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-256 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/opensslconf.h> |
35 | #ifdef OPENSSL_NO_EC_NISTP_64_GCC_128 |
36 | NON_EMPTY_TRANSLATION_UNIT |
37 | #else |
38 | |
39 | # include <stdint.h> |
40 | # include <string.h> |
41 | # include <openssl/err.h> |
42 | # include "ec_local.h" |
43 | |
44 | # if defined(__SIZEOF_INT128__) && __SIZEOF_INT128__==16 |
45 | /* even with gcc, the typedef won't work for 32-bit platforms */ |
46 | typedef __uint128_t uint128_t; /* nonstandard; implemented by gcc on 64-bit |
47 | * platforms */ |
48 | typedef __int128_t int128_t; |
49 | # else |
50 | # error "Your compiler doesn't appear to support 128-bit integer types" |
51 | # endif |
52 | |
53 | typedef uint8_t u8; |
54 | typedef uint32_t u32; |
55 | typedef uint64_t u64; |
56 | |
57 | /* |
58 | * The underlying field. P256 operates over GF(2^256-2^224+2^192+2^96-1). We |
59 | * can serialise an element of this field into 32 bytes. We call this an |
60 | * felem_bytearray. |
61 | */ |
62 | |
63 | typedef u8 felem_bytearray[32]; |
64 | |
65 | /* |
66 | * These are the parameters of P256, taken from FIPS 186-3, page 86. These |
67 | * values are big-endian. |
68 | */ |
69 | static const felem_bytearray nistp256_curve_params[5] = { |
70 | {0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01, /* p */ |
71 | 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, |
72 | 0x00, 0x00, 0x00, 0x00, 0xff, 0xff, 0xff, 0xff, |
73 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff}, |
74 | {0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01, /* a = -3 */ |
75 | 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, |
76 | 0x00, 0x00, 0x00, 0x00, 0xff, 0xff, 0xff, 0xff, |
77 | 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xfc}, /* b */ |
78 | {0x5a, 0xc6, 0x35, 0xd8, 0xaa, 0x3a, 0x93, 0xe7, |
79 | 0xb3, 0xeb, 0xbd, 0x55, 0x76, 0x98, 0x86, 0xbc, |
80 | 0x65, 0x1d, 0x06, 0xb0, 0xcc, 0x53, 0xb0, 0xf6, |
81 | 0x3b, 0xce, 0x3c, 0x3e, 0x27, 0xd2, 0x60, 0x4b}, |
82 | {0x6b, 0x17, 0xd1, 0xf2, 0xe1, 0x2c, 0x42, 0x47, /* x */ |
83 | 0xf8, 0xbc, 0xe6, 0xe5, 0x63, 0xa4, 0x40, 0xf2, |
84 | 0x77, 0x03, 0x7d, 0x81, 0x2d, 0xeb, 0x33, 0xa0, |
85 | 0xf4, 0xa1, 0x39, 0x45, 0xd8, 0x98, 0xc2, 0x96}, |
86 | {0x4f, 0xe3, 0x42, 0xe2, 0xfe, 0x1a, 0x7f, 0x9b, /* y */ |
87 | 0x8e, 0xe7, 0xeb, 0x4a, 0x7c, 0x0f, 0x9e, 0x16, |
88 | 0x2b, 0xce, 0x33, 0x57, 0x6b, 0x31, 0x5e, 0xce, |
89 | 0xcb, 0xb6, 0x40, 0x68, 0x37, 0xbf, 0x51, 0xf5} |
90 | }; |
91 | |
92 | /*- |
93 | * The representation of field elements. |
94 | * ------------------------------------ |
95 | * |
96 | * We represent field elements with either four 128-bit values, eight 128-bit |
97 | * values, or four 64-bit values. The field element represented is: |
98 | * v[0]*2^0 + v[1]*2^64 + v[2]*2^128 + v[3]*2^192 (mod p) |
99 | * or: |
100 | * v[0]*2^0 + v[1]*2^64 + v[2]*2^128 + ... + v[8]*2^512 (mod p) |
101 | * |
102 | * 128-bit values are called 'limbs'. Since the limbs are spaced only 64 bits |
103 | * apart, but are 128-bits wide, the most significant bits of each limb overlap |
104 | * with the least significant bits of the next. |
105 | * |
106 | * A field element with four limbs is an 'felem'. One with eight limbs is a |
107 | * 'longfelem' |
108 | * |
109 | * A field element with four, 64-bit values is called a 'smallfelem'. Small |
110 | * values are used as intermediate values before multiplication. |
111 | */ |
112 | |
113 | # define NLIMBS 4 |
114 | |
115 | typedef uint128_t limb; |
116 | typedef limb felem[NLIMBS]; |
117 | typedef limb longfelem[NLIMBS * 2]; |
118 | typedef u64 smallfelem[NLIMBS]; |
119 | |
120 | /* This is the value of the prime as four 64-bit words, little-endian. */ |
121 | static const u64 kPrime[4] = |
122 | { 0xfffffffffffffffful, 0xffffffff, 0, 0xffffffff00000001ul }; |
123 | static const u64 bottom63bits = 0x7ffffffffffffffful; |
124 | |
125 | /* |
126 | * bin32_to_felem takes a little-endian byte array and converts it into felem |
127 | * form. This assumes that the CPU is little-endian. |
128 | */ |
129 | static void bin32_to_felem(felem out, const u8 in[32]) |
130 | { |
131 | out[0] = *((u64 *)&in[0]); |
132 | out[1] = *((u64 *)&in[8]); |
133 | out[2] = *((u64 *)&in[16]); |
134 | out[3] = *((u64 *)&in[24]); |
135 | } |
136 | |
137 | /* |
138 | * smallfelem_to_bin32 takes a smallfelem and serialises into a little |
139 | * endian, 32 byte array. This assumes that the CPU is little-endian. |
140 | */ |
141 | static void smallfelem_to_bin32(u8 out[32], const smallfelem in) |
142 | { |
143 | *((u64 *)&out[0]) = in[0]; |
144 | *((u64 *)&out[8]) = in[1]; |
145 | *((u64 *)&out[16]) = in[2]; |
146 | *((u64 *)&out[24]) = in[3]; |
147 | } |
148 | |
149 | /* BN_to_felem converts an OpenSSL BIGNUM into an felem */ |
150 | static int BN_to_felem(felem out, const BIGNUM *bn) |
151 | { |
152 | felem_bytearray b_out; |
153 | int num_bytes; |
154 | |
155 | if (BN_is_negative(bn)) { |
156 | ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE); |
157 | return 0; |
158 | } |
159 | num_bytes = BN_bn2lebinpad(bn, b_out, sizeof(b_out)); |
160 | if (num_bytes < 0) { |
161 | ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE); |
162 | return 0; |
163 | } |
164 | bin32_to_felem(out, b_out); |
165 | return 1; |
166 | } |
167 | |
168 | /* felem_to_BN converts an felem into an OpenSSL BIGNUM */ |
169 | static BIGNUM *smallfelem_to_BN(BIGNUM *out, const smallfelem in) |
170 | { |
171 | felem_bytearray b_out; |
172 | smallfelem_to_bin32(b_out, in); |
173 | return BN_lebin2bn(b_out, sizeof(b_out), out); |
174 | } |
175 | |
176 | /*- |
177 | * Field operations |
178 | * ---------------- |
179 | */ |
180 | |
181 | static void smallfelem_one(smallfelem out) |
182 | { |
183 | out[0] = 1; |
184 | out[1] = 0; |
185 | out[2] = 0; |
186 | out[3] = 0; |
187 | } |
188 | |
189 | static void smallfelem_assign(smallfelem out, const smallfelem in) |
190 | { |
191 | out[0] = in[0]; |
192 | out[1] = in[1]; |
193 | out[2] = in[2]; |
194 | out[3] = in[3]; |
195 | } |
196 | |
197 | static void felem_assign(felem out, const felem in) |
198 | { |
199 | out[0] = in[0]; |
200 | out[1] = in[1]; |
201 | out[2] = in[2]; |
202 | out[3] = in[3]; |
203 | } |
204 | |
205 | /* felem_sum sets out = out + in. */ |
206 | static void felem_sum(felem out, const felem in) |
207 | { |
208 | out[0] += in[0]; |
209 | out[1] += in[1]; |
210 | out[2] += in[2]; |
211 | out[3] += in[3]; |
212 | } |
213 | |
214 | /* felem_small_sum sets out = out + in. */ |
215 | static void felem_small_sum(felem out, const smallfelem in) |
216 | { |
217 | out[0] += in[0]; |
218 | out[1] += in[1]; |
219 | out[2] += in[2]; |
220 | out[3] += in[3]; |
221 | } |
222 | |
223 | /* felem_scalar sets out = out * scalar */ |
224 | static void felem_scalar(felem out, const u64 scalar) |
225 | { |
226 | out[0] *= scalar; |
227 | out[1] *= scalar; |
228 | out[2] *= scalar; |
229 | out[3] *= scalar; |
230 | } |
231 | |
232 | /* longfelem_scalar sets out = out * scalar */ |
233 | static void longfelem_scalar(longfelem out, const u64 scalar) |
234 | { |
235 | out[0] *= scalar; |
236 | out[1] *= scalar; |
237 | out[2] *= scalar; |
238 | out[3] *= scalar; |
239 | out[4] *= scalar; |
240 | out[5] *= scalar; |
241 | out[6] *= scalar; |
242 | out[7] *= scalar; |
243 | } |
244 | |
245 | # define two105m41m9 (((limb)1) << 105) - (((limb)1) << 41) - (((limb)1) << 9) |
246 | # define two105 (((limb)1) << 105) |
247 | # define two105m41p9 (((limb)1) << 105) - (((limb)1) << 41) + (((limb)1) << 9) |
248 | |
249 | /* zero105 is 0 mod p */ |
250 | static const felem zero105 = |
251 | { two105m41m9, two105, two105m41p9, two105m41p9 }; |
252 | |
253 | /*- |
254 | * smallfelem_neg sets |out| to |-small| |
255 | * On exit: |
256 | * out[i] < out[i] + 2^105 |
257 | */ |
258 | static void smallfelem_neg(felem out, const smallfelem small) |
259 | { |
260 | /* In order to prevent underflow, we subtract from 0 mod p. */ |
261 | out[0] = zero105[0] - small[0]; |
262 | out[1] = zero105[1] - small[1]; |
263 | out[2] = zero105[2] - small[2]; |
264 | out[3] = zero105[3] - small[3]; |
265 | } |
266 | |
267 | /*- |
268 | * felem_diff subtracts |in| from |out| |
269 | * On entry: |
270 | * in[i] < 2^104 |
271 | * On exit: |
272 | * out[i] < out[i] + 2^105 |
273 | */ |
274 | static void felem_diff(felem out, const felem in) |
275 | { |
276 | /* |
277 | * In order to prevent underflow, we add 0 mod p before subtracting. |
278 | */ |
279 | out[0] += zero105[0]; |
280 | out[1] += zero105[1]; |
281 | out[2] += zero105[2]; |
282 | out[3] += zero105[3]; |
283 | |
284 | out[0] -= in[0]; |
285 | out[1] -= in[1]; |
286 | out[2] -= in[2]; |
287 | out[3] -= in[3]; |
288 | } |
289 | |
290 | # define two107m43m11 (((limb)1) << 107) - (((limb)1) << 43) - (((limb)1) << 11) |
291 | # define two107 (((limb)1) << 107) |
292 | # define two107m43p11 (((limb)1) << 107) - (((limb)1) << 43) + (((limb)1) << 11) |
293 | |
294 | /* zero107 is 0 mod p */ |
295 | static const felem zero107 = |
296 | { two107m43m11, two107, two107m43p11, two107m43p11 }; |
297 | |
298 | /*- |
299 | * An alternative felem_diff for larger inputs |in| |
300 | * felem_diff_zero107 subtracts |in| from |out| |
301 | * On entry: |
302 | * in[i] < 2^106 |
303 | * On exit: |
304 | * out[i] < out[i] + 2^107 |
305 | */ |
306 | static void felem_diff_zero107(felem out, const felem in) |
307 | { |
308 | /* |
309 | * In order to prevent underflow, we add 0 mod p before subtracting. |
310 | */ |
311 | out[0] += zero107[0]; |
312 | out[1] += zero107[1]; |
313 | out[2] += zero107[2]; |
314 | out[3] += zero107[3]; |
315 | |
316 | out[0] -= in[0]; |
317 | out[1] -= in[1]; |
318 | out[2] -= in[2]; |
319 | out[3] -= in[3]; |
320 | } |
321 | |
322 | /*- |
323 | * longfelem_diff subtracts |in| from |out| |
324 | * On entry: |
325 | * in[i] < 7*2^67 |
326 | * On exit: |
327 | * out[i] < out[i] + 2^70 + 2^40 |
328 | */ |
329 | static void longfelem_diff(longfelem out, const longfelem in) |
330 | { |
331 | static const limb two70m8p6 = |
332 | (((limb) 1) << 70) - (((limb) 1) << 8) + (((limb) 1) << 6); |
333 | static const limb two70p40 = (((limb) 1) << 70) + (((limb) 1) << 40); |
334 | static const limb two70 = (((limb) 1) << 70); |
335 | static const limb two70m40m38p6 = |
336 | (((limb) 1) << 70) - (((limb) 1) << 40) - (((limb) 1) << 38) + |
337 | (((limb) 1) << 6); |
338 | static const limb two70m6 = (((limb) 1) << 70) - (((limb) 1) << 6); |
339 | |
340 | /* add 0 mod p to avoid underflow */ |
341 | out[0] += two70m8p6; |
342 | out[1] += two70p40; |
343 | out[2] += two70; |
344 | out[3] += two70m40m38p6; |
345 | out[4] += two70m6; |
346 | out[5] += two70m6; |
347 | out[6] += two70m6; |
348 | out[7] += two70m6; |
349 | |
350 | /* in[i] < 7*2^67 < 2^70 - 2^40 - 2^38 + 2^6 */ |
351 | out[0] -= in[0]; |
352 | out[1] -= in[1]; |
353 | out[2] -= in[2]; |
354 | out[3] -= in[3]; |
355 | out[4] -= in[4]; |
356 | out[5] -= in[5]; |
357 | out[6] -= in[6]; |
358 | out[7] -= in[7]; |
359 | } |
360 | |
361 | # define two64m0 (((limb)1) << 64) - 1 |
362 | # define two110p32m0 (((limb)1) << 110) + (((limb)1) << 32) - 1 |
363 | # define two64m46 (((limb)1) << 64) - (((limb)1) << 46) |
364 | # define two64m32 (((limb)1) << 64) - (((limb)1) << 32) |
365 | |
366 | /* zero110 is 0 mod p */ |
367 | static const felem zero110 = { two64m0, two110p32m0, two64m46, two64m32 }; |
368 | |
369 | /*- |
370 | * felem_shrink converts an felem into a smallfelem. The result isn't quite |
371 | * minimal as the value may be greater than p. |
372 | * |
373 | * On entry: |
374 | * in[i] < 2^109 |
375 | * On exit: |
376 | * out[i] < 2^64 |
377 | */ |
378 | static void felem_shrink(smallfelem out, const felem in) |
379 | { |
380 | felem tmp; |
381 | u64 a, b, mask; |
382 | u64 high, low; |
383 | static const u64 kPrime3Test = 0x7fffffff00000001ul; /* 2^63 - 2^32 + 1 */ |
384 | |
385 | /* Carry 2->3 */ |
386 | tmp[3] = zero110[3] + in[3] + ((u64)(in[2] >> 64)); |
387 | /* tmp[3] < 2^110 */ |
388 | |
389 | tmp[2] = zero110[2] + (u64)in[2]; |
390 | tmp[0] = zero110[0] + in[0]; |
391 | tmp[1] = zero110[1] + in[1]; |
392 | /* tmp[0] < 2**110, tmp[1] < 2^111, tmp[2] < 2**65 */ |
393 | |
394 | /* |
395 | * We perform two partial reductions where we eliminate the high-word of |
396 | * tmp[3]. We don't update the other words till the end. |
397 | */ |
398 | a = tmp[3] >> 64; /* a < 2^46 */ |
399 | tmp[3] = (u64)tmp[3]; |
400 | tmp[3] -= a; |
401 | tmp[3] += ((limb) a) << 32; |
402 | /* tmp[3] < 2^79 */ |
403 | |
404 | b = a; |
405 | a = tmp[3] >> 64; /* a < 2^15 */ |
406 | b += a; /* b < 2^46 + 2^15 < 2^47 */ |
407 | tmp[3] = (u64)tmp[3]; |
408 | tmp[3] -= a; |
409 | tmp[3] += ((limb) a) << 32; |
410 | /* tmp[3] < 2^64 + 2^47 */ |
411 | |
412 | /* |
413 | * This adjusts the other two words to complete the two partial |
414 | * reductions. |
415 | */ |
416 | tmp[0] += b; |
417 | tmp[1] -= (((limb) b) << 32); |
418 | |
419 | /* |
420 | * In order to make space in tmp[3] for the carry from 2 -> 3, we |
421 | * conditionally subtract kPrime if tmp[3] is large enough. |
422 | */ |
423 | high = (u64)(tmp[3] >> 64); |
424 | /* As tmp[3] < 2^65, high is either 1 or 0 */ |
425 | high = 0 - high; |
426 | /*- |
427 | * high is: |
428 | * all ones if the high word of tmp[3] is 1 |
429 | * all zeros if the high word of tmp[3] if 0 |
430 | */ |
431 | low = (u64)tmp[3]; |
432 | mask = 0 - (low >> 63); |
433 | /*- |
434 | * mask is: |
435 | * all ones if the MSB of low is 1 |
436 | * all zeros if the MSB of low if 0 |
437 | */ |
438 | low &= bottom63bits; |
439 | low -= kPrime3Test; |
440 | /* if low was greater than kPrime3Test then the MSB is zero */ |
441 | low = ~low; |
442 | low = 0 - (low >> 63); |
443 | /*- |
444 | * low is: |
445 | * all ones if low was > kPrime3Test |
446 | * all zeros if low was <= kPrime3Test |
447 | */ |
448 | mask = (mask & low) | high; |
449 | tmp[0] -= mask & kPrime[0]; |
450 | tmp[1] -= mask & kPrime[1]; |
451 | /* kPrime[2] is zero, so omitted */ |
452 | tmp[3] -= mask & kPrime[3]; |
453 | /* tmp[3] < 2**64 - 2**32 + 1 */ |
454 | |
455 | tmp[1] += ((u64)(tmp[0] >> 64)); |
456 | tmp[0] = (u64)tmp[0]; |
457 | tmp[2] += ((u64)(tmp[1] >> 64)); |
458 | tmp[1] = (u64)tmp[1]; |
459 | tmp[3] += ((u64)(tmp[2] >> 64)); |
460 | tmp[2] = (u64)tmp[2]; |
461 | /* tmp[i] < 2^64 */ |
462 | |
463 | out[0] = tmp[0]; |
464 | out[1] = tmp[1]; |
465 | out[2] = tmp[2]; |
466 | out[3] = tmp[3]; |
467 | } |
468 | |
469 | /* smallfelem_expand converts a smallfelem to an felem */ |
470 | static void smallfelem_expand(felem out, const smallfelem in) |
471 | { |
472 | out[0] = in[0]; |
473 | out[1] = in[1]; |
474 | out[2] = in[2]; |
475 | out[3] = in[3]; |
476 | } |
477 | |
478 | /*- |
479 | * smallfelem_square sets |out| = |small|^2 |
480 | * On entry: |
481 | * small[i] < 2^64 |
482 | * On exit: |
483 | * out[i] < 7 * 2^64 < 2^67 |
484 | */ |
485 | static void smallfelem_square(longfelem out, const smallfelem small) |
486 | { |
487 | limb a; |
488 | u64 high, low; |
489 | |
490 | a = ((uint128_t) small[0]) * small[0]; |
491 | low = a; |
492 | high = a >> 64; |
493 | out[0] = low; |
494 | out[1] = high; |
495 | |
496 | a = ((uint128_t) small[0]) * small[1]; |
497 | low = a; |
498 | high = a >> 64; |
499 | out[1] += low; |
500 | out[1] += low; |
501 | out[2] = high; |
502 | |
503 | a = ((uint128_t) small[0]) * small[2]; |
504 | low = a; |
505 | high = a >> 64; |
506 | out[2] += low; |
507 | out[2] *= 2; |
508 | out[3] = high; |
509 | |
510 | a = ((uint128_t) small[0]) * small[3]; |
511 | low = a; |
512 | high = a >> 64; |
513 | out[3] += low; |
514 | out[4] = high; |
515 | |
516 | a = ((uint128_t) small[1]) * small[2]; |
517 | low = a; |
518 | high = a >> 64; |
519 | out[3] += low; |
520 | out[3] *= 2; |
521 | out[4] += high; |
522 | |
523 | a = ((uint128_t) small[1]) * small[1]; |
524 | low = a; |
525 | high = a >> 64; |
526 | out[2] += low; |
527 | out[3] += high; |
528 | |
529 | a = ((uint128_t) small[1]) * small[3]; |
530 | low = a; |
531 | high = a >> 64; |
532 | out[4] += low; |
533 | out[4] *= 2; |
534 | out[5] = high; |
535 | |
536 | a = ((uint128_t) small[2]) * small[3]; |
537 | low = a; |
538 | high = a >> 64; |
539 | out[5] += low; |
540 | out[5] *= 2; |
541 | out[6] = high; |
542 | out[6] += high; |
543 | |
544 | a = ((uint128_t) small[2]) * small[2]; |
545 | low = a; |
546 | high = a >> 64; |
547 | out[4] += low; |
548 | out[5] += high; |
549 | |
550 | a = ((uint128_t) small[3]) * small[3]; |
551 | low = a; |
552 | high = a >> 64; |
553 | out[6] += low; |
554 | out[7] = high; |
555 | } |
556 | |
557 | /*- |
558 | * felem_square sets |out| = |in|^2 |
559 | * On entry: |
560 | * in[i] < 2^109 |
561 | * On exit: |
562 | * out[i] < 7 * 2^64 < 2^67 |
563 | */ |
564 | static void felem_square(longfelem out, const felem in) |
565 | { |
566 | u64 small[4]; |
567 | felem_shrink(small, in); |
568 | smallfelem_square(out, small); |
569 | } |
570 | |
571 | /*- |
572 | * smallfelem_mul sets |out| = |small1| * |small2| |
573 | * On entry: |
574 | * small1[i] < 2^64 |
575 | * small2[i] < 2^64 |
576 | * On exit: |
577 | * out[i] < 7 * 2^64 < 2^67 |
578 | */ |
579 | static void smallfelem_mul(longfelem out, const smallfelem small1, |
580 | const smallfelem small2) |
581 | { |
582 | limb a; |
583 | u64 high, low; |
584 | |
585 | a = ((uint128_t) small1[0]) * small2[0]; |
586 | low = a; |
587 | high = a >> 64; |
588 | out[0] = low; |
589 | out[1] = high; |
590 | |
591 | a = ((uint128_t) small1[0]) * small2[1]; |
592 | low = a; |
593 | high = a >> 64; |
594 | out[1] += low; |
595 | out[2] = high; |
596 | |
597 | a = ((uint128_t) small1[1]) * small2[0]; |
598 | low = a; |
599 | high = a >> 64; |
600 | out[1] += low; |
601 | out[2] += high; |
602 | |
603 | a = ((uint128_t) small1[0]) * small2[2]; |
604 | low = a; |
605 | high = a >> 64; |
606 | out[2] += low; |
607 | out[3] = high; |
608 | |
609 | a = ((uint128_t) small1[1]) * small2[1]; |
610 | low = a; |
611 | high = a >> 64; |
612 | out[2] += low; |
613 | out[3] += high; |
614 | |
615 | a = ((uint128_t) small1[2]) * small2[0]; |
616 | low = a; |
617 | high = a >> 64; |
618 | out[2] += low; |
619 | out[3] += high; |
620 | |
621 | a = ((uint128_t) small1[0]) * small2[3]; |
622 | low = a; |
623 | high = a >> 64; |
624 | out[3] += low; |
625 | out[4] = high; |
626 | |
627 | a = ((uint128_t) small1[1]) * small2[2]; |
628 | low = a; |
629 | high = a >> 64; |
630 | out[3] += low; |
631 | out[4] += high; |
632 | |
633 | a = ((uint128_t) small1[2]) * small2[1]; |
634 | low = a; |
635 | high = a >> 64; |
636 | out[3] += low; |
637 | out[4] += high; |
638 | |
639 | a = ((uint128_t) small1[3]) * small2[0]; |
640 | low = a; |
641 | high = a >> 64; |
642 | out[3] += low; |
643 | out[4] += high; |
644 | |
645 | a = ((uint128_t) small1[1]) * small2[3]; |
646 | low = a; |
647 | high = a >> 64; |
648 | out[4] += low; |
649 | out[5] = high; |
650 | |
651 | a = ((uint128_t) small1[2]) * small2[2]; |
652 | low = a; |
653 | high = a >> 64; |
654 | out[4] += low; |
655 | out[5] += high; |
656 | |
657 | a = ((uint128_t) small1[3]) * small2[1]; |
658 | low = a; |
659 | high = a >> 64; |
660 | out[4] += low; |
661 | out[5] += high; |
662 | |
663 | a = ((uint128_t) small1[2]) * small2[3]; |
664 | low = a; |
665 | high = a >> 64; |
666 | out[5] += low; |
667 | out[6] = high; |
668 | |
669 | a = ((uint128_t) small1[3]) * small2[2]; |
670 | low = a; |
671 | high = a >> 64; |
672 | out[5] += low; |
673 | out[6] += high; |
674 | |
675 | a = ((uint128_t) small1[3]) * small2[3]; |
676 | low = a; |
677 | high = a >> 64; |
678 | out[6] += low; |
679 | out[7] = high; |
680 | } |
681 | |
682 | /*- |
683 | * felem_mul sets |out| = |in1| * |in2| |
684 | * On entry: |
685 | * in1[i] < 2^109 |
686 | * in2[i] < 2^109 |
687 | * On exit: |
688 | * out[i] < 7 * 2^64 < 2^67 |
689 | */ |
690 | static void felem_mul(longfelem out, const felem in1, const felem in2) |
691 | { |
692 | smallfelem small1, small2; |
693 | felem_shrink(small1, in1); |
694 | felem_shrink(small2, in2); |
695 | smallfelem_mul(out, small1, small2); |
696 | } |
697 | |
698 | /*- |
699 | * felem_small_mul sets |out| = |small1| * |in2| |
700 | * On entry: |
701 | * small1[i] < 2^64 |
702 | * in2[i] < 2^109 |
703 | * On exit: |
704 | * out[i] < 7 * 2^64 < 2^67 |
705 | */ |
706 | static void felem_small_mul(longfelem out, const smallfelem small1, |
707 | const felem in2) |
708 | { |
709 | smallfelem small2; |
710 | felem_shrink(small2, in2); |
711 | smallfelem_mul(out, small1, small2); |
712 | } |
713 | |
714 | # define two100m36m4 (((limb)1) << 100) - (((limb)1) << 36) - (((limb)1) << 4) |
715 | # define two100 (((limb)1) << 100) |
716 | # define two100m36p4 (((limb)1) << 100) - (((limb)1) << 36) + (((limb)1) << 4) |
717 | /* zero100 is 0 mod p */ |
718 | static const felem zero100 = |
719 | { two100m36m4, two100, two100m36p4, two100m36p4 }; |
720 | |
721 | /*- |
722 | * Internal function for the different flavours of felem_reduce. |
723 | * felem_reduce_ reduces the higher coefficients in[4]-in[7]. |
724 | * On entry: |
725 | * out[0] >= in[6] + 2^32*in[6] + in[7] + 2^32*in[7] |
726 | * out[1] >= in[7] + 2^32*in[4] |
727 | * out[2] >= in[5] + 2^32*in[5] |
728 | * out[3] >= in[4] + 2^32*in[5] + 2^32*in[6] |
729 | * On exit: |
730 | * out[0] <= out[0] + in[4] + 2^32*in[5] |
731 | * out[1] <= out[1] + in[5] + 2^33*in[6] |
732 | * out[2] <= out[2] + in[7] + 2*in[6] + 2^33*in[7] |
733 | * out[3] <= out[3] + 2^32*in[4] + 3*in[7] |
734 | */ |
735 | static void felem_reduce_(felem out, const longfelem in) |
736 | { |
737 | int128_t c; |
738 | /* combine common terms from below */ |
739 | c = in[4] + (in[5] << 32); |
740 | out[0] += c; |
741 | out[3] -= c; |
742 | |
743 | c = in[5] - in[7]; |
744 | out[1] += c; |
745 | out[2] -= c; |
746 | |
747 | /* the remaining terms */ |
748 | /* 256: [(0,1),(96,-1),(192,-1),(224,1)] */ |
749 | out[1] -= (in[4] << 32); |
750 | out[3] += (in[4] << 32); |
751 | |
752 | /* 320: [(32,1),(64,1),(128,-1),(160,-1),(224,-1)] */ |
753 | out[2] -= (in[5] << 32); |
754 | |
755 | /* 384: [(0,-1),(32,-1),(96,2),(128,2),(224,-1)] */ |
756 | out[0] -= in[6]; |
757 | out[0] -= (in[6] << 32); |
758 | out[1] += (in[6] << 33); |
759 | out[2] += (in[6] * 2); |
760 | out[3] -= (in[6] << 32); |
761 | |
762 | /* 448: [(0,-1),(32,-1),(64,-1),(128,1),(160,2),(192,3)] */ |
763 | out[0] -= in[7]; |
764 | out[0] -= (in[7] << 32); |
765 | out[2] += (in[7] << 33); |
766 | out[3] += (in[7] * 3); |
767 | } |
768 | |
769 | /*- |
770 | * felem_reduce converts a longfelem into an felem. |
771 | * To be called directly after felem_square or felem_mul. |
772 | * On entry: |
773 | * in[0] < 2^64, in[1] < 3*2^64, in[2] < 5*2^64, in[3] < 7*2^64 |
774 | * in[4] < 7*2^64, in[5] < 5*2^64, in[6] < 3*2^64, in[7] < 2*64 |
775 | * On exit: |
776 | * out[i] < 2^101 |
777 | */ |
778 | static void felem_reduce(felem out, const longfelem in) |
779 | { |
780 | out[0] = zero100[0] + in[0]; |
781 | out[1] = zero100[1] + in[1]; |
782 | out[2] = zero100[2] + in[2]; |
783 | out[3] = zero100[3] + in[3]; |
784 | |
785 | felem_reduce_(out, in); |
786 | |
787 | /*- |
788 | * out[0] > 2^100 - 2^36 - 2^4 - 3*2^64 - 3*2^96 - 2^64 - 2^96 > 0 |
789 | * out[1] > 2^100 - 2^64 - 7*2^96 > 0 |
790 | * out[2] > 2^100 - 2^36 + 2^4 - 5*2^64 - 5*2^96 > 0 |
791 | * out[3] > 2^100 - 2^36 + 2^4 - 7*2^64 - 5*2^96 - 3*2^96 > 0 |
792 | * |
793 | * out[0] < 2^100 + 2^64 + 7*2^64 + 5*2^96 < 2^101 |
794 | * out[1] < 2^100 + 3*2^64 + 5*2^64 + 3*2^97 < 2^101 |
795 | * out[2] < 2^100 + 5*2^64 + 2^64 + 3*2^65 + 2^97 < 2^101 |
796 | * out[3] < 2^100 + 7*2^64 + 7*2^96 + 3*2^64 < 2^101 |
797 | */ |
798 | } |
799 | |
800 | /*- |
801 | * felem_reduce_zero105 converts a larger longfelem into an felem. |
802 | * On entry: |
803 | * in[0] < 2^71 |
804 | * On exit: |
805 | * out[i] < 2^106 |
806 | */ |
807 | static void felem_reduce_zero105(felem out, const longfelem in) |
808 | { |
809 | out[0] = zero105[0] + in[0]; |
810 | out[1] = zero105[1] + in[1]; |
811 | out[2] = zero105[2] + in[2]; |
812 | out[3] = zero105[3] + in[3]; |
813 | |
814 | felem_reduce_(out, in); |
815 | |
816 | /*- |
817 | * out[0] > 2^105 - 2^41 - 2^9 - 2^71 - 2^103 - 2^71 - 2^103 > 0 |
818 | * out[1] > 2^105 - 2^71 - 2^103 > 0 |
819 | * out[2] > 2^105 - 2^41 + 2^9 - 2^71 - 2^103 > 0 |
820 | * out[3] > 2^105 - 2^41 + 2^9 - 2^71 - 2^103 - 2^103 > 0 |
821 | * |
822 | * out[0] < 2^105 + 2^71 + 2^71 + 2^103 < 2^106 |
823 | * out[1] < 2^105 + 2^71 + 2^71 + 2^103 < 2^106 |
824 | * out[2] < 2^105 + 2^71 + 2^71 + 2^71 + 2^103 < 2^106 |
825 | * out[3] < 2^105 + 2^71 + 2^103 + 2^71 < 2^106 |
826 | */ |
827 | } |
828 | |
829 | /* |
830 | * subtract_u64 sets *result = *result - v and *carry to one if the |
831 | * subtraction underflowed. |
832 | */ |
833 | static void subtract_u64(u64 *result, u64 *carry, u64 v) |
834 | { |
835 | uint128_t r = *result; |
836 | r -= v; |
837 | *carry = (r >> 64) & 1; |
838 | *result = (u64)r; |
839 | } |
840 | |
841 | /* |
842 | * felem_contract converts |in| to its unique, minimal representation. On |
843 | * entry: in[i] < 2^109 |
844 | */ |
845 | static void felem_contract(smallfelem out, const felem in) |
846 | { |
847 | unsigned i; |
848 | u64 all_equal_so_far = 0, result = 0, carry; |
849 | |
850 | felem_shrink(out, in); |
851 | /* small is minimal except that the value might be > p */ |
852 | |
853 | all_equal_so_far--; |
854 | /* |
855 | * We are doing a constant time test if out >= kPrime. We need to compare |
856 | * each u64, from most-significant to least significant. For each one, if |
857 | * all words so far have been equal (m is all ones) then a non-equal |
858 | * result is the answer. Otherwise we continue. |
859 | */ |
860 | for (i = 3; i < 4; i--) { |
861 | u64 equal; |
862 | uint128_t a = ((uint128_t) kPrime[i]) - out[i]; |
863 | /* |
864 | * if out[i] > kPrime[i] then a will underflow and the high 64-bits |
865 | * will all be set. |
866 | */ |
867 | result |= all_equal_so_far & ((u64)(a >> 64)); |
868 | |
869 | /* |
870 | * if kPrime[i] == out[i] then |equal| will be all zeros and the |
871 | * decrement will make it all ones. |
872 | */ |
873 | equal = kPrime[i] ^ out[i]; |
874 | equal--; |
875 | equal &= equal << 32; |
876 | equal &= equal << 16; |
877 | equal &= equal << 8; |
878 | equal &= equal << 4; |
879 | equal &= equal << 2; |
880 | equal &= equal << 1; |
881 | equal = 0 - (equal >> 63); |
882 | |
883 | all_equal_so_far &= equal; |
884 | } |
885 | |
886 | /* |
887 | * if all_equal_so_far is still all ones then the two values are equal |
888 | * and so out >= kPrime is true. |
889 | */ |
890 | result |= all_equal_so_far; |
891 | |
892 | /* if out >= kPrime then we subtract kPrime. */ |
893 | subtract_u64(&out[0], &carry, result & kPrime[0]); |
894 | subtract_u64(&out[1], &carry, carry); |
895 | subtract_u64(&out[2], &carry, carry); |
896 | subtract_u64(&out[3], &carry, carry); |
897 | |
898 | subtract_u64(&out[1], &carry, result & kPrime[1]); |
899 | subtract_u64(&out[2], &carry, carry); |
900 | subtract_u64(&out[3], &carry, carry); |
901 | |
902 | subtract_u64(&out[2], &carry, result & kPrime[2]); |
903 | subtract_u64(&out[3], &carry, carry); |
904 | |
905 | subtract_u64(&out[3], &carry, result & kPrime[3]); |
906 | } |
907 | |
908 | static void smallfelem_square_contract(smallfelem out, const smallfelem in) |
909 | { |
910 | longfelem longtmp; |
911 | felem tmp; |
912 | |
913 | smallfelem_square(longtmp, in); |
914 | felem_reduce(tmp, longtmp); |
915 | felem_contract(out, tmp); |
916 | } |
917 | |
918 | static void smallfelem_mul_contract(smallfelem out, const smallfelem in1, |
919 | const smallfelem in2) |
920 | { |
921 | longfelem longtmp; |
922 | felem tmp; |
923 | |
924 | smallfelem_mul(longtmp, in1, in2); |
925 | felem_reduce(tmp, longtmp); |
926 | felem_contract(out, tmp); |
927 | } |
928 | |
929 | /*- |
930 | * felem_is_zero returns a limb with all bits set if |in| == 0 (mod p) and 0 |
931 | * otherwise. |
932 | * On entry: |
933 | * small[i] < 2^64 |
934 | */ |
935 | static limb smallfelem_is_zero(const smallfelem small) |
936 | { |
937 | limb result; |
938 | u64 is_p; |
939 | |
940 | u64 is_zero = small[0] | small[1] | small[2] | small[3]; |
941 | is_zero--; |
942 | is_zero &= is_zero << 32; |
943 | is_zero &= is_zero << 16; |
944 | is_zero &= is_zero << 8; |
945 | is_zero &= is_zero << 4; |
946 | is_zero &= is_zero << 2; |
947 | is_zero &= is_zero << 1; |
948 | is_zero = 0 - (is_zero >> 63); |
949 | |
950 | is_p = (small[0] ^ kPrime[0]) | |
951 | (small[1] ^ kPrime[1]) | |
952 | (small[2] ^ kPrime[2]) | (small[3] ^ kPrime[3]); |
953 | is_p--; |
954 | is_p &= is_p << 32; |
955 | is_p &= is_p << 16; |
956 | is_p &= is_p << 8; |
957 | is_p &= is_p << 4; |
958 | is_p &= is_p << 2; |
959 | is_p &= is_p << 1; |
960 | is_p = 0 - (is_p >> 63); |
961 | |
962 | is_zero |= is_p; |
963 | |
964 | result = is_zero; |
965 | result |= ((limb) is_zero) << 64; |
966 | return result; |
967 | } |
968 | |
969 | static int smallfelem_is_zero_int(const void *small) |
970 | { |
971 | return (int)(smallfelem_is_zero(small) & ((limb) 1)); |
972 | } |
973 | |
974 | /*- |
975 | * felem_inv calculates |out| = |in|^{-1} |
976 | * |
977 | * Based on Fermat's Little Theorem: |
978 | * a^p = a (mod p) |
979 | * a^{p-1} = 1 (mod p) |
980 | * a^{p-2} = a^{-1} (mod p) |
981 | */ |
982 | static void felem_inv(felem out, const felem in) |
983 | { |
984 | felem ftmp, ftmp2; |
985 | /* each e_I will hold |in|^{2^I - 1} */ |
986 | felem e2, e4, e8, e16, e32, e64; |
987 | longfelem tmp; |
988 | unsigned i; |
989 | |
990 | felem_square(tmp, in); |
991 | felem_reduce(ftmp, tmp); /* 2^1 */ |
992 | felem_mul(tmp, in, ftmp); |
993 | felem_reduce(ftmp, tmp); /* 2^2 - 2^0 */ |
994 | felem_assign(e2, ftmp); |
995 | felem_square(tmp, ftmp); |
996 | felem_reduce(ftmp, tmp); /* 2^3 - 2^1 */ |
997 | felem_square(tmp, ftmp); |
998 | felem_reduce(ftmp, tmp); /* 2^4 - 2^2 */ |
999 | felem_mul(tmp, ftmp, e2); |
1000 | felem_reduce(ftmp, tmp); /* 2^4 - 2^0 */ |
1001 | felem_assign(e4, ftmp); |
1002 | felem_square(tmp, ftmp); |
1003 | felem_reduce(ftmp, tmp); /* 2^5 - 2^1 */ |
1004 | felem_square(tmp, ftmp); |
1005 | felem_reduce(ftmp, tmp); /* 2^6 - 2^2 */ |
1006 | felem_square(tmp, ftmp); |
1007 | felem_reduce(ftmp, tmp); /* 2^7 - 2^3 */ |
1008 | felem_square(tmp, ftmp); |
1009 | felem_reduce(ftmp, tmp); /* 2^8 - 2^4 */ |
1010 | felem_mul(tmp, ftmp, e4); |
1011 | felem_reduce(ftmp, tmp); /* 2^8 - 2^0 */ |
1012 | felem_assign(e8, ftmp); |
1013 | for (i = 0; i < 8; i++) { |
1014 | felem_square(tmp, ftmp); |
1015 | felem_reduce(ftmp, tmp); |
1016 | } /* 2^16 - 2^8 */ |
1017 | felem_mul(tmp, ftmp, e8); |
1018 | felem_reduce(ftmp, tmp); /* 2^16 - 2^0 */ |
1019 | felem_assign(e16, ftmp); |
1020 | for (i = 0; i < 16; i++) { |
1021 | felem_square(tmp, ftmp); |
1022 | felem_reduce(ftmp, tmp); |
1023 | } /* 2^32 - 2^16 */ |
1024 | felem_mul(tmp, ftmp, e16); |
1025 | felem_reduce(ftmp, tmp); /* 2^32 - 2^0 */ |
1026 | felem_assign(e32, ftmp); |
1027 | for (i = 0; i < 32; i++) { |
1028 | felem_square(tmp, ftmp); |
1029 | felem_reduce(ftmp, tmp); |
1030 | } /* 2^64 - 2^32 */ |
1031 | felem_assign(e64, ftmp); |
1032 | felem_mul(tmp, ftmp, in); |
1033 | felem_reduce(ftmp, tmp); /* 2^64 - 2^32 + 2^0 */ |
1034 | for (i = 0; i < 192; i++) { |
1035 | felem_square(tmp, ftmp); |
1036 | felem_reduce(ftmp, tmp); |
1037 | } /* 2^256 - 2^224 + 2^192 */ |
1038 | |
1039 | felem_mul(tmp, e64, e32); |
1040 | felem_reduce(ftmp2, tmp); /* 2^64 - 2^0 */ |
1041 | for (i = 0; i < 16; i++) { |
1042 | felem_square(tmp, ftmp2); |
1043 | felem_reduce(ftmp2, tmp); |
1044 | } /* 2^80 - 2^16 */ |
1045 | felem_mul(tmp, ftmp2, e16); |
1046 | felem_reduce(ftmp2, tmp); /* 2^80 - 2^0 */ |
1047 | for (i = 0; i < 8; i++) { |
1048 | felem_square(tmp, ftmp2); |
1049 | felem_reduce(ftmp2, tmp); |
1050 | } /* 2^88 - 2^8 */ |
1051 | felem_mul(tmp, ftmp2, e8); |
1052 | felem_reduce(ftmp2, tmp); /* 2^88 - 2^0 */ |
1053 | for (i = 0; i < 4; i++) { |
1054 | felem_square(tmp, ftmp2); |
1055 | felem_reduce(ftmp2, tmp); |
1056 | } /* 2^92 - 2^4 */ |
1057 | felem_mul(tmp, ftmp2, e4); |
1058 | felem_reduce(ftmp2, tmp); /* 2^92 - 2^0 */ |
1059 | felem_square(tmp, ftmp2); |
1060 | felem_reduce(ftmp2, tmp); /* 2^93 - 2^1 */ |
1061 | felem_square(tmp, ftmp2); |
1062 | felem_reduce(ftmp2, tmp); /* 2^94 - 2^2 */ |
1063 | felem_mul(tmp, ftmp2, e2); |
1064 | felem_reduce(ftmp2, tmp); /* 2^94 - 2^0 */ |
1065 | felem_square(tmp, ftmp2); |
1066 | felem_reduce(ftmp2, tmp); /* 2^95 - 2^1 */ |
1067 | felem_square(tmp, ftmp2); |
1068 | felem_reduce(ftmp2, tmp); /* 2^96 - 2^2 */ |
1069 | felem_mul(tmp, ftmp2, in); |
1070 | felem_reduce(ftmp2, tmp); /* 2^96 - 3 */ |
1071 | |
1072 | felem_mul(tmp, ftmp2, ftmp); |
1073 | felem_reduce(out, tmp); /* 2^256 - 2^224 + 2^192 + 2^96 - 3 */ |
1074 | } |
1075 | |
1076 | static void smallfelem_inv_contract(smallfelem out, const smallfelem in) |
1077 | { |
1078 | felem tmp; |
1079 | |
1080 | smallfelem_expand(tmp, in); |
1081 | felem_inv(tmp, tmp); |
1082 | felem_contract(out, tmp); |
1083 | } |
1084 | |
1085 | /*- |
1086 | * Group operations |
1087 | * ---------------- |
1088 | * |
1089 | * Building on top of the field operations we have the operations on the |
1090 | * elliptic curve group itself. Points on the curve are represented in Jacobian |
1091 | * coordinates |
1092 | */ |
1093 | |
1094 | /*- |
1095 | * point_double calculates 2*(x_in, y_in, z_in) |
1096 | * |
1097 | * The method is taken from: |
1098 | * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b |
1099 | * |
1100 | * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed. |
1101 | * while x_out == y_in is not (maybe this works, but it's not tested). |
1102 | */ |
1103 | static void |
1104 | point_double(felem x_out, felem y_out, felem z_out, |
1105 | const felem x_in, const felem y_in, const felem z_in) |
1106 | { |
1107 | longfelem tmp, tmp2; |
1108 | felem delta, gamma, beta, alpha, ftmp, ftmp2; |
1109 | smallfelem small1, small2; |
1110 | |
1111 | felem_assign(ftmp, x_in); |
1112 | /* ftmp[i] < 2^106 */ |
1113 | felem_assign(ftmp2, x_in); |
1114 | /* ftmp2[i] < 2^106 */ |
1115 | |
1116 | /* delta = z^2 */ |
1117 | felem_square(tmp, z_in); |
1118 | felem_reduce(delta, tmp); |
1119 | /* delta[i] < 2^101 */ |
1120 | |
1121 | /* gamma = y^2 */ |
1122 | felem_square(tmp, y_in); |
1123 | felem_reduce(gamma, tmp); |
1124 | /* gamma[i] < 2^101 */ |
1125 | felem_shrink(small1, gamma); |
1126 | |
1127 | /* beta = x*gamma */ |
1128 | felem_small_mul(tmp, small1, x_in); |
1129 | felem_reduce(beta, tmp); |
1130 | /* beta[i] < 2^101 */ |
1131 | |
1132 | /* alpha = 3*(x-delta)*(x+delta) */ |
1133 | felem_diff(ftmp, delta); |
1134 | /* ftmp[i] < 2^105 + 2^106 < 2^107 */ |
1135 | felem_sum(ftmp2, delta); |
1136 | /* ftmp2[i] < 2^105 + 2^106 < 2^107 */ |
1137 | felem_scalar(ftmp2, 3); |
1138 | /* ftmp2[i] < 3 * 2^107 < 2^109 */ |
1139 | felem_mul(tmp, ftmp, ftmp2); |
1140 | felem_reduce(alpha, tmp); |
1141 | /* alpha[i] < 2^101 */ |
1142 | felem_shrink(small2, alpha); |
1143 | |
1144 | /* x' = alpha^2 - 8*beta */ |
1145 | smallfelem_square(tmp, small2); |
1146 | felem_reduce(x_out, tmp); |
1147 | felem_assign(ftmp, beta); |
1148 | felem_scalar(ftmp, 8); |
1149 | /* ftmp[i] < 8 * 2^101 = 2^104 */ |
1150 | felem_diff(x_out, ftmp); |
1151 | /* x_out[i] < 2^105 + 2^101 < 2^106 */ |
1152 | |
1153 | /* z' = (y + z)^2 - gamma - delta */ |
1154 | felem_sum(delta, gamma); |
1155 | /* delta[i] < 2^101 + 2^101 = 2^102 */ |
1156 | felem_assign(ftmp, y_in); |
1157 | felem_sum(ftmp, z_in); |
1158 | /* ftmp[i] < 2^106 + 2^106 = 2^107 */ |
1159 | felem_square(tmp, ftmp); |
1160 | felem_reduce(z_out, tmp); |
1161 | felem_diff(z_out, delta); |
1162 | /* z_out[i] < 2^105 + 2^101 < 2^106 */ |
1163 | |
1164 | /* y' = alpha*(4*beta - x') - 8*gamma^2 */ |
1165 | felem_scalar(beta, 4); |
1166 | /* beta[i] < 4 * 2^101 = 2^103 */ |
1167 | felem_diff_zero107(beta, x_out); |
1168 | /* beta[i] < 2^107 + 2^103 < 2^108 */ |
1169 | felem_small_mul(tmp, small2, beta); |
1170 | /* tmp[i] < 7 * 2^64 < 2^67 */ |
1171 | smallfelem_square(tmp2, small1); |
1172 | /* tmp2[i] < 7 * 2^64 */ |
1173 | longfelem_scalar(tmp2, 8); |
1174 | /* tmp2[i] < 8 * 7 * 2^64 = 7 * 2^67 */ |
1175 | longfelem_diff(tmp, tmp2); |
1176 | /* tmp[i] < 2^67 + 2^70 + 2^40 < 2^71 */ |
1177 | felem_reduce_zero105(y_out, tmp); |
1178 | /* y_out[i] < 2^106 */ |
1179 | } |
1180 | |
1181 | /* |
1182 | * point_double_small is the same as point_double, except that it operates on |
1183 | * smallfelems |
1184 | */ |
1185 | static void |
1186 | point_double_small(smallfelem x_out, smallfelem y_out, smallfelem z_out, |
1187 | const smallfelem x_in, const smallfelem y_in, |
1188 | const smallfelem z_in) |
1189 | { |
1190 | felem felem_x_out, felem_y_out, felem_z_out; |
1191 | felem felem_x_in, felem_y_in, felem_z_in; |
1192 | |
1193 | smallfelem_expand(felem_x_in, x_in); |
1194 | smallfelem_expand(felem_y_in, y_in); |
1195 | smallfelem_expand(felem_z_in, z_in); |
1196 | point_double(felem_x_out, felem_y_out, felem_z_out, |
1197 | felem_x_in, felem_y_in, felem_z_in); |
1198 | felem_shrink(x_out, felem_x_out); |
1199 | felem_shrink(y_out, felem_y_out); |
1200 | felem_shrink(z_out, felem_z_out); |
1201 | } |
1202 | |
1203 | /* copy_conditional copies in to out iff mask is all ones. */ |
1204 | static void copy_conditional(felem out, const felem in, limb mask) |
1205 | { |
1206 | unsigned i; |
1207 | for (i = 0; i < NLIMBS; ++i) { |
1208 | const limb tmp = mask & (in[i] ^ out[i]); |
1209 | out[i] ^= tmp; |
1210 | } |
1211 | } |
1212 | |
1213 | /* copy_small_conditional copies in to out iff mask is all ones. */ |
1214 | static void copy_small_conditional(felem out, const smallfelem in, limb mask) |
1215 | { |
1216 | unsigned i; |
1217 | const u64 mask64 = mask; |
1218 | for (i = 0; i < NLIMBS; ++i) { |
1219 | out[i] = ((limb) (in[i] & mask64)) | (out[i] & ~mask); |
1220 | } |
1221 | } |
1222 | |
1223 | /*- |
1224 | * point_add calculates (x1, y1, z1) + (x2, y2, z2) |
1225 | * |
1226 | * The method is taken from: |
1227 | * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl, |
1228 | * adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity). |
1229 | * |
1230 | * This function includes a branch for checking whether the two input points |
1231 | * are equal, (while not equal to the point at infinity). This case never |
1232 | * happens during single point multiplication, so there is no timing leak for |
1233 | * ECDH or ECDSA signing. |
1234 | */ |
1235 | static void point_add(felem x3, felem y3, felem z3, |
1236 | const felem x1, const felem y1, const felem z1, |
1237 | const int mixed, const smallfelem x2, |
1238 | const smallfelem y2, const smallfelem z2) |
1239 | { |
1240 | felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, ftmp6, x_out, y_out, z_out; |
1241 | longfelem tmp, tmp2; |
1242 | smallfelem small1, small2, small3, small4, small5; |
1243 | limb x_equal, y_equal, z1_is_zero, z2_is_zero; |
1244 | |
1245 | felem_shrink(small3, z1); |
1246 | |
1247 | z1_is_zero = smallfelem_is_zero(small3); |
1248 | z2_is_zero = smallfelem_is_zero(z2); |
1249 | |
1250 | /* ftmp = z1z1 = z1**2 */ |
1251 | smallfelem_square(tmp, small3); |
1252 | felem_reduce(ftmp, tmp); |
1253 | /* ftmp[i] < 2^101 */ |
1254 | felem_shrink(small1, ftmp); |
1255 | |
1256 | if (!mixed) { |
1257 | /* ftmp2 = z2z2 = z2**2 */ |
1258 | smallfelem_square(tmp, z2); |
1259 | felem_reduce(ftmp2, tmp); |
1260 | /* ftmp2[i] < 2^101 */ |
1261 | felem_shrink(small2, ftmp2); |
1262 | |
1263 | felem_shrink(small5, x1); |
1264 | |
1265 | /* u1 = ftmp3 = x1*z2z2 */ |
1266 | smallfelem_mul(tmp, small5, small2); |
1267 | felem_reduce(ftmp3, tmp); |
1268 | /* ftmp3[i] < 2^101 */ |
1269 | |
1270 | /* ftmp5 = z1 + z2 */ |
1271 | felem_assign(ftmp5, z1); |
1272 | felem_small_sum(ftmp5, z2); |
1273 | /* ftmp5[i] < 2^107 */ |
1274 | |
1275 | /* ftmp5 = (z1 + z2)**2 - (z1z1 + z2z2) = 2z1z2 */ |
1276 | felem_square(tmp, ftmp5); |
1277 | felem_reduce(ftmp5, tmp); |
1278 | /* ftmp2 = z2z2 + z1z1 */ |
1279 | felem_sum(ftmp2, ftmp); |
1280 | /* ftmp2[i] < 2^101 + 2^101 = 2^102 */ |
1281 | felem_diff(ftmp5, ftmp2); |
1282 | /* ftmp5[i] < 2^105 + 2^101 < 2^106 */ |
1283 | |
1284 | /* ftmp2 = z2 * z2z2 */ |
1285 | smallfelem_mul(tmp, small2, z2); |
1286 | felem_reduce(ftmp2, tmp); |
1287 | |
1288 | /* s1 = ftmp2 = y1 * z2**3 */ |
1289 | felem_mul(tmp, y1, ftmp2); |
1290 | felem_reduce(ftmp6, tmp); |
1291 | /* ftmp6[i] < 2^101 */ |
1292 | } else { |
1293 | /* |
1294 | * We'll assume z2 = 1 (special case z2 = 0 is handled later) |
1295 | */ |
1296 | |
1297 | /* u1 = ftmp3 = x1*z2z2 */ |
1298 | felem_assign(ftmp3, x1); |
1299 | /* ftmp3[i] < 2^106 */ |
1300 | |
1301 | /* ftmp5 = 2z1z2 */ |
1302 | felem_assign(ftmp5, z1); |
1303 | felem_scalar(ftmp5, 2); |
1304 | /* ftmp5[i] < 2*2^106 = 2^107 */ |
1305 | |
1306 | /* s1 = ftmp2 = y1 * z2**3 */ |
1307 | felem_assign(ftmp6, y1); |
1308 | /* ftmp6[i] < 2^106 */ |
1309 | } |
1310 | |
1311 | /* u2 = x2*z1z1 */ |
1312 | smallfelem_mul(tmp, x2, small1); |
1313 | felem_reduce(ftmp4, tmp); |
1314 | |
1315 | /* h = ftmp4 = u2 - u1 */ |
1316 | felem_diff_zero107(ftmp4, ftmp3); |
1317 | /* ftmp4[i] < 2^107 + 2^101 < 2^108 */ |
1318 | felem_shrink(small4, ftmp4); |
1319 | |
1320 | x_equal = smallfelem_is_zero(small4); |
1321 | |
1322 | /* z_out = ftmp5 * h */ |
1323 | felem_small_mul(tmp, small4, ftmp5); |
1324 | felem_reduce(z_out, tmp); |
1325 | /* z_out[i] < 2^101 */ |
1326 | |
1327 | /* ftmp = z1 * z1z1 */ |
1328 | smallfelem_mul(tmp, small1, small3); |
1329 | felem_reduce(ftmp, tmp); |
1330 | |
1331 | /* s2 = tmp = y2 * z1**3 */ |
1332 | felem_small_mul(tmp, y2, ftmp); |
1333 | felem_reduce(ftmp5, tmp); |
1334 | |
1335 | /* r = ftmp5 = (s2 - s1)*2 */ |
1336 | felem_diff_zero107(ftmp5, ftmp6); |
1337 | /* ftmp5[i] < 2^107 + 2^107 = 2^108 */ |
1338 | felem_scalar(ftmp5, 2); |
1339 | /* ftmp5[i] < 2^109 */ |
1340 | felem_shrink(small1, ftmp5); |
1341 | y_equal = smallfelem_is_zero(small1); |
1342 | |
1343 | if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) { |
1344 | point_double(x3, y3, z3, x1, y1, z1); |
1345 | return; |
1346 | } |
1347 | |
1348 | /* I = ftmp = (2h)**2 */ |
1349 | felem_assign(ftmp, ftmp4); |
1350 | felem_scalar(ftmp, 2); |
1351 | /* ftmp[i] < 2*2^108 = 2^109 */ |
1352 | felem_square(tmp, ftmp); |
1353 | felem_reduce(ftmp, tmp); |
1354 | |
1355 | /* J = ftmp2 = h * I */ |
1356 | felem_mul(tmp, ftmp4, ftmp); |
1357 | felem_reduce(ftmp2, tmp); |
1358 | |
1359 | /* V = ftmp4 = U1 * I */ |
1360 | felem_mul(tmp, ftmp3, ftmp); |
1361 | felem_reduce(ftmp4, tmp); |
1362 | |
1363 | /* x_out = r**2 - J - 2V */ |
1364 | smallfelem_square(tmp, small1); |
1365 | felem_reduce(x_out, tmp); |
1366 | felem_assign(ftmp3, ftmp4); |
1367 | felem_scalar(ftmp4, 2); |
1368 | felem_sum(ftmp4, ftmp2); |
1369 | /* ftmp4[i] < 2*2^101 + 2^101 < 2^103 */ |
1370 | felem_diff(x_out, ftmp4); |
1371 | /* x_out[i] < 2^105 + 2^101 */ |
1372 | |
1373 | /* y_out = r(V-x_out) - 2 * s1 * J */ |
1374 | felem_diff_zero107(ftmp3, x_out); |
1375 | /* ftmp3[i] < 2^107 + 2^101 < 2^108 */ |
1376 | felem_small_mul(tmp, small1, ftmp3); |
1377 | felem_mul(tmp2, ftmp6, ftmp2); |
1378 | longfelem_scalar(tmp2, 2); |
1379 | /* tmp2[i] < 2*2^67 = 2^68 */ |
1380 | longfelem_diff(tmp, tmp2); |
1381 | /* tmp[i] < 2^67 + 2^70 + 2^40 < 2^71 */ |
1382 | felem_reduce_zero105(y_out, tmp); |
1383 | /* y_out[i] < 2^106 */ |
1384 | |
1385 | copy_small_conditional(x_out, x2, z1_is_zero); |
1386 | copy_conditional(x_out, x1, z2_is_zero); |
1387 | copy_small_conditional(y_out, y2, z1_is_zero); |
1388 | copy_conditional(y_out, y1, z2_is_zero); |
1389 | copy_small_conditional(z_out, z2, z1_is_zero); |
1390 | copy_conditional(z_out, z1, z2_is_zero); |
1391 | felem_assign(x3, x_out); |
1392 | felem_assign(y3, y_out); |
1393 | felem_assign(z3, z_out); |
1394 | } |
1395 | |
1396 | /* |
1397 | * point_add_small is the same as point_add, except that it operates on |
1398 | * smallfelems |
1399 | */ |
1400 | static void point_add_small(smallfelem x3, smallfelem y3, smallfelem z3, |
1401 | smallfelem x1, smallfelem y1, smallfelem z1, |
1402 | smallfelem x2, smallfelem y2, smallfelem z2) |
1403 | { |
1404 | felem felem_x3, felem_y3, felem_z3; |
1405 | felem felem_x1, felem_y1, felem_z1; |
1406 | smallfelem_expand(felem_x1, x1); |
1407 | smallfelem_expand(felem_y1, y1); |
1408 | smallfelem_expand(felem_z1, z1); |
1409 | point_add(felem_x3, felem_y3, felem_z3, felem_x1, felem_y1, felem_z1, 0, |
1410 | x2, y2, z2); |
1411 | felem_shrink(x3, felem_x3); |
1412 | felem_shrink(y3, felem_y3); |
1413 | felem_shrink(z3, felem_z3); |
1414 | } |
1415 | |
1416 | /*- |
1417 | * Base point pre computation |
1418 | * -------------------------- |
1419 | * |
1420 | * Two different sorts of precomputed tables are used in the following code. |
1421 | * Each contain various points on the curve, where each point is three field |
1422 | * elements (x, y, z). |
1423 | * |
1424 | * For the base point table, z is usually 1 (0 for the point at infinity). |
1425 | * This table has 2 * 16 elements, starting with the following: |
1426 | * index | bits | point |
1427 | * ------+---------+------------------------------ |
1428 | * 0 | 0 0 0 0 | 0G |
1429 | * 1 | 0 0 0 1 | 1G |
1430 | * 2 | 0 0 1 0 | 2^64G |
1431 | * 3 | 0 0 1 1 | (2^64 + 1)G |
1432 | * 4 | 0 1 0 0 | 2^128G |
1433 | * 5 | 0 1 0 1 | (2^128 + 1)G |
1434 | * 6 | 0 1 1 0 | (2^128 + 2^64)G |
1435 | * 7 | 0 1 1 1 | (2^128 + 2^64 + 1)G |
1436 | * 8 | 1 0 0 0 | 2^192G |
1437 | * 9 | 1 0 0 1 | (2^192 + 1)G |
1438 | * 10 | 1 0 1 0 | (2^192 + 2^64)G |
1439 | * 11 | 1 0 1 1 | (2^192 + 2^64 + 1)G |
1440 | * 12 | 1 1 0 0 | (2^192 + 2^128)G |
1441 | * 13 | 1 1 0 1 | (2^192 + 2^128 + 1)G |
1442 | * 14 | 1 1 1 0 | (2^192 + 2^128 + 2^64)G |
1443 | * 15 | 1 1 1 1 | (2^192 + 2^128 + 2^64 + 1)G |
1444 | * followed by a copy of this with each element multiplied by 2^32. |
1445 | * |
1446 | * The reason for this is so that we can clock bits into four different |
1447 | * locations when doing simple scalar multiplies against the base point, |
1448 | * and then another four locations using the second 16 elements. |
1449 | * |
1450 | * Tables for other points have table[i] = iG for i in 0 .. 16. */ |
1451 | |
1452 | /* gmul is the table of precomputed base points */ |
1453 | static const smallfelem gmul[2][16][3] = { |
1454 | {{{0, 0, 0, 0}, |
1455 | {0, 0, 0, 0}, |
1456 | {0, 0, 0, 0}}, |
1457 | {{0xf4a13945d898c296, 0x77037d812deb33a0, 0xf8bce6e563a440f2, |
1458 | 0x6b17d1f2e12c4247}, |
1459 | {0xcbb6406837bf51f5, 0x2bce33576b315ece, 0x8ee7eb4a7c0f9e16, |
1460 | 0x4fe342e2fe1a7f9b}, |
1461 | {1, 0, 0, 0}}, |
1462 | {{0x90e75cb48e14db63, 0x29493baaad651f7e, 0x8492592e326e25de, |
1463 | 0x0fa822bc2811aaa5}, |
1464 | {0xe41124545f462ee7, 0x34b1a65050fe82f5, 0x6f4ad4bcb3df188b, |
1465 | 0xbff44ae8f5dba80d}, |
1466 | {1, 0, 0, 0}}, |
1467 | {{0x93391ce2097992af, 0xe96c98fd0d35f1fa, 0xb257c0de95e02789, |
1468 | 0x300a4bbc89d6726f}, |
1469 | {0xaa54a291c08127a0, 0x5bb1eeada9d806a5, 0x7f1ddb25ff1e3c6f, |
1470 | 0x72aac7e0d09b4644}, |
1471 | {1, 0, 0, 0}}, |
1472 | {{0x57c84fc9d789bd85, 0xfc35ff7dc297eac3, 0xfb982fd588c6766e, |
1473 | 0x447d739beedb5e67}, |
1474 | {0x0c7e33c972e25b32, 0x3d349b95a7fae500, 0xe12e9d953a4aaff7, |
1475 | 0x2d4825ab834131ee}, |
1476 | {1, 0, 0, 0}}, |
1477 | {{0x13949c932a1d367f, 0xef7fbd2b1a0a11b7, 0xddc6068bb91dfc60, |
1478 | 0xef9519328a9c72ff}, |
1479 | {0x196035a77376d8a8, 0x23183b0895ca1740, 0xc1ee9807022c219c, |
1480 | 0x611e9fc37dbb2c9b}, |
1481 | {1, 0, 0, 0}}, |
1482 | {{0xcae2b1920b57f4bc, 0x2936df5ec6c9bc36, 0x7dea6482e11238bf, |
1483 | 0x550663797b51f5d8}, |
1484 | {0x44ffe216348a964c, 0x9fb3d576dbdefbe1, 0x0afa40018d9d50e5, |
1485 | 0x157164848aecb851}, |
1486 | {1, 0, 0, 0}}, |
1487 | {{0xe48ecafffc5cde01, 0x7ccd84e70d715f26, 0xa2e8f483f43e4391, |
1488 | 0xeb5d7745b21141ea}, |
1489 | {0xcac917e2731a3479, 0x85f22cfe2844b645, 0x0990e6a158006cee, |
1490 | 0xeafd72ebdbecc17b}, |
1491 | {1, 0, 0, 0}}, |
1492 | {{0x6cf20ffb313728be, 0x96439591a3c6b94a, 0x2736ff8344315fc5, |
1493 | 0xa6d39677a7849276}, |
1494 | {0xf2bab833c357f5f4, 0x824a920c2284059b, 0x66b8babd2d27ecdf, |
1495 | 0x674f84749b0b8816}, |
1496 | {1, 0, 0, 0}}, |
1497 | {{0x2df48c04677c8a3e, 0x74e02f080203a56b, 0x31855f7db8c7fedb, |
1498 | 0x4e769e7672c9ddad}, |
1499 | {0xa4c36165b824bbb0, 0xfb9ae16f3b9122a5, 0x1ec0057206947281, |
1500 | 0x42b99082de830663}, |
1501 | {1, 0, 0, 0}}, |
1502 | {{0x6ef95150dda868b9, 0xd1f89e799c0ce131, 0x7fdc1ca008a1c478, |
1503 | 0x78878ef61c6ce04d}, |
1504 | {0x9c62b9121fe0d976, 0x6ace570ebde08d4f, 0xde53142c12309def, |
1505 | 0xb6cb3f5d7b72c321}, |
1506 | {1, 0, 0, 0}}, |
1507 | {{0x7f991ed2c31a3573, 0x5b82dd5bd54fb496, 0x595c5220812ffcae, |
1508 | 0x0c88bc4d716b1287}, |
1509 | {0x3a57bf635f48aca8, 0x7c8181f4df2564f3, 0x18d1b5b39c04e6aa, |
1510 | 0xdd5ddea3f3901dc6}, |
1511 | {1, 0, 0, 0}}, |
1512 | {{0xe96a79fb3e72ad0c, 0x43a0a28c42ba792f, 0xefe0a423083e49f3, |
1513 | 0x68f344af6b317466}, |
1514 | {0xcdfe17db3fb24d4a, 0x668bfc2271f5c626, 0x604ed93c24d67ff3, |
1515 | 0x31b9c405f8540a20}, |
1516 | {1, 0, 0, 0}}, |
1517 | {{0xd36b4789a2582e7f, 0x0d1a10144ec39c28, 0x663c62c3edbad7a0, |
1518 | 0x4052bf4b6f461db9}, |
1519 | {0x235a27c3188d25eb, 0xe724f33999bfcc5b, 0x862be6bd71d70cc8, |
1520 | 0xfecf4d5190b0fc61}, |
1521 | {1, 0, 0, 0}}, |
1522 | {{0x74346c10a1d4cfac, 0xafdf5cc08526a7a4, 0x123202a8f62bff7a, |
1523 | 0x1eddbae2c802e41a}, |
1524 | {0x8fa0af2dd603f844, 0x36e06b7e4c701917, 0x0c45f45273db33a0, |
1525 | 0x43104d86560ebcfc}, |
1526 | {1, 0, 0, 0}}, |
1527 | {{0x9615b5110d1d78e5, 0x66b0de3225c4744b, 0x0a4a46fb6aaf363a, |
1528 | 0xb48e26b484f7a21c}, |
1529 | {0x06ebb0f621a01b2d, 0xc004e4048b7b0f98, 0x64131bcdfed6f668, |
1530 | 0xfac015404d4d3dab}, |
1531 | {1, 0, 0, 0}}}, |
1532 | {{{0, 0, 0, 0}, |
1533 | {0, 0, 0, 0}, |
1534 | {0, 0, 0, 0}}, |
1535 | {{0x3a5a9e22185a5943, 0x1ab919365c65dfb6, 0x21656b32262c71da, |
1536 | 0x7fe36b40af22af89}, |
1537 | {0xd50d152c699ca101, 0x74b3d5867b8af212, 0x9f09f40407dca6f1, |
1538 | 0xe697d45825b63624}, |
1539 | {1, 0, 0, 0}}, |
1540 | {{0xa84aa9397512218e, 0xe9a521b074ca0141, 0x57880b3a18a2e902, |
1541 | 0x4a5b506612a677a6}, |
1542 | {0x0beada7a4c4f3840, 0x626db15419e26d9d, 0xc42604fbe1627d40, |
1543 | 0xeb13461ceac089f1}, |
1544 | {1, 0, 0, 0}}, |
1545 | {{0xf9faed0927a43281, 0x5e52c4144103ecbc, 0xc342967aa815c857, |
1546 | 0x0781b8291c6a220a}, |
1547 | {0x5a8343ceeac55f80, 0x88f80eeee54a05e3, 0x97b2a14f12916434, |
1548 | 0x690cde8df0151593}, |
1549 | {1, 0, 0, 0}}, |
1550 | {{0xaee9c75df7f82f2a, 0x9e4c35874afdf43a, 0xf5622df437371326, |
1551 | 0x8a535f566ec73617}, |
1552 | {0xc5f9a0ac223094b7, 0xcde533864c8c7669, 0x37e02819085a92bf, |
1553 | 0x0455c08468b08bd7}, |
1554 | {1, 0, 0, 0}}, |
1555 | {{0x0c0a6e2c9477b5d9, 0xf9a4bf62876dc444, 0x5050a949b6cdc279, |
1556 | 0x06bada7ab77f8276}, |
1557 | {0xc8b4aed1ea48dac9, 0xdebd8a4b7ea1070f, 0x427d49101366eb70, |
1558 | 0x5b476dfd0e6cb18a}, |
1559 | {1, 0, 0, 0}}, |
1560 | {{0x7c5c3e44278c340a, 0x4d54606812d66f3b, 0x29a751b1ae23c5d8, |
1561 | 0x3e29864e8a2ec908}, |
1562 | {0x142d2a6626dbb850, 0xad1744c4765bd780, 0x1f150e68e322d1ed, |
1563 | 0x239b90ea3dc31e7e}, |
1564 | {1, 0, 0, 0}}, |
1565 | {{0x78c416527a53322a, 0x305dde6709776f8e, 0xdbcab759f8862ed4, |
1566 | 0x820f4dd949f72ff7}, |
1567 | {0x6cc544a62b5debd4, 0x75be5d937b4e8cc4, 0x1b481b1b215c14d3, |
1568 | 0x140406ec783a05ec}, |
1569 | {1, 0, 0, 0}}, |
1570 | {{0x6a703f10e895df07, 0xfd75f3fa01876bd8, 0xeb5b06e70ce08ffe, |
1571 | 0x68f6b8542783dfee}, |
1572 | {0x90c76f8a78712655, 0xcf5293d2f310bf7f, 0xfbc8044dfda45028, |
1573 | 0xcbe1feba92e40ce6}, |
1574 | {1, 0, 0, 0}}, |
1575 | {{0xe998ceea4396e4c1, 0xfc82ef0b6acea274, 0x230f729f2250e927, |
1576 | 0xd0b2f94d2f420109}, |
1577 | {0x4305adddb38d4966, 0x10b838f8624c3b45, 0x7db2636658954e7a, |
1578 | 0x971459828b0719e5}, |
1579 | {1, 0, 0, 0}}, |
1580 | {{0x4bd6b72623369fc9, 0x57f2929e53d0b876, 0xc2d5cba4f2340687, |
1581 | 0x961610004a866aba}, |
1582 | {0x49997bcd2e407a5e, 0x69ab197d92ddcb24, 0x2cf1f2438fe5131c, |
1583 | 0x7acb9fadcee75e44}, |
1584 | {1, 0, 0, 0}}, |
1585 | {{0x254e839423d2d4c0, 0xf57f0c917aea685b, 0xa60d880f6f75aaea, |
1586 | 0x24eb9acca333bf5b}, |
1587 | {0xe3de4ccb1cda5dea, 0xfeef9341c51a6b4f, 0x743125f88bac4c4d, |
1588 | 0x69f891c5acd079cc}, |
1589 | {1, 0, 0, 0}}, |
1590 | {{0xeee44b35702476b5, 0x7ed031a0e45c2258, 0xb422d1e7bd6f8514, |
1591 | 0xe51f547c5972a107}, |
1592 | {0xa25bcd6fc9cf343d, 0x8ca922ee097c184e, 0xa62f98b3a9fe9a06, |
1593 | 0x1c309a2b25bb1387}, |
1594 | {1, 0, 0, 0}}, |
1595 | {{0x9295dbeb1967c459, 0xb00148833472c98e, 0xc504977708011828, |
1596 | 0x20b87b8aa2c4e503}, |
1597 | {0x3063175de057c277, 0x1bd539338fe582dd, 0x0d11adef5f69a044, |
1598 | 0xf5c6fa49919776be}, |
1599 | {1, 0, 0, 0}}, |
1600 | {{0x8c944e760fd59e11, 0x3876cba1102fad5f, 0xa454c3fad83faa56, |
1601 | 0x1ed7d1b9332010b9}, |
1602 | {0xa1011a270024b889, 0x05e4d0dcac0cd344, 0x52b520f0eb6a2a24, |
1603 | 0x3a2b03f03217257a}, |
1604 | {1, 0, 0, 0}}, |
1605 | {{0xf20fc2afdf1d043d, 0xf330240db58d5a62, 0xfc7d229ca0058c3b, |
1606 | 0x15fee545c78dd9f6}, |
1607 | {0x501e82885bc98cda, 0x41ef80e5d046ac04, 0x557d9f49461210fb, |
1608 | 0x4ab5b6b2b8753f81}, |
1609 | {1, 0, 0, 0}}} |
1610 | }; |
1611 | |
1612 | /* |
1613 | * select_point selects the |idx|th point from a precomputation table and |
1614 | * copies it to out. |
1615 | */ |
1616 | static void select_point(const u64 idx, unsigned int size, |
1617 | const smallfelem pre_comp[16][3], smallfelem out[3]) |
1618 | { |
1619 | unsigned i, j; |
1620 | u64 *outlimbs = &out[0][0]; |
1621 | |
1622 | memset(out, 0, sizeof(*out) * 3); |
1623 | |
1624 | for (i = 0; i < size; i++) { |
1625 | const u64 *inlimbs = (u64 *)&pre_comp[i][0][0]; |
1626 | u64 mask = i ^ idx; |
1627 | mask |= mask >> 4; |
1628 | mask |= mask >> 2; |
1629 | mask |= mask >> 1; |
1630 | mask &= 1; |
1631 | mask--; |
1632 | for (j = 0; j < NLIMBS * 3; j++) |
1633 | outlimbs[j] |= inlimbs[j] & mask; |
1634 | } |
1635 | } |
1636 | |
1637 | /* get_bit returns the |i|th bit in |in| */ |
1638 | static char get_bit(const felem_bytearray in, int i) |
1639 | { |
1640 | if ((i < 0) || (i >= 256)) |
1641 | return 0; |
1642 | return (in[i >> 3] >> (i & 7)) & 1; |
1643 | } |
1644 | |
1645 | /* |
1646 | * Interleaved point multiplication using precomputed point multiples: The |
1647 | * small point multiples 0*P, 1*P, ..., 17*P are in pre_comp[], the scalars |
1648 | * in scalars[]. If g_scalar is non-NULL, we also add this multiple of the |
1649 | * generator, using certain (large) precomputed multiples in g_pre_comp. |
1650 | * Output point (X, Y, Z) is stored in x_out, y_out, z_out |
1651 | */ |
1652 | static void batch_mul(felem x_out, felem y_out, felem z_out, |
1653 | const felem_bytearray scalars[], |
1654 | const unsigned num_points, const u8 *g_scalar, |
1655 | const int mixed, const smallfelem pre_comp[][17][3], |
1656 | const smallfelem g_pre_comp[2][16][3]) |
1657 | { |
1658 | int i, skip; |
1659 | unsigned num, gen_mul = (g_scalar != NULL); |
1660 | felem nq[3], ftmp; |
1661 | smallfelem tmp[3]; |
1662 | u64 bits; |
1663 | u8 sign, digit; |
1664 | |
1665 | /* set nq to the point at infinity */ |
1666 | memset(nq, 0, sizeof(nq)); |
1667 | |
1668 | /* |
1669 | * Loop over all scalars msb-to-lsb, interleaving additions of multiples |
1670 | * of the generator (two in each of the last 32 rounds) and additions of |
1671 | * other points multiples (every 5th round). |
1672 | */ |
1673 | skip = 1; /* save two point operations in the first |
1674 | * round */ |
1675 | for (i = (num_points ? 255 : 31); i >= 0; --i) { |
1676 | /* double */ |
1677 | if (!skip) |
1678 | point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]); |
1679 | |
1680 | /* add multiples of the generator */ |
1681 | if (gen_mul && (i <= 31)) { |
1682 | /* first, look 32 bits upwards */ |
1683 | bits = get_bit(g_scalar, i + 224) << 3; |
1684 | bits |= get_bit(g_scalar, i + 160) << 2; |
1685 | bits |= get_bit(g_scalar, i + 96) << 1; |
1686 | bits |= get_bit(g_scalar, i + 32); |
1687 | /* select the point to add, in constant time */ |
1688 | select_point(bits, 16, g_pre_comp[1], tmp); |
1689 | |
1690 | if (!skip) { |
1691 | /* Arg 1 below is for "mixed" */ |
1692 | point_add(nq[0], nq[1], nq[2], |
1693 | nq[0], nq[1], nq[2], 1, tmp[0], tmp[1], tmp[2]); |
1694 | } else { |
1695 | smallfelem_expand(nq[0], tmp[0]); |
1696 | smallfelem_expand(nq[1], tmp[1]); |
1697 | smallfelem_expand(nq[2], tmp[2]); |
1698 | skip = 0; |
1699 | } |
1700 | |
1701 | /* second, look at the current position */ |
1702 | bits = get_bit(g_scalar, i + 192) << 3; |
1703 | bits |= get_bit(g_scalar, i + 128) << 2; |
1704 | bits |= get_bit(g_scalar, i + 64) << 1; |
1705 | bits |= get_bit(g_scalar, i); |
1706 | /* select the point to add, in constant time */ |
1707 | select_point(bits, 16, g_pre_comp[0], tmp); |
1708 | /* Arg 1 below is for "mixed" */ |
1709 | point_add(nq[0], nq[1], nq[2], |
1710 | nq[0], nq[1], nq[2], 1, tmp[0], tmp[1], tmp[2]); |
1711 | } |
1712 | |
1713 | /* do other additions every 5 doublings */ |
1714 | if (num_points && (i % 5 == 0)) { |
1715 | /* loop over all scalars */ |
1716 | for (num = 0; num < num_points; ++num) { |
1717 | bits = get_bit(scalars[num], i + 4) << 5; |
1718 | bits |= get_bit(scalars[num], i + 3) << 4; |
1719 | bits |= get_bit(scalars[num], i + 2) << 3; |
1720 | bits |= get_bit(scalars[num], i + 1) << 2; |
1721 | bits |= get_bit(scalars[num], i) << 1; |
1722 | bits |= get_bit(scalars[num], i - 1); |
1723 | ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits); |
1724 | |
1725 | /* |
1726 | * select the point to add or subtract, in constant time |
1727 | */ |
1728 | select_point(digit, 17, pre_comp[num], tmp); |
1729 | smallfelem_neg(ftmp, tmp[1]); /* (X, -Y, Z) is the negative |
1730 | * point */ |
1731 | copy_small_conditional(ftmp, tmp[1], (((limb) sign) - 1)); |
1732 | felem_contract(tmp[1], ftmp); |
1733 | |
1734 | if (!skip) { |
1735 | point_add(nq[0], nq[1], nq[2], |
1736 | nq[0], nq[1], nq[2], |
1737 | mixed, tmp[0], tmp[1], tmp[2]); |
1738 | } else { |
1739 | smallfelem_expand(nq[0], tmp[0]); |
1740 | smallfelem_expand(nq[1], tmp[1]); |
1741 | smallfelem_expand(nq[2], tmp[2]); |
1742 | skip = 0; |
1743 | } |
1744 | } |
1745 | } |
1746 | } |
1747 | felem_assign(x_out, nq[0]); |
1748 | felem_assign(y_out, nq[1]); |
1749 | felem_assign(z_out, nq[2]); |
1750 | } |
1751 | |
1752 | /* Precomputation for the group generator. */ |
1753 | struct nistp256_pre_comp_st { |
1754 | smallfelem g_pre_comp[2][16][3]; |
1755 | CRYPTO_REF_COUNT references; |
1756 | CRYPTO_RWLOCK *lock; |
1757 | }; |
1758 | |
1759 | const EC_METHOD *EC_GFp_nistp256_method(void) |
1760 | { |
1761 | static const EC_METHOD ret = { |
1762 | EC_FLAGS_DEFAULT_OCT, |
1763 | NID_X9_62_prime_field, |
1764 | ec_GFp_nistp256_group_init, |
1765 | ec_GFp_simple_group_finish, |
1766 | ec_GFp_simple_group_clear_finish, |
1767 | ec_GFp_nist_group_copy, |
1768 | ec_GFp_nistp256_group_set_curve, |
1769 | ec_GFp_simple_group_get_curve, |
1770 | ec_GFp_simple_group_get_degree, |
1771 | ec_group_simple_order_bits, |
1772 | ec_GFp_simple_group_check_discriminant, |
1773 | ec_GFp_simple_point_init, |
1774 | ec_GFp_simple_point_finish, |
1775 | ec_GFp_simple_point_clear_finish, |
1776 | ec_GFp_simple_point_copy, |
1777 | ec_GFp_simple_point_set_to_infinity, |
1778 | ec_GFp_simple_set_Jprojective_coordinates_GFp, |
1779 | ec_GFp_simple_get_Jprojective_coordinates_GFp, |
1780 | ec_GFp_simple_point_set_affine_coordinates, |
1781 | ec_GFp_nistp256_point_get_affine_coordinates, |
1782 | 0 /* point_set_compressed_coordinates */ , |
1783 | 0 /* point2oct */ , |
1784 | 0 /* oct2point */ , |
1785 | ec_GFp_simple_add, |
1786 | ec_GFp_simple_dbl, |
1787 | ec_GFp_simple_invert, |
1788 | ec_GFp_simple_is_at_infinity, |
1789 | ec_GFp_simple_is_on_curve, |
1790 | ec_GFp_simple_cmp, |
1791 | ec_GFp_simple_make_affine, |
1792 | ec_GFp_simple_points_make_affine, |
1793 | ec_GFp_nistp256_points_mul, |
1794 | ec_GFp_nistp256_precompute_mult, |
1795 | ec_GFp_nistp256_have_precompute_mult, |
1796 | ec_GFp_nist_field_mul, |
1797 | ec_GFp_nist_field_sqr, |
1798 | 0 /* field_div */ , |
1799 | ec_GFp_simple_field_inv, |
1800 | 0 /* field_encode */ , |
1801 | 0 /* field_decode */ , |
1802 | 0, /* field_set_to_one */ |
1803 | ec_key_simple_priv2oct, |
1804 | ec_key_simple_oct2priv, |
1805 | 0, /* set private */ |
1806 | ec_key_simple_generate_key, |
1807 | ec_key_simple_check_key, |
1808 | ec_key_simple_generate_public_key, |
1809 | 0, /* keycopy */ |
1810 | 0, /* keyfinish */ |
1811 | ecdh_simple_compute_key, |
1812 | ecdsa_simple_sign_setup, |
1813 | ecdsa_simple_sign_sig, |
1814 | ecdsa_simple_verify_sig, |
1815 | 0, /* field_inverse_mod_ord */ |
1816 | 0, /* blind_coordinates */ |
1817 | 0, /* ladder_pre */ |
1818 | 0, /* ladder_step */ |
1819 | 0 /* ladder_post */ |
1820 | }; |
1821 | |
1822 | return &ret; |
1823 | } |
1824 | |
1825 | /******************************************************************************/ |
1826 | /* |
1827 | * FUNCTIONS TO MANAGE PRECOMPUTATION |
1828 | */ |
1829 | |
1830 | static NISTP256_PRE_COMP *nistp256_pre_comp_new(void) |
1831 | { |
1832 | NISTP256_PRE_COMP *ret = OPENSSL_zalloc(sizeof(*ret)); |
1833 | |
1834 | if (ret == NULL) { |
1835 | ECerr(EC_F_NISTP256_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE); |
1836 | return ret; |
1837 | } |
1838 | |
1839 | ret->references = 1; |
1840 | |
1841 | ret->lock = CRYPTO_THREAD_lock_new(); |
1842 | if (ret->lock == NULL) { |
1843 | ECerr(EC_F_NISTP256_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE); |
1844 | OPENSSL_free(ret); |
1845 | return NULL; |
1846 | } |
1847 | return ret; |
1848 | } |
1849 | |
1850 | NISTP256_PRE_COMP *EC_nistp256_pre_comp_dup(NISTP256_PRE_COMP *p) |
1851 | { |
1852 | int i; |
1853 | if (p != NULL) |
1854 | CRYPTO_UP_REF(&p->references, &i, p->lock); |
1855 | return p; |
1856 | } |
1857 | |
1858 | void EC_nistp256_pre_comp_free(NISTP256_PRE_COMP *pre) |
1859 | { |
1860 | int i; |
1861 | |
1862 | if (pre == NULL) |
1863 | return; |
1864 | |
1865 | CRYPTO_DOWN_REF(&pre->references, &i, pre->lock); |
1866 | REF_PRINT_COUNT("EC_nistp256" , x); |
1867 | if (i > 0) |
1868 | return; |
1869 | REF_ASSERT_ISNT(i < 0); |
1870 | |
1871 | CRYPTO_THREAD_lock_free(pre->lock); |
1872 | OPENSSL_free(pre); |
1873 | } |
1874 | |
1875 | /******************************************************************************/ |
1876 | /* |
1877 | * OPENSSL EC_METHOD FUNCTIONS |
1878 | */ |
1879 | |
1880 | int ec_GFp_nistp256_group_init(EC_GROUP *group) |
1881 | { |
1882 | int ret; |
1883 | ret = ec_GFp_simple_group_init(group); |
1884 | group->a_is_minus3 = 1; |
1885 | return ret; |
1886 | } |
1887 | |
1888 | int ec_GFp_nistp256_group_set_curve(EC_GROUP *group, const BIGNUM *p, |
1889 | const BIGNUM *a, const BIGNUM *b, |
1890 | BN_CTX *ctx) |
1891 | { |
1892 | int ret = 0; |
1893 | BIGNUM *curve_p, *curve_a, *curve_b; |
1894 | #ifndef FIPS_MODE |
1895 | BN_CTX *new_ctx = NULL; |
1896 | |
1897 | if (ctx == NULL) |
1898 | ctx = new_ctx = BN_CTX_new(); |
1899 | #endif |
1900 | if (ctx == NULL) |
1901 | return 0; |
1902 | |
1903 | BN_CTX_start(ctx); |
1904 | curve_p = BN_CTX_get(ctx); |
1905 | curve_a = BN_CTX_get(ctx); |
1906 | curve_b = BN_CTX_get(ctx); |
1907 | if (curve_b == NULL) |
1908 | goto err; |
1909 | BN_bin2bn(nistp256_curve_params[0], sizeof(felem_bytearray), curve_p); |
1910 | BN_bin2bn(nistp256_curve_params[1], sizeof(felem_bytearray), curve_a); |
1911 | BN_bin2bn(nistp256_curve_params[2], sizeof(felem_bytearray), curve_b); |
1912 | if ((BN_cmp(curve_p, p)) || (BN_cmp(curve_a, a)) || (BN_cmp(curve_b, b))) { |
1913 | ECerr(EC_F_EC_GFP_NISTP256_GROUP_SET_CURVE, |
1914 | EC_R_WRONG_CURVE_PARAMETERS); |
1915 | goto err; |
1916 | } |
1917 | group->field_mod_func = BN_nist_mod_256; |
1918 | ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx); |
1919 | err: |
1920 | BN_CTX_end(ctx); |
1921 | #ifndef FIPS_MODE |
1922 | BN_CTX_free(new_ctx); |
1923 | #endif |
1924 | return ret; |
1925 | } |
1926 | |
1927 | /* |
1928 | * Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') = |
1929 | * (X/Z^2, Y/Z^3) |
1930 | */ |
1931 | int ec_GFp_nistp256_point_get_affine_coordinates(const EC_GROUP *group, |
1932 | const EC_POINT *point, |
1933 | BIGNUM *x, BIGNUM *y, |
1934 | BN_CTX *ctx) |
1935 | { |
1936 | felem z1, z2, x_in, y_in; |
1937 | smallfelem x_out, y_out; |
1938 | longfelem tmp; |
1939 | |
1940 | if (EC_POINT_is_at_infinity(group, point)) { |
1941 | ECerr(EC_F_EC_GFP_NISTP256_POINT_GET_AFFINE_COORDINATES, |
1942 | EC_R_POINT_AT_INFINITY); |
1943 | return 0; |
1944 | } |
1945 | if ((!BN_to_felem(x_in, point->X)) || (!BN_to_felem(y_in, point->Y)) || |
1946 | (!BN_to_felem(z1, point->Z))) |
1947 | return 0; |
1948 | felem_inv(z2, z1); |
1949 | felem_square(tmp, z2); |
1950 | felem_reduce(z1, tmp); |
1951 | felem_mul(tmp, x_in, z1); |
1952 | felem_reduce(x_in, tmp); |
1953 | felem_contract(x_out, x_in); |
1954 | if (x != NULL) { |
1955 | if (!smallfelem_to_BN(x, x_out)) { |
1956 | ECerr(EC_F_EC_GFP_NISTP256_POINT_GET_AFFINE_COORDINATES, |
1957 | ERR_R_BN_LIB); |
1958 | return 0; |
1959 | } |
1960 | } |
1961 | felem_mul(tmp, z1, z2); |
1962 | felem_reduce(z1, tmp); |
1963 | felem_mul(tmp, y_in, z1); |
1964 | felem_reduce(y_in, tmp); |
1965 | felem_contract(y_out, y_in); |
1966 | if (y != NULL) { |
1967 | if (!smallfelem_to_BN(y, y_out)) { |
1968 | ECerr(EC_F_EC_GFP_NISTP256_POINT_GET_AFFINE_COORDINATES, |
1969 | ERR_R_BN_LIB); |
1970 | return 0; |
1971 | } |
1972 | } |
1973 | return 1; |
1974 | } |
1975 | |
1976 | /* points below is of size |num|, and tmp_smallfelems is of size |num+1| */ |
1977 | static void make_points_affine(size_t num, smallfelem points[][3], |
1978 | smallfelem tmp_smallfelems[]) |
1979 | { |
1980 | /* |
1981 | * Runs in constant time, unless an input is the point at infinity (which |
1982 | * normally shouldn't happen). |
1983 | */ |
1984 | ec_GFp_nistp_points_make_affine_internal(num, |
1985 | points, |
1986 | sizeof(smallfelem), |
1987 | tmp_smallfelems, |
1988 | (void (*)(void *))smallfelem_one, |
1989 | smallfelem_is_zero_int, |
1990 | (void (*)(void *, const void *)) |
1991 | smallfelem_assign, |
1992 | (void (*)(void *, const void *)) |
1993 | smallfelem_square_contract, |
1994 | (void (*) |
1995 | (void *, const void *, |
1996 | const void *)) |
1997 | smallfelem_mul_contract, |
1998 | (void (*)(void *, const void *)) |
1999 | smallfelem_inv_contract, |
2000 | /* nothing to contract */ |
2001 | (void (*)(void *, const void *)) |
2002 | smallfelem_assign); |
2003 | } |
2004 | |
2005 | /* |
2006 | * Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL |
2007 | * values Result is stored in r (r can equal one of the inputs). |
2008 | */ |
2009 | int ec_GFp_nistp256_points_mul(const EC_GROUP *group, EC_POINT *r, |
2010 | const BIGNUM *scalar, size_t num, |
2011 | const EC_POINT *points[], |
2012 | const BIGNUM *scalars[], BN_CTX *ctx) |
2013 | { |
2014 | int ret = 0; |
2015 | int j; |
2016 | int mixed = 0; |
2017 | BIGNUM *x, *y, *z, *tmp_scalar; |
2018 | felem_bytearray g_secret; |
2019 | felem_bytearray *secrets = NULL; |
2020 | smallfelem (*pre_comp)[17][3] = NULL; |
2021 | smallfelem *tmp_smallfelems = NULL; |
2022 | unsigned i; |
2023 | int num_bytes; |
2024 | int have_pre_comp = 0; |
2025 | size_t num_points = num; |
2026 | smallfelem x_in, y_in, z_in; |
2027 | felem x_out, y_out, z_out; |
2028 | NISTP256_PRE_COMP *pre = NULL; |
2029 | const smallfelem(*g_pre_comp)[16][3] = NULL; |
2030 | EC_POINT *generator = NULL; |
2031 | const EC_POINT *p = NULL; |
2032 | const BIGNUM *p_scalar = NULL; |
2033 | |
2034 | BN_CTX_start(ctx); |
2035 | x = BN_CTX_get(ctx); |
2036 | y = BN_CTX_get(ctx); |
2037 | z = BN_CTX_get(ctx); |
2038 | tmp_scalar = BN_CTX_get(ctx); |
2039 | if (tmp_scalar == NULL) |
2040 | goto err; |
2041 | |
2042 | if (scalar != NULL) { |
2043 | pre = group->pre_comp.nistp256; |
2044 | if (pre) |
2045 | /* we have precomputation, try to use it */ |
2046 | g_pre_comp = (const smallfelem(*)[16][3])pre->g_pre_comp; |
2047 | else |
2048 | /* try to use the standard precomputation */ |
2049 | g_pre_comp = &gmul[0]; |
2050 | generator = EC_POINT_new(group); |
2051 | if (generator == NULL) |
2052 | goto err; |
2053 | /* get the generator from precomputation */ |
2054 | if (!smallfelem_to_BN(x, g_pre_comp[0][1][0]) || |
2055 | !smallfelem_to_BN(y, g_pre_comp[0][1][1]) || |
2056 | !smallfelem_to_BN(z, g_pre_comp[0][1][2])) { |
2057 | ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL, ERR_R_BN_LIB); |
2058 | goto err; |
2059 | } |
2060 | if (!EC_POINT_set_Jprojective_coordinates_GFp(group, |
2061 | generator, x, y, z, |
2062 | ctx)) |
2063 | goto err; |
2064 | if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) |
2065 | /* precomputation matches generator */ |
2066 | have_pre_comp = 1; |
2067 | else |
2068 | /* |
2069 | * we don't have valid precomputation: treat the generator as a |
2070 | * random point |
2071 | */ |
2072 | num_points++; |
2073 | } |
2074 | if (num_points > 0) { |
2075 | if (num_points >= 3) { |
2076 | /* |
2077 | * unless we precompute multiples for just one or two points, |
2078 | * converting those into affine form is time well spent |
2079 | */ |
2080 | mixed = 1; |
2081 | } |
2082 | secrets = OPENSSL_malloc(sizeof(*secrets) * num_points); |
2083 | pre_comp = OPENSSL_malloc(sizeof(*pre_comp) * num_points); |
2084 | if (mixed) |
2085 | tmp_smallfelems = |
2086 | OPENSSL_malloc(sizeof(*tmp_smallfelems) * (num_points * 17 + 1)); |
2087 | if ((secrets == NULL) || (pre_comp == NULL) |
2088 | || (mixed && (tmp_smallfelems == NULL))) { |
2089 | ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL, ERR_R_MALLOC_FAILURE); |
2090 | goto err; |
2091 | } |
2092 | |
2093 | /* |
2094 | * we treat NULL scalars as 0, and NULL points as points at infinity, |
2095 | * i.e., they contribute nothing to the linear combination |
2096 | */ |
2097 | memset(secrets, 0, sizeof(*secrets) * num_points); |
2098 | memset(pre_comp, 0, sizeof(*pre_comp) * num_points); |
2099 | for (i = 0; i < num_points; ++i) { |
2100 | if (i == num) { |
2101 | /* |
2102 | * we didn't have a valid precomputation, so we pick the |
2103 | * generator |
2104 | */ |
2105 | p = EC_GROUP_get0_generator(group); |
2106 | p_scalar = scalar; |
2107 | } else { |
2108 | /* the i^th point */ |
2109 | p = points[i]; |
2110 | p_scalar = scalars[i]; |
2111 | } |
2112 | if ((p_scalar != NULL) && (p != NULL)) { |
2113 | /* reduce scalar to 0 <= scalar < 2^256 */ |
2114 | if ((BN_num_bits(p_scalar) > 256) |
2115 | || (BN_is_negative(p_scalar))) { |
2116 | /* |
2117 | * this is an unusual input, and we don't guarantee |
2118 | * constant-timeness |
2119 | */ |
2120 | if (!BN_nnmod(tmp_scalar, p_scalar, group->order, ctx)) { |
2121 | ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL, ERR_R_BN_LIB); |
2122 | goto err; |
2123 | } |
2124 | num_bytes = BN_bn2lebinpad(tmp_scalar, |
2125 | secrets[i], sizeof(secrets[i])); |
2126 | } else { |
2127 | num_bytes = BN_bn2lebinpad(p_scalar, |
2128 | secrets[i], sizeof(secrets[i])); |
2129 | } |
2130 | if (num_bytes < 0) { |
2131 | ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL, ERR_R_BN_LIB); |
2132 | goto err; |
2133 | } |
2134 | /* precompute multiples */ |
2135 | if ((!BN_to_felem(x_out, p->X)) || |
2136 | (!BN_to_felem(y_out, p->Y)) || |
2137 | (!BN_to_felem(z_out, p->Z))) |
2138 | goto err; |
2139 | felem_shrink(pre_comp[i][1][0], x_out); |
2140 | felem_shrink(pre_comp[i][1][1], y_out); |
2141 | felem_shrink(pre_comp[i][1][2], z_out); |
2142 | for (j = 2; j <= 16; ++j) { |
2143 | if (j & 1) { |
2144 | point_add_small(pre_comp[i][j][0], pre_comp[i][j][1], |
2145 | pre_comp[i][j][2], pre_comp[i][1][0], |
2146 | pre_comp[i][1][1], pre_comp[i][1][2], |
2147 | pre_comp[i][j - 1][0], |
2148 | pre_comp[i][j - 1][1], |
2149 | pre_comp[i][j - 1][2]); |
2150 | } else { |
2151 | point_double_small(pre_comp[i][j][0], |
2152 | pre_comp[i][j][1], |
2153 | pre_comp[i][j][2], |
2154 | pre_comp[i][j / 2][0], |
2155 | pre_comp[i][j / 2][1], |
2156 | pre_comp[i][j / 2][2]); |
2157 | } |
2158 | } |
2159 | } |
2160 | } |
2161 | if (mixed) |
2162 | make_points_affine(num_points * 17, pre_comp[0], tmp_smallfelems); |
2163 | } |
2164 | |
2165 | /* the scalar for the generator */ |
2166 | if ((scalar != NULL) && (have_pre_comp)) { |
2167 | memset(g_secret, 0, sizeof(g_secret)); |
2168 | /* reduce scalar to 0 <= scalar < 2^256 */ |
2169 | if ((BN_num_bits(scalar) > 256) || (BN_is_negative(scalar))) { |
2170 | /* |
2171 | * this is an unusual input, and we don't guarantee |
2172 | * constant-timeness |
2173 | */ |
2174 | if (!BN_nnmod(tmp_scalar, scalar, group->order, ctx)) { |
2175 | ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL, ERR_R_BN_LIB); |
2176 | goto err; |
2177 | } |
2178 | num_bytes = BN_bn2lebinpad(tmp_scalar, g_secret, sizeof(g_secret)); |
2179 | } else { |
2180 | num_bytes = BN_bn2lebinpad(scalar, g_secret, sizeof(g_secret)); |
2181 | } |
2182 | /* do the multiplication with generator precomputation */ |
2183 | batch_mul(x_out, y_out, z_out, |
2184 | (const felem_bytearray(*))secrets, num_points, |
2185 | g_secret, |
2186 | mixed, (const smallfelem(*)[17][3])pre_comp, g_pre_comp); |
2187 | } else { |
2188 | /* do the multiplication without generator precomputation */ |
2189 | batch_mul(x_out, y_out, z_out, |
2190 | (const felem_bytearray(*))secrets, num_points, |
2191 | NULL, mixed, (const smallfelem(*)[17][3])pre_comp, NULL); |
2192 | } |
2193 | /* reduce the output to its unique minimal representation */ |
2194 | felem_contract(x_in, x_out); |
2195 | felem_contract(y_in, y_out); |
2196 | felem_contract(z_in, z_out); |
2197 | if ((!smallfelem_to_BN(x, x_in)) || (!smallfelem_to_BN(y, y_in)) || |
2198 | (!smallfelem_to_BN(z, z_in))) { |
2199 | ECerr(EC_F_EC_GFP_NISTP256_POINTS_MUL, ERR_R_BN_LIB); |
2200 | goto err; |
2201 | } |
2202 | ret = EC_POINT_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx); |
2203 | |
2204 | err: |
2205 | BN_CTX_end(ctx); |
2206 | EC_POINT_free(generator); |
2207 | OPENSSL_free(secrets); |
2208 | OPENSSL_free(pre_comp); |
2209 | OPENSSL_free(tmp_smallfelems); |
2210 | return ret; |
2211 | } |
2212 | |
2213 | int ec_GFp_nistp256_precompute_mult(EC_GROUP *group, BN_CTX *ctx) |
2214 | { |
2215 | int ret = 0; |
2216 | NISTP256_PRE_COMP *pre = NULL; |
2217 | int i, j; |
2218 | BIGNUM *x, *y; |
2219 | EC_POINT *generator = NULL; |
2220 | smallfelem tmp_smallfelems[32]; |
2221 | felem x_tmp, y_tmp, z_tmp; |
2222 | #ifndef FIPS_MODE |
2223 | BN_CTX *new_ctx = NULL; |
2224 | #endif |
2225 | |
2226 | /* throw away old precomputation */ |
2227 | EC_pre_comp_free(group); |
2228 | |
2229 | #ifndef FIPS_MODE |
2230 | if (ctx == NULL) |
2231 | ctx = new_ctx = BN_CTX_new(); |
2232 | #endif |
2233 | if (ctx == NULL) |
2234 | return 0; |
2235 | |
2236 | BN_CTX_start(ctx); |
2237 | x = BN_CTX_get(ctx); |
2238 | y = BN_CTX_get(ctx); |
2239 | if (y == NULL) |
2240 | goto err; |
2241 | /* get the generator */ |
2242 | if (group->generator == NULL) |
2243 | goto err; |
2244 | generator = EC_POINT_new(group); |
2245 | if (generator == NULL) |
2246 | goto err; |
2247 | BN_bin2bn(nistp256_curve_params[3], sizeof(felem_bytearray), x); |
2248 | BN_bin2bn(nistp256_curve_params[4], sizeof(felem_bytearray), y); |
2249 | if (!EC_POINT_set_affine_coordinates(group, generator, x, y, ctx)) |
2250 | goto err; |
2251 | if ((pre = nistp256_pre_comp_new()) == NULL) |
2252 | goto err; |
2253 | /* |
2254 | * if the generator is the standard one, use built-in precomputation |
2255 | */ |
2256 | if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) { |
2257 | memcpy(pre->g_pre_comp, gmul, sizeof(pre->g_pre_comp)); |
2258 | goto done; |
2259 | } |
2260 | if ((!BN_to_felem(x_tmp, group->generator->X)) || |
2261 | (!BN_to_felem(y_tmp, group->generator->Y)) || |
2262 | (!BN_to_felem(z_tmp, group->generator->Z))) |
2263 | goto err; |
2264 | felem_shrink(pre->g_pre_comp[0][1][0], x_tmp); |
2265 | felem_shrink(pre->g_pre_comp[0][1][1], y_tmp); |
2266 | felem_shrink(pre->g_pre_comp[0][1][2], z_tmp); |
2267 | /* |
2268 | * compute 2^64*G, 2^128*G, 2^192*G for the first table, 2^32*G, 2^96*G, |
2269 | * 2^160*G, 2^224*G for the second one |
2270 | */ |
2271 | for (i = 1; i <= 8; i <<= 1) { |
2272 | point_double_small(pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1], |
2273 | pre->g_pre_comp[1][i][2], pre->g_pre_comp[0][i][0], |
2274 | pre->g_pre_comp[0][i][1], |
2275 | pre->g_pre_comp[0][i][2]); |
2276 | for (j = 0; j < 31; ++j) { |
2277 | point_double_small(pre->g_pre_comp[1][i][0], |
2278 | pre->g_pre_comp[1][i][1], |
2279 | pre->g_pre_comp[1][i][2], |
2280 | pre->g_pre_comp[1][i][0], |
2281 | pre->g_pre_comp[1][i][1], |
2282 | pre->g_pre_comp[1][i][2]); |
2283 | } |
2284 | if (i == 8) |
2285 | break; |
2286 | point_double_small(pre->g_pre_comp[0][2 * i][0], |
2287 | pre->g_pre_comp[0][2 * i][1], |
2288 | pre->g_pre_comp[0][2 * i][2], |
2289 | pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1], |
2290 | pre->g_pre_comp[1][i][2]); |
2291 | for (j = 0; j < 31; ++j) { |
2292 | point_double_small(pre->g_pre_comp[0][2 * i][0], |
2293 | pre->g_pre_comp[0][2 * i][1], |
2294 | pre->g_pre_comp[0][2 * i][2], |
2295 | pre->g_pre_comp[0][2 * i][0], |
2296 | pre->g_pre_comp[0][2 * i][1], |
2297 | pre->g_pre_comp[0][2 * i][2]); |
2298 | } |
2299 | } |
2300 | for (i = 0; i < 2; i++) { |
2301 | /* g_pre_comp[i][0] is the point at infinity */ |
2302 | memset(pre->g_pre_comp[i][0], 0, sizeof(pre->g_pre_comp[i][0])); |
2303 | /* the remaining multiples */ |
2304 | /* 2^64*G + 2^128*G resp. 2^96*G + 2^160*G */ |
2305 | point_add_small(pre->g_pre_comp[i][6][0], pre->g_pre_comp[i][6][1], |
2306 | pre->g_pre_comp[i][6][2], pre->g_pre_comp[i][4][0], |
2307 | pre->g_pre_comp[i][4][1], pre->g_pre_comp[i][4][2], |
2308 | pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1], |
2309 | pre->g_pre_comp[i][2][2]); |
2310 | /* 2^64*G + 2^192*G resp. 2^96*G + 2^224*G */ |
2311 | point_add_small(pre->g_pre_comp[i][10][0], pre->g_pre_comp[i][10][1], |
2312 | pre->g_pre_comp[i][10][2], pre->g_pre_comp[i][8][0], |
2313 | pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2], |
2314 | pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1], |
2315 | pre->g_pre_comp[i][2][2]); |
2316 | /* 2^128*G + 2^192*G resp. 2^160*G + 2^224*G */ |
2317 | point_add_small(pre->g_pre_comp[i][12][0], pre->g_pre_comp[i][12][1], |
2318 | pre->g_pre_comp[i][12][2], pre->g_pre_comp[i][8][0], |
2319 | pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2], |
2320 | pre->g_pre_comp[i][4][0], pre->g_pre_comp[i][4][1], |
2321 | pre->g_pre_comp[i][4][2]); |
2322 | /* |
2323 | * 2^64*G + 2^128*G + 2^192*G resp. 2^96*G + 2^160*G + 2^224*G |
2324 | */ |
2325 | point_add_small(pre->g_pre_comp[i][14][0], pre->g_pre_comp[i][14][1], |
2326 | pre->g_pre_comp[i][14][2], pre->g_pre_comp[i][12][0], |
2327 | pre->g_pre_comp[i][12][1], pre->g_pre_comp[i][12][2], |
2328 | pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1], |
2329 | pre->g_pre_comp[i][2][2]); |
2330 | for (j = 1; j < 8; ++j) { |
2331 | /* odd multiples: add G resp. 2^32*G */ |
2332 | point_add_small(pre->g_pre_comp[i][2 * j + 1][0], |
2333 | pre->g_pre_comp[i][2 * j + 1][1], |
2334 | pre->g_pre_comp[i][2 * j + 1][2], |
2335 | pre->g_pre_comp[i][2 * j][0], |
2336 | pre->g_pre_comp[i][2 * j][1], |
2337 | pre->g_pre_comp[i][2 * j][2], |
2338 | pre->g_pre_comp[i][1][0], |
2339 | pre->g_pre_comp[i][1][1], |
2340 | pre->g_pre_comp[i][1][2]); |
2341 | } |
2342 | } |
2343 | make_points_affine(31, &(pre->g_pre_comp[0][1]), tmp_smallfelems); |
2344 | |
2345 | done: |
2346 | SETPRECOMP(group, nistp256, pre); |
2347 | pre = NULL; |
2348 | ret = 1; |
2349 | |
2350 | err: |
2351 | BN_CTX_end(ctx); |
2352 | EC_POINT_free(generator); |
2353 | #ifndef FIPS_MODE |
2354 | BN_CTX_free(new_ctx); |
2355 | #endif |
2356 | EC_nistp256_pre_comp_free(pre); |
2357 | return ret; |
2358 | } |
2359 | |
2360 | int ec_GFp_nistp256_have_precompute_mult(const EC_GROUP *group) |
2361 | { |
2362 | return HAVEPRECOMP(group, nistp256); |
2363 | } |
2364 | #endif |
2365 | |