1 /*
2 * Copyright 2016-2024 The OpenSSL Project Authors. All Rights Reserved.
3 *
4 * Licensed under the Apache License 2.0 (the "License"). You may not use
5 * this file except in compliance with the License. You can obtain a copy
6 * in the file LICENSE in the source distribution or at
7 * https://www.openssl.org/source/license.html
8 */
9
10 /*
11 * ECDSA low level APIs are deprecated for public use, but still ok for
12 * internal use.
13 */
14 #include "internal/deprecated.h"
15
16 #include <string.h>
17 #include "crypto/ecx.h"
18 #include "ec_local.h"
19 #include <openssl/evp.h>
20 #include <openssl/sha.h>
21
22 #include "internal/numbers.h"
23
24 #if defined(X25519_ASM) && (defined(__x86_64) || defined(__x86_64__) || \
25 defined(_M_AMD64) || defined(_M_X64))
26
27 # define BASE_2_64_IMPLEMENTED
28
29 typedef uint64_t fe64[4];
30
31 int x25519_fe64_eligible(void);
32
33 /*
34 * Following subroutines perform corresponding operations modulo
35 * 2^256-38, i.e. double the curve modulus. However, inputs and
36 * outputs are permitted to be partially reduced, i.e. to remain
37 * in [0..2^256) range. It's all tied up in final fe64_tobytes
38 * that performs full reduction modulo 2^255-19.
39 *
40 * There are no reference C implementations for these.
41 */
42 void x25519_fe64_mul(fe64 h, const fe64 f, const fe64 g);
43 void x25519_fe64_sqr(fe64 h, const fe64 f);
44 void x25519_fe64_mul121666(fe64 h, fe64 f);
45 void x25519_fe64_add(fe64 h, const fe64 f, const fe64 g);
46 void x25519_fe64_sub(fe64 h, const fe64 f, const fe64 g);
47 void x25519_fe64_tobytes(uint8_t *s, const fe64 f);
48 # define fe64_mul x25519_fe64_mul
49 # define fe64_sqr x25519_fe64_sqr
50 # define fe64_mul121666 x25519_fe64_mul121666
51 # define fe64_add x25519_fe64_add
52 # define fe64_sub x25519_fe64_sub
53 # define fe64_tobytes x25519_fe64_tobytes
54
load_8(const uint8_t * in)55 static uint64_t load_8(const uint8_t *in)
56 {
57 uint64_t result;
58
59 result = in[0];
60 result |= ((uint64_t)in[1]) << 8;
61 result |= ((uint64_t)in[2]) << 16;
62 result |= ((uint64_t)in[3]) << 24;
63 result |= ((uint64_t)in[4]) << 32;
64 result |= ((uint64_t)in[5]) << 40;
65 result |= ((uint64_t)in[6]) << 48;
66 result |= ((uint64_t)in[7]) << 56;
67
68 return result;
69 }
70
fe64_frombytes(fe64 h,const uint8_t * s)71 static void fe64_frombytes(fe64 h, const uint8_t *s)
72 {
73 h[0] = load_8(s);
74 h[1] = load_8(s + 8);
75 h[2] = load_8(s + 16);
76 h[3] = load_8(s + 24) & 0x7fffffffffffffff;
77 }
78
fe64_0(fe64 h)79 static void fe64_0(fe64 h)
80 {
81 h[0] = 0;
82 h[1] = 0;
83 h[2] = 0;
84 h[3] = 0;
85 }
86
fe64_1(fe64 h)87 static void fe64_1(fe64 h)
88 {
89 h[0] = 1;
90 h[1] = 0;
91 h[2] = 0;
92 h[3] = 0;
93 }
94
fe64_copy(fe64 h,const fe64 f)95 static void fe64_copy(fe64 h, const fe64 f)
96 {
97 h[0] = f[0];
98 h[1] = f[1];
99 h[2] = f[2];
100 h[3] = f[3];
101 }
102
fe64_cswap(fe64 f,fe64 g,unsigned int b)103 static void fe64_cswap(fe64 f, fe64 g, unsigned int b)
104 {
105 int i;
106 uint64_t mask = 0 - (uint64_t)b;
107
108 for (i = 0; i < 4; i++) {
109 uint64_t x = f[i] ^ g[i];
110 x &= mask;
111 f[i] ^= x;
112 g[i] ^= x;
113 }
114 }
115
fe64_invert(fe64 out,const fe64 z)116 static void fe64_invert(fe64 out, const fe64 z)
117 {
118 fe64 t0;
119 fe64 t1;
120 fe64 t2;
121 fe64 t3;
122 int i;
123
124 /*
125 * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as
126 * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11.
127 */
128
129 /* t0 = z ** 2 */
130 fe64_sqr(t0, z);
131
132 /* t1 = t0 ** (2 ** 2) = z ** 8 */
133 fe64_sqr(t1, t0);
134 fe64_sqr(t1, t1);
135
136 /* t1 = z * t1 = z ** 9 */
137 fe64_mul(t1, z, t1);
138 /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */
139 fe64_mul(t0, t0, t1);
140
141 /* t2 = t0 ** 2 = z ** 22 */
142 fe64_sqr(t2, t0);
143
144 /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */
145 fe64_mul(t1, t1, t2);
146
147 /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */
148 fe64_sqr(t2, t1);
149 for (i = 1; i < 5; ++i)
150 fe64_sqr(t2, t2);
151
152 /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */
153 fe64_mul(t1, t2, t1);
154
155 /* Continuing similarly... */
156
157 /* t2 = z ** (2 ** 20 - 1) */
158 fe64_sqr(t2, t1);
159 for (i = 1; i < 10; ++i)
160 fe64_sqr(t2, t2);
161
162 fe64_mul(t2, t2, t1);
163
164 /* t2 = z ** (2 ** 40 - 1) */
165 fe64_sqr(t3, t2);
166 for (i = 1; i < 20; ++i)
167 fe64_sqr(t3, t3);
168
169 fe64_mul(t2, t3, t2);
170
171 /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */
172 for (i = 0; i < 10; ++i)
173 fe64_sqr(t2, t2);
174
175 /* t1 = z ** (2 ** 50 - 1) */
176 fe64_mul(t1, t2, t1);
177
178 /* t2 = z ** (2 ** 100 - 1) */
179 fe64_sqr(t2, t1);
180 for (i = 1; i < 50; ++i)
181 fe64_sqr(t2, t2);
182
183 fe64_mul(t2, t2, t1);
184
185 /* t2 = z ** (2 ** 200 - 1) */
186 fe64_sqr(t3, t2);
187 for (i = 1; i < 100; ++i)
188 fe64_sqr(t3, t3);
189
190 fe64_mul(t2, t3, t2);
191
192 /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */
193 for (i = 0; i < 50; ++i)
194 fe64_sqr(t2, t2);
195
196 /* t1 = z ** (2 ** 250 - 1) */
197 fe64_mul(t1, t2, t1);
198
199 /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */
200 for (i = 0; i < 5; ++i)
201 fe64_sqr(t1, t1);
202
203 /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */
204 fe64_mul(out, t1, t0);
205 }
206
207 /*
208 * Duplicate of original x25519_scalar_mult_generic, but using
209 * fe64_* subroutines.
210 */
x25519_scalar_mulx(uint8_t out[32],const uint8_t scalar[32],const uint8_t point[32])211 static void x25519_scalar_mulx(uint8_t out[32], const uint8_t scalar[32],
212 const uint8_t point[32])
213 {
214 fe64 x1, x2, z2, x3, z3, tmp0, tmp1;
215 uint8_t e[32];
216 unsigned swap = 0;
217 int pos;
218
219 memcpy(e, scalar, 32);
220 e[0] &= 0xf8;
221 e[31] &= 0x7f;
222 e[31] |= 0x40;
223 fe64_frombytes(x1, point);
224 fe64_1(x2);
225 fe64_0(z2);
226 fe64_copy(x3, x1);
227 fe64_1(z3);
228
229 for (pos = 254; pos >= 0; --pos) {
230 unsigned int b = 1 & (e[pos / 8] >> (pos & 7));
231
232 swap ^= b;
233 fe64_cswap(x2, x3, swap);
234 fe64_cswap(z2, z3, swap);
235 swap = b;
236 fe64_sub(tmp0, x3, z3);
237 fe64_sub(tmp1, x2, z2);
238 fe64_add(x2, x2, z2);
239 fe64_add(z2, x3, z3);
240 fe64_mul(z3, x2, tmp0);
241 fe64_mul(z2, z2, tmp1);
242 fe64_sqr(tmp0, tmp1);
243 fe64_sqr(tmp1, x2);
244 fe64_add(x3, z3, z2);
245 fe64_sub(z2, z3, z2);
246 fe64_mul(x2, tmp1, tmp0);
247 fe64_sub(tmp1, tmp1, tmp0);
248 fe64_sqr(z2, z2);
249 fe64_mul121666(z3, tmp1);
250 fe64_sqr(x3, x3);
251 fe64_add(tmp0, tmp0, z3);
252 fe64_mul(z3, x1, z2);
253 fe64_mul(z2, tmp1, tmp0);
254 }
255
256 fe64_invert(z2, z2);
257 fe64_mul(x2, x2, z2);
258 fe64_tobytes(out, x2);
259
260 OPENSSL_cleanse(e, sizeof(e));
261 }
262 #endif
263
264 #if defined(X25519_ASM) \
265 || ( defined(INT128_MAX) \
266 && !defined(__sparc__) \
267 && (!defined(__SIZEOF_LONG__) || (__SIZEOF_LONG__ == 8)) \
268 && !(defined(__ANDROID__) && !defined(__clang__)) )
269 /*
270 * Base 2^51 implementation. It's virtually no different from reference
271 * base 2^25.5 implementation in respect to lax boundary conditions for
272 * intermediate values and even individual limbs. So that whatever you
273 * know about the reference, applies even here...
274 */
275 # define BASE_2_51_IMPLEMENTED
276
277 typedef uint64_t fe51[5];
278
279 static const uint64_t MASK51 = 0x7ffffffffffff;
280
load_7(const uint8_t * in)281 static uint64_t load_7(const uint8_t *in)
282 {
283 uint64_t result;
284
285 result = in[0];
286 result |= ((uint64_t)in[1]) << 8;
287 result |= ((uint64_t)in[2]) << 16;
288 result |= ((uint64_t)in[3]) << 24;
289 result |= ((uint64_t)in[4]) << 32;
290 result |= ((uint64_t)in[5]) << 40;
291 result |= ((uint64_t)in[6]) << 48;
292
293 return result;
294 }
295
load_6(const uint8_t * in)296 static uint64_t load_6(const uint8_t *in)
297 {
298 uint64_t result;
299
300 result = in[0];
301 result |= ((uint64_t)in[1]) << 8;
302 result |= ((uint64_t)in[2]) << 16;
303 result |= ((uint64_t)in[3]) << 24;
304 result |= ((uint64_t)in[4]) << 32;
305 result |= ((uint64_t)in[5]) << 40;
306
307 return result;
308 }
309
fe51_frombytes(fe51 h,const uint8_t * s)310 static void fe51_frombytes(fe51 h, const uint8_t *s)
311 {
312 uint64_t h0 = load_7(s); /* 56 bits */
313 uint64_t h1 = load_6(s + 7) << 5; /* 53 bits */
314 uint64_t h2 = load_7(s + 13) << 2; /* 58 bits */
315 uint64_t h3 = load_6(s + 20) << 7; /* 55 bits */
316 uint64_t h4 = (load_6(s + 26) & 0x7fffffffffff) << 4; /* 51 bits */
317
318 h1 |= h0 >> 51; h0 &= MASK51;
319 h2 |= h1 >> 51; h1 &= MASK51;
320 h3 |= h2 >> 51; h2 &= MASK51;
321 h4 |= h3 >> 51; h3 &= MASK51;
322
323 h[0] = h0;
324 h[1] = h1;
325 h[2] = h2;
326 h[3] = h3;
327 h[4] = h4;
328 }
329
fe51_tobytes(uint8_t * s,const fe51 h)330 static void fe51_tobytes(uint8_t *s, const fe51 h)
331 {
332 uint64_t h0 = h[0];
333 uint64_t h1 = h[1];
334 uint64_t h2 = h[2];
335 uint64_t h3 = h[3];
336 uint64_t h4 = h[4];
337 uint64_t q;
338
339 /* compare to modulus */
340 q = (h0 + 19) >> 51;
341 q = (h1 + q) >> 51;
342 q = (h2 + q) >> 51;
343 q = (h3 + q) >> 51;
344 q = (h4 + q) >> 51;
345
346 /* full reduce */
347 h0 += 19 * q;
348 h1 += h0 >> 51; h0 &= MASK51;
349 h2 += h1 >> 51; h1 &= MASK51;
350 h3 += h2 >> 51; h2 &= MASK51;
351 h4 += h3 >> 51; h3 &= MASK51;
352 h4 &= MASK51;
353
354 /* smash */
355 s[0] = (uint8_t)(h0 >> 0);
356 s[1] = (uint8_t)(h0 >> 8);
357 s[2] = (uint8_t)(h0 >> 16);
358 s[3] = (uint8_t)(h0 >> 24);
359 s[4] = (uint8_t)(h0 >> 32);
360 s[5] = (uint8_t)(h0 >> 40);
361 s[6] = (uint8_t)((h0 >> 48) | ((uint32_t)h1 << 3));
362 s[7] = (uint8_t)(h1 >> 5);
363 s[8] = (uint8_t)(h1 >> 13);
364 s[9] = (uint8_t)(h1 >> 21);
365 s[10] = (uint8_t)(h1 >> 29);
366 s[11] = (uint8_t)(h1 >> 37);
367 s[12] = (uint8_t)((h1 >> 45) | ((uint32_t)h2 << 6));
368 s[13] = (uint8_t)(h2 >> 2);
369 s[14] = (uint8_t)(h2 >> 10);
370 s[15] = (uint8_t)(h2 >> 18);
371 s[16] = (uint8_t)(h2 >> 26);
372 s[17] = (uint8_t)(h2 >> 34);
373 s[18] = (uint8_t)(h2 >> 42);
374 s[19] = (uint8_t)((h2 >> 50) | ((uint32_t)h3 << 1));
375 s[20] = (uint8_t)(h3 >> 7);
376 s[21] = (uint8_t)(h3 >> 15);
377 s[22] = (uint8_t)(h3 >> 23);
378 s[23] = (uint8_t)(h3 >> 31);
379 s[24] = (uint8_t)(h3 >> 39);
380 s[25] = (uint8_t)((h3 >> 47) | ((uint32_t)h4 << 4));
381 s[26] = (uint8_t)(h4 >> 4);
382 s[27] = (uint8_t)(h4 >> 12);
383 s[28] = (uint8_t)(h4 >> 20);
384 s[29] = (uint8_t)(h4 >> 28);
385 s[30] = (uint8_t)(h4 >> 36);
386 s[31] = (uint8_t)(h4 >> 44);
387 }
388
389 # if defined(X25519_ASM)
390 void x25519_fe51_mul(fe51 h, const fe51 f, const fe51 g);
391 void x25519_fe51_sqr(fe51 h, const fe51 f);
392 void x25519_fe51_mul121666(fe51 h, fe51 f);
393 # define fe51_mul x25519_fe51_mul
394 # define fe51_sq x25519_fe51_sqr
395 # define fe51_mul121666 x25519_fe51_mul121666
396 # else
397
398 typedef uint128_t u128;
399
fe51_mul(fe51 h,const fe51 f,const fe51 g)400 static void fe51_mul(fe51 h, const fe51 f, const fe51 g)
401 {
402 u128 h0, h1, h2, h3, h4;
403 uint64_t f_i, g0, g1, g2, g3, g4;
404
405 f_i = f[0];
406 h0 = (u128)f_i * (g0 = g[0]);
407 h1 = (u128)f_i * (g1 = g[1]);
408 h2 = (u128)f_i * (g2 = g[2]);
409 h3 = (u128)f_i * (g3 = g[3]);
410 h4 = (u128)f_i * (g4 = g[4]);
411
412 f_i = f[1];
413 h0 += (u128)f_i * (g4 *= 19);
414 h1 += (u128)f_i * g0;
415 h2 += (u128)f_i * g1;
416 h3 += (u128)f_i * g2;
417 h4 += (u128)f_i * g3;
418
419 f_i = f[2];
420 h0 += (u128)f_i * (g3 *= 19);
421 h1 += (u128)f_i * g4;
422 h2 += (u128)f_i * g0;
423 h3 += (u128)f_i * g1;
424 h4 += (u128)f_i * g2;
425
426 f_i = f[3];
427 h0 += (u128)f_i * (g2 *= 19);
428 h1 += (u128)f_i * g3;
429 h2 += (u128)f_i * g4;
430 h3 += (u128)f_i * g0;
431 h4 += (u128)f_i * g1;
432
433 f_i = f[4];
434 h0 += (u128)f_i * (g1 *= 19);
435 h1 += (u128)f_i * g2;
436 h2 += (u128)f_i * g3;
437 h3 += (u128)f_i * g4;
438 h4 += (u128)f_i * g0;
439
440 /* partial [lazy] reduction */
441 h3 += (uint64_t)(h2 >> 51); g2 = (uint64_t)h2 & MASK51;
442 h1 += (uint64_t)(h0 >> 51); g0 = (uint64_t)h0 & MASK51;
443
444 h4 += (uint64_t)(h3 >> 51); g3 = (uint64_t)h3 & MASK51;
445 g2 += (uint64_t)(h1 >> 51); g1 = (uint64_t)h1 & MASK51;
446
447 g0 += (uint64_t)(h4 >> 51) * 19; g4 = (uint64_t)h4 & MASK51;
448 g3 += g2 >> 51; g2 &= MASK51;
449 g1 += g0 >> 51; g0 &= MASK51;
450
451 h[0] = g0;
452 h[1] = g1;
453 h[2] = g2;
454 h[3] = g3;
455 h[4] = g4;
456 }
457
fe51_sq(fe51 h,const fe51 f)458 static void fe51_sq(fe51 h, const fe51 f)
459 {
460 # if defined(OPENSSL_SMALL_FOOTPRINT)
461 fe51_mul(h, f, f);
462 # else
463 /* dedicated squaring gives 16-25% overall improvement */
464 uint64_t g0 = f[0];
465 uint64_t g1 = f[1];
466 uint64_t g2 = f[2];
467 uint64_t g3 = f[3];
468 uint64_t g4 = f[4];
469 u128 h0, h1, h2, h3, h4;
470
471 h0 = (u128)g0 * g0; g0 *= 2;
472 h1 = (u128)g0 * g1;
473 h2 = (u128)g0 * g2;
474 h3 = (u128)g0 * g3;
475 h4 = (u128)g0 * g4;
476
477 g0 = g4; /* borrow g0 */
478 h3 += (u128)g0 * (g4 *= 19);
479
480 h2 += (u128)g1 * g1; g1 *= 2;
481 h3 += (u128)g1 * g2;
482 h4 += (u128)g1 * g3;
483 h0 += (u128)g1 * g4;
484
485 g0 = g3; /* borrow g0 */
486 h1 += (u128)g0 * (g3 *= 19);
487 h2 += (u128)(g0 * 2) * g4;
488
489 h4 += (u128)g2 * g2; g2 *= 2;
490 h0 += (u128)g2 * g3;
491 h1 += (u128)g2 * g4;
492
493 /* partial [lazy] reduction */
494 h3 += (uint64_t)(h2 >> 51); g2 = (uint64_t)h2 & MASK51;
495 h1 += (uint64_t)(h0 >> 51); g0 = (uint64_t)h0 & MASK51;
496
497 h4 += (uint64_t)(h3 >> 51); g3 = (uint64_t)h3 & MASK51;
498 g2 += (uint64_t)(h1 >> 51); g1 = (uint64_t)h1 & MASK51;
499
500 g0 += (uint64_t)(h4 >> 51) * 19; g4 = (uint64_t)h4 & MASK51;
501 g3 += g2 >> 51; g2 &= MASK51;
502 g1 += g0 >> 51; g0 &= MASK51;
503
504 h[0] = g0;
505 h[1] = g1;
506 h[2] = g2;
507 h[3] = g3;
508 h[4] = g4;
509 # endif
510 }
511
fe51_mul121666(fe51 h,fe51 f)512 static void fe51_mul121666(fe51 h, fe51 f)
513 {
514 u128 h0 = f[0] * (u128)121666;
515 u128 h1 = f[1] * (u128)121666;
516 u128 h2 = f[2] * (u128)121666;
517 u128 h3 = f[3] * (u128)121666;
518 u128 h4 = f[4] * (u128)121666;
519 uint64_t g0, g1, g2, g3, g4;
520
521 h3 += (uint64_t)(h2 >> 51); g2 = (uint64_t)h2 & MASK51;
522 h1 += (uint64_t)(h0 >> 51); g0 = (uint64_t)h0 & MASK51;
523
524 h4 += (uint64_t)(h3 >> 51); g3 = (uint64_t)h3 & MASK51;
525 g2 += (uint64_t)(h1 >> 51); g1 = (uint64_t)h1 & MASK51;
526
527 g0 += (uint64_t)(h4 >> 51) * 19; g4 = (uint64_t)h4 & MASK51;
528 g3 += g2 >> 51; g2 &= MASK51;
529 g1 += g0 >> 51; g0 &= MASK51;
530
531 h[0] = g0;
532 h[1] = g1;
533 h[2] = g2;
534 h[3] = g3;
535 h[4] = g4;
536 }
537 # endif
538
fe51_add(fe51 h,const fe51 f,const fe51 g)539 static void fe51_add(fe51 h, const fe51 f, const fe51 g)
540 {
541 h[0] = f[0] + g[0];
542 h[1] = f[1] + g[1];
543 h[2] = f[2] + g[2];
544 h[3] = f[3] + g[3];
545 h[4] = f[4] + g[4];
546 }
547
fe51_sub(fe51 h,const fe51 f,const fe51 g)548 static void fe51_sub(fe51 h, const fe51 f, const fe51 g)
549 {
550 /*
551 * Add 2*modulus to ensure that result remains positive
552 * even if subtrahend is partially reduced.
553 */
554 h[0] = (f[0] + 0xfffffffffffda) - g[0];
555 h[1] = (f[1] + 0xffffffffffffe) - g[1];
556 h[2] = (f[2] + 0xffffffffffffe) - g[2];
557 h[3] = (f[3] + 0xffffffffffffe) - g[3];
558 h[4] = (f[4] + 0xffffffffffffe) - g[4];
559 }
560
fe51_0(fe51 h)561 static void fe51_0(fe51 h)
562 {
563 h[0] = 0;
564 h[1] = 0;
565 h[2] = 0;
566 h[3] = 0;
567 h[4] = 0;
568 }
569
fe51_1(fe51 h)570 static void fe51_1(fe51 h)
571 {
572 h[0] = 1;
573 h[1] = 0;
574 h[2] = 0;
575 h[3] = 0;
576 h[4] = 0;
577 }
578
fe51_copy(fe51 h,const fe51 f)579 static void fe51_copy(fe51 h, const fe51 f)
580 {
581 h[0] = f[0];
582 h[1] = f[1];
583 h[2] = f[2];
584 h[3] = f[3];
585 h[4] = f[4];
586 }
587
fe51_cswap(fe51 f,fe51 g,unsigned int b)588 static void fe51_cswap(fe51 f, fe51 g, unsigned int b)
589 {
590 int i;
591 uint64_t mask = 0 - (uint64_t)b;
592
593 for (i = 0; i < 5; i++) {
594 int64_t x = f[i] ^ g[i];
595 x &= mask;
596 f[i] ^= x;
597 g[i] ^= x;
598 }
599 }
600
fe51_invert(fe51 out,const fe51 z)601 static void fe51_invert(fe51 out, const fe51 z)
602 {
603 fe51 t0;
604 fe51 t1;
605 fe51 t2;
606 fe51 t3;
607 int i;
608
609 /*
610 * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as
611 * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11.
612 */
613
614 /* t0 = z ** 2 */
615 fe51_sq(t0, z);
616
617 /* t1 = t0 ** (2 ** 2) = z ** 8 */
618 fe51_sq(t1, t0);
619 fe51_sq(t1, t1);
620
621 /* t1 = z * t1 = z ** 9 */
622 fe51_mul(t1, z, t1);
623 /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */
624 fe51_mul(t0, t0, t1);
625
626 /* t2 = t0 ** 2 = z ** 22 */
627 fe51_sq(t2, t0);
628
629 /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */
630 fe51_mul(t1, t1, t2);
631
632 /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */
633 fe51_sq(t2, t1);
634 for (i = 1; i < 5; ++i)
635 fe51_sq(t2, t2);
636
637 /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */
638 fe51_mul(t1, t2, t1);
639
640 /* Continuing similarly... */
641
642 /* t2 = z ** (2 ** 20 - 1) */
643 fe51_sq(t2, t1);
644 for (i = 1; i < 10; ++i)
645 fe51_sq(t2, t2);
646
647 fe51_mul(t2, t2, t1);
648
649 /* t2 = z ** (2 ** 40 - 1) */
650 fe51_sq(t3, t2);
651 for (i = 1; i < 20; ++i)
652 fe51_sq(t3, t3);
653
654 fe51_mul(t2, t3, t2);
655
656 /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */
657 for (i = 0; i < 10; ++i)
658 fe51_sq(t2, t2);
659
660 /* t1 = z ** (2 ** 50 - 1) */
661 fe51_mul(t1, t2, t1);
662
663 /* t2 = z ** (2 ** 100 - 1) */
664 fe51_sq(t2, t1);
665 for (i = 1; i < 50; ++i)
666 fe51_sq(t2, t2);
667
668 fe51_mul(t2, t2, t1);
669
670 /* t2 = z ** (2 ** 200 - 1) */
671 fe51_sq(t3, t2);
672 for (i = 1; i < 100; ++i)
673 fe51_sq(t3, t3);
674
675 fe51_mul(t2, t3, t2);
676
677 /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */
678 for (i = 0; i < 50; ++i)
679 fe51_sq(t2, t2);
680
681 /* t1 = z ** (2 ** 250 - 1) */
682 fe51_mul(t1, t2, t1);
683
684 /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */
685 for (i = 0; i < 5; ++i)
686 fe51_sq(t1, t1);
687
688 /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */
689 fe51_mul(out, t1, t0);
690 }
691
692 /*
693 * Duplicate of original x25519_scalar_mult_generic, but using
694 * fe51_* subroutines.
695 */
x25519_scalar_mult(uint8_t out[32],const uint8_t scalar[32],const uint8_t point[32])696 static void x25519_scalar_mult(uint8_t out[32], const uint8_t scalar[32],
697 const uint8_t point[32])
698 {
699 fe51 x1, x2, z2, x3, z3, tmp0, tmp1;
700 uint8_t e[32];
701 unsigned swap = 0;
702 int pos;
703
704 # ifdef BASE_2_64_IMPLEMENTED
705 if (x25519_fe64_eligible()) {
706 x25519_scalar_mulx(out, scalar, point);
707 return;
708 }
709 # endif
710
711 memcpy(e, scalar, 32);
712 e[0] &= 0xf8;
713 e[31] &= 0x7f;
714 e[31] |= 0x40;
715 fe51_frombytes(x1, point);
716 fe51_1(x2);
717 fe51_0(z2);
718 fe51_copy(x3, x1);
719 fe51_1(z3);
720
721 for (pos = 254; pos >= 0; --pos) {
722 unsigned int b = 1 & (e[pos / 8] >> (pos & 7));
723
724 swap ^= b;
725 fe51_cswap(x2, x3, swap);
726 fe51_cswap(z2, z3, swap);
727 swap = b;
728 fe51_sub(tmp0, x3, z3);
729 fe51_sub(tmp1, x2, z2);
730 fe51_add(x2, x2, z2);
731 fe51_add(z2, x3, z3);
732 fe51_mul(z3, tmp0, x2);
733 fe51_mul(z2, z2, tmp1);
734 fe51_sq(tmp0, tmp1);
735 fe51_sq(tmp1, x2);
736 fe51_add(x3, z3, z2);
737 fe51_sub(z2, z3, z2);
738 fe51_mul(x2, tmp1, tmp0);
739 fe51_sub(tmp1, tmp1, tmp0);
740 fe51_sq(z2, z2);
741 fe51_mul121666(z3, tmp1);
742 fe51_sq(x3, x3);
743 fe51_add(tmp0, tmp0, z3);
744 fe51_mul(z3, x1, z2);
745 fe51_mul(z2, tmp1, tmp0);
746 }
747
748 fe51_invert(z2, z2);
749 fe51_mul(x2, x2, z2);
750 fe51_tobytes(out, x2);
751
752 OPENSSL_cleanse(e, sizeof(e));
753 }
754 #endif
755
756 /*
757 * Reference base 2^25.5 implementation.
758 *
759 * This code is mostly taken from the ref10 version of Ed25519 in SUPERCOP
760 * 20141124 (http://bench.cr.yp.to/supercop.html).
761 *
762 * The field functions are shared by Ed25519 and X25519 where possible.
763 */
764
765 /*
766 * fe means field element. Here the field is \Z/(2^255-19). An element t,
767 * entries t[0]...t[9], represents the integer t[0]+2^26 t[1]+2^51 t[2]+2^77
768 * t[3]+2^102 t[4]+...+2^230 t[9]. Bounds on each t[i] vary depending on
769 * context.
770 */
771 typedef int32_t fe[10];
772
773 static const int64_t kBottom21Bits = 0x1fffffLL;
774 static const int64_t kBottom25Bits = 0x1ffffffLL;
775 static const int64_t kBottom26Bits = 0x3ffffffLL;
776 static const int64_t kTop39Bits = 0xfffffffffe000000LL;
777 static const int64_t kTop38Bits = 0xfffffffffc000000LL;
778
load_3(const uint8_t * in)779 static uint64_t load_3(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 return result;
787 }
788
load_4(const uint8_t * in)789 static uint64_t load_4(const uint8_t *in)
790 {
791 uint64_t result;
792
793 result = ((uint64_t)in[0]);
794 result |= ((uint64_t)in[1]) << 8;
795 result |= ((uint64_t)in[2]) << 16;
796 result |= ((uint64_t)in[3]) << 24;
797 return result;
798 }
799
fe_frombytes(fe h,const uint8_t * s)800 static void fe_frombytes(fe h, const uint8_t *s)
801 {
802 /* Ignores top bit of h. */
803 int64_t h0 = load_4(s);
804 int64_t h1 = load_3(s + 4) << 6;
805 int64_t h2 = load_3(s + 7) << 5;
806 int64_t h3 = load_3(s + 10) << 3;
807 int64_t h4 = load_3(s + 13) << 2;
808 int64_t h5 = load_4(s + 16);
809 int64_t h6 = load_3(s + 20) << 7;
810 int64_t h7 = load_3(s + 23) << 5;
811 int64_t h8 = load_3(s + 26) << 4;
812 int64_t h9 = (load_3(s + 29) & 0x7fffff) << 2;
813 int64_t carry0;
814 int64_t carry1;
815 int64_t carry2;
816 int64_t carry3;
817 int64_t carry4;
818 int64_t carry5;
819 int64_t carry6;
820 int64_t carry7;
821 int64_t carry8;
822 int64_t carry9;
823
824 carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
825 carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
826 carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
827 carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
828 carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;
829
830 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
831 carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
832 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
833 carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
834 carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;
835
836 h[0] = (int32_t)h0;
837 h[1] = (int32_t)h1;
838 h[2] = (int32_t)h2;
839 h[3] = (int32_t)h3;
840 h[4] = (int32_t)h4;
841 h[5] = (int32_t)h5;
842 h[6] = (int32_t)h6;
843 h[7] = (int32_t)h7;
844 h[8] = (int32_t)h8;
845 h[9] = (int32_t)h9;
846 }
847
848 /*
849 * Preconditions:
850 * |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
851 *
852 * Write p=2^255-19; q=floor(h/p).
853 * Basic claim: q = floor(2^(-255)(h + 19 2^(-25)h9 + 2^(-1))).
854 *
855 * Proof:
856 * Have |h|<=p so |q|<=1 so |19^2 2^(-255) q|<1/4.
857 * Also have |h-2^230 h9|<2^231 so |19 2^(-255)(h-2^230 h9)|<1/4.
858 *
859 * Write y=2^(-1)-19^2 2^(-255)q-19 2^(-255)(h-2^230 h9).
860 * Then 0<y<1.
861 *
862 * Write r=h-pq.
863 * Have 0<=r<=p-1=2^255-20.
864 * Thus 0<=r+19(2^-255)r<r+19(2^-255)2^255<=2^255-1.
865 *
866 * Write x=r+19(2^-255)r+y.
867 * Then 0<x<2^255 so floor(2^(-255)x) = 0 so floor(q+2^(-255)x) = q.
868 *
869 * Have q+2^(-255)x = 2^(-255)(h + 19 2^(-25) h9 + 2^(-1))
870 * so floor(2^(-255)(h + 19 2^(-25) h9 + 2^(-1))) = q.
871 */
fe_tobytes(uint8_t * s,const fe h)872 static void fe_tobytes(uint8_t *s, const fe h)
873 {
874 int32_t h0 = h[0];
875 int32_t h1 = h[1];
876 int32_t h2 = h[2];
877 int32_t h3 = h[3];
878 int32_t h4 = h[4];
879 int32_t h5 = h[5];
880 int32_t h6 = h[6];
881 int32_t h7 = h[7];
882 int32_t h8 = h[8];
883 int32_t h9 = h[9];
884 int32_t q;
885
886 q = (19 * h9 + (((int32_t) 1) << 24)) >> 25;
887 q = (h0 + q) >> 26;
888 q = (h1 + q) >> 25;
889 q = (h2 + q) >> 26;
890 q = (h3 + q) >> 25;
891 q = (h4 + q) >> 26;
892 q = (h5 + q) >> 25;
893 q = (h6 + q) >> 26;
894 q = (h7 + q) >> 25;
895 q = (h8 + q) >> 26;
896 q = (h9 + q) >> 25;
897
898 /* Goal: Output h-(2^255-19)q, which is between 0 and 2^255-20. */
899 h0 += 19 * q;
900 /* Goal: Output h-2^255 q, which is between 0 and 2^255-20. */
901
902 h1 += h0 >> 26; h0 &= kBottom26Bits;
903 h2 += h1 >> 25; h1 &= kBottom25Bits;
904 h3 += h2 >> 26; h2 &= kBottom26Bits;
905 h4 += h3 >> 25; h3 &= kBottom25Bits;
906 h5 += h4 >> 26; h4 &= kBottom26Bits;
907 h6 += h5 >> 25; h5 &= kBottom25Bits;
908 h7 += h6 >> 26; h6 &= kBottom26Bits;
909 h8 += h7 >> 25; h7 &= kBottom25Bits;
910 h9 += h8 >> 26; h8 &= kBottom26Bits;
911 h9 &= kBottom25Bits;
912 /* h10 = carry9 */
913
914 /*
915 * Goal: Output h0+...+2^255 h10-2^255 q, which is between 0 and 2^255-20.
916 * Have h0+...+2^230 h9 between 0 and 2^255-1;
917 * evidently 2^255 h10-2^255 q = 0.
918 * Goal: Output h0+...+2^230 h9.
919 */
920 s[ 0] = (uint8_t) (h0 >> 0);
921 s[ 1] = (uint8_t) (h0 >> 8);
922 s[ 2] = (uint8_t) (h0 >> 16);
923 s[ 3] = (uint8_t)((h0 >> 24) | ((uint32_t)(h1) << 2));
924 s[ 4] = (uint8_t) (h1 >> 6);
925 s[ 5] = (uint8_t) (h1 >> 14);
926 s[ 6] = (uint8_t)((h1 >> 22) | ((uint32_t)(h2) << 3));
927 s[ 7] = (uint8_t) (h2 >> 5);
928 s[ 8] = (uint8_t) (h2 >> 13);
929 s[ 9] = (uint8_t)((h2 >> 21) | ((uint32_t)(h3) << 5));
930 s[10] = (uint8_t) (h3 >> 3);
931 s[11] = (uint8_t) (h3 >> 11);
932 s[12] = (uint8_t)((h3 >> 19) | ((uint32_t)(h4) << 6));
933 s[13] = (uint8_t) (h4 >> 2);
934 s[14] = (uint8_t) (h4 >> 10);
935 s[15] = (uint8_t) (h4 >> 18);
936 s[16] = (uint8_t) (h5 >> 0);
937 s[17] = (uint8_t) (h5 >> 8);
938 s[18] = (uint8_t) (h5 >> 16);
939 s[19] = (uint8_t)((h5 >> 24) | ((uint32_t)(h6) << 1));
940 s[20] = (uint8_t) (h6 >> 7);
941 s[21] = (uint8_t) (h6 >> 15);
942 s[22] = (uint8_t)((h6 >> 23) | ((uint32_t)(h7) << 3));
943 s[23] = (uint8_t) (h7 >> 5);
944 s[24] = (uint8_t) (h7 >> 13);
945 s[25] = (uint8_t)((h7 >> 21) | ((uint32_t)(h8) << 4));
946 s[26] = (uint8_t) (h8 >> 4);
947 s[27] = (uint8_t) (h8 >> 12);
948 s[28] = (uint8_t)((h8 >> 20) | ((uint32_t)(h9) << 6));
949 s[29] = (uint8_t) (h9 >> 2);
950 s[30] = (uint8_t) (h9 >> 10);
951 s[31] = (uint8_t) (h9 >> 18);
952 }
953
954 /* h = f */
fe_copy(fe h,const fe f)955 static void fe_copy(fe h, const fe f)
956 {
957 memmove(h, f, sizeof(int32_t) * 10);
958 }
959
960 /* h = 0 */
fe_0(fe h)961 static void fe_0(fe h)
962 {
963 memset(h, 0, sizeof(int32_t) * 10);
964 }
965
966 /* h = 1 */
fe_1(fe h)967 static void fe_1(fe h)
968 {
969 memset(h, 0, sizeof(int32_t) * 10);
970 h[0] = 1;
971 }
972
973 /*
974 * h = f + g
975 *
976 * Can overlap h with f or g.
977 *
978 * Preconditions:
979 * |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
980 * |g| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
981 *
982 * Postconditions:
983 * |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
984 */
fe_add(fe h,const fe f,const fe g)985 static void fe_add(fe h, const fe f, const fe g)
986 {
987 unsigned i;
988
989 for (i = 0; i < 10; i++) {
990 h[i] = f[i] + g[i];
991 }
992 }
993
994 /*
995 * h = f - g
996 *
997 * Can overlap h with f or g.
998 *
999 * Preconditions:
1000 * |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
1001 * |g| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
1002 *
1003 * Postconditions:
1004 * |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
1005 */
fe_sub(fe h,const fe f,const fe g)1006 static void fe_sub(fe h, const fe f, const fe g)
1007 {
1008 unsigned i;
1009
1010 for (i = 0; i < 10; i++) {
1011 h[i] = f[i] - g[i];
1012 }
1013 }
1014
1015 /*
1016 * h = f * g
1017 *
1018 * Can overlap h with f or g.
1019 *
1020 * Preconditions:
1021 * |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
1022 * |g| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
1023 *
1024 * Postconditions:
1025 * |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
1026 *
1027 * Notes on implementation strategy:
1028 *
1029 * Using schoolbook multiplication.
1030 * Karatsuba would save a little in some cost models.
1031 *
1032 * Most multiplications by 2 and 19 are 32-bit precomputations;
1033 * cheaper than 64-bit postcomputations.
1034 *
1035 * There is one remaining multiplication by 19 in the carry chain;
1036 * one *19 precomputation can be merged into this,
1037 * but the resulting data flow is considerably less clean.
1038 *
1039 * There are 12 carries below.
1040 * 10 of them are 2-way parallelizable and vectorizable.
1041 * Can get away with 11 carries, but then data flow is much deeper.
1042 *
1043 * With tighter constraints on inputs can squeeze carries into int32.
1044 */
fe_mul(fe h,const fe f,const fe g)1045 static void fe_mul(fe h, const fe f, const fe g)
1046 {
1047 int32_t f0 = f[0];
1048 int32_t f1 = f[1];
1049 int32_t f2 = f[2];
1050 int32_t f3 = f[3];
1051 int32_t f4 = f[4];
1052 int32_t f5 = f[5];
1053 int32_t f6 = f[6];
1054 int32_t f7 = f[7];
1055 int32_t f8 = f[8];
1056 int32_t f9 = f[9];
1057 int32_t g0 = g[0];
1058 int32_t g1 = g[1];
1059 int32_t g2 = g[2];
1060 int32_t g3 = g[3];
1061 int32_t g4 = g[4];
1062 int32_t g5 = g[5];
1063 int32_t g6 = g[6];
1064 int32_t g7 = g[7];
1065 int32_t g8 = g[8];
1066 int32_t g9 = g[9];
1067 int32_t g1_19 = 19 * g1; /* 1.959375*2^29 */
1068 int32_t g2_19 = 19 * g2; /* 1.959375*2^30; still ok */
1069 int32_t g3_19 = 19 * g3;
1070 int32_t g4_19 = 19 * g4;
1071 int32_t g5_19 = 19 * g5;
1072 int32_t g6_19 = 19 * g6;
1073 int32_t g7_19 = 19 * g7;
1074 int32_t g8_19 = 19 * g8;
1075 int32_t g9_19 = 19 * g9;
1076 int32_t f1_2 = 2 * f1;
1077 int32_t f3_2 = 2 * f3;
1078 int32_t f5_2 = 2 * f5;
1079 int32_t f7_2 = 2 * f7;
1080 int32_t f9_2 = 2 * f9;
1081 int64_t f0g0 = f0 * (int64_t) g0;
1082 int64_t f0g1 = f0 * (int64_t) g1;
1083 int64_t f0g2 = f0 * (int64_t) g2;
1084 int64_t f0g3 = f0 * (int64_t) g3;
1085 int64_t f0g4 = f0 * (int64_t) g4;
1086 int64_t f0g5 = f0 * (int64_t) g5;
1087 int64_t f0g6 = f0 * (int64_t) g6;
1088 int64_t f0g7 = f0 * (int64_t) g7;
1089 int64_t f0g8 = f0 * (int64_t) g8;
1090 int64_t f0g9 = f0 * (int64_t) g9;
1091 int64_t f1g0 = f1 * (int64_t) g0;
1092 int64_t f1g1_2 = f1_2 * (int64_t) g1;
1093 int64_t f1g2 = f1 * (int64_t) g2;
1094 int64_t f1g3_2 = f1_2 * (int64_t) g3;
1095 int64_t f1g4 = f1 * (int64_t) g4;
1096 int64_t f1g5_2 = f1_2 * (int64_t) g5;
1097 int64_t f1g6 = f1 * (int64_t) g6;
1098 int64_t f1g7_2 = f1_2 * (int64_t) g7;
1099 int64_t f1g8 = f1 * (int64_t) g8;
1100 int64_t f1g9_38 = f1_2 * (int64_t) g9_19;
1101 int64_t f2g0 = f2 * (int64_t) g0;
1102 int64_t f2g1 = f2 * (int64_t) g1;
1103 int64_t f2g2 = f2 * (int64_t) g2;
1104 int64_t f2g3 = f2 * (int64_t) g3;
1105 int64_t f2g4 = f2 * (int64_t) g4;
1106 int64_t f2g5 = f2 * (int64_t) g5;
1107 int64_t f2g6 = f2 * (int64_t) g6;
1108 int64_t f2g7 = f2 * (int64_t) g7;
1109 int64_t f2g8_19 = f2 * (int64_t) g8_19;
1110 int64_t f2g9_19 = f2 * (int64_t) g9_19;
1111 int64_t f3g0 = f3 * (int64_t) g0;
1112 int64_t f3g1_2 = f3_2 * (int64_t) g1;
1113 int64_t f3g2 = f3 * (int64_t) g2;
1114 int64_t f3g3_2 = f3_2 * (int64_t) g3;
1115 int64_t f3g4 = f3 * (int64_t) g4;
1116 int64_t f3g5_2 = f3_2 * (int64_t) g5;
1117 int64_t f3g6 = f3 * (int64_t) g6;
1118 int64_t f3g7_38 = f3_2 * (int64_t) g7_19;
1119 int64_t f3g8_19 = f3 * (int64_t) g8_19;
1120 int64_t f3g9_38 = f3_2 * (int64_t) g9_19;
1121 int64_t f4g0 = f4 * (int64_t) g0;
1122 int64_t f4g1 = f4 * (int64_t) g1;
1123 int64_t f4g2 = f4 * (int64_t) g2;
1124 int64_t f4g3 = f4 * (int64_t) g3;
1125 int64_t f4g4 = f4 * (int64_t) g4;
1126 int64_t f4g5 = f4 * (int64_t) g5;
1127 int64_t f4g6_19 = f4 * (int64_t) g6_19;
1128 int64_t f4g7_19 = f4 * (int64_t) g7_19;
1129 int64_t f4g8_19 = f4 * (int64_t) g8_19;
1130 int64_t f4g9_19 = f4 * (int64_t) g9_19;
1131 int64_t f5g0 = f5 * (int64_t) g0;
1132 int64_t f5g1_2 = f5_2 * (int64_t) g1;
1133 int64_t f5g2 = f5 * (int64_t) g2;
1134 int64_t f5g3_2 = f5_2 * (int64_t) g3;
1135 int64_t f5g4 = f5 * (int64_t) g4;
1136 int64_t f5g5_38 = f5_2 * (int64_t) g5_19;
1137 int64_t f5g6_19 = f5 * (int64_t) g6_19;
1138 int64_t f5g7_38 = f5_2 * (int64_t) g7_19;
1139 int64_t f5g8_19 = f5 * (int64_t) g8_19;
1140 int64_t f5g9_38 = f5_2 * (int64_t) g9_19;
1141 int64_t f6g0 = f6 * (int64_t) g0;
1142 int64_t f6g1 = f6 * (int64_t) g1;
1143 int64_t f6g2 = f6 * (int64_t) g2;
1144 int64_t f6g3 = f6 * (int64_t) g3;
1145 int64_t f6g4_19 = f6 * (int64_t) g4_19;
1146 int64_t f6g5_19 = f6 * (int64_t) g5_19;
1147 int64_t f6g6_19 = f6 * (int64_t) g6_19;
1148 int64_t f6g7_19 = f6 * (int64_t) g7_19;
1149 int64_t f6g8_19 = f6 * (int64_t) g8_19;
1150 int64_t f6g9_19 = f6 * (int64_t) g9_19;
1151 int64_t f7g0 = f7 * (int64_t) g0;
1152 int64_t f7g1_2 = f7_2 * (int64_t) g1;
1153 int64_t f7g2 = f7 * (int64_t) g2;
1154 int64_t f7g3_38 = f7_2 * (int64_t) g3_19;
1155 int64_t f7g4_19 = f7 * (int64_t) g4_19;
1156 int64_t f7g5_38 = f7_2 * (int64_t) g5_19;
1157 int64_t f7g6_19 = f7 * (int64_t) g6_19;
1158 int64_t f7g7_38 = f7_2 * (int64_t) g7_19;
1159 int64_t f7g8_19 = f7 * (int64_t) g8_19;
1160 int64_t f7g9_38 = f7_2 * (int64_t) g9_19;
1161 int64_t f8g0 = f8 * (int64_t) g0;
1162 int64_t f8g1 = f8 * (int64_t) g1;
1163 int64_t f8g2_19 = f8 * (int64_t) g2_19;
1164 int64_t f8g3_19 = f8 * (int64_t) g3_19;
1165 int64_t f8g4_19 = f8 * (int64_t) g4_19;
1166 int64_t f8g5_19 = f8 * (int64_t) g5_19;
1167 int64_t f8g6_19 = f8 * (int64_t) g6_19;
1168 int64_t f8g7_19 = f8 * (int64_t) g7_19;
1169 int64_t f8g8_19 = f8 * (int64_t) g8_19;
1170 int64_t f8g9_19 = f8 * (int64_t) g9_19;
1171 int64_t f9g0 = f9 * (int64_t) g0;
1172 int64_t f9g1_38 = f9_2 * (int64_t) g1_19;
1173 int64_t f9g2_19 = f9 * (int64_t) g2_19;
1174 int64_t f9g3_38 = f9_2 * (int64_t) g3_19;
1175 int64_t f9g4_19 = f9 * (int64_t) g4_19;
1176 int64_t f9g5_38 = f9_2 * (int64_t) g5_19;
1177 int64_t f9g6_19 = f9 * (int64_t) g6_19;
1178 int64_t f9g7_38 = f9_2 * (int64_t) g7_19;
1179 int64_t f9g8_19 = f9 * (int64_t) g8_19;
1180 int64_t f9g9_38 = f9_2 * (int64_t) g9_19;
1181 int64_t h0 = f0g0 + f1g9_38 + f2g8_19 + f3g7_38 + f4g6_19 + f5g5_38 + f6g4_19 + f7g3_38 + f8g2_19 + f9g1_38;
1182 int64_t h1 = f0g1 + f1g0 + f2g9_19 + f3g8_19 + f4g7_19 + f5g6_19 + f6g5_19 + f7g4_19 + f8g3_19 + f9g2_19;
1183 int64_t h2 = f0g2 + f1g1_2 + f2g0 + f3g9_38 + f4g8_19 + f5g7_38 + f6g6_19 + f7g5_38 + f8g4_19 + f9g3_38;
1184 int64_t h3 = f0g3 + f1g2 + f2g1 + f3g0 + f4g9_19 + f5g8_19 + f6g7_19 + f7g6_19 + f8g5_19 + f9g4_19;
1185 int64_t h4 = f0g4 + f1g3_2 + f2g2 + f3g1_2 + f4g0 + f5g9_38 + f6g8_19 + f7g7_38 + f8g6_19 + f9g5_38;
1186 int64_t h5 = f0g5 + f1g4 + f2g3 + f3g2 + f4g1 + f5g0 + f6g9_19 + f7g8_19 + f8g7_19 + f9g6_19;
1187 int64_t h6 = f0g6 + f1g5_2 + f2g4 + f3g3_2 + f4g2 + f5g1_2 + f6g0 + f7g9_38 + f8g8_19 + f9g7_38;
1188 int64_t h7 = f0g7 + f1g6 + f2g5 + f3g4 + f4g3 + f5g2 + f6g1 + f7g0 + f8g9_19 + f9g8_19;
1189 int64_t h8 = f0g8 + f1g7_2 + f2g6 + f3g5_2 + f4g4 + f5g3_2 + f6g2 + f7g1_2 + f8g0 + f9g9_38;
1190 int64_t h9 = f0g9 + f1g8 + f2g7 + f3g6 + f4g5 + f5g4 + f6g3 + f7g2 + f8g1 + f9g0 ;
1191 int64_t carry0;
1192 int64_t carry1;
1193 int64_t carry2;
1194 int64_t carry3;
1195 int64_t carry4;
1196 int64_t carry5;
1197 int64_t carry6;
1198 int64_t carry7;
1199 int64_t carry8;
1200 int64_t carry9;
1201
1202 /* |h0| <= (1.65*1.65*2^52*(1+19+19+19+19)+1.65*1.65*2^50*(38+38+38+38+38))
1203 * i.e. |h0| <= 1.4*2^60; narrower ranges for h2, h4, h6, h8
1204 * |h1| <= (1.65*1.65*2^51*(1+1+19+19+19+19+19+19+19+19))
1205 * i.e. |h1| <= 1.7*2^59; narrower ranges for h3, h5, h7, h9 */
1206
1207 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
1208 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
1209 /* |h0| <= 2^25 */
1210 /* |h4| <= 2^25 */
1211 /* |h1| <= 1.71*2^59 */
1212 /* |h5| <= 1.71*2^59 */
1213
1214 carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
1215 carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
1216 /* |h1| <= 2^24; from now on fits into int32 */
1217 /* |h5| <= 2^24; from now on fits into int32 */
1218 /* |h2| <= 1.41*2^60 */
1219 /* |h6| <= 1.41*2^60 */
1220
1221 carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
1222 carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
1223 /* |h2| <= 2^25; from now on fits into int32 unchanged */
1224 /* |h6| <= 2^25; from now on fits into int32 unchanged */
1225 /* |h3| <= 1.71*2^59 */
1226 /* |h7| <= 1.71*2^59 */
1227
1228 carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
1229 carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;
1230 /* |h3| <= 2^24; from now on fits into int32 unchanged */
1231 /* |h7| <= 2^24; from now on fits into int32 unchanged */
1232 /* |h4| <= 1.72*2^34 */
1233 /* |h8| <= 1.41*2^60 */
1234
1235 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
1236 carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;
1237 /* |h4| <= 2^25; from now on fits into int32 unchanged */
1238 /* |h8| <= 2^25; from now on fits into int32 unchanged */
1239 /* |h5| <= 1.01*2^24 */
1240 /* |h9| <= 1.71*2^59 */
1241
1242 carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
1243 /* |h9| <= 2^24; from now on fits into int32 unchanged */
1244 /* |h0| <= 1.1*2^39 */
1245
1246 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
1247 /* |h0| <= 2^25; from now on fits into int32 unchanged */
1248 /* |h1| <= 1.01*2^24 */
1249
1250 h[0] = (int32_t)h0;
1251 h[1] = (int32_t)h1;
1252 h[2] = (int32_t)h2;
1253 h[3] = (int32_t)h3;
1254 h[4] = (int32_t)h4;
1255 h[5] = (int32_t)h5;
1256 h[6] = (int32_t)h6;
1257 h[7] = (int32_t)h7;
1258 h[8] = (int32_t)h8;
1259 h[9] = (int32_t)h9;
1260 }
1261
1262 /*
1263 * h = f * f
1264 *
1265 * Can overlap h with f.
1266 *
1267 * Preconditions:
1268 * |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
1269 *
1270 * Postconditions:
1271 * |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
1272 *
1273 * See fe_mul.c for discussion of implementation strategy.
1274 */
fe_sq(fe h,const fe f)1275 static void fe_sq(fe h, const fe f)
1276 {
1277 int32_t f0 = f[0];
1278 int32_t f1 = f[1];
1279 int32_t f2 = f[2];
1280 int32_t f3 = f[3];
1281 int32_t f4 = f[4];
1282 int32_t f5 = f[5];
1283 int32_t f6 = f[6];
1284 int32_t f7 = f[7];
1285 int32_t f8 = f[8];
1286 int32_t f9 = f[9];
1287 int32_t f0_2 = 2 * f0;
1288 int32_t f1_2 = 2 * f1;
1289 int32_t f2_2 = 2 * f2;
1290 int32_t f3_2 = 2 * f3;
1291 int32_t f4_2 = 2 * f4;
1292 int32_t f5_2 = 2 * f5;
1293 int32_t f6_2 = 2 * f6;
1294 int32_t f7_2 = 2 * f7;
1295 int32_t f5_38 = 38 * f5; /* 1.959375*2^30 */
1296 int32_t f6_19 = 19 * f6; /* 1.959375*2^30 */
1297 int32_t f7_38 = 38 * f7; /* 1.959375*2^30 */
1298 int32_t f8_19 = 19 * f8; /* 1.959375*2^30 */
1299 int32_t f9_38 = 38 * f9; /* 1.959375*2^30 */
1300 int64_t f0f0 = f0 * (int64_t) f0;
1301 int64_t f0f1_2 = f0_2 * (int64_t) f1;
1302 int64_t f0f2_2 = f0_2 * (int64_t) f2;
1303 int64_t f0f3_2 = f0_2 * (int64_t) f3;
1304 int64_t f0f4_2 = f0_2 * (int64_t) f4;
1305 int64_t f0f5_2 = f0_2 * (int64_t) f5;
1306 int64_t f0f6_2 = f0_2 * (int64_t) f6;
1307 int64_t f0f7_2 = f0_2 * (int64_t) f7;
1308 int64_t f0f8_2 = f0_2 * (int64_t) f8;
1309 int64_t f0f9_2 = f0_2 * (int64_t) f9;
1310 int64_t f1f1_2 = f1_2 * (int64_t) f1;
1311 int64_t f1f2_2 = f1_2 * (int64_t) f2;
1312 int64_t f1f3_4 = f1_2 * (int64_t) f3_2;
1313 int64_t f1f4_2 = f1_2 * (int64_t) f4;
1314 int64_t f1f5_4 = f1_2 * (int64_t) f5_2;
1315 int64_t f1f6_2 = f1_2 * (int64_t) f6;
1316 int64_t f1f7_4 = f1_2 * (int64_t) f7_2;
1317 int64_t f1f8_2 = f1_2 * (int64_t) f8;
1318 int64_t f1f9_76 = f1_2 * (int64_t) f9_38;
1319 int64_t f2f2 = f2 * (int64_t) f2;
1320 int64_t f2f3_2 = f2_2 * (int64_t) f3;
1321 int64_t f2f4_2 = f2_2 * (int64_t) f4;
1322 int64_t f2f5_2 = f2_2 * (int64_t) f5;
1323 int64_t f2f6_2 = f2_2 * (int64_t) f6;
1324 int64_t f2f7_2 = f2_2 * (int64_t) f7;
1325 int64_t f2f8_38 = f2_2 * (int64_t) f8_19;
1326 int64_t f2f9_38 = f2 * (int64_t) f9_38;
1327 int64_t f3f3_2 = f3_2 * (int64_t) f3;
1328 int64_t f3f4_2 = f3_2 * (int64_t) f4;
1329 int64_t f3f5_4 = f3_2 * (int64_t) f5_2;
1330 int64_t f3f6_2 = f3_2 * (int64_t) f6;
1331 int64_t f3f7_76 = f3_2 * (int64_t) f7_38;
1332 int64_t f3f8_38 = f3_2 * (int64_t) f8_19;
1333 int64_t f3f9_76 = f3_2 * (int64_t) f9_38;
1334 int64_t f4f4 = f4 * (int64_t) f4;
1335 int64_t f4f5_2 = f4_2 * (int64_t) f5;
1336 int64_t f4f6_38 = f4_2 * (int64_t) f6_19;
1337 int64_t f4f7_38 = f4 * (int64_t) f7_38;
1338 int64_t f4f8_38 = f4_2 * (int64_t) f8_19;
1339 int64_t f4f9_38 = f4 * (int64_t) f9_38;
1340 int64_t f5f5_38 = f5 * (int64_t) f5_38;
1341 int64_t f5f6_38 = f5_2 * (int64_t) f6_19;
1342 int64_t f5f7_76 = f5_2 * (int64_t) f7_38;
1343 int64_t f5f8_38 = f5_2 * (int64_t) f8_19;
1344 int64_t f5f9_76 = f5_2 * (int64_t) f9_38;
1345 int64_t f6f6_19 = f6 * (int64_t) f6_19;
1346 int64_t f6f7_38 = f6 * (int64_t) f7_38;
1347 int64_t f6f8_38 = f6_2 * (int64_t) f8_19;
1348 int64_t f6f9_38 = f6 * (int64_t) f9_38;
1349 int64_t f7f7_38 = f7 * (int64_t) f7_38;
1350 int64_t f7f8_38 = f7_2 * (int64_t) f8_19;
1351 int64_t f7f9_76 = f7_2 * (int64_t) f9_38;
1352 int64_t f8f8_19 = f8 * (int64_t) f8_19;
1353 int64_t f8f9_38 = f8 * (int64_t) f9_38;
1354 int64_t f9f9_38 = f9 * (int64_t) f9_38;
1355 int64_t h0 = f0f0 + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38;
1356 int64_t h1 = f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38;
1357 int64_t h2 = f0f2_2 + f1f1_2 + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19;
1358 int64_t h3 = f0f3_2 + f1f2_2 + f4f9_38 + f5f8_38 + f6f7_38;
1359 int64_t h4 = f0f4_2 + f1f3_4 + f2f2 + f5f9_76 + f6f8_38 + f7f7_38;
1360 int64_t h5 = f0f5_2 + f1f4_2 + f2f3_2 + f6f9_38 + f7f8_38;
1361 int64_t h6 = f0f6_2 + f1f5_4 + f2f4_2 + f3f3_2 + f7f9_76 + f8f8_19;
1362 int64_t h7 = f0f7_2 + f1f6_2 + f2f5_2 + f3f4_2 + f8f9_38;
1363 int64_t h8 = f0f8_2 + f1f7_4 + f2f6_2 + f3f5_4 + f4f4 + f9f9_38;
1364 int64_t h9 = f0f9_2 + f1f8_2 + f2f7_2 + f3f6_2 + f4f5_2;
1365 int64_t carry0;
1366 int64_t carry1;
1367 int64_t carry2;
1368 int64_t carry3;
1369 int64_t carry4;
1370 int64_t carry5;
1371 int64_t carry6;
1372 int64_t carry7;
1373 int64_t carry8;
1374 int64_t carry9;
1375
1376 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
1377 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
1378
1379 carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
1380 carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
1381
1382 carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
1383 carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
1384
1385 carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
1386 carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;
1387
1388 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
1389 carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;
1390
1391 carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
1392
1393 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
1394
1395 h[0] = (int32_t)h0;
1396 h[1] = (int32_t)h1;
1397 h[2] = (int32_t)h2;
1398 h[3] = (int32_t)h3;
1399 h[4] = (int32_t)h4;
1400 h[5] = (int32_t)h5;
1401 h[6] = (int32_t)h6;
1402 h[7] = (int32_t)h7;
1403 h[8] = (int32_t)h8;
1404 h[9] = (int32_t)h9;
1405 }
1406
fe_invert(fe out,const fe z)1407 static void fe_invert(fe out, const fe z)
1408 {
1409 fe t0;
1410 fe t1;
1411 fe t2;
1412 fe t3;
1413 int i;
1414
1415 /*
1416 * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as
1417 * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11.
1418 */
1419
1420 /* t0 = z ** 2 */
1421 fe_sq(t0, z);
1422
1423 /* t1 = t0 ** (2 ** 2) = z ** 8 */
1424 fe_sq(t1, t0);
1425 fe_sq(t1, t1);
1426
1427 /* t1 = z * t1 = z ** 9 */
1428 fe_mul(t1, z, t1);
1429 /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */
1430 fe_mul(t0, t0, t1);
1431
1432 /* t2 = t0 ** 2 = z ** 22 */
1433 fe_sq(t2, t0);
1434
1435 /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */
1436 fe_mul(t1, t1, t2);
1437
1438 /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */
1439 fe_sq(t2, t1);
1440 for (i = 1; i < 5; ++i) {
1441 fe_sq(t2, t2);
1442 }
1443
1444 /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */
1445 fe_mul(t1, t2, t1);
1446
1447 /* Continuing similarly... */
1448
1449 /* t2 = z ** (2 ** 20 - 1) */
1450 fe_sq(t2, t1);
1451 for (i = 1; i < 10; ++i) {
1452 fe_sq(t2, t2);
1453 }
1454 fe_mul(t2, t2, t1);
1455
1456 /* t2 = z ** (2 ** 40 - 1) */
1457 fe_sq(t3, t2);
1458 for (i = 1; i < 20; ++i) {
1459 fe_sq(t3, t3);
1460 }
1461 fe_mul(t2, t3, t2);
1462
1463 /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */
1464 for (i = 0; i < 10; ++i) {
1465 fe_sq(t2, t2);
1466 }
1467 /* t1 = z ** (2 ** 50 - 1) */
1468 fe_mul(t1, t2, t1);
1469
1470 /* t2 = z ** (2 ** 100 - 1) */
1471 fe_sq(t2, t1);
1472 for (i = 1; i < 50; ++i) {
1473 fe_sq(t2, t2);
1474 }
1475 fe_mul(t2, t2, t1);
1476
1477 /* t2 = z ** (2 ** 200 - 1) */
1478 fe_sq(t3, t2);
1479 for (i = 1; i < 100; ++i) {
1480 fe_sq(t3, t3);
1481 }
1482 fe_mul(t2, t3, t2);
1483
1484 /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */
1485 fe_sq(t2, t2);
1486 for (i = 1; i < 50; ++i) {
1487 fe_sq(t2, t2);
1488 }
1489
1490 /* t1 = z ** (2 ** 250 - 1) */
1491 fe_mul(t1, t2, t1);
1492
1493 /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */
1494 fe_sq(t1, t1);
1495 for (i = 1; i < 5; ++i) {
1496 fe_sq(t1, t1);
1497 }
1498
1499 /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */
1500 fe_mul(out, t1, t0);
1501 }
1502
1503 /*
1504 * h = -f
1505 *
1506 * Preconditions:
1507 * |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
1508 *
1509 * Postconditions:
1510 * |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
1511 */
fe_neg(fe h,const fe f)1512 static void fe_neg(fe h, const fe f)
1513 {
1514 unsigned i;
1515
1516 for (i = 0; i < 10; i++) {
1517 h[i] = -f[i];
1518 }
1519 }
1520
1521 /*
1522 * Replace (f,g) with (g,g) if b == 1;
1523 * replace (f,g) with (f,g) if b == 0.
1524 *
1525 * Preconditions: b in {0,1}.
1526 */
fe_cmov(fe f,const fe g,unsigned b)1527 static void fe_cmov(fe f, const fe g, unsigned b)
1528 {
1529 size_t i;
1530
1531 b = 0-b;
1532 for (i = 0; i < 10; i++) {
1533 int32_t x = f[i] ^ g[i];
1534 x &= b;
1535 f[i] ^= x;
1536 }
1537 }
1538
1539 /*
1540 * return 0 if f == 0
1541 * return 1 if f != 0
1542 *
1543 * Preconditions:
1544 * |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
1545 */
fe_isnonzero(const fe f)1546 static int fe_isnonzero(const fe f)
1547 {
1548 uint8_t s[32];
1549 static const uint8_t zero[32] = {0};
1550
1551 fe_tobytes(s, f);
1552
1553 return CRYPTO_memcmp(s, zero, sizeof(zero)) != 0;
1554 }
1555
1556 /*
1557 * return 1 if f is in {1,3,5,...,q-2}
1558 * return 0 if f is in {0,2,4,...,q-1}
1559 *
1560 * Preconditions:
1561 * |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
1562 */
fe_isnegative(const fe f)1563 static int fe_isnegative(const fe f)
1564 {
1565 uint8_t s[32];
1566
1567 fe_tobytes(s, f);
1568 return s[0] & 1;
1569 }
1570
1571 /*
1572 * h = 2 * f * f
1573 *
1574 * Can overlap h with f.
1575 *
1576 * Preconditions:
1577 * |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
1578 *
1579 * Postconditions:
1580 * |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
1581 *
1582 * See fe_mul.c for discussion of implementation strategy.
1583 */
fe_sq2(fe h,const fe f)1584 static void fe_sq2(fe h, const fe f)
1585 {
1586 int32_t f0 = f[0];
1587 int32_t f1 = f[1];
1588 int32_t f2 = f[2];
1589 int32_t f3 = f[3];
1590 int32_t f4 = f[4];
1591 int32_t f5 = f[5];
1592 int32_t f6 = f[6];
1593 int32_t f7 = f[7];
1594 int32_t f8 = f[8];
1595 int32_t f9 = f[9];
1596 int32_t f0_2 = 2 * f0;
1597 int32_t f1_2 = 2 * f1;
1598 int32_t f2_2 = 2 * f2;
1599 int32_t f3_2 = 2 * f3;
1600 int32_t f4_2 = 2 * f4;
1601 int32_t f5_2 = 2 * f5;
1602 int32_t f6_2 = 2 * f6;
1603 int32_t f7_2 = 2 * f7;
1604 int32_t f5_38 = 38 * f5; /* 1.959375*2^30 */
1605 int32_t f6_19 = 19 * f6; /* 1.959375*2^30 */
1606 int32_t f7_38 = 38 * f7; /* 1.959375*2^30 */
1607 int32_t f8_19 = 19 * f8; /* 1.959375*2^30 */
1608 int32_t f9_38 = 38 * f9; /* 1.959375*2^30 */
1609 int64_t f0f0 = f0 * (int64_t) f0;
1610 int64_t f0f1_2 = f0_2 * (int64_t) f1;
1611 int64_t f0f2_2 = f0_2 * (int64_t) f2;
1612 int64_t f0f3_2 = f0_2 * (int64_t) f3;
1613 int64_t f0f4_2 = f0_2 * (int64_t) f4;
1614 int64_t f0f5_2 = f0_2 * (int64_t) f5;
1615 int64_t f0f6_2 = f0_2 * (int64_t) f6;
1616 int64_t f0f7_2 = f0_2 * (int64_t) f7;
1617 int64_t f0f8_2 = f0_2 * (int64_t) f8;
1618 int64_t f0f9_2 = f0_2 * (int64_t) f9;
1619 int64_t f1f1_2 = f1_2 * (int64_t) f1;
1620 int64_t f1f2_2 = f1_2 * (int64_t) f2;
1621 int64_t f1f3_4 = f1_2 * (int64_t) f3_2;
1622 int64_t f1f4_2 = f1_2 * (int64_t) f4;
1623 int64_t f1f5_4 = f1_2 * (int64_t) f5_2;
1624 int64_t f1f6_2 = f1_2 * (int64_t) f6;
1625 int64_t f1f7_4 = f1_2 * (int64_t) f7_2;
1626 int64_t f1f8_2 = f1_2 * (int64_t) f8;
1627 int64_t f1f9_76 = f1_2 * (int64_t) f9_38;
1628 int64_t f2f2 = f2 * (int64_t) f2;
1629 int64_t f2f3_2 = f2_2 * (int64_t) f3;
1630 int64_t f2f4_2 = f2_2 * (int64_t) f4;
1631 int64_t f2f5_2 = f2_2 * (int64_t) f5;
1632 int64_t f2f6_2 = f2_2 * (int64_t) f6;
1633 int64_t f2f7_2 = f2_2 * (int64_t) f7;
1634 int64_t f2f8_38 = f2_2 * (int64_t) f8_19;
1635 int64_t f2f9_38 = f2 * (int64_t) f9_38;
1636 int64_t f3f3_2 = f3_2 * (int64_t) f3;
1637 int64_t f3f4_2 = f3_2 * (int64_t) f4;
1638 int64_t f3f5_4 = f3_2 * (int64_t) f5_2;
1639 int64_t f3f6_2 = f3_2 * (int64_t) f6;
1640 int64_t f3f7_76 = f3_2 * (int64_t) f7_38;
1641 int64_t f3f8_38 = f3_2 * (int64_t) f8_19;
1642 int64_t f3f9_76 = f3_2 * (int64_t) f9_38;
1643 int64_t f4f4 = f4 * (int64_t) f4;
1644 int64_t f4f5_2 = f4_2 * (int64_t) f5;
1645 int64_t f4f6_38 = f4_2 * (int64_t) f6_19;
1646 int64_t f4f7_38 = f4 * (int64_t) f7_38;
1647 int64_t f4f8_38 = f4_2 * (int64_t) f8_19;
1648 int64_t f4f9_38 = f4 * (int64_t) f9_38;
1649 int64_t f5f5_38 = f5 * (int64_t) f5_38;
1650 int64_t f5f6_38 = f5_2 * (int64_t) f6_19;
1651 int64_t f5f7_76 = f5_2 * (int64_t) f7_38;
1652 int64_t f5f8_38 = f5_2 * (int64_t) f8_19;
1653 int64_t f5f9_76 = f5_2 * (int64_t) f9_38;
1654 int64_t f6f6_19 = f6 * (int64_t) f6_19;
1655 int64_t f6f7_38 = f6 * (int64_t) f7_38;
1656 int64_t f6f8_38 = f6_2 * (int64_t) f8_19;
1657 int64_t f6f9_38 = f6 * (int64_t) f9_38;
1658 int64_t f7f7_38 = f7 * (int64_t) f7_38;
1659 int64_t f7f8_38 = f7_2 * (int64_t) f8_19;
1660 int64_t f7f9_76 = f7_2 * (int64_t) f9_38;
1661 int64_t f8f8_19 = f8 * (int64_t) f8_19;
1662 int64_t f8f9_38 = f8 * (int64_t) f9_38;
1663 int64_t f9f9_38 = f9 * (int64_t) f9_38;
1664 int64_t h0 = f0f0 + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38;
1665 int64_t h1 = f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38;
1666 int64_t h2 = f0f2_2 + f1f1_2 + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19;
1667 int64_t h3 = f0f3_2 + f1f2_2 + f4f9_38 + f5f8_38 + f6f7_38;
1668 int64_t h4 = f0f4_2 + f1f3_4 + f2f2 + f5f9_76 + f6f8_38 + f7f7_38;
1669 int64_t h5 = f0f5_2 + f1f4_2 + f2f3_2 + f6f9_38 + f7f8_38;
1670 int64_t h6 = f0f6_2 + f1f5_4 + f2f4_2 + f3f3_2 + f7f9_76 + f8f8_19;
1671 int64_t h7 = f0f7_2 + f1f6_2 + f2f5_2 + f3f4_2 + f8f9_38;
1672 int64_t h8 = f0f8_2 + f1f7_4 + f2f6_2 + f3f5_4 + f4f4 + f9f9_38;
1673 int64_t h9 = f0f9_2 + f1f8_2 + f2f7_2 + f3f6_2 + f4f5_2;
1674 int64_t carry0;
1675 int64_t carry1;
1676 int64_t carry2;
1677 int64_t carry3;
1678 int64_t carry4;
1679 int64_t carry5;
1680 int64_t carry6;
1681 int64_t carry7;
1682 int64_t carry8;
1683 int64_t carry9;
1684
1685 h0 += h0;
1686 h1 += h1;
1687 h2 += h2;
1688 h3 += h3;
1689 h4 += h4;
1690 h5 += h5;
1691 h6 += h6;
1692 h7 += h7;
1693 h8 += h8;
1694 h9 += h9;
1695
1696 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
1697 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
1698
1699 carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
1700 carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
1701
1702 carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
1703 carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
1704
1705 carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
1706 carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;
1707
1708 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
1709 carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;
1710
1711 carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
1712
1713 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
1714
1715 h[0] = (int32_t)h0;
1716 h[1] = (int32_t)h1;
1717 h[2] = (int32_t)h2;
1718 h[3] = (int32_t)h3;
1719 h[4] = (int32_t)h4;
1720 h[5] = (int32_t)h5;
1721 h[6] = (int32_t)h6;
1722 h[7] = (int32_t)h7;
1723 h[8] = (int32_t)h8;
1724 h[9] = (int32_t)h9;
1725 }
1726
fe_pow22523(fe out,const fe z)1727 static void fe_pow22523(fe out, const fe z)
1728 {
1729 fe t0;
1730 fe t1;
1731 fe t2;
1732 int i;
1733
1734 fe_sq(t0, z);
1735 fe_sq(t1, t0);
1736 for (i = 1; i < 2; ++i) {
1737 fe_sq(t1, t1);
1738 }
1739 fe_mul(t1, z, t1);
1740 fe_mul(t0, t0, t1);
1741 fe_sq(t0, t0);
1742 fe_mul(t0, t1, t0);
1743 fe_sq(t1, t0);
1744 for (i = 1; i < 5; ++i) {
1745 fe_sq(t1, t1);
1746 }
1747 fe_mul(t0, t1, t0);
1748 fe_sq(t1, t0);
1749 for (i = 1; i < 10; ++i) {
1750 fe_sq(t1, t1);
1751 }
1752 fe_mul(t1, t1, t0);
1753 fe_sq(t2, t1);
1754 for (i = 1; i < 20; ++i) {
1755 fe_sq(t2, t2);
1756 }
1757 fe_mul(t1, t2, t1);
1758 fe_sq(t1, t1);
1759 for (i = 1; i < 10; ++i) {
1760 fe_sq(t1, t1);
1761 }
1762 fe_mul(t0, t1, t0);
1763 fe_sq(t1, t0);
1764 for (i = 1; i < 50; ++i) {
1765 fe_sq(t1, t1);
1766 }
1767 fe_mul(t1, t1, t0);
1768 fe_sq(t2, t1);
1769 for (i = 1; i < 100; ++i) {
1770 fe_sq(t2, t2);
1771 }
1772 fe_mul(t1, t2, t1);
1773 fe_sq(t1, t1);
1774 for (i = 1; i < 50; ++i) {
1775 fe_sq(t1, t1);
1776 }
1777 fe_mul(t0, t1, t0);
1778 fe_sq(t0, t0);
1779 for (i = 1; i < 2; ++i) {
1780 fe_sq(t0, t0);
1781 }
1782 fe_mul(out, t0, z);
1783 }
1784
1785 /*
1786 * ge means group element.
1787 *
1788 * Here the group is the set of pairs (x,y) of field elements (see fe.h)
1789 * satisfying -x^2 + y^2 = 1 + d x^2y^2
1790 * where d = -121665/121666.
1791 *
1792 * Representations:
1793 * ge_p2 (projective): (X:Y:Z) satisfying x=X/Z, y=Y/Z
1794 * ge_p3 (extended): (X:Y:Z:T) satisfying x=X/Z, y=Y/Z, XY=ZT
1795 * ge_p1p1 (completed): ((X:Z),(Y:T)) satisfying x=X/Z, y=Y/T
1796 * ge_precomp (Duif): (y+x,y-x,2dxy)
1797 */
1798 typedef struct {
1799 fe X;
1800 fe Y;
1801 fe Z;
1802 } ge_p2;
1803
1804 typedef struct {
1805 fe X;
1806 fe Y;
1807 fe Z;
1808 fe T;
1809 } ge_p3;
1810
1811 typedef struct {
1812 fe X;
1813 fe Y;
1814 fe Z;
1815 fe T;
1816 } ge_p1p1;
1817
1818 typedef struct {
1819 fe yplusx;
1820 fe yminusx;
1821 fe xy2d;
1822 } ge_precomp;
1823
1824 typedef struct {
1825 fe YplusX;
1826 fe YminusX;
1827 fe Z;
1828 fe T2d;
1829 } ge_cached;
1830
ge_tobytes(uint8_t * s,const ge_p2 * h)1831 static void ge_tobytes(uint8_t *s, const ge_p2 *h)
1832 {
1833 fe recip;
1834 fe x;
1835 fe y;
1836
1837 fe_invert(recip, h->Z);
1838 fe_mul(x, h->X, recip);
1839 fe_mul(y, h->Y, recip);
1840 fe_tobytes(s, y);
1841 s[31] ^= fe_isnegative(x) << 7;
1842 }
1843
ge_p3_tobytes(uint8_t * s,const ge_p3 * h)1844 static void ge_p3_tobytes(uint8_t *s, const ge_p3 *h)
1845 {
1846 fe recip;
1847 fe x;
1848 fe y;
1849
1850 fe_invert(recip, h->Z);
1851 fe_mul(x, h->X, recip);
1852 fe_mul(y, h->Y, recip);
1853 fe_tobytes(s, y);
1854 s[31] ^= fe_isnegative(x) << 7;
1855 }
1856
1857 static const fe d = {
1858 -10913610, 13857413, -15372611, 6949391, 114729,
1859 -8787816, -6275908, -3247719, -18696448, -12055116
1860 };
1861
1862 static const fe sqrtm1 = {
1863 -32595792, -7943725, 9377950, 3500415, 12389472,
1864 -272473, -25146209, -2005654, 326686, 11406482
1865 };
1866
ge_frombytes_vartime(ge_p3 * h,const uint8_t * s)1867 static int ge_frombytes_vartime(ge_p3 *h, const uint8_t *s)
1868 {
1869 fe u;
1870 fe v;
1871 fe w;
1872 fe vxx;
1873 fe check;
1874
1875 fe_frombytes(h->Y, s);
1876 fe_1(h->Z);
1877 fe_sq(u, h->Y);
1878 fe_mul(v, u, d);
1879 fe_sub(u, u, h->Z); /* u = y^2-1 */
1880 fe_add(v, v, h->Z); /* v = dy^2+1 */
1881
1882 fe_mul(w, u, v); /* w = u*v */
1883
1884 fe_pow22523(h->X, w); /* x = w^((q-5)/8) */
1885 fe_mul(h->X, h->X, u); /* x = u * w^((q-5)/8) */
1886
1887 fe_sq(vxx, h->X);
1888 fe_mul(vxx, vxx, v);
1889 fe_sub(check, vxx, u); /* vx^2-u */
1890 if (fe_isnonzero(check)) {
1891 fe_add(check, vxx, u); /* vx^2+u */
1892 if (fe_isnonzero(check)) {
1893 return -1;
1894 }
1895 fe_mul(h->X, h->X, sqrtm1);
1896 }
1897
1898 if (fe_isnegative(h->X) != (s[31] >> 7)) {
1899 fe_neg(h->X, h->X);
1900 }
1901
1902 fe_mul(h->T, h->X, h->Y);
1903 return 0;
1904 }
1905
ge_p2_0(ge_p2 * h)1906 static void ge_p2_0(ge_p2 *h)
1907 {
1908 fe_0(h->X);
1909 fe_1(h->Y);
1910 fe_1(h->Z);
1911 }
1912
ge_p3_0(ge_p3 * h)1913 static void ge_p3_0(ge_p3 *h)
1914 {
1915 fe_0(h->X);
1916 fe_1(h->Y);
1917 fe_1(h->Z);
1918 fe_0(h->T);
1919 }
1920
ge_precomp_0(ge_precomp * h)1921 static void ge_precomp_0(ge_precomp *h)
1922 {
1923 fe_1(h->yplusx);
1924 fe_1(h->yminusx);
1925 fe_0(h->xy2d);
1926 }
1927
1928 /* r = p */
ge_p3_to_p2(ge_p2 * r,const ge_p3 * p)1929 static void ge_p3_to_p2(ge_p2 *r, const ge_p3 *p)
1930 {
1931 fe_copy(r->X, p->X);
1932 fe_copy(r->Y, p->Y);
1933 fe_copy(r->Z, p->Z);
1934 }
1935
1936 static const fe d2 = {
1937 -21827239, -5839606, -30745221, 13898782, 229458,
1938 15978800, -12551817, -6495438, 29715968, 9444199
1939 };
1940
1941 /* r = p */
ge_p3_to_cached(ge_cached * r,const ge_p3 * p)1942 static void ge_p3_to_cached(ge_cached *r, const ge_p3 *p)
1943 {
1944 fe_add(r->YplusX, p->Y, p->X);
1945 fe_sub(r->YminusX, p->Y, p->X);
1946 fe_copy(r->Z, p->Z);
1947 fe_mul(r->T2d, p->T, d2);
1948 }
1949
1950 /* r = p */
ge_p1p1_to_p2(ge_p2 * r,const ge_p1p1 * p)1951 static void ge_p1p1_to_p2(ge_p2 *r, const ge_p1p1 *p)
1952 {
1953 fe_mul(r->X, p->X, p->T);
1954 fe_mul(r->Y, p->Y, p->Z);
1955 fe_mul(r->Z, p->Z, p->T);
1956 }
1957
1958 /* r = p */
ge_p1p1_to_p3(ge_p3 * r,const ge_p1p1 * p)1959 static void ge_p1p1_to_p3(ge_p3 *r, const ge_p1p1 *p)
1960 {
1961 fe_mul(r->X, p->X, p->T);
1962 fe_mul(r->Y, p->Y, p->Z);
1963 fe_mul(r->Z, p->Z, p->T);
1964 fe_mul(r->T, p->X, p->Y);
1965 }
1966
1967 /* r = 2 * p */
ge_p2_dbl(ge_p1p1 * r,const ge_p2 * p)1968 static void ge_p2_dbl(ge_p1p1 *r, const ge_p2 *p)
1969 {
1970 fe t0;
1971
1972 fe_sq(r->X, p->X);
1973 fe_sq(r->Z, p->Y);
1974 fe_sq2(r->T, p->Z);
1975 fe_add(r->Y, p->X, p->Y);
1976 fe_sq(t0, r->Y);
1977 fe_add(r->Y, r->Z, r->X);
1978 fe_sub(r->Z, r->Z, r->X);
1979 fe_sub(r->X, t0, r->Y);
1980 fe_sub(r->T, r->T, r->Z);
1981 }
1982
1983 /* r = 2 * p */
ge_p3_dbl(ge_p1p1 * r,const ge_p3 * p)1984 static void ge_p3_dbl(ge_p1p1 *r, const ge_p3 *p)
1985 {
1986 ge_p2 q;
1987 ge_p3_to_p2(&q, p);
1988 ge_p2_dbl(r, &q);
1989 }
1990
1991 /* r = p + q */
ge_madd(ge_p1p1 * r,const ge_p3 * p,const ge_precomp * q)1992 static void ge_madd(ge_p1p1 *r, const ge_p3 *p, const ge_precomp *q)
1993 {
1994 fe t0;
1995
1996 fe_add(r->X, p->Y, p->X);
1997 fe_sub(r->Y, p->Y, p->X);
1998 fe_mul(r->Z, r->X, q->yplusx);
1999 fe_mul(r->Y, r->Y, q->yminusx);
2000 fe_mul(r->T, q->xy2d, p->T);
2001 fe_add(t0, p->Z, p->Z);
2002 fe_sub(r->X, r->Z, r->Y);
2003 fe_add(r->Y, r->Z, r->Y);
2004 fe_add(r->Z, t0, r->T);
2005 fe_sub(r->T, t0, r->T);
2006 }
2007
2008 /* r = p - q */
ge_msub(ge_p1p1 * r,const ge_p3 * p,const ge_precomp * q)2009 static void ge_msub(ge_p1p1 *r, const ge_p3 *p, const ge_precomp *q)
2010 {
2011 fe t0;
2012
2013 fe_add(r->X, p->Y, p->X);
2014 fe_sub(r->Y, p->Y, p->X);
2015 fe_mul(r->Z, r->X, q->yminusx);
2016 fe_mul(r->Y, r->Y, q->yplusx);
2017 fe_mul(r->T, q->xy2d, p->T);
2018 fe_add(t0, p->Z, p->Z);
2019 fe_sub(r->X, r->Z, r->Y);
2020 fe_add(r->Y, r->Z, r->Y);
2021 fe_sub(r->Z, t0, r->T);
2022 fe_add(r->T, t0, r->T);
2023 }
2024
2025 /* r = p + q */
ge_add(ge_p1p1 * r,const ge_p3 * p,const ge_cached * q)2026 static void ge_add(ge_p1p1 *r, const ge_p3 *p, const ge_cached *q)
2027 {
2028 fe t0;
2029
2030 fe_add(r->X, p->Y, p->X);
2031 fe_sub(r->Y, p->Y, p->X);
2032 fe_mul(r->Z, r->X, q->YplusX);
2033 fe_mul(r->Y, r->Y, q->YminusX);
2034 fe_mul(r->T, q->T2d, p->T);
2035 fe_mul(r->X, p->Z, q->Z);
2036 fe_add(t0, r->X, r->X);
2037 fe_sub(r->X, r->Z, r->Y);
2038 fe_add(r->Y, r->Z, r->Y);
2039 fe_add(r->Z, t0, r->T);
2040 fe_sub(r->T, t0, r->T);
2041 }
2042
2043 /* r = p - q */
ge_sub(ge_p1p1 * r,const ge_p3 * p,const ge_cached * q)2044 static void ge_sub(ge_p1p1 *r, const ge_p3 *p, const ge_cached *q)
2045 {
2046 fe t0;
2047
2048 fe_add(r->X, p->Y, p->X);
2049 fe_sub(r->Y, p->Y, p->X);
2050 fe_mul(r->Z, r->X, q->YminusX);
2051 fe_mul(r->Y, r->Y, q->YplusX);
2052 fe_mul(r->T, q->T2d, p->T);
2053 fe_mul(r->X, p->Z, q->Z);
2054 fe_add(t0, r->X, r->X);
2055 fe_sub(r->X, r->Z, r->Y);
2056 fe_add(r->Y, r->Z, r->Y);
2057 fe_sub(r->Z, t0, r->T);
2058 fe_add(r->T, t0, r->T);
2059 }
2060
equal(signed char b,signed char c)2061 static uint8_t equal(signed char b, signed char c)
2062 {
2063 uint8_t ub = b;
2064 uint8_t uc = c;
2065 uint8_t x = ub ^ uc; /* 0: yes; 1..255: no */
2066 uint32_t y = x; /* 0: yes; 1..255: no */
2067 y -= 1; /* 4294967295: yes; 0..254: no */
2068 y >>= 31; /* 1: yes; 0: no */
2069 return y;
2070 }
2071
cmov(ge_precomp * t,const ge_precomp * u,uint8_t b)2072 static void cmov(ge_precomp *t, const ge_precomp *u, uint8_t b)
2073 {
2074 fe_cmov(t->yplusx, u->yplusx, b);
2075 fe_cmov(t->yminusx, u->yminusx, b);
2076 fe_cmov(t->xy2d, u->xy2d, b);
2077 }
2078
2079 /* k25519Precomp[i][j] = (j+1)*256^i*B */
2080 static const ge_precomp k25519Precomp[32][8] = {
2081 {
2082 {
2083 {25967493, -14356035, 29566456, 3660896, -12694345, 4014787,
2084 27544626, -11754271, -6079156, 2047605},
2085 {-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692,
2086 5043384, 19500929, -15469378},
2087 {-8738181, 4489570, 9688441, -14785194, 10184609, -12363380,
2088 29287919, 11864899, -24514362, -4438546},
2089 },
2090 {
2091 {-12815894, -12976347, -21581243, 11784320, -25355658, -2750717,
2092 -11717903, -3814571, -358445, -10211303},
2093 {-21703237, 6903825, 27185491, 6451973, -29577724, -9554005,
2094 -15616551, 11189268, -26829678, -5319081},
2095 {26966642, 11152617, 32442495, 15396054, 14353839, -12752335,
2096 -3128826, -9541118, -15472047, -4166697},
2097 },
2098 {
2099 {15636291, -9688557, 24204773, -7912398, 616977, -16685262,
2100 27787600, -14772189, 28944400, -1550024},
2101 {16568933, 4717097, -11556148, -1102322, 15682896, -11807043,
2102 16354577, -11775962, 7689662, 11199574},
2103 {30464156, -5976125, -11779434, -15670865, 23220365, 15915852,
2104 7512774, 10017326, -17749093, -9920357},
2105 },
2106 {
2107 {-17036878, 13921892, 10945806, -6033431, 27105052, -16084379,
2108 -28926210, 15006023, 3284568, -6276540},
2109 {23599295, -8306047, -11193664, -7687416, 13236774, 10506355,
2110 7464579, 9656445, 13059162, 10374397},
2111 {7798556, 16710257, 3033922, 2874086, 28997861, 2835604, 32406664,
2112 -3839045, -641708, -101325},
2113 },
2114 {
2115 {10861363, 11473154, 27284546, 1981175, -30064349, 12577861,
2116 32867885, 14515107, -15438304, 10819380},
2117 {4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668,
2118 12483688, -12668491, 5581306},
2119 {19563160, 16186464, -29386857, 4097519, 10237984, -4348115,
2120 28542350, 13850243, -23678021, -15815942},
2121 },
2122 {
2123 {-15371964, -12862754, 32573250, 4720197, -26436522, 5875511,
2124 -19188627, -15224819, -9818940, -12085777},
2125 {-8549212, 109983, 15149363, 2178705, 22900618, 4543417, 3044240,
2126 -15689887, 1762328, 14866737},
2127 {-18199695, -15951423, -10473290, 1707278, -17185920, 3916101,
2128 -28236412, 3959421, 27914454, 4383652},
2129 },
2130 {
2131 {5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852,
2132 5230134, -23952439, -15175766},
2133 {-30269007, -3463509, 7665486, 10083793, 28475525, 1649722,
2134 20654025, 16520125, 30598449, 7715701},
2135 {28881845, 14381568, 9657904, 3680757, -20181635, 7843316,
2136 -31400660, 1370708, 29794553, -1409300},
2137 },
2138 {
2139 {14499471, -2729599, -33191113, -4254652, 28494862, 14271267,
2140 30290735, 10876454, -33154098, 2381726},
2141 {-7195431, -2655363, -14730155, 462251, -27724326, 3941372,
2142 -6236617, 3696005, -32300832, 15351955},
2143 {27431194, 8222322, 16448760, -3907995, -18707002, 11938355,
2144 -32961401, -2970515, 29551813, 10109425},
2145 },
2146 },
2147 {
2148 {
2149 {-13657040, -13155431, -31283750, 11777098, 21447386, 6519384,
2150 -2378284, -1627556, 10092783, -4764171},
2151 {27939166, 14210322, 4677035, 16277044, -22964462, -12398139,
2152 -32508754, 12005538, -17810127, 12803510},
2153 {17228999, -15661624, -1233527, 300140, -1224870, -11714777,
2154 30364213, -9038194, 18016357, 4397660},
2155 },
2156 {
2157 {-10958843, -7690207, 4776341, -14954238, 27850028, -15602212,
2158 -26619106, 14544525, -17477504, 982639},
2159 {29253598, 15796703, -2863982, -9908884, 10057023, 3163536, 7332899,
2160 -4120128, -21047696, 9934963},
2161 {5793303, 16271923, -24131614, -10116404, 29188560, 1206517,
2162 -14747930, 4559895, -30123922, -10897950},
2163 },
2164 {
2165 {-27643952, -11493006, 16282657, -11036493, 28414021, -15012264,
2166 24191034, 4541697, -13338309, 5500568},
2167 {12650548, -1497113, 9052871, 11355358, -17680037, -8400164,
2168 -17430592, 12264343, 10874051, 13524335},
2169 {25556948, -3045990, 714651, 2510400, 23394682, -10415330, 33119038,
2170 5080568, -22528059, 5376628},
2171 },
2172 {
2173 {-26088264, -4011052, -17013699, -3537628, -6726793, 1920897,
2174 -22321305, -9447443, 4535768, 1569007},
2175 {-2255422, 14606630, -21692440, -8039818, 28430649, 8775819,
2176 -30494562, 3044290, 31848280, 12543772},
2177 {-22028579, 2943893, -31857513, 6777306, 13784462, -4292203,
2178 -27377195, -2062731, 7718482, 14474653},
2179 },
2180 {
2181 {2385315, 2454213, -22631320, 46603, -4437935, -15680415, 656965,
2182 -7236665, 24316168, -5253567},
2183 {13741529, 10911568, -33233417, -8603737, -20177830, -1033297,
2184 33040651, -13424532, -20729456, 8321686},
2185 {21060490, -2212744, 15712757, -4336099, 1639040, 10656336,
2186 23845965, -11874838, -9984458, 608372},
2187 },
2188 {
2189 {-13672732, -15087586, -10889693, -7557059, -6036909, 11305547,
2190 1123968, -6780577, 27229399, 23887},
2191 {-23244140, -294205, -11744728, 14712571, -29465699, -2029617,
2192 12797024, -6440308, -1633405, 16678954},
2193 {-29500620, 4770662, -16054387, 14001338, 7830047, 9564805,
2194 -1508144, -4795045, -17169265, 4904953},
2195 },
2196 {
2197 {24059557, 14617003, 19037157, -15039908, 19766093, -14906429,
2198 5169211, 16191880, 2128236, -4326833},
2199 {-16981152, 4124966, -8540610, -10653797, 30336522, -14105247,
2200 -29806336, 916033, -6882542, -2986532},
2201 {-22630907, 12419372, -7134229, -7473371, -16478904, 16739175,
2202 285431, 2763829, 15736322, 4143876},
2203 },
2204 {
2205 {2379352, 11839345, -4110402, -5988665, 11274298, 794957, 212801,
2206 -14594663, 23527084, -16458268},
2207 {33431127, -11130478, -17838966, -15626900, 8909499, 8376530,
2208 -32625340, 4087881, -15188911, -14416214},
2209 {1767683, 7197987, -13205226, -2022635, -13091350, 448826, 5799055,
2210 4357868, -4774191, -16323038},
2211 },
2212 },
2213 {
2214 {
2215 {6721966, 13833823, -23523388, -1551314, 26354293, -11863321,
2216 23365147, -3949732, 7390890, 2759800},
2217 {4409041, 2052381, 23373853, 10530217, 7676779, -12885954, 21302353,
2218 -4264057, 1244380, -12919645},
2219 {-4421239, 7169619, 4982368, -2957590, 30256825, -2777540, 14086413,
2220 9208236, 15886429, 16489664},
2221 },
2222 {
2223 {1996075, 10375649, 14346367, 13311202, -6874135, -16438411,
2224 -13693198, 398369, -30606455, -712933},
2225 {-25307465, 9795880, -2777414, 14878809, -33531835, 14780363,
2226 13348553, 12076947, -30836462, 5113182},
2227 {-17770784, 11797796, 31950843, 13929123, -25888302, 12288344,
2228 -30341101, -7336386, 13847711, 5387222},
2229 },
2230 {
2231 {-18582163, -3416217, 17824843, -2340966, 22744343, -10442611,
2232 8763061, 3617786, -19600662, 10370991},
2233 {20246567, -14369378, 22358229, -543712, 18507283, -10413996,
2234 14554437, -8746092, 32232924, 16763880},
2235 {9648505, 10094563, 26416693, 14745928, -30374318, -6472621,
2236 11094161, 15689506, 3140038, -16510092},
2237 },
2238 {
2239 {-16160072, 5472695, 31895588, 4744994, 8823515, 10365685,
2240 -27224800, 9448613, -28774454, 366295},
2241 {19153450, 11523972, -11096490, -6503142, -24647631, 5420647,
2242 28344573, 8041113, 719605, 11671788},
2243 {8678025, 2694440, -6808014, 2517372, 4964326, 11152271, -15432916,
2244 -15266516, 27000813, -10195553},
2245 },
2246 {
2247 {-15157904, 7134312, 8639287, -2814877, -7235688, 10421742, 564065,
2248 5336097, 6750977, -14521026},
2249 {11836410, -3979488, 26297894, 16080799, 23455045, 15735944,
2250 1695823, -8819122, 8169720, 16220347},
2251 {-18115838, 8653647, 17578566, -6092619, -8025777, -16012763,
2252 -11144307, -2627664, -5990708, -14166033},
2253 },
2254 {
2255 {-23308498, -10968312, 15213228, -10081214, -30853605, -11050004,
2256 27884329, 2847284, 2655861, 1738395},
2257 {-27537433, -14253021, -25336301, -8002780, -9370762, 8129821,
2258 21651608, -3239336, -19087449, -11005278},
2259 {1533110, 3437855, 23735889, 459276, 29970501, 11335377, 26030092,
2260 5821408, 10478196, 8544890},
2261 },
2262 {
2263 {32173121, -16129311, 24896207, 3921497, 22579056, -3410854,
2264 19270449, 12217473, 17789017, -3395995},
2265 {-30552961, -2228401, -15578829, -10147201, 13243889, 517024,
2266 15479401, -3853233, 30460520, 1052596},
2267 {-11614875, 13323618, 32618793, 8175907, -15230173, 12596687,
2268 27491595, -4612359, 3179268, -9478891},
2269 },
2270 {
2271 {31947069, -14366651, -4640583, -15339921, -15125977, -6039709,
2272 -14756777, -16411740, 19072640, -9511060},
2273 {11685058, 11822410, 3158003, -13952594, 33402194, -4165066,
2274 5977896, -5215017, 473099, 5040608},
2275 {-20290863, 8198642, -27410132, 11602123, 1290375, -2799760,
2276 28326862, 1721092, -19558642, -3131606},
2277 },
2278 },
2279 {
2280 {
2281 {7881532, 10687937, 7578723, 7738378, -18951012, -2553952, 21820786,
2282 8076149, -27868496, 11538389},
2283 {-19935666, 3899861, 18283497, -6801568, -15728660, -11249211,
2284 8754525, 7446702, -5676054, 5797016},
2285 {-11295600, -3793569, -15782110, -7964573, 12708869, -8456199,
2286 2014099, -9050574, -2369172, -5877341},
2287 },
2288 {
2289 {-22472376, -11568741, -27682020, 1146375, 18956691, 16640559,
2290 1192730, -3714199, 15123619, 10811505},
2291 {14352098, -3419715, -18942044, 10822655, 32750596, 4699007, -70363,
2292 15776356, -28886779, -11974553},
2293 {-28241164, -8072475, -4978962, -5315317, 29416931, 1847569,
2294 -20654173, -16484855, 4714547, -9600655},
2295 },
2296 {
2297 {15200332, 8368572, 19679101, 15970074, -31872674, 1959451,
2298 24611599, -4543832, -11745876, 12340220},
2299 {12876937, -10480056, 33134381, 6590940, -6307776, 14872440,
2300 9613953, 8241152, 15370987, 9608631},
2301 {-4143277, -12014408, 8446281, -391603, 4407738, 13629032, -7724868,
2302 15866074, -28210621, -8814099},
2303 },
2304 {
2305 {26660628, -15677655, 8393734, 358047, -7401291, 992988, -23904233,
2306 858697, 20571223, 8420556},
2307 {14620715, 13067227, -15447274, 8264467, 14106269, 15080814,
2308 33531827, 12516406, -21574435, -12476749},
2309 {236881, 10476226, 57258, -14677024, 6472998, 2466984, 17258519,
2310 7256740, 8791136, 15069930},
2311 },
2312 {
2313 {1276410, -9371918, 22949635, -16322807, -23493039, -5702186,
2314 14711875, 4874229, -30663140, -2331391},
2315 {5855666, 4990204, -13711848, 7294284, -7804282, 1924647, -1423175,
2316 -7912378, -33069337, 9234253},
2317 {20590503, -9018988, 31529744, -7352666, -2706834, 10650548,
2318 31559055, -11609587, 18979186, 13396066},
2319 },
2320 {
2321 {24474287, 4968103, 22267082, 4407354, 24063882, -8325180,
2322 -18816887, 13594782, 33514650, 7021958},
2323 {-11566906, -6565505, -21365085, 15928892, -26158305, 4315421,
2324 -25948728, -3916677, -21480480, 12868082},
2325 {-28635013, 13504661, 19988037, -2132761, 21078225, 6443208,
2326 -21446107, 2244500, -12455797, -8089383},
2327 },
2328 {
2329 {-30595528, 13793479, -5852820, 319136, -25723172, -6263899,
2330 33086546, 8957937, -15233648, 5540521},
2331 {-11630176, -11503902, -8119500, -7643073, 2620056, 1022908,
2332 -23710744, -1568984, -16128528, -14962807},
2333 {23152971, 775386, 27395463, 14006635, -9701118, 4649512, 1689819,
2334 892185, -11513277, -15205948},
2335 },
2336 {
2337 {9770129, 9586738, 26496094, 4324120, 1556511, -3550024, 27453819,
2338 4763127, -19179614, 5867134},
2339 {-32765025, 1927590, 31726409, -4753295, 23962434, -16019500,
2340 27846559, 5931263, -29749703, -16108455},
2341 {27461885, -2977536, 22380810, 1815854, -23033753, -3031938,
2342 7283490, -15148073, -19526700, 7734629},
2343 },
2344 },
2345 {
2346 {
2347 {-8010264, -9590817, -11120403, 6196038, 29344158, -13430885,
2348 7585295, -3176626, 18549497, 15302069},
2349 {-32658337, -6171222, -7672793, -11051681, 6258878, 13504381,
2350 10458790, -6418461, -8872242, 8424746},
2351 {24687205, 8613276, -30667046, -3233545, 1863892, -1830544,
2352 19206234, 7134917, -11284482, -828919},
2353 },
2354 {
2355 {11334899, -9218022, 8025293, 12707519, 17523892, -10476071,
2356 10243738, -14685461, -5066034, 16498837},
2357 {8911542, 6887158, -9584260, -6958590, 11145641, -9543680, 17303925,
2358 -14124238, 6536641, 10543906},
2359 {-28946384, 15479763, -17466835, 568876, -1497683, 11223454,
2360 -2669190, -16625574, -27235709, 8876771},
2361 },
2362 {
2363 {-25742899, -12566864, -15649966, -846607, -33026686, -796288,
2364 -33481822, 15824474, -604426, -9039817},
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3871 },
3872 {
3873 {31429822, -13959116, 29173532, 15632448, 12174511, -2760094,
3874 32808831, 3977186, 26143136, -3148876},
3875 {22648901, 1402143, -22799984, 13746059, 7936347, 365344, -8668633,
3876 -1674433, -3758243, -2304625},
3877 {-15491917, 8012313, -2514730, -12702462, -23965846, -10254029,
3878 -1612713, -1535569, -16664475, 8194478},
3879 },
3880 {
3881 {27338066, -7507420, -7414224, 10140405, -19026427, -6589889,
3882 27277191, 8855376, 28572286, 3005164},
3883 {26287124, 4821776, 25476601, -4145903, -3764513, -15788984,
3884 -18008582, 1182479, -26094821, -13079595},
3885 {-7171154, 3178080, 23970071, 6201893, -17195577, -4489192,
3886 -21876275, -13982627, 32208683, -1198248},
3887 },
3888 {
3889 {-16657702, 2817643, -10286362, 14811298, 6024667, 13349505,
3890 -27315504, -10497842, -27672585, -11539858},
3891 {15941029, -9405932, -21367050, 8062055, 31876073, -238629,
3892 -15278393, -1444429, 15397331, -4130193},
3893 {8934485, -13485467, -23286397, -13423241, -32446090, 14047986,
3894 31170398, -1441021, -27505566, 15087184},
3895 },
3896 {
3897 {-18357243, -2156491, 24524913, -16677868, 15520427, -6360776,
3898 -15502406, 11461896, 16788528, -5868942},
3899 {-1947386, 16013773, 21750665, 3714552, -17401782, -16055433,
3900 -3770287, -10323320, 31322514, -11615635},
3901 {21426655, -5650218, -13648287, -5347537, -28812189, -4920970,
3902 -18275391, -14621414, 13040862, -12112948},
3903 },
3904 {
3905 {11293895, 12478086, -27136401, 15083750, -29307421, 14748872,
3906 14555558, -13417103, 1613711, 4896935},
3907 {-25894883, 15323294, -8489791, -8057900, 25967126, -13425460,
3908 2825960, -4897045, -23971776, -11267415},
3909 {-15924766, -5229880, -17443532, 6410664, 3622847, 10243618,
3910 20615400, 12405433, -23753030, -8436416},
3911 },
3912 {
3913 {-7091295, 12556208, -20191352, 9025187, -17072479, 4333801,
3914 4378436, 2432030, 23097949, -566018},
3915 {4565804, -16025654, 20084412, -7842817, 1724999, 189254, 24767264,
3916 10103221, -18512313, 2424778},
3917 {366633, -11976806, 8173090, -6890119, 30788634, 5745705, -7168678,
3918 1344109, -3642553, 12412659},
3919 },
3920 {
3921 {-24001791, 7690286, 14929416, -168257, -32210835, -13412986,
3922 24162697, -15326504, -3141501, 11179385},
3923 {18289522, -14724954, 8056945, 16430056, -21729724, 7842514,
3924 -6001441, -1486897, -18684645, -11443503},
3925 {476239, 6601091, -6152790, -9723375, 17503545, -4863900, 27672959,
3926 13403813, 11052904, 5219329},
3927 },
3928 },
3929 {
3930 {
3931 {20678546, -8375738, -32671898, 8849123, -5009758, 14574752,
3932 31186971, -3973730, 9014762, -8579056},
3933 {-13644050, -10350239, -15962508, 5075808, -1514661, -11534600,
3934 -33102500, 9160280, 8473550, -3256838},
3935 {24900749, 14435722, 17209120, -15292541, -22592275, 9878983,
3936 -7689309, -16335821, -24568481, 11788948},
3937 },
3938 {
3939 {-3118155, -11395194, -13802089, 14797441, 9652448, -6845904,
3940 -20037437, 10410733, -24568470, -1458691},
3941 {-15659161, 16736706, -22467150, 10215878, -9097177, 7563911,
3942 11871841, -12505194, -18513325, 8464118},
3943 {-23400612, 8348507, -14585951, -861714, -3950205, -6373419,
3944 14325289, 8628612, 33313881, -8370517},
3945 },
3946 {
3947 {-20186973, -4967935, 22367356, 5271547, -1097117, -4788838,
3948 -24805667, -10236854, -8940735, -5818269},
3949 {-6948785, -1795212, -32625683, -16021179, 32635414, -7374245,
3950 15989197, -12838188, 28358192, -4253904},
3951 {-23561781, -2799059, -32351682, -1661963, -9147719, 10429267,
3952 -16637684, 4072016, -5351664, 5596589},
3953 },
3954 {
3955 {-28236598, -3390048, 12312896, 6213178, 3117142, 16078565,
3956 29266239, 2557221, 1768301, 15373193},
3957 {-7243358, -3246960, -4593467, -7553353, -127927, -912245, -1090902,
3958 -4504991, -24660491, 3442910},
3959 {-30210571, 5124043, 14181784, 8197961, 18964734, -11939093,
3960 22597931, 7176455, -18585478, 13365930},
3961 },
3962 {
3963 {-7877390, -1499958, 8324673, 4690079, 6261860, 890446, 24538107,
3964 -8570186, -9689599, -3031667},
3965 {25008904, -10771599, -4305031, -9638010, 16265036, 15721635,
3966 683793, -11823784, 15723479, -15163481},
3967 {-9660625, 12374379, -27006999, -7026148, -7724114, -12314514,
3968 11879682, 5400171, 519526, -1235876},
3969 },
3970 {
3971 {22258397, -16332233, -7869817, 14613016, -22520255, -2950923,
3972 -20353881, 7315967, 16648397, 7605640},
3973 {-8081308, -8464597, -8223311, 9719710, 19259459, -15348212,
3974 23994942, -5281555, -9468848, 4763278},
3975 {-21699244, 9220969, -15730624, 1084137, -25476107, -2852390,
3976 31088447, -7764523, -11356529, 728112},
3977 },
3978 {
3979 {26047220, -11751471, -6900323, -16521798, 24092068, 9158119,
3980 -4273545, -12555558, -29365436, -5498272},
3981 {17510331, -322857, 5854289, 8403524, 17133918, -3112612, -28111007,
3982 12327945, 10750447, 10014012},
3983 {-10312768, 3936952, 9156313, -8897683, 16498692, -994647,
3984 -27481051, -666732, 3424691, 7540221},
3985 },
3986 {
3987 {30322361, -6964110, 11361005, -4143317, 7433304, 4989748, -7071422,
3988 -16317219, -9244265, 15258046},
3989 {13054562, -2779497, 19155474, 469045, -12482797, 4566042, 5631406,
3990 2711395, 1062915, -5136345},
3991 {-19240248, -11254599, -29509029, -7499965, -5835763, 13005411,
3992 -6066489, 12194497, 32960380, 1459310},
3993 },
3994 },
3995 {
3996 {
3997 {19852034, 7027924, 23669353, 10020366, 8586503, -6657907, 394197,
3998 -6101885, 18638003, -11174937},
3999 {31395534, 15098109, 26581030, 8030562, -16527914, -5007134,
4000 9012486, -7584354, -6643087, -5442636},
4001 {-9192165, -2347377, -1997099, 4529534, 25766844, 607986, -13222,
4002 9677543, -32294889, -6456008},
4003 },
4004 {
4005 {-2444496, -149937, 29348902, 8186665, 1873760, 12489863, -30934579,
4006 -7839692, -7852844, -8138429},
4007 {-15236356, -15433509, 7766470, 746860, 26346930, -10221762,
4008 -27333451, 10754588, -9431476, 5203576},
4009 {31834314, 14135496, -770007, 5159118, 20917671, -16768096,
4010 -7467973, -7337524, 31809243, 7347066},
4011 },
4012 {
4013 {-9606723, -11874240, 20414459, 13033986, 13716524, -11691881,
4014 19797970, -12211255, 15192876, -2087490},
4015 {-12663563, -2181719, 1168162, -3804809, 26747877, -14138091,
4016 10609330, 12694420, 33473243, -13382104},
4017 {33184999, 11180355, 15832085, -11385430, -1633671, 225884,
4018 15089336, -11023903, -6135662, 14480053},
4019 },
4020 {
4021 {31308717, -5619998, 31030840, -1897099, 15674547, -6582883,
4022 5496208, 13685227, 27595050, 8737275},
4023 {-20318852, -15150239, 10933843, -16178022, 8335352, -7546022,
4024 -31008351, -12610604, 26498114, 66511},
4025 {22644454, -8761729, -16671776, 4884562, -3105614, -13559366,
4026 30540766, -4286747, -13327787, -7515095},
4027 },
4028 {
4029 {-28017847, 9834845, 18617207, -2681312, -3401956, -13307506,
4030 8205540, 13585437, -17127465, 15115439},
4031 {23711543, -672915, 31206561, -8362711, 6164647, -9709987,
4032 -33535882, -1426096, 8236921, 16492939},
4033 {-23910559, -13515526, -26299483, -4503841, 25005590, -7687270,
4034 19574902, 10071562, 6708380, -6222424},
4035 },
4036 {
4037 {2101391, -4930054, 19702731, 2367575, -15427167, 1047675, 5301017,
4038 9328700, 29955601, -11678310},
4039 {3096359, 9271816, -21620864, -15521844, -14847996, -7592937,
4040 -25892142, -12635595, -9917575, 6216608},
4041 {-32615849, 338663, -25195611, 2510422, -29213566, -13820213,
4042 24822830, -6146567, -26767480, 7525079},
4043 },
4044 {
4045 {-23066649, -13985623, 16133487, -7896178, -3389565, 778788,
4046 -910336, -2782495, -19386633, 11994101},
4047 {21691500, -13624626, -641331, -14367021, 3285881, -3483596,
4048 -25064666, 9718258, -7477437, 13381418},
4049 {18445390, -4202236, 14979846, 11622458, -1727110, -3582980,
4050 23111648, -6375247, 28535282, 15779576},
4051 },
4052 {
4053 {30098053, 3089662, -9234387, 16662135, -21306940, 11308411,
4054 -14068454, 12021730, 9955285, -16303356},
4055 {9734894, -14576830, -7473633, -9138735, 2060392, 11313496,
4056 -18426029, 9924399, 20194861, 13380996},
4057 {-26378102, -7965207, -22167821, 15789297, -18055342, -6168792,
4058 -1984914, 15707771, 26342023, 10146099},
4059 },
4060 },
4061 {
4062 {
4063 {-26016874, -219943, 21339191, -41388, 19745256, -2878700,
4064 -29637280, 2227040, 21612326, -545728},
4065 {-13077387, 1184228, 23562814, -5970442, -20351244, -6348714,
4066 25764461, 12243797, -20856566, 11649658},
4067 {-10031494, 11262626, 27384172, 2271902, 26947504, -15997771, 39944,
4068 6114064, 33514190, 2333242},
4069 },
4070 {
4071 {-21433588, -12421821, 8119782, 7219913, -21830522, -9016134,
4072 -6679750, -12670638, 24350578, -13450001},
4073 {-4116307, -11271533, -23886186, 4843615, -30088339, 690623,
4074 -31536088, -10406836, 8317860, 12352766},
4075 {18200138, -14475911, -33087759, -2696619, -23702521, -9102511,
4076 -23552096, -2287550, 20712163, 6719373},
4077 },
4078 {
4079 {26656208, 6075253, -7858556, 1886072, -28344043, 4262326, 11117530,
4080 -3763210, 26224235, -3297458},
4081 {-17168938, -14854097, -3395676, -16369877, -19954045, 14050420,
4082 21728352, 9493610, 18620611, -16428628},
4083 {-13323321, 13325349, 11432106, 5964811, 18609221, 6062965,
4084 -5269471, -9725556, -30701573, -16479657},
4085 },
4086 {
4087 {-23860538, -11233159, 26961357, 1640861, -32413112, -16737940,
4088 12248509, -5240639, 13735342, 1934062},
4089 {25089769, 6742589, 17081145, -13406266, 21909293, -16067981,
4090 -15136294, -3765346, -21277997, 5473616},
4091 {31883677, -7961101, 1083432, -11572403, 22828471, 13290673,
4092 -7125085, 12469656, 29111212, -5451014},
4093 },
4094 {
4095 {24244947, -15050407, -26262976, 2791540, -14997599, 16666678,
4096 24367466, 6388839, -10295587, 452383},
4097 {-25640782, -3417841, 5217916, 16224624, 19987036, -4082269,
4098 -24236251, -5915248, 15766062, 8407814},
4099 {-20406999, 13990231, 15495425, 16395525, 5377168, 15166495,
4100 -8917023, -4388953, -8067909, 2276718},
4101 },
4102 {
4103 {30157918, 12924066, -17712050, 9245753, 19895028, 3368142,
4104 -23827587, 5096219, 22740376, -7303417},
4105 {2041139, -14256350, 7783687, 13876377, -25946985, -13352459,
4106 24051124, 13742383, -15637599, 13295222},
4107 {33338237, -8505733, 12532113, 7977527, 9106186, -1715251,
4108 -17720195, -4612972, -4451357, -14669444},
4109 },
4110 {
4111 {-20045281, 5454097, -14346548, 6447146, 28862071, 1883651,
4112 -2469266, -4141880, 7770569, 9620597},
4113 {23208068, 7979712, 33071466, 8149229, 1758231, -10834995, 30945528,
4114 -1694323, -33502340, -14767970},
4115 {1439958, -16270480, -1079989, -793782, 4625402, 10647766, -5043801,
4116 1220118, 30494170, -11440799},
4117 },
4118 {
4119 {-5037580, -13028295, -2970559, -3061767, 15640974, -6701666,
4120 -26739026, 926050, -1684339, -13333647},
4121 {13908495, -3549272, 30919928, -6273825, -21521863, 7989039,
4122 9021034, 9078865, 3353509, 4033511},
4123 {-29663431, -15113610, 32259991, -344482, 24295849, -12912123,
4124 23161163, 8839127, 27485041, 7356032},
4125 },
4126 },
4127 {
4128 {
4129 {9661027, 705443, 11980065, -5370154, -1628543, 14661173, -6346142,
4130 2625015, 28431036, -16771834},
4131 {-23839233, -8311415, -25945511, 7480958, -17681669, -8354183,
4132 -22545972, 14150565, 15970762, 4099461},
4133 {29262576, 16756590, 26350592, -8793563, 8529671, -11208050,
4134 13617293, -9937143, 11465739, 8317062},
4135 },
4136 {
4137 {-25493081, -6962928, 32500200, -9419051, -23038724, -2302222,
4138 14898637, 3848455, 20969334, -5157516},
4139 {-20384450, -14347713, -18336405, 13884722, -33039454, 2842114,
4140 -21610826, -3649888, 11177095, 14989547},
4141 {-24496721, -11716016, 16959896, 2278463, 12066309, 10137771,
4142 13515641, 2581286, -28487508, 9930240},
4143 },
4144 {
4145 {-17751622, -2097826, 16544300, -13009300, -15914807, -14949081,
4146 18345767, -13403753, 16291481, -5314038},
4147 {-33229194, 2553288, 32678213, 9875984, 8534129, 6889387, -9676774,
4148 6957617, 4368891, 9788741},
4149 {16660756, 7281060, -10830758, 12911820, 20108584, -8101676,
4150 -21722536, -8613148, 16250552, -11111103},
4151 },
4152 {
4153 {-19765507, 2390526, -16551031, 14161980, 1905286, 6414907, 4689584,
4154 10604807, -30190403, 4782747},
4155 {-1354539, 14736941, -7367442, -13292886, 7710542, -14155590,
4156 -9981571, 4383045, 22546403, 437323},
4157 {31665577, -12180464, -16186830, 1491339, -18368625, 3294682,
4158 27343084, 2786261, -30633590, -14097016},
4159 },
4160 {
4161 {-14467279, -683715, -33374107, 7448552, 19294360, 14334329,
4162 -19690631, 2355319, -19284671, -6114373},
4163 {15121312, -15796162, 6377020, -6031361, -10798111, -12957845,
4164 18952177, 15496498, -29380133, 11754228},
4165 {-2637277, -13483075, 8488727, -14303896, 12728761, -1622493,
4166 7141596, 11724556, 22761615, -10134141},
4167 },
4168 {
4169 {16918416, 11729663, -18083579, 3022987, -31015732, -13339659,
4170 -28741185, -12227393, 32851222, 11717399},
4171 {11166634, 7338049, -6722523, 4531520, -29468672, -7302055,
4172 31474879, 3483633, -1193175, -4030831},
4173 {-185635, 9921305, 31456609, -13536438, -12013818, 13348923,
4174 33142652, 6546660, -19985279, -3948376},
4175 },
4176 {
4177 {-32460596, 11266712, -11197107, -7899103, 31703694, 3855903,
4178 -8537131, -12833048, -30772034, -15486313},
4179 {-18006477, 12709068, 3991746, -6479188, -21491523, -10550425,
4180 -31135347, -16049879, 10928917, 3011958},
4181 {-6957757, -15594337, 31696059, 334240, 29576716, 14796075,
4182 -30831056, -12805180, 18008031, 10258577},
4183 },
4184 {
4185 {-22448644, 15655569, 7018479, -4410003, -30314266, -1201591,
4186 -1853465, 1367120, 25127874, 6671743},
4187 {29701166, -14373934, -10878120, 9279288, -17568, 13127210,
4188 21382910, 11042292, 25838796, 4642684},
4189 {-20430234, 14955537, -24126347, 8124619, -5369288, -5990470,
4190 30468147, -13900640, 18423289, 4177476},
4191 },
4192 },
4193 };
4194
negative(signed char b)4195 static uint8_t negative(signed char b)
4196 {
4197 uint32_t x = b;
4198
4199 x >>= 31; /* 1: yes; 0: no */
4200 return x;
4201 }
4202
table_select(ge_precomp * t,int pos,signed char b)4203 static void table_select(ge_precomp *t, int pos, signed char b)
4204 {
4205 ge_precomp minust;
4206 uint8_t bnegative = negative(b);
4207 uint8_t babs = b - ((uint8_t)((-bnegative) & b) << 1);
4208
4209 ge_precomp_0(t);
4210 cmov(t, &k25519Precomp[pos][0], equal(babs, 1));
4211 cmov(t, &k25519Precomp[pos][1], equal(babs, 2));
4212 cmov(t, &k25519Precomp[pos][2], equal(babs, 3));
4213 cmov(t, &k25519Precomp[pos][3], equal(babs, 4));
4214 cmov(t, &k25519Precomp[pos][4], equal(babs, 5));
4215 cmov(t, &k25519Precomp[pos][5], equal(babs, 6));
4216 cmov(t, &k25519Precomp[pos][6], equal(babs, 7));
4217 cmov(t, &k25519Precomp[pos][7], equal(babs, 8));
4218 fe_copy(minust.yplusx, t->yminusx);
4219 fe_copy(minust.yminusx, t->yplusx);
4220 fe_neg(minust.xy2d, t->xy2d);
4221 cmov(t, &minust, bnegative);
4222 }
4223
4224 /*
4225 * h = a * B
4226 *
4227 * where a = a[0]+256*a[1]+...+256^31 a[31]
4228 * B is the Ed25519 base point (x,4/5) with x positive.
4229 *
4230 * Preconditions:
4231 * a[31] <= 127
4232 */
ge_scalarmult_base(ge_p3 * h,const uint8_t * a)4233 static void ge_scalarmult_base(ge_p3 *h, const uint8_t *a)
4234 {
4235 signed char e[64];
4236 signed char carry;
4237 ge_p1p1 r;
4238 ge_p2 s;
4239 ge_precomp t;
4240 int i;
4241
4242 for (i = 0; i < 32; ++i) {
4243 e[2 * i + 0] = (a[i] >> 0) & 15;
4244 e[2 * i + 1] = (a[i] >> 4) & 15;
4245 }
4246 /* each e[i] is between 0 and 15 */
4247 /* e[63] is between 0 and 7 */
4248
4249 carry = 0;
4250 for (i = 0; i < 63; ++i) {
4251 e[i] += carry;
4252 carry = e[i] + 8;
4253 carry >>= 4;
4254 e[i] -= carry << 4;
4255 }
4256 e[63] += carry;
4257 /* each e[i] is between -8 and 8 */
4258
4259 ge_p3_0(h);
4260 for (i = 1; i < 64; i += 2) {
4261 table_select(&t, i / 2, e[i]);
4262 ge_madd(&r, h, &t);
4263 ge_p1p1_to_p3(h, &r);
4264 }
4265
4266 ge_p3_dbl(&r, h);
4267 ge_p1p1_to_p2(&s, &r);
4268 ge_p2_dbl(&r, &s);
4269 ge_p1p1_to_p2(&s, &r);
4270 ge_p2_dbl(&r, &s);
4271 ge_p1p1_to_p2(&s, &r);
4272 ge_p2_dbl(&r, &s);
4273 ge_p1p1_to_p3(h, &r);
4274
4275 for (i = 0; i < 64; i += 2) {
4276 table_select(&t, i / 2, e[i]);
4277 ge_madd(&r, h, &t);
4278 ge_p1p1_to_p3(h, &r);
4279 }
4280
4281 OPENSSL_cleanse(e, sizeof(e));
4282 }
4283
4284 #if !defined(BASE_2_51_IMPLEMENTED)
4285 /*
4286 * Replace (f,g) with (g,f) if b == 1;
4287 * replace (f,g) with (f,g) if b == 0.
4288 *
4289 * Preconditions: b in {0,1}.
4290 */
fe_cswap(fe f,fe g,unsigned int b)4291 static void fe_cswap(fe f, fe g, unsigned int b)
4292 {
4293 size_t i;
4294
4295 b = 0-b;
4296 for (i = 0; i < 10; i++) {
4297 int32_t x = f[i] ^ g[i];
4298 x &= b;
4299 f[i] ^= x;
4300 g[i] ^= x;
4301 }
4302 }
4303
4304 /*
4305 * h = f * 121666
4306 *
4307 * Can overlap h with f.
4308 *
4309 * Preconditions:
4310 * |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
4311 *
4312 * Postconditions:
4313 * |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
4314 */
fe_mul121666(fe h,fe f)4315 static void fe_mul121666(fe h, fe f)
4316 {
4317 int32_t f0 = f[0];
4318 int32_t f1 = f[1];
4319 int32_t f2 = f[2];
4320 int32_t f3 = f[3];
4321 int32_t f4 = f[4];
4322 int32_t f5 = f[5];
4323 int32_t f6 = f[6];
4324 int32_t f7 = f[7];
4325 int32_t f8 = f[8];
4326 int32_t f9 = f[9];
4327 int64_t h0 = f0 * (int64_t) 121666;
4328 int64_t h1 = f1 * (int64_t) 121666;
4329 int64_t h2 = f2 * (int64_t) 121666;
4330 int64_t h3 = f3 * (int64_t) 121666;
4331 int64_t h4 = f4 * (int64_t) 121666;
4332 int64_t h5 = f5 * (int64_t) 121666;
4333 int64_t h6 = f6 * (int64_t) 121666;
4334 int64_t h7 = f7 * (int64_t) 121666;
4335 int64_t h8 = f8 * (int64_t) 121666;
4336 int64_t h9 = f9 * (int64_t) 121666;
4337 int64_t carry0;
4338 int64_t carry1;
4339 int64_t carry2;
4340 int64_t carry3;
4341 int64_t carry4;
4342 int64_t carry5;
4343 int64_t carry6;
4344 int64_t carry7;
4345 int64_t carry8;
4346 int64_t carry9;
4347
4348 carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
4349 carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
4350 carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
4351 carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
4352 carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;
4353
4354 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
4355 carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
4356 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
4357 carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
4358 carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;
4359
4360 h[0] = (int32_t)h0;
4361 h[1] = (int32_t)h1;
4362 h[2] = (int32_t)h2;
4363 h[3] = (int32_t)h3;
4364 h[4] = (int32_t)h4;
4365 h[5] = (int32_t)h5;
4366 h[6] = (int32_t)h6;
4367 h[7] = (int32_t)h7;
4368 h[8] = (int32_t)h8;
4369 h[9] = (int32_t)h9;
4370 }
4371
x25519_scalar_mult_generic(uint8_t out[32],const uint8_t scalar[32],const uint8_t point[32])4372 static void x25519_scalar_mult_generic(uint8_t out[32],
4373 const uint8_t scalar[32],
4374 const uint8_t point[32]) {
4375 fe x1, x2, z2, x3, z3, tmp0, tmp1;
4376 uint8_t e[32];
4377 unsigned swap = 0;
4378 int pos;
4379
4380 memcpy(e, scalar, 32);
4381 e[0] &= 248;
4382 e[31] &= 127;
4383 e[31] |= 64;
4384 fe_frombytes(x1, point);
4385 fe_1(x2);
4386 fe_0(z2);
4387 fe_copy(x3, x1);
4388 fe_1(z3);
4389
4390 for (pos = 254; pos >= 0; --pos) {
4391 unsigned b = 1 & (e[pos / 8] >> (pos & 7));
4392 swap ^= b;
4393 fe_cswap(x2, x3, swap);
4394 fe_cswap(z2, z3, swap);
4395 swap = b;
4396 fe_sub(tmp0, x3, z3);
4397 fe_sub(tmp1, x2, z2);
4398 fe_add(x2, x2, z2);
4399 fe_add(z2, x3, z3);
4400 fe_mul(z3, tmp0, x2);
4401 fe_mul(z2, z2, tmp1);
4402 fe_sq(tmp0, tmp1);
4403 fe_sq(tmp1, x2);
4404 fe_add(x3, z3, z2);
4405 fe_sub(z2, z3, z2);
4406 fe_mul(x2, tmp1, tmp0);
4407 fe_sub(tmp1, tmp1, tmp0);
4408 fe_sq(z2, z2);
4409 fe_mul121666(z3, tmp1);
4410 fe_sq(x3, x3);
4411 fe_add(tmp0, tmp0, z3);
4412 fe_mul(z3, x1, z2);
4413 fe_mul(z2, tmp1, tmp0);
4414 }
4415
4416 fe_invert(z2, z2);
4417 fe_mul(x2, x2, z2);
4418 fe_tobytes(out, x2);
4419
4420 OPENSSL_cleanse(e, sizeof(e));
4421 }
4422
x25519_scalar_mult(uint8_t out[32],const uint8_t scalar[32],const uint8_t point[32])4423 static void x25519_scalar_mult(uint8_t out[32], const uint8_t scalar[32],
4424 const uint8_t point[32]) {
4425 x25519_scalar_mult_generic(out, scalar, point);
4426 }
4427 #endif
4428
slide(signed char * r,const uint8_t * a)4429 static void slide(signed char *r, const uint8_t *a)
4430 {
4431 int i;
4432 int b;
4433 int k;
4434
4435 for (i = 0; i < 256; ++i) {
4436 r[i] = 1 & (a[i >> 3] >> (i & 7));
4437 }
4438
4439 for (i = 0; i < 256; ++i) {
4440 if (r[i]) {
4441 for (b = 1; b <= 6 && i + b < 256; ++b) {
4442 if (r[i + b]) {
4443 if (r[i] + (r[i + b] << b) <= 15) {
4444 r[i] += r[i + b] << b;
4445 r[i + b] = 0;
4446 } else if (r[i] - (r[i + b] << b) >= -15) {
4447 r[i] -= r[i + b] << b;
4448 for (k = i + b; k < 256; ++k) {
4449 if (!r[k]) {
4450 r[k] = 1;
4451 break;
4452 }
4453 r[k] = 0;
4454 }
4455 } else {
4456 break;
4457 }
4458 }
4459 }
4460 }
4461 }
4462 }
4463
4464 static const ge_precomp Bi[8] = {
4465 {
4466 {25967493, -14356035, 29566456, 3660896, -12694345, 4014787, 27544626,
4467 -11754271, -6079156, 2047605},
4468 {-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692,
4469 5043384, 19500929, -15469378},
4470 {-8738181, 4489570, 9688441, -14785194, 10184609, -12363380, 29287919,
4471 11864899, -24514362, -4438546},
4472 },
4473 {
4474 {15636291, -9688557, 24204773, -7912398, 616977, -16685262, 27787600,
4475 -14772189, 28944400, -1550024},
4476 {16568933, 4717097, -11556148, -1102322, 15682896, -11807043, 16354577,
4477 -11775962, 7689662, 11199574},
4478 {30464156, -5976125, -11779434, -15670865, 23220365, 15915852, 7512774,
4479 10017326, -17749093, -9920357},
4480 },
4481 {
4482 {10861363, 11473154, 27284546, 1981175, -30064349, 12577861, 32867885,
4483 14515107, -15438304, 10819380},
4484 {4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668,
4485 12483688, -12668491, 5581306},
4486 {19563160, 16186464, -29386857, 4097519, 10237984, -4348115, 28542350,
4487 13850243, -23678021, -15815942},
4488 },
4489 {
4490 {5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852,
4491 5230134, -23952439, -15175766},
4492 {-30269007, -3463509, 7665486, 10083793, 28475525, 1649722, 20654025,
4493 16520125, 30598449, 7715701},
4494 {28881845, 14381568, 9657904, 3680757, -20181635, 7843316, -31400660,
4495 1370708, 29794553, -1409300},
4496 },
4497 {
4498 {-22518993, -6692182, 14201702, -8745502, -23510406, 8844726, 18474211,
4499 -1361450, -13062696, 13821877},
4500 {-6455177, -7839871, 3374702, -4740862, -27098617, -10571707, 31655028,
4501 -7212327, 18853322, -14220951},
4502 {4566830, -12963868, -28974889, -12240689, -7602672, -2830569, -8514358,
4503 -10431137, 2207753, -3209784},
4504 },
4505 {
4506 {-25154831, -4185821, 29681144, 7868801, -6854661, -9423865, -12437364,
4507 -663000, -31111463, -16132436},
4508 {25576264, -2703214, 7349804, -11814844, 16472782, 9300885, 3844789,
4509 15725684, 171356, 6466918},
4510 {23103977, 13316479, 9739013, -16149481, 817875, -15038942, 8965339,
4511 -14088058, -30714912, 16193877},
4512 },
4513 {
4514 {-33521811, 3180713, -2394130, 14003687, -16903474, -16270840, 17238398,
4515 4729455, -18074513, 9256800},
4516 {-25182317, -4174131, 32336398, 5036987, -21236817, 11360617, 22616405,
4517 9761698, -19827198, 630305},
4518 {-13720693, 2639453, -24237460, -7406481, 9494427, -5774029, -6554551,
4519 -15960994, -2449256, -14291300},
4520 },
4521 {
4522 {-3151181, -5046075, 9282714, 6866145, -31907062, -863023, -18940575,
4523 15033784, 25105118, -7894876},
4524 {-24326370, 15950226, -31801215, -14592823, -11662737, -5090925,
4525 1573892, -2625887, 2198790, -15804619},
4526 {-3099351, 10324967, -2241613, 7453183, -5446979, -2735503, -13812022,
4527 -16236442, -32461234, -12290683},
4528 },
4529 };
4530
4531 /*
4532 * r = a * A + b * B
4533 *
4534 * where a = a[0]+256*a[1]+...+256^31 a[31].
4535 * and b = b[0]+256*b[1]+...+256^31 b[31].
4536 * B is the Ed25519 base point (x,4/5) with x positive.
4537 */
ge_double_scalarmult_vartime(ge_p2 * r,const uint8_t * a,const ge_p3 * A,const uint8_t * b)4538 static void ge_double_scalarmult_vartime(ge_p2 *r, const uint8_t *a,
4539 const ge_p3 *A, const uint8_t *b)
4540 {
4541 signed char aslide[256];
4542 signed char bslide[256];
4543 ge_cached Ai[8]; /* A,3A,5A,7A,9A,11A,13A,15A */
4544 ge_p1p1 t;
4545 ge_p3 u;
4546 ge_p3 A2;
4547 int i;
4548
4549 slide(aslide, a);
4550 slide(bslide, b);
4551
4552 ge_p3_to_cached(&Ai[0], A);
4553 ge_p3_dbl(&t, A);
4554 ge_p1p1_to_p3(&A2, &t);
4555 ge_add(&t, &A2, &Ai[0]);
4556 ge_p1p1_to_p3(&u, &t);
4557 ge_p3_to_cached(&Ai[1], &u);
4558 ge_add(&t, &A2, &Ai[1]);
4559 ge_p1p1_to_p3(&u, &t);
4560 ge_p3_to_cached(&Ai[2], &u);
4561 ge_add(&t, &A2, &Ai[2]);
4562 ge_p1p1_to_p3(&u, &t);
4563 ge_p3_to_cached(&Ai[3], &u);
4564 ge_add(&t, &A2, &Ai[3]);
4565 ge_p1p1_to_p3(&u, &t);
4566 ge_p3_to_cached(&Ai[4], &u);
4567 ge_add(&t, &A2, &Ai[4]);
4568 ge_p1p1_to_p3(&u, &t);
4569 ge_p3_to_cached(&Ai[5], &u);
4570 ge_add(&t, &A2, &Ai[5]);
4571 ge_p1p1_to_p3(&u, &t);
4572 ge_p3_to_cached(&Ai[6], &u);
4573 ge_add(&t, &A2, &Ai[6]);
4574 ge_p1p1_to_p3(&u, &t);
4575 ge_p3_to_cached(&Ai[7], &u);
4576
4577 ge_p2_0(r);
4578
4579 for (i = 255; i >= 0; --i) {
4580 if (aslide[i] || bslide[i]) {
4581 break;
4582 }
4583 }
4584
4585 for (; i >= 0; --i) {
4586 ge_p2_dbl(&t, r);
4587
4588 if (aslide[i] > 0) {
4589 ge_p1p1_to_p3(&u, &t);
4590 ge_add(&t, &u, &Ai[aslide[i] / 2]);
4591 } else if (aslide[i] < 0) {
4592 ge_p1p1_to_p3(&u, &t);
4593 ge_sub(&t, &u, &Ai[(-aslide[i]) / 2]);
4594 }
4595
4596 if (bslide[i] > 0) {
4597 ge_p1p1_to_p3(&u, &t);
4598 ge_madd(&t, &u, &Bi[bslide[i] / 2]);
4599 } else if (bslide[i] < 0) {
4600 ge_p1p1_to_p3(&u, &t);
4601 ge_msub(&t, &u, &Bi[(-bslide[i]) / 2]);
4602 }
4603
4604 ge_p1p1_to_p2(r, &t);
4605 }
4606 }
4607
4608 /*
4609 * The set of scalars is \Z/l
4610 * where l = 2^252 + 27742317777372353535851937790883648493.
4611 *
4612 * Input:
4613 * s[0]+256*s[1]+...+256^63*s[63] = s
4614 *
4615 * Output:
4616 * s[0]+256*s[1]+...+256^31*s[31] = s mod l
4617 * where l = 2^252 + 27742317777372353535851937790883648493.
4618 * Overwrites s in place.
4619 */
x25519_sc_reduce(uint8_t * s)4620 static void x25519_sc_reduce(uint8_t *s)
4621 {
4622 int64_t s0 = kBottom21Bits & load_3(s);
4623 int64_t s1 = kBottom21Bits & (load_4(s + 2) >> 5);
4624 int64_t s2 = kBottom21Bits & (load_3(s + 5) >> 2);
4625 int64_t s3 = kBottom21Bits & (load_4(s + 7) >> 7);
4626 int64_t s4 = kBottom21Bits & (load_4(s + 10) >> 4);
4627 int64_t s5 = kBottom21Bits & (load_3(s + 13) >> 1);
4628 int64_t s6 = kBottom21Bits & (load_4(s + 15) >> 6);
4629 int64_t s7 = kBottom21Bits & (load_3(s + 18) >> 3);
4630 int64_t s8 = kBottom21Bits & load_3(s + 21);
4631 int64_t s9 = kBottom21Bits & (load_4(s + 23) >> 5);
4632 int64_t s10 = kBottom21Bits & (load_3(s + 26) >> 2);
4633 int64_t s11 = kBottom21Bits & (load_4(s + 28) >> 7);
4634 int64_t s12 = kBottom21Bits & (load_4(s + 31) >> 4);
4635 int64_t s13 = kBottom21Bits & (load_3(s + 34) >> 1);
4636 int64_t s14 = kBottom21Bits & (load_4(s + 36) >> 6);
4637 int64_t s15 = kBottom21Bits & (load_3(s + 39) >> 3);
4638 int64_t s16 = kBottom21Bits & load_3(s + 42);
4639 int64_t s17 = kBottom21Bits & (load_4(s + 44) >> 5);
4640 int64_t s18 = kBottom21Bits & (load_3(s + 47) >> 2);
4641 int64_t s19 = kBottom21Bits & (load_4(s + 49) >> 7);
4642 int64_t s20 = kBottom21Bits & (load_4(s + 52) >> 4);
4643 int64_t s21 = kBottom21Bits & (load_3(s + 55) >> 1);
4644 int64_t s22 = kBottom21Bits & (load_4(s + 57) >> 6);
4645 int64_t s23 = (load_4(s + 60) >> 3);
4646 int64_t carry0;
4647 int64_t carry1;
4648 int64_t carry2;
4649 int64_t carry3;
4650 int64_t carry4;
4651 int64_t carry5;
4652 int64_t carry6;
4653 int64_t carry7;
4654 int64_t carry8;
4655 int64_t carry9;
4656 int64_t carry10;
4657 int64_t carry11;
4658 int64_t carry12;
4659 int64_t carry13;
4660 int64_t carry14;
4661 int64_t carry15;
4662 int64_t carry16;
4663
4664 s11 += s23 * 666643;
4665 s12 += s23 * 470296;
4666 s13 += s23 * 654183;
4667 s14 -= s23 * 997805;
4668 s15 += s23 * 136657;
4669 s16 -= s23 * 683901;
4670 s23 = 0;
4671
4672 s10 += s22 * 666643;
4673 s11 += s22 * 470296;
4674 s12 += s22 * 654183;
4675 s13 -= s22 * 997805;
4676 s14 += s22 * 136657;
4677 s15 -= s22 * 683901;
4678 s22 = 0;
4679
4680 s9 += s21 * 666643;
4681 s10 += s21 * 470296;
4682 s11 += s21 * 654183;
4683 s12 -= s21 * 997805;
4684 s13 += s21 * 136657;
4685 s14 -= s21 * 683901;
4686 s21 = 0;
4687
4688 s8 += s20 * 666643;
4689 s9 += s20 * 470296;
4690 s10 += s20 * 654183;
4691 s11 -= s20 * 997805;
4692 s12 += s20 * 136657;
4693 s13 -= s20 * 683901;
4694 s20 = 0;
4695
4696 s7 += s19 * 666643;
4697 s8 += s19 * 470296;
4698 s9 += s19 * 654183;
4699 s10 -= s19 * 997805;
4700 s11 += s19 * 136657;
4701 s12 -= s19 * 683901;
4702 s19 = 0;
4703
4704 s6 += s18 * 666643;
4705 s7 += s18 * 470296;
4706 s8 += s18 * 654183;
4707 s9 -= s18 * 997805;
4708 s10 += s18 * 136657;
4709 s11 -= s18 * 683901;
4710 s18 = 0;
4711
4712 carry6 = (s6 + (1 << 20)) >> 21;
4713 s7 += carry6;
4714 s6 -= carry6 * (1 << 21);
4715 carry8 = (s8 + (1 << 20)) >> 21;
4716 s9 += carry8;
4717 s8 -= carry8 * (1 << 21);
4718 carry10 = (s10 + (1 << 20)) >> 21;
4719 s11 += carry10;
4720 s10 -= carry10 * (1 << 21);
4721 carry12 = (s12 + (1 << 20)) >> 21;
4722 s13 += carry12;
4723 s12 -= carry12 * (1 << 21);
4724 carry14 = (s14 + (1 << 20)) >> 21;
4725 s15 += carry14;
4726 s14 -= carry14 * (1 << 21);
4727 carry16 = (s16 + (1 << 20)) >> 21;
4728 s17 += carry16;
4729 s16 -= carry16 * (1 << 21);
4730
4731 carry7 = (s7 + (1 << 20)) >> 21;
4732 s8 += carry7;
4733 s7 -= carry7 * (1 << 21);
4734 carry9 = (s9 + (1 << 20)) >> 21;
4735 s10 += carry9;
4736 s9 -= carry9 * (1 << 21);
4737 carry11 = (s11 + (1 << 20)) >> 21;
4738 s12 += carry11;
4739 s11 -= carry11 * (1 << 21);
4740 carry13 = (s13 + (1 << 20)) >> 21;
4741 s14 += carry13;
4742 s13 -= carry13 * (1 << 21);
4743 carry15 = (s15 + (1 << 20)) >> 21;
4744 s16 += carry15;
4745 s15 -= carry15 * (1 << 21);
4746
4747 s5 += s17 * 666643;
4748 s6 += s17 * 470296;
4749 s7 += s17 * 654183;
4750 s8 -= s17 * 997805;
4751 s9 += s17 * 136657;
4752 s10 -= s17 * 683901;
4753 s17 = 0;
4754
4755 s4 += s16 * 666643;
4756 s5 += s16 * 470296;
4757 s6 += s16 * 654183;
4758 s7 -= s16 * 997805;
4759 s8 += s16 * 136657;
4760 s9 -= s16 * 683901;
4761 s16 = 0;
4762
4763 s3 += s15 * 666643;
4764 s4 += s15 * 470296;
4765 s5 += s15 * 654183;
4766 s6 -= s15 * 997805;
4767 s7 += s15 * 136657;
4768 s8 -= s15 * 683901;
4769 s15 = 0;
4770
4771 s2 += s14 * 666643;
4772 s3 += s14 * 470296;
4773 s4 += s14 * 654183;
4774 s5 -= s14 * 997805;
4775 s6 += s14 * 136657;
4776 s7 -= s14 * 683901;
4777 s14 = 0;
4778
4779 s1 += s13 * 666643;
4780 s2 += s13 * 470296;
4781 s3 += s13 * 654183;
4782 s4 -= s13 * 997805;
4783 s5 += s13 * 136657;
4784 s6 -= s13 * 683901;
4785 s13 = 0;
4786
4787 s0 += s12 * 666643;
4788 s1 += s12 * 470296;
4789 s2 += s12 * 654183;
4790 s3 -= s12 * 997805;
4791 s4 += s12 * 136657;
4792 s5 -= s12 * 683901;
4793 s12 = 0;
4794
4795 carry0 = (s0 + (1 << 20)) >> 21;
4796 s1 += carry0;
4797 s0 -= carry0 * (1 << 21);
4798 carry2 = (s2 + (1 << 20)) >> 21;
4799 s3 += carry2;
4800 s2 -= carry2 * (1 << 21);
4801 carry4 = (s4 + (1 << 20)) >> 21;
4802 s5 += carry4;
4803 s4 -= carry4 * (1 << 21);
4804 carry6 = (s6 + (1 << 20)) >> 21;
4805 s7 += carry6;
4806 s6 -= carry6 * (1 << 21);
4807 carry8 = (s8 + (1 << 20)) >> 21;
4808 s9 += carry8;
4809 s8 -= carry8 * (1 << 21);
4810 carry10 = (s10 + (1 << 20)) >> 21;
4811 s11 += carry10;
4812 s10 -= carry10 * (1 << 21);
4813
4814 carry1 = (s1 + (1 << 20)) >> 21;
4815 s2 += carry1;
4816 s1 -= carry1 * (1 << 21);
4817 carry3 = (s3 + (1 << 20)) >> 21;
4818 s4 += carry3;
4819 s3 -= carry3 * (1 << 21);
4820 carry5 = (s5 + (1 << 20)) >> 21;
4821 s6 += carry5;
4822 s5 -= carry5 * (1 << 21);
4823 carry7 = (s7 + (1 << 20)) >> 21;
4824 s8 += carry7;
4825 s7 -= carry7 * (1 << 21);
4826 carry9 = (s9 + (1 << 20)) >> 21;
4827 s10 += carry9;
4828 s9 -= carry9 * (1 << 21);
4829 carry11 = (s11 + (1 << 20)) >> 21;
4830 s12 += carry11;
4831 s11 -= carry11 * (1 << 21);
4832
4833 s0 += s12 * 666643;
4834 s1 += s12 * 470296;
4835 s2 += s12 * 654183;
4836 s3 -= s12 * 997805;
4837 s4 += s12 * 136657;
4838 s5 -= s12 * 683901;
4839 s12 = 0;
4840
4841 carry0 = s0 >> 21;
4842 s1 += carry0;
4843 s0 -= carry0 * (1 << 21);
4844 carry1 = s1 >> 21;
4845 s2 += carry1;
4846 s1 -= carry1 * (1 << 21);
4847 carry2 = s2 >> 21;
4848 s3 += carry2;
4849 s2 -= carry2 * (1 << 21);
4850 carry3 = s3 >> 21;
4851 s4 += carry3;
4852 s3 -= carry3 * (1 << 21);
4853 carry4 = s4 >> 21;
4854 s5 += carry4;
4855 s4 -= carry4 * (1 << 21);
4856 carry5 = s5 >> 21;
4857 s6 += carry5;
4858 s5 -= carry5 * (1 << 21);
4859 carry6 = s6 >> 21;
4860 s7 += carry6;
4861 s6 -= carry6 * (1 << 21);
4862 carry7 = s7 >> 21;
4863 s8 += carry7;
4864 s7 -= carry7 * (1 << 21);
4865 carry8 = s8 >> 21;
4866 s9 += carry8;
4867 s8 -= carry8 * (1 << 21);
4868 carry9 = s9 >> 21;
4869 s10 += carry9;
4870 s9 -= carry9 * (1 << 21);
4871 carry10 = s10 >> 21;
4872 s11 += carry10;
4873 s10 -= carry10 * (1 << 21);
4874 carry11 = s11 >> 21;
4875 s12 += carry11;
4876 s11 -= carry11 * (1 << 21);
4877
4878 s0 += s12 * 666643;
4879 s1 += s12 * 470296;
4880 s2 += s12 * 654183;
4881 s3 -= s12 * 997805;
4882 s4 += s12 * 136657;
4883 s5 -= s12 * 683901;
4884 s12 = 0;
4885
4886 carry0 = s0 >> 21;
4887 s1 += carry0;
4888 s0 -= carry0 * (1 << 21);
4889 carry1 = s1 >> 21;
4890 s2 += carry1;
4891 s1 -= carry1 * (1 << 21);
4892 carry2 = s2 >> 21;
4893 s3 += carry2;
4894 s2 -= carry2 * (1 << 21);
4895 carry3 = s3 >> 21;
4896 s4 += carry3;
4897 s3 -= carry3 * (1 << 21);
4898 carry4 = s4 >> 21;
4899 s5 += carry4;
4900 s4 -= carry4 * (1 << 21);
4901 carry5 = s5 >> 21;
4902 s6 += carry5;
4903 s5 -= carry5 * (1 << 21);
4904 carry6 = s6 >> 21;
4905 s7 += carry6;
4906 s6 -= carry6 * (1 << 21);
4907 carry7 = s7 >> 21;
4908 s8 += carry7;
4909 s7 -= carry7 * (1 << 21);
4910 carry8 = s8 >> 21;
4911 s9 += carry8;
4912 s8 -= carry8 * (1 << 21);
4913 carry9 = s9 >> 21;
4914 s10 += carry9;
4915 s9 -= carry9 * (1 << 21);
4916 carry10 = s10 >> 21;
4917 s11 += carry10;
4918 s10 -= carry10 * (1 << 21);
4919
4920 s[ 0] = (uint8_t) (s0 >> 0);
4921 s[ 1] = (uint8_t) (s0 >> 8);
4922 s[ 2] = (uint8_t)((s0 >> 16) | (s1 << 5));
4923 s[ 3] = (uint8_t) (s1 >> 3);
4924 s[ 4] = (uint8_t) (s1 >> 11);
4925 s[ 5] = (uint8_t)((s1 >> 19) | (s2 << 2));
4926 s[ 6] = (uint8_t) (s2 >> 6);
4927 s[ 7] = (uint8_t)((s2 >> 14) | (s3 << 7));
4928 s[ 8] = (uint8_t) (s3 >> 1);
4929 s[ 9] = (uint8_t) (s3 >> 9);
4930 s[10] = (uint8_t)((s3 >> 17) | (s4 << 4));
4931 s[11] = (uint8_t) (s4 >> 4);
4932 s[12] = (uint8_t) (s4 >> 12);
4933 s[13] = (uint8_t)((s4 >> 20) | (s5 << 1));
4934 s[14] = (uint8_t) (s5 >> 7);
4935 s[15] = (uint8_t)((s5 >> 15) | (s6 << 6));
4936 s[16] = (uint8_t) (s6 >> 2);
4937 s[17] = (uint8_t) (s6 >> 10);
4938 s[18] = (uint8_t)((s6 >> 18) | (s7 << 3));
4939 s[19] = (uint8_t) (s7 >> 5);
4940 s[20] = (uint8_t) (s7 >> 13);
4941 s[21] = (uint8_t) (s8 >> 0);
4942 s[22] = (uint8_t) (s8 >> 8);
4943 s[23] = (uint8_t)((s8 >> 16) | (s9 << 5));
4944 s[24] = (uint8_t) (s9 >> 3);
4945 s[25] = (uint8_t) (s9 >> 11);
4946 s[26] = (uint8_t)((s9 >> 19) | (s10 << 2));
4947 s[27] = (uint8_t) (s10 >> 6);
4948 s[28] = (uint8_t)((s10 >> 14) | (s11 << 7));
4949 s[29] = (uint8_t) (s11 >> 1);
4950 s[30] = (uint8_t) (s11 >> 9);
4951 s[31] = (uint8_t) (s11 >> 17);
4952 }
4953
4954 /*
4955 * Input:
4956 * a[0]+256*a[1]+...+256^31*a[31] = a
4957 * b[0]+256*b[1]+...+256^31*b[31] = b
4958 * c[0]+256*c[1]+...+256^31*c[31] = c
4959 *
4960 * Output:
4961 * s[0]+256*s[1]+...+256^31*s[31] = (ab+c) mod l
4962 * where l = 2^252 + 27742317777372353535851937790883648493.
4963 */
sc_muladd(uint8_t * s,const uint8_t * a,const uint8_t * b,const uint8_t * c)4964 static void sc_muladd(uint8_t *s, const uint8_t *a, const uint8_t *b,
4965 const uint8_t *c)
4966 {
4967 int64_t a0 = kBottom21Bits & load_3(a);
4968 int64_t a1 = kBottom21Bits & (load_4(a + 2) >> 5);
4969 int64_t a2 = kBottom21Bits & (load_3(a + 5) >> 2);
4970 int64_t a3 = kBottom21Bits & (load_4(a + 7) >> 7);
4971 int64_t a4 = kBottom21Bits & (load_4(a + 10) >> 4);
4972 int64_t a5 = kBottom21Bits & (load_3(a + 13) >> 1);
4973 int64_t a6 = kBottom21Bits & (load_4(a + 15) >> 6);
4974 int64_t a7 = kBottom21Bits & (load_3(a + 18) >> 3);
4975 int64_t a8 = kBottom21Bits & load_3(a + 21);
4976 int64_t a9 = kBottom21Bits & (load_4(a + 23) >> 5);
4977 int64_t a10 = kBottom21Bits & (load_3(a + 26) >> 2);
4978 int64_t a11 = (load_4(a + 28) >> 7);
4979 int64_t b0 = kBottom21Bits & load_3(b);
4980 int64_t b1 = kBottom21Bits & (load_4(b + 2) >> 5);
4981 int64_t b2 = kBottom21Bits & (load_3(b + 5) >> 2);
4982 int64_t b3 = kBottom21Bits & (load_4(b + 7) >> 7);
4983 int64_t b4 = kBottom21Bits & (load_4(b + 10) >> 4);
4984 int64_t b5 = kBottom21Bits & (load_3(b + 13) >> 1);
4985 int64_t b6 = kBottom21Bits & (load_4(b + 15) >> 6);
4986 int64_t b7 = kBottom21Bits & (load_3(b + 18) >> 3);
4987 int64_t b8 = kBottom21Bits & load_3(b + 21);
4988 int64_t b9 = kBottom21Bits & (load_4(b + 23) >> 5);
4989 int64_t b10 = kBottom21Bits & (load_3(b + 26) >> 2);
4990 int64_t b11 = (load_4(b + 28) >> 7);
4991 int64_t c0 = kBottom21Bits & load_3(c);
4992 int64_t c1 = kBottom21Bits & (load_4(c + 2) >> 5);
4993 int64_t c2 = kBottom21Bits & (load_3(c + 5) >> 2);
4994 int64_t c3 = kBottom21Bits & (load_4(c + 7) >> 7);
4995 int64_t c4 = kBottom21Bits & (load_4(c + 10) >> 4);
4996 int64_t c5 = kBottom21Bits & (load_3(c + 13) >> 1);
4997 int64_t c6 = kBottom21Bits & (load_4(c + 15) >> 6);
4998 int64_t c7 = kBottom21Bits & (load_3(c + 18) >> 3);
4999 int64_t c8 = kBottom21Bits & load_3(c + 21);
5000 int64_t c9 = kBottom21Bits & (load_4(c + 23) >> 5);
5001 int64_t c10 = kBottom21Bits & (load_3(c + 26) >> 2);
5002 int64_t c11 = (load_4(c + 28) >> 7);
5003 int64_t s0;
5004 int64_t s1;
5005 int64_t s2;
5006 int64_t s3;
5007 int64_t s4;
5008 int64_t s5;
5009 int64_t s6;
5010 int64_t s7;
5011 int64_t s8;
5012 int64_t s9;
5013 int64_t s10;
5014 int64_t s11;
5015 int64_t s12;
5016 int64_t s13;
5017 int64_t s14;
5018 int64_t s15;
5019 int64_t s16;
5020 int64_t s17;
5021 int64_t s18;
5022 int64_t s19;
5023 int64_t s20;
5024 int64_t s21;
5025 int64_t s22;
5026 int64_t s23;
5027 int64_t carry0;
5028 int64_t carry1;
5029 int64_t carry2;
5030 int64_t carry3;
5031 int64_t carry4;
5032 int64_t carry5;
5033 int64_t carry6;
5034 int64_t carry7;
5035 int64_t carry8;
5036 int64_t carry9;
5037 int64_t carry10;
5038 int64_t carry11;
5039 int64_t carry12;
5040 int64_t carry13;
5041 int64_t carry14;
5042 int64_t carry15;
5043 int64_t carry16;
5044 int64_t carry17;
5045 int64_t carry18;
5046 int64_t carry19;
5047 int64_t carry20;
5048 int64_t carry21;
5049 int64_t carry22;
5050
5051 s0 = c0 + a0 * b0;
5052 s1 = c1 + a0 * b1 + a1 * b0;
5053 s2 = c2 + a0 * b2 + a1 * b1 + a2 * b0;
5054 s3 = c3 + a0 * b3 + a1 * b2 + a2 * b1 + a3 * b0;
5055 s4 = c4 + a0 * b4 + a1 * b3 + a2 * b2 + a3 * b1 + a4 * b0;
5056 s5 = c5 + a0 * b5 + a1 * b4 + a2 * b3 + a3 * b2 + a4 * b1 + a5 * b0;
5057 s6 = c6 + a0 * b6 + a1 * b5 + a2 * b4 + a3 * b3 + a4 * b2 + a5 * b1 + a6 * b0;
5058 s7 = c7 + a0 * b7 + a1 * b6 + a2 * b5 + a3 * b4 + a4 * b3 + a5 * b2 + a6 * b1 + a7 * b0;
5059 s8 = c8 + a0 * b8 + a1 * b7 + a2 * b6 + a3 * b5 + a4 * b4 + a5 * b3 + a6 * b2 + a7 * b1 + a8 * b0;
5060 s9 = c9 + a0 * b9 + a1 * b8 + a2 * b7 + a3 * b6 + a4 * b5 + a5 * b4 + a6 * b3 + a7 * b2 + a8 * b1 + a9 * b0;
5061 s10 = c10 + a0 * b10 + a1 * b9 + a2 * b8 + a3 * b7 + a4 * b6 + a5 * b5 + a6 * b4 + a7 * b3 + a8 * b2 + a9 * b1 + a10 * b0;
5062 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;
5063 s12 = a1 * b11 + a2 * b10 + a3 * b9 + a4 * b8 + a5 * b7 + a6 * b6 + a7 * b5 + a8 * b4 + a9 * b3 + a10 * b2 + a11 * b1;
5064 s13 = a2 * b11 + a3 * b10 + a4 * b9 + a5 * b8 + a6 * b7 + a7 * b6 + a8 * b5 + a9 * b4 + a10 * b3 + a11 * b2;
5065 s14 = a3 * b11 + a4 * b10 + a5 * b9 + a6 * b8 + a7 * b7 + a8 * b6 + a9 * b5 + a10 * b4 + a11 * b3;
5066 s15 = a4 * b11 + a5 * b10 + a6 * b9 + a7 * b8 + a8 * b7 + a9 * b6 + a10 * b5 + a11 * b4;
5067 s16 = a5 * b11 + a6 * b10 + a7 * b9 + a8 * b8 + a9 * b7 + a10 * b6 + a11 * b5;
5068 s17 = a6 * b11 + a7 * b10 + a8 * b9 + a9 * b8 + a10 * b7 + a11 * b6;
5069 s18 = a7 * b11 + a8 * b10 + a9 * b9 + a10 * b8 + a11 * b7;
5070 s19 = a8 * b11 + a9 * b10 + a10 * b9 + a11 * b8;
5071 s20 = a9 * b11 + a10 * b10 + a11 * b9;
5072 s21 = a10 * b11 + a11 * b10;
5073 s22 = a11 * b11;
5074 s23 = 0;
5075
5076 carry0 = (s0 + (1 << 20)) >> 21;
5077 s1 += carry0;
5078 s0 -= carry0 * (1 << 21);
5079 carry2 = (s2 + (1 << 20)) >> 21;
5080 s3 += carry2;
5081 s2 -= carry2 * (1 << 21);
5082 carry4 = (s4 + (1 << 20)) >> 21;
5083 s5 += carry4;
5084 s4 -= carry4 * (1 << 21);
5085 carry6 = (s6 + (1 << 20)) >> 21;
5086 s7 += carry6;
5087 s6 -= carry6 * (1 << 21);
5088 carry8 = (s8 + (1 << 20)) >> 21;
5089 s9 += carry8;
5090 s8 -= carry8 * (1 << 21);
5091 carry10 = (s10 + (1 << 20)) >> 21;
5092 s11 += carry10;
5093 s10 -= carry10 * (1 << 21);
5094 carry12 = (s12 + (1 << 20)) >> 21;
5095 s13 += carry12;
5096 s12 -= carry12 * (1 << 21);
5097 carry14 = (s14 + (1 << 20)) >> 21;
5098 s15 += carry14;
5099 s14 -= carry14 * (1 << 21);
5100 carry16 = (s16 + (1 << 20)) >> 21;
5101 s17 += carry16;
5102 s16 -= carry16 * (1 << 21);
5103 carry18 = (s18 + (1 << 20)) >> 21;
5104 s19 += carry18;
5105 s18 -= carry18 * (1 << 21);
5106 carry20 = (s20 + (1 << 20)) >> 21;
5107 s21 += carry20;
5108 s20 -= carry20 * (1 << 21);
5109 carry22 = (s22 + (1 << 20)) >> 21;
5110 s23 += carry22;
5111 s22 -= carry22 * (1 << 21);
5112
5113 carry1 = (s1 + (1 << 20)) >> 21;
5114 s2 += carry1;
5115 s1 -= carry1 * (1 << 21);
5116 carry3 = (s3 + (1 << 20)) >> 21;
5117 s4 += carry3;
5118 s3 -= carry3 * (1 << 21);
5119 carry5 = (s5 + (1 << 20)) >> 21;
5120 s6 += carry5;
5121 s5 -= carry5 * (1 << 21);
5122 carry7 = (s7 + (1 << 20)) >> 21;
5123 s8 += carry7;
5124 s7 -= carry7 * (1 << 21);
5125 carry9 = (s9 + (1 << 20)) >> 21;
5126 s10 += carry9;
5127 s9 -= carry9 * (1 << 21);
5128 carry11 = (s11 + (1 << 20)) >> 21;
5129 s12 += carry11;
5130 s11 -= carry11 * (1 << 21);
5131 carry13 = (s13 + (1 << 20)) >> 21;
5132 s14 += carry13;
5133 s13 -= carry13 * (1 << 21);
5134 carry15 = (s15 + (1 << 20)) >> 21;
5135 s16 += carry15;
5136 s15 -= carry15 * (1 << 21);
5137 carry17 = (s17 + (1 << 20)) >> 21;
5138 s18 += carry17;
5139 s17 -= carry17 * (1 << 21);
5140 carry19 = (s19 + (1 << 20)) >> 21;
5141 s20 += carry19;
5142 s19 -= carry19 * (1 << 21);
5143 carry21 = (s21 + (1 << 20)) >> 21;
5144 s22 += carry21;
5145 s21 -= carry21 * (1 << 21);
5146
5147 s11 += s23 * 666643;
5148 s12 += s23 * 470296;
5149 s13 += s23 * 654183;
5150 s14 -= s23 * 997805;
5151 s15 += s23 * 136657;
5152 s16 -= s23 * 683901;
5153 s23 = 0;
5154
5155 s10 += s22 * 666643;
5156 s11 += s22 * 470296;
5157 s12 += s22 * 654183;
5158 s13 -= s22 * 997805;
5159 s14 += s22 * 136657;
5160 s15 -= s22 * 683901;
5161 s22 = 0;
5162
5163 s9 += s21 * 666643;
5164 s10 += s21 * 470296;
5165 s11 += s21 * 654183;
5166 s12 -= s21 * 997805;
5167 s13 += s21 * 136657;
5168 s14 -= s21 * 683901;
5169 s21 = 0;
5170
5171 s8 += s20 * 666643;
5172 s9 += s20 * 470296;
5173 s10 += s20 * 654183;
5174 s11 -= s20 * 997805;
5175 s12 += s20 * 136657;
5176 s13 -= s20 * 683901;
5177 s20 = 0;
5178
5179 s7 += s19 * 666643;
5180 s8 += s19 * 470296;
5181 s9 += s19 * 654183;
5182 s10 -= s19 * 997805;
5183 s11 += s19 * 136657;
5184 s12 -= s19 * 683901;
5185 s19 = 0;
5186
5187 s6 += s18 * 666643;
5188 s7 += s18 * 470296;
5189 s8 += s18 * 654183;
5190 s9 -= s18 * 997805;
5191 s10 += s18 * 136657;
5192 s11 -= s18 * 683901;
5193 s18 = 0;
5194
5195 carry6 = (s6 + (1 << 20)) >> 21;
5196 s7 += carry6;
5197 s6 -= carry6 * (1 << 21);
5198 carry8 = (s8 + (1 << 20)) >> 21;
5199 s9 += carry8;
5200 s8 -= carry8 * (1 << 21);
5201 carry10 = (s10 + (1 << 20)) >> 21;
5202 s11 += carry10;
5203 s10 -= carry10 * (1 << 21);
5204 carry12 = (s12 + (1 << 20)) >> 21;
5205 s13 += carry12;
5206 s12 -= carry12 * (1 << 21);
5207 carry14 = (s14 + (1 << 20)) >> 21;
5208 s15 += carry14;
5209 s14 -= carry14 * (1 << 21);
5210 carry16 = (s16 + (1 << 20)) >> 21;
5211 s17 += carry16;
5212 s16 -= carry16 * (1 << 21);
5213
5214 carry7 = (s7 + (1 << 20)) >> 21;
5215 s8 += carry7;
5216 s7 -= carry7 * (1 << 21);
5217 carry9 = (s9 + (1 << 20)) >> 21;
5218 s10 += carry9;
5219 s9 -= carry9 * (1 << 21);
5220 carry11 = (s11 + (1 << 20)) >> 21;
5221 s12 += carry11;
5222 s11 -= carry11 * (1 << 21);
5223 carry13 = (s13 + (1 << 20)) >> 21;
5224 s14 += carry13;
5225 s13 -= carry13 * (1 << 21);
5226 carry15 = (s15 + (1 << 20)) >> 21;
5227 s16 += carry15;
5228 s15 -= carry15 * (1 << 21);
5229
5230 s5 += s17 * 666643;
5231 s6 += s17 * 470296;
5232 s7 += s17 * 654183;
5233 s8 -= s17 * 997805;
5234 s9 += s17 * 136657;
5235 s10 -= s17 * 683901;
5236 s17 = 0;
5237
5238 s4 += s16 * 666643;
5239 s5 += s16 * 470296;
5240 s6 += s16 * 654183;
5241 s7 -= s16 * 997805;
5242 s8 += s16 * 136657;
5243 s9 -= s16 * 683901;
5244 s16 = 0;
5245
5246 s3 += s15 * 666643;
5247 s4 += s15 * 470296;
5248 s5 += s15 * 654183;
5249 s6 -= s15 * 997805;
5250 s7 += s15 * 136657;
5251 s8 -= s15 * 683901;
5252 s15 = 0;
5253
5254 s2 += s14 * 666643;
5255 s3 += s14 * 470296;
5256 s4 += s14 * 654183;
5257 s5 -= s14 * 997805;
5258 s6 += s14 * 136657;
5259 s7 -= s14 * 683901;
5260 s14 = 0;
5261
5262 s1 += s13 * 666643;
5263 s2 += s13 * 470296;
5264 s3 += s13 * 654183;
5265 s4 -= s13 * 997805;
5266 s5 += s13 * 136657;
5267 s6 -= s13 * 683901;
5268 s13 = 0;
5269
5270 s0 += s12 * 666643;
5271 s1 += s12 * 470296;
5272 s2 += s12 * 654183;
5273 s3 -= s12 * 997805;
5274 s4 += s12 * 136657;
5275 s5 -= s12 * 683901;
5276 s12 = 0;
5277
5278 carry0 = (s0 + (1 << 20)) >> 21;
5279 s1 += carry0;
5280 s0 -= carry0 * (1 << 21);
5281 carry2 = (s2 + (1 << 20)) >> 21;
5282 s3 += carry2;
5283 s2 -= carry2 * (1 << 21);
5284 carry4 = (s4 + (1 << 20)) >> 21;
5285 s5 += carry4;
5286 s4 -= carry4 * (1 << 21);
5287 carry6 = (s6 + (1 << 20)) >> 21;
5288 s7 += carry6;
5289 s6 -= carry6 * (1 << 21);
5290 carry8 = (s8 + (1 << 20)) >> 21;
5291 s9 += carry8;
5292 s8 -= carry8 * (1 << 21);
5293 carry10 = (s10 + (1 << 20)) >> 21;
5294 s11 += carry10;
5295 s10 -= carry10 * (1 << 21);
5296
5297 carry1 = (s1 + (1 << 20)) >> 21;
5298 s2 += carry1;
5299 s1 -= carry1 * (1 << 21);
5300 carry3 = (s3 + (1 << 20)) >> 21;
5301 s4 += carry3;
5302 s3 -= carry3 * (1 << 21);
5303 carry5 = (s5 + (1 << 20)) >> 21;
5304 s6 += carry5;
5305 s5 -= carry5 * (1 << 21);
5306 carry7 = (s7 + (1 << 20)) >> 21;
5307 s8 += carry7;
5308 s7 -= carry7 * (1 << 21);
5309 carry9 = (s9 + (1 << 20)) >> 21;
5310 s10 += carry9;
5311 s9 -= carry9 * (1 << 21);
5312 carry11 = (s11 + (1 << 20)) >> 21;
5313 s12 += carry11;
5314 s11 -= carry11 * (1 << 21);
5315
5316 s0 += s12 * 666643;
5317 s1 += s12 * 470296;
5318 s2 += s12 * 654183;
5319 s3 -= s12 * 997805;
5320 s4 += s12 * 136657;
5321 s5 -= s12 * 683901;
5322 s12 = 0;
5323
5324 carry0 = s0 >> 21;
5325 s1 += carry0;
5326 s0 -= carry0 * (1 << 21);
5327 carry1 = s1 >> 21;
5328 s2 += carry1;
5329 s1 -= carry1 * (1 << 21);
5330 carry2 = s2 >> 21;
5331 s3 += carry2;
5332 s2 -= carry2 * (1 << 21);
5333 carry3 = s3 >> 21;
5334 s4 += carry3;
5335 s3 -= carry3 * (1 << 21);
5336 carry4 = s4 >> 21;
5337 s5 += carry4;
5338 s4 -= carry4 * (1 << 21);
5339 carry5 = s5 >> 21;
5340 s6 += carry5;
5341 s5 -= carry5 * (1 << 21);
5342 carry6 = s6 >> 21;
5343 s7 += carry6;
5344 s6 -= carry6 * (1 << 21);
5345 carry7 = s7 >> 21;
5346 s8 += carry7;
5347 s7 -= carry7 * (1 << 21);
5348 carry8 = s8 >> 21;
5349 s9 += carry8;
5350 s8 -= carry8 * (1 << 21);
5351 carry9 = s9 >> 21;
5352 s10 += carry9;
5353 s9 -= carry9 * (1 << 21);
5354 carry10 = s10 >> 21;
5355 s11 += carry10;
5356 s10 -= carry10 * (1 << 21);
5357 carry11 = s11 >> 21;
5358 s12 += carry11;
5359 s11 -= carry11 * (1 << 21);
5360
5361 s0 += s12 * 666643;
5362 s1 += s12 * 470296;
5363 s2 += s12 * 654183;
5364 s3 -= s12 * 997805;
5365 s4 += s12 * 136657;
5366 s5 -= s12 * 683901;
5367 s12 = 0;
5368
5369 carry0 = s0 >> 21;
5370 s1 += carry0;
5371 s0 -= carry0 * (1 << 21);
5372 carry1 = s1 >> 21;
5373 s2 += carry1;
5374 s1 -= carry1 * (1 << 21);
5375 carry2 = s2 >> 21;
5376 s3 += carry2;
5377 s2 -= carry2 * (1 << 21);
5378 carry3 = s3 >> 21;
5379 s4 += carry3;
5380 s3 -= carry3 * (1 << 21);
5381 carry4 = s4 >> 21;
5382 s5 += carry4;
5383 s4 -= carry4 * (1 << 21);
5384 carry5 = s5 >> 21;
5385 s6 += carry5;
5386 s5 -= carry5 * (1 << 21);
5387 carry6 = s6 >> 21;
5388 s7 += carry6;
5389 s6 -= carry6 * (1 << 21);
5390 carry7 = s7 >> 21;
5391 s8 += carry7;
5392 s7 -= carry7 * (1 << 21);
5393 carry8 = s8 >> 21;
5394 s9 += carry8;
5395 s8 -= carry8 * (1 << 21);
5396 carry9 = s9 >> 21;
5397 s10 += carry9;
5398 s9 -= carry9 * (1 << 21);
5399 carry10 = s10 >> 21;
5400 s11 += carry10;
5401 s10 -= carry10 * (1 << 21);
5402
5403 s[ 0] = (uint8_t) (s0 >> 0);
5404 s[ 1] = (uint8_t) (s0 >> 8);
5405 s[ 2] = (uint8_t)((s0 >> 16) | (s1 << 5));
5406 s[ 3] = (uint8_t) (s1 >> 3);
5407 s[ 4] = (uint8_t) (s1 >> 11);
5408 s[ 5] = (uint8_t)((s1 >> 19) | (s2 << 2));
5409 s[ 6] = (uint8_t) (s2 >> 6);
5410 s[ 7] = (uint8_t)((s2 >> 14) | (s3 << 7));
5411 s[ 8] = (uint8_t) (s3 >> 1);
5412 s[ 9] = (uint8_t) (s3 >> 9);
5413 s[10] = (uint8_t)((s3 >> 17) | (s4 << 4));
5414 s[11] = (uint8_t) (s4 >> 4);
5415 s[12] = (uint8_t) (s4 >> 12);
5416 s[13] = (uint8_t)((s4 >> 20) | (s5 << 1));
5417 s[14] = (uint8_t) (s5 >> 7);
5418 s[15] = (uint8_t)((s5 >> 15) | (s6 << 6));
5419 s[16] = (uint8_t) (s6 >> 2);
5420 s[17] = (uint8_t) (s6 >> 10);
5421 s[18] = (uint8_t)((s6 >> 18) | (s7 << 3));
5422 s[19] = (uint8_t) (s7 >> 5);
5423 s[20] = (uint8_t) (s7 >> 13);
5424 s[21] = (uint8_t) (s8 >> 0);
5425 s[22] = (uint8_t) (s8 >> 8);
5426 s[23] = (uint8_t)((s8 >> 16) | (s9 << 5));
5427 s[24] = (uint8_t) (s9 >> 3);
5428 s[25] = (uint8_t) (s9 >> 11);
5429 s[26] = (uint8_t)((s9 >> 19) | (s10 << 2));
5430 s[27] = (uint8_t) (s10 >> 6);
5431 s[28] = (uint8_t)((s10 >> 14) | (s11 << 7));
5432 s[29] = (uint8_t) (s11 >> 1);
5433 s[30] = (uint8_t) (s11 >> 9);
5434 s[31] = (uint8_t) (s11 >> 17);
5435 }
5436
hash_init_with_dom(EVP_MD_CTX * hash_ctx,EVP_MD * sha512,const uint8_t dom2flag,const uint8_t phflag,const uint8_t * context,const size_t context_len)5437 static int hash_init_with_dom(EVP_MD_CTX *hash_ctx,
5438 EVP_MD *sha512,
5439 const uint8_t dom2flag,
5440 const uint8_t phflag,
5441 const uint8_t *context,
5442 const size_t context_len)
5443 {
5444 /* ASCII: "SigEd25519 no Ed25519 collisions", in hex for EBCDIC compatibility */
5445 const char dom_s[] =
5446 "\x53\x69\x67\x45\x64\x32\x35\x35\x31\x39\x20\x6e"
5447 "\x6f\x20\x45\x64\x32\x35\x35\x31\x39\x20\x63\x6f"
5448 "\x6c\x6c\x69\x73\x69\x6f\x6e\x73";
5449 uint8_t dom[2];
5450
5451 if (!EVP_DigestInit_ex(hash_ctx, sha512, NULL))
5452 return 0;
5453
5454 /* return early if dom2flag is not set */
5455 if (!dom2flag)
5456 return 1;
5457
5458 if (context_len > UINT8_MAX)
5459 return 0;
5460
5461 dom[0] = (uint8_t)(phflag >= 1 ? 1 : 0);
5462 dom[1] = (uint8_t)context_len;
5463
5464 if (!EVP_DigestUpdate(hash_ctx, dom_s, sizeof(dom_s)-1)
5465 || !EVP_DigestUpdate(hash_ctx, dom, sizeof(dom))
5466 || !EVP_DigestUpdate(hash_ctx, context, context_len)) {
5467 return 0;
5468 }
5469
5470 return 1;
5471 }
5472
5473 int
ossl_ed25519_sign(uint8_t * out_sig,const uint8_t * tbs,size_t tbs_len,const uint8_t public_key[32],const uint8_t private_key[32],const uint8_t dom2flag,const uint8_t phflag,const uint8_t csflag,const uint8_t * context,size_t context_len,OSSL_LIB_CTX * libctx,const char * propq)5474 ossl_ed25519_sign(uint8_t *out_sig, const uint8_t *tbs, size_t tbs_len,
5475 const uint8_t public_key[32], const uint8_t private_key[32],
5476 const uint8_t dom2flag, const uint8_t phflag, const uint8_t csflag,
5477 const uint8_t *context, size_t context_len,
5478 OSSL_LIB_CTX *libctx, const char *propq)
5479 {
5480 uint8_t az[SHA512_DIGEST_LENGTH];
5481 uint8_t nonce[SHA512_DIGEST_LENGTH];
5482 ge_p3 R;
5483 uint8_t hram[SHA512_DIGEST_LENGTH];
5484 EVP_MD *sha512 = EVP_MD_fetch(libctx, SN_sha512, propq);
5485 EVP_MD_CTX *hash_ctx = EVP_MD_CTX_new();
5486 unsigned int sz;
5487 int res = 0;
5488
5489 if (context == NULL)
5490 context_len = 0;
5491
5492 /* if csflag is set, then a non-empty context-string is required */
5493 if (csflag && context_len == 0)
5494 goto err;
5495
5496 /* if dom2flag is not set, then an empty context-string is required */
5497 if (!dom2flag && context_len > 0)
5498 goto err;
5499
5500 if (sha512 == NULL || hash_ctx == NULL)
5501 goto err;
5502
5503 if (!EVP_DigestInit_ex(hash_ctx, sha512, NULL)
5504 || !EVP_DigestUpdate(hash_ctx, private_key, 32)
5505 || !EVP_DigestFinal_ex(hash_ctx, az, &sz))
5506 goto err;
5507
5508 az[0] &= 248;
5509 az[31] &= 63;
5510 az[31] |= 64;
5511
5512 if (!hash_init_with_dom(hash_ctx, sha512, dom2flag, phflag, context, context_len)
5513 || !EVP_DigestUpdate(hash_ctx, az + 32, 32)
5514 || !EVP_DigestUpdate(hash_ctx, tbs, tbs_len)
5515 || !EVP_DigestFinal_ex(hash_ctx, nonce, &sz))
5516 goto err;
5517
5518 x25519_sc_reduce(nonce);
5519 ge_scalarmult_base(&R, nonce);
5520 ge_p3_tobytes(out_sig, &R);
5521
5522 if (!hash_init_with_dom(hash_ctx, sha512, dom2flag, phflag, context, context_len)
5523 || !EVP_DigestUpdate(hash_ctx, out_sig, 32)
5524 || !EVP_DigestUpdate(hash_ctx, public_key, 32)
5525 || !EVP_DigestUpdate(hash_ctx, tbs, tbs_len)
5526 || !EVP_DigestFinal_ex(hash_ctx, hram, &sz))
5527 goto err;
5528
5529 x25519_sc_reduce(hram);
5530 sc_muladd(out_sig + 32, hram, az, nonce);
5531
5532 res = 1;
5533 err:
5534 OPENSSL_cleanse(nonce, sizeof(nonce));
5535 OPENSSL_cleanse(az, sizeof(az));
5536 EVP_MD_free(sha512);
5537 EVP_MD_CTX_free(hash_ctx);
5538 return res;
5539 }
5540
5541 /*
5542 * This function should not be necessary since ossl_ed25519_verify() already
5543 * does this check internally.
5544 * For some reason the FIPS ACVP requires a EDDSA KeyVer test.
5545 */
5546 int
ossl_ed25519_pubkey_verify(const uint8_t * pub,size_t pub_len)5547 ossl_ed25519_pubkey_verify(const uint8_t *pub, size_t pub_len)
5548 {
5549 ge_p3 A;
5550
5551 if (pub_len != ED25519_KEYLEN)
5552 return 0;
5553 return (ge_frombytes_vartime(&A, pub) == 0);
5554 }
5555
5556 static const char allzeroes[15];
5557
5558 int
ossl_ed25519_verify(const uint8_t * tbs,size_t tbs_len,const uint8_t signature[64],const uint8_t public_key[32],const uint8_t dom2flag,const uint8_t phflag,const uint8_t csflag,const uint8_t * context,size_t context_len,OSSL_LIB_CTX * libctx,const char * propq)5559 ossl_ed25519_verify(const uint8_t *tbs, size_t tbs_len,
5560 const uint8_t signature[64], const uint8_t public_key[32],
5561 const uint8_t dom2flag, const uint8_t phflag, const uint8_t csflag,
5562 const uint8_t *context, size_t context_len,
5563 OSSL_LIB_CTX *libctx, const char *propq)
5564 {
5565 int i;
5566 ge_p3 A;
5567 const uint8_t *r, *s;
5568 EVP_MD *sha512;
5569 EVP_MD_CTX *hash_ctx = NULL;
5570 unsigned int sz;
5571 int res = 0;
5572 ge_p2 R;
5573 uint8_t rcheck[32];
5574 uint8_t h[SHA512_DIGEST_LENGTH];
5575 /* 27742317777372353535851937790883648493 in little endian format */
5576 const uint8_t l_low[16] = {
5577 0xED, 0xD3, 0xF5, 0x5C, 0x1A, 0x63, 0x12, 0x58, 0xD6, 0x9C, 0xF7, 0xA2,
5578 0xDE, 0xF9, 0xDE, 0x14
5579 };
5580
5581 if (context == NULL)
5582 context_len = 0;
5583
5584 /* if csflag is set, then a non-empty context-string is required */
5585 if (csflag && context_len == 0)
5586 return 0;
5587
5588 /* if dom2flag is not set, then an empty context-string is required */
5589 if (!dom2flag && context_len > 0)
5590 return 0;
5591
5592 r = signature;
5593 s = signature + 32;
5594
5595 /*
5596 * Check 0 <= s < L where L = 2^252 + 27742317777372353535851937790883648493
5597 *
5598 * If not the signature is publicly invalid. Since it's public we can do the
5599 * check in variable time.
5600 *
5601 * First check the most significant byte
5602 */
5603 if (s[31] > 0x10)
5604 return 0;
5605 if (s[31] == 0x10) {
5606 /*
5607 * Most significant byte indicates a value close to 2^252 so check the
5608 * rest
5609 */
5610 if (memcmp(s + 16, allzeroes, sizeof(allzeroes)) != 0)
5611 return 0;
5612 for (i = 15; i >= 0; i--) {
5613 if (s[i] < l_low[i])
5614 break;
5615 if (s[i] > l_low[i])
5616 return 0;
5617 }
5618 if (i < 0)
5619 return 0;
5620 }
5621
5622 if (ge_frombytes_vartime(&A, public_key) != 0) {
5623 return 0;
5624 }
5625
5626 fe_neg(A.X, A.X);
5627 fe_neg(A.T, A.T);
5628
5629 sha512 = EVP_MD_fetch(libctx, SN_sha512, propq);
5630 if (sha512 == NULL)
5631 return 0;
5632 hash_ctx = EVP_MD_CTX_new();
5633 if (hash_ctx == NULL)
5634 goto err;
5635
5636 if (!hash_init_with_dom(hash_ctx, sha512, dom2flag, phflag, context, context_len)
5637 || !EVP_DigestUpdate(hash_ctx, r, 32)
5638 || !EVP_DigestUpdate(hash_ctx, public_key, 32)
5639 || !EVP_DigestUpdate(hash_ctx, tbs, tbs_len)
5640 || !EVP_DigestFinal_ex(hash_ctx, h, &sz))
5641 goto err;
5642
5643 x25519_sc_reduce(h);
5644
5645 ge_double_scalarmult_vartime(&R, h, &A, s);
5646
5647 ge_tobytes(rcheck, &R);
5648
5649 res = CRYPTO_memcmp(rcheck, r, sizeof(rcheck)) == 0;
5650
5651 /* note that we have used the strict verification equation here.
5652 * we checked that ENC( [h](-A) + [s]B ) == r
5653 * B is the base point.
5654 *
5655 * the less strict verification equation uses the curve cofactor:
5656 * [h*8](-A) + [s*8]B == [8]R
5657 */
5658 err:
5659 EVP_MD_free(sha512);
5660 EVP_MD_CTX_free(hash_ctx);
5661 return res;
5662 }
5663
5664 int
ossl_ed25519_public_from_private(OSSL_LIB_CTX * ctx,uint8_t out_public_key[32],const uint8_t private_key[32],const char * propq)5665 ossl_ed25519_public_from_private(OSSL_LIB_CTX *ctx, uint8_t out_public_key[32],
5666 const uint8_t private_key[32],
5667 const char *propq)
5668 {
5669 uint8_t az[SHA512_DIGEST_LENGTH];
5670 ge_p3 A;
5671 int r;
5672 EVP_MD *sha512 = NULL;
5673
5674 sha512 = EVP_MD_fetch(ctx, SN_sha512, propq);
5675 if (sha512 == NULL)
5676 return 0;
5677 r = EVP_Digest(private_key, 32, az, NULL, sha512, NULL);
5678 EVP_MD_free(sha512);
5679 if (!r) {
5680 OPENSSL_cleanse(az, sizeof(az));
5681 return 0;
5682 }
5683
5684 az[0] &= 248;
5685 az[31] &= 63;
5686 az[31] |= 64;
5687
5688 ge_scalarmult_base(&A, az);
5689 ge_p3_tobytes(out_public_key, &A);
5690
5691 OPENSSL_cleanse(az, sizeof(az));
5692 return 1;
5693 }
5694
5695 int
ossl_x25519(uint8_t out_shared_key[32],const uint8_t private_key[32],const uint8_t peer_public_value[32])5696 ossl_x25519(uint8_t out_shared_key[32], const uint8_t private_key[32],
5697 const uint8_t peer_public_value[32])
5698 {
5699 static const uint8_t kZeros[32] = {0};
5700 x25519_scalar_mult(out_shared_key, private_key, peer_public_value);
5701 /* The all-zero output results when the input is a point of small order. */
5702 return CRYPTO_memcmp(kZeros, out_shared_key, 32) != 0;
5703 }
5704
5705 void
ossl_x25519_public_from_private(uint8_t out_public_value[32],const uint8_t private_key[32])5706 ossl_x25519_public_from_private(uint8_t out_public_value[32],
5707 const uint8_t private_key[32])
5708 {
5709 uint8_t e[32];
5710 ge_p3 A;
5711 fe zplusy, zminusy, zminusy_inv;
5712
5713 memcpy(e, private_key, 32);
5714 e[0] &= 248;
5715 e[31] &= 127;
5716 e[31] |= 64;
5717
5718 ge_scalarmult_base(&A, e);
5719
5720 /*
5721 * We only need the u-coordinate of the curve25519 point.
5722 * The map is u=(y+1)/(1-y). Since y=Y/Z, this gives
5723 * u=(Z+Y)/(Z-Y).
5724 */
5725 fe_add(zplusy, A.Z, A.Y);
5726 fe_sub(zminusy, A.Z, A.Y);
5727 fe_invert(zminusy_inv, zminusy);
5728 fe_mul(zplusy, zplusy, zminusy_inv);
5729 fe_tobytes(out_public_value, zplusy);
5730
5731 OPENSSL_cleanse(e, sizeof(e));
5732 }
5733