2 * Copyright 2010-2018 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 * 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 ecdsa_simple_sign_setup,
296 ecdsa_simple_sign_sig,
297 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 uint64_t *)(in)) & 0x00ffffffffffffff;
314 out[1] = (*((const uint64_t *)(in + 7))) & 0x00ffffffffffffff;
315 out[2] = (*((const uint64_t *)(in + 14))) & 0x00ffffffffffffff;
316 out[3] = (*((const uint64_t *)(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 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
340 num_bytes = BN_bn2lebinpad(bn, b_out, sizeof(b_out));
342 ECerr(EC_F_BN_TO_FELEM, 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;
918 felem_square(tmp, z2);
919 felem_reduce(ftmp2, tmp);
922 felem_mul(tmp, ftmp2, z2);
923 felem_reduce(ftmp4, tmp);
925 /* ftmp4 = z2^3*y1 */
926 felem_mul(tmp2, ftmp4, y1);
927 felem_reduce(ftmp4, tmp2);
929 /* ftmp2 = z2^2*x1 */
930 felem_mul(tmp2, ftmp2, x1);
931 felem_reduce(ftmp2, tmp2);
934 * We'll assume z2 = 1 (special case z2 = 0 is handled later)
937 /* ftmp4 = z2^3*y1 */
938 felem_assign(ftmp4, y1);
940 /* ftmp2 = z2^2*x1 */
941 felem_assign(ftmp2, x1);
945 felem_square(tmp, z1);
946 felem_reduce(ftmp, tmp);
949 felem_mul(tmp, ftmp, z1);
950 felem_reduce(ftmp3, tmp);
953 felem_mul(tmp, ftmp3, y2);
954 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
956 /* ftmp3 = z1^3*y2 - z2^3*y1 */
957 felem_diff_128_64(tmp, ftmp4);
958 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
959 felem_reduce(ftmp3, tmp);
962 felem_mul(tmp, ftmp, x2);
963 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
965 /* ftmp = z1^2*x2 - z2^2*x1 */
966 felem_diff_128_64(tmp, ftmp2);
967 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
968 felem_reduce(ftmp, tmp);
971 * the formulae are incorrect if the points are equal so we check for
972 * this and do doubling if this happens
974 x_equal = felem_is_zero(ftmp);
975 y_equal = felem_is_zero(ftmp3);
976 z1_is_zero = felem_is_zero(z1);
977 z2_is_zero = felem_is_zero(z2);
978 /* In affine coordinates, (X_1, Y_1) == (X_2, Y_2) */
979 if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) {
980 point_double(x3, y3, z3, x1, y1, z1);
986 felem_mul(tmp, z1, z2);
987 felem_reduce(ftmp5, tmp);
989 /* special case z2 = 0 is handled later */
990 felem_assign(ftmp5, z1);
993 /* z_out = (z1^2*x2 - z2^2*x1)*(z1*z2) */
994 felem_mul(tmp, ftmp, ftmp5);
995 felem_reduce(z_out, tmp);
997 /* ftmp = (z1^2*x2 - z2^2*x1)^2 */
998 felem_assign(ftmp5, ftmp);
999 felem_square(tmp, ftmp);
1000 felem_reduce(ftmp, tmp);
1002 /* ftmp5 = (z1^2*x2 - z2^2*x1)^3 */
1003 felem_mul(tmp, ftmp, ftmp5);
1004 felem_reduce(ftmp5, tmp);
1006 /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
1007 felem_mul(tmp, ftmp2, ftmp);
1008 felem_reduce(ftmp2, tmp);
1010 /* tmp = z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */
1011 felem_mul(tmp, ftmp4, ftmp5);
1012 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
1014 /* tmp2 = (z1^3*y2 - z2^3*y1)^2 */
1015 felem_square(tmp2, ftmp3);
1016 /* tmp2[i] < 4 * 2^57 * 2^57 < 2^116 */
1018 /* tmp2 = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 */
1019 felem_diff_128_64(tmp2, ftmp5);
1020 /* tmp2[i] < 2^116 + 2^64 + 8 < 2^117 */
1022 /* ftmp5 = 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
1023 felem_assign(ftmp5, ftmp2);
1024 felem_scalar(ftmp5, 2);
1025 /* ftmp5[i] < 2 * 2^57 = 2^58 */
1028 * x_out = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 -
1029 * 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2
1031 felem_diff_128_64(tmp2, ftmp5);
1032 /* tmp2[i] < 2^117 + 2^64 + 8 < 2^118 */
1033 felem_reduce(x_out, tmp2);
1035 /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out */
1036 felem_diff(ftmp2, x_out);
1037 /* ftmp2[i] < 2^57 + 2^58 + 2 < 2^59 */
1040 * tmp2 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out)
1042 felem_mul(tmp2, ftmp3, ftmp2);
1043 /* tmp2[i] < 4 * 2^57 * 2^59 = 2^118 */
1046 * y_out = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) -
1047 * z2^3*y1*(z1^2*x2 - z2^2*x1)^3
1049 widefelem_diff(tmp2, tmp);
1050 /* tmp2[i] < 2^118 + 2^120 < 2^121 */
1051 felem_reduce(y_out, tmp2);
1054 * the result (x_out, y_out, z_out) is incorrect if one of the inputs is
1055 * the point at infinity, so we need to check for this separately
1059 * if point 1 is at infinity, copy point 2 to output, and vice versa
1061 copy_conditional(x_out, x2, z1_is_zero);
1062 copy_conditional(x_out, x1, z2_is_zero);
1063 copy_conditional(y_out, y2, z1_is_zero);
1064 copy_conditional(y_out, y1, z2_is_zero);
1065 copy_conditional(z_out, z2, z1_is_zero);
1066 copy_conditional(z_out, z1, z2_is_zero);
1067 felem_assign(x3, x_out);
1068 felem_assign(y3, y_out);
1069 felem_assign(z3, z_out);
1073 * select_point selects the |idx|th point from a precomputation table and
1075 * The pre_comp array argument should be size of |size| argument
1077 static void select_point(const u64 idx, unsigned int size,
1078 const felem pre_comp[][3], felem out[3])
1081 limb *outlimbs = &out[0][0];
1083 memset(out, 0, sizeof(*out) * 3);
1084 for (i = 0; i < size; i++) {
1085 const limb *inlimbs = &pre_comp[i][0][0];
1092 for (j = 0; j < 4 * 3; j++)
1093 outlimbs[j] |= inlimbs[j] & mask;
1097 /* get_bit returns the |i|th bit in |in| */
1098 static char get_bit(const felem_bytearray in, unsigned i)
1102 return (in[i >> 3] >> (i & 7)) & 1;
1106 * Interleaved point multiplication using precomputed point multiples: The
1107 * small point multiples 0*P, 1*P, ..., 16*P are in pre_comp[], the scalars
1108 * in scalars[]. If g_scalar is non-NULL, we also add this multiple of the
1109 * generator, using certain (large) precomputed multiples in g_pre_comp.
1110 * Output point (X, Y, Z) is stored in x_out, y_out, z_out
1112 static void batch_mul(felem x_out, felem y_out, felem z_out,
1113 const felem_bytearray scalars[],
1114 const unsigned num_points, const u8 *g_scalar,
1115 const int mixed, const felem pre_comp[][17][3],
1116 const felem g_pre_comp[2][16][3])
1120 unsigned gen_mul = (g_scalar != NULL);
1121 felem nq[3], tmp[4];
1125 /* set nq to the point at infinity */
1126 memset(nq, 0, sizeof(nq));
1129 * Loop over all scalars msb-to-lsb, interleaving additions of multiples
1130 * of the generator (two in each of the last 28 rounds) and additions of
1131 * other points multiples (every 5th round).
1133 skip = 1; /* save two point operations in the first
1135 for (i = (num_points ? 220 : 27); i >= 0; --i) {
1138 point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
1140 /* add multiples of the generator */
1141 if (gen_mul && (i <= 27)) {
1142 /* first, look 28 bits upwards */
1143 bits = get_bit(g_scalar, i + 196) << 3;
1144 bits |= get_bit(g_scalar, i + 140) << 2;
1145 bits |= get_bit(g_scalar, i + 84) << 1;
1146 bits |= get_bit(g_scalar, i + 28);
1147 /* select the point to add, in constant time */
1148 select_point(bits, 16, g_pre_comp[1], tmp);
1151 /* value 1 below is argument for "mixed" */
1152 point_add(nq[0], nq[1], nq[2],
1153 nq[0], nq[1], nq[2], 1, tmp[0], tmp[1], tmp[2]);
1155 memcpy(nq, tmp, 3 * sizeof(felem));
1159 /* second, look at the current position */
1160 bits = get_bit(g_scalar, i + 168) << 3;
1161 bits |= get_bit(g_scalar, i + 112) << 2;
1162 bits |= get_bit(g_scalar, i + 56) << 1;
1163 bits |= get_bit(g_scalar, i);
1164 /* select the point to add, in constant time */
1165 select_point(bits, 16, g_pre_comp[0], tmp);
1166 point_add(nq[0], nq[1], nq[2],
1167 nq[0], nq[1], nq[2],
1168 1 /* mixed */ , tmp[0], tmp[1], tmp[2]);
1171 /* do other additions every 5 doublings */
1172 if (num_points && (i % 5 == 0)) {
1173 /* loop over all scalars */
1174 for (num = 0; num < num_points; ++num) {
1175 bits = get_bit(scalars[num], i + 4) << 5;
1176 bits |= get_bit(scalars[num], i + 3) << 4;
1177 bits |= get_bit(scalars[num], i + 2) << 3;
1178 bits |= get_bit(scalars[num], i + 1) << 2;
1179 bits |= get_bit(scalars[num], i) << 1;
1180 bits |= get_bit(scalars[num], i - 1);
1181 ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits);
1183 /* select the point to add or subtract */
1184 select_point(digit, 17, pre_comp[num], tmp);
1185 felem_neg(tmp[3], tmp[1]); /* (X, -Y, Z) is the negative
1187 copy_conditional(tmp[1], tmp[3], sign);
1190 point_add(nq[0], nq[1], nq[2],
1191 nq[0], nq[1], nq[2],
1192 mixed, tmp[0], tmp[1], tmp[2]);
1194 memcpy(nq, tmp, 3 * sizeof(felem));
1200 felem_assign(x_out, nq[0]);
1201 felem_assign(y_out, nq[1]);
1202 felem_assign(z_out, nq[2]);
1205 /******************************************************************************/
1207 * FUNCTIONS TO MANAGE PRECOMPUTATION
1210 static NISTP224_PRE_COMP *nistp224_pre_comp_new(void)
1212 NISTP224_PRE_COMP *ret = OPENSSL_zalloc(sizeof(*ret));
1215 ECerr(EC_F_NISTP224_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1219 ret->references = 1;
1221 ret->lock = CRYPTO_THREAD_lock_new();
1222 if (ret->lock == NULL) {
1223 ECerr(EC_F_NISTP224_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1230 NISTP224_PRE_COMP *EC_nistp224_pre_comp_dup(NISTP224_PRE_COMP *p)
1234 CRYPTO_UP_REF(&p->references, &i, p->lock);
1238 void EC_nistp224_pre_comp_free(NISTP224_PRE_COMP *p)
1245 CRYPTO_DOWN_REF(&p->references, &i, p->lock);
1246 REF_PRINT_COUNT("EC_nistp224", x);
1249 REF_ASSERT_ISNT(i < 0);
1251 CRYPTO_THREAD_lock_free(p->lock);
1255 /******************************************************************************/
1257 * OPENSSL EC_METHOD FUNCTIONS
1260 int ec_GFp_nistp224_group_init(EC_GROUP *group)
1263 ret = ec_GFp_simple_group_init(group);
1264 group->a_is_minus3 = 1;
1268 int ec_GFp_nistp224_group_set_curve(EC_GROUP *group, const BIGNUM *p,
1269 const BIGNUM *a, const BIGNUM *b,
1273 BIGNUM *curve_p, *curve_a, *curve_b;
1275 BN_CTX *new_ctx = NULL;
1278 ctx = new_ctx = BN_CTX_new();
1284 curve_p = BN_CTX_get(ctx);
1285 curve_a = BN_CTX_get(ctx);
1286 curve_b = BN_CTX_get(ctx);
1287 if (curve_b == NULL)
1289 BN_bin2bn(nistp224_curve_params[0], sizeof(felem_bytearray), curve_p);
1290 BN_bin2bn(nistp224_curve_params[1], sizeof(felem_bytearray), curve_a);
1291 BN_bin2bn(nistp224_curve_params[2], sizeof(felem_bytearray), curve_b);
1292 if ((BN_cmp(curve_p, p)) || (BN_cmp(curve_a, a)) || (BN_cmp(curve_b, b))) {
1293 ECerr(EC_F_EC_GFP_NISTP224_GROUP_SET_CURVE,
1294 EC_R_WRONG_CURVE_PARAMETERS);
1297 group->field_mod_func = BN_nist_mod_224;
1298 ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx);
1302 BN_CTX_free(new_ctx);
1308 * Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') =
1311 int ec_GFp_nistp224_point_get_affine_coordinates(const EC_GROUP *group,
1312 const EC_POINT *point,
1313 BIGNUM *x, BIGNUM *y,
1316 felem z1, z2, x_in, y_in, x_out, y_out;
1319 if (EC_POINT_is_at_infinity(group, point)) {
1320 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1321 EC_R_POINT_AT_INFINITY);
1324 if ((!BN_to_felem(x_in, point->X)) || (!BN_to_felem(y_in, point->Y)) ||
1325 (!BN_to_felem(z1, point->Z)))
1328 felem_square(tmp, z2);
1329 felem_reduce(z1, tmp);
1330 felem_mul(tmp, x_in, z1);
1331 felem_reduce(x_in, tmp);
1332 felem_contract(x_out, x_in);
1334 if (!felem_to_BN(x, x_out)) {
1335 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1340 felem_mul(tmp, z1, z2);
1341 felem_reduce(z1, tmp);
1342 felem_mul(tmp, y_in, z1);
1343 felem_reduce(y_in, tmp);
1344 felem_contract(y_out, y_in);
1346 if (!felem_to_BN(y, y_out)) {
1347 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1355 static void make_points_affine(size_t num, felem points[ /* num */ ][3],
1356 felem tmp_felems[ /* num+1 */ ])
1359 * Runs in constant time, unless an input is the point at infinity (which
1360 * normally shouldn't happen).
1362 ec_GFp_nistp_points_make_affine_internal(num,
1366 (void (*)(void *))felem_one,
1368 (void (*)(void *, const void *))
1370 (void (*)(void *, const void *))
1371 felem_square_reduce, (void (*)
1378 (void (*)(void *, const void *))
1380 (void (*)(void *, const void *))
1385 * Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL
1386 * values Result is stored in r (r can equal one of the inputs).
1388 int ec_GFp_nistp224_points_mul(const EC_GROUP *group, EC_POINT *r,
1389 const BIGNUM *scalar, size_t num,
1390 const EC_POINT *points[],
1391 const BIGNUM *scalars[], BN_CTX *ctx)
1397 BIGNUM *x, *y, *z, *tmp_scalar;
1398 felem_bytearray g_secret;
1399 felem_bytearray *secrets = NULL;
1400 felem (*pre_comp)[17][3] = NULL;
1401 felem *tmp_felems = NULL;
1403 int have_pre_comp = 0;
1404 size_t num_points = num;
1405 felem x_in, y_in, z_in, x_out, y_out, z_out;
1406 NISTP224_PRE_COMP *pre = NULL;
1407 const felem(*g_pre_comp)[16][3] = NULL;
1408 EC_POINT *generator = NULL;
1409 const EC_POINT *p = NULL;
1410 const BIGNUM *p_scalar = NULL;
1413 x = BN_CTX_get(ctx);
1414 y = BN_CTX_get(ctx);
1415 z = BN_CTX_get(ctx);
1416 tmp_scalar = BN_CTX_get(ctx);
1417 if (tmp_scalar == NULL)
1420 if (scalar != NULL) {
1421 pre = group->pre_comp.nistp224;
1423 /* we have precomputation, try to use it */
1424 g_pre_comp = (const felem(*)[16][3])pre->g_pre_comp;
1426 /* try to use the standard precomputation */
1427 g_pre_comp = &gmul[0];
1428 generator = EC_POINT_new(group);
1429 if (generator == NULL)
1431 /* get the generator from precomputation */
1432 if (!felem_to_BN(x, g_pre_comp[0][1][0]) ||
1433 !felem_to_BN(y, g_pre_comp[0][1][1]) ||
1434 !felem_to_BN(z, g_pre_comp[0][1][2])) {
1435 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1438 if (!EC_POINT_set_Jprojective_coordinates_GFp(group,
1442 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
1443 /* precomputation matches generator */
1447 * we don't have valid precomputation: treat the generator as a
1450 num_points = num_points + 1;
1453 if (num_points > 0) {
1454 if (num_points >= 3) {
1456 * unless we precompute multiples for just one or two points,
1457 * converting those into affine form is time well spent
1461 secrets = OPENSSL_zalloc(sizeof(*secrets) * num_points);
1462 pre_comp = OPENSSL_zalloc(sizeof(*pre_comp) * num_points);
1465 OPENSSL_malloc(sizeof(felem) * (num_points * 17 + 1));
1466 if ((secrets == NULL) || (pre_comp == NULL)
1467 || (mixed && (tmp_felems == NULL))) {
1468 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_MALLOC_FAILURE);
1473 * we treat NULL scalars as 0, and NULL points as points at infinity,
1474 * i.e., they contribute nothing to the linear combination
1476 for (i = 0; i < num_points; ++i) {
1479 p = EC_GROUP_get0_generator(group);
1482 /* the i^th point */
1484 p_scalar = scalars[i];
1486 if ((p_scalar != NULL) && (p != NULL)) {
1487 /* reduce scalar to 0 <= scalar < 2^224 */
1488 if ((BN_num_bits(p_scalar) > 224)
1489 || (BN_is_negative(p_scalar))) {
1491 * this is an unusual input, and we don't guarantee
1494 if (!BN_nnmod(tmp_scalar, p_scalar, group->order, ctx)) {
1495 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1498 num_bytes = BN_bn2lebinpad(tmp_scalar,
1499 secrets[i], sizeof(secrets[i]));
1501 num_bytes = BN_bn2lebinpad(p_scalar,
1502 secrets[i], sizeof(secrets[i]));
1504 if (num_bytes < 0) {
1505 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
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_bn2lebinpad(tmp_scalar, g_secret, sizeof(g_secret));
1552 num_bytes = BN_bn2lebinpad(scalar, g_secret, sizeof(g_secret));
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);
1565 /* reduce the output to its unique minimal representation */
1566 felem_contract(x_in, x_out);
1567 felem_contract(y_in, y_out);
1568 felem_contract(z_in, z_out);
1569 if ((!felem_to_BN(x, x_in)) || (!felem_to_BN(y, y_in)) ||
1570 (!felem_to_BN(z, z_in))) {
1571 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1574 ret = EC_POINT_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx);
1578 EC_POINT_free(generator);
1579 OPENSSL_free(secrets);
1580 OPENSSL_free(pre_comp);
1581 OPENSSL_free(tmp_felems);
1585 int ec_GFp_nistp224_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
1588 NISTP224_PRE_COMP *pre = NULL;
1591 EC_POINT *generator = NULL;
1592 felem tmp_felems[32];
1594 BN_CTX *new_ctx = NULL;
1597 /* throw away old precomputation */
1598 EC_pre_comp_free(group);
1602 ctx = new_ctx = BN_CTX_new();
1608 x = BN_CTX_get(ctx);
1609 y = BN_CTX_get(ctx);
1612 /* get the generator */
1613 if (group->generator == NULL)
1615 generator = EC_POINT_new(group);
1616 if (generator == NULL)
1618 BN_bin2bn(nistp224_curve_params[3], sizeof(felem_bytearray), x);
1619 BN_bin2bn(nistp224_curve_params[4], sizeof(felem_bytearray), y);
1620 if (!EC_POINT_set_affine_coordinates(group, generator, x, y, ctx))
1622 if ((pre = nistp224_pre_comp_new()) == NULL)
1625 * if the generator is the standard one, use built-in precomputation
1627 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) {
1628 memcpy(pre->g_pre_comp, gmul, sizeof(pre->g_pre_comp));
1631 if ((!BN_to_felem(pre->g_pre_comp[0][1][0], group->generator->X)) ||
1632 (!BN_to_felem(pre->g_pre_comp[0][1][1], group->generator->Y)) ||
1633 (!BN_to_felem(pre->g_pre_comp[0][1][2], group->generator->Z)))
1636 * compute 2^56*G, 2^112*G, 2^168*G for the first table, 2^28*G, 2^84*G,
1637 * 2^140*G, 2^196*G for the second one
1639 for (i = 1; i <= 8; i <<= 1) {
1640 point_double(pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1],
1641 pre->g_pre_comp[1][i][2], pre->g_pre_comp[0][i][0],
1642 pre->g_pre_comp[0][i][1], pre->g_pre_comp[0][i][2]);
1643 for (j = 0; j < 27; ++j) {
1644 point_double(pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1],
1645 pre->g_pre_comp[1][i][2], pre->g_pre_comp[1][i][0],
1646 pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2]);
1650 point_double(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], pre->g_pre_comp[1][i][0],
1653 pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2]);
1654 for (j = 0; j < 27; ++j) {
1655 point_double(pre->g_pre_comp[0][2 * i][0],
1656 pre->g_pre_comp[0][2 * i][1],
1657 pre->g_pre_comp[0][2 * i][2],
1658 pre->g_pre_comp[0][2 * i][0],
1659 pre->g_pre_comp[0][2 * i][1],
1660 pre->g_pre_comp[0][2 * i][2]);
1663 for (i = 0; i < 2; i++) {
1664 /* g_pre_comp[i][0] is the point at infinity */
1665 memset(pre->g_pre_comp[i][0], 0, sizeof(pre->g_pre_comp[i][0]));
1666 /* the remaining multiples */
1667 /* 2^56*G + 2^112*G resp. 2^84*G + 2^140*G */
1668 point_add(pre->g_pre_comp[i][6][0], pre->g_pre_comp[i][6][1],
1669 pre->g_pre_comp[i][6][2], pre->g_pre_comp[i][4][0],
1670 pre->g_pre_comp[i][4][1], pre->g_pre_comp[i][4][2],
1671 0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
1672 pre->g_pre_comp[i][2][2]);
1673 /* 2^56*G + 2^168*G resp. 2^84*G + 2^196*G */
1674 point_add(pre->g_pre_comp[i][10][0], pre->g_pre_comp[i][10][1],
1675 pre->g_pre_comp[i][10][2], pre->g_pre_comp[i][8][0],
1676 pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2],
1677 0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
1678 pre->g_pre_comp[i][2][2]);
1679 /* 2^112*G + 2^168*G resp. 2^140*G + 2^196*G */
1680 point_add(pre->g_pre_comp[i][12][0], pre->g_pre_comp[i][12][1],
1681 pre->g_pre_comp[i][12][2], pre->g_pre_comp[i][8][0],
1682 pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2],
1683 0, pre->g_pre_comp[i][4][0], pre->g_pre_comp[i][4][1],
1684 pre->g_pre_comp[i][4][2]);
1686 * 2^56*G + 2^112*G + 2^168*G resp. 2^84*G + 2^140*G + 2^196*G
1688 point_add(pre->g_pre_comp[i][14][0], pre->g_pre_comp[i][14][1],
1689 pre->g_pre_comp[i][14][2], pre->g_pre_comp[i][12][0],
1690 pre->g_pre_comp[i][12][1], pre->g_pre_comp[i][12][2],
1691 0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
1692 pre->g_pre_comp[i][2][2]);
1693 for (j = 1; j < 8; ++j) {
1694 /* odd multiples: add G resp. 2^28*G */
1695 point_add(pre->g_pre_comp[i][2 * j + 1][0],
1696 pre->g_pre_comp[i][2 * j + 1][1],
1697 pre->g_pre_comp[i][2 * j + 1][2],
1698 pre->g_pre_comp[i][2 * j][0],
1699 pre->g_pre_comp[i][2 * j][1],
1700 pre->g_pre_comp[i][2 * j][2], 0,
1701 pre->g_pre_comp[i][1][0], pre->g_pre_comp[i][1][1],
1702 pre->g_pre_comp[i][1][2]);
1705 make_points_affine(31, &(pre->g_pre_comp[0][1]), tmp_felems);
1708 SETPRECOMP(group, nistp224, pre);
1713 EC_POINT_free(generator);
1715 BN_CTX_free(new_ctx);
1717 EC_nistp224_pre_comp_free(pre);
1721 int ec_GFp_nistp224_have_precompute_mult(const EC_GROUP *group)
1723 return HAVEPRECOMP(group, nistp224);