xref: /freebsd/crypto/openssl/crypto/ec/curve25519.c (revision f25b8c9fb4f58cf61adb47d7570abe7caa6d385d) 
1 /*
2  * Copyright 2016-2024 The OpenSSL Project Authors. All Rights Reserved.
3  *
4  * Licensed under the Apache License 2.0 (the "License").  You may not use
5  * this file except in compliance with the License.  You can obtain a copy
6  * in the file LICENSE in the source distribution or at
7  * https://www.openssl.org/source/license.html
8  */
9 
10 /*
11  * ECDSA low level APIs are deprecated for public use, but still ok for
12  * internal use.
13  */
14 #include "internal/deprecated.h"
15 
16 #include <string.h>
17 #include "crypto/ecx.h"
18 #include "ec_local.h"
19 #include <openssl/evp.h>
20 #include <openssl/sha.h>
21 
22 #include "internal/numbers.h"
23 
24 #if defined(X25519_ASM) && (defined(__x86_64) || defined(__x86_64__) || defined(_M_AMD64) || defined(_M_X64))
25 
26 #define BASE_2_64_IMPLEMENTED
27 
28 typedef uint64_t fe64[4];
29 
30 int x25519_fe64_eligible(void);
31 
32 /*
33  * Following subroutines perform corresponding operations modulo
34  * 2^256-38, i.e. double the curve modulus. However, inputs and
35  * outputs are permitted to be partially reduced, i.e. to remain
36  * in [0..2^256) range. It's all tied up in final fe64_tobytes
37  * that performs full reduction modulo 2^255-19.
38  *
39  * There are no reference C implementations for these.
40  */
41 void x25519_fe64_mul(fe64 h, const fe64 f, const fe64 g);
42 void x25519_fe64_sqr(fe64 h, const fe64 f);
43 void x25519_fe64_mul121666(fe64 h, fe64 f);
44 void x25519_fe64_add(fe64 h, const fe64 f, const fe64 g);
45 void x25519_fe64_sub(fe64 h, const fe64 f, const fe64 g);
46 void x25519_fe64_tobytes(uint8_t *s, const fe64 f);
47 #define fe64_mul x25519_fe64_mul
48 #define fe64_sqr x25519_fe64_sqr
49 #define fe64_mul121666 x25519_fe64_mul121666
50 #define fe64_add x25519_fe64_add
51 #define fe64_sub x25519_fe64_sub
52 #define fe64_tobytes x25519_fe64_tobytes
53 
load_8(const uint8_t * in)54 static uint64_t load_8(const uint8_t *in)
55 {
56     uint64_t result;
57 
58     result = in[0];
59     result |= ((uint64_t)in[1]) << 8;
60     result |= ((uint64_t)in[2]) << 16;
61     result |= ((uint64_t)in[3]) << 24;
62     result |= ((uint64_t)in[4]) << 32;
63     result |= ((uint64_t)in[5]) << 40;
64     result |= ((uint64_t)in[6]) << 48;
65     result |= ((uint64_t)in[7]) << 56;
66 
67     return result;
68 }
69 
fe64_frombytes(fe64 h,const uint8_t * s)70 static void fe64_frombytes(fe64 h, const uint8_t *s)
71 {
72     h[0] = load_8(s);
73     h[1] = load_8(s + 8);
74     h[2] = load_8(s + 16);
75     h[3] = load_8(s + 24) & 0x7fffffffffffffff;
76 }
77 
fe64_0(fe64 h)78 static void fe64_0(fe64 h)
79 {
80     h[0] = 0;
81     h[1] = 0;
82     h[2] = 0;
83     h[3] = 0;
84 }
85 
fe64_1(fe64 h)86 static void fe64_1(fe64 h)
87 {
88     h[0] = 1;
89     h[1] = 0;
90     h[2] = 0;
91     h[3] = 0;
92 }
93 
fe64_copy(fe64 h,const fe64 f)94 static void fe64_copy(fe64 h, const fe64 f)
95 {
96     h[0] = f[0];
97     h[1] = f[1];
98     h[2] = f[2];
99     h[3] = f[3];
100 }
101 
fe64_cswap(fe64 f,fe64 g,unsigned int b)102 static void fe64_cswap(fe64 f, fe64 g, unsigned int b)
103 {
104     int i;
105     uint64_t mask = 0 - (uint64_t)b;
106 
107     for (i = 0; i < 4; i++) {
108         uint64_t x = f[i] ^ g[i];
109         x &= mask;
110         f[i] ^= x;
111         g[i] ^= x;
112     }
113 }
114 
fe64_invert(fe64 out,const fe64 z)115 static void fe64_invert(fe64 out, const fe64 z)
116 {
117     fe64 t0;
118     fe64 t1;
119     fe64 t2;
120     fe64 t3;
121     int i;
122 
123     /*
124      * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as
125      * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11.
126      */
127 
128     /* t0 = z ** 2 */
129     fe64_sqr(t0, z);
130 
131     /* t1 = t0 ** (2 ** 2) = z ** 8 */
132     fe64_sqr(t1, t0);
133     fe64_sqr(t1, t1);
134 
135     /* t1 = z * t1 = z ** 9 */
136     fe64_mul(t1, z, t1);
137     /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */
138     fe64_mul(t0, t0, t1);
139 
140     /* t2 = t0 ** 2 = z ** 22 */
141     fe64_sqr(t2, t0);
142 
143     /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */
144     fe64_mul(t1, t1, t2);
145 
146     /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */
147     fe64_sqr(t2, t1);
148     for (i = 1; i < 5; ++i)
149         fe64_sqr(t2, t2);
150 
151     /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */
152     fe64_mul(t1, t2, t1);
153 
154     /* Continuing similarly... */
155 
156     /* t2 = z ** (2 ** 20 - 1) */
157     fe64_sqr(t2, t1);
158     for (i = 1; i < 10; ++i)
159         fe64_sqr(t2, t2);
160 
161     fe64_mul(t2, t2, t1);
162 
163     /* t2 = z ** (2 ** 40 - 1) */
164     fe64_sqr(t3, t2);
165     for (i = 1; i < 20; ++i)
166         fe64_sqr(t3, t3);
167 
168     fe64_mul(t2, t3, t2);
169 
170     /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */
171     for (i = 0; i < 10; ++i)
172         fe64_sqr(t2, t2);
173 
174     /* t1 = z ** (2 ** 50 - 1) */
175     fe64_mul(t1, t2, t1);
176 
177     /* t2 = z ** (2 ** 100 - 1) */
178     fe64_sqr(t2, t1);
179     for (i = 1; i < 50; ++i)
180         fe64_sqr(t2, t2);
181 
182     fe64_mul(t2, t2, t1);
183 
184     /* t2 = z ** (2 ** 200 - 1) */
185     fe64_sqr(t3, t2);
186     for (i = 1; i < 100; ++i)
187         fe64_sqr(t3, t3);
188 
189     fe64_mul(t2, t3, t2);
190 
191     /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */
192     for (i = 0; i < 50; ++i)
193         fe64_sqr(t2, t2);
194 
195     /* t1 = z ** (2 ** 250 - 1) */
196     fe64_mul(t1, t2, t1);
197 
198     /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */
199     for (i = 0; i < 5; ++i)
200         fe64_sqr(t1, t1);
201 
202     /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */
203     fe64_mul(out, t1, t0);
204 }
205 
206 /*
207  * Duplicate of original x25519_scalar_mult_generic, but using
208  * fe64_* subroutines.
209  */
x25519_scalar_mulx(uint8_t out[32],const uint8_t scalar[32],const uint8_t point[32])210 static void x25519_scalar_mulx(uint8_t out[32], const uint8_t scalar[32],
211     const uint8_t point[32])
212 {
213     fe64 x1, x2, z2, x3, z3, tmp0, tmp1;
214     uint8_t e[32];
215     unsigned swap = 0;
216     int pos;
217 
218     memcpy(e, scalar, 32);
219     e[0] &= 0xf8;
220     e[31] &= 0x7f;
221     e[31] |= 0x40;
222     fe64_frombytes(x1, point);
223     fe64_1(x2);
224     fe64_0(z2);
225     fe64_copy(x3, x1);
226     fe64_1(z3);
227 
228     for (pos = 254; pos >= 0; --pos) {
229         unsigned int b = 1 & (e[pos / 8] >> (pos & 7));
230 
231         swap ^= b;
232         fe64_cswap(x2, x3, swap);
233         fe64_cswap(z2, z3, swap);
234         swap = b;
235         fe64_sub(tmp0, x3, z3);
236         fe64_sub(tmp1, x2, z2);
237         fe64_add(x2, x2, z2);
238         fe64_add(z2, x3, z3);
239         fe64_mul(z3, x2, tmp0);
240         fe64_mul(z2, z2, tmp1);
241         fe64_sqr(tmp0, tmp1);
242         fe64_sqr(tmp1, x2);
243         fe64_add(x3, z3, z2);
244         fe64_sub(z2, z3, z2);
245         fe64_mul(x2, tmp1, tmp0);
246         fe64_sub(tmp1, tmp1, tmp0);
247         fe64_sqr(z2, z2);
248         fe64_mul121666(z3, tmp1);
249         fe64_sqr(x3, x3);
250         fe64_add(tmp0, tmp0, z3);
251         fe64_mul(z3, x1, z2);
252         fe64_mul(z2, tmp1, tmp0);
253     }
254 
255     fe64_invert(z2, z2);
256     fe64_mul(x2, x2, z2);
257     fe64_tobytes(out, x2);
258 
259     OPENSSL_cleanse(e, sizeof(e));
260 }
261 #endif
262 
263 #if defined(X25519_ASM)                                          \
264     || (defined(INT128_MAX)                                      \
265         && !defined(__sparc__)                                   \
266         && (!defined(__SIZEOF_LONG__) || (__SIZEOF_LONG__ == 8)) \
267         && !(defined(__ANDROID__) && !defined(__clang__)))
268 /*
269  * Base 2^51 implementation. It's virtually no different from reference
270  * base 2^25.5 implementation in respect to lax boundary conditions for
271  * intermediate values and even individual limbs. So that whatever you
272  * know about the reference, applies even here...
273  */
274 #define BASE_2_51_IMPLEMENTED
275 
276 typedef uint64_t fe51[5];
277 
278 static const uint64_t MASK51 = 0x7ffffffffffff;
279 
load_7(const uint8_t * in)280 static uint64_t load_7(const uint8_t *in)
281 {
282     uint64_t result;
283 
284     result = in[0];
285     result |= ((uint64_t)in[1]) << 8;
286     result |= ((uint64_t)in[2]) << 16;
287     result |= ((uint64_t)in[3]) << 24;
288     result |= ((uint64_t)in[4]) << 32;
289     result |= ((uint64_t)in[5]) << 40;
290     result |= ((uint64_t)in[6]) << 48;
291 
292     return result;
293 }
294 
load_6(const uint8_t * in)295 static uint64_t load_6(const uint8_t *in)
296 {
297     uint64_t result;
298 
299     result = in[0];
300     result |= ((uint64_t)in[1]) << 8;
301     result |= ((uint64_t)in[2]) << 16;
302     result |= ((uint64_t)in[3]) << 24;
303     result |= ((uint64_t)in[4]) << 32;
304     result |= ((uint64_t)in[5]) << 40;
305 
306     return result;
307 }
308 
fe51_frombytes(fe51 h,const uint8_t * s)309 static void fe51_frombytes(fe51 h, const uint8_t *s)
310 {
311     uint64_t h0 = load_7(s); /* 56 bits */
312     uint64_t h1 = load_6(s + 7) << 5; /* 53 bits */
313     uint64_t h2 = load_7(s + 13) << 2; /* 58 bits */
314     uint64_t h3 = load_6(s + 20) << 7; /* 55 bits */
315     uint64_t h4 = (load_6(s + 26) & 0x7fffffffffff) << 4; /* 51 bits */
316 
317     h1 |= h0 >> 51;
318     h0 &= MASK51;
319     h2 |= h1 >> 51;
320     h1 &= MASK51;
321     h3 |= h2 >> 51;
322     h2 &= MASK51;
323     h4 |= h3 >> 51;
324     h3 &= MASK51;
325 
326     h[0] = h0;
327     h[1] = h1;
328     h[2] = h2;
329     h[3] = h3;
330     h[4] = h4;
331 }
332 
fe51_tobytes(uint8_t * s,const fe51 h)333 static void fe51_tobytes(uint8_t *s, const fe51 h)
334 {
335     uint64_t h0 = h[0];
336     uint64_t h1 = h[1];
337     uint64_t h2 = h[2];
338     uint64_t h3 = h[3];
339     uint64_t h4 = h[4];
340     uint64_t q;
341 
342     /* compare to modulus */
343     q = (h0 + 19) >> 51;
344     q = (h1 + q) >> 51;
345     q = (h2 + q) >> 51;
346     q = (h3 + q) >> 51;
347     q = (h4 + q) >> 51;
348 
349     /* full reduce */
350     h0 += 19 * q;
351     h1 += h0 >> 51;
352     h0 &= MASK51;
353     h2 += h1 >> 51;
354     h1 &= MASK51;
355     h3 += h2 >> 51;
356     h2 &= MASK51;
357     h4 += h3 >> 51;
358     h3 &= MASK51;
359     h4 &= MASK51;
360 
361     /* smash */
362     s[0] = (uint8_t)(h0 >> 0);
363     s[1] = (uint8_t)(h0 >> 8);
364     s[2] = (uint8_t)(h0 >> 16);
365     s[3] = (uint8_t)(h0 >> 24);
366     s[4] = (uint8_t)(h0 >> 32);
367     s[5] = (uint8_t)(h0 >> 40);
368     s[6] = (uint8_t)((h0 >> 48) | ((uint32_t)h1 << 3));
369     s[7] = (uint8_t)(h1 >> 5);
370     s[8] = (uint8_t)(h1 >> 13);
371     s[9] = (uint8_t)(h1 >> 21);
372     s[10] = (uint8_t)(h1 >> 29);
373     s[11] = (uint8_t)(h1 >> 37);
374     s[12] = (uint8_t)((h1 >> 45) | ((uint32_t)h2 << 6));
375     s[13] = (uint8_t)(h2 >> 2);
376     s[14] = (uint8_t)(h2 >> 10);
377     s[15] = (uint8_t)(h2 >> 18);
378     s[16] = (uint8_t)(h2 >> 26);
379     s[17] = (uint8_t)(h2 >> 34);
380     s[18] = (uint8_t)(h2 >> 42);
381     s[19] = (uint8_t)((h2 >> 50) | ((uint32_t)h3 << 1));
382     s[20] = (uint8_t)(h3 >> 7);
383     s[21] = (uint8_t)(h3 >> 15);
384     s[22] = (uint8_t)(h3 >> 23);
385     s[23] = (uint8_t)(h3 >> 31);
386     s[24] = (uint8_t)(h3 >> 39);
387     s[25] = (uint8_t)((h3 >> 47) | ((uint32_t)h4 << 4));
388     s[26] = (uint8_t)(h4 >> 4);
389     s[27] = (uint8_t)(h4 >> 12);
390     s[28] = (uint8_t)(h4 >> 20);
391     s[29] = (uint8_t)(h4 >> 28);
392     s[30] = (uint8_t)(h4 >> 36);
393     s[31] = (uint8_t)(h4 >> 44);
394 }
395 
396 #if defined(X25519_ASM)
397 void x25519_fe51_mul(fe51 h, const fe51 f, const fe51 g);
398 void x25519_fe51_sqr(fe51 h, const fe51 f);
399 void x25519_fe51_mul121666(fe51 h, fe51 f);
400 #define fe51_mul x25519_fe51_mul
401 #define fe51_sq x25519_fe51_sqr
402 #define fe51_mul121666 x25519_fe51_mul121666
403 #else
404 
405 typedef uint128_t u128;
406 
fe51_mul(fe51 h,const fe51 f,const fe51 g)407 static void fe51_mul(fe51 h, const fe51 f, const fe51 g)
408 {
409     u128 h0, h1, h2, h3, h4;
410     uint64_t f_i, g0, g1, g2, g3, g4;
411 
412     f_i = f[0];
413     h0 = (u128)f_i * (g0 = g[0]);
414     h1 = (u128)f_i * (g1 = g[1]);
415     h2 = (u128)f_i * (g2 = g[2]);
416     h3 = (u128)f_i * (g3 = g[3]);
417     h4 = (u128)f_i * (g4 = g[4]);
418 
419     f_i = f[1];
420     h0 += (u128)f_i * (g4 *= 19);
421     h1 += (u128)f_i * g0;
422     h2 += (u128)f_i * g1;
423     h3 += (u128)f_i * g2;
424     h4 += (u128)f_i * g3;
425 
426     f_i = f[2];
427     h0 += (u128)f_i * (g3 *= 19);
428     h1 += (u128)f_i * g4;
429     h2 += (u128)f_i * g0;
430     h3 += (u128)f_i * g1;
431     h4 += (u128)f_i * g2;
432 
433     f_i = f[3];
434     h0 += (u128)f_i * (g2 *= 19);
435     h1 += (u128)f_i * g3;
436     h2 += (u128)f_i * g4;
437     h3 += (u128)f_i * g0;
438     h4 += (u128)f_i * g1;
439 
440     f_i = f[4];
441     h0 += (u128)f_i * (g1 *= 19);
442     h1 += (u128)f_i * g2;
443     h2 += (u128)f_i * g3;
444     h3 += (u128)f_i * g4;
445     h4 += (u128)f_i * g0;
446 
447     /* partial [lazy] reduction */
448     h3 += (uint64_t)(h2 >> 51);
449     g2 = (uint64_t)h2 & MASK51;
450     h1 += (uint64_t)(h0 >> 51);
451     g0 = (uint64_t)h0 & MASK51;
452 
453     h4 += (uint64_t)(h3 >> 51);
454     g3 = (uint64_t)h3 & MASK51;
455     g2 += (uint64_t)(h1 >> 51);
456     g1 = (uint64_t)h1 & MASK51;
457 
458     g0 += (uint64_t)(h4 >> 51) * 19;
459     g4 = (uint64_t)h4 & MASK51;
460     g3 += g2 >> 51;
461     g2 &= MASK51;
462     g1 += g0 >> 51;
463     g0 &= MASK51;
464 
465     h[0] = g0;
466     h[1] = g1;
467     h[2] = g2;
468     h[3] = g3;
469     h[4] = g4;
470 }
471 
fe51_sq(fe51 h,const fe51 f)472 static void fe51_sq(fe51 h, const fe51 f)
473 {
474 #if defined(OPENSSL_SMALL_FOOTPRINT)
475     fe51_mul(h, f, f);
476 #else
477     /* dedicated squaring gives 16-25% overall improvement */
478     uint64_t g0 = f[0];
479     uint64_t g1 = f[1];
480     uint64_t g2 = f[2];
481     uint64_t g3 = f[3];
482     uint64_t g4 = f[4];
483     u128 h0, h1, h2, h3, h4;
484 
485     h0 = (u128)g0 * g0;
486     g0 *= 2;
487     h1 = (u128)g0 * g1;
488     h2 = (u128)g0 * g2;
489     h3 = (u128)g0 * g3;
490     h4 = (u128)g0 * g4;
491 
492     g0 = g4; /* borrow g0 */
493     h3 += (u128)g0 * (g4 *= 19);
494 
495     h2 += (u128)g1 * g1;
496     g1 *= 2;
497     h3 += (u128)g1 * g2;
498     h4 += (u128)g1 * g3;
499     h0 += (u128)g1 * g4;
500 
501     g0 = g3; /* borrow g0 */
502     h1 += (u128)g0 * (g3 *= 19);
503     h2 += (u128)(g0 * 2) * g4;
504 
505     h4 += (u128)g2 * g2;
506     g2 *= 2;
507     h0 += (u128)g2 * g3;
508     h1 += (u128)g2 * g4;
509 
510     /* partial [lazy] reduction */
511     h3 += (uint64_t)(h2 >> 51);
512     g2 = (uint64_t)h2 & MASK51;
513     h1 += (uint64_t)(h0 >> 51);
514     g0 = (uint64_t)h0 & MASK51;
515 
516     h4 += (uint64_t)(h3 >> 51);
517     g3 = (uint64_t)h3 & MASK51;
518     g2 += (uint64_t)(h1 >> 51);
519     g1 = (uint64_t)h1 & MASK51;
520 
521     g0 += (uint64_t)(h4 >> 51) * 19;
522     g4 = (uint64_t)h4 & MASK51;
523     g3 += g2 >> 51;
524     g2 &= MASK51;
525     g1 += g0 >> 51;
526     g0 &= MASK51;
527 
528     h[0] = g0;
529     h[1] = g1;
530     h[2] = g2;
531     h[3] = g3;
532     h[4] = g4;
533 #endif
534 }
535 
fe51_mul121666(fe51 h,fe51 f)536 static void fe51_mul121666(fe51 h, fe51 f)
537 {
538     u128 h0 = f[0] * (u128)121666;
539     u128 h1 = f[1] * (u128)121666;
540     u128 h2 = f[2] * (u128)121666;
541     u128 h3 = f[3] * (u128)121666;
542     u128 h4 = f[4] * (u128)121666;
543     uint64_t g0, g1, g2, g3, g4;
544 
545     h3 += (uint64_t)(h2 >> 51);
546     g2 = (uint64_t)h2 & MASK51;
547     h1 += (uint64_t)(h0 >> 51);
548     g0 = (uint64_t)h0 & MASK51;
549 
550     h4 += (uint64_t)(h3 >> 51);
551     g3 = (uint64_t)h3 & MASK51;
552     g2 += (uint64_t)(h1 >> 51);
553     g1 = (uint64_t)h1 & MASK51;
554 
555     g0 += (uint64_t)(h4 >> 51) * 19;
556     g4 = (uint64_t)h4 & MASK51;
557     g3 += g2 >> 51;
558     g2 &= MASK51;
559     g1 += g0 >> 51;
560     g0 &= MASK51;
561 
562     h[0] = g0;
563     h[1] = g1;
564     h[2] = g2;
565     h[3] = g3;
566     h[4] = g4;
567 }
568 #endif
569 
fe51_add(fe51 h,const fe51 f,const fe51 g)570 static void fe51_add(fe51 h, const fe51 f, const fe51 g)
571 {
572     h[0] = f[0] + g[0];
573     h[1] = f[1] + g[1];
574     h[2] = f[2] + g[2];
575     h[3] = f[3] + g[3];
576     h[4] = f[4] + g[4];
577 }
578 
fe51_sub(fe51 h,const fe51 f,const fe51 g)579 static void fe51_sub(fe51 h, const fe51 f, const fe51 g)
580 {
581     /*
582      * Add 2*modulus to ensure that result remains positive
583      * even if subtrahend is partially reduced.
584      */
585     h[0] = (f[0] + 0xfffffffffffda) - g[0];
586     h[1] = (f[1] + 0xffffffffffffe) - g[1];
587     h[2] = (f[2] + 0xffffffffffffe) - g[2];
588     h[3] = (f[3] + 0xffffffffffffe) - g[3];
589     h[4] = (f[4] + 0xffffffffffffe) - g[4];
590 }
591 
fe51_0(fe51 h)592 static void fe51_0(fe51 h)
593 {
594     h[0] = 0;
595     h[1] = 0;
596     h[2] = 0;
597     h[3] = 0;
598     h[4] = 0;
599 }
600 
fe51_1(fe51 h)601 static void fe51_1(fe51 h)
602 {
603     h[0] = 1;
604     h[1] = 0;
605     h[2] = 0;
606     h[3] = 0;
607     h[4] = 0;
608 }
609 
fe51_copy(fe51 h,const fe51 f)610 static void fe51_copy(fe51 h, const fe51 f)
611 {
612     h[0] = f[0];
613     h[1] = f[1];
614     h[2] = f[2];
615     h[3] = f[3];
616     h[4] = f[4];
617 }
618 
fe51_cswap(fe51 f,fe51 g,unsigned int b)619 static void fe51_cswap(fe51 f, fe51 g, unsigned int b)
620 {
621     int i;
622     uint64_t mask = 0 - (uint64_t)b;
623 
624     for (i = 0; i < 5; i++) {
625         int64_t x = f[i] ^ g[i];
626         x &= mask;
627         f[i] ^= x;
628         g[i] ^= x;
629     }
630 }
631 
fe51_invert(fe51 out,const fe51 z)632 static void fe51_invert(fe51 out, const fe51 z)
633 {
634     fe51 t0;
635     fe51 t1;
636     fe51 t2;
637     fe51 t3;
638     int i;
639 
640     /*
641      * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as
642      * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11.
643      */
644 
645     /* t0 = z ** 2 */
646     fe51_sq(t0, z);
647 
648     /* t1 = t0 ** (2 ** 2) = z ** 8 */
649     fe51_sq(t1, t0);
650     fe51_sq(t1, t1);
651 
652     /* t1 = z * t1 = z ** 9 */
653     fe51_mul(t1, z, t1);
654     /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */
655     fe51_mul(t0, t0, t1);
656 
657     /* t2 = t0 ** 2 = z ** 22 */
658     fe51_sq(t2, t0);
659 
660     /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */
661     fe51_mul(t1, t1, t2);
662 
663     /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */
664     fe51_sq(t2, t1);
665     for (i = 1; i < 5; ++i)
666         fe51_sq(t2, t2);
667 
668     /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */
669     fe51_mul(t1, t2, t1);
670 
671     /* Continuing similarly... */
672 
673     /* t2 = z ** (2 ** 20 - 1) */
674     fe51_sq(t2, t1);
675     for (i = 1; i < 10; ++i)
676         fe51_sq(t2, t2);
677 
678     fe51_mul(t2, t2, t1);
679 
680     /* t2 = z ** (2 ** 40 - 1) */
681     fe51_sq(t3, t2);
682     for (i = 1; i < 20; ++i)
683         fe51_sq(t3, t3);
684 
685     fe51_mul(t2, t3, t2);
686 
687     /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */
688     for (i = 0; i < 10; ++i)
689         fe51_sq(t2, t2);
690 
691     /* t1 = z ** (2 ** 50 - 1) */
692     fe51_mul(t1, t2, t1);
693 
694     /* t2 = z ** (2 ** 100 - 1) */
695     fe51_sq(t2, t1);
696     for (i = 1; i < 50; ++i)
697         fe51_sq(t2, t2);
698 
699     fe51_mul(t2, t2, t1);
700 
701     /* t2 = z ** (2 ** 200 - 1) */
702     fe51_sq(t3, t2);
703     for (i = 1; i < 100; ++i)
704         fe51_sq(t3, t3);
705 
706     fe51_mul(t2, t3, t2);
707 
708     /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */
709     for (i = 0; i < 50; ++i)
710         fe51_sq(t2, t2);
711 
712     /* t1 = z ** (2 ** 250 - 1) */
713     fe51_mul(t1, t2, t1);
714 
715     /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */
716     for (i = 0; i < 5; ++i)
717         fe51_sq(t1, t1);
718 
719     /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */
720     fe51_mul(out, t1, t0);
721 }
722 
723 /*
724  * Duplicate of original x25519_scalar_mult_generic, but using
725  * fe51_* subroutines.
726  */
x25519_scalar_mult(uint8_t out[32],const uint8_t scalar[32],const uint8_t point[32])727 static void x25519_scalar_mult(uint8_t out[32], const uint8_t scalar[32],
728     const uint8_t point[32])
729 {
730     fe51 x1, x2, z2, x3, z3, tmp0, tmp1;
731     uint8_t e[32];
732     unsigned swap = 0;
733     int pos;
734 
735 #ifdef BASE_2_64_IMPLEMENTED
736     if (x25519_fe64_eligible()) {
737         x25519_scalar_mulx(out, scalar, point);
738         return;
739     }
740 #endif
741 
742     memcpy(e, scalar, 32);
743     e[0] &= 0xf8;
744     e[31] &= 0x7f;
745     e[31] |= 0x40;
746     fe51_frombytes(x1, point);
747     fe51_1(x2);
748     fe51_0(z2);
749     fe51_copy(x3, x1);
750     fe51_1(z3);
751 
752     for (pos = 254; pos >= 0; --pos) {
753         unsigned int b = 1 & (e[pos / 8] >> (pos & 7));
754 
755         swap ^= b;
756         fe51_cswap(x2, x3, swap);
757         fe51_cswap(z2, z3, swap);
758         swap = b;
759         fe51_sub(tmp0, x3, z3);
760         fe51_sub(tmp1, x2, z2);
761         fe51_add(x2, x2, z2);
762         fe51_add(z2, x3, z3);
763         fe51_mul(z3, tmp0, x2);
764         fe51_mul(z2, z2, tmp1);
765         fe51_sq(tmp0, tmp1);
766         fe51_sq(tmp1, x2);
767         fe51_add(x3, z3, z2);
768         fe51_sub(z2, z3, z2);
769         fe51_mul(x2, tmp1, tmp0);
770         fe51_sub(tmp1, tmp1, tmp0);
771         fe51_sq(z2, z2);
772         fe51_mul121666(z3, tmp1);
773         fe51_sq(x3, x3);
774         fe51_add(tmp0, tmp0, z3);
775         fe51_mul(z3, x1, z2);
776         fe51_mul(z2, tmp1, tmp0);
777     }
778 
779     fe51_invert(z2, z2);
780     fe51_mul(x2, x2, z2);
781     fe51_tobytes(out, x2);
782 
783     OPENSSL_cleanse(e, sizeof(e));
784 }
785 #endif
786 
787 /*
788  * Reference base 2^25.5 implementation.
789  *
790  * This code is mostly taken from the ref10 version of Ed25519 in SUPERCOP
791  * 20141124 (http://bench.cr.yp.to/supercop.html).
792  *
793  * The field functions are shared by Ed25519 and X25519 where possible.
794  */
795 
796 /*
797  * fe means field element. Here the field is \Z/(2^255-19). An element t,
798  * entries t[0]...t[9], represents the integer t[0]+2^26 t[1]+2^51 t[2]+2^77
799  * t[3]+2^102 t[4]+...+2^230 t[9]. Bounds on each t[i] vary depending on
800  * context.
801  */
802 typedef int32_t fe[10];
803 
804 static const int64_t kBottom21Bits = 0x1fffffLL;
805 static const int64_t kBottom25Bits = 0x1ffffffLL;
806 static const int64_t kBottom26Bits = 0x3ffffffLL;
807 static const int64_t kTop39Bits = 0xfffffffffe000000LL;
808 static const int64_t kTop38Bits = 0xfffffffffc000000LL;
809 
load_3(const uint8_t * in)810 static uint64_t load_3(const uint8_t *in)
811 {
812     uint64_t result;
813 
814     result = ((uint64_t)in[0]);
815     result |= ((uint64_t)in[1]) << 8;
816     result |= ((uint64_t)in[2]) << 16;
817     return result;
818 }
819 
load_4(const uint8_t * in)820 static uint64_t load_4(const uint8_t *in)
821 {
822     uint64_t result;
823 
824     result = ((uint64_t)in[0]);
825     result |= ((uint64_t)in[1]) << 8;
826     result |= ((uint64_t)in[2]) << 16;
827     result |= ((uint64_t)in[3]) << 24;
828     return result;
829 }
830 
fe_frombytes(fe h,const uint8_t * s)831 static void fe_frombytes(fe h, const uint8_t *s)
832 {
833     /* Ignores top bit of h. */
834     int64_t h0 = load_4(s);
835     int64_t h1 = load_3(s + 4) << 6;
836     int64_t h2 = load_3(s + 7) << 5;
837     int64_t h3 = load_3(s + 10) << 3;
838     int64_t h4 = load_3(s + 13) << 2;
839     int64_t h5 = load_4(s + 16);
840     int64_t h6 = load_3(s + 20) << 7;
841     int64_t h7 = load_3(s + 23) << 5;
842     int64_t h8 = load_3(s + 26) << 4;
843     int64_t h9 = (load_3(s + 29) & 0x7fffff) << 2;
844     int64_t carry0;
845     int64_t carry1;
846     int64_t carry2;
847     int64_t carry3;
848     int64_t carry4;
849     int64_t carry5;
850     int64_t carry6;
851     int64_t carry7;
852     int64_t carry8;
853     int64_t carry9;
854 
855     carry9 = h9 + (1 << 24);
856     h0 += (carry9 >> 25) * 19;
857     h9 -= carry9 & kTop39Bits;
858     carry1 = h1 + (1 << 24);
859     h2 += carry1 >> 25;
860     h1 -= carry1 & kTop39Bits;
861     carry3 = h3 + (1 << 24);
862     h4 += carry3 >> 25;
863     h3 -= carry3 & kTop39Bits;
864     carry5 = h5 + (1 << 24);
865     h6 += carry5 >> 25;
866     h5 -= carry5 & kTop39Bits;
867     carry7 = h7 + (1 << 24);
868     h8 += carry7 >> 25;
869     h7 -= carry7 & kTop39Bits;
870 
871     carry0 = h0 + (1 << 25);
872     h1 += carry0 >> 26;
873     h0 -= carry0 & kTop38Bits;
874     carry2 = h2 + (1 << 25);
875     h3 += carry2 >> 26;
876     h2 -= carry2 & kTop38Bits;
877     carry4 = h4 + (1 << 25);
878     h5 += carry4 >> 26;
879     h4 -= carry4 & kTop38Bits;
880     carry6 = h6 + (1 << 25);
881     h7 += carry6 >> 26;
882     h6 -= carry6 & kTop38Bits;
883     carry8 = h8 + (1 << 25);
884     h9 += carry8 >> 26;
885     h8 -= carry8 & kTop38Bits;
886 
887     h[0] = (int32_t)h0;
888     h[1] = (int32_t)h1;
889     h[2] = (int32_t)h2;
890     h[3] = (int32_t)h3;
891     h[4] = (int32_t)h4;
892     h[5] = (int32_t)h5;
893     h[6] = (int32_t)h6;
894     h[7] = (int32_t)h7;
895     h[8] = (int32_t)h8;
896     h[9] = (int32_t)h9;
897 }
898 
899 /*
900  * Preconditions:
901  *   |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
902  *
903  * Write p=2^255-19; q=floor(h/p).
904  * Basic claim: q = floor(2^(-255)(h + 19 2^(-25)h9 + 2^(-1))).
905  *
906  * Proof:
907  *   Have |h|<=p so |q|<=1 so |19^2 2^(-255) q|<1/4.
908  *   Also have |h-2^230 h9|<2^231 so |19 2^(-255)(h-2^230 h9)|<1/4.
909  *
910  *   Write y=2^(-1)-19^2 2^(-255)q-19 2^(-255)(h-2^230 h9).
911  *   Then 0<y<1.
912  *
913  *   Write r=h-pq.
914  *   Have 0<=r<=p-1=2^255-20.
915  *   Thus 0<=r+19(2^-255)r<r+19(2^-255)2^255<=2^255-1.
916  *
917  *   Write x=r+19(2^-255)r+y.
918  *   Then 0<x<2^255 so floor(2^(-255)x) = 0 so floor(q+2^(-255)x) = q.
919  *
920  *   Have q+2^(-255)x = 2^(-255)(h + 19 2^(-25) h9 + 2^(-1))
921  *   so floor(2^(-255)(h + 19 2^(-25) h9 + 2^(-1))) = q.
922  */
fe_tobytes(uint8_t * s,const fe h)923 static void fe_tobytes(uint8_t *s, const fe h)
924 {
925     int32_t h0 = h[0];
926     int32_t h1 = h[1];
927     int32_t h2 = h[2];
928     int32_t h3 = h[3];
929     int32_t h4 = h[4];
930     int32_t h5 = h[5];
931     int32_t h6 = h[6];
932     int32_t h7 = h[7];
933     int32_t h8 = h[8];
934     int32_t h9 = h[9];
935     int32_t q;
936 
937     q = (19 * h9 + (((int32_t)1) << 24)) >> 25;
938     q = (h0 + q) >> 26;
939     q = (h1 + q) >> 25;
940     q = (h2 + q) >> 26;
941     q = (h3 + q) >> 25;
942     q = (h4 + q) >> 26;
943     q = (h5 + q) >> 25;
944     q = (h6 + q) >> 26;
945     q = (h7 + q) >> 25;
946     q = (h8 + q) >> 26;
947     q = (h9 + q) >> 25;
948 
949     /* Goal: Output h-(2^255-19)q, which is between 0 and 2^255-20. */
950     h0 += 19 * q;
951     /* Goal: Output h-2^255 q, which is between 0 and 2^255-20. */
952 
953     h1 += h0 >> 26;
954     h0 &= kBottom26Bits;
955     h2 += h1 >> 25;
956     h1 &= kBottom25Bits;
957     h3 += h2 >> 26;
958     h2 &= kBottom26Bits;
959     h4 += h3 >> 25;
960     h3 &= kBottom25Bits;
961     h5 += h4 >> 26;
962     h4 &= kBottom26Bits;
963     h6 += h5 >> 25;
964     h5 &= kBottom25Bits;
965     h7 += h6 >> 26;
966     h6 &= kBottom26Bits;
967     h8 += h7 >> 25;
968     h7 &= kBottom25Bits;
969     h9 += h8 >> 26;
970     h8 &= kBottom26Bits;
971     h9 &= kBottom25Bits;
972     /* h10 = carry9 */
973 
974     /*
975      * Goal: Output h0+...+2^255 h10-2^255 q, which is between 0 and 2^255-20.
976      * Have h0+...+2^230 h9 between 0 and 2^255-1;
977      * evidently 2^255 h10-2^255 q = 0.
978      * Goal: Output h0+...+2^230 h9.
979      */
980     s[0] = (uint8_t)(h0 >> 0);
981     s[1] = (uint8_t)(h0 >> 8);
982     s[2] = (uint8_t)(h0 >> 16);
983     s[3] = (uint8_t)((h0 >> 24) | ((uint32_t)(h1) << 2));
984     s[4] = (uint8_t)(h1 >> 6);
985     s[5] = (uint8_t)(h1 >> 14);
986     s[6] = (uint8_t)((h1 >> 22) | ((uint32_t)(h2) << 3));
987     s[7] = (uint8_t)(h2 >> 5);
988     s[8] = (uint8_t)(h2 >> 13);
989     s[9] = (uint8_t)((h2 >> 21) | ((uint32_t)(h3) << 5));
990     s[10] = (uint8_t)(h3 >> 3);
991     s[11] = (uint8_t)(h3 >> 11);
992     s[12] = (uint8_t)((h3 >> 19) | ((uint32_t)(h4) << 6));
993     s[13] = (uint8_t)(h4 >> 2);
994     s[14] = (uint8_t)(h4 >> 10);
995     s[15] = (uint8_t)(h4 >> 18);
996     s[16] = (uint8_t)(h5 >> 0);
997     s[17] = (uint8_t)(h5 >> 8);
998     s[18] = (uint8_t)(h5 >> 16);
999     s[19] = (uint8_t)((h5 >> 24) | ((uint32_t)(h6) << 1));
1000     s[20] = (uint8_t)(h6 >> 7);
1001     s[21] = (uint8_t)(h6 >> 15);
1002     s[22] = (uint8_t)((h6 >> 23) | ((uint32_t)(h7) << 3));
1003     s[23] = (uint8_t)(h7 >> 5);
1004     s[24] = (uint8_t)(h7 >> 13);
1005     s[25] = (uint8_t)((h7 >> 21) | ((uint32_t)(h8) << 4));
1006     s[26] = (uint8_t)(h8 >> 4);
1007     s[27] = (uint8_t)(h8 >> 12);
1008     s[28] = (uint8_t)((h8 >> 20) | ((uint32_t)(h9) << 6));
1009     s[29] = (uint8_t)(h9 >> 2);
1010     s[30] = (uint8_t)(h9 >> 10);
1011     s[31] = (uint8_t)(h9 >> 18);
1012 }
1013 
1014 /* h = f */
fe_copy(fe h,const fe f)1015 static void fe_copy(fe h, const fe f)
1016 {
1017     memmove(h, f, sizeof(int32_t) * 10);
1018 }
1019 
1020 /* h = 0 */
fe_0(fe h)1021 static void fe_0(fe h)
1022 {
1023     memset(h, 0, sizeof(int32_t) * 10);
1024 }
1025 
1026 /* h = 1 */
fe_1(fe h)1027 static void fe_1(fe h)
1028 {
1029     memset(h, 0, sizeof(int32_t) * 10);
1030     h[0] = 1;
1031 }
1032 
1033 /*
1034  * h = f + g
1035  *
1036  * Can overlap h with f or g.
1037  *
1038  * Preconditions:
1039  *    |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
1040  *    |g| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
1041  *
1042  * Postconditions:
1043  *    |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
1044  */
fe_add(fe h,const fe f,const fe g)1045 static void fe_add(fe h, const fe f, const fe g)
1046 {
1047     unsigned i;
1048 
1049     for (i = 0; i < 10; i++) {
1050         h[i] = f[i] + g[i];
1051     }
1052 }
1053 
1054 /*
1055  * h = f - g
1056  *
1057  * Can overlap h with f or g.
1058  *
1059  * Preconditions:
1060  *    |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
1061  *    |g| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
1062  *
1063  * Postconditions:
1064  *    |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
1065  */
fe_sub(fe h,const fe f,const fe g)1066 static void fe_sub(fe h, const fe f, const fe g)
1067 {
1068     unsigned i;
1069 
1070     for (i = 0; i < 10; i++) {
1071         h[i] = f[i] - g[i];
1072     }
1073 }
1074 
1075 /*
1076  * h = f * g
1077  *
1078  * Can overlap h with f or g.
1079  *
1080  * Preconditions:
1081  *    |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
1082  *    |g| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
1083  *
1084  * Postconditions:
1085  *    |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
1086  *
1087  * Notes on implementation strategy:
1088  *
1089  * Using schoolbook multiplication.
1090  * Karatsuba would save a little in some cost models.
1091  *
1092  * Most multiplications by 2 and 19 are 32-bit precomputations;
1093  * cheaper than 64-bit postcomputations.
1094  *
1095  * There is one remaining multiplication by 19 in the carry chain;
1096  * one *19 precomputation can be merged into this,
1097  * but the resulting data flow is considerably less clean.
1098  *
1099  * There are 12 carries below.
1100  * 10 of them are 2-way parallelizable and vectorizable.
1101  * Can get away with 11 carries, but then data flow is much deeper.
1102  *
1103  * With tighter constraints on inputs can squeeze carries into int32.
1104  */
fe_mul(fe h,const fe f,const fe g)1105 static void fe_mul(fe h, const fe f, const fe g)
1106 {
1107     int32_t f0 = f[0];
1108     int32_t f1 = f[1];
1109     int32_t f2 = f[2];
1110     int32_t f3 = f[3];
1111     int32_t f4 = f[4];
1112     int32_t f5 = f[5];
1113     int32_t f6 = f[6];
1114     int32_t f7 = f[7];
1115     int32_t f8 = f[8];
1116     int32_t f9 = f[9];
1117     int32_t g0 = g[0];
1118     int32_t g1 = g[1];
1119     int32_t g2 = g[2];
1120     int32_t g3 = g[3];
1121     int32_t g4 = g[4];
1122     int32_t g5 = g[5];
1123     int32_t g6 = g[6];
1124     int32_t g7 = g[7];
1125     int32_t g8 = g[8];
1126     int32_t g9 = g[9];
1127     int32_t g1_19 = 19 * g1; /* 1.959375*2^29 */
1128     int32_t g2_19 = 19 * g2; /* 1.959375*2^30; still ok */
1129     int32_t g3_19 = 19 * g3;
1130     int32_t g4_19 = 19 * g4;
1131     int32_t g5_19 = 19 * g5;
1132     int32_t g6_19 = 19 * g6;
1133     int32_t g7_19 = 19 * g7;
1134     int32_t g8_19 = 19 * g8;
1135     int32_t g9_19 = 19 * g9;
1136     int32_t f1_2 = 2 * f1;
1137     int32_t f3_2 = 2 * f3;
1138     int32_t f5_2 = 2 * f5;
1139     int32_t f7_2 = 2 * f7;
1140     int32_t f9_2 = 2 * f9;
1141     int64_t f0g0 = f0 * (int64_t)g0;
1142     int64_t f0g1 = f0 * (int64_t)g1;
1143     int64_t f0g2 = f0 * (int64_t)g2;
1144     int64_t f0g3 = f0 * (int64_t)g3;
1145     int64_t f0g4 = f0 * (int64_t)g4;
1146     int64_t f0g5 = f0 * (int64_t)g5;
1147     int64_t f0g6 = f0 * (int64_t)g6;
1148     int64_t f0g7 = f0 * (int64_t)g7;
1149     int64_t f0g8 = f0 * (int64_t)g8;
1150     int64_t f0g9 = f0 * (int64_t)g9;
1151     int64_t f1g0 = f1 * (int64_t)g0;
1152     int64_t f1g1_2 = f1_2 * (int64_t)g1;
1153     int64_t f1g2 = f1 * (int64_t)g2;
1154     int64_t f1g3_2 = f1_2 * (int64_t)g3;
1155     int64_t f1g4 = f1 * (int64_t)g4;
1156     int64_t f1g5_2 = f1_2 * (int64_t)g5;
1157     int64_t f1g6 = f1 * (int64_t)g6;
1158     int64_t f1g7_2 = f1_2 * (int64_t)g7;
1159     int64_t f1g8 = f1 * (int64_t)g8;
1160     int64_t f1g9_38 = f1_2 * (int64_t)g9_19;
1161     int64_t f2g0 = f2 * (int64_t)g0;
1162     int64_t f2g1 = f2 * (int64_t)g1;
1163     int64_t f2g2 = f2 * (int64_t)g2;
1164     int64_t f2g3 = f2 * (int64_t)g3;
1165     int64_t f2g4 = f2 * (int64_t)g4;
1166     int64_t f2g5 = f2 * (int64_t)g5;
1167     int64_t f2g6 = f2 * (int64_t)g6;
1168     int64_t f2g7 = f2 * (int64_t)g7;
1169     int64_t f2g8_19 = f2 * (int64_t)g8_19;
1170     int64_t f2g9_19 = f2 * (int64_t)g9_19;
1171     int64_t f3g0 = f3 * (int64_t)g0;
1172     int64_t f3g1_2 = f3_2 * (int64_t)g1;
1173     int64_t f3g2 = f3 * (int64_t)g2;
1174     int64_t f3g3_2 = f3_2 * (int64_t)g3;
1175     int64_t f3g4 = f3 * (int64_t)g4;
1176     int64_t f3g5_2 = f3_2 * (int64_t)g5;
1177     int64_t f3g6 = f3 * (int64_t)g6;
1178     int64_t f3g7_38 = f3_2 * (int64_t)g7_19;
1179     int64_t f3g8_19 = f3 * (int64_t)g8_19;
1180     int64_t f3g9_38 = f3_2 * (int64_t)g9_19;
1181     int64_t f4g0 = f4 * (int64_t)g0;
1182     int64_t f4g1 = f4 * (int64_t)g1;
1183     int64_t f4g2 = f4 * (int64_t)g2;
1184     int64_t f4g3 = f4 * (int64_t)g3;
1185     int64_t f4g4 = f4 * (int64_t)g4;
1186     int64_t f4g5 = f4 * (int64_t)g5;
1187     int64_t f4g6_19 = f4 * (int64_t)g6_19;
1188     int64_t f4g7_19 = f4 * (int64_t)g7_19;
1189     int64_t f4g8_19 = f4 * (int64_t)g8_19;
1190     int64_t f4g9_19 = f4 * (int64_t)g9_19;
1191     int64_t f5g0 = f5 * (int64_t)g0;
1192     int64_t f5g1_2 = f5_2 * (int64_t)g1;
1193     int64_t f5g2 = f5 * (int64_t)g2;
1194     int64_t f5g3_2 = f5_2 * (int64_t)g3;
1195     int64_t f5g4 = f5 * (int64_t)g4;
1196     int64_t f5g5_38 = f5_2 * (int64_t)g5_19;
1197     int64_t f5g6_19 = f5 * (int64_t)g6_19;
1198     int64_t f5g7_38 = f5_2 * (int64_t)g7_19;
1199     int64_t f5g8_19 = f5 * (int64_t)g8_19;
1200     int64_t f5g9_38 = f5_2 * (int64_t)g9_19;
1201     int64_t f6g0 = f6 * (int64_t)g0;
1202     int64_t f6g1 = f6 * (int64_t)g1;
1203     int64_t f6g2 = f6 * (int64_t)g2;
1204     int64_t f6g3 = f6 * (int64_t)g3;
1205     int64_t f6g4_19 = f6 * (int64_t)g4_19;
1206     int64_t f6g5_19 = f6 * (int64_t)g5_19;
1207     int64_t f6g6_19 = f6 * (int64_t)g6_19;
1208     int64_t f6g7_19 = f6 * (int64_t)g7_19;
1209     int64_t f6g8_19 = f6 * (int64_t)g8_19;
1210     int64_t f6g9_19 = f6 * (int64_t)g9_19;
1211     int64_t f7g0 = f7 * (int64_t)g0;
1212     int64_t f7g1_2 = f7_2 * (int64_t)g1;
1213     int64_t f7g2 = f7 * (int64_t)g2;
1214     int64_t f7g3_38 = f7_2 * (int64_t)g3_19;
1215     int64_t f7g4_19 = f7 * (int64_t)g4_19;
1216     int64_t f7g5_38 = f7_2 * (int64_t)g5_19;
1217     int64_t f7g6_19 = f7 * (int64_t)g6_19;
1218     int64_t f7g7_38 = f7_2 * (int64_t)g7_19;
1219     int64_t f7g8_19 = f7 * (int64_t)g8_19;
1220     int64_t f7g9_38 = f7_2 * (int64_t)g9_19;
1221     int64_t f8g0 = f8 * (int64_t)g0;
1222     int64_t f8g1 = f8 * (int64_t)g1;
1223     int64_t f8g2_19 = f8 * (int64_t)g2_19;
1224     int64_t f8g3_19 = f8 * (int64_t)g3_19;
1225     int64_t f8g4_19 = f8 * (int64_t)g4_19;
1226     int64_t f8g5_19 = f8 * (int64_t)g5_19;
1227     int64_t f8g6_19 = f8 * (int64_t)g6_19;
1228     int64_t f8g7_19 = f8 * (int64_t)g7_19;
1229     int64_t f8g8_19 = f8 * (int64_t)g8_19;
1230     int64_t f8g9_19 = f8 * (int64_t)g9_19;
1231     int64_t f9g0 = f9 * (int64_t)g0;
1232     int64_t f9g1_38 = f9_2 * (int64_t)g1_19;
1233     int64_t f9g2_19 = f9 * (int64_t)g2_19;
1234     int64_t f9g3_38 = f9_2 * (int64_t)g3_19;
1235     int64_t f9g4_19 = f9 * (int64_t)g4_19;
1236     int64_t f9g5_38 = f9_2 * (int64_t)g5_19;
1237     int64_t f9g6_19 = f9 * (int64_t)g6_19;
1238     int64_t f9g7_38 = f9_2 * (int64_t)g7_19;
1239     int64_t f9g8_19 = f9 * (int64_t)g8_19;
1240     int64_t f9g9_38 = f9_2 * (int64_t)g9_19;
1241     int64_t h0 = f0g0 + f1g9_38 + f2g8_19 + f3g7_38 + f4g6_19 + f5g5_38 + f6g4_19 + f7g3_38 + f8g2_19 + f9g1_38;
1242     int64_t h1 = f0g1 + f1g0 + f2g9_19 + f3g8_19 + f4g7_19 + f5g6_19 + f6g5_19 + f7g4_19 + f8g3_19 + f9g2_19;
1243     int64_t h2 = f0g2 + f1g1_2 + f2g0 + f3g9_38 + f4g8_19 + f5g7_38 + f6g6_19 + f7g5_38 + f8g4_19 + f9g3_38;
1244     int64_t h3 = f0g3 + f1g2 + f2g1 + f3g0 + f4g9_19 + f5g8_19 + f6g7_19 + f7g6_19 + f8g5_19 + f9g4_19;
1245     int64_t h4 = f0g4 + f1g3_2 + f2g2 + f3g1_2 + f4g0 + f5g9_38 + f6g8_19 + f7g7_38 + f8g6_19 + f9g5_38;
1246     int64_t h5 = f0g5 + f1g4 + f2g3 + f3g2 + f4g1 + f5g0 + f6g9_19 + f7g8_19 + f8g7_19 + f9g6_19;
1247     int64_t h6 = f0g6 + f1g5_2 + f2g4 + f3g3_2 + f4g2 + f5g1_2 + f6g0 + f7g9_38 + f8g8_19 + f9g7_38;
1248     int64_t h7 = f0g7 + f1g6 + f2g5 + f3g4 + f4g3 + f5g2 + f6g1 + f7g0 + f8g9_19 + f9g8_19;
1249     int64_t h8 = f0g8 + f1g7_2 + f2g6 + f3g5_2 + f4g4 + f5g3_2 + f6g2 + f7g1_2 + f8g0 + f9g9_38;
1250     int64_t h9 = f0g9 + f1g8 + f2g7 + f3g6 + f4g5 + f5g4 + f6g3 + f7g2 + f8g1 + f9g0;
1251     int64_t carry0;
1252     int64_t carry1;
1253     int64_t carry2;
1254     int64_t carry3;
1255     int64_t carry4;
1256     int64_t carry5;
1257     int64_t carry6;
1258     int64_t carry7;
1259     int64_t carry8;
1260     int64_t carry9;
1261 
1262     /* |h0| <= (1.65*1.65*2^52*(1+19+19+19+19)+1.65*1.65*2^50*(38+38+38+38+38))
1263      *   i.e. |h0| <= 1.4*2^60; narrower ranges for h2, h4, h6, h8
1264      * |h1| <= (1.65*1.65*2^51*(1+1+19+19+19+19+19+19+19+19))
1265      *   i.e. |h1| <= 1.7*2^59; narrower ranges for h3, h5, h7, h9 */
1266 
1267     carry0 = h0 + (1 << 25);
1268     h1 += carry0 >> 26;
1269     h0 -= carry0 & kTop38Bits;
1270     carry4 = h4 + (1 << 25);
1271     h5 += carry4 >> 26;
1272     h4 -= carry4 & kTop38Bits;
1273     /* |h0| <= 2^25 */
1274     /* |h4| <= 2^25 */
1275     /* |h1| <= 1.71*2^59 */
1276     /* |h5| <= 1.71*2^59 */
1277 
1278     carry1 = h1 + (1 << 24);
1279     h2 += carry1 >> 25;
1280     h1 -= carry1 & kTop39Bits;
1281     carry5 = h5 + (1 << 24);
1282     h6 += carry5 >> 25;
1283     h5 -= carry5 & kTop39Bits;
1284     /* |h1| <= 2^24; from now on fits into int32 */
1285     /* |h5| <= 2^24; from now on fits into int32 */
1286     /* |h2| <= 1.41*2^60 */
1287     /* |h6| <= 1.41*2^60 */
1288 
1289     carry2 = h2 + (1 << 25);
1290     h3 += carry2 >> 26;
1291     h2 -= carry2 & kTop38Bits;
1292     carry6 = h6 + (1 << 25);
1293     h7 += carry6 >> 26;
1294     h6 -= carry6 & kTop38Bits;
1295     /* |h2| <= 2^25; from now on fits into int32 unchanged */
1296     /* |h6| <= 2^25; from now on fits into int32 unchanged */
1297     /* |h3| <= 1.71*2^59 */
1298     /* |h7| <= 1.71*2^59 */
1299 
1300     carry3 = h3 + (1 << 24);
1301     h4 += carry3 >> 25;
1302     h3 -= carry3 & kTop39Bits;
1303     carry7 = h7 + (1 << 24);
1304     h8 += carry7 >> 25;
1305     h7 -= carry7 & kTop39Bits;
1306     /* |h3| <= 2^24; from now on fits into int32 unchanged */
1307     /* |h7| <= 2^24; from now on fits into int32 unchanged */
1308     /* |h4| <= 1.72*2^34 */
1309     /* |h8| <= 1.41*2^60 */
1310 
1311     carry4 = h4 + (1 << 25);
1312     h5 += carry4 >> 26;
1313     h4 -= carry4 & kTop38Bits;
1314     carry8 = h8 + (1 << 25);
1315     h9 += carry8 >> 26;
1316     h8 -= carry8 & kTop38Bits;
1317     /* |h4| <= 2^25; from now on fits into int32 unchanged */
1318     /* |h8| <= 2^25; from now on fits into int32 unchanged */
1319     /* |h5| <= 1.01*2^24 */
1320     /* |h9| <= 1.71*2^59 */
1321 
1322     carry9 = h9 + (1 << 24);
1323     h0 += (carry9 >> 25) * 19;
1324     h9 -= carry9 & kTop39Bits;
1325     /* |h9| <= 2^24; from now on fits into int32 unchanged */
1326     /* |h0| <= 1.1*2^39 */
1327 
1328     carry0 = h0 + (1 << 25);
1329     h1 += carry0 >> 26;
1330     h0 -= carry0 & kTop38Bits;
1331     /* |h0| <= 2^25; from now on fits into int32 unchanged */
1332     /* |h1| <= 1.01*2^24 */
1333 
1334     h[0] = (int32_t)h0;
1335     h[1] = (int32_t)h1;
1336     h[2] = (int32_t)h2;
1337     h[3] = (int32_t)h3;
1338     h[4] = (int32_t)h4;
1339     h[5] = (int32_t)h5;
1340     h[6] = (int32_t)h6;
1341     h[7] = (int32_t)h7;
1342     h[8] = (int32_t)h8;
1343     h[9] = (int32_t)h9;
1344 }
1345 
1346 /*
1347  * h = f * f
1348  *
1349  * Can overlap h with f.
1350  *
1351  * Preconditions:
1352  *    |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
1353  *
1354  * Postconditions:
1355  *    |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
1356  *
1357  * See fe_mul.c for discussion of implementation strategy.
1358  */
fe_sq(fe h,const fe f)1359 static void fe_sq(fe h, const fe f)
1360 {
1361     int32_t f0 = f[0];
1362     int32_t f1 = f[1];
1363     int32_t f2 = f[2];
1364     int32_t f3 = f[3];
1365     int32_t f4 = f[4];
1366     int32_t f5 = f[5];
1367     int32_t f6 = f[6];
1368     int32_t f7 = f[7];
1369     int32_t f8 = f[8];
1370     int32_t f9 = f[9];
1371     int32_t f0_2 = 2 * f0;
1372     int32_t f1_2 = 2 * f1;
1373     int32_t f2_2 = 2 * f2;
1374     int32_t f3_2 = 2 * f3;
1375     int32_t f4_2 = 2 * f4;
1376     int32_t f5_2 = 2 * f5;
1377     int32_t f6_2 = 2 * f6;
1378     int32_t f7_2 = 2 * f7;
1379     int32_t f5_38 = 38 * f5; /* 1.959375*2^30 */
1380     int32_t f6_19 = 19 * f6; /* 1.959375*2^30 */
1381     int32_t f7_38 = 38 * f7; /* 1.959375*2^30 */
1382     int32_t f8_19 = 19 * f8; /* 1.959375*2^30 */
1383     int32_t f9_38 = 38 * f9; /* 1.959375*2^30 */
1384     int64_t f0f0 = f0 * (int64_t)f0;
1385     int64_t f0f1_2 = f0_2 * (int64_t)f1;
1386     int64_t f0f2_2 = f0_2 * (int64_t)f2;
1387     int64_t f0f3_2 = f0_2 * (int64_t)f3;
1388     int64_t f0f4_2 = f0_2 * (int64_t)f4;
1389     int64_t f0f5_2 = f0_2 * (int64_t)f5;
1390     int64_t f0f6_2 = f0_2 * (int64_t)f6;
1391     int64_t f0f7_2 = f0_2 * (int64_t)f7;
1392     int64_t f0f8_2 = f0_2 * (int64_t)f8;
1393     int64_t f0f9_2 = f0_2 * (int64_t)f9;
1394     int64_t f1f1_2 = f1_2 * (int64_t)f1;
1395     int64_t f1f2_2 = f1_2 * (int64_t)f2;
1396     int64_t f1f3_4 = f1_2 * (int64_t)f3_2;
1397     int64_t f1f4_2 = f1_2 * (int64_t)f4;
1398     int64_t f1f5_4 = f1_2 * (int64_t)f5_2;
1399     int64_t f1f6_2 = f1_2 * (int64_t)f6;
1400     int64_t f1f7_4 = f1_2 * (int64_t)f7_2;
1401     int64_t f1f8_2 = f1_2 * (int64_t)f8;
1402     int64_t f1f9_76 = f1_2 * (int64_t)f9_38;
1403     int64_t f2f2 = f2 * (int64_t)f2;
1404     int64_t f2f3_2 = f2_2 * (int64_t)f3;
1405     int64_t f2f4_2 = f2_2 * (int64_t)f4;
1406     int64_t f2f5_2 = f2_2 * (int64_t)f5;
1407     int64_t f2f6_2 = f2_2 * (int64_t)f6;
1408     int64_t f2f7_2 = f2_2 * (int64_t)f7;
1409     int64_t f2f8_38 = f2_2 * (int64_t)f8_19;
1410     int64_t f2f9_38 = f2 * (int64_t)f9_38;
1411     int64_t f3f3_2 = f3_2 * (int64_t)f3;
1412     int64_t f3f4_2 = f3_2 * (int64_t)f4;
1413     int64_t f3f5_4 = f3_2 * (int64_t)f5_2;
1414     int64_t f3f6_2 = f3_2 * (int64_t)f6;
1415     int64_t f3f7_76 = f3_2 * (int64_t)f7_38;
1416     int64_t f3f8_38 = f3_2 * (int64_t)f8_19;
1417     int64_t f3f9_76 = f3_2 * (int64_t)f9_38;
1418     int64_t f4f4 = f4 * (int64_t)f4;
1419     int64_t f4f5_2 = f4_2 * (int64_t)f5;
1420     int64_t f4f6_38 = f4_2 * (int64_t)f6_19;
1421     int64_t f4f7_38 = f4 * (int64_t)f7_38;
1422     int64_t f4f8_38 = f4_2 * (int64_t)f8_19;
1423     int64_t f4f9_38 = f4 * (int64_t)f9_38;
1424     int64_t f5f5_38 = f5 * (int64_t)f5_38;
1425     int64_t f5f6_38 = f5_2 * (int64_t)f6_19;
1426     int64_t f5f7_76 = f5_2 * (int64_t)f7_38;
1427     int64_t f5f8_38 = f5_2 * (int64_t)f8_19;
1428     int64_t f5f9_76 = f5_2 * (int64_t)f9_38;
1429     int64_t f6f6_19 = f6 * (int64_t)f6_19;
1430     int64_t f6f7_38 = f6 * (int64_t)f7_38;
1431     int64_t f6f8_38 = f6_2 * (int64_t)f8_19;
1432     int64_t f6f9_38 = f6 * (int64_t)f9_38;
1433     int64_t f7f7_38 = f7 * (int64_t)f7_38;
1434     int64_t f7f8_38 = f7_2 * (int64_t)f8_19;
1435     int64_t f7f9_76 = f7_2 * (int64_t)f9_38;
1436     int64_t f8f8_19 = f8 * (int64_t)f8_19;
1437     int64_t f8f9_38 = f8 * (int64_t)f9_38;
1438     int64_t f9f9_38 = f9 * (int64_t)f9_38;
1439     int64_t h0 = f0f0 + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38;
1440     int64_t h1 = f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38;
1441     int64_t h2 = f0f2_2 + f1f1_2 + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19;
1442     int64_t h3 = f0f3_2 + f1f2_2 + f4f9_38 + f5f8_38 + f6f7_38;
1443     int64_t h4 = f0f4_2 + f1f3_4 + f2f2 + f5f9_76 + f6f8_38 + f7f7_38;
1444     int64_t h5 = f0f5_2 + f1f4_2 + f2f3_2 + f6f9_38 + f7f8_38;
1445     int64_t h6 = f0f6_2 + f1f5_4 + f2f4_2 + f3f3_2 + f7f9_76 + f8f8_19;
1446     int64_t h7 = f0f7_2 + f1f6_2 + f2f5_2 + f3f4_2 + f8f9_38;
1447     int64_t h8 = f0f8_2 + f1f7_4 + f2f6_2 + f3f5_4 + f4f4 + f9f9_38;
1448     int64_t h9 = f0f9_2 + f1f8_2 + f2f7_2 + f3f6_2 + f4f5_2;
1449     int64_t carry0;
1450     int64_t carry1;
1451     int64_t carry2;
1452     int64_t carry3;
1453     int64_t carry4;
1454     int64_t carry5;
1455     int64_t carry6;
1456     int64_t carry7;
1457     int64_t carry8;
1458     int64_t carry9;
1459 
1460     carry0 = h0 + (1 << 25);
1461     h1 += carry0 >> 26;
1462     h0 -= carry0 & kTop38Bits;
1463     carry4 = h4 + (1 << 25);
1464     h5 += carry4 >> 26;
1465     h4 -= carry4 & kTop38Bits;
1466 
1467     carry1 = h1 + (1 << 24);
1468     h2 += carry1 >> 25;
1469     h1 -= carry1 & kTop39Bits;
1470     carry5 = h5 + (1 << 24);
1471     h6 += carry5 >> 25;
1472     h5 -= carry5 & kTop39Bits;
1473 
1474     carry2 = h2 + (1 << 25);
1475     h3 += carry2 >> 26;
1476     h2 -= carry2 & kTop38Bits;
1477     carry6 = h6 + (1 << 25);
1478     h7 += carry6 >> 26;
1479     h6 -= carry6 & kTop38Bits;
1480 
1481     carry3 = h3 + (1 << 24);
1482     h4 += carry3 >> 25;
1483     h3 -= carry3 & kTop39Bits;
1484     carry7 = h7 + (1 << 24);
1485     h8 += carry7 >> 25;
1486     h7 -= carry7 & kTop39Bits;
1487 
1488     carry4 = h4 + (1 << 25);
1489     h5 += carry4 >> 26;
1490     h4 -= carry4 & kTop38Bits;
1491     carry8 = h8 + (1 << 25);
1492     h9 += carry8 >> 26;
1493     h8 -= carry8 & kTop38Bits;
1494 
1495     carry9 = h9 + (1 << 24);
1496     h0 += (carry9 >> 25) * 19;
1497     h9 -= carry9 & kTop39Bits;
1498 
1499     carry0 = h0 + (1 << 25);
1500     h1 += carry0 >> 26;
1501     h0 -= carry0 & kTop38Bits;
1502 
1503     h[0] = (int32_t)h0;
1504     h[1] = (int32_t)h1;
1505     h[2] = (int32_t)h2;
1506     h[3] = (int32_t)h3;
1507     h[4] = (int32_t)h4;
1508     h[5] = (int32_t)h5;
1509     h[6] = (int32_t)h6;
1510     h[7] = (int32_t)h7;
1511     h[8] = (int32_t)h8;
1512     h[9] = (int32_t)h9;
1513 }
1514 
fe_invert(fe out,const fe z)1515 static void fe_invert(fe out, const fe z)
1516 {
1517     fe t0;
1518     fe t1;
1519     fe t2;
1520     fe t3;
1521     int i;
1522 
1523     /*
1524      * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as
1525      * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11.
1526      */
1527 
1528     /* t0 = z ** 2 */
1529     fe_sq(t0, z);
1530 
1531     /* t1 = t0 ** (2 ** 2) = z ** 8 */
1532     fe_sq(t1, t0);
1533     fe_sq(t1, t1);
1534 
1535     /* t1 = z * t1 = z ** 9 */
1536     fe_mul(t1, z, t1);
1537     /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */
1538     fe_mul(t0, t0, t1);
1539 
1540     /* t2 = t0 ** 2 = z ** 22 */
1541     fe_sq(t2, t0);
1542 
1543     /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */
1544     fe_mul(t1, t1, t2);
1545 
1546     /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */
1547     fe_sq(t2, t1);
1548     for (i = 1; i < 5; ++i) {
1549         fe_sq(t2, t2);
1550     }
1551 
1552     /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */
1553     fe_mul(t1, t2, t1);
1554 
1555     /* Continuing similarly... */
1556 
1557     /* t2 = z ** (2 ** 20 - 1) */
1558     fe_sq(t2, t1);
1559     for (i = 1; i < 10; ++i) {
1560         fe_sq(t2, t2);
1561     }
1562     fe_mul(t2, t2, t1);
1563 
1564     /* t2 = z ** (2 ** 40 - 1) */
1565     fe_sq(t3, t2);
1566     for (i = 1; i < 20; ++i) {
1567         fe_sq(t3, t3);
1568     }
1569     fe_mul(t2, t3, t2);
1570 
1571     /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */
1572     for (i = 0; i < 10; ++i) {
1573         fe_sq(t2, t2);
1574     }
1575     /* t1 = z ** (2 ** 50 - 1) */
1576     fe_mul(t1, t2, t1);
1577 
1578     /* t2 = z ** (2 ** 100 - 1) */
1579     fe_sq(t2, t1);
1580     for (i = 1; i < 50; ++i) {
1581         fe_sq(t2, t2);
1582     }
1583     fe_mul(t2, t2, t1);
1584 
1585     /* t2 = z ** (2 ** 200 - 1) */
1586     fe_sq(t3, t2);
1587     for (i = 1; i < 100; ++i) {
1588         fe_sq(t3, t3);
1589     }
1590     fe_mul(t2, t3, t2);
1591 
1592     /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */
1593     fe_sq(t2, t2);
1594     for (i = 1; i < 50; ++i) {
1595         fe_sq(t2, t2);
1596     }
1597 
1598     /* t1 = z ** (2 ** 250 - 1) */
1599     fe_mul(t1, t2, t1);
1600 
1601     /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */
1602     fe_sq(t1, t1);
1603     for (i = 1; i < 5; ++i) {
1604         fe_sq(t1, t1);
1605     }
1606 
1607     /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */
1608     fe_mul(out, t1, t0);
1609 }
1610 
1611 /*
1612  * h = -f
1613  *
1614  * Preconditions:
1615  *    |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
1616  *
1617  * Postconditions:
1618  *    |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
1619  */
fe_neg(fe h,const fe f)1620 static void fe_neg(fe h, const fe f)
1621 {
1622     unsigned i;
1623 
1624     for (i = 0; i < 10; i++) {
1625         h[i] = -f[i];
1626     }
1627 }
1628 
1629 /*
1630  * Replace (f,g) with (g,g) if b == 1;
1631  * replace (f,g) with (f,g) if b == 0.
1632  *
1633  * Preconditions: b in {0,1}.
1634  */
fe_cmov(fe f,const fe g,unsigned b)1635 static void fe_cmov(fe f, const fe g, unsigned b)
1636 {
1637     size_t i;
1638 
1639     b = 0 - b;
1640     for (i = 0; i < 10; i++) {
1641         int32_t x = f[i] ^ g[i];
1642         x &= b;
1643         f[i] ^= x;
1644     }
1645 }
1646 
1647 /*
1648  * return 0 if f == 0
1649  * return 1 if f != 0
1650  *
1651  * Preconditions:
1652  *    |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
1653  */
fe_isnonzero(const fe f)1654 static int fe_isnonzero(const fe f)
1655 {
1656     uint8_t s[32];
1657     static const uint8_t zero[32] = { 0 };
1658 
1659     fe_tobytes(s, f);
1660 
1661     return CRYPTO_memcmp(s, zero, sizeof(zero)) != 0;
1662 }
1663 
1664 /*
1665  * return 1 if f is in {1,3,5,...,q-2}
1666  * return 0 if f is in {0,2,4,...,q-1}
1667  *
1668  * Preconditions:
1669  *    |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
1670  */
fe_isnegative(const fe f)1671 static int fe_isnegative(const fe f)
1672 {
1673     uint8_t s[32];
1674 
1675     fe_tobytes(s, f);
1676     return s[0] & 1;
1677 }
1678 
1679 /*
1680  * h = 2 * f * f
1681  *
1682  * Can overlap h with f.
1683  *
1684  * Preconditions:
1685  *    |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
1686  *
1687  * Postconditions:
1688  *    |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
1689  *
1690  * See fe_mul.c for discussion of implementation strategy.
1691  */
fe_sq2(fe h,const fe f)1692 static void fe_sq2(fe h, const fe f)
1693 {
1694     int32_t f0 = f[0];
1695     int32_t f1 = f[1];
1696     int32_t f2 = f[2];
1697     int32_t f3 = f[3];
1698     int32_t f4 = f[4];
1699     int32_t f5 = f[5];
1700     int32_t f6 = f[6];
1701     int32_t f7 = f[7];
1702     int32_t f8 = f[8];
1703     int32_t f9 = f[9];
1704     int32_t f0_2 = 2 * f0;
1705     int32_t f1_2 = 2 * f1;
1706     int32_t f2_2 = 2 * f2;
1707     int32_t f3_2 = 2 * f3;
1708     int32_t f4_2 = 2 * f4;
1709     int32_t f5_2 = 2 * f5;
1710     int32_t f6_2 = 2 * f6;
1711     int32_t f7_2 = 2 * f7;
1712     int32_t f5_38 = 38 * f5; /* 1.959375*2^30 */
1713     int32_t f6_19 = 19 * f6; /* 1.959375*2^30 */
1714     int32_t f7_38 = 38 * f7; /* 1.959375*2^30 */
1715     int32_t f8_19 = 19 * f8; /* 1.959375*2^30 */
1716     int32_t f9_38 = 38 * f9; /* 1.959375*2^30 */
1717     int64_t f0f0 = f0 * (int64_t)f0;
1718     int64_t f0f1_2 = f0_2 * (int64_t)f1;
1719     int64_t f0f2_2 = f0_2 * (int64_t)f2;
1720     int64_t f0f3_2 = f0_2 * (int64_t)f3;
1721     int64_t f0f4_2 = f0_2 * (int64_t)f4;
1722     int64_t f0f5_2 = f0_2 * (int64_t)f5;
1723     int64_t f0f6_2 = f0_2 * (int64_t)f6;
1724     int64_t f0f7_2 = f0_2 * (int64_t)f7;
1725     int64_t f0f8_2 = f0_2 * (int64_t)f8;
1726     int64_t f0f9_2 = f0_2 * (int64_t)f9;
1727     int64_t f1f1_2 = f1_2 * (int64_t)f1;
1728     int64_t f1f2_2 = f1_2 * (int64_t)f2;
1729     int64_t f1f3_4 = f1_2 * (int64_t)f3_2;
1730     int64_t f1f4_2 = f1_2 * (int64_t)f4;
1731     int64_t f1f5_4 = f1_2 * (int64_t)f5_2;
1732     int64_t f1f6_2 = f1_2 * (int64_t)f6;
1733     int64_t f1f7_4 = f1_2 * (int64_t)f7_2;
1734     int64_t f1f8_2 = f1_2 * (int64_t)f8;
1735     int64_t f1f9_76 = f1_2 * (int64_t)f9_38;
1736     int64_t f2f2 = f2 * (int64_t)f2;
1737     int64_t f2f3_2 = f2_2 * (int64_t)f3;
1738     int64_t f2f4_2 = f2_2 * (int64_t)f4;
1739     int64_t f2f5_2 = f2_2 * (int64_t)f5;
1740     int64_t f2f6_2 = f2_2 * (int64_t)f6;
1741     int64_t f2f7_2 = f2_2 * (int64_t)f7;
1742     int64_t f2f8_38 = f2_2 * (int64_t)f8_19;
1743     int64_t f2f9_38 = f2 * (int64_t)f9_38;
1744     int64_t f3f3_2 = f3_2 * (int64_t)f3;
1745     int64_t f3f4_2 = f3_2 * (int64_t)f4;
1746     int64_t f3f5_4 = f3_2 * (int64_t)f5_2;
1747     int64_t f3f6_2 = f3_2 * (int64_t)f6;
1748     int64_t f3f7_76 = f3_2 * (int64_t)f7_38;
1749     int64_t f3f8_38 = f3_2 * (int64_t)f8_19;
1750     int64_t f3f9_76 = f3_2 * (int64_t)f9_38;
1751     int64_t f4f4 = f4 * (int64_t)f4;
1752     int64_t f4f5_2 = f4_2 * (int64_t)f5;
1753     int64_t f4f6_38 = f4_2 * (int64_t)f6_19;
1754     int64_t f4f7_38 = f4 * (int64_t)f7_38;
1755     int64_t f4f8_38 = f4_2 * (int64_t)f8_19;
1756     int64_t f4f9_38 = f4 * (int64_t)f9_38;
1757     int64_t f5f5_38 = f5 * (int64_t)f5_38;
1758     int64_t f5f6_38 = f5_2 * (int64_t)f6_19;
1759     int64_t f5f7_76 = f5_2 * (int64_t)f7_38;
1760     int64_t f5f8_38 = f5_2 * (int64_t)f8_19;
1761     int64_t f5f9_76 = f5_2 * (int64_t)f9_38;
1762     int64_t f6f6_19 = f6 * (int64_t)f6_19;
1763     int64_t f6f7_38 = f6 * (int64_t)f7_38;
1764     int64_t f6f8_38 = f6_2 * (int64_t)f8_19;
1765     int64_t f6f9_38 = f6 * (int64_t)f9_38;
1766     int64_t f7f7_38 = f7 * (int64_t)f7_38;
1767     int64_t f7f8_38 = f7_2 * (int64_t)f8_19;
1768     int64_t f7f9_76 = f7_2 * (int64_t)f9_38;
1769     int64_t f8f8_19 = f8 * (int64_t)f8_19;
1770     int64_t f8f9_38 = f8 * (int64_t)f9_38;
1771     int64_t f9f9_38 = f9 * (int64_t)f9_38;
1772     int64_t h0 = f0f0 + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38;
1773     int64_t h1 = f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38;
1774     int64_t h2 = f0f2_2 + f1f1_2 + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19;
1775     int64_t h3 = f0f3_2 + f1f2_2 + f4f9_38 + f5f8_38 + f6f7_38;
1776     int64_t h4 = f0f4_2 + f1f3_4 + f2f2 + f5f9_76 + f6f8_38 + f7f7_38;
1777     int64_t h5 = f0f5_2 + f1f4_2 + f2f3_2 + f6f9_38 + f7f8_38;
1778     int64_t h6 = f0f6_2 + f1f5_4 + f2f4_2 + f3f3_2 + f7f9_76 + f8f8_19;
1779     int64_t h7 = f0f7_2 + f1f6_2 + f2f5_2 + f3f4_2 + f8f9_38;
1780     int64_t h8 = f0f8_2 + f1f7_4 + f2f6_2 + f3f5_4 + f4f4 + f9f9_38;
1781     int64_t h9 = f0f9_2 + f1f8_2 + f2f7_2 + f3f6_2 + f4f5_2;
1782     int64_t carry0;
1783     int64_t carry1;
1784     int64_t carry2;
1785     int64_t carry3;
1786     int64_t carry4;
1787     int64_t carry5;
1788     int64_t carry6;
1789     int64_t carry7;
1790     int64_t carry8;
1791     int64_t carry9;
1792 
1793     h0 += h0;
1794     h1 += h1;
1795     h2 += h2;
1796     h3 += h3;
1797     h4 += h4;
1798     h5 += h5;
1799     h6 += h6;
1800     h7 += h7;
1801     h8 += h8;
1802     h9 += h9;
1803 
1804     carry0 = h0 + (1 << 25);
1805     h1 += carry0 >> 26;
1806     h0 -= carry0 & kTop38Bits;
1807     carry4 = h4 + (1 << 25);
1808     h5 += carry4 >> 26;
1809     h4 -= carry4 & kTop38Bits;
1810 
1811     carry1 = h1 + (1 << 24);
1812     h2 += carry1 >> 25;
1813     h1 -= carry1 & kTop39Bits;
1814     carry5 = h5 + (1 << 24);
1815     h6 += carry5 >> 25;
1816     h5 -= carry5 & kTop39Bits;
1817 
1818     carry2 = h2 + (1 << 25);
1819     h3 += carry2 >> 26;
1820     h2 -= carry2 & kTop38Bits;
1821     carry6 = h6 + (1 << 25);
1822     h7 += carry6 >> 26;
1823     h6 -= carry6 & kTop38Bits;
1824 
1825     carry3 = h3 + (1 << 24);
1826     h4 += carry3 >> 25;
1827     h3 -= carry3 & kTop39Bits;
1828     carry7 = h7 + (1 << 24);
1829     h8 += carry7 >> 25;
1830     h7 -= carry7 & kTop39Bits;
1831 
1832     carry4 = h4 + (1 << 25);
1833     h5 += carry4 >> 26;
1834     h4 -= carry4 & kTop38Bits;
1835     carry8 = h8 + (1 << 25);
1836     h9 += carry8 >> 26;
1837     h8 -= carry8 & kTop38Bits;
1838 
1839     carry9 = h9 + (1 << 24);
1840     h0 += (carry9 >> 25) * 19;
1841     h9 -= carry9 & kTop39Bits;
1842 
1843     carry0 = h0 + (1 << 25);
1844     h1 += carry0 >> 26;
1845     h0 -= carry0 & kTop38Bits;
1846 
1847     h[0] = (int32_t)h0;
1848     h[1] = (int32_t)h1;
1849     h[2] = (int32_t)h2;
1850     h[3] = (int32_t)h3;
1851     h[4] = (int32_t)h4;
1852     h[5] = (int32_t)h5;
1853     h[6] = (int32_t)h6;
1854     h[7] = (int32_t)h7;
1855     h[8] = (int32_t)h8;
1856     h[9] = (int32_t)h9;
1857 }
1858 
fe_pow22523(fe out,const fe z)1859 static void fe_pow22523(fe out, const fe z)
1860 {
1861     fe t0;
1862     fe t1;
1863     fe t2;
1864     int i;
1865 
1866     fe_sq(t0, z);
1867     fe_sq(t1, t0);
1868     for (i = 1; i < 2; ++i) {
1869         fe_sq(t1, t1);
1870     }
1871     fe_mul(t1, z, t1);
1872     fe_mul(t0, t0, t1);
1873     fe_sq(t0, t0);
1874     fe_mul(t0, t1, t0);
1875     fe_sq(t1, t0);
1876     for (i = 1; i < 5; ++i) {
1877         fe_sq(t1, t1);
1878     }
1879     fe_mul(t0, t1, t0);
1880     fe_sq(t1, t0);
1881     for (i = 1; i < 10; ++i) {
1882         fe_sq(t1, t1);
1883     }
1884     fe_mul(t1, t1, t0);
1885     fe_sq(t2, t1);
1886     for (i = 1; i < 20; ++i) {
1887         fe_sq(t2, t2);
1888     }
1889     fe_mul(t1, t2, t1);
1890     fe_sq(t1, t1);
1891     for (i = 1; i < 10; ++i) {
1892         fe_sq(t1, t1);
1893     }
1894     fe_mul(t0, t1, t0);
1895     fe_sq(t1, t0);
1896     for (i = 1; i < 50; ++i) {
1897         fe_sq(t1, t1);
1898     }
1899     fe_mul(t1, t1, t0);
1900     fe_sq(t2, t1);
1901     for (i = 1; i < 100; ++i) {
1902         fe_sq(t2, t2);
1903     }
1904     fe_mul(t1, t2, t1);
1905     fe_sq(t1, t1);
1906     for (i = 1; i < 50; ++i) {
1907         fe_sq(t1, t1);
1908     }
1909     fe_mul(t0, t1, t0);
1910     fe_sq(t0, t0);
1911     for (i = 1; i < 2; ++i) {
1912         fe_sq(t0, t0);
1913     }
1914     fe_mul(out, t0, z);
1915 }
1916 
1917 /*
1918  * ge means group element.
1919  *
1920  * Here the group is the set of pairs (x,y) of field elements (see fe.h)
1921  * satisfying -x^2 + y^2 = 1 + d x^2y^2
1922  * where d = -121665/121666.
1923  *
1924  * Representations:
1925  *   ge_p2 (projective): (X:Y:Z) satisfying x=X/Z, y=Y/Z
1926  *   ge_p3 (extended): (X:Y:Z:T) satisfying x=X/Z, y=Y/Z, XY=ZT
1927  *   ge_p1p1 (completed): ((X:Z),(Y:T)) satisfying x=X/Z, y=Y/T
1928  *   ge_precomp (Duif): (y+x,y-x,2dxy)
1929  */
1930 typedef struct {
1931     fe X;
1932     fe Y;
1933     fe Z;
1934 } ge_p2;
1935 
1936 typedef struct {
1937     fe X;
1938     fe Y;
1939     fe Z;
1940     fe T;
1941 } ge_p3;
1942 
1943 typedef struct {
1944     fe X;
1945     fe Y;
1946     fe Z;
1947     fe T;
1948 } ge_p1p1;
1949 
1950 typedef struct {
1951     fe yplusx;
1952     fe yminusx;
1953     fe xy2d;
1954 } ge_precomp;
1955 
1956 typedef struct {
1957     fe YplusX;
1958     fe YminusX;
1959     fe Z;
1960     fe T2d;
1961 } ge_cached;
1962 
ge_tobytes(uint8_t * s,const ge_p2 * h)1963 static void ge_tobytes(uint8_t *s, const ge_p2 *h)
1964 {
1965     fe recip;
1966     fe x;
1967     fe y;
1968 
1969     fe_invert(recip, h->Z);
1970     fe_mul(x, h->X, recip);
1971     fe_mul(y, h->Y, recip);
1972     fe_tobytes(s, y);
1973     s[31] ^= fe_isnegative(x) << 7;
1974 }
1975 
ge_p3_tobytes(uint8_t * s,const ge_p3 * h)1976 static void ge_p3_tobytes(uint8_t *s, const ge_p3 *h)
1977 {
1978     fe recip;
1979     fe x;
1980     fe y;
1981 
1982     fe_invert(recip, h->Z);
1983     fe_mul(x, h->X, recip);
1984     fe_mul(y, h->Y, recip);
1985     fe_tobytes(s, y);
1986     s[31] ^= fe_isnegative(x) << 7;
1987 }
1988 
1989 static const fe d = {
1990     -10913610, 13857413, -15372611, 6949391, 114729,
1991     -8787816, -6275908, -3247719, -18696448, -12055116
1992 };
1993 
1994 static const fe sqrtm1 = {
1995     -32595792, -7943725, 9377950, 3500415, 12389472,
1996     -272473, -25146209, -2005654, 326686, 11406482
1997 };
1998 
ge_frombytes_vartime(ge_p3 * h,const uint8_t * s)1999 static int ge_frombytes_vartime(ge_p3 *h, const uint8_t *s)
2000 {
2001     fe u;
2002     fe v;
2003     fe w;
2004     fe vxx;
2005     fe check;
2006 
2007     fe_frombytes(h->Y, s);
2008     fe_1(h->Z);
2009     fe_sq(u, h->Y);
2010     fe_mul(v, u, d);
2011     fe_sub(u, u, h->Z); /* u = y^2-1 */
2012     fe_add(v, v, h->Z); /* v = dy^2+1 */
2013 
2014     fe_mul(w, u, v); /* w = u*v */
2015 
2016     fe_pow22523(h->X, w); /* x = w^((q-5)/8) */
2017     fe_mul(h->X, h->X, u); /* x = u * w^((q-5)/8) */
2018 
2019     fe_sq(vxx, h->X);
2020     fe_mul(vxx, vxx, v);
2021     fe_sub(check, vxx, u); /* vx^2-u */
2022     if (fe_isnonzero(check)) {
2023         fe_add(check, vxx, u); /* vx^2+u */
2024         if (fe_isnonzero(check)) {
2025             return -1;
2026         }
2027         fe_mul(h->X, h->X, sqrtm1);
2028     }
2029 
2030     if (fe_isnegative(h->X) != (s[31] >> 7)) {
2031         fe_neg(h->X, h->X);
2032     }
2033 
2034     fe_mul(h->T, h->X, h->Y);
2035     return 0;
2036 }
2037 
ge_p2_0(ge_p2 * h)2038 static void ge_p2_0(ge_p2 *h)
2039 {
2040     fe_0(h->X);
2041     fe_1(h->Y);
2042     fe_1(h->Z);
2043 }
2044 
ge_p3_0(ge_p3 * h)2045 static void ge_p3_0(ge_p3 *h)
2046 {
2047     fe_0(h->X);
2048     fe_1(h->Y);
2049     fe_1(h->Z);
2050     fe_0(h->T);
2051 }
2052 
ge_precomp_0(ge_precomp * h)2053 static void ge_precomp_0(ge_precomp *h)
2054 {
2055     fe_1(h->yplusx);
2056     fe_1(h->yminusx);
2057     fe_0(h->xy2d);
2058 }
2059 
2060 /* r = p */
ge_p3_to_p2(ge_p2 * r,const ge_p3 * p)2061 static void ge_p3_to_p2(ge_p2 *r, const ge_p3 *p)
2062 {
2063     fe_copy(r->X, p->X);
2064     fe_copy(r->Y, p->Y);
2065     fe_copy(r->Z, p->Z);
2066 }
2067 
2068 static const fe d2 = {
2069     -21827239, -5839606, -30745221, 13898782, 229458,
2070     15978800, -12551817, -6495438, 29715968, 9444199
2071 };
2072 
2073 /* r = p */
ge_p3_to_cached(ge_cached * r,const ge_p3 * p)2074 static void ge_p3_to_cached(ge_cached *r, const ge_p3 *p)
2075 {
2076     fe_add(r->YplusX, p->Y, p->X);
2077     fe_sub(r->YminusX, p->Y, p->X);
2078     fe_copy(r->Z, p->Z);
2079     fe_mul(r->T2d, p->T, d2);
2080 }
2081 
2082 /* r = p */
ge_p1p1_to_p2(ge_p2 * r,const ge_p1p1 * p)2083 static void ge_p1p1_to_p2(ge_p2 *r, const ge_p1p1 *p)
2084 {
2085     fe_mul(r->X, p->X, p->T);
2086     fe_mul(r->Y, p->Y, p->Z);
2087     fe_mul(r->Z, p->Z, p->T);
2088 }
2089 
2090 /* r = p */
ge_p1p1_to_p3(ge_p3 * r,const ge_p1p1 * p)2091 static void ge_p1p1_to_p3(ge_p3 *r, const ge_p1p1 *p)
2092 {
2093     fe_mul(r->X, p->X, p->T);
2094     fe_mul(r->Y, p->Y, p->Z);
2095     fe_mul(r->Z, p->Z, p->T);
2096     fe_mul(r->T, p->X, p->Y);
2097 }
2098 
2099 /* r = 2 * p */
ge_p2_dbl(ge_p1p1 * r,const ge_p2 * p)2100 static void ge_p2_dbl(ge_p1p1 *r, const ge_p2 *p)
2101 {
2102     fe t0;
2103 
2104     fe_sq(r->X, p->X);
2105     fe_sq(r->Z, p->Y);
2106     fe_sq2(r->T, p->Z);
2107     fe_add(r->Y, p->X, p->Y);
2108     fe_sq(t0, r->Y);
2109     fe_add(r->Y, r->Z, r->X);
2110     fe_sub(r->Z, r->Z, r->X);
2111     fe_sub(r->X, t0, r->Y);
2112     fe_sub(r->T, r->T, r->Z);
2113 }
2114 
2115 /* r = 2 * p */
ge_p3_dbl(ge_p1p1 * r,const ge_p3 * p)2116 static void ge_p3_dbl(ge_p1p1 *r, const ge_p3 *p)
2117 {
2118     ge_p2 q;
2119     ge_p3_to_p2(&q, p);
2120     ge_p2_dbl(r, &q);
2121 }
2122 
2123 /* r = p + q */
ge_madd(ge_p1p1 * r,const ge_p3 * p,const ge_precomp * q)2124 static void ge_madd(ge_p1p1 *r, const ge_p3 *p, const ge_precomp *q)
2125 {
2126     fe t0;
2127 
2128     fe_add(r->X, p->Y, p->X);
2129     fe_sub(r->Y, p->Y, p->X);
2130     fe_mul(r->Z, r->X, q->yplusx);
2131     fe_mul(r->Y, r->Y, q->yminusx);
2132     fe_mul(r->T, q->xy2d, p->T);
2133     fe_add(t0, p->Z, p->Z);
2134     fe_sub(r->X, r->Z, r->Y);
2135     fe_add(r->Y, r->Z, r->Y);
2136     fe_add(r->Z, t0, r->T);
2137     fe_sub(r->T, t0, r->T);
2138 }
2139 
2140 /* r = p - q */
ge_msub(ge_p1p1 * r,const ge_p3 * p,const ge_precomp * q)2141 static void ge_msub(ge_p1p1 *r, const ge_p3 *p, const ge_precomp *q)
2142 {
2143     fe t0;
2144 
2145     fe_add(r->X, p->Y, p->X);
2146     fe_sub(r->Y, p->Y, p->X);
2147     fe_mul(r->Z, r->X, q->yminusx);
2148     fe_mul(r->Y, r->Y, q->yplusx);
2149     fe_mul(r->T, q->xy2d, p->T);
2150     fe_add(t0, p->Z, p->Z);
2151     fe_sub(r->X, r->Z, r->Y);
2152     fe_add(r->Y, r->Z, r->Y);
2153     fe_sub(r->Z, t0, r->T);
2154     fe_add(r->T, t0, r->T);
2155 }
2156 
2157 /* r = p + q */
ge_add(ge_p1p1 * r,const ge_p3 * p,const ge_cached * q)2158 static void ge_add(ge_p1p1 *r, const ge_p3 *p, const ge_cached *q)
2159 {
2160     fe t0;
2161 
2162     fe_add(r->X, p->Y, p->X);
2163     fe_sub(r->Y, p->Y, p->X);
2164     fe_mul(r->Z, r->X, q->YplusX);
2165     fe_mul(r->Y, r->Y, q->YminusX);
2166     fe_mul(r->T, q->T2d, p->T);
2167     fe_mul(r->X, p->Z, q->Z);
2168     fe_add(t0, r->X, r->X);
2169     fe_sub(r->X, r->Z, r->Y);
2170     fe_add(r->Y, r->Z, r->Y);
2171     fe_add(r->Z, t0, r->T);
2172     fe_sub(r->T, t0, r->T);
2173 }
2174 
2175 /* r = p - q */
ge_sub(ge_p1p1 * r,const ge_p3 * p,const ge_cached * q)2176 static void ge_sub(ge_p1p1 *r, const ge_p3 *p, const ge_cached *q)
2177 {
2178     fe t0;
2179 
2180     fe_add(r->X, p->Y, p->X);
2181     fe_sub(r->Y, p->Y, p->X);
2182     fe_mul(r->Z, r->X, q->YminusX);
2183     fe_mul(r->Y, r->Y, q->YplusX);
2184     fe_mul(r->T, q->T2d, p->T);
2185     fe_mul(r->X, p->Z, q->Z);
2186     fe_add(t0, r->X, r->X);
2187     fe_sub(r->X, r->Z, r->Y);
2188     fe_add(r->Y, r->Z, r->Y);
2189     fe_sub(r->Z, t0, r->T);
2190     fe_add(r->T, t0, r->T);
2191 }
2192 
equal(signed char b,signed char c)2193 static uint8_t equal(signed char b, signed char c)
2194 {
2195     uint8_t ub = b;
2196     uint8_t uc = c;
2197     uint8_t x = ub ^ uc; /* 0: yes; 1..255: no */
2198     uint32_t y = x; /* 0: yes; 1..255: no */
2199     y -= 1; /* 4294967295: yes; 0..254: no */
2200     y >>= 31; /* 1: yes; 0: no */
2201     return y;
2202 }
2203 
cmov(ge_precomp * t,const ge_precomp * u,uint8_t b)2204 static void cmov(ge_precomp *t, const ge_precomp *u, uint8_t b)
2205 {
2206     fe_cmov(t->yplusx, u->yplusx, b);
2207     fe_cmov(t->yminusx, u->yminusx, b);
2208     fe_cmov(t->xy2d, u->xy2d, b);
2209 }
2210 
2211 /* k25519Precomp[i][j] = (j+1)*256^i*B */
2212 static const ge_precomp k25519Precomp[32][8] = {
2213     {
2214         {
2215             { 25967493, -14356035, 29566456, 3660896, -12694345, 4014787,
2216                 27544626, -11754271, -6079156, 2047605 },
2217             { -12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692,
2218                 5043384, 19500929, -15469378 },
2219             { -8738181, 4489570, 9688441, -14785194, 10184609, -12363380,
2220                 29287919, 11864899, -24514362, -4438546 },
2221         },
2222         {
2223             { -12815894, -12976347, -21581243, 11784320, -25355658, -2750717,
2224                 -11717903, -3814571, -358445, -10211303 },
2225             { -21703237, 6903825, 27185491, 6451973, -29577724, -9554005,
2226                 -15616551, 11189268, -26829678, -5319081 },
2227             { 26966642, 11152617, 32442495, 15396054, 14353839, -12752335,
2228                 -3128826, -9541118, -15472047, -4166697 },
2229         },
2230         {
2231             { 15636291, -9688557, 24204773, -7912398, 616977, -16685262,
2232                 27787600, -14772189, 28944400, -1550024 },
2233             { 16568933, 4717097, -11556148, -1102322, 15682896, -11807043,
2234                 16354577, -11775962, 7689662, 11199574 },
2235             { 30464156, -5976125, -11779434, -15670865, 23220365, 15915852,
2236                 7512774, 10017326, -17749093, -9920357 },
2237         },
2238         {
2239             { -17036878, 13921892, 10945806, -6033431, 27105052, -16084379,
2240                 -28926210, 15006023, 3284568, -6276540 },
2241             { 23599295, -8306047, -11193664, -7687416, 13236774, 10506355,
2242                 7464579, 9656445, 13059162, 10374397 },
2243             { 7798556, 16710257, 3033922, 2874086, 28997861, 2835604, 32406664,
2244                 -3839045, -641708, -101325 },
2245         },
2246         {
2247             { 10861363, 11473154, 27284546, 1981175, -30064349, 12577861,
2248                 32867885, 14515107, -15438304, 10819380 },
2249             { 4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668,
2250                 12483688, -12668491, 5581306 },
2251             { 19563160, 16186464, -29386857, 4097519, 10237984, -4348115,
2252                 28542350, 13850243, -23678021, -15815942 },
2253         },
2254         {
2255             { -15371964, -12862754, 32573250, 4720197, -26436522, 5875511,
2256                 -19188627, -15224819, -9818940, -12085777 },
2257             { -8549212, 109983, 15149363, 2178705, 22900618, 4543417, 3044240,
2258                 -15689887, 1762328, 14866737 },
2259             { -18199695, -15951423, -10473290, 1707278, -17185920, 3916101,
2260                 -28236412, 3959421, 27914454, 4383652 },
2261         },
2262         {
2263             { 5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852,
2264                 5230134, -23952439, -15175766 },
2265             { -30269007, -3463509, 7665486, 10083793, 28475525, 1649722,
2266                 20654025, 16520125, 30598449, 7715701 },
2267             { 28881845, 14381568, 9657904, 3680757, -20181635, 7843316,
2268                 -31400660, 1370708, 29794553, -1409300 },
2269         },
2270         {
2271             { 14499471, -2729599, -33191113, -4254652, 28494862, 14271267,
2272                 30290735, 10876454, -33154098, 2381726 },
2273             { -7195431, -2655363, -14730155, 462251, -27724326, 3941372,
2274                 -6236617, 3696005, -32300832, 15351955 },
2275             { 27431194, 8222322, 16448760, -3907995, -18707002, 11938355,
2276                 -32961401, -2970515, 29551813, 10109425 },
2277         },
2278     },
2279     {
2280         {
2281             { -13657040, -13155431, -31283750, 11777098, 21447386, 6519384,
2282                 -2378284, -1627556, 10092783, -4764171 },
2283             { 27939166, 14210322, 4677035, 16277044, -22964462, -12398139,
2284                 -32508754, 12005538, -17810127, 12803510 },
2285             { 17228999, -15661624, -1233527, 300140, -1224870, -11714777,
2286                 30364213, -9038194, 18016357, 4397660 },
2287         },
2288         {
2289             { -10958843, -7690207, 4776341, -14954238, 27850028, -15602212,
2290                 -26619106, 14544525, -17477504, 982639 },
2291             { 29253598, 15796703, -2863982, -9908884, 10057023, 3163536, 7332899,
2292                 -4120128, -21047696, 9934963 },
2293             { 5793303, 16271923, -24131614, -10116404, 29188560, 1206517,
2294                 -14747930, 4559895, -30123922, -10897950 },
2295         },
2296         {
2297             { -27643952, -11493006, 16282657, -11036493, 28414021, -15012264,
2298                 24191034, 4541697, -13338309, 5500568 },
2299             { 12650548, -1497113, 9052871, 11355358, -17680037, -8400164,
2300                 -17430592, 12264343, 10874051, 13524335 },
2301             { 25556948, -3045990, 714651, 2510400, 23394682, -10415330, 33119038,
2302                 5080568, -22528059, 5376628 },
2303         },
2304         {
2305             { -26088264, -4011052, -17013699, -3537628, -6726793, 1920897,
2306                 -22321305, -9447443, 4535768, 1569007 },
2307             { -2255422, 14606630, -21692440, -8039818, 28430649, 8775819,
2308                 -30494562, 3044290, 31848280, 12543772 },
2309             { -22028579, 2943893, -31857513, 6777306, 13784462, -4292203,
2310                 -27377195, -2062731, 7718482, 14474653 },
2311         },
2312         {
2313             { 2385315, 2454213, -22631320, 46603, -4437935, -15680415, 656965,
2314                 -7236665, 24316168, -5253567 },
2315             { 13741529, 10911568, -33233417, -8603737, -20177830, -1033297,
2316                 33040651, -13424532, -20729456, 8321686 },
2317             { 21060490, -2212744, 15712757, -4336099, 1639040, 10656336,
2318                 23845965, -11874838, -9984458, 608372 },
2319         },
2320         {
2321             { -13672732, -15087586, -10889693, -7557059, -6036909, 11305547,
2322                 1123968, -6780577, 27229399, 23887 },
2323             { -23244140, -294205, -11744728, 14712571, -29465699, -2029617,
2324                 12797024, -6440308, -1633405, 16678954 },
2325             { -29500620, 4770662, -16054387, 14001338, 7830047, 9564805,
2326                 -1508144, -4795045, -17169265, 4904953 },
2327         },
2328         {
2329             { 24059557, 14617003, 19037157, -15039908, 19766093, -14906429,
2330                 5169211, 16191880, 2128236, -4326833 },
2331             { -16981152, 4124966, -8540610, -10653797, 30336522, -14105247,
2332                 -29806336, 916033, -6882542, -2986532 },
2333             { -22630907, 12419372, -7134229, -7473371, -16478904, 16739175,
2334                 285431, 2763829, 15736322, 4143876 },
2335         },
2336         {
2337             { 2379352, 11839345, -4110402, -5988665, 11274298, 794957, 212801,
2338                 -14594663, 23527084, -16458268 },
2339             { 33431127, -11130478, -17838966, -15626900, 8909499, 8376530,
2340                 -32625340, 4087881, -15188911, -14416214 },
2341             { 1767683, 7197987, -13205226, -2022635, -13091350, 448826, 5799055,
2342                 4357868, -4774191, -16323038 },
2343         },
2344     },
2345     {
2346         {
2347             { 6721966, 13833823, -23523388, -1551314, 26354293, -11863321,
2348                 23365147, -3949732, 7390890, 2759800 },
2349             { 4409041, 2052381, 23373853, 10530217, 7676779, -12885954, 21302353,
2350                 -4264057, 1244380, -12919645 },
2351             { -4421239, 7169619, 4982368, -2957590, 30256825, -2777540, 14086413,
2352                 9208236, 15886429, 16489664 },
2353         },
2354         {
2355             { 1996075, 10375649, 14346367, 13311202, -6874135, -16438411,
2356                 -13693198, 398369, -30606455, -712933 },
2357             { -25307465, 9795880, -2777414, 14878809, -33531835, 14780363,
2358                 13348553, 12076947, -30836462, 5113182 },
2359             { -17770784, 11797796, 31950843, 13929123, -25888302, 12288344,
2360                 -30341101, -7336386, 13847711, 5387222 },
2361         },
2362         {
2363             { -18582163, -3416217, 17824843, -2340966, 22744343, -10442611,
2364                 8763061, 3617786, -19600662, 10370991 },
2365             { 20246567, -14369378, 22358229, -543712, 18507283, -10413996,
2366                 14554437, -8746092, 32232924, 16763880 },
2367             { 9648505, 10094563, 26416693, 14745928, -30374318, -6472621,
2368                 11094161, 15689506, 3140038, -16510092 },
2369         },
2370         {
2371             { -16160072, 5472695, 31895588, 4744994, 8823515, 10365685,
2372                 -27224800, 9448613, -28774454, 366295 },
2373             { 19153450, 11523972, -11096490, -6503142, -24647631, 5420647,
2374                 28344573, 8041113, 719605, 11671788 },
2375             { 8678025, 2694440, -6808014, 2517372, 4964326, 11152271, -15432916,
2376                 -15266516, 27000813, -10195553 },
2377         },
2378         {
2379             { -15157904, 7134312, 8639287, -2814877, -7235688, 10421742, 564065,
2380                 5336097, 6750977, -14521026 },
2381             { 11836410, -3979488, 26297894, 16080799, 23455045, 15735944,
2382                 1695823, -8819122, 8169720, 16220347 },
2383             { -18115838, 8653647, 17578566, -6092619, -8025777, -16012763,
2384                 -11144307, -2627664, -5990708, -14166033 },
2385         },
2386         {
2387             { -23308498, -10968312, 15213228, -10081214, -30853605, -11050004,
2388                 27884329, 2847284, 2655861, 1738395 },
2389             { -27537433, -14253021, -25336301, -8002780, -9370762, 8129821,
2390                 21651608, -3239336, -19087449, -11005278 },
2391             { 1533110, 3437855, 23735889, 459276, 29970501, 11335377, 26030092,
2392                 5821408, 10478196, 8544890 },
2393         },
2394         {
2395             { 32173121, -16129311, 24896207, 3921497, 22579056, -3410854,
2396                 19270449, 12217473, 17789017, -3395995 },
2397             { -30552961, -2228401, -15578829, -10147201, 13243889, 517024,
2398                 15479401, -3853233, 30460520, 1052596 },
2399             { -11614875, 13323618, 32618793, 8175907, -15230173, 12596687,
2400                 27491595, -4612359, 3179268, -9478891 },
2401         },
2402         {
2403             { 31947069, -14366651, -4640583, -15339921, -15125977, -6039709,
2404                 -14756777, -16411740, 19072640, -9511060 },
2405             { 11685058, 11822410, 3158003, -13952594, 33402194, -4165066,
2406                 5977896, -5215017, 473099, 5040608 },
2407             { -20290863, 8198642, -27410132, 11602123, 1290375, -2799760,
2408                 28326862, 1721092, -19558642, -3131606 },
2409         },
2410     },
2411     {
2412         {
2413             { 7881532, 10687937, 7578723, 7738378, -18951012, -2553952, 21820786,
2414                 8076149, -27868496, 11538389 },
2415             { -19935666, 3899861, 18283497, -6801568, -15728660, -11249211,
2416                 8754525, 7446702, -5676054, 5797016 },
2417             { -11295600, -3793569, -15782110, -7964573, 12708869, -8456199,
2418                 2014099, -9050574, -2369172, -5877341 },
2419         },
2420         {
2421             { -22472376, -11568741, -27682020, 1146375, 18956691, 16640559,
2422                 1192730, -3714199, 15123619, 10811505 },
2423             { 14352098, -3419715, -18942044, 10822655, 32750596, 4699007, -70363,
2424                 15776356, -28886779, -11974553 },
2425             { -28241164, -8072475, -4978962, -5315317, 29416931, 1847569,
2426                 -20654173, -16484855, 4714547, -9600655 },
2427         },
2428         {
2429             { 15200332, 8368572, 19679101, 15970074, -31872674, 1959451,
2430                 24611599, -4543832, -11745876, 12340220 },
2431             { 12876937, -10480056, 33134381, 6590940, -6307776, 14872440,
2432                 9613953, 8241152, 15370987, 9608631 },
2433             { -4143277, -12014408, 8446281, -391603, 4407738, 13629032, -7724868,
2434                 15866074, -28210621, -8814099 },
2435         },
2436         {
2437             { 26660628, -15677655, 8393734, 358047, -7401291, 992988, -23904233,
2438                 858697, 20571223, 8420556 },
2439             { 14620715, 13067227, -15447274, 8264467, 14106269, 15080814,
2440                 33531827, 12516406, -21574435, -12476749 },
2441             { 236881, 10476226, 57258, -14677024, 6472998, 2466984, 17258519,
2442                 7256740, 8791136, 15069930 },
2443         },
2444         {
2445             { 1276410, -9371918, 22949635, -16322807, -23493039, -5702186,
2446                 14711875, 4874229, -30663140, -2331391 },
2447             { 5855666, 4990204, -13711848, 7294284, -7804282, 1924647, -1423175,
2448                 -7912378, -33069337, 9234253 },
2449             { 20590503, -9018988, 31529744, -7352666, -2706834, 10650548,
2450                 31559055, -11609587, 18979186, 13396066 },
2451         },
2452         {
2453             { 24474287, 4968103, 22267082, 4407354, 24063882, -8325180,
2454                 -18816887, 13594782, 33514650, 7021958 },
2455             { -11566906, -6565505, -21365085, 15928892, -26158305, 4315421,
2456                 -25948728, -3916677, -21480480, 12868082 },
2457             { -28635013, 13504661, 19988037, -2132761, 21078225, 6443208,
2458                 -21446107, 2244500, -12455797, -8089383 },
2459         },
2460         {
2461             { -30595528, 13793479, -5852820, 319136, -25723172, -6263899,
2462                 33086546, 8957937, -15233648, 5540521 },
2463             { -11630176, -11503902, -8119500, -7643073, 2620056, 1022908,
2464                 -23710744, -1568984, -16128528, -14962807 },
2465             { 23152971, 775386, 27395463, 14006635, -9701118, 4649512, 1689819,
2466                 892185, -11513277, -15205948 },
2467         },
2468         {
2469             { 9770129, 9586738, 26496094, 4324120, 1556511, -3550024, 27453819,
2470                 4763127, -19179614, 5867134 },
2471             { -32765025, 1927590, 31726409, -4753295, 23962434, -16019500,
2472                 27846559, 5931263, -29749703, -16108455 },
2473             { 27461885, -2977536, 22380810, 1815854, -23033753, -3031938,
2474                 7283490, -15148073, -19526700, 7734629 },
2475         },
2476     },
2477     {
2478         {
2479             { -8010264, -9590817, -11120403, 6196038, 29344158, -13430885,
2480                 7585295, -3176626, 18549497, 15302069 },
2481             { -32658337, -6171222, -7672793, -11051681, 6258878, 13504381,
2482                 10458790, -6418461, -8872242, 8424746 },
2483             { 24687205, 8613276, -30667046, -3233545, 1863892, -1830544,
2484                 19206234, 7134917, -11284482, -828919 },
2485         },
2486         {
2487             { 11334899, -9218022, 8025293, 12707519, 17523892, -10476071,
2488                 10243738, -14685461, -5066034, 16498837 },
2489             { 8911542, 6887158, -9584260, -6958590, 11145641, -9543680, 17303925,
2490                 -14124238, 6536641, 10543906 },
2491             { -28946384, 15479763, -17466835, 568876, -1497683, 11223454,
2492                 -2669190, -16625574, -27235709, 8876771 },
2493         },
2494         {
2495             { -25742899, -12566864, -15649966, -846607, -33026686, -796288,
2496                 -33481822, 15824474, -604426, -9039817 },
2497             { 10330056, 70051, 7957388, -9002667, 9764902, 15609756, 27698697,
2498                 -4890037, 1657394, 3084098 },
2499             { 10477963, -7470260, 12119566, -13250805, 29016247, -5365589,
2500                 31280319, 14396151, -30233575, 15272409 },
2501         },
2502         {
2503             { -12288309, 3169463, 28813183, 16658753, 25116432, -5630466,
2504                 -25173957, -12636138, -25014757, 1950504 },
2505             { -26180358, 9489187, 11053416, -14746161, -31053720, 5825630,
2506                 -8384306, -8767532, 15341279, 8373727 },
2507             { 28685821, 7759505, -14378516, -12002860, -31971820, 4079242,
2508                 298136, -10232602, -2878207, 15190420 },
2509         },
2510         {
2511             { -32932876, 13806336, -14337485, -15794431, -24004620, 10940928,
2512                 8669718, 2742393, -26033313, -6875003 },
2513             { -1580388, -11729417, -25979658, -11445023, -17411874, -10912854,
2514                 9291594, -16247779, -12154742, 6048605 },
2515             { -30305315, 14843444, 1539301, 11864366, 20201677, 1900163,
2516                 13934231, 5128323, 11213262, 9168384 },
2517         },
2518         {
2519             { -26280513, 11007847, 19408960, -940758, -18592965, -4328580,
2520                 -5088060, -11105150, 20470157, -16398701 },
2521             { -23136053, 9282192, 14855179, -15390078, -7362815, -14408560,
2522                 -22783952, 14461608, 14042978, 5230683 },
2523             { 29969567, -2741594, -16711867, -8552442, 9175486, -2468974,
2524                 21556951, 3506042, -5933891, -12449708 },
2525         },
2526         {
2527             { -3144746, 8744661, 19704003, 4581278, -20430686, 6830683,
2528                 -21284170, 8971513, -28539189, 15326563 },
2529             { -19464629, 10110288, -17262528, -3503892, -23500387, 1355669,
2530                 -15523050, 15300988, -20514118, 9168260 },
2531             { -5353335, 4488613, -23803248, 16314347, 7780487, -15638939,
2532                 -28948358, 9601605, 33087103, -9011387 },
2533         },
2534         {
2535             { -19443170, -15512900, -20797467, -12445323, -29824447, 10229461,
2536                 -27444329, -15000531, -5996870, 15664672 },
2537             { 23294591, -16632613, -22650781, -8470978, 27844204, 11461195,
2538                 13099750, -2460356, 18151676, 13417686 },
2539             { -24722913, -4176517, -31150679, 5988919, -26858785, 6685065,
2540                 1661597, -12551441, 15271676, -15452665 },
2541         },
2542     },
2543     {
2544         {
2545             { 11433042, -13228665, 8239631, -5279517, -1985436, -725718,
2546                 -18698764, 2167544, -6921301, -13440182 },
2547             { -31436171, 15575146, 30436815, 12192228, -22463353, 9395379,
2548                 -9917708, -8638997, 12215110, 12028277 },
2549             { 14098400, 6555944, 23007258, 5757252, -15427832, -12950502,
2550                 30123440, 4617780, -16900089, -655628 },
2551         },
2552         {
2553             { -4026201, -15240835, 11893168, 13718664, -14809462, 1847385,
2554                 -15819999, 10154009, 23973261, -12684474 },
2555             { -26531820, -3695990, -1908898, 2534301, -31870557, -16550355,
2556                 18341390, -11419951, 32013174, -10103539 },
2557             { -25479301, 10876443, -11771086, -14625140, -12369567, 1838104,
2558                 21911214, 6354752, 4425632, -837822 },
2559         },
2560         {
2561             { -10433389, -14612966, 22229858, -3091047, -13191166, 776729,
2562                 -17415375, -12020462, 4725005, 14044970 },
2563             { 19268650, -7304421, 1555349, 8692754, -21474059, -9910664, 6347390,
2564                 -1411784, -19522291, -16109756 },
2565             { -24864089, 12986008, -10898878, -5558584, -11312371, -148526,
2566                 19541418, 8180106, 9282262, 10282508 },
2567         },
2568         {
2569             { -26205082, 4428547, -8661196, -13194263, 4098402, -14165257,
2570                 15522535, 8372215, 5542595, -10702683 },
2571             { -10562541, 14895633, 26814552, -16673850, -17480754, -2489360,
2572                 -2781891, 6993761, -18093885, 10114655 },
2573             { -20107055, -929418, 31422704, 10427861, -7110749, 6150669,
2574                 -29091755, -11529146, 25953725, -106158 },
2575         },
2576         {
2577             { -4234397, -8039292, -9119125, 3046000, 2101609, -12607294,
2578                 19390020, 6094296, -3315279, 12831125 },
2579             { -15998678, 7578152, 5310217, 14408357, -33548620, -224739,
2580                 31575954, 6326196, 7381791, -2421839 },
2581             { -20902779, 3296811, 24736065, -16328389, 18374254, 7318640,
2582                 6295303, 8082724, -15362489, 12339664 },
2583         },
2584         {
2585             { 27724736, 2291157, 6088201, -14184798, 1792727, 5857634, 13848414,
2586                 15768922, 25091167, 14856294 },
2587             { -18866652, 8331043, 24373479, 8541013, -701998, -9269457, 12927300,
2588                 -12695493, -22182473, -9012899 },
2589             { -11423429, -5421590, 11632845, 3405020, 30536730, -11674039,
2590                 -27260765, 13866390, 30146206, 9142070 },
2591         },
2592         {
2593             { 3924129, -15307516, -13817122, -10054960, 12291820, -668366,
2594                 -27702774, 9326384, -8237858, 4171294 },
2595             { -15921940, 16037937, 6713787, 16606682, -21612135, 2790944,
2596                 26396185, 3731949, 345228, -5462949 },
2597             { -21327538, 13448259, 25284571, 1143661, 20614966, -8849387,
2598                 2031539, -12391231, -16253183, -13582083 },
2599         },
2600         {
2601             { 31016211, -16722429, 26371392, -14451233, -5027349, 14854137,
2602                 17477601, 3842657, 28012650, -16405420 },
2603             { -5075835, 9368966, -8562079, -4600902, -15249953, 6970560,
2604                 -9189873, 16292057, -8867157, 3507940 },
2605             { 29439664, 3537914, 23333589, 6997794, -17555561, -11018068,
2606                 -15209202, -15051267, -9164929, 6580396 },
2607         },
2608     },
2609     {
2610         {
2611             { -12185861, -7679788, 16438269, 10826160, -8696817, -6235611,
2612                 17860444, -9273846, -2095802, 9304567 },
2613             { 20714564, -4336911, 29088195, 7406487, 11426967, -5095705,
2614                 14792667, -14608617, 5289421, -477127 },
2615             { -16665533, -10650790, -6160345, -13305760, 9192020, -1802462,
2616                 17271490, 12349094, 26939669, -3752294 },
2617         },
2618         {
2619             { -12889898, 9373458, 31595848, 16374215, 21471720, 13221525,
2620                 -27283495, -12348559, -3698806, 117887 },
2621             { 22263325, -6560050, 3984570, -11174646, -15114008, -566785,
2622                 28311253, 5358056, -23319780, 541964 },
2623             { 16259219, 3261970, 2309254, -15534474, -16885711, -4581916,
2624                 24134070, -16705829, -13337066, -13552195 },
2625         },
2626         {
2627             { 9378160, -13140186, -22845982, -12745264, 28198281, -7244098,
2628                 -2399684, -717351, 690426, 14876244 },
2629             { 24977353, -314384, -8223969, -13465086, 28432343, -1176353,
2630                 -13068804, -12297348, -22380984, 6618999 },
2631             { -1538174, 11685646, 12944378, 13682314, -24389511, -14413193,
2632                 8044829, -13817328, 32239829, -5652762 },
2633         },
2634         {
2635             { -18603066, 4762990, -926250, 8885304, -28412480, -3187315, 9781647,
2636                 -10350059, 32779359, 5095274 },
2637             { -33008130, -5214506, -32264887, -3685216, 9460461, -9327423,
2638                 -24601656, 14506724, 21639561, -2630236 },
2639             { -16400943, -13112215, 25239338, 15531969, 3987758, -4499318,
2640                 -1289502, -6863535, 17874574, 558605 },
2641         },
2642         {
2643             { -13600129, 10240081, 9171883, 16131053, -20869254, 9599700,
2644                 33499487, 5080151, 2085892, 5119761 },
2645             { -22205145, -2519528, -16381601, 414691, -25019550, 2170430,
2646                 30634760, -8363614, -31999993, -5759884 },
2647             { -6845704, 15791202, 8550074, -1312654, 29928809, -12092256,
2648                 27534430, -7192145, -22351378, 12961482 },
2649         },
2650         {
2651             { -24492060, -9570771, 10368194, 11582341, -23397293, -2245287,
2652                 16533930, 8206996, -30194652, -5159638 },
2653             { -11121496, -3382234, 2307366, 6362031, -135455, 8868177, -16835630,
2654                 7031275, 7589640, 8945490 },
2655             { -32152748, 8917967, 6661220, -11677616, -1192060, -15793393,
2656                 7251489, -11182180, 24099109, -14456170 },
2657         },
2658         {
2659             { 5019558, -7907470, 4244127, -14714356, -26933272, 6453165,
2660                 -19118182, -13289025, -6231896, -10280736 },
2661             { 10853594, 10721687, 26480089, 5861829, -22995819, 1972175,
2662                 -1866647, -10557898, -3363451, -6441124 },
2663             { -17002408, 5906790, 221599, -6563147, 7828208, -13248918, 24362661,
2664                 -2008168, -13866408, 7421392 },
2665         },
2666         {
2667             { 8139927, -6546497, 32257646, -5890546, 30375719, 1886181,
2668                 -21175108, 15441252, 28826358, -4123029 },
2669             { 6267086, 9695052, 7709135, -16603597, -32869068, -1886135,
2670                 14795160, -7840124, 13746021, -1742048 },
2671             { 28584902, 7787108, -6732942, -15050729, 22846041, -7571236,
2672                 -3181936, -363524, 4771362, -8419958 },
2673         },
2674     },
2675     {
2676         {
2677             { 24949256, 6376279, -27466481, -8174608, -18646154, -9930606,
2678                 33543569, -12141695, 3569627, 11342593 },
2679             { 26514989, 4740088, 27912651, 3697550, 19331575, -11472339, 6809886,
2680                 4608608, 7325975, -14801071 },
2681             { -11618399, -14554430, -24321212, 7655128, -1369274, 5214312,
2682                 -27400540, 10258390, -17646694, -8186692 },
2683         },
2684         {
2685             { 11431204, 15823007, 26570245, 14329124, 18029990, 4796082,
2686                 -31446179, 15580664, 9280358, -3973687 },
2687             { -160783, -10326257, -22855316, -4304997, -20861367, -13621002,
2688                 -32810901, -11181622, -15545091, 4387441 },
2689             { -20799378, 12194512, 3937617, -5805892, -27154820, 9340370,
2690                 -24513992, 8548137, 20617071, -7482001 },
2691         },
2692         {
2693             { -938825, -3930586, -8714311, 16124718, 24603125, -6225393,
2694                 -13775352, -11875822, 24345683, 10325460 },
2695             { -19855277, -1568885, -22202708, 8714034, 14007766, 6928528,
2696                 16318175, -1010689, 4766743, 3552007 },
2697             { -21751364, -16730916, 1351763, -803421, -4009670, 3950935, 3217514,
2698                 14481909, 10988822, -3994762 },
2699         },
2700         {
2701             { 15564307, -14311570, 3101243, 5684148, 30446780, -8051356,
2702                 12677127, -6505343, -8295852, 13296005 },
2703             { -9442290, 6624296, -30298964, -11913677, -4670981, -2057379,
2704                 31521204, 9614054, -30000824, 12074674 },
2705             { 4771191, -135239, 14290749, -13089852, 27992298, 14998318,
2706                 -1413936, -1556716, 29832613, -16391035 },
2707         },
2708         {
2709             { 7064884, -7541174, -19161962, -5067537, -18891269, -2912736,
2710                 25825242, 5293297, -27122660, 13101590 },
2711             { -2298563, 2439670, -7466610, 1719965, -27267541, -16328445,
2712                 32512469, -5317593, -30356070, -4190957 },
2713             { -30006540, 10162316, -33180176, 3981723, -16482138, -13070044,
2714                 14413974, 9515896, 19568978, 9628812 },
2715         },
2716         {
2717             { 33053803, 199357, 15894591, 1583059, 27380243, -4580435, -17838894,
2718                 -6106839, -6291786, 3437740 },
2719             { -18978877, 3884493, 19469877, 12726490, 15913552, 13614290,
2720                 -22961733, 70104, 7463304, 4176122 },
2721             { -27124001, 10659917, 11482427, -16070381, 12771467, -6635117,
2722                 -32719404, -5322751, 24216882, 5944158 },
2723         },
2724         {
2725             { 8894125, 7450974, -2664149, -9765752, -28080517, -12389115,
2726                 19345746, 14680796, 11632993, 5847885 },
2727             { 26942781, -2315317, 9129564, -4906607, 26024105, 11769399,
2728                 -11518837, 6367194, -9727230, 4782140 },
2729             { 19916461, -4828410, -22910704, -11414391, 25606324, -5972441,
2730                 33253853, 8220911, 6358847, -1873857 },
2731         },
2732         {
2733             { 801428, -2081702, 16569428, 11065167, 29875704, 96627, 7908388,
2734                 -4480480, -13538503, 1387155 },
2735             { 19646058, 5720633, -11416706, 12814209, 11607948, 12749789,
2736                 14147075, 15156355, -21866831, 11835260 },
2737             { 19299512, 1155910, 28703737, 14890794, 2925026, 7269399, 26121523,
2738                 15467869, -26560550, 5052483 },
2739         },
2740     },
2741     {
2742         {
2743             { -3017432, 10058206, 1980837, 3964243, 22160966, 12322533, -6431123,
2744                 -12618185, 12228557, -7003677 },
2745             { 32944382, 14922211, -22844894, 5188528, 21913450, -8719943,
2746                 4001465, 13238564, -6114803, 8653815 },
2747             { 22865569, -4652735, 27603668, -12545395, 14348958, 8234005,
2748                 24808405, 5719875, 28483275, 2841751 },
2749         },
2750         {
2751             { -16420968, -1113305, -327719, -12107856, 21886282, -15552774,
2752                 -1887966, -315658, 19932058, -12739203 },
2753             { -11656086, 10087521, -8864888, -5536143, -19278573, -3055912,
2754                 3999228, 13239134, -4777469, -13910208 },
2755             { 1382174, -11694719, 17266790, 9194690, -13324356, 9720081,
2756                 20403944, 11284705, -14013818, 3093230 },
2757         },
2758         {
2759             { 16650921, -11037932, -1064178, 1570629, -8329746, 7352753, -302424,
2760                 16271225, -24049421, -6691850 },
2761             { -21911077, -5927941, -4611316, -5560156, -31744103, -10785293,
2762                 24123614, 15193618, -21652117, -16739389 },
2763             { -9935934, -4289447, -25279823, 4372842, 2087473, 10399484,
2764                 31870908, 14690798, 17361620, 11864968 },
2765         },
2766         {
2767             { -11307610, 6210372, 13206574, 5806320, -29017692, -13967200,
2768                 -12331205, -7486601, -25578460, -16240689 },
2769             { 14668462, -12270235, 26039039, 15305210, 25515617, 4542480,
2770                 10453892, 6577524, 9145645, -6443880 },
2771             { 5974874, 3053895, -9433049, -10385191, -31865124, 3225009,
2772                 -7972642, 3936128, -5652273, -3050304 },
2773         },
2774         {
2775             { 30625386, -4729400, -25555961, -12792866, -20484575, 7695099,
2776                 17097188, -16303496, -27999779, 1803632 },
2777             { -3553091, 9865099, -5228566, 4272701, -5673832, -16689700,
2778                 14911344, 12196514, -21405489, 7047412 },
2779             { 20093277, 9920966, -11138194, -5343857, 13161587, 12044805,
2780                 -32856851, 4124601, -32343828, -10257566 },
2781         },
2782         {
2783             { -20788824, 14084654, -13531713, 7842147, 19119038, -13822605,
2784                 4752377, -8714640, -21679658, 2288038 },
2785             { -26819236, -3283715, 29965059, 3039786, -14473765, 2540457,
2786                 29457502, 14625692, -24819617, 12570232 },
2787             { -1063558, -11551823, 16920318, 12494842, 1278292, -5869109,
2788                 -21159943, -3498680, -11974704, 4724943 },
2789         },
2790         {
2791             { 17960970, -11775534, -4140968, -9702530, -8876562, -1410617,
2792                 -12907383, -8659932, -29576300, 1903856 },
2793             { 23134274, -14279132, -10681997, -1611936, 20684485, 15770816,
2794                 -12989750, 3190296, 26955097, 14109738 },
2795             { 15308788, 5320727, -30113809, -14318877, 22902008, 7767164,
2796                 29425325, -11277562, 31960942, 11934971 },
2797         },
2798         {
2799             { -27395711, 8435796, 4109644, 12222639, -24627868, 14818669,
2800                 20638173, 4875028, 10491392, 1379718 },
2801             { -13159415, 9197841, 3875503, -8936108, -1383712, -5879801,
2802                 33518459, 16176658, 21432314, 12180697 },
2803             { -11787308, 11500838, 13787581, -13832590, -22430679, 10140205,
2804                 1465425, 12689540, -10301319, -13872883 },
2805         },
2806     },
2807     {
2808         {
2809             { 5414091, -15386041, -21007664, 9643570, 12834970, 1186149,
2810                 -2622916, -1342231, 26128231, 6032912 },
2811             { -26337395, -13766162, 32496025, -13653919, 17847801, -12669156,
2812                 3604025, 8316894, -25875034, -10437358 },
2813             { 3296484, 6223048, 24680646, -12246460, -23052020, 5903205,
2814                 -8862297, -4639164, 12376617, 3188849 },
2815         },
2816         {
2817             { 29190488, -14659046, 27549113, -1183516, 3520066, -10697301,
2818                 32049515, -7309113, -16109234, -9852307 },
2819             { -14744486, -9309156, 735818, -598978, -20407687, -5057904,
2820                 25246078, -15795669, 18640741, -960977 },
2821             { -6928835, -16430795, 10361374, 5642961, 4910474, 12345252,
2822                 -31638386, -494430, 10530747, 1053335 },
2823         },
2824         {
2825             { -29265967, -14186805, -13538216, -12117373, -19457059, -10655384,
2826                 -31462369, -2948985, 24018831, 15026644 },
2827             { -22592535, -3145277, -2289276, 5953843, -13440189, 9425631,
2828                 25310643, 13003497, -2314791, -15145616 },
2829             { -27419985, -603321, -8043984, -1669117, -26092265, 13987819,
2830                 -27297622, 187899, -23166419, -2531735 },
2831         },
2832         {
2833             { -21744398, -13810475, 1844840, 5021428, -10434399, -15911473,
2834                 9716667, 16266922, -5070217, 726099 },
2835             { 29370922, -6053998, 7334071, -15342259, 9385287, 2247707,
2836                 -13661962, -4839461, 30007388, -15823341 },
2837             { -936379, 16086691, 23751945, -543318, -1167538, -5189036, 9137109,
2838                 730663, 9835848, 4555336 },
2839         },
2840         {
2841             { -23376435, 1410446, -22253753, -12899614, 30867635, 15826977,
2842                 17693930, 544696, -11985298, 12422646 },
2843             { 31117226, -12215734, -13502838, 6561947, -9876867, -12757670,
2844                 -5118685, -4096706, 29120153, 13924425 },
2845             { -17400879, -14233209, 19675799, -2734756, -11006962, -5858820,
2846                 -9383939, -11317700, 7240931, -237388 },
2847         },
2848         {
2849             { -31361739, -11346780, -15007447, -5856218, -22453340, -12152771,
2850                 1222336, 4389483, 3293637, -15551743 },
2851             { -16684801, -14444245, 11038544, 11054958, -13801175, -3338533,
2852                 -24319580, 7733547, 12796905, -6335822 },
2853             { -8759414, -10817836, -25418864, 10783769, -30615557, -9746811,
2854                 -28253339, 3647836, 3222231, -11160462 },
2855         },
2856         {
2857             { 18606113, 1693100, -25448386, -15170272, 4112353, 10045021,
2858                 23603893, -2048234, -7550776, 2484985 },
2859             { 9255317, -3131197, -12156162, -1004256, 13098013, -9214866,
2860                 16377220, -2102812, -19802075, -3034702 },
2861             { -22729289, 7496160, -5742199, 11329249, 19991973, -3347502,
2862                 -31718148, 9936966, -30097688, -10618797 },
2863         },
2864         {
2865             { 21878590, -5001297, 4338336, 13643897, -3036865, 13160960,
2866                 19708896, 5415497, -7360503, -4109293 },
2867             { 27736861, 10103576, 12500508, 8502413, -3413016, -9633558,
2868                 10436918, -1550276, -23659143, -8132100 },
2869             { 19492550, -12104365, -29681976, -852630, -3208171, 12403437,
2870                 30066266, 8367329, 13243957, 8709688 },
2871         },
2872     },
2873     {
2874         {
2875             { 12015105, 2801261, 28198131, 10151021, 24818120, -4743133,
2876                 -11194191, -5645734, 5150968, 7274186 },
2877             { 2831366, -12492146, 1478975, 6122054, 23825128, -12733586,
2878                 31097299, 6083058, 31021603, -9793610 },
2879             { -2529932, -2229646, 445613, 10720828, -13849527, -11505937,
2880                 -23507731, 16354465, 15067285, -14147707 },
2881         },
2882         {
2883             { 7840942, 14037873, -33364863, 15934016, -728213, -3642706,
2884                 21403988, 1057586, -19379462, -12403220 },
2885             { 915865, -16469274, 15608285, -8789130, -24357026, 6060030,
2886                 -17371319, 8410997, -7220461, 16527025 },
2887             { 32922597, -556987, 20336074, -16184568, 10903705, -5384487,
2888                 16957574, 52992, 23834301, 6588044 },
2889         },
2890         {
2891             { 32752030, 11232950, 3381995, -8714866, 22652988, -10744103,
2892                 17159699, 16689107, -20314580, -1305992 },
2893             { -4689649, 9166776, -25710296, -10847306, 11576752, 12733943,
2894                 7924251, -2752281, 1976123, -7249027 },
2895             { 21251222, 16309901, -2983015, -6783122, 30810597, 12967303, 156041,
2896                 -3371252, 12331345, -8237197 },
2897         },
2898         {
2899             { 8651614, -4477032, -16085636, -4996994, 13002507, 2950805,
2900                 29054427, -5106970, 10008136, -4667901 },
2901             { 31486080, 15114593, -14261250, 12951354, 14369431, -7387845,
2902                 16347321, -13662089, 8684155, -10532952 },
2903             { 19443825, 11385320, 24468943, -9659068, -23919258, 2187569,
2904                 -26263207, -6086921, 31316348, 14219878 },
2905         },
2906         {
2907             { -28594490, 1193785, 32245219, 11392485, 31092169, 15722801,
2908                 27146014, 6992409, 29126555, 9207390 },
2909             { 32382935, 1110093, 18477781, 11028262, -27411763, -7548111,
2910                 -4980517, 10843782, -7957600, -14435730 },
2911             { 2814918, 7836403, 27519878, -7868156, -20894015, -11553689,
2912                 -21494559, 8550130, 28346258, 1994730 },
2913         },
2914         {
2915             { -19578299, 8085545, -14000519, -3948622, 2785838, -16231307,
2916                 -19516951, 7174894, 22628102, 8115180 },
2917             { -30405132, 955511, -11133838, -15078069, -32447087, -13278079,
2918                 -25651578, 3317160, -9943017, 930272 },
2919             { -15303681, -6833769, 28856490, 1357446, 23421993, 1057177,
2920                 24091212, -1388970, -22765376, -10650715 },
2921         },
2922         {
2923             { -22751231, -5303997, -12907607, -12768866, -15811511, -7797053,
2924                 -14839018, -16554220, -1867018, 8398970 },
2925             { -31969310, 2106403, -4736360, 1362501, 12813763, 16200670,
2926                 22981545, -6291273, 18009408, -15772772 },
2927             { -17220923, -9545221, -27784654, 14166835, 29815394, 7444469,
2928                 29551787, -3727419, 19288549, 1325865 },
2929         },
2930         {
2931             { 15100157, -15835752, -23923978, -1005098, -26450192, 15509408,
2932                 12376730, -3479146, 33166107, -8042750 },
2933             { 20909231, 13023121, -9209752, 16251778, -5778415, -8094914,
2934                 12412151, 10018715, 2213263, -13878373 },
2935             { 32529814, -11074689, 30361439, -16689753, -9135940, 1513226,
2936                 22922121, 6382134, -5766928, 8371348 },
2937         },
2938     },
2939     {
2940         {
2941             { 9923462, 11271500, 12616794, 3544722, -29998368, -1721626,
2942                 12891687, -8193132, -26442943, 10486144 },
2943             { -22597207, -7012665, 8587003, -8257861, 4084309, -12970062, 361726,
2944                 2610596, -23921530, -11455195 },
2945             { 5408411, -1136691, -4969122, 10561668, 24145918, 14240566,
2946                 31319731, -4235541, 19985175, -3436086 },
2947         },
2948         {
2949             { -13994457, 16616821, 14549246, 3341099, 32155958, 13648976,
2950                 -17577068, 8849297, 65030, 8370684 },
2951             { -8320926, -12049626, 31204563, 5839400, -20627288, -1057277,
2952                 -19442942, 6922164, 12743482, -9800518 },
2953             { -2361371, 12678785, 28815050, 4759974, -23893047, 4884717,
2954                 23783145, 11038569, 18800704, 255233 },
2955         },
2956         {
2957             { -5269658, -1773886, 13957886, 7990715, 23132995, 728773, 13393847,
2958                 9066957, 19258688, -14753793 },
2959             { -2936654, -10827535, -10432089, 14516793, -3640786, 4372541,
2960                 -31934921, 2209390, -1524053, 2055794 },
2961             { 580882, 16705327, 5468415, -2683018, -30926419, -14696000,
2962                 -7203346, -8994389, -30021019, 7394435 },
2963         },
2964         {
2965             { 23838809, 1822728, -15738443, 15242727, 8318092, -3733104,
2966                 -21672180, -3492205, -4821741, 14799921 },
2967             { 13345610, 9759151, 3371034, -16137791, 16353039, 8577942, 31129804,
2968                 13496856, -9056018, 7402518 },
2969             { 2286874, -4435931, -20042458, -2008336, -13696227, 5038122,
2970                 11006906, -15760352, 8205061, 1607563 },
2971         },
2972         {
2973             { 14414086, -8002132, 3331830, -3208217, 22249151, -5594188,
2974                 18364661, -2906958, 30019587, -9029278 },
2975             { -27688051, 1585953, -10775053, 931069, -29120221, -11002319,
2976                 -14410829, 12029093, 9944378, 8024 },
2977             { 4368715, -3709630, 29874200, -15022983, -20230386, -11410704,
2978                 -16114594, -999085, -8142388, 5640030 },
2979         },
2980         {
2981             { 10299610, 13746483, 11661824, 16234854, 7630238, 5998374, 9809887,
2982                 -16694564, 15219798, -14327783 },
2983             { 27425505, -5719081, 3055006, 10660664, 23458024, 595578, -15398605,
2984                 -1173195, -18342183, 9742717 },
2985             { 6744077, 2427284, 26042789, 2720740, -847906, 1118974, 32324614,
2986                 7406442, 12420155, 1994844 },
2987         },
2988         {
2989             { 14012521, -5024720, -18384453, -9578469, -26485342, -3936439,
2990                 -13033478, -10909803, 24319929, -6446333 },
2991             { 16412690, -4507367, 10772641, 15929391, -17068788, -4658621,
2992                 10555945, -10484049, -30102368, -4739048 },
2993             { 22397382, -7767684, -9293161, -12792868, 17166287, -9755136,
2994                 -27333065, 6199366, 21880021, -12250760 },
2995         },
2996         {
2997             { -4283307, 5368523, -31117018, 8163389, -30323063, 3209128,
2998                 16557151, 8890729, 8840445, 4957760 },
2999             { -15447727, 709327, -6919446, -10870178, -29777922, 6522332,
3000                 -21720181, 12130072, -14796503, 5005757 },
3001             { -2114751, -14308128, 23019042, 15765735, -25269683, 6002752,
3002                 10183197, -13239326, -16395286, -2176112 },
3003         },
3004     },
3005     {
3006         {
3007             { -19025756, 1632005, 13466291, -7995100, -23640451, 16573537,
3008                 -32013908, -3057104, 22208662, 2000468 },
3009             { 3065073, -1412761, -25598674, -361432, -17683065, -5703415,
3010                 -8164212, 11248527, -3691214, -7414184 },
3011             { 10379208, -6045554, 8877319, 1473647, -29291284, -12507580,
3012                 16690915, 2553332, -3132688, 16400289 },
3013         },
3014         {
3015             { 15716668, 1254266, -18472690, 7446274, -8448918, 6344164,
3016                 -22097271, -7285580, 26894937, 9132066 },
3017             { 24158887, 12938817, 11085297, -8177598, -28063478, -4457083,
3018                 -30576463, 64452, -6817084, -2692882 },
3019             { 13488534, 7794716, 22236231, 5989356, 25426474, -12578208, 2350710,
3020                 -3418511, -4688006, 2364226 },
3021         },
3022         {
3023             { 16335052, 9132434, 25640582, 6678888, 1725628, 8517937, -11807024,
3024                 -11697457, 15445875, -7798101 },
3025             { 29004207, -7867081, 28661402, -640412, -12794003, -7943086,
3026                 31863255, -4135540, -278050, -15759279 },
3027             { -6122061, -14866665, -28614905, 14569919, -10857999, -3591829,
3028                 10343412, -6976290, -29828287, -10815811 },
3029         },
3030         {
3031             { 27081650, 3463984, 14099042, -4517604, 1616303, -6205604, 29542636,
3032                 15372179, 17293797, 960709 },
3033             { 20263915, 11434237, -5765435, 11236810, 13505955, -10857102,
3034                 -16111345, 6493122, -19384511, 7639714 },
3035             { -2830798, -14839232, 25403038, -8215196, -8317012, -16173699,
3036                 18006287, -16043750, 29994677, -15808121 },
3037         },
3038         {
3039             { 9769828, 5202651, -24157398, -13631392, -28051003, -11561624,
3040                 -24613141, -13860782, -31184575, 709464 },
3041             { 12286395, 13076066, -21775189, -1176622, -25003198, 4057652,
3042                 -32018128, -8890874, 16102007, 13205847 },
3043             { 13733362, 5599946, 10557076, 3195751, -5557991, 8536970, -25540170,
3044                 8525972, 10151379, 10394400 },
3045         },
3046         {
3047             { 4024660, -16137551, 22436262, 12276534, -9099015, -2686099,
3048                 19698229, 11743039, -33302334, 8934414 },
3049             { -15879800, -4525240, -8580747, -2934061, 14634845, -698278,
3050                 -9449077, 3137094, -11536886, 11721158 },
3051             { 17555939, -5013938, 8268606, 2331751, -22738815, 9761013, 9319229,
3052                 8835153, -9205489, -1280045 },
3053         },
3054         {
3055             { -461409, -7830014, 20614118, 16688288, -7514766, -4807119,
3056                 22300304, 505429, 6108462, -6183415 },
3057             { -5070281, 12367917, -30663534, 3234473, 32617080, -8422642,
3058                 29880583, -13483331, -26898490, -7867459 },
3059             { -31975283, 5726539, 26934134, 10237677, -3173717, -605053,
3060                 24199304, 3795095, 7592688, -14992079 },
3061         },
3062         {
3063             { 21594432, -14964228, 17466408, -4077222, 32537084, 2739898,
3064                 6407723, 12018833, -28256052, 4298412 },
3065             { -20650503, -11961496, -27236275, 570498, 3767144, -1717540,
3066                 13891942, -1569194, 13717174, 10805743 },
3067             { -14676630, -15644296, 15287174, 11927123, 24177847, -8175568,
3068                 -796431, 14860609, -26938930, -5863836 },
3069         },
3070     },
3071     {
3072         {
3073             { 12962541, 5311799, -10060768, 11658280, 18855286, -7954201,
3074                 13286263, -12808704, -4381056, 9882022 },
3075             { 18512079, 11319350, -20123124, 15090309, 18818594, 5271736,
3076                 -22727904, 3666879, -23967430, -3299429 },
3077             { -6789020, -3146043, 16192429, 13241070, 15898607, -14206114,
3078                 -10084880, -6661110, -2403099, 5276065 },
3079         },
3080         {
3081             { 30169808, -5317648, 26306206, -11750859, 27814964, 7069267,
3082                 7152851, 3684982, 1449224, 13082861 },
3083             { 10342826, 3098505, 2119311, 193222, 25702612, 12233820, 23697382,
3084                 15056736, -21016438, -8202000 },
3085             { -33150110, 3261608, 22745853, 7948688, 19370557, -15177665,
3086                 -26171976, 6482814, -10300080, -11060101 },
3087         },
3088         {
3089             { 32869458, -5408545, 25609743, 15678670, -10687769, -15471071,
3090                 26112421, 2521008, -22664288, 6904815 },
3091             { 29506923, 4457497, 3377935, -9796444, -30510046, 12935080, 1561737,
3092                 3841096, -29003639, -6657642 },
3093             { 10340844, -6630377, -18656632, -2278430, 12621151, -13339055,
3094                 30878497, -11824370, -25584551, 5181966 },
3095         },
3096         {
3097             { 25940115, -12658025, 17324188, -10307374, -8671468, 15029094,
3098                 24396252, -16450922, -2322852, -12388574 },
3099             { -21765684, 9916823, -1300409, 4079498, -1028346, 11909559, 1782390,
3100                 12641087, 20603771, -6561742 },
3101             { -18882287, -11673380, 24849422, 11501709, 13161720, -4768874,
3102                 1925523, 11914390, 4662781, 7820689 },
3103         },
3104         {
3105             { 12241050, -425982, 8132691, 9393934, 32846760, -1599620, 29749456,
3106                 12172924, 16136752, 15264020 },
3107             { -10349955, -14680563, -8211979, 2330220, -17662549, -14545780,
3108                 10658213, 6671822, 19012087, 3772772 },
3109             { 3753511, -3421066, 10617074, 2028709, 14841030, -6721664, 28718732,
3110                 -15762884, 20527771, 12988982 },
3111         },
3112         {
3113             { -14822485, -5797269, -3707987, 12689773, -898983, -10914866,
3114                 -24183046, -10564943, 3299665, -12424953 },
3115             { -16777703, -15253301, -9642417, 4978983, 3308785, 8755439, 6943197,
3116                 6461331, -25583147, 8991218 },
3117             { -17226263, 1816362, -1673288, -6086439, 31783888, -8175991,
3118                 -32948145, 7417950, -30242287, 1507265 },
3119         },
3120         {
3121             { 29692663, 6829891, -10498800, 4334896, 20945975, -11906496,
3122                 -28887608, 8209391, 14606362, -10647073 },
3123             { -3481570, 8707081, 32188102, 5672294, 22096700, 1711240, -33020695,
3124                 9761487, 4170404, -2085325 },
3125             { -11587470, 14855945, -4127778, -1531857, -26649089, 15084046,
3126                 22186522, 16002000, -14276837, -8400798 },
3127         },
3128         {
3129             { -4811456, 13761029, -31703877, -2483919, -3312471, 7869047,
3130                 -7113572, -9620092, 13240845, 10965870 },
3131             { -7742563, -8256762, -14768334, -13656260, -23232383, 12387166,
3132                 4498947, 14147411, 29514390, 4302863 },
3133             { -13413405, -12407859, 20757302, -13801832, 14785143, 8976368,
3134                 -5061276, -2144373, 17846988, -13971927 },
3135         },
3136     },
3137     {
3138         {
3139             { -2244452, -754728, -4597030, -1066309, -6247172, 1455299,
3140                 -21647728, -9214789, -5222701, 12650267 },
3141             { -9906797, -16070310, 21134160, 12198166, -27064575, 708126, 387813,
3142                 13770293, -19134326, 10958663 },
3143             { 22470984, 12369526, 23446014, -5441109, -21520802, -9698723,
3144                 -11772496, -11574455, -25083830, 4271862 },
3145         },
3146         {
3147             { -25169565, -10053642, -19909332, 15361595, -5984358, 2159192,
3148                 75375, -4278529, -32526221, 8469673 },
3149             { 15854970, 4148314, -8893890, 7259002, 11666551, 13824734,
3150                 -30531198, 2697372, 24154791, -9460943 },
3151             { 15446137, -15806644, 29759747, 14019369, 30811221, -9610191,
3152                 -31582008, 12840104, 24913809, 9815020 },
3153         },
3154         {
3155             { -4709286, -5614269, -31841498, -12288893, -14443537, 10799414,
3156                 -9103676, 13438769, 18735128, 9466238 },
3157             { 11933045, 9281483, 5081055, -5183824, -2628162, -4905629, -7727821,
3158                 -10896103, -22728655, 16199064 },
3159             { 14576810, 379472, -26786533, -8317236, -29426508, -10812974,
3160                 -102766, 1876699, 30801119, 2164795 },
3161         },
3162         {
3163             { 15995086, 3199873, 13672555, 13712240, -19378835, -4647646,
3164                 -13081610, -15496269, -13492807, 1268052 },
3165             { -10290614, -3659039, -3286592, 10948818, 23037027, 3794475,
3166                 -3470338, -12600221, -17055369, 3565904 },
3167             { 29210088, -9419337, -5919792, -4952785, 10834811, -13327726,
3168                 -16512102, -10820713, -27162222, -14030531 },
3169         },
3170         {
3171             { -13161890, 15508588, 16663704, -8156150, -28349942, 9019123,
3172                 -29183421, -3769423, 2244111, -14001979 },
3173             { -5152875, -3800936, -9306475, -6071583, 16243069, 14684434,
3174                 -25673088, -16180800, 13491506, 4641841 },
3175             { 10813417, 643330, -19188515, -728916, 30292062, -16600078,
3176                 27548447, -7721242, 14476989, -12767431 },
3177         },
3178         {
3179             { 10292079, 9984945, 6481436, 8279905, -7251514, 7032743, 27282937,
3180                 -1644259, -27912810, 12651324 },
3181             { -31185513, -813383, 22271204, 11835308, 10201545, 15351028,
3182                 17099662, 3988035, 21721536, -3148940 },
3183             { 10202177, -6545839, -31373232, -9574638, -32150642, -8119683,
3184                 -12906320, 3852694, 13216206, 14842320 },
3185         },
3186         {
3187             { -15815640, -10601066, -6538952, -7258995, -6984659, -6581778,
3188                 -31500847, 13765824, -27434397, 9900184 },
3189             { 14465505, -13833331, -32133984, -14738873, -27443187, 12990492,
3190                 33046193, 15796406, -7051866, -8040114 },
3191             { 30924417, -8279620, 6359016, -12816335, 16508377, 9071735,
3192                 -25488601, 15413635, 9524356, -7018878 },
3193         },
3194         {
3195             { 12274201, -13175547, 32627641, -1785326, 6736625, 13267305,
3196                 5237659, -5109483, 15663516, 4035784 },
3197             { -2951309, 8903985, 17349946, 601635, -16432815, -4612556,
3198                 -13732739, -15889334, -22258478, 4659091 },
3199             { -16916263, -4952973, -30393711, -15158821, 20774812, 15897498,
3200                 5736189, 15026997, -2178256, -13455585 },
3201         },
3202     },
3203     {
3204         {
3205             { -8858980, -2219056, 28571666, -10155518, -474467, -10105698,
3206                 -3801496, 278095, 23440562, -290208 },
3207             { 10226241, -5928702, 15139956, 120818, -14867693, 5218603, 32937275,
3208                 11551483, -16571960, -7442864 },
3209             { 17932739, -12437276, -24039557, 10749060, 11316803, 7535897,
3210                 22503767, 5561594, -3646624, 3898661 },
3211         },
3212         {
3213             { 7749907, -969567, -16339731, -16464, -25018111, 15122143, -1573531,
3214                 7152530, 21831162, 1245233 },
3215             { 26958459, -14658026, 4314586, 8346991, -5677764, 11960072,
3216                 -32589295, -620035, -30402091, -16716212 },
3217             { -12165896, 9166947, 33491384, 13673479, 29787085, 13096535,
3218                 6280834, 14587357, -22338025, 13987525 },
3219         },
3220         {
3221             { -24349909, 7778775, 21116000, 15572597, -4833266, -5357778,
3222                 -4300898, -5124639, -7469781, -2858068 },
3223             { 9681908, -6737123, -31951644, 13591838, -6883821, 386950, 31622781,
3224                 6439245, -14581012, 4091397 },
3225             { -8426427, 1470727, -28109679, -1596990, 3978627, -5123623,
3226                 -19622683, 12092163, 29077877, -14741988 },
3227         },
3228         {
3229             { 5269168, -6859726, -13230211, -8020715, 25932563, 1763552,
3230                 -5606110, -5505881, -20017847, 2357889 },
3231             { 32264008, -15407652, -5387735, -1160093, -2091322, -3946900,
3232                 23104804, -12869908, 5727338, 189038 },
3233             { 14609123, -8954470, -6000566, -16622781, -14577387, -7743898,
3234                 -26745169, 10942115, -25888931, -14884697 },
3235         },
3236         {
3237             { 20513500, 5557931, -15604613, 7829531, 26413943, -2019404,
3238                 -21378968, 7471781, 13913677, -5137875 },
3239             { -25574376, 11967826, 29233242, 12948236, -6754465, 4713227,
3240                 -8940970, 14059180, 12878652, 8511905 },
3241             { -25656801, 3393631, -2955415, -7075526, -2250709, 9366908,
3242                 -30223418, 6812974, 5568676, -3127656 },
3243         },
3244         {
3245             { 11630004, 12144454, 2116339, 13606037, 27378885, 15676917,
3246                 -17408753, -13504373, -14395196, 8070818 },
3247             { 27117696, -10007378, -31282771, -5570088, 1127282, 12772488,
3248                 -29845906, 10483306, -11552749, -1028714 },
3249             { 10637467, -5688064, 5674781, 1072708, -26343588, -6982302,
3250                 -1683975, 9177853, -27493162, 15431203 },
3251         },
3252         {
3253             { 20525145, 10892566, -12742472, 12779443, -29493034, 16150075,
3254                 -28240519, 14943142, -15056790, -7935931 },
3255             { -30024462, 5626926, -551567, -9981087, 753598, 11981191, 25244767,
3256                 -3239766, -3356550, 9594024 },
3257             { -23752644, 2636870, -5163910, -10103818, 585134, 7877383, 11345683,
3258                 -6492290, 13352335, -10977084 },
3259         },
3260         {
3261             { -1931799, -5407458, 3304649, -12884869, 17015806, -4877091,
3262                 -29783850, -7752482, -13215537, -319204 },
3263             { 20239939, 6607058, 6203985, 3483793, -18386976, -779229, -20723742,
3264                 15077870, -22750759, 14523817 },
3265             { 27406042, -6041657, 27423596, -4497394, 4996214, 10002360,
3266                 -28842031, -4545494, -30172742, -4805667 },
3267         },
3268     },
3269     {
3270         {
3271             { 11374242, 12660715, 17861383, -12540833, 10935568, 1099227,
3272                 -13886076, -9091740, -27727044, 11358504 },
3273             { -12730809, 10311867, 1510375, 10778093, -2119455, -9145702,
3274                 32676003, 11149336, -26123651, 4985768 },
3275             { -19096303, 341147, -6197485, -239033, 15756973, -8796662, -983043,
3276                 13794114, -19414307, -15621255 },
3277         },
3278         {
3279             { 6490081, 11940286, 25495923, -7726360, 8668373, -8751316, 3367603,
3280                 6970005, -1691065, -9004790 },
3281             { 1656497, 13457317, 15370807, 6364910, 13605745, 8362338, -19174622,
3282                 -5475723, -16796596, -5031438 },
3283             { -22273315, -13524424, -64685, -4334223, -18605636, -10921968,
3284                 -20571065, -7007978, -99853, -10237333 },
3285         },
3286         {
3287             { 17747465, 10039260, 19368299, -4050591, -20630635, -16041286,
3288                 31992683, -15857976, -29260363, -5511971 },
3289             { 31932027, -4986141, -19612382, 16366580, 22023614, 88450, 11371999,
3290                 -3744247, 4882242, -10626905 },
3291             { 29796507, 37186, 19818052, 10115756, -11829032, 3352736, 18551198,
3292                 3272828, -5190932, -4162409 },
3293         },
3294         {
3295             { 12501286, 4044383, -8612957, -13392385, -32430052, 5136599,
3296                 -19230378, -3529697, 330070, -3659409 },
3297             { 6384877, 2899513, 17807477, 7663917, -2358888, 12363165, 25366522,
3298                 -8573892, -271295, 12071499 },
3299             { -8365515, -4042521, 25133448, -4517355, -6211027, 2265927,
3300                 -32769618, 1936675, -5159697, 3829363 },
3301         },
3302         {
3303             { 28425966, -5835433, -577090, -4697198, -14217555, 6870930, 7921550,
3304                 -6567787, 26333140, 14267664 },
3305             { -11067219, 11871231, 27385719, -10559544, -4585914, -11189312,
3306                 10004786, -8709488, -21761224, 8930324 },
3307             { -21197785, -16396035, 25654216, -1725397, 12282012, 11008919,
3308                 1541940, 4757911, -26491501, -16408940 },
3309         },
3310         {
3311             { 13537262, -7759490, -20604840, 10961927, -5922820, -13218065,
3312                 -13156584, 6217254, -15943699, 13814990 },
3313             { -17422573, 15157790, 18705543, 29619, 24409717, -260476, 27361681,
3314                 9257833, -1956526, -1776914 },
3315             { -25045300, -10191966, 15366585, 15166509, -13105086, 8423556,
3316                 -29171540, 12361135, -18685978, 4578290 },
3317         },
3318         {
3319             { 24579768, 3711570, 1342322, -11180126, -27005135, 14124956,
3320                 -22544529, 14074919, 21964432, 8235257 },
3321             { -6528613, -2411497, 9442966, -5925588, 12025640, -1487420,
3322                 -2981514, -1669206, 13006806, 2355433 },
3323             { -16304899, -13605259, -6632427, -5142349, 16974359, -10911083,
3324                 27202044, 1719366, 1141648, -12796236 },
3325         },
3326         {
3327             { -12863944, -13219986, -8318266, -11018091, -6810145, -4843894,
3328                 13475066, -3133972, 32674895, 13715045 },
3329             { 11423335, -5468059, 32344216, 8962751, 24989809, 9241752,
3330                 -13265253, 16086212, -28740881, -15642093 },
3331             { -1409668, 12530728, -6368726, 10847387, 19531186, -14132160,
3332                 -11709148, 7791794, -27245943, 4383347 },
3333         },
3334     },
3335     {
3336         {
3337             { -28970898, 5271447, -1266009, -9736989, -12455236, 16732599,
3338                 -4862407, -4906449, 27193557, 6245191 },
3339             { -15193956, 5362278, -1783893, 2695834, 4960227, 12840725, 23061898,
3340                 3260492, 22510453, 8577507 },
3341             { -12632451, 11257346, -32692994, 13548177, -721004, 10879011,
3342                 31168030, 13952092, -29571492, -3635906 },
3343         },
3344         {
3345             { 3877321, -9572739, 32416692, 5405324, -11004407, -13656635,
3346                 3759769, 11935320, 5611860, 8164018 },
3347             { -16275802, 14667797, 15906460, 12155291, -22111149, -9039718,
3348                 32003002, -8832289, 5773085, -8422109 },
3349             { -23788118, -8254300, 1950875, 8937633, 18686727, 16459170, -905725,
3350                 12376320, 31632953, 190926 },
3351         },
3352         {
3353             { -24593607, -16138885, -8423991, 13378746, 14162407, 6901328,
3354                 -8288749, 4508564, -25341555, -3627528 },
3355             { 8884438, -5884009, 6023974, 10104341, -6881569, -4941533, 18722941,
3356                 -14786005, -1672488, 827625 },
3357             { -32720583, -16289296, -32503547, 7101210, 13354605, 2659080,
3358                 -1800575, -14108036, -24878478, 1541286 },
3359         },
3360         {
3361             { 2901347, -1117687, 3880376, -10059388, -17620940, -3612781,
3362                 -21802117, -3567481, 20456845, -1885033 },
3363             { 27019610, 12299467, -13658288, -1603234, -12861660, -4861471,
3364                 -19540150, -5016058, 29439641, 15138866 },
3365             { 21536104, -6626420, -32447818, -10690208, -22408077, 5175814,
3366                 -5420040, -16361163, 7779328, 109896 },
3367         },
3368         {
3369             { 30279744, 14648750, -8044871, 6425558, 13639621, -743509, 28698390,
3370                 12180118, 23177719, -554075 },
3371             { 26572847, 3405927, -31701700, 12890905, -19265668, 5335866,
3372                 -6493768, 2378492, 4439158, -13279347 },
3373             { -22716706, 3489070, -9225266, -332753, 18875722, -1140095,
3374                 14819434, -12731527, -17717757, -5461437 },
3375         },
3376         {
3377             { -5056483, 16566551, 15953661, 3767752, -10436499, 15627060,
3378                 -820954, 2177225, 8550082, -15114165 },
3379             { -18473302, 16596775, -381660, 15663611, 22860960, 15585581,
3380                 -27844109, -3582739, -23260460, -8428588 },
3381             { -32480551, 15707275, -8205912, -5652081, 29464558, 2713815,
3382                 -22725137, 15860482, -21902570, 1494193 },
3383         },
3384         {
3385             { -19562091, -14087393, -25583872, -9299552, 13127842, 759709,
3386                 21923482, 16529112, 8742704, 12967017 },
3387             { -28464899, 1553205, 32536856, -10473729, -24691605, -406174,
3388                 -8914625, -2933896, -29903758, 15553883 },
3389             { 21877909, 3230008, 9881174, 10539357, -4797115, 2841332, 11543572,
3390                 14513274, 19375923, -12647961 },
3391         },
3392         {
3393             { 8832269, -14495485, 13253511, 5137575, 5037871, 4078777, 24880818,
3394                 -6222716, 2862653, 9455043 },
3395             { 29306751, 5123106, 20245049, -14149889, 9592566, 8447059, -2077124,
3396                 -2990080, 15511449, 4789663 },
3397             { -20679756, 7004547, 8824831, -9434977, -4045704, -3750736,
3398                 -5754762, 108893, 23513200, 16652362 },
3399         },
3400     },
3401     {
3402         {
3403             { -33256173, 4144782, -4476029, -6579123, 10770039, -7155542,
3404                 -6650416, -12936300, -18319198, 10212860 },
3405             { 2756081, 8598110, 7383731, -6859892, 22312759, -1105012, 21179801,
3406                 2600940, -9988298, -12506466 },
3407             { -24645692, 13317462, -30449259, -15653928, 21365574, -10869657,
3408                 11344424, 864440, -2499677, -16710063 },
3409         },
3410         {
3411             { -26432803, 6148329, -17184412, -14474154, 18782929, -275997,
3412                 -22561534, 211300, 2719757, 4940997 },
3413             { -1323882, 3911313, -6948744, 14759765, -30027150, 7851207,
3414                 21690126, 8518463, 26699843, 5276295 },
3415             { -13149873, -6429067, 9396249, 365013, 24703301, -10488939, 1321586,
3416                 149635, -15452774, 7159369 },
3417         },
3418         {
3419             { 9987780, -3404759, 17507962, 9505530, 9731535, -2165514, 22356009,
3420                 8312176, 22477218, -8403385 },
3421             { 18155857, -16504990, 19744716, 9006923, 15154154, -10538976,
3422                 24256460, -4864995, -22548173, 9334109 },
3423             { 2986088, -4911893, 10776628, -3473844, 10620590, -7083203,
3424                 -21413845, 14253545, -22587149, 536906 },
3425         },
3426         {
3427             { 4377756, 8115836, 24567078, 15495314, 11625074, 13064599, 7390551,
3428                 10589625, 10838060, -15420424 },
3429             { -19342404, 867880, 9277171, -3218459, -14431572, -1986443,
3430                 19295826, -15796950, 6378260, 699185 },
3431             { 7895026, 4057113, -7081772, -13077756, -17886831, -323126, -716039,
3432                 15693155, -5045064, -13373962 },
3433         },
3434         {
3435             { -7737563, -5869402, -14566319, -7406919, 11385654, 13201616,
3436                 31730678, -10962840, -3918636, -9669325 },
3437             { 10188286, -15770834, -7336361, 13427543, 22223443, 14896287,
3438                 30743455, 7116568, -21786507, 5427593 },
3439             { 696102, 13206899, 27047647, -10632082, 15285305, -9853179,
3440                 10798490, -4578720, 19236243, 12477404 },
3441         },
3442         {
3443             { -11229439, 11243796, -17054270, -8040865, -788228, -8167967,
3444                 -3897669, 11180504, -23169516, 7733644 },
3445             { 17800790, -14036179, -27000429, -11766671, 23887827, 3149671,
3446                 23466177, -10538171, 10322027, 15313801 },
3447             { 26246234, 11968874, 32263343, -5468728, 6830755, -13323031,
3448                 -15794704, -101982, -24449242, 10890804 },
3449         },
3450         {
3451             { -31365647, 10271363, -12660625, -6267268, 16690207, -13062544,
3452                 -14982212, 16484931, 25180797, -5334884 },
3453             { -586574, 10376444, -32586414, -11286356, 19801893, 10997610,
3454                 2276632, 9482883, 316878, 13820577 },
3455             { -9882808, -4510367, -2115506, 16457136, -11100081, 11674996,
3456                 30756178, -7515054, 30696930, -3712849 },
3457         },
3458         {
3459             { 32988917, -9603412, 12499366, 7910787, -10617257, -11931514,
3460                 -7342816, -9985397, -32349517, 7392473 },
3461             { -8855661, 15927861, 9866406, -3649411, -2396914, -16655781,
3462                 -30409476, -9134995, 25112947, -2926644 },
3463             { -2504044, -436966, 25621774, -5678772, 15085042, -5479877,
3464                 -24884878, -13526194, 5537438, -13914319 },
3465         },
3466     },
3467     {
3468         {
3469             { -11225584, 2320285, -9584280, 10149187, -33444663, 5808648,
3470                 -14876251, -1729667, 31234590, 6090599 },
3471             { -9633316, 116426, 26083934, 2897444, -6364437, -2688086, 609721,
3472                 15878753, -6970405, -9034768 },
3473             { -27757857, 247744, -15194774, -9002551, 23288161, -10011936,
3474                 -23869595, 6503646, 20650474, 1804084 },
3475         },
3476         {
3477             { -27589786, 15456424, 8972517, 8469608, 15640622, 4439847, 3121995,
3478                 -10329713, 27842616, -202328 },
3479             { -15306973, 2839644, 22530074, 10026331, 4602058, 5048462, 28248656,
3480                 5031932, -11375082, 12714369 },
3481             { 20807691, -7270825, 29286141, 11421711, -27876523, -13868230,
3482                 -21227475, 1035546, -19733229, 12796920 },
3483         },
3484         {
3485             { 12076899, -14301286, -8785001, -11848922, -25012791, 16400684,
3486                 -17591495, -12899438, 3480665, -15182815 },
3487             { -32361549, 5457597, 28548107, 7833186, 7303070, -11953545,
3488                 -24363064, -15921875, -33374054, 2771025 },
3489             { -21389266, 421932, 26597266, 6860826, 22486084, -6737172,
3490                 -17137485, -4210226, -24552282, 15673397 },
3491         },
3492         {
3493             { -20184622, 2338216, 19788685, -9620956, -4001265, -8740893,
3494                 -20271184, 4733254, 3727144, -12934448 },
3495             { 6120119, 814863, -11794402, -622716, 6812205, -15747771, 2019594,
3496                 7975683, 31123697, -10958981 },
3497             { 30069250, -11435332, 30434654, 2958439, 18399564, -976289,
3498                 12296869, 9204260, -16432438, 9648165 },
3499         },
3500         {
3501             { 32705432, -1550977, 30705658, 7451065, -11805606, 9631813, 3305266,
3502                 5248604, -26008332, -11377501 },
3503             { 17219865, 2375039, -31570947, -5575615, -19459679, 9219903, 294711,
3504                 15298639, 2662509, -16297073 },
3505             { -1172927, -7558695, -4366770, -4287744, -21346413, -8434326,
3506                 32087529, -1222777, 32247248, -14389861 },
3507         },
3508         {
3509             { 14312628, 1221556, 17395390, -8700143, -4945741, -8684635,
3510                 -28197744, -9637817, -16027623, -13378845 },
3511             { -1428825, -9678990, -9235681, 6549687, -7383069, -468664, 23046502,
3512                 9803137, 17597934, 2346211 },
3513             { 18510800, 15337574, 26171504, 981392, -22241552, 7827556,
3514                 -23491134, -11323352, 3059833, -11782870 },
3515         },
3516         {
3517             { 10141598, 6082907, 17829293, -1947643, 9830092, 13613136,
3518                 -25556636, -5544586, -33502212, 3592096 },
3519             { 33114168, -15889352, -26525686, -13343397, 33076705, 8716171,
3520                 1151462, 1521897, -982665, -6837803 },
3521             { -32939165, -4255815, 23947181, -324178, -33072974, -12305637,
3522                 -16637686, 3891704, 26353178, 693168 },
3523         },
3524         {
3525             { 30374239, 1595580, -16884039, 13186931, 4600344, 406904, 9585294,
3526                 -400668, 31375464, 14369965 },
3527             { -14370654, -7772529, 1510301, 6434173, -18784789, -6262728,
3528                 32732230, -13108839, 17901441, 16011505 },
3529             { 18171223, -11934626, -12500402, 15197122, -11038147, -15230035,
3530                 -19172240, -16046376, 8764035, 12309598 },
3531         },
3532     },
3533     {
3534         {
3535             { 5975908, -5243188, -19459362, -9681747, -11541277, 14015782,
3536                 -23665757, 1228319, 17544096, -10593782 },
3537             { 5811932, -1715293, 3442887, -2269310, -18367348, -8359541,
3538                 -18044043, -15410127, -5565381, 12348900 },
3539             { -31399660, 11407555, 25755363, 6891399, -3256938, 14872274,
3540                 -24849353, 8141295, -10632534, -585479 },
3541         },
3542         {
3543             { -12675304, 694026, -5076145, 13300344, 14015258, -14451394,
3544                 -9698672, -11329050, 30944593, 1130208 },
3545             { 8247766, -6710942, -26562381, -7709309, -14401939, -14648910,
3546                 4652152, 2488540, 23550156, -271232 },
3547             { 17294316, -3788438, 7026748, 15626851, 22990044, 113481, 2267737,
3548                 -5908146, -408818, -137719 },
3549         },
3550         {
3551             { 16091085, -16253926, 18599252, 7340678, 2137637, -1221657,
3552                 -3364161, 14550936, 3260525, -7166271 },
3553             { -4910104, -13332887, 18550887, 10864893, -16459325, -7291596,
3554                 -23028869, -13204905, -12748722, 2701326 },
3555             { -8574695, 16099415, 4629974, -16340524, -20786213, -6005432,
3556                 -10018363, 9276971, 11329923, 1862132 },
3557         },
3558         {
3559             { 14763076, -15903608, -30918270, 3689867, 3511892, 10313526,
3560                 -21951088, 12219231, -9037963, -940300 },
3561             { 8894987, -3446094, 6150753, 3013931, 301220, 15693451, -31981216,
3562                 -2909717, -15438168, 11595570 },
3563             { 15214962, 3537601, -26238722, -14058872, 4418657, -15230761,
3564                 13947276, 10730794, -13489462, -4363670 },
3565         },
3566         {
3567             { -2538306, 7682793, 32759013, 263109, -29984731, -7955452,
3568                 -22332124, -10188635, 977108, 699994 },
3569             { -12466472, 4195084, -9211532, 550904, -15565337, 12917920,
3570                 19118110, -439841, -30534533, -14337913 },
3571             { 31788461, -14507657, 4799989, 7372237, 8808585, -14747943, 9408237,
3572                 -10051775, 12493932, -5409317 },
3573         },
3574         {
3575             { -25680606, 5260744, -19235809, -6284470, -3695942, 16566087,
3576                 27218280, 2607121, 29375955, 6024730 },
3577             { 842132, -2794693, -4763381, -8722815, 26332018, -12405641,
3578                 11831880, 6985184, -9940361, 2854096 },
3579             { -4847262, -7969331, 2516242, -5847713, 9695691, -7221186, 16512645,
3580                 960770, 12121869, 16648078 },
3581         },
3582         {
3583             { -15218652, 14667096, -13336229, 2013717, 30598287, -464137,
3584                 -31504922, -7882064, 20237806, 2838411 },
3585             { -19288047, 4453152, 15298546, -16178388, 22115043, -15972604,
3586                 12544294, -13470457, 1068881, -12499905 },
3587             { -9558883, -16518835, 33238498, 13506958, 30505848, -1114596,
3588                 -8486907, -2630053, 12521378, 4845654 },
3589         },
3590         {
3591             { -28198521, 10744108, -2958380, 10199664, 7759311, -13088600,
3592                 3409348, -873400, -6482306, -12885870 },
3593             { -23561822, 6230156, -20382013, 10655314, -24040585, -11621172,
3594                 10477734, -1240216, -3113227, 13974498 },
3595             { 12966261, 15550616, -32038948, -1615346, 21025980, -629444,
3596                 5642325, 7188737, 18895762, 12629579 },
3597         },
3598     },
3599     {
3600         {
3601             { 14741879, -14946887, 22177208, -11721237, 1279741, 8058600,
3602                 11758140, 789443, 32195181, 3895677 },
3603             { 10758205, 15755439, -4509950, 9243698, -4879422, 6879879, -2204575,
3604                 -3566119, -8982069, 4429647 },
3605             { -2453894, 15725973, -20436342, -10410672, -5803908, -11040220,
3606                 -7135870, -11642895, 18047436, -15281743 },
3607         },
3608         {
3609             { -25173001, -11307165, 29759956, 11776784, -22262383, -15820455,
3610                 10993114, -12850837, -17620701, -9408468 },
3611             { 21987233, 700364, -24505048, 14972008, -7774265, -5718395,
3612                 32155026, 2581431, -29958985, 8773375 },
3613             { -25568350, 454463, -13211935, 16126715, 25240068, 8594567,
3614                 20656846, 12017935, -7874389, -13920155 },
3615         },
3616         {
3617             { 6028182, 6263078, -31011806, -11301710, -818919, 2461772,
3618                 -31841174, -5468042, -1721788, -2776725 },
3619             { -12278994, 16624277, 987579, -5922598, 32908203, 1248608, 7719845,
3620                 -4166698, 28408820, 6816612 },
3621             { -10358094, -8237829, 19549651, -12169222, 22082623, 16147817,
3622                 20613181, 13982702, -10339570, 5067943 },
3623         },
3624         {
3625             { -30505967, -3821767, 12074681, 13582412, -19877972, 2443951,
3626                 -19719286, 12746132, 5331210, -10105944 },
3627             { 30528811, 3601899, -1957090, 4619785, -27361822, -15436388,
3628                 24180793, -12570394, 27679908, -1648928 },
3629             { 9402404, -13957065, 32834043, 10838634, -26580150, -13237195,
3630                 26653274, -8685565, 22611444, -12715406 },
3631         },
3632         {
3633             { 22190590, 1118029, 22736441, 15130463, -30460692, -5991321,
3634                 19189625, -4648942, 4854859, 6622139 },
3635             { -8310738, -2953450, -8262579, -3388049, -10401731, -271929,
3636                 13424426, -3567227, 26404409, 13001963 },
3637             { -31241838, -15415700, -2994250, 8939346, 11562230, -12840670,
3638                 -26064365, -11621720, -15405155, 11020693 },
3639         },
3640         {
3641             { 1866042, -7949489, -7898649, -10301010, 12483315, 13477547,
3642                 3175636, -12424163, 28761762, 1406734 },
3643             { -448555, -1777666, 13018551, 3194501, -9580420, -11161737,
3644                 24760585, -4347088, 25577411, -13378680 },
3645             { -24290378, 4759345, -690653, -1852816, 2066747, 10693769,
3646                 -29595790, 9884936, -9368926, 4745410 },
3647         },
3648         {
3649             { -9141284, 6049714, -19531061, -4341411, -31260798, 9944276,
3650                 -15462008, -11311852, 10931924, -11931931 },
3651             { -16561513, 14112680, -8012645, 4817318, -8040464, -11414606,
3652                 -22853429, 10856641, -20470770, 13434654 },
3653             { 22759489, -10073434, -16766264, -1871422, 13637442, -10168091,
3654                 1765144, -12654326, 28445307, -5364710 },
3655         },
3656         {
3657             { 29875063, 12493613, 2795536, -3786330, 1710620, 15181182,
3658                 -10195717, -8788675, 9074234, 1167180 },
3659             { -26205683, 11014233, -9842651, -2635485, -26908120, 7532294,
3660                 -18716888, -9535498, 3843903, 9367684 },
3661             { -10969595, -6403711, 9591134, 9582310, 11349256, 108879, 16235123,
3662                 8601684, -139197, 4242895 },
3663         },
3664     },
3665     {
3666         {
3667             { 22092954, -13191123, -2042793, -11968512, 32186753, -11517388,
3668                 -6574341, 2470660, -27417366, 16625501 },
3669             { -11057722, 3042016, 13770083, -9257922, 584236, -544855, -7770857,
3670                 2602725, -27351616, 14247413 },
3671             { 6314175, -10264892, -32772502, 15957557, -10157730, 168750,
3672                 -8618807, 14290061, 27108877, -1180880 },
3673         },
3674         {
3675             { -8586597, -7170966, 13241782, 10960156, -32991015, -13794596,
3676                 33547976, -11058889, -27148451, 981874 },
3677             { 22833440, 9293594, -32649448, -13618667, -9136966, 14756819,
3678                 -22928859, -13970780, -10479804, -16197962 },
3679             { -7768587, 3326786, -28111797, 10783824, 19178761, 14905060,
3680                 22680049, 13906969, -15933690, 3797899 },
3681         },
3682         {
3683             { 21721356, -4212746, -12206123, 9310182, -3882239, -13653110,
3684                 23740224, -2709232, 20491983, -8042152 },
3685             { 9209270, -15135055, -13256557, -6167798, -731016, 15289673,
3686                 25947805, 15286587, 30997318, -6703063 },
3687             { 7392032, 16618386, 23946583, -8039892, -13265164, -1533858,
3688                 -14197445, -2321576, 17649998, -250080 },
3689         },
3690         {
3691             { -9301088, -14193827, 30609526, -3049543, -25175069, -1283752,
3692                 -15241566, -9525724, -2233253, 7662146 },
3693             { -17558673, 1763594, -33114336, 15908610, -30040870, -12174295,
3694                 7335080, -8472199, -3174674, 3440183 },
3695             { -19889700, -5977008, -24111293, -9688870, 10799743, -16571957,
3696                 40450, -4431835, 4862400, 1133 },
3697         },
3698         {
3699             { -32856209, -7873957, -5422389, 14860950, -16319031, 7956142,
3700                 7258061, 311861, -30594991, -7379421 },
3701             { -3773428, -1565936, 28985340, 7499440, 24445838, 9325937, 29727763,
3702                 16527196, 18278453, 15405622 },
3703             { -4381906, 8508652, -19898366, -3674424, -5984453, 15149970,
3704                 -13313598, 843523, -21875062, 13626197 },
3705         },
3706         {
3707             { 2281448, -13487055, -10915418, -2609910, 1879358, 16164207,
3708                 -10783882, 3953792, 13340839, 15928663 },
3709             { 31727126, -7179855, -18437503, -8283652, 2875793, -16390330,
3710                 -25269894, -7014826, -23452306, 5964753 },
3711             { 4100420, -5959452, -17179337, 6017714, -18705837, 12227141,
3712                 -26684835, 11344144, 2538215, -7570755 },
3713         },
3714         {
3715             { -9433605, 6123113, 11159803, -2156608, 30016280, 14966241,
3716                 -20474983, 1485421, -629256, -15958862 },
3717             { -26804558, 4260919, 11851389, 9658551, -32017107, 16367492,
3718                 -20205425, -13191288, 11659922, -11115118 },
3719             { 26180396, 10015009, -30844224, -8581293, 5418197, 9480663, 2231568,
3720                 -10170080, 33100372, -1306171 },
3721         },
3722         {
3723             { 15121113, -5201871, -10389905, 15427821, -27509937, -15992507,
3724                 21670947, 4486675, -5931810, -14466380 },
3725             { 16166486, -9483733, -11104130, 6023908, -31926798, -1364923,
3726                 2340060, -16254968, -10735770, -10039824 },
3727             { 28042865, -3557089, -12126526, 12259706, -3717498, -6945899,
3728                 6766453, -8689599, 18036436, 5803270 },
3729         },
3730     },
3731     {
3732         {
3733             { -817581, 6763912, 11803561, 1585585, 10958447, -2671165, 23855391,
3734                 4598332, -6159431, -14117438 },
3735             { -31031306, -14256194, 17332029, -2383520, 31312682, -5967183,
3736                 696309, 50292, -20095739, 11763584 },
3737             { -594563, -2514283, -32234153, 12643980, 12650761, 14811489, 665117,
3738                 -12613632, -19773211, -10713562 },
3739         },
3740         {
3741             { 30464590, -11262872, -4127476, -12734478, 19835327, -7105613,
3742                 -24396175, 2075773, -17020157, 992471 },
3743             { 18357185, -6994433, 7766382, 16342475, -29324918, 411174, 14578841,
3744                 8080033, -11574335, -10601610 },
3745             { 19598397, 10334610, 12555054, 2555664, 18821899, -10339780,
3746                 21873263, 16014234, 26224780, 16452269 },
3747         },
3748         {
3749             { -30223925, 5145196, 5944548, 16385966, 3976735, 2009897, -11377804,
3750                 -7618186, -20533829, 3698650 },
3751             { 14187449, 3448569, -10636236, -10810935, -22663880, -3433596,
3752                 7268410, -10890444, 27394301, 12015369 },
3753             { 19695761, 16087646, 28032085, 12999827, 6817792, 11427614,
3754                 20244189, -1312777, -13259127, -3402461 },
3755         },
3756         {
3757             { 30860103, 12735208, -1888245, -4699734, -16974906, 2256940,
3758                 -8166013, 12298312, -8550524, -10393462 },
3759             { -5719826, -11245325, -1910649, 15569035, 26642876, -7587760,
3760                 -5789354, -15118654, -4976164, 12651793 },
3761             { -2848395, 9953421, 11531313, -5282879, 26895123, -12697089,
3762                 -13118820, -16517902, 9768698, -2533218 },
3763         },
3764         {
3765             { -24719459, 1894651, -287698, -4704085, 15348719, -8156530,
3766                 32767513, 12765450, 4940095, 10678226 },
3767             { 18860224, 15980149, -18987240, -1562570, -26233012, -11071856,
3768                 -7843882, 13944024, -24372348, 16582019 },
3769             { -15504260, 4970268, -29893044, 4175593, -20993212, -2199756,
3770                 -11704054, 15444560, -11003761, 7989037 },
3771         },
3772         {
3773             { 31490452, 5568061, -2412803, 2182383, -32336847, 4531686,
3774                 -32078269, 6200206, -19686113, -14800171 },
3775             { -17308668, -15879940, -31522777, -2831, -32887382, 16375549,
3776                 8680158, -16371713, 28550068, -6857132 },
3777             { -28126887, -5688091, 16837845, -1820458, -6850681, 12700016,
3778                 -30039981, 4364038, 1155602, 5988841 },
3779         },
3780         {
3781             { 21890435, -13272907, -12624011, 12154349, -7831873, 15300496,
3782                 23148983, -4470481, 24618407, 8283181 },
3783             { -33136107, -10512751, 9975416, 6841041, -31559793, 16356536,
3784                 3070187, -7025928, 1466169, 10740210 },
3785             { -1509399, -15488185, -13503385, -10655916, 32799044, 909394,
3786                 -13938903, -5779719, -32164649, -15327040 },
3787         },
3788         {
3789             { 3960823, -14267803, -28026090, -15918051, -19404858, 13146868,
3790                 15567327, 951507, -3260321, -573935 },
3791             { 24740841, 5052253, -30094131, 8961361, 25877428, 6165135,
3792                 -24368180, 14397372, -7380369, -6144105 },
3793             { -28888365, 3510803, -28103278, -1158478, -11238128, -10631454,
3794                 -15441463, -14453128, -1625486, -6494814 },
3795         },
3796     },
3797     {
3798         {
3799             { 793299, -9230478, 8836302, -6235707, -27360908, -2369593, 33152843,
3800                 -4885251, -9906200, -621852 },
3801             { 5666233, 525582, 20782575, -8038419, -24538499, 14657740, 16099374,
3802                 1468826, -6171428, -15186581 },
3803             { -4859255, -3779343, -2917758, -6748019, 7778750, 11688288,
3804                 -30404353, -9871238, -1558923, -9863646 },
3805         },
3806         {
3807             { 10896332, -7719704, 824275, 472601, -19460308, 3009587, 25248958,
3808                 14783338, -30581476, -15757844 },
3809             { 10566929, 12612572, -31944212, 11118703, -12633376, 12362879,
3810                 21752402, 8822496, 24003793, 14264025 },
3811             { 27713862, -7355973, -11008240, 9227530, 27050101, 2504721,
3812                 23886875, -13117525, 13958495, -5732453 },
3813         },
3814         {
3815             { -23481610, 4867226, -27247128, 3900521, 29838369, -8212291,
3816                 -31889399, -10041781, 7340521, -15410068 },
3817             { 4646514, -8011124, -22766023, -11532654, 23184553, 8566613,
3818                 31366726, -1381061, -15066784, -10375192 },
3819             { -17270517, 12723032, -16993061, 14878794, 21619651, -6197576,
3820                 27584817, 3093888, -8843694, 3849921 },
3821         },
3822         {
3823             { -9064912, 2103172, 25561640, -15125738, -5239824, 9582958,
3824                 32477045, -9017955, 5002294, -15550259 },
3825             { -12057553, -11177906, 21115585, -13365155, 8808712, -12030708,
3826                 16489530, 13378448, -25845716, 12741426 },
3827             { -5946367, 10645103, -30911586, 15390284, -3286982, -7118677,
3828                 24306472, 15852464, 28834118, -7646072 },
3829         },
3830         {
3831             { -17335748, -9107057, -24531279, 9434953, -8472084, -583362,
3832                 -13090771, 455841, 20461858, 5491305 },
3833             { 13669248, -16095482, -12481974, -10203039, -14569770, -11893198,
3834                 -24995986, 11293807, -28588204, -9421832 },
3835             { 28497928, 6272777, -33022994, 14470570, 8906179, -1225630,
3836                 18504674, -14165166, 29867745, -8795943 },
3837         },
3838         {
3839             { -16207023, 13517196, -27799630, -13697798, 24009064, -6373891,
3840                 -6367600, -13175392, 22853429, -4012011 },
3841             { 24191378, 16712145, -13931797, 15217831, 14542237, 1646131,
3842                 18603514, -11037887, 12876623, -2112447 },
3843             { 17902668, 4518229, -411702, -2829247, 26878217, 5258055, -12860753,
3844                 608397, 16031844, 3723494 },
3845         },
3846         {
3847             { -28632773, 12763728, -20446446, 7577504, 33001348, -13017745,
3848                 17558842, -7872890, 23896954, -4314245 },
3849             { -20005381, -12011952, 31520464, 605201, 2543521, 5991821, -2945064,
3850                 7229064, -9919646, -8826859 },
3851             { 28816045, 298879, -28165016, -15920938, 19000928, -1665890,
3852                 -12680833, -2949325, -18051778, -2082915 },
3853         },
3854         {
3855             { 16000882, -344896, 3493092, -11447198, -29504595, -13159789,
3856                 12577740, 16041268, -19715240, 7847707 },
3857             { 10151868, 10572098, 27312476, 7922682, 14825339, 4723128,
3858                 -32855931, -6519018, -10020567, 3852848 },
3859             { -11430470, 15697596, -21121557, -4420647, 5386314, 15063598,
3860                 16514493, -15932110, 29330899, -15076224 },
3861         },
3862     },
3863     {
3864         {
3865             { -25499735, -4378794, -15222908, -6901211, 16615731, 2051784,
3866                 3303702, 15490, -27548796, 12314391 },
3867             { 15683520, -6003043, 18109120, -9980648, 15337968, -5997823,
3868                 -16717435, 15921866, 16103996, -3731215 },
3869             { -23169824, -10781249, 13588192, -1628807, -3798557, -1074929,
3870                 -19273607, 5402699, -29815713, -9841101 },
3871         },
3872         {
3873             { 23190676, 2384583, -32714340, 3462154, -29903655, -1529132,
3874                 -11266856, 8911517, -25205859, 2739713 },
3875             { 21374101, -3554250, -33524649, 9874411, 15377179, 11831242,
3876                 -33529904, 6134907, 4931255, 11987849 },
3877             { -7732, -2978858, -16223486, 7277597, 105524, -322051, -31480539,
3878                 13861388, -30076310, 10117930 },
3879         },
3880         {
3881             { -29501170, -10744872, -26163768, 13051539, -25625564, 5089643,
3882                 -6325503, 6704079, 12890019, 15728940 },
3883             { -21972360, -11771379, -951059, -4418840, 14704840, 2695116, 903376,
3884                 -10428139, 12885167, 8311031 },
3885             { -17516482, 5352194, 10384213, -13811658, 7506451, 13453191,
3886                 26423267, 4384730, 1888765, -5435404 },
3887         },
3888         {
3889             { -25817338, -3107312, -13494599, -3182506, 30896459, -13921729,
3890                 -32251644, -12707869, -19464434, -3340243 },
3891             { -23607977, -2665774, -526091, 4651136, 5765089, 4618330, 6092245,
3892                 14845197, 17151279, -9854116 },
3893             { -24830458, -12733720, -15165978, 10367250, -29530908, -265356,
3894                 22825805, -7087279, -16866484, 16176525 },
3895         },
3896         {
3897             { -23583256, 6564961, 20063689, 3798228, -4740178, 7359225, 2006182,
3898                 -10363426, -28746253, -10197509 },
3899             { -10626600, -4486402, -13320562, -5125317, 3432136, -6393229,
3900                 23632037, -1940610, 32808310, 1099883 },
3901             { 15030977, 5768825, -27451236, -2887299, -6427378, -15361371,
3902                 -15277896, -6809350, 2051441, -15225865 },
3903         },
3904         {
3905             { -3362323, -7239372, 7517890, 9824992, 23555850, 295369, 5148398,
3906                 -14154188, -22686354, 16633660 },
3907             { 4577086, -16752288, 13249841, -15304328, 19958763, -14537274,
3908                 18559670, -10759549, 8402478, -9864273 },
3909             { -28406330, -1051581, -26790155, -907698, -17212414, -11030789,
3910                 9453451, -14980072, 17983010, 9967138 },
3911         },
3912         {
3913             { -25762494, 6524722, 26585488, 9969270, 24709298, 1220360, -1677990,
3914                 7806337, 17507396, 3651560 },
3915             { -10420457, -4118111, 14584639, 15971087, -15768321, 8861010,
3916                 26556809, -5574557, -18553322, -11357135 },
3917             { 2839101, 14284142, 4029895, 3472686, 14402957, 12689363, -26642121,
3918                 8459447, -5605463, -7621941 },
3919         },
3920         {
3921             { -4839289, -3535444, 9744961, 2871048, 25113978, 3187018, -25110813,
3922                 -849066, 17258084, -7977739 },
3923             { 18164541, -10595176, -17154882, -1542417, 19237078, -9745295,
3924                 23357533, -15217008, 26908270, 12150756 },
3925             { -30264870, -7647865, 5112249, -7036672, -1499807, -6974257, 43168,
3926                 -5537701, -32302074, 16215819 },
3927         },
3928     },
3929     {
3930         {
3931             { -6898905, 9824394, -12304779, -4401089, -31397141, -6276835,
3932                 32574489, 12532905, -7503072, -8675347 },
3933             { -27343522, -16515468, -27151524, -10722951, 946346, 16291093,
3934                 254968, 7168080, 21676107, -1943028 },
3935             { 21260961, -8424752, -16831886, -11920822, -23677961, 3968121,
3936                 -3651949, -6215466, -3556191, -7913075 },
3937         },
3938         {
3939             { 16544754, 13250366, -16804428, 15546242, -4583003, 12757258,
3940                 -2462308, -8680336, -18907032, -9662799 },
3941             { -2415239, -15577728, 18312303, 4964443, -15272530, -12653564,
3942                 26820651, 16690659, 25459437, -4564609 },
3943             { -25144690, 11425020, 28423002, -11020557, -6144921, -15826224,
3944                 9142795, -2391602, -6432418, -1644817 },
3945         },
3946         {
3947             { -23104652, 6253476, 16964147, -3768872, -25113972, -12296437,
3948                 -27457225, -16344658, 6335692, 7249989 },
3949             { -30333227, 13979675, 7503222, -12368314, -11956721, -4621693,
3950                 -30272269, 2682242, 25993170, -12478523 },
3951             { 4364628, 5930691, 32304656, -10044554, -8054781, 15091131,
3952                 22857016, -10598955, 31820368, 15075278 },
3953         },
3954         {
3955             { 31879134, -8918693, 17258761, 90626, -8041836, -4917709, 24162788,
3956                 -9650886, -17970238, 12833045 },
3957             { 19073683, 14851414, -24403169, -11860168, 7625278, 11091125,
3958                 -19619190, 2074449, -9413939, 14905377 },
3959             { 24483667, -11935567, -2518866, -11547418, -1553130, 15355506,
3960                 -25282080, 9253129, 27628530, -7555480 },
3961         },
3962         {
3963             { 17597607, 8340603, 19355617, 552187, 26198470, -3176583, 4593324,
3964                 -9157582, -14110875, 15297016 },
3965             { 510886, 14337390, -31785257, 16638632, 6328095, 2713355, -20217417,
3966                 -11864220, 8683221, 2921426 },
3967             { 18606791, 11874196, 27155355, -5281482, -24031742, 6265446,
3968                 -25178240, -1278924, 4674690, 13890525 },
3969         },
3970         {
3971             { 13609624, 13069022, -27372361, -13055908, 24360586, 9592974,
3972                 14977157, 9835105, 4389687, 288396 },
3973             { 9922506, -519394, 13613107, 5883594, -18758345, -434263, -12304062,
3974                 8317628, 23388070, 16052080 },
3975             { 12720016, 11937594, -31970060, -5028689, 26900120, 8561328,
3976                 -20155687, -11632979, -14754271, -10812892 },
3977         },
3978         {
3979             { 15961858, 14150409, 26716931, -665832, -22794328, 13603569,
3980                 11829573, 7467844, -28822128, 929275 },
3981             { 11038231, -11582396, -27310482, -7316562, -10498527, -16307831,
3982                 -23479533, -9371869, -21393143, 2465074 },
3983             { 20017163, -4323226, 27915242, 1529148, 12396362, 15675764,
3984                 13817261, -9658066, 2463391, -4622140 },
3985         },
3986         {
3987             { -16358878, -12663911, -12065183, 4996454, -1256422, 1073572,
3988                 9583558, 12851107, 4003896, 12673717 },
3989             { -1731589, -15155870, -3262930, 16143082, 19294135, 13385325,
3990                 14741514, -9103726, 7903886, 2348101 },
3991             { 24536016, -16515207, 12715592, -3862155, 1511293, 10047386,
3992                 -3842346, -7129159, -28377538, 10048127 },
3993         },
3994     },
3995     {
3996         {
3997             { -12622226, -6204820, 30718825, 2591312, -10617028, 12192840,
3998                 18873298, -7297090, -32297756, 15221632 },
3999             { -26478122, -11103864, 11546244, -1852483, 9180880, 7656409,
4000                 -21343950, 2095755, 29769758, 6593415 },
4001             { -31994208, -2907461, 4176912, 3264766, 12538965, -868111, 26312345,
4002                 -6118678, 30958054, 8292160 },
4003         },
4004         {
4005             { 31429822, -13959116, 29173532, 15632448, 12174511, -2760094,
4006                 32808831, 3977186, 26143136, -3148876 },
4007             { 22648901, 1402143, -22799984, 13746059, 7936347, 365344, -8668633,
4008                 -1674433, -3758243, -2304625 },
4009             { -15491917, 8012313, -2514730, -12702462, -23965846, -10254029,
4010                 -1612713, -1535569, -16664475, 8194478 },
4011         },
4012         {
4013             { 27338066, -7507420, -7414224, 10140405, -19026427, -6589889,
4014                 27277191, 8855376, 28572286, 3005164 },
4015             { 26287124, 4821776, 25476601, -4145903, -3764513, -15788984,
4016                 -18008582, 1182479, -26094821, -13079595 },
4017             { -7171154, 3178080, 23970071, 6201893, -17195577, -4489192,
4018                 -21876275, -13982627, 32208683, -1198248 },
4019         },
4020         {
4021             { -16657702, 2817643, -10286362, 14811298, 6024667, 13349505,
4022                 -27315504, -10497842, -27672585, -11539858 },
4023             { 15941029, -9405932, -21367050, 8062055, 31876073, -238629,
4024                 -15278393, -1444429, 15397331, -4130193 },
4025             { 8934485, -13485467, -23286397, -13423241, -32446090, 14047986,
4026                 31170398, -1441021, -27505566, 15087184 },
4027         },
4028         {
4029             { -18357243, -2156491, 24524913, -16677868, 15520427, -6360776,
4030                 -15502406, 11461896, 16788528, -5868942 },
4031             { -1947386, 16013773, 21750665, 3714552, -17401782, -16055433,
4032                 -3770287, -10323320, 31322514, -11615635 },
4033             { 21426655, -5650218, -13648287, -5347537, -28812189, -4920970,
4034                 -18275391, -14621414, 13040862, -12112948 },
4035         },
4036         {
4037             { 11293895, 12478086, -27136401, 15083750, -29307421, 14748872,
4038                 14555558, -13417103, 1613711, 4896935 },
4039             { -25894883, 15323294, -8489791, -8057900, 25967126, -13425460,
4040                 2825960, -4897045, -23971776, -11267415 },
4041             { -15924766, -5229880, -17443532, 6410664, 3622847, 10243618,
4042                 20615400, 12405433, -23753030, -8436416 },
4043         },
4044         {
4045             { -7091295, 12556208, -20191352, 9025187, -17072479, 4333801,
4046                 4378436, 2432030, 23097949, -566018 },
4047             { 4565804, -16025654, 20084412, -7842817, 1724999, 189254, 24767264,
4048                 10103221, -18512313, 2424778 },
4049             { 366633, -11976806, 8173090, -6890119, 30788634, 5745705, -7168678,
4050                 1344109, -3642553, 12412659 },
4051         },
4052         {
4053             { -24001791, 7690286, 14929416, -168257, -32210835, -13412986,
4054                 24162697, -15326504, -3141501, 11179385 },
4055             { 18289522, -14724954, 8056945, 16430056, -21729724, 7842514,
4056                 -6001441, -1486897, -18684645, -11443503 },
4057             { 476239, 6601091, -6152790, -9723375, 17503545, -4863900, 27672959,
4058                 13403813, 11052904, 5219329 },
4059         },
4060     },
4061     {
4062         {
4063             { 20678546, -8375738, -32671898, 8849123, -5009758, 14574752,
4064                 31186971, -3973730, 9014762, -8579056 },
4065             { -13644050, -10350239, -15962508, 5075808, -1514661, -11534600,
4066                 -33102500, 9160280, 8473550, -3256838 },
4067             { 24900749, 14435722, 17209120, -15292541, -22592275, 9878983,
4068                 -7689309, -16335821, -24568481, 11788948 },
4069         },
4070         {
4071             { -3118155, -11395194, -13802089, 14797441, 9652448, -6845904,
4072                 -20037437, 10410733, -24568470, -1458691 },
4073             { -15659161, 16736706, -22467150, 10215878, -9097177, 7563911,
4074                 11871841, -12505194, -18513325, 8464118 },
4075             { -23400612, 8348507, -14585951, -861714, -3950205, -6373419,
4076                 14325289, 8628612, 33313881, -8370517 },
4077         },
4078         {
4079             { -20186973, -4967935, 22367356, 5271547, -1097117, -4788838,
4080                 -24805667, -10236854, -8940735, -5818269 },
4081             { -6948785, -1795212, -32625683, -16021179, 32635414, -7374245,
4082                 15989197, -12838188, 28358192, -4253904 },
4083             { -23561781, -2799059, -32351682, -1661963, -9147719, 10429267,
4084                 -16637684, 4072016, -5351664, 5596589 },
4085         },
4086         {
4087             { -28236598, -3390048, 12312896, 6213178, 3117142, 16078565,
4088                 29266239, 2557221, 1768301, 15373193 },
4089             { -7243358, -3246960, -4593467, -7553353, -127927, -912245, -1090902,
4090                 -4504991, -24660491, 3442910 },
4091             { -30210571, 5124043, 14181784, 8197961, 18964734, -11939093,
4092                 22597931, 7176455, -18585478, 13365930 },
4093         },
4094         {
4095             { -7877390, -1499958, 8324673, 4690079, 6261860, 890446, 24538107,
4096                 -8570186, -9689599, -3031667 },
4097             { 25008904, -10771599, -4305031, -9638010, 16265036, 15721635,
4098                 683793, -11823784, 15723479, -15163481 },
4099             { -9660625, 12374379, -27006999, -7026148, -7724114, -12314514,
4100                 11879682, 5400171, 519526, -1235876 },
4101         },
4102         {
4103             { 22258397, -16332233, -7869817, 14613016, -22520255, -2950923,
4104                 -20353881, 7315967, 16648397, 7605640 },
4105             { -8081308, -8464597, -8223311, 9719710, 19259459, -15348212,
4106                 23994942, -5281555, -9468848, 4763278 },
4107             { -21699244, 9220969, -15730624, 1084137, -25476107, -2852390,
4108                 31088447, -7764523, -11356529, 728112 },
4109         },
4110         {
4111             { 26047220, -11751471, -6900323, -16521798, 24092068, 9158119,
4112                 -4273545, -12555558, -29365436, -5498272 },
4113             { 17510331, -322857, 5854289, 8403524, 17133918, -3112612, -28111007,
4114                 12327945, 10750447, 10014012 },
4115             { -10312768, 3936952, 9156313, -8897683, 16498692, -994647,
4116                 -27481051, -666732, 3424691, 7540221 },
4117         },
4118         {
4119             { 30322361, -6964110, 11361005, -4143317, 7433304, 4989748, -7071422,
4120                 -16317219, -9244265, 15258046 },
4121             { 13054562, -2779497, 19155474, 469045, -12482797, 4566042, 5631406,
4122                 2711395, 1062915, -5136345 },
4123             { -19240248, -11254599, -29509029, -7499965, -5835763, 13005411,
4124                 -6066489, 12194497, 32960380, 1459310 },
4125         },
4126     },
4127     {
4128         {
4129             { 19852034, 7027924, 23669353, 10020366, 8586503, -6657907, 394197,
4130                 -6101885, 18638003, -11174937 },
4131             { 31395534, 15098109, 26581030, 8030562, -16527914, -5007134,
4132                 9012486, -7584354, -6643087, -5442636 },
4133             { -9192165, -2347377, -1997099, 4529534, 25766844, 607986, -13222,
4134                 9677543, -32294889, -6456008 },
4135         },
4136         {
4137             { -2444496, -149937, 29348902, 8186665, 1873760, 12489863, -30934579,
4138                 -7839692, -7852844, -8138429 },
4139             { -15236356, -15433509, 7766470, 746860, 26346930, -10221762,
4140                 -27333451, 10754588, -9431476, 5203576 },
4141             { 31834314, 14135496, -770007, 5159118, 20917671, -16768096,
4142                 -7467973, -7337524, 31809243, 7347066 },
4143         },
4144         {
4145             { -9606723, -11874240, 20414459, 13033986, 13716524, -11691881,
4146                 19797970, -12211255, 15192876, -2087490 },
4147             { -12663563, -2181719, 1168162, -3804809, 26747877, -14138091,
4148                 10609330, 12694420, 33473243, -13382104 },
4149             { 33184999, 11180355, 15832085, -11385430, -1633671, 225884,
4150                 15089336, -11023903, -6135662, 14480053 },
4151         },
4152         {
4153             { 31308717, -5619998, 31030840, -1897099, 15674547, -6582883,
4154                 5496208, 13685227, 27595050, 8737275 },
4155             { -20318852, -15150239, 10933843, -16178022, 8335352, -7546022,
4156                 -31008351, -12610604, 26498114, 66511 },
4157             { 22644454, -8761729, -16671776, 4884562, -3105614, -13559366,
4158                 30540766, -4286747, -13327787, -7515095 },
4159         },
4160         {
4161             { -28017847, 9834845, 18617207, -2681312, -3401956, -13307506,
4162                 8205540, 13585437, -17127465, 15115439 },
4163             { 23711543, -672915, 31206561, -8362711, 6164647, -9709987,
4164                 -33535882, -1426096, 8236921, 16492939 },
4165             { -23910559, -13515526, -26299483, -4503841, 25005590, -7687270,
4166                 19574902, 10071562, 6708380, -6222424 },
4167         },
4168         {
4169             { 2101391, -4930054, 19702731, 2367575, -15427167, 1047675, 5301017,
4170                 9328700, 29955601, -11678310 },
4171             { 3096359, 9271816, -21620864, -15521844, -14847996, -7592937,
4172                 -25892142, -12635595, -9917575, 6216608 },
4173             { -32615849, 338663, -25195611, 2510422, -29213566, -13820213,
4174                 24822830, -6146567, -26767480, 7525079 },
4175         },
4176         {
4177             { -23066649, -13985623, 16133487, -7896178, -3389565, 778788,
4178                 -910336, -2782495, -19386633, 11994101 },
4179             { 21691500, -13624626, -641331, -14367021, 3285881, -3483596,
4180                 -25064666, 9718258, -7477437, 13381418 },
4181             { 18445390, -4202236, 14979846, 11622458, -1727110, -3582980,
4182                 23111648, -6375247, 28535282, 15779576 },
4183         },
4184         {
4185             { 30098053, 3089662, -9234387, 16662135, -21306940, 11308411,
4186                 -14068454, 12021730, 9955285, -16303356 },
4187             { 9734894, -14576830, -7473633, -9138735, 2060392, 11313496,
4188                 -18426029, 9924399, 20194861, 13380996 },
4189             { -26378102, -7965207, -22167821, 15789297, -18055342, -6168792,
4190                 -1984914, 15707771, 26342023, 10146099 },
4191         },
4192     },
4193     {
4194         {
4195             { -26016874, -219943, 21339191, -41388, 19745256, -2878700,
4196                 -29637280, 2227040, 21612326, -545728 },
4197             { -13077387, 1184228, 23562814, -5970442, -20351244, -6348714,
4198                 25764461, 12243797, -20856566, 11649658 },
4199             { -10031494, 11262626, 27384172, 2271902, 26947504, -15997771, 39944,
4200                 6114064, 33514190, 2333242 },
4201         },
4202         {
4203             { -21433588, -12421821, 8119782, 7219913, -21830522, -9016134,
4204                 -6679750, -12670638, 24350578, -13450001 },
4205             { -4116307, -11271533, -23886186, 4843615, -30088339, 690623,
4206                 -31536088, -10406836, 8317860, 12352766 },
4207             { 18200138, -14475911, -33087759, -2696619, -23702521, -9102511,
4208                 -23552096, -2287550, 20712163, 6719373 },
4209         },
4210         {
4211             { 26656208, 6075253, -7858556, 1886072, -28344043, 4262326, 11117530,
4212                 -3763210, 26224235, -3297458 },
4213             { -17168938, -14854097, -3395676, -16369877, -19954045, 14050420,
4214                 21728352, 9493610, 18620611, -16428628 },
4215             { -13323321, 13325349, 11432106, 5964811, 18609221, 6062965,
4216                 -5269471, -9725556, -30701573, -16479657 },
4217         },
4218         {
4219             { -23860538, -11233159, 26961357, 1640861, -32413112, -16737940,
4220                 12248509, -5240639, 13735342, 1934062 },
4221             { 25089769, 6742589, 17081145, -13406266, 21909293, -16067981,
4222                 -15136294, -3765346, -21277997, 5473616 },
4223             { 31883677, -7961101, 1083432, -11572403, 22828471, 13290673,
4224                 -7125085, 12469656, 29111212, -5451014 },
4225         },
4226         {
4227             { 24244947, -15050407, -26262976, 2791540, -14997599, 16666678,
4228                 24367466, 6388839, -10295587, 452383 },
4229             { -25640782, -3417841, 5217916, 16224624, 19987036, -4082269,
4230                 -24236251, -5915248, 15766062, 8407814 },
4231             { -20406999, 13990231, 15495425, 16395525, 5377168, 15166495,
4232                 -8917023, -4388953, -8067909, 2276718 },
4233         },
4234         {
4235             { 30157918, 12924066, -17712050, 9245753, 19895028, 3368142,
4236                 -23827587, 5096219, 22740376, -7303417 },
4237             { 2041139, -14256350, 7783687, 13876377, -25946985, -13352459,
4238                 24051124, 13742383, -15637599, 13295222 },
4239             { 33338237, -8505733, 12532113, 7977527, 9106186, -1715251,
4240                 -17720195, -4612972, -4451357, -14669444 },
4241         },
4242         {
4243             { -20045281, 5454097, -14346548, 6447146, 28862071, 1883651,
4244                 -2469266, -4141880, 7770569, 9620597 },
4245             { 23208068, 7979712, 33071466, 8149229, 1758231, -10834995, 30945528,
4246                 -1694323, -33502340, -14767970 },
4247             { 1439958, -16270480, -1079989, -793782, 4625402, 10647766, -5043801,
4248                 1220118, 30494170, -11440799 },
4249         },
4250         {
4251             { -5037580, -13028295, -2970559, -3061767, 15640974, -6701666,
4252                 -26739026, 926050, -1684339, -13333647 },
4253             { 13908495, -3549272, 30919928, -6273825, -21521863, 7989039,
4254                 9021034, 9078865, 3353509, 4033511 },
4255             { -29663431, -15113610, 32259991, -344482, 24295849, -12912123,
4256                 23161163, 8839127, 27485041, 7356032 },
4257         },
4258     },
4259     {
4260         {
4261             { 9661027, 705443, 11980065, -5370154, -1628543, 14661173, -6346142,
4262                 2625015, 28431036, -16771834 },
4263             { -23839233, -8311415, -25945511, 7480958, -17681669, -8354183,
4264                 -22545972, 14150565, 15970762, 4099461 },
4265             { 29262576, 16756590, 26350592, -8793563, 8529671, -11208050,
4266                 13617293, -9937143, 11465739, 8317062 },
4267         },
4268         {
4269             { -25493081, -6962928, 32500200, -9419051, -23038724, -2302222,
4270                 14898637, 3848455, 20969334, -5157516 },
4271             { -20384450, -14347713, -18336405, 13884722, -33039454, 2842114,
4272                 -21610826, -3649888, 11177095, 14989547 },
4273             { -24496721, -11716016, 16959896, 2278463, 12066309, 10137771,
4274                 13515641, 2581286, -28487508, 9930240 },
4275         },
4276         {
4277             { -17751622, -2097826, 16544300, -13009300, -15914807, -14949081,
4278                 18345767, -13403753, 16291481, -5314038 },
4279             { -33229194, 2553288, 32678213, 9875984, 8534129, 6889387, -9676774,
4280                 6957617, 4368891, 9788741 },
4281             { 16660756, 7281060, -10830758, 12911820, 20108584, -8101676,
4282                 -21722536, -8613148, 16250552, -11111103 },
4283         },
4284         {
4285             { -19765507, 2390526, -16551031, 14161980, 1905286, 6414907, 4689584,
4286                 10604807, -30190403, 4782747 },
4287             { -1354539, 14736941, -7367442, -13292886, 7710542, -14155590,
4288                 -9981571, 4383045, 22546403, 437323 },
4289             { 31665577, -12180464, -16186830, 1491339, -18368625, 3294682,
4290                 27343084, 2786261, -30633590, -14097016 },
4291         },
4292         {
4293             { -14467279, -683715, -33374107, 7448552, 19294360, 14334329,
4294                 -19690631, 2355319, -19284671, -6114373 },
4295             { 15121312, -15796162, 6377020, -6031361, -10798111, -12957845,
4296                 18952177, 15496498, -29380133, 11754228 },
4297             { -2637277, -13483075, 8488727, -14303896, 12728761, -1622493,
4298                 7141596, 11724556, 22761615, -10134141 },
4299         },
4300         {
4301             { 16918416, 11729663, -18083579, 3022987, -31015732, -13339659,
4302                 -28741185, -12227393, 32851222, 11717399 },
4303             { 11166634, 7338049, -6722523, 4531520, -29468672, -7302055,
4304                 31474879, 3483633, -1193175, -4030831 },
4305             { -185635, 9921305, 31456609, -13536438, -12013818, 13348923,
4306                 33142652, 6546660, -19985279, -3948376 },
4307         },
4308         {
4309             { -32460596, 11266712, -11197107, -7899103, 31703694, 3855903,
4310                 -8537131, -12833048, -30772034, -15486313 },
4311             { -18006477, 12709068, 3991746, -6479188, -21491523, -10550425,
4312                 -31135347, -16049879, 10928917, 3011958 },
4313             { -6957757, -15594337, 31696059, 334240, 29576716, 14796075,
4314                 -30831056, -12805180, 18008031, 10258577 },
4315         },
4316         {
4317             { -22448644, 15655569, 7018479, -4410003, -30314266, -1201591,
4318                 -1853465, 1367120, 25127874, 6671743 },
4319             { 29701166, -14373934, -10878120, 9279288, -17568, 13127210,
4320                 21382910, 11042292, 25838796, 4642684 },
4321             { -20430234, 14955537, -24126347, 8124619, -5369288, -5990470,
4322                 30468147, -13900640, 18423289, 4177476 },
4323         },
4324     },
4325 };
4326 
negative(signed char b)4327 static uint8_t negative(signed char b)
4328 {
4329     uint32_t x = b;
4330 
4331     x >>= 31; /* 1: yes; 0: no */
4332     return x;
4333 }
4334 
table_select(ge_precomp * t,int pos,signed char b)4335 static void table_select(ge_precomp *t, int pos, signed char b)
4336 {
4337     ge_precomp minust;
4338     uint8_t bnegative = negative(b);
4339     uint8_t babs = b - ((uint8_t)((-bnegative) & b) << 1);
4340 
4341     ge_precomp_0(t);
4342     cmov(t, &k25519Precomp[pos][0], equal(babs, 1));
4343     cmov(t, &k25519Precomp[pos][1], equal(babs, 2));
4344     cmov(t, &k25519Precomp[pos][2], equal(babs, 3));
4345     cmov(t, &k25519Precomp[pos][3], equal(babs, 4));
4346     cmov(t, &k25519Precomp[pos][4], equal(babs, 5));
4347     cmov(t, &k25519Precomp[pos][5], equal(babs, 6));
4348     cmov(t, &k25519Precomp[pos][6], equal(babs, 7));
4349     cmov(t, &k25519Precomp[pos][7], equal(babs, 8));
4350     fe_copy(minust.yplusx, t->yminusx);
4351     fe_copy(minust.yminusx, t->yplusx);
4352     fe_neg(minust.xy2d, t->xy2d);
4353     cmov(t, &minust, bnegative);
4354 }
4355 
4356 /*
4357  * h = a * B
4358  *
4359  * where a = a[0]+256*a[1]+...+256^31 a[31]
4360  * B is the Ed25519 base point (x,4/5) with x positive.
4361  *
4362  * Preconditions:
4363  *   a[31] <= 127
4364  */
ge_scalarmult_base(ge_p3 * h,const uint8_t * a)4365 static void ge_scalarmult_base(ge_p3 *h, const uint8_t *a)
4366 {
4367     signed char e[64];
4368     signed char carry;
4369     ge_p1p1 r;
4370     ge_p2 s;
4371     ge_precomp t;
4372     int i;
4373 
4374     for (i = 0; i < 32; ++i) {
4375         e[2 * i + 0] = (a[i] >> 0) & 15;
4376         e[2 * i + 1] = (a[i] >> 4) & 15;
4377     }
4378     /* each e[i] is between 0 and 15 */
4379     /* e[63] is between 0 and 7 */
4380 
4381     carry = 0;
4382     for (i = 0; i < 63; ++i) {
4383         e[i] += carry;
4384         carry = e[i] + 8;
4385         carry >>= 4;
4386         e[i] -= carry << 4;
4387     }
4388     e[63] += carry;
4389     /* each e[i] is between -8 and 8 */
4390 
4391     ge_p3_0(h);
4392     for (i = 1; i < 64; i += 2) {
4393         table_select(&t, i / 2, e[i]);
4394         ge_madd(&r, h, &t);
4395         ge_p1p1_to_p3(h, &r);
4396     }
4397 
4398     ge_p3_dbl(&r, h);
4399     ge_p1p1_to_p2(&s, &r);
4400     ge_p2_dbl(&r, &s);
4401     ge_p1p1_to_p2(&s, &r);
4402     ge_p2_dbl(&r, &s);
4403     ge_p1p1_to_p2(&s, &r);
4404     ge_p2_dbl(&r, &s);
4405     ge_p1p1_to_p3(h, &r);
4406 
4407     for (i = 0; i < 64; i += 2) {
4408         table_select(&t, i / 2, e[i]);
4409         ge_madd(&r, h, &t);
4410         ge_p1p1_to_p3(h, &r);
4411     }
4412 
4413     OPENSSL_cleanse(e, sizeof(e));
4414 }
4415 
4416 #if !defined(BASE_2_51_IMPLEMENTED)
4417 /*
4418  * Replace (f,g) with (g,f) if b == 1;
4419  * replace (f,g) with (f,g) if b == 0.
4420  *
4421  * Preconditions: b in {0,1}.
4422  */
fe_cswap(fe f,fe g,unsigned int b)4423 static void fe_cswap(fe f, fe g, unsigned int b)
4424 {
4425     size_t i;
4426 
4427     b = 0 - b;
4428     for (i = 0; i < 10; i++) {
4429         int32_t x = f[i] ^ g[i];
4430         x &= b;
4431         f[i] ^= x;
4432         g[i] ^= x;
4433     }
4434 }
4435 
4436 /*
4437  * h = f * 121666
4438  *
4439  * Can overlap h with f.
4440  *
4441  * Preconditions:
4442  *    |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
4443  *
4444  * Postconditions:
4445  *    |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
4446  */
fe_mul121666(fe h,fe f)4447 static void fe_mul121666(fe h, fe f)
4448 {
4449     int32_t f0 = f[0];
4450     int32_t f1 = f[1];
4451     int32_t f2 = f[2];
4452     int32_t f3 = f[3];
4453     int32_t f4 = f[4];
4454     int32_t f5 = f[5];
4455     int32_t f6 = f[6];
4456     int32_t f7 = f[7];
4457     int32_t f8 = f[8];
4458     int32_t f9 = f[9];
4459     int64_t h0 = f0 * (int64_t)121666;
4460     int64_t h1 = f1 * (int64_t)121666;
4461     int64_t h2 = f2 * (int64_t)121666;
4462     int64_t h3 = f3 * (int64_t)121666;
4463     int64_t h4 = f4 * (int64_t)121666;
4464     int64_t h5 = f5 * (int64_t)121666;
4465     int64_t h6 = f6 * (int64_t)121666;
4466     int64_t h7 = f7 * (int64_t)121666;
4467     int64_t h8 = f8 * (int64_t)121666;
4468     int64_t h9 = f9 * (int64_t)121666;
4469     int64_t carry0;
4470     int64_t carry1;
4471     int64_t carry2;
4472     int64_t carry3;
4473     int64_t carry4;
4474     int64_t carry5;
4475     int64_t carry6;
4476     int64_t carry7;
4477     int64_t carry8;
4478     int64_t carry9;
4479 
4480     carry9 = h9 + (1 << 24);
4481     h0 += (carry9 >> 25) * 19;
4482     h9 -= carry9 & kTop39Bits;
4483     carry1 = h1 + (1 << 24);
4484     h2 += carry1 >> 25;
4485     h1 -= carry1 & kTop39Bits;
4486     carry3 = h3 + (1 << 24);
4487     h4 += carry3 >> 25;
4488     h3 -= carry3 & kTop39Bits;
4489     carry5 = h5 + (1 << 24);
4490     h6 += carry5 >> 25;
4491     h5 -= carry5 & kTop39Bits;
4492     carry7 = h7 + (1 << 24);
4493     h8 += carry7 >> 25;
4494     h7 -= carry7 & kTop39Bits;
4495 
4496     carry0 = h0 + (1 << 25);
4497     h1 += carry0 >> 26;
4498     h0 -= carry0 & kTop38Bits;
4499     carry2 = h2 + (1 << 25);
4500     h3 += carry2 >> 26;
4501     h2 -= carry2 & kTop38Bits;
4502     carry4 = h4 + (1 << 25);
4503     h5 += carry4 >> 26;
4504     h4 -= carry4 & kTop38Bits;
4505     carry6 = h6 + (1 << 25);
4506     h7 += carry6 >> 26;
4507     h6 -= carry6 & kTop38Bits;
4508     carry8 = h8 + (1 << 25);
4509     h9 += carry8 >> 26;
4510     h8 -= carry8 & kTop38Bits;
4511 
4512     h[0] = (int32_t)h0;
4513     h[1] = (int32_t)h1;
4514     h[2] = (int32_t)h2;
4515     h[3] = (int32_t)h3;
4516     h[4] = (int32_t)h4;
4517     h[5] = (int32_t)h5;
4518     h[6] = (int32_t)h6;
4519     h[7] = (int32_t)h7;
4520     h[8] = (int32_t)h8;
4521     h[9] = (int32_t)h9;
4522 }
4523 
x25519_scalar_mult_generic(uint8_t out[32],const uint8_t scalar[32],const uint8_t point[32])4524 static void x25519_scalar_mult_generic(uint8_t out[32],
4525     const uint8_t scalar[32],
4526     const uint8_t point[32])
4527 {
4528     fe x1, x2, z2, x3, z3, tmp0, tmp1;
4529     uint8_t e[32];
4530     unsigned swap = 0;
4531     int pos;
4532 
4533     memcpy(e, scalar, 32);
4534     e[0] &= 248;
4535     e[31] &= 127;
4536     e[31] |= 64;
4537     fe_frombytes(x1, point);
4538     fe_1(x2);
4539     fe_0(z2);
4540     fe_copy(x3, x1);
4541     fe_1(z3);
4542 
4543     for (pos = 254; pos >= 0; --pos) {
4544         unsigned b = 1 & (e[pos / 8] >> (pos & 7));
4545         swap ^= b;
4546         fe_cswap(x2, x3, swap);
4547         fe_cswap(z2, z3, swap);
4548         swap = b;
4549         fe_sub(tmp0, x3, z3);
4550         fe_sub(tmp1, x2, z2);
4551         fe_add(x2, x2, z2);
4552         fe_add(z2, x3, z3);
4553         fe_mul(z3, tmp0, x2);
4554         fe_mul(z2, z2, tmp1);
4555         fe_sq(tmp0, tmp1);
4556         fe_sq(tmp1, x2);
4557         fe_add(x3, z3, z2);
4558         fe_sub(z2, z3, z2);
4559         fe_mul(x2, tmp1, tmp0);
4560         fe_sub(tmp1, tmp1, tmp0);
4561         fe_sq(z2, z2);
4562         fe_mul121666(z3, tmp1);
4563         fe_sq(x3, x3);
4564         fe_add(tmp0, tmp0, z3);
4565         fe_mul(z3, x1, z2);
4566         fe_mul(z2, tmp1, tmp0);
4567     }
4568 
4569     fe_invert(z2, z2);
4570     fe_mul(x2, x2, z2);
4571     fe_tobytes(out, x2);
4572 
4573     OPENSSL_cleanse(e, sizeof(e));
4574 }
4575 
x25519_scalar_mult(uint8_t out[32],const uint8_t scalar[32],const uint8_t point[32])4576 static void x25519_scalar_mult(uint8_t out[32], const uint8_t scalar[32],
4577     const uint8_t point[32])
4578 {
4579     x25519_scalar_mult_generic(out, scalar, point);
4580 }
4581 #endif
4582 
slide(signed char * r,const uint8_t * a)4583 static void slide(signed char *r, const uint8_t *a)
4584 {
4585     int i;
4586     int b;
4587     int k;
4588 
4589     for (i = 0; i < 256; ++i) {
4590         r[i] = 1 & (a[i >> 3] >> (i & 7));
4591     }
4592 
4593     for (i = 0; i < 256; ++i) {
4594         if (r[i]) {
4595             for (b = 1; b <= 6 && i + b < 256; ++b) {
4596                 if (r[i + b]) {
4597                     if (r[i] + (r[i + b] << b) <= 15) {
4598                         r[i] += r[i + b] << b;
4599                         r[i + b] = 0;
4600                     } else if (r[i] - (r[i + b] << b) >= -15) {
4601                         r[i] -= r[i + b] << b;
4602                         for (k = i + b; k < 256; ++k) {
4603                             if (!r[k]) {
4604                                 r[k] = 1;
4605                                 break;
4606                             }
4607                             r[k] = 0;
4608                         }
4609                     } else {
4610                         break;
4611                     }
4612                 }
4613             }
4614         }
4615     }
4616 }
4617 
4618 static const ge_precomp Bi[8] = {
4619     {
4620         { 25967493, -14356035, 29566456, 3660896, -12694345, 4014787, 27544626,
4621             -11754271, -6079156, 2047605 },
4622         { -12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692,
4623             5043384, 19500929, -15469378 },
4624         { -8738181, 4489570, 9688441, -14785194, 10184609, -12363380, 29287919,
4625             11864899, -24514362, -4438546 },
4626     },
4627     {
4628         { 15636291, -9688557, 24204773, -7912398, 616977, -16685262, 27787600,
4629             -14772189, 28944400, -1550024 },
4630         { 16568933, 4717097, -11556148, -1102322, 15682896, -11807043, 16354577,
4631             -11775962, 7689662, 11199574 },
4632         { 30464156, -5976125, -11779434, -15670865, 23220365, 15915852, 7512774,
4633             10017326, -17749093, -9920357 },
4634     },
4635     {
4636         { 10861363, 11473154, 27284546, 1981175, -30064349, 12577861, 32867885,
4637             14515107, -15438304, 10819380 },
4638         { 4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668,
4639             12483688, -12668491, 5581306 },
4640         { 19563160, 16186464, -29386857, 4097519, 10237984, -4348115, 28542350,
4641             13850243, -23678021, -15815942 },
4642     },
4643     {
4644         { 5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852,
4645             5230134, -23952439, -15175766 },
4646         { -30269007, -3463509, 7665486, 10083793, 28475525, 1649722, 20654025,
4647             16520125, 30598449, 7715701 },
4648         { 28881845, 14381568, 9657904, 3680757, -20181635, 7843316, -31400660,
4649             1370708, 29794553, -1409300 },
4650     },
4651     {
4652         { -22518993, -6692182, 14201702, -8745502, -23510406, 8844726, 18474211,
4653             -1361450, -13062696, 13821877 },
4654         { -6455177, -7839871, 3374702, -4740862, -27098617, -10571707, 31655028,
4655             -7212327, 18853322, -14220951 },
4656         { 4566830, -12963868, -28974889, -12240689, -7602672, -2830569, -8514358,
4657             -10431137, 2207753, -3209784 },
4658     },
4659     {
4660         { -25154831, -4185821, 29681144, 7868801, -6854661, -9423865, -12437364,
4661             -663000, -31111463, -16132436 },
4662         { 25576264, -2703214, 7349804, -11814844, 16472782, 9300885, 3844789,
4663             15725684, 171356, 6466918 },
4664         { 23103977, 13316479, 9739013, -16149481, 817875, -15038942, 8965339,
4665             -14088058, -30714912, 16193877 },
4666     },
4667     {
4668         { -33521811, 3180713, -2394130, 14003687, -16903474, -16270840, 17238398,
4669             4729455, -18074513, 9256800 },
4670         { -25182317, -4174131, 32336398, 5036987, -21236817, 11360617, 22616405,
4671             9761698, -19827198, 630305 },
4672         { -13720693, 2639453, -24237460, -7406481, 9494427, -5774029, -6554551,
4673             -15960994, -2449256, -14291300 },
4674     },
4675     {
4676         { -3151181, -5046075, 9282714, 6866145, -31907062, -863023, -18940575,
4677             15033784, 25105118, -7894876 },
4678         { -24326370, 15950226, -31801215, -14592823, -11662737, -5090925,
4679             1573892, -2625887, 2198790, -15804619 },
4680         { -3099351, 10324967, -2241613, 7453183, -5446979, -2735503, -13812022,
4681             -16236442, -32461234, -12290683 },
4682     },
4683 };
4684 
4685 /*
4686  * r = a * A + b * B
4687  *
4688  * where a = a[0]+256*a[1]+...+256^31 a[31].
4689  * and b = b[0]+256*b[1]+...+256^31 b[31].
4690  * B is the Ed25519 base point (x,4/5) with x positive.
4691  */
ge_double_scalarmult_vartime(ge_p2 * r,const uint8_t * a,const ge_p3 * A,const uint8_t * b)4692 static void ge_double_scalarmult_vartime(ge_p2 *r, const uint8_t *a,
4693     const ge_p3 *A, const uint8_t *b)
4694 {
4695     signed char aslide[256];
4696     signed char bslide[256];
4697     ge_cached Ai[8]; /* A,3A,5A,7A,9A,11A,13A,15A */
4698     ge_p1p1 t;
4699     ge_p3 u;
4700     ge_p3 A2;
4701     int i;
4702 
4703     slide(aslide, a);
4704     slide(bslide, b);
4705 
4706     ge_p3_to_cached(&Ai[0], A);
4707     ge_p3_dbl(&t, A);
4708     ge_p1p1_to_p3(&A2, &t);
4709     ge_add(&t, &A2, &Ai[0]);
4710     ge_p1p1_to_p3(&u, &t);
4711     ge_p3_to_cached(&Ai[1], &u);
4712     ge_add(&t, &A2, &Ai[1]);
4713     ge_p1p1_to_p3(&u, &t);
4714     ge_p3_to_cached(&Ai[2], &u);
4715     ge_add(&t, &A2, &Ai[2]);
4716     ge_p1p1_to_p3(&u, &t);
4717     ge_p3_to_cached(&Ai[3], &u);
4718     ge_add(&t, &A2, &Ai[3]);
4719     ge_p1p1_to_p3(&u, &t);
4720     ge_p3_to_cached(&Ai[4], &u);
4721     ge_add(&t, &A2, &Ai[4]);
4722     ge_p1p1_to_p3(&u, &t);
4723     ge_p3_to_cached(&Ai[5], &u);
4724     ge_add(&t, &A2, &Ai[5]);
4725     ge_p1p1_to_p3(&u, &t);
4726     ge_p3_to_cached(&Ai[6], &u);
4727     ge_add(&t, &A2, &Ai[6]);
4728     ge_p1p1_to_p3(&u, &t);
4729     ge_p3_to_cached(&Ai[7], &u);
4730 
4731     ge_p2_0(r);
4732 
4733     for (i = 255; i >= 0; --i) {
4734         if (aslide[i] || bslide[i]) {
4735             break;
4736         }
4737     }
4738 
4739     for (; i >= 0; --i) {
4740         ge_p2_dbl(&t, r);
4741 
4742         if (aslide[i] > 0) {
4743             ge_p1p1_to_p3(&u, &t);
4744             ge_add(&t, &u, &Ai[aslide[i] / 2]);
4745         } else if (aslide[i] < 0) {
4746             ge_p1p1_to_p3(&u, &t);
4747             ge_sub(&t, &u, &Ai[(-aslide[i]) / 2]);
4748         }
4749 
4750         if (bslide[i] > 0) {
4751             ge_p1p1_to_p3(&u, &t);
4752             ge_madd(&t, &u, &Bi[bslide[i] / 2]);
4753         } else if (bslide[i] < 0) {
4754             ge_p1p1_to_p3(&u, &t);
4755             ge_msub(&t, &u, &Bi[(-bslide[i]) / 2]);
4756         }
4757 
4758         ge_p1p1_to_p2(r, &t);
4759     }
4760 }
4761 
4762 /*
4763  * The set of scalars is \Z/l
4764  * where l = 2^252 + 27742317777372353535851937790883648493.
4765  *
4766  * Input:
4767  *   s[0]+256*s[1]+...+256^63*s[63] = s
4768  *
4769  * Output:
4770  *   s[0]+256*s[1]+...+256^31*s[31] = s mod l
4771  *   where l = 2^252 + 27742317777372353535851937790883648493.
4772  *   Overwrites s in place.
4773  */
x25519_sc_reduce(uint8_t * s)4774 static void x25519_sc_reduce(uint8_t *s)
4775 {
4776     int64_t s0 = kBottom21Bits & load_3(s);
4777     int64_t s1 = kBottom21Bits & (load_4(s + 2) >> 5);
4778     int64_t s2 = kBottom21Bits & (load_3(s + 5) >> 2);
4779     int64_t s3 = kBottom21Bits & (load_4(s + 7) >> 7);
4780     int64_t s4 = kBottom21Bits & (load_4(s + 10) >> 4);
4781     int64_t s5 = kBottom21Bits & (load_3(s + 13) >> 1);
4782     int64_t s6 = kBottom21Bits & (load_4(s + 15) >> 6);
4783     int64_t s7 = kBottom21Bits & (load_3(s + 18) >> 3);
4784     int64_t s8 = kBottom21Bits & load_3(s + 21);
4785     int64_t s9 = kBottom21Bits & (load_4(s + 23) >> 5);
4786     int64_t s10 = kBottom21Bits & (load_3(s + 26) >> 2);
4787     int64_t s11 = kBottom21Bits & (load_4(s + 28) >> 7);
4788     int64_t s12 = kBottom21Bits & (load_4(s + 31) >> 4);
4789     int64_t s13 = kBottom21Bits & (load_3(s + 34) >> 1);
4790     int64_t s14 = kBottom21Bits & (load_4(s + 36) >> 6);
4791     int64_t s15 = kBottom21Bits & (load_3(s + 39) >> 3);
4792     int64_t s16 = kBottom21Bits & load_3(s + 42);
4793     int64_t s17 = kBottom21Bits & (load_4(s + 44) >> 5);
4794     int64_t s18 = kBottom21Bits & (load_3(s + 47) >> 2);
4795     int64_t s19 = kBottom21Bits & (load_4(s + 49) >> 7);
4796     int64_t s20 = kBottom21Bits & (load_4(s + 52) >> 4);
4797     int64_t s21 = kBottom21Bits & (load_3(s + 55) >> 1);
4798     int64_t s22 = kBottom21Bits & (load_4(s + 57) >> 6);
4799     int64_t s23 = (load_4(s + 60) >> 3);
4800     int64_t carry0;
4801     int64_t carry1;
4802     int64_t carry2;
4803     int64_t carry3;
4804     int64_t carry4;
4805     int64_t carry5;
4806     int64_t carry6;
4807     int64_t carry7;
4808     int64_t carry8;
4809     int64_t carry9;
4810     int64_t carry10;
4811     int64_t carry11;
4812     int64_t carry12;
4813     int64_t carry13;
4814     int64_t carry14;
4815     int64_t carry15;
4816     int64_t carry16;
4817 
4818     s11 += s23 * 666643;
4819     s12 += s23 * 470296;
4820     s13 += s23 * 654183;
4821     s14 -= s23 * 997805;
4822     s15 += s23 * 136657;
4823     s16 -= s23 * 683901;
4824     s23 = 0;
4825 
4826     s10 += s22 * 666643;
4827     s11 += s22 * 470296;
4828     s12 += s22 * 654183;
4829     s13 -= s22 * 997805;
4830     s14 += s22 * 136657;
4831     s15 -= s22 * 683901;
4832     s22 = 0;
4833 
4834     s9 += s21 * 666643;
4835     s10 += s21 * 470296;
4836     s11 += s21 * 654183;
4837     s12 -= s21 * 997805;
4838     s13 += s21 * 136657;
4839     s14 -= s21 * 683901;
4840     s21 = 0;
4841 
4842     s8 += s20 * 666643;
4843     s9 += s20 * 470296;
4844     s10 += s20 * 654183;
4845     s11 -= s20 * 997805;
4846     s12 += s20 * 136657;
4847     s13 -= s20 * 683901;
4848     s20 = 0;
4849 
4850     s7 += s19 * 666643;
4851     s8 += s19 * 470296;
4852     s9 += s19 * 654183;
4853     s10 -= s19 * 997805;
4854     s11 += s19 * 136657;
4855     s12 -= s19 * 683901;
4856     s19 = 0;
4857 
4858     s6 += s18 * 666643;
4859     s7 += s18 * 470296;
4860     s8 += s18 * 654183;
4861     s9 -= s18 * 997805;
4862     s10 += s18 * 136657;
4863     s11 -= s18 * 683901;
4864     s18 = 0;
4865 
4866     carry6 = (s6 + (1 << 20)) >> 21;
4867     s7 += carry6;
4868     s6 -= carry6 * (1 << 21);
4869     carry8 = (s8 + (1 << 20)) >> 21;
4870     s9 += carry8;
4871     s8 -= carry8 * (1 << 21);
4872     carry10 = (s10 + (1 << 20)) >> 21;
4873     s11 += carry10;
4874     s10 -= carry10 * (1 << 21);
4875     carry12 = (s12 + (1 << 20)) >> 21;
4876     s13 += carry12;
4877     s12 -= carry12 * (1 << 21);
4878     carry14 = (s14 + (1 << 20)) >> 21;
4879     s15 += carry14;
4880     s14 -= carry14 * (1 << 21);
4881     carry16 = (s16 + (1 << 20)) >> 21;
4882     s17 += carry16;
4883     s16 -= carry16 * (1 << 21);
4884 
4885     carry7 = (s7 + (1 << 20)) >> 21;
4886     s8 += carry7;
4887     s7 -= carry7 * (1 << 21);
4888     carry9 = (s9 + (1 << 20)) >> 21;
4889     s10 += carry9;
4890     s9 -= carry9 * (1 << 21);
4891     carry11 = (s11 + (1 << 20)) >> 21;
4892     s12 += carry11;
4893     s11 -= carry11 * (1 << 21);
4894     carry13 = (s13 + (1 << 20)) >> 21;
4895     s14 += carry13;
4896     s13 -= carry13 * (1 << 21);
4897     carry15 = (s15 + (1 << 20)) >> 21;
4898     s16 += carry15;
4899     s15 -= carry15 * (1 << 21);
4900 
4901     s5 += s17 * 666643;
4902     s6 += s17 * 470296;
4903     s7 += s17 * 654183;
4904     s8 -= s17 * 997805;
4905     s9 += s17 * 136657;
4906     s10 -= s17 * 683901;
4907     s17 = 0;
4908 
4909     s4 += s16 * 666643;
4910     s5 += s16 * 470296;
4911     s6 += s16 * 654183;
4912     s7 -= s16 * 997805;
4913     s8 += s16 * 136657;
4914     s9 -= s16 * 683901;
4915     s16 = 0;
4916 
4917     s3 += s15 * 666643;
4918     s4 += s15 * 470296;
4919     s5 += s15 * 654183;
4920     s6 -= s15 * 997805;
4921     s7 += s15 * 136657;
4922     s8 -= s15 * 683901;
4923     s15 = 0;
4924 
4925     s2 += s14 * 666643;
4926     s3 += s14 * 470296;
4927     s4 += s14 * 654183;
4928     s5 -= s14 * 997805;
4929     s6 += s14 * 136657;
4930     s7 -= s14 * 683901;
4931     s14 = 0;
4932 
4933     s1 += s13 * 666643;
4934     s2 += s13 * 470296;
4935     s3 += s13 * 654183;
4936     s4 -= s13 * 997805;
4937     s5 += s13 * 136657;
4938     s6 -= s13 * 683901;
4939     s13 = 0;
4940 
4941     s0 += s12 * 666643;
4942     s1 += s12 * 470296;
4943     s2 += s12 * 654183;
4944     s3 -= s12 * 997805;
4945     s4 += s12 * 136657;
4946     s5 -= s12 * 683901;
4947     s12 = 0;
4948 
4949     carry0 = (s0 + (1 << 20)) >> 21;
4950     s1 += carry0;
4951     s0 -= carry0 * (1 << 21);
4952     carry2 = (s2 + (1 << 20)) >> 21;
4953     s3 += carry2;
4954     s2 -= carry2 * (1 << 21);
4955     carry4 = (s4 + (1 << 20)) >> 21;
4956     s5 += carry4;
4957     s4 -= carry4 * (1 << 21);
4958     carry6 = (s6 + (1 << 20)) >> 21;
4959     s7 += carry6;
4960     s6 -= carry6 * (1 << 21);
4961     carry8 = (s8 + (1 << 20)) >> 21;
4962     s9 += carry8;
4963     s8 -= carry8 * (1 << 21);
4964     carry10 = (s10 + (1 << 20)) >> 21;
4965     s11 += carry10;
4966     s10 -= carry10 * (1 << 21);
4967 
4968     carry1 = (s1 + (1 << 20)) >> 21;
4969     s2 += carry1;
4970     s1 -= carry1 * (1 << 21);
4971     carry3 = (s3 + (1 << 20)) >> 21;
4972     s4 += carry3;
4973     s3 -= carry3 * (1 << 21);
4974     carry5 = (s5 + (1 << 20)) >> 21;
4975     s6 += carry5;
4976     s5 -= carry5 * (1 << 21);
4977     carry7 = (s7 + (1 << 20)) >> 21;
4978     s8 += carry7;
4979     s7 -= carry7 * (1 << 21);
4980     carry9 = (s9 + (1 << 20)) >> 21;
4981     s10 += carry9;
4982     s9 -= carry9 * (1 << 21);
4983     carry11 = (s11 + (1 << 20)) >> 21;
4984     s12 += carry11;
4985     s11 -= carry11 * (1 << 21);
4986 
4987     s0 += s12 * 666643;
4988     s1 += s12 * 470296;
4989     s2 += s12 * 654183;
4990     s3 -= s12 * 997805;
4991     s4 += s12 * 136657;
4992     s5 -= s12 * 683901;
4993     s12 = 0;
4994 
4995     carry0 = s0 >> 21;
4996     s1 += carry0;
4997     s0 -= carry0 * (1 << 21);
4998     carry1 = s1 >> 21;
4999     s2 += carry1;
5000     s1 -= carry1 * (1 << 21);
5001     carry2 = s2 >> 21;
5002     s3 += carry2;
5003     s2 -= carry2 * (1 << 21);
5004     carry3 = s3 >> 21;
5005     s4 += carry3;
5006     s3 -= carry3 * (1 << 21);
5007     carry4 = s4 >> 21;
5008     s5 += carry4;
5009     s4 -= carry4 * (1 << 21);
5010     carry5 = s5 >> 21;
5011     s6 += carry5;
5012     s5 -= carry5 * (1 << 21);
5013     carry6 = s6 >> 21;
5014     s7 += carry6;
5015     s6 -= carry6 * (1 << 21);
5016     carry7 = s7 >> 21;
5017     s8 += carry7;
5018     s7 -= carry7 * (1 << 21);
5019     carry8 = s8 >> 21;
5020     s9 += carry8;
5021     s8 -= carry8 * (1 << 21);
5022     carry9 = s9 >> 21;
5023     s10 += carry9;
5024     s9 -= carry9 * (1 << 21);
5025     carry10 = s10 >> 21;
5026     s11 += carry10;
5027     s10 -= carry10 * (1 << 21);
5028     carry11 = s11 >> 21;
5029     s12 += carry11;
5030     s11 -= carry11 * (1 << 21);
5031 
5032     s0 += s12 * 666643;
5033     s1 += s12 * 470296;
5034     s2 += s12 * 654183;
5035     s3 -= s12 * 997805;
5036     s4 += s12 * 136657;
5037     s5 -= s12 * 683901;
5038     s12 = 0;
5039 
5040     carry0 = s0 >> 21;
5041     s1 += carry0;
5042     s0 -= carry0 * (1 << 21);
5043     carry1 = s1 >> 21;
5044     s2 += carry1;
5045     s1 -= carry1 * (1 << 21);
5046     carry2 = s2 >> 21;
5047     s3 += carry2;
5048     s2 -= carry2 * (1 << 21);
5049     carry3 = s3 >> 21;
5050     s4 += carry3;
5051     s3 -= carry3 * (1 << 21);
5052     carry4 = s4 >> 21;
5053     s5 += carry4;
5054     s4 -= carry4 * (1 << 21);
5055     carry5 = s5 >> 21;
5056     s6 += carry5;
5057     s5 -= carry5 * (1 << 21);
5058     carry6 = s6 >> 21;
5059     s7 += carry6;
5060     s6 -= carry6 * (1 << 21);
5061     carry7 = s7 >> 21;
5062     s8 += carry7;
5063     s7 -= carry7 * (1 << 21);
5064     carry8 = s8 >> 21;
5065     s9 += carry8;
5066     s8 -= carry8 * (1 << 21);
5067     carry9 = s9 >> 21;
5068     s10 += carry9;
5069     s9 -= carry9 * (1 << 21);
5070     carry10 = s10 >> 21;
5071     s11 += carry10;
5072     s10 -= carry10 * (1 << 21);
5073 
5074     s[0] = (uint8_t)(s0 >> 0);
5075     s[1] = (uint8_t)(s0 >> 8);
5076     s[2] = (uint8_t)((s0 >> 16) | (s1 << 5));
5077     s[3] = (uint8_t)(s1 >> 3);
5078     s[4] = (uint8_t)(s1 >> 11);
5079     s[5] = (uint8_t)((s1 >> 19) | (s2 << 2));
5080     s[6] = (uint8_t)(s2 >> 6);
5081     s[7] = (uint8_t)((s2 >> 14) | (s3 << 7));
5082     s[8] = (uint8_t)(s3 >> 1);
5083     s[9] = (uint8_t)(s3 >> 9);
5084     s[10] = (uint8_t)((s3 >> 17) | (s4 << 4));
5085     s[11] = (uint8_t)(s4 >> 4);
5086     s[12] = (uint8_t)(s4 >> 12);
5087     s[13] = (uint8_t)((s4 >> 20) | (s5 << 1));
5088     s[14] = (uint8_t)(s5 >> 7);
5089     s[15] = (uint8_t)((s5 >> 15) | (s6 << 6));
5090     s[16] = (uint8_t)(s6 >> 2);
5091     s[17] = (uint8_t)(s6 >> 10);
5092     s[18] = (uint8_t)((s6 >> 18) | (s7 << 3));
5093     s[19] = (uint8_t)(s7 >> 5);
5094     s[20] = (uint8_t)(s7 >> 13);
5095     s[21] = (uint8_t)(s8 >> 0);
5096     s[22] = (uint8_t)(s8 >> 8);
5097     s[23] = (uint8_t)((s8 >> 16) | (s9 << 5));
5098     s[24] = (uint8_t)(s9 >> 3);
5099     s[25] = (uint8_t)(s9 >> 11);
5100     s[26] = (uint8_t)((s9 >> 19) | (s10 << 2));
5101     s[27] = (uint8_t)(s10 >> 6);
5102     s[28] = (uint8_t)((s10 >> 14) | (s11 << 7));
5103     s[29] = (uint8_t)(s11 >> 1);
5104     s[30] = (uint8_t)(s11 >> 9);
5105     s[31] = (uint8_t)(s11 >> 17);
5106 }
5107 
5108 /*
5109  * Input:
5110  *   a[0]+256*a[1]+...+256^31*a[31] = a
5111  *   b[0]+256*b[1]+...+256^31*b[31] = b
5112  *   c[0]+256*c[1]+...+256^31*c[31] = c
5113  *
5114  * Output:
5115  *   s[0]+256*s[1]+...+256^31*s[31] = (ab+c) mod l
5116  *   where l = 2^252 + 27742317777372353535851937790883648493.
5117  */
sc_muladd(uint8_t * s,const uint8_t * a,const uint8_t * b,const uint8_t * c)5118 static void sc_muladd(uint8_t *s, const uint8_t *a, const uint8_t *b,
5119     const uint8_t *c)
5120 {
5121     int64_t a0 = kBottom21Bits & load_3(a);
5122     int64_t a1 = kBottom21Bits & (load_4(a + 2) >> 5);
5123     int64_t a2 = kBottom21Bits & (load_3(a + 5) >> 2);
5124     int64_t a3 = kBottom21Bits & (load_4(a + 7) >> 7);
5125     int64_t a4 = kBottom21Bits & (load_4(a + 10) >> 4);
5126     int64_t a5 = kBottom21Bits & (load_3(a + 13) >> 1);
5127     int64_t a6 = kBottom21Bits & (load_4(a + 15) >> 6);
5128     int64_t a7 = kBottom21Bits & (load_3(a + 18) >> 3);
5129     int64_t a8 = kBottom21Bits & load_3(a + 21);
5130     int64_t a9 = kBottom21Bits & (load_4(a + 23) >> 5);
5131     int64_t a10 = kBottom21Bits & (load_3(a + 26) >> 2);
5132     int64_t a11 = (load_4(a + 28) >> 7);
5133     int64_t b0 = kBottom21Bits & load_3(b);
5134     int64_t b1 = kBottom21Bits & (load_4(b + 2) >> 5);
5135     int64_t b2 = kBottom21Bits & (load_3(b + 5) >> 2);
5136     int64_t b3 = kBottom21Bits & (load_4(b + 7) >> 7);
5137     int64_t b4 = kBottom21Bits & (load_4(b + 10) >> 4);
5138     int64_t b5 = kBottom21Bits & (load_3(b + 13) >> 1);
5139     int64_t b6 = kBottom21Bits & (load_4(b + 15) >> 6);
5140     int64_t b7 = kBottom21Bits & (load_3(b + 18) >> 3);
5141     int64_t b8 = kBottom21Bits & load_3(b + 21);
5142     int64_t b9 = kBottom21Bits & (load_4(b + 23) >> 5);
5143     int64_t b10 = kBottom21Bits & (load_3(b + 26) >> 2);
5144     int64_t b11 = (load_4(b + 28) >> 7);
5145     int64_t c0 = kBottom21Bits & load_3(c);
5146     int64_t c1 = kBottom21Bits & (load_4(c + 2) >> 5);
5147     int64_t c2 = kBottom21Bits & (load_3(c + 5) >> 2);
5148     int64_t c3 = kBottom21Bits & (load_4(c + 7) >> 7);
5149     int64_t c4 = kBottom21Bits & (load_4(c + 10) >> 4);
5150     int64_t c5 = kBottom21Bits & (load_3(c + 13) >> 1);
5151     int64_t c6 = kBottom21Bits & (load_4(c + 15) >> 6);
5152     int64_t c7 = kBottom21Bits & (load_3(c + 18) >> 3);
5153     int64_t c8 = kBottom21Bits & load_3(c + 21);
5154     int64_t c9 = kBottom21Bits & (load_4(c + 23) >> 5);
5155     int64_t c10 = kBottom21Bits & (load_3(c + 26) >> 2);
5156     int64_t c11 = (load_4(c + 28) >> 7);
5157     int64_t s0;
5158     int64_t s1;
5159     int64_t s2;
5160     int64_t s3;
5161     int64_t s4;
5162     int64_t s5;
5163     int64_t s6;
5164     int64_t s7;
5165     int64_t s8;
5166     int64_t s9;
5167     int64_t s10;
5168     int64_t s11;
5169     int64_t s12;
5170     int64_t s13;
5171     int64_t s14;
5172     int64_t s15;
5173     int64_t s16;
5174     int64_t s17;
5175     int64_t s18;
5176     int64_t s19;
5177     int64_t s20;
5178     int64_t s21;
5179     int64_t s22;
5180     int64_t s23;
5181     int64_t carry0;
5182     int64_t carry1;
5183     int64_t carry2;
5184     int64_t carry3;
5185     int64_t carry4;
5186     int64_t carry5;
5187     int64_t carry6;
5188     int64_t carry7;
5189     int64_t carry8;
5190     int64_t carry9;
5191     int64_t carry10;
5192     int64_t carry11;
5193     int64_t carry12;
5194     int64_t carry13;
5195     int64_t carry14;
5196     int64_t carry15;
5197     int64_t carry16;
5198     int64_t carry17;
5199     int64_t carry18;
5200     int64_t carry19;
5201     int64_t carry20;
5202     int64_t carry21;
5203     int64_t carry22;
5204 
5205     s0 = c0 + a0 * b0;
5206     s1 = c1 + a0 * b1 + a1 * b0;
5207     s2 = c2 + a0 * b2 + a1 * b1 + a2 * b0;
5208     s3 = c3 + a0 * b3 + a1 * b2 + a2 * b1 + a3 * b0;
5209     s4 = c4 + a0 * b4 + a1 * b3 + a2 * b2 + a3 * b1 + a4 * b0;
5210     s5 = c5 + a0 * b5 + a1 * b4 + a2 * b3 + a3 * b2 + a4 * b1 + a5 * b0;
5211     s6 = c6 + a0 * b6 + a1 * b5 + a2 * b4 + a3 * b3 + a4 * b2 + a5 * b1 + a6 * b0;
5212     s7 = c7 + a0 * b7 + a1 * b6 + a2 * b5 + a3 * b4 + a4 * b3 + a5 * b2 + a6 * b1 + a7 * b0;
5213     s8 = c8 + a0 * b8 + a1 * b7 + a2 * b6 + a3 * b5 + a4 * b4 + a5 * b3 + a6 * b2 + a7 * b1 + a8 * b0;
5214     s9 = c9 + a0 * b9 + a1 * b8 + a2 * b7 + a3 * b6 + a4 * b5 + a5 * b4 + a6 * b3 + a7 * b2 + a8 * b1 + a9 * b0;
5215     s10 = c10 + a0 * b10 + a1 * b9 + a2 * b8 + a3 * b7 + a4 * b6 + a5 * b5 + a6 * b4 + a7 * b3 + a8 * b2 + a9 * b1 + a10 * b0;
5216     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;
5217     s12 = a1 * b11 + a2 * b10 + a3 * b9 + a4 * b8 + a5 * b7 + a6 * b6 + a7 * b5 + a8 * b4 + a9 * b3 + a10 * b2 + a11 * b1;
5218     s13 = a2 * b11 + a3 * b10 + a4 * b9 + a5 * b8 + a6 * b7 + a7 * b6 + a8 * b5 + a9 * b4 + a10 * b3 + a11 * b2;
5219     s14 = a3 * b11 + a4 * b10 + a5 * b9 + a6 * b8 + a7 * b7 + a8 * b6 + a9 * b5 + a10 * b4 + a11 * b3;
5220     s15 = a4 * b11 + a5 * b10 + a6 * b9 + a7 * b8 + a8 * b7 + a9 * b6 + a10 * b5 + a11 * b4;
5221     s16 = a5 * b11 + a6 * b10 + a7 * b9 + a8 * b8 + a9 * b7 + a10 * b6 + a11 * b5;
5222     s17 = a6 * b11 + a7 * b10 + a8 * b9 + a9 * b8 + a10 * b7 + a11 * b6;
5223     s18 = a7 * b11 + a8 * b10 + a9 * b9 + a10 * b8 + a11 * b7;
5224     s19 = a8 * b11 + a9 * b10 + a10 * b9 + a11 * b8;
5225     s20 = a9 * b11 + a10 * b10 + a11 * b9;
5226     s21 = a10 * b11 + a11 * b10;
5227     s22 = a11 * b11;
5228     s23 = 0;
5229 
5230     carry0 = (s0 + (1 << 20)) >> 21;
5231     s1 += carry0;
5232     s0 -= carry0 * (1 << 21);
5233     carry2 = (s2 + (1 << 20)) >> 21;
5234     s3 += carry2;
5235     s2 -= carry2 * (1 << 21);
5236     carry4 = (s4 + (1 << 20)) >> 21;
5237     s5 += carry4;
5238     s4 -= carry4 * (1 << 21);
5239     carry6 = (s6 + (1 << 20)) >> 21;
5240     s7 += carry6;
5241     s6 -= carry6 * (1 << 21);
5242     carry8 = (s8 + (1 << 20)) >> 21;
5243     s9 += carry8;
5244     s8 -= carry8 * (1 << 21);
5245     carry10 = (s10 + (1 << 20)) >> 21;
5246     s11 += carry10;
5247     s10 -= carry10 * (1 << 21);
5248     carry12 = (s12 + (1 << 20)) >> 21;
5249     s13 += carry12;
5250     s12 -= carry12 * (1 << 21);
5251     carry14 = (s14 + (1 << 20)) >> 21;
5252     s15 += carry14;
5253     s14 -= carry14 * (1 << 21);
5254     carry16 = (s16 + (1 << 20)) >> 21;
5255     s17 += carry16;
5256     s16 -= carry16 * (1 << 21);
5257     carry18 = (s18 + (1 << 20)) >> 21;
5258     s19 += carry18;
5259     s18 -= carry18 * (1 << 21);
5260     carry20 = (s20 + (1 << 20)) >> 21;
5261     s21 += carry20;
5262     s20 -= carry20 * (1 << 21);
5263     carry22 = (s22 + (1 << 20)) >> 21;
5264     s23 += carry22;
5265     s22 -= carry22 * (1 << 21);
5266 
5267     carry1 = (s1 + (1 << 20)) >> 21;
5268     s2 += carry1;
5269     s1 -= carry1 * (1 << 21);
5270     carry3 = (s3 + (1 << 20)) >> 21;
5271     s4 += carry3;
5272     s3 -= carry3 * (1 << 21);
5273     carry5 = (s5 + (1 << 20)) >> 21;
5274     s6 += carry5;
5275     s5 -= carry5 * (1 << 21);
5276     carry7 = (s7 + (1 << 20)) >> 21;
5277     s8 += carry7;
5278     s7 -= carry7 * (1 << 21);
5279     carry9 = (s9 + (1 << 20)) >> 21;
5280     s10 += carry9;
5281     s9 -= carry9 * (1 << 21);
5282     carry11 = (s11 + (1 << 20)) >> 21;
5283     s12 += carry11;
5284     s11 -= carry11 * (1 << 21);
5285     carry13 = (s13 + (1 << 20)) >> 21;
5286     s14 += carry13;
5287     s13 -= carry13 * (1 << 21);
5288     carry15 = (s15 + (1 << 20)) >> 21;
5289     s16 += carry15;
5290     s15 -= carry15 * (1 << 21);
5291     carry17 = (s17 + (1 << 20)) >> 21;
5292     s18 += carry17;
5293     s17 -= carry17 * (1 << 21);
5294     carry19 = (s19 + (1 << 20)) >> 21;
5295     s20 += carry19;
5296     s19 -= carry19 * (1 << 21);
5297     carry21 = (s21 + (1 << 20)) >> 21;
5298     s22 += carry21;
5299     s21 -= carry21 * (1 << 21);
5300 
5301     s11 += s23 * 666643;
5302     s12 += s23 * 470296;
5303     s13 += s23 * 654183;
5304     s14 -= s23 * 997805;
5305     s15 += s23 * 136657;
5306     s16 -= s23 * 683901;
5307     s23 = 0;
5308 
5309     s10 += s22 * 666643;
5310     s11 += s22 * 470296;
5311     s12 += s22 * 654183;
5312     s13 -= s22 * 997805;
5313     s14 += s22 * 136657;
5314     s15 -= s22 * 683901;
5315     s22 = 0;
5316 
5317     s9 += s21 * 666643;
5318     s10 += s21 * 470296;
5319     s11 += s21 * 654183;
5320     s12 -= s21 * 997805;
5321     s13 += s21 * 136657;
5322     s14 -= s21 * 683901;
5323     s21 = 0;
5324 
5325     s8 += s20 * 666643;
5326     s9 += s20 * 470296;
5327     s10 += s20 * 654183;
5328     s11 -= s20 * 997805;
5329     s12 += s20 * 136657;
5330     s13 -= s20 * 683901;
5331     s20 = 0;
5332 
5333     s7 += s19 * 666643;
5334     s8 += s19 * 470296;
5335     s9 += s19 * 654183;
5336     s10 -= s19 * 997805;
5337     s11 += s19 * 136657;
5338     s12 -= s19 * 683901;
5339     s19 = 0;
5340 
5341     s6 += s18 * 666643;
5342     s7 += s18 * 470296;
5343     s8 += s18 * 654183;
5344     s9 -= s18 * 997805;
5345     s10 += s18 * 136657;
5346     s11 -= s18 * 683901;
5347     s18 = 0;
5348 
5349     carry6 = (s6 + (1 << 20)) >> 21;
5350     s7 += carry6;
5351     s6 -= carry6 * (1 << 21);
5352     carry8 = (s8 + (1 << 20)) >> 21;
5353     s9 += carry8;
5354     s8 -= carry8 * (1 << 21);
5355     carry10 = (s10 + (1 << 20)) >> 21;
5356     s11 += carry10;
5357     s10 -= carry10 * (1 << 21);
5358     carry12 = (s12 + (1 << 20)) >> 21;
5359     s13 += carry12;
5360     s12 -= carry12 * (1 << 21);
5361     carry14 = (s14 + (1 << 20)) >> 21;
5362     s15 += carry14;
5363     s14 -= carry14 * (1 << 21);
5364     carry16 = (s16 + (1 << 20)) >> 21;
5365     s17 += carry16;
5366     s16 -= carry16 * (1 << 21);
5367 
5368     carry7 = (s7 + (1 << 20)) >> 21;
5369     s8 += carry7;
5370     s7 -= carry7 * (1 << 21);
5371     carry9 = (s9 + (1 << 20)) >> 21;
5372     s10 += carry9;
5373     s9 -= carry9 * (1 << 21);
5374     carry11 = (s11 + (1 << 20)) >> 21;
5375     s12 += carry11;
5376     s11 -= carry11 * (1 << 21);
5377     carry13 = (s13 + (1 << 20)) >> 21;
5378     s14 += carry13;
5379     s13 -= carry13 * (1 << 21);
5380     carry15 = (s15 + (1 << 20)) >> 21;
5381     s16 += carry15;
5382     s15 -= carry15 * (1 << 21);
5383 
5384     s5 += s17 * 666643;
5385     s6 += s17 * 470296;
5386     s7 += s17 * 654183;
5387     s8 -= s17 * 997805;
5388     s9 += s17 * 136657;
5389     s10 -= s17 * 683901;
5390     s17 = 0;
5391 
5392     s4 += s16 * 666643;
5393     s5 += s16 * 470296;
5394     s6 += s16 * 654183;
5395     s7 -= s16 * 997805;
5396     s8 += s16 * 136657;
5397     s9 -= s16 * 683901;
5398     s16 = 0;
5399 
5400     s3 += s15 * 666643;
5401     s4 += s15 * 470296;
5402     s5 += s15 * 654183;
5403     s6 -= s15 * 997805;
5404     s7 += s15 * 136657;
5405     s8 -= s15 * 683901;
5406     s15 = 0;
5407 
5408     s2 += s14 * 666643;
5409     s3 += s14 * 470296;
5410     s4 += s14 * 654183;
5411     s5 -= s14 * 997805;
5412     s6 += s14 * 136657;
5413     s7 -= s14 * 683901;
5414     s14 = 0;
5415 
5416     s1 += s13 * 666643;
5417     s2 += s13 * 470296;
5418     s3 += s13 * 654183;
5419     s4 -= s13 * 997805;
5420     s5 += s13 * 136657;
5421     s6 -= s13 * 683901;
5422     s13 = 0;
5423 
5424     s0 += s12 * 666643;
5425     s1 += s12 * 470296;
5426     s2 += s12 * 654183;
5427     s3 -= s12 * 997805;
5428     s4 += s12 * 136657;
5429     s5 -= s12 * 683901;
5430     s12 = 0;
5431 
5432     carry0 = (s0 + (1 << 20)) >> 21;
5433     s1 += carry0;
5434     s0 -= carry0 * (1 << 21);
5435     carry2 = (s2 + (1 << 20)) >> 21;
5436     s3 += carry2;
5437     s2 -= carry2 * (1 << 21);
5438     carry4 = (s4 + (1 << 20)) >> 21;
5439     s5 += carry4;
5440     s4 -= carry4 * (1 << 21);
5441     carry6 = (s6 + (1 << 20)) >> 21;
5442     s7 += carry6;
5443     s6 -= carry6 * (1 << 21);
5444     carry8 = (s8 + (1 << 20)) >> 21;
5445     s9 += carry8;
5446     s8 -= carry8 * (1 << 21);
5447     carry10 = (s10 + (1 << 20)) >> 21;
5448     s11 += carry10;
5449     s10 -= carry10 * (1 << 21);
5450 
5451     carry1 = (s1 + (1 << 20)) >> 21;
5452     s2 += carry1;
5453     s1 -= carry1 * (1 << 21);
5454     carry3 = (s3 + (1 << 20)) >> 21;
5455     s4 += carry3;
5456     s3 -= carry3 * (1 << 21);
5457     carry5 = (s5 + (1 << 20)) >> 21;
5458     s6 += carry5;
5459     s5 -= carry5 * (1 << 21);
5460     carry7 = (s7 + (1 << 20)) >> 21;
5461     s8 += carry7;
5462     s7 -= carry7 * (1 << 21);
5463     carry9 = (s9 + (1 << 20)) >> 21;
5464     s10 += carry9;
5465     s9 -= carry9 * (1 << 21);
5466     carry11 = (s11 + (1 << 20)) >> 21;
5467     s12 += carry11;
5468     s11 -= carry11 * (1 << 21);
5469 
5470     s0 += s12 * 666643;
5471     s1 += s12 * 470296;
5472     s2 += s12 * 654183;
5473     s3 -= s12 * 997805;
5474     s4 += s12 * 136657;
5475     s5 -= s12 * 683901;
5476     s12 = 0;
5477 
5478     carry0 = s0 >> 21;
5479     s1 += carry0;
5480     s0 -= carry0 * (1 << 21);
5481     carry1 = s1 >> 21;
5482     s2 += carry1;
5483     s1 -= carry1 * (1 << 21);
5484     carry2 = s2 >> 21;
5485     s3 += carry2;
5486     s2 -= carry2 * (1 << 21);
5487     carry3 = s3 >> 21;
5488     s4 += carry3;
5489     s3 -= carry3 * (1 << 21);
5490     carry4 = s4 >> 21;
5491     s5 += carry4;
5492     s4 -= carry4 * (1 << 21);
5493     carry5 = s5 >> 21;
5494     s6 += carry5;
5495     s5 -= carry5 * (1 << 21);
5496     carry6 = s6 >> 21;
5497     s7 += carry6;
5498     s6 -= carry6 * (1 << 21);
5499     carry7 = s7 >> 21;
5500     s8 += carry7;
5501     s7 -= carry7 * (1 << 21);
5502     carry8 = s8 >> 21;
5503     s9 += carry8;
5504     s8 -= carry8 * (1 << 21);
5505     carry9 = s9 >> 21;
5506     s10 += carry9;
5507     s9 -= carry9 * (1 << 21);
5508     carry10 = s10 >> 21;
5509     s11 += carry10;
5510     s10 -= carry10 * (1 << 21);
5511     carry11 = s11 >> 21;
5512     s12 += carry11;
5513     s11 -= carry11 * (1 << 21);
5514 
5515     s0 += s12 * 666643;
5516     s1 += s12 * 470296;
5517     s2 += s12 * 654183;
5518     s3 -= s12 * 997805;
5519     s4 += s12 * 136657;
5520     s5 -= s12 * 683901;
5521     s12 = 0;
5522 
5523     carry0 = s0 >> 21;
5524     s1 += carry0;
5525     s0 -= carry0 * (1 << 21);
5526     carry1 = s1 >> 21;
5527     s2 += carry1;
5528     s1 -= carry1 * (1 << 21);
5529     carry2 = s2 >> 21;
5530     s3 += carry2;
5531     s2 -= carry2 * (1 << 21);
5532     carry3 = s3 >> 21;
5533     s4 += carry3;
5534     s3 -= carry3 * (1 << 21);
5535     carry4 = s4 >> 21;
5536     s5 += carry4;
5537     s4 -= carry4 * (1 << 21);
5538     carry5 = s5 >> 21;
5539     s6 += carry5;
5540     s5 -= carry5 * (1 << 21);
5541     carry6 = s6 >> 21;
5542     s7 += carry6;
5543     s6 -= carry6 * (1 << 21);
5544     carry7 = s7 >> 21;
5545     s8 += carry7;
5546     s7 -= carry7 * (1 << 21);
5547     carry8 = s8 >> 21;
5548     s9 += carry8;
5549     s8 -= carry8 * (1 << 21);
5550     carry9 = s9 >> 21;
5551     s10 += carry9;
5552     s9 -= carry9 * (1 << 21);
5553     carry10 = s10 >> 21;
5554     s11 += carry10;
5555     s10 -= carry10 * (1 << 21);
5556 
5557     s[0] = (uint8_t)(s0 >> 0);
5558     s[1] = (uint8_t)(s0 >> 8);
5559     s[2] = (uint8_t)((s0 >> 16) | (s1 << 5));
5560     s[3] = (uint8_t)(s1 >> 3);
5561     s[4] = (uint8_t)(s1 >> 11);
5562     s[5] = (uint8_t)((s1 >> 19) | (s2 << 2));
5563     s[6] = (uint8_t)(s2 >> 6);
5564     s[7] = (uint8_t)((s2 >> 14) | (s3 << 7));
5565     s[8] = (uint8_t)(s3 >> 1);
5566     s[9] = (uint8_t)(s3 >> 9);
5567     s[10] = (uint8_t)((s3 >> 17) | (s4 << 4));
5568     s[11] = (uint8_t)(s4 >> 4);
5569     s[12] = (uint8_t)(s4 >> 12);
5570     s[13] = (uint8_t)((s4 >> 20) | (s5 << 1));
5571     s[14] = (uint8_t)(s5 >> 7);
5572     s[15] = (uint8_t)((s5 >> 15) | (s6 << 6));
5573     s[16] = (uint8_t)(s6 >> 2);
5574     s[17] = (uint8_t)(s6 >> 10);
5575     s[18] = (uint8_t)((s6 >> 18) | (s7 << 3));
5576     s[19] = (uint8_t)(s7 >> 5);
5577     s[20] = (uint8_t)(s7 >> 13);
5578     s[21] = (uint8_t)(s8 >> 0);
5579     s[22] = (uint8_t)(s8 >> 8);
5580     s[23] = (uint8_t)((s8 >> 16) | (s9 << 5));
5581     s[24] = (uint8_t)(s9 >> 3);
5582     s[25] = (uint8_t)(s9 >> 11);
5583     s[26] = (uint8_t)((s9 >> 19) | (s10 << 2));
5584     s[27] = (uint8_t)(s10 >> 6);
5585     s[28] = (uint8_t)((s10 >> 14) | (s11 << 7));
5586     s[29] = (uint8_t)(s11 >> 1);
5587     s[30] = (uint8_t)(s11 >> 9);
5588     s[31] = (uint8_t)(s11 >> 17);
5589 }
5590 
hash_init_with_dom(EVP_MD_CTX * hash_ctx,EVP_MD * sha512,const uint8_t dom2flag,const uint8_t phflag,const uint8_t * context,const size_t context_len)5591 static int hash_init_with_dom(EVP_MD_CTX *hash_ctx,
5592     EVP_MD *sha512,
5593     const uint8_t dom2flag,
5594     const uint8_t phflag,
5595     const uint8_t *context,
5596     const size_t context_len)
5597 {
5598     /* ASCII: "SigEd25519 no Ed25519 collisions", in hex for EBCDIC compatibility */
5599     const char dom_s[] = "\x53\x69\x67\x45\x64\x32\x35\x35\x31\x39\x20\x6e"
5600                          "\x6f\x20\x45\x64\x32\x35\x35\x31\x39\x20\x63\x6f"
5601                          "\x6c\x6c\x69\x73\x69\x6f\x6e\x73";
5602     uint8_t dom[2];
5603 
5604     if (!EVP_DigestInit_ex(hash_ctx, sha512, NULL))
5605         return 0;
5606 
5607     /* return early if dom2flag is not set */
5608     if (!dom2flag)
5609         return 1;
5610 
5611     if (context_len > UINT8_MAX)
5612         return 0;
5613 
5614     dom[0] = (uint8_t)(phflag >= 1 ? 1 : 0);
5615     dom[1] = (uint8_t)context_len;
5616 
5617     if (!EVP_DigestUpdate(hash_ctx, dom_s, sizeof(dom_s) - 1)
5618         || !EVP_DigestUpdate(hash_ctx, dom, sizeof(dom))
5619         || !EVP_DigestUpdate(hash_ctx, context, context_len)) {
5620         return 0;
5621     }
5622 
5623     return 1;
5624 }
5625 
ossl_ed25519_sign(uint8_t * out_sig,const uint8_t * tbs,size_t tbs_len,const uint8_t public_key[32],const uint8_t private_key[32],const uint8_t dom2flag,const uint8_t phflag,const uint8_t csflag,const uint8_t * context,size_t context_len,OSSL_LIB_CTX * libctx,const char * propq)5626 int ossl_ed25519_sign(uint8_t *out_sig, const uint8_t *tbs, size_t tbs_len,
5627     const uint8_t public_key[32], const uint8_t private_key[32],
5628     const uint8_t dom2flag, const uint8_t phflag, const uint8_t csflag,
5629     const uint8_t *context, size_t context_len,
5630     OSSL_LIB_CTX *libctx, const char *propq)
5631 {
5632     uint8_t az[SHA512_DIGEST_LENGTH];
5633     uint8_t nonce[SHA512_DIGEST_LENGTH];
5634     ge_p3 R;
5635     uint8_t hram[SHA512_DIGEST_LENGTH];
5636     EVP_MD *sha512 = EVP_MD_fetch(libctx, SN_sha512, propq);
5637     EVP_MD_CTX *hash_ctx = EVP_MD_CTX_new();
5638     unsigned int sz;
5639     int res = 0;
5640 
5641     if (context == NULL)
5642         context_len = 0;
5643 
5644     /* if csflag is set, then a non-empty context-string is required */
5645     if (csflag && context_len == 0)
5646         goto err;
5647 
5648     /* if dom2flag is not set, then an empty context-string is required */
5649     if (!dom2flag && context_len > 0)
5650         goto err;
5651 
5652     if (sha512 == NULL || hash_ctx == NULL)
5653         goto err;
5654 
5655     if (!EVP_DigestInit_ex(hash_ctx, sha512, NULL)
5656         || !EVP_DigestUpdate(hash_ctx, private_key, 32)
5657         || !EVP_DigestFinal_ex(hash_ctx, az, &sz))
5658         goto err;
5659 
5660     az[0] &= 248;
5661     az[31] &= 63;
5662     az[31] |= 64;
5663 
5664     if (!hash_init_with_dom(hash_ctx, sha512, dom2flag, phflag, context, context_len)
5665         || !EVP_DigestUpdate(hash_ctx, az + 32, 32)
5666         || !EVP_DigestUpdate(hash_ctx, tbs, tbs_len)
5667         || !EVP_DigestFinal_ex(hash_ctx, nonce, &sz))
5668         goto err;
5669 
5670     x25519_sc_reduce(nonce);
5671     ge_scalarmult_base(&R, nonce);
5672     ge_p3_tobytes(out_sig, &R);
5673 
5674     if (!hash_init_with_dom(hash_ctx, sha512, dom2flag, phflag, context, context_len)
5675         || !EVP_DigestUpdate(hash_ctx, out_sig, 32)
5676         || !EVP_DigestUpdate(hash_ctx, public_key, 32)
5677         || !EVP_DigestUpdate(hash_ctx, tbs, tbs_len)
5678         || !EVP_DigestFinal_ex(hash_ctx, hram, &sz))
5679         goto err;
5680 
5681     x25519_sc_reduce(hram);
5682     sc_muladd(out_sig + 32, hram, az, nonce);
5683 
5684     res = 1;
5685 err:
5686     OPENSSL_cleanse(nonce, sizeof(nonce));
5687     OPENSSL_cleanse(az, sizeof(az));
5688     EVP_MD_free(sha512);
5689     EVP_MD_CTX_free(hash_ctx);
5690     return res;
5691 }
5692 
5693 /*
5694  * This function should not be necessary since ossl_ed25519_verify() already
5695  * does this check internally.
5696  * For some reason the FIPS ACVP requires a EDDSA KeyVer test.
5697  */
ossl_ed25519_pubkey_verify(const uint8_t * pub,size_t pub_len)5698 int ossl_ed25519_pubkey_verify(const uint8_t *pub, size_t pub_len)
5699 {
5700     ge_p3 A;
5701 
5702     if (pub_len != ED25519_KEYLEN)
5703         return 0;
5704     return (ge_frombytes_vartime(&A, pub) == 0);
5705 }
5706 
5707 static const char allzeroes[15];
5708 
ossl_ed25519_verify(const uint8_t * tbs,size_t tbs_len,const uint8_t signature[64],const uint8_t public_key[32],const uint8_t dom2flag,const uint8_t phflag,const uint8_t csflag,const uint8_t * context,size_t context_len,OSSL_LIB_CTX * libctx,const char * propq)5709 int ossl_ed25519_verify(const uint8_t *tbs, size_t tbs_len,
5710     const uint8_t signature[64], const uint8_t public_key[32],
5711     const uint8_t dom2flag, const uint8_t phflag, const uint8_t csflag,
5712     const uint8_t *context, size_t context_len,
5713     OSSL_LIB_CTX *libctx, const char *propq)
5714 {
5715     int i;
5716     ge_p3 A;
5717     const uint8_t *r, *s;
5718     EVP_MD *sha512;
5719     EVP_MD_CTX *hash_ctx = NULL;
5720     unsigned int sz;
5721     int res = 0;
5722     ge_p2 R;
5723     uint8_t rcheck[32];
5724     uint8_t h[SHA512_DIGEST_LENGTH];
5725     /* 27742317777372353535851937790883648493 in little endian format */
5726     const uint8_t l_low[16] = {
5727         0xED, 0xD3, 0xF5, 0x5C, 0x1A, 0x63, 0x12, 0x58, 0xD6, 0x9C, 0xF7, 0xA2,
5728         0xDE, 0xF9, 0xDE, 0x14
5729     };
5730 
5731     if (context == NULL)
5732         context_len = 0;
5733 
5734     /* if csflag is set, then a non-empty context-string is required */
5735     if (csflag && context_len == 0)
5736         return 0;
5737 
5738     /* if dom2flag is not set, then an empty context-string is required */
5739     if (!dom2flag && context_len > 0)
5740         return 0;
5741 
5742     r = signature;
5743     s = signature + 32;
5744 
5745     /*
5746      * Check 0 <= s < L where L = 2^252 + 27742317777372353535851937790883648493
5747      *
5748      * If not the signature is publicly invalid. Since it's public we can do the
5749      * check in variable time.
5750      *
5751      * First check the most significant byte
5752      */
5753     if (s[31] > 0x10)
5754         return 0;
5755     if (s[31] == 0x10) {
5756         /*
5757          * Most significant byte indicates a value close to 2^252 so check the
5758          * rest
5759          */
5760         if (memcmp(s + 16, allzeroes, sizeof(allzeroes)) != 0)
5761             return 0;
5762         for (i = 15; i >= 0; i--) {
5763             if (s[i] < l_low[i])
5764                 break;
5765             if (s[i] > l_low[i])
5766                 return 0;
5767         }
5768         if (i < 0)
5769             return 0;
5770     }
5771 
5772     if (ge_frombytes_vartime(&A, public_key) != 0) {
5773         return 0;
5774     }
5775 
5776     fe_neg(A.X, A.X);
5777     fe_neg(A.T, A.T);
5778 
5779     sha512 = EVP_MD_fetch(libctx, SN_sha512, propq);
5780     if (sha512 == NULL)
5781         return 0;
5782     hash_ctx = EVP_MD_CTX_new();
5783     if (hash_ctx == NULL)
5784         goto err;
5785 
5786     if (!hash_init_with_dom(hash_ctx, sha512, dom2flag, phflag, context, context_len)
5787         || !EVP_DigestUpdate(hash_ctx, r, 32)
5788         || !EVP_DigestUpdate(hash_ctx, public_key, 32)
5789         || !EVP_DigestUpdate(hash_ctx, tbs, tbs_len)
5790         || !EVP_DigestFinal_ex(hash_ctx, h, &sz))
5791         goto err;
5792 
5793     x25519_sc_reduce(h);
5794 
5795     ge_double_scalarmult_vartime(&R, h, &A, s);
5796 
5797     ge_tobytes(rcheck, &R);
5798 
5799     res = CRYPTO_memcmp(rcheck, r, sizeof(rcheck)) == 0;
5800 
5801     /* note that we have used the strict verification equation here.
5802      * we checked that  ENC( [h](-A) + [s]B ) == r
5803      * B is the base point.
5804      *
5805      * the less strict verification equation uses the curve cofactor:
5806      *          [h*8](-A) + [s*8]B == [8]R
5807      */
5808 err:
5809     EVP_MD_free(sha512);
5810     EVP_MD_CTX_free(hash_ctx);
5811     return res;
5812 }
5813 
ossl_ed25519_public_from_private(OSSL_LIB_CTX * ctx,uint8_t out_public_key[32],const uint8_t private_key[32],const char * propq)5814 int ossl_ed25519_public_from_private(OSSL_LIB_CTX *ctx, uint8_t out_public_key[32],
5815     const uint8_t private_key[32],
5816     const char *propq)
5817 {
5818     uint8_t az[SHA512_DIGEST_LENGTH];
5819     ge_p3 A;
5820     int r;
5821     EVP_MD *sha512 = NULL;
5822 
5823     sha512 = EVP_MD_fetch(ctx, SN_sha512, propq);
5824     if (sha512 == NULL)
5825         return 0;
5826     r = EVP_Digest(private_key, 32, az, NULL, sha512, NULL);
5827     EVP_MD_free(sha512);
5828     if (!r) {
5829         OPENSSL_cleanse(az, sizeof(az));
5830         return 0;
5831     }
5832 
5833     az[0] &= 248;
5834     az[31] &= 63;
5835     az[31] |= 64;
5836 
5837     ge_scalarmult_base(&A, az);
5838     ge_p3_tobytes(out_public_key, &A);
5839 
5840     OPENSSL_cleanse(az, sizeof(az));
5841     return 1;
5842 }
5843 
ossl_x25519(uint8_t out_shared_key[32],const uint8_t private_key[32],const uint8_t peer_public_value[32])5844 int ossl_x25519(uint8_t out_shared_key[32], const uint8_t private_key[32],
5845     const uint8_t peer_public_value[32])
5846 {
5847     static const uint8_t kZeros[32] = { 0 };
5848     x25519_scalar_mult(out_shared_key, private_key, peer_public_value);
5849     /* The all-zero output results when the input is a point of small order. */
5850     return CRYPTO_memcmp(kZeros, out_shared_key, 32) != 0;
5851 }
5852 
ossl_x25519_public_from_private(uint8_t out_public_value[32],const uint8_t private_key[32])5853 void ossl_x25519_public_from_private(uint8_t out_public_value[32],
5854     const uint8_t private_key[32])
5855 {
5856     uint8_t e[32];
5857     ge_p3 A;
5858     fe zplusy, zminusy, zminusy_inv;
5859 
5860     memcpy(e, private_key, 32);
5861     e[0] &= 248;
5862     e[31] &= 127;
5863     e[31] |= 64;
5864 
5865     ge_scalarmult_base(&A, e);
5866 
5867     /*
5868      * We only need the u-coordinate of the curve25519 point.
5869      * The map is u=(y+1)/(1-y). Since y=Y/Z, this gives
5870      * u=(Z+Y)/(Z-Y).
5871      */
5872     fe_add(zplusy, A.Z, A.Y);
5873     fe_sub(zminusy, A.Z, A.Y);
5874     fe_invert(zminusy_inv, zminusy);
5875     fe_mul(zplusy, zplusy, zminusy_inv);
5876     fe_tobytes(out_public_value, zplusy);
5877 
5878     OPENSSL_cleanse(e, sizeof(e));
5879 }
5880