pre_comp, NULL);
1588 /* reduce the output to its unique minimal representation */
1589 felem_contract(x_in, x_out);
1590 felem_contract(y_in, y_out);
1591 felem_contract(z_in, z_out);
1592 if ((!felem_to_BN(x, x_in)) || (!felem_to_BN(y, y_in)) ||
1593 (!felem_to_BN(z, z_in))) {
1594 ERR_raise(ERR_LIB_EC, ERR_R_BN_LIB);
1597 ret = ossl_ec_GFp_simple_set_Jprojective_coordinates_GFp(group, r, x, y, z,
1602 EC_POINT_free(generator);
1603 OPENSSL_free(secrets);
1604 OPENSSL_free(pre_comp);
1605 OPENSSL_free(tmp_felems);
1609 int ossl_ec_GFp_nistp224_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
1612 NISTP224_PRE_COMP *pre = NULL;
1615 EC_POINT *generator = NULL;
1616 felem tmp_felems[32];
1618 BN_CTX *new_ctx = NULL;
1621 /* throw away old precomputation */
1622 EC_pre_comp_free(group);
1626 ctx = new_ctx = BN_CTX_new();
1632 x = BN_CTX_get(ctx);
1633 y = BN_CTX_get(ctx);
1636 /* get the generator */
1637 if (group->generator == NULL)
1639 generator = EC_POINT_new(group);
1640 if (generator == NULL)
1642 BN_bin2bn(nistp224_curve_params[3], sizeof(felem_bytearray), x);
1643 BN_bin2bn(nistp224_curve_params[4], sizeof(felem_bytearray), y);
1644 if (!EC_POINT_set_affine_coordinates(group, generator, x, y, ctx))
1646 if ((pre = nistp224_pre_comp_new()) == NULL)
1649 * if the generator is the standard one, use built-in precomputation
1651 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) {
1652 memcpy(pre->g_pre_comp, gmul, sizeof(pre->g_pre_comp));
1655 if ((!BN_to_felem(pre->g_pre_comp[0][1][0], group->generator->X)) ||
1656 (!BN_to_felem(pre->g_pre_comp[0][1][1], group->generator->Y)) ||
1657 (!BN_to_felem(pre->g_pre_comp[0][1][2], group->generator->Z)))
1660 * compute 2^56*G, 2^112*G, 2^168*G for the first table, 2^28*G, 2^84*G,
1661 * 2^140*G, 2^196*G for the second one
1663 for (i = 1; i <= 8; i <<= 1) {
1664 point_double(pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1],
1665 pre->g_pre_comp[1][i][2], pre->g_pre_comp[0][i][0],
1666 pre->g_pre_comp[0][i][1], pre->g_pre_comp[0][i][2]);
1667 for (j = 0; j < 27; ++j) {
1668 point_double(pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1],
1669 pre->g_pre_comp[1][i][2], pre->g_pre_comp[1][i][0],
1670 pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2]);
1674 point_double(pre->g_pre_comp[0][2 * i][0],
1675 pre->g_pre_comp[0][2 * i][1],
1676 pre->g_pre_comp[0][2 * i][2], pre->g_pre_comp[1][i][0],
1677 pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2]);
1678 for (j = 0; j < 27; ++j) {
1679 point_double(pre->g_pre_comp[0][2 * i][0],
1680 pre->g_pre_comp[0][2 * i][1],
1681 pre->g_pre_comp[0][2 * i][2],
1682 pre->g_pre_comp[0][2 * i][0],
1683 pre->g_pre_comp[0][2 * i][1],
1684 pre->g_pre_comp[0][2 * i][2]);
1687 for (i = 0; i < 2; i++) {
1688 /* g_pre_comp[i][0] is the point at infinity */
1689 memset(pre->g_pre_comp[i][0], 0, sizeof(pre->g_pre_comp[i][0]));
1690 /* the remaining multiples */
1691 /* 2^56*G + 2^112*G resp. 2^84*G + 2^140*G */
1692 point_add(pre->g_pre_comp[i][6][0], pre->g_pre_comp[i][6][1],
1693 pre->g_pre_comp[i][6][2], pre->g_pre_comp[i][4][0],
1694 pre->g_pre_comp[i][4][1], pre->g_pre_comp[i][4][2],
1695 0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
1696 pre->g_pre_comp[i][2][2]);
1697 /* 2^56*G + 2^168*G resp. 2^84*G + 2^196*G */
1698 point_add(pre->g_pre_comp[i][10][0], pre->g_pre_comp[i][10][1],
1699 pre->g_pre_comp[i][10][2], pre->g_pre_comp[i][8][0],
1700 pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2],
1701 0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
1702 pre->g_pre_comp[i][2][2]);
1703 /* 2^112*G + 2^168*G resp. 2^140*G + 2^196*G */
1704 point_add(pre->g_pre_comp[i][12][0], pre->g_pre_comp[i][12][1],
1705 pre->g_pre_comp[i][12][2], pre->g_pre_comp[i][8][0],
1706 pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2],
1707 0, pre->g_pre_comp[i][4][0], pre->g_pre_comp[i][4][1],
1708 pre->g_pre_comp[i][4][2]);
1710 * 2^56*G + 2^112*G + 2^168*G resp. 2^84*G + 2^140*G + 2^196*G
1712 point_add(pre->g_pre_comp[i][14][0], pre->g_pre_comp[i][14][1],
1713 pre->g_pre_comp[i][14][2], pre->g_pre_comp[i][12][0],
1714 pre->g_pre_comp[i][12][1], pre->g_pre_comp[i][12][2],
1715 0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
1716 pre->g_pre_comp[i][2][2]);
1717 for (j = 1; j < 8; ++j) {
1718 /* odd multiples: add G resp. 2^28*G */
1719 point_add(pre->g_pre_comp[i][2 * j + 1][0],
1720 pre->g_pre_comp[i][2 * j + 1][1],
1721 pre->g_pre_comp[i][2 * j + 1][2],
1722 pre->g_pre_comp[i][2 * j][0],
1723 pre->g_pre_comp[i][2 * j][1],
1724 pre->g_pre_comp[i][2 * j][2], 0,
1725 pre->g_pre_comp[i][1][0], pre->g_pre_comp[i][1][1],
1726 pre->g_pre_comp[i][1][2]);
1729 make_points_affine(31, &(pre->g_pre_comp[0][1]), tmp_felems);
1732 SETPRECOMP(group, nistp224, pre);
1737 EC_POINT_free(generator);
1739 BN_CTX_free(new_ctx);
1741 EC_nistp224_pre_comp_free(pre);
1745 int ossl_ec_GFp_nistp224_have_precompute_mult(const EC_GROUP *group)
1747 return HAVEPRECOMP(group, nistp224);