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192.168.1.100 / T106_DC / 856028caa88fd3719f8c0252225ed8d64a78149c / . / ap / lib / libssl / openssl-1.1.1o / crypto / ec / curve25519.c
blob: 952da0e653853c46161c2d834cf8dc826e2fdc67 [file] [log] [blame]
lh9ed821d2023-04-07 01:36:19 -0700[diff] [blame]1/*
2 * Copyright 2016-2019 The OpenSSL Project Authors. All Rights Reserved.
3 *
4 * Licensed under the OpenSSL license (the "License"). You may not use
5 * this file except in compliance with the License. You can obtain a copy
6 * in the file LICENSE in the source distribution or at
7 * https://www.openssl.org/source/license.html
8 */
9
10#include <string.h>
11#include "ec_local.h"
12#include <openssl/sha.h>
13
14#if defined(X25519_ASM) && (defined(__x86_64) || defined(__x86_64__) || \
15 defined(_M_AMD64) || defined(_M_X64))
16
17# define BASE_2_64_IMPLEMENTED
18
19typedef uint64_t fe64[4];
20
21int x25519_fe64_eligible(void);
22
23/*
24 * Following subroutines perform corresponding operations modulo
25 * 2^256-38, i.e. double the curve modulus. However, inputs and
26 * outputs are permitted to be partially reduced, i.e. to remain
27 * in [0..2^256) range. It's all tied up in final fe64_tobytes
28 * that performs full reduction modulo 2^255-19.
29 *
30 * There are no reference C implementations for these.
31 */
32void x25519_fe64_mul(fe64 h, const fe64 f, const fe64 g);
33void x25519_fe64_sqr(fe64 h, const fe64 f);
34void x25519_fe64_mul121666(fe64 h, fe64 f);
35void x25519_fe64_add(fe64 h, const fe64 f, const fe64 g);
36void x25519_fe64_sub(fe64 h, const fe64 f, const fe64 g);
37void x25519_fe64_tobytes(uint8_t *s, const fe64 f);
38# define fe64_mul x25519_fe64_mul
39# define fe64_sqr x25519_fe64_sqr
40# define fe64_mul121666 x25519_fe64_mul121666
41# define fe64_add x25519_fe64_add
42# define fe64_sub x25519_fe64_sub
43# define fe64_tobytes x25519_fe64_tobytes
44
45static uint64_t load_8(const uint8_t *in)
46{
47 uint64_t result;
48
49 result = in[0];
50 result |= ((uint64_t)in[1]) << 8;
51 result |= ((uint64_t)in[2]) << 16;
52 result |= ((uint64_t)in[3]) << 24;
53 result |= ((uint64_t)in[4]) << 32;
54 result |= ((uint64_t)in[5]) << 40;
55 result |= ((uint64_t)in[6]) << 48;
56 result |= ((uint64_t)in[7]) << 56;
57
58 return result;
59}
60
61static void fe64_frombytes(fe64 h, const uint8_t *s)
62{
63 h[0] = load_8(s);
64 h[1] = load_8(s + 8);
65 h[2] = load_8(s + 16);
66 h[3] = load_8(s + 24) & 0x7fffffffffffffff;
67}
68
69static void fe64_0(fe64 h)
70{
71 h[0] = 0;
72 h[1] = 0;
73 h[2] = 0;
74 h[3] = 0;
75}
76
77static void fe64_1(fe64 h)
78{
79 h[0] = 1;
80 h[1] = 0;
81 h[2] = 0;
82 h[3] = 0;
83}
84
85static void fe64_copy(fe64 h, const fe64 f)
86{
87 h[0] = f[0];
88 h[1] = f[1];
89 h[2] = f[2];
90 h[3] = f[3];
91}
92
93static void fe64_cswap(fe64 f, fe64 g, unsigned int b)
94{
95 int i;
96 uint64_t mask = 0 - (uint64_t)b;
97
98 for (i = 0; i < 4; i++) {
99 uint64_t x = f[i] ^ g[i];
100 x &= mask;
101 f[i] ^= x;
102 g[i] ^= x;
103 }
104}
105
106static void fe64_invert(fe64 out, const fe64 z)
107{
108 fe64 t0;
109 fe64 t1;
110 fe64 t2;
111 fe64 t3;
112 int i;
113
114 /*
115 * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as
116 * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11.
117 */
118
119 /* t0 = z ** 2 */
120 fe64_sqr(t0, z);
121
122 /* t1 = t0 ** (2 ** 2) = z ** 8 */
123 fe64_sqr(t1, t0);
124 fe64_sqr(t1, t1);
125
126 /* t1 = z * t1 = z ** 9 */
127 fe64_mul(t1, z, t1);
128 /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */
129 fe64_mul(t0, t0, t1);
130
131 /* t2 = t0 ** 2 = z ** 22 */
132 fe64_sqr(t2, t0);
133
134 /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */
135 fe64_mul(t1, t1, t2);
136
137 /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */
138 fe64_sqr(t2, t1);
139 for (i = 1; i < 5; ++i)
140 fe64_sqr(t2, t2);
141
142 /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */
143 fe64_mul(t1, t2, t1);
144
145 /* Continuing similarly... */
146
147 /* t2 = z ** (2 ** 20 - 1) */
148 fe64_sqr(t2, t1);
149 for (i = 1; i < 10; ++i)
150 fe64_sqr(t2, t2);
151
152 fe64_mul(t2, t2, t1);
153
154 /* t2 = z ** (2 ** 40 - 1) */
155 fe64_sqr(t3, t2);
156 for (i = 1; i < 20; ++i)
157 fe64_sqr(t3, t3);
158
159 fe64_mul(t2, t3, t2);
160
161 /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */
162 for (i = 0; i < 10; ++i)
163 fe64_sqr(t2, t2);
164
165 /* t1 = z ** (2 ** 50 - 1) */
166 fe64_mul(t1, t2, t1);
167
168 /* t2 = z ** (2 ** 100 - 1) */
169 fe64_sqr(t2, t1);
170 for (i = 1; i < 50; ++i)
171 fe64_sqr(t2, t2);
172
173 fe64_mul(t2, t2, t1);
174
175 /* t2 = z ** (2 ** 200 - 1) */
176 fe64_sqr(t3, t2);
177 for (i = 1; i < 100; ++i)
178 fe64_sqr(t3, t3);
179
180 fe64_mul(t2, t3, t2);
181
182 /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */
183 for (i = 0; i < 50; ++i)
184 fe64_sqr(t2, t2);
185
186 /* t1 = z ** (2 ** 250 - 1) */
187 fe64_mul(t1, t2, t1);
188
189 /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */
190 for (i = 0; i < 5; ++i)
191 fe64_sqr(t1, t1);
192
193 /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */
194 fe64_mul(out, t1, t0);
195}
196
197/*
198 * Duplicate of original x25519_scalar_mult_generic, but using
199 * fe64_* subroutines.
200 */
201static void x25519_scalar_mulx(uint8_t out[32], const uint8_t scalar[32],
202 const uint8_t point[32])
203{
204 fe64 x1, x2, z2, x3, z3, tmp0, tmp1;
205 uint8_t e[32];
206 unsigned swap = 0;
207 int pos;
208
209 memcpy(e, scalar, 32);
210 e[0] &= 0xf8;
211 e[31] &= 0x7f;
212 e[31] |= 0x40;
213 fe64_frombytes(x1, point);
214 fe64_1(x2);
215 fe64_0(z2);
216 fe64_copy(x3, x1);
217 fe64_1(z3);
218
219 for (pos = 254; pos >= 0; --pos) {
220 unsigned int b = 1 & (e[pos / 8] >> (pos & 7));
221
222 swap ^= b;
223 fe64_cswap(x2, x3, swap);
224 fe64_cswap(z2, z3, swap);
225 swap = b;
226 fe64_sub(tmp0, x3, z3);
227 fe64_sub(tmp1, x2, z2);
228 fe64_add(x2, x2, z2);
229 fe64_add(z2, x3, z3);
230 fe64_mul(z3, x2, tmp0);
231 fe64_mul(z2, z2, tmp1);
232 fe64_sqr(tmp0, tmp1);
233 fe64_sqr(tmp1, x2);
234 fe64_add(x3, z3, z2);
235 fe64_sub(z2, z3, z2);
236 fe64_mul(x2, tmp1, tmp0);
237 fe64_sub(tmp1, tmp1, tmp0);
238 fe64_sqr(z2, z2);
239 fe64_mul121666(z3, tmp1);
240 fe64_sqr(x3, x3);
241 fe64_add(tmp0, tmp0, z3);
242 fe64_mul(z3, x1, z2);
243 fe64_mul(z2, tmp1, tmp0);
244 }
245
246 fe64_invert(z2, z2);
247 fe64_mul(x2, x2, z2);
248 fe64_tobytes(out, x2);
249
250 OPENSSL_cleanse(e, sizeof(e));
251}
252#endif
253
254#if defined(X25519_ASM) \
255 || ( (defined(__SIZEOF_INT128__) && __SIZEOF_INT128__ == 16) \
256 && !defined(__sparc__) \
257 && (!defined(__SIZEOF_LONG__) || (__SIZEOF_LONG__ == 8)) \
258 && !(defined(__ANDROID__) && !defined(__clang__)) )
259/*
260 * Base 2^51 implementation. It's virtually no different from reference
261 * base 2^25.5 implementation in respect to lax boundary conditions for
262 * intermediate values and even individual limbs. So that whatever you
263 * know about the reference, applies even here...
264 */
265# define BASE_2_51_IMPLEMENTED
266
267typedef uint64_t fe51[5];
268
269static const uint64_t MASK51 = 0x7ffffffffffff;
270
271static uint64_t load_7(const uint8_t *in)
272{
273 uint64_t result;
274
275 result = in[0];
276 result |= ((uint64_t)in[1]) << 8;
277 result |= ((uint64_t)in[2]) << 16;
278 result |= ((uint64_t)in[3]) << 24;
279 result |= ((uint64_t)in[4]) << 32;
280 result |= ((uint64_t)in[5]) << 40;
281 result |= ((uint64_t)in[6]) << 48;
282
283 return result;
284}
285
286static uint64_t load_6(const uint8_t *in)
287{
288 uint64_t result;
289
290 result = in[0];
291 result |= ((uint64_t)in[1]) << 8;
292 result |= ((uint64_t)in[2]) << 16;
293 result |= ((uint64_t)in[3]) << 24;
294 result |= ((uint64_t)in[4]) << 32;
295 result |= ((uint64_t)in[5]) << 40;
296
297 return result;
298}
299
300static void fe51_frombytes(fe51 h, const uint8_t *s)
301{
302 uint64_t h0 = load_7(s); /* 56 bits */
303 uint64_t h1 = load_6(s + 7) << 5; /* 53 bits */
304 uint64_t h2 = load_7(s + 13) << 2; /* 58 bits */
305 uint64_t h3 = load_6(s + 20) << 7; /* 55 bits */
306 uint64_t h4 = (load_6(s + 26) & 0x7fffffffffff) << 4; /* 51 bits */
307
308 h1 |= h0 >> 51; h0 &= MASK51;
309 h2 |= h1 >> 51; h1 &= MASK51;
310 h3 |= h2 >> 51; h2 &= MASK51;
311 h4 |= h3 >> 51; h3 &= MASK51;
312
313 h[0] = h0;
314 h[1] = h1;
315 h[2] = h2;
316 h[3] = h3;
317 h[4] = h4;
318}
319
320static void fe51_tobytes(uint8_t *s, const fe51 h)
321{
322 uint64_t h0 = h[0];
323 uint64_t h1 = h[1];
324 uint64_t h2 = h[2];
325 uint64_t h3 = h[3];
326 uint64_t h4 = h[4];
327 uint64_t q;
328
329 /* compare to modulus */
330 q = (h0 + 19) >> 51;
331 q = (h1 + q) >> 51;
332 q = (h2 + q) >> 51;
333 q = (h3 + q) >> 51;
334 q = (h4 + q) >> 51;
335
336 /* full reduce */
337 h0 += 19 * q;
338 h1 += h0 >> 51; h0 &= MASK51;
339 h2 += h1 >> 51; h1 &= MASK51;
340 h3 += h2 >> 51; h2 &= MASK51;
341 h4 += h3 >> 51; h3 &= MASK51;
342 h4 &= MASK51;
343
344 /* smash */
345 s[0] = (uint8_t)(h0 >> 0);
346 s[1] = (uint8_t)(h0 >> 8);
347 s[2] = (uint8_t)(h0 >> 16);
348 s[3] = (uint8_t)(h0 >> 24);
349 s[4] = (uint8_t)(h0 >> 32);
350 s[5] = (uint8_t)(h0 >> 40);
351 s[6] = (uint8_t)((h0 >> 48) | ((uint32_t)h1 << 3));
352 s[7] = (uint8_t)(h1 >> 5);
353 s[8] = (uint8_t)(h1 >> 13);
354 s[9] = (uint8_t)(h1 >> 21);
355 s[10] = (uint8_t)(h1 >> 29);
356 s[11] = (uint8_t)(h1 >> 37);
357 s[12] = (uint8_t)((h1 >> 45) | ((uint32_t)h2 << 6));
358 s[13] = (uint8_t)(h2 >> 2);
359 s[14] = (uint8_t)(h2 >> 10);
360 s[15] = (uint8_t)(h2 >> 18);
361 s[16] = (uint8_t)(h2 >> 26);
362 s[17] = (uint8_t)(h2 >> 34);
363 s[18] = (uint8_t)(h2 >> 42);
364 s[19] = (uint8_t)((h2 >> 50) | ((uint32_t)h3 << 1));
365 s[20] = (uint8_t)(h3 >> 7);
366 s[21] = (uint8_t)(h3 >> 15);
367 s[22] = (uint8_t)(h3 >> 23);
368 s[23] = (uint8_t)(h3 >> 31);
369 s[24] = (uint8_t)(h3 >> 39);
370 s[25] = (uint8_t)((h3 >> 47) | ((uint32_t)h4 << 4));
371 s[26] = (uint8_t)(h4 >> 4);
372 s[27] = (uint8_t)(h4 >> 12);
373 s[28] = (uint8_t)(h4 >> 20);
374 s[29] = (uint8_t)(h4 >> 28);
375 s[30] = (uint8_t)(h4 >> 36);
376 s[31] = (uint8_t)(h4 >> 44);
377}
378
379# if defined(X25519_ASM)
380void x25519_fe51_mul(fe51 h, const fe51 f, const fe51 g);
381void x25519_fe51_sqr(fe51 h, const fe51 f);
382void x25519_fe51_mul121666(fe51 h, fe51 f);
383# define fe51_mul x25519_fe51_mul
384# define fe51_sq x25519_fe51_sqr
385# define fe51_mul121666 x25519_fe51_mul121666
386# else
387
388typedef __uint128_t u128;
389
390static void fe51_mul(fe51 h, const fe51 f, const fe51 g)
391{
392 u128 h0, h1, h2, h3, h4;
393 uint64_t f_i, g0, g1, g2, g3, g4;
394
395 f_i = f[0];
396 h0 = (u128)f_i * (g0 = g[0]);
397 h1 = (u128)f_i * (g1 = g[1]);
398 h2 = (u128)f_i * (g2 = g[2]);
399 h3 = (u128)f_i * (g3 = g[3]);
400 h4 = (u128)f_i * (g4 = g[4]);
401
402 f_i = f[1];
403 h0 += (u128)f_i * (g4 *= 19);
404 h1 += (u128)f_i * g0;
405 h2 += (u128)f_i * g1;
406 h3 += (u128)f_i * g2;
407 h4 += (u128)f_i * g3;
408
409 f_i = f[2];
410 h0 += (u128)f_i * (g3 *= 19);
411 h1 += (u128)f_i * g4;
412 h2 += (u128)f_i * g0;
413 h3 += (u128)f_i * g1;
414 h4 += (u128)f_i * g2;
415
416 f_i = f[3];
417 h0 += (u128)f_i * (g2 *= 19);
418 h1 += (u128)f_i * g3;
419 h2 += (u128)f_i * g4;
420 h3 += (u128)f_i * g0;
421 h4 += (u128)f_i * g1;
422
423 f_i = f[4];
424 h0 += (u128)f_i * (g1 *= 19);
425 h1 += (u128)f_i * g2;
426 h2 += (u128)f_i * g3;
427 h3 += (u128)f_i * g4;
428 h4 += (u128)f_i * g0;
429
430 /* partial [lazy] reduction */
431 h3 += (uint64_t)(h2 >> 51); g2 = (uint64_t)h2 & MASK51;
432 h1 += (uint64_t)(h0 >> 51); g0 = (uint64_t)h0 & MASK51;
433
434 h4 += (uint64_t)(h3 >> 51); g3 = (uint64_t)h3 & MASK51;
435 g2 += (uint64_t)(h1 >> 51); g1 = (uint64_t)h1 & MASK51;
436
437 g0 += (uint64_t)(h4 >> 51) * 19; g4 = (uint64_t)h4 & MASK51;
438 g3 += g2 >> 51; g2 &= MASK51;
439 g1 += g0 >> 51; g0 &= MASK51;
440
441 h[0] = g0;
442 h[1] = g1;
443 h[2] = g2;
444 h[3] = g3;
445 h[4] = g4;
446}
447
448static void fe51_sq(fe51 h, const fe51 f)
449{
450# if defined(OPENSSL_SMALL_FOOTPRINT)
451 fe51_mul(h, f, f);
452# else
453 /* dedicated squaring gives 16-25% overall improvement */
454 uint64_t g0 = f[0];
455 uint64_t g1 = f[1];
456 uint64_t g2 = f[2];
457 uint64_t g3 = f[3];
458 uint64_t g4 = f[4];
459 u128 h0, h1, h2, h3, h4;
460
461 h0 = (u128)g0 * g0; g0 *= 2;
462 h1 = (u128)g0 * g1;
463 h2 = (u128)g0 * g2;
464 h3 = (u128)g0 * g3;
465 h4 = (u128)g0 * g4;
466
467 g0 = g4; /* borrow g0 */
468 h3 += (u128)g0 * (g4 *= 19);
469
470 h2 += (u128)g1 * g1; g1 *= 2;
471 h3 += (u128)g1 * g2;
472 h4 += (u128)g1 * g3;
473 h0 += (u128)g1 * g4;
474
475 g0 = g3; /* borrow g0 */
476 h1 += (u128)g0 * (g3 *= 19);
477 h2 += (u128)(g0 * 2) * g4;
478
479 h4 += (u128)g2 * g2; g2 *= 2;
480 h0 += (u128)g2 * g3;
481 h1 += (u128)g2 * g4;
482
483 /* partial [lazy] reduction */
484 h3 += (uint64_t)(h2 >> 51); g2 = (uint64_t)h2 & MASK51;
485 h1 += (uint64_t)(h0 >> 51); g0 = (uint64_t)h0 & MASK51;
486
487 h4 += (uint64_t)(h3 >> 51); g3 = (uint64_t)h3 & MASK51;
488 g2 += (uint64_t)(h1 >> 51); g1 = (uint64_t)h1 & MASK51;
489
490 g0 += (uint64_t)(h4 >> 51) * 19; g4 = (uint64_t)h4 & MASK51;
491 g3 += g2 >> 51; g2 &= MASK51;
492 g1 += g0 >> 51; g0 &= MASK51;
493
494 h[0] = g0;
495 h[1] = g1;
496 h[2] = g2;
497 h[3] = g3;
498 h[4] = g4;
499# endif
500}
501
502static void fe51_mul121666(fe51 h, fe51 f)
503{
504 u128 h0 = f[0] * (u128)121666;
505 u128 h1 = f[1] * (u128)121666;
506 u128 h2 = f[2] * (u128)121666;
507 u128 h3 = f[3] * (u128)121666;
508 u128 h4 = f[4] * (u128)121666;
509 uint64_t g0, g1, g2, g3, g4;
510
511 h3 += (uint64_t)(h2 >> 51); g2 = (uint64_t)h2 & MASK51;
512 h1 += (uint64_t)(h0 >> 51); g0 = (uint64_t)h0 & MASK51;
513
514 h4 += (uint64_t)(h3 >> 51); g3 = (uint64_t)h3 & MASK51;
515 g2 += (uint64_t)(h1 >> 51); g1 = (uint64_t)h1 & MASK51;
516
517 g0 += (uint64_t)(h4 >> 51) * 19; g4 = (uint64_t)h4 & MASK51;
518 g3 += g2 >> 51; g2 &= MASK51;
519 g1 += g0 >> 51; g0 &= MASK51;
520
521 h[0] = g0;
522 h[1] = g1;
523 h[2] = g2;
524 h[3] = g3;
525 h[4] = g4;
526}
527# endif
528
529static void fe51_add(fe51 h, const fe51 f, const fe51 g)
530{
531 h[0] = f[0] + g[0];
532 h[1] = f[1] + g[1];
533 h[2] = f[2] + g[2];
534 h[3] = f[3] + g[3];
535 h[4] = f[4] + g[4];
536}
537
538static void fe51_sub(fe51 h, const fe51 f, const fe51 g)
539{
540 /*
541 * Add 2*modulus to ensure that result remains positive
542 * even if subtrahend is partially reduced.
543 */
544 h[0] = (f[0] + 0xfffffffffffda) - g[0];
545 h[1] = (f[1] + 0xffffffffffffe) - g[1];
546 h[2] = (f[2] + 0xffffffffffffe) - g[2];
547 h[3] = (f[3] + 0xffffffffffffe) - g[3];
548 h[4] = (f[4] + 0xffffffffffffe) - g[4];
549}
550
551static void fe51_0(fe51 h)
552{
553 h[0] = 0;
554 h[1] = 0;
555 h[2] = 0;
556 h[3] = 0;
557 h[4] = 0;
558}
559
560static void fe51_1(fe51 h)
561{
562 h[0] = 1;
563 h[1] = 0;
564 h[2] = 0;
565 h[3] = 0;
566 h[4] = 0;
567}
568
569static void fe51_copy(fe51 h, const fe51 f)
570{
571 h[0] = f[0];
572 h[1] = f[1];
573 h[2] = f[2];
574 h[3] = f[3];
575 h[4] = f[4];
576}
577
578static void fe51_cswap(fe51 f, fe51 g, unsigned int b)
579{
580 int i;
581 uint64_t mask = 0 - (uint64_t)b;
582
583 for (i = 0; i < 5; i++) {
584 int64_t x = f[i] ^ g[i];
585 x &= mask;
586 f[i] ^= x;
587 g[i] ^= x;
588 }
589}
590
591static void fe51_invert(fe51 out, const fe51 z)
592{
593 fe51 t0;
594 fe51 t1;
595 fe51 t2;
596 fe51 t3;
597 int i;
598
599 /*
600 * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as
601 * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11.
602 */
603
604 /* t0 = z ** 2 */
605 fe51_sq(t0, z);
606
607 /* t1 = t0 ** (2 ** 2) = z ** 8 */
608 fe51_sq(t1, t0);
609 fe51_sq(t1, t1);
610
611 /* t1 = z * t1 = z ** 9 */
612 fe51_mul(t1, z, t1);
613 /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */
614 fe51_mul(t0, t0, t1);
615
616 /* t2 = t0 ** 2 = z ** 22 */
617 fe51_sq(t2, t0);
618
619 /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */
620 fe51_mul(t1, t1, t2);
621
622 /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */
623 fe51_sq(t2, t1);
624 for (i = 1; i < 5; ++i)
625 fe51_sq(t2, t2);
626
627 /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */
628 fe51_mul(t1, t2, t1);
629
630 /* Continuing similarly... */
631
632 /* t2 = z ** (2 ** 20 - 1) */
633 fe51_sq(t2, t1);
634 for (i = 1; i < 10; ++i)
635 fe51_sq(t2, t2);
636
637 fe51_mul(t2, t2, t1);
638
639 /* t2 = z ** (2 ** 40 - 1) */
640 fe51_sq(t3, t2);
641 for (i = 1; i < 20; ++i)
642 fe51_sq(t3, t3);
643
644 fe51_mul(t2, t3, t2);
645
646 /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */
647 for (i = 0; i < 10; ++i)
648 fe51_sq(t2, t2);
649
650 /* t1 = z ** (2 ** 50 - 1) */
651 fe51_mul(t1, t2, t1);
652
653 /* t2 = z ** (2 ** 100 - 1) */
654 fe51_sq(t2, t1);
655 for (i = 1; i < 50; ++i)
656 fe51_sq(t2, t2);
657
658 fe51_mul(t2, t2, t1);
659
660 /* t2 = z ** (2 ** 200 - 1) */
661 fe51_sq(t3, t2);
662 for (i = 1; i < 100; ++i)
663 fe51_sq(t3, t3);
664
665 fe51_mul(t2, t3, t2);
666
667 /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */
668 for (i = 0; i < 50; ++i)
669 fe51_sq(t2, t2);
670
671 /* t1 = z ** (2 ** 250 - 1) */
672 fe51_mul(t1, t2, t1);
673
674 /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */
675 for (i = 0; i < 5; ++i)
676 fe51_sq(t1, t1);
677
678 /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */
679 fe51_mul(out, t1, t0);
680}
681
682/*
683 * Duplicate of original x25519_scalar_mult_generic, but using
684 * fe51_* subroutines.
685 */
686static void x25519_scalar_mult(uint8_t out[32], const uint8_t scalar[32],
687 const uint8_t point[32])
688{
689 fe51 x1, x2, z2, x3, z3, tmp0, tmp1;
690 uint8_t e[32];
691 unsigned swap = 0;
692 int pos;
693
694# ifdef BASE_2_64_IMPLEMENTED
695 if (x25519_fe64_eligible()) {
696 x25519_scalar_mulx(out, scalar, point);
697 return;
698 }
699# endif
700
701 memcpy(e, scalar, 32);
702 e[0] &= 0xf8;
703 e[31] &= 0x7f;
704 e[31] |= 0x40;
705 fe51_frombytes(x1, point);
706 fe51_1(x2);
707 fe51_0(z2);
708 fe51_copy(x3, x1);
709 fe51_1(z3);
710
711 for (pos = 254; pos >= 0; --pos) {
712 unsigned int b = 1 & (e[pos / 8] >> (pos & 7));
713
714 swap ^= b;
715 fe51_cswap(x2, x3, swap);
716 fe51_cswap(z2, z3, swap);
717 swap = b;
718 fe51_sub(tmp0, x3, z3);
719 fe51_sub(tmp1, x2, z2);
720 fe51_add(x2, x2, z2);
721 fe51_add(z2, x3, z3);
722 fe51_mul(z3, tmp0, x2);
723 fe51_mul(z2, z2, tmp1);
724 fe51_sq(tmp0, tmp1);
725 fe51_sq(tmp1, x2);
726 fe51_add(x3, z3, z2);
727 fe51_sub(z2, z3, z2);
728 fe51_mul(x2, tmp1, tmp0);
729 fe51_sub(tmp1, tmp1, tmp0);
730 fe51_sq(z2, z2);
731 fe51_mul121666(z3, tmp1);
732 fe51_sq(x3, x3);
733 fe51_add(tmp0, tmp0, z3);
734 fe51_mul(z3, x1, z2);
735 fe51_mul(z2, tmp1, tmp0);
736 }
737
738 fe51_invert(z2, z2);
739 fe51_mul(x2, x2, z2);
740 fe51_tobytes(out, x2);
741
742 OPENSSL_cleanse(e, sizeof(e));
743}
744#endif
745
746/*
747 * Reference base 2^25.5 implementation.
748 *
749 * This code is mostly taken from the ref10 version of Ed25519 in SUPERCOP
750 * 20141124 (http://bench.cr.yp.to/supercop.html).
751 *
752 * The field functions are shared by Ed25519 and X25519 where possible.
753 */
754
755/*
756 * fe means field element. Here the field is \Z/(2^255-19). An element t,
757 * entries t[0]...t[9], represents the integer t[0]+2^26 t[1]+2^51 t[2]+2^77
758 * t[3]+2^102 t[4]+...+2^230 t[9]. Bounds on each t[i] vary depending on
759 * context.
760 */
761typedef int32_t fe[10];
762
763static const int64_t kBottom21Bits = 0x1fffffLL;
764static const int64_t kBottom25Bits = 0x1ffffffLL;
765static const int64_t kBottom26Bits = 0x3ffffffLL;
766static const int64_t kTop39Bits = 0xfffffffffe000000LL;
767static const int64_t kTop38Bits = 0xfffffffffc000000LL;
768
769static uint64_t load_3(const uint8_t *in)
770{
771 uint64_t result;
772
773 result = ((uint64_t)in[0]);
774 result |= ((uint64_t)in[1]) << 8;
775 result |= ((uint64_t)in[2]) << 16;
776 return result;
777}
778
779static uint64_t load_4(const uint8_t *in)
780{
781 uint64_t result;
782
783 result = ((uint64_t)in[0]);
784 result |= ((uint64_t)in[1]) << 8;
785 result |= ((uint64_t)in[2]) << 16;
786 result |= ((uint64_t)in[3]) << 24;
787 return result;
788}
789
790static void fe_frombytes(fe h, const uint8_t *s)
791{
792 /* Ignores top bit of h. */
793 int64_t h0 = load_4(s);
794 int64_t h1 = load_3(s + 4) << 6;
795 int64_t h2 = load_3(s + 7) << 5;
796 int64_t h3 = load_3(s + 10) << 3;
797 int64_t h4 = load_3(s + 13) << 2;
798 int64_t h5 = load_4(s + 16);
799 int64_t h6 = load_3(s + 20) << 7;
800 int64_t h7 = load_3(s + 23) << 5;
801 int64_t h8 = load_3(s + 26) << 4;
802 int64_t h9 = (load_3(s + 29) & 0x7fffff) << 2;
803 int64_t carry0;
804 int64_t carry1;
805 int64_t carry2;
806 int64_t carry3;
807 int64_t carry4;
808 int64_t carry5;
809 int64_t carry6;
810 int64_t carry7;
811 int64_t carry8;
812 int64_t carry9;
813
814 carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
815 carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
816 carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
817 carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
818 carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;
819
820 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
821 carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
822 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
823 carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
824 carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;
825
826 h[0] = (int32_t)h0;
827 h[1] = (int32_t)h1;
828 h[2] = (int32_t)h2;
829 h[3] = (int32_t)h3;
830 h[4] = (int32_t)h4;
831 h[5] = (int32_t)h5;
832 h[6] = (int32_t)h6;
833 h[7] = (int32_t)h7;
834 h[8] = (int32_t)h8;
835 h[9] = (int32_t)h9;
836}
837
838/*
839 * Preconditions:
840 * |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
841 *
842 * Write p=2^255-19; q=floor(h/p).
843 * Basic claim: q = floor(2^(-255)(h + 19 2^(-25)h9 + 2^(-1))).
844 *
845 * Proof:
846 * Have |h|<=p so |q|<=1 so |19^2 2^(-255) q|<1/4.
847 * Also have |h-2^230 h9|<2^231 so |19 2^(-255)(h-2^230 h9)|<1/4.
848 *
849 * Write y=2^(-1)-19^2 2^(-255)q-19 2^(-255)(h-2^230 h9).
850 * Then 0<y<1.
851 *
852 * Write r=h-pq.
853 * Have 0<=r<=p-1=2^255-20.
854 * Thus 0<=r+19(2^-255)r<r+19(2^-255)2^255<=2^255-1.
855 *
856 * Write x=r+19(2^-255)r+y.
857 * Then 0<x<2^255 so floor(2^(-255)x) = 0 so floor(q+2^(-255)x) = q.
858 *
859 * Have q+2^(-255)x = 2^(-255)(h + 19 2^(-25) h9 + 2^(-1))
860 * so floor(2^(-255)(h + 19 2^(-25) h9 + 2^(-1))) = q.
861 */
862static void fe_tobytes(uint8_t *s, const fe h)
863{
864 int32_t h0 = h[0];
865 int32_t h1 = h[1];
866 int32_t h2 = h[2];
867 int32_t h3 = h[3];
868 int32_t h4 = h[4];
869 int32_t h5 = h[5];
870 int32_t h6 = h[6];
871 int32_t h7 = h[7];
872 int32_t h8 = h[8];
873 int32_t h9 = h[9];
874 int32_t q;
875
876 q = (19 * h9 + (((int32_t) 1) << 24)) >> 25;
877 q = (h0 + q) >> 26;
878 q = (h1 + q) >> 25;
879 q = (h2 + q) >> 26;
880 q = (h3 + q) >> 25;
881 q = (h4 + q) >> 26;
882 q = (h5 + q) >> 25;
883 q = (h6 + q) >> 26;
884 q = (h7 + q) >> 25;
885 q = (h8 + q) >> 26;
886 q = (h9 + q) >> 25;
887
888 /* Goal: Output h-(2^255-19)q, which is between 0 and 2^255-20. */
889 h0 += 19 * q;
890 /* Goal: Output h-2^255 q, which is between 0 and 2^255-20. */
891
892 h1 += h0 >> 26; h0 &= kBottom26Bits;
893 h2 += h1 >> 25; h1 &= kBottom25Bits;
894 h3 += h2 >> 26; h2 &= kBottom26Bits;
895 h4 += h3 >> 25; h3 &= kBottom25Bits;
896 h5 += h4 >> 26; h4 &= kBottom26Bits;
897 h6 += h5 >> 25; h5 &= kBottom25Bits;
898 h7 += h6 >> 26; h6 &= kBottom26Bits;
899 h8 += h7 >> 25; h7 &= kBottom25Bits;
900 h9 += h8 >> 26; h8 &= kBottom26Bits;
901 h9 &= kBottom25Bits;
902 /* h10 = carry9 */
903
904 /*
905 * Goal: Output h0+...+2^255 h10-2^255 q, which is between 0 and 2^255-20.
906 * Have h0+...+2^230 h9 between 0 and 2^255-1;
907 * evidently 2^255 h10-2^255 q = 0.
908 * Goal: Output h0+...+2^230 h9.
909 */
910 s[ 0] = (uint8_t) (h0 >> 0);
911 s[ 1] = (uint8_t) (h0 >> 8);
912 s[ 2] = (uint8_t) (h0 >> 16);
913 s[ 3] = (uint8_t)((h0 >> 24) | ((uint32_t)(h1) << 2));
914 s[ 4] = (uint8_t) (h1 >> 6);
915 s[ 5] = (uint8_t) (h1 >> 14);
916 s[ 6] = (uint8_t)((h1 >> 22) | ((uint32_t)(h2) << 3));
917 s[ 7] = (uint8_t) (h2 >> 5);
918 s[ 8] = (uint8_t) (h2 >> 13);
919 s[ 9] = (uint8_t)((h2 >> 21) | ((uint32_t)(h3) << 5));
920 s[10] = (uint8_t) (h3 >> 3);
921 s[11] = (uint8_t) (h3 >> 11);
922 s[12] = (uint8_t)((h3 >> 19) | ((uint32_t)(h4) << 6));
923 s[13] = (uint8_t) (h4 >> 2);
924 s[14] = (uint8_t) (h4 >> 10);
925 s[15] = (uint8_t) (h4 >> 18);
926 s[16] = (uint8_t) (h5 >> 0);
927 s[17] = (uint8_t) (h5 >> 8);
928 s[18] = (uint8_t) (h5 >> 16);
929 s[19] = (uint8_t)((h5 >> 24) | ((uint32_t)(h6) << 1));
930 s[20] = (uint8_t) (h6 >> 7);
931 s[21] = (uint8_t) (h6 >> 15);
932 s[22] = (uint8_t)((h6 >> 23) | ((uint32_t)(h7) << 3));
933 s[23] = (uint8_t) (h7 >> 5);
934 s[24] = (uint8_t) (h7 >> 13);
935 s[25] = (uint8_t)((h7 >> 21) | ((uint32_t)(h8) << 4));
936 s[26] = (uint8_t) (h8 >> 4);
937 s[27] = (uint8_t) (h8 >> 12);
938 s[28] = (uint8_t)((h8 >> 20) | ((uint32_t)(h9) << 6));
939 s[29] = (uint8_t) (h9 >> 2);
940 s[30] = (uint8_t) (h9 >> 10);
941 s[31] = (uint8_t) (h9 >> 18);
942}
943
944/* h = f */
945static void fe_copy(fe h, const fe f)
946{
947 memmove(h, f, sizeof(int32_t) * 10);
948}
949
950/* h = 0 */
951static void fe_0(fe h)
952{
953 memset(h, 0, sizeof(int32_t) * 10);
954}
955
956/* h = 1 */
957static void fe_1(fe h)
958{
959 memset(h, 0, sizeof(int32_t) * 10);
960 h[0] = 1;
961}
962
963/*
964 * h = f + g
965 *
966 * Can overlap h with f or g.
967 *
968 * Preconditions:
969 * |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
970 * |g| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
971 *
972 * Postconditions:
973 * |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
974 */
975static void fe_add(fe h, const fe f, const fe g)
976{
977 unsigned i;
978
979 for (i = 0; i < 10; i++) {
980 h[i] = f[i] + g[i];
981 }
982}
983
984/*
985 * h = f - g
986 *
987 * Can overlap h with f or g.
988 *
989 * Preconditions:
990 * |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
991 * |g| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
992 *
993 * Postconditions:
994 * |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
995 */
996static void fe_sub(fe h, const fe f, const fe g)
997{
998 unsigned i;
999
1000 for (i = 0; i < 10; i++) {
1001 h[i] = f[i] - g[i];
1002 }
1003}
1004
1005/*
1006 * h = f * g
1007 *
1008 * Can overlap h with f or g.
1009 *
1010 * Preconditions:
1011 * |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
1012 * |g| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
1013 *
1014 * Postconditions:
1015 * |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
1016 *
1017 * Notes on implementation strategy:
1018 *
1019 * Using schoolbook multiplication.
1020 * Karatsuba would save a little in some cost models.
1021 *
1022 * Most multiplications by 2 and 19 are 32-bit precomputations;
1023 * cheaper than 64-bit postcomputations.
1024 *
1025 * There is one remaining multiplication by 19 in the carry chain;
1026 * one *19 precomputation can be merged into this,
1027 * but the resulting data flow is considerably less clean.
1028 *
1029 * There are 12 carries below.
1030 * 10 of them are 2-way parallelizable and vectorizable.
1031 * Can get away with 11 carries, but then data flow is much deeper.
1032 *
1033 * With tighter constraints on inputs can squeeze carries into int32.
1034 */
1035static void fe_mul(fe h, const fe f, const fe g)
1036{
1037 int32_t f0 = f[0];
1038 int32_t f1 = f[1];
1039 int32_t f2 = f[2];
1040 int32_t f3 = f[3];
1041 int32_t f4 = f[4];
1042 int32_t f5 = f[5];
1043 int32_t f6 = f[6];
1044 int32_t f7 = f[7];
1045 int32_t f8 = f[8];
1046 int32_t f9 = f[9];
1047 int32_t g0 = g[0];
1048 int32_t g1 = g[1];
1049 int32_t g2 = g[2];
1050 int32_t g3 = g[3];
1051 int32_t g4 = g[4];
1052 int32_t g5 = g[5];
1053 int32_t g6 = g[6];
1054 int32_t g7 = g[7];
1055 int32_t g8 = g[8];
1056 int32_t g9 = g[9];
1057 int32_t g1_19 = 19 * g1; /* 1.959375*2^29 */
1058 int32_t g2_19 = 19 * g2; /* 1.959375*2^30; still ok */
1059 int32_t g3_19 = 19 * g3;
1060 int32_t g4_19 = 19 * g4;
1061 int32_t g5_19 = 19 * g5;
1062 int32_t g6_19 = 19 * g6;
1063 int32_t g7_19 = 19 * g7;
1064 int32_t g8_19 = 19 * g8;
1065 int32_t g9_19 = 19 * g9;
1066 int32_t f1_2 = 2 * f1;
1067 int32_t f3_2 = 2 * f3;
1068 int32_t f5_2 = 2 * f5;
1069 int32_t f7_2 = 2 * f7;
1070 int32_t f9_2 = 2 * f9;
1071 int64_t f0g0 = f0 * (int64_t) g0;
1072 int64_t f0g1 = f0 * (int64_t) g1;
1073 int64_t f0g2 = f0 * (int64_t) g2;
1074 int64_t f0g3 = f0 * (int64_t) g3;
1075 int64_t f0g4 = f0 * (int64_t) g4;
1076 int64_t f0g5 = f0 * (int64_t) g5;
1077 int64_t f0g6 = f0 * (int64_t) g6;
1078 int64_t f0g7 = f0 * (int64_t) g7;
1079 int64_t f0g8 = f0 * (int64_t) g8;
1080 int64_t f0g9 = f0 * (int64_t) g9;
1081 int64_t f1g0 = f1 * (int64_t) g0;
1082 int64_t f1g1_2 = f1_2 * (int64_t) g1;
1083 int64_t f1g2 = f1 * (int64_t) g2;
1084 int64_t f1g3_2 = f1_2 * (int64_t) g3;
1085 int64_t f1g4 = f1 * (int64_t) g4;
1086 int64_t f1g5_2 = f1_2 * (int64_t) g5;
1087 int64_t f1g6 = f1 * (int64_t) g6;
1088 int64_t f1g7_2 = f1_2 * (int64_t) g7;
1089 int64_t f1g8 = f1 * (int64_t) g8;
1090 int64_t f1g9_38 = f1_2 * (int64_t) g9_19;
1091 int64_t f2g0 = f2 * (int64_t) g0;
1092 int64_t f2g1 = f2 * (int64_t) g1;
1093 int64_t f2g2 = f2 * (int64_t) g2;
1094 int64_t f2g3 = f2 * (int64_t) g3;
1095 int64_t f2g4 = f2 * (int64_t) g4;
1096 int64_t f2g5 = f2 * (int64_t) g5;
1097 int64_t f2g6 = f2 * (int64_t) g6;
1098 int64_t f2g7 = f2 * (int64_t) g7;
1099 int64_t f2g8_19 = f2 * (int64_t) g8_19;
1100 int64_t f2g9_19 = f2 * (int64_t) g9_19;
1101 int64_t f3g0 = f3 * (int64_t) g0;
1102 int64_t f3g1_2 = f3_2 * (int64_t) g1;
1103 int64_t f3g2 = f3 * (int64_t) g2;
1104 int64_t f3g3_2 = f3_2 * (int64_t) g3;
1105 int64_t f3g4 = f3 * (int64_t) g4;
1106 int64_t f3g5_2 = f3_2 * (int64_t) g5;
1107 int64_t f3g6 = f3 * (int64_t) g6;
1108 int64_t f3g7_38 = f3_2 * (int64_t) g7_19;
1109 int64_t f3g8_19 = f3 * (int64_t) g8_19;
1110 int64_t f3g9_38 = f3_2 * (int64_t) g9_19;
1111 int64_t f4g0 = f4 * (int64_t) g0;
1112 int64_t f4g1 = f4 * (int64_t) g1;
1113 int64_t f4g2 = f4 * (int64_t) g2;
1114 int64_t f4g3 = f4 * (int64_t) g3;
1115 int64_t f4g4 = f4 * (int64_t) g4;
1116 int64_t f4g5 = f4 * (int64_t) g5;
1117 int64_t f4g6_19 = f4 * (int64_t) g6_19;
1118 int64_t f4g7_19 = f4 * (int64_t) g7_19;
1119 int64_t f4g8_19 = f4 * (int64_t) g8_19;
1120 int64_t f4g9_19 = f4 * (int64_t) g9_19;
1121 int64_t f5g0 = f5 * (int64_t) g0;
1122 int64_t f5g1_2 = f5_2 * (int64_t) g1;
1123 int64_t f5g2 = f5 * (int64_t) g2;
1124 int64_t f5g3_2 = f5_2 * (int64_t) g3;
1125 int64_t f5g4 = f5 * (int64_t) g4;
1126 int64_t f5g5_38 = f5_2 * (int64_t) g5_19;
1127 int64_t f5g6_19 = f5 * (int64_t) g6_19;
1128 int64_t f5g7_38 = f5_2 * (int64_t) g7_19;
1129 int64_t f5g8_19 = f5 * (int64_t) g8_19;
1130 int64_t f5g9_38 = f5_2 * (int64_t) g9_19;
1131 int64_t f6g0 = f6 * (int64_t) g0;
1132 int64_t f6g1 = f6 * (int64_t) g1;
1133 int64_t f6g2 = f6 * (int64_t) g2;
1134 int64_t f6g3 = f6 * (int64_t) g3;
1135 int64_t f6g4_19 = f6 * (int64_t) g4_19;
1136 int64_t f6g5_19 = f6 * (int64_t) g5_19;
1137 int64_t f6g6_19 = f6 * (int64_t) g6_19;
1138 int64_t f6g7_19 = f6 * (int64_t) g7_19;
1139 int64_t f6g8_19 = f6 * (int64_t) g8_19;
1140 int64_t f6g9_19 = f6 * (int64_t) g9_19;
1141 int64_t f7g0 = f7 * (int64_t) g0;
1142 int64_t f7g1_2 = f7_2 * (int64_t) g1;
1143 int64_t f7g2 = f7 * (int64_t) g2;
1144 int64_t f7g3_38 = f7_2 * (int64_t) g3_19;
1145 int64_t f7g4_19 = f7 * (int64_t) g4_19;
1146 int64_t f7g5_38 = f7_2 * (int64_t) g5_19;
1147 int64_t f7g6_19 = f7 * (int64_t) g6_19;
1148 int64_t f7g7_38 = f7_2 * (int64_t) g7_19;
1149 int64_t f7g8_19 = f7 * (int64_t) g8_19;
1150 int64_t f7g9_38 = f7_2 * (int64_t) g9_19;
1151 int64_t f8g0 = f8 * (int64_t) g0;
1152 int64_t f8g1 = f8 * (int64_t) g1;
1153 int64_t f8g2_19 = f8 * (int64_t) g2_19;
1154 int64_t f8g3_19 = f8 * (int64_t) g3_19;
1155 int64_t f8g4_19 = f8 * (int64_t) g4_19;
1156 int64_t f8g5_19 = f8 * (int64_t) g5_19;
1157 int64_t f8g6_19 = f8 * (int64_t) g6_19;
1158 int64_t f8g7_19 = f8 * (int64_t) g7_19;
1159 int64_t f8g8_19 = f8 * (int64_t) g8_19;
1160 int64_t f8g9_19 = f8 * (int64_t) g9_19;
1161 int64_t f9g0 = f9 * (int64_t) g0;
1162 int64_t f9g1_38 = f9_2 * (int64_t) g1_19;
1163 int64_t f9g2_19 = f9 * (int64_t) g2_19;
1164 int64_t f9g3_38 = f9_2 * (int64_t) g3_19;
1165 int64_t f9g4_19 = f9 * (int64_t) g4_19;
1166 int64_t f9g5_38 = f9_2 * (int64_t) g5_19;
1167 int64_t f9g6_19 = f9 * (int64_t) g6_19;
1168 int64_t f9g7_38 = f9_2 * (int64_t) g7_19;
1169 int64_t f9g8_19 = f9 * (int64_t) g8_19;
1170 int64_t f9g9_38 = f9_2 * (int64_t) g9_19;
1171 int64_t h0 = f0g0 + f1g9_38 + f2g8_19 + f3g7_38 + f4g6_19 + f5g5_38 + f6g4_19 + f7g3_38 + f8g2_19 + f9g1_38;
1172 int64_t h1 = f0g1 + f1g0 + f2g9_19 + f3g8_19 + f4g7_19 + f5g6_19 + f6g5_19 + f7g4_19 + f8g3_19 + f9g2_19;
1173 int64_t h2 = f0g2 + f1g1_2 + f2g0 + f3g9_38 + f4g8_19 + f5g7_38 + f6g6_19 + f7g5_38 + f8g4_19 + f9g3_38;
1174 int64_t h3 = f0g3 + f1g2 + f2g1 + f3g0 + f4g9_19 + f5g8_19 + f6g7_19 + f7g6_19 + f8g5_19 + f9g4_19;
1175 int64_t h4 = f0g4 + f1g3_2 + f2g2 + f3g1_2 + f4g0 + f5g9_38 + f6g8_19 + f7g7_38 + f8g6_19 + f9g5_38;
1176 int64_t h5 = f0g5 + f1g4 + f2g3 + f3g2 + f4g1 + f5g0 + f6g9_19 + f7g8_19 + f8g7_19 + f9g6_19;
1177 int64_t h6 = f0g6 + f1g5_2 + f2g4 + f3g3_2 + f4g2 + f5g1_2 + f6g0 + f7g9_38 + f8g8_19 + f9g7_38;
1178 int64_t h7 = f0g7 + f1g6 + f2g5 + f3g4 + f4g3 + f5g2 + f6g1 + f7g0 + f8g9_19 + f9g8_19;
1179 int64_t h8 = f0g8 + f1g7_2 + f2g6 + f3g5_2 + f4g4 + f5g3_2 + f6g2 + f7g1_2 + f8g0 + f9g9_38;
1180 int64_t h9 = f0g9 + f1g8 + f2g7 + f3g6 + f4g5 + f5g4 + f6g3 + f7g2 + f8g1 + f9g0 ;
1181 int64_t carry0;
1182 int64_t carry1;
1183 int64_t carry2;
1184 int64_t carry3;
1185 int64_t carry4;
1186 int64_t carry5;
1187 int64_t carry6;
1188 int64_t carry7;
1189 int64_t carry8;
1190 int64_t carry9;
1191
1192 /* |h0| <= (1.65*1.65*2^52*(1+19+19+19+19)+1.65*1.65*2^50*(38+38+38+38+38))
1193 * i.e. |h0| <= 1.4*2^60; narrower ranges for h2, h4, h6, h8
1194 * |h1| <= (1.65*1.65*2^51*(1+1+19+19+19+19+19+19+19+19))
1195 * i.e. |h1| <= 1.7*2^59; narrower ranges for h3, h5, h7, h9 */
1196
1197 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
1198 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
1199 /* |h0| <= 2^25 */
1200 /* |h4| <= 2^25 */
1201 /* |h1| <= 1.71*2^59 */
1202 /* |h5| <= 1.71*2^59 */
1203
1204 carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
1205 carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
1206 /* |h1| <= 2^24; from now on fits into int32 */
1207 /* |h5| <= 2^24; from now on fits into int32 */
1208 /* |h2| <= 1.41*2^60 */
1209 /* |h6| <= 1.41*2^60 */
1210
1211 carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
1212 carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
1213 /* |h2| <= 2^25; from now on fits into int32 unchanged */
1214 /* |h6| <= 2^25; from now on fits into int32 unchanged */
1215 /* |h3| <= 1.71*2^59 */
1216 /* |h7| <= 1.71*2^59 */
1217
1218 carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
1219 carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;
1220 /* |h3| <= 2^24; from now on fits into int32 unchanged */
1221 /* |h7| <= 2^24; from now on fits into int32 unchanged */
1222 /* |h4| <= 1.72*2^34 */
1223 /* |h8| <= 1.41*2^60 */
1224
1225 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
1226 carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;
1227 /* |h4| <= 2^25; from now on fits into int32 unchanged */
1228 /* |h8| <= 2^25; from now on fits into int32 unchanged */
1229 /* |h5| <= 1.01*2^24 */
1230 /* |h9| <= 1.71*2^59 */
1231
1232 carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
1233 /* |h9| <= 2^24; from now on fits into int32 unchanged */
1234 /* |h0| <= 1.1*2^39 */
1235
1236 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
1237 /* |h0| <= 2^25; from now on fits into int32 unchanged */
1238 /* |h1| <= 1.01*2^24 */
1239
1240 h[0] = (int32_t)h0;
1241 h[1] = (int32_t)h1;
1242 h[2] = (int32_t)h2;
1243 h[3] = (int32_t)h3;
1244 h[4] = (int32_t)h4;
1245 h[5] = (int32_t)h5;
1246 h[6] = (int32_t)h6;
1247 h[7] = (int32_t)h7;
1248 h[8] = (int32_t)h8;
1249 h[9] = (int32_t)h9;
1250}
1251
1252/*
1253 * h = f * f
1254 *
1255 * Can overlap h with f.
1256 *
1257 * Preconditions:
1258 * |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
1259 *
1260 * Postconditions:
1261 * |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
1262 *
1263 * See fe_mul.c for discussion of implementation strategy.
1264 */
1265static void fe_sq(fe h, const fe f)
1266{
1267 int32_t f0 = f[0];
1268 int32_t f1 = f[1];
1269 int32_t f2 = f[2];
1270 int32_t f3 = f[3];
1271 int32_t f4 = f[4];
1272 int32_t f5 = f[5];
1273 int32_t f6 = f[6];
1274 int32_t f7 = f[7];
1275 int32_t f8 = f[8];
1276 int32_t f9 = f[9];
1277 int32_t f0_2 = 2 * f0;
1278 int32_t f1_2 = 2 * f1;
1279 int32_t f2_2 = 2 * f2;
1280 int32_t f3_2 = 2 * f3;
1281 int32_t f4_2 = 2 * f4;
1282 int32_t f5_2 = 2 * f5;
1283 int32_t f6_2 = 2 * f6;
1284 int32_t f7_2 = 2 * f7;
1285 int32_t f5_38 = 38 * f5; /* 1.959375*2^30 */
1286 int32_t f6_19 = 19 * f6; /* 1.959375*2^30 */
1287 int32_t f7_38 = 38 * f7; /* 1.959375*2^30 */
1288 int32_t f8_19 = 19 * f8; /* 1.959375*2^30 */
1289 int32_t f9_38 = 38 * f9; /* 1.959375*2^30 */
1290 int64_t f0f0 = f0 * (int64_t) f0;
1291 int64_t f0f1_2 = f0_2 * (int64_t) f1;
1292 int64_t f0f2_2 = f0_2 * (int64_t) f2;
1293 int64_t f0f3_2 = f0_2 * (int64_t) f3;
1294 int64_t f0f4_2 = f0_2 * (int64_t) f4;
1295 int64_t f0f5_2 = f0_2 * (int64_t) f5;
1296 int64_t f0f6_2 = f0_2 * (int64_t) f6;
1297 int64_t f0f7_2 = f0_2 * (int64_t) f7;
1298 int64_t f0f8_2 = f0_2 * (int64_t) f8;
1299 int64_t f0f9_2 = f0_2 * (int64_t) f9;
1300 int64_t f1f1_2 = f1_2 * (int64_t) f1;
1301 int64_t f1f2_2 = f1_2 * (int64_t) f2;
1302 int64_t f1f3_4 = f1_2 * (int64_t) f3_2;
1303 int64_t f1f4_2 = f1_2 * (int64_t) f4;
1304 int64_t f1f5_4 = f1_2 * (int64_t) f5_2;
1305 int64_t f1f6_2 = f1_2 * (int64_t) f6;
1306 int64_t f1f7_4 = f1_2 * (int64_t) f7_2;
1307 int64_t f1f8_2 = f1_2 * (int64_t) f8;
1308 int64_t f1f9_76 = f1_2 * (int64_t) f9_38;
1309 int64_t f2f2 = f2 * (int64_t) f2;
1310 int64_t f2f3_2 = f2_2 * (int64_t) f3;
1311 int64_t f2f4_2 = f2_2 * (int64_t) f4;
1312 int64_t f2f5_2 = f2_2 * (int64_t) f5;
1313 int64_t f2f6_2 = f2_2 * (int64_t) f6;
1314 int64_t f2f7_2 = f2_2 * (int64_t) f7;
1315 int64_t f2f8_38 = f2_2 * (int64_t) f8_19;
1316 int64_t f2f9_38 = f2 * (int64_t) f9_38;
1317 int64_t f3f3_2 = f3_2 * (int64_t) f3;
1318 int64_t f3f4_2 = f3_2 * (int64_t) f4;
1319 int64_t f3f5_4 = f3_2 * (int64_t) f5_2;
1320 int64_t f3f6_2 = f3_2 * (int64_t) f6;
1321 int64_t f3f7_76 = f3_2 * (int64_t) f7_38;
1322 int64_t f3f8_38 = f3_2 * (int64_t) f8_19;
1323 int64_t f3f9_76 = f3_2 * (int64_t) f9_38;
1324 int64_t f4f4 = f4 * (int64_t) f4;
1325 int64_t f4f5_2 = f4_2 * (int64_t) f5;
1326 int64_t f4f6_38 = f4_2 * (int64_t) f6_19;
1327 int64_t f4f7_38 = f4 * (int64_t) f7_38;
1328 int64_t f4f8_38 = f4_2 * (int64_t) f8_19;
1329 int64_t f4f9_38 = f4 * (int64_t) f9_38;
1330 int64_t f5f5_38 = f5 * (int64_t) f5_38;
1331 int64_t f5f6_38 = f5_2 * (int64_t) f6_19;
1332 int64_t f5f7_76 = f5_2 * (int64_t) f7_38;
1333 int64_t f5f8_38 = f5_2 * (int64_t) f8_19;
1334 int64_t f5f9_76 = f5_2 * (int64_t) f9_38;
1335 int64_t f6f6_19 = f6 * (int64_t) f6_19;
1336 int64_t f6f7_38 = f6 * (int64_t) f7_38;
1337 int64_t f6f8_38 = f6_2 * (int64_t) f8_19;
1338 int64_t f6f9_38 = f6 * (int64_t) f9_38;
1339 int64_t f7f7_38 = f7 * (int64_t) f7_38;
1340 int64_t f7f8_38 = f7_2 * (int64_t) f8_19;
1341 int64_t f7f9_76 = f7_2 * (int64_t) f9_38;
1342 int64_t f8f8_19 = f8 * (int64_t) f8_19;
1343 int64_t f8f9_38 = f8 * (int64_t) f9_38;
1344 int64_t f9f9_38 = f9 * (int64_t) f9_38;
1345 int64_t h0 = f0f0 + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38;
1346 int64_t h1 = f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38;
1347 int64_t h2 = f0f2_2 + f1f1_2 + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19;
1348 int64_t h3 = f0f3_2 + f1f2_2 + f4f9_38 + f5f8_38 + f6f7_38;
1349 int64_t h4 = f0f4_2 + f1f3_4 + f2f2 + f5f9_76 + f6f8_38 + f7f7_38;
1350 int64_t h5 = f0f5_2 + f1f4_2 + f2f3_2 + f6f9_38 + f7f8_38;
1351 int64_t h6 = f0f6_2 + f1f5_4 + f2f4_2 + f3f3_2 + f7f9_76 + f8f8_19;
1352 int64_t h7 = f0f7_2 + f1f6_2 + f2f5_2 + f3f4_2 + f8f9_38;
1353 int64_t h8 = f0f8_2 + f1f7_4 + f2f6_2 + f3f5_4 + f4f4 + f9f9_38;
1354 int64_t h9 = f0f9_2 + f1f8_2 + f2f7_2 + f3f6_2 + f4f5_2;
1355 int64_t carry0;
1356 int64_t carry1;
1357 int64_t carry2;
1358 int64_t carry3;
1359 int64_t carry4;
1360 int64_t carry5;
1361 int64_t carry6;
1362 int64_t carry7;
1363 int64_t carry8;
1364 int64_t carry9;
1365
1366 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
1367 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
1368
1369 carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
1370 carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
1371
1372 carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
1373 carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
1374
1375 carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
1376 carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;
1377
1378 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
1379 carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;
1380
1381 carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
1382
1383 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
1384
1385 h[0] = (int32_t)h0;
1386 h[1] = (int32_t)h1;
1387 h[2] = (int32_t)h2;
1388 h[3] = (int32_t)h3;
1389 h[4] = (int32_t)h4;
1390 h[5] = (int32_t)h5;
1391 h[6] = (int32_t)h6;
1392 h[7] = (int32_t)h7;
1393 h[8] = (int32_t)h8;
1394 h[9] = (int32_t)h9;
1395}
1396
1397static void fe_invert(fe out, const fe z)
1398{
1399 fe t0;
1400 fe t1;
1401 fe t2;
1402 fe t3;
1403 int i;
1404
1405 /*
1406 * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as
1407 * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11.
1408 */
1409
1410 /* t0 = z ** 2 */
1411 fe_sq(t0, z);
1412
1413 /* t1 = t0 ** (2 ** 2) = z ** 8 */
1414 fe_sq(t1, t0);
1415 fe_sq(t1, t1);
1416
1417 /* t1 = z * t1 = z ** 9 */
1418 fe_mul(t1, z, t1);
1419 /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */
1420 fe_mul(t0, t0, t1);
1421
1422 /* t2 = t0 ** 2 = z ** 22 */
1423 fe_sq(t2, t0);
1424
1425 /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */
1426 fe_mul(t1, t1, t2);
1427
1428 /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */
1429 fe_sq(t2, t1);
1430 for (i = 1; i < 5; ++i) {
1431 fe_sq(t2, t2);
1432 }
1433
1434 /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */
1435 fe_mul(t1, t2, t1);
1436
1437 /* Continuing similarly... */
1438
1439 /* t2 = z ** (2 ** 20 - 1) */
1440 fe_sq(t2, t1);
1441 for (i = 1; i < 10; ++i) {
1442 fe_sq(t2, t2);
1443 }
1444 fe_mul(t2, t2, t1);
1445
1446 /* t2 = z ** (2 ** 40 - 1) */
1447 fe_sq(t3, t2);
1448 for (i = 1; i < 20; ++i) {
1449 fe_sq(t3, t3);
1450 }
1451 fe_mul(t2, t3, t2);
1452
1453 /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */
1454 for (i = 0; i < 10; ++i) {
1455 fe_sq(t2, t2);
1456 }
1457 /* t1 = z ** (2 ** 50 - 1) */
1458 fe_mul(t1, t2, t1);
1459
1460 /* t2 = z ** (2 ** 100 - 1) */
1461 fe_sq(t2, t1);
1462 for (i = 1; i < 50; ++i) {
1463 fe_sq(t2, t2);
1464 }
1465 fe_mul(t2, t2, t1);
1466
1467 /* t2 = z ** (2 ** 200 - 1) */
1468 fe_sq(t3, t2);
1469 for (i = 1; i < 100; ++i) {
1470 fe_sq(t3, t3);
1471 }
1472 fe_mul(t2, t3, t2);
1473
1474 /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */
1475 fe_sq(t2, t2);
1476 for (i = 1; i < 50; ++i) {
1477 fe_sq(t2, t2);
1478 }
1479
1480 /* t1 = z ** (2 ** 250 - 1) */
1481 fe_mul(t1, t2, t1);
1482
1483 /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */
1484 fe_sq(t1, t1);
1485 for (i = 1; i < 5; ++i) {
1486 fe_sq(t1, t1);
1487 }
1488
1489 /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */
1490 fe_mul(out, t1, t0);
1491}
1492
1493/*
1494 * h = -f
1495 *
1496 * Preconditions:
1497 * |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
1498 *
1499 * Postconditions:
1500 * |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
1501 */
1502static void fe_neg(fe h, const fe f)
1503{
1504 unsigned i;
1505
1506 for (i = 0; i < 10; i++) {
1507 h[i] = -f[i];
1508 }
1509}
1510
1511/*
1512 * Replace (f,g) with (g,g) if b == 1;
1513 * replace (f,g) with (f,g) if b == 0.
1514 *
1515 * Preconditions: b in {0,1}.
1516 */
1517static void fe_cmov(fe f, const fe g, unsigned b)
1518{
1519 size_t i;
1520
1521 b = 0-b;
1522 for (i = 0; i < 10; i++) {
1523 int32_t x = f[i] ^ g[i];
1524 x &= b;
1525 f[i] ^= x;
1526 }
1527}
1528
1529/*
1530 * return 0 if f == 0
1531 * return 1 if f != 0
1532 *
1533 * Preconditions:
1534 * |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
1535 */
1536static int fe_isnonzero(const fe f)
1537{
1538 uint8_t s[32];
1539 static const uint8_t zero[32] = {0};
1540
1541 fe_tobytes(s, f);
1542
1543 return CRYPTO_memcmp(s, zero, sizeof(zero)) != 0;
1544}
1545
1546/*
1547 * return 1 if f is in {1,3,5,...,q-2}
1548 * return 0 if f is in {0,2,4,...,q-1}
1549 *
1550 * Preconditions:
1551 * |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
1552 */
1553static int fe_isnegative(const fe f)
1554{
1555 uint8_t s[32];
1556
1557 fe_tobytes(s, f);
1558 return s[0] & 1;
1559}
1560
1561/*
1562 * h = 2 * f * f
1563 *
1564 * Can overlap h with f.
1565 *
1566 * Preconditions:
1567 * |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
1568 *
1569 * Postconditions:
1570 * |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
1571 *
1572 * See fe_mul.c for discussion of implementation strategy.
1573 */
1574static void fe_sq2(fe h, const fe f)
1575{
1576 int32_t f0 = f[0];
1577 int32_t f1 = f[1];
1578 int32_t f2 = f[2];
1579 int32_t f3 = f[3];
1580 int32_t f4 = f[4];
1581 int32_t f5 = f[5];
1582 int32_t f6 = f[6];
1583 int32_t f7 = f[7];
1584 int32_t f8 = f[8];
1585 int32_t f9 = f[9];
1586 int32_t f0_2 = 2 * f0;
1587 int32_t f1_2 = 2 * f1;
1588 int32_t f2_2 = 2 * f2;
1589 int32_t f3_2 = 2 * f3;
1590 int32_t f4_2 = 2 * f4;
1591 int32_t f5_2 = 2 * f5;
1592 int32_t f6_2 = 2 * f6;
1593 int32_t f7_2 = 2 * f7;
1594 int32_t f5_38 = 38 * f5; /* 1.959375*2^30 */
1595 int32_t f6_19 = 19 * f6; /* 1.959375*2^30 */
1596 int32_t f7_38 = 38 * f7; /* 1.959375*2^30 */
1597 int32_t f8_19 = 19 * f8; /* 1.959375*2^30 */
1598 int32_t f9_38 = 38 * f9; /* 1.959375*2^30 */
1599 int64_t f0f0 = f0 * (int64_t) f0;
1600 int64_t f0f1_2 = f0_2 * (int64_t) f1;
1601 int64_t f0f2_2 = f0_2 * (int64_t) f2;
1602 int64_t f0f3_2 = f0_2 * (int64_t) f3;
1603 int64_t f0f4_2 = f0_2 * (int64_t) f4;
1604 int64_t f0f5_2 = f0_2 * (int64_t) f5;
1605 int64_t f0f6_2 = f0_2 * (int64_t) f6;
1606 int64_t f0f7_2 = f0_2 * (int64_t) f7;
1607 int64_t f0f8_2 = f0_2 * (int64_t) f8;
1608 int64_t f0f9_2 = f0_2 * (int64_t) f9;
1609 int64_t f1f1_2 = f1_2 * (int64_t) f1;
1610 int64_t f1f2_2 = f1_2 * (int64_t) f2;
1611 int64_t f1f3_4 = f1_2 * (int64_t) f3_2;
1612 int64_t f1f4_2 = f1_2 * (int64_t) f4;
1613 int64_t f1f5_4 = f1_2 * (int64_t) f5_2;
1614 int64_t f1f6_2 = f1_2 * (int64_t) f6;
1615 int64_t f1f7_4 = f1_2 * (int64_t) f7_2;
1616 int64_t f1f8_2 = f1_2 * (int64_t) f8;
1617 int64_t f1f9_76 = f1_2 * (int64_t) f9_38;
1618 int64_t f2f2 = f2 * (int64_t) f2;
1619 int64_t f2f3_2 = f2_2 * (int64_t) f3;
1620 int64_t f2f4_2 = f2_2 * (int64_t) f4;
1621 int64_t f2f5_2 = f2_2 * (int64_t) f5;
1622 int64_t f2f6_2 = f2_2 * (int64_t) f6;
1623 int64_t f2f7_2 = f2_2 * (int64_t) f7;
1624 int64_t f2f8_38 = f2_2 * (int64_t) f8_19;
1625 int64_t f2f9_38 = f2 * (int64_t) f9_38;
1626 int64_t f3f3_2 = f3_2 * (int64_t) f3;
1627 int64_t f3f4_2 = f3_2 * (int64_t) f4;
1628 int64_t f3f5_4 = f3_2 * (int64_t) f5_2;
1629 int64_t f3f6_2 = f3_2 * (int64_t) f6;
1630 int64_t f3f7_76 = f3_2 * (int64_t) f7_38;
1631 int64_t f3f8_38 = f3_2 * (int64_t) f8_19;
1632 int64_t f3f9_76 = f3_2 * (int64_t) f9_38;
1633 int64_t f4f4 = f4 * (int64_t) f4;
1634 int64_t f4f5_2 = f4_2 * (int64_t) f5;
1635 int64_t f4f6_38 = f4_2 * (int64_t) f6_19;
1636 int64_t f4f7_38 = f4 * (int64_t) f7_38;
1637 int64_t f4f8_38 = f4_2 * (int64_t) f8_19;
1638 int64_t f4f9_38 = f4 * (int64_t) f9_38;
1639 int64_t f5f5_38 = f5 * (int64_t) f5_38;
1640 int64_t f5f6_38 = f5_2 * (int64_t) f6_19;
1641 int64_t f5f7_76 = f5_2 * (int64_t) f7_38;
1642 int64_t f5f8_38 = f5_2 * (int64_t) f8_19;
1643 int64_t f5f9_76 = f5_2 * (int64_t) f9_38;
1644 int64_t f6f6_19 = f6 * (int64_t) f6_19;
1645 int64_t f6f7_38 = f6 * (int64_t) f7_38;
1646 int64_t f6f8_38 = f6_2 * (int64_t) f8_19;
1647 int64_t f6f9_38 = f6 * (int64_t) f9_38;
1648 int64_t f7f7_38 = f7 * (int64_t) f7_38;
1649 int64_t f7f8_38 = f7_2 * (int64_t) f8_19;
1650 int64_t f7f9_76 = f7_2 * (int64_t) f9_38;
1651 int64_t f8f8_19 = f8 * (int64_t) f8_19;
1652 int64_t f8f9_38 = f8 * (int64_t) f9_38;
1653 int64_t f9f9_38 = f9 * (int64_t) f9_38;
1654 int64_t h0 = f0f0 + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38;
1655 int64_t h1 = f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38;
1656 int64_t h2 = f0f2_2 + f1f1_2 + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19;
1657 int64_t h3 = f0f3_2 + f1f2_2 + f4f9_38 + f5f8_38 + f6f7_38;
1658 int64_t h4 = f0f4_2 + f1f3_4 + f2f2 + f5f9_76 + f6f8_38 + f7f7_38;
1659 int64_t h5 = f0f5_2 + f1f4_2 + f2f3_2 + f6f9_38 + f7f8_38;
1660 int64_t h6 = f0f6_2 + f1f5_4 + f2f4_2 + f3f3_2 + f7f9_76 + f8f8_19;
1661 int64_t h7 = f0f7_2 + f1f6_2 + f2f5_2 + f3f4_2 + f8f9_38;
1662 int64_t h8 = f0f8_2 + f1f7_4 + f2f6_2 + f3f5_4 + f4f4 + f9f9_38;
1663 int64_t h9 = f0f9_2 + f1f8_2 + f2f7_2 + f3f6_2 + f4f5_2;
1664 int64_t carry0;
1665 int64_t carry1;
1666 int64_t carry2;
1667 int64_t carry3;
1668 int64_t carry4;
1669 int64_t carry5;
1670 int64_t carry6;
1671 int64_t carry7;
1672 int64_t carry8;
1673 int64_t carry9;
1674
1675 h0 += h0;
1676 h1 += h1;
1677 h2 += h2;
1678 h3 += h3;
1679 h4 += h4;
1680 h5 += h5;
1681 h6 += h6;
1682 h7 += h7;
1683 h8 += h8;
1684 h9 += h9;
1685
1686 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
1687 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
1688
1689 carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
1690 carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
1691
1692 carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
1693 carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
1694
1695 carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
1696 carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;
1697
1698 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
1699 carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;
1700
1701 carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
1702
1703 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
1704
1705 h[0] = (int32_t)h0;
1706 h[1] = (int32_t)h1;
1707 h[2] = (int32_t)h2;
1708 h[3] = (int32_t)h3;
1709 h[4] = (int32_t)h4;
1710 h[5] = (int32_t)h5;
1711 h[6] = (int32_t)h6;
1712 h[7] = (int32_t)h7;
1713 h[8] = (int32_t)h8;
1714 h[9] = (int32_t)h9;
1715}
1716
1717static void fe_pow22523(fe out, const fe z)
1718{
1719 fe t0;
1720 fe t1;
1721 fe t2;
1722 int i;
1723
1724 fe_sq(t0, z);
1725 fe_sq(t1, t0);
1726 for (i = 1; i < 2; ++i) {
1727 fe_sq(t1, t1);
1728 }
1729 fe_mul(t1, z, t1);
1730 fe_mul(t0, t0, t1);
1731 fe_sq(t0, t0);
1732 fe_mul(t0, t1, t0);
1733 fe_sq(t1, t0);
1734 for (i = 1; i < 5; ++i) {
1735 fe_sq(t1, t1);
1736 }
1737 fe_mul(t0, t1, t0);
1738 fe_sq(t1, t0);
1739 for (i = 1; i < 10; ++i) {
1740 fe_sq(t1, t1);
1741 }
1742 fe_mul(t1, t1, t0);
1743 fe_sq(t2, t1);
1744 for (i = 1; i < 20; ++i) {
1745 fe_sq(t2, t2);
1746 }
1747 fe_mul(t1, t2, t1);
1748 fe_sq(t1, t1);
1749 for (i = 1; i < 10; ++i) {
1750 fe_sq(t1, t1);
1751 }
1752 fe_mul(t0, t1, t0);
1753 fe_sq(t1, t0);
1754 for (i = 1; i < 50; ++i) {
1755 fe_sq(t1, t1);
1756 }
1757 fe_mul(t1, t1, t0);
1758 fe_sq(t2, t1);
1759 for (i = 1; i < 100; ++i) {
1760 fe_sq(t2, t2);
1761 }
1762 fe_mul(t1, t2, t1);
1763 fe_sq(t1, t1);
1764 for (i = 1; i < 50; ++i) {
1765 fe_sq(t1, t1);
1766 }
1767 fe_mul(t0, t1, t0);
1768 fe_sq(t0, t0);
1769 for (i = 1; i < 2; ++i) {
1770 fe_sq(t0, t0);
1771 }
1772 fe_mul(out, t0, z);
1773}
1774
1775/*
1776 * ge means group element.
1777 *
1778 * Here the group is the set of pairs (x,y) of field elements (see fe.h)
1779 * satisfying -x^2 + y^2 = 1 + d x^2y^2
1780 * where d = -121665/121666.
1781 *
1782 * Representations:
1783 * ge_p2 (projective): (X:Y:Z) satisfying x=X/Z, y=Y/Z
1784 * ge_p3 (extended): (X:Y:Z:T) satisfying x=X/Z, y=Y/Z, XY=ZT
1785 * ge_p1p1 (completed): ((X:Z),(Y:T)) satisfying x=X/Z, y=Y/T
1786 * ge_precomp (Duif): (y+x,y-x,2dxy)
1787 */
1788typedef struct {
1789 fe X;
1790 fe Y;
1791 fe Z;
1792} ge_p2;
1793
1794typedef struct {
1795 fe X;
1796 fe Y;
1797 fe Z;
1798 fe T;
1799} ge_p3;
1800
1801typedef struct {
1802 fe X;
1803 fe Y;
1804 fe Z;
1805 fe T;
1806} ge_p1p1;
1807
1808typedef struct {
1809 fe yplusx;
1810 fe yminusx;
1811 fe xy2d;
1812} ge_precomp;
1813
1814typedef struct {
1815 fe YplusX;
1816 fe YminusX;
1817 fe Z;
1818 fe T2d;
1819} ge_cached;
1820
1821static void ge_tobytes(uint8_t *s, const ge_p2 *h)
1822{
1823 fe recip;
1824 fe x;
1825 fe y;
1826
1827 fe_invert(recip, h->Z);
1828 fe_mul(x, h->X, recip);
1829 fe_mul(y, h->Y, recip);
1830 fe_tobytes(s, y);
1831 s[31] ^= fe_isnegative(x) << 7;
1832}
1833
1834static void ge_p3_tobytes(uint8_t *s, const ge_p3 *h)
1835{
1836 fe recip;
1837 fe x;
1838 fe y;
1839
1840 fe_invert(recip, h->Z);
1841 fe_mul(x, h->X, recip);
1842 fe_mul(y, h->Y, recip);
1843 fe_tobytes(s, y);
1844 s[31] ^= fe_isnegative(x) << 7;
1845}
1846
1847static const fe d = {
1848 -10913610, 13857413, -15372611, 6949391, 114729,
1849 -8787816, -6275908, -3247719, -18696448, -12055116
1850};
1851
1852static const fe sqrtm1 = {
1853 -32595792, -7943725, 9377950, 3500415, 12389472,
1854 -272473, -25146209, -2005654, 326686, 11406482
1855};
1856
1857static int ge_frombytes_vartime(ge_p3 *h, const uint8_t *s)
1858{
1859 fe u;
1860 fe v;
1861 fe v3;
1862 fe vxx;
1863 fe check;
1864
1865 fe_frombytes(h->Y, s);
1866 fe_1(h->Z);
1867 fe_sq(u, h->Y);
1868 fe_mul(v, u, d);
1869 fe_sub(u, u, h->Z); /* u = y^2-1 */
1870 fe_add(v, v, h->Z); /* v = dy^2+1 */
1871
1872 fe_sq(v3, v);
1873 fe_mul(v3, v3, v); /* v3 = v^3 */
1874 fe_sq(h->X, v3);
1875 fe_mul(h->X, h->X, v);
1876 fe_mul(h->X, h->X, u); /* x = uv^7 */
1877
1878 fe_pow22523(h->X, h->X); /* x = (uv^7)^((q-5)/8) */
1879 fe_mul(h->X, h->X, v3);
1880 fe_mul(h->X, h->X, u); /* x = uv^3(uv^7)^((q-5)/8) */
1881
1882 fe_sq(vxx, h->X);
1883 fe_mul(vxx, vxx, v);
1884 fe_sub(check, vxx, u); /* vx^2-u */
1885 if (fe_isnonzero(check)) {
1886 fe_add(check, vxx, u); /* vx^2+u */
1887 if (fe_isnonzero(check)) {
1888 return -1;
1889 }
1890 fe_mul(h->X, h->X, sqrtm1);
1891 }
1892
1893 if (fe_isnegative(h->X) != (s[31] >> 7)) {
1894 fe_neg(h->X, h->X);
1895 }
1896
1897 fe_mul(h->T, h->X, h->Y);
1898 return 0;
1899}
1900
1901static void ge_p2_0(ge_p2 *h)
1902{
1903 fe_0(h->X);
1904 fe_1(h->Y);
1905 fe_1(h->Z);
1906}
1907
1908static void ge_p3_0(ge_p3 *h)
1909{
1910 fe_0(h->X);
1911 fe_1(h->Y);
1912 fe_1(h->Z);
1913 fe_0(h->T);
1914}
1915
1916static void ge_precomp_0(ge_precomp *h)
1917{
1918 fe_1(h->yplusx);
1919 fe_1(h->yminusx);
1920 fe_0(h->xy2d);
1921}
1922
1923/* r = p */
1924static void ge_p3_to_p2(ge_p2 *r, const ge_p3 *p)
1925{
1926 fe_copy(r->X, p->X);
1927 fe_copy(r->Y, p->Y);
1928 fe_copy(r->Z, p->Z);
1929}
1930
1931static const fe d2 = {
1932 -21827239, -5839606, -30745221, 13898782, 229458,
1933 15978800, -12551817, -6495438, 29715968, 9444199
1934};
1935
1936/* r = p */
1937static void ge_p3_to_cached(ge_cached *r, const ge_p3 *p)
1938{
1939 fe_add(r->YplusX, p->Y, p->X);
1940 fe_sub(r->YminusX, p->Y, p->X);
1941 fe_copy(r->Z, p->Z);
1942 fe_mul(r->T2d, p->T, d2);
1943}
1944
1945/* r = p */
1946static void ge_p1p1_to_p2(ge_p2 *r, const ge_p1p1 *p)
1947{
1948 fe_mul(r->X, p->X, p->T);
1949 fe_mul(r->Y, p->Y, p->Z);
1950 fe_mul(r->Z, p->Z, p->T);
1951}
1952
1953/* r = p */
1954static void ge_p1p1_to_p3(ge_p3 *r, const ge_p1p1 *p)
1955{
1956 fe_mul(r->X, p->X, p->T);
1957 fe_mul(r->Y, p->Y, p->Z);
1958 fe_mul(r->Z, p->Z, p->T);
1959 fe_mul(r->T, p->X, p->Y);
1960}
1961
1962/* r = 2 * p */
1963static void ge_p2_dbl(ge_p1p1 *r, const ge_p2 *p)
1964{
1965 fe t0;
1966
1967 fe_sq(r->X, p->X);
1968 fe_sq(r->Z, p->Y);
1969 fe_sq2(r->T, p->Z);
1970 fe_add(r->Y, p->X, p->Y);
1971 fe_sq(t0, r->Y);
1972 fe_add(r->Y, r->Z, r->X);
1973 fe_sub(r->Z, r->Z, r->X);
1974 fe_sub(r->X, t0, r->Y);
1975 fe_sub(r->T, r->T, r->Z);
1976}
1977
1978/* r = 2 * p */
1979static void ge_p3_dbl(ge_p1p1 *r, const ge_p3 *p)
1980{
1981 ge_p2 q;
1982 ge_p3_to_p2(&q, p);
1983 ge_p2_dbl(r, &q);
1984}
1985
1986/* r = p + q */
1987static void ge_madd(ge_p1p1 *r, const ge_p3 *p, const ge_precomp *q)
1988{
1989 fe t0;
1990
1991 fe_add(r->X, p->Y, p->X);
1992 fe_sub(r->Y, p->Y, p->X);
1993 fe_mul(r->Z, r->X, q->yplusx);
1994 fe_mul(r->Y, r->Y, q->yminusx);
1995 fe_mul(r->T, q->xy2d, p->T);
1996 fe_add(t0, p->Z, p->Z);
1997 fe_sub(r->X, r->Z, r->Y);
1998 fe_add(r->Y, r->Z, r->Y);
1999 fe_add(r->Z, t0, r->T);
2000 fe_sub(r->T, t0, r->T);
2001}
2002
2003/* r = p - q */
2004static void ge_msub(ge_p1p1 *r, const ge_p3 *p, const ge_precomp *q)
2005{
2006 fe t0;
2007
2008 fe_add(r->X, p->Y, p->X);
2009 fe_sub(r->Y, p->Y, p->X);
2010 fe_mul(r->Z, r->X, q->yminusx);
2011 fe_mul(r->Y, r->Y, q->yplusx);
2012 fe_mul(r->T, q->xy2d, p->T);
2013 fe_add(t0, p->Z, p->Z);
2014 fe_sub(r->X, r->Z, r->Y);
2015 fe_add(r->Y, r->Z, r->Y);
2016 fe_sub(r->Z, t0, r->T);
2017 fe_add(r->T, t0, r->T);
2018}
2019
2020/* r = p + q */
2021static void ge_add(ge_p1p1 *r, const ge_p3 *p, const ge_cached *q)
2022{
2023 fe t0;
2024
2025 fe_add(r->X, p->Y, p->X);
2026 fe_sub(r->Y, p->Y, p->X);
2027 fe_mul(r->Z, r->X, q->YplusX);
2028 fe_mul(r->Y, r->Y, q->YminusX);
2029 fe_mul(r->T, q->T2d, p->T);
2030 fe_mul(r->X, p->Z, q->Z);
2031 fe_add(t0, r->X, r->X);
2032 fe_sub(r->X, r->Z, r->Y);
2033 fe_add(r->Y, r->Z, r->Y);
2034 fe_add(r->Z, t0, r->T);
2035 fe_sub(r->T, t0, r->T);
2036}
2037
2038/* r = p - q */
2039static void ge_sub(ge_p1p1 *r, const ge_p3 *p, const ge_cached *q)
2040{
2041 fe t0;
2042
2043 fe_add(r->X, p->Y, p->X);
2044 fe_sub(r->Y, p->Y, p->X);
2045 fe_mul(r->Z, r->X, q->YminusX);
2046 fe_mul(r->Y, r->Y, q->YplusX);
2047 fe_mul(r->T, q->T2d, p->T);
2048 fe_mul(r->X, p->Z, q->Z);
2049 fe_add(t0, r->X, r->X);
2050 fe_sub(r->X, r->Z, r->Y);
2051 fe_add(r->Y, r->Z, r->Y);
2052 fe_sub(r->Z, t0, r->T);
2053 fe_add(r->T, t0, r->T);
2054}
2055
2056static uint8_t equal(signed char b, signed char c)
2057{
2058 uint8_t ub = b;
2059 uint8_t uc = c;
2060 uint8_t x = ub ^ uc; /* 0: yes; 1..255: no */
2061 uint32_t y = x; /* 0: yes; 1..255: no */
2062 y -= 1; /* 4294967295: yes; 0..254: no */
2063 y >>= 31; /* 1: yes; 0: no */
2064 return y;
2065}
2066
2067static void cmov(ge_precomp *t, const ge_precomp *u, uint8_t b)
2068{
2069 fe_cmov(t->yplusx, u->yplusx, b);
2070 fe_cmov(t->yminusx, u->yminusx, b);
2071 fe_cmov(t->xy2d, u->xy2d, b);
2072}
2073
2074/* k25519Precomp[i][j] = (j+1)*256^i*B */
2075static const ge_precomp k25519Precomp[32][8] = {
2076 {
2077 {
2078 {25967493, -14356035, 29566456, 3660896, -12694345, 4014787,
2079 27544626, -11754271, -6079156, 2047605},
2080 {-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692,
2081 5043384, 19500929, -15469378},
2082 {-8738181, 4489570, 9688441, -14785194, 10184609, -12363380,
2083 29287919, 11864899, -24514362, -4438546},
2084 },
2085 {
2086 {-12815894, -12976347, -21581243, 11784320, -25355658, -2750717,
2087 -11717903, -3814571, -358445, -10211303},
2088 {-21703237, 6903825, 27185491, 6451973, -29577724, -9554005,
2089 -15616551, 11189268, -26829678, -5319081},
2090 {26966642, 11152617, 32442495, 15396054, 14353839, -12752335,
2091 -3128826, -9541118, -15472047, -4166697},
2092 },
2093 {
2094 {15636291, -9688557, 24204773, -7912398, 616977, -16685262,
2095 27787600, -14772189, 28944400, -1550024},
2096 {16568933, 4717097, -11556148, -1102322, 15682896, -11807043,
2097 16354577, -11775962, 7689662, 11199574},
2098 {30464156, -5976125, -11779434, -15670865, 23220365, 15915852,
2099 7512774, 10017326, -17749093, -9920357},
2100 },
2101 {
2102 {-17036878, 13921892, 10945806, -6033431, 27105052, -16084379,
2103 -28926210, 15006023, 3284568, -6276540},
2104 {23599295, -8306047, -11193664, -7687416, 13236774, 10506355,
2105 7464579, 9656445, 13059162, 10374397},
2106 {7798556, 16710257, 3033922, 2874086, 28997861, 2835604, 32406664,
2107 -3839045, -641708, -101325},
2108 },
2109 {
2110 {10861363, 11473154, 27284546, 1981175, -30064349, 12577861,
2111 32867885, 14515107, -15438304, 10819380},
2112 {4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668,
2113 12483688, -12668491, 5581306},
2114 {19563160, 16186464, -29386857, 4097519, 10237984, -4348115,
2115 28542350, 13850243, -23678021, -15815942},
2116 },
2117 {
2118 {-15371964, -12862754, 32573250, 4720197, -26436522, 5875511,
2119 -19188627, -15224819, -9818940, -12085777},
2120 {-8549212, 109983, 15149363, 2178705, 22900618, 4543417, 3044240,
2121 -15689887, 1762328, 14866737},
2122 {-18199695, -15951423, -10473290, 1707278, -17185920, 3916101,
2123 -28236412, 3959421, 27914454, 4383652},
2124 },
2125 {
2126 {5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852,
2127 5230134, -23952439, -15175766},
2128 {-30269007, -3463509, 7665486, 10083793, 28475525, 1649722,
2129 20654025, 16520125, 30598449, 7715701},
2130 {28881845, 14381568, 9657904, 3680757, -20181635, 7843316,
2131 -31400660, 1370708, 29794553, -1409300},
2132 },
2133 {
2134 {14499471, -2729599, -33191113, -4254652, 28494862, 14271267,
2135 30290735, 10876454, -33154098, 2381726},
2136 {-7195431, -2655363, -14730155, 462251, -27724326, 3941372,
2137 -6236617, 3696005, -32300832, 15351955},
2138 {27431194, 8222322, 16448760, -3907995, -18707002, 11938355,
2139 -32961401, -2970515, 29551813, 10109425},
2140 },
2141 },
2142 {
2143 {
2144 {-13657040, -13155431, -31283750, 11777098, 21447386, 6519384,
2145 -2378284, -1627556, 10092783, -4764171},
2146 {27939166, 14210322, 4677035, 16277044, -22964462, -12398139,
2147 -32508754, 12005538, -17810127, 12803510},
2148 {17228999, -15661624, -1233527, 300140, -1224870, -11714777,
2149 30364213, -9038194, 18016357, 4397660},
2150 },
2151 {
2152 {-10958843, -7690207, 4776341, -14954238, 27850028, -15602212,
2153 -26619106, 14544525, -17477504, 982639},
2154 {29253598, 15796703, -2863982, -9908884, 10057023, 3163536, 7332899,
2155 -4120128, -21047696, 9934963},
2156 {5793303, 16271923, -24131614, -10116404, 29188560, 1206517,
2157 -14747930, 4559895, -30123922, -10897950},
2158 },
2159 {
2160 {-27643952, -11493006, 16282657, -11036493, 28414021, -15012264,
2161 24191034, 4541697, -13338309, 5500568},
2162 {12650548, -1497113, 9052871, 11355358, -17680037, -8400164,
2163 -17430592, 12264343, 10874051, 13524335},
2164 {25556948, -3045990, 714651, 2510400, 23394682, -10415330, 33119038,
2165 5080568, -22528059, 5376628},
2166 },
2167 {
2168 {-26088264, -4011052, -17013699, -3537628, -6726793, 1920897,
2169 -22321305, -9447443, 4535768, 1569007},
2170 {-2255422, 14606630, -21692440, -8039818, 28430649, 8775819,
2171 -30494562, 3044290, 31848280, 12543772},
2172 {-22028579, 2943893, -31857513, 6777306, 13784462, -4292203,
2173 -27377195, -2062731, 7718482, 14474653},
2174 },
2175 {
2176 {2385315, 2454213, -22631320, 46603, -4437935, -15680415, 656965,
2177 -7236665, 24316168, -5253567},
2178 {13741529, 10911568, -33233417, -8603737, -20177830, -1033297,
2179 33040651, -13424532, -20729456, 8321686},
2180 {21060490, -2212744, 15712757, -4336099, 1639040, 10656336,
2181 23845965, -11874838, -9984458, 608372},
2182 },
2183 {
2184 {-13672732, -15087586, -10889693, -7557059, -6036909, 11305547,
2185 1123968, -6780577, 27229399, 23887},
2186 {-23244140, -294205, -11744728, 14712571, -29465699, -2029617,
2187 12797024, -6440308, -1633405, 16678954},
2188 {-29500620, 4770662, -16054387, 14001338, 7830047, 9564805,
2189 -1508144, -4795045, -17169265, 4904953},
2190 },
2191 {
2192 {24059557, 14617003, 19037157, -15039908, 19766093, -14906429,
2193 5169211, 16191880, 2128236, -4326833},
2194 {-16981152, 4124966, -8540610, -10653797, 30336522, -14105247,
2195 -29806336, 916033, -6882542, -2986532},
2196 {-22630907, 12419372, -7134229, -7473371, -16478904, 16739175,
2197 285431, 2763829, 15736322, 4143876},
2198 },
2199 {
2200 {2379352, 11839345, -4110402, -5988665, 11274298, 794957, 212801,
2201 -14594663, 23527084, -16458268},
2202 {33431127, -11130478, -17838966, -15626900, 8909499, 8376530,
2203 -32625340, 4087881, -15188911, -14416214},
2204 {1767683, 7197987, -13205226, -2022635, -13091350, 448826, 5799055,
2205 4357868, -4774191, -16323038},
2206 },
2207 },
2208 {
2209 {
2210 {6721966, 13833823, -23523388, -1551314, 26354293, -11863321,
2211 23365147, -3949732, 7390890, 2759800},
2212 {4409041, 2052381, 23373853, 10530217, 7676779, -12885954, 21302353,
2213 -4264057, 1244380, -12919645},
2214 {-4421239, 7169619, 4982368, -2957590, 30256825, -2777540, 14086413,
2215 9208236, 15886429, 16489664},
2216 },
2217 {
2218 {1996075, 10375649, 14346367, 13311202, -6874135, -16438411,
2219 -13693198, 398369, -30606455, -712933},
2220 {-25307465, 9795880, -2777414, 14878809, -33531835, 14780363,
2221 13348553, 12076947, -30836462, 5113182},
2222 {-17770784, 11797796, 31950843, 13929123, -25888302, 12288344,
2223 -30341101, -7336386, 13847711, 5387222},
2224 },
2225 {
2226 {-18582163, -3416217, 17824843, -2340966, 22744343, -10442611,
2227 8763061, 3617786, -19600662, 10370991},
2228 {20246567, -14369378, 22358229, -543712, 18507283, -10413996,
2229 14554437, -8746092, 32232924, 16763880},
2230 {9648505, 10094563, 26416693, 14745928, -30374318, -6472621,
2231 11094161, 15689506, 3140038, -16510092},
2232 },
2233 {
2234 {-16160072, 5472695, 31895588, 4744994, 8823515, 10365685,
2235 -27224800, 9448613, -28774454, 366295},
2236 {19153450, 11523972, -11096490, -6503142, -24647631, 5420647,
2237 28344573, 8041113, 719605, 11671788},
2238 {8678025, 2694440, -6808014, 2517372, 4964326, 11152271, -15432916,
2239 -15266516, 27000813, -10195553},
2240 },
2241 {
2242 {-15157904, 7134312, 8639287, -2814877, -7235688, 10421742, 564065,
2243 5336097, 6750977, -14521026},
2244 {11836410, -3979488, 26297894, 16080799, 23455045, 15735944,
2245 1695823, -8819122, 8169720, 16220347},
2246 {-18115838, 8653647, 17578566, -6092619, -8025777, -16012763,
2247 -11144307, -2627664, -5990708, -14166033},
2248 },
2249 {
2250 {-23308498, -10968312, 15213228, -10081214, -30853605, -11050004,
2251 27884329, 2847284, 2655861, 1738395},
2252 {-27537433, -14253021, -25336301, -8002780, -9370762, 8129821,
2253 21651608, -3239336, -19087449, -11005278},
2254 {1533110, 3437855, 23735889, 459276, 29970501, 11335377, 26030092,
2255 5821408, 10478196, 8544890},
2256 },
2257 {
2258 {32173121, -16129311, 24896207, 3921497, 22579056, -3410854,
2259 19270449, 12217473, 17789017, -3395995},
2260 {-30552961, -2228401, -15578829, -10147201, 13243889, 517024,
2261 15479401, -3853233, 30460520, 1052596},
2262 {-11614875, 13323618, 32618793, 8175907, -15230173, 12596687,
2263 27491595, -4612359, 3179268, -9478891},
2264 },
2265 {
2266 {31947069, -14366651, -4640583, -15339921, -15125977, -6039709,
2267 -14756777, -16411740, 19072640, -9511060},
2268 {11685058, 11822410, 3158003, -13952594, 33402194, -4165066,
2269 5977896, -5215017, 473099, 5040608},
2270 {-20290863, 8198642, -27410132, 11602123, 1290375, -2799760,
2271 28326862, 1721092, -19558642, -3131606},
2272 },
2273 },
2274 {
2275 {
2276 {7881532, 10687937, 7578723, 7738378, -18951012, -2553952, 21820786,
2277 8076149, -27868496, 11538389},
2278 {-19935666, 3899861, 18283497, -6801568, -15728660, -11249211,
2279 8754525, 7446702, -5676054, 5797016},
2280 {-11295600, -3793569, -15782110, -7964573, 12708869, -8456199,
2281 2014099, -9050574, -2369172, -5877341},
2282 },
2283 {
2284 {-22472376, -11568741, -27682020, 1146375, 18956691, 16640559,
2285 1192730, -3714199, 15123619, 10811505},
2286 {14352098, -3419715, -18942044, 10822655, 32750596, 4699007, -70363,
2287 15776356, -28886779, -11974553},
2288 {-28241164, -8072475, -4978962, -5315317, 29416931, 1847569,
2289 -20654173, -16484855, 4714547, -9600655},
2290 },
2291 {
2292 {15200332, 8368572, 19679101, 15970074, -31872674, 1959451,
2293 24611599, -4543832, -11745876, 12340220},
2294 {12876937, -10480056, 33134381, 6590940, -6307776, 14872440,
2295 9613953, 8241152, 15370987, 9608631},
2296 {-4143277, -12014408, 8446281, -391603, 4407738, 13629032, -7724868,
2297 15866074, -28210621, -8814099},
2298 },
2299 {
2300 {26660628, -15677655, 8393734, 358047, -7401291, 992988, -23904233,
2301 858697, 20571223, 8420556},
2302 {14620715, 13067227, -15447274, 8264467, 14106269, 15080814,
2303 33531827, 12516406, -21574435, -12476749},
2304 {236881, 10476226, 57258, -14677024, 6472998, 2466984, 17258519,
2305 7256740, 8791136, 15069930},
2306 },
2307 {
2308 {1276410, -9371918, 22949635, -16322807, -23493039, -5702186,
2309 14711875, 4874229, -30663140, -2331391},
2310 {5855666, 4990204, -13711848, 7294284, -7804282, 1924647, -1423175,
2311 -7912378, -33069337, 9234253},
2312 {20590503, -9018988, 31529744, -7352666, -2706834, 10650548,
2313 31559055, -11609587, 18979186, 13396066},
2314 },
2315 {
2316 {24474287, 4968103, 22267082, 4407354, 24063882, -8325180,
2317 -18816887, 13594782, 33514650, 7021958},
2318 {-11566906, -6565505, -21365085, 15928892, -26158305, 4315421,
2319 -25948728, -3916677, -21480480, 12868082},
2320 {-28635013, 13504661, 19988037, -2132761, 21078225, 6443208,
2321 -21446107, 2244500, -12455797, -8089383},
2322 },
2323 {
2324 {-30595528, 13793479, -5852820, 319136, -25723172, -6263899,
2325 33086546, 8957937, -15233648, 5540521},
2326 {-11630176, -11503902, -8119500, -7643073, 2620056, 1022908,
2327 -23710744, -1568984, -16128528, -14962807},
2328 {23152971, 775386, 27395463, 14006635, -9701118, 4649512, 1689819,
2329 892185, -11513277, -15205948},
2330 },
2331 {
2332 {9770129, 9586738, 26496094, 4324120, 1556511, -3550024, 27453819,
2333 4763127, -19179614, 5867134},
2334 {-32765025, 1927590, 31726409, -4753295, 23962434, -16019500,
2335 27846559, 5931263, -29749703, -16108455},
2336 {27461885, -2977536, 22380810, 1815854, -23033753, -3031938,
2337 7283490, -15148073, -19526700, 7734629},
2338 },
2339 },
2340 {
2341 {
2342 {-8010264, -9590817, -11120403, 6196038, 29344158, -13430885,
2343 7585295, -3176626, 18549497, 15302069},
2344 {-32658337, -6171222, -7672793, -11051681, 6258878, 13504381,
2345 10458790, -6418461, -8872242, 8424746},
2346 {24687205, 8613276, -30667046, -3233545, 1863892, -1830544,
2347 19206234, 7134917, -11284482, -828919},
2348 },
2349 {
2350 {11334899, -9218022, 8025293, 12707519, 17523892, -10476071,
2351 10243738, -14685461, -5066034, 16498837},
2352 {8911542, 6887158, -9584260, -6958590, 11145641, -9543680, 17303925,
2353 -14124238, 6536641, 10543906},
2354 {-28946384, 15479763, -17466835, 568876, -1497683, 11223454,
2355 -2669190, -16625574, -27235709, 8876771},
2356 },
2357 {
2358 {-25742899, -12566864, -15649966, -846607, -33026686, -796288,
2359 -33481822, 15824474, -604426, -9039817},
2360 {10330056, 70051, 7957388, -9002667, 9764902, 15609756, 27698697,
2361 -4890037, 1657394, 3084098},
2362 {10477963, -7470260, 12119566, -13250805, 29016247, -5365589,
2363 31280319, 14396151, -30233575, 15272409},
2364 },
2365 {
2366 {-12288309, 3169463, 28813183, 16658753, 25116432, -5630466,
2367 -25173957, -12636138, -25014757, 1950504},
2368 {-26180358, 9489187, 11053416, -14746161, -31053720, 5825630,
2369 -8384306, -8767532, 15341279, 8373727},
2370 {28685821, 7759505, -14378516, -12002860, -31971820, 4079242,
2371 298136, -10232602, -2878207, 15190420},
2372 },
2373 {
2374 {-32932876, 13806336, -14337485, -15794431, -24004620, 10940928,
2375 8669718, 2742393, -26033313, -6875003},
2376 {-1580388, -11729417, -25979658, -11445023, -17411874, -10912854,
2377 9291594, -16247779, -12154742, 6048605},
2378 {-30305315, 14843444, 1539301, 11864366, 20201677, 1900163,
2379 13934231, 5128323, 11213262, 9168384},
2380 },
2381 {
2382 {-26280513, 11007847, 19408960, -940758, -18592965, -4328580,
2383 -5088060, -11105150, 20470157, -16398701},
2384 {-23136053, 9282192, 14855179, -15390078, -7362815, -14408560,
2385 -22783952, 14461608, 14042978, 5230683},
2386 {29969567, -2741594, -16711867, -8552442, 9175486, -2468974,
2387 21556951, 3506042, -5933891, -12449708},
2388 },
2389 {
2390 {-3144746, 8744661, 19704003, 4581278, -20430686, 6830683,
2391 -21284170, 8971513, -28539189, 15326563},
2392 {-19464629, 10110288, -17262528, -3503892, -23500387, 1355669,
2393 -15523050, 15300988, -20514118, 9168260},
2394 {-5353335, 4488613, -23803248, 16314347, 7780487, -15638939,
2395 -28948358, 9601605, 33087103, -9011387},
2396 },
2397 {
2398 {-19443170, -15512900, -20797467, -12445323, -29824447, 10229461,
2399 -27444329, -15000531, -5996870, 15664672},
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3902 {-25894883, 15323294, -8489791, -8057900, 25967126, -13425460,
3903 2825960, -4897045, -23971776, -11267415},
3904 {-15924766, -5229880, -17443532, 6410664, 3622847, 10243618,
3905 20615400, 12405433, -23753030, -8436416},
3906 },
3907 {
3908 {-7091295, 12556208, -20191352, 9025187, -17072479, 4333801,
3909 4378436, 2432030, 23097949, -566018},
3910 {4565804, -16025654, 20084412, -7842817, 1724999, 189254, 24767264,
3911 10103221, -18512313, 2424778},
3912 {366633, -11976806, 8173090, -6890119, 30788634, 5745705, -7168678,
3913 1344109, -3642553, 12412659},
3914 },
3915 {
3916 {-24001791, 7690286, 14929416, -168257, -32210835, -13412986,
3917 24162697, -15326504, -3141501, 11179385},
3918 {18289522, -14724954, 8056945, 16430056, -21729724, 7842514,
3919 -6001441, -1486897, -18684645, -11443503},
3920 {476239, 6601091, -6152790, -9723375, 17503545, -4863900, 27672959,
3921 13403813, 11052904, 5219329},
3922 },
3923 },
3924 {
3925 {
3926 {20678546, -8375738, -32671898, 8849123, -5009758, 14574752,
3927 31186971, -3973730, 9014762, -8579056},
3928 {-13644050, -10350239, -15962508, 5075808, -1514661, -11534600,
3929 -33102500, 9160280, 8473550, -3256838},
3930 {24900749, 14435722, 17209120, -15292541, -22592275, 9878983,
3931 -7689309, -16335821, -24568481, 11788948},
3932 },
3933 {
3934 {-3118155, -11395194, -13802089, 14797441, 9652448, -6845904,
3935 -20037437, 10410733, -24568470, -1458691},
3936 {-15659161, 16736706, -22467150, 10215878, -9097177, 7563911,
3937 11871841, -12505194, -18513325, 8464118},
3938 {-23400612, 8348507, -14585951, -861714, -3950205, -6373419,
3939 14325289, 8628612, 33313881, -8370517},
3940 },
3941 {
3942 {-20186973, -4967935, 22367356, 5271547, -1097117, -4788838,
3943 -24805667, -10236854, -8940735, -5818269},
3944 {-6948785, -1795212, -32625683, -16021179, 32635414, -7374245,
3945 15989197, -12838188, 28358192, -4253904},
3946 {-23561781, -2799059, -32351682, -1661963, -9147719, 10429267,
3947 -16637684, 4072016, -5351664, 5596589},
3948 },
3949 {
3950 {-28236598, -3390048, 12312896, 6213178, 3117142, 16078565,
3951 29266239, 2557221, 1768301, 15373193},
3952 {-7243358, -3246960, -4593467, -7553353, -127927, -912245, -1090902,
3953 -4504991, -24660491, 3442910},
3954 {-30210571, 5124043, 14181784, 8197961, 18964734, -11939093,
3955 22597931, 7176455, -18585478, 13365930},
3956 },
3957 {
3958 {-7877390, -1499958, 8324673, 4690079, 6261860, 890446, 24538107,
3959 -8570186, -9689599, -3031667},
3960 {25008904, -10771599, -4305031, -9638010, 16265036, 15721635,
3961 683793, -11823784, 15723479, -15163481},
3962 {-9660625, 12374379, -27006999, -7026148, -7724114, -12314514,
3963 11879682, 5400171, 519526, -1235876},
3964 },
3965 {
3966 {22258397, -16332233, -7869817, 14613016, -22520255, -2950923,
3967 -20353881, 7315967, 16648397, 7605640},
3968 {-8081308, -8464597, -8223311, 9719710, 19259459, -15348212,
3969 23994942, -5281555, -9468848, 4763278},
3970 {-21699244, 9220969, -15730624, 1084137, -25476107, -2852390,
3971 31088447, -7764523, -11356529, 728112},
3972 },
3973 {
3974 {26047220, -11751471, -6900323, -16521798, 24092068, 9158119,
3975 -4273545, -12555558, -29365436, -5498272},
3976 {17510331, -322857, 5854289, 8403524, 17133918, -3112612, -28111007,
3977 12327945, 10750447, 10014012},
3978 {-10312768, 3936952, 9156313, -8897683, 16498692, -994647,
3979 -27481051, -666732, 3424691, 7540221},
3980 },
3981 {
3982 {30322361, -6964110, 11361005, -4143317, 7433304, 4989748, -7071422,
3983 -16317219, -9244265, 15258046},
3984 {13054562, -2779497, 19155474, 469045, -12482797, 4566042, 5631406,
3985 2711395, 1062915, -5136345},
3986 {-19240248, -11254599, -29509029, -7499965, -5835763, 13005411,
3987 -6066489, 12194497, 32960380, 1459310},
3988 },
3989 },
3990 {
3991 {
3992 {19852034, 7027924, 23669353, 10020366, 8586503, -6657907, 394197,
3993 -6101885, 18638003, -11174937},
3994 {31395534, 15098109, 26581030, 8030562, -16527914, -5007134,
3995 9012486, -7584354, -6643087, -5442636},
3996 {-9192165, -2347377, -1997099, 4529534, 25766844, 607986, -13222,
3997 9677543, -32294889, -6456008},
3998 },
3999 {
4000 {-2444496, -149937, 29348902, 8186665, 1873760, 12489863, -30934579,
4001 -7839692, -7852844, -8138429},
4002 {-15236356, -15433509, 7766470, 746860, 26346930, -10221762,
4003 -27333451, 10754588, -9431476, 5203576},
4004 {31834314, 14135496, -770007, 5159118, 20917671, -16768096,
4005 -7467973, -7337524, 31809243, 7347066},
4006 },
4007 {
4008 {-9606723, -11874240, 20414459, 13033986, 13716524, -11691881,
4009 19797970, -12211255, 15192876, -2087490},
4010 {-12663563, -2181719, 1168162, -3804809, 26747877, -14138091,
4011 10609330, 12694420, 33473243, -13382104},
4012 {33184999, 11180355, 15832085, -11385430, -1633671, 225884,
4013 15089336, -11023903, -6135662, 14480053},
4014 },
4015 {
4016 {31308717, -5619998, 31030840, -1897099, 15674547, -6582883,
4017 5496208, 13685227, 27595050, 8737275},
4018 {-20318852, -15150239, 10933843, -16178022, 8335352, -7546022,
4019 -31008351, -12610604, 26498114, 66511},
4020 {22644454, -8761729, -16671776, 4884562, -3105614, -13559366,
4021 30540766, -4286747, -13327787, -7515095},
4022 },
4023 {
4024 {-28017847, 9834845, 18617207, -2681312, -3401956, -13307506,
4025 8205540, 13585437, -17127465, 15115439},
4026 {23711543, -672915, 31206561, -8362711, 6164647, -9709987,
4027 -33535882, -1426096, 8236921, 16492939},
4028 {-23910559, -13515526, -26299483, -4503841, 25005590, -7687270,
4029 19574902, 10071562, 6708380, -6222424},
4030 },
4031 {
4032 {2101391, -4930054, 19702731, 2367575, -15427167, 1047675, 5301017,
4033 9328700, 29955601, -11678310},
4034 {3096359, 9271816, -21620864, -15521844, -14847996, -7592937,
4035 -25892142, -12635595, -9917575, 6216608},
4036 {-32615849, 338663, -25195611, 2510422, -29213566, -13820213,
4037 24822830, -6146567, -26767480, 7525079},
4038 },
4039 {
4040 {-23066649, -13985623, 16133487, -7896178, -3389565, 778788,
4041 -910336, -2782495, -19386633, 11994101},
4042 {21691500, -13624626, -641331, -14367021, 3285881, -3483596,
4043 -25064666, 9718258, -7477437, 13381418},
4044 {18445390, -4202236, 14979846, 11622458, -1727110, -3582980,
4045 23111648, -6375247, 28535282, 15779576},
4046 },
4047 {
4048 {30098053, 3089662, -9234387, 16662135, -21306940, 11308411,
4049 -14068454, 12021730, 9955285, -16303356},
4050 {9734894, -14576830, -7473633, -9138735, 2060392, 11313496,
4051 -18426029, 9924399, 20194861, 13380996},
4052 {-26378102, -7965207, -22167821, 15789297, -18055342, -6168792,
4053 -1984914, 15707771, 26342023, 10146099},
4054 },
4055 },
4056 {
4057 {
4058 {-26016874, -219943, 21339191, -41388, 19745256, -2878700,
4059 -29637280, 2227040, 21612326, -545728},
4060 {-13077387, 1184228, 23562814, -5970442, -20351244, -6348714,
4061 25764461, 12243797, -20856566, 11649658},
4062 {-10031494, 11262626, 27384172, 2271902, 26947504, -15997771, 39944,
4063 6114064, 33514190, 2333242},
4064 },
4065 {
4066 {-21433588, -12421821, 8119782, 7219913, -21830522, -9016134,
4067 -6679750, -12670638, 24350578, -13450001},
4068 {-4116307, -11271533, -23886186, 4843615, -30088339, 690623,
4069 -31536088, -10406836, 8317860, 12352766},
4070 {18200138, -14475911, -33087759, -2696619, -23702521, -9102511,
4071 -23552096, -2287550, 20712163, 6719373},
4072 },
4073 {
4074 {26656208, 6075253, -7858556, 1886072, -28344043, 4262326, 11117530,
4075 -3763210, 26224235, -3297458},
4076 {-17168938, -14854097, -3395676, -16369877, -19954045, 14050420,
4077 21728352, 9493610, 18620611, -16428628},
4078 {-13323321, 13325349, 11432106, 5964811, 18609221, 6062965,
4079 -5269471, -9725556, -30701573, -16479657},
4080 },
4081 {
4082 {-23860538, -11233159, 26961357, 1640861, -32413112, -16737940,
4083 12248509, -5240639, 13735342, 1934062},
4084 {25089769, 6742589, 17081145, -13406266, 21909293, -16067981,
4085 -15136294, -3765346, -21277997, 5473616},
4086 {31883677, -7961101, 1083432, -11572403, 22828471, 13290673,
4087 -7125085, 12469656, 29111212, -5451014},
4088 },
4089 {
4090 {24244947, -15050407, -26262976, 2791540, -14997599, 16666678,
4091 24367466, 6388839, -10295587, 452383},
4092 {-25640782, -3417841, 5217916, 16224624, 19987036, -4082269,
4093 -24236251, -5915248, 15766062, 8407814},
4094 {-20406999, 13990231, 15495425, 16395525, 5377168, 15166495,
4095 -8917023, -4388953, -8067909, 2276718},
4096 },
4097 {
4098 {30157918, 12924066, -17712050, 9245753, 19895028, 3368142,
4099 -23827587, 5096219, 22740376, -7303417},
4100 {2041139, -14256350, 7783687, 13876377, -25946985, -13352459,
4101 24051124, 13742383, -15637599, 13295222},
4102 {33338237, -8505733, 12532113, 7977527, 9106186, -1715251,
4103 -17720195, -4612972, -4451357, -14669444},
4104 },
4105 {
4106 {-20045281, 5454097, -14346548, 6447146, 28862071, 1883651,
4107 -2469266, -4141880, 7770569, 9620597},
4108 {23208068, 7979712, 33071466, 8149229, 1758231, -10834995, 30945528,
4109 -1694323, -33502340, -14767970},
4110 {1439958, -16270480, -1079989, -793782, 4625402, 10647766, -5043801,
4111 1220118, 30494170, -11440799},
4112 },
4113 {
4114 {-5037580, -13028295, -2970559, -3061767, 15640974, -6701666,
4115 -26739026, 926050, -1684339, -13333647},
4116 {13908495, -3549272, 30919928, -6273825, -21521863, 7989039,
4117 9021034, 9078865, 3353509, 4033511},
4118 {-29663431, -15113610, 32259991, -344482, 24295849, -12912123,
4119 23161163, 8839127, 27485041, 7356032},
4120 },
4121 },
4122 {
4123 {
4124 {9661027, 705443, 11980065, -5370154, -1628543, 14661173, -6346142,
4125 2625015, 28431036, -16771834},
4126 {-23839233, -8311415, -25945511, 7480958, -17681669, -8354183,
4127 -22545972, 14150565, 15970762, 4099461},
4128 {29262576, 16756590, 26350592, -8793563, 8529671, -11208050,
4129 13617293, -9937143, 11465739, 8317062},
4130 },
4131 {
4132 {-25493081, -6962928, 32500200, -9419051, -23038724, -2302222,
4133 14898637, 3848455, 20969334, -5157516},
4134 {-20384450, -14347713, -18336405, 13884722, -33039454, 2842114,
4135 -21610826, -3649888, 11177095, 14989547},
4136 {-24496721, -11716016, 16959896, 2278463, 12066309, 10137771,
4137 13515641, 2581286, -28487508, 9930240},
4138 },
4139 {
4140 {-17751622, -2097826, 16544300, -13009300, -15914807, -14949081,
4141 18345767, -13403753, 16291481, -5314038},
4142 {-33229194, 2553288, 32678213, 9875984, 8534129, 6889387, -9676774,
4143 6957617, 4368891, 9788741},
4144 {16660756, 7281060, -10830758, 12911820, 20108584, -8101676,
4145 -21722536, -8613148, 16250552, -11111103},
4146 },
4147 {
4148 {-19765507, 2390526, -16551031, 14161980, 1905286, 6414907, 4689584,
4149 10604807, -30190403, 4782747},
4150 {-1354539, 14736941, -7367442, -13292886, 7710542, -14155590,
4151 -9981571, 4383045, 22546403, 437323},
4152 {31665577, -12180464, -16186830, 1491339, -18368625, 3294682,
4153 27343084, 2786261, -30633590, -14097016},
4154 },
4155 {
4156 {-14467279, -683715, -33374107, 7448552, 19294360, 14334329,
4157 -19690631, 2355319, -19284671, -6114373},
4158 {15121312, -15796162, 6377020, -6031361, -10798111, -12957845,
4159 18952177, 15496498, -29380133, 11754228},
4160 {-2637277, -13483075, 8488727, -14303896, 12728761, -1622493,
4161 7141596, 11724556, 22761615, -10134141},
4162 },
4163 {
4164 {16918416, 11729663, -18083579, 3022987, -31015732, -13339659,
4165 -28741185, -12227393, 32851222, 11717399},
4166 {11166634, 7338049, -6722523, 4531520, -29468672, -7302055,
4167 31474879, 3483633, -1193175, -4030831},
4168 {-185635, 9921305, 31456609, -13536438, -12013818, 13348923,
4169 33142652, 6546660, -19985279, -3948376},
4170 },
4171 {
4172 {-32460596, 11266712, -11197107, -7899103, 31703694, 3855903,
4173 -8537131, -12833048, -30772034, -15486313},
4174 {-18006477, 12709068, 3991746, -6479188, -21491523, -10550425,
4175 -31135347, -16049879, 10928917, 3011958},
4176 {-6957757, -15594337, 31696059, 334240, 29576716, 14796075,
4177 -30831056, -12805180, 18008031, 10258577},
4178 },
4179 {
4180 {-22448644, 15655569, 7018479, -4410003, -30314266, -1201591,
4181 -1853465, 1367120, 25127874, 6671743},
4182 {29701166, -14373934, -10878120, 9279288, -17568, 13127210,
4183 21382910, 11042292, 25838796, 4642684},
4184 {-20430234, 14955537, -24126347, 8124619, -5369288, -5990470,
4185 30468147, -13900640, 18423289, 4177476},
4186 },
4187 },
4188};
4189
4190static uint8_t negative(signed char b)
4191{
4192 uint32_t x = b;
4193
4194 x >>= 31; /* 1: yes; 0: no */
4195 return x;
4196}
4197
4198static void table_select(ge_precomp *t, int pos, signed char b)
4199{
4200 ge_precomp minust;
4201 uint8_t bnegative = negative(b);
4202 uint8_t babs = b - ((uint8_t)((-bnegative) & b) << 1);
4203
4204 ge_precomp_0(t);
4205 cmov(t, &k25519Precomp[pos][0], equal(babs, 1));
4206 cmov(t, &k25519Precomp[pos][1], equal(babs, 2));
4207 cmov(t, &k25519Precomp[pos][2], equal(babs, 3));
4208 cmov(t, &k25519Precomp[pos][3], equal(babs, 4));
4209 cmov(t, &k25519Precomp[pos][4], equal(babs, 5));
4210 cmov(t, &k25519Precomp[pos][5], equal(babs, 6));
4211 cmov(t, &k25519Precomp[pos][6], equal(babs, 7));
4212 cmov(t, &k25519Precomp[pos][7], equal(babs, 8));
4213 fe_copy(minust.yplusx, t->yminusx);
4214 fe_copy(minust.yminusx, t->yplusx);
4215 fe_neg(minust.xy2d, t->xy2d);
4216 cmov(t, &minust, bnegative);
4217}
4218
4219/*
4220 * h = a * B
4221 *
4222 * where a = a[0]+256*a[1]+...+256^31 a[31]
4223 * B is the Ed25519 base point (x,4/5) with x positive.
4224 *
4225 * Preconditions:
4226 * a[31] <= 127
4227 */
4228static void ge_scalarmult_base(ge_p3 *h, const uint8_t *a)
4229{
4230 signed char e[64];
4231 signed char carry;
4232 ge_p1p1 r;
4233 ge_p2 s;
4234 ge_precomp t;
4235 int i;
4236
4237 for (i = 0; i < 32; ++i) {
4238 e[2 * i + 0] = (a[i] >> 0) & 15;
4239 e[2 * i + 1] = (a[i] >> 4) & 15;
4240 }
4241 /* each e[i] is between 0 and 15 */
4242 /* e[63] is between 0 and 7 */
4243
4244 carry = 0;
4245 for (i = 0; i < 63; ++i) {
4246 e[i] += carry;
4247 carry = e[i] + 8;
4248 carry >>= 4;
4249 e[i] -= carry << 4;
4250 }
4251 e[63] += carry;
4252 /* each e[i] is between -8 and 8 */
4253
4254 ge_p3_0(h);
4255 for (i = 1; i < 64; i += 2) {
4256 table_select(&t, i / 2, e[i]);
4257 ge_madd(&r, h, &t);
4258 ge_p1p1_to_p3(h, &r);
4259 }
4260
4261 ge_p3_dbl(&r, h);
4262 ge_p1p1_to_p2(&s, &r);
4263 ge_p2_dbl(&r, &s);
4264 ge_p1p1_to_p2(&s, &r);
4265 ge_p2_dbl(&r, &s);
4266 ge_p1p1_to_p2(&s, &r);
4267 ge_p2_dbl(&r, &s);
4268 ge_p1p1_to_p3(h, &r);
4269
4270 for (i = 0; i < 64; i += 2) {
4271 table_select(&t, i / 2, e[i]);
4272 ge_madd(&r, h, &t);
4273 ge_p1p1_to_p3(h, &r);
4274 }
4275
4276 OPENSSL_cleanse(e, sizeof(e));
4277}
4278
4279#if !defined(BASE_2_51_IMPLEMENTED)
4280/*
4281 * Replace (f,g) with (g,f) if b == 1;
4282 * replace (f,g) with (f,g) if b == 0.
4283 *
4284 * Preconditions: b in {0,1}.
4285 */
4286static void fe_cswap(fe f, fe g, unsigned int b)
4287{
4288 size_t i;
4289
4290 b = 0-b;
4291 for (i = 0; i < 10; i++) {
4292 int32_t x = f[i] ^ g[i];
4293 x &= b;
4294 f[i] ^= x;
4295 g[i] ^= x;
4296 }
4297}
4298
4299/*
4300 * h = f * 121666
4301 *
4302 * Can overlap h with f.
4303 *
4304 * Preconditions:
4305 * |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
4306 *
4307 * Postconditions:
4308 * |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
4309 */
4310static void fe_mul121666(fe h, fe f)
4311{
4312 int32_t f0 = f[0];
4313 int32_t f1 = f[1];
4314 int32_t f2 = f[2];
4315 int32_t f3 = f[3];
4316 int32_t f4 = f[4];
4317 int32_t f5 = f[5];
4318 int32_t f6 = f[6];
4319 int32_t f7 = f[7];
4320 int32_t f8 = f[8];
4321 int32_t f9 = f[9];
4322 int64_t h0 = f0 * (int64_t) 121666;
4323 int64_t h1 = f1 * (int64_t) 121666;
4324 int64_t h2 = f2 * (int64_t) 121666;
4325 int64_t h3 = f3 * (int64_t) 121666;
4326 int64_t h4 = f4 * (int64_t) 121666;
4327 int64_t h5 = f5 * (int64_t) 121666;
4328 int64_t h6 = f6 * (int64_t) 121666;
4329 int64_t h7 = f7 * (int64_t) 121666;
4330 int64_t h8 = f8 * (int64_t) 121666;
4331 int64_t h9 = f9 * (int64_t) 121666;
4332 int64_t carry0;
4333 int64_t carry1;
4334 int64_t carry2;
4335 int64_t carry3;
4336 int64_t carry4;
4337 int64_t carry5;
4338 int64_t carry6;
4339 int64_t carry7;
4340 int64_t carry8;
4341 int64_t carry9;
4342
4343 carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
4344 carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
4345 carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
4346 carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
4347 carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;
4348
4349 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
4350 carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
4351 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
4352 carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
4353 carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;
4354
4355 h[0] = (int32_t)h0;
4356 h[1] = (int32_t)h1;
4357 h[2] = (int32_t)h2;
4358 h[3] = (int32_t)h3;
4359 h[4] = (int32_t)h4;
4360 h[5] = (int32_t)h5;
4361 h[6] = (int32_t)h6;
4362 h[7] = (int32_t)h7;
4363 h[8] = (int32_t)h8;
4364 h[9] = (int32_t)h9;
4365}
4366
4367static void x25519_scalar_mult_generic(uint8_t out[32],
4368 const uint8_t scalar[32],
4369 const uint8_t point[32]) {
4370 fe x1, x2, z2, x3, z3, tmp0, tmp1;
4371 uint8_t e[32];
4372 unsigned swap = 0;
4373 int pos;
4374
4375 memcpy(e, scalar, 32);
4376 e[0] &= 248;
4377 e[31] &= 127;
4378 e[31] |= 64;
4379 fe_frombytes(x1, point);
4380 fe_1(x2);
4381 fe_0(z2);
4382 fe_copy(x3, x1);
4383 fe_1(z3);
4384
4385 for (pos = 254; pos >= 0; --pos) {
4386 unsigned b = 1 & (e[pos / 8] >> (pos & 7));
4387 swap ^= b;
4388 fe_cswap(x2, x3, swap);
4389 fe_cswap(z2, z3, swap);
4390 swap = b;
4391 fe_sub(tmp0, x3, z3);
4392 fe_sub(tmp1, x2, z2);
4393 fe_add(x2, x2, z2);
4394 fe_add(z2, x3, z3);
4395 fe_mul(z3, tmp0, x2);
4396 fe_mul(z2, z2, tmp1);
4397 fe_sq(tmp0, tmp1);
4398 fe_sq(tmp1, x2);
4399 fe_add(x3, z3, z2);
4400 fe_sub(z2, z3, z2);
4401 fe_mul(x2, tmp1, tmp0);
4402 fe_sub(tmp1, tmp1, tmp0);
4403 fe_sq(z2, z2);
4404 fe_mul121666(z3, tmp1);
4405 fe_sq(x3, x3);
4406 fe_add(tmp0, tmp0, z3);
4407 fe_mul(z3, x1, z2);
4408 fe_mul(z2, tmp1, tmp0);
4409 }
4410
4411 fe_invert(z2, z2);
4412 fe_mul(x2, x2, z2);
4413 fe_tobytes(out, x2);
4414
4415 OPENSSL_cleanse(e, sizeof(e));
4416}
4417
4418static void x25519_scalar_mult(uint8_t out[32], const uint8_t scalar[32],
4419 const uint8_t point[32]) {
4420 x25519_scalar_mult_generic(out, scalar, point);
4421}
4422#endif
4423
4424static void slide(signed char *r, const uint8_t *a)
4425{
4426 int i;
4427 int b;
4428 int k;
4429
4430 for (i = 0; i < 256; ++i) {
4431 r[i] = 1 & (a[i >> 3] >> (i & 7));
4432 }
4433
4434 for (i = 0; i < 256; ++i) {
4435 if (r[i]) {
4436 for (b = 1; b <= 6 && i + b < 256; ++b) {
4437 if (r[i + b]) {
4438 if (r[i] + (r[i + b] << b) <= 15) {
4439 r[i] += r[i + b] << b;
4440 r[i + b] = 0;
4441 } else if (r[i] - (r[i + b] << b) >= -15) {
4442 r[i] -= r[i + b] << b;
4443 for (k = i + b; k < 256; ++k) {
4444 if (!r[k]) {
4445 r[k] = 1;
4446 break;
4447 }
4448 r[k] = 0;
4449 }
4450 } else {
4451 break;
4452 }
4453 }
4454 }
4455 }
4456 }
4457}
4458
4459static const ge_precomp Bi[8] = {
4460 {
4461 {25967493, -14356035, 29566456, 3660896, -12694345, 4014787, 27544626,
4462 -11754271, -6079156, 2047605},
4463 {-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692,
4464 5043384, 19500929, -15469378},
4465 {-8738181, 4489570, 9688441, -14785194, 10184609, -12363380, 29287919,
4466 11864899, -24514362, -4438546},
4467 },
4468 {
4469 {15636291, -9688557, 24204773, -7912398, 616977, -16685262, 27787600,
4470 -14772189, 28944400, -1550024},
4471 {16568933, 4717097, -11556148, -1102322, 15682896, -11807043, 16354577,
4472 -11775962, 7689662, 11199574},
4473 {30464156, -5976125, -11779434, -15670865, 23220365, 15915852, 7512774,
4474 10017326, -17749093, -9920357},
4475 },
4476 {
4477 {10861363, 11473154, 27284546, 1981175, -30064349, 12577861, 32867885,
4478 14515107, -15438304, 10819380},
4479 {4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668,
4480 12483688, -12668491, 5581306},
4481 {19563160, 16186464, -29386857, 4097519, 10237984, -4348115, 28542350,
4482 13850243, -23678021, -15815942},
4483 },
4484 {
4485 {5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852,
4486 5230134, -23952439, -15175766},
4487 {-30269007, -3463509, 7665486, 10083793, 28475525, 1649722, 20654025,
4488 16520125, 30598449, 7715701},
4489 {28881845, 14381568, 9657904, 3680757, -20181635, 7843316, -31400660,
4490 1370708, 29794553, -1409300},
4491 },
4492 {
4493 {-22518993, -6692182, 14201702, -8745502, -23510406, 8844726, 18474211,
4494 -1361450, -13062696, 13821877},
4495 {-6455177, -7839871, 3374702, -4740862, -27098617, -10571707, 31655028,
4496 -7212327, 18853322, -14220951},
4497 {4566830, -12963868, -28974889, -12240689, -7602672, -2830569, -8514358,
4498 -10431137, 2207753, -3209784},
4499 },
4500 {
4501 {-25154831, -4185821, 29681144, 7868801, -6854661, -9423865, -12437364,
4502 -663000, -31111463, -16132436},
4503 {25576264, -2703214, 7349804, -11814844, 16472782, 9300885, 3844789,
4504 15725684, 171356, 6466918},
4505 {23103977, 13316479, 9739013, -16149481, 817875, -15038942, 8965339,
4506 -14088058, -30714912, 16193877},
4507 },
4508 {
4509 {-33521811, 3180713, -2394130, 14003687, -16903474, -16270840, 17238398,
4510 4729455, -18074513, 9256800},
4511 {-25182317, -4174131, 32336398, 5036987, -21236817, 11360617, 22616405,
4512 9761698, -19827198, 630305},
4513 {-13720693, 2639453, -24237460, -7406481, 9494427, -5774029, -6554551,
4514 -15960994, -2449256, -14291300},
4515 },
4516 {
4517 {-3151181, -5046075, 9282714, 6866145, -31907062, -863023, -18940575,
4518 15033784, 25105118, -7894876},
4519 {-24326370, 15950226, -31801215, -14592823, -11662737, -5090925,
4520 1573892, -2625887, 2198790, -15804619},
4521 {-3099351, 10324967, -2241613, 7453183, -5446979, -2735503, -13812022,
4522 -16236442, -32461234, -12290683},
4523 },
4524};
4525
4526/*
4527 * r = a * A + b * B
4528 *
4529 * where a = a[0]+256*a[1]+...+256^31 a[31].
4530 * and b = b[0]+256*b[1]+...+256^31 b[31].
4531 * B is the Ed25519 base point (x,4/5) with x positive.
4532 */
4533static void ge_double_scalarmult_vartime(ge_p2 *r, const uint8_t *a,
4534 const ge_p3 *A, const uint8_t *b)
4535{
4536 signed char aslide[256];
4537 signed char bslide[256];
4538 ge_cached Ai[8]; /* A,3A,5A,7A,9A,11A,13A,15A */
4539 ge_p1p1 t;
4540 ge_p3 u;
4541 ge_p3 A2;
4542 int i;
4543
4544 slide(aslide, a);
4545 slide(bslide, b);
4546
4547 ge_p3_to_cached(&Ai[0], A);
4548 ge_p3_dbl(&t, A);
4549 ge_p1p1_to_p3(&A2, &t);
4550 ge_add(&t, &A2, &Ai[0]);
4551 ge_p1p1_to_p3(&u, &t);
4552 ge_p3_to_cached(&Ai[1], &u);
4553 ge_add(&t, &A2, &Ai[1]);
4554 ge_p1p1_to_p3(&u, &t);
4555 ge_p3_to_cached(&Ai[2], &u);
4556 ge_add(&t, &A2, &Ai[2]);
4557 ge_p1p1_to_p3(&u, &t);
4558 ge_p3_to_cached(&Ai[3], &u);
4559 ge_add(&t, &A2, &Ai[3]);
4560 ge_p1p1_to_p3(&u, &t);
4561 ge_p3_to_cached(&Ai[4], &u);
4562 ge_add(&t, &A2, &Ai[4]);
4563 ge_p1p1_to_p3(&u, &t);
4564 ge_p3_to_cached(&Ai[5], &u);
4565 ge_add(&t, &A2, &Ai[5]);
4566 ge_p1p1_to_p3(&u, &t);
4567 ge_p3_to_cached(&Ai[6], &u);
4568 ge_add(&t, &A2, &Ai[6]);
4569 ge_p1p1_to_p3(&u, &t);
4570 ge_p3_to_cached(&Ai[7], &u);
4571
4572 ge_p2_0(r);
4573
4574 for (i = 255; i >= 0; --i) {
4575 if (aslide[i] || bslide[i]) {
4576 break;
4577 }
4578 }
4579
4580 for (; i >= 0; --i) {
4581 ge_p2_dbl(&t, r);
4582
4583 if (aslide[i] > 0) {
4584 ge_p1p1_to_p3(&u, &t);
4585 ge_add(&t, &u, &Ai[aslide[i] / 2]);
4586 } else if (aslide[i] < 0) {
4587 ge_p1p1_to_p3(&u, &t);
4588 ge_sub(&t, &u, &Ai[(-aslide[i]) / 2]);
4589 }
4590
4591 if (bslide[i] > 0) {
4592 ge_p1p1_to_p3(&u, &t);
4593 ge_madd(&t, &u, &Bi[bslide[i] / 2]);
4594 } else if (bslide[i] < 0) {
4595 ge_p1p1_to_p3(&u, &t);
4596 ge_msub(&t, &u, &Bi[(-bslide[i]) / 2]);
4597 }
4598
4599 ge_p1p1_to_p2(r, &t);
4600 }
4601}
4602
4603/*
4604 * The set of scalars is \Z/l
4605 * where l = 2^252 + 27742317777372353535851937790883648493.
4606 *
4607 * Input:
4608 * s[0]+256*s[1]+...+256^63*s[63] = s
4609 *
4610 * Output:
4611 * s[0]+256*s[1]+...+256^31*s[31] = s mod l
4612 * where l = 2^252 + 27742317777372353535851937790883648493.
4613 * Overwrites s in place.
4614*/
4615static void x25519_sc_reduce(uint8_t *s)
4616{
4617 int64_t s0 = kBottom21Bits & load_3(s);
4618 int64_t s1 = kBottom21Bits & (load_4(s + 2) >> 5);
4619 int64_t s2 = kBottom21Bits & (load_3(s + 5) >> 2);
4620 int64_t s3 = kBottom21Bits & (load_4(s + 7) >> 7);
4621 int64_t s4 = kBottom21Bits & (load_4(s + 10) >> 4);
4622 int64_t s5 = kBottom21Bits & (load_3(s + 13) >> 1);
4623 int64_t s6 = kBottom21Bits & (load_4(s + 15) >> 6);
4624 int64_t s7 = kBottom21Bits & (load_3(s + 18) >> 3);
4625 int64_t s8 = kBottom21Bits & load_3(s + 21);
4626 int64_t s9 = kBottom21Bits & (load_4(s + 23) >> 5);
4627 int64_t s10 = kBottom21Bits & (load_3(s + 26) >> 2);
4628 int64_t s11 = kBottom21Bits & (load_4(s + 28) >> 7);
4629 int64_t s12 = kBottom21Bits & (load_4(s + 31) >> 4);
4630 int64_t s13 = kBottom21Bits & (load_3(s + 34) >> 1);
4631 int64_t s14 = kBottom21Bits & (load_4(s + 36) >> 6);
4632 int64_t s15 = kBottom21Bits & (load_3(s + 39) >> 3);
4633 int64_t s16 = kBottom21Bits & load_3(s + 42);
4634 int64_t s17 = kBottom21Bits & (load_4(s + 44) >> 5);
4635 int64_t s18 = kBottom21Bits & (load_3(s + 47) >> 2);
4636 int64_t s19 = kBottom21Bits & (load_4(s + 49) >> 7);
4637 int64_t s20 = kBottom21Bits & (load_4(s + 52) >> 4);
4638 int64_t s21 = kBottom21Bits & (load_3(s + 55) >> 1);
4639 int64_t s22 = kBottom21Bits & (load_4(s + 57) >> 6);
4640 int64_t s23 = (load_4(s + 60) >> 3);
4641 int64_t carry0;
4642 int64_t carry1;
4643 int64_t carry2;
4644 int64_t carry3;
4645 int64_t carry4;
4646 int64_t carry5;
4647 int64_t carry6;
4648 int64_t carry7;
4649 int64_t carry8;
4650 int64_t carry9;
4651 int64_t carry10;
4652 int64_t carry11;
4653 int64_t carry12;
4654 int64_t carry13;
4655 int64_t carry14;
4656 int64_t carry15;
4657 int64_t carry16;
4658
4659 s11 += s23 * 666643;
4660 s12 += s23 * 470296;
4661 s13 += s23 * 654183;
4662 s14 -= s23 * 997805;
4663 s15 += s23 * 136657;
4664 s16 -= s23 * 683901;
4665 s23 = 0;
4666
4667 s10 += s22 * 666643;
4668 s11 += s22 * 470296;
4669 s12 += s22 * 654183;
4670 s13 -= s22 * 997805;
4671 s14 += s22 * 136657;
4672 s15 -= s22 * 683901;
4673 s22 = 0;
4674
4675 s9 += s21 * 666643;
4676 s10 += s21 * 470296;
4677 s11 += s21 * 654183;
4678 s12 -= s21 * 997805;
4679 s13 += s21 * 136657;
4680 s14 -= s21 * 683901;
4681 s21 = 0;
4682
4683 s8 += s20 * 666643;
4684 s9 += s20 * 470296;
4685 s10 += s20 * 654183;
4686 s11 -= s20 * 997805;
4687 s12 += s20 * 136657;
4688 s13 -= s20 * 683901;
4689 s20 = 0;
4690
4691 s7 += s19 * 666643;
4692 s8 += s19 * 470296;
4693 s9 += s19 * 654183;
4694 s10 -= s19 * 997805;
4695 s11 += s19 * 136657;
4696 s12 -= s19 * 683901;
4697 s19 = 0;
4698
4699 s6 += s18 * 666643;
4700 s7 += s18 * 470296;
4701 s8 += s18 * 654183;
4702 s9 -= s18 * 997805;
4703 s10 += s18 * 136657;
4704 s11 -= s18 * 683901;
4705 s18 = 0;
4706
4707 carry6 = (s6 + (1 << 20)) >> 21;
4708 s7 += carry6;
4709 s6 -= carry6 * (1 << 21);
4710 carry8 = (s8 + (1 << 20)) >> 21;
4711 s9 += carry8;
4712 s8 -= carry8 * (1 << 21);
4713 carry10 = (s10 + (1 << 20)) >> 21;
4714 s11 += carry10;
4715 s10 -= carry10 * (1 << 21);
4716 carry12 = (s12 + (1 << 20)) >> 21;
4717 s13 += carry12;
4718 s12 -= carry12 * (1 << 21);
4719 carry14 = (s14 + (1 << 20)) >> 21;
4720 s15 += carry14;
4721 s14 -= carry14 * (1 << 21);
4722 carry16 = (s16 + (1 << 20)) >> 21;
4723 s17 += carry16;
4724 s16 -= carry16 * (1 << 21);
4725
4726 carry7 = (s7 + (1 << 20)) >> 21;
4727 s8 += carry7;
4728 s7 -= carry7 * (1 << 21);
4729 carry9 = (s9 + (1 << 20)) >> 21;
4730 s10 += carry9;
4731 s9 -= carry9 * (1 << 21);
4732 carry11 = (s11 + (1 << 20)) >> 21;
4733 s12 += carry11;
4734 s11 -= carry11 * (1 << 21);
4735 carry13 = (s13 + (1 << 20)) >> 21;
4736 s14 += carry13;
4737 s13 -= carry13 * (1 << 21);
4738 carry15 = (s15 + (1 << 20)) >> 21;
4739 s16 += carry15;
4740 s15 -= carry15 * (1 << 21);
4741
4742 s5 += s17 * 666643;
4743 s6 += s17 * 470296;
4744 s7 += s17 * 654183;
4745 s8 -= s17 * 997805;
4746 s9 += s17 * 136657;
4747 s10 -= s17 * 683901;
4748 s17 = 0;
4749
4750 s4 += s16 * 666643;
4751 s5 += s16 * 470296;
4752 s6 += s16 * 654183;
4753 s7 -= s16 * 997805;
4754 s8 += s16 * 136657;
4755 s9 -= s16 * 683901;
4756 s16 = 0;
4757
4758 s3 += s15 * 666643;
4759 s4 += s15 * 470296;
4760 s5 += s15 * 654183;
4761 s6 -= s15 * 997805;
4762 s7 += s15 * 136657;
4763 s8 -= s15 * 683901;
4764 s15 = 0;
4765
4766 s2 += s14 * 666643;
4767 s3 += s14 * 470296;
4768 s4 += s14 * 654183;
4769 s5 -= s14 * 997805;
4770 s6 += s14 * 136657;
4771 s7 -= s14 * 683901;
4772 s14 = 0;
4773
4774 s1 += s13 * 666643;
4775 s2 += s13 * 470296;
4776 s3 += s13 * 654183;
4777 s4 -= s13 * 997805;
4778 s5 += s13 * 136657;
4779 s6 -= s13 * 683901;
4780 s13 = 0;
4781
4782 s0 += s12 * 666643;
4783 s1 += s12 * 470296;
4784 s2 += s12 * 654183;
4785 s3 -= s12 * 997805;
4786 s4 += s12 * 136657;
4787 s5 -= s12 * 683901;
4788 s12 = 0;
4789
4790 carry0 = (s0 + (1 << 20)) >> 21;
4791 s1 += carry0;
4792 s0 -= carry0 * (1 << 21);
4793 carry2 = (s2 + (1 << 20)) >> 21;
4794 s3 += carry2;
4795 s2 -= carry2 * (1 << 21);
4796 carry4 = (s4 + (1 << 20)) >> 21;
4797 s5 += carry4;
4798 s4 -= carry4 * (1 << 21);
4799 carry6 = (s6 + (1 << 20)) >> 21;
4800 s7 += carry6;
4801 s6 -= carry6 * (1 << 21);
4802 carry8 = (s8 + (1 << 20)) >> 21;
4803 s9 += carry8;
4804 s8 -= carry8 * (1 << 21);
4805 carry10 = (s10 + (1 << 20)) >> 21;
4806 s11 += carry10;
4807 s10 -= carry10 * (1 << 21);
4808
4809 carry1 = (s1 + (1 << 20)) >> 21;
4810 s2 += carry1;
4811 s1 -= carry1 * (1 << 21);
4812 carry3 = (s3 + (1 << 20)) >> 21;
4813 s4 += carry3;
4814 s3 -= carry3 * (1 << 21);
4815 carry5 = (s5 + (1 << 20)) >> 21;
4816 s6 += carry5;
4817 s5 -= carry5 * (1 << 21);
4818 carry7 = (s7 + (1 << 20)) >> 21;
4819 s8 += carry7;
4820 s7 -= carry7 * (1 << 21);
4821 carry9 = (s9 + (1 << 20)) >> 21;
4822 s10 += carry9;
4823 s9 -= carry9 * (1 << 21);
4824 carry11 = (s11 + (1 << 20)) >> 21;
4825 s12 += carry11;
4826 s11 -= carry11 * (1 << 21);
4827
4828 s0 += s12 * 666643;
4829 s1 += s12 * 470296;
4830 s2 += s12 * 654183;
4831 s3 -= s12 * 997805;
4832 s4 += s12 * 136657;
4833 s5 -= s12 * 683901;
4834 s12 = 0;
4835
4836 carry0 = s0 >> 21;
4837 s1 += carry0;
4838 s0 -= carry0 * (1 << 21);
4839 carry1 = s1 >> 21;
4840 s2 += carry1;
4841 s1 -= carry1 * (1 << 21);
4842 carry2 = s2 >> 21;
4843 s3 += carry2;
4844 s2 -= carry2 * (1 << 21);
4845 carry3 = s3 >> 21;
4846 s4 += carry3;
4847 s3 -= carry3 * (1 << 21);
4848 carry4 = s4 >> 21;
4849 s5 += carry4;
4850 s4 -= carry4 * (1 << 21);
4851 carry5 = s5 >> 21;
4852 s6 += carry5;
4853 s5 -= carry5 * (1 << 21);
4854 carry6 = s6 >> 21;
4855 s7 += carry6;
4856 s6 -= carry6 * (1 << 21);
4857 carry7 = s7 >> 21;
4858 s8 += carry7;
4859 s7 -= carry7 * (1 << 21);
4860 carry8 = s8 >> 21;
4861 s9 += carry8;
4862 s8 -= carry8 * (1 << 21);
4863 carry9 = s9 >> 21;
4864 s10 += carry9;
4865 s9 -= carry9 * (1 << 21);
4866 carry10 = s10 >> 21;
4867 s11 += carry10;
4868 s10 -= carry10 * (1 << 21);
4869 carry11 = s11 >> 21;
4870 s12 += carry11;
4871 s11 -= carry11 * (1 << 21);
4872
4873 s0 += s12 * 666643;
4874 s1 += s12 * 470296;
4875 s2 += s12 * 654183;
4876 s3 -= s12 * 997805;
4877 s4 += s12 * 136657;
4878 s5 -= s12 * 683901;
4879 s12 = 0;
4880
4881 carry0 = s0 >> 21;
4882 s1 += carry0;
4883 s0 -= carry0 * (1 << 21);
4884 carry1 = s1 >> 21;
4885 s2 += carry1;
4886 s1 -= carry1 * (1 << 21);
4887 carry2 = s2 >> 21;
4888 s3 += carry2;
4889 s2 -= carry2 * (1 << 21);
4890 carry3 = s3 >> 21;
4891 s4 += carry3;
4892 s3 -= carry3 * (1 << 21);
4893 carry4 = s4 >> 21;
4894 s5 += carry4;
4895 s4 -= carry4 * (1 << 21);
4896 carry5 = s5 >> 21;
4897 s6 += carry5;
4898 s5 -= carry5 * (1 << 21);
4899 carry6 = s6 >> 21;
4900 s7 += carry6;
4901 s6 -= carry6 * (1 << 21);
4902 carry7 = s7 >> 21;
4903 s8 += carry7;
4904 s7 -= carry7 * (1 << 21);
4905 carry8 = s8 >> 21;
4906 s9 += carry8;
4907 s8 -= carry8 * (1 << 21);
4908 carry9 = s9 >> 21;
4909 s10 += carry9;
4910 s9 -= carry9 * (1 << 21);
4911 carry10 = s10 >> 21;
4912 s11 += carry10;
4913 s10 -= carry10 * (1 << 21);
4914
4915 s[ 0] = (uint8_t) (s0 >> 0);
4916 s[ 1] = (uint8_t) (s0 >> 8);
4917 s[ 2] = (uint8_t)((s0 >> 16) | (s1 << 5));
4918 s[ 3] = (uint8_t) (s1 >> 3);
4919 s[ 4] = (uint8_t) (s1 >> 11);
4920 s[ 5] = (uint8_t)((s1 >> 19) | (s2 << 2));
4921 s[ 6] = (uint8_t) (s2 >> 6);
4922 s[ 7] = (uint8_t)((s2 >> 14) | (s3 << 7));
4923 s[ 8] = (uint8_t) (s3 >> 1);
4924 s[ 9] = (uint8_t) (s3 >> 9);
4925 s[10] = (uint8_t)((s3 >> 17) | (s4 << 4));
4926 s[11] = (uint8_t) (s4 >> 4);
4927 s[12] = (uint8_t) (s4 >> 12);
4928 s[13] = (uint8_t)((s4 >> 20) | (s5 << 1));
4929 s[14] = (uint8_t) (s5 >> 7);
4930 s[15] = (uint8_t)((s5 >> 15) | (s6 << 6));
4931 s[16] = (uint8_t) (s6 >> 2);
4932 s[17] = (uint8_t) (s6 >> 10);
4933 s[18] = (uint8_t)((s6 >> 18) | (s7 << 3));
4934 s[19] = (uint8_t) (s7 >> 5);
4935 s[20] = (uint8_t) (s7 >> 13);
4936 s[21] = (uint8_t) (s8 >> 0);
4937 s[22] = (uint8_t) (s8 >> 8);
4938 s[23] = (uint8_t)((s8 >> 16) | (s9 << 5));
4939 s[24] = (uint8_t) (s9 >> 3);
4940 s[25] = (uint8_t) (s9 >> 11);
4941 s[26] = (uint8_t)((s9 >> 19) | (s10 << 2));
4942 s[27] = (uint8_t) (s10 >> 6);
4943 s[28] = (uint8_t)((s10 >> 14) | (s11 << 7));
4944 s[29] = (uint8_t) (s11 >> 1);
4945 s[30] = (uint8_t) (s11 >> 9);
4946 s[31] = (uint8_t) (s11 >> 17);
4947}
4948
4949/*
4950 * Input:
4951 * a[0]+256*a[1]+...+256^31*a[31] = a
4952 * b[0]+256*b[1]+...+256^31*b[31] = b
4953 * c[0]+256*c[1]+...+256^31*c[31] = c
4954 *
4955 * Output:
4956 * s[0]+256*s[1]+...+256^31*s[31] = (ab+c) mod l
4957 * where l = 2^252 + 27742317777372353535851937790883648493.
4958 */
4959static void sc_muladd(uint8_t *s, const uint8_t *a, const uint8_t *b,
4960 const uint8_t *c)
4961{
4962 int64_t a0 = kBottom21Bits & load_3(a);
4963 int64_t a1 = kBottom21Bits & (load_4(a + 2) >> 5);
4964 int64_t a2 = kBottom21Bits & (load_3(a + 5) >> 2);
4965 int64_t a3 = kBottom21Bits & (load_4(a + 7) >> 7);
4966 int64_t a4 = kBottom21Bits & (load_4(a + 10) >> 4);
4967 int64_t a5 = kBottom21Bits & (load_3(a + 13) >> 1);
4968 int64_t a6 = kBottom21Bits & (load_4(a + 15) >> 6);
4969 int64_t a7 = kBottom21Bits & (load_3(a + 18) >> 3);
4970 int64_t a8 = kBottom21Bits & load_3(a + 21);
4971 int64_t a9 = kBottom21Bits & (load_4(a + 23) >> 5);
4972 int64_t a10 = kBottom21Bits & (load_3(a + 26) >> 2);
4973 int64_t a11 = (load_4(a + 28) >> 7);
4974 int64_t b0 = kBottom21Bits & load_3(b);
4975 int64_t b1 = kBottom21Bits & (load_4(b + 2) >> 5);
4976 int64_t b2 = kBottom21Bits & (load_3(b + 5) >> 2);
4977 int64_t b3 = kBottom21Bits & (load_4(b + 7) >> 7);
4978 int64_t b4 = kBottom21Bits & (load_4(b + 10) >> 4);
4979 int64_t b5 = kBottom21Bits & (load_3(b + 13) >> 1);
4980 int64_t b6 = kBottom21Bits & (load_4(b + 15) >> 6);
4981 int64_t b7 = kBottom21Bits & (load_3(b + 18) >> 3);
4982 int64_t b8 = kBottom21Bits & load_3(b + 21);
4983 int64_t b9 = kBottom21Bits & (load_4(b + 23) >> 5);
4984 int64_t b10 = kBottom21Bits & (load_3(b + 26) >> 2);
4985 int64_t b11 = (load_4(b + 28) >> 7);
4986 int64_t c0 = kBottom21Bits & load_3(c);
4987 int64_t c1 = kBottom21Bits & (load_4(c + 2) >> 5);
4988 int64_t c2 = kBottom21Bits & (load_3(c + 5) >> 2);
4989 int64_t c3 = kBottom21Bits & (load_4(c + 7) >> 7);
4990 int64_t c4 = kBottom21Bits & (load_4(c + 10) >> 4);
4991 int64_t c5 = kBottom21Bits & (load_3(c + 13) >> 1);
4992 int64_t c6 = kBottom21Bits & (load_4(c + 15) >> 6);
4993 int64_t c7 = kBottom21Bits & (load_3(c + 18) >> 3);
4994 int64_t c8 = kBottom21Bits & load_3(c + 21);
4995 int64_t c9 = kBottom21Bits & (load_4(c + 23) >> 5);
4996 int64_t c10 = kBottom21Bits & (load_3(c + 26) >> 2);
4997 int64_t c11 = (load_4(c + 28) >> 7);
4998 int64_t s0;
4999 int64_t s1;
5000 int64_t s2;
5001 int64_t s3;
5002 int64_t s4;
5003 int64_t s5;
5004 int64_t s6;
5005 int64_t s7;
5006 int64_t s8;
5007 int64_t s9;
5008 int64_t s10;
5009 int64_t s11;
5010 int64_t s12;
5011 int64_t s13;
5012 int64_t s14;
5013 int64_t s15;
5014 int64_t s16;
5015 int64_t s17;
5016 int64_t s18;
5017 int64_t s19;
5018 int64_t s20;
5019 int64_t s21;
5020 int64_t s22;
5021 int64_t s23;
5022 int64_t carry0;
5023 int64_t carry1;
5024 int64_t carry2;
5025 int64_t carry3;
5026 int64_t carry4;
5027 int64_t carry5;
5028 int64_t carry6;
5029 int64_t carry7;
5030 int64_t carry8;
5031 int64_t carry9;
5032 int64_t carry10;
5033 int64_t carry11;
5034 int64_t carry12;
5035 int64_t carry13;
5036 int64_t carry14;
5037 int64_t carry15;
5038 int64_t carry16;
5039 int64_t carry17;
5040 int64_t carry18;
5041 int64_t carry19;
5042 int64_t carry20;
5043 int64_t carry21;
5044 int64_t carry22;
5045
5046 s0 = c0 + a0 * b0;
5047 s1 = c1 + a0 * b1 + a1 * b0;
5048 s2 = c2 + a0 * b2 + a1 * b1 + a2 * b0;
5049 s3 = c3 + a0 * b3 + a1 * b2 + a2 * b1 + a3 * b0;
5050 s4 = c4 + a0 * b4 + a1 * b3 + a2 * b2 + a3 * b1 + a4 * b0;
5051 s5 = c5 + a0 * b5 + a1 * b4 + a2 * b3 + a3 * b2 + a4 * b1 + a5 * b0;
5052 s6 = c6 + a0 * b6 + a1 * b5 + a2 * b4 + a3 * b3 + a4 * b2 + a5 * b1 + a6 * b0;
5053 s7 = c7 + a0 * b7 + a1 * b6 + a2 * b5 + a3 * b4 + a4 * b3 + a5 * b2 + a6 * b1 + a7 * b0;
5054 s8 = c8 + a0 * b8 + a1 * b7 + a2 * b6 + a3 * b5 + a4 * b4 + a5 * b3 + a6 * b2 + a7 * b1 + a8 * b0;
5055 s9 = c9 + a0 * b9 + a1 * b8 + a2 * b7 + a3 * b6 + a4 * b5 + a5 * b4 + a6 * b3 + a7 * b2 + a8 * b1 + a9 * b0;
5056 s10 = c10 + a0 * b10 + a1 * b9 + a2 * b8 + a3 * b7 + a4 * b6 + a5 * b5 + a6 * b4 + a7 * b3 + a8 * b2 + a9 * b1 + a10 * b0;
5057 s11 = c11 + a0 * b11 + a1 * b10 + a2 * b9 + a3 * b8 + a4 * b7 + a5 * b6 + a6 * b5 + a7 * b4 + a8 * b3 + a9 * b2 + a10 * b1 + a11 * b0;
5058 s12 = a1 * b11 + a2 * b10 + a3 * b9 + a4 * b8 + a5 * b7 + a6 * b6 + a7 * b5 + a8 * b4 + a9 * b3 + a10 * b2 + a11 * b1;
5059 s13 = a2 * b11 + a3 * b10 + a4 * b9 + a5 * b8 + a6 * b7 + a7 * b6 + a8 * b5 + a9 * b4 + a10 * b3 + a11 * b2;
5060 s14 = a3 * b11 + a4 * b10 + a5 * b9 + a6 * b8 + a7 * b7 + a8 * b6 + a9 * b5 + a10 * b4 + a11 * b3;
5061 s15 = a4 * b11 + a5 * b10 + a6 * b9 + a7 * b8 + a8 * b7 + a9 * b6 + a10 * b5 + a11 * b4;
5062 s16 = a5 * b11 + a6 * b10 + a7 * b9 + a8 * b8 + a9 * b7 + a10 * b6 + a11 * b5;
5063 s17 = a6 * b11 + a7 * b10 + a8 * b9 + a9 * b8 + a10 * b7 + a11 * b6;
5064 s18 = a7 * b11 + a8 * b10 + a9 * b9 + a10 * b8 + a11 * b7;
5065 s19 = a8 * b11 + a9 * b10 + a10 * b9 + a11 * b8;
5066 s20 = a9 * b11 + a10 * b10 + a11 * b9;
5067 s21 = a10 * b11 + a11 * b10;
5068 s22 = a11 * b11;
5069 s23 = 0;
5070
5071 carry0 = (s0 + (1 << 20)) >> 21;
5072 s1 += carry0;
5073 s0 -= carry0 * (1 << 21);
5074 carry2 = (s2 + (1 << 20)) >> 21;
5075 s3 += carry2;
5076 s2 -= carry2 * (1 << 21);
5077 carry4 = (s4 + (1 << 20)) >> 21;
5078 s5 += carry4;
5079 s4 -= carry4 * (1 << 21);
5080 carry6 = (s6 + (1 << 20)) >> 21;
5081 s7 += carry6;
5082 s6 -= carry6 * (1 << 21);
5083 carry8 = (s8 + (1 << 20)) >> 21;
5084 s9 += carry8;
5085 s8 -= carry8 * (1 << 21);
5086 carry10 = (s10 + (1 << 20)) >> 21;
5087 s11 += carry10;
5088 s10 -= carry10 * (1 << 21);
5089 carry12 = (s12 + (1 << 20)) >> 21;
5090 s13 += carry12;
5091 s12 -= carry12 * (1 << 21);
5092 carry14 = (s14 + (1 << 20)) >> 21;
5093 s15 += carry14;
5094 s14 -= carry14 * (1 << 21);
5095 carry16 = (s16 + (1 << 20)) >> 21;
5096 s17 += carry16;
5097 s16 -= carry16 * (1 << 21);
5098 carry18 = (s18 + (1 << 20)) >> 21;
5099 s19 += carry18;
5100 s18 -= carry18 * (1 << 21);
5101 carry20 = (s20 + (1 << 20)) >> 21;
5102 s21 += carry20;
5103 s20 -= carry20 * (1 << 21);
5104 carry22 = (s22 + (1 << 20)) >> 21;
5105 s23 += carry22;
5106 s22 -= carry22 * (1 << 21);
5107
5108 carry1 = (s1 + (1 << 20)) >> 21;
5109 s2 += carry1;
5110 s1 -= carry1 * (1 << 21);
5111 carry3 = (s3 + (1 << 20)) >> 21;
5112 s4 += carry3;
5113 s3 -= carry3 * (1 << 21);
5114 carry5 = (s5 + (1 << 20)) >> 21;
5115 s6 += carry5;
5116 s5 -= carry5 * (1 << 21);
5117 carry7 = (s7 + (1 << 20)) >> 21;
5118 s8 += carry7;
5119 s7 -= carry7 * (1 << 21);
5120 carry9 = (s9 + (1 << 20)) >> 21;
5121 s10 += carry9;
5122 s9 -= carry9 * (1 << 21);
5123 carry11 = (s11 + (1 << 20)) >> 21;
5124 s12 += carry11;
5125 s11 -= carry11 * (1 << 21);
5126 carry13 = (s13 + (1 << 20)) >> 21;
5127 s14 += carry13;
5128 s13 -= carry13 * (1 << 21);
5129 carry15 = (s15 + (1 << 20)) >> 21;
5130 s16 += carry15;
5131 s15 -= carry15 * (1 << 21);
5132 carry17 = (s17 + (1 << 20)) >> 21;
5133 s18 += carry17;
5134 s17 -= carry17 * (1 << 21);
5135 carry19 = (s19 + (1 << 20)) >> 21;
5136 s20 += carry19;
5137 s19 -= carry19 * (1 << 21);
5138 carry21 = (s21 + (1 << 20)) >> 21;
5139 s22 += carry21;
5140 s21 -= carry21 * (1 << 21);
5141
5142 s11 += s23 * 666643;
5143 s12 += s23 * 470296;
5144 s13 += s23 * 654183;
5145 s14 -= s23 * 997805;
5146 s15 += s23 * 136657;
5147 s16 -= s23 * 683901;
5148 s23 = 0;
5149
5150 s10 += s22 * 666643;
5151 s11 += s22 * 470296;
5152 s12 += s22 * 654183;
5153 s13 -= s22 * 997805;
5154 s14 += s22 * 136657;
5155 s15 -= s22 * 683901;
5156 s22 = 0;
5157
5158 s9 += s21 * 666643;
5159 s10 += s21 * 470296;
5160 s11 += s21 * 654183;
5161 s12 -= s21 * 997805;
5162 s13 += s21 * 136657;
5163 s14 -= s21 * 683901;
5164 s21 = 0;
5165
5166 s8 += s20 * 666643;
5167 s9 += s20 * 470296;
5168 s10 += s20 * 654183;
5169 s11 -= s20 * 997805;
5170 s12 += s20 * 136657;
5171 s13 -= s20 * 683901;
5172 s20 = 0;
5173
5174 s7 += s19 * 666643;
5175 s8 += s19 * 470296;
5176 s9 += s19 * 654183;
5177 s10 -= s19 * 997805;
5178 s11 += s19 * 136657;
5179 s12 -= s19 * 683901;
5180 s19 = 0;
5181
5182 s6 += s18 * 666643;
5183 s7 += s18 * 470296;
5184 s8 += s18 * 654183;
5185 s9 -= s18 * 997805;
5186 s10 += s18 * 136657;
5187 s11 -= s18 * 683901;
5188 s18 = 0;
5189
5190 carry6 = (s6 + (1 << 20)) >> 21;
5191 s7 += carry6;
5192 s6 -= carry6 * (1 << 21);
5193 carry8 = (s8 + (1 << 20)) >> 21;
5194 s9 += carry8;
5195 s8 -= carry8 * (1 << 21);
5196 carry10 = (s10 + (1 << 20)) >> 21;
5197 s11 += carry10;
5198 s10 -= carry10 * (1 << 21);
5199 carry12 = (s12 + (1 << 20)) >> 21;
5200 s13 += carry12;
5201 s12 -= carry12 * (1 << 21);
5202 carry14 = (s14 + (1 << 20)) >> 21;
5203 s15 += carry14;
5204 s14 -= carry14 * (1 << 21);
5205 carry16 = (s16 + (1 << 20)) >> 21;
5206 s17 += carry16;
5207 s16 -= carry16 * (1 << 21);
5208
5209 carry7 = (s7 + (1 << 20)) >> 21;
5210 s8 += carry7;
5211 s7 -= carry7 * (1 << 21);
5212 carry9 = (s9 + (1 << 20)) >> 21;
5213 s10 += carry9;
5214 s9 -= carry9 * (1 << 21);
5215 carry11 = (s11 + (1 << 20)) >> 21;
5216 s12 += carry11;
5217 s11 -= carry11 * (1 << 21);
5218 carry13 = (s13 + (1 << 20)) >> 21;
5219 s14 += carry13;
5220 s13 -= carry13 * (1 << 21);
5221 carry15 = (s15 + (1 << 20)) >> 21;
5222 s16 += carry15;
5223 s15 -= carry15 * (1 << 21);
5224
5225 s5 += s17 * 666643;
5226 s6 += s17 * 470296;
5227 s7 += s17 * 654183;
5228 s8 -= s17 * 997805;
5229 s9 += s17 * 136657;
5230 s10 -= s17 * 683901;
5231 s17 = 0;
5232
5233 s4 += s16 * 666643;
5234 s5 += s16 * 470296;
5235 s6 += s16 * 654183;
5236 s7 -= s16 * 997805;
5237 s8 += s16 * 136657;
5238 s9 -= s16 * 683901;
5239 s16 = 0;
5240
5241 s3 += s15 * 666643;
5242 s4 += s15 * 470296;
5243 s5 += s15 * 654183;
5244 s6 -= s15 * 997805;
5245 s7 += s15 * 136657;
5246 s8 -= s15 * 683901;
5247 s15 = 0;
5248
5249 s2 += s14 * 666643;
5250 s3 += s14 * 470296;
5251 s4 += s14 * 654183;
5252 s5 -= s14 * 997805;
5253 s6 += s14 * 136657;
5254 s7 -= s14 * 683901;
5255 s14 = 0;
5256
5257 s1 += s13 * 666643;
5258 s2 += s13 * 470296;
5259 s3 += s13 * 654183;
5260 s4 -= s13 * 997805;
5261 s5 += s13 * 136657;
5262 s6 -= s13 * 683901;
5263 s13 = 0;
5264
5265 s0 += s12 * 666643;
5266 s1 += s12 * 470296;
5267 s2 += s12 * 654183;
5268 s3 -= s12 * 997805;
5269 s4 += s12 * 136657;
5270 s5 -= s12 * 683901;
5271 s12 = 0;
5272
5273 carry0 = (s0 + (1 << 20)) >> 21;
5274 s1 += carry0;
5275 s0 -= carry0 * (1 << 21);
5276 carry2 = (s2 + (1 << 20)) >> 21;
5277 s3 += carry2;
5278 s2 -= carry2 * (1 << 21);
5279 carry4 = (s4 + (1 << 20)) >> 21;
5280 s5 += carry4;
5281 s4 -= carry4 * (1 << 21);
5282 carry6 = (s6 + (1 << 20)) >> 21;
5283 s7 += carry6;
5284 s6 -= carry6 * (1 << 21);
5285 carry8 = (s8 + (1 << 20)) >> 21;
5286 s9 += carry8;
5287 s8 -= carry8 * (1 << 21);
5288 carry10 = (s10 + (1 << 20)) >> 21;
5289 s11 += carry10;
5290 s10 -= carry10 * (1 << 21);
5291
5292 carry1 = (s1 + (1 << 20)) >> 21;
5293 s2 += carry1;
5294 s1 -= carry1 * (1 << 21);
5295 carry3 = (s3 + (1 << 20)) >> 21;
5296 s4 += carry3;
5297 s3 -= carry3 * (1 << 21);
5298 carry5 = (s5 + (1 << 20)) >> 21;
5299 s6 += carry5;
5300 s5 -= carry5 * (1 << 21);
5301 carry7 = (s7 + (1 << 20)) >> 21;
5302 s8 += carry7;
5303 s7 -= carry7 * (1 << 21);
5304 carry9 = (s9 + (1 << 20)) >> 21;
5305 s10 += carry9;
5306 s9 -= carry9 * (1 << 21);
5307 carry11 = (s11 + (1 << 20)) >> 21;
5308 s12 += carry11;
5309 s11 -= carry11 * (1 << 21);
5310
5311 s0 += s12 * 666643;
5312 s1 += s12 * 470296;
5313 s2 += s12 * 654183;
5314 s3 -= s12 * 997805;
5315 s4 += s12 * 136657;
5316 s5 -= s12 * 683901;
5317 s12 = 0;
5318
5319 carry0 = s0 >> 21;
5320 s1 += carry0;
5321 s0 -= carry0 * (1 << 21);
5322 carry1 = s1 >> 21;
5323 s2 += carry1;
5324 s1 -= carry1 * (1 << 21);
5325 carry2 = s2 >> 21;
5326 s3 += carry2;
5327 s2 -= carry2 * (1 << 21);
5328 carry3 = s3 >> 21;
5329 s4 += carry3;
5330 s3 -= carry3 * (1 << 21);
5331 carry4 = s4 >> 21;
5332 s5 += carry4;
5333 s4 -= carry4 * (1 << 21);
5334 carry5 = s5 >> 21;
5335 s6 += carry5;
5336 s5 -= carry5 * (1 << 21);
5337 carry6 = s6 >> 21;
5338 s7 += carry6;
5339 s6 -= carry6 * (1 << 21);
5340 carry7 = s7 >> 21;
5341 s8 += carry7;
5342 s7 -= carry7 * (1 << 21);
5343 carry8 = s8 >> 21;
5344 s9 += carry8;
5345 s8 -= carry8 * (1 << 21);
5346 carry9 = s9 >> 21;
5347 s10 += carry9;
5348 s9 -= carry9 * (1 << 21);
5349 carry10 = s10 >> 21;
5350 s11 += carry10;
5351 s10 -= carry10 * (1 << 21);
5352 carry11 = s11 >> 21;
5353 s12 += carry11;
5354 s11 -= carry11 * (1 << 21);
5355
5356 s0 += s12 * 666643;
5357 s1 += s12 * 470296;
5358 s2 += s12 * 654183;
5359 s3 -= s12 * 997805;
5360 s4 += s12 * 136657;
5361 s5 -= s12 * 683901;
5362 s12 = 0;
5363
5364 carry0 = s0 >> 21;
5365 s1 += carry0;
5366 s0 -= carry0 * (1 << 21);
5367 carry1 = s1 >> 21;
5368 s2 += carry1;
5369 s1 -= carry1 * (1 << 21);
5370 carry2 = s2 >> 21;
5371 s3 += carry2;
5372 s2 -= carry2 * (1 << 21);
5373 carry3 = s3 >> 21;
5374 s4 += carry3;
5375 s3 -= carry3 * (1 << 21);
5376 carry4 = s4 >> 21;
5377 s5 += carry4;
5378 s4 -= carry4 * (1 << 21);
5379 carry5 = s5 >> 21;
5380 s6 += carry5;
5381 s5 -= carry5 * (1 << 21);
5382 carry6 = s6 >> 21;
5383 s7 += carry6;
5384 s6 -= carry6 * (1 << 21);
5385 carry7 = s7 >> 21;
5386 s8 += carry7;
5387 s7 -= carry7 * (1 << 21);
5388 carry8 = s8 >> 21;
5389 s9 += carry8;
5390 s8 -= carry8 * (1 << 21);
5391 carry9 = s9 >> 21;
5392 s10 += carry9;
5393 s9 -= carry9 * (1 << 21);
5394 carry10 = s10 >> 21;
5395 s11 += carry10;
5396 s10 -= carry10 * (1 << 21);
5397
5398 s[ 0] = (uint8_t) (s0 >> 0);
5399 s[ 1] = (uint8_t) (s0 >> 8);
5400 s[ 2] = (uint8_t)((s0 >> 16) | (s1 << 5));
5401 s[ 3] = (uint8_t) (s1 >> 3);
5402 s[ 4] = (uint8_t) (s1 >> 11);
5403 s[ 5] = (uint8_t)((s1 >> 19) | (s2 << 2));
5404 s[ 6] = (uint8_t) (s2 >> 6);
5405 s[ 7] = (uint8_t)((s2 >> 14) | (s3 << 7));
5406 s[ 8] = (uint8_t) (s3 >> 1);
5407 s[ 9] = (uint8_t) (s3 >> 9);
5408 s[10] = (uint8_t)((s3 >> 17) | (s4 << 4));
5409 s[11] = (uint8_t) (s4 >> 4);
5410 s[12] = (uint8_t) (s4 >> 12);
5411 s[13] = (uint8_t)((s4 >> 20) | (s5 << 1));
5412 s[14] = (uint8_t) (s5 >> 7);
5413 s[15] = (uint8_t)((s5 >> 15) | (s6 << 6));
5414 s[16] = (uint8_t) (s6 >> 2);
5415 s[17] = (uint8_t) (s6 >> 10);
5416 s[18] = (uint8_t)((s6 >> 18) | (s7 << 3));
5417 s[19] = (uint8_t) (s7 >> 5);
5418 s[20] = (uint8_t) (s7 >> 13);
5419 s[21] = (uint8_t) (s8 >> 0);
5420 s[22] = (uint8_t) (s8 >> 8);
5421 s[23] = (uint8_t)((s8 >> 16) | (s9 << 5));
5422 s[24] = (uint8_t) (s9 >> 3);
5423 s[25] = (uint8_t) (s9 >> 11);
5424 s[26] = (uint8_t)((s9 >> 19) | (s10 << 2));
5425 s[27] = (uint8_t) (s10 >> 6);
5426 s[28] = (uint8_t)((s10 >> 14) | (s11 << 7));
5427 s[29] = (uint8_t) (s11 >> 1);
5428 s[30] = (uint8_t) (s11 >> 9);
5429 s[31] = (uint8_t) (s11 >> 17);
5430}
5431
5432int ED25519_sign(uint8_t *out_sig, const uint8_t *message, size_t message_len,
5433 const uint8_t public_key[32], const uint8_t private_key[32])
5434{
5435 uint8_t az[SHA512_DIGEST_LENGTH];
5436 uint8_t nonce[SHA512_DIGEST_LENGTH];
5437 ge_p3 R;
5438 uint8_t hram[SHA512_DIGEST_LENGTH];
5439 SHA512_CTX hash_ctx;
5440
5441 SHA512_Init(&hash_ctx);
5442 SHA512_Update(&hash_ctx, private_key, 32);
5443 SHA512_Final(az, &hash_ctx);
5444
5445 az[0] &= 248;
5446 az[31] &= 63;
5447 az[31] |= 64;
5448
5449 SHA512_Init(&hash_ctx);
5450 SHA512_Update(&hash_ctx, az + 32, 32);
5451 SHA512_Update(&hash_ctx, message, message_len);
5452 SHA512_Final(nonce, &hash_ctx);
5453
5454 x25519_sc_reduce(nonce);
5455 ge_scalarmult_base(&R, nonce);
5456 ge_p3_tobytes(out_sig, &R);
5457
5458 SHA512_Init(&hash_ctx);
5459 SHA512_Update(&hash_ctx, out_sig, 32);
5460 SHA512_Update(&hash_ctx, public_key, 32);
5461 SHA512_Update(&hash_ctx, message, message_len);
5462 SHA512_Final(hram, &hash_ctx);
5463
5464 x25519_sc_reduce(hram);
5465 sc_muladd(out_sig + 32, hram, az, nonce);
5466
5467 OPENSSL_cleanse(&hash_ctx, sizeof(hash_ctx));
5468 OPENSSL_cleanse(nonce, sizeof(nonce));
5469 OPENSSL_cleanse(az, sizeof(az));
5470
5471 return 1;
5472}
5473
5474static const char allzeroes[15];
5475
5476int ED25519_verify(const uint8_t *message, size_t message_len,
5477 const uint8_t signature[64], const uint8_t public_key[32])
5478{
5479 int i;
5480 ge_p3 A;
5481 const uint8_t *r, *s;
5482 SHA512_CTX hash_ctx;
5483 ge_p2 R;
5484 uint8_t rcheck[32];
5485 uint8_t h[SHA512_DIGEST_LENGTH];
5486 /* 27742317777372353535851937790883648493 in little endian format */
5487 const uint8_t l_low[16] = {
5488 0xED, 0xD3, 0xF5, 0x5C, 0x1A, 0x63, 0x12, 0x58, 0xD6, 0x9C, 0xF7, 0xA2,
5489 0xDE, 0xF9, 0xDE, 0x14
5490 };
5491
5492 r = signature;
5493 s = signature + 32;
5494
5495 /*
5496 * Check 0 <= s < L where L = 2^252 + 27742317777372353535851937790883648493
5497 *
5498 * If not the signature is publicly invalid. Since it's public we can do the
5499 * check in variable time.
5500 *
5501 * First check the most significant byte
5502 */
5503 if (s[31] > 0x10)
5504 return 0;
5505 if (s[31] == 0x10) {
5506 /*
5507 * Most significant byte indicates a value close to 2^252 so check the
5508 * rest
5509 */
5510 if (memcmp(s + 16, allzeroes, sizeof(allzeroes)) != 0)
5511 return 0;
5512 for (i = 15; i >= 0; i--) {
5513 if (s[i] < l_low[i])
5514 break;
5515 if (s[i] > l_low[i])
5516 return 0;
5517 }
5518 if (i < 0)
5519 return 0;
5520 }
5521
5522 if (ge_frombytes_vartime(&A, public_key) != 0) {
5523 return 0;
5524 }
5525
5526 fe_neg(A.X, A.X);
5527 fe_neg(A.T, A.T);
5528
5529 SHA512_Init(&hash_ctx);
5530 SHA512_Update(&hash_ctx, r, 32);
5531 SHA512_Update(&hash_ctx, public_key, 32);
5532 SHA512_Update(&hash_ctx, message, message_len);
5533 SHA512_Final(h, &hash_ctx);
5534
5535 x25519_sc_reduce(h);
5536
5537 ge_double_scalarmult_vartime(&R, h, &A, s);
5538
5539 ge_tobytes(rcheck, &R);
5540
5541 return CRYPTO_memcmp(rcheck, r, sizeof(rcheck)) == 0;
5542}
5543
5544void ED25519_public_from_private(uint8_t out_public_key[32],
5545 const uint8_t private_key[32])
5546{
5547 uint8_t az[SHA512_DIGEST_LENGTH];
5548 ge_p3 A;
5549
5550 SHA512(private_key, 32, az);
5551
5552 az[0] &= 248;
5553 az[31] &= 63;
5554 az[31] |= 64;
5555
5556 ge_scalarmult_base(&A, az);
5557 ge_p3_tobytes(out_public_key, &A);
5558
5559 OPENSSL_cleanse(az, sizeof(az));
5560}
5561
5562int X25519(uint8_t out_shared_key[32], const uint8_t private_key[32],
5563 const uint8_t peer_public_value[32])
5564{
5565 static const uint8_t kZeros[32] = {0};
5566 x25519_scalar_mult(out_shared_key, private_key, peer_public_value);
5567 /* The all-zero output results when the input is a point of small order. */
5568 return CRYPTO_memcmp(kZeros, out_shared_key, 32) != 0;
5569}
5570
5571void X25519_public_from_private(uint8_t out_public_value[32],
5572 const uint8_t private_key[32])
5573{
5574 uint8_t e[32];
5575 ge_p3 A;
5576 fe zplusy, zminusy, zminusy_inv;
5577
5578 memcpy(e, private_key, 32);
5579 e[0] &= 248;
5580 e[31] &= 127;
5581 e[31] |= 64;
5582
5583 ge_scalarmult_base(&A, e);
5584
5585 /*
5586 * We only need the u-coordinate of the curve25519 point.
5587 * The map is u=(y+1)/(1-y). Since y=Y/Z, this gives
5588 * u=(Z+Y)/(Z-Y).
5589 */
5590 fe_add(zplusy, A.Z, A.Y);
5591 fe_sub(zminusy, A.Z, A.Y);
5592 fe_invert(zminusy_inv, zminusy);
5593 fe_mul(zplusy, zplusy, zminusy_inv);
5594 fe_tobytes(out_public_value, zplusy);
5595
5596 OPENSSL_cleanse(e, sizeof(e));
5597}