curve25519.c 198 KB

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  1. /*
  2. * Copyright 2016-2021 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. * ECDSA low level APIs are deprecated for public use, but still ok for
  11. * internal use.
  12. */
  13. #include "internal/deprecated.h"
  14. #include <string.h>
  15. #include "crypto/ecx.h"
  16. #include "ec_local.h"
  17. #include <openssl/evp.h>
  18. #include <openssl/sha.h>
  19. #include "internal/numbers.h"
  20. #if defined(X25519_ASM) && (defined(__x86_64) || defined(__x86_64__) || \
  21. defined(_M_AMD64) || defined(_M_X64))
  22. # define BASE_2_64_IMPLEMENTED
  23. typedef uint64_t fe64[4];
  24. int x25519_fe64_eligible(void);
  25. /*
  26. * Following subroutines perform corresponding operations modulo
  27. * 2^256-38, i.e. double the curve modulus. However, inputs and
  28. * outputs are permitted to be partially reduced, i.e. to remain
  29. * in [0..2^256) range. It's all tied up in final fe64_tobytes
  30. * that performs full reduction modulo 2^255-19.
  31. *
  32. * There are no reference C implementations for these.
  33. */
  34. void x25519_fe64_mul(fe64 h, const fe64 f, const fe64 g);
  35. void x25519_fe64_sqr(fe64 h, const fe64 f);
  36. void x25519_fe64_mul121666(fe64 h, fe64 f);
  37. void x25519_fe64_add(fe64 h, const fe64 f, const fe64 g);
  38. void x25519_fe64_sub(fe64 h, const fe64 f, const fe64 g);
  39. void x25519_fe64_tobytes(uint8_t *s, const fe64 f);
  40. # define fe64_mul x25519_fe64_mul
  41. # define fe64_sqr x25519_fe64_sqr
  42. # define fe64_mul121666 x25519_fe64_mul121666
  43. # define fe64_add x25519_fe64_add
  44. # define fe64_sub x25519_fe64_sub
  45. # define fe64_tobytes x25519_fe64_tobytes
  46. static uint64_t load_8(const uint8_t *in)
  47. {
  48. uint64_t result;
  49. result = in[0];
  50. result |= ((uint64_t)in[1]) << 8;
  51. result |= ((uint64_t)in[2]) << 16;
  52. result |= ((uint64_t)in[3]) << 24;
  53. result |= ((uint64_t)in[4]) << 32;
  54. result |= ((uint64_t)in[5]) << 40;
  55. result |= ((uint64_t)in[6]) << 48;
  56. result |= ((uint64_t)in[7]) << 56;
  57. return result;
  58. }
  59. static void fe64_frombytes(fe64 h, const uint8_t *s)
  60. {
  61. h[0] = load_8(s);
  62. h[1] = load_8(s + 8);
  63. h[2] = load_8(s + 16);
  64. h[3] = load_8(s + 24) & 0x7fffffffffffffff;
  65. }
  66. static void fe64_0(fe64 h)
  67. {
  68. h[0] = 0;
  69. h[1] = 0;
  70. h[2] = 0;
  71. h[3] = 0;
  72. }
  73. static void fe64_1(fe64 h)
  74. {
  75. h[0] = 1;
  76. h[1] = 0;
  77. h[2] = 0;
  78. h[3] = 0;
  79. }
  80. static void fe64_copy(fe64 h, const fe64 f)
  81. {
  82. h[0] = f[0];
  83. h[1] = f[1];
  84. h[2] = f[2];
  85. h[3] = f[3];
  86. }
  87. static void fe64_cswap(fe64 f, fe64 g, unsigned int b)
  88. {
  89. int i;
  90. uint64_t mask = 0 - (uint64_t)b;
  91. for (i = 0; i < 4; i++) {
  92. uint64_t x = f[i] ^ g[i];
  93. x &= mask;
  94. f[i] ^= x;
  95. g[i] ^= x;
  96. }
  97. }
  98. static void fe64_invert(fe64 out, const fe64 z)
  99. {
  100. fe64 t0;
  101. fe64 t1;
  102. fe64 t2;
  103. fe64 t3;
  104. int i;
  105. /*
  106. * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as
  107. * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11.
  108. */
  109. /* t0 = z ** 2 */
  110. fe64_sqr(t0, z);
  111. /* t1 = t0 ** (2 ** 2) = z ** 8 */
  112. fe64_sqr(t1, t0);
  113. fe64_sqr(t1, t1);
  114. /* t1 = z * t1 = z ** 9 */
  115. fe64_mul(t1, z, t1);
  116. /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */
  117. fe64_mul(t0, t0, t1);
  118. /* t2 = t0 ** 2 = z ** 22 */
  119. fe64_sqr(t2, t0);
  120. /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */
  121. fe64_mul(t1, t1, t2);
  122. /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */
  123. fe64_sqr(t2, t1);
  124. for (i = 1; i < 5; ++i)
  125. fe64_sqr(t2, t2);
  126. /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */
  127. fe64_mul(t1, t2, t1);
  128. /* Continuing similarly... */
  129. /* t2 = z ** (2 ** 20 - 1) */
  130. fe64_sqr(t2, t1);
  131. for (i = 1; i < 10; ++i)
  132. fe64_sqr(t2, t2);
  133. fe64_mul(t2, t2, t1);
  134. /* t2 = z ** (2 ** 40 - 1) */
  135. fe64_sqr(t3, t2);
  136. for (i = 1; i < 20; ++i)
  137. fe64_sqr(t3, t3);
  138. fe64_mul(t2, t3, t2);
  139. /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */
  140. for (i = 0; i < 10; ++i)
  141. fe64_sqr(t2, t2);
  142. /* t1 = z ** (2 ** 50 - 1) */
  143. fe64_mul(t1, t2, t1);
  144. /* t2 = z ** (2 ** 100 - 1) */
  145. fe64_sqr(t2, t1);
  146. for (i = 1; i < 50; ++i)
  147. fe64_sqr(t2, t2);
  148. fe64_mul(t2, t2, t1);
  149. /* t2 = z ** (2 ** 200 - 1) */
  150. fe64_sqr(t3, t2);
  151. for (i = 1; i < 100; ++i)
  152. fe64_sqr(t3, t3);
  153. fe64_mul(t2, t3, t2);
  154. /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */
  155. for (i = 0; i < 50; ++i)
  156. fe64_sqr(t2, t2);
  157. /* t1 = z ** (2 ** 250 - 1) */
  158. fe64_mul(t1, t2, t1);
  159. /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */
  160. for (i = 0; i < 5; ++i)
  161. fe64_sqr(t1, t1);
  162. /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */
  163. fe64_mul(out, t1, t0);
  164. }
  165. /*
  166. * Duplicate of original x25519_scalar_mult_generic, but using
  167. * fe64_* subroutines.
  168. */
  169. static void x25519_scalar_mulx(uint8_t out[32], const uint8_t scalar[32],
  170. const uint8_t point[32])
  171. {
  172. fe64 x1, x2, z2, x3, z3, tmp0, tmp1;
  173. uint8_t e[32];
  174. unsigned swap = 0;
  175. int pos;
  176. memcpy(e, scalar, 32);
  177. e[0] &= 0xf8;
  178. e[31] &= 0x7f;
  179. e[31] |= 0x40;
  180. fe64_frombytes(x1, point);
  181. fe64_1(x2);
  182. fe64_0(z2);
  183. fe64_copy(x3, x1);
  184. fe64_1(z3);
  185. for (pos = 254; pos >= 0; --pos) {
  186. unsigned int b = 1 & (e[pos / 8] >> (pos & 7));
  187. swap ^= b;
  188. fe64_cswap(x2, x3, swap);
  189. fe64_cswap(z2, z3, swap);
  190. swap = b;
  191. fe64_sub(tmp0, x3, z3);
  192. fe64_sub(tmp1, x2, z2);
  193. fe64_add(x2, x2, z2);
  194. fe64_add(z2, x3, z3);
  195. fe64_mul(z3, x2, tmp0);
  196. fe64_mul(z2, z2, tmp1);
  197. fe64_sqr(tmp0, tmp1);
  198. fe64_sqr(tmp1, x2);
  199. fe64_add(x3, z3, z2);
  200. fe64_sub(z2, z3, z2);
  201. fe64_mul(x2, tmp1, tmp0);
  202. fe64_sub(tmp1, tmp1, tmp0);
  203. fe64_sqr(z2, z2);
  204. fe64_mul121666(z3, tmp1);
  205. fe64_sqr(x3, x3);
  206. fe64_add(tmp0, tmp0, z3);
  207. fe64_mul(z3, x1, z2);
  208. fe64_mul(z2, tmp1, tmp0);
  209. }
  210. fe64_invert(z2, z2);
  211. fe64_mul(x2, x2, z2);
  212. fe64_tobytes(out, x2);
  213. OPENSSL_cleanse(e, sizeof(e));
  214. }
  215. #endif
  216. #if defined(X25519_ASM) \
  217. || ( defined(INT128_MAX) \
  218. && !defined(__sparc__) \
  219. && (!defined(__SIZEOF_LONG__) || (__SIZEOF_LONG__ == 8)) \
  220. && !(defined(__ANDROID__) && !defined(__clang__)) )
  221. /*
  222. * Base 2^51 implementation. It's virtually no different from reference
  223. * base 2^25.5 implementation in respect to lax boundary conditions for
  224. * intermediate values and even individual limbs. So that whatever you
  225. * know about the reference, applies even here...
  226. */
  227. # define BASE_2_51_IMPLEMENTED
  228. typedef uint64_t fe51[5];
  229. static const uint64_t MASK51 = 0x7ffffffffffff;
  230. static uint64_t load_7(const uint8_t *in)
  231. {
  232. uint64_t result;
  233. result = in[0];
  234. result |= ((uint64_t)in[1]) << 8;
  235. result |= ((uint64_t)in[2]) << 16;
  236. result |= ((uint64_t)in[3]) << 24;
  237. result |= ((uint64_t)in[4]) << 32;
  238. result |= ((uint64_t)in[5]) << 40;
  239. result |= ((uint64_t)in[6]) << 48;
  240. return result;
  241. }
  242. static uint64_t load_6(const uint8_t *in)
  243. {
  244. uint64_t result;
  245. result = in[0];
  246. result |= ((uint64_t)in[1]) << 8;
  247. result |= ((uint64_t)in[2]) << 16;
  248. result |= ((uint64_t)in[3]) << 24;
  249. result |= ((uint64_t)in[4]) << 32;
  250. result |= ((uint64_t)in[5]) << 40;
  251. return result;
  252. }
  253. static void fe51_frombytes(fe51 h, const uint8_t *s)
  254. {
  255. uint64_t h0 = load_7(s); /* 56 bits */
  256. uint64_t h1 = load_6(s + 7) << 5; /* 53 bits */
  257. uint64_t h2 = load_7(s + 13) << 2; /* 58 bits */
  258. uint64_t h3 = load_6(s + 20) << 7; /* 55 bits */
  259. uint64_t h4 = (load_6(s + 26) & 0x7fffffffffff) << 4; /* 51 bits */
  260. h1 |= h0 >> 51; h0 &= MASK51;
  261. h2 |= h1 >> 51; h1 &= MASK51;
  262. h3 |= h2 >> 51; h2 &= MASK51;
  263. h4 |= h3 >> 51; h3 &= MASK51;
  264. h[0] = h0;
  265. h[1] = h1;
  266. h[2] = h2;
  267. h[3] = h3;
  268. h[4] = h4;
  269. }
  270. static void fe51_tobytes(uint8_t *s, const fe51 h)
  271. {
  272. uint64_t h0 = h[0];
  273. uint64_t h1 = h[1];
  274. uint64_t h2 = h[2];
  275. uint64_t h3 = h[3];
  276. uint64_t h4 = h[4];
  277. uint64_t q;
  278. /* compare to modulus */
  279. q = (h0 + 19) >> 51;
  280. q = (h1 + q) >> 51;
  281. q = (h2 + q) >> 51;
  282. q = (h3 + q) >> 51;
  283. q = (h4 + q) >> 51;
  284. /* full reduce */
  285. h0 += 19 * q;
  286. h1 += h0 >> 51; h0 &= MASK51;
  287. h2 += h1 >> 51; h1 &= MASK51;
  288. h3 += h2 >> 51; h2 &= MASK51;
  289. h4 += h3 >> 51; h3 &= MASK51;
  290. h4 &= MASK51;
  291. /* smash */
  292. s[0] = (uint8_t)(h0 >> 0);
  293. s[1] = (uint8_t)(h0 >> 8);
  294. s[2] = (uint8_t)(h0 >> 16);
  295. s[3] = (uint8_t)(h0 >> 24);
  296. s[4] = (uint8_t)(h0 >> 32);
  297. s[5] = (uint8_t)(h0 >> 40);
  298. s[6] = (uint8_t)((h0 >> 48) | ((uint32_t)h1 << 3));
  299. s[7] = (uint8_t)(h1 >> 5);
  300. s[8] = (uint8_t)(h1 >> 13);
  301. s[9] = (uint8_t)(h1 >> 21);
  302. s[10] = (uint8_t)(h1 >> 29);
  303. s[11] = (uint8_t)(h1 >> 37);
  304. s[12] = (uint8_t)((h1 >> 45) | ((uint32_t)h2 << 6));
  305. s[13] = (uint8_t)(h2 >> 2);
  306. s[14] = (uint8_t)(h2 >> 10);
  307. s[15] = (uint8_t)(h2 >> 18);
  308. s[16] = (uint8_t)(h2 >> 26);
  309. s[17] = (uint8_t)(h2 >> 34);
  310. s[18] = (uint8_t)(h2 >> 42);
  311. s[19] = (uint8_t)((h2 >> 50) | ((uint32_t)h3 << 1));
  312. s[20] = (uint8_t)(h3 >> 7);
  313. s[21] = (uint8_t)(h3 >> 15);
  314. s[22] = (uint8_t)(h3 >> 23);
  315. s[23] = (uint8_t)(h3 >> 31);
  316. s[24] = (uint8_t)(h3 >> 39);
  317. s[25] = (uint8_t)((h3 >> 47) | ((uint32_t)h4 << 4));
  318. s[26] = (uint8_t)(h4 >> 4);
  319. s[27] = (uint8_t)(h4 >> 12);
  320. s[28] = (uint8_t)(h4 >> 20);
  321. s[29] = (uint8_t)(h4 >> 28);
  322. s[30] = (uint8_t)(h4 >> 36);
  323. s[31] = (uint8_t)(h4 >> 44);
  324. }
  325. # if defined(X25519_ASM)
  326. void x25519_fe51_mul(fe51 h, const fe51 f, const fe51 g);
  327. void x25519_fe51_sqr(fe51 h, const fe51 f);
  328. void x25519_fe51_mul121666(fe51 h, fe51 f);
  329. # define fe51_mul x25519_fe51_mul
  330. # define fe51_sq x25519_fe51_sqr
  331. # define fe51_mul121666 x25519_fe51_mul121666
  332. # else
  333. typedef uint128_t u128;
  334. static void fe51_mul(fe51 h, const fe51 f, const fe51 g)
  335. {
  336. u128 h0, h1, h2, h3, h4;
  337. uint64_t f_i, g0, g1, g2, g3, g4;
  338. f_i = f[0];
  339. h0 = (u128)f_i * (g0 = g[0]);
  340. h1 = (u128)f_i * (g1 = g[1]);
  341. h2 = (u128)f_i * (g2 = g[2]);
  342. h3 = (u128)f_i * (g3 = g[3]);
  343. h4 = (u128)f_i * (g4 = g[4]);
  344. f_i = f[1];
  345. h0 += (u128)f_i * (g4 *= 19);
  346. h1 += (u128)f_i * g0;
  347. h2 += (u128)f_i * g1;
  348. h3 += (u128)f_i * g2;
  349. h4 += (u128)f_i * g3;
  350. f_i = f[2];
  351. h0 += (u128)f_i * (g3 *= 19);
  352. h1 += (u128)f_i * g4;
  353. h2 += (u128)f_i * g0;
  354. h3 += (u128)f_i * g1;
  355. h4 += (u128)f_i * g2;
  356. f_i = f[3];
  357. h0 += (u128)f_i * (g2 *= 19);
  358. h1 += (u128)f_i * g3;
  359. h2 += (u128)f_i * g4;
  360. h3 += (u128)f_i * g0;
  361. h4 += (u128)f_i * g1;
  362. f_i = f[4];
  363. h0 += (u128)f_i * (g1 *= 19);
  364. h1 += (u128)f_i * g2;
  365. h2 += (u128)f_i * g3;
  366. h3 += (u128)f_i * g4;
  367. h4 += (u128)f_i * g0;
  368. /* partial [lazy] reduction */
  369. h3 += (uint64_t)(h2 >> 51); g2 = (uint64_t)h2 & MASK51;
  370. h1 += (uint64_t)(h0 >> 51); g0 = (uint64_t)h0 & MASK51;
  371. h4 += (uint64_t)(h3 >> 51); g3 = (uint64_t)h3 & MASK51;
  372. g2 += (uint64_t)(h1 >> 51); g1 = (uint64_t)h1 & MASK51;
  373. g0 += (uint64_t)(h4 >> 51) * 19; g4 = (uint64_t)h4 & MASK51;
  374. g3 += g2 >> 51; g2 &= MASK51;
  375. g1 += g0 >> 51; g0 &= MASK51;
  376. h[0] = g0;
  377. h[1] = g1;
  378. h[2] = g2;
  379. h[3] = g3;
  380. h[4] = g4;
  381. }
  382. static void fe51_sq(fe51 h, const fe51 f)
  383. {
  384. # if defined(OPENSSL_SMALL_FOOTPRINT)
  385. fe51_mul(h, f, f);
  386. # else
  387. /* dedicated squaring gives 16-25% overall improvement */
  388. uint64_t g0 = f[0];
  389. uint64_t g1 = f[1];
  390. uint64_t g2 = f[2];
  391. uint64_t g3 = f[3];
  392. uint64_t g4 = f[4];
  393. u128 h0, h1, h2, h3, h4;
  394. h0 = (u128)g0 * g0; g0 *= 2;
  395. h1 = (u128)g0 * g1;
  396. h2 = (u128)g0 * g2;
  397. h3 = (u128)g0 * g3;
  398. h4 = (u128)g0 * g4;
  399. g0 = g4; /* borrow g0 */
  400. h3 += (u128)g0 * (g4 *= 19);
  401. h2 += (u128)g1 * g1; g1 *= 2;
  402. h3 += (u128)g1 * g2;
  403. h4 += (u128)g1 * g3;
  404. h0 += (u128)g1 * g4;
  405. g0 = g3; /* borrow g0 */
  406. h1 += (u128)g0 * (g3 *= 19);
  407. h2 += (u128)(g0 * 2) * g4;
  408. h4 += (u128)g2 * g2; g2 *= 2;
  409. h0 += (u128)g2 * g3;
  410. h1 += (u128)g2 * g4;
  411. /* partial [lazy] reduction */
  412. h3 += (uint64_t)(h2 >> 51); g2 = (uint64_t)h2 & MASK51;
  413. h1 += (uint64_t)(h0 >> 51); g0 = (uint64_t)h0 & MASK51;
  414. h4 += (uint64_t)(h3 >> 51); g3 = (uint64_t)h3 & MASK51;
  415. g2 += (uint64_t)(h1 >> 51); g1 = (uint64_t)h1 & MASK51;
  416. g0 += (uint64_t)(h4 >> 51) * 19; g4 = (uint64_t)h4 & MASK51;
  417. g3 += g2 >> 51; g2 &= MASK51;
  418. g1 += g0 >> 51; g0 &= MASK51;
  419. h[0] = g0;
  420. h[1] = g1;
  421. h[2] = g2;
  422. h[3] = g3;
  423. h[4] = g4;
  424. # endif
  425. }
  426. static void fe51_mul121666(fe51 h, fe51 f)
  427. {
  428. u128 h0 = f[0] * (u128)121666;
  429. u128 h1 = f[1] * (u128)121666;
  430. u128 h2 = f[2] * (u128)121666;
  431. u128 h3 = f[3] * (u128)121666;
  432. u128 h4 = f[4] * (u128)121666;
  433. uint64_t g0, g1, g2, g3, g4;
  434. h3 += (uint64_t)(h2 >> 51); g2 = (uint64_t)h2 & MASK51;
  435. h1 += (uint64_t)(h0 >> 51); g0 = (uint64_t)h0 & MASK51;
  436. h4 += (uint64_t)(h3 >> 51); g3 = (uint64_t)h3 & MASK51;
  437. g2 += (uint64_t)(h1 >> 51); g1 = (uint64_t)h1 & MASK51;
  438. g0 += (uint64_t)(h4 >> 51) * 19; g4 = (uint64_t)h4 & MASK51;
  439. g3 += g2 >> 51; g2 &= MASK51;
  440. g1 += g0 >> 51; g0 &= MASK51;
  441. h[0] = g0;
  442. h[1] = g1;
  443. h[2] = g2;
  444. h[3] = g3;
  445. h[4] = g4;
  446. }
  447. # endif
  448. static void fe51_add(fe51 h, const fe51 f, const fe51 g)
  449. {
  450. h[0] = f[0] + g[0];
  451. h[1] = f[1] + g[1];
  452. h[2] = f[2] + g[2];
  453. h[3] = f[3] + g[3];
  454. h[4] = f[4] + g[4];
  455. }
  456. static void fe51_sub(fe51 h, const fe51 f, const fe51 g)
  457. {
  458. /*
  459. * Add 2*modulus to ensure that result remains positive
  460. * even if subtrahend is partially reduced.
  461. */
  462. h[0] = (f[0] + 0xfffffffffffda) - g[0];
  463. h[1] = (f[1] + 0xffffffffffffe) - g[1];
  464. h[2] = (f[2] + 0xffffffffffffe) - g[2];
  465. h[3] = (f[3] + 0xffffffffffffe) - g[3];
  466. h[4] = (f[4] + 0xffffffffffffe) - g[4];
  467. }
  468. static void fe51_0(fe51 h)
  469. {
  470. h[0] = 0;
  471. h[1] = 0;
  472. h[2] = 0;
  473. h[3] = 0;
  474. h[4] = 0;
  475. }
  476. static void fe51_1(fe51 h)
  477. {
  478. h[0] = 1;
  479. h[1] = 0;
  480. h[2] = 0;
  481. h[3] = 0;
  482. h[4] = 0;
  483. }
  484. static void fe51_copy(fe51 h, const fe51 f)
  485. {
  486. h[0] = f[0];
  487. h[1] = f[1];
  488. h[2] = f[2];
  489. h[3] = f[3];
  490. h[4] = f[4];
  491. }
  492. static void fe51_cswap(fe51 f, fe51 g, unsigned int b)
  493. {
  494. int i;
  495. uint64_t mask = 0 - (uint64_t)b;
  496. for (i = 0; i < 5; i++) {
  497. int64_t x = f[i] ^ g[i];
  498. x &= mask;
  499. f[i] ^= x;
  500. g[i] ^= x;
  501. }
  502. }
  503. static void fe51_invert(fe51 out, const fe51 z)
  504. {
  505. fe51 t0;
  506. fe51 t1;
  507. fe51 t2;
  508. fe51 t3;
  509. int i;
  510. /*
  511. * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as
  512. * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11.
  513. */
  514. /* t0 = z ** 2 */
  515. fe51_sq(t0, z);
  516. /* t1 = t0 ** (2 ** 2) = z ** 8 */
  517. fe51_sq(t1, t0);
  518. fe51_sq(t1, t1);
  519. /* t1 = z * t1 = z ** 9 */
  520. fe51_mul(t1, z, t1);
  521. /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */
  522. fe51_mul(t0, t0, t1);
  523. /* t2 = t0 ** 2 = z ** 22 */
  524. fe51_sq(t2, t0);
  525. /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */
  526. fe51_mul(t1, t1, t2);
  527. /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */
  528. fe51_sq(t2, t1);
  529. for (i = 1; i < 5; ++i)
  530. fe51_sq(t2, t2);
  531. /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */
  532. fe51_mul(t1, t2, t1);
  533. /* Continuing similarly... */
  534. /* t2 = z ** (2 ** 20 - 1) */
  535. fe51_sq(t2, t1);
  536. for (i = 1; i < 10; ++i)
  537. fe51_sq(t2, t2);
  538. fe51_mul(t2, t2, t1);
  539. /* t2 = z ** (2 ** 40 - 1) */
  540. fe51_sq(t3, t2);
  541. for (i = 1; i < 20; ++i)
  542. fe51_sq(t3, t3);
  543. fe51_mul(t2, t3, t2);
  544. /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */
  545. for (i = 0; i < 10; ++i)
  546. fe51_sq(t2, t2);
  547. /* t1 = z ** (2 ** 50 - 1) */
  548. fe51_mul(t1, t2, t1);
  549. /* t2 = z ** (2 ** 100 - 1) */
  550. fe51_sq(t2, t1);
  551. for (i = 1; i < 50; ++i)
  552. fe51_sq(t2, t2);
  553. fe51_mul(t2, t2, t1);
  554. /* t2 = z ** (2 ** 200 - 1) */
  555. fe51_sq(t3, t2);
  556. for (i = 1; i < 100; ++i)
  557. fe51_sq(t3, t3);
  558. fe51_mul(t2, t3, t2);
  559. /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */
  560. for (i = 0; i < 50; ++i)
  561. fe51_sq(t2, t2);
  562. /* t1 = z ** (2 ** 250 - 1) */
  563. fe51_mul(t1, t2, t1);
  564. /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */
  565. for (i = 0; i < 5; ++i)
  566. fe51_sq(t1, t1);
  567. /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */
  568. fe51_mul(out, t1, t0);
  569. }
  570. /*
  571. * Duplicate of original x25519_scalar_mult_generic, but using
  572. * fe51_* subroutines.
  573. */
  574. static void x25519_scalar_mult(uint8_t out[32], const uint8_t scalar[32],
  575. const uint8_t point[32])
  576. {
  577. fe51 x1, x2, z2, x3, z3, tmp0, tmp1;
  578. uint8_t e[32];
  579. unsigned swap = 0;
  580. int pos;
  581. # ifdef BASE_2_64_IMPLEMENTED
  582. if (x25519_fe64_eligible()) {
  583. x25519_scalar_mulx(out, scalar, point);
  584. return;
  585. }
  586. # endif
  587. memcpy(e, scalar, 32);
  588. e[0] &= 0xf8;
  589. e[31] &= 0x7f;
  590. e[31] |= 0x40;
  591. fe51_frombytes(x1, point);
  592. fe51_1(x2);
  593. fe51_0(z2);
  594. fe51_copy(x3, x1);
  595. fe51_1(z3);
  596. for (pos = 254; pos >= 0; --pos) {
  597. unsigned int b = 1 & (e[pos / 8] >> (pos & 7));
  598. swap ^= b;
  599. fe51_cswap(x2, x3, swap);
  600. fe51_cswap(z2, z3, swap);
  601. swap = b;
  602. fe51_sub(tmp0, x3, z3);
  603. fe51_sub(tmp1, x2, z2);
  604. fe51_add(x2, x2, z2);
  605. fe51_add(z2, x3, z3);
  606. fe51_mul(z3, tmp0, x2);
  607. fe51_mul(z2, z2, tmp1);
  608. fe51_sq(tmp0, tmp1);
  609. fe51_sq(tmp1, x2);
  610. fe51_add(x3, z3, z2);
  611. fe51_sub(z2, z3, z2);
  612. fe51_mul(x2, tmp1, tmp0);
  613. fe51_sub(tmp1, tmp1, tmp0);
  614. fe51_sq(z2, z2);
  615. fe51_mul121666(z3, tmp1);
  616. fe51_sq(x3, x3);
  617. fe51_add(tmp0, tmp0, z3);
  618. fe51_mul(z3, x1, z2);
  619. fe51_mul(z2, tmp1, tmp0);
  620. }
  621. fe51_invert(z2, z2);
  622. fe51_mul(x2, x2, z2);
  623. fe51_tobytes(out, x2);
  624. OPENSSL_cleanse(e, sizeof(e));
  625. }
  626. #endif
  627. /*
  628. * Reference base 2^25.5 implementation.
  629. *
  630. * This code is mostly taken from the ref10 version of Ed25519 in SUPERCOP
  631. * 20141124 (http://bench.cr.yp.to/supercop.html).
  632. *
  633. * The field functions are shared by Ed25519 and X25519 where possible.
  634. */
  635. /*
  636. * fe means field element. Here the field is \Z/(2^255-19). An element t,
  637. * entries t[0]...t[9], represents the integer t[0]+2^26 t[1]+2^51 t[2]+2^77
  638. * t[3]+2^102 t[4]+...+2^230 t[9]. Bounds on each t[i] vary depending on
  639. * context.
  640. */
  641. typedef int32_t fe[10];
  642. static const int64_t kBottom21Bits = 0x1fffffLL;
  643. static const int64_t kBottom25Bits = 0x1ffffffLL;
  644. static const int64_t kBottom26Bits = 0x3ffffffLL;
  645. static const int64_t kTop39Bits = 0xfffffffffe000000LL;
  646. static const int64_t kTop38Bits = 0xfffffffffc000000LL;
  647. static uint64_t load_3(const uint8_t *in)
  648. {
  649. uint64_t result;
  650. result = ((uint64_t)in[0]);
  651. result |= ((uint64_t)in[1]) << 8;
  652. result |= ((uint64_t)in[2]) << 16;
  653. return result;
  654. }
  655. static uint64_t load_4(const uint8_t *in)
  656. {
  657. uint64_t result;
  658. result = ((uint64_t)in[0]);
  659. result |= ((uint64_t)in[1]) << 8;
  660. result |= ((uint64_t)in[2]) << 16;
  661. result |= ((uint64_t)in[3]) << 24;
  662. return result;
  663. }
  664. static void fe_frombytes(fe h, const uint8_t *s)
  665. {
  666. /* Ignores top bit of h. */
  667. int64_t h0 = load_4(s);
  668. int64_t h1 = load_3(s + 4) << 6;
  669. int64_t h2 = load_3(s + 7) << 5;
  670. int64_t h3 = load_3(s + 10) << 3;
  671. int64_t h4 = load_3(s + 13) << 2;
  672. int64_t h5 = load_4(s + 16);
  673. int64_t h6 = load_3(s + 20) << 7;
  674. int64_t h7 = load_3(s + 23) << 5;
  675. int64_t h8 = load_3(s + 26) << 4;
  676. int64_t h9 = (load_3(s + 29) & 0x7fffff) << 2;
  677. int64_t carry0;
  678. int64_t carry1;
  679. int64_t carry2;
  680. int64_t carry3;
  681. int64_t carry4;
  682. int64_t carry5;
  683. int64_t carry6;
  684. int64_t carry7;
  685. int64_t carry8;
  686. int64_t carry9;
  687. carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
  688. carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
  689. carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
  690. carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
  691. carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;
  692. carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
  693. carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
  694. carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
  695. carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
  696. carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;
  697. h[0] = (int32_t)h0;
  698. h[1] = (int32_t)h1;
  699. h[2] = (int32_t)h2;
  700. h[3] = (int32_t)h3;
  701. h[4] = (int32_t)h4;
  702. h[5] = (int32_t)h5;
  703. h[6] = (int32_t)h6;
  704. h[7] = (int32_t)h7;
  705. h[8] = (int32_t)h8;
  706. h[9] = (int32_t)h9;
  707. }
  708. /*
  709. * Preconditions:
  710. * |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
  711. *
  712. * Write p=2^255-19; q=floor(h/p).
  713. * Basic claim: q = floor(2^(-255)(h + 19 2^(-25)h9 + 2^(-1))).
  714. *
  715. * Proof:
  716. * Have |h|<=p so |q|<=1 so |19^2 2^(-255) q|<1/4.
  717. * Also have |h-2^230 h9|<2^231 so |19 2^(-255)(h-2^230 h9)|<1/4.
  718. *
  719. * Write y=2^(-1)-19^2 2^(-255)q-19 2^(-255)(h-2^230 h9).
  720. * Then 0<y<1.
  721. *
  722. * Write r=h-pq.
  723. * Have 0<=r<=p-1=2^255-20.
  724. * Thus 0<=r+19(2^-255)r<r+19(2^-255)2^255<=2^255-1.
  725. *
  726. * Write x=r+19(2^-255)r+y.
  727. * Then 0<x<2^255 so floor(2^(-255)x) = 0 so floor(q+2^(-255)x) = q.
  728. *
  729. * Have q+2^(-255)x = 2^(-255)(h + 19 2^(-25) h9 + 2^(-1))
  730. * so floor(2^(-255)(h + 19 2^(-25) h9 + 2^(-1))) = q.
  731. */
  732. static void fe_tobytes(uint8_t *s, const fe h)
  733. {
  734. int32_t h0 = h[0];
  735. int32_t h1 = h[1];
  736. int32_t h2 = h[2];
  737. int32_t h3 = h[3];
  738. int32_t h4 = h[4];
  739. int32_t h5 = h[5];
  740. int32_t h6 = h[6];
  741. int32_t h7 = h[7];
  742. int32_t h8 = h[8];
  743. int32_t h9 = h[9];
  744. int32_t q;
  745. q = (19 * h9 + (((int32_t) 1) << 24)) >> 25;
  746. q = (h0 + q) >> 26;
  747. q = (h1 + q) >> 25;
  748. q = (h2 + q) >> 26;
  749. q = (h3 + q) >> 25;
  750. q = (h4 + q) >> 26;
  751. q = (h5 + q) >> 25;
  752. q = (h6 + q) >> 26;
  753. q = (h7 + q) >> 25;
  754. q = (h8 + q) >> 26;
  755. q = (h9 + q) >> 25;
  756. /* Goal: Output h-(2^255-19)q, which is between 0 and 2^255-20. */
  757. h0 += 19 * q;
  758. /* Goal: Output h-2^255 q, which is between 0 and 2^255-20. */
  759. h1 += h0 >> 26; h0 &= kBottom26Bits;
  760. h2 += h1 >> 25; h1 &= kBottom25Bits;
  761. h3 += h2 >> 26; h2 &= kBottom26Bits;
  762. h4 += h3 >> 25; h3 &= kBottom25Bits;
  763. h5 += h4 >> 26; h4 &= kBottom26Bits;
  764. h6 += h5 >> 25; h5 &= kBottom25Bits;
  765. h7 += h6 >> 26; h6 &= kBottom26Bits;
  766. h8 += h7 >> 25; h7 &= kBottom25Bits;
  767. h9 += h8 >> 26; h8 &= kBottom26Bits;
  768. h9 &= kBottom25Bits;
  769. /* h10 = carry9 */
  770. /*
  771. * Goal: Output h0+...+2^255 h10-2^255 q, which is between 0 and 2^255-20.
  772. * Have h0+...+2^230 h9 between 0 and 2^255-1;
  773. * evidently 2^255 h10-2^255 q = 0.
  774. * Goal: Output h0+...+2^230 h9.
  775. */
  776. s[ 0] = (uint8_t) (h0 >> 0);
  777. s[ 1] = (uint8_t) (h0 >> 8);
  778. s[ 2] = (uint8_t) (h0 >> 16);
  779. s[ 3] = (uint8_t)((h0 >> 24) | ((uint32_t)(h1) << 2));
  780. s[ 4] = (uint8_t) (h1 >> 6);
  781. s[ 5] = (uint8_t) (h1 >> 14);
  782. s[ 6] = (uint8_t)((h1 >> 22) | ((uint32_t)(h2) << 3));
  783. s[ 7] = (uint8_t) (h2 >> 5);
  784. s[ 8] = (uint8_t) (h2 >> 13);
  785. s[ 9] = (uint8_t)((h2 >> 21) | ((uint32_t)(h3) << 5));
  786. s[10] = (uint8_t) (h3 >> 3);
  787. s[11] = (uint8_t) (h3 >> 11);
  788. s[12] = (uint8_t)((h3 >> 19) | ((uint32_t)(h4) << 6));
  789. s[13] = (uint8_t) (h4 >> 2);
  790. s[14] = (uint8_t) (h4 >> 10);
  791. s[15] = (uint8_t) (h4 >> 18);
  792. s[16] = (uint8_t) (h5 >> 0);
  793. s[17] = (uint8_t) (h5 >> 8);
  794. s[18] = (uint8_t) (h5 >> 16);
  795. s[19] = (uint8_t)((h5 >> 24) | ((uint32_t)(h6) << 1));
  796. s[20] = (uint8_t) (h6 >> 7);
  797. s[21] = (uint8_t) (h6 >> 15);
  798. s[22] = (uint8_t)((h6 >> 23) | ((uint32_t)(h7) << 3));
  799. s[23] = (uint8_t) (h7 >> 5);
  800. s[24] = (uint8_t) (h7 >> 13);
  801. s[25] = (uint8_t)((h7 >> 21) | ((uint32_t)(h8) << 4));
  802. s[26] = (uint8_t) (h8 >> 4);
  803. s[27] = (uint8_t) (h8 >> 12);
  804. s[28] = (uint8_t)((h8 >> 20) | ((uint32_t)(h9) << 6));
  805. s[29] = (uint8_t) (h9 >> 2);
  806. s[30] = (uint8_t) (h9 >> 10);
  807. s[31] = (uint8_t) (h9 >> 18);
  808. }
  809. /* h = f */
  810. static void fe_copy(fe h, const fe f)
  811. {
  812. memmove(h, f, sizeof(int32_t) * 10);
  813. }
  814. /* h = 0 */
  815. static void fe_0(fe h)
  816. {
  817. memset(h, 0, sizeof(int32_t) * 10);
  818. }
  819. /* h = 1 */
  820. static void fe_1(fe h)
  821. {
  822. memset(h, 0, sizeof(int32_t) * 10);
  823. h[0] = 1;
  824. }
  825. /*
  826. * h = f + g
  827. *
  828. * Can overlap h with f or g.
  829. *
  830. * Preconditions:
  831. * |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
  832. * |g| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
  833. *
  834. * Postconditions:
  835. * |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
  836. */
  837. static void fe_add(fe h, const fe f, const fe g)
  838. {
  839. unsigned i;
  840. for (i = 0; i < 10; i++) {
  841. h[i] = f[i] + g[i];
  842. }
  843. }
  844. /*
  845. * h = f - g
  846. *
  847. * Can overlap h with f or g.
  848. *
  849. * Preconditions:
  850. * |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
  851. * |g| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
  852. *
  853. * Postconditions:
  854. * |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
  855. */
  856. static void fe_sub(fe h, const fe f, const fe g)
  857. {
  858. unsigned i;
  859. for (i = 0; i < 10; i++) {
  860. h[i] = f[i] - g[i];
  861. }
  862. }
  863. /*
  864. * h = f * g
  865. *
  866. * Can overlap h with f or g.
  867. *
  868. * Preconditions:
  869. * |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
  870. * |g| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
  871. *
  872. * Postconditions:
  873. * |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
  874. *
  875. * Notes on implementation strategy:
  876. *
  877. * Using schoolbook multiplication.
  878. * Karatsuba would save a little in some cost models.
  879. *
  880. * Most multiplications by 2 and 19 are 32-bit precomputations;
  881. * cheaper than 64-bit postcomputations.
  882. *
  883. * There is one remaining multiplication by 19 in the carry chain;
  884. * one *19 precomputation can be merged into this,
  885. * but the resulting data flow is considerably less clean.
  886. *
  887. * There are 12 carries below.
  888. * 10 of them are 2-way parallelizable and vectorizable.
  889. * Can get away with 11 carries, but then data flow is much deeper.
  890. *
  891. * With tighter constraints on inputs can squeeze carries into int32.
  892. */
  893. static void fe_mul(fe h, const fe f, const fe g)
  894. {
  895. int32_t f0 = f[0];
  896. int32_t f1 = f[1];
  897. int32_t f2 = f[2];
  898. int32_t f3 = f[3];
  899. int32_t f4 = f[4];
  900. int32_t f5 = f[5];
  901. int32_t f6 = f[6];
  902. int32_t f7 = f[7];
  903. int32_t f8 = f[8];
  904. int32_t f9 = f[9];
  905. int32_t g0 = g[0];
  906. int32_t g1 = g[1];
  907. int32_t g2 = g[2];
  908. int32_t g3 = g[3];
  909. int32_t g4 = g[4];
  910. int32_t g5 = g[5];
  911. int32_t g6 = g[6];
  912. int32_t g7 = g[7];
  913. int32_t g8 = g[8];
  914. int32_t g9 = g[9];
  915. int32_t g1_19 = 19 * g1; /* 1.959375*2^29 */
  916. int32_t g2_19 = 19 * g2; /* 1.959375*2^30; still ok */
  917. int32_t g3_19 = 19 * g3;
  918. int32_t g4_19 = 19 * g4;
  919. int32_t g5_19 = 19 * g5;
  920. int32_t g6_19 = 19 * g6;
  921. int32_t g7_19 = 19 * g7;
  922. int32_t g8_19 = 19 * g8;
  923. int32_t g9_19 = 19 * g9;
  924. int32_t f1_2 = 2 * f1;
  925. int32_t f3_2 = 2 * f3;
  926. int32_t f5_2 = 2 * f5;
  927. int32_t f7_2 = 2 * f7;
  928. int32_t f9_2 = 2 * f9;
  929. int64_t f0g0 = f0 * (int64_t) g0;
  930. int64_t f0g1 = f0 * (int64_t) g1;
  931. int64_t f0g2 = f0 * (int64_t) g2;
  932. int64_t f0g3 = f0 * (int64_t) g3;
  933. int64_t f0g4 = f0 * (int64_t) g4;
  934. int64_t f0g5 = f0 * (int64_t) g5;
  935. int64_t f0g6 = f0 * (int64_t) g6;
  936. int64_t f0g7 = f0 * (int64_t) g7;
  937. int64_t f0g8 = f0 * (int64_t) g8;
  938. int64_t f0g9 = f0 * (int64_t) g9;
  939. int64_t f1g0 = f1 * (int64_t) g0;
  940. int64_t f1g1_2 = f1_2 * (int64_t) g1;
  941. int64_t f1g2 = f1 * (int64_t) g2;
  942. int64_t f1g3_2 = f1_2 * (int64_t) g3;
  943. int64_t f1g4 = f1 * (int64_t) g4;
  944. int64_t f1g5_2 = f1_2 * (int64_t) g5;
  945. int64_t f1g6 = f1 * (int64_t) g6;
  946. int64_t f1g7_2 = f1_2 * (int64_t) g7;
  947. int64_t f1g8 = f1 * (int64_t) g8;
  948. int64_t f1g9_38 = f1_2 * (int64_t) g9_19;
  949. int64_t f2g0 = f2 * (int64_t) g0;
  950. int64_t f2g1 = f2 * (int64_t) g1;
  951. int64_t f2g2 = f2 * (int64_t) g2;
  952. int64_t f2g3 = f2 * (int64_t) g3;
  953. int64_t f2g4 = f2 * (int64_t) g4;
  954. int64_t f2g5 = f2 * (int64_t) g5;
  955. int64_t f2g6 = f2 * (int64_t) g6;
  956. int64_t f2g7 = f2 * (int64_t) g7;
  957. int64_t f2g8_19 = f2 * (int64_t) g8_19;
  958. int64_t f2g9_19 = f2 * (int64_t) g9_19;
  959. int64_t f3g0 = f3 * (int64_t) g0;
  960. int64_t f3g1_2 = f3_2 * (int64_t) g1;
  961. int64_t f3g2 = f3 * (int64_t) g2;
  962. int64_t f3g3_2 = f3_2 * (int64_t) g3;
  963. int64_t f3g4 = f3 * (int64_t) g4;
  964. int64_t f3g5_2 = f3_2 * (int64_t) g5;
  965. int64_t f3g6 = f3 * (int64_t) g6;
  966. int64_t f3g7_38 = f3_2 * (int64_t) g7_19;
  967. int64_t f3g8_19 = f3 * (int64_t) g8_19;
  968. int64_t f3g9_38 = f3_2 * (int64_t) g9_19;
  969. int64_t f4g0 = f4 * (int64_t) g0;
  970. int64_t f4g1 = f4 * (int64_t) g1;
  971. int64_t f4g2 = f4 * (int64_t) g2;
  972. int64_t f4g3 = f4 * (int64_t) g3;
  973. int64_t f4g4 = f4 * (int64_t) g4;
  974. int64_t f4g5 = f4 * (int64_t) g5;
  975. int64_t f4g6_19 = f4 * (int64_t) g6_19;
  976. int64_t f4g7_19 = f4 * (int64_t) g7_19;
  977. int64_t f4g8_19 = f4 * (int64_t) g8_19;
  978. int64_t f4g9_19 = f4 * (int64_t) g9_19;
  979. int64_t f5g0 = f5 * (int64_t) g0;
  980. int64_t f5g1_2 = f5_2 * (int64_t) g1;
  981. int64_t f5g2 = f5 * (int64_t) g2;
  982. int64_t f5g3_2 = f5_2 * (int64_t) g3;
  983. int64_t f5g4 = f5 * (int64_t) g4;
  984. int64_t f5g5_38 = f5_2 * (int64_t) g5_19;
  985. int64_t f5g6_19 = f5 * (int64_t) g6_19;
  986. int64_t f5g7_38 = f5_2 * (int64_t) g7_19;
  987. int64_t f5g8_19 = f5 * (int64_t) g8_19;
  988. int64_t f5g9_38 = f5_2 * (int64_t) g9_19;
  989. int64_t f6g0 = f6 * (int64_t) g0;
  990. int64_t f6g1 = f6 * (int64_t) g1;
  991. int64_t f6g2 = f6 * (int64_t) g2;
  992. int64_t f6g3 = f6 * (int64_t) g3;
  993. int64_t f6g4_19 = f6 * (int64_t) g4_19;
  994. int64_t f6g5_19 = f6 * (int64_t) g5_19;
  995. int64_t f6g6_19 = f6 * (int64_t) g6_19;
  996. int64_t f6g7_19 = f6 * (int64_t) g7_19;
  997. int64_t f6g8_19 = f6 * (int64_t) g8_19;
  998. int64_t f6g9_19 = f6 * (int64_t) g9_19;
  999. int64_t f7g0 = f7 * (int64_t) g0;
  1000. int64_t f7g1_2 = f7_2 * (int64_t) g1;
  1001. int64_t f7g2 = f7 * (int64_t) g2;
  1002. int64_t f7g3_38 = f7_2 * (int64_t) g3_19;
  1003. int64_t f7g4_19 = f7 * (int64_t) g4_19;
  1004. int64_t f7g5_38 = f7_2 * (int64_t) g5_19;
  1005. int64_t f7g6_19 = f7 * (int64_t) g6_19;
  1006. int64_t f7g7_38 = f7_2 * (int64_t) g7_19;
  1007. int64_t f7g8_19 = f7 * (int64_t) g8_19;
  1008. int64_t f7g9_38 = f7_2 * (int64_t) g9_19;
  1009. int64_t f8g0 = f8 * (int64_t) g0;
  1010. int64_t f8g1 = f8 * (int64_t) g1;
  1011. int64_t f8g2_19 = f8 * (int64_t) g2_19;
  1012. int64_t f8g3_19 = f8 * (int64_t) g3_19;
  1013. int64_t f8g4_19 = f8 * (int64_t) g4_19;
  1014. int64_t f8g5_19 = f8 * (int64_t) g5_19;
  1015. int64_t f8g6_19 = f8 * (int64_t) g6_19;
  1016. int64_t f8g7_19 = f8 * (int64_t) g7_19;
  1017. int64_t f8g8_19 = f8 * (int64_t) g8_19;
  1018. int64_t f8g9_19 = f8 * (int64_t) g9_19;
  1019. int64_t f9g0 = f9 * (int64_t) g0;
  1020. int64_t f9g1_38 = f9_2 * (int64_t) g1_19;
  1021. int64_t f9g2_19 = f9 * (int64_t) g2_19;
  1022. int64_t f9g3_38 = f9_2 * (int64_t) g3_19;
  1023. int64_t f9g4_19 = f9 * (int64_t) g4_19;
  1024. int64_t f9g5_38 = f9_2 * (int64_t) g5_19;
  1025. int64_t f9g6_19 = f9 * (int64_t) g6_19;
  1026. int64_t f9g7_38 = f9_2 * (int64_t) g7_19;
  1027. int64_t f9g8_19 = f9 * (int64_t) g8_19;
  1028. int64_t f9g9_38 = f9_2 * (int64_t) g9_19;
  1029. int64_t h0 = f0g0 + f1g9_38 + f2g8_19 + f3g7_38 + f4g6_19 + f5g5_38 + f6g4_19 + f7g3_38 + f8g2_19 + f9g1_38;
  1030. int64_t h1 = f0g1 + f1g0 + f2g9_19 + f3g8_19 + f4g7_19 + f5g6_19 + f6g5_19 + f7g4_19 + f8g3_19 + f9g2_19;
  1031. int64_t h2 = f0g2 + f1g1_2 + f2g0 + f3g9_38 + f4g8_19 + f5g7_38 + f6g6_19 + f7g5_38 + f8g4_19 + f9g3_38;
  1032. int64_t h3 = f0g3 + f1g2 + f2g1 + f3g0 + f4g9_19 + f5g8_19 + f6g7_19 + f7g6_19 + f8g5_19 + f9g4_19;
  1033. int64_t h4 = f0g4 + f1g3_2 + f2g2 + f3g1_2 + f4g0 + f5g9_38 + f6g8_19 + f7g7_38 + f8g6_19 + f9g5_38;
  1034. int64_t h5 = f0g5 + f1g4 + f2g3 + f3g2 + f4g1 + f5g0 + f6g9_19 + f7g8_19 + f8g7_19 + f9g6_19;
  1035. int64_t h6 = f0g6 + f1g5_2 + f2g4 + f3g3_2 + f4g2 + f5g1_2 + f6g0 + f7g9_38 + f8g8_19 + f9g7_38;
  1036. int64_t h7 = f0g7 + f1g6 + f2g5 + f3g4 + f4g3 + f5g2 + f6g1 + f7g0 + f8g9_19 + f9g8_19;
  1037. int64_t h8 = f0g8 + f1g7_2 + f2g6 + f3g5_2 + f4g4 + f5g3_2 + f6g2 + f7g1_2 + f8g0 + f9g9_38;
  1038. int64_t h9 = f0g9 + f1g8 + f2g7 + f3g6 + f4g5 + f5g4 + f6g3 + f7g2 + f8g1 + f9g0 ;
  1039. int64_t carry0;
  1040. int64_t carry1;
  1041. int64_t carry2;
  1042. int64_t carry3;
  1043. int64_t carry4;
  1044. int64_t carry5;
  1045. int64_t carry6;
  1046. int64_t carry7;
  1047. int64_t carry8;
  1048. int64_t carry9;
  1049. /* |h0| <= (1.65*1.65*2^52*(1+19+19+19+19)+1.65*1.65*2^50*(38+38+38+38+38))
  1050. * i.e. |h0| <= 1.4*2^60; narrower ranges for h2, h4, h6, h8
  1051. * |h1| <= (1.65*1.65*2^51*(1+1+19+19+19+19+19+19+19+19))
  1052. * i.e. |h1| <= 1.7*2^59; narrower ranges for h3, h5, h7, h9 */
  1053. carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
  1054. carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
  1055. /* |h0| <= 2^25 */
  1056. /* |h4| <= 2^25 */
  1057. /* |h1| <= 1.71*2^59 */
  1058. /* |h5| <= 1.71*2^59 */
  1059. carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
  1060. carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
  1061. /* |h1| <= 2^24; from now on fits into int32 */
  1062. /* |h5| <= 2^24; from now on fits into int32 */
  1063. /* |h2| <= 1.41*2^60 */
  1064. /* |h6| <= 1.41*2^60 */
  1065. carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
  1066. carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
  1067. /* |h2| <= 2^25; from now on fits into int32 unchanged */
  1068. /* |h6| <= 2^25; from now on fits into int32 unchanged */
  1069. /* |h3| <= 1.71*2^59 */
  1070. /* |h7| <= 1.71*2^59 */
  1071. carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
  1072. carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;
  1073. /* |h3| <= 2^24; from now on fits into int32 unchanged */
  1074. /* |h7| <= 2^24; from now on fits into int32 unchanged */
  1075. /* |h4| <= 1.72*2^34 */
  1076. /* |h8| <= 1.41*2^60 */
  1077. carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
  1078. carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;
  1079. /* |h4| <= 2^25; from now on fits into int32 unchanged */
  1080. /* |h8| <= 2^25; from now on fits into int32 unchanged */
  1081. /* |h5| <= 1.01*2^24 */
  1082. /* |h9| <= 1.71*2^59 */
  1083. carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
  1084. /* |h9| <= 2^24; from now on fits into int32 unchanged */
  1085. /* |h0| <= 1.1*2^39 */
  1086. carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
  1087. /* |h0| <= 2^25; from now on fits into int32 unchanged */
  1088. /* |h1| <= 1.01*2^24 */
  1089. h[0] = (int32_t)h0;
  1090. h[1] = (int32_t)h1;
  1091. h[2] = (int32_t)h2;
  1092. h[3] = (int32_t)h3;
  1093. h[4] = (int32_t)h4;
  1094. h[5] = (int32_t)h5;
  1095. h[6] = (int32_t)h6;
  1096. h[7] = (int32_t)h7;
  1097. h[8] = (int32_t)h8;
  1098. h[9] = (int32_t)h9;
  1099. }
  1100. /*
  1101. * h = f * f
  1102. *
  1103. * Can overlap h with f.
  1104. *
  1105. * Preconditions:
  1106. * |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
  1107. *
  1108. * Postconditions:
  1109. * |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
  1110. *
  1111. * See fe_mul.c for discussion of implementation strategy.
  1112. */
  1113. static void fe_sq(fe h, const fe f)
  1114. {
  1115. int32_t f0 = f[0];
  1116. int32_t f1 = f[1];
  1117. int32_t f2 = f[2];
  1118. int32_t f3 = f[3];
  1119. int32_t f4 = f[4];
  1120. int32_t f5 = f[5];
  1121. int32_t f6 = f[6];
  1122. int32_t f7 = f[7];
  1123. int32_t f8 = f[8];
  1124. int32_t f9 = f[9];
  1125. int32_t f0_2 = 2 * f0;
  1126. int32_t f1_2 = 2 * f1;
  1127. int32_t f2_2 = 2 * f2;
  1128. int32_t f3_2 = 2 * f3;
  1129. int32_t f4_2 = 2 * f4;
  1130. int32_t f5_2 = 2 * f5;
  1131. int32_t f6_2 = 2 * f6;
  1132. int32_t f7_2 = 2 * f7;
  1133. int32_t f5_38 = 38 * f5; /* 1.959375*2^30 */
  1134. int32_t f6_19 = 19 * f6; /* 1.959375*2^30 */
  1135. int32_t f7_38 = 38 * f7; /* 1.959375*2^30 */
  1136. int32_t f8_19 = 19 * f8; /* 1.959375*2^30 */
  1137. int32_t f9_38 = 38 * f9; /* 1.959375*2^30 */
  1138. int64_t f0f0 = f0 * (int64_t) f0;
  1139. int64_t f0f1_2 = f0_2 * (int64_t) f1;
  1140. int64_t f0f2_2 = f0_2 * (int64_t) f2;
  1141. int64_t f0f3_2 = f0_2 * (int64_t) f3;
  1142. int64_t f0f4_2 = f0_2 * (int64_t) f4;
  1143. int64_t f0f5_2 = f0_2 * (int64_t) f5;
  1144. int64_t f0f6_2 = f0_2 * (int64_t) f6;
  1145. int64_t f0f7_2 = f0_2 * (int64_t) f7;
  1146. int64_t f0f8_2 = f0_2 * (int64_t) f8;
  1147. int64_t f0f9_2 = f0_2 * (int64_t) f9;
  1148. int64_t f1f1_2 = f1_2 * (int64_t) f1;
  1149. int64_t f1f2_2 = f1_2 * (int64_t) f2;
  1150. int64_t f1f3_4 = f1_2 * (int64_t) f3_2;
  1151. int64_t f1f4_2 = f1_2 * (int64_t) f4;
  1152. int64_t f1f5_4 = f1_2 * (int64_t) f5_2;
  1153. int64_t f1f6_2 = f1_2 * (int64_t) f6;
  1154. int64_t f1f7_4 = f1_2 * (int64_t) f7_2;
  1155. int64_t f1f8_2 = f1_2 * (int64_t) f8;
  1156. int64_t f1f9_76 = f1_2 * (int64_t) f9_38;
  1157. int64_t f2f2 = f2 * (int64_t) f2;
  1158. int64_t f2f3_2 = f2_2 * (int64_t) f3;
  1159. int64_t f2f4_2 = f2_2 * (int64_t) f4;
  1160. int64_t f2f5_2 = f2_2 * (int64_t) f5;
  1161. int64_t f2f6_2 = f2_2 * (int64_t) f6;
  1162. int64_t f2f7_2 = f2_2 * (int64_t) f7;
  1163. int64_t f2f8_38 = f2_2 * (int64_t) f8_19;
  1164. int64_t f2f9_38 = f2 * (int64_t) f9_38;
  1165. int64_t f3f3_2 = f3_2 * (int64_t) f3;
  1166. int64_t f3f4_2 = f3_2 * (int64_t) f4;
  1167. int64_t f3f5_4 = f3_2 * (int64_t) f5_2;
  1168. int64_t f3f6_2 = f3_2 * (int64_t) f6;
  1169. int64_t f3f7_76 = f3_2 * (int64_t) f7_38;
  1170. int64_t f3f8_38 = f3_2 * (int64_t) f8_19;
  1171. int64_t f3f9_76 = f3_2 * (int64_t) f9_38;
  1172. int64_t f4f4 = f4 * (int64_t) f4;
  1173. int64_t f4f5_2 = f4_2 * (int64_t) f5;
  1174. int64_t f4f6_38 = f4_2 * (int64_t) f6_19;
  1175. int64_t f4f7_38 = f4 * (int64_t) f7_38;
  1176. int64_t f4f8_38 = f4_2 * (int64_t) f8_19;
  1177. int64_t f4f9_38 = f4 * (int64_t) f9_38;
  1178. int64_t f5f5_38 = f5 * (int64_t) f5_38;
  1179. int64_t f5f6_38 = f5_2 * (int64_t) f6_19;
  1180. int64_t f5f7_76 = f5_2 * (int64_t) f7_38;
  1181. int64_t f5f8_38 = f5_2 * (int64_t) f8_19;
  1182. int64_t f5f9_76 = f5_2 * (int64_t) f9_38;
  1183. int64_t f6f6_19 = f6 * (int64_t) f6_19;
  1184. int64_t f6f7_38 = f6 * (int64_t) f7_38;
  1185. int64_t f6f8_38 = f6_2 * (int64_t) f8_19;
  1186. int64_t f6f9_38 = f6 * (int64_t) f9_38;
  1187. int64_t f7f7_38 = f7 * (int64_t) f7_38;
  1188. int64_t f7f8_38 = f7_2 * (int64_t) f8_19;
  1189. int64_t f7f9_76 = f7_2 * (int64_t) f9_38;
  1190. int64_t f8f8_19 = f8 * (int64_t) f8_19;
  1191. int64_t f8f9_38 = f8 * (int64_t) f9_38;
  1192. int64_t f9f9_38 = f9 * (int64_t) f9_38;
  1193. int64_t h0 = f0f0 + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38;
  1194. int64_t h1 = f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38;
  1195. int64_t h2 = f0f2_2 + f1f1_2 + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19;
  1196. int64_t h3 = f0f3_2 + f1f2_2 + f4f9_38 + f5f8_38 + f6f7_38;
  1197. int64_t h4 = f0f4_2 + f1f3_4 + f2f2 + f5f9_76 + f6f8_38 + f7f7_38;
  1198. int64_t h5 = f0f5_2 + f1f4_2 + f2f3_2 + f6f9_38 + f7f8_38;
  1199. int64_t h6 = f0f6_2 + f1f5_4 + f2f4_2 + f3f3_2 + f7f9_76 + f8f8_19;
  1200. int64_t h7 = f0f7_2 + f1f6_2 + f2f5_2 + f3f4_2 + f8f9_38;
  1201. int64_t h8 = f0f8_2 + f1f7_4 + f2f6_2 + f3f5_4 + f4f4 + f9f9_38;
  1202. int64_t h9 = f0f9_2 + f1f8_2 + f2f7_2 + f3f6_2 + f4f5_2;
  1203. int64_t carry0;
  1204. int64_t carry1;
  1205. int64_t carry2;
  1206. int64_t carry3;
  1207. int64_t carry4;
  1208. int64_t carry5;
  1209. int64_t carry6;
  1210. int64_t carry7;
  1211. int64_t carry8;
  1212. int64_t carry9;
  1213. carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
  1214. carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
  1215. carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
  1216. carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
  1217. carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
  1218. carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
  1219. carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
  1220. carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;
  1221. carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
  1222. carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;
  1223. carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
  1224. carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
  1225. h[0] = (int32_t)h0;
  1226. h[1] = (int32_t)h1;
  1227. h[2] = (int32_t)h2;
  1228. h[3] = (int32_t)h3;
  1229. h[4] = (int32_t)h4;
  1230. h[5] = (int32_t)h5;
  1231. h[6] = (int32_t)h6;
  1232. h[7] = (int32_t)h7;
  1233. h[8] = (int32_t)h8;
  1234. h[9] = (int32_t)h9;
  1235. }
  1236. static void fe_invert(fe out, const fe z)
  1237. {
  1238. fe t0;
  1239. fe t1;
  1240. fe t2;
  1241. fe t3;
  1242. int i;
  1243. /*
  1244. * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as
  1245. * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11.
  1246. */
  1247. /* t0 = z ** 2 */
  1248. fe_sq(t0, z);
  1249. /* t1 = t0 ** (2 ** 2) = z ** 8 */
  1250. fe_sq(t1, t0);
  1251. fe_sq(t1, t1);
  1252. /* t1 = z * t1 = z ** 9 */
  1253. fe_mul(t1, z, t1);
  1254. /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */
  1255. fe_mul(t0, t0, t1);
  1256. /* t2 = t0 ** 2 = z ** 22 */
  1257. fe_sq(t2, t0);
  1258. /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */
  1259. fe_mul(t1, t1, t2);
  1260. /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */
  1261. fe_sq(t2, t1);
  1262. for (i = 1; i < 5; ++i) {
  1263. fe_sq(t2, t2);
  1264. }
  1265. /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */
  1266. fe_mul(t1, t2, t1);
  1267. /* Continuing similarly... */
  1268. /* t2 = z ** (2 ** 20 - 1) */
  1269. fe_sq(t2, t1);
  1270. for (i = 1; i < 10; ++i) {
  1271. fe_sq(t2, t2);
  1272. }
  1273. fe_mul(t2, t2, t1);
  1274. /* t2 = z ** (2 ** 40 - 1) */
  1275. fe_sq(t3, t2);
  1276. for (i = 1; i < 20; ++i) {
  1277. fe_sq(t3, t3);
  1278. }
  1279. fe_mul(t2, t3, t2);
  1280. /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */
  1281. for (i = 0; i < 10; ++i) {
  1282. fe_sq(t2, t2);
  1283. }
  1284. /* t1 = z ** (2 ** 50 - 1) */
  1285. fe_mul(t1, t2, t1);
  1286. /* t2 = z ** (2 ** 100 - 1) */
  1287. fe_sq(t2, t1);
  1288. for (i = 1; i < 50; ++i) {
  1289. fe_sq(t2, t2);
  1290. }
  1291. fe_mul(t2, t2, t1);
  1292. /* t2 = z ** (2 ** 200 - 1) */
  1293. fe_sq(t3, t2);
  1294. for (i = 1; i < 100; ++i) {
  1295. fe_sq(t3, t3);
  1296. }
  1297. fe_mul(t2, t3, t2);
  1298. /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */
  1299. fe_sq(t2, t2);
  1300. for (i = 1; i < 50; ++i) {
  1301. fe_sq(t2, t2);
  1302. }
  1303. /* t1 = z ** (2 ** 250 - 1) */
  1304. fe_mul(t1, t2, t1);
  1305. /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */
  1306. fe_sq(t1, t1);
  1307. for (i = 1; i < 5; ++i) {
  1308. fe_sq(t1, t1);
  1309. }
  1310. /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */
  1311. fe_mul(out, t1, t0);
  1312. }
  1313. /*
  1314. * h = -f
  1315. *
  1316. * Preconditions:
  1317. * |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
  1318. *
  1319. * Postconditions:
  1320. * |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
  1321. */
  1322. static void fe_neg(fe h, const fe f)
  1323. {
  1324. unsigned i;
  1325. for (i = 0; i < 10; i++) {
  1326. h[i] = -f[i];
  1327. }
  1328. }
  1329. /*
  1330. * Replace (f,g) with (g,g) if b == 1;
  1331. * replace (f,g) with (f,g) if b == 0.
  1332. *
  1333. * Preconditions: b in {0,1}.
  1334. */
  1335. static void fe_cmov(fe f, const fe g, unsigned b)
  1336. {
  1337. size_t i;
  1338. b = 0-b;
  1339. for (i = 0; i < 10; i++) {
  1340. int32_t x = f[i] ^ g[i];
  1341. x &= b;
  1342. f[i] ^= x;
  1343. }
  1344. }
  1345. /*
  1346. * return 0 if f == 0
  1347. * return 1 if f != 0
  1348. *
  1349. * Preconditions:
  1350. * |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
  1351. */
  1352. static int fe_isnonzero(const fe f)
  1353. {
  1354. uint8_t s[32];
  1355. static const uint8_t zero[32] = {0};
  1356. fe_tobytes(s, f);
  1357. return CRYPTO_memcmp(s, zero, sizeof(zero)) != 0;
  1358. }
  1359. /*
  1360. * return 1 if f is in {1,3,5,...,q-2}
  1361. * return 0 if f is in {0,2,4,...,q-1}
  1362. *
  1363. * Preconditions:
  1364. * |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
  1365. */
  1366. static int fe_isnegative(const fe f)
  1367. {
  1368. uint8_t s[32];
  1369. fe_tobytes(s, f);
  1370. return s[0] & 1;
  1371. }
  1372. /*
  1373. * h = 2 * f * f
  1374. *
  1375. * Can overlap h with f.
  1376. *
  1377. * Preconditions:
  1378. * |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
  1379. *
  1380. * Postconditions:
  1381. * |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
  1382. *
  1383. * See fe_mul.c for discussion of implementation strategy.
  1384. */
  1385. static void fe_sq2(fe h, const fe f)
  1386. {
  1387. int32_t f0 = f[0];
  1388. int32_t f1 = f[1];
  1389. int32_t f2 = f[2];
  1390. int32_t f3 = f[3];
  1391. int32_t f4 = f[4];
  1392. int32_t f5 = f[5];
  1393. int32_t f6 = f[6];
  1394. int32_t f7 = f[7];
  1395. int32_t f8 = f[8];
  1396. int32_t f9 = f[9];
  1397. int32_t f0_2 = 2 * f0;
  1398. int32_t f1_2 = 2 * f1;
  1399. int32_t f2_2 = 2 * f2;
  1400. int32_t f3_2 = 2 * f3;
  1401. int32_t f4_2 = 2 * f4;
  1402. int32_t f5_2 = 2 * f5;
  1403. int32_t f6_2 = 2 * f6;
  1404. int32_t f7_2 = 2 * f7;
  1405. int32_t f5_38 = 38 * f5; /* 1.959375*2^30 */
  1406. int32_t f6_19 = 19 * f6; /* 1.959375*2^30 */
  1407. int32_t f7_38 = 38 * f7; /* 1.959375*2^30 */
  1408. int32_t f8_19 = 19 * f8; /* 1.959375*2^30 */
  1409. int32_t f9_38 = 38 * f9; /* 1.959375*2^30 */
  1410. int64_t f0f0 = f0 * (int64_t) f0;
  1411. int64_t f0f1_2 = f0_2 * (int64_t) f1;
  1412. int64_t f0f2_2 = f0_2 * (int64_t) f2;
  1413. int64_t f0f3_2 = f0_2 * (int64_t) f3;
  1414. int64_t f0f4_2 = f0_2 * (int64_t) f4;
  1415. int64_t f0f5_2 = f0_2 * (int64_t) f5;
  1416. int64_t f0f6_2 = f0_2 * (int64_t) f6;
  1417. int64_t f0f7_2 = f0_2 * (int64_t) f7;
  1418. int64_t f0f8_2 = f0_2 * (int64_t) f8;
  1419. int64_t f0f9_2 = f0_2 * (int64_t) f9;
  1420. int64_t f1f1_2 = f1_2 * (int64_t) f1;
  1421. int64_t f1f2_2 = f1_2 * (int64_t) f2;
  1422. int64_t f1f3_4 = f1_2 * (int64_t) f3_2;
  1423. int64_t f1f4_2 = f1_2 * (int64_t) f4;
  1424. int64_t f1f5_4 = f1_2 * (int64_t) f5_2;
  1425. int64_t f1f6_2 = f1_2 * (int64_t) f6;
  1426. int64_t f1f7_4 = f1_2 * (int64_t) f7_2;
  1427. int64_t f1f8_2 = f1_2 * (int64_t) f8;
  1428. int64_t f1f9_76 = f1_2 * (int64_t) f9_38;
  1429. int64_t f2f2 = f2 * (int64_t) f2;
  1430. int64_t f2f3_2 = f2_2 * (int64_t) f3;
  1431. int64_t f2f4_2 = f2_2 * (int64_t) f4;
  1432. int64_t f2f5_2 = f2_2 * (int64_t) f5;
  1433. int64_t f2f6_2 = f2_2 * (int64_t) f6;
  1434. int64_t f2f7_2 = f2_2 * (int64_t) f7;
  1435. int64_t f2f8_38 = f2_2 * (int64_t) f8_19;
  1436. int64_t f2f9_38 = f2 * (int64_t) f9_38;
  1437. int64_t f3f3_2 = f3_2 * (int64_t) f3;
  1438. int64_t f3f4_2 = f3_2 * (int64_t) f4;
  1439. int64_t f3f5_4 = f3_2 * (int64_t) f5_2;
  1440. int64_t f3f6_2 = f3_2 * (int64_t) f6;
  1441. int64_t f3f7_76 = f3_2 * (int64_t) f7_38;
  1442. int64_t f3f8_38 = f3_2 * (int64_t) f8_19;
  1443. int64_t f3f9_76 = f3_2 * (int64_t) f9_38;
  1444. int64_t f4f4 = f4 * (int64_t) f4;
  1445. int64_t f4f5_2 = f4_2 * (int64_t) f5;
  1446. int64_t f4f6_38 = f4_2 * (int64_t) f6_19;
  1447. int64_t f4f7_38 = f4 * (int64_t) f7_38;
  1448. int64_t f4f8_38 = f4_2 * (int64_t) f8_19;
  1449. int64_t f4f9_38 = f4 * (int64_t) f9_38;
  1450. int64_t f5f5_38 = f5 * (int64_t) f5_38;
  1451. int64_t f5f6_38 = f5_2 * (int64_t) f6_19;
  1452. int64_t f5f7_76 = f5_2 * (int64_t) f7_38;
  1453. int64_t f5f8_38 = f5_2 * (int64_t) f8_19;
  1454. int64_t f5f9_76 = f5_2 * (int64_t) f9_38;
  1455. int64_t f6f6_19 = f6 * (int64_t) f6_19;
  1456. int64_t f6f7_38 = f6 * (int64_t) f7_38;
  1457. int64_t f6f8_38 = f6_2 * (int64_t) f8_19;
  1458. int64_t f6f9_38 = f6 * (int64_t) f9_38;
  1459. int64_t f7f7_38 = f7 * (int64_t) f7_38;
  1460. int64_t f7f8_38 = f7_2 * (int64_t) f8_19;
  1461. int64_t f7f9_76 = f7_2 * (int64_t) f9_38;
  1462. int64_t f8f8_19 = f8 * (int64_t) f8_19;
  1463. int64_t f8f9_38 = f8 * (int64_t) f9_38;
  1464. int64_t f9f9_38 = f9 * (int64_t) f9_38;
  1465. int64_t h0 = f0f0 + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38;
  1466. int64_t h1 = f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38;
  1467. int64_t h2 = f0f2_2 + f1f1_2 + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19;
  1468. int64_t h3 = f0f3_2 + f1f2_2 + f4f9_38 + f5f8_38 + f6f7_38;
  1469. int64_t h4 = f0f4_2 + f1f3_4 + f2f2 + f5f9_76 + f6f8_38 + f7f7_38;
  1470. int64_t h5 = f0f5_2 + f1f4_2 + f2f3_2 + f6f9_38 + f7f8_38;
  1471. int64_t h6 = f0f6_2 + f1f5_4 + f2f4_2 + f3f3_2 + f7f9_76 + f8f8_19;
  1472. int64_t h7 = f0f7_2 + f1f6_2 + f2f5_2 + f3f4_2 + f8f9_38;
  1473. int64_t h8 = f0f8_2 + f1f7_4 + f2f6_2 + f3f5_4 + f4f4 + f9f9_38;
  1474. int64_t h9 = f0f9_2 + f1f8_2 + f2f7_2 + f3f6_2 + f4f5_2;
  1475. int64_t carry0;
  1476. int64_t carry1;
  1477. int64_t carry2;
  1478. int64_t carry3;
  1479. int64_t carry4;
  1480. int64_t carry5;
  1481. int64_t carry6;
  1482. int64_t carry7;
  1483. int64_t carry8;
  1484. int64_t carry9;
  1485. h0 += h0;
  1486. h1 += h1;
  1487. h2 += h2;
  1488. h3 += h3;
  1489. h4 += h4;
  1490. h5 += h5;
  1491. h6 += h6;
  1492. h7 += h7;
  1493. h8 += h8;
  1494. h9 += h9;
  1495. carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
  1496. carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
  1497. carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
  1498. carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
  1499. carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
  1500. carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
  1501. carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
  1502. carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;
  1503. carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
  1504. carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;
  1505. carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
  1506. carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
  1507. h[0] = (int32_t)h0;
  1508. h[1] = (int32_t)h1;
  1509. h[2] = (int32_t)h2;
  1510. h[3] = (int32_t)h3;
  1511. h[4] = (int32_t)h4;
  1512. h[5] = (int32_t)h5;
  1513. h[6] = (int32_t)h6;
  1514. h[7] = (int32_t)h7;
  1515. h[8] = (int32_t)h8;
  1516. h[9] = (int32_t)h9;
  1517. }
  1518. static void fe_pow22523(fe out, const fe z)
  1519. {
  1520. fe t0;
  1521. fe t1;
  1522. fe t2;
  1523. int i;
  1524. fe_sq(t0, z);
  1525. fe_sq(t1, t0);
  1526. for (i = 1; i < 2; ++i) {
  1527. fe_sq(t1, t1);
  1528. }
  1529. fe_mul(t1, z, t1);
  1530. fe_mul(t0, t0, t1);
  1531. fe_sq(t0, t0);
  1532. fe_mul(t0, t1, t0);
  1533. fe_sq(t1, t0);
  1534. for (i = 1; i < 5; ++i) {
  1535. fe_sq(t1, t1);
  1536. }
  1537. fe_mul(t0, t1, t0);
  1538. fe_sq(t1, t0);
  1539. for (i = 1; i < 10; ++i) {
  1540. fe_sq(t1, t1);
  1541. }
  1542. fe_mul(t1, t1, t0);
  1543. fe_sq(t2, t1);
  1544. for (i = 1; i < 20; ++i) {
  1545. fe_sq(t2, t2);
  1546. }
  1547. fe_mul(t1, t2, t1);
  1548. fe_sq(t1, t1);
  1549. for (i = 1; i < 10; ++i) {
  1550. fe_sq(t1, t1);
  1551. }
  1552. fe_mul(t0, t1, t0);
  1553. fe_sq(t1, t0);
  1554. for (i = 1; i < 50; ++i) {
  1555. fe_sq(t1, t1);
  1556. }
  1557. fe_mul(t1, t1, t0);
  1558. fe_sq(t2, t1);
  1559. for (i = 1; i < 100; ++i) {
  1560. fe_sq(t2, t2);
  1561. }
  1562. fe_mul(t1, t2, t1);
  1563. fe_sq(t1, t1);
  1564. for (i = 1; i < 50; ++i) {
  1565. fe_sq(t1, t1);
  1566. }
  1567. fe_mul(t0, t1, t0);
  1568. fe_sq(t0, t0);
  1569. for (i = 1; i < 2; ++i) {
  1570. fe_sq(t0, t0);
  1571. }
  1572. fe_mul(out, t0, z);
  1573. }
  1574. /*
  1575. * ge means group element.
  1576. *
  1577. * Here the group is the set of pairs (x,y) of field elements (see fe.h)
  1578. * satisfying -x^2 + y^2 = 1 + d x^2y^2
  1579. * where d = -121665/121666.
  1580. *
  1581. * Representations:
  1582. * ge_p2 (projective): (X:Y:Z) satisfying x=X/Z, y=Y/Z
  1583. * ge_p3 (extended): (X:Y:Z:T) satisfying x=X/Z, y=Y/Z, XY=ZT
  1584. * ge_p1p1 (completed): ((X:Z),(Y:T)) satisfying x=X/Z, y=Y/T
  1585. * ge_precomp (Duif): (y+x,y-x,2dxy)
  1586. */
  1587. typedef struct {
  1588. fe X;
  1589. fe Y;
  1590. fe Z;
  1591. } ge_p2;
  1592. typedef struct {
  1593. fe X;
  1594. fe Y;
  1595. fe Z;
  1596. fe T;
  1597. } ge_p3;
  1598. typedef struct {
  1599. fe X;
  1600. fe Y;
  1601. fe Z;
  1602. fe T;
  1603. } ge_p1p1;
  1604. typedef struct {
  1605. fe yplusx;
  1606. fe yminusx;
  1607. fe xy2d;
  1608. } ge_precomp;
  1609. typedef struct {
  1610. fe YplusX;
  1611. fe YminusX;
  1612. fe Z;
  1613. fe T2d;
  1614. } ge_cached;
  1615. static void ge_tobytes(uint8_t *s, const ge_p2 *h)
  1616. {
  1617. fe recip;
  1618. fe x;
  1619. fe y;
  1620. fe_invert(recip, h->Z);
  1621. fe_mul(x, h->X, recip);
  1622. fe_mul(y, h->Y, recip);
  1623. fe_tobytes(s, y);
  1624. s[31] ^= fe_isnegative(x) << 7;
  1625. }
  1626. static void ge_p3_tobytes(uint8_t *s, const ge_p3 *h)
  1627. {
  1628. fe recip;
  1629. fe x;
  1630. fe y;
  1631. fe_invert(recip, h->Z);
  1632. fe_mul(x, h->X, recip);
  1633. fe_mul(y, h->Y, recip);
  1634. fe_tobytes(s, y);
  1635. s[31] ^= fe_isnegative(x) << 7;
  1636. }
  1637. static const fe d = {
  1638. -10913610, 13857413, -15372611, 6949391, 114729,
  1639. -8787816, -6275908, -3247719, -18696448, -12055116
  1640. };
  1641. static const fe sqrtm1 = {
  1642. -32595792, -7943725, 9377950, 3500415, 12389472,
  1643. -272473, -25146209, -2005654, 326686, 11406482
  1644. };
  1645. static int ge_frombytes_vartime(ge_p3 *h, const uint8_t *s)
  1646. {
  1647. fe u;
  1648. fe v;
  1649. fe v3;
  1650. fe vxx;
  1651. fe check;
  1652. fe_frombytes(h->Y, s);
  1653. fe_1(h->Z);
  1654. fe_sq(u, h->Y);
  1655. fe_mul(v, u, d);
  1656. fe_sub(u, u, h->Z); /* u = y^2-1 */
  1657. fe_add(v, v, h->Z); /* v = dy^2+1 */
  1658. fe_sq(v3, v);
  1659. fe_mul(v3, v3, v); /* v3 = v^3 */
  1660. fe_sq(h->X, v3);
  1661. fe_mul(h->X, h->X, v);
  1662. fe_mul(h->X, h->X, u); /* x = uv^7 */
  1663. fe_pow22523(h->X, h->X); /* x = (uv^7)^((q-5)/8) */
  1664. fe_mul(h->X, h->X, v3);
  1665. fe_mul(h->X, h->X, u); /* x = uv^3(uv^7)^((q-5)/8) */
  1666. fe_sq(vxx, h->X);
  1667. fe_mul(vxx, vxx, v);
  1668. fe_sub(check, vxx, u); /* vx^2-u */
  1669. if (fe_isnonzero(check)) {
  1670. fe_add(check, vxx, u); /* vx^2+u */
  1671. if (fe_isnonzero(check)) {
  1672. return -1;
  1673. }
  1674. fe_mul(h->X, h->X, sqrtm1);
  1675. }
  1676. if (fe_isnegative(h->X) != (s[31] >> 7)) {
  1677. fe_neg(h->X, h->X);
  1678. }
  1679. fe_mul(h->T, h->X, h->Y);
  1680. return 0;
  1681. }
  1682. static void ge_p2_0(ge_p2 *h)
  1683. {
  1684. fe_0(h->X);
  1685. fe_1(h->Y);
  1686. fe_1(h->Z);
  1687. }
  1688. static void ge_p3_0(ge_p3 *h)
  1689. {
  1690. fe_0(h->X);
  1691. fe_1(h->Y);
  1692. fe_1(h->Z);
  1693. fe_0(h->T);
  1694. }
  1695. static void ge_precomp_0(ge_precomp *h)
  1696. {
  1697. fe_1(h->yplusx);
  1698. fe_1(h->yminusx);
  1699. fe_0(h->xy2d);
  1700. }
  1701. /* r = p */
  1702. static void ge_p3_to_p2(ge_p2 *r, const ge_p3 *p)
  1703. {
  1704. fe_copy(r->X, p->X);
  1705. fe_copy(r->Y, p->Y);
  1706. fe_copy(r->Z, p->Z);
  1707. }
  1708. static const fe d2 = {
  1709. -21827239, -5839606, -30745221, 13898782, 229458,
  1710. 15978800, -12551817, -6495438, 29715968, 9444199
  1711. };
  1712. /* r = p */
  1713. static void ge_p3_to_cached(ge_cached *r, const ge_p3 *p)
  1714. {
  1715. fe_add(r->YplusX, p->Y, p->X);
  1716. fe_sub(r->YminusX, p->Y, p->X);
  1717. fe_copy(r->Z, p->Z);
  1718. fe_mul(r->T2d, p->T, d2);
  1719. }
  1720. /* r = p */
  1721. static void ge_p1p1_to_p2(ge_p2 *r, const ge_p1p1 *p)
  1722. {
  1723. fe_mul(r->X, p->X, p->T);
  1724. fe_mul(r->Y, p->Y, p->Z);
  1725. fe_mul(r->Z, p->Z, p->T);
  1726. }
  1727. /* r = p */
  1728. static void ge_p1p1_to_p3(ge_p3 *r, const ge_p1p1 *p)
  1729. {
  1730. fe_mul(r->X, p->X, p->T);
  1731. fe_mul(r->Y, p->Y, p->Z);
  1732. fe_mul(r->Z, p->Z, p->T);
  1733. fe_mul(r->T, p->X, p->Y);
  1734. }
  1735. /* r = 2 * p */
  1736. static void ge_p2_dbl(ge_p1p1 *r, const ge_p2 *p)
  1737. {
  1738. fe t0;
  1739. fe_sq(r->X, p->X);
  1740. fe_sq(r->Z, p->Y);
  1741. fe_sq2(r->T, p->Z);
  1742. fe_add(r->Y, p->X, p->Y);
  1743. fe_sq(t0, r->Y);
  1744. fe_add(r->Y, r->Z, r->X);
  1745. fe_sub(r->Z, r->Z, r->X);
  1746. fe_sub(r->X, t0, r->Y);
  1747. fe_sub(r->T, r->T, r->Z);
  1748. }
  1749. /* r = 2 * p */
  1750. static void ge_p3_dbl(ge_p1p1 *r, const ge_p3 *p)
  1751. {
  1752. ge_p2 q;
  1753. ge_p3_to_p2(&q, p);
  1754. ge_p2_dbl(r, &q);
  1755. }
  1756. /* r = p + q */
  1757. static void ge_madd(ge_p1p1 *r, const ge_p3 *p, const ge_precomp *q)
  1758. {
  1759. fe t0;
  1760. fe_add(r->X, p->Y, p->X);
  1761. fe_sub(r->Y, p->Y, p->X);
  1762. fe_mul(r->Z, r->X, q->yplusx);
  1763. fe_mul(r->Y, r->Y, q->yminusx);
  1764. fe_mul(r->T, q->xy2d, p->T);
  1765. fe_add(t0, p->Z, p->Z);
  1766. fe_sub(r->X, r->Z, r->Y);
  1767. fe_add(r->Y, r->Z, r->Y);
  1768. fe_add(r->Z, t0, r->T);
  1769. fe_sub(r->T, t0, r->T);
  1770. }
  1771. /* r = p - q */
  1772. static void ge_msub(ge_p1p1 *r, const ge_p3 *p, const ge_precomp *q)
  1773. {
  1774. fe t0;
  1775. fe_add(r->X, p->Y, p->X);
  1776. fe_sub(r->Y, p->Y, p->X);
  1777. fe_mul(r->Z, r->X, q->yminusx);
  1778. fe_mul(r->Y, r->Y, q->yplusx);
  1779. fe_mul(r->T, q->xy2d, p->T);
  1780. fe_add(t0, p->Z, p->Z);
  1781. fe_sub(r->X, r->Z, r->Y);
  1782. fe_add(r->Y, r->Z, r->Y);
  1783. fe_sub(r->Z, t0, r->T);
  1784. fe_add(r->T, t0, r->T);
  1785. }
  1786. /* r = p + q */
  1787. static void ge_add(ge_p1p1 *r, const ge_p3 *p, const ge_cached *q)
  1788. {
  1789. fe t0;
  1790. fe_add(r->X, p->Y, p->X);
  1791. fe_sub(r->Y, p->Y, p->X);
  1792. fe_mul(r->Z, r->X, q->YplusX);
  1793. fe_mul(r->Y, r->Y, q->YminusX);
  1794. fe_mul(r->T, q->T2d, p->T);
  1795. fe_mul(r->X, p->Z, q->Z);
  1796. fe_add(t0, r->X, r->X);
  1797. fe_sub(r->X, r->Z, r->Y);
  1798. fe_add(r->Y, r->Z, r->Y);
  1799. fe_add(r->Z, t0, r->T);
  1800. fe_sub(r->T, t0, r->T);
  1801. }
  1802. /* r = p - q */
  1803. static void ge_sub(ge_p1p1 *r, const ge_p3 *p, const ge_cached *q)
  1804. {
  1805. fe t0;
  1806. fe_add(r->X, p->Y, p->X);
  1807. fe_sub(r->Y, p->Y, p->X);
  1808. fe_mul(r->Z, r->X, q->YminusX);
  1809. fe_mul(r->Y, r->Y, q->YplusX);
  1810. fe_mul(r->T, q->T2d, p->T);
  1811. fe_mul(r->X, p->Z, q->Z);
  1812. fe_add(t0, r->X, r->X);
  1813. fe_sub(r->X, r->Z, r->Y);
  1814. fe_add(r->Y, r->Z, r->Y);
  1815. fe_sub(r->Z, t0, r->T);
  1816. fe_add(r->T, t0, r->T);
  1817. }
  1818. static uint8_t equal(signed char b, signed char c)
  1819. {
  1820. uint8_t ub = b;
  1821. uint8_t uc = c;
  1822. uint8_t x = ub ^ uc; /* 0: yes; 1..255: no */
  1823. uint32_t y = x; /* 0: yes; 1..255: no */
  1824. y -= 1; /* 4294967295: yes; 0..254: no */
  1825. y >>= 31; /* 1: yes; 0: no */
  1826. return y;
  1827. }
  1828. static void cmov(ge_precomp *t, const ge_precomp *u, uint8_t b)
  1829. {
  1830. fe_cmov(t->yplusx, u->yplusx, b);
  1831. fe_cmov(t->yminusx, u->yminusx, b);
  1832. fe_cmov(t->xy2d, u->xy2d, b);
  1833. }
  1834. /* k25519Precomp[i][j] = (j+1)*256^i*B */
  1835. static const ge_precomp k25519Precomp[32][8] = {
  1836. {
  1837. {
  1838. {25967493, -14356035, 29566456, 3660896, -12694345, 4014787,
  1839. 27544626, -11754271, -6079156, 2047605},
  1840. {-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692,
  1841. 5043384, 19500929, -15469378},
  1842. {-8738181, 4489570, 9688441, -14785194, 10184609, -12363380,
  1843. 29287919, 11864899, -24514362, -4438546},
  1844. },
  1845. {
  1846. {-12815894, -12976347, -21581243, 11784320, -25355658, -2750717,
  1847. -11717903, -3814571, -358445, -10211303},
  1848. {-21703237, 6903825, 27185491, 6451973, -29577724, -9554005,
  1849. -15616551, 11189268, -26829678, -5319081},
  1850. {26966642, 11152617, 32442495, 15396054, 14353839, -12752335,
  1851. -3128826, -9541118, -15472047, -4166697},
  1852. },
  1853. {
  1854. {15636291, -9688557, 24204773, -7912398, 616977, -16685262,
  1855. 27787600, -14772189, 28944400, -1550024},
  1856. {16568933, 4717097, -11556148, -1102322, 15682896, -11807043,
  1857. 16354577, -11775962, 7689662, 11199574},
  1858. {30464156, -5976125, -11779434, -15670865, 23220365, 15915852,
  1859. 7512774, 10017326, -17749093, -9920357},
  1860. },
  1861. {
  1862. {-17036878, 13921892, 10945806, -6033431, 27105052, -16084379,
  1863. -28926210, 15006023, 3284568, -6276540},
  1864. {23599295, -8306047, -11193664, -7687416, 13236774, 10506355,
  1865. 7464579, 9656445, 13059162, 10374397},
  1866. {7798556, 16710257, 3033922, 2874086, 28997861, 2835604, 32406664,
  1867. -3839045, -641708, -101325},
  1868. },
  1869. {
  1870. {10861363, 11473154, 27284546, 1981175, -30064349, 12577861,
  1871. 32867885, 14515107, -15438304, 10819380},
  1872. {4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668,
  1873. 12483688, -12668491, 5581306},
  1874. {19563160, 16186464, -29386857, 4097519, 10237984, -4348115,
  1875. 28542350, 13850243, -23678021, -15815942},
  1876. },
  1877. {
  1878. {-15371964, -12862754, 32573250, 4720197, -26436522, 5875511,
  1879. -19188627, -15224819, -9818940, -12085777},
  1880. {-8549212, 109983, 15149363, 2178705, 22900618, 4543417, 3044240,
  1881. -15689887, 1762328, 14866737},
  1882. {-18199695, -15951423, -10473290, 1707278, -17185920, 3916101,
  1883. -28236412, 3959421, 27914454, 4383652},
  1884. },
  1885. {
  1886. {5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852,
  1887. 5230134, -23952439, -15175766},
  1888. {-30269007, -3463509, 7665486, 10083793, 28475525, 1649722,
  1889. 20654025, 16520125, 30598449, 7715701},
  1890. {28881845, 14381568, 9657904, 3680757, -20181635, 7843316,
  1891. -31400660, 1370708, 29794553, -1409300},
  1892. },
  1893. {
  1894. {14499471, -2729599, -33191113, -4254652, 28494862, 14271267,
  1895. 30290735, 10876454, -33154098, 2381726},
  1896. {-7195431, -2655363, -14730155, 462251, -27724326, 3941372,
  1897. -6236617, 3696005, -32300832, 15351955},
  1898. {27431194, 8222322, 16448760, -3907995, -18707002, 11938355,
  1899. -32961401, -2970515, 29551813, 10109425},
  1900. },
  1901. },
  1902. {
  1903. {
  1904. {-13657040, -13155431, -31283750, 11777098, 21447386, 6519384,
  1905. -2378284, -1627556, 10092783, -4764171},
  1906. {27939166, 14210322, 4677035, 16277044, -22964462, -12398139,
  1907. -32508754, 12005538, -17810127, 12803510},
  1908. {17228999, -15661624, -1233527, 300140, -1224870, -11714777,
  1909. 30364213, -9038194, 18016357, 4397660},
  1910. },
  1911. {
  1912. {-10958843, -7690207, 4776341, -14954238, 27850028, -15602212,
  1913. -26619106, 14544525, -17477504, 982639},
  1914. {29253598, 15796703, -2863982, -9908884, 10057023, 3163536, 7332899,
  1915. -4120128, -21047696, 9934963},
  1916. {5793303, 16271923, -24131614, -10116404, 29188560, 1206517,
  1917. -14747930, 4559895, -30123922, -10897950},
  1918. },
  1919. {
  1920. {-27643952, -11493006, 16282657, -11036493, 28414021, -15012264,
  1921. 24191034, 4541697, -13338309, 5500568},
  1922. {12650548, -1497113, 9052871, 11355358, -17680037, -8400164,
  1923. -17430592, 12264343, 10874051, 13524335},
  1924. {25556948, -3045990, 714651, 2510400, 23394682, -10415330, 33119038,
  1925. 5080568, -22528059, 5376628},
  1926. },
  1927. {
  1928. {-26088264, -4011052, -17013699, -3537628, -6726793, 1920897,
  1929. -22321305, -9447443, 4535768, 1569007},
  1930. {-2255422, 14606630, -21692440, -8039818, 28430649, 8775819,
  1931. -30494562, 3044290, 31848280, 12543772},
  1932. {-22028579, 2943893, -31857513, 6777306, 13784462, -4292203,
  1933. -27377195, -2062731, 7718482, 14474653},
  1934. },
  1935. {
  1936. {2385315, 2454213, -22631320, 46603, -4437935, -15680415, 656965,
  1937. -7236665, 24316168, -5253567},
  1938. {13741529, 10911568, -33233417, -8603737, -20177830, -1033297,
  1939. 33040651, -13424532, -20729456, 8321686},
  1940. {21060490, -2212744, 15712757, -4336099, 1639040, 10656336,
  1941. 23845965, -11874838, -9984458, 608372},
  1942. },
  1943. {
  1944. {-13672732, -15087586, -10889693, -7557059, -6036909, 11305547,
  1945. 1123968, -6780577, 27229399, 23887},
  1946. {-23244140, -294205, -11744728, 14712571, -29465699, -2029617,
  1947. 12797024, -6440308, -1633405, 16678954},
  1948. {-29500620, 4770662, -16054387, 14001338, 7830047, 9564805,
  1949. -1508144, -4795045, -17169265, 4904953},
  1950. },
  1951. {
  1952. {24059557, 14617003, 19037157, -15039908, 19766093, -14906429,
  1953. 5169211, 16191880, 2128236, -4326833},
  1954. {-16981152, 4124966, -8540610, -10653797, 30336522, -14105247,
  1955. -29806336, 916033, -6882542, -2986532},
  1956. {-22630907, 12419372, -7134229, -7473371, -16478904, 16739175,
  1957. 285431, 2763829, 15736322, 4143876},
  1958. },
  1959. {
  1960. {2379352, 11839345, -4110402, -5988665, 11274298, 794957, 212801,
  1961. -14594663, 23527084, -16458268},
  1962. {33431127, -11130478, -17838966, -15626900, 8909499, 8376530,
  1963. -32625340, 4087881, -15188911, -14416214},
  1964. {1767683, 7197987, -13205226, -2022635, -13091350, 448826, 5799055,
  1965. 4357868, -4774191, -16323038},
  1966. },
  1967. },
  1968. {
  1969. {
  1970. {6721966, 13833823, -23523388, -1551314, 26354293, -11863321,
  1971. 23365147, -3949732, 7390890, 2759800},
  1972. {4409041, 2052381, 23373853, 10530217, 7676779, -12885954, 21302353,
  1973. -4264057, 1244380, -12919645},
  1974. {-4421239, 7169619, 4982368, -2957590, 30256825, -2777540, 14086413,
  1975. 9208236, 15886429, 16489664},
  1976. },
  1977. {
  1978. {1996075, 10375649, 14346367, 13311202, -6874135, -16438411,
  1979. -13693198, 398369, -30606455, -712933},
  1980. {-25307465, 9795880, -2777414, 14878809, -33531835, 14780363,
  1981. 13348553, 12076947, -30836462, 5113182},
  1982. {-17770784, 11797796, 31950843, 13929123, -25888302, 12288344,
  1983. -30341101, -7336386, 13847711, 5387222},
  1984. },
  1985. {
  1986. {-18582163, -3416217, 17824843, -2340966, 22744343, -10442611,
  1987. 8763061, 3617786, -19600662, 10370991},
  1988. {20246567, -14369378, 22358229, -543712, 18507283, -10413996,
  1989. 14554437, -8746092, 32232924, 16763880},
  1990. {9648505, 10094563, 26416693, 14745928, -30374318, -6472621,
  1991. 11094161, 15689506, 3140038, -16510092},
  1992. },
  1993. {
  1994. {-16160072, 5472695, 31895588, 4744994, 8823515, 10365685,
  1995. -27224800, 9448613, -28774454, 366295},
  1996. {19153450, 11523972, -11096490, -6503142, -24647631, 5420647,
  1997. 28344573, 8041113, 719605, 11671788},
  1998. {8678025, 2694440, -6808014, 2517372, 4964326, 11152271, -15432916,
  1999. -15266516, 27000813, -10195553},
  2000. },
  2001. {
  2002. {-15157904, 7134312, 8639287, -2814877, -7235688, 10421742, 564065,
  2003. 5336097, 6750977, -14521026},
  2004. {11836410, -3979488, 26297894, 16080799, 23455045, 15735944,
  2005. 1695823, -8819122, 8169720, 16220347},
  2006. {-18115838, 8653647, 17578566, -6092619, -8025777, -16012763,
  2007. -11144307, -2627664, -5990708, -14166033},
  2008. },
  2009. {
  2010. {-23308498, -10968312, 15213228, -10081214, -30853605, -11050004,
  2011. 27884329, 2847284, 2655861, 1738395},
  2012. {-27537433, -14253021, -25336301, -8002780, -9370762, 8129821,
  2013. 21651608, -3239336, -19087449, -11005278},
  2014. {1533110, 3437855, 23735889, 459276, 29970501, 11335377, 26030092,
  2015. 5821408, 10478196, 8544890},
  2016. },
  2017. {
  2018. {32173121, -16129311, 24896207, 3921497, 22579056, -3410854,
  2019. 19270449, 12217473, 17789017, -3395995},
  2020. {-30552961, -2228401, -15578829, -10147201, 13243889, 517024,
  2021. 15479401, -3853233, 30460520, 1052596},
  2022. {-11614875, 13323618, 32618793, 8175907, -15230173, 12596687,
  2023. 27491595, -4612359, 3179268, -9478891},
  2024. },
  2025. {
  2026. {31947069, -14366651, -4640583, -15339921, -15125977, -6039709,
  2027. -14756777, -16411740, 19072640, -9511060},
  2028. {11685058, 11822410, 3158003, -13952594, 33402194, -4165066,
  2029. 5977896, -5215017, 473099, 5040608},
  2030. {-20290863, 8198642, -27410132, 11602123, 1290375, -2799760,
  2031. 28326862, 1721092, -19558642, -3131606},
  2032. },
  2033. },
  2034. {
  2035. {
  2036. {7881532, 10687937, 7578723, 7738378, -18951012, -2553952, 21820786,
  2037. 8076149, -27868496, 11538389},
  2038. {-19935666, 3899861, 18283497, -6801568, -15728660, -11249211,
  2039. 8754525, 7446702, -5676054, 5797016},
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  3761. -7839692, -7852844, -8138429},
  3762. {-15236356, -15433509, 7766470, 746860, 26346930, -10221762,
  3763. -27333451, 10754588, -9431476, 5203576},
  3764. {31834314, 14135496, -770007, 5159118, 20917671, -16768096,
  3765. -7467973, -7337524, 31809243, 7347066},
  3766. },
  3767. {
  3768. {-9606723, -11874240, 20414459, 13033986, 13716524, -11691881,
  3769. 19797970, -12211255, 15192876, -2087490},
  3770. {-12663563, -2181719, 1168162, -3804809, 26747877, -14138091,
  3771. 10609330, 12694420, 33473243, -13382104},
  3772. {33184999, 11180355, 15832085, -11385430, -1633671, 225884,
  3773. 15089336, -11023903, -6135662, 14480053},
  3774. },
  3775. {
  3776. {31308717, -5619998, 31030840, -1897099, 15674547, -6582883,
  3777. 5496208, 13685227, 27595050, 8737275},
  3778. {-20318852, -15150239, 10933843, -16178022, 8335352, -7546022,
  3779. -31008351, -12610604, 26498114, 66511},
  3780. {22644454, -8761729, -16671776, 4884562, -3105614, -13559366,
  3781. 30540766, -4286747, -13327787, -7515095},
  3782. },
  3783. {
  3784. {-28017847, 9834845, 18617207, -2681312, -3401956, -13307506,
  3785. 8205540, 13585437, -17127465, 15115439},
  3786. {23711543, -672915, 31206561, -8362711, 6164647, -9709987,
  3787. -33535882, -1426096, 8236921, 16492939},
  3788. {-23910559, -13515526, -26299483, -4503841, 25005590, -7687270,
  3789. 19574902, 10071562, 6708380, -6222424},
  3790. },
  3791. {
  3792. {2101391, -4930054, 19702731, 2367575, -15427167, 1047675, 5301017,
  3793. 9328700, 29955601, -11678310},
  3794. {3096359, 9271816, -21620864, -15521844, -14847996, -7592937,
  3795. -25892142, -12635595, -9917575, 6216608},
  3796. {-32615849, 338663, -25195611, 2510422, -29213566, -13820213,
  3797. 24822830, -6146567, -26767480, 7525079},
  3798. },
  3799. {
  3800. {-23066649, -13985623, 16133487, -7896178, -3389565, 778788,
  3801. -910336, -2782495, -19386633, 11994101},
  3802. {21691500, -13624626, -641331, -14367021, 3285881, -3483596,
  3803. -25064666, 9718258, -7477437, 13381418},
  3804. {18445390, -4202236, 14979846, 11622458, -1727110, -3582980,
  3805. 23111648, -6375247, 28535282, 15779576},
  3806. },
  3807. {
  3808. {30098053, 3089662, -9234387, 16662135, -21306940, 11308411,
  3809. -14068454, 12021730, 9955285, -16303356},
  3810. {9734894, -14576830, -7473633, -9138735, 2060392, 11313496,
  3811. -18426029, 9924399, 20194861, 13380996},
  3812. {-26378102, -7965207, -22167821, 15789297, -18055342, -6168792,
  3813. -1984914, 15707771, 26342023, 10146099},
  3814. },
  3815. },
  3816. {
  3817. {
  3818. {-26016874, -219943, 21339191, -41388, 19745256, -2878700,
  3819. -29637280, 2227040, 21612326, -545728},
  3820. {-13077387, 1184228, 23562814, -5970442, -20351244, -6348714,
  3821. 25764461, 12243797, -20856566, 11649658},
  3822. {-10031494, 11262626, 27384172, 2271902, 26947504, -15997771, 39944,
  3823. 6114064, 33514190, 2333242},
  3824. },
  3825. {
  3826. {-21433588, -12421821, 8119782, 7219913, -21830522, -9016134,
  3827. -6679750, -12670638, 24350578, -13450001},
  3828. {-4116307, -11271533, -23886186, 4843615, -30088339, 690623,
  3829. -31536088, -10406836, 8317860, 12352766},
  3830. {18200138, -14475911, -33087759, -2696619, -23702521, -9102511,
  3831. -23552096, -2287550, 20712163, 6719373},
  3832. },
  3833. {
  3834. {26656208, 6075253, -7858556, 1886072, -28344043, 4262326, 11117530,
  3835. -3763210, 26224235, -3297458},
  3836. {-17168938, -14854097, -3395676, -16369877, -19954045, 14050420,
  3837. 21728352, 9493610, 18620611, -16428628},
  3838. {-13323321, 13325349, 11432106, 5964811, 18609221, 6062965,
  3839. -5269471, -9725556, -30701573, -16479657},
  3840. },
  3841. {
  3842. {-23860538, -11233159, 26961357, 1640861, -32413112, -16737940,
  3843. 12248509, -5240639, 13735342, 1934062},
  3844. {25089769, 6742589, 17081145, -13406266, 21909293, -16067981,
  3845. -15136294, -3765346, -21277997, 5473616},
  3846. {31883677, -7961101, 1083432, -11572403, 22828471, 13290673,
  3847. -7125085, 12469656, 29111212, -5451014},
  3848. },
  3849. {
  3850. {24244947, -15050407, -26262976, 2791540, -14997599, 16666678,
  3851. 24367466, 6388839, -10295587, 452383},
  3852. {-25640782, -3417841, 5217916, 16224624, 19987036, -4082269,
  3853. -24236251, -5915248, 15766062, 8407814},
  3854. {-20406999, 13990231, 15495425, 16395525, 5377168, 15166495,
  3855. -8917023, -4388953, -8067909, 2276718},
  3856. },
  3857. {
  3858. {30157918, 12924066, -17712050, 9245753, 19895028, 3368142,
  3859. -23827587, 5096219, 22740376, -7303417},
  3860. {2041139, -14256350, 7783687, 13876377, -25946985, -13352459,
  3861. 24051124, 13742383, -15637599, 13295222},
  3862. {33338237, -8505733, 12532113, 7977527, 9106186, -1715251,
  3863. -17720195, -4612972, -4451357, -14669444},
  3864. },
  3865. {
  3866. {-20045281, 5454097, -14346548, 6447146, 28862071, 1883651,
  3867. -2469266, -4141880, 7770569, 9620597},
  3868. {23208068, 7979712, 33071466, 8149229, 1758231, -10834995, 30945528,
  3869. -1694323, -33502340, -14767970},
  3870. {1439958, -16270480, -1079989, -793782, 4625402, 10647766, -5043801,
  3871. 1220118, 30494170, -11440799},
  3872. },
  3873. {
  3874. {-5037580, -13028295, -2970559, -3061767, 15640974, -6701666,
  3875. -26739026, 926050, -1684339, -13333647},
  3876. {13908495, -3549272, 30919928, -6273825, -21521863, 7989039,
  3877. 9021034, 9078865, 3353509, 4033511},
  3878. {-29663431, -15113610, 32259991, -344482, 24295849, -12912123,
  3879. 23161163, 8839127, 27485041, 7356032},
  3880. },
  3881. },
  3882. {
  3883. {
  3884. {9661027, 705443, 11980065, -5370154, -1628543, 14661173, -6346142,
  3885. 2625015, 28431036, -16771834},
  3886. {-23839233, -8311415, -25945511, 7480958, -17681669, -8354183,
  3887. -22545972, 14150565, 15970762, 4099461},
  3888. {29262576, 16756590, 26350592, -8793563, 8529671, -11208050,
  3889. 13617293, -9937143, 11465739, 8317062},
  3890. },
  3891. {
  3892. {-25493081, -6962928, 32500200, -9419051, -23038724, -2302222,
  3893. 14898637, 3848455, 20969334, -5157516},
  3894. {-20384450, -14347713, -18336405, 13884722, -33039454, 2842114,
  3895. -21610826, -3649888, 11177095, 14989547},
  3896. {-24496721, -11716016, 16959896, 2278463, 12066309, 10137771,
  3897. 13515641, 2581286, -28487508, 9930240},
  3898. },
  3899. {
  3900. {-17751622, -2097826, 16544300, -13009300, -15914807, -14949081,
  3901. 18345767, -13403753, 16291481, -5314038},
  3902. {-33229194, 2553288, 32678213, 9875984, 8534129, 6889387, -9676774,
  3903. 6957617, 4368891, 9788741},
  3904. {16660756, 7281060, -10830758, 12911820, 20108584, -8101676,
  3905. -21722536, -8613148, 16250552, -11111103},
  3906. },
  3907. {
  3908. {-19765507, 2390526, -16551031, 14161980, 1905286, 6414907, 4689584,
  3909. 10604807, -30190403, 4782747},
  3910. {-1354539, 14736941, -7367442, -13292886, 7710542, -14155590,
  3911. -9981571, 4383045, 22546403, 437323},
  3912. {31665577, -12180464, -16186830, 1491339, -18368625, 3294682,
  3913. 27343084, 2786261, -30633590, -14097016},
  3914. },
  3915. {
  3916. {-14467279, -683715, -33374107, 7448552, 19294360, 14334329,
  3917. -19690631, 2355319, -19284671, -6114373},
  3918. {15121312, -15796162, 6377020, -6031361, -10798111, -12957845,
  3919. 18952177, 15496498, -29380133, 11754228},
  3920. {-2637277, -13483075, 8488727, -14303896, 12728761, -1622493,
  3921. 7141596, 11724556, 22761615, -10134141},
  3922. },
  3923. {
  3924. {16918416, 11729663, -18083579, 3022987, -31015732, -13339659,
  3925. -28741185, -12227393, 32851222, 11717399},
  3926. {11166634, 7338049, -6722523, 4531520, -29468672, -7302055,
  3927. 31474879, 3483633, -1193175, -4030831},
  3928. {-185635, 9921305, 31456609, -13536438, -12013818, 13348923,
  3929. 33142652, 6546660, -19985279, -3948376},
  3930. },
  3931. {
  3932. {-32460596, 11266712, -11197107, -7899103, 31703694, 3855903,
  3933. -8537131, -12833048, -30772034, -15486313},
  3934. {-18006477, 12709068, 3991746, -6479188, -21491523, -10550425,
  3935. -31135347, -16049879, 10928917, 3011958},
  3936. {-6957757, -15594337, 31696059, 334240, 29576716, 14796075,
  3937. -30831056, -12805180, 18008031, 10258577},
  3938. },
  3939. {
  3940. {-22448644, 15655569, 7018479, -4410003, -30314266, -1201591,
  3941. -1853465, 1367120, 25127874, 6671743},
  3942. {29701166, -14373934, -10878120, 9279288, -17568, 13127210,
  3943. 21382910, 11042292, 25838796, 4642684},
  3944. {-20430234, 14955537, -24126347, 8124619, -5369288, -5990470,
  3945. 30468147, -13900640, 18423289, 4177476},
  3946. },
  3947. },
  3948. };
  3949. static uint8_t negative(signed char b)
  3950. {
  3951. uint32_t x = b;
  3952. x >>= 31; /* 1: yes; 0: no */
  3953. return x;
  3954. }
  3955. static void table_select(ge_precomp *t, int pos, signed char b)
  3956. {
  3957. ge_precomp minust;
  3958. uint8_t bnegative = negative(b);
  3959. uint8_t babs = b - ((uint8_t)((-bnegative) & b) << 1);
  3960. ge_precomp_0(t);
  3961. cmov(t, &k25519Precomp[pos][0], equal(babs, 1));
  3962. cmov(t, &k25519Precomp[pos][1], equal(babs, 2));
  3963. cmov(t, &k25519Precomp[pos][2], equal(babs, 3));
  3964. cmov(t, &k25519Precomp[pos][3], equal(babs, 4));
  3965. cmov(t, &k25519Precomp[pos][4], equal(babs, 5));
  3966. cmov(t, &k25519Precomp[pos][5], equal(babs, 6));
  3967. cmov(t, &k25519Precomp[pos][6], equal(babs, 7));
  3968. cmov(t, &k25519Precomp[pos][7], equal(babs, 8));
  3969. fe_copy(minust.yplusx, t->yminusx);
  3970. fe_copy(minust.yminusx, t->yplusx);
  3971. fe_neg(minust.xy2d, t->xy2d);
  3972. cmov(t, &minust, bnegative);
  3973. }
  3974. /*
  3975. * h = a * B
  3976. *
  3977. * where a = a[0]+256*a[1]+...+256^31 a[31]
  3978. * B is the Ed25519 base point (x,4/5) with x positive.
  3979. *
  3980. * Preconditions:
  3981. * a[31] <= 127
  3982. */
  3983. static void ge_scalarmult_base(ge_p3 *h, const uint8_t *a)
  3984. {
  3985. signed char e[64];
  3986. signed char carry;
  3987. ge_p1p1 r;
  3988. ge_p2 s;
  3989. ge_precomp t;
  3990. int i;
  3991. for (i = 0; i < 32; ++i) {
  3992. e[2 * i + 0] = (a[i] >> 0) & 15;
  3993. e[2 * i + 1] = (a[i] >> 4) & 15;
  3994. }
  3995. /* each e[i] is between 0 and 15 */
  3996. /* e[63] is between 0 and 7 */
  3997. carry = 0;
  3998. for (i = 0; i < 63; ++i) {
  3999. e[i] += carry;
  4000. carry = e[i] + 8;
  4001. carry >>= 4;
  4002. e[i] -= carry << 4;
  4003. }
  4004. e[63] += carry;
  4005. /* each e[i] is between -8 and 8 */
  4006. ge_p3_0(h);
  4007. for (i = 1; i < 64; i += 2) {
  4008. table_select(&t, i / 2, e[i]);
  4009. ge_madd(&r, h, &t);
  4010. ge_p1p1_to_p3(h, &r);
  4011. }
  4012. ge_p3_dbl(&r, h);
  4013. ge_p1p1_to_p2(&s, &r);
  4014. ge_p2_dbl(&r, &s);
  4015. ge_p1p1_to_p2(&s, &r);
  4016. ge_p2_dbl(&r, &s);
  4017. ge_p1p1_to_p2(&s, &r);
  4018. ge_p2_dbl(&r, &s);
  4019. ge_p1p1_to_p3(h, &r);
  4020. for (i = 0; i < 64; i += 2) {
  4021. table_select(&t, i / 2, e[i]);
  4022. ge_madd(&r, h, &t);
  4023. ge_p1p1_to_p3(h, &r);
  4024. }
  4025. OPENSSL_cleanse(e, sizeof(e));
  4026. }
  4027. #if !defined(BASE_2_51_IMPLEMENTED)
  4028. /*
  4029. * Replace (f,g) with (g,f) if b == 1;
  4030. * replace (f,g) with (f,g) if b == 0.
  4031. *
  4032. * Preconditions: b in {0,1}.
  4033. */
  4034. static void fe_cswap(fe f, fe g, unsigned int b)
  4035. {
  4036. size_t i;
  4037. b = 0-b;
  4038. for (i = 0; i < 10; i++) {
  4039. int32_t x = f[i] ^ g[i];
  4040. x &= b;
  4041. f[i] ^= x;
  4042. g[i] ^= x;
  4043. }
  4044. }
  4045. /*
  4046. * h = f * 121666
  4047. *
  4048. * Can overlap h with f.
  4049. *
  4050. * Preconditions:
  4051. * |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
  4052. *
  4053. * Postconditions:
  4054. * |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
  4055. */
  4056. static void fe_mul121666(fe h, fe f)
  4057. {
  4058. int32_t f0 = f[0];
  4059. int32_t f1 = f[1];
  4060. int32_t f2 = f[2];
  4061. int32_t f3 = f[3];
  4062. int32_t f4 = f[4];
  4063. int32_t f5 = f[5];
  4064. int32_t f6 = f[6];
  4065. int32_t f7 = f[7];
  4066. int32_t f8 = f[8];
  4067. int32_t f9 = f[9];
  4068. int64_t h0 = f0 * (int64_t) 121666;
  4069. int64_t h1 = f1 * (int64_t) 121666;
  4070. int64_t h2 = f2 * (int64_t) 121666;
  4071. int64_t h3 = f3 * (int64_t) 121666;
  4072. int64_t h4 = f4 * (int64_t) 121666;
  4073. int64_t h5 = f5 * (int64_t) 121666;
  4074. int64_t h6 = f6 * (int64_t) 121666;
  4075. int64_t h7 = f7 * (int64_t) 121666;
  4076. int64_t h8 = f8 * (int64_t) 121666;
  4077. int64_t h9 = f9 * (int64_t) 121666;
  4078. int64_t carry0;
  4079. int64_t carry1;
  4080. int64_t carry2;
  4081. int64_t carry3;
  4082. int64_t carry4;
  4083. int64_t carry5;
  4084. int64_t carry6;
  4085. int64_t carry7;
  4086. int64_t carry8;
  4087. int64_t carry9;
  4088. carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
  4089. carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
  4090. carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
  4091. carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
  4092. carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;
  4093. carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
  4094. carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
  4095. carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
  4096. carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
  4097. carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;
  4098. h[0] = (int32_t)h0;
  4099. h[1] = (int32_t)h1;
  4100. h[2] = (int32_t)h2;
  4101. h[3] = (int32_t)h3;
  4102. h[4] = (int32_t)h4;
  4103. h[5] = (int32_t)h5;
  4104. h[6] = (int32_t)h6;
  4105. h[7] = (int32_t)h7;
  4106. h[8] = (int32_t)h8;
  4107. h[9] = (int32_t)h9;
  4108. }
  4109. static void x25519_scalar_mult_generic(uint8_t out[32],
  4110. const uint8_t scalar[32],
  4111. const uint8_t point[32]) {
  4112. fe x1, x2, z2, x3, z3, tmp0, tmp1;
  4113. uint8_t e[32];
  4114. unsigned swap = 0;
  4115. int pos;
  4116. memcpy(e, scalar, 32);
  4117. e[0] &= 248;
  4118. e[31] &= 127;
  4119. e[31] |= 64;
  4120. fe_frombytes(x1, point);
  4121. fe_1(x2);
  4122. fe_0(z2);
  4123. fe_copy(x3, x1);
  4124. fe_1(z3);
  4125. for (pos = 254; pos >= 0; --pos) {
  4126. unsigned b = 1 & (e[pos / 8] >> (pos & 7));
  4127. swap ^= b;
  4128. fe_cswap(x2, x3, swap);
  4129. fe_cswap(z2, z3, swap);
  4130. swap = b;
  4131. fe_sub(tmp0, x3, z3);
  4132. fe_sub(tmp1, x2, z2);
  4133. fe_add(x2, x2, z2);
  4134. fe_add(z2, x3, z3);
  4135. fe_mul(z3, tmp0, x2);
  4136. fe_mul(z2, z2, tmp1);
  4137. fe_sq(tmp0, tmp1);
  4138. fe_sq(tmp1, x2);
  4139. fe_add(x3, z3, z2);
  4140. fe_sub(z2, z3, z2);
  4141. fe_mul(x2, tmp1, tmp0);
  4142. fe_sub(tmp1, tmp1, tmp0);
  4143. fe_sq(z2, z2);
  4144. fe_mul121666(z3, tmp1);
  4145. fe_sq(x3, x3);
  4146. fe_add(tmp0, tmp0, z3);
  4147. fe_mul(z3, x1, z2);
  4148. fe_mul(z2, tmp1, tmp0);
  4149. }
  4150. fe_invert(z2, z2);
  4151. fe_mul(x2, x2, z2);
  4152. fe_tobytes(out, x2);
  4153. OPENSSL_cleanse(e, sizeof(e));
  4154. }
  4155. static void x25519_scalar_mult(uint8_t out[32], const uint8_t scalar[32],
  4156. const uint8_t point[32]) {
  4157. x25519_scalar_mult_generic(out, scalar, point);
  4158. }
  4159. #endif
  4160. static void slide(signed char *r, const uint8_t *a)
  4161. {
  4162. int i;
  4163. int b;
  4164. int k;
  4165. for (i = 0; i < 256; ++i) {
  4166. r[i] = 1 & (a[i >> 3] >> (i & 7));
  4167. }
  4168. for (i = 0; i < 256; ++i) {
  4169. if (r[i]) {
  4170. for (b = 1; b <= 6 && i + b < 256; ++b) {
  4171. if (r[i + b]) {
  4172. if (r[i] + (r[i + b] << b) <= 15) {
  4173. r[i] += r[i + b] << b;
  4174. r[i + b] = 0;
  4175. } else if (r[i] - (r[i + b] << b) >= -15) {
  4176. r[i] -= r[i + b] << b;
  4177. for (k = i + b; k < 256; ++k) {
  4178. if (!r[k]) {
  4179. r[k] = 1;
  4180. break;
  4181. }
  4182. r[k] = 0;
  4183. }
  4184. } else {
  4185. break;
  4186. }
  4187. }
  4188. }
  4189. }
  4190. }
  4191. }
  4192. static const ge_precomp Bi[8] = {
  4193. {
  4194. {25967493, -14356035, 29566456, 3660896, -12694345, 4014787, 27544626,
  4195. -11754271, -6079156, 2047605},
  4196. {-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692,
  4197. 5043384, 19500929, -15469378},
  4198. {-8738181, 4489570, 9688441, -14785194, 10184609, -12363380, 29287919,
  4199. 11864899, -24514362, -4438546},
  4200. },
  4201. {
  4202. {15636291, -9688557, 24204773, -7912398, 616977, -16685262, 27787600,
  4203. -14772189, 28944400, -1550024},
  4204. {16568933, 4717097, -11556148, -1102322, 15682896, -11807043, 16354577,
  4205. -11775962, 7689662, 11199574},
  4206. {30464156, -5976125, -11779434, -15670865, 23220365, 15915852, 7512774,
  4207. 10017326, -17749093, -9920357},
  4208. },
  4209. {
  4210. {10861363, 11473154, 27284546, 1981175, -30064349, 12577861, 32867885,
  4211. 14515107, -15438304, 10819380},
  4212. {4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668,
  4213. 12483688, -12668491, 5581306},
  4214. {19563160, 16186464, -29386857, 4097519, 10237984, -4348115, 28542350,
  4215. 13850243, -23678021, -15815942},
  4216. },
  4217. {
  4218. {5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852,
  4219. 5230134, -23952439, -15175766},
  4220. {-30269007, -3463509, 7665486, 10083793, 28475525, 1649722, 20654025,
  4221. 16520125, 30598449, 7715701},
  4222. {28881845, 14381568, 9657904, 3680757, -20181635, 7843316, -31400660,
  4223. 1370708, 29794553, -1409300},
  4224. },
  4225. {
  4226. {-22518993, -6692182, 14201702, -8745502, -23510406, 8844726, 18474211,
  4227. -1361450, -13062696, 13821877},
  4228. {-6455177, -7839871, 3374702, -4740862, -27098617, -10571707, 31655028,
  4229. -7212327, 18853322, -14220951},
  4230. {4566830, -12963868, -28974889, -12240689, -7602672, -2830569, -8514358,
  4231. -10431137, 2207753, -3209784},
  4232. },
  4233. {
  4234. {-25154831, -4185821, 29681144, 7868801, -6854661, -9423865, -12437364,
  4235. -663000, -31111463, -16132436},
  4236. {25576264, -2703214, 7349804, -11814844, 16472782, 9300885, 3844789,
  4237. 15725684, 171356, 6466918},
  4238. {23103977, 13316479, 9739013, -16149481, 817875, -15038942, 8965339,
  4239. -14088058, -30714912, 16193877},
  4240. },
  4241. {
  4242. {-33521811, 3180713, -2394130, 14003687, -16903474, -16270840, 17238398,
  4243. 4729455, -18074513, 9256800},
  4244. {-25182317, -4174131, 32336398, 5036987, -21236817, 11360617, 22616405,
  4245. 9761698, -19827198, 630305},
  4246. {-13720693, 2639453, -24237460, -7406481, 9494427, -5774029, -6554551,
  4247. -15960994, -2449256, -14291300},
  4248. },
  4249. {
  4250. {-3151181, -5046075, 9282714, 6866145, -31907062, -863023, -18940575,
  4251. 15033784, 25105118, -7894876},
  4252. {-24326370, 15950226, -31801215, -14592823, -11662737, -5090925,
  4253. 1573892, -2625887, 2198790, -15804619},
  4254. {-3099351, 10324967, -2241613, 7453183, -5446979, -2735503, -13812022,
  4255. -16236442, -32461234, -12290683},
  4256. },
  4257. };
  4258. /*
  4259. * r = a * A + b * B
  4260. *
  4261. * where a = a[0]+256*a[1]+...+256^31 a[31].
  4262. * and b = b[0]+256*b[1]+...+256^31 b[31].
  4263. * B is the Ed25519 base point (x,4/5) with x positive.
  4264. */
  4265. static void ge_double_scalarmult_vartime(ge_p2 *r, const uint8_t *a,
  4266. const ge_p3 *A, const uint8_t *b)
  4267. {
  4268. signed char aslide[256];
  4269. signed char bslide[256];
  4270. ge_cached Ai[8]; /* A,3A,5A,7A,9A,11A,13A,15A */
  4271. ge_p1p1 t;
  4272. ge_p3 u;
  4273. ge_p3 A2;
  4274. int i;
  4275. slide(aslide, a);
  4276. slide(bslide, b);
  4277. ge_p3_to_cached(&Ai[0], A);
  4278. ge_p3_dbl(&t, A);
  4279. ge_p1p1_to_p3(&A2, &t);
  4280. ge_add(&t, &A2, &Ai[0]);
  4281. ge_p1p1_to_p3(&u, &t);
  4282. ge_p3_to_cached(&Ai[1], &u);
  4283. ge_add(&t, &A2, &Ai[1]);
  4284. ge_p1p1_to_p3(&u, &t);
  4285. ge_p3_to_cached(&Ai[2], &u);
  4286. ge_add(&t, &A2, &Ai[2]);
  4287. ge_p1p1_to_p3(&u, &t);
  4288. ge_p3_to_cached(&Ai[3], &u);
  4289. ge_add(&t, &A2, &Ai[3]);
  4290. ge_p1p1_to_p3(&u, &t);
  4291. ge_p3_to_cached(&Ai[4], &u);
  4292. ge_add(&t, &A2, &Ai[4]);
  4293. ge_p1p1_to_p3(&u, &t);
  4294. ge_p3_to_cached(&Ai[5], &u);
  4295. ge_add(&t, &A2, &Ai[5]);
  4296. ge_p1p1_to_p3(&u, &t);
  4297. ge_p3_to_cached(&Ai[6], &u);
  4298. ge_add(&t, &A2, &Ai[6]);
  4299. ge_p1p1_to_p3(&u, &t);
  4300. ge_p3_to_cached(&Ai[7], &u);
  4301. ge_p2_0(r);
  4302. for (i = 255; i >= 0; --i) {
  4303. if (aslide[i] || bslide[i]) {
  4304. break;
  4305. }
  4306. }
  4307. for (; i >= 0; --i) {
  4308. ge_p2_dbl(&t, r);
  4309. if (aslide[i] > 0) {
  4310. ge_p1p1_to_p3(&u, &t);
  4311. ge_add(&t, &u, &Ai[aslide[i] / 2]);
  4312. } else if (aslide[i] < 0) {
  4313. ge_p1p1_to_p3(&u, &t);
  4314. ge_sub(&t, &u, &Ai[(-aslide[i]) / 2]);
  4315. }
  4316. if (bslide[i] > 0) {
  4317. ge_p1p1_to_p3(&u, &t);
  4318. ge_madd(&t, &u, &Bi[bslide[i] / 2]);
  4319. } else if (bslide[i] < 0) {
  4320. ge_p1p1_to_p3(&u, &t);
  4321. ge_msub(&t, &u, &Bi[(-bslide[i]) / 2]);
  4322. }
  4323. ge_p1p1_to_p2(r, &t);
  4324. }
  4325. }
  4326. /*
  4327. * The set of scalars is \Z/l
  4328. * where l = 2^252 + 27742317777372353535851937790883648493.
  4329. *
  4330. * Input:
  4331. * s[0]+256*s[1]+...+256^63*s[63] = s
  4332. *
  4333. * Output:
  4334. * s[0]+256*s[1]+...+256^31*s[31] = s mod l
  4335. * where l = 2^252 + 27742317777372353535851937790883648493.
  4336. * Overwrites s in place.
  4337. */
  4338. static void x25519_sc_reduce(uint8_t *s)
  4339. {
  4340. int64_t s0 = kBottom21Bits & load_3(s);
  4341. int64_t s1 = kBottom21Bits & (load_4(s + 2) >> 5);
  4342. int64_t s2 = kBottom21Bits & (load_3(s + 5) >> 2);
  4343. int64_t s3 = kBottom21Bits & (load_4(s + 7) >> 7);
  4344. int64_t s4 = kBottom21Bits & (load_4(s + 10) >> 4);
  4345. int64_t s5 = kBottom21Bits & (load_3(s + 13) >> 1);
  4346. int64_t s6 = kBottom21Bits & (load_4(s + 15) >> 6);
  4347. int64_t s7 = kBottom21Bits & (load_3(s + 18) >> 3);
  4348. int64_t s8 = kBottom21Bits & load_3(s + 21);
  4349. int64_t s9 = kBottom21Bits & (load_4(s + 23) >> 5);
  4350. int64_t s10 = kBottom21Bits & (load_3(s + 26) >> 2);
  4351. int64_t s11 = kBottom21Bits & (load_4(s + 28) >> 7);
  4352. int64_t s12 = kBottom21Bits & (load_4(s + 31) >> 4);
  4353. int64_t s13 = kBottom21Bits & (load_3(s + 34) >> 1);
  4354. int64_t s14 = kBottom21Bits & (load_4(s + 36) >> 6);
  4355. int64_t s15 = kBottom21Bits & (load_3(s + 39) >> 3);
  4356. int64_t s16 = kBottom21Bits & load_3(s + 42);
  4357. int64_t s17 = kBottom21Bits & (load_4(s + 44) >> 5);
  4358. int64_t s18 = kBottom21Bits & (load_3(s + 47) >> 2);
  4359. int64_t s19 = kBottom21Bits & (load_4(s + 49) >> 7);
  4360. int64_t s20 = kBottom21Bits & (load_4(s + 52) >> 4);
  4361. int64_t s21 = kBottom21Bits & (load_3(s + 55) >> 1);
  4362. int64_t s22 = kBottom21Bits & (load_4(s + 57) >> 6);
  4363. int64_t s23 = (load_4(s + 60) >> 3);
  4364. int64_t carry0;
  4365. int64_t carry1;
  4366. int64_t carry2;
  4367. int64_t carry3;
  4368. int64_t carry4;
  4369. int64_t carry5;
  4370. int64_t carry6;
  4371. int64_t carry7;
  4372. int64_t carry8;
  4373. int64_t carry9;
  4374. int64_t carry10;
  4375. int64_t carry11;
  4376. int64_t carry12;
  4377. int64_t carry13;
  4378. int64_t carry14;
  4379. int64_t carry15;
  4380. int64_t carry16;
  4381. s11 += s23 * 666643;
  4382. s12 += s23 * 470296;
  4383. s13 += s23 * 654183;
  4384. s14 -= s23 * 997805;
  4385. s15 += s23 * 136657;
  4386. s16 -= s23 * 683901;
  4387. s23 = 0;
  4388. s10 += s22 * 666643;
  4389. s11 += s22 * 470296;
  4390. s12 += s22 * 654183;
  4391. s13 -= s22 * 997805;
  4392. s14 += s22 * 136657;
  4393. s15 -= s22 * 683901;
  4394. s22 = 0;
  4395. s9 += s21 * 666643;
  4396. s10 += s21 * 470296;
  4397. s11 += s21 * 654183;
  4398. s12 -= s21 * 997805;
  4399. s13 += s21 * 136657;
  4400. s14 -= s21 * 683901;
  4401. s21 = 0;
  4402. s8 += s20 * 666643;
  4403. s9 += s20 * 470296;
  4404. s10 += s20 * 654183;
  4405. s11 -= s20 * 997805;
  4406. s12 += s20 * 136657;
  4407. s13 -= s20 * 683901;
  4408. s20 = 0;
  4409. s7 += s19 * 666643;
  4410. s8 += s19 * 470296;
  4411. s9 += s19 * 654183;
  4412. s10 -= s19 * 997805;
  4413. s11 += s19 * 136657;
  4414. s12 -= s19 * 683901;
  4415. s19 = 0;
  4416. s6 += s18 * 666643;
  4417. s7 += s18 * 470296;
  4418. s8 += s18 * 654183;
  4419. s9 -= s18 * 997805;
  4420. s10 += s18 * 136657;
  4421. s11 -= s18 * 683901;
  4422. s18 = 0;
  4423. carry6 = (s6 + (1 << 20)) >> 21;
  4424. s7 += carry6;
  4425. s6 -= carry6 * (1 << 21);
  4426. carry8 = (s8 + (1 << 20)) >> 21;
  4427. s9 += carry8;
  4428. s8 -= carry8 * (1 << 21);
  4429. carry10 = (s10 + (1 << 20)) >> 21;
  4430. s11 += carry10;
  4431. s10 -= carry10 * (1 << 21);
  4432. carry12 = (s12 + (1 << 20)) >> 21;
  4433. s13 += carry12;
  4434. s12 -= carry12 * (1 << 21);
  4435. carry14 = (s14 + (1 << 20)) >> 21;
  4436. s15 += carry14;
  4437. s14 -= carry14 * (1 << 21);
  4438. carry16 = (s16 + (1 << 20)) >> 21;
  4439. s17 += carry16;
  4440. s16 -= carry16 * (1 << 21);
  4441. carry7 = (s7 + (1 << 20)) >> 21;
  4442. s8 += carry7;
  4443. s7 -= carry7 * (1 << 21);
  4444. carry9 = (s9 + (1 << 20)) >> 21;
  4445. s10 += carry9;
  4446. s9 -= carry9 * (1 << 21);
  4447. carry11 = (s11 + (1 << 20)) >> 21;
  4448. s12 += carry11;
  4449. s11 -= carry11 * (1 << 21);
  4450. carry13 = (s13 + (1 << 20)) >> 21;
  4451. s14 += carry13;
  4452. s13 -= carry13 * (1 << 21);
  4453. carry15 = (s15 + (1 << 20)) >> 21;
  4454. s16 += carry15;
  4455. s15 -= carry15 * (1 << 21);
  4456. s5 += s17 * 666643;
  4457. s6 += s17 * 470296;
  4458. s7 += s17 * 654183;
  4459. s8 -= s17 * 997805;
  4460. s9 += s17 * 136657;
  4461. s10 -= s17 * 683901;
  4462. s17 = 0;
  4463. s4 += s16 * 666643;
  4464. s5 += s16 * 470296;
  4465. s6 += s16 * 654183;
  4466. s7 -= s16 * 997805;
  4467. s8 += s16 * 136657;
  4468. s9 -= s16 * 683901;
  4469. s16 = 0;
  4470. s3 += s15 * 666643;
  4471. s4 += s15 * 470296;
  4472. s5 += s15 * 654183;
  4473. s6 -= s15 * 997805;
  4474. s7 += s15 * 136657;
  4475. s8 -= s15 * 683901;
  4476. s15 = 0;
  4477. s2 += s14 * 666643;
  4478. s3 += s14 * 470296;
  4479. s4 += s14 * 654183;
  4480. s5 -= s14 * 997805;
  4481. s6 += s14 * 136657;
  4482. s7 -= s14 * 683901;
  4483. s14 = 0;
  4484. s1 += s13 * 666643;
  4485. s2 += s13 * 470296;
  4486. s3 += s13 * 654183;
  4487. s4 -= s13 * 997805;
  4488. s5 += s13 * 136657;
  4489. s6 -= s13 * 683901;
  4490. s13 = 0;
  4491. s0 += s12 * 666643;
  4492. s1 += s12 * 470296;
  4493. s2 += s12 * 654183;
  4494. s3 -= s12 * 997805;
  4495. s4 += s12 * 136657;
  4496. s5 -= s12 * 683901;
  4497. s12 = 0;
  4498. carry0 = (s0 + (1 << 20)) >> 21;
  4499. s1 += carry0;
  4500. s0 -= carry0 * (1 << 21);
  4501. carry2 = (s2 + (1 << 20)) >> 21;
  4502. s3 += carry2;
  4503. s2 -= carry2 * (1 << 21);
  4504. carry4 = (s4 + (1 << 20)) >> 21;
  4505. s5 += carry4;
  4506. s4 -= carry4 * (1 << 21);
  4507. carry6 = (s6 + (1 << 20)) >> 21;
  4508. s7 += carry6;
  4509. s6 -= carry6 * (1 << 21);
  4510. carry8 = (s8 + (1 << 20)) >> 21;
  4511. s9 += carry8;
  4512. s8 -= carry8 * (1 << 21);
  4513. carry10 = (s10 + (1 << 20)) >> 21;
  4514. s11 += carry10;
  4515. s10 -= carry10 * (1 << 21);
  4516. carry1 = (s1 + (1 << 20)) >> 21;
  4517. s2 += carry1;
  4518. s1 -= carry1 * (1 << 21);
  4519. carry3 = (s3 + (1 << 20)) >> 21;
  4520. s4 += carry3;
  4521. s3 -= carry3 * (1 << 21);
  4522. carry5 = (s5 + (1 << 20)) >> 21;
  4523. s6 += carry5;
  4524. s5 -= carry5 * (1 << 21);
  4525. carry7 = (s7 + (1 << 20)) >> 21;
  4526. s8 += carry7;
  4527. s7 -= carry7 * (1 << 21);
  4528. carry9 = (s9 + (1 << 20)) >> 21;
  4529. s10 += carry9;
  4530. s9 -= carry9 * (1 << 21);
  4531. carry11 = (s11 + (1 << 20)) >> 21;
  4532. s12 += carry11;
  4533. s11 -= carry11 * (1 << 21);
  4534. s0 += s12 * 666643;
  4535. s1 += s12 * 470296;
  4536. s2 += s12 * 654183;
  4537. s3 -= s12 * 997805;
  4538. s4 += s12 * 136657;
  4539. s5 -= s12 * 683901;
  4540. s12 = 0;
  4541. carry0 = s0 >> 21;
  4542. s1 += carry0;
  4543. s0 -= carry0 * (1 << 21);
  4544. carry1 = s1 >> 21;
  4545. s2 += carry1;
  4546. s1 -= carry1 * (1 << 21);
  4547. carry2 = s2 >> 21;
  4548. s3 += carry2;
  4549. s2 -= carry2 * (1 << 21);
  4550. carry3 = s3 >> 21;
  4551. s4 += carry3;
  4552. s3 -= carry3 * (1 << 21);
  4553. carry4 = s4 >> 21;
  4554. s5 += carry4;
  4555. s4 -= carry4 * (1 << 21);
  4556. carry5 = s5 >> 21;
  4557. s6 += carry5;
  4558. s5 -= carry5 * (1 << 21);
  4559. carry6 = s6 >> 21;
  4560. s7 += carry6;
  4561. s6 -= carry6 * (1 << 21);
  4562. carry7 = s7 >> 21;
  4563. s8 += carry7;
  4564. s7 -= carry7 * (1 << 21);
  4565. carry8 = s8 >> 21;
  4566. s9 += carry8;
  4567. s8 -= carry8 * (1 << 21);
  4568. carry9 = s9 >> 21;
  4569. s10 += carry9;
  4570. s9 -= carry9 * (1 << 21);
  4571. carry10 = s10 >> 21;
  4572. s11 += carry10;
  4573. s10 -= carry10 * (1 << 21);
  4574. carry11 = s11 >> 21;
  4575. s12 += carry11;
  4576. s11 -= carry11 * (1 << 21);
  4577. s0 += s12 * 666643;
  4578. s1 += s12 * 470296;
  4579. s2 += s12 * 654183;
  4580. s3 -= s12 * 997805;
  4581. s4 += s12 * 136657;
  4582. s5 -= s12 * 683901;
  4583. s12 = 0;
  4584. carry0 = s0 >> 21;
  4585. s1 += carry0;
  4586. s0 -= carry0 * (1 << 21);
  4587. carry1 = s1 >> 21;
  4588. s2 += carry1;
  4589. s1 -= carry1 * (1 << 21);
  4590. carry2 = s2 >> 21;
  4591. s3 += carry2;
  4592. s2 -= carry2 * (1 << 21);
  4593. carry3 = s3 >> 21;
  4594. s4 += carry3;
  4595. s3 -= carry3 * (1 << 21);
  4596. carry4 = s4 >> 21;
  4597. s5 += carry4;
  4598. s4 -= carry4 * (1 << 21);
  4599. carry5 = s5 >> 21;
  4600. s6 += carry5;
  4601. s5 -= carry5 * (1 << 21);
  4602. carry6 = s6 >> 21;
  4603. s7 += carry6;
  4604. s6 -= carry6 * (1 << 21);
  4605. carry7 = s7 >> 21;
  4606. s8 += carry7;
  4607. s7 -= carry7 * (1 << 21);
  4608. carry8 = s8 >> 21;
  4609. s9 += carry8;
  4610. s8 -= carry8 * (1 << 21);
  4611. carry9 = s9 >> 21;
  4612. s10 += carry9;
  4613. s9 -= carry9 * (1 << 21);
  4614. carry10 = s10 >> 21;
  4615. s11 += carry10;
  4616. s10 -= carry10 * (1 << 21);
  4617. s[ 0] = (uint8_t) (s0 >> 0);
  4618. s[ 1] = (uint8_t) (s0 >> 8);
  4619. s[ 2] = (uint8_t)((s0 >> 16) | (s1 << 5));
  4620. s[ 3] = (uint8_t) (s1 >> 3);
  4621. s[ 4] = (uint8_t) (s1 >> 11);
  4622. s[ 5] = (uint8_t)((s1 >> 19) | (s2 << 2));
  4623. s[ 6] = (uint8_t) (s2 >> 6);
  4624. s[ 7] = (uint8_t)((s2 >> 14) | (s3 << 7));
  4625. s[ 8] = (uint8_t) (s3 >> 1);
  4626. s[ 9] = (uint8_t) (s3 >> 9);
  4627. s[10] = (uint8_t)((s3 >> 17) | (s4 << 4));
  4628. s[11] = (uint8_t) (s4 >> 4);
  4629. s[12] = (uint8_t) (s4 >> 12);
  4630. s[13] = (uint8_t)((s4 >> 20) | (s5 << 1));
  4631. s[14] = (uint8_t) (s5 >> 7);
  4632. s[15] = (uint8_t)((s5 >> 15) | (s6 << 6));
  4633. s[16] = (uint8_t) (s6 >> 2);
  4634. s[17] = (uint8_t) (s6 >> 10);
  4635. s[18] = (uint8_t)((s6 >> 18) | (s7 << 3));
  4636. s[19] = (uint8_t) (s7 >> 5);
  4637. s[20] = (uint8_t) (s7 >> 13);
  4638. s[21] = (uint8_t) (s8 >> 0);
  4639. s[22] = (uint8_t) (s8 >> 8);
  4640. s[23] = (uint8_t)((s8 >> 16) | (s9 << 5));
  4641. s[24] = (uint8_t) (s9 >> 3);
  4642. s[25] = (uint8_t) (s9 >> 11);
  4643. s[26] = (uint8_t)((s9 >> 19) | (s10 << 2));
  4644. s[27] = (uint8_t) (s10 >> 6);
  4645. s[28] = (uint8_t)((s10 >> 14) | (s11 << 7));
  4646. s[29] = (uint8_t) (s11 >> 1);
  4647. s[30] = (uint8_t) (s11 >> 9);
  4648. s[31] = (uint8_t) (s11 >> 17);
  4649. }
  4650. /*
  4651. * Input:
  4652. * a[0]+256*a[1]+...+256^31*a[31] = a
  4653. * b[0]+256*b[1]+...+256^31*b[31] = b
  4654. * c[0]+256*c[1]+...+256^31*c[31] = c
  4655. *
  4656. * Output:
  4657. * s[0]+256*s[1]+...+256^31*s[31] = (ab+c) mod l
  4658. * where l = 2^252 + 27742317777372353535851937790883648493.
  4659. */
  4660. static void sc_muladd(uint8_t *s, const uint8_t *a, const uint8_t *b,
  4661. const uint8_t *c)
  4662. {
  4663. int64_t a0 = kBottom21Bits & load_3(a);
  4664. int64_t a1 = kBottom21Bits & (load_4(a + 2) >> 5);
  4665. int64_t a2 = kBottom21Bits & (load_3(a + 5) >> 2);
  4666. int64_t a3 = kBottom21Bits & (load_4(a + 7) >> 7);
  4667. int64_t a4 = kBottom21Bits & (load_4(a + 10) >> 4);
  4668. int64_t a5 = kBottom21Bits & (load_3(a + 13) >> 1);
  4669. int64_t a6 = kBottom21Bits & (load_4(a + 15) >> 6);
  4670. int64_t a7 = kBottom21Bits & (load_3(a + 18) >> 3);
  4671. int64_t a8 = kBottom21Bits & load_3(a + 21);
  4672. int64_t a9 = kBottom21Bits & (load_4(a + 23) >> 5);
  4673. int64_t a10 = kBottom21Bits & (load_3(a + 26) >> 2);
  4674. int64_t a11 = (load_4(a + 28) >> 7);
  4675. int64_t b0 = kBottom21Bits & load_3(b);
  4676. int64_t b1 = kBottom21Bits & (load_4(b + 2) >> 5);
  4677. int64_t b2 = kBottom21Bits & (load_3(b + 5) >> 2);
  4678. int64_t b3 = kBottom21Bits & (load_4(b + 7) >> 7);
  4679. int64_t b4 = kBottom21Bits & (load_4(b + 10) >> 4);
  4680. int64_t b5 = kBottom21Bits & (load_3(b + 13) >> 1);
  4681. int64_t b6 = kBottom21Bits & (load_4(b + 15) >> 6);
  4682. int64_t b7 = kBottom21Bits & (load_3(b + 18) >> 3);
  4683. int64_t b8 = kBottom21Bits & load_3(b + 21);
  4684. int64_t b9 = kBottom21Bits & (load_4(b + 23) >> 5);
  4685. int64_t b10 = kBottom21Bits & (load_3(b + 26) >> 2);
  4686. int64_t b11 = (load_4(b + 28) >> 7);
  4687. int64_t c0 = kBottom21Bits & load_3(c);
  4688. int64_t c1 = kBottom21Bits & (load_4(c + 2) >> 5);
  4689. int64_t c2 = kBottom21Bits & (load_3(c + 5) >> 2);
  4690. int64_t c3 = kBottom21Bits & (load_4(c + 7) >> 7);
  4691. int64_t c4 = kBottom21Bits & (load_4(c + 10) >> 4);
  4692. int64_t c5 = kBottom21Bits & (load_3(c + 13) >> 1);
  4693. int64_t c6 = kBottom21Bits & (load_4(c + 15) >> 6);
  4694. int64_t c7 = kBottom21Bits & (load_3(c + 18) >> 3);
  4695. int64_t c8 = kBottom21Bits & load_3(c + 21);
  4696. int64_t c9 = kBottom21Bits & (load_4(c + 23) >> 5);
  4697. int64_t c10 = kBottom21Bits & (load_3(c + 26) >> 2);
  4698. int64_t c11 = (load_4(c + 28) >> 7);
  4699. int64_t s0;
  4700. int64_t s1;
  4701. int64_t s2;
  4702. int64_t s3;
  4703. int64_t s4;
  4704. int64_t s5;
  4705. int64_t s6;
  4706. int64_t s7;
  4707. int64_t s8;
  4708. int64_t s9;
  4709. int64_t s10;
  4710. int64_t s11;
  4711. int64_t s12;
  4712. int64_t s13;
  4713. int64_t s14;
  4714. int64_t s15;
  4715. int64_t s16;
  4716. int64_t s17;
  4717. int64_t s18;
  4718. int64_t s19;
  4719. int64_t s20;
  4720. int64_t s21;
  4721. int64_t s22;
  4722. int64_t s23;
  4723. int64_t carry0;
  4724. int64_t carry1;
  4725. int64_t carry2;
  4726. int64_t carry3;
  4727. int64_t carry4;
  4728. int64_t carry5;
  4729. int64_t carry6;
  4730. int64_t carry7;
  4731. int64_t carry8;
  4732. int64_t carry9;
  4733. int64_t carry10;
  4734. int64_t carry11;
  4735. int64_t carry12;
  4736. int64_t carry13;
  4737. int64_t carry14;
  4738. int64_t carry15;
  4739. int64_t carry16;
  4740. int64_t carry17;
  4741. int64_t carry18;
  4742. int64_t carry19;
  4743. int64_t carry20;
  4744. int64_t carry21;
  4745. int64_t carry22;
  4746. s0 = c0 + a0 * b0;
  4747. s1 = c1 + a0 * b1 + a1 * b0;
  4748. s2 = c2 + a0 * b2 + a1 * b1 + a2 * b0;
  4749. s3 = c3 + a0 * b3 + a1 * b2 + a2 * b1 + a3 * b0;
  4750. s4 = c4 + a0 * b4 + a1 * b3 + a2 * b2 + a3 * b1 + a4 * b0;
  4751. s5 = c5 + a0 * b5 + a1 * b4 + a2 * b3 + a3 * b2 + a4 * b1 + a5 * b0;
  4752. s6 = c6 + a0 * b6 + a1 * b5 + a2 * b4 + a3 * b3 + a4 * b2 + a5 * b1 + a6 * b0;
  4753. s7 = c7 + a0 * b7 + a1 * b6 + a2 * b5 + a3 * b4 + a4 * b3 + a5 * b2 + a6 * b1 + a7 * b0;
  4754. s8 = c8 + a0 * b8 + a1 * b7 + a2 * b6 + a3 * b5 + a4 * b4 + a5 * b3 + a6 * b2 + a7 * b1 + a8 * b0;
  4755. s9 = c9 + a0 * b9 + a1 * b8 + a2 * b7 + a3 * b6 + a4 * b5 + a5 * b4 + a6 * b3 + a7 * b2 + a8 * b1 + a9 * b0;
  4756. s10 = c10 + a0 * b10 + a1 * b9 + a2 * b8 + a3 * b7 + a4 * b6 + a5 * b5 + a6 * b4 + a7 * b3 + a8 * b2 + a9 * b1 + a10 * b0;
  4757. 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;
  4758. s12 = a1 * b11 + a2 * b10 + a3 * b9 + a4 * b8 + a5 * b7 + a6 * b6 + a7 * b5 + a8 * b4 + a9 * b3 + a10 * b2 + a11 * b1;
  4759. s13 = a2 * b11 + a3 * b10 + a4 * b9 + a5 * b8 + a6 * b7 + a7 * b6 + a8 * b5 + a9 * b4 + a10 * b3 + a11 * b2;
  4760. s14 = a3 * b11 + a4 * b10 + a5 * b9 + a6 * b8 + a7 * b7 + a8 * b6 + a9 * b5 + a10 * b4 + a11 * b3;
  4761. s15 = a4 * b11 + a5 * b10 + a6 * b9 + a7 * b8 + a8 * b7 + a9 * b6 + a10 * b5 + a11 * b4;
  4762. s16 = a5 * b11 + a6 * b10 + a7 * b9 + a8 * b8 + a9 * b7 + a10 * b6 + a11 * b5;
  4763. s17 = a6 * b11 + a7 * b10 + a8 * b9 + a9 * b8 + a10 * b7 + a11 * b6;
  4764. s18 = a7 * b11 + a8 * b10 + a9 * b9 + a10 * b8 + a11 * b7;
  4765. s19 = a8 * b11 + a9 * b10 + a10 * b9 + a11 * b8;
  4766. s20 = a9 * b11 + a10 * b10 + a11 * b9;
  4767. s21 = a10 * b11 + a11 * b10;
  4768. s22 = a11 * b11;
  4769. s23 = 0;
  4770. carry0 = (s0 + (1 << 20)) >> 21;
  4771. s1 += carry0;
  4772. s0 -= carry0 * (1 << 21);
  4773. carry2 = (s2 + (1 << 20)) >> 21;
  4774. s3 += carry2;
  4775. s2 -= carry2 * (1 << 21);
  4776. carry4 = (s4 + (1 << 20)) >> 21;
  4777. s5 += carry4;
  4778. s4 -= carry4 * (1 << 21);
  4779. carry6 = (s6 + (1 << 20)) >> 21;
  4780. s7 += carry6;
  4781. s6 -= carry6 * (1 << 21);
  4782. carry8 = (s8 + (1 << 20)) >> 21;
  4783. s9 += carry8;
  4784. s8 -= carry8 * (1 << 21);
  4785. carry10 = (s10 + (1 << 20)) >> 21;
  4786. s11 += carry10;
  4787. s10 -= carry10 * (1 << 21);
  4788. carry12 = (s12 + (1 << 20)) >> 21;
  4789. s13 += carry12;
  4790. s12 -= carry12 * (1 << 21);
  4791. carry14 = (s14 + (1 << 20)) >> 21;
  4792. s15 += carry14;
  4793. s14 -= carry14 * (1 << 21);
  4794. carry16 = (s16 + (1 << 20)) >> 21;
  4795. s17 += carry16;
  4796. s16 -= carry16 * (1 << 21);
  4797. carry18 = (s18 + (1 << 20)) >> 21;
  4798. s19 += carry18;
  4799. s18 -= carry18 * (1 << 21);
  4800. carry20 = (s20 + (1 << 20)) >> 21;
  4801. s21 += carry20;
  4802. s20 -= carry20 * (1 << 21);
  4803. carry22 = (s22 + (1 << 20)) >> 21;
  4804. s23 += carry22;
  4805. s22 -= carry22 * (1 << 21);
  4806. carry1 = (s1 + (1 << 20)) >> 21;
  4807. s2 += carry1;
  4808. s1 -= carry1 * (1 << 21);
  4809. carry3 = (s3 + (1 << 20)) >> 21;
  4810. s4 += carry3;
  4811. s3 -= carry3 * (1 << 21);
  4812. carry5 = (s5 + (1 << 20)) >> 21;
  4813. s6 += carry5;
  4814. s5 -= carry5 * (1 << 21);
  4815. carry7 = (s7 + (1 << 20)) >> 21;
  4816. s8 += carry7;
  4817. s7 -= carry7 * (1 << 21);
  4818. carry9 = (s9 + (1 << 20)) >> 21;
  4819. s10 += carry9;
  4820. s9 -= carry9 * (1 << 21);
  4821. carry11 = (s11 + (1 << 20)) >> 21;
  4822. s12 += carry11;
  4823. s11 -= carry11 * (1 << 21);
  4824. carry13 = (s13 + (1 << 20)) >> 21;
  4825. s14 += carry13;
  4826. s13 -= carry13 * (1 << 21);
  4827. carry15 = (s15 + (1 << 20)) >> 21;
  4828. s16 += carry15;
  4829. s15 -= carry15 * (1 << 21);
  4830. carry17 = (s17 + (1 << 20)) >> 21;
  4831. s18 += carry17;
  4832. s17 -= carry17 * (1 << 21);
  4833. carry19 = (s19 + (1 << 20)) >> 21;
  4834. s20 += carry19;
  4835. s19 -= carry19 * (1 << 21);
  4836. carry21 = (s21 + (1 << 20)) >> 21;
  4837. s22 += carry21;
  4838. s21 -= carry21 * (1 << 21);
  4839. s11 += s23 * 666643;
  4840. s12 += s23 * 470296;
  4841. s13 += s23 * 654183;
  4842. s14 -= s23 * 997805;
  4843. s15 += s23 * 136657;
  4844. s16 -= s23 * 683901;
  4845. s23 = 0;
  4846. s10 += s22 * 666643;
  4847. s11 += s22 * 470296;
  4848. s12 += s22 * 654183;
  4849. s13 -= s22 * 997805;
  4850. s14 += s22 * 136657;
  4851. s15 -= s22 * 683901;
  4852. s22 = 0;
  4853. s9 += s21 * 666643;
  4854. s10 += s21 * 470296;
  4855. s11 += s21 * 654183;
  4856. s12 -= s21 * 997805;
  4857. s13 += s21 * 136657;
  4858. s14 -= s21 * 683901;
  4859. s21 = 0;
  4860. s8 += s20 * 666643;
  4861. s9 += s20 * 470296;
  4862. s10 += s20 * 654183;
  4863. s11 -= s20 * 997805;
  4864. s12 += s20 * 136657;
  4865. s13 -= s20 * 683901;
  4866. s20 = 0;
  4867. s7 += s19 * 666643;
  4868. s8 += s19 * 470296;
  4869. s9 += s19 * 654183;
  4870. s10 -= s19 * 997805;
  4871. s11 += s19 * 136657;
  4872. s12 -= s19 * 683901;
  4873. s19 = 0;
  4874. s6 += s18 * 666643;
  4875. s7 += s18 * 470296;
  4876. s8 += s18 * 654183;
  4877. s9 -= s18 * 997805;
  4878. s10 += s18 * 136657;
  4879. s11 -= s18 * 683901;
  4880. s18 = 0;
  4881. carry6 = (s6 + (1 << 20)) >> 21;
  4882. s7 += carry6;
  4883. s6 -= carry6 * (1 << 21);
  4884. carry8 = (s8 + (1 << 20)) >> 21;
  4885. s9 += carry8;
  4886. s8 -= carry8 * (1 << 21);
  4887. carry10 = (s10 + (1 << 20)) >> 21;
  4888. s11 += carry10;
  4889. s10 -= carry10 * (1 << 21);
  4890. carry12 = (s12 + (1 << 20)) >> 21;
  4891. s13 += carry12;
  4892. s12 -= carry12 * (1 << 21);
  4893. carry14 = (s14 + (1 << 20)) >> 21;
  4894. s15 += carry14;
  4895. s14 -= carry14 * (1 << 21);
  4896. carry16 = (s16 + (1 << 20)) >> 21;
  4897. s17 += carry16;
  4898. s16 -= carry16 * (1 << 21);
  4899. carry7 = (s7 + (1 << 20)) >> 21;
  4900. s8 += carry7;
  4901. s7 -= carry7 * (1 << 21);
  4902. carry9 = (s9 + (1 << 20)) >> 21;
  4903. s10 += carry9;
  4904. s9 -= carry9 * (1 << 21);
  4905. carry11 = (s11 + (1 << 20)) >> 21;
  4906. s12 += carry11;
  4907. s11 -= carry11 * (1 << 21);
  4908. carry13 = (s13 + (1 << 20)) >> 21;
  4909. s14 += carry13;
  4910. s13 -= carry13 * (1 << 21);
  4911. carry15 = (s15 + (1 << 20)) >> 21;
  4912. s16 += carry15;
  4913. s15 -= carry15 * (1 << 21);
  4914. s5 += s17 * 666643;
  4915. s6 += s17 * 470296;
  4916. s7 += s17 * 654183;
  4917. s8 -= s17 * 997805;
  4918. s9 += s17 * 136657;
  4919. s10 -= s17 * 683901;
  4920. s17 = 0;
  4921. s4 += s16 * 666643;
  4922. s5 += s16 * 470296;
  4923. s6 += s16 * 654183;
  4924. s7 -= s16 * 997805;
  4925. s8 += s16 * 136657;
  4926. s9 -= s16 * 683901;
  4927. s16 = 0;
  4928. s3 += s15 * 666643;
  4929. s4 += s15 * 470296;
  4930. s5 += s15 * 654183;
  4931. s6 -= s15 * 997805;
  4932. s7 += s15 * 136657;
  4933. s8 -= s15 * 683901;
  4934. s15 = 0;
  4935. s2 += s14 * 666643;
  4936. s3 += s14 * 470296;
  4937. s4 += s14 * 654183;
  4938. s5 -= s14 * 997805;
  4939. s6 += s14 * 136657;
  4940. s7 -= s14 * 683901;
  4941. s14 = 0;
  4942. s1 += s13 * 666643;
  4943. s2 += s13 * 470296;
  4944. s3 += s13 * 654183;
  4945. s4 -= s13 * 997805;
  4946. s5 += s13 * 136657;
  4947. s6 -= s13 * 683901;
  4948. s13 = 0;
  4949. s0 += s12 * 666643;
  4950. s1 += s12 * 470296;
  4951. s2 += s12 * 654183;
  4952. s3 -= s12 * 997805;
  4953. s4 += s12 * 136657;
  4954. s5 -= s12 * 683901;
  4955. s12 = 0;
  4956. carry0 = (s0 + (1 << 20)) >> 21;
  4957. s1 += carry0;
  4958. s0 -= carry0 * (1 << 21);
  4959. carry2 = (s2 + (1 << 20)) >> 21;
  4960. s3 += carry2;
  4961. s2 -= carry2 * (1 << 21);
  4962. carry4 = (s4 + (1 << 20)) >> 21;
  4963. s5 += carry4;
  4964. s4 -= carry4 * (1 << 21);
  4965. carry6 = (s6 + (1 << 20)) >> 21;
  4966. s7 += carry6;
  4967. s6 -= carry6 * (1 << 21);
  4968. carry8 = (s8 + (1 << 20)) >> 21;
  4969. s9 += carry8;
  4970. s8 -= carry8 * (1 << 21);
  4971. carry10 = (s10 + (1 << 20)) >> 21;
  4972. s11 += carry10;
  4973. s10 -= carry10 * (1 << 21);
  4974. carry1 = (s1 + (1 << 20)) >> 21;
  4975. s2 += carry1;
  4976. s1 -= carry1 * (1 << 21);
  4977. carry3 = (s3 + (1 << 20)) >> 21;
  4978. s4 += carry3;
  4979. s3 -= carry3 * (1 << 21);
  4980. carry5 = (s5 + (1 << 20)) >> 21;
  4981. s6 += carry5;
  4982. s5 -= carry5 * (1 << 21);
  4983. carry7 = (s7 + (1 << 20)) >> 21;
  4984. s8 += carry7;
  4985. s7 -= carry7 * (1 << 21);
  4986. carry9 = (s9 + (1 << 20)) >> 21;
  4987. s10 += carry9;
  4988. s9 -= carry9 * (1 << 21);
  4989. carry11 = (s11 + (1 << 20)) >> 21;
  4990. s12 += carry11;
  4991. s11 -= carry11 * (1 << 21);
  4992. s0 += s12 * 666643;
  4993. s1 += s12 * 470296;
  4994. s2 += s12 * 654183;
  4995. s3 -= s12 * 997805;
  4996. s4 += s12 * 136657;
  4997. s5 -= s12 * 683901;
  4998. s12 = 0;
  4999. carry0 = s0 >> 21;
  5000. s1 += carry0;
  5001. s0 -= carry0 * (1 << 21);
  5002. carry1 = s1 >> 21;
  5003. s2 += carry1;
  5004. s1 -= carry1 * (1 << 21);
  5005. carry2 = s2 >> 21;
  5006. s3 += carry2;
  5007. s2 -= carry2 * (1 << 21);
  5008. carry3 = s3 >> 21;
  5009. s4 += carry3;
  5010. s3 -= carry3 * (1 << 21);
  5011. carry4 = s4 >> 21;
  5012. s5 += carry4;
  5013. s4 -= carry4 * (1 << 21);
  5014. carry5 = s5 >> 21;
  5015. s6 += carry5;
  5016. s5 -= carry5 * (1 << 21);
  5017. carry6 = s6 >> 21;
  5018. s7 += carry6;
  5019. s6 -= carry6 * (1 << 21);
  5020. carry7 = s7 >> 21;
  5021. s8 += carry7;
  5022. s7 -= carry7 * (1 << 21);
  5023. carry8 = s8 >> 21;
  5024. s9 += carry8;
  5025. s8 -= carry8 * (1 << 21);
  5026. carry9 = s9 >> 21;
  5027. s10 += carry9;
  5028. s9 -= carry9 * (1 << 21);
  5029. carry10 = s10 >> 21;
  5030. s11 += carry10;
  5031. s10 -= carry10 * (1 << 21);
  5032. carry11 = s11 >> 21;
  5033. s12 += carry11;
  5034. s11 -= carry11 * (1 << 21);
  5035. s0 += s12 * 666643;
  5036. s1 += s12 * 470296;
  5037. s2 += s12 * 654183;
  5038. s3 -= s12 * 997805;
  5039. s4 += s12 * 136657;
  5040. s5 -= s12 * 683901;
  5041. s12 = 0;
  5042. carry0 = s0 >> 21;
  5043. s1 += carry0;
  5044. s0 -= carry0 * (1 << 21);
  5045. carry1 = s1 >> 21;
  5046. s2 += carry1;
  5047. s1 -= carry1 * (1 << 21);
  5048. carry2 = s2 >> 21;
  5049. s3 += carry2;
  5050. s2 -= carry2 * (1 << 21);
  5051. carry3 = s3 >> 21;
  5052. s4 += carry3;
  5053. s3 -= carry3 * (1 << 21);
  5054. carry4 = s4 >> 21;
  5055. s5 += carry4;
  5056. s4 -= carry4 * (1 << 21);
  5057. carry5 = s5 >> 21;
  5058. s6 += carry5;
  5059. s5 -= carry5 * (1 << 21);
  5060. carry6 = s6 >> 21;
  5061. s7 += carry6;
  5062. s6 -= carry6 * (1 << 21);
  5063. carry7 = s7 >> 21;
  5064. s8 += carry7;
  5065. s7 -= carry7 * (1 << 21);
  5066. carry8 = s8 >> 21;
  5067. s9 += carry8;
  5068. s8 -= carry8 * (1 << 21);
  5069. carry9 = s9 >> 21;
  5070. s10 += carry9;
  5071. s9 -= carry9 * (1 << 21);
  5072. carry10 = s10 >> 21;
  5073. s11 += carry10;
  5074. s10 -= carry10 * (1 << 21);
  5075. s[ 0] = (uint8_t) (s0 >> 0);
  5076. s[ 1] = (uint8_t) (s0 >> 8);
  5077. s[ 2] = (uint8_t)((s0 >> 16) | (s1 << 5));
  5078. s[ 3] = (uint8_t) (s1 >> 3);
  5079. s[ 4] = (uint8_t) (s1 >> 11);
  5080. s[ 5] = (uint8_t)((s1 >> 19) | (s2 << 2));
  5081. s[ 6] = (uint8_t) (s2 >> 6);
  5082. s[ 7] = (uint8_t)((s2 >> 14) | (s3 << 7));
  5083. s[ 8] = (uint8_t) (s3 >> 1);
  5084. s[ 9] = (uint8_t) (s3 >> 9);
  5085. s[10] = (uint8_t)((s3 >> 17) | (s4 << 4));
  5086. s[11] = (uint8_t) (s4 >> 4);
  5087. s[12] = (uint8_t) (s4 >> 12);
  5088. s[13] = (uint8_t)((s4 >> 20) | (s5 << 1));
  5089. s[14] = (uint8_t) (s5 >> 7);
  5090. s[15] = (uint8_t)((s5 >> 15) | (s6 << 6));
  5091. s[16] = (uint8_t) (s6 >> 2);
  5092. s[17] = (uint8_t) (s6 >> 10);
  5093. s[18] = (uint8_t)((s6 >> 18) | (s7 << 3));
  5094. s[19] = (uint8_t) (s7 >> 5);
  5095. s[20] = (uint8_t) (s7 >> 13);
  5096. s[21] = (uint8_t) (s8 >> 0);
  5097. s[22] = (uint8_t) (s8 >> 8);
  5098. s[23] = (uint8_t)((s8 >> 16) | (s9 << 5));
  5099. s[24] = (uint8_t) (s9 >> 3);
  5100. s[25] = (uint8_t) (s9 >> 11);
  5101. s[26] = (uint8_t)((s9 >> 19) | (s10 << 2));
  5102. s[27] = (uint8_t) (s10 >> 6);
  5103. s[28] = (uint8_t)((s10 >> 14) | (s11 << 7));
  5104. s[29] = (uint8_t) (s11 >> 1);
  5105. s[30] = (uint8_t) (s11 >> 9);
  5106. s[31] = (uint8_t) (s11 >> 17);
  5107. }
  5108. int
  5109. ossl_ed25519_sign(uint8_t *out_sig, const uint8_t *message, size_t message_len,
  5110. const uint8_t public_key[32], const uint8_t private_key[32],
  5111. OSSL_LIB_CTX *libctx, const char *propq)
  5112. {
  5113. uint8_t az[SHA512_DIGEST_LENGTH];
  5114. uint8_t nonce[SHA512_DIGEST_LENGTH];
  5115. ge_p3 R;
  5116. uint8_t hram[SHA512_DIGEST_LENGTH];
  5117. EVP_MD *sha512 = EVP_MD_fetch(libctx, SN_sha512, propq);
  5118. EVP_MD_CTX *hash_ctx = EVP_MD_CTX_new();
  5119. unsigned int sz;
  5120. int res = 0;
  5121. if (sha512 == NULL || hash_ctx == NULL)
  5122. goto err;
  5123. if (!EVP_DigestInit_ex(hash_ctx, sha512, NULL)
  5124. || !EVP_DigestUpdate(hash_ctx, private_key, 32)
  5125. || !EVP_DigestFinal_ex(hash_ctx, az, &sz))
  5126. goto err;
  5127. az[0] &= 248;
  5128. az[31] &= 63;
  5129. az[31] |= 64;
  5130. if (!EVP_DigestInit_ex(hash_ctx, sha512, NULL)
  5131. || !EVP_DigestUpdate(hash_ctx, az + 32, 32)
  5132. || !EVP_DigestUpdate(hash_ctx, message, message_len)
  5133. || !EVP_DigestFinal_ex(hash_ctx, nonce, &sz))
  5134. goto err;
  5135. x25519_sc_reduce(nonce);
  5136. ge_scalarmult_base(&R, nonce);
  5137. ge_p3_tobytes(out_sig, &R);
  5138. if (!EVP_DigestInit_ex(hash_ctx, sha512, NULL)
  5139. || !EVP_DigestUpdate(hash_ctx, out_sig, 32)
  5140. || !EVP_DigestUpdate(hash_ctx, public_key, 32)
  5141. || !EVP_DigestUpdate(hash_ctx, message, message_len)
  5142. || !EVP_DigestFinal_ex(hash_ctx, hram, &sz))
  5143. goto err;
  5144. x25519_sc_reduce(hram);
  5145. sc_muladd(out_sig + 32, hram, az, nonce);
  5146. res = 1;
  5147. err:
  5148. OPENSSL_cleanse(nonce, sizeof(nonce));
  5149. OPENSSL_cleanse(az, sizeof(az));
  5150. EVP_MD_free(sha512);
  5151. EVP_MD_CTX_free(hash_ctx);
  5152. return res;
  5153. }
  5154. static const char allzeroes[15];
  5155. int
  5156. ossl_ed25519_verify(const uint8_t *message, size_t message_len,
  5157. const uint8_t signature[64], const uint8_t public_key[32],
  5158. OSSL_LIB_CTX *libctx, const char *propq)
  5159. {
  5160. int i;
  5161. ge_p3 A;
  5162. const uint8_t *r, *s;
  5163. EVP_MD *sha512;
  5164. EVP_MD_CTX *hash_ctx = NULL;
  5165. unsigned int sz;
  5166. int res = 0;
  5167. ge_p2 R;
  5168. uint8_t rcheck[32];
  5169. uint8_t h[SHA512_DIGEST_LENGTH];
  5170. /* 27742317777372353535851937790883648493 in little endian format */
  5171. const uint8_t l_low[16] = {
  5172. 0xED, 0xD3, 0xF5, 0x5C, 0x1A, 0x63, 0x12, 0x58, 0xD6, 0x9C, 0xF7, 0xA2,
  5173. 0xDE, 0xF9, 0xDE, 0x14
  5174. };
  5175. r = signature;
  5176. s = signature + 32;
  5177. /*
  5178. * Check 0 <= s < L where L = 2^252 + 27742317777372353535851937790883648493
  5179. *
  5180. * If not the signature is publicly invalid. Since it's public we can do the
  5181. * check in variable time.
  5182. *
  5183. * First check the most significant byte
  5184. */
  5185. if (s[31] > 0x10)
  5186. return 0;
  5187. if (s[31] == 0x10) {
  5188. /*
  5189. * Most significant byte indicates a value close to 2^252 so check the
  5190. * rest
  5191. */
  5192. if (memcmp(s + 16, allzeroes, sizeof(allzeroes)) != 0)
  5193. return 0;
  5194. for (i = 15; i >= 0; i--) {
  5195. if (s[i] < l_low[i])
  5196. break;
  5197. if (s[i] > l_low[i])
  5198. return 0;
  5199. }
  5200. if (i < 0)
  5201. return 0;
  5202. }
  5203. if (ge_frombytes_vartime(&A, public_key) != 0) {
  5204. return 0;
  5205. }
  5206. fe_neg(A.X, A.X);
  5207. fe_neg(A.T, A.T);
  5208. sha512 = EVP_MD_fetch(libctx, SN_sha512, propq);
  5209. if (sha512 == NULL)
  5210. return 0;
  5211. hash_ctx = EVP_MD_CTX_new();
  5212. if (hash_ctx == NULL)
  5213. goto err;
  5214. if (!EVP_DigestInit_ex(hash_ctx, sha512, NULL)
  5215. || !EVP_DigestUpdate(hash_ctx, r, 32)
  5216. || !EVP_DigestUpdate(hash_ctx, public_key, 32)
  5217. || !EVP_DigestUpdate(hash_ctx, message, message_len)
  5218. || !EVP_DigestFinal_ex(hash_ctx, h, &sz))
  5219. goto err;
  5220. x25519_sc_reduce(h);
  5221. ge_double_scalarmult_vartime(&R, h, &A, s);
  5222. ge_tobytes(rcheck, &R);
  5223. res = CRYPTO_memcmp(rcheck, r, sizeof(rcheck)) == 0;
  5224. err:
  5225. EVP_MD_free(sha512);
  5226. EVP_MD_CTX_free(hash_ctx);
  5227. return res;
  5228. }
  5229. int
  5230. ossl_ed25519_public_from_private(OSSL_LIB_CTX *ctx, uint8_t out_public_key[32],
  5231. const uint8_t private_key[32],
  5232. const char *propq)
  5233. {
  5234. uint8_t az[SHA512_DIGEST_LENGTH];
  5235. ge_p3 A;
  5236. int r;
  5237. EVP_MD *sha512 = NULL;
  5238. sha512 = EVP_MD_fetch(ctx, SN_sha512, propq);
  5239. if (sha512 == NULL)
  5240. return 0;
  5241. r = EVP_Digest(private_key, 32, az, NULL, sha512, NULL);
  5242. EVP_MD_free(sha512);
  5243. if (!r) {
  5244. OPENSSL_cleanse(az, sizeof(az));
  5245. return 0;
  5246. }
  5247. az[0] &= 248;
  5248. az[31] &= 63;
  5249. az[31] |= 64;
  5250. ge_scalarmult_base(&A, az);
  5251. ge_p3_tobytes(out_public_key, &A);
  5252. OPENSSL_cleanse(az, sizeof(az));
  5253. return 1;
  5254. }
  5255. int
  5256. ossl_x25519(uint8_t out_shared_key[32], const uint8_t private_key[32],
  5257. const uint8_t peer_public_value[32])
  5258. {
  5259. static const uint8_t kZeros[32] = {0};
  5260. x25519_scalar_mult(out_shared_key, private_key, peer_public_value);
  5261. /* The all-zero output results when the input is a point of small order. */
  5262. return CRYPTO_memcmp(kZeros, out_shared_key, 32) != 0;
  5263. }
  5264. void
  5265. ossl_x25519_public_from_private(uint8_t out_public_value[32],
  5266. const uint8_t private_key[32])
  5267. {
  5268. uint8_t e[32];
  5269. ge_p3 A;
  5270. fe zplusy, zminusy, zminusy_inv;
  5271. memcpy(e, private_key, 32);
  5272. e[0] &= 248;
  5273. e[31] &= 127;
  5274. e[31] |= 64;
  5275. ge_scalarmult_base(&A, e);
  5276. /*
  5277. * We only need the u-coordinate of the curve25519 point.
  5278. * The map is u=(y+1)/(1-y). Since y=Y/Z, this gives
  5279. * u=(Z+Y)/(Z-Y).
  5280. */
  5281. fe_add(zplusy, A.Z, A.Y);
  5282. fe_sub(zminusy, A.Z, A.Y);
  5283. fe_invert(zminusy_inv, zminusy);
  5284. fe_mul(zplusy, zplusy, zminusy_inv);
  5285. fe_tobytes(out_public_value, zplusy);
  5286. OPENSSL_cleanse(e, sizeof(e));
  5287. }