1 | /* |
2 | * Copyright 2011-2018 The OpenSSL Project Authors. All Rights Reserved. |
3 | * |
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 |
8 | */ |
9 | |
10 | /* Copyright 2011 Google Inc. |
11 | * |
12 | * Licensed under the Apache License, Version 2.0 (the "License"); |
13 | * |
14 | * you may not use this file except in compliance with the License. |
15 | * You may obtain a copy of the License at |
16 | * |
17 | * http://www.apache.org/licenses/LICENSE-2.0 |
18 | * |
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. |
24 | */ |
25 | |
26 | /* |
27 | * A 64-bit implementation of the NIST P-521 elliptic curve point multiplication |
28 | * |
29 | * OpenSSL integration was taken from Emilia Kasper's work in ecp_nistp224.c. |
30 | * Otherwise based on Emilia's P224 work, which was inspired by my curve25519 |
31 | * work which got its smarts from Daniel J. Bernstein's work on the same. |
32 | */ |
33 | |
34 | #include <openssl/e_os2.h> |
35 | #ifdef OPENSSL_NO_EC_NISTP_64_GCC_128 |
36 | NON_EMPTY_TRANSLATION_UNIT |
37 | #else |
38 | |
39 | # include <string.h> |
40 | # include <openssl/err.h> |
41 | # include "ec_local.h" |
42 | |
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 |
46 | * platforms */ |
47 | # else |
48 | # error "Your compiler doesn't appear to support 128-bit integer types" |
49 | # endif |
50 | |
51 | typedef uint8_t u8; |
52 | typedef uint64_t u64; |
53 | |
54 | /* |
55 | * The underlying field. P521 operates over GF(2^521-1). We can serialise an |
56 | * element of this field into 66 bytes where the most significant byte |
57 | * contains only a single bit. We call this an felem_bytearray. |
58 | */ |
59 | |
60 | typedef u8 felem_bytearray[66]; |
61 | |
62 | /* |
63 | * These are the parameters of P521, taken from FIPS 186-3, section D.1.2.5. |
64 | * These values are big-endian. |
65 | */ |
66 | static const felem_bytearray nistp521_curve_params[5] = { |
67 | {0x01, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, /* p */ |
68 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, |
69 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, |
70 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, |
71 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, |
72 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, |
73 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, |
74 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, |
75 | 0xff, 0xff}, |
76 | {0x01, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, /* a = -3 */ |
77 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, |
78 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, |
79 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, |
80 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, |
81 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, |
82 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, |
83 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, |
84 | 0xff, 0xfc}, |
85 | {0x00, 0x51, 0x95, 0x3e, 0xb9, 0x61, 0x8e, 0x1c, /* b */ |
86 | 0x9a, 0x1f, 0x92, 0x9a, 0x21, 0xa0, 0xb6, 0x85, |
87 | 0x40, 0xee, 0xa2, 0xda, 0x72, 0x5b, 0x99, 0xb3, |
88 | 0x15, 0xf3, 0xb8, 0xb4, 0x89, 0x91, 0x8e, 0xf1, |
89 | 0x09, 0xe1, 0x56, 0x19, 0x39, 0x51, 0xec, 0x7e, |
90 | 0x93, 0x7b, 0x16, 0x52, 0xc0, 0xbd, 0x3b, 0xb1, |
91 | 0xbf, 0x07, 0x35, 0x73, 0xdf, 0x88, 0x3d, 0x2c, |
92 | 0x34, 0xf1, 0xef, 0x45, 0x1f, 0xd4, 0x6b, 0x50, |
93 | 0x3f, 0x00}, |
94 | {0x00, 0xc6, 0x85, 0x8e, 0x06, 0xb7, 0x04, 0x04, /* x */ |
95 | 0xe9, 0xcd, 0x9e, 0x3e, 0xcb, 0x66, 0x23, 0x95, |
96 | 0xb4, 0x42, 0x9c, 0x64, 0x81, 0x39, 0x05, 0x3f, |
97 | 0xb5, 0x21, 0xf8, 0x28, 0xaf, 0x60, 0x6b, 0x4d, |
98 | 0x3d, 0xba, 0xa1, 0x4b, 0x5e, 0x77, 0xef, 0xe7, |
99 | 0x59, 0x28, 0xfe, 0x1d, 0xc1, 0x27, 0xa2, 0xff, |
100 | 0xa8, 0xde, 0x33, 0x48, 0xb3, 0xc1, 0x85, 0x6a, |
101 | 0x42, 0x9b, 0xf9, 0x7e, 0x7e, 0x31, 0xc2, 0xe5, |
102 | 0xbd, 0x66}, |
103 | {0x01, 0x18, 0x39, 0x29, 0x6a, 0x78, 0x9a, 0x3b, /* y */ |
104 | 0xc0, 0x04, 0x5c, 0x8a, 0x5f, 0xb4, 0x2c, 0x7d, |
105 | 0x1b, 0xd9, 0x98, 0xf5, 0x44, 0x49, 0x57, 0x9b, |
106 | 0x44, 0x68, 0x17, 0xaf, 0xbd, 0x17, 0x27, 0x3e, |
107 | 0x66, 0x2c, 0x97, 0xee, 0x72, 0x99, 0x5e, 0xf4, |
108 | 0x26, 0x40, 0xc5, 0x50, 0xb9, 0x01, 0x3f, 0xad, |
109 | 0x07, 0x61, 0x35, 0x3c, 0x70, 0x86, 0xa2, 0x72, |
110 | 0xc2, 0x40, 0x88, 0xbe, 0x94, 0x76, 0x9f, 0xd1, |
111 | 0x66, 0x50} |
112 | }; |
113 | |
114 | /*- |
115 | * The representation of field elements. |
116 | * ------------------------------------ |
117 | * |
118 | * We represent field elements with nine values. These values are either 64 or |
119 | * 128 bits and the field element represented is: |
120 | * v[0]*2^0 + v[1]*2^58 + v[2]*2^116 + ... + v[8]*2^464 (mod p) |
121 | * Each of the nine values is called a 'limb'. Since the limbs are spaced only |
122 | * 58 bits apart, but are greater than 58 bits in length, the most significant |
123 | * bits of each limb overlap with the least significant bits of the next. |
124 | * |
125 | * A field element with 64-bit limbs is an 'felem'. One with 128-bit limbs is a |
126 | * 'largefelem' */ |
127 | |
128 | # define NLIMBS 9 |
129 | |
130 | typedef uint64_t limb; |
131 | typedef limb felem[NLIMBS]; |
132 | typedef uint128_t largefelem[NLIMBS]; |
133 | |
134 | static const limb bottom57bits = 0x1ffffffffffffff; |
135 | static const limb bottom58bits = 0x3ffffffffffffff; |
136 | |
137 | /* |
138 | * bin66_to_felem takes a little-endian byte array and converts it into felem |
139 | * form. This assumes that the CPU is little-endian. |
140 | */ |
141 | static void bin66_to_felem(felem out, const u8 in[66]) |
142 | { |
143 | out[0] = (*((limb *) & in[0])) & bottom58bits; |
144 | out[1] = (*((limb *) & in[7]) >> 2) & bottom58bits; |
145 | out[2] = (*((limb *) & in[14]) >> 4) & bottom58bits; |
146 | out[3] = (*((limb *) & in[21]) >> 6) & bottom58bits; |
147 | out[4] = (*((limb *) & in[29])) & bottom58bits; |
148 | out[5] = (*((limb *) & in[36]) >> 2) & bottom58bits; |
149 | out[6] = (*((limb *) & in[43]) >> 4) & bottom58bits; |
150 | out[7] = (*((limb *) & in[50]) >> 6) & bottom58bits; |
151 | out[8] = (*((limb *) & in[58])) & bottom57bits; |
152 | } |
153 | |
154 | /* |
155 | * felem_to_bin66 takes an felem and serialises into a little endian, 66 byte |
156 | * array. This assumes that the CPU is little-endian. |
157 | */ |
158 | static void felem_to_bin66(u8 out[66], const felem in) |
159 | { |
160 | memset(out, 0, 66); |
161 | (*((limb *) & out[0])) = in[0]; |
162 | (*((limb *) & out[7])) |= in[1] << 2; |
163 | (*((limb *) & out[14])) |= in[2] << 4; |
164 | (*((limb *) & out[21])) |= in[3] << 6; |
165 | (*((limb *) & out[29])) = in[4]; |
166 | (*((limb *) & out[36])) |= in[5] << 2; |
167 | (*((limb *) & out[43])) |= in[6] << 4; |
168 | (*((limb *) & out[50])) |= in[7] << 6; |
169 | (*((limb *) & out[58])) = in[8]; |
170 | } |
171 | |
172 | /* BN_to_felem converts an OpenSSL BIGNUM into an felem */ |
173 | static int BN_to_felem(felem out, const BIGNUM *bn) |
174 | { |
175 | felem_bytearray b_out; |
176 | int num_bytes; |
177 | |
178 | if (BN_is_negative(bn)) { |
179 | ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE); |
180 | return 0; |
181 | } |
182 | num_bytes = BN_bn2lebinpad(bn, b_out, sizeof(b_out)); |
183 | if (num_bytes < 0) { |
184 | ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE); |
185 | return 0; |
186 | } |
187 | bin66_to_felem(out, b_out); |
188 | return 1; |
189 | } |
190 | |
191 | /* felem_to_BN converts an felem into an OpenSSL BIGNUM */ |
192 | static BIGNUM *felem_to_BN(BIGNUM *out, const felem in) |
193 | { |
194 | felem_bytearray b_out; |
195 | felem_to_bin66(b_out, in); |
196 | return BN_lebin2bn(b_out, sizeof(b_out), out); |
197 | } |
198 | |
199 | /*- |
200 | * Field operations |
201 | * ---------------- |
202 | */ |
203 | |
204 | static void felem_one(felem out) |
205 | { |
206 | out[0] = 1; |
207 | out[1] = 0; |
208 | out[2] = 0; |
209 | out[3] = 0; |
210 | out[4] = 0; |
211 | out[5] = 0; |
212 | out[6] = 0; |
213 | out[7] = 0; |
214 | out[8] = 0; |
215 | } |
216 | |
217 | static void felem_assign(felem out, const felem in) |
218 | { |
219 | out[0] = in[0]; |
220 | out[1] = in[1]; |
221 | out[2] = in[2]; |
222 | out[3] = in[3]; |
223 | out[4] = in[4]; |
224 | out[5] = in[5]; |
225 | out[6] = in[6]; |
226 | out[7] = in[7]; |
227 | out[8] = in[8]; |
228 | } |
229 | |
230 | /* felem_sum64 sets out = out + in. */ |
231 | static void felem_sum64(felem out, const felem in) |
232 | { |
233 | out[0] += in[0]; |
234 | out[1] += in[1]; |
235 | out[2] += in[2]; |
236 | out[3] += in[3]; |
237 | out[4] += in[4]; |
238 | out[5] += in[5]; |
239 | out[6] += in[6]; |
240 | out[7] += in[7]; |
241 | out[8] += in[8]; |
242 | } |
243 | |
244 | /* felem_scalar sets out = in * scalar */ |
245 | static void felem_scalar(felem out, const felem in, limb scalar) |
246 | { |
247 | out[0] = in[0] * scalar; |
248 | out[1] = in[1] * scalar; |
249 | out[2] = in[2] * scalar; |
250 | out[3] = in[3] * scalar; |
251 | out[4] = in[4] * scalar; |
252 | out[5] = in[5] * scalar; |
253 | out[6] = in[6] * scalar; |
254 | out[7] = in[7] * scalar; |
255 | out[8] = in[8] * scalar; |
256 | } |
257 | |
258 | /* felem_scalar64 sets out = out * scalar */ |
259 | static void felem_scalar64(felem out, limb scalar) |
260 | { |
261 | out[0] *= scalar; |
262 | out[1] *= scalar; |
263 | out[2] *= scalar; |
264 | out[3] *= scalar; |
265 | out[4] *= scalar; |
266 | out[5] *= scalar; |
267 | out[6] *= scalar; |
268 | out[7] *= scalar; |
269 | out[8] *= scalar; |
270 | } |
271 | |
272 | /* felem_scalar128 sets out = out * scalar */ |
273 | static void felem_scalar128(largefelem out, limb scalar) |
274 | { |
275 | out[0] *= scalar; |
276 | out[1] *= scalar; |
277 | out[2] *= scalar; |
278 | out[3] *= scalar; |
279 | out[4] *= scalar; |
280 | out[5] *= scalar; |
281 | out[6] *= scalar; |
282 | out[7] *= scalar; |
283 | out[8] *= scalar; |
284 | } |
285 | |
286 | /*- |
287 | * felem_neg sets |out| to |-in| |
288 | * On entry: |
289 | * in[i] < 2^59 + 2^14 |
290 | * On exit: |
291 | * out[i] < 2^62 |
292 | */ |
293 | static void felem_neg(felem out, const felem in) |
294 | { |
295 | /* In order to prevent underflow, we subtract from 0 mod p. */ |
296 | static const limb two62m3 = (((limb) 1) << 62) - (((limb) 1) << 5); |
297 | static const limb two62m2 = (((limb) 1) << 62) - (((limb) 1) << 4); |
298 | |
299 | out[0] = two62m3 - in[0]; |
300 | out[1] = two62m2 - in[1]; |
301 | out[2] = two62m2 - in[2]; |
302 | out[3] = two62m2 - in[3]; |
303 | out[4] = two62m2 - in[4]; |
304 | out[5] = two62m2 - in[5]; |
305 | out[6] = two62m2 - in[6]; |
306 | out[7] = two62m2 - in[7]; |
307 | out[8] = two62m2 - in[8]; |
308 | } |
309 | |
310 | /*- |
311 | * felem_diff64 subtracts |in| from |out| |
312 | * On entry: |
313 | * in[i] < 2^59 + 2^14 |
314 | * On exit: |
315 | * out[i] < out[i] + 2^62 |
316 | */ |
317 | static void felem_diff64(felem out, const felem in) |
318 | { |
319 | /* |
320 | * In order to prevent underflow, we add 0 mod p before subtracting. |
321 | */ |
322 | static const limb two62m3 = (((limb) 1) << 62) - (((limb) 1) << 5); |
323 | static const limb two62m2 = (((limb) 1) << 62) - (((limb) 1) << 4); |
324 | |
325 | out[0] += two62m3 - in[0]; |
326 | out[1] += two62m2 - in[1]; |
327 | out[2] += two62m2 - in[2]; |
328 | out[3] += two62m2 - in[3]; |
329 | out[4] += two62m2 - in[4]; |
330 | out[5] += two62m2 - in[5]; |
331 | out[6] += two62m2 - in[6]; |
332 | out[7] += two62m2 - in[7]; |
333 | out[8] += two62m2 - in[8]; |
334 | } |
335 | |
336 | /*- |
337 | * felem_diff_128_64 subtracts |in| from |out| |
338 | * On entry: |
339 | * in[i] < 2^62 + 2^17 |
340 | * On exit: |
341 | * out[i] < out[i] + 2^63 |
342 | */ |
343 | static void felem_diff_128_64(largefelem out, const felem in) |
344 | { |
345 | /* |
346 | * In order to prevent underflow, we add 64p mod p (which is equivalent |
347 | * to 0 mod p) before subtracting. p is 2^521 - 1, i.e. in binary a 521 |
348 | * digit number with all bits set to 1. See "The representation of field |
349 | * elements" comment above for a description of how limbs are used to |
350 | * represent a number. 64p is represented with 8 limbs containing a number |
351 | * with 58 bits set and one limb with a number with 57 bits set. |
352 | */ |
353 | static const limb two63m6 = (((limb) 1) << 63) - (((limb) 1) << 6); |
354 | static const limb two63m5 = (((limb) 1) << 63) - (((limb) 1) << 5); |
355 | |
356 | out[0] += two63m6 - in[0]; |
357 | out[1] += two63m5 - in[1]; |
358 | out[2] += two63m5 - in[2]; |
359 | out[3] += two63m5 - in[3]; |
360 | out[4] += two63m5 - in[4]; |
361 | out[5] += two63m5 - in[5]; |
362 | out[6] += two63m5 - in[6]; |
363 | out[7] += two63m5 - in[7]; |
364 | out[8] += two63m5 - in[8]; |
365 | } |
366 | |
367 | /*- |
368 | * felem_diff_128_64 subtracts |in| from |out| |
369 | * On entry: |
370 | * in[i] < 2^126 |
371 | * On exit: |
372 | * out[i] < out[i] + 2^127 - 2^69 |
373 | */ |
374 | static void felem_diff128(largefelem out, const largefelem in) |
375 | { |
376 | /* |
377 | * In order to prevent underflow, we add 0 mod p before subtracting. |
378 | */ |
379 | static const uint128_t two127m70 = |
380 | (((uint128_t) 1) << 127) - (((uint128_t) 1) << 70); |
381 | static const uint128_t two127m69 = |
382 | (((uint128_t) 1) << 127) - (((uint128_t) 1) << 69); |
383 | |
384 | out[0] += (two127m70 - in[0]); |
385 | out[1] += (two127m69 - in[1]); |
386 | out[2] += (two127m69 - in[2]); |
387 | out[3] += (two127m69 - in[3]); |
388 | out[4] += (two127m69 - in[4]); |
389 | out[5] += (two127m69 - in[5]); |
390 | out[6] += (two127m69 - in[6]); |
391 | out[7] += (two127m69 - in[7]); |
392 | out[8] += (two127m69 - in[8]); |
393 | } |
394 | |
395 | /*- |
396 | * felem_square sets |out| = |in|^2 |
397 | * On entry: |
398 | * in[i] < 2^62 |
399 | * On exit: |
400 | * out[i] < 17 * max(in[i]) * max(in[i]) |
401 | */ |
402 | static void felem_square(largefelem out, const felem in) |
403 | { |
404 | felem inx2, inx4; |
405 | felem_scalar(inx2, in, 2); |
406 | felem_scalar(inx4, in, 4); |
407 | |
408 | /*- |
409 | * We have many cases were we want to do |
410 | * in[x] * in[y] + |
411 | * in[y] * in[x] |
412 | * This is obviously just |
413 | * 2 * in[x] * in[y] |
414 | * However, rather than do the doubling on the 128 bit result, we |
415 | * double one of the inputs to the multiplication by reading from |
416 | * |inx2| |
417 | */ |
418 | |
419 | out[0] = ((uint128_t) in[0]) * in[0]; |
420 | out[1] = ((uint128_t) in[0]) * inx2[1]; |
421 | out[2] = ((uint128_t) in[0]) * inx2[2] + ((uint128_t) in[1]) * in[1]; |
422 | out[3] = ((uint128_t) in[0]) * inx2[3] + ((uint128_t) in[1]) * inx2[2]; |
423 | out[4] = ((uint128_t) in[0]) * inx2[4] + |
424 | ((uint128_t) in[1]) * inx2[3] + ((uint128_t) in[2]) * in[2]; |
425 | out[5] = ((uint128_t) in[0]) * inx2[5] + |
426 | ((uint128_t) in[1]) * inx2[4] + ((uint128_t) in[2]) * inx2[3]; |
427 | out[6] = ((uint128_t) in[0]) * inx2[6] + |
428 | ((uint128_t) in[1]) * inx2[5] + |
429 | ((uint128_t) in[2]) * inx2[4] + ((uint128_t) in[3]) * in[3]; |
430 | out[7] = ((uint128_t) in[0]) * inx2[7] + |
431 | ((uint128_t) in[1]) * inx2[6] + |
432 | ((uint128_t) in[2]) * inx2[5] + ((uint128_t) in[3]) * inx2[4]; |
433 | out[8] = ((uint128_t) in[0]) * inx2[8] + |
434 | ((uint128_t) in[1]) * inx2[7] + |
435 | ((uint128_t) in[2]) * inx2[6] + |
436 | ((uint128_t) in[3]) * inx2[5] + ((uint128_t) in[4]) * in[4]; |
437 | |
438 | /* |
439 | * The remaining limbs fall above 2^521, with the first falling at 2^522. |
440 | * They correspond to locations one bit up from the limbs produced above |
441 | * so we would have to multiply by two to align them. Again, rather than |
442 | * operate on the 128-bit result, we double one of the inputs to the |
443 | * multiplication. If we want to double for both this reason, and the |
444 | * reason above, then we end up multiplying by four. |
445 | */ |
446 | |
447 | /* 9 */ |
448 | out[0] += ((uint128_t) in[1]) * inx4[8] + |
449 | ((uint128_t) in[2]) * inx4[7] + |
450 | ((uint128_t) in[3]) * inx4[6] + ((uint128_t) in[4]) * inx4[5]; |
451 | |
452 | /* 10 */ |
453 | out[1] += ((uint128_t) in[2]) * inx4[8] + |
454 | ((uint128_t) in[3]) * inx4[7] + |
455 | ((uint128_t) in[4]) * inx4[6] + ((uint128_t) in[5]) * inx2[5]; |
456 | |
457 | /* 11 */ |
458 | out[2] += ((uint128_t) in[3]) * inx4[8] + |
459 | ((uint128_t) in[4]) * inx4[7] + ((uint128_t) in[5]) * inx4[6]; |
460 | |
461 | /* 12 */ |
462 | out[3] += ((uint128_t) in[4]) * inx4[8] + |
463 | ((uint128_t) in[5]) * inx4[7] + ((uint128_t) in[6]) * inx2[6]; |
464 | |
465 | /* 13 */ |
466 | out[4] += ((uint128_t) in[5]) * inx4[8] + ((uint128_t) in[6]) * inx4[7]; |
467 | |
468 | /* 14 */ |
469 | out[5] += ((uint128_t) in[6]) * inx4[8] + ((uint128_t) in[7]) * inx2[7]; |
470 | |
471 | /* 15 */ |
472 | out[6] += ((uint128_t) in[7]) * inx4[8]; |
473 | |
474 | /* 16 */ |
475 | out[7] += ((uint128_t) in[8]) * inx2[8]; |
476 | } |
477 | |
478 | /*- |
479 | * felem_mul sets |out| = |in1| * |in2| |
480 | * On entry: |
481 | * in1[i] < 2^64 |
482 | * in2[i] < 2^63 |
483 | * On exit: |
484 | * out[i] < 17 * max(in1[i]) * max(in2[i]) |
485 | */ |
486 | static void felem_mul(largefelem out, const felem in1, const felem in2) |
487 | { |
488 | felem in2x2; |
489 | felem_scalar(in2x2, in2, 2); |
490 | |
491 | out[0] = ((uint128_t) in1[0]) * in2[0]; |
492 | |
493 | out[1] = ((uint128_t) in1[0]) * in2[1] + |
494 | ((uint128_t) in1[1]) * in2[0]; |
495 | |
496 | out[2] = ((uint128_t) in1[0]) * in2[2] + |
497 | ((uint128_t) in1[1]) * in2[1] + |
498 | ((uint128_t) in1[2]) * in2[0]; |
499 | |
500 | out[3] = ((uint128_t) in1[0]) * in2[3] + |
501 | ((uint128_t) in1[1]) * in2[2] + |
502 | ((uint128_t) in1[2]) * in2[1] + |
503 | ((uint128_t) in1[3]) * in2[0]; |
504 | |
505 | out[4] = ((uint128_t) in1[0]) * in2[4] + |
506 | ((uint128_t) in1[1]) * in2[3] + |
507 | ((uint128_t) in1[2]) * in2[2] + |
508 | ((uint128_t) in1[3]) * in2[1] + |
509 | ((uint128_t) in1[4]) * in2[0]; |
510 | |
511 | out[5] = ((uint128_t) in1[0]) * in2[5] + |
512 | ((uint128_t) in1[1]) * in2[4] + |
513 | ((uint128_t) in1[2]) * in2[3] + |
514 | ((uint128_t) in1[3]) * in2[2] + |
515 | ((uint128_t) in1[4]) * in2[1] + |
516 | ((uint128_t) in1[5]) * in2[0]; |
517 | |
518 | out[6] = ((uint128_t) in1[0]) * in2[6] + |
519 | ((uint128_t) in1[1]) * in2[5] + |
520 | ((uint128_t) in1[2]) * in2[4] + |
521 | ((uint128_t) in1[3]) * in2[3] + |
522 | ((uint128_t) in1[4]) * in2[2] + |
523 | ((uint128_t) in1[5]) * in2[1] + |
524 | ((uint128_t) in1[6]) * in2[0]; |
525 | |
526 | out[7] = ((uint128_t) in1[0]) * in2[7] + |
527 | ((uint128_t) in1[1]) * in2[6] + |
528 | ((uint128_t) in1[2]) * in2[5] + |
529 | ((uint128_t) in1[3]) * in2[4] + |
530 | ((uint128_t) in1[4]) * in2[3] + |
531 | ((uint128_t) in1[5]) * in2[2] + |
532 | ((uint128_t) in1[6]) * in2[1] + |
533 | ((uint128_t) in1[7]) * in2[0]; |
534 | |
535 | out[8] = ((uint128_t) in1[0]) * in2[8] + |
536 | ((uint128_t) in1[1]) * in2[7] + |
537 | ((uint128_t) in1[2]) * in2[6] + |
538 | ((uint128_t) in1[3]) * in2[5] + |
539 | ((uint128_t) in1[4]) * in2[4] + |
540 | ((uint128_t) in1[5]) * in2[3] + |
541 | ((uint128_t) in1[6]) * in2[2] + |
542 | ((uint128_t) in1[7]) * in2[1] + |
543 | ((uint128_t) in1[8]) * in2[0]; |
544 | |
545 | /* See comment in felem_square about the use of in2x2 here */ |
546 | |
547 | out[0] += ((uint128_t) in1[1]) * in2x2[8] + |
548 | ((uint128_t) in1[2]) * in2x2[7] + |
549 | ((uint128_t) in1[3]) * in2x2[6] + |
550 | ((uint128_t) in1[4]) * in2x2[5] + |
551 | ((uint128_t) in1[5]) * in2x2[4] + |
552 | ((uint128_t) in1[6]) * in2x2[3] + |
553 | ((uint128_t) in1[7]) * in2x2[2] + |
554 | ((uint128_t) in1[8]) * in2x2[1]; |
555 | |
556 | out[1] += ((uint128_t) in1[2]) * in2x2[8] + |
557 | ((uint128_t) in1[3]) * in2x2[7] + |
558 | ((uint128_t) in1[4]) * in2x2[6] + |
559 | ((uint128_t) in1[5]) * in2x2[5] + |
560 | ((uint128_t) in1[6]) * in2x2[4] + |
561 | ((uint128_t) in1[7]) * in2x2[3] + |
562 | ((uint128_t) in1[8]) * in2x2[2]; |
563 | |
564 | out[2] += ((uint128_t) in1[3]) * in2x2[8] + |
565 | ((uint128_t) in1[4]) * in2x2[7] + |
566 | ((uint128_t) in1[5]) * in2x2[6] + |
567 | ((uint128_t) in1[6]) * in2x2[5] + |
568 | ((uint128_t) in1[7]) * in2x2[4] + |
569 | ((uint128_t) in1[8]) * in2x2[3]; |
570 | |
571 | out[3] += ((uint128_t) in1[4]) * in2x2[8] + |
572 | ((uint128_t) in1[5]) * in2x2[7] + |
573 | ((uint128_t) in1[6]) * in2x2[6] + |
574 | ((uint128_t) in1[7]) * in2x2[5] + |
575 | ((uint128_t) in1[8]) * in2x2[4]; |
576 | |
577 | out[4] += ((uint128_t) in1[5]) * in2x2[8] + |
578 | ((uint128_t) in1[6]) * in2x2[7] + |
579 | ((uint128_t) in1[7]) * in2x2[6] + |
580 | ((uint128_t) in1[8]) * in2x2[5]; |
581 | |
582 | out[5] += ((uint128_t) in1[6]) * in2x2[8] + |
583 | ((uint128_t) in1[7]) * in2x2[7] + |
584 | ((uint128_t) in1[8]) * in2x2[6]; |
585 | |
586 | out[6] += ((uint128_t) in1[7]) * in2x2[8] + |
587 | ((uint128_t) in1[8]) * in2x2[7]; |
588 | |
589 | out[7] += ((uint128_t) in1[8]) * in2x2[8]; |
590 | } |
591 | |
592 | static const limb bottom52bits = 0xfffffffffffff; |
593 | |
594 | /*- |
595 | * felem_reduce converts a largefelem to an felem. |
596 | * On entry: |
597 | * in[i] < 2^128 |
598 | * On exit: |
599 | * out[i] < 2^59 + 2^14 |
600 | */ |
601 | static void felem_reduce(felem out, const largefelem in) |
602 | { |
603 | u64 overflow1, overflow2; |
604 | |
605 | out[0] = ((limb) in[0]) & bottom58bits; |
606 | out[1] = ((limb) in[1]) & bottom58bits; |
607 | out[2] = ((limb) in[2]) & bottom58bits; |
608 | out[3] = ((limb) in[3]) & bottom58bits; |
609 | out[4] = ((limb) in[4]) & bottom58bits; |
610 | out[5] = ((limb) in[5]) & bottom58bits; |
611 | out[6] = ((limb) in[6]) & bottom58bits; |
612 | out[7] = ((limb) in[7]) & bottom58bits; |
613 | out[8] = ((limb) in[8]) & bottom58bits; |
614 | |
615 | /* out[i] < 2^58 */ |
616 | |
617 | out[1] += ((limb) in[0]) >> 58; |
618 | out[1] += (((limb) (in[0] >> 64)) & bottom52bits) << 6; |
619 | /*- |
620 | * out[1] < 2^58 + 2^6 + 2^58 |
621 | * = 2^59 + 2^6 |
622 | */ |
623 | out[2] += ((limb) (in[0] >> 64)) >> 52; |
624 | |
625 | out[2] += ((limb) in[1]) >> 58; |
626 | out[2] += (((limb) (in[1] >> 64)) & bottom52bits) << 6; |
627 | out[3] += ((limb) (in[1] >> 64)) >> 52; |
628 | |
629 | out[3] += ((limb) in[2]) >> 58; |
630 | out[3] += (((limb) (in[2] >> 64)) & bottom52bits) << 6; |
631 | out[4] += ((limb) (in[2] >> 64)) >> 52; |
632 | |
633 | out[4] += ((limb) in[3]) >> 58; |
634 | out[4] += (((limb) (in[3] >> 64)) & bottom52bits) << 6; |
635 | out[5] += ((limb) (in[3] >> 64)) >> 52; |
636 | |
637 | out[5] += ((limb) in[4]) >> 58; |
638 | out[5] += (((limb) (in[4] >> 64)) & bottom52bits) << 6; |
639 | out[6] += ((limb) (in[4] >> 64)) >> 52; |
640 | |
641 | out[6] += ((limb) in[5]) >> 58; |
642 | out[6] += (((limb) (in[5] >> 64)) & bottom52bits) << 6; |
643 | out[7] += ((limb) (in[5] >> 64)) >> 52; |
644 | |
645 | out[7] += ((limb) in[6]) >> 58; |
646 | out[7] += (((limb) (in[6] >> 64)) & bottom52bits) << 6; |
647 | out[8] += ((limb) (in[6] >> 64)) >> 52; |
648 | |
649 | out[8] += ((limb) in[7]) >> 58; |
650 | out[8] += (((limb) (in[7] >> 64)) & bottom52bits) << 6; |
651 | /*- |
652 | * out[x > 1] < 2^58 + 2^6 + 2^58 + 2^12 |
653 | * < 2^59 + 2^13 |
654 | */ |
655 | overflow1 = ((limb) (in[7] >> 64)) >> 52; |
656 | |
657 | overflow1 += ((limb) in[8]) >> 58; |
658 | overflow1 += (((limb) (in[8] >> 64)) & bottom52bits) << 6; |
659 | overflow2 = ((limb) (in[8] >> 64)) >> 52; |
660 | |
661 | overflow1 <<= 1; /* overflow1 < 2^13 + 2^7 + 2^59 */ |
662 | overflow2 <<= 1; /* overflow2 < 2^13 */ |
663 | |
664 | out[0] += overflow1; /* out[0] < 2^60 */ |
665 | out[1] += overflow2; /* out[1] < 2^59 + 2^6 + 2^13 */ |
666 | |
667 | out[1] += out[0] >> 58; |
668 | out[0] &= bottom58bits; |
669 | /*- |
670 | * out[0] < 2^58 |
671 | * out[1] < 2^59 + 2^6 + 2^13 + 2^2 |
672 | * < 2^59 + 2^14 |
673 | */ |
674 | } |
675 | |
676 | static void felem_square_reduce(felem out, const felem in) |
677 | { |
678 | largefelem tmp; |
679 | felem_square(tmp, in); |
680 | felem_reduce(out, tmp); |
681 | } |
682 | |
683 | static void felem_mul_reduce(felem out, const felem in1, const felem in2) |
684 | { |
685 | largefelem tmp; |
686 | felem_mul(tmp, in1, in2); |
687 | felem_reduce(out, tmp); |
688 | } |
689 | |
690 | /*- |
691 | * felem_inv calculates |out| = |in|^{-1} |
692 | * |
693 | * Based on Fermat's Little Theorem: |
694 | * a^p = a (mod p) |
695 | * a^{p-1} = 1 (mod p) |
696 | * a^{p-2} = a^{-1} (mod p) |
697 | */ |
698 | static void felem_inv(felem out, const felem in) |
699 | { |
700 | felem ftmp, ftmp2, ftmp3, ftmp4; |
701 | largefelem tmp; |
702 | unsigned i; |
703 | |
704 | felem_square(tmp, in); |
705 | felem_reduce(ftmp, tmp); /* 2^1 */ |
706 | felem_mul(tmp, in, ftmp); |
707 | felem_reduce(ftmp, tmp); /* 2^2 - 2^0 */ |
708 | felem_assign(ftmp2, ftmp); |
709 | felem_square(tmp, ftmp); |
710 | felem_reduce(ftmp, tmp); /* 2^3 - 2^1 */ |
711 | felem_mul(tmp, in, ftmp); |
712 | felem_reduce(ftmp, tmp); /* 2^3 - 2^0 */ |
713 | felem_square(tmp, ftmp); |
714 | felem_reduce(ftmp, tmp); /* 2^4 - 2^1 */ |
715 | |
716 | felem_square(tmp, ftmp2); |
717 | felem_reduce(ftmp3, tmp); /* 2^3 - 2^1 */ |
718 | felem_square(tmp, ftmp3); |
719 | felem_reduce(ftmp3, tmp); /* 2^4 - 2^2 */ |
720 | felem_mul(tmp, ftmp3, ftmp2); |
721 | felem_reduce(ftmp3, tmp); /* 2^4 - 2^0 */ |
722 | |
723 | felem_assign(ftmp2, ftmp3); |
724 | felem_square(tmp, ftmp3); |
725 | felem_reduce(ftmp3, tmp); /* 2^5 - 2^1 */ |
726 | felem_square(tmp, ftmp3); |
727 | felem_reduce(ftmp3, tmp); /* 2^6 - 2^2 */ |
728 | felem_square(tmp, ftmp3); |
729 | felem_reduce(ftmp3, tmp); /* 2^7 - 2^3 */ |
730 | felem_square(tmp, ftmp3); |
731 | felem_reduce(ftmp3, tmp); /* 2^8 - 2^4 */ |
732 | felem_assign(ftmp4, ftmp3); |
733 | felem_mul(tmp, ftmp3, ftmp); |
734 | felem_reduce(ftmp4, tmp); /* 2^8 - 2^1 */ |
735 | felem_square(tmp, ftmp4); |
736 | felem_reduce(ftmp4, tmp); /* 2^9 - 2^2 */ |
737 | felem_mul(tmp, ftmp3, ftmp2); |
738 | felem_reduce(ftmp3, tmp); /* 2^8 - 2^0 */ |
739 | felem_assign(ftmp2, ftmp3); |
740 | |
741 | for (i = 0; i < 8; i++) { |
742 | felem_square(tmp, ftmp3); |
743 | felem_reduce(ftmp3, tmp); /* 2^16 - 2^8 */ |
744 | } |
745 | felem_mul(tmp, ftmp3, ftmp2); |
746 | felem_reduce(ftmp3, tmp); /* 2^16 - 2^0 */ |
747 | felem_assign(ftmp2, ftmp3); |
748 | |
749 | for (i = 0; i < 16; i++) { |
750 | felem_square(tmp, ftmp3); |
751 | felem_reduce(ftmp3, tmp); /* 2^32 - 2^16 */ |
752 | } |
753 | felem_mul(tmp, ftmp3, ftmp2); |
754 | felem_reduce(ftmp3, tmp); /* 2^32 - 2^0 */ |
755 | felem_assign(ftmp2, ftmp3); |
756 | |
757 | for (i = 0; i < 32; i++) { |
758 | felem_square(tmp, ftmp3); |
759 | felem_reduce(ftmp3, tmp); /* 2^64 - 2^32 */ |
760 | } |
761 | felem_mul(tmp, ftmp3, ftmp2); |
762 | felem_reduce(ftmp3, tmp); /* 2^64 - 2^0 */ |
763 | felem_assign(ftmp2, ftmp3); |
764 | |
765 | for (i = 0; i < 64; i++) { |
766 | felem_square(tmp, ftmp3); |
767 | felem_reduce(ftmp3, tmp); /* 2^128 - 2^64 */ |
768 | } |
769 | felem_mul(tmp, ftmp3, ftmp2); |
770 | felem_reduce(ftmp3, tmp); /* 2^128 - 2^0 */ |
771 | felem_assign(ftmp2, ftmp3); |
772 | |
773 | for (i = 0; i < 128; i++) { |
774 | felem_square(tmp, ftmp3); |
775 | felem_reduce(ftmp3, tmp); /* 2^256 - 2^128 */ |
776 | } |
777 | felem_mul(tmp, ftmp3, ftmp2); |
778 | felem_reduce(ftmp3, tmp); /* 2^256 - 2^0 */ |
779 | felem_assign(ftmp2, ftmp3); |
780 | |
781 | for (i = 0; i < 256; i++) { |
782 | felem_square(tmp, ftmp3); |
783 | felem_reduce(ftmp3, tmp); /* 2^512 - 2^256 */ |
784 | } |
785 | felem_mul(tmp, ftmp3, ftmp2); |
786 | felem_reduce(ftmp3, tmp); /* 2^512 - 2^0 */ |
787 | |
788 | for (i = 0; i < 9; i++) { |
789 | felem_square(tmp, ftmp3); |
790 | felem_reduce(ftmp3, tmp); /* 2^521 - 2^9 */ |
791 | } |
792 | felem_mul(tmp, ftmp3, ftmp4); |
793 | felem_reduce(ftmp3, tmp); /* 2^512 - 2^2 */ |
794 | felem_mul(tmp, ftmp3, in); |
795 | felem_reduce(out, tmp); /* 2^512 - 3 */ |
796 | } |
797 | |
798 | /* This is 2^521-1, expressed as an felem */ |
799 | static const felem kPrime = { |
800 | 0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff, |
801 | 0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff, |
802 | 0x03ffffffffffffff, 0x03ffffffffffffff, 0x01ffffffffffffff |
803 | }; |
804 | |
805 | /*- |
806 | * felem_is_zero returns a limb with all bits set if |in| == 0 (mod p) and 0 |
807 | * otherwise. |
808 | * On entry: |
809 | * in[i] < 2^59 + 2^14 |
810 | */ |
811 | static limb felem_is_zero(const felem in) |
812 | { |
813 | felem ftmp; |
814 | limb is_zero, is_p; |
815 | felem_assign(ftmp, in); |
816 | |
817 | ftmp[0] += ftmp[8] >> 57; |
818 | ftmp[8] &= bottom57bits; |
819 | /* ftmp[8] < 2^57 */ |
820 | ftmp[1] += ftmp[0] >> 58; |
821 | ftmp[0] &= bottom58bits; |
822 | ftmp[2] += ftmp[1] >> 58; |
823 | ftmp[1] &= bottom58bits; |
824 | ftmp[3] += ftmp[2] >> 58; |
825 | ftmp[2] &= bottom58bits; |
826 | ftmp[4] += ftmp[3] >> 58; |
827 | ftmp[3] &= bottom58bits; |
828 | ftmp[5] += ftmp[4] >> 58; |
829 | ftmp[4] &= bottom58bits; |
830 | ftmp[6] += ftmp[5] >> 58; |
831 | ftmp[5] &= bottom58bits; |
832 | ftmp[7] += ftmp[6] >> 58; |
833 | ftmp[6] &= bottom58bits; |
834 | ftmp[8] += ftmp[7] >> 58; |
835 | ftmp[7] &= bottom58bits; |
836 | /* ftmp[8] < 2^57 + 4 */ |
837 | |
838 | /* |
839 | * The ninth limb of 2*(2^521-1) is 0x03ffffffffffffff, which is greater |
840 | * than our bound for ftmp[8]. Therefore we only have to check if the |
841 | * zero is zero or 2^521-1. |
842 | */ |
843 | |
844 | is_zero = 0; |
845 | is_zero |= ftmp[0]; |
846 | is_zero |= ftmp[1]; |
847 | is_zero |= ftmp[2]; |
848 | is_zero |= ftmp[3]; |
849 | is_zero |= ftmp[4]; |
850 | is_zero |= ftmp[5]; |
851 | is_zero |= ftmp[6]; |
852 | is_zero |= ftmp[7]; |
853 | is_zero |= ftmp[8]; |
854 | |
855 | is_zero--; |
856 | /* |
857 | * We know that ftmp[i] < 2^63, therefore the only way that the top bit |
858 | * can be set is if is_zero was 0 before the decrement. |
859 | */ |
860 | is_zero = 0 - (is_zero >> 63); |
861 | |
862 | is_p = ftmp[0] ^ kPrime[0]; |
863 | is_p |= ftmp[1] ^ kPrime[1]; |
864 | is_p |= ftmp[2] ^ kPrime[2]; |
865 | is_p |= ftmp[3] ^ kPrime[3]; |
866 | is_p |= ftmp[4] ^ kPrime[4]; |
867 | is_p |= ftmp[5] ^ kPrime[5]; |
868 | is_p |= ftmp[6] ^ kPrime[6]; |
869 | is_p |= ftmp[7] ^ kPrime[7]; |
870 | is_p |= ftmp[8] ^ kPrime[8]; |
871 | |
872 | is_p--; |
873 | is_p = 0 - (is_p >> 63); |
874 | |
875 | is_zero |= is_p; |
876 | return is_zero; |
877 | } |
878 | |
879 | static int felem_is_zero_int(const void *in) |
880 | { |
881 | return (int)(felem_is_zero(in) & ((limb) 1)); |
882 | } |
883 | |
884 | /*- |
885 | * felem_contract converts |in| to its unique, minimal representation. |
886 | * On entry: |
887 | * in[i] < 2^59 + 2^14 |
888 | */ |
889 | static void felem_contract(felem out, const felem in) |
890 | { |
891 | limb is_p, is_greater, sign; |
892 | static const limb two58 = ((limb) 1) << 58; |
893 | |
894 | felem_assign(out, in); |
895 | |
896 | out[0] += out[8] >> 57; |
897 | out[8] &= bottom57bits; |
898 | /* out[8] < 2^57 */ |
899 | out[1] += out[0] >> 58; |
900 | out[0] &= bottom58bits; |
901 | out[2] += out[1] >> 58; |
902 | out[1] &= bottom58bits; |
903 | out[3] += out[2] >> 58; |
904 | out[2] &= bottom58bits; |
905 | out[4] += out[3] >> 58; |
906 | out[3] &= bottom58bits; |
907 | out[5] += out[4] >> 58; |
908 | out[4] &= bottom58bits; |
909 | out[6] += out[5] >> 58; |
910 | out[5] &= bottom58bits; |
911 | out[7] += out[6] >> 58; |
912 | out[6] &= bottom58bits; |
913 | out[8] += out[7] >> 58; |
914 | out[7] &= bottom58bits; |
915 | /* out[8] < 2^57 + 4 */ |
916 | |
917 | /* |
918 | * If the value is greater than 2^521-1 then we have to subtract 2^521-1 |
919 | * out. See the comments in felem_is_zero regarding why we don't test for |
920 | * other multiples of the prime. |
921 | */ |
922 | |
923 | /* |
924 | * First, if |out| is equal to 2^521-1, we subtract it out to get zero. |
925 | */ |
926 | |
927 | is_p = out[0] ^ kPrime[0]; |
928 | is_p |= out[1] ^ kPrime[1]; |
929 | is_p |= out[2] ^ kPrime[2]; |
930 | is_p |= out[3] ^ kPrime[3]; |
931 | is_p |= out[4] ^ kPrime[4]; |
932 | is_p |= out[5] ^ kPrime[5]; |
933 | is_p |= out[6] ^ kPrime[6]; |
934 | is_p |= out[7] ^ kPrime[7]; |
935 | is_p |= out[8] ^ kPrime[8]; |
936 | |
937 | is_p--; |
938 | is_p &= is_p << 32; |
939 | is_p &= is_p << 16; |
940 | is_p &= is_p << 8; |
941 | is_p &= is_p << 4; |
942 | is_p &= is_p << 2; |
943 | is_p &= is_p << 1; |
944 | is_p = 0 - (is_p >> 63); |
945 | is_p = ~is_p; |
946 | |
947 | /* is_p is 0 iff |out| == 2^521-1 and all ones otherwise */ |
948 | |
949 | out[0] &= is_p; |
950 | out[1] &= is_p; |
951 | out[2] &= is_p; |
952 | out[3] &= is_p; |
953 | out[4] &= is_p; |
954 | out[5] &= is_p; |
955 | out[6] &= is_p; |
956 | out[7] &= is_p; |
957 | out[8] &= is_p; |
958 | |
959 | /* |
960 | * In order to test that |out| >= 2^521-1 we need only test if out[8] >> |
961 | * 57 is greater than zero as (2^521-1) + x >= 2^522 |
962 | */ |
963 | is_greater = out[8] >> 57; |
964 | is_greater |= is_greater << 32; |
965 | is_greater |= is_greater << 16; |
966 | is_greater |= is_greater << 8; |
967 | is_greater |= is_greater << 4; |
968 | is_greater |= is_greater << 2; |
969 | is_greater |= is_greater << 1; |
970 | is_greater = 0 - (is_greater >> 63); |
971 | |
972 | out[0] -= kPrime[0] & is_greater; |
973 | out[1] -= kPrime[1] & is_greater; |
974 | out[2] -= kPrime[2] & is_greater; |
975 | out[3] -= kPrime[3] & is_greater; |
976 | out[4] -= kPrime[4] & is_greater; |
977 | out[5] -= kPrime[5] & is_greater; |
978 | out[6] -= kPrime[6] & is_greater; |
979 | out[7] -= kPrime[7] & is_greater; |
980 | out[8] -= kPrime[8] & is_greater; |
981 | |
982 | /* Eliminate negative coefficients */ |
983 | sign = -(out[0] >> 63); |
984 | out[0] += (two58 & sign); |
985 | out[1] -= (1 & sign); |
986 | sign = -(out[1] >> 63); |
987 | out[1] += (two58 & sign); |
988 | out[2] -= (1 & sign); |
989 | sign = -(out[2] >> 63); |
990 | out[2] += (two58 & sign); |
991 | out[3] -= (1 & sign); |
992 | sign = -(out[3] >> 63); |
993 | out[3] += (two58 & sign); |
994 | out[4] -= (1 & sign); |
995 | sign = -(out[4] >> 63); |
996 | out[4] += (two58 & sign); |
997 | out[5] -= (1 & sign); |
998 | sign = -(out[0] >> 63); |
999 | out[5] += (two58 & sign); |
1000 | out[6] -= (1 & sign); |
1001 | sign = -(out[6] >> 63); |
1002 | out[6] += (two58 & sign); |
1003 | out[7] -= (1 & sign); |
1004 | sign = -(out[7] >> 63); |
1005 | out[7] += (two58 & sign); |
1006 | out[8] -= (1 & sign); |
1007 | sign = -(out[5] >> 63); |
1008 | out[5] += (two58 & sign); |
1009 | out[6] -= (1 & sign); |
1010 | sign = -(out[6] >> 63); |
1011 | out[6] += (two58 & sign); |
1012 | out[7] -= (1 & sign); |
1013 | sign = -(out[7] >> 63); |
1014 | out[7] += (two58 & sign); |
1015 | out[8] -= (1 & sign); |
1016 | } |
1017 | |
1018 | /*- |
1019 | * Group operations |
1020 | * ---------------- |
1021 | * |
1022 | * Building on top of the field operations we have the operations on the |
1023 | * elliptic curve group itself. Points on the curve are represented in Jacobian |
1024 | * coordinates */ |
1025 | |
1026 | /*- |
1027 | * point_double calculates 2*(x_in, y_in, z_in) |
1028 | * |
1029 | * The method is taken from: |
1030 | * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b |
1031 | * |
1032 | * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed. |
1033 | * while x_out == y_in is not (maybe this works, but it's not tested). */ |
1034 | static void |
1035 | point_double(felem x_out, felem y_out, felem z_out, |
1036 | const felem x_in, const felem y_in, const felem z_in) |
1037 | { |
1038 | largefelem tmp, tmp2; |
1039 | felem delta, gamma, beta, alpha, ftmp, ftmp2; |
1040 | |
1041 | felem_assign(ftmp, x_in); |
1042 | felem_assign(ftmp2, x_in); |
1043 | |
1044 | /* delta = z^2 */ |
1045 | felem_square(tmp, z_in); |
1046 | felem_reduce(delta, tmp); /* delta[i] < 2^59 + 2^14 */ |
1047 | |
1048 | /* gamma = y^2 */ |
1049 | felem_square(tmp, y_in); |
1050 | felem_reduce(gamma, tmp); /* gamma[i] < 2^59 + 2^14 */ |
1051 | |
1052 | /* beta = x*gamma */ |
1053 | felem_mul(tmp, x_in, gamma); |
1054 | felem_reduce(beta, tmp); /* beta[i] < 2^59 + 2^14 */ |
1055 | |
1056 | /* alpha = 3*(x-delta)*(x+delta) */ |
1057 | felem_diff64(ftmp, delta); |
1058 | /* ftmp[i] < 2^61 */ |
1059 | felem_sum64(ftmp2, delta); |
1060 | /* ftmp2[i] < 2^60 + 2^15 */ |
1061 | felem_scalar64(ftmp2, 3); |
1062 | /* ftmp2[i] < 3*2^60 + 3*2^15 */ |
1063 | felem_mul(tmp, ftmp, ftmp2); |
1064 | /*- |
1065 | * tmp[i] < 17(3*2^121 + 3*2^76) |
1066 | * = 61*2^121 + 61*2^76 |
1067 | * < 64*2^121 + 64*2^76 |
1068 | * = 2^127 + 2^82 |
1069 | * < 2^128 |
1070 | */ |
1071 | felem_reduce(alpha, tmp); |
1072 | |
1073 | /* x' = alpha^2 - 8*beta */ |
1074 | felem_square(tmp, alpha); |
1075 | /* |
1076 | * tmp[i] < 17*2^120 < 2^125 |
1077 | */ |
1078 | felem_assign(ftmp, beta); |
1079 | felem_scalar64(ftmp, 8); |
1080 | /* ftmp[i] < 2^62 + 2^17 */ |
1081 | felem_diff_128_64(tmp, ftmp); |
1082 | /* tmp[i] < 2^125 + 2^63 + 2^62 + 2^17 */ |
1083 | felem_reduce(x_out, tmp); |
1084 | |
1085 | /* z' = (y + z)^2 - gamma - delta */ |
1086 | felem_sum64(delta, gamma); |
1087 | /* delta[i] < 2^60 + 2^15 */ |
1088 | felem_assign(ftmp, y_in); |
1089 | felem_sum64(ftmp, z_in); |
1090 | /* ftmp[i] < 2^60 + 2^15 */ |
1091 | felem_square(tmp, ftmp); |
1092 | /* |
1093 | * tmp[i] < 17(2^122) < 2^127 |
1094 | */ |
1095 | felem_diff_128_64(tmp, delta); |
1096 | /* tmp[i] < 2^127 + 2^63 */ |
1097 | felem_reduce(z_out, tmp); |
1098 | |
1099 | /* y' = alpha*(4*beta - x') - 8*gamma^2 */ |
1100 | felem_scalar64(beta, 4); |
1101 | /* beta[i] < 2^61 + 2^16 */ |
1102 | felem_diff64(beta, x_out); |
1103 | /* beta[i] < 2^61 + 2^60 + 2^16 */ |
1104 | felem_mul(tmp, alpha, beta); |
1105 | /*- |
1106 | * tmp[i] < 17*((2^59 + 2^14)(2^61 + 2^60 + 2^16)) |
1107 | * = 17*(2^120 + 2^75 + 2^119 + 2^74 + 2^75 + 2^30) |
1108 | * = 17*(2^120 + 2^119 + 2^76 + 2^74 + 2^30) |
1109 | * < 2^128 |
1110 | */ |
1111 | felem_square(tmp2, gamma); |
1112 | /*- |
1113 | * tmp2[i] < 17*(2^59 + 2^14)^2 |
1114 | * = 17*(2^118 + 2^74 + 2^28) |
1115 | */ |
1116 | felem_scalar128(tmp2, 8); |
1117 | /*- |
1118 | * tmp2[i] < 8*17*(2^118 + 2^74 + 2^28) |
1119 | * = 2^125 + 2^121 + 2^81 + 2^77 + 2^35 + 2^31 |
1120 | * < 2^126 |
1121 | */ |
1122 | felem_diff128(tmp, tmp2); |
1123 | /*- |
1124 | * tmp[i] < 2^127 - 2^69 + 17(2^120 + 2^119 + 2^76 + 2^74 + 2^30) |
1125 | * = 2^127 + 2^124 + 2^122 + 2^120 + 2^118 + 2^80 + 2^78 + 2^76 + |
1126 | * 2^74 + 2^69 + 2^34 + 2^30 |
1127 | * < 2^128 |
1128 | */ |
1129 | felem_reduce(y_out, tmp); |
1130 | } |
1131 | |
1132 | /* copy_conditional copies in to out iff mask is all ones. */ |
1133 | static void copy_conditional(felem out, const felem in, limb mask) |
1134 | { |
1135 | unsigned i; |
1136 | for (i = 0; i < NLIMBS; ++i) { |
1137 | const limb tmp = mask & (in[i] ^ out[i]); |
1138 | out[i] ^= tmp; |
1139 | } |
1140 | } |
1141 | |
1142 | /*- |
1143 | * point_add calculates (x1, y1, z1) + (x2, y2, z2) |
1144 | * |
1145 | * The method is taken from |
1146 | * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl, |
1147 | * adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity). |
1148 | * |
1149 | * This function includes a branch for checking whether the two input points |
1150 | * are equal (while not equal to the point at infinity). See comment below |
1151 | * on constant-time. |
1152 | */ |
1153 | static void point_add(felem x3, felem y3, felem z3, |
1154 | const felem x1, const felem y1, const felem z1, |
1155 | const int mixed, const felem x2, const felem y2, |
1156 | const felem z2) |
1157 | { |
1158 | felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, ftmp6, x_out, y_out, z_out; |
1159 | largefelem tmp, tmp2; |
1160 | limb x_equal, y_equal, z1_is_zero, z2_is_zero; |
1161 | |
1162 | z1_is_zero = felem_is_zero(z1); |
1163 | z2_is_zero = felem_is_zero(z2); |
1164 | |
1165 | /* ftmp = z1z1 = z1**2 */ |
1166 | felem_square(tmp, z1); |
1167 | felem_reduce(ftmp, tmp); |
1168 | |
1169 | if (!mixed) { |
1170 | /* ftmp2 = z2z2 = z2**2 */ |
1171 | felem_square(tmp, z2); |
1172 | felem_reduce(ftmp2, tmp); |
1173 | |
1174 | /* u1 = ftmp3 = x1*z2z2 */ |
1175 | felem_mul(tmp, x1, ftmp2); |
1176 | felem_reduce(ftmp3, tmp); |
1177 | |
1178 | /* ftmp5 = z1 + z2 */ |
1179 | felem_assign(ftmp5, z1); |
1180 | felem_sum64(ftmp5, z2); |
1181 | /* ftmp5[i] < 2^61 */ |
1182 | |
1183 | /* ftmp5 = (z1 + z2)**2 - z1z1 - z2z2 = 2*z1z2 */ |
1184 | felem_square(tmp, ftmp5); |
1185 | /* tmp[i] < 17*2^122 */ |
1186 | felem_diff_128_64(tmp, ftmp); |
1187 | /* tmp[i] < 17*2^122 + 2^63 */ |
1188 | felem_diff_128_64(tmp, ftmp2); |
1189 | /* tmp[i] < 17*2^122 + 2^64 */ |
1190 | felem_reduce(ftmp5, tmp); |
1191 | |
1192 | /* ftmp2 = z2 * z2z2 */ |
1193 | felem_mul(tmp, ftmp2, z2); |
1194 | felem_reduce(ftmp2, tmp); |
1195 | |
1196 | /* s1 = ftmp6 = y1 * z2**3 */ |
1197 | felem_mul(tmp, y1, ftmp2); |
1198 | felem_reduce(ftmp6, tmp); |
1199 | } else { |
1200 | /* |
1201 | * We'll assume z2 = 1 (special case z2 = 0 is handled later) |
1202 | */ |
1203 | |
1204 | /* u1 = ftmp3 = x1*z2z2 */ |
1205 | felem_assign(ftmp3, x1); |
1206 | |
1207 | /* ftmp5 = 2*z1z2 */ |
1208 | felem_scalar(ftmp5, z1, 2); |
1209 | |
1210 | /* s1 = ftmp6 = y1 * z2**3 */ |
1211 | felem_assign(ftmp6, y1); |
1212 | } |
1213 | |
1214 | /* u2 = x2*z1z1 */ |
1215 | felem_mul(tmp, x2, ftmp); |
1216 | /* tmp[i] < 17*2^120 */ |
1217 | |
1218 | /* h = ftmp4 = u2 - u1 */ |
1219 | felem_diff_128_64(tmp, ftmp3); |
1220 | /* tmp[i] < 17*2^120 + 2^63 */ |
1221 | felem_reduce(ftmp4, tmp); |
1222 | |
1223 | x_equal = felem_is_zero(ftmp4); |
1224 | |
1225 | /* z_out = ftmp5 * h */ |
1226 | felem_mul(tmp, ftmp5, ftmp4); |
1227 | felem_reduce(z_out, tmp); |
1228 | |
1229 | /* ftmp = z1 * z1z1 */ |
1230 | felem_mul(tmp, ftmp, z1); |
1231 | felem_reduce(ftmp, tmp); |
1232 | |
1233 | /* s2 = tmp = y2 * z1**3 */ |
1234 | felem_mul(tmp, y2, ftmp); |
1235 | /* tmp[i] < 17*2^120 */ |
1236 | |
1237 | /* r = ftmp5 = (s2 - s1)*2 */ |
1238 | felem_diff_128_64(tmp, ftmp6); |
1239 | /* tmp[i] < 17*2^120 + 2^63 */ |
1240 | felem_reduce(ftmp5, tmp); |
1241 | y_equal = felem_is_zero(ftmp5); |
1242 | felem_scalar64(ftmp5, 2); |
1243 | /* ftmp5[i] < 2^61 */ |
1244 | |
1245 | if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) { |
1246 | /* |
1247 | * This is obviously not constant-time but it will almost-never happen |
1248 | * for ECDH / ECDSA. The case where it can happen is during scalar-mult |
1249 | * where the intermediate value gets very close to the group order. |
1250 | * Since |ec_GFp_nistp_recode_scalar_bits| produces signed digits for |
1251 | * the scalar, it's possible for the intermediate value to be a small |
1252 | * negative multiple of the base point, and for the final signed digit |
1253 | * to be the same value. We believe that this only occurs for the scalar |
1254 | * 1fffffffffffffffffffffffffffffffffffffffffffffffffffffffffff |
1255 | * ffffffa51868783bf2f966b7fcc0148f709a5d03bb5c9b8899c47aebb6fb |
1256 | * 71e913863f7, in that case the penultimate intermediate is -9G and |
1257 | * the final digit is also -9G. Since this only happens for a single |
1258 | * scalar, the timing leak is irrelevant. (Any attacker who wanted to |
1259 | * check whether a secret scalar was that exact value, can already do |
1260 | * so.) |
1261 | */ |
1262 | point_double(x3, y3, z3, x1, y1, z1); |
1263 | return; |
1264 | } |
1265 | |
1266 | /* I = ftmp = (2h)**2 */ |
1267 | felem_assign(ftmp, ftmp4); |
1268 | felem_scalar64(ftmp, 2); |
1269 | /* ftmp[i] < 2^61 */ |
1270 | felem_square(tmp, ftmp); |
1271 | /* tmp[i] < 17*2^122 */ |
1272 | felem_reduce(ftmp, tmp); |
1273 | |
1274 | /* J = ftmp2 = h * I */ |
1275 | felem_mul(tmp, ftmp4, ftmp); |
1276 | felem_reduce(ftmp2, tmp); |
1277 | |
1278 | /* V = ftmp4 = U1 * I */ |
1279 | felem_mul(tmp, ftmp3, ftmp); |
1280 | felem_reduce(ftmp4, tmp); |
1281 | |
1282 | /* x_out = r**2 - J - 2V */ |
1283 | felem_square(tmp, ftmp5); |
1284 | /* tmp[i] < 17*2^122 */ |
1285 | felem_diff_128_64(tmp, ftmp2); |
1286 | /* tmp[i] < 17*2^122 + 2^63 */ |
1287 | felem_assign(ftmp3, ftmp4); |
1288 | felem_scalar64(ftmp4, 2); |
1289 | /* ftmp4[i] < 2^61 */ |
1290 | felem_diff_128_64(tmp, ftmp4); |
1291 | /* tmp[i] < 17*2^122 + 2^64 */ |
1292 | felem_reduce(x_out, tmp); |
1293 | |
1294 | /* y_out = r(V-x_out) - 2 * s1 * J */ |
1295 | felem_diff64(ftmp3, x_out); |
1296 | /* |
1297 | * ftmp3[i] < 2^60 + 2^60 = 2^61 |
1298 | */ |
1299 | felem_mul(tmp, ftmp5, ftmp3); |
1300 | /* tmp[i] < 17*2^122 */ |
1301 | felem_mul(tmp2, ftmp6, ftmp2); |
1302 | /* tmp2[i] < 17*2^120 */ |
1303 | felem_scalar128(tmp2, 2); |
1304 | /* tmp2[i] < 17*2^121 */ |
1305 | felem_diff128(tmp, tmp2); |
1306 | /*- |
1307 | * tmp[i] < 2^127 - 2^69 + 17*2^122 |
1308 | * = 2^126 - 2^122 - 2^6 - 2^2 - 1 |
1309 | * < 2^127 |
1310 | */ |
1311 | felem_reduce(y_out, tmp); |
1312 | |
1313 | copy_conditional(x_out, x2, z1_is_zero); |
1314 | copy_conditional(x_out, x1, z2_is_zero); |
1315 | copy_conditional(y_out, y2, z1_is_zero); |
1316 | copy_conditional(y_out, y1, z2_is_zero); |
1317 | copy_conditional(z_out, z2, z1_is_zero); |
1318 | copy_conditional(z_out, z1, z2_is_zero); |
1319 | felem_assign(x3, x_out); |
1320 | felem_assign(y3, y_out); |
1321 | felem_assign(z3, z_out); |
1322 | } |
1323 | |
1324 | /*- |
1325 | * Base point pre computation |
1326 | * -------------------------- |
1327 | * |
1328 | * Two different sorts of precomputed tables are used in the following code. |
1329 | * Each contain various points on the curve, where each point is three field |
1330 | * elements (x, y, z). |
1331 | * |
1332 | * For the base point table, z is usually 1 (0 for the point at infinity). |
1333 | * This table has 16 elements: |
1334 | * index | bits | point |
1335 | * ------+---------+------------------------------ |
1336 | * 0 | 0 0 0 0 | 0G |
1337 | * 1 | 0 0 0 1 | 1G |
1338 | * 2 | 0 0 1 0 | 2^130G |
1339 | * 3 | 0 0 1 1 | (2^130 + 1)G |
1340 | * 4 | 0 1 0 0 | 2^260G |
1341 | * 5 | 0 1 0 1 | (2^260 + 1)G |
1342 | * 6 | 0 1 1 0 | (2^260 + 2^130)G |
1343 | * 7 | 0 1 1 1 | (2^260 + 2^130 + 1)G |
1344 | * 8 | 1 0 0 0 | 2^390G |
1345 | * 9 | 1 0 0 1 | (2^390 + 1)G |
1346 | * 10 | 1 0 1 0 | (2^390 + 2^130)G |
1347 | * 11 | 1 0 1 1 | (2^390 + 2^130 + 1)G |
1348 | * 12 | 1 1 0 0 | (2^390 + 2^260)G |
1349 | * 13 | 1 1 0 1 | (2^390 + 2^260 + 1)G |
1350 | * 14 | 1 1 1 0 | (2^390 + 2^260 + 2^130)G |
1351 | * 15 | 1 1 1 1 | (2^390 + 2^260 + 2^130 + 1)G |
1352 | * |
1353 | * The reason for this is so that we can clock bits into four different |
1354 | * locations when doing simple scalar multiplies against the base point. |
1355 | * |
1356 | * Tables for other points have table[i] = iG for i in 0 .. 16. */ |
1357 | |
1358 | /* gmul is the table of precomputed base points */ |
1359 | static const felem gmul[16][3] = { |
1360 | {{0, 0, 0, 0, 0, 0, 0, 0, 0}, |
1361 | {0, 0, 0, 0, 0, 0, 0, 0, 0}, |
1362 | {0, 0, 0, 0, 0, 0, 0, 0, 0}}, |
1363 | {{0x017e7e31c2e5bd66, 0x022cf0615a90a6fe, 0x00127a2ffa8de334, |
1364 | 0x01dfbf9d64a3f877, 0x006b4d3dbaa14b5e, 0x014fed487e0a2bd8, |
1365 | 0x015b4429c6481390, 0x03a73678fb2d988e, 0x00c6858e06b70404}, |
1366 | {0x00be94769fd16650, 0x031c21a89cb09022, 0x039013fad0761353, |
1367 | 0x02657bd099031542, 0x03273e662c97ee72, 0x01e6d11a05ebef45, |
1368 | 0x03d1bd998f544495, 0x03001172297ed0b1, 0x011839296a789a3b}, |
1369 | {1, 0, 0, 0, 0, 0, 0, 0, 0}}, |
1370 | {{0x0373faacbc875bae, 0x00f325023721c671, 0x00f666fd3dbde5ad, |
1371 | 0x01a6932363f88ea7, 0x01fc6d9e13f9c47b, 0x03bcbffc2bbf734e, |
1372 | 0x013ee3c3647f3a92, 0x029409fefe75d07d, 0x00ef9199963d85e5}, |
1373 | {0x011173743ad5b178, 0x02499c7c21bf7d46, 0x035beaeabb8b1a58, |
1374 | 0x00f989c4752ea0a3, 0x0101e1de48a9c1a3, 0x01a20076be28ba6c, |
1375 | 0x02f8052e5eb2de95, 0x01bfe8f82dea117c, 0x0160074d3c36ddb7}, |
1376 | {1, 0, 0, 0, 0, 0, 0, 0, 0}}, |
1377 | {{0x012f3fc373393b3b, 0x03d3d6172f1419fa, 0x02adc943c0b86873, |
1378 | 0x00d475584177952b, 0x012a4d1673750ee2, 0x00512517a0f13b0c, |
1379 | 0x02b184671a7b1734, 0x0315b84236f1a50a, 0x00a4afc472edbdb9}, |
1380 | {0x00152a7077f385c4, 0x03044007d8d1c2ee, 0x0065829d61d52b52, |
1381 | 0x00494ff6b6631d0d, 0x00a11d94d5f06bcf, 0x02d2f89474d9282e, |
1382 | 0x0241c5727c06eeb9, 0x0386928710fbdb9d, 0x01f883f727b0dfbe}, |
1383 | {1, 0, 0, 0, 0, 0, 0, 0, 0}}, |
1384 | {{0x019b0c3c9185544d, 0x006243a37c9d97db, 0x02ee3cbe030a2ad2, |
1385 | 0x00cfdd946bb51e0d, 0x0271c00932606b91, 0x03f817d1ec68c561, |
1386 | 0x03f37009806a369c, 0x03c1f30baf184fd5, 0x01091022d6d2f065}, |
1387 | {0x0292c583514c45ed, 0x0316fca51f9a286c, 0x00300af507c1489a, |
1388 | 0x0295f69008298cf1, 0x02c0ed8274943d7b, 0x016509b9b47a431e, |
1389 | 0x02bc9de9634868ce, 0x005b34929bffcb09, 0x000c1a0121681524}, |
1390 | {1, 0, 0, 0, 0, 0, 0, 0, 0}}, |
1391 | {{0x0286abc0292fb9f2, 0x02665eee9805b3f7, 0x01ed7455f17f26d6, |
1392 | 0x0346355b83175d13, 0x006284944cd0a097, 0x0191895bcdec5e51, |
1393 | 0x02e288370afda7d9, 0x03b22312bfefa67a, 0x01d104d3fc0613fe}, |
1394 | {0x0092421a12f7e47f, 0x0077a83fa373c501, 0x03bd25c5f696bd0d, |
1395 | 0x035c41e4d5459761, 0x01ca0d1742b24f53, 0x00aaab27863a509c, |
1396 | 0x018b6de47df73917, 0x025c0b771705cd01, 0x01fd51d566d760a7}, |
1397 | {1, 0, 0, 0, 0, 0, 0, 0, 0}}, |
1398 | {{0x01dd92ff6b0d1dbd, 0x039c5e2e8f8afa69, 0x0261ed13242c3b27, |
1399 | 0x0382c6e67026e6a0, 0x01d60b10be2089f9, 0x03c15f3dce86723f, |
1400 | 0x03c764a32d2a062d, 0x017307eac0fad056, 0x018207c0b96c5256}, |
1401 | {0x0196a16d60e13154, 0x03e6ce74c0267030, 0x00ddbf2b4e52a5aa, |
1402 | 0x012738241bbf31c8, 0x00ebe8dc04685a28, 0x024c2ad6d380d4a2, |
1403 | 0x035ee062a6e62d0e, 0x0029ed74af7d3a0f, 0x00eef32aec142ebd}, |
1404 | {1, 0, 0, 0, 0, 0, 0, 0, 0}}, |
1405 | {{0x00c31ec398993b39, 0x03a9f45bcda68253, 0x00ac733c24c70890, |
1406 | 0x00872b111401ff01, 0x01d178c23195eafb, 0x03bca2c816b87f74, |
1407 | 0x0261a9af46fbad7a, 0x0324b2a8dd3d28f9, 0x00918121d8f24e23}, |
1408 | {0x032bc8c1ca983cd7, 0x00d869dfb08fc8c6, 0x01693cb61fce1516, |
1409 | 0x012a5ea68f4e88a8, 0x010869cab88d7ae3, 0x009081ad277ceee1, |
1410 | 0x033a77166d064cdc, 0x03955235a1fb3a95, 0x01251a4a9b25b65e}, |
1411 | {1, 0, 0, 0, 0, 0, 0, 0, 0}}, |
1412 | {{0x00148a3a1b27f40b, 0x0123186df1b31fdc, 0x00026e7beaad34ce, |
1413 | 0x01db446ac1d3dbba, 0x0299c1a33437eaec, 0x024540610183cbb7, |
1414 | 0x0173bb0e9ce92e46, 0x02b937e43921214b, 0x01ab0436a9bf01b5}, |
1415 | {0x0383381640d46948, 0x008dacbf0e7f330f, 0x03602122bcc3f318, |
1416 | 0x01ee596b200620d6, 0x03bd0585fda430b3, 0x014aed77fd123a83, |
1417 | 0x005ace749e52f742, 0x0390fe041da2b842, 0x0189a8ceb3299242}, |
1418 | {1, 0, 0, 0, 0, 0, 0, 0, 0}}, |
1419 | {{0x012a19d6b3282473, 0x00c0915918b423ce, 0x023a954eb94405ae, |
1420 | 0x00529f692be26158, 0x0289fa1b6fa4b2aa, 0x0198ae4ceea346ef, |
1421 | 0x0047d8cdfbdedd49, 0x00cc8c8953f0f6b8, 0x001424abbff49203}, |
1422 | {0x0256732a1115a03a, 0x0351bc38665c6733, 0x03f7b950fb4a6447, |
1423 | 0x000afffa94c22155, 0x025763d0a4dab540, 0x000511e92d4fc283, |
1424 | 0x030a7e9eda0ee96c, 0x004c3cd93a28bf0a, 0x017edb3a8719217f}, |
1425 | {1, 0, 0, 0, 0, 0, 0, 0, 0}}, |
1426 | {{0x011de5675a88e673, 0x031d7d0f5e567fbe, 0x0016b2062c970ae5, |
1427 | 0x03f4a2be49d90aa7, 0x03cef0bd13822866, 0x03f0923dcf774a6c, |
1428 | 0x0284bebc4f322f72, 0x016ab2645302bb2c, 0x01793f95dace0e2a}, |
1429 | {0x010646e13527a28f, 0x01ca1babd59dc5e7, 0x01afedfd9a5595df, |
1430 | 0x01f15785212ea6b1, 0x0324e5d64f6ae3f4, 0x02d680f526d00645, |
1431 | 0x0127920fadf627a7, 0x03b383f75df4f684, 0x0089e0057e783b0a}, |
1432 | {1, 0, 0, 0, 0, 0, 0, 0, 0}}, |
1433 | {{0x00f334b9eb3c26c6, 0x0298fdaa98568dce, 0x01c2d24843a82292, |
1434 | 0x020bcb24fa1b0711, 0x02cbdb3d2b1875e6, 0x0014907598f89422, |
1435 | 0x03abe3aa43b26664, 0x02cbf47f720bc168, 0x0133b5e73014b79b}, |
1436 | {0x034aab5dab05779d, 0x00cdc5d71fee9abb, 0x0399f16bd4bd9d30, |
1437 | 0x03582fa592d82647, 0x02be1cdfb775b0e9, 0x0034f7cea32e94cb, |
1438 | 0x0335a7f08f56f286, 0x03b707e9565d1c8b, 0x0015c946ea5b614f}, |
1439 | {1, 0, 0, 0, 0, 0, 0, 0, 0}}, |
1440 | {{0x024676f6cff72255, 0x00d14625cac96378, 0x00532b6008bc3767, |
1441 | 0x01fc16721b985322, 0x023355ea1b091668, 0x029de7afdc0317c3, |
1442 | 0x02fc8a7ca2da037c, 0x02de1217d74a6f30, 0x013f7173175b73bf}, |
1443 | {0x0344913f441490b5, 0x0200f9e272b61eca, 0x0258a246b1dd55d2, |
1444 | 0x03753db9ea496f36, 0x025e02937a09c5ef, 0x030cbd3d14012692, |
1445 | 0x01793a67e70dc72a, 0x03ec1d37048a662e, 0x006550f700c32a8d}, |
1446 | {1, 0, 0, 0, 0, 0, 0, 0, 0}}, |
1447 | {{0x00d3f48a347eba27, 0x008e636649b61bd8, 0x00d3b93716778fb3, |
1448 | 0x004d1915757bd209, 0x019d5311a3da44e0, 0x016d1afcbbe6aade, |
1449 | 0x0241bf5f73265616, 0x0384672e5d50d39b, 0x005009fee522b684}, |
1450 | {0x029b4fab064435fe, 0x018868ee095bbb07, 0x01ea3d6936cc92b8, |
1451 | 0x000608b00f78a2f3, 0x02db911073d1c20f, 0x018205938470100a, |
1452 | 0x01f1e4964cbe6ff2, 0x021a19a29eed4663, 0x01414485f42afa81}, |
1453 | {1, 0, 0, 0, 0, 0, 0, 0, 0}}, |
1454 | {{0x01612b3a17f63e34, 0x03813992885428e6, 0x022b3c215b5a9608, |
1455 | 0x029b4057e19f2fcb, 0x0384059a587af7e6, 0x02d6400ace6fe610, |
1456 | 0x029354d896e8e331, 0x00c047ee6dfba65e, 0x0037720542e9d49d}, |
1457 | {0x02ce9eed7c5e9278, 0x0374ed703e79643b, 0x01316c54c4072006, |
1458 | 0x005aaa09054b2ee8, 0x002824000c840d57, 0x03d4eba24771ed86, |
1459 | 0x0189c50aabc3bdae, 0x0338c01541e15510, 0x00466d56e38eed42}, |
1460 | {1, 0, 0, 0, 0, 0, 0, 0, 0}}, |
1461 | {{0x007efd8330ad8bd6, 0x02465ed48047710b, 0x0034c6606b215e0c, |
1462 | 0x016ae30c53cbf839, 0x01fa17bd37161216, 0x018ead4e61ce8ab9, |
1463 | 0x005482ed5f5dee46, 0x037543755bba1d7f, 0x005e5ac7e70a9d0f}, |
1464 | {0x0117e1bb2fdcb2a2, 0x03deea36249f40c4, 0x028d09b4a6246cb7, |
1465 | 0x03524b8855bcf756, 0x023d7d109d5ceb58, 0x0178e43e3223ef9c, |
1466 | 0x0154536a0c6e966a, 0x037964d1286ee9fe, 0x0199bcd90e125055}, |
1467 | {1, 0, 0, 0, 0, 0, 0, 0, 0}} |
1468 | }; |
1469 | |
1470 | /* |
1471 | * select_point selects the |idx|th point from a precomputation table and |
1472 | * copies it to out. |
1473 | */ |
1474 | /* pre_comp below is of the size provided in |size| */ |
1475 | static void select_point(const limb idx, unsigned int size, |
1476 | const felem pre_comp[][3], felem out[3]) |
1477 | { |
1478 | unsigned i, j; |
1479 | limb *outlimbs = &out[0][0]; |
1480 | |
1481 | memset(out, 0, sizeof(*out) * 3); |
1482 | |
1483 | for (i = 0; i < size; i++) { |
1484 | const limb *inlimbs = &pre_comp[i][0][0]; |
1485 | limb mask = i ^ idx; |
1486 | mask |= mask >> 4; |
1487 | mask |= mask >> 2; |
1488 | mask |= mask >> 1; |
1489 | mask &= 1; |
1490 | mask--; |
1491 | for (j = 0; j < NLIMBS * 3; j++) |
1492 | outlimbs[j] |= inlimbs[j] & mask; |
1493 | } |
1494 | } |
1495 | |
1496 | /* get_bit returns the |i|th bit in |in| */ |
1497 | static char get_bit(const felem_bytearray in, int i) |
1498 | { |
1499 | if (i < 0) |
1500 | return 0; |
1501 | return (in[i >> 3] >> (i & 7)) & 1; |
1502 | } |
1503 | |
1504 | /* |
1505 | * Interleaved point multiplication using precomputed point multiples: The |
1506 | * small point multiples 0*P, 1*P, ..., 16*P are in pre_comp[], the scalars |
1507 | * in scalars[]. If g_scalar is non-NULL, we also add this multiple of the |
1508 | * generator, using certain (large) precomputed multiples in g_pre_comp. |
1509 | * Output point (X, Y, Z) is stored in x_out, y_out, z_out |
1510 | */ |
1511 | static void batch_mul(felem x_out, felem y_out, felem z_out, |
1512 | const felem_bytearray scalars[], |
1513 | const unsigned num_points, const u8 *g_scalar, |
1514 | const int mixed, const felem pre_comp[][17][3], |
1515 | const felem g_pre_comp[16][3]) |
1516 | { |
1517 | int i, skip; |
1518 | unsigned num, gen_mul = (g_scalar != NULL); |
1519 | felem nq[3], tmp[4]; |
1520 | limb bits; |
1521 | u8 sign, digit; |
1522 | |
1523 | /* set nq to the point at infinity */ |
1524 | memset(nq, 0, sizeof(nq)); |
1525 | |
1526 | /* |
1527 | * Loop over all scalars msb-to-lsb, interleaving additions of multiples |
1528 | * of the generator (last quarter of rounds) and additions of other |
1529 | * points multiples (every 5th round). |
1530 | */ |
1531 | skip = 1; /* save two point operations in the first |
1532 | * round */ |
1533 | for (i = (num_points ? 520 : 130); i >= 0; --i) { |
1534 | /* double */ |
1535 | if (!skip) |
1536 | point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]); |
1537 | |
1538 | /* add multiples of the generator */ |
1539 | if (gen_mul && (i <= 130)) { |
1540 | bits = get_bit(g_scalar, i + 390) << 3; |
1541 | if (i < 130) { |
1542 | bits |= get_bit(g_scalar, i + 260) << 2; |
1543 | bits |= get_bit(g_scalar, i + 130) << 1; |
1544 | bits |= get_bit(g_scalar, i); |
1545 | } |
1546 | /* select the point to add, in constant time */ |
1547 | select_point(bits, 16, g_pre_comp, tmp); |
1548 | if (!skip) { |
1549 | /* The 1 argument below is for "mixed" */ |
1550 | point_add(nq[0], nq[1], nq[2], |
1551 | nq[0], nq[1], nq[2], 1, tmp[0], tmp[1], tmp[2]); |
1552 | } else { |
1553 | memcpy(nq, tmp, 3 * sizeof(felem)); |
1554 | skip = 0; |
1555 | } |
1556 | } |
1557 | |
1558 | /* do other additions every 5 doublings */ |
1559 | if (num_points && (i % 5 == 0)) { |
1560 | /* loop over all scalars */ |
1561 | for (num = 0; num < num_points; ++num) { |
1562 | bits = get_bit(scalars[num], i + 4) << 5; |
1563 | bits |= get_bit(scalars[num], i + 3) << 4; |
1564 | bits |= get_bit(scalars[num], i + 2) << 3; |
1565 | bits |= get_bit(scalars[num], i + 1) << 2; |
1566 | bits |= get_bit(scalars[num], i) << 1; |
1567 | bits |= get_bit(scalars[num], i - 1); |
1568 | ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits); |
1569 | |
1570 | /* |
1571 | * select the point to add or subtract, in constant time |
1572 | */ |
1573 | select_point(digit, 17, pre_comp[num], tmp); |
1574 | felem_neg(tmp[3], tmp[1]); /* (X, -Y, Z) is the negative |
1575 | * point */ |
1576 | copy_conditional(tmp[1], tmp[3], (-(limb) sign)); |
1577 | |
1578 | if (!skip) { |
1579 | point_add(nq[0], nq[1], nq[2], |
1580 | nq[0], nq[1], nq[2], |
1581 | mixed, tmp[0], tmp[1], tmp[2]); |
1582 | } else { |
1583 | memcpy(nq, tmp, 3 * sizeof(felem)); |
1584 | skip = 0; |
1585 | } |
1586 | } |
1587 | } |
1588 | } |
1589 | felem_assign(x_out, nq[0]); |
1590 | felem_assign(y_out, nq[1]); |
1591 | felem_assign(z_out, nq[2]); |
1592 | } |
1593 | |
1594 | /* Precomputation for the group generator. */ |
1595 | struct nistp521_pre_comp_st { |
1596 | felem g_pre_comp[16][3]; |
1597 | CRYPTO_REF_COUNT references; |
1598 | CRYPTO_RWLOCK *lock; |
1599 | }; |
1600 | |
1601 | const EC_METHOD *EC_GFp_nistp521_method(void) |
1602 | { |
1603 | static const EC_METHOD ret = { |
1604 | EC_FLAGS_DEFAULT_OCT, |
1605 | NID_X9_62_prime_field, |
1606 | ec_GFp_nistp521_group_init, |
1607 | ec_GFp_simple_group_finish, |
1608 | ec_GFp_simple_group_clear_finish, |
1609 | ec_GFp_nist_group_copy, |
1610 | ec_GFp_nistp521_group_set_curve, |
1611 | ec_GFp_simple_group_get_curve, |
1612 | ec_GFp_simple_group_get_degree, |
1613 | ec_group_simple_order_bits, |
1614 | ec_GFp_simple_group_check_discriminant, |
1615 | ec_GFp_simple_point_init, |
1616 | ec_GFp_simple_point_finish, |
1617 | ec_GFp_simple_point_clear_finish, |
1618 | ec_GFp_simple_point_copy, |
1619 | ec_GFp_simple_point_set_to_infinity, |
1620 | ec_GFp_simple_set_Jprojective_coordinates_GFp, |
1621 | ec_GFp_simple_get_Jprojective_coordinates_GFp, |
1622 | ec_GFp_simple_point_set_affine_coordinates, |
1623 | ec_GFp_nistp521_point_get_affine_coordinates, |
1624 | 0 /* point_set_compressed_coordinates */ , |
1625 | 0 /* point2oct */ , |
1626 | 0 /* oct2point */ , |
1627 | ec_GFp_simple_add, |
1628 | ec_GFp_simple_dbl, |
1629 | ec_GFp_simple_invert, |
1630 | ec_GFp_simple_is_at_infinity, |
1631 | ec_GFp_simple_is_on_curve, |
1632 | ec_GFp_simple_cmp, |
1633 | ec_GFp_simple_make_affine, |
1634 | ec_GFp_simple_points_make_affine, |
1635 | ec_GFp_nistp521_points_mul, |
1636 | ec_GFp_nistp521_precompute_mult, |
1637 | ec_GFp_nistp521_have_precompute_mult, |
1638 | ec_GFp_nist_field_mul, |
1639 | ec_GFp_nist_field_sqr, |
1640 | 0 /* field_div */ , |
1641 | ec_GFp_simple_field_inv, |
1642 | 0 /* field_encode */ , |
1643 | 0 /* field_decode */ , |
1644 | 0, /* field_set_to_one */ |
1645 | ec_key_simple_priv2oct, |
1646 | ec_key_simple_oct2priv, |
1647 | 0, /* set private */ |
1648 | ec_key_simple_generate_key, |
1649 | ec_key_simple_check_key, |
1650 | ec_key_simple_generate_public_key, |
1651 | 0, /* keycopy */ |
1652 | 0, /* keyfinish */ |
1653 | ecdh_simple_compute_key, |
1654 | ecdsa_simple_sign_setup, |
1655 | ecdsa_simple_sign_sig, |
1656 | ecdsa_simple_verify_sig, |
1657 | 0, /* field_inverse_mod_ord */ |
1658 | 0, /* blind_coordinates */ |
1659 | 0, /* ladder_pre */ |
1660 | 0, /* ladder_step */ |
1661 | 0 /* ladder_post */ |
1662 | }; |
1663 | |
1664 | return &ret; |
1665 | } |
1666 | |
1667 | /******************************************************************************/ |
1668 | /* |
1669 | * FUNCTIONS TO MANAGE PRECOMPUTATION |
1670 | */ |
1671 | |
1672 | static NISTP521_PRE_COMP *nistp521_pre_comp_new(void) |
1673 | { |
1674 | NISTP521_PRE_COMP *ret = OPENSSL_zalloc(sizeof(*ret)); |
1675 | |
1676 | if (ret == NULL) { |
1677 | ECerr(EC_F_NISTP521_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE); |
1678 | return ret; |
1679 | } |
1680 | |
1681 | ret->references = 1; |
1682 | |
1683 | ret->lock = CRYPTO_THREAD_lock_new(); |
1684 | if (ret->lock == NULL) { |
1685 | ECerr(EC_F_NISTP521_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE); |
1686 | OPENSSL_free(ret); |
1687 | return NULL; |
1688 | } |
1689 | return ret; |
1690 | } |
1691 | |
1692 | NISTP521_PRE_COMP *EC_nistp521_pre_comp_dup(NISTP521_PRE_COMP *p) |
1693 | { |
1694 | int i; |
1695 | if (p != NULL) |
1696 | CRYPTO_UP_REF(&p->references, &i, p->lock); |
1697 | return p; |
1698 | } |
1699 | |
1700 | void EC_nistp521_pre_comp_free(NISTP521_PRE_COMP *p) |
1701 | { |
1702 | int i; |
1703 | |
1704 | if (p == NULL) |
1705 | return; |
1706 | |
1707 | CRYPTO_DOWN_REF(&p->references, &i, p->lock); |
1708 | REF_PRINT_COUNT("EC_nistp521" , x); |
1709 | if (i > 0) |
1710 | return; |
1711 | REF_ASSERT_ISNT(i < 0); |
1712 | |
1713 | CRYPTO_THREAD_lock_free(p->lock); |
1714 | OPENSSL_free(p); |
1715 | } |
1716 | |
1717 | /******************************************************************************/ |
1718 | /* |
1719 | * OPENSSL EC_METHOD FUNCTIONS |
1720 | */ |
1721 | |
1722 | int ec_GFp_nistp521_group_init(EC_GROUP *group) |
1723 | { |
1724 | int ret; |
1725 | ret = ec_GFp_simple_group_init(group); |
1726 | group->a_is_minus3 = 1; |
1727 | return ret; |
1728 | } |
1729 | |
1730 | int ec_GFp_nistp521_group_set_curve(EC_GROUP *group, const BIGNUM *p, |
1731 | const BIGNUM *a, const BIGNUM *b, |
1732 | BN_CTX *ctx) |
1733 | { |
1734 | int ret = 0; |
1735 | BIGNUM *curve_p, *curve_a, *curve_b; |
1736 | #ifndef FIPS_MODE |
1737 | BN_CTX *new_ctx = NULL; |
1738 | |
1739 | if (ctx == NULL) |
1740 | ctx = new_ctx = BN_CTX_new(); |
1741 | #endif |
1742 | if (ctx == NULL) |
1743 | return 0; |
1744 | |
1745 | BN_CTX_start(ctx); |
1746 | curve_p = BN_CTX_get(ctx); |
1747 | curve_a = BN_CTX_get(ctx); |
1748 | curve_b = BN_CTX_get(ctx); |
1749 | if (curve_b == NULL) |
1750 | goto err; |
1751 | BN_bin2bn(nistp521_curve_params[0], sizeof(felem_bytearray), curve_p); |
1752 | BN_bin2bn(nistp521_curve_params[1], sizeof(felem_bytearray), curve_a); |
1753 | BN_bin2bn(nistp521_curve_params[2], sizeof(felem_bytearray), curve_b); |
1754 | if ((BN_cmp(curve_p, p)) || (BN_cmp(curve_a, a)) || (BN_cmp(curve_b, b))) { |
1755 | ECerr(EC_F_EC_GFP_NISTP521_GROUP_SET_CURVE, |
1756 | EC_R_WRONG_CURVE_PARAMETERS); |
1757 | goto err; |
1758 | } |
1759 | group->field_mod_func = BN_nist_mod_521; |
1760 | ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx); |
1761 | err: |
1762 | BN_CTX_end(ctx); |
1763 | #ifndef FIPS_MODE |
1764 | BN_CTX_free(new_ctx); |
1765 | #endif |
1766 | return ret; |
1767 | } |
1768 | |
1769 | /* |
1770 | * Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') = |
1771 | * (X/Z^2, Y/Z^3) |
1772 | */ |
1773 | int ec_GFp_nistp521_point_get_affine_coordinates(const EC_GROUP *group, |
1774 | const EC_POINT *point, |
1775 | BIGNUM *x, BIGNUM *y, |
1776 | BN_CTX *ctx) |
1777 | { |
1778 | felem z1, z2, x_in, y_in, x_out, y_out; |
1779 | largefelem tmp; |
1780 | |
1781 | if (EC_POINT_is_at_infinity(group, point)) { |
1782 | ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES, |
1783 | EC_R_POINT_AT_INFINITY); |
1784 | return 0; |
1785 | } |
1786 | if ((!BN_to_felem(x_in, point->X)) || (!BN_to_felem(y_in, point->Y)) || |
1787 | (!BN_to_felem(z1, point->Z))) |
1788 | return 0; |
1789 | felem_inv(z2, z1); |
1790 | felem_square(tmp, z2); |
1791 | felem_reduce(z1, tmp); |
1792 | felem_mul(tmp, x_in, z1); |
1793 | felem_reduce(x_in, tmp); |
1794 | felem_contract(x_out, x_in); |
1795 | if (x != NULL) { |
1796 | if (!felem_to_BN(x, x_out)) { |
1797 | ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES, |
1798 | ERR_R_BN_LIB); |
1799 | return 0; |
1800 | } |
1801 | } |
1802 | felem_mul(tmp, z1, z2); |
1803 | felem_reduce(z1, tmp); |
1804 | felem_mul(tmp, y_in, z1); |
1805 | felem_reduce(y_in, tmp); |
1806 | felem_contract(y_out, y_in); |
1807 | if (y != NULL) { |
1808 | if (!felem_to_BN(y, y_out)) { |
1809 | ECerr(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES, |
1810 | ERR_R_BN_LIB); |
1811 | return 0; |
1812 | } |
1813 | } |
1814 | return 1; |
1815 | } |
1816 | |
1817 | /* points below is of size |num|, and tmp_felems is of size |num+1/ */ |
1818 | static void make_points_affine(size_t num, felem points[][3], |
1819 | felem tmp_felems[]) |
1820 | { |
1821 | /* |
1822 | * Runs in constant time, unless an input is the point at infinity (which |
1823 | * normally shouldn't happen). |
1824 | */ |
1825 | ec_GFp_nistp_points_make_affine_internal(num, |
1826 | points, |
1827 | sizeof(felem), |
1828 | tmp_felems, |
1829 | (void (*)(void *))felem_one, |
1830 | felem_is_zero_int, |
1831 | (void (*)(void *, const void *)) |
1832 | felem_assign, |
1833 | (void (*)(void *, const void *)) |
1834 | felem_square_reduce, (void (*) |
1835 | (void *, |
1836 | const void |
1837 | *, |
1838 | const void |
1839 | *)) |
1840 | felem_mul_reduce, |
1841 | (void (*)(void *, const void *)) |
1842 | felem_inv, |
1843 | (void (*)(void *, const void *)) |
1844 | felem_contract); |
1845 | } |
1846 | |
1847 | /* |
1848 | * Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL |
1849 | * values Result is stored in r (r can equal one of the inputs). |
1850 | */ |
1851 | int ec_GFp_nistp521_points_mul(const EC_GROUP *group, EC_POINT *r, |
1852 | const BIGNUM *scalar, size_t num, |
1853 | const EC_POINT *points[], |
1854 | const BIGNUM *scalars[], BN_CTX *ctx) |
1855 | { |
1856 | int ret = 0; |
1857 | int j; |
1858 | int mixed = 0; |
1859 | BIGNUM *x, *y, *z, *tmp_scalar; |
1860 | felem_bytearray g_secret; |
1861 | felem_bytearray *secrets = NULL; |
1862 | felem (*pre_comp)[17][3] = NULL; |
1863 | felem *tmp_felems = NULL; |
1864 | unsigned i; |
1865 | int num_bytes; |
1866 | int have_pre_comp = 0; |
1867 | size_t num_points = num; |
1868 | felem x_in, y_in, z_in, x_out, y_out, z_out; |
1869 | NISTP521_PRE_COMP *pre = NULL; |
1870 | felem(*g_pre_comp)[3] = NULL; |
1871 | EC_POINT *generator = NULL; |
1872 | const EC_POINT *p = NULL; |
1873 | const BIGNUM *p_scalar = NULL; |
1874 | |
1875 | BN_CTX_start(ctx); |
1876 | x = BN_CTX_get(ctx); |
1877 | y = BN_CTX_get(ctx); |
1878 | z = BN_CTX_get(ctx); |
1879 | tmp_scalar = BN_CTX_get(ctx); |
1880 | if (tmp_scalar == NULL) |
1881 | goto err; |
1882 | |
1883 | if (scalar != NULL) { |
1884 | pre = group->pre_comp.nistp521; |
1885 | if (pre) |
1886 | /* we have precomputation, try to use it */ |
1887 | g_pre_comp = &pre->g_pre_comp[0]; |
1888 | else |
1889 | /* try to use the standard precomputation */ |
1890 | g_pre_comp = (felem(*)[3]) gmul; |
1891 | generator = EC_POINT_new(group); |
1892 | if (generator == NULL) |
1893 | goto err; |
1894 | /* get the generator from precomputation */ |
1895 | if (!felem_to_BN(x, g_pre_comp[1][0]) || |
1896 | !felem_to_BN(y, g_pre_comp[1][1]) || |
1897 | !felem_to_BN(z, g_pre_comp[1][2])) { |
1898 | ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB); |
1899 | goto err; |
1900 | } |
1901 | if (!EC_POINT_set_Jprojective_coordinates_GFp(group, |
1902 | generator, x, y, z, |
1903 | ctx)) |
1904 | goto err; |
1905 | if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) |
1906 | /* precomputation matches generator */ |
1907 | have_pre_comp = 1; |
1908 | else |
1909 | /* |
1910 | * we don't have valid precomputation: treat the generator as a |
1911 | * random point |
1912 | */ |
1913 | num_points++; |
1914 | } |
1915 | |
1916 | if (num_points > 0) { |
1917 | if (num_points >= 2) { |
1918 | /* |
1919 | * unless we precompute multiples for just one point, converting |
1920 | * those into affine form is time well spent |
1921 | */ |
1922 | mixed = 1; |
1923 | } |
1924 | secrets = OPENSSL_zalloc(sizeof(*secrets) * num_points); |
1925 | pre_comp = OPENSSL_zalloc(sizeof(*pre_comp) * num_points); |
1926 | if (mixed) |
1927 | tmp_felems = |
1928 | OPENSSL_malloc(sizeof(*tmp_felems) * (num_points * 17 + 1)); |
1929 | if ((secrets == NULL) || (pre_comp == NULL) |
1930 | || (mixed && (tmp_felems == NULL))) { |
1931 | ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_MALLOC_FAILURE); |
1932 | goto err; |
1933 | } |
1934 | |
1935 | /* |
1936 | * we treat NULL scalars as 0, and NULL points as points at infinity, |
1937 | * i.e., they contribute nothing to the linear combination |
1938 | */ |
1939 | for (i = 0; i < num_points; ++i) { |
1940 | if (i == num) { |
1941 | /* |
1942 | * we didn't have a valid precomputation, so we pick the |
1943 | * generator |
1944 | */ |
1945 | p = EC_GROUP_get0_generator(group); |
1946 | p_scalar = scalar; |
1947 | } else { |
1948 | /* the i^th point */ |
1949 | p = points[i]; |
1950 | p_scalar = scalars[i]; |
1951 | } |
1952 | if ((p_scalar != NULL) && (p != NULL)) { |
1953 | /* reduce scalar to 0 <= scalar < 2^521 */ |
1954 | if ((BN_num_bits(p_scalar) > 521) |
1955 | || (BN_is_negative(p_scalar))) { |
1956 | /* |
1957 | * this is an unusual input, and we don't guarantee |
1958 | * constant-timeness |
1959 | */ |
1960 | if (!BN_nnmod(tmp_scalar, p_scalar, group->order, ctx)) { |
1961 | ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB); |
1962 | goto err; |
1963 | } |
1964 | num_bytes = BN_bn2lebinpad(tmp_scalar, |
1965 | secrets[i], sizeof(secrets[i])); |
1966 | } else { |
1967 | num_bytes = BN_bn2lebinpad(p_scalar, |
1968 | secrets[i], sizeof(secrets[i])); |
1969 | } |
1970 | if (num_bytes < 0) { |
1971 | ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB); |
1972 | goto err; |
1973 | } |
1974 | /* precompute multiples */ |
1975 | if ((!BN_to_felem(x_out, p->X)) || |
1976 | (!BN_to_felem(y_out, p->Y)) || |
1977 | (!BN_to_felem(z_out, p->Z))) |
1978 | goto err; |
1979 | memcpy(pre_comp[i][1][0], x_out, sizeof(felem)); |
1980 | memcpy(pre_comp[i][1][1], y_out, sizeof(felem)); |
1981 | memcpy(pre_comp[i][1][2], z_out, sizeof(felem)); |
1982 | for (j = 2; j <= 16; ++j) { |
1983 | if (j & 1) { |
1984 | point_add(pre_comp[i][j][0], pre_comp[i][j][1], |
1985 | pre_comp[i][j][2], pre_comp[i][1][0], |
1986 | pre_comp[i][1][1], pre_comp[i][1][2], 0, |
1987 | pre_comp[i][j - 1][0], |
1988 | pre_comp[i][j - 1][1], |
1989 | pre_comp[i][j - 1][2]); |
1990 | } else { |
1991 | point_double(pre_comp[i][j][0], pre_comp[i][j][1], |
1992 | pre_comp[i][j][2], pre_comp[i][j / 2][0], |
1993 | pre_comp[i][j / 2][1], |
1994 | pre_comp[i][j / 2][2]); |
1995 | } |
1996 | } |
1997 | } |
1998 | } |
1999 | if (mixed) |
2000 | make_points_affine(num_points * 17, pre_comp[0], tmp_felems); |
2001 | } |
2002 | |
2003 | /* the scalar for the generator */ |
2004 | if ((scalar != NULL) && (have_pre_comp)) { |
2005 | memset(g_secret, 0, sizeof(g_secret)); |
2006 | /* reduce scalar to 0 <= scalar < 2^521 */ |
2007 | if ((BN_num_bits(scalar) > 521) || (BN_is_negative(scalar))) { |
2008 | /* |
2009 | * this is an unusual input, and we don't guarantee |
2010 | * constant-timeness |
2011 | */ |
2012 | if (!BN_nnmod(tmp_scalar, scalar, group->order, ctx)) { |
2013 | ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB); |
2014 | goto err; |
2015 | } |
2016 | num_bytes = BN_bn2lebinpad(tmp_scalar, g_secret, sizeof(g_secret)); |
2017 | } else { |
2018 | num_bytes = BN_bn2lebinpad(scalar, g_secret, sizeof(g_secret)); |
2019 | } |
2020 | /* do the multiplication with generator precomputation */ |
2021 | batch_mul(x_out, y_out, z_out, |
2022 | (const felem_bytearray(*))secrets, num_points, |
2023 | g_secret, |
2024 | mixed, (const felem(*)[17][3])pre_comp, |
2025 | (const felem(*)[3])g_pre_comp); |
2026 | } else { |
2027 | /* do the multiplication without generator precomputation */ |
2028 | batch_mul(x_out, y_out, z_out, |
2029 | (const felem_bytearray(*))secrets, num_points, |
2030 | NULL, mixed, (const felem(*)[17][3])pre_comp, NULL); |
2031 | } |
2032 | /* reduce the output to its unique minimal representation */ |
2033 | felem_contract(x_in, x_out); |
2034 | felem_contract(y_in, y_out); |
2035 | felem_contract(z_in, z_out); |
2036 | if ((!felem_to_BN(x, x_in)) || (!felem_to_BN(y, y_in)) || |
2037 | (!felem_to_BN(z, z_in))) { |
2038 | ECerr(EC_F_EC_GFP_NISTP521_POINTS_MUL, ERR_R_BN_LIB); |
2039 | goto err; |
2040 | } |
2041 | ret = EC_POINT_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx); |
2042 | |
2043 | err: |
2044 | BN_CTX_end(ctx); |
2045 | EC_POINT_free(generator); |
2046 | OPENSSL_free(secrets); |
2047 | OPENSSL_free(pre_comp); |
2048 | OPENSSL_free(tmp_felems); |
2049 | return ret; |
2050 | } |
2051 | |
2052 | int ec_GFp_nistp521_precompute_mult(EC_GROUP *group, BN_CTX *ctx) |
2053 | { |
2054 | int ret = 0; |
2055 | NISTP521_PRE_COMP *pre = NULL; |
2056 | int i, j; |
2057 | BIGNUM *x, *y; |
2058 | EC_POINT *generator = NULL; |
2059 | felem tmp_felems[16]; |
2060 | #ifndef FIPS_MODE |
2061 | BN_CTX *new_ctx = NULL; |
2062 | #endif |
2063 | |
2064 | /* throw away old precomputation */ |
2065 | EC_pre_comp_free(group); |
2066 | |
2067 | #ifndef FIPS_MODE |
2068 | if (ctx == NULL) |
2069 | ctx = new_ctx = BN_CTX_new(); |
2070 | #endif |
2071 | if (ctx == NULL) |
2072 | return 0; |
2073 | |
2074 | BN_CTX_start(ctx); |
2075 | x = BN_CTX_get(ctx); |
2076 | y = BN_CTX_get(ctx); |
2077 | if (y == NULL) |
2078 | goto err; |
2079 | /* get the generator */ |
2080 | if (group->generator == NULL) |
2081 | goto err; |
2082 | generator = EC_POINT_new(group); |
2083 | if (generator == NULL) |
2084 | goto err; |
2085 | BN_bin2bn(nistp521_curve_params[3], sizeof(felem_bytearray), x); |
2086 | BN_bin2bn(nistp521_curve_params[4], sizeof(felem_bytearray), y); |
2087 | if (!EC_POINT_set_affine_coordinates(group, generator, x, y, ctx)) |
2088 | goto err; |
2089 | if ((pre = nistp521_pre_comp_new()) == NULL) |
2090 | goto err; |
2091 | /* |
2092 | * if the generator is the standard one, use built-in precomputation |
2093 | */ |
2094 | if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) { |
2095 | memcpy(pre->g_pre_comp, gmul, sizeof(pre->g_pre_comp)); |
2096 | goto done; |
2097 | } |
2098 | if ((!BN_to_felem(pre->g_pre_comp[1][0], group->generator->X)) || |
2099 | (!BN_to_felem(pre->g_pre_comp[1][1], group->generator->Y)) || |
2100 | (!BN_to_felem(pre->g_pre_comp[1][2], group->generator->Z))) |
2101 | goto err; |
2102 | /* compute 2^130*G, 2^260*G, 2^390*G */ |
2103 | for (i = 1; i <= 4; i <<= 1) { |
2104 | point_double(pre->g_pre_comp[2 * i][0], pre->g_pre_comp[2 * i][1], |
2105 | pre->g_pre_comp[2 * i][2], pre->g_pre_comp[i][0], |
2106 | pre->g_pre_comp[i][1], pre->g_pre_comp[i][2]); |
2107 | for (j = 0; j < 129; ++j) { |
2108 | point_double(pre->g_pre_comp[2 * i][0], |
2109 | pre->g_pre_comp[2 * i][1], |
2110 | pre->g_pre_comp[2 * i][2], |
2111 | pre->g_pre_comp[2 * i][0], |
2112 | pre->g_pre_comp[2 * i][1], |
2113 | pre->g_pre_comp[2 * i][2]); |
2114 | } |
2115 | } |
2116 | /* g_pre_comp[0] is the point at infinity */ |
2117 | memset(pre->g_pre_comp[0], 0, sizeof(pre->g_pre_comp[0])); |
2118 | /* the remaining multiples */ |
2119 | /* 2^130*G + 2^260*G */ |
2120 | point_add(pre->g_pre_comp[6][0], pre->g_pre_comp[6][1], |
2121 | pre->g_pre_comp[6][2], pre->g_pre_comp[4][0], |
2122 | pre->g_pre_comp[4][1], pre->g_pre_comp[4][2], |
2123 | 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1], |
2124 | pre->g_pre_comp[2][2]); |
2125 | /* 2^130*G + 2^390*G */ |
2126 | point_add(pre->g_pre_comp[10][0], pre->g_pre_comp[10][1], |
2127 | pre->g_pre_comp[10][2], pre->g_pre_comp[8][0], |
2128 | pre->g_pre_comp[8][1], pre->g_pre_comp[8][2], |
2129 | 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1], |
2130 | pre->g_pre_comp[2][2]); |
2131 | /* 2^260*G + 2^390*G */ |
2132 | point_add(pre->g_pre_comp[12][0], pre->g_pre_comp[12][1], |
2133 | pre->g_pre_comp[12][2], pre->g_pre_comp[8][0], |
2134 | pre->g_pre_comp[8][1], pre->g_pre_comp[8][2], |
2135 | 0, pre->g_pre_comp[4][0], pre->g_pre_comp[4][1], |
2136 | pre->g_pre_comp[4][2]); |
2137 | /* 2^130*G + 2^260*G + 2^390*G */ |
2138 | point_add(pre->g_pre_comp[14][0], pre->g_pre_comp[14][1], |
2139 | pre->g_pre_comp[14][2], pre->g_pre_comp[12][0], |
2140 | pre->g_pre_comp[12][1], pre->g_pre_comp[12][2], |
2141 | 0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1], |
2142 | pre->g_pre_comp[2][2]); |
2143 | for (i = 1; i < 8; ++i) { |
2144 | /* odd multiples: add G */ |
2145 | point_add(pre->g_pre_comp[2 * i + 1][0], |
2146 | pre->g_pre_comp[2 * i + 1][1], |
2147 | pre->g_pre_comp[2 * i + 1][2], pre->g_pre_comp[2 * i][0], |
2148 | pre->g_pre_comp[2 * i][1], pre->g_pre_comp[2 * i][2], 0, |
2149 | pre->g_pre_comp[1][0], pre->g_pre_comp[1][1], |
2150 | pre->g_pre_comp[1][2]); |
2151 | } |
2152 | make_points_affine(15, &(pre->g_pre_comp[1]), tmp_felems); |
2153 | |
2154 | done: |
2155 | SETPRECOMP(group, nistp521, pre); |
2156 | ret = 1; |
2157 | pre = NULL; |
2158 | err: |
2159 | BN_CTX_end(ctx); |
2160 | EC_POINT_free(generator); |
2161 | #ifndef FIPS_MODE |
2162 | BN_CTX_free(new_ctx); |
2163 | #endif |
2164 | EC_nistp521_pre_comp_free(pre); |
2165 | return ret; |
2166 | } |
2167 | |
2168 | int ec_GFp_nistp521_have_precompute_mult(const EC_GROUP *group) |
2169 | { |
2170 | return HAVEPRECOMP(group, nistp521); |
2171 | } |
2172 | |
2173 | #endif |
2174 | |