2 * Copyright 2010-2019 The OpenSSL Project Authors. All Rights Reserved.
4 * Licensed under the OpenSSL license (the "License"). You may not use
5 * this file except in compliance with the License. You can obtain a copy
6 * in the file LICENSE in the source distribution or at
7 * https://www.openssl.org/source/license.html
10 /* Copyright 2011 Google Inc.
12 * Licensed under the Apache License, Version 2.0 (the "License");
14 * you may not use this file except in compliance with the License.
15 * You may obtain a copy of the License at
17 * http://www.apache.org/licenses/LICENSE-2.0
19 * Unless required by applicable law or agreed to in writing, software
20 * distributed under the License is distributed on an "AS IS" BASIS,
21 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
22 * See the License for the specific language governing permissions and
23 * limitations under the License.
27 * A 64-bit implementation of the NIST P-224 elliptic curve point multiplication
29 * Inspired by Daniel J. Bernstein's public domain nistp224 implementation
30 * and Adam Langley's public domain 64-bit C implementation of curve25519
33 #include <openssl/opensslconf.h>
34 #ifdef OPENSSL_NO_EC_NISTP_64_GCC_128
35 NON_EMPTY_TRANSLATION_UNIT
40 # include <openssl/err.h>
43 # if defined(__SIZEOF_INT128__) && __SIZEOF_INT128__==16
44 /* even with gcc, the typedef won't work for 32-bit platforms */
45 typedef __uint128_t uint128_t; /* nonstandard; implemented by gcc on 64-bit
48 # error "Your compiler doesn't appear to support 128-bit integer types"
54 /******************************************************************************/
56 * INTERNAL REPRESENTATION OF FIELD ELEMENTS
58 * Field elements are represented as a_0 + 2^56*a_1 + 2^112*a_2 + 2^168*a_3
59 * using 64-bit coefficients called 'limbs',
60 * and sometimes (for multiplication results) as
61 * b_0 + 2^56*b_1 + 2^112*b_2 + 2^168*b_3 + 2^224*b_4 + 2^280*b_5 + 2^336*b_6
62 * using 128-bit coefficients called 'widelimbs'.
63 * A 4-limb representation is an 'felem';
64 * a 7-widelimb representation is a 'widefelem'.
65 * Even within felems, bits of adjacent limbs overlap, and we don't always
66 * reduce the representations: we ensure that inputs to each felem
67 * multiplication satisfy a_i < 2^60, so outputs satisfy b_i < 4*2^60*2^60,
68 * and fit into a 128-bit word without overflow. The coefficients are then
69 * again partially reduced to obtain an felem satisfying a_i < 2^57.
70 * We only reduce to the unique minimal representation at the end of the
74 typedef uint64_t limb;
75 typedef uint128_t widelimb;
77 typedef limb felem[4];
78 typedef widelimb widefelem[7];
81 * Field element represented as a byte array. 28*8 = 224 bits is also the
82 * group order size for the elliptic curve, and we also use this type for
83 * scalars for point multiplication.
85 typedef u8 felem_bytearray[28];
87 static const felem_bytearray nistp224_curve_params[5] = {
88 {0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, /* p */
89 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0x00, 0x00, 0x00, 0x00,
90 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x01},
91 {0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, /* a */
92 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFE, 0xFF, 0xFF, 0xFF, 0xFF,
93 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFE},
94 {0xB4, 0x05, 0x0A, 0x85, 0x0C, 0x04, 0xB3, 0xAB, 0xF5, 0x41, /* b */
95 0x32, 0x56, 0x50, 0x44, 0xB0, 0xB7, 0xD7, 0xBF, 0xD8, 0xBA,
96 0x27, 0x0B, 0x39, 0x43, 0x23, 0x55, 0xFF, 0xB4},
97 {0xB7, 0x0E, 0x0C, 0xBD, 0x6B, 0xB4, 0xBF, 0x7F, 0x32, 0x13, /* x */
98 0x90, 0xB9, 0x4A, 0x03, 0xC1, 0xD3, 0x56, 0xC2, 0x11, 0x22,
99 0x34, 0x32, 0x80, 0xD6, 0x11, 0x5C, 0x1D, 0x21},
100 {0xbd, 0x37, 0x63, 0x88, 0xb5, 0xf7, 0x23, 0xfb, 0x4c, 0x22, /* y */
101 0xdf, 0xe6, 0xcd, 0x43, 0x75, 0xa0, 0x5a, 0x07, 0x47, 0x64,
102 0x44, 0xd5, 0x81, 0x99, 0x85, 0x00, 0x7e, 0x34}
106 * Precomputed multiples of the standard generator
107 * Points are given in coordinates (X, Y, Z) where Z normally is 1
108 * (0 for the point at infinity).
109 * For each field element, slice a_0 is word 0, etc.
111 * The table has 2 * 16 elements, starting with the following:
112 * index | bits | point
113 * ------+---------+------------------------------
116 * 2 | 0 0 1 0 | 2^56G
117 * 3 | 0 0 1 1 | (2^56 + 1)G
118 * 4 | 0 1 0 0 | 2^112G
119 * 5 | 0 1 0 1 | (2^112 + 1)G
120 * 6 | 0 1 1 0 | (2^112 + 2^56)G
121 * 7 | 0 1 1 1 | (2^112 + 2^56 + 1)G
122 * 8 | 1 0 0 0 | 2^168G
123 * 9 | 1 0 0 1 | (2^168 + 1)G
124 * 10 | 1 0 1 0 | (2^168 + 2^56)G
125 * 11 | 1 0 1 1 | (2^168 + 2^56 + 1)G
126 * 12 | 1 1 0 0 | (2^168 + 2^112)G
127 * 13 | 1 1 0 1 | (2^168 + 2^112 + 1)G
128 * 14 | 1 1 1 0 | (2^168 + 2^112 + 2^56)G
129 * 15 | 1 1 1 1 | (2^168 + 2^112 + 2^56 + 1)G
130 * followed by a copy of this with each element multiplied by 2^28.
132 * The reason for this is so that we can clock bits into four different
133 * locations when doing simple scalar multiplies against the base point,
134 * and then another four locations using the second 16 elements.
136 static const felem gmul[2][16][3] = {
140 {{0x3280d6115c1d21, 0xc1d356c2112234, 0x7f321390b94a03, 0xb70e0cbd6bb4bf},
141 {0xd5819985007e34, 0x75a05a07476444, 0xfb4c22dfe6cd43, 0xbd376388b5f723},
143 {{0xfd9675666ebbe9, 0xbca7664d40ce5e, 0x2242df8d8a2a43, 0x1f49bbb0f99bc5},
144 {0x29e0b892dc9c43, 0xece8608436e662, 0xdc858f185310d0, 0x9812dd4eb8d321},
146 {{0x6d3e678d5d8eb8, 0x559eed1cb362f1, 0x16e9a3bbce8a3f, 0xeedcccd8c2a748},
147 {0xf19f90ed50266d, 0xabf2b4bf65f9df, 0x313865468fafec, 0x5cb379ba910a17},
149 {{0x0641966cab26e3, 0x91fb2991fab0a0, 0xefec27a4e13a0b, 0x0499aa8a5f8ebe},
150 {0x7510407766af5d, 0x84d929610d5450, 0x81d77aae82f706, 0x6916f6d4338c5b},
152 {{0xea95ac3b1f15c6, 0x086000905e82d4, 0xdd323ae4d1c8b1, 0x932b56be7685a3},
153 {0x9ef93dea25dbbf, 0x41665960f390f0, 0xfdec76dbe2a8a7, 0x523e80f019062a},
155 {{0x822fdd26732c73, 0xa01c83531b5d0f, 0x363f37347c1ba4, 0xc391b45c84725c},
156 {0xbbd5e1b2d6ad24, 0xddfbcde19dfaec, 0xc393da7e222a7f, 0x1efb7890ede244},
158 {{0x4c9e90ca217da1, 0xd11beca79159bb, 0xff8d33c2c98b7c, 0x2610b39409f849},
159 {0x44d1352ac64da0, 0xcdbb7b2c46b4fb, 0x966c079b753c89, 0xfe67e4e820b112},
161 {{0xe28cae2df5312d, 0xc71b61d16f5c6e, 0x79b7619a3e7c4c, 0x05c73240899b47},
162 {0x9f7f6382c73e3a, 0x18615165c56bda, 0x641fab2116fd56, 0x72855882b08394},
164 {{0x0469182f161c09, 0x74a98ca8d00fb5, 0xb89da93489a3e0, 0x41c98768fb0c1d},
165 {0xe5ea05fb32da81, 0x3dce9ffbca6855, 0x1cfe2d3fbf59e6, 0x0e5e03408738a7},
167 {{0xdab22b2333e87f, 0x4430137a5dd2f6, 0xe03ab9f738beb8, 0xcb0c5d0dc34f24},
168 {0x764a7df0c8fda5, 0x185ba5c3fa2044, 0x9281d688bcbe50, 0xc40331df893881},
170 {{0xb89530796f0f60, 0xade92bd26909a3, 0x1a0c83fb4884da, 0x1765bf22a5a984},
171 {0x772a9ee75db09e, 0x23bc6c67cec16f, 0x4c1edba8b14e2f, 0xe2a215d9611369},
173 {{0x571e509fb5efb3, 0xade88696410552, 0xc8ae85fada74fe, 0x6c7e4be83bbde3},
174 {0xff9f51160f4652, 0xb47ce2495a6539, 0xa2946c53b582f4, 0x286d2db3ee9a60},
176 {{0x40bbd5081a44af, 0x0995183b13926c, 0xbcefba6f47f6d0, 0x215619e9cc0057},
177 {0x8bc94d3b0df45e, 0xf11c54a3694f6f, 0x8631b93cdfe8b5, 0xe7e3f4b0982db9},
179 {{0xb17048ab3e1c7b, 0xac38f36ff8a1d8, 0x1c29819435d2c6, 0xc813132f4c07e9},
180 {0x2891425503b11f, 0x08781030579fea, 0xf5426ba5cc9674, 0x1e28ebf18562bc},
182 {{0x9f31997cc864eb, 0x06cd91d28b5e4c, 0xff17036691a973, 0xf1aef351497c58},
183 {0xdd1f2d600564ff, 0xdead073b1402db, 0x74a684435bd693, 0xeea7471f962558},
188 {{0x9665266dddf554, 0x9613d78b60ef2d, 0xce27a34cdba417, 0xd35ab74d6afc31},
189 {0x85ccdd22deb15e, 0x2137e5783a6aab, 0xa141cffd8c93c6, 0x355a1830e90f2d},
191 {{0x1a494eadaade65, 0xd6da4da77fe53c, 0xe7992996abec86, 0x65c3553c6090e3},
192 {0xfa610b1fb09346, 0xf1c6540b8a4aaf, 0xc51a13ccd3cbab, 0x02995b1b18c28a},
194 {{0x7874568e7295ef, 0x86b419fbe38d04, 0xdc0690a7550d9a, 0xd3966a44beac33},
195 {0x2b7280ec29132f, 0xbeaa3b6a032df3, 0xdc7dd88ae41200, 0xd25e2513e3a100},
197 {{0x924857eb2efafd, 0xac2bce41223190, 0x8edaa1445553fc, 0x825800fd3562d5},
198 {0x8d79148ea96621, 0x23a01c3dd9ed8d, 0xaf8b219f9416b5, 0xd8db0cc277daea},
200 {{0x76a9c3b1a700f0, 0xe9acd29bc7e691, 0x69212d1a6b0327, 0x6322e97fe154be},
201 {0x469fc5465d62aa, 0x8d41ed18883b05, 0x1f8eae66c52b88, 0xe4fcbe9325be51},
203 {{0x825fdf583cac16, 0x020b857c7b023a, 0x683c17744b0165, 0x14ffd0a2daf2f1},
204 {0x323b36184218f9, 0x4944ec4e3b47d4, 0xc15b3080841acf, 0x0bced4b01a28bb},
206 {{0x92ac22230df5c4, 0x52f33b4063eda8, 0xcb3f19870c0c93, 0x40064f2ba65233},
207 {0xfe16f0924f8992, 0x012da25af5b517, 0x1a57bb24f723a6, 0x06f8bc76760def},
209 {{0x4a7084f7817cb9, 0xbcab0738ee9a78, 0x3ec11e11d9c326, 0xdc0fe90e0f1aae},
210 {0xcf639ea5f98390, 0x5c350aa22ffb74, 0x9afae98a4047b7, 0x956ec2d617fc45},
212 {{0x4306d648c1be6a, 0x9247cd8bc9a462, 0xf5595e377d2f2e, 0xbd1c3caff1a52e},
213 {0x045e14472409d0, 0x29f3e17078f773, 0x745a602b2d4f7d, 0x191837685cdfbb},
215 {{0x5b6ee254a8cb79, 0x4953433f5e7026, 0xe21faeb1d1def4, 0xc4c225785c09de},
216 {0x307ce7bba1e518, 0x31b125b1036db8, 0x47e91868839e8f, 0xc765866e33b9f3},
218 {{0x3bfece24f96906, 0x4794da641e5093, 0xde5df64f95db26, 0x297ecd89714b05},
219 {0x701bd3ebb2c3aa, 0x7073b4f53cb1d5, 0x13c5665658af16, 0x9895089d66fe58},
221 {{0x0fef05f78c4790, 0x2d773633b05d2e, 0x94229c3a951c94, 0xbbbd70df4911bb},
222 {0xb2c6963d2c1168, 0x105f47a72b0d73, 0x9fdf6111614080, 0x7b7e94b39e67b0},
224 {{0xad1a7d6efbe2b3, 0xf012482c0da69d, 0x6b3bdf12438345, 0x40d7558d7aa4d9},
225 {0x8a09fffb5c6d3d, 0x9a356e5d9ffd38, 0x5973f15f4f9b1c, 0xdcd5f59f63c3ea},
227 {{0xacf39f4c5ca7ab, 0x4c8071cc5fd737, 0xc64e3602cd1184, 0x0acd4644c9abba},
228 {0x6c011a36d8bf6e, 0xfecd87ba24e32a, 0x19f6f56574fad8, 0x050b204ced9405},
230 {{0xed4f1cae7d9a96, 0x5ceef7ad94c40a, 0x778e4a3bf3ef9b, 0x7405783dc3b55e},
231 {0x32477c61b6e8c6, 0xb46a97570f018b, 0x91176d0a7e95d1, 0x3df90fbc4c7d0e},
235 /* Precomputation for the group generator. */
236 struct nistp224_pre_comp_st {
237 felem g_pre_comp[2][16][3];
238 CRYPTO_REF_COUNT references;
242 const EC_METHOD *EC_GFp_nistp224_method(void)
244 static const EC_METHOD ret = {
245 EC_FLAGS_DEFAULT_OCT,
246 NID_X9_62_prime_field,
247 ec_GFp_nistp224_group_init,
248 ec_GFp_simple_group_finish,
249 ec_GFp_simple_group_clear_finish,
250 ec_GFp_nist_group_copy,
251 ec_GFp_nistp224_group_set_curve,
252 ec_GFp_simple_group_get_curve,
253 ec_GFp_simple_group_get_degree,
254 ec_group_simple_order_bits,
255 ec_GFp_simple_group_check_discriminant,
256 ec_GFp_simple_point_init,
257 ec_GFp_simple_point_finish,
258 ec_GFp_simple_point_clear_finish,
259 ec_GFp_simple_point_copy,
260 ec_GFp_simple_point_set_to_infinity,
261 ec_GFp_simple_set_Jprojective_coordinates_GFp,
262 ec_GFp_simple_get_Jprojective_coordinates_GFp,
263 ec_GFp_simple_point_set_affine_coordinates,
264 ec_GFp_nistp224_point_get_affine_coordinates,
265 0 /* point_set_compressed_coordinates */ ,
270 ec_GFp_simple_invert,
271 ec_GFp_simple_is_at_infinity,
272 ec_GFp_simple_is_on_curve,
274 ec_GFp_simple_make_affine,
275 ec_GFp_simple_points_make_affine,
276 ec_GFp_nistp224_points_mul,
277 ec_GFp_nistp224_precompute_mult,
278 ec_GFp_nistp224_have_precompute_mult,
279 ec_GFp_nist_field_mul,
280 ec_GFp_nist_field_sqr,
282 ec_GFp_simple_field_inv,
283 0 /* field_encode */ ,
284 0 /* field_decode */ ,
285 0, /* field_set_to_one */
286 ec_key_simple_priv2oct,
287 ec_key_simple_oct2priv,
289 ec_key_simple_generate_key,
290 ec_key_simple_check_key,
291 ec_key_simple_generate_public_key,
294 ecdh_simple_compute_key,
295 0, /* field_inverse_mod_ord */
296 0, /* blind_coordinates */
306 * Helper functions to convert field elements to/from internal representation
308 static void bin28_to_felem(felem out, const u8 in[28])
310 out[0] = *((const uint64_t *)(in)) & 0x00ffffffffffffff;
311 out[1] = (*((const uint64_t *)(in + 7))) & 0x00ffffffffffffff;
312 out[2] = (*((const uint64_t *)(in + 14))) & 0x00ffffffffffffff;
313 out[3] = (*((const uint64_t *)(in+20))) >> 8;
316 static void felem_to_bin28(u8 out[28], const felem in)
319 for (i = 0; i < 7; ++i) {
320 out[i] = in[0] >> (8 * i);
321 out[i + 7] = in[1] >> (8 * i);
322 out[i + 14] = in[2] >> (8 * i);
323 out[i + 21] = in[3] >> (8 * i);
327 /* To preserve endianness when using BN_bn2bin and BN_bin2bn */
328 static void flip_endian(u8 *out, const u8 *in, unsigned len)
331 for (i = 0; i < len; ++i)
332 out[i] = in[len - 1 - i];
335 /* From OpenSSL BIGNUM to internal representation */
336 static int BN_to_felem(felem out, const BIGNUM *bn)
338 felem_bytearray b_in;
339 felem_bytearray b_out;
342 num_bytes = BN_num_bytes(bn);
343 if (num_bytes > sizeof(b_out)) {
344 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
347 if (BN_is_negative(bn)) {
348 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
351 num_bytes = BN_bn2binpad(bn, b_in, sizeof(b_in));
352 flip_endian(b_out, b_in, num_bytes);
353 bin28_to_felem(out, b_out);
357 /* From internal representation to OpenSSL BIGNUM */
358 static BIGNUM *felem_to_BN(BIGNUM *out, const felem in)
360 felem_bytearray b_in, b_out;
361 felem_to_bin28(b_in, in);
362 flip_endian(b_out, b_in, sizeof(b_out));
363 return BN_bin2bn(b_out, sizeof(b_out), out);
366 /******************************************************************************/
370 * Field operations, using the internal representation of field elements.
371 * NB! These operations are specific to our point multiplication and cannot be
372 * expected to be correct in general - e.g., multiplication with a large scalar
373 * will cause an overflow.
377 static void felem_one(felem out)
385 static void felem_assign(felem out, const felem in)
393 /* Sum two field elements: out += in */
394 static void felem_sum(felem out, const felem in)
402 /* Subtract field elements: out -= in */
403 /* Assumes in[i] < 2^57 */
404 static void felem_diff(felem out, const felem in)
406 static const limb two58p2 = (((limb) 1) << 58) + (((limb) 1) << 2);
407 static const limb two58m2 = (((limb) 1) << 58) - (((limb) 1) << 2);
408 static const limb two58m42m2 = (((limb) 1) << 58) -
409 (((limb) 1) << 42) - (((limb) 1) << 2);
411 /* Add 0 mod 2^224-2^96+1 to ensure out > in */
413 out[1] += two58m42m2;
423 /* Subtract in unreduced 128-bit mode: out -= in */
424 /* Assumes in[i] < 2^119 */
425 static void widefelem_diff(widefelem out, const widefelem in)
427 static const widelimb two120 = ((widelimb) 1) << 120;
428 static const widelimb two120m64 = (((widelimb) 1) << 120) -
429 (((widelimb) 1) << 64);
430 static const widelimb two120m104m64 = (((widelimb) 1) << 120) -
431 (((widelimb) 1) << 104) - (((widelimb) 1) << 64);
433 /* Add 0 mod 2^224-2^96+1 to ensure out > in */
438 out[4] += two120m104m64;
451 /* Subtract in mixed mode: out128 -= in64 */
453 static void felem_diff_128_64(widefelem out, const felem in)
455 static const widelimb two64p8 = (((widelimb) 1) << 64) +
456 (((widelimb) 1) << 8);
457 static const widelimb two64m8 = (((widelimb) 1) << 64) -
458 (((widelimb) 1) << 8);
459 static const widelimb two64m48m8 = (((widelimb) 1) << 64) -
460 (((widelimb) 1) << 48) - (((widelimb) 1) << 8);
462 /* Add 0 mod 2^224-2^96+1 to ensure out > in */
464 out[1] += two64m48m8;
475 * Multiply a field element by a scalar: out = out * scalar The scalars we
476 * actually use are small, so results fit without overflow
478 static void felem_scalar(felem out, const limb scalar)
487 * Multiply an unreduced field element by a scalar: out = out * scalar The
488 * scalars we actually use are small, so results fit without overflow
490 static void widefelem_scalar(widefelem out, const widelimb scalar)
501 /* Square a field element: out = in^2 */
502 static void felem_square(widefelem out, const felem in)
504 limb tmp0, tmp1, tmp2;
508 out[0] = ((widelimb) in[0]) * in[0];
509 out[1] = ((widelimb) in[0]) * tmp1;
510 out[2] = ((widelimb) in[0]) * tmp2 + ((widelimb) in[1]) * in[1];
511 out[3] = ((widelimb) in[3]) * tmp0 + ((widelimb) in[1]) * tmp2;
512 out[4] = ((widelimb) in[3]) * tmp1 + ((widelimb) in[2]) * in[2];
513 out[5] = ((widelimb) in[3]) * tmp2;
514 out[6] = ((widelimb) in[3]) * in[3];
517 /* Multiply two field elements: out = in1 * in2 */
518 static void felem_mul(widefelem out, const felem in1, const felem in2)
520 out[0] = ((widelimb) in1[0]) * in2[0];
521 out[1] = ((widelimb) in1[0]) * in2[1] + ((widelimb) in1[1]) * in2[0];
522 out[2] = ((widelimb) in1[0]) * in2[2] + ((widelimb) in1[1]) * in2[1] +
523 ((widelimb) in1[2]) * in2[0];
524 out[3] = ((widelimb) in1[0]) * in2[3] + ((widelimb) in1[1]) * in2[2] +
525 ((widelimb) in1[2]) * in2[1] + ((widelimb) in1[3]) * in2[0];
526 out[4] = ((widelimb) in1[1]) * in2[3] + ((widelimb) in1[2]) * in2[2] +
527 ((widelimb) in1[3]) * in2[1];
528 out[5] = ((widelimb) in1[2]) * in2[3] + ((widelimb) in1[3]) * in2[2];
529 out[6] = ((widelimb) in1[3]) * in2[3];
533 * Reduce seven 128-bit coefficients to four 64-bit coefficients.
534 * Requires in[i] < 2^126,
535 * ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] <= 2^56 + 2^16 */
536 static void felem_reduce(felem out, const widefelem in)
538 static const widelimb two127p15 = (((widelimb) 1) << 127) +
539 (((widelimb) 1) << 15);
540 static const widelimb two127m71 = (((widelimb) 1) << 127) -
541 (((widelimb) 1) << 71);
542 static const widelimb two127m71m55 = (((widelimb) 1) << 127) -
543 (((widelimb) 1) << 71) - (((widelimb) 1) << 55);
546 /* Add 0 mod 2^224-2^96+1 to ensure all differences are positive */
547 output[0] = in[0] + two127p15;
548 output[1] = in[1] + two127m71m55;
549 output[2] = in[2] + two127m71;
553 /* Eliminate in[4], in[5], in[6] */
554 output[4] += in[6] >> 16;
555 output[3] += (in[6] & 0xffff) << 40;
558 output[3] += in[5] >> 16;
559 output[2] += (in[5] & 0xffff) << 40;
562 output[2] += output[4] >> 16;
563 output[1] += (output[4] & 0xffff) << 40;
564 output[0] -= output[4];
566 /* Carry 2 -> 3 -> 4 */
567 output[3] += output[2] >> 56;
568 output[2] &= 0x00ffffffffffffff;
570 output[4] = output[3] >> 56;
571 output[3] &= 0x00ffffffffffffff;
573 /* Now output[2] < 2^56, output[3] < 2^56, output[4] < 2^72 */
575 /* Eliminate output[4] */
576 output[2] += output[4] >> 16;
577 /* output[2] < 2^56 + 2^56 = 2^57 */
578 output[1] += (output[4] & 0xffff) << 40;
579 output[0] -= output[4];
581 /* Carry 0 -> 1 -> 2 -> 3 */
582 output[1] += output[0] >> 56;
583 out[0] = output[0] & 0x00ffffffffffffff;
585 output[2] += output[1] >> 56;
586 /* output[2] < 2^57 + 2^72 */
587 out[1] = output[1] & 0x00ffffffffffffff;
588 output[3] += output[2] >> 56;
589 /* output[3] <= 2^56 + 2^16 */
590 out[2] = output[2] & 0x00ffffffffffffff;
593 * out[0] < 2^56, out[1] < 2^56, out[2] < 2^56,
594 * out[3] <= 2^56 + 2^16 (due to final carry),
600 static void felem_square_reduce(felem out, const felem in)
603 felem_square(tmp, in);
604 felem_reduce(out, tmp);
607 static void felem_mul_reduce(felem out, const felem in1, const felem in2)
610 felem_mul(tmp, in1, in2);
611 felem_reduce(out, tmp);
615 * Reduce to unique minimal representation. Requires 0 <= in < 2*p (always
616 * call felem_reduce first)
618 static void felem_contract(felem out, const felem in)
620 static const int64_t two56 = ((limb) 1) << 56;
621 /* 0 <= in < 2*p, p = 2^224 - 2^96 + 1 */
622 /* if in > p , reduce in = in - 2^224 + 2^96 - 1 */
628 /* Case 1: a = 1 iff in >= 2^224 */
632 tmp[3] &= 0x00ffffffffffffff;
634 * Case 2: a = 0 iff p <= in < 2^224, i.e., the high 128 bits are all 1
635 * and the lower part is non-zero
637 a = ((in[3] & in[2] & (in[1] | 0x000000ffffffffff)) + 1) |
638 (((int64_t) (in[0] + (in[1] & 0x000000ffffffffff)) - 1) >> 63);
639 a &= 0x00ffffffffffffff;
640 /* turn a into an all-one mask (if a = 0) or an all-zero mask */
642 /* subtract 2^224 - 2^96 + 1 if a is all-one */
643 tmp[3] &= a ^ 0xffffffffffffffff;
644 tmp[2] &= a ^ 0xffffffffffffffff;
645 tmp[1] &= (a ^ 0xffffffffffffffff) | 0x000000ffffffffff;
649 * eliminate negative coefficients: if tmp[0] is negative, tmp[1] must be
650 * non-zero, so we only need one step
656 /* carry 1 -> 2 -> 3 */
657 tmp[2] += tmp[1] >> 56;
658 tmp[1] &= 0x00ffffffffffffff;
660 tmp[3] += tmp[2] >> 56;
661 tmp[2] &= 0x00ffffffffffffff;
663 /* Now 0 <= out < p */
671 * Get negative value: out = -in
672 * Requires in[i] < 2^63,
673 * ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] <= 2^56 + 2^16
675 static void felem_neg(felem out, const felem in)
678 felem_diff_128_64(tmp, in);
679 felem_reduce(out, tmp);
683 * Zero-check: returns 1 if input is 0, and 0 otherwise. We know that field
684 * elements are reduced to in < 2^225, so we only need to check three cases:
685 * 0, 2^224 - 2^96 + 1, and 2^225 - 2^97 + 2
687 static limb felem_is_zero(const felem in)
689 limb zero, two224m96p1, two225m97p2;
691 zero = in[0] | in[1] | in[2] | in[3];
692 zero = (((int64_t) (zero) - 1) >> 63) & 1;
693 two224m96p1 = (in[0] ^ 1) | (in[1] ^ 0x00ffff0000000000)
694 | (in[2] ^ 0x00ffffffffffffff) | (in[3] ^ 0x00ffffffffffffff);
695 two224m96p1 = (((int64_t) (two224m96p1) - 1) >> 63) & 1;
696 two225m97p2 = (in[0] ^ 2) | (in[1] ^ 0x00fffe0000000000)
697 | (in[2] ^ 0x00ffffffffffffff) | (in[3] ^ 0x01ffffffffffffff);
698 two225m97p2 = (((int64_t) (two225m97p2) - 1) >> 63) & 1;
699 return (zero | two224m96p1 | two225m97p2);
702 static int felem_is_zero_int(const void *in)
704 return (int)(felem_is_zero(in) & ((limb) 1));
707 /* Invert a field element */
708 /* Computation chain copied from djb's code */
709 static void felem_inv(felem out, const felem in)
711 felem ftmp, ftmp2, ftmp3, ftmp4;
715 felem_square(tmp, in);
716 felem_reduce(ftmp, tmp); /* 2 */
717 felem_mul(tmp, in, ftmp);
718 felem_reduce(ftmp, tmp); /* 2^2 - 1 */
719 felem_square(tmp, ftmp);
720 felem_reduce(ftmp, tmp); /* 2^3 - 2 */
721 felem_mul(tmp, in, ftmp);
722 felem_reduce(ftmp, tmp); /* 2^3 - 1 */
723 felem_square(tmp, ftmp);
724 felem_reduce(ftmp2, tmp); /* 2^4 - 2 */
725 felem_square(tmp, ftmp2);
726 felem_reduce(ftmp2, tmp); /* 2^5 - 4 */
727 felem_square(tmp, ftmp2);
728 felem_reduce(ftmp2, tmp); /* 2^6 - 8 */
729 felem_mul(tmp, ftmp2, ftmp);
730 felem_reduce(ftmp, tmp); /* 2^6 - 1 */
731 felem_square(tmp, ftmp);
732 felem_reduce(ftmp2, tmp); /* 2^7 - 2 */
733 for (i = 0; i < 5; ++i) { /* 2^12 - 2^6 */
734 felem_square(tmp, ftmp2);
735 felem_reduce(ftmp2, tmp);
737 felem_mul(tmp, ftmp2, ftmp);
738 felem_reduce(ftmp2, tmp); /* 2^12 - 1 */
739 felem_square(tmp, ftmp2);
740 felem_reduce(ftmp3, tmp); /* 2^13 - 2 */
741 for (i = 0; i < 11; ++i) { /* 2^24 - 2^12 */
742 felem_square(tmp, ftmp3);
743 felem_reduce(ftmp3, tmp);
745 felem_mul(tmp, ftmp3, ftmp2);
746 felem_reduce(ftmp2, tmp); /* 2^24 - 1 */
747 felem_square(tmp, ftmp2);
748 felem_reduce(ftmp3, tmp); /* 2^25 - 2 */
749 for (i = 0; i < 23; ++i) { /* 2^48 - 2^24 */
750 felem_square(tmp, ftmp3);
751 felem_reduce(ftmp3, tmp);
753 felem_mul(tmp, ftmp3, ftmp2);
754 felem_reduce(ftmp3, tmp); /* 2^48 - 1 */
755 felem_square(tmp, ftmp3);
756 felem_reduce(ftmp4, tmp); /* 2^49 - 2 */
757 for (i = 0; i < 47; ++i) { /* 2^96 - 2^48 */
758 felem_square(tmp, ftmp4);
759 felem_reduce(ftmp4, tmp);
761 felem_mul(tmp, ftmp3, ftmp4);
762 felem_reduce(ftmp3, tmp); /* 2^96 - 1 */
763 felem_square(tmp, ftmp3);
764 felem_reduce(ftmp4, tmp); /* 2^97 - 2 */
765 for (i = 0; i < 23; ++i) { /* 2^120 - 2^24 */
766 felem_square(tmp, ftmp4);
767 felem_reduce(ftmp4, tmp);
769 felem_mul(tmp, ftmp2, ftmp4);
770 felem_reduce(ftmp2, tmp); /* 2^120 - 1 */
771 for (i = 0; i < 6; ++i) { /* 2^126 - 2^6 */
772 felem_square(tmp, ftmp2);
773 felem_reduce(ftmp2, tmp);
775 felem_mul(tmp, ftmp2, ftmp);
776 felem_reduce(ftmp, tmp); /* 2^126 - 1 */
777 felem_square(tmp, ftmp);
778 felem_reduce(ftmp, tmp); /* 2^127 - 2 */
779 felem_mul(tmp, ftmp, in);
780 felem_reduce(ftmp, tmp); /* 2^127 - 1 */
781 for (i = 0; i < 97; ++i) { /* 2^224 - 2^97 */
782 felem_square(tmp, ftmp);
783 felem_reduce(ftmp, tmp);
785 felem_mul(tmp, ftmp, ftmp3);
786 felem_reduce(out, tmp); /* 2^224 - 2^96 - 1 */
790 * Copy in constant time: if icopy == 1, copy in to out, if icopy == 0, copy
793 static void copy_conditional(felem out, const felem in, limb icopy)
797 * icopy is a (64-bit) 0 or 1, so copy is either all-zero or all-one
799 const limb copy = -icopy;
800 for (i = 0; i < 4; ++i) {
801 const limb tmp = copy & (in[i] ^ out[i]);
806 /******************************************************************************/
808 * ELLIPTIC CURVE POINT OPERATIONS
810 * Points are represented in Jacobian projective coordinates:
811 * (X, Y, Z) corresponds to the affine point (X/Z^2, Y/Z^3),
812 * or to the point at infinity if Z == 0.
817 * Double an elliptic curve point:
818 * (X', Y', Z') = 2 * (X, Y, Z), where
819 * X' = (3 * (X - Z^2) * (X + Z^2))^2 - 8 * X * Y^2
820 * Y' = 3 * (X - Z^2) * (X + Z^2) * (4 * X * Y^2 - X') - 8 * Y^4
821 * Z' = (Y + Z)^2 - Y^2 - Z^2 = 2 * Y * Z
822 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed,
823 * while x_out == y_in is not (maybe this works, but it's not tested).
826 point_double(felem x_out, felem y_out, felem z_out,
827 const felem x_in, const felem y_in, const felem z_in)
830 felem delta, gamma, beta, alpha, ftmp, ftmp2;
832 felem_assign(ftmp, x_in);
833 felem_assign(ftmp2, x_in);
836 felem_square(tmp, z_in);
837 felem_reduce(delta, tmp);
840 felem_square(tmp, y_in);
841 felem_reduce(gamma, tmp);
844 felem_mul(tmp, x_in, gamma);
845 felem_reduce(beta, tmp);
847 /* alpha = 3*(x-delta)*(x+delta) */
848 felem_diff(ftmp, delta);
849 /* ftmp[i] < 2^57 + 2^58 + 2 < 2^59 */
850 felem_sum(ftmp2, delta);
851 /* ftmp2[i] < 2^57 + 2^57 = 2^58 */
852 felem_scalar(ftmp2, 3);
853 /* ftmp2[i] < 3 * 2^58 < 2^60 */
854 felem_mul(tmp, ftmp, ftmp2);
855 /* tmp[i] < 2^60 * 2^59 * 4 = 2^121 */
856 felem_reduce(alpha, tmp);
858 /* x' = alpha^2 - 8*beta */
859 felem_square(tmp, alpha);
860 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
861 felem_assign(ftmp, beta);
862 felem_scalar(ftmp, 8);
863 /* ftmp[i] < 8 * 2^57 = 2^60 */
864 felem_diff_128_64(tmp, ftmp);
865 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
866 felem_reduce(x_out, tmp);
868 /* z' = (y + z)^2 - gamma - delta */
869 felem_sum(delta, gamma);
870 /* delta[i] < 2^57 + 2^57 = 2^58 */
871 felem_assign(ftmp, y_in);
872 felem_sum(ftmp, z_in);
873 /* ftmp[i] < 2^57 + 2^57 = 2^58 */
874 felem_square(tmp, ftmp);
875 /* tmp[i] < 4 * 2^58 * 2^58 = 2^118 */
876 felem_diff_128_64(tmp, delta);
877 /* tmp[i] < 2^118 + 2^64 + 8 < 2^119 */
878 felem_reduce(z_out, tmp);
880 /* y' = alpha*(4*beta - x') - 8*gamma^2 */
881 felem_scalar(beta, 4);
882 /* beta[i] < 4 * 2^57 = 2^59 */
883 felem_diff(beta, x_out);
884 /* beta[i] < 2^59 + 2^58 + 2 < 2^60 */
885 felem_mul(tmp, alpha, beta);
886 /* tmp[i] < 4 * 2^57 * 2^60 = 2^119 */
887 felem_square(tmp2, gamma);
888 /* tmp2[i] < 4 * 2^57 * 2^57 = 2^116 */
889 widefelem_scalar(tmp2, 8);
890 /* tmp2[i] < 8 * 2^116 = 2^119 */
891 widefelem_diff(tmp, tmp2);
892 /* tmp[i] < 2^119 + 2^120 < 2^121 */
893 felem_reduce(y_out, tmp);
897 * Add two elliptic curve points:
898 * (X_1, Y_1, Z_1) + (X_2, Y_2, Z_2) = (X_3, Y_3, Z_3), where
899 * X_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1)^2 - (Z_1^2 * X_2 - Z_2^2 * X_1)^3 -
900 * 2 * Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^2
901 * 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) -
902 * Z_2^3 * Y_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^3
903 * Z_3 = (Z_1^2 * X_2 - Z_2^2 * X_1) * (Z_1 * Z_2)
905 * This runs faster if 'mixed' is set, which requires Z_2 = 1 or Z_2 = 0.
909 * This function is not entirely constant-time: it includes a branch for
910 * checking whether the two input points are equal, (while not equal to the
911 * point at infinity). This case never happens during single point
912 * multiplication, so there is no timing leak for ECDH or ECDSA signing.
914 static void point_add(felem x3, felem y3, felem z3,
915 const felem x1, const felem y1, const felem z1,
916 const int mixed, const felem x2, const felem y2,
919 felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, x_out, y_out, z_out;
921 limb z1_is_zero, z2_is_zero, x_equal, y_equal;
925 felem_square(tmp, z2);
926 felem_reduce(ftmp2, tmp);
929 felem_mul(tmp, ftmp2, z2);
930 felem_reduce(ftmp4, tmp);
932 /* ftmp4 = z2^3*y1 */
933 felem_mul(tmp2, ftmp4, y1);
934 felem_reduce(ftmp4, tmp2);
936 /* ftmp2 = z2^2*x1 */
937 felem_mul(tmp2, ftmp2, x1);
938 felem_reduce(ftmp2, tmp2);
941 * We'll assume z2 = 1 (special case z2 = 0 is handled later)
944 /* ftmp4 = z2^3*y1 */
945 felem_assign(ftmp4, y1);
947 /* ftmp2 = z2^2*x1 */
948 felem_assign(ftmp2, x1);
952 felem_square(tmp, z1);
953 felem_reduce(ftmp, tmp);
956 felem_mul(tmp, ftmp, z1);
957 felem_reduce(ftmp3, tmp);
960 felem_mul(tmp, ftmp3, y2);
961 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
963 /* ftmp3 = z1^3*y2 - z2^3*y1 */
964 felem_diff_128_64(tmp, ftmp4);
965 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
966 felem_reduce(ftmp3, tmp);
969 felem_mul(tmp, ftmp, x2);
970 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
972 /* ftmp = z1^2*x2 - z2^2*x1 */
973 felem_diff_128_64(tmp, ftmp2);
974 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
975 felem_reduce(ftmp, tmp);
978 * the formulae are incorrect if the points are equal so we check for
979 * this and do doubling if this happens
981 x_equal = felem_is_zero(ftmp);
982 y_equal = felem_is_zero(ftmp3);
983 z1_is_zero = felem_is_zero(z1);
984 z2_is_zero = felem_is_zero(z2);
985 /* In affine coordinates, (X_1, Y_1) == (X_2, Y_2) */
986 if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) {
987 point_double(x3, y3, z3, x1, y1, z1);
993 felem_mul(tmp, z1, z2);
994 felem_reduce(ftmp5, tmp);
996 /* special case z2 = 0 is handled later */
997 felem_assign(ftmp5, z1);
1000 /* z_out = (z1^2*x2 - z2^2*x1)*(z1*z2) */
1001 felem_mul(tmp, ftmp, ftmp5);
1002 felem_reduce(z_out, tmp);
1004 /* ftmp = (z1^2*x2 - z2^2*x1)^2 */
1005 felem_assign(ftmp5, ftmp);
1006 felem_square(tmp, ftmp);
1007 felem_reduce(ftmp, tmp);
1009 /* ftmp5 = (z1^2*x2 - z2^2*x1)^3 */
1010 felem_mul(tmp, ftmp, ftmp5);
1011 felem_reduce(ftmp5, tmp);
1013 /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
1014 felem_mul(tmp, ftmp2, ftmp);
1015 felem_reduce(ftmp2, tmp);
1017 /* tmp = z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */
1018 felem_mul(tmp, ftmp4, ftmp5);
1019 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
1021 /* tmp2 = (z1^3*y2 - z2^3*y1)^2 */
1022 felem_square(tmp2, ftmp3);
1023 /* tmp2[i] < 4 * 2^57 * 2^57 < 2^116 */
1025 /* tmp2 = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 */
1026 felem_diff_128_64(tmp2, ftmp5);
1027 /* tmp2[i] < 2^116 + 2^64 + 8 < 2^117 */
1029 /* ftmp5 = 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
1030 felem_assign(ftmp5, ftmp2);
1031 felem_scalar(ftmp5, 2);
1032 /* ftmp5[i] < 2 * 2^57 = 2^58 */
1035 * x_out = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 -
1036 * 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2
1038 felem_diff_128_64(tmp2, ftmp5);
1039 /* tmp2[i] < 2^117 + 2^64 + 8 < 2^118 */
1040 felem_reduce(x_out, tmp2);
1042 /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out */
1043 felem_diff(ftmp2, x_out);
1044 /* ftmp2[i] < 2^57 + 2^58 + 2 < 2^59 */
1047 * tmp2 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out)
1049 felem_mul(tmp2, ftmp3, ftmp2);
1050 /* tmp2[i] < 4 * 2^57 * 2^59 = 2^118 */
1053 * y_out = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) -
1054 * z2^3*y1*(z1^2*x2 - z2^2*x1)^3
1056 widefelem_diff(tmp2, tmp);
1057 /* tmp2[i] < 2^118 + 2^120 < 2^121 */
1058 felem_reduce(y_out, tmp2);
1061 * the result (x_out, y_out, z_out) is incorrect if one of the inputs is
1062 * the point at infinity, so we need to check for this separately
1066 * if point 1 is at infinity, copy point 2 to output, and vice versa
1068 copy_conditional(x_out, x2, z1_is_zero);
1069 copy_conditional(x_out, x1, z2_is_zero);
1070 copy_conditional(y_out, y2, z1_is_zero);
1071 copy_conditional(y_out, y1, z2_is_zero);
1072 copy_conditional(z_out, z2, z1_is_zero);
1073 copy_conditional(z_out, z1, z2_is_zero);
1074 felem_assign(x3, x_out);
1075 felem_assign(y3, y_out);
1076 felem_assign(z3, z_out);
1080 * select_point selects the |idx|th point from a precomputation table and
1082 * The pre_comp array argument should be size of |size| argument
1084 static void select_point(const u64 idx, unsigned int size,
1085 const felem pre_comp[][3], felem out[3])
1088 limb *outlimbs = &out[0][0];
1090 memset(out, 0, sizeof(*out) * 3);
1091 for (i = 0; i < size; i++) {
1092 const limb *inlimbs = &pre_comp[i][0][0];
1099 for (j = 0; j < 4 * 3; j++)
1100 outlimbs[j] |= inlimbs[j] & mask;
1104 /* get_bit returns the |i|th bit in |in| */
1105 static char get_bit(const felem_bytearray in, unsigned i)
1109 return (in[i >> 3] >> (i & 7)) & 1;
1113 * Interleaved point multiplication using precomputed point multiples: The
1114 * small point multiples 0*P, 1*P, ..., 16*P are in pre_comp[], the scalars
1115 * in scalars[]. If g_scalar is non-NULL, we also add this multiple of the
1116 * generator, using certain (large) precomputed multiples in g_pre_comp.
1117 * Output point (X, Y, Z) is stored in x_out, y_out, z_out
1119 static void batch_mul(felem x_out, felem y_out, felem z_out,
1120 const felem_bytearray scalars[],
1121 const unsigned num_points, const u8 *g_scalar,
1122 const int mixed, const felem pre_comp[][17][3],
1123 const felem g_pre_comp[2][16][3])
1127 unsigned gen_mul = (g_scalar != NULL);
1128 felem nq[3], tmp[4];
1132 /* set nq to the point at infinity */
1133 memset(nq, 0, sizeof(nq));
1136 * Loop over all scalars msb-to-lsb, interleaving additions of multiples
1137 * of the generator (two in each of the last 28 rounds) and additions of
1138 * other points multiples (every 5th round).
1140 skip = 1; /* save two point operations in the first
1142 for (i = (num_points ? 220 : 27); i >= 0; --i) {
1145 point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
1147 /* add multiples of the generator */
1148 if (gen_mul && (i <= 27)) {
1149 /* first, look 28 bits upwards */
1150 bits = get_bit(g_scalar, i + 196) << 3;
1151 bits |= get_bit(g_scalar, i + 140) << 2;
1152 bits |= get_bit(g_scalar, i + 84) << 1;
1153 bits |= get_bit(g_scalar, i + 28);
1154 /* select the point to add, in constant time */
1155 select_point(bits, 16, g_pre_comp[1], tmp);
1158 /* value 1 below is argument for "mixed" */
1159 point_add(nq[0], nq[1], nq[2],
1160 nq[0], nq[1], nq[2], 1, tmp[0], tmp[1], tmp[2]);
1162 memcpy(nq, tmp, 3 * sizeof(felem));
1166 /* second, look at the current position */
1167 bits = get_bit(g_scalar, i + 168) << 3;
1168 bits |= get_bit(g_scalar, i + 112) << 2;
1169 bits |= get_bit(g_scalar, i + 56) << 1;
1170 bits |= get_bit(g_scalar, i);
1171 /* select the point to add, in constant time */
1172 select_point(bits, 16, g_pre_comp[0], tmp);
1173 point_add(nq[0], nq[1], nq[2],
1174 nq[0], nq[1], nq[2],
1175 1 /* mixed */ , tmp[0], tmp[1], tmp[2]);
1178 /* do other additions every 5 doublings */
1179 if (num_points && (i % 5 == 0)) {
1180 /* loop over all scalars */
1181 for (num = 0; num < num_points; ++num) {
1182 bits = get_bit(scalars[num], i + 4) << 5;
1183 bits |= get_bit(scalars[num], i + 3) << 4;
1184 bits |= get_bit(scalars[num], i + 2) << 3;
1185 bits |= get_bit(scalars[num], i + 1) << 2;
1186 bits |= get_bit(scalars[num], i) << 1;
1187 bits |= get_bit(scalars[num], i - 1);
1188 ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits);
1190 /* select the point to add or subtract */
1191 select_point(digit, 17, pre_comp[num], tmp);
1192 felem_neg(tmp[3], tmp[1]); /* (X, -Y, Z) is the negative
1194 copy_conditional(tmp[1], tmp[3], sign);
1197 point_add(nq[0], nq[1], nq[2],
1198 nq[0], nq[1], nq[2],
1199 mixed, tmp[0], tmp[1], tmp[2]);
1201 memcpy(nq, tmp, 3 * sizeof(felem));
1207 felem_assign(x_out, nq[0]);
1208 felem_assign(y_out, nq[1]);
1209 felem_assign(z_out, nq[2]);
1212 /******************************************************************************/
1214 * FUNCTIONS TO MANAGE PRECOMPUTATION
1217 static NISTP224_PRE_COMP *nistp224_pre_comp_new(void)
1219 NISTP224_PRE_COMP *ret = OPENSSL_zalloc(sizeof(*ret));
1222 ECerr(EC_F_NISTP224_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1226 ret->references = 1;
1228 ret->lock = CRYPTO_THREAD_lock_new();
1229 if (ret->lock == NULL) {
1230 ECerr(EC_F_NISTP224_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1237 NISTP224_PRE_COMP *EC_nistp224_pre_comp_dup(NISTP224_PRE_COMP *p)
1241 CRYPTO_UP_REF(&p->references, &i, p->lock);
1245 void EC_nistp224_pre_comp_free(NISTP224_PRE_COMP *p)
1252 CRYPTO_DOWN_REF(&p->references, &i, p->lock);
1253 REF_PRINT_COUNT("EC_nistp224", x);
1256 REF_ASSERT_ISNT(i < 0);
1258 CRYPTO_THREAD_lock_free(p->lock);
1262 /******************************************************************************/
1264 * OPENSSL EC_METHOD FUNCTIONS
1267 int ec_GFp_nistp224_group_init(EC_GROUP *group)
1270 ret = ec_GFp_simple_group_init(group);
1271 group->a_is_minus3 = 1;
1275 int ec_GFp_nistp224_group_set_curve(EC_GROUP *group, const BIGNUM *p,
1276 const BIGNUM *a, const BIGNUM *b,
1280 BN_CTX *new_ctx = NULL;
1281 BIGNUM *curve_p, *curve_a, *curve_b;
1284 if ((ctx = new_ctx = BN_CTX_new()) == NULL)
1287 curve_p = BN_CTX_get(ctx);
1288 curve_a = BN_CTX_get(ctx);
1289 curve_b = BN_CTX_get(ctx);
1290 if (curve_b == NULL)
1292 BN_bin2bn(nistp224_curve_params[0], sizeof(felem_bytearray), curve_p);
1293 BN_bin2bn(nistp224_curve_params[1], sizeof(felem_bytearray), curve_a);
1294 BN_bin2bn(nistp224_curve_params[2], sizeof(felem_bytearray), curve_b);
1295 if ((BN_cmp(curve_p, p)) || (BN_cmp(curve_a, a)) || (BN_cmp(curve_b, b))) {
1296 ECerr(EC_F_EC_GFP_NISTP224_GROUP_SET_CURVE,
1297 EC_R_WRONG_CURVE_PARAMETERS);
1300 group->field_mod_func = BN_nist_mod_224;
1301 ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx);
1304 BN_CTX_free(new_ctx);
1309 * Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') =
1312 int ec_GFp_nistp224_point_get_affine_coordinates(const EC_GROUP *group,
1313 const EC_POINT *point,
1314 BIGNUM *x, BIGNUM *y,
1317 felem z1, z2, x_in, y_in, x_out, y_out;
1320 if (EC_POINT_is_at_infinity(group, point)) {
1321 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1322 EC_R_POINT_AT_INFINITY);
1325 if ((!BN_to_felem(x_in, point->X)) || (!BN_to_felem(y_in, point->Y)) ||
1326 (!BN_to_felem(z1, point->Z)))
1329 felem_square(tmp, z2);
1330 felem_reduce(z1, tmp);
1331 felem_mul(tmp, x_in, z1);
1332 felem_reduce(x_in, tmp);
1333 felem_contract(x_out, x_in);
1335 if (!felem_to_BN(x, x_out)) {
1336 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1341 felem_mul(tmp, z1, z2);
1342 felem_reduce(z1, tmp);
1343 felem_mul(tmp, y_in, z1);
1344 felem_reduce(y_in, tmp);
1345 felem_contract(y_out, y_in);
1347 if (!felem_to_BN(y, y_out)) {
1348 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1356 static void make_points_affine(size_t num, felem points[ /* num */ ][3],
1357 felem tmp_felems[ /* num+1 */ ])
1360 * Runs in constant time, unless an input is the point at infinity (which
1361 * normally shouldn't happen).
1363 ec_GFp_nistp_points_make_affine_internal(num,
1367 (void (*)(void *))felem_one,
1369 (void (*)(void *, const void *))
1371 (void (*)(void *, const void *))
1372 felem_square_reduce, (void (*)
1379 (void (*)(void *, const void *))
1381 (void (*)(void *, const void *))
1386 * Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL
1387 * values Result is stored in r (r can equal one of the inputs).
1389 int ec_GFp_nistp224_points_mul(const EC_GROUP *group, EC_POINT *r,
1390 const BIGNUM *scalar, size_t num,
1391 const EC_POINT *points[],
1392 const BIGNUM *scalars[], BN_CTX *ctx)
1398 BIGNUM *x, *y, *z, *tmp_scalar;
1399 felem_bytearray g_secret;
1400 felem_bytearray *secrets = NULL;
1401 felem (*pre_comp)[17][3] = NULL;
1402 felem *tmp_felems = NULL;
1403 felem_bytearray tmp;
1405 int have_pre_comp = 0;
1406 size_t num_points = num;
1407 felem x_in, y_in, z_in, x_out, y_out, z_out;
1408 NISTP224_PRE_COMP *pre = NULL;
1409 const felem(*g_pre_comp)[16][3] = NULL;
1410 EC_POINT *generator = NULL;
1411 const EC_POINT *p = NULL;
1412 const BIGNUM *p_scalar = NULL;
1415 x = BN_CTX_get(ctx);
1416 y = BN_CTX_get(ctx);
1417 z = BN_CTX_get(ctx);
1418 tmp_scalar = BN_CTX_get(ctx);
1419 if (tmp_scalar == NULL)
1422 if (scalar != NULL) {
1423 pre = group->pre_comp.nistp224;
1425 /* we have precomputation, try to use it */
1426 g_pre_comp = (const felem(*)[16][3])pre->g_pre_comp;
1428 /* try to use the standard precomputation */
1429 g_pre_comp = &gmul[0];
1430 generator = EC_POINT_new(group);
1431 if (generator == NULL)
1433 /* get the generator from precomputation */
1434 if (!felem_to_BN(x, g_pre_comp[0][1][0]) ||
1435 !felem_to_BN(y, g_pre_comp[0][1][1]) ||
1436 !felem_to_BN(z, g_pre_comp[0][1][2])) {
1437 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1440 if (!EC_POINT_set_Jprojective_coordinates_GFp(group,
1444 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
1445 /* precomputation matches generator */
1449 * we don't have valid precomputation: treat the generator as a
1452 num_points = num_points + 1;
1455 if (num_points > 0) {
1456 if (num_points >= 3) {
1458 * unless we precompute multiples for just one or two points,
1459 * converting those into affine form is time well spent
1463 secrets = OPENSSL_zalloc(sizeof(*secrets) * num_points);
1464 pre_comp = OPENSSL_zalloc(sizeof(*pre_comp) * num_points);
1467 OPENSSL_malloc(sizeof(felem) * (num_points * 17 + 1));
1468 if ((secrets == NULL) || (pre_comp == NULL)
1469 || (mixed && (tmp_felems == NULL))) {
1470 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_MALLOC_FAILURE);
1475 * we treat NULL scalars as 0, and NULL points as points at infinity,
1476 * i.e., they contribute nothing to the linear combination
1478 for (i = 0; i < num_points; ++i) {
1482 p = EC_GROUP_get0_generator(group);
1485 /* the i^th point */
1488 p_scalar = scalars[i];
1490 if ((p_scalar != NULL) && (p != NULL)) {
1491 /* reduce scalar to 0 <= scalar < 2^224 */
1492 if ((BN_num_bits(p_scalar) > 224)
1493 || (BN_is_negative(p_scalar))) {
1495 * this is an unusual input, and we don't guarantee
1498 if (!BN_nnmod(tmp_scalar, p_scalar, group->order, ctx)) {
1499 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1502 num_bytes = BN_bn2binpad(tmp_scalar, tmp, sizeof(tmp));
1504 num_bytes = BN_bn2binpad(p_scalar, tmp, sizeof(tmp));
1505 flip_endian(secrets[i], tmp, num_bytes);
1506 /* precompute multiples */
1507 if ((!BN_to_felem(x_out, p->X)) ||
1508 (!BN_to_felem(y_out, p->Y)) ||
1509 (!BN_to_felem(z_out, p->Z)))
1511 felem_assign(pre_comp[i][1][0], x_out);
1512 felem_assign(pre_comp[i][1][1], y_out);
1513 felem_assign(pre_comp[i][1][2], z_out);
1514 for (j = 2; j <= 16; ++j) {
1516 point_add(pre_comp[i][j][0], pre_comp[i][j][1],
1517 pre_comp[i][j][2], pre_comp[i][1][0],
1518 pre_comp[i][1][1], pre_comp[i][1][2], 0,
1519 pre_comp[i][j - 1][0],
1520 pre_comp[i][j - 1][1],
1521 pre_comp[i][j - 1][2]);
1523 point_double(pre_comp[i][j][0], pre_comp[i][j][1],
1524 pre_comp[i][j][2], pre_comp[i][j / 2][0],
1525 pre_comp[i][j / 2][1],
1526 pre_comp[i][j / 2][2]);
1532 make_points_affine(num_points * 17, pre_comp[0], tmp_felems);
1535 /* the scalar for the generator */
1536 if ((scalar != NULL) && (have_pre_comp)) {
1537 memset(g_secret, 0, sizeof(g_secret));
1538 /* reduce scalar to 0 <= scalar < 2^224 */
1539 if ((BN_num_bits(scalar) > 224) || (BN_is_negative(scalar))) {
1541 * this is an unusual input, and we don't guarantee
1544 if (!BN_nnmod(tmp_scalar, scalar, group->order, ctx)) {
1545 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1548 num_bytes = BN_bn2binpad(tmp_scalar, tmp, sizeof(tmp));
1550 num_bytes = BN_bn2binpad(scalar, tmp, sizeof(tmp));
1551 flip_endian(g_secret, tmp, num_bytes);
1552 /* do the multiplication with generator precomputation */
1553 batch_mul(x_out, y_out, z_out,
1554 (const felem_bytearray(*))secrets, num_points,
1556 mixed, (const felem(*)[17][3])pre_comp, g_pre_comp);
1558 /* do the multiplication without generator precomputation */
1559 batch_mul(x_out, y_out, z_out,
1560 (const felem_bytearray(*))secrets, num_points,
1561 NULL, mixed, (const felem(*)[17][3])pre_comp, NULL);
1562 /* reduce the output to its unique minimal representation */
1563 felem_contract(x_in, x_out);
1564 felem_contract(y_in, y_out);
1565 felem_contract(z_in, z_out);
1566 if ((!felem_to_BN(x, x_in)) || (!felem_to_BN(y, y_in)) ||
1567 (!felem_to_BN(z, z_in))) {
1568 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1571 ret = EC_POINT_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx);
1575 EC_POINT_free(generator);
1576 OPENSSL_free(secrets);
1577 OPENSSL_free(pre_comp);
1578 OPENSSL_free(tmp_felems);
1582 int ec_GFp_nistp224_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
1585 NISTP224_PRE_COMP *pre = NULL;
1587 BN_CTX *new_ctx = NULL;
1589 EC_POINT *generator = NULL;
1590 felem tmp_felems[32];
1592 /* throw away old precomputation */
1593 EC_pre_comp_free(group);
1595 if ((ctx = new_ctx = BN_CTX_new()) == NULL)
1598 x = BN_CTX_get(ctx);
1599 y = BN_CTX_get(ctx);
1602 /* get the generator */
1603 if (group->generator == NULL)
1605 generator = EC_POINT_new(group);
1606 if (generator == NULL)
1608 BN_bin2bn(nistp224_curve_params[3], sizeof(felem_bytearray), x);
1609 BN_bin2bn(nistp224_curve_params[4], sizeof(felem_bytearray), y);
1610 if (!EC_POINT_set_affine_coordinates(group, generator, x, y, ctx))
1612 if ((pre = nistp224_pre_comp_new()) == NULL)
1615 * if the generator is the standard one, use built-in precomputation
1617 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) {
1618 memcpy(pre->g_pre_comp, gmul, sizeof(pre->g_pre_comp));
1621 if ((!BN_to_felem(pre->g_pre_comp[0][1][0], group->generator->X)) ||
1622 (!BN_to_felem(pre->g_pre_comp[0][1][1], group->generator->Y)) ||
1623 (!BN_to_felem(pre->g_pre_comp[0][1][2], group->generator->Z)))
1626 * compute 2^56*G, 2^112*G, 2^168*G for the first table, 2^28*G, 2^84*G,
1627 * 2^140*G, 2^196*G for the second one
1629 for (i = 1; i <= 8; i <<= 1) {
1630 point_double(pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1],
1631 pre->g_pre_comp[1][i][2], pre->g_pre_comp[0][i][0],
1632 pre->g_pre_comp[0][i][1], pre->g_pre_comp[0][i][2]);
1633 for (j = 0; j < 27; ++j) {
1634 point_double(pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1],
1635 pre->g_pre_comp[1][i][2], pre->g_pre_comp[1][i][0],
1636 pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2]);
1640 point_double(pre->g_pre_comp[0][2 * i][0],
1641 pre->g_pre_comp[0][2 * i][1],
1642 pre->g_pre_comp[0][2 * i][2], pre->g_pre_comp[1][i][0],
1643 pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2]);
1644 for (j = 0; j < 27; ++j) {
1645 point_double(pre->g_pre_comp[0][2 * i][0],
1646 pre->g_pre_comp[0][2 * i][1],
1647 pre->g_pre_comp[0][2 * i][2],
1648 pre->g_pre_comp[0][2 * i][0],
1649 pre->g_pre_comp[0][2 * i][1],
1650 pre->g_pre_comp[0][2 * i][2]);
1653 for (i = 0; i < 2; i++) {
1654 /* g_pre_comp[i][0] is the point at infinity */
1655 memset(pre->g_pre_comp[i][0], 0, sizeof(pre->g_pre_comp[i][0]));
1656 /* the remaining multiples */
1657 /* 2^56*G + 2^112*G resp. 2^84*G + 2^140*G */
1658 point_add(pre->g_pre_comp[i][6][0], pre->g_pre_comp[i][6][1],
1659 pre->g_pre_comp[i][6][2], pre->g_pre_comp[i][4][0],
1660 pre->g_pre_comp[i][4][1], pre->g_pre_comp[i][4][2],
1661 0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
1662 pre->g_pre_comp[i][2][2]);
1663 /* 2^56*G + 2^168*G resp. 2^84*G + 2^196*G */
1664 point_add(pre->g_pre_comp[i][10][0], pre->g_pre_comp[i][10][1],
1665 pre->g_pre_comp[i][10][2], pre->g_pre_comp[i][8][0],
1666 pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2],
1667 0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
1668 pre->g_pre_comp[i][2][2]);
1669 /* 2^112*G + 2^168*G resp. 2^140*G + 2^196*G */
1670 point_add(pre->g_pre_comp[i][12][0], pre->g_pre_comp[i][12][1],
1671 pre->g_pre_comp[i][12][2], pre->g_pre_comp[i][8][0],
1672 pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2],
1673 0, pre->g_pre_comp[i][4][0], pre->g_pre_comp[i][4][1],
1674 pre->g_pre_comp[i][4][2]);
1676 * 2^56*G + 2^112*G + 2^168*G resp. 2^84*G + 2^140*G + 2^196*G
1678 point_add(pre->g_pre_comp[i][14][0], pre->g_pre_comp[i][14][1],
1679 pre->g_pre_comp[i][14][2], pre->g_pre_comp[i][12][0],
1680 pre->g_pre_comp[i][12][1], pre->g_pre_comp[i][12][2],
1681 0, pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1],
1682 pre->g_pre_comp[i][2][2]);
1683 for (j = 1; j < 8; ++j) {
1684 /* odd multiples: add G resp. 2^28*G */
1685 point_add(pre->g_pre_comp[i][2 * j + 1][0],
1686 pre->g_pre_comp[i][2 * j + 1][1],
1687 pre->g_pre_comp[i][2 * j + 1][2],
1688 pre->g_pre_comp[i][2 * j][0],
1689 pre->g_pre_comp[i][2 * j][1],
1690 pre->g_pre_comp[i][2 * j][2], 0,
1691 pre->g_pre_comp[i][1][0], pre->g_pre_comp[i][1][1],
1692 pre->g_pre_comp[i][1][2]);
1695 make_points_affine(31, &(pre->g_pre_comp[0][1]), tmp_felems);
1698 SETPRECOMP(group, nistp224, pre);
1703 EC_POINT_free(generator);
1704 BN_CTX_free(new_ctx);
1705 EC_nistp224_pre_comp_free(pre);
1709 int ec_GFp_nistp224_have_precompute_mult(const EC_GROUP *group)
1711 return HAVEPRECOMP(group, nistp224);