1 /* Copyright (c) 2015, Google Inc.
2 *
3 * Permission to use, copy, modify, and/or distribute this software for any
4 * purpose with or without fee is hereby granted, provided that the above
5 * copyright notice and this permission notice appear in all copies.
6 *
7 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
8 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
9 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY
10 * SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
11 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION
12 * OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN
13 * CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. */
14
15 /* A 64-bit implementation of the NIST P-256 elliptic curve point
16 * multiplication
17 *
18 * OpenSSL integration was taken from Emilia Kasper's work in ecp_nistp224.c.
19 * Otherwise based on Emilia's P224 work, which was inspired by my curve25519
20 * work which got its smarts from Daniel J. Bernstein's work on the same. */
21
22 #include <openssl/base.h>
23
24 #if defined(OPENSSL_64_BIT) && !defined(OPENSSL_WINDOWS)
25
26 #include <openssl/bn.h>
27 #include <openssl/ec.h>
28 #include <openssl/err.h>
29 #include <openssl/mem.h>
30 #include <openssl/obj.h>
31
32 #include <string.h>
33
34 #include "internal.h"
35
36
37 typedef uint8_t u8;
38 typedef uint64_t u64;
39 typedef int64_t s64;
40 typedef __uint128_t uint128_t;
41 typedef __int128_t int128_t;
42
43 /* The underlying field. P256 operates over GF(2^256-2^224+2^192+2^96-1). We
44 * can serialise an element of this field into 32 bytes. We call this an
45 * felem_bytearray. */
46 typedef u8 felem_bytearray[32];
47
48 /* These are the parameters of P256, taken from FIPS 186-3, page 86. These
49 * values are big-endian. */
50 static const felem_bytearray nistp256_curve_params[5] = {
51 {0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01, /* p */
52 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
53 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff},
54 {0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01, /* a = -3 */
55 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
56 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
57 0xfc}, /* b */
58 {0x5a, 0xc6, 0x35, 0xd8, 0xaa, 0x3a, 0x93, 0xe7, 0xb3, 0xeb, 0xbd, 0x55,
59 0x76, 0x98, 0x86, 0xbc, 0x65, 0x1d, 0x06, 0xb0, 0xcc, 0x53, 0xb0, 0xf6,
60 0x3b, 0xce, 0x3c, 0x3e, 0x27, 0xd2, 0x60, 0x4b},
61 {0x6b, 0x17, 0xd1, 0xf2, 0xe1, 0x2c, 0x42, 0x47, /* x */
62 0xf8, 0xbc, 0xe6, 0xe5, 0x63, 0xa4, 0x40, 0xf2, 0x77, 0x03, 0x7d, 0x81,
63 0x2d, 0xeb, 0x33, 0xa0, 0xf4, 0xa1, 0x39, 0x45, 0xd8, 0x98, 0xc2, 0x96},
64 {0x4f, 0xe3, 0x42, 0xe2, 0xfe, 0x1a, 0x7f, 0x9b, /* y */
65 0x8e, 0xe7, 0xeb, 0x4a, 0x7c, 0x0f, 0x9e, 0x16, 0x2b, 0xce, 0x33, 0x57,
66 0x6b, 0x31, 0x5e, 0xce, 0xcb, 0xb6, 0x40, 0x68, 0x37, 0xbf, 0x51, 0xf5}};
67
68 /* The representation of field elements.
69 * ------------------------------------
70 *
71 * We represent field elements with either four 128-bit values, eight 128-bit
72 * values, or four 64-bit values. The field element represented is:
73 * v[0]*2^0 + v[1]*2^64 + v[2]*2^128 + v[3]*2^192 (mod p)
74 * or:
75 * v[0]*2^0 + v[1]*2^64 + v[2]*2^128 + ... + v[8]*2^512 (mod p)
76 *
77 * 128-bit values are called 'limbs'. Since the limbs are spaced only 64 bits
78 * apart, but are 128-bits wide, the most significant bits of each limb overlap
79 * with the least significant bits of the next.
80 *
81 * A field element with four limbs is an 'felem'. One with eight limbs is a
82 * 'longfelem'
83 *
84 * A field element with four, 64-bit values is called a 'smallfelem'. Small
85 * values are used as intermediate values before multiplication. */
86
87 #define NLIMBS 4
88
89 typedef uint128_t limb;
90 typedef limb felem[NLIMBS];
91 typedef limb longfelem[NLIMBS * 2];
92 typedef u64 smallfelem[NLIMBS];
93
94 /* This is the value of the prime as four 64-bit words, little-endian. */
95 static const u64 kPrime[4] = {0xfffffffffffffffful, 0xffffffff, 0,
96 0xffffffff00000001ul};
97 static const u64 bottom63bits = 0x7ffffffffffffffful;
98
99 /* bin32_to_felem takes a little-endian byte array and converts it into felem
100 * form. This assumes that the CPU is little-endian. */
bin32_to_felem(felem out,const u8 in[32])101 static void bin32_to_felem(felem out, const u8 in[32]) {
102 out[0] = *((u64 *)&in[0]);
103 out[1] = *((u64 *)&in[8]);
104 out[2] = *((u64 *)&in[16]);
105 out[3] = *((u64 *)&in[24]);
106 }
107
108 /* smallfelem_to_bin32 takes a smallfelem and serialises into a little endian,
109 * 32 byte array. This assumes that the CPU is little-endian. */
smallfelem_to_bin32(u8 out[32],const smallfelem in)110 static void smallfelem_to_bin32(u8 out[32], const smallfelem in) {
111 *((u64 *)&out[0]) = in[0];
112 *((u64 *)&out[8]) = in[1];
113 *((u64 *)&out[16]) = in[2];
114 *((u64 *)&out[24]) = in[3];
115 }
116
117 /* To preserve endianness when using BN_bn2bin and BN_bin2bn. */
flip_endian(u8 * out,const u8 * in,unsigned len)118 static void flip_endian(u8 *out, const u8 *in, unsigned len) {
119 unsigned i;
120 for (i = 0; i < len; ++i) {
121 out[i] = in[len - 1 - i];
122 }
123 }
124
125 /* BN_to_felem converts an OpenSSL BIGNUM into an felem. */
BN_to_felem(felem out,const BIGNUM * bn)126 static int BN_to_felem(felem out, const BIGNUM *bn) {
127 if (BN_is_negative(bn)) {
128 OPENSSL_PUT_ERROR(EC, BN_to_felem, EC_R_BIGNUM_OUT_OF_RANGE);
129 return 0;
130 }
131
132 felem_bytearray b_out;
133 /* BN_bn2bin eats leading zeroes */
134 memset(b_out, 0, sizeof(b_out));
135 unsigned num_bytes = BN_num_bytes(bn);
136 if (num_bytes > sizeof(b_out)) {
137 OPENSSL_PUT_ERROR(EC, BN_to_felem, EC_R_BIGNUM_OUT_OF_RANGE);
138 return 0;
139 }
140
141 felem_bytearray b_in;
142 num_bytes = BN_bn2bin(bn, b_in);
143 flip_endian(b_out, b_in, num_bytes);
144 bin32_to_felem(out, b_out);
145 return 1;
146 }
147
148 /* felem_to_BN converts an felem into an OpenSSL BIGNUM. */
smallfelem_to_BN(BIGNUM * out,const smallfelem in)149 static BIGNUM *smallfelem_to_BN(BIGNUM *out, const smallfelem in) {
150 felem_bytearray b_in, b_out;
151 smallfelem_to_bin32(b_in, in);
152 flip_endian(b_out, b_in, sizeof(b_out));
153 return BN_bin2bn(b_out, sizeof(b_out), out);
154 }
155
156 /* Field operations. */
157
smallfelem_one(smallfelem out)158 static void smallfelem_one(smallfelem out) {
159 out[0] = 1;
160 out[1] = 0;
161 out[2] = 0;
162 out[3] = 0;
163 }
164
smallfelem_assign(smallfelem out,const smallfelem in)165 static void smallfelem_assign(smallfelem out, const smallfelem in) {
166 out[0] = in[0];
167 out[1] = in[1];
168 out[2] = in[2];
169 out[3] = in[3];
170 }
171
felem_assign(felem out,const felem in)172 static void felem_assign(felem out, const felem in) {
173 out[0] = in[0];
174 out[1] = in[1];
175 out[2] = in[2];
176 out[3] = in[3];
177 }
178
179 /* felem_sum sets out = out + in. */
felem_sum(felem out,const felem in)180 static void felem_sum(felem out, const felem in) {
181 out[0] += in[0];
182 out[1] += in[1];
183 out[2] += in[2];
184 out[3] += in[3];
185 }
186
187 /* felem_small_sum sets out = out + in. */
felem_small_sum(felem out,const smallfelem in)188 static void felem_small_sum(felem out, const smallfelem in) {
189 out[0] += in[0];
190 out[1] += in[1];
191 out[2] += in[2];
192 out[3] += in[3];
193 }
194
195 /* felem_scalar sets out = out * scalar */
felem_scalar(felem out,const u64 scalar)196 static void felem_scalar(felem out, const u64 scalar) {
197 out[0] *= scalar;
198 out[1] *= scalar;
199 out[2] *= scalar;
200 out[3] *= scalar;
201 }
202
203 /* longfelem_scalar sets out = out * scalar */
longfelem_scalar(longfelem out,const u64 scalar)204 static void longfelem_scalar(longfelem out, const u64 scalar) {
205 out[0] *= scalar;
206 out[1] *= scalar;
207 out[2] *= scalar;
208 out[3] *= scalar;
209 out[4] *= scalar;
210 out[5] *= scalar;
211 out[6] *= scalar;
212 out[7] *= scalar;
213 }
214
215 #define two105m41m9 (((limb)1) << 105) - (((limb)1) << 41) - (((limb)1) << 9)
216 #define two105 (((limb)1) << 105)
217 #define two105m41p9 (((limb)1) << 105) - (((limb)1) << 41) + (((limb)1) << 9)
218
219 /* zero105 is 0 mod p */
220 static const felem zero105 = {two105m41m9, two105, two105m41p9, two105m41p9};
221
222 /* smallfelem_neg sets |out| to |-small|
223 * On exit:
224 * out[i] < out[i] + 2^105 */
smallfelem_neg(felem out,const smallfelem small)225 static void smallfelem_neg(felem out, const smallfelem small) {
226 /* In order to prevent underflow, we subtract from 0 mod p. */
227 out[0] = zero105[0] - small[0];
228 out[1] = zero105[1] - small[1];
229 out[2] = zero105[2] - small[2];
230 out[3] = zero105[3] - small[3];
231 }
232
233 /* felem_diff subtracts |in| from |out|
234 * On entry:
235 * in[i] < 2^104
236 * On exit:
237 * out[i] < out[i] + 2^105. */
felem_diff(felem out,const felem in)238 static void felem_diff(felem out, const felem in) {
239 /* In order to prevent underflow, we add 0 mod p before subtracting. */
240 out[0] += zero105[0];
241 out[1] += zero105[1];
242 out[2] += zero105[2];
243 out[3] += zero105[3];
244
245 out[0] -= in[0];
246 out[1] -= in[1];
247 out[2] -= in[2];
248 out[3] -= in[3];
249 }
250
251 #define two107m43m11 (((limb)1) << 107) - (((limb)1) << 43) - (((limb)1) << 11)
252 #define two107 (((limb)1) << 107)
253 #define two107m43p11 (((limb)1) << 107) - (((limb)1) << 43) + (((limb)1) << 11)
254
255 /* zero107 is 0 mod p */
256 static const felem zero107 = {two107m43m11, two107, two107m43p11, two107m43p11};
257
258 /* An alternative felem_diff for larger inputs |in|
259 * felem_diff_zero107 subtracts |in| from |out|
260 * On entry:
261 * in[i] < 2^106
262 * On exit:
263 * out[i] < out[i] + 2^107. */
felem_diff_zero107(felem out,const felem in)264 static void felem_diff_zero107(felem out, const felem in) {
265 /* In order to prevent underflow, we add 0 mod p before subtracting. */
266 out[0] += zero107[0];
267 out[1] += zero107[1];
268 out[2] += zero107[2];
269 out[3] += zero107[3];
270
271 out[0] -= in[0];
272 out[1] -= in[1];
273 out[2] -= in[2];
274 out[3] -= in[3];
275 }
276
277 /* longfelem_diff subtracts |in| from |out|
278 * On entry:
279 * in[i] < 7*2^67
280 * On exit:
281 * out[i] < out[i] + 2^70 + 2^40. */
longfelem_diff(longfelem out,const longfelem in)282 static void longfelem_diff(longfelem out, const longfelem in) {
283 static const limb two70m8p6 =
284 (((limb)1) << 70) - (((limb)1) << 8) + (((limb)1) << 6);
285 static const limb two70p40 = (((limb)1) << 70) + (((limb)1) << 40);
286 static const limb two70 = (((limb)1) << 70);
287 static const limb two70m40m38p6 = (((limb)1) << 70) - (((limb)1) << 40) -
288 (((limb)1) << 38) + (((limb)1) << 6);
289 static const limb two70m6 = (((limb)1) << 70) - (((limb)1) << 6);
290
291 /* add 0 mod p to avoid underflow */
292 out[0] += two70m8p6;
293 out[1] += two70p40;
294 out[2] += two70;
295 out[3] += two70m40m38p6;
296 out[4] += two70m6;
297 out[5] += two70m6;
298 out[6] += two70m6;
299 out[7] += two70m6;
300
301 /* in[i] < 7*2^67 < 2^70 - 2^40 - 2^38 + 2^6 */
302 out[0] -= in[0];
303 out[1] -= in[1];
304 out[2] -= in[2];
305 out[3] -= in[3];
306 out[4] -= in[4];
307 out[5] -= in[5];
308 out[6] -= in[6];
309 out[7] -= in[7];
310 }
311
312 #define two64m0 (((limb)1) << 64) - 1
313 #define two110p32m0 (((limb)1) << 110) + (((limb)1) << 32) - 1
314 #define two64m46 (((limb)1) << 64) - (((limb)1) << 46)
315 #define two64m32 (((limb)1) << 64) - (((limb)1) << 32)
316
317 /* zero110 is 0 mod p. */
318 static const felem zero110 = {two64m0, two110p32m0, two64m46, two64m32};
319
320 /* felem_shrink converts an felem into a smallfelem. The result isn't quite
321 * minimal as the value may be greater than p.
322 *
323 * On entry:
324 * in[i] < 2^109
325 * On exit:
326 * out[i] < 2^64. */
felem_shrink(smallfelem out,const felem in)327 static void felem_shrink(smallfelem out, const felem in) {
328 felem tmp;
329 u64 a, b, mask;
330 s64 high, low;
331 static const u64 kPrime3Test = 0x7fffffff00000001ul; /* 2^63 - 2^32 + 1 */
332
333 /* Carry 2->3 */
334 tmp[3] = zero110[3] + in[3] + ((u64)(in[2] >> 64));
335 /* tmp[3] < 2^110 */
336
337 tmp[2] = zero110[2] + (u64)in[2];
338 tmp[0] = zero110[0] + in[0];
339 tmp[1] = zero110[1] + in[1];
340 /* tmp[0] < 2**110, tmp[1] < 2^111, tmp[2] < 2**65 */
341
342 /* We perform two partial reductions where we eliminate the high-word of
343 * tmp[3]. We don't update the other words till the end. */
344 a = tmp[3] >> 64; /* a < 2^46 */
345 tmp[3] = (u64)tmp[3];
346 tmp[3] -= a;
347 tmp[3] += ((limb)a) << 32;
348 /* tmp[3] < 2^79 */
349
350 b = a;
351 a = tmp[3] >> 64; /* a < 2^15 */
352 b += a; /* b < 2^46 + 2^15 < 2^47 */
353 tmp[3] = (u64)tmp[3];
354 tmp[3] -= a;
355 tmp[3] += ((limb)a) << 32;
356 /* tmp[3] < 2^64 + 2^47 */
357
358 /* This adjusts the other two words to complete the two partial
359 * reductions. */
360 tmp[0] += b;
361 tmp[1] -= (((limb)b) << 32);
362
363 /* In order to make space in tmp[3] for the carry from 2 -> 3, we
364 * conditionally subtract kPrime if tmp[3] is large enough. */
365 high = tmp[3] >> 64;
366 /* As tmp[3] < 2^65, high is either 1 or 0 */
367 high <<= 63;
368 high >>= 63;
369 /* high is:
370 * all ones if the high word of tmp[3] is 1
371 * all zeros if the high word of tmp[3] if 0 */
372 low = tmp[3];
373 mask = low >> 63;
374 /* mask is:
375 * all ones if the MSB of low is 1
376 * all zeros if the MSB of low if 0 */
377 low &= bottom63bits;
378 low -= kPrime3Test;
379 /* if low was greater than kPrime3Test then the MSB is zero */
380 low = ~low;
381 low >>= 63;
382 /* low is:
383 * all ones if low was > kPrime3Test
384 * all zeros if low was <= kPrime3Test */
385 mask = (mask & low) | high;
386 tmp[0] -= mask & kPrime[0];
387 tmp[1] -= mask & kPrime[1];
388 /* kPrime[2] is zero, so omitted */
389 tmp[3] -= mask & kPrime[3];
390 /* tmp[3] < 2**64 - 2**32 + 1 */
391
392 tmp[1] += ((u64)(tmp[0] >> 64));
393 tmp[0] = (u64)tmp[0];
394 tmp[2] += ((u64)(tmp[1] >> 64));
395 tmp[1] = (u64)tmp[1];
396 tmp[3] += ((u64)(tmp[2] >> 64));
397 tmp[2] = (u64)tmp[2];
398 /* tmp[i] < 2^64 */
399
400 out[0] = tmp[0];
401 out[1] = tmp[1];
402 out[2] = tmp[2];
403 out[3] = tmp[3];
404 }
405
406 /* smallfelem_expand converts a smallfelem to an felem */
smallfelem_expand(felem out,const smallfelem in)407 static void smallfelem_expand(felem out, const smallfelem in) {
408 out[0] = in[0];
409 out[1] = in[1];
410 out[2] = in[2];
411 out[3] = in[3];
412 }
413
414 /* smallfelem_square sets |out| = |small|^2
415 * On entry:
416 * small[i] < 2^64
417 * On exit:
418 * out[i] < 7 * 2^64 < 2^67 */
smallfelem_square(longfelem out,const smallfelem small)419 static void smallfelem_square(longfelem out, const smallfelem small) {
420 limb a;
421 u64 high, low;
422
423 a = ((uint128_t)small[0]) * small[0];
424 low = a;
425 high = a >> 64;
426 out[0] = low;
427 out[1] = high;
428
429 a = ((uint128_t)small[0]) * small[1];
430 low = a;
431 high = a >> 64;
432 out[1] += low;
433 out[1] += low;
434 out[2] = high;
435
436 a = ((uint128_t)small[0]) * small[2];
437 low = a;
438 high = a >> 64;
439 out[2] += low;
440 out[2] *= 2;
441 out[3] = high;
442
443 a = ((uint128_t)small[0]) * small[3];
444 low = a;
445 high = a >> 64;
446 out[3] += low;
447 out[4] = high;
448
449 a = ((uint128_t)small[1]) * small[2];
450 low = a;
451 high = a >> 64;
452 out[3] += low;
453 out[3] *= 2;
454 out[4] += high;
455
456 a = ((uint128_t)small[1]) * small[1];
457 low = a;
458 high = a >> 64;
459 out[2] += low;
460 out[3] += high;
461
462 a = ((uint128_t)small[1]) * small[3];
463 low = a;
464 high = a >> 64;
465 out[4] += low;
466 out[4] *= 2;
467 out[5] = high;
468
469 a = ((uint128_t)small[2]) * small[3];
470 low = a;
471 high = a >> 64;
472 out[5] += low;
473 out[5] *= 2;
474 out[6] = high;
475 out[6] += high;
476
477 a = ((uint128_t)small[2]) * small[2];
478 low = a;
479 high = a >> 64;
480 out[4] += low;
481 out[5] += high;
482
483 a = ((uint128_t)small[3]) * small[3];
484 low = a;
485 high = a >> 64;
486 out[6] += low;
487 out[7] = high;
488 }
489
490 /*felem_square sets |out| = |in|^2
491 * On entry:
492 * in[i] < 2^109
493 * On exit:
494 * out[i] < 7 * 2^64 < 2^67. */
felem_square(longfelem out,const felem in)495 static void felem_square(longfelem out, const felem in) {
496 u64 small[4];
497 felem_shrink(small, in);
498 smallfelem_square(out, small);
499 }
500
501 /* smallfelem_mul sets |out| = |small1| * |small2|
502 * On entry:
503 * small1[i] < 2^64
504 * small2[i] < 2^64
505 * On exit:
506 * out[i] < 7 * 2^64 < 2^67. */
smallfelem_mul(longfelem out,const smallfelem small1,const smallfelem small2)507 static void smallfelem_mul(longfelem out, const smallfelem small1,
508 const smallfelem small2) {
509 limb a;
510 u64 high, low;
511
512 a = ((uint128_t)small1[0]) * small2[0];
513 low = a;
514 high = a >> 64;
515 out[0] = low;
516 out[1] = high;
517
518 a = ((uint128_t)small1[0]) * small2[1];
519 low = a;
520 high = a >> 64;
521 out[1] += low;
522 out[2] = high;
523
524 a = ((uint128_t)small1[1]) * small2[0];
525 low = a;
526 high = a >> 64;
527 out[1] += low;
528 out[2] += high;
529
530 a = ((uint128_t)small1[0]) * small2[2];
531 low = a;
532 high = a >> 64;
533 out[2] += low;
534 out[3] = high;
535
536 a = ((uint128_t)small1[1]) * small2[1];
537 low = a;
538 high = a >> 64;
539 out[2] += low;
540 out[3] += high;
541
542 a = ((uint128_t)small1[2]) * small2[0];
543 low = a;
544 high = a >> 64;
545 out[2] += low;
546 out[3] += high;
547
548 a = ((uint128_t)small1[0]) * small2[3];
549 low = a;
550 high = a >> 64;
551 out[3] += low;
552 out[4] = high;
553
554 a = ((uint128_t)small1[1]) * small2[2];
555 low = a;
556 high = a >> 64;
557 out[3] += low;
558 out[4] += high;
559
560 a = ((uint128_t)small1[2]) * small2[1];
561 low = a;
562 high = a >> 64;
563 out[3] += low;
564 out[4] += high;
565
566 a = ((uint128_t)small1[3]) * small2[0];
567 low = a;
568 high = a >> 64;
569 out[3] += low;
570 out[4] += high;
571
572 a = ((uint128_t)small1[1]) * small2[3];
573 low = a;
574 high = a >> 64;
575 out[4] += low;
576 out[5] = high;
577
578 a = ((uint128_t)small1[2]) * small2[2];
579 low = a;
580 high = a >> 64;
581 out[4] += low;
582 out[5] += high;
583
584 a = ((uint128_t)small1[3]) * small2[1];
585 low = a;
586 high = a >> 64;
587 out[4] += low;
588 out[5] += high;
589
590 a = ((uint128_t)small1[2]) * small2[3];
591 low = a;
592 high = a >> 64;
593 out[5] += low;
594 out[6] = high;
595
596 a = ((uint128_t)small1[3]) * small2[2];
597 low = a;
598 high = a >> 64;
599 out[5] += low;
600 out[6] += high;
601
602 a = ((uint128_t)small1[3]) * small2[3];
603 low = a;
604 high = a >> 64;
605 out[6] += low;
606 out[7] = high;
607 }
608
609 /* felem_mul sets |out| = |in1| * |in2|
610 * On entry:
611 * in1[i] < 2^109
612 * in2[i] < 2^109
613 * On exit:
614 * out[i] < 7 * 2^64 < 2^67 */
felem_mul(longfelem out,const felem in1,const felem in2)615 static void felem_mul(longfelem out, const felem in1, const felem in2) {
616 smallfelem small1, small2;
617 felem_shrink(small1, in1);
618 felem_shrink(small2, in2);
619 smallfelem_mul(out, small1, small2);
620 }
621
622 /* felem_small_mul sets |out| = |small1| * |in2|
623 * On entry:
624 * small1[i] < 2^64
625 * in2[i] < 2^109
626 * On exit:
627 * out[i] < 7 * 2^64 < 2^67 */
felem_small_mul(longfelem out,const smallfelem small1,const felem in2)628 static void felem_small_mul(longfelem out, const smallfelem small1,
629 const felem in2) {
630 smallfelem small2;
631 felem_shrink(small2, in2);
632 smallfelem_mul(out, small1, small2);
633 }
634
635 #define two100m36m4 (((limb)1) << 100) - (((limb)1) << 36) - (((limb)1) << 4)
636 #define two100 (((limb)1) << 100)
637 #define two100m36p4 (((limb)1) << 100) - (((limb)1) << 36) + (((limb)1) << 4)
638
639 /* zero100 is 0 mod p */
640 static const felem zero100 = {two100m36m4, two100, two100m36p4, two100m36p4};
641
642 /* Internal function for the different flavours of felem_reduce.
643 * felem_reduce_ reduces the higher coefficients in[4]-in[7].
644 * On entry:
645 * out[0] >= in[6] + 2^32*in[6] + in[7] + 2^32*in[7]
646 * out[1] >= in[7] + 2^32*in[4]
647 * out[2] >= in[5] + 2^32*in[5]
648 * out[3] >= in[4] + 2^32*in[5] + 2^32*in[6]
649 * On exit:
650 * out[0] <= out[0] + in[4] + 2^32*in[5]
651 * out[1] <= out[1] + in[5] + 2^33*in[6]
652 * out[2] <= out[2] + in[7] + 2*in[6] + 2^33*in[7]
653 * out[3] <= out[3] + 2^32*in[4] + 3*in[7] */
felem_reduce_(felem out,const longfelem in)654 static void felem_reduce_(felem out, const longfelem in) {
655 int128_t c;
656 /* combine common terms from below */
657 c = in[4] + (in[5] << 32);
658 out[0] += c;
659 out[3] -= c;
660
661 c = in[5] - in[7];
662 out[1] += c;
663 out[2] -= c;
664
665 /* the remaining terms */
666 /* 256: [(0,1),(96,-1),(192,-1),(224,1)] */
667 out[1] -= (in[4] << 32);
668 out[3] += (in[4] << 32);
669
670 /* 320: [(32,1),(64,1),(128,-1),(160,-1),(224,-1)] */
671 out[2] -= (in[5] << 32);
672
673 /* 384: [(0,-1),(32,-1),(96,2),(128,2),(224,-1)] */
674 out[0] -= in[6];
675 out[0] -= (in[6] << 32);
676 out[1] += (in[6] << 33);
677 out[2] += (in[6] * 2);
678 out[3] -= (in[6] << 32);
679
680 /* 448: [(0,-1),(32,-1),(64,-1),(128,1),(160,2),(192,3)] */
681 out[0] -= in[7];
682 out[0] -= (in[7] << 32);
683 out[2] += (in[7] << 33);
684 out[3] += (in[7] * 3);
685 }
686
687 /* felem_reduce converts a longfelem into an felem.
688 * To be called directly after felem_square or felem_mul.
689 * On entry:
690 * in[0] < 2^64, in[1] < 3*2^64, in[2] < 5*2^64, in[3] < 7*2^64
691 * in[4] < 7*2^64, in[5] < 5*2^64, in[6] < 3*2^64, in[7] < 2*64
692 * On exit:
693 * out[i] < 2^101 */
felem_reduce(felem out,const longfelem in)694 static void felem_reduce(felem out, const longfelem in) {
695 out[0] = zero100[0] + in[0];
696 out[1] = zero100[1] + in[1];
697 out[2] = zero100[2] + in[2];
698 out[3] = zero100[3] + in[3];
699
700 felem_reduce_(out, in);
701
702 /* out[0] > 2^100 - 2^36 - 2^4 - 3*2^64 - 3*2^96 - 2^64 - 2^96 > 0
703 * out[1] > 2^100 - 2^64 - 7*2^96 > 0
704 * out[2] > 2^100 - 2^36 + 2^4 - 5*2^64 - 5*2^96 > 0
705 * out[3] > 2^100 - 2^36 + 2^4 - 7*2^64 - 5*2^96 - 3*2^96 > 0
706 *
707 * out[0] < 2^100 + 2^64 + 7*2^64 + 5*2^96 < 2^101
708 * out[1] < 2^100 + 3*2^64 + 5*2^64 + 3*2^97 < 2^101
709 * out[2] < 2^100 + 5*2^64 + 2^64 + 3*2^65 + 2^97 < 2^101
710 * out[3] < 2^100 + 7*2^64 + 7*2^96 + 3*2^64 < 2^101 */
711 }
712
713 /* felem_reduce_zero105 converts a larger longfelem into an felem.
714 * On entry:
715 * in[0] < 2^71
716 * On exit:
717 * out[i] < 2^106 */
felem_reduce_zero105(felem out,const longfelem in)718 static void felem_reduce_zero105(felem out, const longfelem in) {
719 out[0] = zero105[0] + in[0];
720 out[1] = zero105[1] + in[1];
721 out[2] = zero105[2] + in[2];
722 out[3] = zero105[3] + in[3];
723
724 felem_reduce_(out, in);
725
726 /* out[0] > 2^105 - 2^41 - 2^9 - 2^71 - 2^103 - 2^71 - 2^103 > 0
727 * out[1] > 2^105 - 2^71 - 2^103 > 0
728 * out[2] > 2^105 - 2^41 + 2^9 - 2^71 - 2^103 > 0
729 * out[3] > 2^105 - 2^41 + 2^9 - 2^71 - 2^103 - 2^103 > 0
730 *
731 * out[0] < 2^105 + 2^71 + 2^71 + 2^103 < 2^106
732 * out[1] < 2^105 + 2^71 + 2^71 + 2^103 < 2^106
733 * out[2] < 2^105 + 2^71 + 2^71 + 2^71 + 2^103 < 2^106
734 * out[3] < 2^105 + 2^71 + 2^103 + 2^71 < 2^106 */
735 }
736
737 /* subtract_u64 sets *result = *result - v and *carry to one if the
738 * subtraction underflowed. */
subtract_u64(u64 * result,u64 * carry,u64 v)739 static void subtract_u64(u64 *result, u64 *carry, u64 v) {
740 uint128_t r = *result;
741 r -= v;
742 *carry = (r >> 64) & 1;
743 *result = (u64)r;
744 }
745
746 /* felem_contract converts |in| to its unique, minimal representation. On
747 * entry: in[i] < 2^109. */
felem_contract(smallfelem out,const felem in)748 static void felem_contract(smallfelem out, const felem in) {
749 u64 all_equal_so_far = 0, result = 0;
750
751 felem_shrink(out, in);
752 /* small is minimal except that the value might be > p */
753
754 all_equal_so_far--;
755 /* We are doing a constant time test if out >= kPrime. We need to compare
756 * each u64, from most-significant to least significant. For each one, if
757 * all words so far have been equal (m is all ones) then a non-equal
758 * result is the answer. Otherwise we continue. */
759 unsigned i;
760 for (i = 3; i < 4; i--) {
761 u64 equal;
762 uint128_t a = ((uint128_t)kPrime[i]) - out[i];
763 /* if out[i] > kPrime[i] then a will underflow and the high 64-bits
764 * will all be set. */
765 result |= all_equal_so_far & ((u64)(a >> 64));
766
767 /* if kPrime[i] == out[i] then |equal| will be all zeros and the
768 * decrement will make it all ones. */
769 equal = kPrime[i] ^ out[i];
770 equal--;
771 equal &= equal << 32;
772 equal &= equal << 16;
773 equal &= equal << 8;
774 equal &= equal << 4;
775 equal &= equal << 2;
776 equal &= equal << 1;
777 equal = ((s64)equal) >> 63;
778
779 all_equal_so_far &= equal;
780 }
781
782 /* if all_equal_so_far is still all ones then the two values are equal
783 * and so out >= kPrime is true. */
784 result |= all_equal_so_far;
785
786 /* if out >= kPrime then we subtract kPrime. */
787 u64 carry;
788 subtract_u64(&out[0], &carry, result & kPrime[0]);
789 subtract_u64(&out[1], &carry, carry);
790 subtract_u64(&out[2], &carry, carry);
791 subtract_u64(&out[3], &carry, carry);
792
793 subtract_u64(&out[1], &carry, result & kPrime[1]);
794 subtract_u64(&out[2], &carry, carry);
795 subtract_u64(&out[3], &carry, carry);
796
797 subtract_u64(&out[2], &carry, result & kPrime[2]);
798 subtract_u64(&out[3], &carry, carry);
799
800 subtract_u64(&out[3], &carry, result & kPrime[3]);
801 }
802
smallfelem_square_contract(smallfelem out,const smallfelem in)803 static void smallfelem_square_contract(smallfelem out, const smallfelem in) {
804 longfelem longtmp;
805 felem tmp;
806
807 smallfelem_square(longtmp, in);
808 felem_reduce(tmp, longtmp);
809 felem_contract(out, tmp);
810 }
811
smallfelem_mul_contract(smallfelem out,const smallfelem in1,const smallfelem in2)812 static void smallfelem_mul_contract(smallfelem out, const smallfelem in1,
813 const smallfelem in2) {
814 longfelem longtmp;
815 felem tmp;
816
817 smallfelem_mul(longtmp, in1, in2);
818 felem_reduce(tmp, longtmp);
819 felem_contract(out, tmp);
820 }
821
822 /* felem_is_zero returns a limb with all bits set if |in| == 0 (mod p) and 0
823 * otherwise.
824 * On entry:
825 * small[i] < 2^64 */
smallfelem_is_zero(const smallfelem small)826 static limb smallfelem_is_zero(const smallfelem small) {
827 limb result;
828 u64 is_p;
829
830 u64 is_zero = small[0] | small[1] | small[2] | small[3];
831 is_zero--;
832 is_zero &= is_zero << 32;
833 is_zero &= is_zero << 16;
834 is_zero &= is_zero << 8;
835 is_zero &= is_zero << 4;
836 is_zero &= is_zero << 2;
837 is_zero &= is_zero << 1;
838 is_zero = ((s64)is_zero) >> 63;
839
840 is_p = (small[0] ^ kPrime[0]) | (small[1] ^ kPrime[1]) |
841 (small[2] ^ kPrime[2]) | (small[3] ^ kPrime[3]);
842 is_p--;
843 is_p &= is_p << 32;
844 is_p &= is_p << 16;
845 is_p &= is_p << 8;
846 is_p &= is_p << 4;
847 is_p &= is_p << 2;
848 is_p &= is_p << 1;
849 is_p = ((s64)is_p) >> 63;
850
851 is_zero |= is_p;
852
853 result = is_zero;
854 result |= ((limb)is_zero) << 64;
855 return result;
856 }
857
smallfelem_is_zero_int(const smallfelem small)858 static int smallfelem_is_zero_int(const smallfelem small) {
859 return (int)(smallfelem_is_zero(small) & ((limb)1));
860 }
861
862 /* felem_inv calculates |out| = |in|^{-1}
863 *
864 * Based on Fermat's Little Theorem:
865 * a^p = a (mod p)
866 * a^{p-1} = 1 (mod p)
867 * a^{p-2} = a^{-1} (mod p) */
felem_inv(felem out,const felem in)868 static void felem_inv(felem out, const felem in) {
869 felem ftmp, ftmp2;
870 /* each e_I will hold |in|^{2^I - 1} */
871 felem e2, e4, e8, e16, e32, e64;
872 longfelem tmp;
873 unsigned i;
874
875 felem_square(tmp, in);
876 felem_reduce(ftmp, tmp); /* 2^1 */
877 felem_mul(tmp, in, ftmp);
878 felem_reduce(ftmp, tmp); /* 2^2 - 2^0 */
879 felem_assign(e2, ftmp);
880 felem_square(tmp, ftmp);
881 felem_reduce(ftmp, tmp); /* 2^3 - 2^1 */
882 felem_square(tmp, ftmp);
883 felem_reduce(ftmp, tmp); /* 2^4 - 2^2 */
884 felem_mul(tmp, ftmp, e2);
885 felem_reduce(ftmp, tmp); /* 2^4 - 2^0 */
886 felem_assign(e4, ftmp);
887 felem_square(tmp, ftmp);
888 felem_reduce(ftmp, tmp); /* 2^5 - 2^1 */
889 felem_square(tmp, ftmp);
890 felem_reduce(ftmp, tmp); /* 2^6 - 2^2 */
891 felem_square(tmp, ftmp);
892 felem_reduce(ftmp, tmp); /* 2^7 - 2^3 */
893 felem_square(tmp, ftmp);
894 felem_reduce(ftmp, tmp); /* 2^8 - 2^4 */
895 felem_mul(tmp, ftmp, e4);
896 felem_reduce(ftmp, tmp); /* 2^8 - 2^0 */
897 felem_assign(e8, ftmp);
898 for (i = 0; i < 8; i++) {
899 felem_square(tmp, ftmp);
900 felem_reduce(ftmp, tmp);
901 } /* 2^16 - 2^8 */
902 felem_mul(tmp, ftmp, e8);
903 felem_reduce(ftmp, tmp); /* 2^16 - 2^0 */
904 felem_assign(e16, ftmp);
905 for (i = 0; i < 16; i++) {
906 felem_square(tmp, ftmp);
907 felem_reduce(ftmp, tmp);
908 } /* 2^32 - 2^16 */
909 felem_mul(tmp, ftmp, e16);
910 felem_reduce(ftmp, tmp); /* 2^32 - 2^0 */
911 felem_assign(e32, ftmp);
912 for (i = 0; i < 32; i++) {
913 felem_square(tmp, ftmp);
914 felem_reduce(ftmp, tmp);
915 } /* 2^64 - 2^32 */
916 felem_assign(e64, ftmp);
917 felem_mul(tmp, ftmp, in);
918 felem_reduce(ftmp, tmp); /* 2^64 - 2^32 + 2^0 */
919 for (i = 0; i < 192; i++) {
920 felem_square(tmp, ftmp);
921 felem_reduce(ftmp, tmp);
922 } /* 2^256 - 2^224 + 2^192 */
923
924 felem_mul(tmp, e64, e32);
925 felem_reduce(ftmp2, tmp); /* 2^64 - 2^0 */
926 for (i = 0; i < 16; i++) {
927 felem_square(tmp, ftmp2);
928 felem_reduce(ftmp2, tmp);
929 } /* 2^80 - 2^16 */
930 felem_mul(tmp, ftmp2, e16);
931 felem_reduce(ftmp2, tmp); /* 2^80 - 2^0 */
932 for (i = 0; i < 8; i++) {
933 felem_square(tmp, ftmp2);
934 felem_reduce(ftmp2, tmp);
935 } /* 2^88 - 2^8 */
936 felem_mul(tmp, ftmp2, e8);
937 felem_reduce(ftmp2, tmp); /* 2^88 - 2^0 */
938 for (i = 0; i < 4; i++) {
939 felem_square(tmp, ftmp2);
940 felem_reduce(ftmp2, tmp);
941 } /* 2^92 - 2^4 */
942 felem_mul(tmp, ftmp2, e4);
943 felem_reduce(ftmp2, tmp); /* 2^92 - 2^0 */
944 felem_square(tmp, ftmp2);
945 felem_reduce(ftmp2, tmp); /* 2^93 - 2^1 */
946 felem_square(tmp, ftmp2);
947 felem_reduce(ftmp2, tmp); /* 2^94 - 2^2 */
948 felem_mul(tmp, ftmp2, e2);
949 felem_reduce(ftmp2, tmp); /* 2^94 - 2^0 */
950 felem_square(tmp, ftmp2);
951 felem_reduce(ftmp2, tmp); /* 2^95 - 2^1 */
952 felem_square(tmp, ftmp2);
953 felem_reduce(ftmp2, tmp); /* 2^96 - 2^2 */
954 felem_mul(tmp, ftmp2, in);
955 felem_reduce(ftmp2, tmp); /* 2^96 - 3 */
956
957 felem_mul(tmp, ftmp2, ftmp);
958 felem_reduce(out, tmp); /* 2^256 - 2^224 + 2^192 + 2^96 - 3 */
959 }
960
smallfelem_inv_contract(smallfelem out,const smallfelem in)961 static void smallfelem_inv_contract(smallfelem out, const smallfelem in) {
962 felem tmp;
963
964 smallfelem_expand(tmp, in);
965 felem_inv(tmp, tmp);
966 felem_contract(out, tmp);
967 }
968
969 /* Group operations
970 * ----------------
971 *
972 * Building on top of the field operations we have the operations on the
973 * elliptic curve group itself. Points on the curve are represented in Jacobian
974 * coordinates. */
975
976 /* point_double calculates 2*(x_in, y_in, z_in)
977 *
978 * The method is taken from:
979 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
980 *
981 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed.
982 * while x_out == y_in is not (maybe this works, but it's not tested). */
point_double(felem x_out,felem y_out,felem z_out,const felem x_in,const felem y_in,const felem z_in)983 static void point_double(felem x_out, felem y_out, felem z_out,
984 const felem x_in, const felem y_in, const felem z_in) {
985 longfelem tmp, tmp2;
986 felem delta, gamma, beta, alpha, ftmp, ftmp2;
987 smallfelem small1, small2;
988
989 felem_assign(ftmp, x_in);
990 /* ftmp[i] < 2^106 */
991 felem_assign(ftmp2, x_in);
992 /* ftmp2[i] < 2^106 */
993
994 /* delta = z^2 */
995 felem_square(tmp, z_in);
996 felem_reduce(delta, tmp);
997 /* delta[i] < 2^101 */
998
999 /* gamma = y^2 */
1000 felem_square(tmp, y_in);
1001 felem_reduce(gamma, tmp);
1002 /* gamma[i] < 2^101 */
1003 felem_shrink(small1, gamma);
1004
1005 /* beta = x*gamma */
1006 felem_small_mul(tmp, small1, x_in);
1007 felem_reduce(beta, tmp);
1008 /* beta[i] < 2^101 */
1009
1010 /* alpha = 3*(x-delta)*(x+delta) */
1011 felem_diff(ftmp, delta);
1012 /* ftmp[i] < 2^105 + 2^106 < 2^107 */
1013 felem_sum(ftmp2, delta);
1014 /* ftmp2[i] < 2^105 + 2^106 < 2^107 */
1015 felem_scalar(ftmp2, 3);
1016 /* ftmp2[i] < 3 * 2^107 < 2^109 */
1017 felem_mul(tmp, ftmp, ftmp2);
1018 felem_reduce(alpha, tmp);
1019 /* alpha[i] < 2^101 */
1020 felem_shrink(small2, alpha);
1021
1022 /* x' = alpha^2 - 8*beta */
1023 smallfelem_square(tmp, small2);
1024 felem_reduce(x_out, tmp);
1025 felem_assign(ftmp, beta);
1026 felem_scalar(ftmp, 8);
1027 /* ftmp[i] < 8 * 2^101 = 2^104 */
1028 felem_diff(x_out, ftmp);
1029 /* x_out[i] < 2^105 + 2^101 < 2^106 */
1030
1031 /* z' = (y + z)^2 - gamma - delta */
1032 felem_sum(delta, gamma);
1033 /* delta[i] < 2^101 + 2^101 = 2^102 */
1034 felem_assign(ftmp, y_in);
1035 felem_sum(ftmp, z_in);
1036 /* ftmp[i] < 2^106 + 2^106 = 2^107 */
1037 felem_square(tmp, ftmp);
1038 felem_reduce(z_out, tmp);
1039 felem_diff(z_out, delta);
1040 /* z_out[i] < 2^105 + 2^101 < 2^106 */
1041
1042 /* y' = alpha*(4*beta - x') - 8*gamma^2 */
1043 felem_scalar(beta, 4);
1044 /* beta[i] < 4 * 2^101 = 2^103 */
1045 felem_diff_zero107(beta, x_out);
1046 /* beta[i] < 2^107 + 2^103 < 2^108 */
1047 felem_small_mul(tmp, small2, beta);
1048 /* tmp[i] < 7 * 2^64 < 2^67 */
1049 smallfelem_square(tmp2, small1);
1050 /* tmp2[i] < 7 * 2^64 */
1051 longfelem_scalar(tmp2, 8);
1052 /* tmp2[i] < 8 * 7 * 2^64 = 7 * 2^67 */
1053 longfelem_diff(tmp, tmp2);
1054 /* tmp[i] < 2^67 + 2^70 + 2^40 < 2^71 */
1055 felem_reduce_zero105(y_out, tmp);
1056 /* y_out[i] < 2^106 */
1057 }
1058
1059 /* point_double_small is the same as point_double, except that it operates on
1060 * smallfelems. */
point_double_small(smallfelem x_out,smallfelem y_out,smallfelem z_out,const smallfelem x_in,const smallfelem y_in,const smallfelem z_in)1061 static void point_double_small(smallfelem x_out, smallfelem y_out,
1062 smallfelem z_out, const smallfelem x_in,
1063 const smallfelem y_in, const smallfelem z_in) {
1064 felem felem_x_out, felem_y_out, felem_z_out;
1065 felem felem_x_in, felem_y_in, felem_z_in;
1066
1067 smallfelem_expand(felem_x_in, x_in);
1068 smallfelem_expand(felem_y_in, y_in);
1069 smallfelem_expand(felem_z_in, z_in);
1070 point_double(felem_x_out, felem_y_out, felem_z_out, felem_x_in, felem_y_in,
1071 felem_z_in);
1072 felem_shrink(x_out, felem_x_out);
1073 felem_shrink(y_out, felem_y_out);
1074 felem_shrink(z_out, felem_z_out);
1075 }
1076
1077 /* copy_conditional copies in to out iff mask is all ones. */
copy_conditional(felem out,const felem in,limb mask)1078 static void copy_conditional(felem out, const felem in, limb mask) {
1079 unsigned i;
1080 for (i = 0; i < NLIMBS; ++i) {
1081 const limb tmp = mask & (in[i] ^ out[i]);
1082 out[i] ^= tmp;
1083 }
1084 }
1085
1086 /* copy_small_conditional copies in to out iff mask is all ones. */
copy_small_conditional(felem out,const smallfelem in,limb mask)1087 static void copy_small_conditional(felem out, const smallfelem in, limb mask) {
1088 unsigned i;
1089 const u64 mask64 = mask;
1090 for (i = 0; i < NLIMBS; ++i) {
1091 out[i] = ((limb)(in[i] & mask64)) | (out[i] & ~mask);
1092 }
1093 }
1094
1095 /* point_add calcuates (x1, y1, z1) + (x2, y2, z2)
1096 *
1097 * The method is taken from:
1098 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl,
1099 * adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity).
1100 *
1101 * This function includes a branch for checking whether the two input points
1102 * are equal, (while not equal to the point at infinity). This case never
1103 * happens during single point multiplication, so there is no timing leak for
1104 * ECDH or ECDSA signing. */
point_add(felem x3,felem y3,felem z3,const felem x1,const felem y1,const felem z1,const int mixed,const smallfelem x2,const smallfelem y2,const smallfelem z2)1105 static void point_add(felem x3, felem y3, felem z3, const felem x1,
1106 const felem y1, const felem z1, const int mixed,
1107 const smallfelem x2, const smallfelem y2,
1108 const smallfelem z2) {
1109 felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, ftmp6, x_out, y_out, z_out;
1110 longfelem tmp, tmp2;
1111 smallfelem small1, small2, small3, small4, small5;
1112 limb x_equal, y_equal, z1_is_zero, z2_is_zero;
1113
1114 felem_shrink(small3, z1);
1115
1116 z1_is_zero = smallfelem_is_zero(small3);
1117 z2_is_zero = smallfelem_is_zero(z2);
1118
1119 /* ftmp = z1z1 = z1**2 */
1120 smallfelem_square(tmp, small3);
1121 felem_reduce(ftmp, tmp);
1122 /* ftmp[i] < 2^101 */
1123 felem_shrink(small1, ftmp);
1124
1125 if (!mixed) {
1126 /* ftmp2 = z2z2 = z2**2 */
1127 smallfelem_square(tmp, z2);
1128 felem_reduce(ftmp2, tmp);
1129 /* ftmp2[i] < 2^101 */
1130 felem_shrink(small2, ftmp2);
1131
1132 felem_shrink(small5, x1);
1133
1134 /* u1 = ftmp3 = x1*z2z2 */
1135 smallfelem_mul(tmp, small5, small2);
1136 felem_reduce(ftmp3, tmp);
1137 /* ftmp3[i] < 2^101 */
1138
1139 /* ftmp5 = z1 + z2 */
1140 felem_assign(ftmp5, z1);
1141 felem_small_sum(ftmp5, z2);
1142 /* ftmp5[i] < 2^107 */
1143
1144 /* ftmp5 = (z1 + z2)**2 - (z1z1 + z2z2) = 2z1z2 */
1145 felem_square(tmp, ftmp5);
1146 felem_reduce(ftmp5, tmp);
1147 /* ftmp2 = z2z2 + z1z1 */
1148 felem_sum(ftmp2, ftmp);
1149 /* ftmp2[i] < 2^101 + 2^101 = 2^102 */
1150 felem_diff(ftmp5, ftmp2);
1151 /* ftmp5[i] < 2^105 + 2^101 < 2^106 */
1152
1153 /* ftmp2 = z2 * z2z2 */
1154 smallfelem_mul(tmp, small2, z2);
1155 felem_reduce(ftmp2, tmp);
1156
1157 /* s1 = ftmp2 = y1 * z2**3 */
1158 felem_mul(tmp, y1, ftmp2);
1159 felem_reduce(ftmp6, tmp);
1160 /* ftmp6[i] < 2^101 */
1161 } else {
1162 /* We'll assume z2 = 1 (special case z2 = 0 is handled later). */
1163
1164 /* u1 = ftmp3 = x1*z2z2 */
1165 felem_assign(ftmp3, x1);
1166 /* ftmp3[i] < 2^106 */
1167
1168 /* ftmp5 = 2z1z2 */
1169 felem_assign(ftmp5, z1);
1170 felem_scalar(ftmp5, 2);
1171 /* ftmp5[i] < 2*2^106 = 2^107 */
1172
1173 /* s1 = ftmp2 = y1 * z2**3 */
1174 felem_assign(ftmp6, y1);
1175 /* ftmp6[i] < 2^106 */
1176 }
1177
1178 /* u2 = x2*z1z1 */
1179 smallfelem_mul(tmp, x2, small1);
1180 felem_reduce(ftmp4, tmp);
1181
1182 /* h = ftmp4 = u2 - u1 */
1183 felem_diff_zero107(ftmp4, ftmp3);
1184 /* ftmp4[i] < 2^107 + 2^101 < 2^108 */
1185 felem_shrink(small4, ftmp4);
1186
1187 x_equal = smallfelem_is_zero(small4);
1188
1189 /* z_out = ftmp5 * h */
1190 felem_small_mul(tmp, small4, ftmp5);
1191 felem_reduce(z_out, tmp);
1192 /* z_out[i] < 2^101 */
1193
1194 /* ftmp = z1 * z1z1 */
1195 smallfelem_mul(tmp, small1, small3);
1196 felem_reduce(ftmp, tmp);
1197
1198 /* s2 = tmp = y2 * z1**3 */
1199 felem_small_mul(tmp, y2, ftmp);
1200 felem_reduce(ftmp5, tmp);
1201
1202 /* r = ftmp5 = (s2 - s1)*2 */
1203 felem_diff_zero107(ftmp5, ftmp6);
1204 /* ftmp5[i] < 2^107 + 2^107 = 2^108 */
1205 felem_scalar(ftmp5, 2);
1206 /* ftmp5[i] < 2^109 */
1207 felem_shrink(small1, ftmp5);
1208 y_equal = smallfelem_is_zero(small1);
1209
1210 if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) {
1211 point_double(x3, y3, z3, x1, y1, z1);
1212 return;
1213 }
1214
1215 /* I = ftmp = (2h)**2 */
1216 felem_assign(ftmp, ftmp4);
1217 felem_scalar(ftmp, 2);
1218 /* ftmp[i] < 2*2^108 = 2^109 */
1219 felem_square(tmp, ftmp);
1220 felem_reduce(ftmp, tmp);
1221
1222 /* J = ftmp2 = h * I */
1223 felem_mul(tmp, ftmp4, ftmp);
1224 felem_reduce(ftmp2, tmp);
1225
1226 /* V = ftmp4 = U1 * I */
1227 felem_mul(tmp, ftmp3, ftmp);
1228 felem_reduce(ftmp4, tmp);
1229
1230 /* x_out = r**2 - J - 2V */
1231 smallfelem_square(tmp, small1);
1232 felem_reduce(x_out, tmp);
1233 felem_assign(ftmp3, ftmp4);
1234 felem_scalar(ftmp4, 2);
1235 felem_sum(ftmp4, ftmp2);
1236 /* ftmp4[i] < 2*2^101 + 2^101 < 2^103 */
1237 felem_diff(x_out, ftmp4);
1238 /* x_out[i] < 2^105 + 2^101 */
1239
1240 /* y_out = r(V-x_out) - 2 * s1 * J */
1241 felem_diff_zero107(ftmp3, x_out);
1242 /* ftmp3[i] < 2^107 + 2^101 < 2^108 */
1243 felem_small_mul(tmp, small1, ftmp3);
1244 felem_mul(tmp2, ftmp6, ftmp2);
1245 longfelem_scalar(tmp2, 2);
1246 /* tmp2[i] < 2*2^67 = 2^68 */
1247 longfelem_diff(tmp, tmp2);
1248 /* tmp[i] < 2^67 + 2^70 + 2^40 < 2^71 */
1249 felem_reduce_zero105(y_out, tmp);
1250 /* y_out[i] < 2^106 */
1251
1252 copy_small_conditional(x_out, x2, z1_is_zero);
1253 copy_conditional(x_out, x1, z2_is_zero);
1254 copy_small_conditional(y_out, y2, z1_is_zero);
1255 copy_conditional(y_out, y1, z2_is_zero);
1256 copy_small_conditional(z_out, z2, z1_is_zero);
1257 copy_conditional(z_out, z1, z2_is_zero);
1258 felem_assign(x3, x_out);
1259 felem_assign(y3, y_out);
1260 felem_assign(z3, z_out);
1261 }
1262
1263 /* point_add_small is the same as point_add, except that it operates on
1264 * smallfelems. */
point_add_small(smallfelem x3,smallfelem y3,smallfelem z3,smallfelem x1,smallfelem y1,smallfelem z1,smallfelem x2,smallfelem y2,smallfelem z2)1265 static void point_add_small(smallfelem x3, smallfelem y3, smallfelem z3,
1266 smallfelem x1, smallfelem y1, smallfelem z1,
1267 smallfelem x2, smallfelem y2, smallfelem z2) {
1268 felem felem_x3, felem_y3, felem_z3;
1269 felem felem_x1, felem_y1, felem_z1;
1270 smallfelem_expand(felem_x1, x1);
1271 smallfelem_expand(felem_y1, y1);
1272 smallfelem_expand(felem_z1, z1);
1273 point_add(felem_x3, felem_y3, felem_z3, felem_x1, felem_y1, felem_z1, 0, x2,
1274 y2, z2);
1275 felem_shrink(x3, felem_x3);
1276 felem_shrink(y3, felem_y3);
1277 felem_shrink(z3, felem_z3);
1278 }
1279
1280 /* Base point pre computation
1281 * --------------------------
1282 *
1283 * Two different sorts of precomputed tables are used in the following code.
1284 * Each contain various points on the curve, where each point is three field
1285 * elements (x, y, z).
1286 *
1287 * For the base point table, z is usually 1 (0 for the point at infinity).
1288 * This table has 2 * 16 elements, starting with the following:
1289 * index | bits | point
1290 * ------+---------+------------------------------
1291 * 0 | 0 0 0 0 | 0G
1292 * 1 | 0 0 0 1 | 1G
1293 * 2 | 0 0 1 0 | 2^64G
1294 * 3 | 0 0 1 1 | (2^64 + 1)G
1295 * 4 | 0 1 0 0 | 2^128G
1296 * 5 | 0 1 0 1 | (2^128 + 1)G
1297 * 6 | 0 1 1 0 | (2^128 + 2^64)G
1298 * 7 | 0 1 1 1 | (2^128 + 2^64 + 1)G
1299 * 8 | 1 0 0 0 | 2^192G
1300 * 9 | 1 0 0 1 | (2^192 + 1)G
1301 * 10 | 1 0 1 0 | (2^192 + 2^64)G
1302 * 11 | 1 0 1 1 | (2^192 + 2^64 + 1)G
1303 * 12 | 1 1 0 0 | (2^192 + 2^128)G
1304 * 13 | 1 1 0 1 | (2^192 + 2^128 + 1)G
1305 * 14 | 1 1 1 0 | (2^192 + 2^128 + 2^64)G
1306 * 15 | 1 1 1 1 | (2^192 + 2^128 + 2^64 + 1)G
1307 * followed by a copy of this with each element multiplied by 2^32.
1308 *
1309 * The reason for this is so that we can clock bits into four different
1310 * locations when doing simple scalar multiplies against the base point,
1311 * and then another four locations using the second 16 elements.
1312 *
1313 * Tables for other points have table[i] = iG for i in 0 .. 16. */
1314
1315 /* gmul is the table of precomputed base points */
1316 static const smallfelem gmul[2][16][3] = {
1317 {{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}},
1318 {{0xf4a13945d898c296, 0x77037d812deb33a0, 0xf8bce6e563a440f2,
1319 0x6b17d1f2e12c4247},
1320 {0xcbb6406837bf51f5, 0x2bce33576b315ece, 0x8ee7eb4a7c0f9e16,
1321 0x4fe342e2fe1a7f9b},
1322 {1, 0, 0, 0}},
1323 {{0x90e75cb48e14db63, 0x29493baaad651f7e, 0x8492592e326e25de,
1324 0x0fa822bc2811aaa5},
1325 {0xe41124545f462ee7, 0x34b1a65050fe82f5, 0x6f4ad4bcb3df188b,
1326 0xbff44ae8f5dba80d},
1327 {1, 0, 0, 0}},
1328 {{0x93391ce2097992af, 0xe96c98fd0d35f1fa, 0xb257c0de95e02789,
1329 0x300a4bbc89d6726f},
1330 {0xaa54a291c08127a0, 0x5bb1eeada9d806a5, 0x7f1ddb25ff1e3c6f,
1331 0x72aac7e0d09b4644},
1332 {1, 0, 0, 0}},
1333 {{0x57c84fc9d789bd85, 0xfc35ff7dc297eac3, 0xfb982fd588c6766e,
1334 0x447d739beedb5e67},
1335 {0x0c7e33c972e25b32, 0x3d349b95a7fae500, 0xe12e9d953a4aaff7,
1336 0x2d4825ab834131ee},
1337 {1, 0, 0, 0}},
1338 {{0x13949c932a1d367f, 0xef7fbd2b1a0a11b7, 0xddc6068bb91dfc60,
1339 0xef9519328a9c72ff},
1340 {0x196035a77376d8a8, 0x23183b0895ca1740, 0xc1ee9807022c219c,
1341 0x611e9fc37dbb2c9b},
1342 {1, 0, 0, 0}},
1343 {{0xcae2b1920b57f4bc, 0x2936df5ec6c9bc36, 0x7dea6482e11238bf,
1344 0x550663797b51f5d8},
1345 {0x44ffe216348a964c, 0x9fb3d576dbdefbe1, 0x0afa40018d9d50e5,
1346 0x157164848aecb851},
1347 {1, 0, 0, 0}},
1348 {{0xe48ecafffc5cde01, 0x7ccd84e70d715f26, 0xa2e8f483f43e4391,
1349 0xeb5d7745b21141ea},
1350 {0xcac917e2731a3479, 0x85f22cfe2844b645, 0x0990e6a158006cee,
1351 0xeafd72ebdbecc17b},
1352 {1, 0, 0, 0}},
1353 {{0x6cf20ffb313728be, 0x96439591a3c6b94a, 0x2736ff8344315fc5,
1354 0xa6d39677a7849276},
1355 {0xf2bab833c357f5f4, 0x824a920c2284059b, 0x66b8babd2d27ecdf,
1356 0x674f84749b0b8816},
1357 {1, 0, 0, 0}},
1358 {{0x2df48c04677c8a3e, 0x74e02f080203a56b, 0x31855f7db8c7fedb,
1359 0x4e769e7672c9ddad},
1360 {0xa4c36165b824bbb0, 0xfb9ae16f3b9122a5, 0x1ec0057206947281,
1361 0x42b99082de830663},
1362 {1, 0, 0, 0}},
1363 {{0x6ef95150dda868b9, 0xd1f89e799c0ce131, 0x7fdc1ca008a1c478,
1364 0x78878ef61c6ce04d},
1365 {0x9c62b9121fe0d976, 0x6ace570ebde08d4f, 0xde53142c12309def,
1366 0xb6cb3f5d7b72c321},
1367 {1, 0, 0, 0}},
1368 {{0x7f991ed2c31a3573, 0x5b82dd5bd54fb496, 0x595c5220812ffcae,
1369 0x0c88bc4d716b1287},
1370 {0x3a57bf635f48aca8, 0x7c8181f4df2564f3, 0x18d1b5b39c04e6aa,
1371 0xdd5ddea3f3901dc6},
1372 {1, 0, 0, 0}},
1373 {{0xe96a79fb3e72ad0c, 0x43a0a28c42ba792f, 0xefe0a423083e49f3,
1374 0x68f344af6b317466},
1375 {0xcdfe17db3fb24d4a, 0x668bfc2271f5c626, 0x604ed93c24d67ff3,
1376 0x31b9c405f8540a20},
1377 {1, 0, 0, 0}},
1378 {{0xd36b4789a2582e7f, 0x0d1a10144ec39c28, 0x663c62c3edbad7a0,
1379 0x4052bf4b6f461db9},
1380 {0x235a27c3188d25eb, 0xe724f33999bfcc5b, 0x862be6bd71d70cc8,
1381 0xfecf4d5190b0fc61},
1382 {1, 0, 0, 0}},
1383 {{0x74346c10a1d4cfac, 0xafdf5cc08526a7a4, 0x123202a8f62bff7a,
1384 0x1eddbae2c802e41a},
1385 {0x8fa0af2dd603f844, 0x36e06b7e4c701917, 0x0c45f45273db33a0,
1386 0x43104d86560ebcfc},
1387 {1, 0, 0, 0}},
1388 {{0x9615b5110d1d78e5, 0x66b0de3225c4744b, 0x0a4a46fb6aaf363a,
1389 0xb48e26b484f7a21c},
1390 {0x06ebb0f621a01b2d, 0xc004e4048b7b0f98, 0x64131bcdfed6f668,
1391 0xfac015404d4d3dab},
1392 {1, 0, 0, 0}}},
1393 {{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}},
1394 {{0x3a5a9e22185a5943, 0x1ab919365c65dfb6, 0x21656b32262c71da,
1395 0x7fe36b40af22af89},
1396 {0xd50d152c699ca101, 0x74b3d5867b8af212, 0x9f09f40407dca6f1,
1397 0xe697d45825b63624},
1398 {1, 0, 0, 0}},
1399 {{0xa84aa9397512218e, 0xe9a521b074ca0141, 0x57880b3a18a2e902,
1400 0x4a5b506612a677a6},
1401 {0x0beada7a4c4f3840, 0x626db15419e26d9d, 0xc42604fbe1627d40,
1402 0xeb13461ceac089f1},
1403 {1, 0, 0, 0}},
1404 {{0xf9faed0927a43281, 0x5e52c4144103ecbc, 0xc342967aa815c857,
1405 0x0781b8291c6a220a},
1406 {0x5a8343ceeac55f80, 0x88f80eeee54a05e3, 0x97b2a14f12916434,
1407 0x690cde8df0151593},
1408 {1, 0, 0, 0}},
1409 {{0xaee9c75df7f82f2a, 0x9e4c35874afdf43a, 0xf5622df437371326,
1410 0x8a535f566ec73617},
1411 {0xc5f9a0ac223094b7, 0xcde533864c8c7669, 0x37e02819085a92bf,
1412 0x0455c08468b08bd7},
1413 {1, 0, 0, 0}},
1414 {{0x0c0a6e2c9477b5d9, 0xf9a4bf62876dc444, 0x5050a949b6cdc279,
1415 0x06bada7ab77f8276},
1416 {0xc8b4aed1ea48dac9, 0xdebd8a4b7ea1070f, 0x427d49101366eb70,
1417 0x5b476dfd0e6cb18a},
1418 {1, 0, 0, 0}},
1419 {{0x7c5c3e44278c340a, 0x4d54606812d66f3b, 0x29a751b1ae23c5d8,
1420 0x3e29864e8a2ec908},
1421 {0x142d2a6626dbb850, 0xad1744c4765bd780, 0x1f150e68e322d1ed,
1422 0x239b90ea3dc31e7e},
1423 {1, 0, 0, 0}},
1424 {{0x78c416527a53322a, 0x305dde6709776f8e, 0xdbcab759f8862ed4,
1425 0x820f4dd949f72ff7},
1426 {0x6cc544a62b5debd4, 0x75be5d937b4e8cc4, 0x1b481b1b215c14d3,
1427 0x140406ec783a05ec},
1428 {1, 0, 0, 0}},
1429 {{0x6a703f10e895df07, 0xfd75f3fa01876bd8, 0xeb5b06e70ce08ffe,
1430 0x68f6b8542783dfee},
1431 {0x90c76f8a78712655, 0xcf5293d2f310bf7f, 0xfbc8044dfda45028,
1432 0xcbe1feba92e40ce6},
1433 {1, 0, 0, 0}},
1434 {{0xe998ceea4396e4c1, 0xfc82ef0b6acea274, 0x230f729f2250e927,
1435 0xd0b2f94d2f420109},
1436 {0x4305adddb38d4966, 0x10b838f8624c3b45, 0x7db2636658954e7a,
1437 0x971459828b0719e5},
1438 {1, 0, 0, 0}},
1439 {{0x4bd6b72623369fc9, 0x57f2929e53d0b876, 0xc2d5cba4f2340687,
1440 0x961610004a866aba},
1441 {0x49997bcd2e407a5e, 0x69ab197d92ddcb24, 0x2cf1f2438fe5131c,
1442 0x7acb9fadcee75e44},
1443 {1, 0, 0, 0}},
1444 {{0x254e839423d2d4c0, 0xf57f0c917aea685b, 0xa60d880f6f75aaea,
1445 0x24eb9acca333bf5b},
1446 {0xe3de4ccb1cda5dea, 0xfeef9341c51a6b4f, 0x743125f88bac4c4d,
1447 0x69f891c5acd079cc},
1448 {1, 0, 0, 0}},
1449 {{0xeee44b35702476b5, 0x7ed031a0e45c2258, 0xb422d1e7bd6f8514,
1450 0xe51f547c5972a107},
1451 {0xa25bcd6fc9cf343d, 0x8ca922ee097c184e, 0xa62f98b3a9fe9a06,
1452 0x1c309a2b25bb1387},
1453 {1, 0, 0, 0}},
1454 {{0x9295dbeb1967c459, 0xb00148833472c98e, 0xc504977708011828,
1455 0x20b87b8aa2c4e503},
1456 {0x3063175de057c277, 0x1bd539338fe582dd, 0x0d11adef5f69a044,
1457 0xf5c6fa49919776be},
1458 {1, 0, 0, 0}},
1459 {{0x8c944e760fd59e11, 0x3876cba1102fad5f, 0xa454c3fad83faa56,
1460 0x1ed7d1b9332010b9},
1461 {0xa1011a270024b889, 0x05e4d0dcac0cd344, 0x52b520f0eb6a2a24,
1462 0x3a2b03f03217257a},
1463 {1, 0, 0, 0}},
1464 {{0xf20fc2afdf1d043d, 0xf330240db58d5a62, 0xfc7d229ca0058c3b,
1465 0x15fee545c78dd9f6},
1466 {0x501e82885bc98cda, 0x41ef80e5d046ac04, 0x557d9f49461210fb,
1467 0x4ab5b6b2b8753f81},
1468 {1, 0, 0, 0}}}};
1469
1470 /* select_point selects the |idx|th point from a precomputation table and
1471 * copies it to out. */
select_point(const u64 idx,unsigned int size,const smallfelem pre_comp[16][3],smallfelem out[3])1472 static void select_point(const u64 idx, unsigned int size,
1473 const smallfelem pre_comp[16][3], smallfelem out[3]) {
1474 unsigned i, j;
1475 u64 *outlimbs = &out[0][0];
1476 memset(outlimbs, 0, 3 * sizeof(smallfelem));
1477
1478 for (i = 0; i < size; i++) {
1479 const u64 *inlimbs = (u64 *)&pre_comp[i][0][0];
1480 u64 mask = i ^ idx;
1481 mask |= mask >> 4;
1482 mask |= mask >> 2;
1483 mask |= mask >> 1;
1484 mask &= 1;
1485 mask--;
1486 for (j = 0; j < NLIMBS * 3; j++) {
1487 outlimbs[j] |= inlimbs[j] & mask;
1488 }
1489 }
1490 }
1491
1492 /* get_bit returns the |i|th bit in |in| */
get_bit(const felem_bytearray in,int i)1493 static char get_bit(const felem_bytearray in, int i) {
1494 if (i < 0 || i >= 256) {
1495 return 0;
1496 }
1497 return (in[i >> 3] >> (i & 7)) & 1;
1498 }
1499
1500 /* Interleaved point multiplication using precomputed point multiples: The
1501 * small point multiples 0*P, 1*P, ..., 17*P are in pre_comp[], the scalars
1502 * in scalars[]. If g_scalar is non-NULL, we also add this multiple of the
1503 * generator, using certain (large) precomputed multiples in g_pre_comp.
1504 * Output point (X, Y, Z) is stored in x_out, y_out, z_out. */
batch_mul(felem x_out,felem y_out,felem z_out,const felem_bytearray scalars[],const unsigned num_points,const u8 * g_scalar,const int mixed,const smallfelem pre_comp[][17][3],const smallfelem g_pre_comp[2][16][3])1505 static void batch_mul(felem x_out, felem y_out, felem z_out,
1506 const felem_bytearray scalars[],
1507 const unsigned num_points, const u8 *g_scalar,
1508 const int mixed, const smallfelem pre_comp[][17][3],
1509 const smallfelem g_pre_comp[2][16][3]) {
1510 int i, skip;
1511 unsigned num, gen_mul = (g_scalar != NULL);
1512 felem nq[3], ftmp;
1513 smallfelem tmp[3];
1514 u64 bits;
1515 u8 sign, digit;
1516
1517 /* set nq to the point at infinity */
1518 memset(nq, 0, 3 * sizeof(felem));
1519
1520 /* Loop over all scalars msb-to-lsb, interleaving additions of multiples
1521 * of the generator (two in each of the last 32 rounds) and additions of
1522 * other points multiples (every 5th round). */
1523
1524 skip = 1; /* save two point operations in the first
1525 * round */
1526 for (i = (num_points ? 255 : 31); i >= 0; --i) {
1527 /* double */
1528 if (!skip) {
1529 point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
1530 }
1531
1532 /* add multiples of the generator */
1533 if (gen_mul && i <= 31) {
1534 /* first, look 32 bits upwards */
1535 bits = get_bit(g_scalar, i + 224) << 3;
1536 bits |= get_bit(g_scalar, i + 160) << 2;
1537 bits |= get_bit(g_scalar, i + 96) << 1;
1538 bits |= get_bit(g_scalar, i + 32);
1539 /* select the point to add, in constant time */
1540 select_point(bits, 16, g_pre_comp[1], tmp);
1541
1542 if (!skip) {
1543 /* Arg 1 below is for "mixed" */
1544 point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1, tmp[0], tmp[1],
1545 tmp[2]);
1546 } else {
1547 smallfelem_expand(nq[0], tmp[0]);
1548 smallfelem_expand(nq[1], tmp[1]);
1549 smallfelem_expand(nq[2], tmp[2]);
1550 skip = 0;
1551 }
1552
1553 /* second, look at the current position */
1554 bits = get_bit(g_scalar, i + 192) << 3;
1555 bits |= get_bit(g_scalar, i + 128) << 2;
1556 bits |= get_bit(g_scalar, i + 64) << 1;
1557 bits |= get_bit(g_scalar, i);
1558 /* select the point to add, in constant time */
1559 select_point(bits, 16, g_pre_comp[0], tmp);
1560 /* Arg 1 below is for "mixed" */
1561 point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1, tmp[0], tmp[1],
1562 tmp[2]);
1563 }
1564
1565 /* do other additions every 5 doublings */
1566 if (num_points && (i % 5 == 0)) {
1567 /* loop over all scalars */
1568 for (num = 0; num < num_points; ++num) {
1569 bits = get_bit(scalars[num], i + 4) << 5;
1570 bits |= get_bit(scalars[num], i + 3) << 4;
1571 bits |= get_bit(scalars[num], i + 2) << 3;
1572 bits |= get_bit(scalars[num], i + 1) << 2;
1573 bits |= get_bit(scalars[num], i) << 1;
1574 bits |= get_bit(scalars[num], i - 1);
1575 ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits);
1576
1577 /* select the point to add or subtract, in constant time. */
1578 select_point(digit, 17, pre_comp[num], tmp);
1579 smallfelem_neg(ftmp, tmp[1]); /* (X, -Y, Z) is the negative
1580 * point */
1581 copy_small_conditional(ftmp, tmp[1], (((limb)sign) - 1));
1582 felem_contract(tmp[1], ftmp);
1583
1584 if (!skip) {
1585 point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], mixed, tmp[0],
1586 tmp[1], tmp[2]);
1587 } else {
1588 smallfelem_expand(nq[0], tmp[0]);
1589 smallfelem_expand(nq[1], tmp[1]);
1590 smallfelem_expand(nq[2], tmp[2]);
1591 skip = 0;
1592 }
1593 }
1594 }
1595 }
1596 felem_assign(x_out, nq[0]);
1597 felem_assign(y_out, nq[1]);
1598 felem_assign(z_out, nq[2]);
1599 }
1600
1601 /* Precomputation for the group generator. */
1602 typedef struct {
1603 smallfelem g_pre_comp[2][16][3];
1604 } NISTP256_PRE_COMP;
1605
1606 /******************************************************************************/
1607 /*
1608 * OPENSSL EC_METHOD FUNCTIONS
1609 */
1610
ec_GFp_nistp256_group_init(EC_GROUP * group)1611 int ec_GFp_nistp256_group_init(EC_GROUP *group) {
1612 int ret = ec_GFp_simple_group_init(group);
1613 group->a_is_minus3 = 1;
1614 return ret;
1615 }
1616
ec_GFp_nistp256_group_set_curve(EC_GROUP * group,const BIGNUM * p,const BIGNUM * a,const BIGNUM * b,BN_CTX * ctx)1617 int ec_GFp_nistp256_group_set_curve(EC_GROUP *group, const BIGNUM *p,
1618 const BIGNUM *a, const BIGNUM *b,
1619 BN_CTX *ctx) {
1620 int ret = 0;
1621 BN_CTX *new_ctx = NULL;
1622 BIGNUM *curve_p, *curve_a, *curve_b;
1623
1624 if (ctx == NULL) {
1625 if ((ctx = new_ctx = BN_CTX_new()) == NULL) {
1626 return 0;
1627 }
1628 }
1629 BN_CTX_start(ctx);
1630 if (((curve_p = BN_CTX_get(ctx)) == NULL) ||
1631 ((curve_a = BN_CTX_get(ctx)) == NULL) ||
1632 ((curve_b = BN_CTX_get(ctx)) == NULL)) {
1633 goto err;
1634 }
1635 BN_bin2bn(nistp256_curve_params[0], sizeof(felem_bytearray), curve_p);
1636 BN_bin2bn(nistp256_curve_params[1], sizeof(felem_bytearray), curve_a);
1637 BN_bin2bn(nistp256_curve_params[2], sizeof(felem_bytearray), curve_b);
1638 if (BN_cmp(curve_p, p) ||
1639 BN_cmp(curve_a, a) ||
1640 BN_cmp(curve_b, b)) {
1641 OPENSSL_PUT_ERROR(EC, ec_GFp_nistp256_group_set_curve,
1642 EC_R_WRONG_CURVE_PARAMETERS);
1643 goto err;
1644 }
1645 ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx);
1646
1647 err:
1648 BN_CTX_end(ctx);
1649 BN_CTX_free(new_ctx);
1650 return ret;
1651 }
1652
1653 /* Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') =
1654 * (X/Z^2, Y/Z^3). */
ec_GFp_nistp256_point_get_affine_coordinates(const EC_GROUP * group,const EC_POINT * point,BIGNUM * x,BIGNUM * y,BN_CTX * ctx)1655 int ec_GFp_nistp256_point_get_affine_coordinates(const EC_GROUP *group,
1656 const EC_POINT *point,
1657 BIGNUM *x, BIGNUM *y,
1658 BN_CTX *ctx) {
1659 felem z1, z2, x_in, y_in;
1660 smallfelem x_out, y_out;
1661 longfelem tmp;
1662
1663 if (EC_POINT_is_at_infinity(group, point)) {
1664 OPENSSL_PUT_ERROR(EC, ec_GFp_nistp256_point_get_affine_coordinates,
1665 EC_R_POINT_AT_INFINITY);
1666 return 0;
1667 }
1668 if (!BN_to_felem(x_in, &point->X) ||
1669 !BN_to_felem(y_in, &point->Y) ||
1670 !BN_to_felem(z1, &point->Z)) {
1671 return 0;
1672 }
1673 felem_inv(z2, z1);
1674 felem_square(tmp, z2);
1675 felem_reduce(z1, tmp);
1676 felem_mul(tmp, x_in, z1);
1677 felem_reduce(x_in, tmp);
1678 felem_contract(x_out, x_in);
1679 if (x != NULL && !smallfelem_to_BN(x, x_out)) {
1680 OPENSSL_PUT_ERROR(EC, ec_GFp_nistp256_point_get_affine_coordinates,
1681 ERR_R_BN_LIB);
1682 return 0;
1683 }
1684 felem_mul(tmp, z1, z2);
1685 felem_reduce(z1, tmp);
1686 felem_mul(tmp, y_in, z1);
1687 felem_reduce(y_in, tmp);
1688 felem_contract(y_out, y_in);
1689 if (y != NULL && !smallfelem_to_BN(y, y_out)) {
1690 OPENSSL_PUT_ERROR(EC, ec_GFp_nistp256_point_get_affine_coordinates,
1691 ERR_R_BN_LIB);
1692 return 0;
1693 }
1694 return 1;
1695 }
1696
1697 /* points below is of size |num|, and tmp_smallfelems is of size |num+1| */
make_points_affine(size_t num,smallfelem points[][3],smallfelem tmp_smallfelems[])1698 static void make_points_affine(size_t num, smallfelem points[][3],
1699 smallfelem tmp_smallfelems[]) {
1700 /* Runs in constant time, unless an input is the point at infinity (which
1701 * normally shouldn't happen). */
1702 ec_GFp_nistp_points_make_affine_internal(
1703 num, points, sizeof(smallfelem), tmp_smallfelems,
1704 (void (*)(void *))smallfelem_one,
1705 (int (*)(const void *))smallfelem_is_zero_int,
1706 (void (*)(void *, const void *))smallfelem_assign,
1707 (void (*)(void *, const void *))smallfelem_square_contract,
1708 (void (*)(void *, const void *, const void *))smallfelem_mul_contract,
1709 (void (*)(void *, const void *))smallfelem_inv_contract,
1710 /* nothing to contract */
1711 (void (*)(void *, const void *))smallfelem_assign);
1712 }
1713
1714 /* Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL
1715 * values Result is stored in r (r can equal one of the inputs). */
ec_GFp_nistp256_points_mul(const EC_GROUP * group,EC_POINT * r,const BIGNUM * scalar,size_t num,const EC_POINT * points[],const BIGNUM * scalars[],BN_CTX * ctx)1716 int ec_GFp_nistp256_points_mul(const EC_GROUP *group, EC_POINT *r,
1717 const BIGNUM *scalar, size_t num,
1718 const EC_POINT *points[],
1719 const BIGNUM *scalars[], BN_CTX *ctx) {
1720 int ret = 0;
1721 int j;
1722 int mixed = 0;
1723 BN_CTX *new_ctx = NULL;
1724 BIGNUM *x, *y, *z, *tmp_scalar;
1725 felem_bytearray g_secret;
1726 felem_bytearray *secrets = NULL;
1727 smallfelem(*pre_comp)[17][3] = NULL;
1728 smallfelem *tmp_smallfelems = NULL;
1729 felem_bytearray tmp;
1730 unsigned i, num_bytes;
1731 int have_pre_comp = 0;
1732 size_t num_points = num;
1733 smallfelem x_in, y_in, z_in;
1734 felem x_out, y_out, z_out;
1735 const smallfelem(*g_pre_comp)[16][3] = NULL;
1736 EC_POINT *generator = NULL;
1737 const EC_POINT *p = NULL;
1738 const BIGNUM *p_scalar = NULL;
1739
1740 if (ctx == NULL) {
1741 ctx = new_ctx = BN_CTX_new();
1742 if (ctx == NULL) {
1743 return 0;
1744 }
1745 }
1746
1747 BN_CTX_start(ctx);
1748 if ((x = BN_CTX_get(ctx)) == NULL ||
1749 (y = BN_CTX_get(ctx)) == NULL ||
1750 (z = BN_CTX_get(ctx)) == NULL ||
1751 (tmp_scalar = BN_CTX_get(ctx)) == NULL) {
1752 goto err;
1753 }
1754
1755 if (scalar != NULL) {
1756 /* try to use the standard precomputation */
1757 g_pre_comp = &gmul[0];
1758 generator = EC_POINT_new(group);
1759 if (generator == NULL) {
1760 goto err;
1761 }
1762 /* get the generator from precomputation */
1763 if (!smallfelem_to_BN(x, g_pre_comp[0][1][0]) ||
1764 !smallfelem_to_BN(y, g_pre_comp[0][1][1]) ||
1765 !smallfelem_to_BN(z, g_pre_comp[0][1][2])) {
1766 OPENSSL_PUT_ERROR(EC, ec_GFp_nistp256_points_mul, ERR_R_BN_LIB);
1767 goto err;
1768 }
1769 if (!ec_point_set_Jprojective_coordinates_GFp(group, generator, x, y, z,
1770 ctx)) {
1771 goto err;
1772 }
1773 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) {
1774 /* precomputation matches generator */
1775 have_pre_comp = 1;
1776 } else {
1777 /* we don't have valid precomputation: treat the generator as a
1778 * random point. */
1779 num_points++;
1780 }
1781 }
1782
1783 if (num_points > 0) {
1784 if (num_points >= 3) {
1785 /* unless we precompute multiples for just one or two points,
1786 * converting those into affine form is time well spent */
1787 mixed = 1;
1788 }
1789 secrets = OPENSSL_malloc(num_points * sizeof(felem_bytearray));
1790 pre_comp = OPENSSL_malloc(num_points * 17 * 3 * sizeof(smallfelem));
1791 if (mixed) {
1792 tmp_smallfelems =
1793 OPENSSL_malloc((num_points * 17 + 1) * sizeof(smallfelem));
1794 }
1795 if (secrets == NULL || pre_comp == NULL ||
1796 (mixed && tmp_smallfelems == NULL)) {
1797 OPENSSL_PUT_ERROR(EC, ec_GFp_nistp256_points_mul, ERR_R_MALLOC_FAILURE);
1798 goto err;
1799 }
1800
1801 /* we treat NULL scalars as 0, and NULL points as points at infinity,
1802 * i.e., they contribute nothing to the linear combination. */
1803 memset(secrets, 0, num_points * sizeof(felem_bytearray));
1804 memset(pre_comp, 0, num_points * 17 * 3 * sizeof(smallfelem));
1805 for (i = 0; i < num_points; ++i) {
1806 if (i == num) {
1807 /* we didn't have a valid precomputation, so we pick the generator. */
1808 p = EC_GROUP_get0_generator(group);
1809 p_scalar = scalar;
1810 } else {
1811 /* the i^th point */
1812 p = points[i];
1813 p_scalar = scalars[i];
1814 }
1815 if (p_scalar != NULL && p != NULL) {
1816 /* reduce scalar to 0 <= scalar < 2^256 */
1817 if (BN_num_bits(p_scalar) > 256 || BN_is_negative(p_scalar)) {
1818 /* this is an unusual input, and we don't guarantee
1819 * constant-timeness. */
1820 if (!BN_nnmod(tmp_scalar, p_scalar, &group->order, ctx)) {
1821 OPENSSL_PUT_ERROR(EC, ec_GFp_nistp256_points_mul, ERR_R_BN_LIB);
1822 goto err;
1823 }
1824 num_bytes = BN_bn2bin(tmp_scalar, tmp);
1825 } else {
1826 num_bytes = BN_bn2bin(p_scalar, tmp);
1827 }
1828 flip_endian(secrets[i], tmp, num_bytes);
1829 /* precompute multiples */
1830 if (!BN_to_felem(x_out, &p->X) ||
1831 !BN_to_felem(y_out, &p->Y) ||
1832 !BN_to_felem(z_out, &p->Z)) {
1833 goto err;
1834 }
1835 felem_shrink(pre_comp[i][1][0], x_out);
1836 felem_shrink(pre_comp[i][1][1], y_out);
1837 felem_shrink(pre_comp[i][1][2], z_out);
1838 for (j = 2; j <= 16; ++j) {
1839 if (j & 1) {
1840 point_add_small(pre_comp[i][j][0], pre_comp[i][j][1],
1841 pre_comp[i][j][2], pre_comp[i][1][0],
1842 pre_comp[i][1][1], pre_comp[i][1][2],
1843 pre_comp[i][j - 1][0], pre_comp[i][j - 1][1],
1844 pre_comp[i][j - 1][2]);
1845 } else {
1846 point_double_small(pre_comp[i][j][0], pre_comp[i][j][1],
1847 pre_comp[i][j][2], pre_comp[i][j / 2][0],
1848 pre_comp[i][j / 2][1], pre_comp[i][j / 2][2]);
1849 }
1850 }
1851 }
1852 }
1853 if (mixed) {
1854 make_points_affine(num_points * 17, pre_comp[0], tmp_smallfelems);
1855 }
1856 }
1857
1858 /* the scalar for the generator */
1859 if (scalar != NULL && have_pre_comp) {
1860 memset(g_secret, 0, sizeof(g_secret));
1861 /* reduce scalar to 0 <= scalar < 2^256 */
1862 if (BN_num_bits(scalar) > 256 || BN_is_negative(scalar)) {
1863 /* this is an unusual input, and we don't guarantee
1864 * constant-timeness. */
1865 if (!BN_nnmod(tmp_scalar, scalar, &group->order, ctx)) {
1866 OPENSSL_PUT_ERROR(EC, ec_GFp_nistp256_points_mul, ERR_R_BN_LIB);
1867 goto err;
1868 }
1869 num_bytes = BN_bn2bin(tmp_scalar, tmp);
1870 } else {
1871 num_bytes = BN_bn2bin(scalar, tmp);
1872 }
1873 flip_endian(g_secret, tmp, num_bytes);
1874 /* do the multiplication with generator precomputation */
1875 batch_mul(x_out, y_out, z_out, (const felem_bytearray(*))secrets,
1876 num_points, g_secret, mixed, (const smallfelem(*)[17][3])pre_comp,
1877 g_pre_comp);
1878 } else {
1879 /* do the multiplication without generator precomputation */
1880 batch_mul(x_out, y_out, z_out, (const felem_bytearray(*))secrets,
1881 num_points, NULL, mixed, (const smallfelem(*)[17][3])pre_comp,
1882 NULL);
1883 }
1884
1885 /* reduce the output to its unique minimal representation */
1886 felem_contract(x_in, x_out);
1887 felem_contract(y_in, y_out);
1888 felem_contract(z_in, z_out);
1889 if (!smallfelem_to_BN(x, x_in) ||
1890 !smallfelem_to_BN(y, y_in) ||
1891 !smallfelem_to_BN(z, z_in)) {
1892 OPENSSL_PUT_ERROR(EC, ec_GFp_nistp256_points_mul, ERR_R_BN_LIB);
1893 goto err;
1894 }
1895 ret = ec_point_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx);
1896
1897 err:
1898 BN_CTX_end(ctx);
1899 EC_POINT_free(generator);
1900 BN_CTX_free(new_ctx);
1901 OPENSSL_free(secrets);
1902 OPENSSL_free(pre_comp);
1903 OPENSSL_free(tmp_smallfelems);
1904 return ret;
1905 }
1906
EC_GFp_nistp256_method(void)1907 const EC_METHOD *EC_GFp_nistp256_method(void) {
1908 static const EC_METHOD ret = {
1909 EC_FLAGS_DEFAULT_OCT,
1910 ec_GFp_nistp256_group_init,
1911 ec_GFp_simple_group_finish,
1912 ec_GFp_simple_group_clear_finish,
1913 ec_GFp_simple_group_copy, ec_GFp_nistp256_group_set_curve,
1914 ec_GFp_simple_group_get_curve, ec_GFp_simple_group_get_degree,
1915 ec_GFp_simple_group_check_discriminant, ec_GFp_simple_point_init,
1916 ec_GFp_simple_point_finish, ec_GFp_simple_point_clear_finish,
1917 ec_GFp_simple_point_copy, ec_GFp_simple_point_set_to_infinity,
1918 ec_GFp_simple_set_Jprojective_coordinates_GFp,
1919 ec_GFp_simple_get_Jprojective_coordinates_GFp,
1920 ec_GFp_simple_point_set_affine_coordinates,
1921 ec_GFp_nistp256_point_get_affine_coordinates,
1922 0 /* point_set_compressed_coordinates */, 0 /* point2oct */,
1923 0 /* oct2point */, ec_GFp_simple_add, ec_GFp_simple_dbl,
1924 ec_GFp_simple_invert, ec_GFp_simple_is_at_infinity,
1925 ec_GFp_simple_is_on_curve, ec_GFp_simple_cmp, ec_GFp_simple_make_affine,
1926 ec_GFp_simple_points_make_affine, ec_GFp_nistp256_points_mul,
1927 0 /* precompute_mult */, 0 /* have_precompute_mult */,
1928 ec_GFp_simple_field_mul, ec_GFp_simple_field_sqr, 0 /* field_div */,
1929 0 /* field_encode */, 0 /* field_decode */, 0 /* field_set_to_one */
1930 };
1931
1932 return &ret;
1933 }
1934
1935 #endif /* 64_BIT && !WINDOWS */
1936