1 /*
2  * Copyright 2016-2019 The OpenSSL Project Authors. All Rights Reserved.
3  *
4  * Licensed under the OpenSSL license (the "License").  You may not use
5  * this file except in compliance with the License.  You can obtain a copy
6  * in the file LICENSE in the source distribution or at
7  * https://www.openssl.org/source/license.html
8  */
9 
10 #include <string.h>
11 #include "ec_local.h"
12 #include <openssl/sha.h>
13 
14 #if defined(X25519_ASM) && (defined(__x86_64) || defined(__x86_64__) || \
15                             defined(_M_AMD64) || defined(_M_X64))
16 
17 # define BASE_2_64_IMPLEMENTED
18 
19 typedef uint64_t fe64[4];
20 
21 int x25519_fe64_eligible(void);
22 
23 /*
24  * Following subroutines perform corresponding operations modulo
25  * 2^256-38, i.e. double the curve modulus. However, inputs and
26  * outputs are permitted to be partially reduced, i.e. to remain
27  * in [0..2^256) range. It's all tied up in final fe64_tobytes
28  * that performs full reduction modulo 2^255-19.
29  *
30  * There are no reference C implementations for these.
31  */
32 void x25519_fe64_mul(fe64 h, const fe64 f, const fe64 g);
33 void x25519_fe64_sqr(fe64 h, const fe64 f);
34 void x25519_fe64_mul121666(fe64 h, fe64 f);
35 void x25519_fe64_add(fe64 h, const fe64 f, const fe64 g);
36 void x25519_fe64_sub(fe64 h, const fe64 f, const fe64 g);
37 void x25519_fe64_tobytes(uint8_t *s, const fe64 f);
38 # define fe64_mul x25519_fe64_mul
39 # define fe64_sqr x25519_fe64_sqr
40 # define fe64_mul121666 x25519_fe64_mul121666
41 # define fe64_add x25519_fe64_add
42 # define fe64_sub x25519_fe64_sub
43 # define fe64_tobytes x25519_fe64_tobytes
44 
load_8(const uint8_t * in)45 static uint64_t load_8(const uint8_t *in)
46 {
47     uint64_t result;
48 
49     result = in[0];
50     result |= ((uint64_t)in[1]) << 8;
51     result |= ((uint64_t)in[2]) << 16;
52     result |= ((uint64_t)in[3]) << 24;
53     result |= ((uint64_t)in[4]) << 32;
54     result |= ((uint64_t)in[5]) << 40;
55     result |= ((uint64_t)in[6]) << 48;
56     result |= ((uint64_t)in[7]) << 56;
57 
58     return result;
59 }
60 
fe64_frombytes(fe64 h,const uint8_t * s)61 static void fe64_frombytes(fe64 h, const uint8_t *s)
62 {
63     h[0] = load_8(s);
64     h[1] = load_8(s + 8);
65     h[2] = load_8(s + 16);
66     h[3] = load_8(s + 24) & 0x7fffffffffffffff;
67 }
68 
fe64_0(fe64 h)69 static void fe64_0(fe64 h)
70 {
71     h[0] = 0;
72     h[1] = 0;
73     h[2] = 0;
74     h[3] = 0;
75 }
76 
fe64_1(fe64 h)77 static void fe64_1(fe64 h)
78 {
79     h[0] = 1;
80     h[1] = 0;
81     h[2] = 0;
82     h[3] = 0;
83 }
84 
fe64_copy(fe64 h,const fe64 f)85 static void fe64_copy(fe64 h, const fe64 f)
86 {
87     h[0] = f[0];
88     h[1] = f[1];
89     h[2] = f[2];
90     h[3] = f[3];
91 }
92 
fe64_cswap(fe64 f,fe64 g,unsigned int b)93 static void fe64_cswap(fe64 f, fe64 g, unsigned int b)
94 {
95     int i;
96     uint64_t mask = 0 - (uint64_t)b;
97 
98     for (i = 0; i < 4; i++) {
99         uint64_t x = f[i] ^ g[i];
100         x &= mask;
101         f[i] ^= x;
102         g[i] ^= x;
103     }
104 }
105 
fe64_invert(fe64 out,const fe64 z)106 static void fe64_invert(fe64 out, const fe64 z)
107 {
108     fe64 t0;
109     fe64 t1;
110     fe64 t2;
111     fe64 t3;
112     int i;
113 
114     /*
115      * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as
116      * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11.
117      */
118 
119     /* t0 = z ** 2 */
120     fe64_sqr(t0, z);
121 
122     /* t1 = t0 ** (2 ** 2) = z ** 8 */
123     fe64_sqr(t1, t0);
124     fe64_sqr(t1, t1);
125 
126     /* t1 = z * t1 = z ** 9 */
127     fe64_mul(t1, z, t1);
128     /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */
129     fe64_mul(t0, t0, t1);
130 
131     /* t2 = t0 ** 2 = z ** 22 */
132     fe64_sqr(t2, t0);
133 
134     /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */
135     fe64_mul(t1, t1, t2);
136 
137     /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */
138     fe64_sqr(t2, t1);
139     for (i = 1; i < 5; ++i)
140         fe64_sqr(t2, t2);
141 
142     /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */
143     fe64_mul(t1, t2, t1);
144 
145     /* Continuing similarly... */
146 
147     /* t2 = z ** (2 ** 20 - 1) */
148     fe64_sqr(t2, t1);
149     for (i = 1; i < 10; ++i)
150         fe64_sqr(t2, t2);
151 
152     fe64_mul(t2, t2, t1);
153 
154     /* t2 = z ** (2 ** 40 - 1) */
155     fe64_sqr(t3, t2);
156     for (i = 1; i < 20; ++i)
157         fe64_sqr(t3, t3);
158 
159     fe64_mul(t2, t3, t2);
160 
161     /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */
162     for (i = 0; i < 10; ++i)
163         fe64_sqr(t2, t2);
164 
165     /* t1 = z ** (2 ** 50 - 1) */
166     fe64_mul(t1, t2, t1);
167 
168     /* t2 = z ** (2 ** 100 - 1) */
169     fe64_sqr(t2, t1);
170     for (i = 1; i < 50; ++i)
171         fe64_sqr(t2, t2);
172 
173     fe64_mul(t2, t2, t1);
174 
175     /* t2 = z ** (2 ** 200 - 1) */
176     fe64_sqr(t3, t2);
177     for (i = 1; i < 100; ++i)
178         fe64_sqr(t3, t3);
179 
180     fe64_mul(t2, t3, t2);
181 
182     /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */
183     for (i = 0; i < 50; ++i)
184         fe64_sqr(t2, t2);
185 
186     /* t1 = z ** (2 ** 250 - 1) */
187     fe64_mul(t1, t2, t1);
188 
189     /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */
190     for (i = 0; i < 5; ++i)
191         fe64_sqr(t1, t1);
192 
193     /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */
194     fe64_mul(out, t1, t0);
195 }
196 
197 /*
198  * Duplicate of original x25519_scalar_mult_generic, but using
199  * fe64_* subroutines.
200  */
x25519_scalar_mulx(uint8_t out[32],const uint8_t scalar[32],const uint8_t point[32])201 static void x25519_scalar_mulx(uint8_t out[32], const uint8_t scalar[32],
202                                const uint8_t point[32])
203 {
204     fe64 x1, x2, z2, x3, z3, tmp0, tmp1;
205     uint8_t e[32];
206     unsigned swap = 0;
207     int pos;
208 
209     memcpy(e, scalar, 32);
210     e[0]  &= 0xf8;
211     e[31] &= 0x7f;
212     e[31] |= 0x40;
213     fe64_frombytes(x1, point);
214     fe64_1(x2);
215     fe64_0(z2);
216     fe64_copy(x3, x1);
217     fe64_1(z3);
218 
219     for (pos = 254; pos >= 0; --pos) {
220         unsigned int b = 1 & (e[pos / 8] >> (pos & 7));
221 
222         swap ^= b;
223         fe64_cswap(x2, x3, swap);
224         fe64_cswap(z2, z3, swap);
225         swap = b;
226         fe64_sub(tmp0, x3, z3);
227         fe64_sub(tmp1, x2, z2);
228         fe64_add(x2, x2, z2);
229         fe64_add(z2, x3, z3);
230         fe64_mul(z3, x2, tmp0);
231         fe64_mul(z2, z2, tmp1);
232         fe64_sqr(tmp0, tmp1);
233         fe64_sqr(tmp1, x2);
234         fe64_add(x3, z3, z2);
235         fe64_sub(z2, z3, z2);
236         fe64_mul(x2, tmp1, tmp0);
237         fe64_sub(tmp1, tmp1, tmp0);
238         fe64_sqr(z2, z2);
239         fe64_mul121666(z3, tmp1);
240         fe64_sqr(x3, x3);
241         fe64_add(tmp0, tmp0, z3);
242         fe64_mul(z3, x1, z2);
243         fe64_mul(z2, tmp1, tmp0);
244     }
245 
246     fe64_invert(z2, z2);
247     fe64_mul(x2, x2, z2);
248     fe64_tobytes(out, x2);
249 
250     OPENSSL_cleanse(e, sizeof(e));
251 }
252 #endif
253 
254 #if defined(X25519_ASM) \
255     || ( (defined(__SIZEOF_INT128__) && __SIZEOF_INT128__ == 16) \
256          && !defined(__sparc__) \
257          && (!defined(__SIZEOF_LONG__) || (__SIZEOF_LONG__ == 8)) \
258          && !(defined(__ANDROID__) && !defined(__clang__)) )
259 /*
260  * Base 2^51 implementation. It's virtually no different from reference
261  * base 2^25.5 implementation in respect to lax boundary conditions for
262  * intermediate values and even individual limbs. So that whatever you
263  * know about the reference, applies even here...
264  */
265 # define BASE_2_51_IMPLEMENTED
266 
267 typedef uint64_t fe51[5];
268 
269 static const uint64_t MASK51 = 0x7ffffffffffff;
270 
load_7(const uint8_t * in)271 static uint64_t load_7(const uint8_t *in)
272 {
273     uint64_t result;
274 
275     result = in[0];
276     result |= ((uint64_t)in[1]) << 8;
277     result |= ((uint64_t)in[2]) << 16;
278     result |= ((uint64_t)in[3]) << 24;
279     result |= ((uint64_t)in[4]) << 32;
280     result |= ((uint64_t)in[5]) << 40;
281     result |= ((uint64_t)in[6]) << 48;
282 
283     return result;
284 }
285 
load_6(const uint8_t * in)286 static uint64_t load_6(const uint8_t *in)
287 {
288     uint64_t result;
289 
290     result = in[0];
291     result |= ((uint64_t)in[1]) << 8;
292     result |= ((uint64_t)in[2]) << 16;
293     result |= ((uint64_t)in[3]) << 24;
294     result |= ((uint64_t)in[4]) << 32;
295     result |= ((uint64_t)in[5]) << 40;
296 
297     return result;
298 }
299 
fe51_frombytes(fe51 h,const uint8_t * s)300 static void fe51_frombytes(fe51 h, const uint8_t *s)
301 {
302     uint64_t h0 = load_7(s);                                /* 56 bits */
303     uint64_t h1 = load_6(s + 7) << 5;                       /* 53 bits */
304     uint64_t h2 = load_7(s + 13) << 2;                      /* 58 bits */
305     uint64_t h3 = load_6(s + 20) << 7;                      /* 55 bits */
306     uint64_t h4 = (load_6(s + 26) & 0x7fffffffffff) << 4;   /* 51 bits */
307 
308     h1 |= h0 >> 51; h0 &= MASK51;
309     h2 |= h1 >> 51; h1 &= MASK51;
310     h3 |= h2 >> 51; h2 &= MASK51;
311     h4 |= h3 >> 51; h3 &= MASK51;
312 
313     h[0] = h0;
314     h[1] = h1;
315     h[2] = h2;
316     h[3] = h3;
317     h[4] = h4;
318 }
319 
fe51_tobytes(uint8_t * s,const fe51 h)320 static void fe51_tobytes(uint8_t *s, const fe51 h)
321 {
322     uint64_t h0 = h[0];
323     uint64_t h1 = h[1];
324     uint64_t h2 = h[2];
325     uint64_t h3 = h[3];
326     uint64_t h4 = h[4];
327     uint64_t q;
328 
329     /* compare to modulus */
330     q = (h0 + 19) >> 51;
331     q = (h1 + q) >> 51;
332     q = (h2 + q) >> 51;
333     q = (h3 + q) >> 51;
334     q = (h4 + q) >> 51;
335 
336     /* full reduce */
337     h0 += 19 * q;
338     h1 += h0 >> 51; h0 &= MASK51;
339     h2 += h1 >> 51; h1 &= MASK51;
340     h3 += h2 >> 51; h2 &= MASK51;
341     h4 += h3 >> 51; h3 &= MASK51;
342                     h4 &= MASK51;
343 
344     /* smash */
345     s[0] = (uint8_t)(h0 >> 0);
346     s[1] = (uint8_t)(h0 >> 8);
347     s[2] = (uint8_t)(h0 >> 16);
348     s[3] = (uint8_t)(h0 >> 24);
349     s[4] = (uint8_t)(h0 >> 32);
350     s[5] = (uint8_t)(h0 >> 40);
351     s[6] = (uint8_t)((h0 >> 48) | ((uint32_t)h1 << 3));
352     s[7] = (uint8_t)(h1 >> 5);
353     s[8] = (uint8_t)(h1 >> 13);
354     s[9] = (uint8_t)(h1 >> 21);
355     s[10] = (uint8_t)(h1 >> 29);
356     s[11] = (uint8_t)(h1 >> 37);
357     s[12] = (uint8_t)((h1 >> 45) | ((uint32_t)h2 << 6));
358     s[13] = (uint8_t)(h2 >> 2);
359     s[14] = (uint8_t)(h2 >> 10);
360     s[15] = (uint8_t)(h2 >> 18);
361     s[16] = (uint8_t)(h2 >> 26);
362     s[17] = (uint8_t)(h2 >> 34);
363     s[18] = (uint8_t)(h2 >> 42);
364     s[19] = (uint8_t)((h2 >> 50) | ((uint32_t)h3 << 1));
365     s[20] = (uint8_t)(h3 >> 7);
366     s[21] = (uint8_t)(h3 >> 15);
367     s[22] = (uint8_t)(h3 >> 23);
368     s[23] = (uint8_t)(h3 >> 31);
369     s[24] = (uint8_t)(h3 >> 39);
370     s[25] = (uint8_t)((h3 >> 47) | ((uint32_t)h4 << 4));
371     s[26] = (uint8_t)(h4 >> 4);
372     s[27] = (uint8_t)(h4 >> 12);
373     s[28] = (uint8_t)(h4 >> 20);
374     s[29] = (uint8_t)(h4 >> 28);
375     s[30] = (uint8_t)(h4 >> 36);
376     s[31] = (uint8_t)(h4 >> 44);
377 }
378 
379 # if defined(X25519_ASM)
380 void x25519_fe51_mul(fe51 h, const fe51 f, const fe51 g);
381 void x25519_fe51_sqr(fe51 h, const fe51 f);
382 void x25519_fe51_mul121666(fe51 h, fe51 f);
383 #  define fe51_mul x25519_fe51_mul
384 #  define fe51_sq  x25519_fe51_sqr
385 #  define fe51_mul121666 x25519_fe51_mul121666
386 # else
387 
388 typedef __uint128_t u128;
389 
fe51_mul(fe51 h,const fe51 f,const fe51 g)390 static void fe51_mul(fe51 h, const fe51 f, const fe51 g)
391 {
392     u128 h0, h1, h2, h3, h4;
393     uint64_t f_i, g0, g1, g2, g3, g4;
394 
395     f_i = f[0];
396     h0 = (u128)f_i * (g0 = g[0]);
397     h1 = (u128)f_i * (g1 = g[1]);
398     h2 = (u128)f_i * (g2 = g[2]);
399     h3 = (u128)f_i * (g3 = g[3]);
400     h4 = (u128)f_i * (g4 = g[4]);
401 
402     f_i = f[1];
403     h0 += (u128)f_i * (g4 *= 19);
404     h1 += (u128)f_i * g0;
405     h2 += (u128)f_i * g1;
406     h3 += (u128)f_i * g2;
407     h4 += (u128)f_i * g3;
408 
409     f_i = f[2];
410     h0 += (u128)f_i * (g3 *= 19);
411     h1 += (u128)f_i * g4;
412     h2 += (u128)f_i * g0;
413     h3 += (u128)f_i * g1;
414     h4 += (u128)f_i * g2;
415 
416     f_i = f[3];
417     h0 += (u128)f_i * (g2 *= 19);
418     h1 += (u128)f_i * g3;
419     h2 += (u128)f_i * g4;
420     h3 += (u128)f_i * g0;
421     h4 += (u128)f_i * g1;
422 
423     f_i = f[4];
424     h0 += (u128)f_i * (g1 *= 19);
425     h1 += (u128)f_i * g2;
426     h2 += (u128)f_i * g3;
427     h3 += (u128)f_i * g4;
428     h4 += (u128)f_i * g0;
429 
430     /* partial [lazy] reduction */
431     h3 += (uint64_t)(h2 >> 51); g2 = (uint64_t)h2 & MASK51;
432     h1 += (uint64_t)(h0 >> 51); g0 = (uint64_t)h0 & MASK51;
433 
434     h4 += (uint64_t)(h3 >> 51); g3 = (uint64_t)h3 & MASK51;
435     g2 += (uint64_t)(h1 >> 51); g1 = (uint64_t)h1 & MASK51;
436 
437     g0 += (uint64_t)(h4 >> 51) * 19; g4 = (uint64_t)h4 & MASK51;
438     g3 += g2 >> 51; g2 &= MASK51;
439     g1 += g0 >> 51; g0 &= MASK51;
440 
441     h[0] = g0;
442     h[1] = g1;
443     h[2] = g2;
444     h[3] = g3;
445     h[4] = g4;
446 }
447 
fe51_sq(fe51 h,const fe51 f)448 static void fe51_sq(fe51 h, const fe51 f)
449 {
450 #  if defined(OPENSSL_SMALL_FOOTPRINT)
451     fe51_mul(h, f, f);
452 #  else
453     /* dedicated squaring gives 16-25% overall improvement */
454     uint64_t g0 = f[0];
455     uint64_t g1 = f[1];
456     uint64_t g2 = f[2];
457     uint64_t g3 = f[3];
458     uint64_t g4 = f[4];
459     u128 h0, h1, h2, h3, h4;
460 
461     h0 = (u128)g0 * g0;     g0 *= 2;
462     h1 = (u128)g0 * g1;
463     h2 = (u128)g0 * g2;
464     h3 = (u128)g0 * g3;
465     h4 = (u128)g0 * g4;
466 
467     g0 = g4;                /* borrow g0 */
468     h3 += (u128)g0 * (g4 *= 19);
469 
470     h2 += (u128)g1 * g1;    g1 *= 2;
471     h3 += (u128)g1 * g2;
472     h4 += (u128)g1 * g3;
473     h0 += (u128)g1 * g4;
474 
475     g0 = g3;                /* borrow g0 */
476     h1 += (u128)g0 * (g3 *= 19);
477     h2 += (u128)(g0 * 2) * g4;
478 
479     h4 += (u128)g2 * g2;    g2 *= 2;
480     h0 += (u128)g2 * g3;
481     h1 += (u128)g2 * g4;
482 
483     /* partial [lazy] reduction */
484     h3 += (uint64_t)(h2 >> 51); g2 = (uint64_t)h2 & MASK51;
485     h1 += (uint64_t)(h0 >> 51); g0 = (uint64_t)h0 & MASK51;
486 
487     h4 += (uint64_t)(h3 >> 51); g3 = (uint64_t)h3 & MASK51;
488     g2 += (uint64_t)(h1 >> 51); g1 = (uint64_t)h1 & MASK51;
489 
490     g0 += (uint64_t)(h4 >> 51) * 19; g4 = (uint64_t)h4 & MASK51;
491     g3 += g2 >> 51; g2 &= MASK51;
492     g1 += g0 >> 51; g0 &= MASK51;
493 
494     h[0] = g0;
495     h[1] = g1;
496     h[2] = g2;
497     h[3] = g3;
498     h[4] = g4;
499 #  endif
500 }
501 
fe51_mul121666(fe51 h,fe51 f)502 static void fe51_mul121666(fe51 h, fe51 f)
503 {
504     u128 h0 = f[0] * (u128)121666;
505     u128 h1 = f[1] * (u128)121666;
506     u128 h2 = f[2] * (u128)121666;
507     u128 h3 = f[3] * (u128)121666;
508     u128 h4 = f[4] * (u128)121666;
509     uint64_t g0, g1, g2, g3, g4;
510 
511     h3 += (uint64_t)(h2 >> 51); g2 = (uint64_t)h2 & MASK51;
512     h1 += (uint64_t)(h0 >> 51); g0 = (uint64_t)h0 & MASK51;
513 
514     h4 += (uint64_t)(h3 >> 51); g3 = (uint64_t)h3 & MASK51;
515     g2 += (uint64_t)(h1 >> 51); g1 = (uint64_t)h1 & MASK51;
516 
517     g0 += (uint64_t)(h4 >> 51) * 19; g4 = (uint64_t)h4 & MASK51;
518     g3 += g2 >> 51; g2 &= MASK51;
519     g1 += g0 >> 51; g0 &= MASK51;
520 
521     h[0] = g0;
522     h[1] = g1;
523     h[2] = g2;
524     h[3] = g3;
525     h[4] = g4;
526 }
527 # endif
528 
fe51_add(fe51 h,const fe51 f,const fe51 g)529 static void fe51_add(fe51 h, const fe51 f, const fe51 g)
530 {
531     h[0] = f[0] + g[0];
532     h[1] = f[1] + g[1];
533     h[2] = f[2] + g[2];
534     h[3] = f[3] + g[3];
535     h[4] = f[4] + g[4];
536 }
537 
fe51_sub(fe51 h,const fe51 f,const fe51 g)538 static void fe51_sub(fe51 h, const fe51 f, const fe51 g)
539 {
540     /*
541      * Add 2*modulus to ensure that result remains positive
542      * even if subtrahend is partially reduced.
543      */
544     h[0] = (f[0] + 0xfffffffffffda) - g[0];
545     h[1] = (f[1] + 0xffffffffffffe) - g[1];
546     h[2] = (f[2] + 0xffffffffffffe) - g[2];
547     h[3] = (f[3] + 0xffffffffffffe) - g[3];
548     h[4] = (f[4] + 0xffffffffffffe) - g[4];
549 }
550 
fe51_0(fe51 h)551 static void fe51_0(fe51 h)
552 {
553     h[0] = 0;
554     h[1] = 0;
555     h[2] = 0;
556     h[3] = 0;
557     h[4] = 0;
558 }
559 
fe51_1(fe51 h)560 static void fe51_1(fe51 h)
561 {
562     h[0] = 1;
563     h[1] = 0;
564     h[2] = 0;
565     h[3] = 0;
566     h[4] = 0;
567 }
568 
fe51_copy(fe51 h,const fe51 f)569 static void fe51_copy(fe51 h, const fe51 f)
570 {
571     h[0] = f[0];
572     h[1] = f[1];
573     h[2] = f[2];
574     h[3] = f[3];
575     h[4] = f[4];
576 }
577 
fe51_cswap(fe51 f,fe51 g,unsigned int b)578 static void fe51_cswap(fe51 f, fe51 g, unsigned int b)
579 {
580     int i;
581     uint64_t mask = 0 - (uint64_t)b;
582 
583     for (i = 0; i < 5; i++) {
584         int64_t x = f[i] ^ g[i];
585         x &= mask;
586         f[i] ^= x;
587         g[i] ^= x;
588     }
589 }
590 
fe51_invert(fe51 out,const fe51 z)591 static void fe51_invert(fe51 out, const fe51 z)
592 {
593     fe51 t0;
594     fe51 t1;
595     fe51 t2;
596     fe51 t3;
597     int i;
598 
599     /*
600      * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as
601      * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11.
602      */
603 
604     /* t0 = z ** 2 */
605     fe51_sq(t0, z);
606 
607     /* t1 = t0 ** (2 ** 2) = z ** 8 */
608     fe51_sq(t1, t0);
609     fe51_sq(t1, t1);
610 
611     /* t1 = z * t1 = z ** 9 */
612     fe51_mul(t1, z, t1);
613     /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */
614     fe51_mul(t0, t0, t1);
615 
616     /* t2 = t0 ** 2 = z ** 22 */
617     fe51_sq(t2, t0);
618 
619     /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */
620     fe51_mul(t1, t1, t2);
621 
622     /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */
623     fe51_sq(t2, t1);
624     for (i = 1; i < 5; ++i)
625         fe51_sq(t2, t2);
626 
627     /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */
628     fe51_mul(t1, t2, t1);
629 
630     /* Continuing similarly... */
631 
632     /* t2 = z ** (2 ** 20 - 1) */
633     fe51_sq(t2, t1);
634     for (i = 1; i < 10; ++i)
635         fe51_sq(t2, t2);
636 
637     fe51_mul(t2, t2, t1);
638 
639     /* t2 = z ** (2 ** 40 - 1) */
640     fe51_sq(t3, t2);
641     for (i = 1; i < 20; ++i)
642         fe51_sq(t3, t3);
643 
644     fe51_mul(t2, t3, t2);
645 
646     /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */
647     for (i = 0; i < 10; ++i)
648         fe51_sq(t2, t2);
649 
650     /* t1 = z ** (2 ** 50 - 1) */
651     fe51_mul(t1, t2, t1);
652 
653     /* t2 = z ** (2 ** 100 - 1) */
654     fe51_sq(t2, t1);
655     for (i = 1; i < 50; ++i)
656         fe51_sq(t2, t2);
657 
658     fe51_mul(t2, t2, t1);
659 
660     /* t2 = z ** (2 ** 200 - 1) */
661     fe51_sq(t3, t2);
662     for (i = 1; i < 100; ++i)
663         fe51_sq(t3, t3);
664 
665     fe51_mul(t2, t3, t2);
666 
667     /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */
668     for (i = 0; i < 50; ++i)
669         fe51_sq(t2, t2);
670 
671     /* t1 = z ** (2 ** 250 - 1) */
672     fe51_mul(t1, t2, t1);
673 
674     /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */
675     for (i = 0; i < 5; ++i)
676         fe51_sq(t1, t1);
677 
678     /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */
679     fe51_mul(out, t1, t0);
680 }
681 
682 /*
683  * Duplicate of original x25519_scalar_mult_generic, but using
684  * fe51_* subroutines.
685  */
x25519_scalar_mult(uint8_t out[32],const uint8_t scalar[32],const uint8_t point[32])686 static void x25519_scalar_mult(uint8_t out[32], const uint8_t scalar[32],
687                                const uint8_t point[32])
688 {
689     fe51 x1, x2, z2, x3, z3, tmp0, tmp1;
690     uint8_t e[32];
691     unsigned swap = 0;
692     int pos;
693 
694 # ifdef BASE_2_64_IMPLEMENTED
695     if (x25519_fe64_eligible()) {
696         x25519_scalar_mulx(out, scalar, point);
697         return;
698     }
699 # endif
700 
701     memcpy(e, scalar, 32);
702     e[0]  &= 0xf8;
703     e[31] &= 0x7f;
704     e[31] |= 0x40;
705     fe51_frombytes(x1, point);
706     fe51_1(x2);
707     fe51_0(z2);
708     fe51_copy(x3, x1);
709     fe51_1(z3);
710 
711     for (pos = 254; pos >= 0; --pos) {
712         unsigned int b = 1 & (e[pos / 8] >> (pos & 7));
713 
714         swap ^= b;
715         fe51_cswap(x2, x3, swap);
716         fe51_cswap(z2, z3, swap);
717         swap = b;
718         fe51_sub(tmp0, x3, z3);
719         fe51_sub(tmp1, x2, z2);
720         fe51_add(x2, x2, z2);
721         fe51_add(z2, x3, z3);
722         fe51_mul(z3, tmp0, x2);
723         fe51_mul(z2, z2, tmp1);
724         fe51_sq(tmp0, tmp1);
725         fe51_sq(tmp1, x2);
726         fe51_add(x3, z3, z2);
727         fe51_sub(z2, z3, z2);
728         fe51_mul(x2, tmp1, tmp0);
729         fe51_sub(tmp1, tmp1, tmp0);
730         fe51_sq(z2, z2);
731         fe51_mul121666(z3, tmp1);
732         fe51_sq(x3, x3);
733         fe51_add(tmp0, tmp0, z3);
734         fe51_mul(z3, x1, z2);
735         fe51_mul(z2, tmp1, tmp0);
736     }
737 
738     fe51_invert(z2, z2);
739     fe51_mul(x2, x2, z2);
740     fe51_tobytes(out, x2);
741 
742     OPENSSL_cleanse(e, sizeof(e));
743 }
744 #endif
745 
746 /*
747  * Reference base 2^25.5 implementation.
748  *
749  * This code is mostly taken from the ref10 version of Ed25519 in SUPERCOP
750  * 20141124 (http://bench.cr.yp.to/supercop.html).
751  *
752  * The field functions are shared by Ed25519 and X25519 where possible.
753  */
754 
755 /*
756  * fe means field element. Here the field is \Z/(2^255-19). An element t,
757  * entries t[0]...t[9], represents the integer t[0]+2^26 t[1]+2^51 t[2]+2^77
758  * t[3]+2^102 t[4]+...+2^230 t[9]. Bounds on each t[i] vary depending on
759  * context.
760  */
761 typedef int32_t fe[10];
762 
763 static const int64_t kBottom21Bits =  0x1fffffLL;
764 static const int64_t kBottom25Bits = 0x1ffffffLL;
765 static const int64_t kBottom26Bits = 0x3ffffffLL;
766 static const int64_t kTop39Bits = 0xfffffffffe000000LL;
767 static const int64_t kTop38Bits = 0xfffffffffc000000LL;
768 
load_3(const uint8_t * in)769 static uint64_t load_3(const uint8_t *in)
770 {
771     uint64_t result;
772 
773     result  = ((uint64_t)in[0]);
774     result |= ((uint64_t)in[1]) << 8;
775     result |= ((uint64_t)in[2]) << 16;
776     return result;
777 }
778 
load_4(const uint8_t * in)779 static uint64_t load_4(const uint8_t *in)
780 {
781     uint64_t result;
782 
783     result  = ((uint64_t)in[0]);
784     result |= ((uint64_t)in[1]) << 8;
785     result |= ((uint64_t)in[2]) << 16;
786     result |= ((uint64_t)in[3]) << 24;
787     return result;
788 }
789 
fe_frombytes(fe h,const uint8_t * s)790 static void fe_frombytes(fe h, const uint8_t *s)
791 {
792     /* Ignores top bit of h. */
793     int64_t h0 =  load_4(s);
794     int64_t h1 =  load_3(s +  4) << 6;
795     int64_t h2 =  load_3(s +  7) << 5;
796     int64_t h3 =  load_3(s + 10) << 3;
797     int64_t h4 =  load_3(s + 13) << 2;
798     int64_t h5 =  load_4(s + 16);
799     int64_t h6 =  load_3(s + 20) << 7;
800     int64_t h7 =  load_3(s + 23) << 5;
801     int64_t h8 =  load_3(s + 26) << 4;
802     int64_t h9 = (load_3(s + 29) & 0x7fffff) << 2;
803     int64_t carry0;
804     int64_t carry1;
805     int64_t carry2;
806     int64_t carry3;
807     int64_t carry4;
808     int64_t carry5;
809     int64_t carry6;
810     int64_t carry7;
811     int64_t carry8;
812     int64_t carry9;
813 
814     carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
815     carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
816     carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
817     carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
818     carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;
819 
820     carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
821     carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
822     carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
823     carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
824     carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;
825 
826     h[0] = (int32_t)h0;
827     h[1] = (int32_t)h1;
828     h[2] = (int32_t)h2;
829     h[3] = (int32_t)h3;
830     h[4] = (int32_t)h4;
831     h[5] = (int32_t)h5;
832     h[6] = (int32_t)h6;
833     h[7] = (int32_t)h7;
834     h[8] = (int32_t)h8;
835     h[9] = (int32_t)h9;
836 }
837 
838 /*
839  * Preconditions:
840  *   |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
841  *
842  * Write p=2^255-19; q=floor(h/p).
843  * Basic claim: q = floor(2^(-255)(h + 19 2^(-25)h9 + 2^(-1))).
844  *
845  * Proof:
846  *   Have |h|<=p so |q|<=1 so |19^2 2^(-255) q|<1/4.
847  *   Also have |h-2^230 h9|<2^231 so |19 2^(-255)(h-2^230 h9)|<1/4.
848  *
849  *   Write y=2^(-1)-19^2 2^(-255)q-19 2^(-255)(h-2^230 h9).
850  *   Then 0<y<1.
851  *
852  *   Write r=h-pq.
853  *   Have 0<=r<=p-1=2^255-20.
854  *   Thus 0<=r+19(2^-255)r<r+19(2^-255)2^255<=2^255-1.
855  *
856  *   Write x=r+19(2^-255)r+y.
857  *   Then 0<x<2^255 so floor(2^(-255)x) = 0 so floor(q+2^(-255)x) = q.
858  *
859  *   Have q+2^(-255)x = 2^(-255)(h + 19 2^(-25) h9 + 2^(-1))
860  *   so floor(2^(-255)(h + 19 2^(-25) h9 + 2^(-1))) = q.
861  */
fe_tobytes(uint8_t * s,const fe h)862 static void fe_tobytes(uint8_t *s, const fe h)
863 {
864     int32_t h0 = h[0];
865     int32_t h1 = h[1];
866     int32_t h2 = h[2];
867     int32_t h3 = h[3];
868     int32_t h4 = h[4];
869     int32_t h5 = h[5];
870     int32_t h6 = h[6];
871     int32_t h7 = h[7];
872     int32_t h8 = h[8];
873     int32_t h9 = h[9];
874     int32_t q;
875 
876     q = (19 * h9 + (((int32_t) 1) << 24)) >> 25;
877     q = (h0 + q) >> 26;
878     q = (h1 + q) >> 25;
879     q = (h2 + q) >> 26;
880     q = (h3 + q) >> 25;
881     q = (h4 + q) >> 26;
882     q = (h5 + q) >> 25;
883     q = (h6 + q) >> 26;
884     q = (h7 + q) >> 25;
885     q = (h8 + q) >> 26;
886     q = (h9 + q) >> 25;
887 
888     /* Goal: Output h-(2^255-19)q, which is between 0 and 2^255-20. */
889     h0 += 19 * q;
890     /* Goal: Output h-2^255 q, which is between 0 and 2^255-20. */
891 
892     h1 += h0 >> 26; h0 &= kBottom26Bits;
893     h2 += h1 >> 25; h1 &= kBottom25Bits;
894     h3 += h2 >> 26; h2 &= kBottom26Bits;
895     h4 += h3 >> 25; h3 &= kBottom25Bits;
896     h5 += h4 >> 26; h4 &= kBottom26Bits;
897     h6 += h5 >> 25; h5 &= kBottom25Bits;
898     h7 += h6 >> 26; h6 &= kBottom26Bits;
899     h8 += h7 >> 25; h7 &= kBottom25Bits;
900     h9 += h8 >> 26; h8 &= kBottom26Bits;
901                     h9 &= kBottom25Bits;
902     /* h10 = carry9 */
903 
904     /*
905      * Goal: Output h0+...+2^255 h10-2^255 q, which is between 0 and 2^255-20.
906      * Have h0+...+2^230 h9 between 0 and 2^255-1;
907      * evidently 2^255 h10-2^255 q = 0.
908      * Goal: Output h0+...+2^230 h9.
909      */
910     s[ 0] = (uint8_t) (h0 >>  0);
911     s[ 1] = (uint8_t) (h0 >>  8);
912     s[ 2] = (uint8_t) (h0 >> 16);
913     s[ 3] = (uint8_t)((h0 >> 24) | ((uint32_t)(h1) << 2));
914     s[ 4] = (uint8_t) (h1 >>  6);
915     s[ 5] = (uint8_t) (h1 >> 14);
916     s[ 6] = (uint8_t)((h1 >> 22) | ((uint32_t)(h2) << 3));
917     s[ 7] = (uint8_t) (h2 >>  5);
918     s[ 8] = (uint8_t) (h2 >> 13);
919     s[ 9] = (uint8_t)((h2 >> 21) | ((uint32_t)(h3) << 5));
920     s[10] = (uint8_t) (h3 >>  3);
921     s[11] = (uint8_t) (h3 >> 11);
922     s[12] = (uint8_t)((h3 >> 19) | ((uint32_t)(h4) << 6));
923     s[13] = (uint8_t) (h4 >>  2);
924     s[14] = (uint8_t) (h4 >> 10);
925     s[15] = (uint8_t) (h4 >> 18);
926     s[16] = (uint8_t) (h5 >>  0);
927     s[17] = (uint8_t) (h5 >>  8);
928     s[18] = (uint8_t) (h5 >> 16);
929     s[19] = (uint8_t)((h5 >> 24) | ((uint32_t)(h6) << 1));
930     s[20] = (uint8_t) (h6 >>  7);
931     s[21] = (uint8_t) (h6 >> 15);
932     s[22] = (uint8_t)((h6 >> 23) | ((uint32_t)(h7) << 3));
933     s[23] = (uint8_t) (h7 >>  5);
934     s[24] = (uint8_t) (h7 >> 13);
935     s[25] = (uint8_t)((h7 >> 21) | ((uint32_t)(h8) << 4));
936     s[26] = (uint8_t) (h8 >>  4);
937     s[27] = (uint8_t) (h8 >> 12);
938     s[28] = (uint8_t)((h8 >> 20) | ((uint32_t)(h9) << 6));
939     s[29] = (uint8_t) (h9 >>  2);
940     s[30] = (uint8_t) (h9 >> 10);
941     s[31] = (uint8_t) (h9 >> 18);
942 }
943 
944 /* h = f */
fe_copy(fe h,const fe f)945 static void fe_copy(fe h, const fe f)
946 {
947     memmove(h, f, sizeof(int32_t) * 10);
948 }
949 
950 /* h = 0 */
fe_0(fe h)951 static void fe_0(fe h)
952 {
953     memset(h, 0, sizeof(int32_t) * 10);
954 }
955 
956 /* h = 1 */
fe_1(fe h)957 static void fe_1(fe h)
958 {
959     memset(h, 0, sizeof(int32_t) * 10);
960     h[0] = 1;
961 }
962 
963 /*
964  * h = f + g
965  *
966  * Can overlap h with f or g.
967  *
968  * Preconditions:
969  *    |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
970  *    |g| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
971  *
972  * Postconditions:
973  *    |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
974  */
fe_add(fe h,const fe f,const fe g)975 static void fe_add(fe h, const fe f, const fe g)
976 {
977     unsigned i;
978 
979     for (i = 0; i < 10; i++) {
980         h[i] = f[i] + g[i];
981     }
982 }
983 
984 /*
985  * h = f - g
986  *
987  * Can overlap h with f or g.
988  *
989  * Preconditions:
990  *    |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
991  *    |g| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
992  *
993  * Postconditions:
994  *    |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
995  */
fe_sub(fe h,const fe f,const fe g)996 static void fe_sub(fe h, const fe f, const fe g)
997 {
998     unsigned i;
999 
1000     for (i = 0; i < 10; i++) {
1001         h[i] = f[i] - g[i];
1002     }
1003 }
1004 
1005 /*
1006  * h = f * g
1007  *
1008  * Can overlap h with f or g.
1009  *
1010  * Preconditions:
1011  *    |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
1012  *    |g| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
1013  *
1014  * Postconditions:
1015  *    |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
1016  *
1017  * Notes on implementation strategy:
1018  *
1019  * Using schoolbook multiplication.
1020  * Karatsuba would save a little in some cost models.
1021  *
1022  * Most multiplications by 2 and 19 are 32-bit precomputations;
1023  * cheaper than 64-bit postcomputations.
1024  *
1025  * There is one remaining multiplication by 19 in the carry chain;
1026  * one *19 precomputation can be merged into this,
1027  * but the resulting data flow is considerably less clean.
1028  *
1029  * There are 12 carries below.
1030  * 10 of them are 2-way parallelizable and vectorizable.
1031  * Can get away with 11 carries, but then data flow is much deeper.
1032  *
1033  * With tighter constraints on inputs can squeeze carries into int32.
1034  */
fe_mul(fe h,const fe f,const fe g)1035 static void fe_mul(fe h, const fe f, const fe g)
1036 {
1037     int32_t f0 = f[0];
1038     int32_t f1 = f[1];
1039     int32_t f2 = f[2];
1040     int32_t f3 = f[3];
1041     int32_t f4 = f[4];
1042     int32_t f5 = f[5];
1043     int32_t f6 = f[6];
1044     int32_t f7 = f[7];
1045     int32_t f8 = f[8];
1046     int32_t f9 = f[9];
1047     int32_t g0 = g[0];
1048     int32_t g1 = g[1];
1049     int32_t g2 = g[2];
1050     int32_t g3 = g[3];
1051     int32_t g4 = g[4];
1052     int32_t g5 = g[5];
1053     int32_t g6 = g[6];
1054     int32_t g7 = g[7];
1055     int32_t g8 = g[8];
1056     int32_t g9 = g[9];
1057     int32_t g1_19 = 19 * g1; /* 1.959375*2^29 */
1058     int32_t g2_19 = 19 * g2; /* 1.959375*2^30; still ok */
1059     int32_t g3_19 = 19 * g3;
1060     int32_t g4_19 = 19 * g4;
1061     int32_t g5_19 = 19 * g5;
1062     int32_t g6_19 = 19 * g6;
1063     int32_t g7_19 = 19 * g7;
1064     int32_t g8_19 = 19 * g8;
1065     int32_t g9_19 = 19 * g9;
1066     int32_t f1_2 = 2 * f1;
1067     int32_t f3_2 = 2 * f3;
1068     int32_t f5_2 = 2 * f5;
1069     int32_t f7_2 = 2 * f7;
1070     int32_t f9_2 = 2 * f9;
1071     int64_t f0g0    = f0   * (int64_t) g0;
1072     int64_t f0g1    = f0   * (int64_t) g1;
1073     int64_t f0g2    = f0   * (int64_t) g2;
1074     int64_t f0g3    = f0   * (int64_t) g3;
1075     int64_t f0g4    = f0   * (int64_t) g4;
1076     int64_t f0g5    = f0   * (int64_t) g5;
1077     int64_t f0g6    = f0   * (int64_t) g6;
1078     int64_t f0g7    = f0   * (int64_t) g7;
1079     int64_t f0g8    = f0   * (int64_t) g8;
1080     int64_t f0g9    = f0   * (int64_t) g9;
1081     int64_t f1g0    = f1   * (int64_t) g0;
1082     int64_t f1g1_2  = f1_2 * (int64_t) g1;
1083     int64_t f1g2    = f1   * (int64_t) g2;
1084     int64_t f1g3_2  = f1_2 * (int64_t) g3;
1085     int64_t f1g4    = f1   * (int64_t) g4;
1086     int64_t f1g5_2  = f1_2 * (int64_t) g5;
1087     int64_t f1g6    = f1   * (int64_t) g6;
1088     int64_t f1g7_2  = f1_2 * (int64_t) g7;
1089     int64_t f1g8    = f1   * (int64_t) g8;
1090     int64_t f1g9_38 = f1_2 * (int64_t) g9_19;
1091     int64_t f2g0    = f2   * (int64_t) g0;
1092     int64_t f2g1    = f2   * (int64_t) g1;
1093     int64_t f2g2    = f2   * (int64_t) g2;
1094     int64_t f2g3    = f2   * (int64_t) g3;
1095     int64_t f2g4    = f2   * (int64_t) g4;
1096     int64_t f2g5    = f2   * (int64_t) g5;
1097     int64_t f2g6    = f2   * (int64_t) g6;
1098     int64_t f2g7    = f2   * (int64_t) g7;
1099     int64_t f2g8_19 = f2   * (int64_t) g8_19;
1100     int64_t f2g9_19 = f2   * (int64_t) g9_19;
1101     int64_t f3g0    = f3   * (int64_t) g0;
1102     int64_t f3g1_2  = f3_2 * (int64_t) g1;
1103     int64_t f3g2    = f3   * (int64_t) g2;
1104     int64_t f3g3_2  = f3_2 * (int64_t) g3;
1105     int64_t f3g4    = f3   * (int64_t) g4;
1106     int64_t f3g5_2  = f3_2 * (int64_t) g5;
1107     int64_t f3g6    = f3   * (int64_t) g6;
1108     int64_t f3g7_38 = f3_2 * (int64_t) g7_19;
1109     int64_t f3g8_19 = f3   * (int64_t) g8_19;
1110     int64_t f3g9_38 = f3_2 * (int64_t) g9_19;
1111     int64_t f4g0    = f4   * (int64_t) g0;
1112     int64_t f4g1    = f4   * (int64_t) g1;
1113     int64_t f4g2    = f4   * (int64_t) g2;
1114     int64_t f4g3    = f4   * (int64_t) g3;
1115     int64_t f4g4    = f4   * (int64_t) g4;
1116     int64_t f4g5    = f4   * (int64_t) g5;
1117     int64_t f4g6_19 = f4   * (int64_t) g6_19;
1118     int64_t f4g7_19 = f4   * (int64_t) g7_19;
1119     int64_t f4g8_19 = f4   * (int64_t) g8_19;
1120     int64_t f4g9_19 = f4   * (int64_t) g9_19;
1121     int64_t f5g0    = f5   * (int64_t) g0;
1122     int64_t f5g1_2  = f5_2 * (int64_t) g1;
1123     int64_t f5g2    = f5   * (int64_t) g2;
1124     int64_t f5g3_2  = f5_2 * (int64_t) g3;
1125     int64_t f5g4    = f5   * (int64_t) g4;
1126     int64_t f5g5_38 = f5_2 * (int64_t) g5_19;
1127     int64_t f5g6_19 = f5   * (int64_t) g6_19;
1128     int64_t f5g7_38 = f5_2 * (int64_t) g7_19;
1129     int64_t f5g8_19 = f5   * (int64_t) g8_19;
1130     int64_t f5g9_38 = f5_2 * (int64_t) g9_19;
1131     int64_t f6g0    = f6   * (int64_t) g0;
1132     int64_t f6g1    = f6   * (int64_t) g1;
1133     int64_t f6g2    = f6   * (int64_t) g2;
1134     int64_t f6g3    = f6   * (int64_t) g3;
1135     int64_t f6g4_19 = f6   * (int64_t) g4_19;
1136     int64_t f6g5_19 = f6   * (int64_t) g5_19;
1137     int64_t f6g6_19 = f6   * (int64_t) g6_19;
1138     int64_t f6g7_19 = f6   * (int64_t) g7_19;
1139     int64_t f6g8_19 = f6   * (int64_t) g8_19;
1140     int64_t f6g9_19 = f6   * (int64_t) g9_19;
1141     int64_t f7g0    = f7   * (int64_t) g0;
1142     int64_t f7g1_2  = f7_2 * (int64_t) g1;
1143     int64_t f7g2    = f7   * (int64_t) g2;
1144     int64_t f7g3_38 = f7_2 * (int64_t) g3_19;
1145     int64_t f7g4_19 = f7   * (int64_t) g4_19;
1146     int64_t f7g5_38 = f7_2 * (int64_t) g5_19;
1147     int64_t f7g6_19 = f7   * (int64_t) g6_19;
1148     int64_t f7g7_38 = f7_2 * (int64_t) g7_19;
1149     int64_t f7g8_19 = f7   * (int64_t) g8_19;
1150     int64_t f7g9_38 = f7_2 * (int64_t) g9_19;
1151     int64_t f8g0    = f8   * (int64_t) g0;
1152     int64_t f8g1    = f8   * (int64_t) g1;
1153     int64_t f8g2_19 = f8   * (int64_t) g2_19;
1154     int64_t f8g3_19 = f8   * (int64_t) g3_19;
1155     int64_t f8g4_19 = f8   * (int64_t) g4_19;
1156     int64_t f8g5_19 = f8   * (int64_t) g5_19;
1157     int64_t f8g6_19 = f8   * (int64_t) g6_19;
1158     int64_t f8g7_19 = f8   * (int64_t) g7_19;
1159     int64_t f8g8_19 = f8   * (int64_t) g8_19;
1160     int64_t f8g9_19 = f8   * (int64_t) g9_19;
1161     int64_t f9g0    = f9   * (int64_t) g0;
1162     int64_t f9g1_38 = f9_2 * (int64_t) g1_19;
1163     int64_t f9g2_19 = f9   * (int64_t) g2_19;
1164     int64_t f9g3_38 = f9_2 * (int64_t) g3_19;
1165     int64_t f9g4_19 = f9   * (int64_t) g4_19;
1166     int64_t f9g5_38 = f9_2 * (int64_t) g5_19;
1167     int64_t f9g6_19 = f9   * (int64_t) g6_19;
1168     int64_t f9g7_38 = f9_2 * (int64_t) g7_19;
1169     int64_t f9g8_19 = f9   * (int64_t) g8_19;
1170     int64_t f9g9_38 = f9_2 * (int64_t) g9_19;
1171     int64_t h0 = f0g0 + f1g9_38 + f2g8_19 + f3g7_38 + f4g6_19 + f5g5_38 + f6g4_19 + f7g3_38 + f8g2_19 + f9g1_38;
1172     int64_t h1 = f0g1 + f1g0    + f2g9_19 + f3g8_19 + f4g7_19 + f5g6_19 + f6g5_19 + f7g4_19 + f8g3_19 + f9g2_19;
1173     int64_t h2 = f0g2 + f1g1_2  + f2g0    + f3g9_38 + f4g8_19 + f5g7_38 + f6g6_19 + f7g5_38 + f8g4_19 + f9g3_38;
1174     int64_t h3 = f0g3 + f1g2    + f2g1    + f3g0    + f4g9_19 + f5g8_19 + f6g7_19 + f7g6_19 + f8g5_19 + f9g4_19;
1175     int64_t h4 = f0g4 + f1g3_2  + f2g2    + f3g1_2  + f4g0    + f5g9_38 + f6g8_19 + f7g7_38 + f8g6_19 + f9g5_38;
1176     int64_t h5 = f0g5 + f1g4    + f2g3    + f3g2    + f4g1    + f5g0    + f6g9_19 + f7g8_19 + f8g7_19 + f9g6_19;
1177     int64_t h6 = f0g6 + f1g5_2  + f2g4    + f3g3_2  + f4g2    + f5g1_2  + f6g0    + f7g9_38 + f8g8_19 + f9g7_38;
1178     int64_t h7 = f0g7 + f1g6    + f2g5    + f3g4    + f4g3    + f5g2    + f6g1    + f7g0    + f8g9_19 + f9g8_19;
1179     int64_t h8 = f0g8 + f1g7_2  + f2g6    + f3g5_2  + f4g4    + f5g3_2  + f6g2    + f7g1_2  + f8g0    + f9g9_38;
1180     int64_t h9 = f0g9 + f1g8    + f2g7    + f3g6    + f4g5    + f5g4    + f6g3    + f7g2    + f8g1    + f9g0   ;
1181     int64_t carry0;
1182     int64_t carry1;
1183     int64_t carry2;
1184     int64_t carry3;
1185     int64_t carry4;
1186     int64_t carry5;
1187     int64_t carry6;
1188     int64_t carry7;
1189     int64_t carry8;
1190     int64_t carry9;
1191 
1192     /* |h0| <= (1.65*1.65*2^52*(1+19+19+19+19)+1.65*1.65*2^50*(38+38+38+38+38))
1193      *   i.e. |h0| <= 1.4*2^60; narrower ranges for h2, h4, h6, h8
1194      * |h1| <= (1.65*1.65*2^51*(1+1+19+19+19+19+19+19+19+19))
1195      *   i.e. |h1| <= 1.7*2^59; narrower ranges for h3, h5, h7, h9 */
1196 
1197     carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
1198     carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
1199     /* |h0| <= 2^25 */
1200     /* |h4| <= 2^25 */
1201     /* |h1| <= 1.71*2^59 */
1202     /* |h5| <= 1.71*2^59 */
1203 
1204     carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
1205     carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
1206     /* |h1| <= 2^24; from now on fits into int32 */
1207     /* |h5| <= 2^24; from now on fits into int32 */
1208     /* |h2| <= 1.41*2^60 */
1209     /* |h6| <= 1.41*2^60 */
1210 
1211     carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
1212     carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
1213     /* |h2| <= 2^25; from now on fits into int32 unchanged */
1214     /* |h6| <= 2^25; from now on fits into int32 unchanged */
1215     /* |h3| <= 1.71*2^59 */
1216     /* |h7| <= 1.71*2^59 */
1217 
1218     carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
1219     carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;
1220     /* |h3| <= 2^24; from now on fits into int32 unchanged */
1221     /* |h7| <= 2^24; from now on fits into int32 unchanged */
1222     /* |h4| <= 1.72*2^34 */
1223     /* |h8| <= 1.41*2^60 */
1224 
1225     carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
1226     carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;
1227     /* |h4| <= 2^25; from now on fits into int32 unchanged */
1228     /* |h8| <= 2^25; from now on fits into int32 unchanged */
1229     /* |h5| <= 1.01*2^24 */
1230     /* |h9| <= 1.71*2^59 */
1231 
1232     carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
1233     /* |h9| <= 2^24; from now on fits into int32 unchanged */
1234     /* |h0| <= 1.1*2^39 */
1235 
1236     carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
1237     /* |h0| <= 2^25; from now on fits into int32 unchanged */
1238     /* |h1| <= 1.01*2^24 */
1239 
1240     h[0] = (int32_t)h0;
1241     h[1] = (int32_t)h1;
1242     h[2] = (int32_t)h2;
1243     h[3] = (int32_t)h3;
1244     h[4] = (int32_t)h4;
1245     h[5] = (int32_t)h5;
1246     h[6] = (int32_t)h6;
1247     h[7] = (int32_t)h7;
1248     h[8] = (int32_t)h8;
1249     h[9] = (int32_t)h9;
1250 }
1251 
1252 /*
1253  * h = f * f
1254  *
1255  * Can overlap h with f.
1256  *
1257  * Preconditions:
1258  *    |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
1259  *
1260  * Postconditions:
1261  *    |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
1262  *
1263  * See fe_mul.c for discussion of implementation strategy.
1264  */
fe_sq(fe h,const fe f)1265 static void fe_sq(fe h, const fe f)
1266 {
1267     int32_t f0 = f[0];
1268     int32_t f1 = f[1];
1269     int32_t f2 = f[2];
1270     int32_t f3 = f[3];
1271     int32_t f4 = f[4];
1272     int32_t f5 = f[5];
1273     int32_t f6 = f[6];
1274     int32_t f7 = f[7];
1275     int32_t f8 = f[8];
1276     int32_t f9 = f[9];
1277     int32_t f0_2 = 2 * f0;
1278     int32_t f1_2 = 2 * f1;
1279     int32_t f2_2 = 2 * f2;
1280     int32_t f3_2 = 2 * f3;
1281     int32_t f4_2 = 2 * f4;
1282     int32_t f5_2 = 2 * f5;
1283     int32_t f6_2 = 2 * f6;
1284     int32_t f7_2 = 2 * f7;
1285     int32_t f5_38 = 38 * f5; /* 1.959375*2^30 */
1286     int32_t f6_19 = 19 * f6; /* 1.959375*2^30 */
1287     int32_t f7_38 = 38 * f7; /* 1.959375*2^30 */
1288     int32_t f8_19 = 19 * f8; /* 1.959375*2^30 */
1289     int32_t f9_38 = 38 * f9; /* 1.959375*2^30 */
1290     int64_t f0f0    = f0   * (int64_t) f0;
1291     int64_t f0f1_2  = f0_2 * (int64_t) f1;
1292     int64_t f0f2_2  = f0_2 * (int64_t) f2;
1293     int64_t f0f3_2  = f0_2 * (int64_t) f3;
1294     int64_t f0f4_2  = f0_2 * (int64_t) f4;
1295     int64_t f0f5_2  = f0_2 * (int64_t) f5;
1296     int64_t f0f6_2  = f0_2 * (int64_t) f6;
1297     int64_t f0f7_2  = f0_2 * (int64_t) f7;
1298     int64_t f0f8_2  = f0_2 * (int64_t) f8;
1299     int64_t f0f9_2  = f0_2 * (int64_t) f9;
1300     int64_t f1f1_2  = f1_2 * (int64_t) f1;
1301     int64_t f1f2_2  = f1_2 * (int64_t) f2;
1302     int64_t f1f3_4  = f1_2 * (int64_t) f3_2;
1303     int64_t f1f4_2  = f1_2 * (int64_t) f4;
1304     int64_t f1f5_4  = f1_2 * (int64_t) f5_2;
1305     int64_t f1f6_2  = f1_2 * (int64_t) f6;
1306     int64_t f1f7_4  = f1_2 * (int64_t) f7_2;
1307     int64_t f1f8_2  = f1_2 * (int64_t) f8;
1308     int64_t f1f9_76 = f1_2 * (int64_t) f9_38;
1309     int64_t f2f2    = f2   * (int64_t) f2;
1310     int64_t f2f3_2  = f2_2 * (int64_t) f3;
1311     int64_t f2f4_2  = f2_2 * (int64_t) f4;
1312     int64_t f2f5_2  = f2_2 * (int64_t) f5;
1313     int64_t f2f6_2  = f2_2 * (int64_t) f6;
1314     int64_t f2f7_2  = f2_2 * (int64_t) f7;
1315     int64_t f2f8_38 = f2_2 * (int64_t) f8_19;
1316     int64_t f2f9_38 = f2   * (int64_t) f9_38;
1317     int64_t f3f3_2  = f3_2 * (int64_t) f3;
1318     int64_t f3f4_2  = f3_2 * (int64_t) f4;
1319     int64_t f3f5_4  = f3_2 * (int64_t) f5_2;
1320     int64_t f3f6_2  = f3_2 * (int64_t) f6;
1321     int64_t f3f7_76 = f3_2 * (int64_t) f7_38;
1322     int64_t f3f8_38 = f3_2 * (int64_t) f8_19;
1323     int64_t f3f9_76 = f3_2 * (int64_t) f9_38;
1324     int64_t f4f4    = f4   * (int64_t) f4;
1325     int64_t f4f5_2  = f4_2 * (int64_t) f5;
1326     int64_t f4f6_38 = f4_2 * (int64_t) f6_19;
1327     int64_t f4f7_38 = f4   * (int64_t) f7_38;
1328     int64_t f4f8_38 = f4_2 * (int64_t) f8_19;
1329     int64_t f4f9_38 = f4   * (int64_t) f9_38;
1330     int64_t f5f5_38 = f5   * (int64_t) f5_38;
1331     int64_t f5f6_38 = f5_2 * (int64_t) f6_19;
1332     int64_t f5f7_76 = f5_2 * (int64_t) f7_38;
1333     int64_t f5f8_38 = f5_2 * (int64_t) f8_19;
1334     int64_t f5f9_76 = f5_2 * (int64_t) f9_38;
1335     int64_t f6f6_19 = f6   * (int64_t) f6_19;
1336     int64_t f6f7_38 = f6   * (int64_t) f7_38;
1337     int64_t f6f8_38 = f6_2 * (int64_t) f8_19;
1338     int64_t f6f9_38 = f6   * (int64_t) f9_38;
1339     int64_t f7f7_38 = f7   * (int64_t) f7_38;
1340     int64_t f7f8_38 = f7_2 * (int64_t) f8_19;
1341     int64_t f7f9_76 = f7_2 * (int64_t) f9_38;
1342     int64_t f8f8_19 = f8   * (int64_t) f8_19;
1343     int64_t f8f9_38 = f8   * (int64_t) f9_38;
1344     int64_t f9f9_38 = f9   * (int64_t) f9_38;
1345     int64_t h0 = f0f0   + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38;
1346     int64_t h1 = f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38;
1347     int64_t h2 = f0f2_2 + f1f1_2  + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19;
1348     int64_t h3 = f0f3_2 + f1f2_2  + f4f9_38 + f5f8_38 + f6f7_38;
1349     int64_t h4 = f0f4_2 + f1f3_4  + f2f2    + f5f9_76 + f6f8_38 + f7f7_38;
1350     int64_t h5 = f0f5_2 + f1f4_2  + f2f3_2  + f6f9_38 + f7f8_38;
1351     int64_t h6 = f0f6_2 + f1f5_4  + f2f4_2  + f3f3_2  + f7f9_76 + f8f8_19;
1352     int64_t h7 = f0f7_2 + f1f6_2  + f2f5_2  + f3f4_2  + f8f9_38;
1353     int64_t h8 = f0f8_2 + f1f7_4  + f2f6_2  + f3f5_4  + f4f4    + f9f9_38;
1354     int64_t h9 = f0f9_2 + f1f8_2  + f2f7_2  + f3f6_2  + f4f5_2;
1355     int64_t carry0;
1356     int64_t carry1;
1357     int64_t carry2;
1358     int64_t carry3;
1359     int64_t carry4;
1360     int64_t carry5;
1361     int64_t carry6;
1362     int64_t carry7;
1363     int64_t carry8;
1364     int64_t carry9;
1365 
1366     carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
1367     carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
1368 
1369     carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
1370     carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
1371 
1372     carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
1373     carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
1374 
1375     carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
1376     carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;
1377 
1378     carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
1379     carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;
1380 
1381     carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
1382 
1383     carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
1384 
1385     h[0] = (int32_t)h0;
1386     h[1] = (int32_t)h1;
1387     h[2] = (int32_t)h2;
1388     h[3] = (int32_t)h3;
1389     h[4] = (int32_t)h4;
1390     h[5] = (int32_t)h5;
1391     h[6] = (int32_t)h6;
1392     h[7] = (int32_t)h7;
1393     h[8] = (int32_t)h8;
1394     h[9] = (int32_t)h9;
1395 }
1396 
fe_invert(fe out,const fe z)1397 static void fe_invert(fe out, const fe z)
1398 {
1399     fe t0;
1400     fe t1;
1401     fe t2;
1402     fe t3;
1403     int i;
1404 
1405     /*
1406      * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as
1407      * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11.
1408      */
1409 
1410     /* t0 = z ** 2 */
1411     fe_sq(t0, z);
1412 
1413     /* t1 = t0 ** (2 ** 2) = z ** 8 */
1414     fe_sq(t1, t0);
1415     fe_sq(t1, t1);
1416 
1417     /* t1 = z * t1 = z ** 9 */
1418     fe_mul(t1, z, t1);
1419     /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */
1420     fe_mul(t0, t0, t1);
1421 
1422     /* t2 = t0 ** 2 = z ** 22 */
1423     fe_sq(t2, t0);
1424 
1425     /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */
1426     fe_mul(t1, t1, t2);
1427 
1428     /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */
1429     fe_sq(t2, t1);
1430     for (i = 1; i < 5; ++i) {
1431         fe_sq(t2, t2);
1432     }
1433 
1434     /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */
1435     fe_mul(t1, t2, t1);
1436 
1437     /* Continuing similarly... */
1438 
1439     /* t2 = z ** (2 ** 20 - 1) */
1440     fe_sq(t2, t1);
1441     for (i = 1; i < 10; ++i) {
1442         fe_sq(t2, t2);
1443     }
1444     fe_mul(t2, t2, t1);
1445 
1446     /* t2 = z ** (2 ** 40 - 1) */
1447     fe_sq(t3, t2);
1448     for (i = 1; i < 20; ++i) {
1449         fe_sq(t3, t3);
1450     }
1451     fe_mul(t2, t3, t2);
1452 
1453     /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */
1454     for (i = 0; i < 10; ++i) {
1455         fe_sq(t2, t2);
1456     }
1457     /* t1 = z ** (2 ** 50 - 1) */
1458     fe_mul(t1, t2, t1);
1459 
1460     /* t2 = z ** (2 ** 100 - 1) */
1461     fe_sq(t2, t1);
1462     for (i = 1; i < 50; ++i) {
1463         fe_sq(t2, t2);
1464     }
1465     fe_mul(t2, t2, t1);
1466 
1467     /* t2 = z ** (2 ** 200 - 1) */
1468     fe_sq(t3, t2);
1469     for (i = 1; i < 100; ++i) {
1470         fe_sq(t3, t3);
1471     }
1472     fe_mul(t2, t3, t2);
1473 
1474     /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */
1475     fe_sq(t2, t2);
1476     for (i = 1; i < 50; ++i) {
1477         fe_sq(t2, t2);
1478     }
1479 
1480     /* t1 = z ** (2 ** 250 - 1) */
1481     fe_mul(t1, t2, t1);
1482 
1483     /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */
1484     fe_sq(t1, t1);
1485     for (i = 1; i < 5; ++i) {
1486         fe_sq(t1, t1);
1487     }
1488 
1489     /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */
1490     fe_mul(out, t1, t0);
1491 }
1492 
1493 /*
1494  * h = -f
1495  *
1496  * Preconditions:
1497  *    |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
1498  *
1499  * Postconditions:
1500  *    |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
1501  */
fe_neg(fe h,const fe f)1502 static void fe_neg(fe h, const fe f)
1503 {
1504     unsigned i;
1505 
1506     for (i = 0; i < 10; i++) {
1507         h[i] = -f[i];
1508     }
1509 }
1510 
1511 /*
1512  * Replace (f,g) with (g,g) if b == 1;
1513  * replace (f,g) with (f,g) if b == 0.
1514  *
1515  * Preconditions: b in {0,1}.
1516  */
fe_cmov(fe f,const fe g,unsigned b)1517 static void fe_cmov(fe f, const fe g, unsigned b)
1518 {
1519     size_t i;
1520 
1521     b = 0-b;
1522     for (i = 0; i < 10; i++) {
1523         int32_t x = f[i] ^ g[i];
1524         x &= b;
1525         f[i] ^= x;
1526     }
1527 }
1528 
1529 /*
1530  * return 0 if f == 0
1531  * return 1 if f != 0
1532  *
1533  * Preconditions:
1534  *    |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
1535  */
fe_isnonzero(const fe f)1536 static int fe_isnonzero(const fe f)
1537 {
1538     uint8_t s[32];
1539     static const uint8_t zero[32] = {0};
1540 
1541     fe_tobytes(s, f);
1542 
1543     return CRYPTO_memcmp(s, zero, sizeof(zero)) != 0;
1544 }
1545 
1546 /*
1547  * return 1 if f is in {1,3,5,...,q-2}
1548  * return 0 if f is in {0,2,4,...,q-1}
1549  *
1550  * Preconditions:
1551  *    |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
1552  */
fe_isnegative(const fe f)1553 static int fe_isnegative(const fe f)
1554 {
1555     uint8_t s[32];
1556 
1557     fe_tobytes(s, f);
1558     return s[0] & 1;
1559 }
1560 
1561 /*
1562  * h = 2 * f * f
1563  *
1564  * Can overlap h with f.
1565  *
1566  * Preconditions:
1567  *    |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
1568  *
1569  * Postconditions:
1570  *    |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
1571  *
1572  * See fe_mul.c for discussion of implementation strategy.
1573  */
fe_sq2(fe h,const fe f)1574 static void fe_sq2(fe h, const fe f)
1575 {
1576     int32_t f0 = f[0];
1577     int32_t f1 = f[1];
1578     int32_t f2 = f[2];
1579     int32_t f3 = f[3];
1580     int32_t f4 = f[4];
1581     int32_t f5 = f[5];
1582     int32_t f6 = f[6];
1583     int32_t f7 = f[7];
1584     int32_t f8 = f[8];
1585     int32_t f9 = f[9];
1586     int32_t f0_2 = 2 * f0;
1587     int32_t f1_2 = 2 * f1;
1588     int32_t f2_2 = 2 * f2;
1589     int32_t f3_2 = 2 * f3;
1590     int32_t f4_2 = 2 * f4;
1591     int32_t f5_2 = 2 * f5;
1592     int32_t f6_2 = 2 * f6;
1593     int32_t f7_2 = 2 * f7;
1594     int32_t f5_38 = 38 * f5; /* 1.959375*2^30 */
1595     int32_t f6_19 = 19 * f6; /* 1.959375*2^30 */
1596     int32_t f7_38 = 38 * f7; /* 1.959375*2^30 */
1597     int32_t f8_19 = 19 * f8; /* 1.959375*2^30 */
1598     int32_t f9_38 = 38 * f9; /* 1.959375*2^30 */
1599     int64_t f0f0    = f0   * (int64_t) f0;
1600     int64_t f0f1_2  = f0_2 * (int64_t) f1;
1601     int64_t f0f2_2  = f0_2 * (int64_t) f2;
1602     int64_t f0f3_2  = f0_2 * (int64_t) f3;
1603     int64_t f0f4_2  = f0_2 * (int64_t) f4;
1604     int64_t f0f5_2  = f0_2 * (int64_t) f5;
1605     int64_t f0f6_2  = f0_2 * (int64_t) f6;
1606     int64_t f0f7_2  = f0_2 * (int64_t) f7;
1607     int64_t f0f8_2  = f0_2 * (int64_t) f8;
1608     int64_t f0f9_2  = f0_2 * (int64_t) f9;
1609     int64_t f1f1_2  = f1_2 * (int64_t) f1;
1610     int64_t f1f2_2  = f1_2 * (int64_t) f2;
1611     int64_t f1f3_4  = f1_2 * (int64_t) f3_2;
1612     int64_t f1f4_2  = f1_2 * (int64_t) f4;
1613     int64_t f1f5_4  = f1_2 * (int64_t) f5_2;
1614     int64_t f1f6_2  = f1_2 * (int64_t) f6;
1615     int64_t f1f7_4  = f1_2 * (int64_t) f7_2;
1616     int64_t f1f8_2  = f1_2 * (int64_t) f8;
1617     int64_t f1f9_76 = f1_2 * (int64_t) f9_38;
1618     int64_t f2f2    = f2   * (int64_t) f2;
1619     int64_t f2f3_2  = f2_2 * (int64_t) f3;
1620     int64_t f2f4_2  = f2_2 * (int64_t) f4;
1621     int64_t f2f5_2  = f2_2 * (int64_t) f5;
1622     int64_t f2f6_2  = f2_2 * (int64_t) f6;
1623     int64_t f2f7_2  = f2_2 * (int64_t) f7;
1624     int64_t f2f8_38 = f2_2 * (int64_t) f8_19;
1625     int64_t f2f9_38 = f2   * (int64_t) f9_38;
1626     int64_t f3f3_2  = f3_2 * (int64_t) f3;
1627     int64_t f3f4_2  = f3_2 * (int64_t) f4;
1628     int64_t f3f5_4  = f3_2 * (int64_t) f5_2;
1629     int64_t f3f6_2  = f3_2 * (int64_t) f6;
1630     int64_t f3f7_76 = f3_2 * (int64_t) f7_38;
1631     int64_t f3f8_38 = f3_2 * (int64_t) f8_19;
1632     int64_t f3f9_76 = f3_2 * (int64_t) f9_38;
1633     int64_t f4f4    = f4   * (int64_t) f4;
1634     int64_t f4f5_2  = f4_2 * (int64_t) f5;
1635     int64_t f4f6_38 = f4_2 * (int64_t) f6_19;
1636     int64_t f4f7_38 = f4   * (int64_t) f7_38;
1637     int64_t f4f8_38 = f4_2 * (int64_t) f8_19;
1638     int64_t f4f9_38 = f4   * (int64_t) f9_38;
1639     int64_t f5f5_38 = f5   * (int64_t) f5_38;
1640     int64_t f5f6_38 = f5_2 * (int64_t) f6_19;
1641     int64_t f5f7_76 = f5_2 * (int64_t) f7_38;
1642     int64_t f5f8_38 = f5_2 * (int64_t) f8_19;
1643     int64_t f5f9_76 = f5_2 * (int64_t) f9_38;
1644     int64_t f6f6_19 = f6   * (int64_t) f6_19;
1645     int64_t f6f7_38 = f6   * (int64_t) f7_38;
1646     int64_t f6f8_38 = f6_2 * (int64_t) f8_19;
1647     int64_t f6f9_38 = f6   * (int64_t) f9_38;
1648     int64_t f7f7_38 = f7   * (int64_t) f7_38;
1649     int64_t f7f8_38 = f7_2 * (int64_t) f8_19;
1650     int64_t f7f9_76 = f7_2 * (int64_t) f9_38;
1651     int64_t f8f8_19 = f8   * (int64_t) f8_19;
1652     int64_t f8f9_38 = f8   * (int64_t) f9_38;
1653     int64_t f9f9_38 = f9   * (int64_t) f9_38;
1654     int64_t h0 = f0f0   + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38;
1655     int64_t h1 = f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38;
1656     int64_t h2 = f0f2_2 + f1f1_2  + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19;
1657     int64_t h3 = f0f3_2 + f1f2_2  + f4f9_38 + f5f8_38 + f6f7_38;
1658     int64_t h4 = f0f4_2 + f1f3_4  + f2f2    + f5f9_76 + f6f8_38 + f7f7_38;
1659     int64_t h5 = f0f5_2 + f1f4_2  + f2f3_2  + f6f9_38 + f7f8_38;
1660     int64_t h6 = f0f6_2 + f1f5_4  + f2f4_2  + f3f3_2  + f7f9_76 + f8f8_19;
1661     int64_t h7 = f0f7_2 + f1f6_2  + f2f5_2  + f3f4_2  + f8f9_38;
1662     int64_t h8 = f0f8_2 + f1f7_4  + f2f6_2  + f3f5_4  + f4f4    + f9f9_38;
1663     int64_t h9 = f0f9_2 + f1f8_2  + f2f7_2  + f3f6_2  + f4f5_2;
1664     int64_t carry0;
1665     int64_t carry1;
1666     int64_t carry2;
1667     int64_t carry3;
1668     int64_t carry4;
1669     int64_t carry5;
1670     int64_t carry6;
1671     int64_t carry7;
1672     int64_t carry8;
1673     int64_t carry9;
1674 
1675     h0 += h0;
1676     h1 += h1;
1677     h2 += h2;
1678     h3 += h3;
1679     h4 += h4;
1680     h5 += h5;
1681     h6 += h6;
1682     h7 += h7;
1683     h8 += h8;
1684     h9 += h9;
1685 
1686     carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
1687     carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
1688 
1689     carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
1690     carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
1691 
1692     carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
1693     carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
1694 
1695     carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
1696     carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;
1697 
1698     carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
1699     carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;
1700 
1701     carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
1702 
1703     carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
1704 
1705     h[0] = (int32_t)h0;
1706     h[1] = (int32_t)h1;
1707     h[2] = (int32_t)h2;
1708     h[3] = (int32_t)h3;
1709     h[4] = (int32_t)h4;
1710     h[5] = (int32_t)h5;
1711     h[6] = (int32_t)h6;
1712     h[7] = (int32_t)h7;
1713     h[8] = (int32_t)h8;
1714     h[9] = (int32_t)h9;
1715 }
1716 
fe_pow22523(fe out,const fe z)1717 static void fe_pow22523(fe out, const fe z)
1718 {
1719     fe t0;
1720     fe t1;
1721     fe t2;
1722     int i;
1723 
1724     fe_sq(t0, z);
1725     fe_sq(t1, t0);
1726     for (i = 1; i < 2; ++i) {
1727         fe_sq(t1, t1);
1728     }
1729     fe_mul(t1, z, t1);
1730     fe_mul(t0, t0, t1);
1731     fe_sq(t0, t0);
1732     fe_mul(t0, t1, t0);
1733     fe_sq(t1, t0);
1734     for (i = 1; i < 5; ++i) {
1735         fe_sq(t1, t1);
1736     }
1737     fe_mul(t0, t1, t0);
1738     fe_sq(t1, t0);
1739     for (i = 1; i < 10; ++i) {
1740         fe_sq(t1, t1);
1741     }
1742     fe_mul(t1, t1, t0);
1743     fe_sq(t2, t1);
1744     for (i = 1; i < 20; ++i) {
1745         fe_sq(t2, t2);
1746     }
1747     fe_mul(t1, t2, t1);
1748     fe_sq(t1, t1);
1749     for (i = 1; i < 10; ++i) {
1750         fe_sq(t1, t1);
1751     }
1752     fe_mul(t0, t1, t0);
1753     fe_sq(t1, t0);
1754     for (i = 1; i < 50; ++i) {
1755         fe_sq(t1, t1);
1756     }
1757     fe_mul(t1, t1, t0);
1758     fe_sq(t2, t1);
1759     for (i = 1; i < 100; ++i) {
1760         fe_sq(t2, t2);
1761     }
1762     fe_mul(t1, t2, t1);
1763     fe_sq(t1, t1);
1764     for (i = 1; i < 50; ++i) {
1765         fe_sq(t1, t1);
1766     }
1767     fe_mul(t0, t1, t0);
1768     fe_sq(t0, t0);
1769     for (i = 1; i < 2; ++i) {
1770         fe_sq(t0, t0);
1771     }
1772     fe_mul(out, t0, z);
1773 }
1774 
1775 /*
1776  * ge means group element.
1777  *
1778  * Here the group is the set of pairs (x,y) of field elements (see fe.h)
1779  * satisfying -x^2 + y^2 = 1 + d x^2y^2
1780  * where d = -121665/121666.
1781  *
1782  * Representations:
1783  *   ge_p2 (projective): (X:Y:Z) satisfying x=X/Z, y=Y/Z
1784  *   ge_p3 (extended): (X:Y:Z:T) satisfying x=X/Z, y=Y/Z, XY=ZT
1785  *   ge_p1p1 (completed): ((X:Z),(Y:T)) satisfying x=X/Z, y=Y/T
1786  *   ge_precomp (Duif): (y+x,y-x,2dxy)
1787  */
1788 typedef struct {
1789     fe X;
1790     fe Y;
1791     fe Z;
1792 } ge_p2;
1793 
1794 typedef struct {
1795     fe X;
1796     fe Y;
1797     fe Z;
1798     fe T;
1799 } ge_p3;
1800 
1801 typedef struct {
1802     fe X;
1803     fe Y;
1804     fe Z;
1805     fe T;
1806 } ge_p1p1;
1807 
1808 typedef struct {
1809     fe yplusx;
1810     fe yminusx;
1811     fe xy2d;
1812 } ge_precomp;
1813 
1814 typedef struct {
1815     fe YplusX;
1816     fe YminusX;
1817     fe Z;
1818     fe T2d;
1819 } ge_cached;
1820 
ge_tobytes(uint8_t * s,const ge_p2 * h)1821 static void ge_tobytes(uint8_t *s, const ge_p2 *h)
1822 {
1823     fe recip;
1824     fe x;
1825     fe y;
1826 
1827     fe_invert(recip, h->Z);
1828     fe_mul(x, h->X, recip);
1829     fe_mul(y, h->Y, recip);
1830     fe_tobytes(s, y);
1831     s[31] ^= fe_isnegative(x) << 7;
1832 }
1833 
ge_p3_tobytes(uint8_t * s,const ge_p3 * h)1834 static void ge_p3_tobytes(uint8_t *s, const ge_p3 *h)
1835 {
1836     fe recip;
1837     fe x;
1838     fe y;
1839 
1840     fe_invert(recip, h->Z);
1841     fe_mul(x, h->X, recip);
1842     fe_mul(y, h->Y, recip);
1843     fe_tobytes(s, y);
1844     s[31] ^= fe_isnegative(x) << 7;
1845 }
1846 
1847 static const fe d = {
1848     -10913610, 13857413, -15372611, 6949391,   114729,
1849     -8787816,  -6275908, -3247719,  -18696448, -12055116
1850 };
1851 
1852 static const fe sqrtm1 = {
1853     -32595792, -7943725,  9377950,  3500415, 12389472,
1854     -272473,   -25146209, -2005654, 326686,  11406482
1855 };
1856 
ge_frombytes_vartime(ge_p3 * h,const uint8_t * s)1857 static int ge_frombytes_vartime(ge_p3 *h, const uint8_t *s)
1858 {
1859     fe u;
1860     fe v;
1861     fe v3;
1862     fe vxx;
1863     fe check;
1864 
1865     fe_frombytes(h->Y, s);
1866     fe_1(h->Z);
1867     fe_sq(u, h->Y);
1868     fe_mul(v, u, d);
1869     fe_sub(u, u, h->Z); /* u = y^2-1 */
1870     fe_add(v, v, h->Z); /* v = dy^2+1 */
1871 
1872     fe_sq(v3, v);
1873     fe_mul(v3, v3, v); /* v3 = v^3 */
1874     fe_sq(h->X, v3);
1875     fe_mul(h->X, h->X, v);
1876     fe_mul(h->X, h->X, u); /* x = uv^7 */
1877 
1878     fe_pow22523(h->X, h->X); /* x = (uv^7)^((q-5)/8) */
1879     fe_mul(h->X, h->X, v3);
1880     fe_mul(h->X, h->X, u); /* x = uv^3(uv^7)^((q-5)/8) */
1881 
1882     fe_sq(vxx, h->X);
1883     fe_mul(vxx, vxx, v);
1884     fe_sub(check, vxx, u); /* vx^2-u */
1885     if (fe_isnonzero(check)) {
1886         fe_add(check, vxx, u); /* vx^2+u */
1887         if (fe_isnonzero(check)) {
1888             return -1;
1889         }
1890         fe_mul(h->X, h->X, sqrtm1);
1891     }
1892 
1893     if (fe_isnegative(h->X) != (s[31] >> 7)) {
1894         fe_neg(h->X, h->X);
1895     }
1896 
1897     fe_mul(h->T, h->X, h->Y);
1898     return 0;
1899 }
1900 
ge_p2_0(ge_p2 * h)1901 static void ge_p2_0(ge_p2 *h)
1902 {
1903     fe_0(h->X);
1904     fe_1(h->Y);
1905     fe_1(h->Z);
1906 }
1907 
ge_p3_0(ge_p3 * h)1908 static void ge_p3_0(ge_p3 *h)
1909 {
1910     fe_0(h->X);
1911     fe_1(h->Y);
1912     fe_1(h->Z);
1913     fe_0(h->T);
1914 }
1915 
ge_precomp_0(ge_precomp * h)1916 static void ge_precomp_0(ge_precomp *h)
1917 {
1918     fe_1(h->yplusx);
1919     fe_1(h->yminusx);
1920     fe_0(h->xy2d);
1921 }
1922 
1923 /* r = p */
ge_p3_to_p2(ge_p2 * r,const ge_p3 * p)1924 static void ge_p3_to_p2(ge_p2 *r, const ge_p3 *p)
1925 {
1926     fe_copy(r->X, p->X);
1927     fe_copy(r->Y, p->Y);
1928     fe_copy(r->Z, p->Z);
1929 }
1930 
1931 static const fe d2 = {
1932     -21827239, -5839606,  -30745221, 13898782, 229458,
1933     15978800,  -12551817, -6495438,  29715968, 9444199
1934 };
1935 
1936 /* r = p */
ge_p3_to_cached(ge_cached * r,const ge_p3 * p)1937 static void ge_p3_to_cached(ge_cached *r, const ge_p3 *p)
1938 {
1939     fe_add(r->YplusX, p->Y, p->X);
1940     fe_sub(r->YminusX, p->Y, p->X);
1941     fe_copy(r->Z, p->Z);
1942     fe_mul(r->T2d, p->T, d2);
1943 }
1944 
1945 /* r = p */
ge_p1p1_to_p2(ge_p2 * r,const ge_p1p1 * p)1946 static void ge_p1p1_to_p2(ge_p2 *r, const ge_p1p1 *p)
1947 {
1948     fe_mul(r->X, p->X, p->T);
1949     fe_mul(r->Y, p->Y, p->Z);
1950     fe_mul(r->Z, p->Z, p->T);
1951 }
1952 
1953 /* r = p */
ge_p1p1_to_p3(ge_p3 * r,const ge_p1p1 * p)1954 static void ge_p1p1_to_p3(ge_p3 *r, const ge_p1p1 *p)
1955 {
1956     fe_mul(r->X, p->X, p->T);
1957     fe_mul(r->Y, p->Y, p->Z);
1958     fe_mul(r->Z, p->Z, p->T);
1959     fe_mul(r->T, p->X, p->Y);
1960 }
1961 
1962 /* r = 2 * p */
ge_p2_dbl(ge_p1p1 * r,const ge_p2 * p)1963 static void ge_p2_dbl(ge_p1p1 *r, const ge_p2 *p)
1964 {
1965     fe t0;
1966 
1967     fe_sq(r->X, p->X);
1968     fe_sq(r->Z, p->Y);
1969     fe_sq2(r->T, p->Z);
1970     fe_add(r->Y, p->X, p->Y);
1971     fe_sq(t0, r->Y);
1972     fe_add(r->Y, r->Z, r->X);
1973     fe_sub(r->Z, r->Z, r->X);
1974     fe_sub(r->X, t0, r->Y);
1975     fe_sub(r->T, r->T, r->Z);
1976 }
1977 
1978 /* r = 2 * p */
ge_p3_dbl(ge_p1p1 * r,const ge_p3 * p)1979 static void ge_p3_dbl(ge_p1p1 *r, const ge_p3 *p)
1980 {
1981     ge_p2 q;
1982     ge_p3_to_p2(&q, p);
1983     ge_p2_dbl(r, &q);
1984 }
1985 
1986 /* r = p + q */
ge_madd(ge_p1p1 * r,const ge_p3 * p,const ge_precomp * q)1987 static void ge_madd(ge_p1p1 *r, const ge_p3 *p, const ge_precomp *q)
1988 {
1989     fe t0;
1990 
1991     fe_add(r->X, p->Y, p->X);
1992     fe_sub(r->Y, p->Y, p->X);
1993     fe_mul(r->Z, r->X, q->yplusx);
1994     fe_mul(r->Y, r->Y, q->yminusx);
1995     fe_mul(r->T, q->xy2d, p->T);
1996     fe_add(t0, p->Z, p->Z);
1997     fe_sub(r->X, r->Z, r->Y);
1998     fe_add(r->Y, r->Z, r->Y);
1999     fe_add(r->Z, t0, r->T);
2000     fe_sub(r->T, t0, r->T);
2001 }
2002 
2003 /* r = p - q */
ge_msub(ge_p1p1 * r,const ge_p3 * p,const ge_precomp * q)2004 static void ge_msub(ge_p1p1 *r, const ge_p3 *p, const ge_precomp *q)
2005 {
2006     fe t0;
2007 
2008     fe_add(r->X, p->Y, p->X);
2009     fe_sub(r->Y, p->Y, p->X);
2010     fe_mul(r->Z, r->X, q->yminusx);
2011     fe_mul(r->Y, r->Y, q->yplusx);
2012     fe_mul(r->T, q->xy2d, p->T);
2013     fe_add(t0, p->Z, p->Z);
2014     fe_sub(r->X, r->Z, r->Y);
2015     fe_add(r->Y, r->Z, r->Y);
2016     fe_sub(r->Z, t0, r->T);
2017     fe_add(r->T, t0, r->T);
2018 }
2019 
2020 /* r = p + q */
ge_add(ge_p1p1 * r,const ge_p3 * p,const ge_cached * q)2021 static void ge_add(ge_p1p1 *r, const ge_p3 *p, const ge_cached *q)
2022 {
2023     fe t0;
2024 
2025     fe_add(r->X, p->Y, p->X);
2026     fe_sub(r->Y, p->Y, p->X);
2027     fe_mul(r->Z, r->X, q->YplusX);
2028     fe_mul(r->Y, r->Y, q->YminusX);
2029     fe_mul(r->T, q->T2d, p->T);
2030     fe_mul(r->X, p->Z, q->Z);
2031     fe_add(t0, r->X, r->X);
2032     fe_sub(r->X, r->Z, r->Y);
2033     fe_add(r->Y, r->Z, r->Y);
2034     fe_add(r->Z, t0, r->T);
2035     fe_sub(r->T, t0, r->T);
2036 }
2037 
2038 /* r = p - q */
ge_sub(ge_p1p1 * r,const ge_p3 * p,const ge_cached * q)2039 static void ge_sub(ge_p1p1 *r, const ge_p3 *p, const ge_cached *q)
2040 {
2041     fe t0;
2042 
2043     fe_add(r->X, p->Y, p->X);
2044     fe_sub(r->Y, p->Y, p->X);
2045     fe_mul(r->Z, r->X, q->YminusX);
2046     fe_mul(r->Y, r->Y, q->YplusX);
2047     fe_mul(r->T, q->T2d, p->T);
2048     fe_mul(r->X, p->Z, q->Z);
2049     fe_add(t0, r->X, r->X);
2050     fe_sub(r->X, r->Z, r->Y);
2051     fe_add(r->Y, r->Z, r->Y);
2052     fe_sub(r->Z, t0, r->T);
2053     fe_add(r->T, t0, r->T);
2054 }
2055 
equal(signed char b,signed char c)2056 static uint8_t equal(signed char b, signed char c)
2057 {
2058     uint8_t ub = b;
2059     uint8_t uc = c;
2060     uint8_t x = ub ^ uc; /* 0: yes; 1..255: no */
2061     uint32_t y = x;      /* 0: yes; 1..255: no */
2062     y -= 1;              /* 4294967295: yes; 0..254: no */
2063     y >>= 31;            /* 1: yes; 0: no */
2064     return y;
2065 }
2066 
cmov(ge_precomp * t,const ge_precomp * u,uint8_t b)2067 static void cmov(ge_precomp *t, const ge_precomp *u, uint8_t b)
2068 {
2069     fe_cmov(t->yplusx, u->yplusx, b);
2070     fe_cmov(t->yminusx, u->yminusx, b);
2071     fe_cmov(t->xy2d, u->xy2d, b);
2072 }
2073 
2074 /* k25519Precomp[i][j] = (j+1)*256^i*B */
2075 static const ge_precomp k25519Precomp[32][8] = {
2076     {
2077         {
2078             {25967493, -14356035, 29566456, 3660896, -12694345, 4014787,
2079              27544626, -11754271, -6079156, 2047605},
2080             {-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692,
2081              5043384, 19500929, -15469378},
2082             {-8738181, 4489570, 9688441, -14785194, 10184609, -12363380,
2083              29287919, 11864899, -24514362, -4438546},
2084         },
2085         {
2086             {-12815894, -12976347, -21581243, 11784320, -25355658, -2750717,
2087              -11717903, -3814571, -358445, -10211303},
2088             {-21703237, 6903825, 27185491, 6451973, -29577724, -9554005,
2089              -15616551, 11189268, -26829678, -5319081},
2090             {26966642, 11152617, 32442495, 15396054, 14353839, -12752335,
2091              -3128826, -9541118, -15472047, -4166697},
2092         },
2093         {
2094             {15636291, -9688557, 24204773, -7912398, 616977, -16685262,
2095              27787600, -14772189, 28944400, -1550024},
2096             {16568933, 4717097, -11556148, -1102322, 15682896, -11807043,
2097              16354577, -11775962, 7689662, 11199574},
2098             {30464156, -5976125, -11779434, -15670865, 23220365, 15915852,
2099              7512774, 10017326, -17749093, -9920357},
2100         },
2101         {
2102             {-17036878, 13921892, 10945806, -6033431, 27105052, -16084379,
2103              -28926210, 15006023, 3284568, -6276540},
2104             {23599295, -8306047, -11193664, -7687416, 13236774, 10506355,
2105              7464579, 9656445, 13059162, 10374397},
2106             {7798556, 16710257, 3033922, 2874086, 28997861, 2835604, 32406664,
2107              -3839045, -641708, -101325},
2108         },
2109         {
2110             {10861363, 11473154, 27284546, 1981175, -30064349, 12577861,
2111              32867885, 14515107, -15438304, 10819380},
2112             {4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668,
2113              12483688, -12668491, 5581306},
2114             {19563160, 16186464, -29386857, 4097519, 10237984, -4348115,
2115              28542350, 13850243, -23678021, -15815942},
2116         },
2117         {
2118             {-15371964, -12862754, 32573250, 4720197, -26436522, 5875511,
2119              -19188627, -15224819, -9818940, -12085777},
2120             {-8549212, 109983, 15149363, 2178705, 22900618, 4543417, 3044240,
2121              -15689887, 1762328, 14866737},
2122             {-18199695, -15951423, -10473290, 1707278, -17185920, 3916101,
2123              -28236412, 3959421, 27914454, 4383652},
2124         },
2125         {
2126             {5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852,
2127              5230134, -23952439, -15175766},
2128             {-30269007, -3463509, 7665486, 10083793, 28475525, 1649722,
2129              20654025, 16520125, 30598449, 7715701},
2130             {28881845, 14381568, 9657904, 3680757, -20181635, 7843316,
2131              -31400660, 1370708, 29794553, -1409300},
2132         },
2133         {
2134             {14499471, -2729599, -33191113, -4254652, 28494862, 14271267,
2135              30290735, 10876454, -33154098, 2381726},
2136             {-7195431, -2655363, -14730155, 462251, -27724326, 3941372,
2137              -6236617, 3696005, -32300832, 15351955},
2138             {27431194, 8222322, 16448760, -3907995, -18707002, 11938355,
2139              -32961401, -2970515, 29551813, 10109425},
2140         },
2141     },
2142     {
2143         {
2144             {-13657040, -13155431, -31283750, 11777098, 21447386, 6519384,
2145              -2378284, -1627556, 10092783, -4764171},
2146             {27939166, 14210322, 4677035, 16277044, -22964462, -12398139,
2147              -32508754, 12005538, -17810127, 12803510},
2148             {17228999, -15661624, -1233527, 300140, -1224870, -11714777,
2149              30364213, -9038194, 18016357, 4397660},
2150         },
2151         {
2152             {-10958843, -7690207, 4776341, -14954238, 27850028, -15602212,
2153              -26619106, 14544525, -17477504, 982639},
2154             {29253598, 15796703, -2863982, -9908884, 10057023, 3163536, 7332899,
2155              -4120128, -21047696, 9934963},
2156             {5793303, 16271923, -24131614, -10116404, 29188560, 1206517,
2157              -14747930, 4559895, -30123922, -10897950},
2158         },
2159         {
2160             {-27643952, -11493006, 16282657, -11036493, 28414021, -15012264,
2161              24191034, 4541697, -13338309, 5500568},
2162             {12650548, -1497113, 9052871, 11355358, -17680037, -8400164,
2163              -17430592, 12264343, 10874051, 13524335},
2164             {25556948, -3045990, 714651, 2510400, 23394682, -10415330, 33119038,
2165              5080568, -22528059, 5376628},
2166         },
2167         {
2168             {-26088264, -4011052, -17013699, -3537628, -6726793, 1920897,
2169              -22321305, -9447443, 4535768, 1569007},
2170             {-2255422, 14606630, -21692440, -8039818, 28430649, 8775819,
2171              -30494562, 3044290, 31848280, 12543772},
2172             {-22028579, 2943893, -31857513, 6777306, 13784462, -4292203,
2173              -27377195, -2062731, 7718482, 14474653},
2174         },
2175         {
2176             {2385315, 2454213, -22631320, 46603, -4437935, -15680415, 656965,
2177              -7236665, 24316168, -5253567},
2178             {13741529, 10911568, -33233417, -8603737, -20177830, -1033297,
2179              33040651, -13424532, -20729456, 8321686},
2180             {21060490, -2212744, 15712757, -4336099, 1639040, 10656336,
2181              23845965, -11874838, -9984458, 608372},
2182         },
2183         {
2184             {-13672732, -15087586, -10889693, -7557059, -6036909, 11305547,
2185              1123968, -6780577, 27229399, 23887},
2186             {-23244140, -294205, -11744728, 14712571, -29465699, -2029617,
2187              12797024, -6440308, -1633405, 16678954},
2188             {-29500620, 4770662, -16054387, 14001338, 7830047, 9564805,
2189              -1508144, -4795045, -17169265, 4904953},
2190         },
2191         {
2192             {24059557, 14617003, 19037157, -15039908, 19766093, -14906429,
2193              5169211, 16191880, 2128236, -4326833},
2194             {-16981152, 4124966, -8540610, -10653797, 30336522, -14105247,
2195              -29806336, 916033, -6882542, -2986532},
2196             {-22630907, 12419372, -7134229, -7473371, -16478904, 16739175,
2197              285431, 2763829, 15736322, 4143876},
2198         },
2199         {
2200             {2379352, 11839345, -4110402, -5988665, 11274298, 794957, 212801,
2201              -14594663, 23527084, -16458268},
2202             {33431127, -11130478, -17838966, -15626900, 8909499, 8376530,
2203              -32625340, 4087881, -15188911, -14416214},
2204             {1767683, 7197987, -13205226, -2022635, -13091350, 448826, 5799055,
2205              4357868, -4774191, -16323038},
2206         },
2207     },
2208     {
2209         {
2210             {6721966, 13833823, -23523388, -1551314, 26354293, -11863321,
2211              23365147, -3949732, 7390890, 2759800},
2212             {4409041, 2052381, 23373853, 10530217, 7676779, -12885954, 21302353,
2213              -4264057, 1244380, -12919645},
2214             {-4421239, 7169619, 4982368, -2957590, 30256825, -2777540, 14086413,
2215              9208236, 15886429, 16489664},
2216         },
2217         {
2218             {1996075, 10375649, 14346367, 13311202, -6874135, -16438411,
2219              -13693198, 398369, -30606455, -712933},
2220             {-25307465, 9795880, -2777414, 14878809, -33531835, 14780363,
2221              13348553, 12076947, -30836462, 5113182},
2222             {-17770784, 11797796, 31950843, 13929123, -25888302, 12288344,
2223              -30341101, -7336386, 13847711, 5387222},
2224         },
2225         {
2226             {-18582163, -3416217, 17824843, -2340966, 22744343, -10442611,
2227              8763061, 3617786, -19600662, 10370991},
2228             {20246567, -14369378, 22358229, -543712, 18507283, -10413996,
2229              14554437, -8746092, 32232924, 16763880},
2230             {9648505, 10094563, 26416693, 14745928, -30374318, -6472621,
2231              11094161, 15689506, 3140038, -16510092},
2232         },
2233         {
2234             {-16160072, 5472695, 31895588, 4744994, 8823515, 10365685,
2235              -27224800, 9448613, -28774454, 366295},
2236             {19153450, 11523972, -11096490, -6503142, -24647631, 5420647,
2237              28344573, 8041113, 719605, 11671788},
2238             {8678025, 2694440, -6808014, 2517372, 4964326, 11152271, -15432916,
2239              -15266516, 27000813, -10195553},
2240         },
2241         {
2242             {-15157904, 7134312, 8639287, -2814877, -7235688, 10421742, 564065,
2243              5336097, 6750977, -14521026},
2244             {11836410, -3979488, 26297894, 16080799, 23455045, 15735944,
2245              1695823, -8819122, 8169720, 16220347},
2246             {-18115838, 8653647, 17578566, -6092619, -8025777, -16012763,
2247              -11144307, -2627664, -5990708, -14166033},
2248         },
2249         {
2250             {-23308498, -10968312, 15213228, -10081214, -30853605, -11050004,
2251              27884329, 2847284, 2655861, 1738395},
2252             {-27537433, -14253021, -25336301, -8002780, -9370762, 8129821,
2253              21651608, -3239336, -19087449, -11005278},
2254             {1533110, 3437855, 23735889, 459276, 29970501, 11335377, 26030092,
2255              5821408, 10478196, 8544890},
2256         },
2257         {
2258             {32173121, -16129311, 24896207, 3921497, 22579056, -3410854,
2259              19270449, 12217473, 17789017, -3395995},
2260             {-30552961, -2228401, -15578829, -10147201, 13243889, 517024,
2261              15479401, -3853233, 30460520, 1052596},
2262             {-11614875, 13323618, 32618793, 8175907, -15230173, 12596687,
2263              27491595, -4612359, 3179268, -9478891},
2264         },
2265         {
2266             {31947069, -14366651, -4640583, -15339921, -15125977, -6039709,
2267              -14756777, -16411740, 19072640, -9511060},
2268             {11685058, 11822410, 3158003, -13952594, 33402194, -4165066,
2269              5977896, -5215017, 473099, 5040608},
2270             {-20290863, 8198642, -27410132, 11602123, 1290375, -2799760,
2271              28326862, 1721092, -19558642, -3131606},
2272         },
2273     },
2274     {
2275         {
2276             {7881532, 10687937, 7578723, 7738378, -18951012, -2553952, 21820786,
2277              8076149, -27868496, 11538389},
2278             {-19935666, 3899861, 18283497, -6801568, -15728660, -11249211,
2279              8754525, 7446702, -5676054, 5797016},
2280             {-11295600, -3793569, -15782110, -7964573, 12708869, -8456199,
2281              2014099, -9050574, -2369172, -5877341},
2282         },
2283         {
2284             {-22472376, -11568741, -27682020, 1146375, 18956691, 16640559,
2285              1192730, -3714199, 15123619, 10811505},
2286             {14352098, -3419715, -18942044, 10822655, 32750596, 4699007, -70363,
2287              15776356, -28886779, -11974553},
2288             {-28241164, -8072475, -4978962, -5315317, 29416931, 1847569,
2289              -20654173, -16484855, 4714547, -9600655},
2290         },
2291         {
2292             {15200332, 8368572, 19679101, 15970074, -31872674, 1959451,
2293              24611599, -4543832, -11745876, 12340220},
2294             {12876937, -10480056, 33134381, 6590940, -6307776, 14872440,
2295              9613953, 8241152, 15370987, 9608631},
2296             {-4143277, -12014408, 8446281, -391603, 4407738, 13629032, -7724868,
2297              15866074, -28210621, -8814099},
2298         },
2299         {
2300             {26660628, -15677655, 8393734, 358047, -7401291, 992988, -23904233,
2301              858697, 20571223, 8420556},
2302             {14620715, 13067227, -15447274, 8264467, 14106269, 15080814,
2303              33531827, 12516406, -21574435, -12476749},
2304             {236881, 10476226, 57258, -14677024, 6472998, 2466984, 17258519,
2305              7256740, 8791136, 15069930},
2306         },
2307         {
2308             {1276410, -9371918, 22949635, -16322807, -23493039, -5702186,
2309              14711875, 4874229, -30663140, -2331391},
2310             {5855666, 4990204, -13711848, 7294284, -7804282, 1924647, -1423175,
2311              -7912378, -33069337, 9234253},
2312             {20590503, -9018988, 31529744, -7352666, -2706834, 10650548,
2313              31559055, -11609587, 18979186, 13396066},
2314         },
2315         {
2316             {24474287, 4968103, 22267082, 4407354, 24063882, -8325180,
2317              -18816887, 13594782, 33514650, 7021958},
2318             {-11566906, -6565505, -21365085, 15928892, -26158305, 4315421,
2319              -25948728, -3916677, -21480480, 12868082},
2320             {-28635013, 13504661, 19988037, -2132761, 21078225, 6443208,
2321              -21446107, 2244500, -12455797, -8089383},
2322         },
2323         {
2324             {-30595528, 13793479, -5852820, 319136, -25723172, -6263899,
2325              33086546, 8957937, -15233648, 5540521},
2326             {-11630176, -11503902, -8119500, -7643073, 2620056, 1022908,
2327              -23710744, -1568984, -16128528, -14962807},
2328             {23152971, 775386, 27395463, 14006635, -9701118, 4649512, 1689819,
2329              892185, -11513277, -15205948},
2330         },
2331         {
2332             {9770129, 9586738, 26496094, 4324120, 1556511, -3550024, 27453819,
2333              4763127, -19179614, 5867134},
2334             {-32765025, 1927590, 31726409, -4753295, 23962434, -16019500,
2335              27846559, 5931263, -29749703, -16108455},
2336             {27461885, -2977536, 22380810, 1815854, -23033753, -3031938,
2337              7283490, -15148073, -19526700, 7734629},
2338         },
2339     },
2340     {
2341         {
2342             {-8010264, -9590817, -11120403, 6196038, 29344158, -13430885,
2343              7585295, -3176626, 18549497, 15302069},
2344             {-32658337, -6171222, -7672793, -11051681, 6258878, 13504381,
2345              10458790, -6418461, -8872242, 8424746},
2346             {24687205, 8613276, -30667046, -3233545, 1863892, -1830544,
2347              19206234, 7134917, -11284482, -828919},
2348         },
2349         {
2350             {11334899, -9218022, 8025293, 12707519, 17523892, -10476071,
2351              10243738, -14685461, -5066034, 16498837},
2352             {8911542, 6887158, -9584260, -6958590, 11145641, -9543680, 17303925,
2353              -14124238, 6536641, 10543906},
2354             {-28946384, 15479763, -17466835, 568876, -1497683, 11223454,
2355              -2669190, -16625574, -27235709, 8876771},
2356         },
2357         {
2358             {-25742899, -12566864, -15649966, -846607, -33026686, -796288,
2359              -33481822, 15824474, -604426, -9039817},
2360             {10330056, 70051, 7957388, -9002667, 9764902, 15609756, 27698697,
2361              -4890037, 1657394, 3084098},
2362             {10477963, -7470260, 12119566, -13250805, 29016247, -5365589,
2363              31280319, 14396151, -30233575, 15272409},
2364         },
2365         {
2366             {-12288309, 3169463, 28813183, 16658753, 25116432, -5630466,
2367              -25173957, -12636138, -25014757, 1950504},
2368             {-26180358, 9489187, 11053416, -14746161, -31053720, 5825630,
2369              -8384306, -8767532, 15341279, 8373727},
2370             {28685821, 7759505, -14378516, -12002860, -31971820, 4079242,
2371              298136, -10232602, -2878207, 15190420},
2372         },
2373         {
2374             {-32932876, 13806336, -14337485, -15794431, -24004620, 10940928,
2375              8669718, 2742393, -26033313, -6875003},
2376             {-1580388, -11729417, -25979658, -11445023, -17411874, -10912854,
2377              9291594, -16247779, -12154742, 6048605},
2378             {-30305315, 14843444, 1539301, 11864366, 20201677, 1900163,
2379              13934231, 5128323, 11213262, 9168384},
2380         },
2381         {
2382             {-26280513, 11007847, 19408960, -940758, -18592965, -4328580,
2383              -5088060, -11105150, 20470157, -16398701},
2384             {-23136053, 9282192, 14855179, -15390078, -7362815, -14408560,
2385              -22783952, 14461608, 14042978, 5230683},
2386             {29969567, -2741594, -16711867, -8552442, 9175486, -2468974,
2387              21556951, 3506042, -5933891, -12449708},
2388         },
2389         {
2390             {-3144746, 8744661, 19704003, 4581278, -20430686, 6830683,
2391              -21284170, 8971513, -28539189, 15326563},
2392             {-19464629, 10110288, -17262528, -3503892, -23500387, 1355669,
2393              -15523050, 15300988, -20514118, 9168260},
2394             {-5353335, 4488613, -23803248, 16314347, 7780487, -15638939,
2395              -28948358, 9601605, 33087103, -9011387},
2396         },
2397         {
2398             {-19443170, -15512900, -20797467, -12445323, -29824447, 10229461,
2399              -27444329, -15000531, -5996870, 15664672},
2400             {23294591, -16632613, -22650781, -8470978, 27844204, 11461195,
2401              13099750, -2460356, 18151676, 13417686},
2402             {-24722913, -4176517, -31150679, 5988919, -26858785, 6685065,
2403              1661597, -12551441, 15271676, -15452665},
2404         },
2405     },
2406     {
2407         {
2408             {11433042, -13228665, 8239631, -5279517, -1985436, -725718,
2409              -18698764, 2167544, -6921301, -13440182},
2410             {-31436171, 15575146, 30436815, 12192228, -22463353, 9395379,
2411              -9917708, -8638997, 12215110, 12028277},
2412             {14098400, 6555944, 23007258, 5757252, -15427832, -12950502,
2413              30123440, 4617780, -16900089, -655628},
2414         },
2415         {
2416             {-4026201, -15240835, 11893168, 13718664, -14809462, 1847385,
2417              -15819999, 10154009, 23973261, -12684474},
2418             {-26531820, -3695990, -1908898, 2534301, -31870557, -16550355,
2419              18341390, -11419951, 32013174, -10103539},
2420             {-25479301, 10876443, -11771086, -14625140, -12369567, 1838104,
2421              21911214, 6354752, 4425632, -837822},
2422         },
2423         {
2424             {-10433389, -14612966, 22229858, -3091047, -13191166, 776729,
2425              -17415375, -12020462, 4725005, 14044970},
2426             {19268650, -7304421, 1555349, 8692754, -21474059, -9910664, 6347390,
2427              -1411784, -19522291, -16109756},
2428             {-24864089, 12986008, -10898878, -5558584, -11312371, -148526,
2429              19541418, 8180106, 9282262, 10282508},
2430         },
2431         {
2432             {-26205082, 4428547, -8661196, -13194263, 4098402, -14165257,
2433              15522535, 8372215, 5542595, -10702683},
2434             {-10562541, 14895633, 26814552, -16673850, -17480754, -2489360,
2435              -2781891, 6993761, -18093885, 10114655},
2436             {-20107055, -929418, 31422704, 10427861, -7110749, 6150669,
2437              -29091755, -11529146, 25953725, -106158},
2438         },
2439         {
2440             {-4234397, -8039292, -9119125, 3046000, 2101609, -12607294,
2441              19390020, 6094296, -3315279, 12831125},
2442             {-15998678, 7578152, 5310217, 14408357, -33548620, -224739,
2443              31575954, 6326196, 7381791, -2421839},
2444             {-20902779, 3296811, 24736065, -16328389, 18374254, 7318640,
2445              6295303, 8082724, -15362489, 12339664},
2446         },
2447         {
2448             {27724736, 2291157, 6088201, -14184798, 1792727, 5857634, 13848414,
2449              15768922, 25091167, 14856294},
2450             {-18866652, 8331043, 24373479, 8541013, -701998, -9269457, 12927300,
2451              -12695493, -22182473, -9012899},
2452             {-11423429, -5421590, 11632845, 3405020, 30536730, -11674039,
2453              -27260765, 13866390, 30146206, 9142070},
2454         },
2455         {
2456             {3924129, -15307516, -13817122, -10054960, 12291820, -668366,
2457              -27702774, 9326384, -8237858, 4171294},
2458             {-15921940, 16037937, 6713787, 16606682, -21612135, 2790944,
2459              26396185, 3731949, 345228, -5462949},
2460             {-21327538, 13448259, 25284571, 1143661, 20614966, -8849387,
2461              2031539, -12391231, -16253183, -13582083},
2462         },
2463         {
2464             {31016211, -16722429, 26371392, -14451233, -5027349, 14854137,
2465              17477601, 3842657, 28012650, -16405420},
2466             {-5075835, 9368966, -8562079, -4600902, -15249953, 6970560,
2467              -9189873, 16292057, -8867157, 3507940},
2468             {29439664, 3537914, 23333589, 6997794, -17555561, -11018068,
2469              -15209202, -15051267, -9164929, 6580396},
2470         },
2471     },
2472     {
2473         {
2474             {-12185861, -7679788, 16438269, 10826160, -8696817, -6235611,
2475              17860444, -9273846, -2095802, 9304567},
2476             {20714564, -4336911, 29088195, 7406487, 11426967, -5095705,
2477              14792667, -14608617, 5289421, -477127},
2478             {-16665533, -10650790, -6160345, -13305760, 9192020, -1802462,
2479              17271490, 12349094, 26939669, -3752294},
2480         },
2481         {
2482             {-12889898, 9373458, 31595848, 16374215, 21471720, 13221525,
2483              -27283495, -12348559, -3698806, 117887},
2484             {22263325, -6560050, 3984570, -11174646, -15114008, -566785,
2485              28311253, 5358056, -23319780, 541964},
2486             {16259219, 3261970, 2309254, -15534474, -16885711, -4581916,
2487              24134070, -16705829, -13337066, -13552195},
2488         },
2489         {
2490             {9378160, -13140186, -22845982, -12745264, 28198281, -7244098,
2491              -2399684, -717351, 690426, 14876244},
2492             {24977353, -314384, -8223969, -13465086, 28432343, -1176353,
2493              -13068804, -12297348, -22380984, 6618999},
2494             {-1538174, 11685646, 12944378, 13682314, -24389511, -14413193,
2495              8044829, -13817328, 32239829, -5652762},
2496         },
2497         {
2498             {-18603066, 4762990, -926250, 8885304, -28412480, -3187315, 9781647,
2499              -10350059, 32779359, 5095274},
2500             {-33008130, -5214506, -32264887, -3685216, 9460461, -9327423,
2501              -24601656, 14506724, 21639561, -2630236},
2502             {-16400943, -13112215, 25239338, 15531969, 3987758, -4499318,
2503              -1289502, -6863535, 17874574, 558605},
2504         },
2505         {
2506             {-13600129, 10240081, 9171883, 16131053, -20869254, 9599700,
2507              33499487, 5080151, 2085892, 5119761},
2508             {-22205145, -2519528, -16381601, 414691, -25019550, 2170430,
2509              30634760, -8363614, -31999993, -5759884},
2510             {-6845704, 15791202, 8550074, -1312654, 29928809, -12092256,
2511              27534430, -7192145, -22351378, 12961482},
2512         },
2513         {
2514             {-24492060, -9570771, 10368194, 11582341, -23397293, -2245287,
2515              16533930, 8206996, -30194652, -5159638},
2516             {-11121496, -3382234, 2307366, 6362031, -135455, 8868177, -16835630,
2517              7031275, 7589640, 8945490},
2518             {-32152748, 8917967, 6661220, -11677616, -1192060, -15793393,
2519              7251489, -11182180, 24099109, -14456170},
2520         },
2521         {
2522             {5019558, -7907470, 4244127, -14714356, -26933272, 6453165,
2523              -19118182, -13289025, -6231896, -10280736},
2524             {10853594, 10721687, 26480089, 5861829, -22995819, 1972175,
2525              -1866647, -10557898, -3363451, -6441124},
2526             {-17002408, 5906790, 221599, -6563147, 7828208, -13248918, 24362661,
2527              -2008168, -13866408, 7421392},
2528         },
2529         {
2530             {8139927, -6546497, 32257646, -5890546, 30375719, 1886181,
2531              -21175108, 15441252, 28826358, -4123029},
2532             {6267086, 9695052, 7709135, -16603597, -32869068, -1886135,
2533              14795160, -7840124, 13746021, -1742048},
2534             {28584902, 7787108, -6732942, -15050729, 22846041, -7571236,
2535              -3181936, -363524, 4771362, -8419958},
2536         },
2537     },
2538     {
2539         {
2540             {24949256, 6376279, -27466481, -8174608, -18646154, -9930606,
2541              33543569, -12141695, 3569627, 11342593},
2542             {26514989, 4740088, 27912651, 3697550, 19331575, -11472339, 6809886,
2543              4608608, 7325975, -14801071},
2544             {-11618399, -14554430, -24321212, 7655128, -1369274, 5214312,
2545              -27400540, 10258390, -17646694, -8186692},
2546         },
2547         {
2548             {11431204, 15823007, 26570245, 14329124, 18029990, 4796082,
2549              -31446179, 15580664, 9280358, -3973687},
2550             {-160783, -10326257, -22855316, -4304997, -20861367, -13621002,
2551              -32810901, -11181622, -15545091, 4387441},
2552             {-20799378, 12194512, 3937617, -5805892, -27154820, 9340370,
2553              -24513992, 8548137, 20617071, -7482001},
2554         },
2555         {
2556             {-938825, -3930586, -8714311, 16124718, 24603125, -6225393,
2557              -13775352, -11875822, 24345683, 10325460},
2558             {-19855277, -1568885, -22202708, 8714034, 14007766, 6928528,
2559              16318175, -1010689, 4766743, 3552007},
2560             {-21751364, -16730916, 1351763, -803421, -4009670, 3950935, 3217514,
2561              14481909, 10988822, -3994762},
2562         },
2563         {
2564             {15564307, -14311570, 3101243, 5684148, 30446780, -8051356,
2565              12677127, -6505343, -8295852, 13296005},
2566             {-9442290, 6624296, -30298964, -11913677, -4670981, -2057379,
2567              31521204, 9614054, -30000824, 12074674},
2568             {4771191, -135239, 14290749, -13089852, 27992298, 14998318,
2569              -1413936, -1556716, 29832613, -16391035},
2570         },
2571         {
2572             {7064884, -7541174, -19161962, -5067537, -18891269, -2912736,
2573              25825242, 5293297, -27122660, 13101590},
2574             {-2298563, 2439670, -7466610, 1719965, -27267541, -16328445,
2575              32512469, -5317593, -30356070, -4190957},
2576             {-30006540, 10162316, -33180176, 3981723, -16482138, -13070044,
2577              14413974, 9515896, 19568978, 9628812},
2578         },
2579         {
2580             {33053803, 199357, 15894591, 1583059, 27380243, -4580435, -17838894,
2581              -6106839, -6291786, 3437740},
2582             {-18978877, 3884493, 19469877, 12726490, 15913552, 13614290,
2583              -22961733, 70104, 7463304, 4176122},
2584             {-27124001, 10659917, 11482427, -16070381, 12771467, -6635117,
2585              -32719404, -5322751, 24216882, 5944158},
2586         },
2587         {
2588             {8894125, 7450974, -2664149, -9765752, -28080517, -12389115,
2589              19345746, 14680796, 11632993, 5847885},
2590             {26942781, -2315317, 9129564, -4906607, 26024105, 11769399,
2591              -11518837, 6367194, -9727230, 4782140},
2592             {19916461, -4828410, -22910704, -11414391, 25606324, -5972441,
2593              33253853, 8220911, 6358847, -1873857},
2594         },
2595         {
2596             {801428, -2081702, 16569428, 11065167, 29875704, 96627, 7908388,
2597              -4480480, -13538503, 1387155},
2598             {19646058, 5720633, -11416706, 12814209, 11607948, 12749789,
2599              14147075, 15156355, -21866831, 11835260},
2600             {19299512, 1155910, 28703737, 14890794, 2925026, 7269399, 26121523,
2601              15467869, -26560550, 5052483},
2602         },
2603     },
2604     {
2605         {
2606             {-3017432, 10058206, 1980837, 3964243, 22160966, 12322533, -6431123,
2607              -12618185, 12228557, -7003677},
2608             {32944382, 14922211, -22844894, 5188528, 21913450, -8719943,
2609              4001465, 13238564, -6114803, 8653815},
2610             {22865569, -4652735, 27603668, -12545395, 14348958, 8234005,
2611              24808405, 5719875, 28483275, 2841751},
2612         },
2613         {
2614             {-16420968, -1113305, -327719, -12107856, 21886282, -15552774,
2615              -1887966, -315658, 19932058, -12739203},
2616             {-11656086, 10087521, -8864888, -5536143, -19278573, -3055912,
2617              3999228, 13239134, -4777469, -13910208},
2618             {1382174, -11694719, 17266790, 9194690, -13324356, 9720081,
2619              20403944, 11284705, -14013818, 3093230},
2620         },
2621         {
2622             {16650921, -11037932, -1064178, 1570629, -8329746, 7352753, -302424,
2623              16271225, -24049421, -6691850},
2624             {-21911077, -5927941, -4611316, -5560156, -31744103, -10785293,
2625              24123614, 15193618, -21652117, -16739389},
2626             {-9935934, -4289447, -25279823, 4372842, 2087473, 10399484,
2627              31870908, 14690798, 17361620, 11864968},
2628         },
2629         {
2630             {-11307610, 6210372, 13206574, 5806320, -29017692, -13967200,
2631              -12331205, -7486601, -25578460, -16240689},
2632             {14668462, -12270235, 26039039, 15305210, 25515617, 4542480,
2633              10453892, 6577524, 9145645, -6443880},
2634             {5974874, 3053895, -9433049, -10385191, -31865124, 3225009,
2635              -7972642, 3936128, -5652273, -3050304},
2636         },
2637         {
2638             {30625386, -4729400, -25555961, -12792866, -20484575, 7695099,
2639              17097188, -16303496, -27999779, 1803632},
2640             {-3553091, 9865099, -5228566, 4272701, -5673832, -16689700,
2641              14911344, 12196514, -21405489, 7047412},
2642             {20093277, 9920966, -11138194, -5343857, 13161587, 12044805,
2643              -32856851, 4124601, -32343828, -10257566},
2644         },
2645         {
2646             {-20788824, 14084654, -13531713, 7842147, 19119038, -13822605,
2647              4752377, -8714640, -21679658, 2288038},
2648             {-26819236, -3283715, 29965059, 3039786, -14473765, 2540457,
2649              29457502, 14625692, -24819617, 12570232},
2650             {-1063558, -11551823, 16920318, 12494842, 1278292, -5869109,
2651              -21159943, -3498680, -11974704, 4724943},
2652         },
2653         {
2654             {17960970, -11775534, -4140968, -9702530, -8876562, -1410617,
2655              -12907383, -8659932, -29576300, 1903856},
2656             {23134274, -14279132, -10681997, -1611936, 20684485, 15770816,
2657              -12989750, 3190296, 26955097, 14109738},
2658             {15308788, 5320727, -30113809, -14318877, 22902008, 7767164,
2659              29425325, -11277562, 31960942, 11934971},
2660         },
2661         {
2662             {-27395711, 8435796, 4109644, 12222639, -24627868, 14818669,
2663              20638173, 4875028, 10491392, 1379718},
2664             {-13159415, 9197841, 3875503, -8936108, -1383712, -5879801,
2665              33518459, 16176658, 21432314, 12180697},
2666             {-11787308, 11500838, 13787581, -13832590, -22430679, 10140205,
2667              1465425, 12689540, -10301319, -13872883},
2668         },
2669     },
2670     {
2671         {
2672             {5414091, -15386041, -21007664, 9643570, 12834970, 1186149,
2673              -2622916, -1342231, 26128231, 6032912},
2674             {-26337395, -13766162, 32496025, -13653919, 17847801, -12669156,
2675              3604025, 8316894, -25875034, -10437358},
2676             {3296484, 6223048, 24680646, -12246460, -23052020, 5903205,
2677              -8862297, -4639164, 12376617, 3188849},
2678         },
2679         {
2680             {29190488, -14659046, 27549113, -1183516, 3520066, -10697301,
2681              32049515, -7309113, -16109234, -9852307},
2682             {-14744486, -9309156, 735818, -598978, -20407687, -5057904,
2683              25246078, -15795669, 18640741, -960977},
2684             {-6928835, -16430795, 10361374, 5642961, 4910474, 12345252,
2685              -31638386, -494430, 10530747, 1053335},
2686         },
2687         {
2688             {-29265967, -14186805, -13538216, -12117373, -19457059, -10655384,
2689              -31462369, -2948985, 24018831, 15026644},
2690             {-22592535, -3145277, -2289276, 5953843, -13440189, 9425631,
2691              25310643, 13003497, -2314791, -15145616},
2692             {-27419985, -603321, -8043984, -1669117, -26092265, 13987819,
2693              -27297622, 187899, -23166419, -2531735},
2694         },
2695         {
2696             {-21744398, -13810475, 1844840, 5021428, -10434399, -15911473,
2697              9716667, 16266922, -5070217, 726099},
2698             {29370922, -6053998, 7334071, -15342259, 9385287, 2247707,
2699              -13661962, -4839461, 30007388, -15823341},
2700             {-936379, 16086691, 23751945, -543318, -1167538, -5189036, 9137109,
2701              730663, 9835848, 4555336},
2702         },
2703         {
2704             {-23376435, 1410446, -22253753, -12899614, 30867635, 15826977,
2705              17693930, 544696, -11985298, 12422646},
2706             {31117226, -12215734, -13502838, 6561947, -9876867, -12757670,
2707              -5118685, -4096706, 29120153, 13924425},
2708             {-17400879, -14233209, 19675799, -2734756, -11006962, -5858820,
2709              -9383939, -11317700, 7240931, -237388},
2710         },
2711         {
2712             {-31361739, -11346780, -15007447, -5856218, -22453340, -12152771,
2713              1222336, 4389483, 3293637, -15551743},
2714             {-16684801, -14444245, 11038544, 11054958, -13801175, -3338533,
2715              -24319580, 7733547, 12796905, -6335822},
2716             {-8759414, -10817836, -25418864, 10783769, -30615557, -9746811,
2717              -28253339, 3647836, 3222231, -11160462},
2718         },
2719         {
2720             {18606113, 1693100, -25448386, -15170272, 4112353, 10045021,
2721              23603893, -2048234, -7550776, 2484985},
2722             {9255317, -3131197, -12156162, -1004256, 13098013, -9214866,
2723              16377220, -2102812, -19802075, -3034702},
2724             {-22729289, 7496160, -5742199, 11329249, 19991973, -3347502,
2725              -31718148, 9936966, -30097688, -10618797},
2726         },
2727         {
2728             {21878590, -5001297, 4338336, 13643897, -3036865, 13160960,
2729              19708896, 5415497, -7360503, -4109293},
2730             {27736861, 10103576, 12500508, 8502413, -3413016, -9633558,
2731              10436918, -1550276, -23659143, -8132100},
2732             {19492550, -12104365, -29681976, -852630, -3208171, 12403437,
2733              30066266, 8367329, 13243957, 8709688},
2734         },
2735     },
2736     {
2737         {
2738             {12015105, 2801261, 28198131, 10151021, 24818120, -4743133,
2739              -11194191, -5645734, 5150968, 7274186},
2740             {2831366, -12492146, 1478975, 6122054, 23825128, -12733586,
2741              31097299, 6083058, 31021603, -9793610},
2742             {-2529932, -2229646, 445613, 10720828, -13849527, -11505937,
2743              -23507731, 16354465, 15067285, -14147707},
2744         },
2745         {
2746             {7840942, 14037873, -33364863, 15934016, -728213, -3642706,
2747              21403988, 1057586, -19379462, -12403220},
2748             {915865, -16469274, 15608285, -8789130, -24357026, 6060030,
2749              -17371319, 8410997, -7220461, 16527025},
2750             {32922597, -556987, 20336074, -16184568, 10903705, -5384487,
2751              16957574, 52992, 23834301, 6588044},
2752         },
2753         {
2754             {32752030, 11232950, 3381995, -8714866, 22652988, -10744103,
2755              17159699, 16689107, -20314580, -1305992},
2756             {-4689649, 9166776, -25710296, -10847306, 11576752, 12733943,
2757              7924251, -2752281, 1976123, -7249027},
2758             {21251222, 16309901, -2983015, -6783122, 30810597, 12967303, 156041,
2759              -3371252, 12331345, -8237197},
2760         },
2761         {
2762             {8651614, -4477032, -16085636, -4996994, 13002507, 2950805,
2763              29054427, -5106970, 10008136, -4667901},
2764             {31486080, 15114593, -14261250, 12951354, 14369431, -7387845,
2765              16347321, -13662089, 8684155, -10532952},
2766             {19443825, 11385320, 24468943, -9659068, -23919258, 2187569,
2767              -26263207, -6086921, 31316348, 14219878},
2768         },
2769         {
2770             {-28594490, 1193785, 32245219, 11392485, 31092169, 15722801,
2771              27146014, 6992409, 29126555, 9207390},
2772             {32382935, 1110093, 18477781, 11028262, -27411763, -7548111,
2773              -4980517, 10843782, -7957600, -14435730},
2774             {2814918, 7836403, 27519878, -7868156, -20894015, -11553689,
2775              -21494559, 8550130, 28346258, 1994730},
2776         },
2777         {
2778             {-19578299, 8085545, -14000519, -3948622, 2785838, -16231307,
2779              -19516951, 7174894, 22628102, 8115180},
2780             {-30405132, 955511, -11133838, -15078069, -32447087, -13278079,
2781              -25651578, 3317160, -9943017, 930272},
2782             {-15303681, -6833769, 28856490, 1357446, 23421993, 1057177,
2783              24091212, -1388970, -22765376, -10650715},
2784         },
2785         {
2786             {-22751231, -5303997, -12907607, -12768866, -15811511, -7797053,
2787              -14839018, -16554220, -1867018, 8398970},
2788             {-31969310, 2106403, -4736360, 1362501, 12813763, 16200670,
2789              22981545, -6291273, 18009408, -15772772},
2790             {-17220923, -9545221, -27784654, 14166835, 29815394, 7444469,
2791              29551787, -3727419, 19288549, 1325865},
2792         },
2793         {
2794             {15100157, -15835752, -23923978, -1005098, -26450192, 15509408,
2795              12376730, -3479146, 33166107, -8042750},
2796             {20909231, 13023121, -9209752, 16251778, -5778415, -8094914,
2797              12412151, 10018715, 2213263, -13878373},
2798             {32529814, -11074689, 30361439, -16689753, -9135940, 1513226,
2799              22922121, 6382134, -5766928, 8371348},
2800         },
2801     },
2802     {
2803         {
2804             {9923462, 11271500, 12616794, 3544722, -29998368, -1721626,
2805              12891687, -8193132, -26442943, 10486144},
2806             {-22597207, -7012665, 8587003, -8257861, 4084309, -12970062, 361726,
2807              2610596, -23921530, -11455195},
2808             {5408411, -1136691, -4969122, 10561668, 24145918, 14240566,
2809              31319731, -4235541, 19985175, -3436086},
2810         },
2811         {
2812             {-13994457, 16616821, 14549246, 3341099, 32155958, 13648976,
2813              -17577068, 8849297, 65030, 8370684},
2814             {-8320926, -12049626, 31204563, 5839400, -20627288, -1057277,
2815              -19442942, 6922164, 12743482, -9800518},
2816             {-2361371, 12678785, 28815050, 4759974, -23893047, 4884717,
2817              23783145, 11038569, 18800704, 255233},
2818         },
2819         {
2820             {-5269658, -1773886, 13957886, 7990715, 23132995, 728773, 13393847,
2821              9066957, 19258688, -14753793},
2822             {-2936654, -10827535, -10432089, 14516793, -3640786, 4372541,
2823              -31934921, 2209390, -1524053, 2055794},
2824             {580882, 16705327, 5468415, -2683018, -30926419, -14696000,
2825              -7203346, -8994389, -30021019, 7394435},
2826         },
2827         {
2828             {23838809, 1822728, -15738443, 15242727, 8318092, -3733104,
2829              -21672180, -3492205, -4821741, 14799921},
2830             {13345610, 9759151, 3371034, -16137791, 16353039, 8577942, 31129804,
2831              13496856, -9056018, 7402518},
2832             {2286874, -4435931, -20042458, -2008336, -13696227, 5038122,
2833              11006906, -15760352, 8205061, 1607563},
2834         },
2835         {
2836             {14414086, -8002132, 3331830, -3208217, 22249151, -5594188,
2837              18364661, -2906958, 30019587, -9029278},
2838             {-27688051, 1585953, -10775053, 931069, -29120221, -11002319,
2839              -14410829, 12029093, 9944378, 8024},
2840             {4368715, -3709630, 29874200, -15022983, -20230386, -11410704,
2841              -16114594, -999085, -8142388, 5640030},
2842         },
2843         {
2844             {10299610, 13746483, 11661824, 16234854, 7630238, 5998374, 9809887,
2845              -16694564, 15219798, -14327783},
2846             {27425505, -5719081, 3055006, 10660664, 23458024, 595578, -15398605,
2847              -1173195, -18342183, 9742717},
2848             {6744077, 2427284, 26042789, 2720740, -847906, 1118974, 32324614,
2849              7406442, 12420155, 1994844},
2850         },
2851         {
2852             {14012521, -5024720, -18384453, -9578469, -26485342, -3936439,
2853              -13033478, -10909803, 24319929, -6446333},
2854             {16412690, -4507367, 10772641, 15929391, -17068788, -4658621,
2855              10555945, -10484049, -30102368, -4739048},
2856             {22397382, -7767684, -9293161, -12792868, 17166287, -9755136,
2857              -27333065, 6199366, 21880021, -12250760},
2858         },
2859         {
2860             {-4283307, 5368523, -31117018, 8163389, -30323063, 3209128,
2861              16557151, 8890729, 8840445, 4957760},
2862             {-15447727, 709327, -6919446, -10870178, -29777922, 6522332,
2863              -21720181, 12130072, -14796503, 5005757},
2864             {-2114751, -14308128, 23019042, 15765735, -25269683, 6002752,
2865              10183197, -13239326, -16395286, -2176112},
2866         },
2867     },
2868     {
2869         {
2870             {-19025756, 1632005, 13466291, -7995100, -23640451, 16573537,
2871              -32013908, -3057104, 22208662, 2000468},
2872             {3065073, -1412761, -25598674, -361432, -17683065, -5703415,
2873              -8164212, 11248527, -3691214, -7414184},
2874             {10379208, -6045554, 8877319, 1473647, -29291284, -12507580,
2875              16690915, 2553332, -3132688, 16400289},
2876         },
2877         {
2878             {15716668, 1254266, -18472690, 7446274, -8448918, 6344164,
2879              -22097271, -7285580, 26894937, 9132066},
2880             {24158887, 12938817, 11085297, -8177598, -28063478, -4457083,
2881              -30576463, 64452, -6817084, -2692882},
2882             {13488534, 7794716, 22236231, 5989356, 25426474, -12578208, 2350710,
2883              -3418511, -4688006, 2364226},
2884         },
2885         {
2886             {16335052, 9132434, 25640582, 6678888, 1725628, 8517937, -11807024,
2887              -11697457, 15445875, -7798101},
2888             {29004207, -7867081, 28661402, -640412, -12794003, -7943086,
2889              31863255, -4135540, -278050, -15759279},
2890             {-6122061, -14866665, -28614905, 14569919, -10857999, -3591829,
2891              10343412, -6976290, -29828287, -10815811},
2892         },
2893         {
2894             {27081650, 3463984, 14099042, -4517604, 1616303, -6205604, 29542636,
2895              15372179, 17293797, 960709},
2896             {20263915, 11434237, -5765435, 11236810, 13505955, -10857102,
2897              -16111345, 6493122, -19384511, 7639714},
2898             {-2830798, -14839232, 25403038, -8215196, -8317012, -16173699,
2899              18006287, -16043750, 29994677, -15808121},
2900         },
2901         {
2902             {9769828, 5202651, -24157398, -13631392, -28051003, -11561624,
2903              -24613141, -13860782, -31184575, 709464},
2904             {12286395, 13076066, -21775189, -1176622, -25003198, 4057652,
2905              -32018128, -8890874, 16102007, 13205847},
2906             {13733362, 5599946, 10557076, 3195751, -5557991, 8536970, -25540170,
2907              8525972, 10151379, 10394400},
2908         },
2909         {
2910             {4024660, -16137551, 22436262, 12276534, -9099015, -2686099,
2911              19698229, 11743039, -33302334, 8934414},
2912             {-15879800, -4525240, -8580747, -2934061, 14634845, -698278,
2913              -9449077, 3137094, -11536886, 11721158},
2914             {17555939, -5013938, 8268606, 2331751, -22738815, 9761013, 9319229,
2915              8835153, -9205489, -1280045},
2916         },
2917         {
2918             {-461409, -7830014, 20614118, 16688288, -7514766, -4807119,
2919              22300304, 505429, 6108462, -6183415},
2920             {-5070281, 12367917, -30663534, 3234473, 32617080, -8422642,
2921              29880583, -13483331, -26898490, -7867459},
2922             {-31975283, 5726539, 26934134, 10237677, -3173717, -605053,
2923              24199304, 3795095, 7592688, -14992079},
2924         },
2925         {
2926             {21594432, -14964228, 17466408, -4077222, 32537084, 2739898,
2927              6407723, 12018833, -28256052, 4298412},
2928             {-20650503, -11961496, -27236275, 570498, 3767144, -1717540,
2929              13891942, -1569194, 13717174, 10805743},
2930             {-14676630, -15644296, 15287174, 11927123, 24177847, -8175568,
2931              -796431, 14860609, -26938930, -5863836},
2932         },
2933     },
2934     {
2935         {
2936             {12962541, 5311799, -10060768, 11658280, 18855286, -7954201,
2937              13286263, -12808704, -4381056, 9882022},
2938             {18512079, 11319350, -20123124, 15090309, 18818594, 5271736,
2939              -22727904, 3666879, -23967430, -3299429},
2940             {-6789020, -3146043, 16192429, 13241070, 15898607, -14206114,
2941              -10084880, -6661110, -2403099, 5276065},
2942         },
2943         {
2944             {30169808, -5317648, 26306206, -11750859, 27814964, 7069267,
2945              7152851, 3684982, 1449224, 13082861},
2946             {10342826, 3098505, 2119311, 193222, 25702612, 12233820, 23697382,
2947              15056736, -21016438, -8202000},
2948             {-33150110, 3261608, 22745853, 7948688, 19370557, -15177665,
2949              -26171976, 6482814, -10300080, -11060101},
2950         },
2951         {
2952             {32869458, -5408545, 25609743, 15678670, -10687769, -15471071,
2953              26112421, 2521008, -22664288, 6904815},
2954             {29506923, 4457497, 3377935, -9796444, -30510046, 12935080, 1561737,
2955              3841096, -29003639, -6657642},
2956             {10340844, -6630377, -18656632, -2278430, 12621151, -13339055,
2957              30878497, -11824370, -25584551, 5181966},
2958         },
2959         {
2960             {25940115, -12658025, 17324188, -10307374, -8671468, 15029094,
2961              24396252, -16450922, -2322852, -12388574},
2962             {-21765684, 9916823, -1300409, 4079498, -1028346, 11909559, 1782390,
2963              12641087, 20603771, -6561742},
2964             {-18882287, -11673380, 24849422, 11501709, 13161720, -4768874,
2965              1925523, 11914390, 4662781, 7820689},
2966         },
2967         {
2968             {12241050, -425982, 8132691, 9393934, 32846760, -1599620, 29749456,
2969              12172924, 16136752, 15264020},
2970             {-10349955, -14680563, -8211979, 2330220, -17662549, -14545780,
2971              10658213, 6671822, 19012087, 3772772},
2972             {3753511, -3421066, 10617074, 2028709, 14841030, -6721664, 28718732,
2973              -15762884, 20527771, 12988982},
2974         },
2975         {
2976             {-14822485, -5797269, -3707987, 12689773, -898983, -10914866,
2977              -24183046, -10564943, 3299665, -12424953},
2978             {-16777703, -15253301, -9642417, 4978983, 3308785, 8755439, 6943197,
2979              6461331, -25583147, 8991218},
2980             {-17226263, 1816362, -1673288, -6086439, 31783888, -8175991,
2981              -32948145, 7417950, -30242287, 1507265},
2982         },
2983         {
2984             {29692663, 6829891, -10498800, 4334896, 20945975, -11906496,
2985              -28887608, 8209391, 14606362, -10647073},
2986             {-3481570, 8707081, 32188102, 5672294, 22096700, 1711240, -33020695,
2987              9761487, 4170404, -2085325},
2988             {-11587470, 14855945, -4127778, -1531857, -26649089, 15084046,
2989              22186522, 16002000, -14276837, -8400798},
2990         },
2991         {
2992             {-4811456, 13761029, -31703877, -2483919, -3312471, 7869047,
2993              -7113572, -9620092, 13240845, 10965870},
2994             {-7742563, -8256762, -14768334, -13656260, -23232383, 12387166,
2995              4498947, 14147411, 29514390, 4302863},
2996             {-13413405, -12407859, 20757302, -13801832, 14785143, 8976368,
2997              -5061276, -2144373, 17846988, -13971927},
2998         },
2999     },
3000     {
3001         {
3002             {-2244452, -754728, -4597030, -1066309, -6247172, 1455299,
3003              -21647728, -9214789, -5222701, 12650267},
3004             {-9906797, -16070310, 21134160, 12198166, -27064575, 708126, 387813,
3005              13770293, -19134326, 10958663},
3006             {22470984, 12369526, 23446014, -5441109, -21520802, -9698723,
3007              -11772496, -11574455, -25083830, 4271862},
3008         },
3009         {
3010             {-25169565, -10053642, -19909332, 15361595, -5984358, 2159192,
3011              75375, -4278529, -32526221, 8469673},
3012             {15854970, 4148314, -8893890, 7259002, 11666551, 13824734,
3013              -30531198, 2697372, 24154791, -9460943},
3014             {15446137, -15806644, 29759747, 14019369, 30811221, -9610191,
3015              -31582008, 12840104, 24913809, 9815020},
3016         },
3017         {
3018             {-4709286, -5614269, -31841498, -12288893, -14443537, 10799414,
3019              -9103676, 13438769, 18735128, 9466238},
3020             {11933045, 9281483, 5081055, -5183824, -2628162, -4905629, -7727821,
3021              -10896103, -22728655, 16199064},
3022             {14576810, 379472, -26786533, -8317236, -29426508, -10812974,
3023              -102766, 1876699, 30801119, 2164795},
3024         },
3025         {
3026             {15995086, 3199873, 13672555, 13712240, -19378835, -4647646,
3027              -13081610, -15496269, -13492807, 1268052},
3028             {-10290614, -3659039, -3286592, 10948818, 23037027, 3794475,
3029              -3470338, -12600221, -17055369, 3565904},
3030             {29210088, -9419337, -5919792, -4952785, 10834811, -13327726,
3031              -16512102, -10820713, -27162222, -14030531},
3032         },
3033         {
3034             {-13161890, 15508588, 16663704, -8156150, -28349942, 9019123,
3035              -29183421, -3769423, 2244111, -14001979},
3036             {-5152875, -3800936, -9306475, -6071583, 16243069, 14684434,
3037              -25673088, -16180800, 13491506, 4641841},
3038             {10813417, 643330, -19188515, -728916, 30292062, -16600078,
3039              27548447, -7721242, 14476989, -12767431},
3040         },
3041         {
3042             {10292079, 9984945, 6481436, 8279905, -7251514, 7032743, 27282937,
3043              -1644259, -27912810, 12651324},
3044             {-31185513, -813383, 22271204, 11835308, 10201545, 15351028,
3045              17099662, 3988035, 21721536, -3148940},
3046             {10202177, -6545839, -31373232, -9574638, -32150642, -8119683,
3047              -12906320, 3852694, 13216206, 14842320},
3048         },
3049         {
3050             {-15815640, -10601066, -6538952, -7258995, -6984659, -6581778,
3051              -31500847, 13765824, -27434397, 9900184},
3052             {14465505, -13833331, -32133984, -14738873, -27443187, 12990492,
3053              33046193, 15796406, -7051866, -8040114},
3054             {30924417, -8279620, 6359016, -12816335, 16508377, 9071735,
3055              -25488601, 15413635, 9524356, -7018878},
3056         },
3057         {
3058             {12274201, -13175547, 32627641, -1785326, 6736625, 13267305,
3059              5237659, -5109483, 15663516, 4035784},
3060             {-2951309, 8903985, 17349946, 601635, -16432815, -4612556,
3061              -13732739, -15889334, -22258478, 4659091},
3062             {-16916263, -4952973, -30393711, -15158821, 20774812, 15897498,
3063              5736189, 15026997, -2178256, -13455585},
3064         },
3065     },
3066     {
3067         {
3068             {-8858980, -2219056, 28571666, -10155518, -474467, -10105698,
3069              -3801496, 278095, 23440562, -290208},
3070             {10226241, -5928702, 15139956, 120818, -14867693, 5218603, 32937275,
3071              11551483, -16571960, -7442864},
3072             {17932739, -12437276, -24039557, 10749060, 11316803, 7535897,
3073              22503767, 5561594, -3646624, 3898661},
3074         },
3075         {
3076             {7749907, -969567, -16339731, -16464, -25018111, 15122143, -1573531,
3077              7152530, 21831162, 1245233},
3078             {26958459, -14658026, 4314586, 8346991, -5677764, 11960072,
3079              -32589295, -620035, -30402091, -16716212},
3080             {-12165896, 9166947, 33491384, 13673479, 29787085, 13096535,
3081              6280834, 14587357, -22338025, 13987525},
3082         },
3083         {
3084             {-24349909, 7778775, 21116000, 15572597, -4833266, -5357778,
3085              -4300898, -5124639, -7469781, -2858068},
3086             {9681908, -6737123, -31951644, 13591838, -6883821, 386950, 31622781,
3087              6439245, -14581012, 4091397},
3088             {-8426427, 1470727, -28109679, -1596990, 3978627, -5123623,
3089              -19622683, 12092163, 29077877, -14741988},
3090         },
3091         {
3092             {5269168, -6859726, -13230211, -8020715, 25932563, 1763552,
3093              -5606110, -5505881, -20017847, 2357889},
3094             {32264008, -15407652, -5387735, -1160093, -2091322, -3946900,
3095              23104804, -12869908, 5727338, 189038},
3096             {14609123, -8954470, -6000566, -16622781, -14577387, -7743898,
3097              -26745169, 10942115, -25888931, -14884697},
3098         },
3099         {
3100             {20513500, 5557931, -15604613, 7829531, 26413943, -2019404,
3101              -21378968, 7471781, 13913677, -5137875},
3102             {-25574376, 11967826, 29233242, 12948236, -6754465, 4713227,
3103              -8940970, 14059180, 12878652, 8511905},
3104             {-25656801, 3393631, -2955415, -7075526, -2250709, 9366908,
3105              -30223418, 6812974, 5568676, -3127656},
3106         },
3107         {
3108             {11630004, 12144454, 2116339, 13606037, 27378885, 15676917,
3109              -17408753, -13504373, -14395196, 8070818},
3110             {27117696, -10007378, -31282771, -5570088, 1127282, 12772488,
3111              -29845906, 10483306, -11552749, -1028714},
3112             {10637467, -5688064, 5674781, 1072708, -26343588, -6982302,
3113              -1683975, 9177853, -27493162, 15431203},
3114         },
3115         {
3116             {20525145, 10892566, -12742472, 12779443, -29493034, 16150075,
3117              -28240519, 14943142, -15056790, -7935931},
3118             {-30024462, 5626926, -551567, -9981087, 753598, 11981191, 25244767,
3119              -3239766, -3356550, 9594024},
3120             {-23752644, 2636870, -5163910, -10103818, 585134, 7877383, 11345683,
3121              -6492290, 13352335, -10977084},
3122         },
3123         {
3124             {-1931799, -5407458, 3304649, -12884869, 17015806, -4877091,
3125              -29783850, -7752482, -13215537, -319204},
3126             {20239939, 6607058, 6203985, 3483793, -18386976, -779229, -20723742,
3127              15077870, -22750759, 14523817},
3128             {27406042, -6041657, 27423596, -4497394, 4996214, 10002360,
3129              -28842031, -4545494, -30172742, -4805667},
3130         },
3131     },
3132     {
3133         {
3134             {11374242, 12660715, 17861383, -12540833, 10935568, 1099227,
3135              -13886076, -9091740, -27727044, 11358504},
3136             {-12730809, 10311867, 1510375, 10778093, -2119455, -9145702,
3137              32676003, 11149336, -26123651, 4985768},
3138             {-19096303, 341147, -6197485, -239033, 15756973, -8796662, -983043,
3139              13794114, -19414307, -15621255},
3140         },
3141         {
3142             {6490081, 11940286, 25495923, -7726360, 8668373, -8751316, 3367603,
3143              6970005, -1691065, -9004790},
3144             {1656497, 13457317, 15370807, 6364910, 13605745, 8362338, -19174622,
3145              -5475723, -16796596, -5031438},
3146             {-22273315, -13524424, -64685, -4334223, -18605636, -10921968,
3147              -20571065, -7007978, -99853, -10237333},
3148         },
3149         {
3150             {17747465, 10039260, 19368299, -4050591, -20630635, -16041286,
3151              31992683, -15857976, -29260363, -5511971},
3152             {31932027, -4986141, -19612382, 16366580, 22023614, 88450, 11371999,
3153              -3744247, 4882242, -10626905},
3154             {29796507, 37186, 19818052, 10115756, -11829032, 3352736, 18551198,
3155              3272828, -5190932, -4162409},
3156         },
3157         {
3158             {12501286, 4044383, -8612957, -13392385, -32430052, 5136599,
3159              -19230378, -3529697, 330070, -3659409},
3160             {6384877, 2899513, 17807477, 7663917, -2358888, 12363165, 25366522,
3161              -8573892, -271295, 12071499},
3162             {-8365515, -4042521, 25133448, -4517355, -6211027, 2265927,
3163              -32769618, 1936675, -5159697, 3829363},
3164         },
3165         {
3166             {28425966, -5835433, -577090, -4697198, -14217555, 6870930, 7921550,
3167              -6567787, 26333140, 14267664},
3168             {-11067219, 11871231, 27385719, -10559544, -4585914, -11189312,
3169              10004786, -8709488, -21761224, 8930324},
3170             {-21197785, -16396035, 25654216, -1725397, 12282012, 11008919,
3171              1541940, 4757911, -26491501, -16408940},
3172         },
3173         {
3174             {13537262, -7759490, -20604840, 10961927, -5922820, -13218065,
3175              -13156584, 6217254, -15943699, 13814990},
3176             {-17422573, 15157790, 18705543, 29619, 24409717, -260476, 27361681,
3177              9257833, -1956526, -1776914},
3178             {-25045300, -10191966, 15366585, 15166509, -13105086, 8423556,
3179              -29171540, 12361135, -18685978, 4578290},
3180         },
3181         {
3182             {24579768, 3711570, 1342322, -11180126, -27005135, 14124956,
3183              -22544529, 14074919, 21964432, 8235257},
3184             {-6528613, -2411497, 9442966, -5925588, 12025640, -1487420,
3185              -2981514, -1669206, 13006806, 2355433},
3186             {-16304899, -13605259, -6632427, -5142349, 16974359, -10911083,
3187              27202044, 1719366, 1141648, -12796236},
3188         },
3189         {
3190             {-12863944, -13219986, -8318266, -11018091, -6810145, -4843894,
3191              13475066, -3133972, 32674895, 13715045},
3192             {11423335, -5468059, 32344216, 8962751, 24989809, 9241752,
3193              -13265253, 16086212, -28740881, -15642093},
3194             {-1409668, 12530728, -6368726, 10847387, 19531186, -14132160,
3195              -11709148, 7791794, -27245943, 4383347},
3196         },
3197     },
3198     {
3199         {
3200             {-28970898, 5271447, -1266009, -9736989, -12455236, 16732599,
3201              -4862407, -4906449, 27193557, 6245191},
3202             {-15193956, 5362278, -1783893, 2695834, 4960227, 12840725, 23061898,
3203              3260492, 22510453, 8577507},
3204             {-12632451, 11257346, -32692994, 13548177, -721004, 10879011,
3205              31168030, 13952092, -29571492, -3635906},
3206         },
3207         {
3208             {3877321, -9572739, 32416692, 5405324, -11004407, -13656635,
3209              3759769, 11935320, 5611860, 8164018},
3210             {-16275802, 14667797, 15906460, 12155291, -22111149, -9039718,
3211              32003002, -8832289, 5773085, -8422109},
3212             {-23788118, -8254300, 1950875, 8937633, 18686727, 16459170, -905725,
3213              12376320, 31632953, 190926},
3214         },
3215         {
3216             {-24593607, -16138885, -8423991, 13378746, 14162407, 6901328,
3217              -8288749, 4508564, -25341555, -3627528},
3218             {8884438, -5884009, 6023974, 10104341, -6881569, -4941533, 18722941,
3219              -14786005, -1672488, 827625},
3220             {-32720583, -16289296, -32503547, 7101210, 13354605, 2659080,
3221              -1800575, -14108036, -24878478, 1541286},
3222         },
3223         {
3224             {2901347, -1117687, 3880376, -10059388, -17620940, -3612781,
3225              -21802117, -3567481, 20456845, -1885033},
3226             {27019610, 12299467, -13658288, -1603234, -12861660, -4861471,
3227              -19540150, -5016058, 29439641, 15138866},
3228             {21536104, -6626420, -32447818, -10690208, -22408077, 5175814,
3229              -5420040, -16361163, 7779328, 109896},
3230         },
3231         {
3232             {30279744, 14648750, -8044871, 6425558, 13639621, -743509, 28698390,
3233              12180118, 23177719, -554075},
3234             {26572847, 3405927, -31701700, 12890905, -19265668, 5335866,
3235              -6493768, 2378492, 4439158, -13279347},
3236             {-22716706, 3489070, -9225266, -332753, 18875722, -1140095,
3237              14819434, -12731527, -17717757, -5461437},
3238         },
3239         {
3240             {-5056483, 16566551, 15953661, 3767752, -10436499, 15627060,
3241              -820954, 2177225, 8550082, -15114165},
3242             {-18473302, 16596775, -381660, 15663611, 22860960, 15585581,
3243              -27844109, -3582739, -23260460, -8428588},
3244             {-32480551, 15707275, -8205912, -5652081, 29464558, 2713815,
3245              -22725137, 15860482, -21902570, 1494193},
3246         },
3247         {
3248             {-19562091, -14087393, -25583872, -9299552, 13127842, 759709,
3249              21923482, 16529112, 8742704, 12967017},
3250             {-28464899, 1553205, 32536856, -10473729, -24691605, -406174,
3251              -8914625, -2933896, -29903758, 15553883},
3252             {21877909, 3230008, 9881174, 10539357, -4797115, 2841332, 11543572,
3253              14513274, 19375923, -12647961},
3254         },
3255         {
3256             {8832269, -14495485, 13253511, 5137575, 5037871, 4078777, 24880818,
3257              -6222716, 2862653, 9455043},
3258             {29306751, 5123106, 20245049, -14149889, 9592566, 8447059, -2077124,
3259              -2990080, 15511449, 4789663},
3260             {-20679756, 7004547, 8824831, -9434977, -4045704, -3750736,
3261              -5754762, 108893, 23513200, 16652362},
3262         },
3263     },
3264     {
3265         {
3266             {-33256173, 4144782, -4476029, -6579123, 10770039, -7155542,
3267              -6650416, -12936300, -18319198, 10212860},
3268             {2756081, 8598110, 7383731, -6859892, 22312759, -1105012, 21179801,
3269              2600940, -9988298, -12506466},
3270             {-24645692, 13317462, -30449259, -15653928, 21365574, -10869657,
3271              11344424, 864440, -2499677, -16710063},
3272         },
3273         {
3274             {-26432803, 6148329, -17184412, -14474154, 18782929, -275997,
3275              -22561534, 211300, 2719757, 4940997},
3276             {-1323882, 3911313, -6948744, 14759765, -30027150, 7851207,
3277              21690126, 8518463, 26699843, 5276295},
3278             {-13149873, -6429067, 9396249, 365013, 24703301, -10488939, 1321586,
3279              149635, -15452774, 7159369},
3280         },
3281         {
3282             {9987780, -3404759, 17507962, 9505530, 9731535, -2165514, 22356009,
3283              8312176, 22477218, -8403385},
3284             {18155857, -16504990, 19744716, 9006923, 15154154, -10538976,
3285              24256460, -4864995, -22548173, 9334109},
3286             {2986088, -4911893, 10776628, -3473844, 10620590, -7083203,
3287              -21413845, 14253545, -22587149, 536906},
3288         },
3289         {
3290             {4377756, 8115836, 24567078, 15495314, 11625074, 13064599, 7390551,
3291              10589625, 10838060, -15420424},
3292             {-19342404, 867880, 9277171, -3218459, -14431572, -1986443,
3293              19295826, -15796950, 6378260, 699185},
3294             {7895026, 4057113, -7081772, -13077756, -17886831, -323126, -716039,
3295              15693155, -5045064, -13373962},
3296         },
3297         {
3298             {-7737563, -5869402, -14566319, -7406919, 11385654, 13201616,
3299              31730678, -10962840, -3918636, -9669325},
3300             {10188286, -15770834, -7336361, 13427543, 22223443, 14896287,
3301              30743455, 7116568, -21786507, 5427593},
3302             {696102, 13206899, 27047647, -10632082, 15285305, -9853179,
3303              10798490, -4578720, 19236243, 12477404},
3304         },
3305         {
3306             {-11229439, 11243796, -17054270, -8040865, -788228, -8167967,
3307              -3897669, 11180504, -23169516, 7733644},
3308             {17800790, -14036179, -27000429, -11766671, 23887827, 3149671,
3309              23466177, -10538171, 10322027, 15313801},
3310             {26246234, 11968874, 32263343, -5468728, 6830755, -13323031,
3311              -15794704, -101982, -24449242, 10890804},
3312         },
3313         {
3314             {-31365647, 10271363, -12660625, -6267268, 16690207, -13062544,
3315              -14982212, 16484931, 25180797, -5334884},
3316             {-586574, 10376444, -32586414, -11286356, 19801893, 10997610,
3317              2276632, 9482883, 316878, 13820577},
3318             {-9882808, -4510367, -2115506, 16457136, -11100081, 11674996,
3319              30756178, -7515054, 30696930, -3712849},
3320         },
3321         {
3322             {32988917, -9603412, 12499366, 7910787, -10617257, -11931514,
3323              -7342816, -9985397, -32349517, 7392473},
3324             {-8855661, 15927861, 9866406, -3649411, -2396914, -16655781,
3325              -30409476, -9134995, 25112947, -2926644},
3326             {-2504044, -436966, 25621774, -5678772, 15085042, -5479877,
3327              -24884878, -13526194, 5537438, -13914319},
3328         },
3329     },
3330     {
3331         {
3332             {-11225584, 2320285, -9584280, 10149187, -33444663, 5808648,
3333              -14876251, -1729667, 31234590, 6090599},
3334             {-9633316, 116426, 26083934, 2897444, -6364437, -2688086, 609721,
3335              15878753, -6970405, -9034768},
3336             {-27757857, 247744, -15194774, -9002551, 23288161, -10011936,
3337              -23869595, 6503646, 20650474, 1804084},
3338         },
3339         {
3340             {-27589786, 15456424, 8972517, 8469608, 15640622, 4439847, 3121995,
3341              -10329713, 27842616, -202328},
3342             {-15306973, 2839644, 22530074, 10026331, 4602058, 5048462, 28248656,
3343              5031932, -11375082, 12714369},
3344             {20807691, -7270825, 29286141, 11421711, -27876523, -13868230,
3345              -21227475, 1035546, -19733229, 12796920},
3346         },
3347         {
3348             {12076899, -14301286, -8785001, -11848922, -25012791, 16400684,
3349              -17591495, -12899438, 3480665, -15182815},
3350             {-32361549, 5457597, 28548107, 7833186, 7303070, -11953545,
3351              -24363064, -15921875, -33374054, 2771025},
3352             {-21389266, 421932, 26597266, 6860826, 22486084, -6737172,
3353              -17137485, -4210226, -24552282, 15673397},
3354         },
3355         {
3356             {-20184622, 2338216, 19788685, -9620956, -4001265, -8740893,
3357              -20271184, 4733254, 3727144, -12934448},
3358             {6120119, 814863, -11794402, -622716, 6812205, -15747771, 2019594,
3359              7975683, 31123697, -10958981},
3360             {30069250, -11435332, 30434654, 2958439, 18399564, -976289,
3361              12296869, 9204260, -16432438, 9648165},
3362         },
3363         {
3364             {32705432, -1550977, 30705658, 7451065, -11805606, 9631813, 3305266,
3365              5248604, -26008332, -11377501},
3366             {17219865, 2375039, -31570947, -5575615, -19459679, 9219903, 294711,
3367              15298639, 2662509, -16297073},
3368             {-1172927, -7558695, -4366770, -4287744, -21346413, -8434326,
3369              32087529, -1222777, 32247248, -14389861},
3370         },
3371         {
3372             {14312628, 1221556, 17395390, -8700143, -4945741, -8684635,
3373              -28197744, -9637817, -16027623, -13378845},
3374             {-1428825, -9678990, -9235681, 6549687, -7383069, -468664, 23046502,
3375              9803137, 17597934, 2346211},
3376             {18510800, 15337574, 26171504, 981392, -22241552, 7827556,
3377              -23491134, -11323352, 3059833, -11782870},
3378         },
3379         {
3380             {10141598, 6082907, 17829293, -1947643, 9830092, 13613136,
3381              -25556636, -5544586, -33502212, 3592096},
3382             {33114168, -15889352, -26525686, -13343397, 33076705, 8716171,
3383              1151462, 1521897, -982665, -6837803},
3384             {-32939165, -4255815, 23947181, -324178, -33072974, -12305637,
3385              -16637686, 3891704, 26353178, 693168},
3386         },
3387         {
3388             {30374239, 1595580, -16884039, 13186931, 4600344, 406904, 9585294,
3389              -400668, 31375464, 14369965},
3390             {-14370654, -7772529, 1510301, 6434173, -18784789, -6262728,
3391              32732230, -13108839, 17901441, 16011505},
3392             {18171223, -11934626, -12500402, 15197122, -11038147, -15230035,
3393              -19172240, -16046376, 8764035, 12309598},
3394         },
3395     },
3396     {
3397         {
3398             {5975908, -5243188, -19459362, -9681747, -11541277, 14015782,
3399              -23665757, 1228319, 17544096, -10593782},
3400             {5811932, -1715293, 3442887, -2269310, -18367348, -8359541,
3401              -18044043, -15410127, -5565381, 12348900},
3402             {-31399660, 11407555, 25755363, 6891399, -3256938, 14872274,
3403              -24849353, 8141295, -10632534, -585479},
3404         },
3405         {
3406             {-12675304, 694026, -5076145, 13300344, 14015258, -14451394,
3407              -9698672, -11329050, 30944593, 1130208},
3408             {8247766, -6710942, -26562381, -7709309, -14401939, -14648910,
3409              4652152, 2488540, 23550156, -271232},
3410             {17294316, -3788438, 7026748, 15626851, 22990044, 113481, 2267737,
3411              -5908146, -408818, -137719},
3412         },
3413         {
3414             {16091085, -16253926, 18599252, 7340678, 2137637, -1221657,
3415              -3364161, 14550936, 3260525, -7166271},
3416             {-4910104, -13332887, 18550887, 10864893, -16459325, -7291596,
3417              -23028869, -13204905, -12748722, 2701326},
3418             {-8574695, 16099415, 4629974, -16340524, -20786213, -6005432,
3419              -10018363, 9276971, 11329923, 1862132},
3420         },
3421         {
3422             {14763076, -15903608, -30918270, 3689867, 3511892, 10313526,
3423              -21951088, 12219231, -9037963, -940300},
3424             {8894987, -3446094, 6150753, 3013931, 301220, 15693451, -31981216,
3425              -2909717, -15438168, 11595570},
3426             {15214962, 3537601, -26238722, -14058872, 4418657, -15230761,
3427              13947276, 10730794, -13489462, -4363670},
3428         },
3429         {
3430             {-2538306, 7682793, 32759013, 263109, -29984731, -7955452,
3431              -22332124, -10188635, 977108, 699994},
3432             {-12466472, 4195084, -9211532, 550904, -15565337, 12917920,
3433              19118110, -439841, -30534533, -14337913},
3434             {31788461, -14507657, 4799989, 7372237, 8808585, -14747943, 9408237,
3435              -10051775, 12493932, -5409317},
3436         },
3437         {
3438             {-25680606, 5260744, -19235809, -6284470, -3695942, 16566087,
3439              27218280, 2607121, 29375955, 6024730},
3440             {842132, -2794693, -4763381, -8722815, 26332018, -12405641,
3441              11831880, 6985184, -9940361, 2854096},
3442             {-4847262, -7969331, 2516242, -5847713, 9695691, -7221186, 16512645,
3443              960770, 12121869, 16648078},
3444         },
3445         {
3446             {-15218652, 14667096, -13336229, 2013717, 30598287, -464137,
3447              -31504922, -7882064, 20237806, 2838411},
3448             {-19288047, 4453152, 15298546, -16178388, 22115043, -15972604,
3449              12544294, -13470457, 1068881, -12499905},
3450             {-9558883, -16518835, 33238498, 13506958, 30505848, -1114596,
3451              -8486907, -2630053, 12521378, 4845654},
3452         },
3453         {
3454             {-28198521, 10744108, -2958380, 10199664, 7759311, -13088600,
3455              3409348, -873400, -6482306, -12885870},
3456             {-23561822, 6230156, -20382013, 10655314, -24040585, -11621172,
3457              10477734, -1240216, -3113227, 13974498},
3458             {12966261, 15550616, -32038948, -1615346, 21025980, -629444,
3459              5642325, 7188737, 18895762, 12629579},
3460         },
3461     },
3462     {
3463         {
3464             {14741879, -14946887, 22177208, -11721237, 1279741, 8058600,
3465              11758140, 789443, 32195181, 3895677},
3466             {10758205, 15755439, -4509950, 9243698, -4879422, 6879879, -2204575,
3467              -3566119, -8982069, 4429647},
3468             {-2453894, 15725973, -20436342, -10410672, -5803908, -11040220,
3469              -7135870, -11642895, 18047436, -15281743},
3470         },
3471         {
3472             {-25173001, -11307165, 29759956, 11776784, -22262383, -15820455,
3473              10993114, -12850837, -17620701, -9408468},
3474             {21987233, 700364, -24505048, 14972008, -7774265, -5718395,
3475              32155026, 2581431, -29958985, 8773375},
3476             {-25568350, 454463, -13211935, 16126715, 25240068, 8594567,
3477              20656846, 12017935, -7874389, -13920155},
3478         },
3479         {
3480             {6028182, 6263078, -31011806, -11301710, -818919, 2461772,
3481              -31841174, -5468042, -1721788, -2776725},
3482             {-12278994, 16624277, 987579, -5922598, 32908203, 1248608, 7719845,
3483              -4166698, 28408820, 6816612},
3484             {-10358094, -8237829, 19549651, -12169222, 22082623, 16147817,
3485              20613181, 13982702, -10339570, 5067943},
3486         },
3487         {
3488             {-30505967, -3821767, 12074681, 13582412, -19877972, 2443951,
3489              -19719286, 12746132, 5331210, -10105944},
3490             {30528811, 3601899, -1957090, 4619785, -27361822, -15436388,
3491              24180793, -12570394, 27679908, -1648928},
3492             {9402404, -13957065, 32834043, 10838634, -26580150, -13237195,
3493              26653274, -8685565, 22611444, -12715406},
3494         },
3495         {
3496             {22190590, 1118029, 22736441, 15130463, -30460692, -5991321,
3497              19189625, -4648942, 4854859, 6622139},
3498             {-8310738, -2953450, -8262579, -3388049, -10401731, -271929,
3499              13424426, -3567227, 26404409, 13001963},
3500             {-31241838, -15415700, -2994250, 8939346, 11562230, -12840670,
3501              -26064365, -11621720, -15405155, 11020693},
3502         },
3503         {
3504             {1866042, -7949489, -7898649, -10301010, 12483315, 13477547,
3505              3175636, -12424163, 28761762, 1406734},
3506             {-448555, -1777666, 13018551, 3194501, -9580420, -11161737,
3507              24760585, -4347088, 25577411, -13378680},
3508             {-24290378, 4759345, -690653, -1852816, 2066747, 10693769,
3509              -29595790, 9884936, -9368926, 4745410},
3510         },
3511         {
3512             {-9141284, 6049714, -19531061, -4341411, -31260798, 9944276,
3513              -15462008, -11311852, 10931924, -11931931},
3514             {-16561513, 14112680, -8012645, 4817318, -8040464, -11414606,
3515              -22853429, 10856641, -20470770, 13434654},
3516             {22759489, -10073434, -16766264, -1871422, 13637442, -10168091,
3517              1765144, -12654326, 28445307, -5364710},
3518         },
3519         {
3520             {29875063, 12493613, 2795536, -3786330, 1710620, 15181182,
3521              -10195717, -8788675, 9074234, 1167180},
3522             {-26205683, 11014233, -9842651, -2635485, -26908120, 7532294,
3523              -18716888, -9535498, 3843903, 9367684},
3524             {-10969595, -6403711, 9591134, 9582310, 11349256, 108879, 16235123,
3525              8601684, -139197, 4242895},
3526         },
3527     },
3528     {
3529         {
3530             {22092954, -13191123, -2042793, -11968512, 32186753, -11517388,
3531              -6574341, 2470660, -27417366, 16625501},
3532             {-11057722, 3042016, 13770083, -9257922, 584236, -544855, -7770857,
3533              2602725, -27351616, 14247413},
3534             {6314175, -10264892, -32772502, 15957557, -10157730, 168750,
3535              -8618807, 14290061, 27108877, -1180880},
3536         },
3537         {
3538             {-8586597, -7170966, 13241782, 10960156, -32991015, -13794596,
3539              33547976, -11058889, -27148451, 981874},
3540             {22833440, 9293594, -32649448, -13618667, -9136966, 14756819,
3541              -22928859, -13970780, -10479804, -16197962},
3542             {-7768587, 3326786, -28111797, 10783824, 19178761, 14905060,
3543              22680049, 13906969, -15933690, 3797899},
3544         },
3545         {
3546             {21721356, -4212746, -12206123, 9310182, -3882239, -13653110,
3547              23740224, -2709232, 20491983, -8042152},
3548             {9209270, -15135055, -13256557, -6167798, -731016, 15289673,
3549              25947805, 15286587, 30997318, -6703063},
3550             {7392032, 16618386, 23946583, -8039892, -13265164, -1533858,
3551              -14197445, -2321576, 17649998, -250080},
3552         },
3553         {
3554             {-9301088, -14193827, 30609526, -3049543, -25175069, -1283752,
3555              -15241566, -9525724, -2233253, 7662146},
3556             {-17558673, 1763594, -33114336, 15908610, -30040870, -12174295,
3557              7335080, -8472199, -3174674, 3440183},
3558             {-19889700, -5977008, -24111293, -9688870, 10799743, -16571957,
3559              40450, -4431835, 4862400, 1133},
3560         },
3561         {
3562             {-32856209, -7873957, -5422389, 14860950, -16319031, 7956142,
3563              7258061, 311861, -30594991, -7379421},
3564             {-3773428, -1565936, 28985340, 7499440, 24445838, 9325937, 29727763,
3565              16527196, 18278453, 15405622},
3566             {-4381906, 8508652, -19898366, -3674424, -5984453, 15149970,
3567              -13313598, 843523, -21875062, 13626197},
3568         },
3569         {
3570             {2281448, -13487055, -10915418, -2609910, 1879358, 16164207,
3571              -10783882, 3953792, 13340839, 15928663},
3572             {31727126, -7179855, -18437503, -8283652, 2875793, -16390330,
3573              -25269894, -7014826, -23452306, 5964753},
3574             {4100420, -5959452, -17179337, 6017714, -18705837, 12227141,
3575              -26684835, 11344144, 2538215, -7570755},
3576         },
3577         {
3578             {-9433605, 6123113, 11159803, -2156608, 30016280, 14966241,
3579              -20474983, 1485421, -629256, -15958862},
3580             {-26804558, 4260919, 11851389, 9658551, -32017107, 16367492,
3581              -20205425, -13191288, 11659922, -11115118},
3582             {26180396, 10015009, -30844224, -8581293, 5418197, 9480663, 2231568,
3583              -10170080, 33100372, -1306171},
3584         },
3585         {
3586             {15121113, -5201871, -10389905, 15427821, -27509937, -15992507,
3587              21670947, 4486675, -5931810, -14466380},
3588             {16166486, -9483733, -11104130, 6023908, -31926798, -1364923,
3589              2340060, -16254968, -10735770, -10039824},
3590             {28042865, -3557089, -12126526, 12259706, -3717498, -6945899,
3591              6766453, -8689599, 18036436, 5803270},
3592         },
3593     },
3594     {
3595         {
3596             {-817581, 6763912, 11803561, 1585585, 10958447, -2671165, 23855391,
3597              4598332, -6159431, -14117438},
3598             {-31031306, -14256194, 17332029, -2383520, 31312682, -5967183,
3599              696309, 50292, -20095739, 11763584},
3600             {-594563, -2514283, -32234153, 12643980, 12650761, 14811489, 665117,
3601              -12613632, -19773211, -10713562},
3602         },
3603         {
3604             {30464590, -11262872, -4127476, -12734478, 19835327, -7105613,
3605              -24396175, 2075773, -17020157, 992471},
3606             {18357185, -6994433, 7766382, 16342475, -29324918, 411174, 14578841,
3607              8080033, -11574335, -10601610},
3608             {19598397, 10334610, 12555054, 2555664, 18821899, -10339780,
3609              21873263, 16014234, 26224780, 16452269},
3610         },
3611         {
3612             {-30223925, 5145196, 5944548, 16385966, 3976735, 2009897, -11377804,
3613              -7618186, -20533829, 3698650},
3614             {14187449, 3448569, -10636236, -10810935, -22663880, -3433596,
3615              7268410, -10890444, 27394301, 12015369},
3616             {19695761, 16087646, 28032085, 12999827, 6817792, 11427614,
3617              20244189, -1312777, -13259127, -3402461},
3618         },
3619         {
3620             {30860103, 12735208, -1888245, -4699734, -16974906, 2256940,
3621              -8166013, 12298312, -8550524, -10393462},
3622             {-5719826, -11245325, -1910649, 15569035, 26642876, -7587760,
3623              -5789354, -15118654, -4976164, 12651793},
3624             {-2848395, 9953421, 11531313, -5282879, 26895123, -12697089,
3625              -13118820, -16517902, 9768698, -2533218},
3626         },
3627         {
3628             {-24719459, 1894651, -287698, -4704085, 15348719, -8156530,
3629              32767513, 12765450, 4940095, 10678226},
3630             {18860224, 15980149, -18987240, -1562570, -26233012, -11071856,
3631              -7843882, 13944024, -24372348, 16582019},
3632             {-15504260, 4970268, -29893044, 4175593, -20993212, -2199756,
3633              -11704054, 15444560, -11003761, 7989037},
3634         },
3635         {
3636             {31490452, 5568061, -2412803, 2182383, -32336847, 4531686,
3637              -32078269, 6200206, -19686113, -14800171},
3638             {-17308668, -15879940, -31522777, -2831, -32887382, 16375549,
3639              8680158, -16371713, 28550068, -6857132},
3640             {-28126887, -5688091, 16837845, -1820458, -6850681, 12700016,
3641              -30039981, 4364038, 1155602, 5988841},
3642         },
3643         {
3644             {21890435, -13272907, -12624011, 12154349, -7831873, 15300496,
3645              23148983, -4470481, 24618407, 8283181},
3646             {-33136107, -10512751, 9975416, 6841041, -31559793, 16356536,
3647              3070187, -7025928, 1466169, 10740210},
3648             {-1509399, -15488185, -13503385, -10655916, 32799044, 909394,
3649              -13938903, -5779719, -32164649, -15327040},
3650         },
3651         {
3652             {3960823, -14267803, -28026090, -15918051, -19404858, 13146868,
3653              15567327, 951507, -3260321, -573935},
3654             {24740841, 5052253, -30094131, 8961361, 25877428, 6165135,
3655              -24368180, 14397372, -7380369, -6144105},
3656             {-28888365, 3510803, -28103278, -1158478, -11238128, -10631454,
3657              -15441463, -14453128, -1625486, -6494814},
3658         },
3659     },
3660     {
3661         {
3662             {793299, -9230478, 8836302, -6235707, -27360908, -2369593, 33152843,
3663              -4885251, -9906200, -621852},
3664             {5666233, 525582, 20782575, -8038419, -24538499, 14657740, 16099374,
3665              1468826, -6171428, -15186581},
3666             {-4859255, -3779343, -2917758, -6748019, 7778750, 11688288,
3667              -30404353, -9871238, -1558923, -9863646},
3668         },
3669         {
3670             {10896332, -7719704, 824275, 472601, -19460308, 3009587, 25248958,
3671              14783338, -30581476, -15757844},
3672             {10566929, 12612572, -31944212, 11118703, -12633376, 12362879,
3673              21752402, 8822496, 24003793, 14264025},
3674             {27713862, -7355973, -11008240, 9227530, 27050101, 2504721,
3675              23886875, -13117525, 13958495, -5732453},
3676         },
3677         {
3678             {-23481610, 4867226, -27247128, 3900521, 29838369, -8212291,
3679              -31889399, -10041781, 7340521, -15410068},
3680             {4646514, -8011124, -22766023, -11532654, 23184553, 8566613,
3681              31366726, -1381061, -15066784, -10375192},
3682             {-17270517, 12723032, -16993061, 14878794, 21619651, -6197576,
3683              27584817, 3093888, -8843694, 3849921},
3684         },
3685         {
3686             {-9064912, 2103172, 25561640, -15125738, -5239824, 9582958,
3687              32477045, -9017955, 5002294, -15550259},
3688             {-12057553, -11177906, 21115585, -13365155, 8808712, -12030708,
3689              16489530, 13378448, -25845716, 12741426},
3690             {-5946367, 10645103, -30911586, 15390284, -3286982, -7118677,
3691              24306472, 15852464, 28834118, -7646072},
3692         },
3693         {
3694             {-17335748, -9107057, -24531279, 9434953, -8472084, -583362,
3695              -13090771, 455841, 20461858, 5491305},
3696             {13669248, -16095482, -12481974, -10203039, -14569770, -11893198,
3697              -24995986, 11293807, -28588204, -9421832},
3698             {28497928, 6272777, -33022994, 14470570, 8906179, -1225630,
3699              18504674, -14165166, 29867745, -8795943},
3700         },
3701         {
3702             {-16207023, 13517196, -27799630, -13697798, 24009064, -6373891,
3703              -6367600, -13175392, 22853429, -4012011},
3704             {24191378, 16712145, -13931797, 15217831, 14542237, 1646131,
3705              18603514, -11037887, 12876623, -2112447},
3706             {17902668, 4518229, -411702, -2829247, 26878217, 5258055, -12860753,
3707              608397, 16031844, 3723494},
3708         },
3709         {
3710             {-28632773, 12763728, -20446446, 7577504, 33001348, -13017745,
3711              17558842, -7872890, 23896954, -4314245},
3712             {-20005381, -12011952, 31520464, 605201, 2543521, 5991821, -2945064,
3713              7229064, -9919646, -8826859},
3714             {28816045, 298879, -28165016, -15920938, 19000928, -1665890,
3715              -12680833, -2949325, -18051778, -2082915},
3716         },
3717         {
3718             {16000882, -344896, 3493092, -11447198, -29504595, -13159789,
3719              12577740, 16041268, -19715240, 7847707},
3720             {10151868, 10572098, 27312476, 7922682, 14825339, 4723128,
3721              -32855931, -6519018, -10020567, 3852848},
3722             {-11430470, 15697596, -21121557, -4420647, 5386314, 15063598,
3723              16514493, -15932110, 29330899, -15076224},
3724         },
3725     },
3726     {
3727         {
3728             {-25499735, -4378794, -15222908, -6901211, 16615731, 2051784,
3729              3303702, 15490, -27548796, 12314391},
3730             {15683520, -6003043, 18109120, -9980648, 15337968, -5997823,
3731              -16717435, 15921866, 16103996, -3731215},
3732             {-23169824, -10781249, 13588192, -1628807, -3798557, -1074929,
3733              -19273607, 5402699, -29815713, -9841101},
3734         },
3735         {
3736             {23190676, 2384583, -32714340, 3462154, -29903655, -1529132,
3737              -11266856, 8911517, -25205859, 2739713},
3738             {21374101, -3554250, -33524649, 9874411, 15377179, 11831242,
3739              -33529904, 6134907, 4931255, 11987849},
3740             {-7732, -2978858, -16223486, 7277597, 105524, -322051, -31480539,
3741              13861388, -30076310, 10117930},
3742         },
3743         {
3744             {-29501170, -10744872, -26163768, 13051539, -25625564, 5089643,
3745              -6325503, 6704079, 12890019, 15728940},
3746             {-21972360, -11771379, -951059, -4418840, 14704840, 2695116, 903376,
3747              -10428139, 12885167, 8311031},
3748             {-17516482, 5352194, 10384213, -13811658, 7506451, 13453191,
3749              26423267, 4384730, 1888765, -5435404},
3750         },
3751         {
3752             {-25817338, -3107312, -13494599, -3182506, 30896459, -13921729,
3753              -32251644, -12707869, -19464434, -3340243},
3754             {-23607977, -2665774, -526091, 4651136, 5765089, 4618330, 6092245,
3755              14845197, 17151279, -9854116},
3756             {-24830458, -12733720, -15165978, 10367250, -29530908, -265356,
3757              22825805, -7087279, -16866484, 16176525},
3758         },
3759         {
3760             {-23583256, 6564961, 20063689, 3798228, -4740178, 7359225, 2006182,
3761              -10363426, -28746253, -10197509},
3762             {-10626600, -4486402, -13320562, -5125317, 3432136, -6393229,
3763              23632037, -1940610, 32808310, 1099883},
3764             {15030977, 5768825, -27451236, -2887299, -6427378, -15361371,
3765              -15277896, -6809350, 2051441, -15225865},
3766         },
3767         {
3768             {-3362323, -7239372, 7517890, 9824992, 23555850, 295369, 5148398,
3769              -14154188, -22686354, 16633660},
3770             {4577086, -16752288, 13249841, -15304328, 19958763, -14537274,
3771              18559670, -10759549, 8402478, -9864273},
3772             {-28406330, -1051581, -26790155, -907698, -17212414, -11030789,
3773              9453451, -14980072, 17983010, 9967138},
3774         },
3775         {
3776             {-25762494, 6524722, 26585488, 9969270, 24709298, 1220360, -1677990,
3777              7806337, 17507396, 3651560},
3778             {-10420457, -4118111, 14584639, 15971087, -15768321, 8861010,
3779              26556809, -5574557, -18553322, -11357135},
3780             {2839101, 14284142, 4029895, 3472686, 14402957, 12689363, -26642121,
3781              8459447, -5605463, -7621941},
3782         },
3783         {
3784             {-4839289, -3535444, 9744961, 2871048, 25113978, 3187018, -25110813,
3785              -849066, 17258084, -7977739},
3786             {18164541, -10595176, -17154882, -1542417, 19237078, -9745295,
3787              23357533, -15217008, 26908270, 12150756},
3788             {-30264870, -7647865, 5112249, -7036672, -1499807, -6974257, 43168,
3789              -5537701, -32302074, 16215819},
3790         },
3791     },
3792     {
3793         {
3794             {-6898905, 9824394, -12304779, -4401089, -31397141, -6276835,
3795              32574489, 12532905, -7503072, -8675347},
3796             {-27343522, -16515468, -27151524, -10722951, 946346, 16291093,
3797              254968, 7168080, 21676107, -1943028},
3798             {21260961, -8424752, -16831886, -11920822, -23677961, 3968121,
3799              -3651949, -6215466, -3556191, -7913075},
3800         },
3801         {
3802             {16544754, 13250366, -16804428, 15546242, -4583003, 12757258,
3803              -2462308, -8680336, -18907032, -9662799},
3804             {-2415239, -15577728, 18312303, 4964443, -15272530, -12653564,
3805              26820651, 16690659, 25459437, -4564609},
3806             {-25144690, 11425020, 28423002, -11020557, -6144921, -15826224,
3807              9142795, -2391602, -6432418, -1644817},
3808         },
3809         {
3810             {-23104652, 6253476, 16964147, -3768872, -25113972, -12296437,
3811              -27457225, -16344658, 6335692, 7249989},
3812             {-30333227, 13979675, 7503222, -12368314, -11956721, -4621693,
3813              -30272269, 2682242, 25993170, -12478523},
3814             {4364628, 5930691, 32304656, -10044554, -8054781, 15091131,
3815              22857016, -10598955, 31820368, 15075278},
3816         },
3817         {
3818             {31879134, -8918693, 17258761, 90626, -8041836, -4917709, 24162788,
3819              -9650886, -17970238, 12833045},
3820             {19073683, 14851414, -24403169, -11860168, 7625278, 11091125,
3821              -19619190, 2074449, -9413939, 14905377},
3822             {24483667, -11935567, -2518866, -11547418, -1553130, 15355506,
3823              -25282080, 9253129, 27628530, -7555480},
3824         },
3825         {
3826             {17597607, 8340603, 19355617, 552187, 26198470, -3176583, 4593324,
3827              -9157582, -14110875, 15297016},
3828             {510886, 14337390, -31785257, 16638632, 6328095, 2713355, -20217417,
3829              -11864220, 8683221, 2921426},
3830             {18606791, 11874196, 27155355, -5281482, -24031742, 6265446,
3831              -25178240, -1278924, 4674690, 13890525},
3832         },
3833         {
3834             {13609624, 13069022, -27372361, -13055908, 24360586, 9592974,
3835              14977157, 9835105, 4389687, 288396},
3836             {9922506, -519394, 13613107, 5883594, -18758345, -434263, -12304062,
3837              8317628, 23388070, 16052080},
3838             {12720016, 11937594, -31970060, -5028689, 26900120, 8561328,
3839              -20155687, -11632979, -14754271, -10812892},
3840         },
3841         {
3842             {15961858, 14150409, 26716931, -665832, -22794328, 13603569,
3843              11829573, 7467844, -28822128, 929275},
3844             {11038231, -11582396, -27310482, -7316562, -10498527, -16307831,
3845              -23479533, -9371869, -21393143, 2465074},
3846             {20017163, -4323226, 27915242, 1529148, 12396362, 15675764,
3847              13817261, -9658066, 2463391, -4622140},
3848         },
3849         {
3850             {-16358878, -12663911, -12065183, 4996454, -1256422, 1073572,
3851              9583558, 12851107, 4003896, 12673717},
3852             {-1731589, -15155870, -3262930, 16143082, 19294135, 13385325,
3853              14741514, -9103726, 7903886, 2348101},
3854             {24536016, -16515207, 12715592, -3862155, 1511293, 10047386,
3855              -3842346, -7129159, -28377538, 10048127},
3856         },
3857     },
3858     {
3859         {
3860             {-12622226, -6204820, 30718825, 2591312, -10617028, 12192840,
3861              18873298, -7297090, -32297756, 15221632},
3862             {-26478122, -11103864, 11546244, -1852483, 9180880, 7656409,
3863              -21343950, 2095755, 29769758, 6593415},
3864             {-31994208, -2907461, 4176912, 3264766, 12538965, -868111, 26312345,
3865              -6118678, 30958054, 8292160},
3866         },
3867         {
3868             {31429822, -13959116, 29173532, 15632448, 12174511, -2760094,
3869              32808831, 3977186, 26143136, -3148876},
3870             {22648901, 1402143, -22799984, 13746059, 7936347, 365344, -8668633,
3871              -1674433, -3758243, -2304625},
3872             {-15491917, 8012313, -2514730, -12702462, -23965846, -10254029,
3873              -1612713, -1535569, -16664475, 8194478},
3874         },
3875         {
3876             {27338066, -7507420, -7414224, 10140405, -19026427, -6589889,
3877              27277191, 8855376, 28572286, 3005164},
3878             {26287124, 4821776, 25476601, -4145903, -3764513, -15788984,
3879              -18008582, 1182479, -26094821, -13079595},
3880             {-7171154, 3178080, 23970071, 6201893, -17195577, -4489192,
3881              -21876275, -13982627, 32208683, -1198248},
3882         },
3883         {
3884             {-16657702, 2817643, -10286362, 14811298, 6024667, 13349505,
3885              -27315504, -10497842, -27672585, -11539858},
3886             {15941029, -9405932, -21367050, 8062055, 31876073, -238629,
3887              -15278393, -1444429, 15397331, -4130193},
3888             {8934485, -13485467, -23286397, -13423241, -32446090, 14047986,
3889              31170398, -1441021, -27505566, 15087184},
3890         },
3891         {
3892             {-18357243, -2156491, 24524913, -16677868, 15520427, -6360776,
3893              -15502406, 11461896, 16788528, -5868942},
3894             {-1947386, 16013773, 21750665, 3714552, -17401782, -16055433,
3895              -3770287, -10323320, 31322514, -11615635},
3896             {21426655, -5650218, -13648287, -5347537, -28812189, -4920970,
3897              -18275391, -14621414, 13040862, -12112948},
3898         },
3899         {
3900             {11293895, 12478086, -27136401, 15083750, -29307421, 14748872,
3901              14555558, -13417103, 1613711, 4896935},
3902             {-25894883, 15323294, -8489791, -8057900, 25967126, -13425460,
3903              2825960, -4897045, -23971776, -11267415},
3904             {-15924766, -5229880, -17443532, 6410664, 3622847, 10243618,
3905              20615400, 12405433, -23753030, -8436416},
3906         },
3907         {
3908             {-7091295, 12556208, -20191352, 9025187, -17072479, 4333801,
3909              4378436, 2432030, 23097949, -566018},
3910             {4565804, -16025654, 20084412, -7842817, 1724999, 189254, 24767264,
3911              10103221, -18512313, 2424778},
3912             {366633, -11976806, 8173090, -6890119, 30788634, 5745705, -7168678,
3913              1344109, -3642553, 12412659},
3914         },
3915         {
3916             {-24001791, 7690286, 14929416, -168257, -32210835, -13412986,
3917              24162697, -15326504, -3141501, 11179385},
3918             {18289522, -14724954, 8056945, 16430056, -21729724, 7842514,
3919              -6001441, -1486897, -18684645, -11443503},
3920             {476239, 6601091, -6152790, -9723375, 17503545, -4863900, 27672959,
3921              13403813, 11052904, 5219329},
3922         },
3923     },
3924     {
3925         {
3926             {20678546, -8375738, -32671898, 8849123, -5009758, 14574752,
3927              31186971, -3973730, 9014762, -8579056},
3928             {-13644050, -10350239, -15962508, 5075808, -1514661, -11534600,
3929              -33102500, 9160280, 8473550, -3256838},
3930             {24900749, 14435722, 17209120, -15292541, -22592275, 9878983,
3931              -7689309, -16335821, -24568481, 11788948},
3932         },
3933         {
3934             {-3118155, -11395194, -13802089, 14797441, 9652448, -6845904,
3935              -20037437, 10410733, -24568470, -1458691},
3936             {-15659161, 16736706, -22467150, 10215878, -9097177, 7563911,
3937              11871841, -12505194, -18513325, 8464118},
3938             {-23400612, 8348507, -14585951, -861714, -3950205, -6373419,
3939              14325289, 8628612, 33313881, -8370517},
3940         },
3941         {
3942             {-20186973, -4967935, 22367356, 5271547, -1097117, -4788838,
3943              -24805667, -10236854, -8940735, -5818269},
3944             {-6948785, -1795212, -32625683, -16021179, 32635414, -7374245,
3945              15989197, -12838188, 28358192, -4253904},
3946             {-23561781, -2799059, -32351682, -1661963, -9147719, 10429267,
3947              -16637684, 4072016, -5351664, 5596589},
3948         },
3949         {
3950             {-28236598, -3390048, 12312896, 6213178, 3117142, 16078565,
3951              29266239, 2557221, 1768301, 15373193},
3952             {-7243358, -3246960, -4593467, -7553353, -127927, -912245, -1090902,
3953              -4504991, -24660491, 3442910},
3954             {-30210571, 5124043, 14181784, 8197961, 18964734, -11939093,
3955              22597931, 7176455, -18585478, 13365930},
3956         },
3957         {
3958             {-7877390, -1499958, 8324673, 4690079, 6261860, 890446, 24538107,
3959              -8570186, -9689599, -3031667},
3960             {25008904, -10771599, -4305031, -9638010, 16265036, 15721635,
3961              683793, -11823784, 15723479, -15163481},
3962             {-9660625, 12374379, -27006999, -7026148, -7724114, -12314514,
3963              11879682, 5400171, 519526, -1235876},
3964         },
3965         {
3966             {22258397, -16332233, -7869817, 14613016, -22520255, -2950923,
3967              -20353881, 7315967, 16648397, 7605640},
3968             {-8081308, -8464597, -8223311, 9719710, 19259459, -15348212,
3969              23994942, -5281555, -9468848, 4763278},
3970             {-21699244, 9220969, -15730624, 1084137, -25476107, -2852390,
3971              31088447, -7764523, -11356529, 728112},
3972         },
3973         {
3974             {26047220, -11751471, -6900323, -16521798, 24092068, 9158119,
3975              -4273545, -12555558, -29365436, -5498272},
3976             {17510331, -322857, 5854289, 8403524, 17133918, -3112612, -28111007,
3977              12327945, 10750447, 10014012},
3978             {-10312768, 3936952, 9156313, -8897683, 16498692, -994647,
3979              -27481051, -666732, 3424691, 7540221},
3980         },
3981         {
3982             {30322361, -6964110, 11361005, -4143317, 7433304, 4989748, -7071422,
3983              -16317219, -9244265, 15258046},
3984             {13054562, -2779497, 19155474, 469045, -12482797, 4566042, 5631406,
3985              2711395, 1062915, -5136345},
3986             {-19240248, -11254599, -29509029, -7499965, -5835763, 13005411,
3987              -6066489, 12194497, 32960380, 1459310},
3988         },
3989     },
3990     {
3991         {
3992             {19852034, 7027924, 23669353, 10020366, 8586503, -6657907, 394197,
3993              -6101885, 18638003, -11174937},
3994             {31395534, 15098109, 26581030, 8030562, -16527914, -5007134,
3995              9012486, -7584354, -6643087, -5442636},
3996             {-9192165, -2347377, -1997099, 4529534, 25766844, 607986, -13222,
3997              9677543, -32294889, -6456008},
3998         },
3999         {
4000             {-2444496, -149937, 29348902, 8186665, 1873760, 12489863, -30934579,
4001              -7839692, -7852844, -8138429},
4002             {-15236356, -15433509, 7766470, 746860, 26346930, -10221762,
4003              -27333451, 10754588, -9431476, 5203576},
4004             {31834314, 14135496, -770007, 5159118, 20917671, -16768096,
4005              -7467973, -7337524, 31809243, 7347066},
4006         },
4007         {
4008             {-9606723, -11874240, 20414459, 13033986, 13716524, -11691881,
4009              19797970, -12211255, 15192876, -2087490},
4010             {-12663563, -2181719, 1168162, -3804809, 26747877, -14138091,
4011              10609330, 12694420, 33473243, -13382104},
4012             {33184999, 11180355, 15832085, -11385430, -1633671, 225884,
4013              15089336, -11023903, -6135662, 14480053},
4014         },
4015         {
4016             {31308717, -5619998, 31030840, -1897099, 15674547, -6582883,
4017              5496208, 13685227, 27595050, 8737275},
4018             {-20318852, -15150239, 10933843, -16178022, 8335352, -7546022,
4019              -31008351, -12610604, 26498114, 66511},
4020             {22644454, -8761729, -16671776, 4884562, -3105614, -13559366,
4021              30540766, -4286747, -13327787, -7515095},
4022         },
4023         {
4024             {-28017847, 9834845, 18617207, -2681312, -3401956, -13307506,
4025              8205540, 13585437, -17127465, 15115439},
4026             {23711543, -672915, 31206561, -8362711, 6164647, -9709987,
4027              -33535882, -1426096, 8236921, 16492939},
4028             {-23910559, -13515526, -26299483, -4503841, 25005590, -7687270,
4029              19574902, 10071562, 6708380, -6222424},
4030         },
4031         {
4032             {2101391, -4930054, 19702731, 2367575, -15427167, 1047675, 5301017,
4033              9328700, 29955601, -11678310},
4034             {3096359, 9271816, -21620864, -15521844, -14847996, -7592937,
4035              -25892142, -12635595, -9917575, 6216608},
4036             {-32615849, 338663, -25195611, 2510422, -29213566, -13820213,
4037              24822830, -6146567, -26767480, 7525079},
4038         },
4039         {
4040             {-23066649, -13985623, 16133487, -7896178, -3389565, 778788,
4041              -910336, -2782495, -19386633, 11994101},
4042             {21691500, -13624626, -641331, -14367021, 3285881, -3483596,
4043              -25064666, 9718258, -7477437, 13381418},
4044             {18445390, -4202236, 14979846, 11622458, -1727110, -3582980,
4045              23111648, -6375247, 28535282, 15779576},
4046         },
4047         {
4048             {30098053, 3089662, -9234387, 16662135, -21306940, 11308411,
4049              -14068454, 12021730, 9955285, -16303356},
4050             {9734894, -14576830, -7473633, -9138735, 2060392, 11313496,
4051              -18426029, 9924399, 20194861, 13380996},
4052             {-26378102, -7965207, -22167821, 15789297, -18055342, -6168792,
4053              -1984914, 15707771, 26342023, 10146099},
4054         },
4055     },
4056     {
4057         {
4058             {-26016874, -219943, 21339191, -41388, 19745256, -2878700,
4059              -29637280, 2227040, 21612326, -545728},
4060             {-13077387, 1184228, 23562814, -5970442, -20351244, -6348714,
4061              25764461, 12243797, -20856566, 11649658},
4062             {-10031494, 11262626, 27384172, 2271902, 26947504, -15997771, 39944,
4063              6114064, 33514190, 2333242},
4064         },
4065         {
4066             {-21433588, -12421821, 8119782, 7219913, -21830522, -9016134,
4067              -6679750, -12670638, 24350578, -13450001},
4068             {-4116307, -11271533, -23886186, 4843615, -30088339, 690623,
4069              -31536088, -10406836, 8317860, 12352766},
4070             {18200138, -14475911, -33087759, -2696619, -23702521, -9102511,
4071              -23552096, -2287550, 20712163, 6719373},
4072         },
4073         {
4074             {26656208, 6075253, -7858556, 1886072, -28344043, 4262326, 11117530,
4075              -3763210, 26224235, -3297458},
4076             {-17168938, -14854097, -3395676, -16369877, -19954045, 14050420,
4077              21728352, 9493610, 18620611, -16428628},
4078             {-13323321, 13325349, 11432106, 5964811, 18609221, 6062965,
4079              -5269471, -9725556, -30701573, -16479657},
4080         },
4081         {
4082             {-23860538, -11233159, 26961357, 1640861, -32413112, -16737940,
4083              12248509, -5240639, 13735342, 1934062},
4084             {25089769, 6742589, 17081145, -13406266, 21909293, -16067981,
4085              -15136294, -3765346, -21277997, 5473616},
4086             {31883677, -7961101, 1083432, -11572403, 22828471, 13290673,
4087              -7125085, 12469656, 29111212, -5451014},
4088         },
4089         {
4090             {24244947, -15050407, -26262976, 2791540, -14997599, 16666678,
4091              24367466, 6388839, -10295587, 452383},
4092             {-25640782, -3417841, 5217916, 16224624, 19987036, -4082269,
4093              -24236251, -5915248, 15766062, 8407814},
4094             {-20406999, 13990231, 15495425, 16395525, 5377168, 15166495,
4095              -8917023, -4388953, -8067909, 2276718},
4096         },
4097         {
4098             {30157918, 12924066, -17712050, 9245753, 19895028, 3368142,
4099              -23827587, 5096219, 22740376, -7303417},
4100             {2041139, -14256350, 7783687, 13876377, -25946985, -13352459,
4101              24051124, 13742383, -15637599, 13295222},
4102             {33338237, -8505733, 12532113, 7977527, 9106186, -1715251,
4103              -17720195, -4612972, -4451357, -14669444},
4104         },
4105         {
4106             {-20045281, 5454097, -14346548, 6447146, 28862071, 1883651,
4107              -2469266, -4141880, 7770569, 9620597},
4108             {23208068, 7979712, 33071466, 8149229, 1758231, -10834995, 30945528,
4109              -1694323, -33502340, -14767970},
4110             {1439958, -16270480, -1079989, -793782, 4625402, 10647766, -5043801,
4111              1220118, 30494170, -11440799},
4112         },
4113         {
4114             {-5037580, -13028295, -2970559, -3061767, 15640974, -6701666,
4115              -26739026, 926050, -1684339, -13333647},
4116             {13908495, -3549272, 30919928, -6273825, -21521863, 7989039,
4117              9021034, 9078865, 3353509, 4033511},
4118             {-29663431, -15113610, 32259991, -344482, 24295849, -12912123,
4119              23161163, 8839127, 27485041, 7356032},
4120         },
4121     },
4122     {
4123         {
4124             {9661027, 705443, 11980065, -5370154, -1628543, 14661173, -6346142,
4125              2625015, 28431036, -16771834},
4126             {-23839233, -8311415, -25945511, 7480958, -17681669, -8354183,
4127              -22545972, 14150565, 15970762, 4099461},
4128             {29262576, 16756590, 26350592, -8793563, 8529671, -11208050,
4129              13617293, -9937143, 11465739, 8317062},
4130         },
4131         {
4132             {-25493081, -6962928, 32500200, -9419051, -23038724, -2302222,
4133              14898637, 3848455, 20969334, -5157516},
4134             {-20384450, -14347713, -18336405, 13884722, -33039454, 2842114,
4135              -21610826, -3649888, 11177095, 14989547},
4136             {-24496721, -11716016, 16959896, 2278463, 12066309, 10137771,
4137              13515641, 2581286, -28487508, 9930240},
4138         },
4139         {
4140             {-17751622, -2097826, 16544300, -13009300, -15914807, -14949081,
4141              18345767, -13403753, 16291481, -5314038},
4142             {-33229194, 2553288, 32678213, 9875984, 8534129, 6889387, -9676774,
4143              6957617, 4368891, 9788741},
4144             {16660756, 7281060, -10830758, 12911820, 20108584, -8101676,
4145              -21722536, -8613148, 16250552, -11111103},
4146         },
4147         {
4148             {-19765507, 2390526, -16551031, 14161980, 1905286, 6414907, 4689584,
4149              10604807, -30190403, 4782747},
4150             {-1354539, 14736941, -7367442, -13292886, 7710542, -14155590,
4151              -9981571, 4383045, 22546403, 437323},
4152             {31665577, -12180464, -16186830, 1491339, -18368625, 3294682,
4153              27343084, 2786261, -30633590, -14097016},
4154         },
4155         {
4156             {-14467279, -683715, -33374107, 7448552, 19294360, 14334329,
4157              -19690631, 2355319, -19284671, -6114373},
4158             {15121312, -15796162, 6377020, -6031361, -10798111, -12957845,
4159              18952177, 15496498, -29380133, 11754228},
4160             {-2637277, -13483075, 8488727, -14303896, 12728761, -1622493,
4161              7141596, 11724556, 22761615, -10134141},
4162         },
4163         {
4164             {16918416, 11729663, -18083579, 3022987, -31015732, -13339659,
4165              -28741185, -12227393, 32851222, 11717399},
4166             {11166634, 7338049, -6722523, 4531520, -29468672, -7302055,
4167              31474879, 3483633, -1193175, -4030831},
4168             {-185635, 9921305, 31456609, -13536438, -12013818, 13348923,
4169              33142652, 6546660, -19985279, -3948376},
4170         },
4171         {
4172             {-32460596, 11266712, -11197107, -7899103, 31703694, 3855903,
4173              -8537131, -12833048, -30772034, -15486313},
4174             {-18006477, 12709068, 3991746, -6479188, -21491523, -10550425,
4175              -31135347, -16049879, 10928917, 3011958},
4176             {-6957757, -15594337, 31696059, 334240, 29576716, 14796075,
4177              -30831056, -12805180, 18008031, 10258577},
4178         },
4179         {
4180             {-22448644, 15655569, 7018479, -4410003, -30314266, -1201591,
4181              -1853465, 1367120, 25127874, 6671743},
4182             {29701166, -14373934, -10878120, 9279288, -17568, 13127210,
4183              21382910, 11042292, 25838796, 4642684},
4184             {-20430234, 14955537, -24126347, 8124619, -5369288, -5990470,
4185              30468147, -13900640, 18423289, 4177476},
4186         },
4187     },
4188 };
4189 
negative(signed char b)4190 static uint8_t negative(signed char b)
4191 {
4192     uint32_t x = b;
4193 
4194     x >>= 31; /* 1: yes; 0: no */
4195     return x;
4196 }
4197 
table_select(ge_precomp * t,int pos,signed char b)4198 static void table_select(ge_precomp *t, int pos, signed char b)
4199 {
4200     ge_precomp minust;
4201     uint8_t bnegative = negative(b);
4202     uint8_t babs = b - ((uint8_t)((-bnegative) & b) << 1);
4203 
4204     ge_precomp_0(t);
4205     cmov(t, &k25519Precomp[pos][0], equal(babs, 1));
4206     cmov(t, &k25519Precomp[pos][1], equal(babs, 2));
4207     cmov(t, &k25519Precomp[pos][2], equal(babs, 3));
4208     cmov(t, &k25519Precomp[pos][3], equal(babs, 4));
4209     cmov(t, &k25519Precomp[pos][4], equal(babs, 5));
4210     cmov(t, &k25519Precomp[pos][5], equal(babs, 6));
4211     cmov(t, &k25519Precomp[pos][6], equal(babs, 7));
4212     cmov(t, &k25519Precomp[pos][7], equal(babs, 8));
4213     fe_copy(minust.yplusx, t->yminusx);
4214     fe_copy(minust.yminusx, t->yplusx);
4215     fe_neg(minust.xy2d, t->xy2d);
4216     cmov(t, &minust, bnegative);
4217 }
4218 
4219 /*
4220  * h = a * B
4221  *
4222  * where a = a[0]+256*a[1]+...+256^31 a[31]
4223  * B is the Ed25519 base point (x,4/5) with x positive.
4224  *
4225  * Preconditions:
4226  *   a[31] <= 127
4227  */
ge_scalarmult_base(ge_p3 * h,const uint8_t * a)4228 static void ge_scalarmult_base(ge_p3 *h, const uint8_t *a)
4229 {
4230     signed char e[64];
4231     signed char carry;
4232     ge_p1p1 r;
4233     ge_p2 s;
4234     ge_precomp t;
4235     int i;
4236 
4237     for (i = 0; i < 32; ++i) {
4238         e[2 * i + 0] = (a[i] >> 0) & 15;
4239         e[2 * i + 1] = (a[i] >> 4) & 15;
4240     }
4241     /* each e[i] is between 0 and 15 */
4242     /* e[63] is between 0 and 7 */
4243 
4244     carry = 0;
4245     for (i = 0; i < 63; ++i) {
4246         e[i] += carry;
4247         carry = e[i] + 8;
4248         carry >>= 4;
4249         e[i] -= carry << 4;
4250     }
4251     e[63] += carry;
4252     /* each e[i] is between -8 and 8 */
4253 
4254     ge_p3_0(h);
4255     for (i = 1; i < 64; i += 2) {
4256         table_select(&t, i / 2, e[i]);
4257         ge_madd(&r, h, &t);
4258         ge_p1p1_to_p3(h, &r);
4259     }
4260 
4261     ge_p3_dbl(&r, h);
4262     ge_p1p1_to_p2(&s, &r);
4263     ge_p2_dbl(&r, &s);
4264     ge_p1p1_to_p2(&s, &r);
4265     ge_p2_dbl(&r, &s);
4266     ge_p1p1_to_p2(&s, &r);
4267     ge_p2_dbl(&r, &s);
4268     ge_p1p1_to_p3(h, &r);
4269 
4270     for (i = 0; i < 64; i += 2) {
4271         table_select(&t, i / 2, e[i]);
4272         ge_madd(&r, h, &t);
4273         ge_p1p1_to_p3(h, &r);
4274     }
4275 
4276     OPENSSL_cleanse(e, sizeof(e));
4277 }
4278 
4279 #if !defined(BASE_2_51_IMPLEMENTED)
4280 /*
4281  * Replace (f,g) with (g,f) if b == 1;
4282  * replace (f,g) with (f,g) if b == 0.
4283  *
4284  * Preconditions: b in {0,1}.
4285  */
fe_cswap(fe f,fe g,unsigned int b)4286 static void fe_cswap(fe f, fe g, unsigned int b)
4287 {
4288     size_t i;
4289 
4290     b = 0-b;
4291     for (i = 0; i < 10; i++) {
4292         int32_t x = f[i] ^ g[i];
4293         x &= b;
4294         f[i] ^= x;
4295         g[i] ^= x;
4296     }
4297 }
4298 
4299 /*
4300  * h = f * 121666
4301  *
4302  * Can overlap h with f.
4303  *
4304  * Preconditions:
4305  *    |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
4306  *
4307  * Postconditions:
4308  *    |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
4309  */
fe_mul121666(fe h,fe f)4310 static void fe_mul121666(fe h, fe f)
4311 {
4312     int32_t f0 = f[0];
4313     int32_t f1 = f[1];
4314     int32_t f2 = f[2];
4315     int32_t f3 = f[3];
4316     int32_t f4 = f[4];
4317     int32_t f5 = f[5];
4318     int32_t f6 = f[6];
4319     int32_t f7 = f[7];
4320     int32_t f8 = f[8];
4321     int32_t f9 = f[9];
4322     int64_t h0 = f0 * (int64_t) 121666;
4323     int64_t h1 = f1 * (int64_t) 121666;
4324     int64_t h2 = f2 * (int64_t) 121666;
4325     int64_t h3 = f3 * (int64_t) 121666;
4326     int64_t h4 = f4 * (int64_t) 121666;
4327     int64_t h5 = f5 * (int64_t) 121666;
4328     int64_t h6 = f6 * (int64_t) 121666;
4329     int64_t h7 = f7 * (int64_t) 121666;
4330     int64_t h8 = f8 * (int64_t) 121666;
4331     int64_t h9 = f9 * (int64_t) 121666;
4332     int64_t carry0;
4333     int64_t carry1;
4334     int64_t carry2;
4335     int64_t carry3;
4336     int64_t carry4;
4337     int64_t carry5;
4338     int64_t carry6;
4339     int64_t carry7;
4340     int64_t carry8;
4341     int64_t carry9;
4342 
4343     carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
4344     carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
4345     carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
4346     carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
4347     carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;
4348 
4349     carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
4350     carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
4351     carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
4352     carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
4353     carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;
4354 
4355     h[0] = (int32_t)h0;
4356     h[1] = (int32_t)h1;
4357     h[2] = (int32_t)h2;
4358     h[3] = (int32_t)h3;
4359     h[4] = (int32_t)h4;
4360     h[5] = (int32_t)h5;
4361     h[6] = (int32_t)h6;
4362     h[7] = (int32_t)h7;
4363     h[8] = (int32_t)h8;
4364     h[9] = (int32_t)h9;
4365 }
4366 
x25519_scalar_mult_generic(uint8_t out[32],const uint8_t scalar[32],const uint8_t point[32])4367 static void x25519_scalar_mult_generic(uint8_t out[32],
4368                                        const uint8_t scalar[32],
4369                                        const uint8_t point[32]) {
4370     fe x1, x2, z2, x3, z3, tmp0, tmp1;
4371     uint8_t e[32];
4372     unsigned swap = 0;
4373     int pos;
4374 
4375     memcpy(e, scalar, 32);
4376     e[0] &= 248;
4377     e[31] &= 127;
4378     e[31] |= 64;
4379     fe_frombytes(x1, point);
4380     fe_1(x2);
4381     fe_0(z2);
4382     fe_copy(x3, x1);
4383     fe_1(z3);
4384 
4385     for (pos = 254; pos >= 0; --pos) {
4386         unsigned b = 1 & (e[pos / 8] >> (pos & 7));
4387         swap ^= b;
4388         fe_cswap(x2, x3, swap);
4389         fe_cswap(z2, z3, swap);
4390         swap = b;
4391         fe_sub(tmp0, x3, z3);
4392         fe_sub(tmp1, x2, z2);
4393         fe_add(x2, x2, z2);
4394         fe_add(z2, x3, z3);
4395         fe_mul(z3, tmp0, x2);
4396         fe_mul(z2, z2, tmp1);
4397         fe_sq(tmp0, tmp1);
4398         fe_sq(tmp1, x2);
4399         fe_add(x3, z3, z2);
4400         fe_sub(z2, z3, z2);
4401         fe_mul(x2, tmp1, tmp0);
4402         fe_sub(tmp1, tmp1, tmp0);
4403         fe_sq(z2, z2);
4404         fe_mul121666(z3, tmp1);
4405         fe_sq(x3, x3);
4406         fe_add(tmp0, tmp0, z3);
4407         fe_mul(z3, x1, z2);
4408         fe_mul(z2, tmp1, tmp0);
4409     }
4410 
4411     fe_invert(z2, z2);
4412     fe_mul(x2, x2, z2);
4413     fe_tobytes(out, x2);
4414 
4415     OPENSSL_cleanse(e, sizeof(e));
4416 }
4417 
x25519_scalar_mult(uint8_t out[32],const uint8_t scalar[32],const uint8_t point[32])4418 static void x25519_scalar_mult(uint8_t out[32], const uint8_t scalar[32],
4419                                const uint8_t point[32]) {
4420     x25519_scalar_mult_generic(out, scalar, point);
4421 }
4422 #endif
4423 
slide(signed char * r,const uint8_t * a)4424 static void slide(signed char *r, const uint8_t *a)
4425 {
4426     int i;
4427     int b;
4428     int k;
4429 
4430     for (i = 0; i < 256; ++i) {
4431         r[i] = 1 & (a[i >> 3] >> (i & 7));
4432     }
4433 
4434     for (i = 0; i < 256; ++i) {
4435         if (r[i]) {
4436             for (b = 1; b <= 6 && i + b < 256; ++b) {
4437                 if (r[i + b]) {
4438                     if (r[i] + (r[i + b] << b) <= 15) {
4439                         r[i] += r[i + b] << b;
4440                         r[i + b] = 0;
4441                     } else if (r[i] - (r[i + b] << b) >= -15) {
4442                         r[i] -= r[i + b] << b;
4443                         for (k = i + b; k < 256; ++k) {
4444                             if (!r[k]) {
4445                                 r[k] = 1;
4446                                 break;
4447                             }
4448                             r[k] = 0;
4449                         }
4450                     } else {
4451                         break;
4452                     }
4453                 }
4454             }
4455         }
4456     }
4457 }
4458 
4459 static const ge_precomp Bi[8] = {
4460     {
4461         {25967493, -14356035, 29566456, 3660896, -12694345, 4014787, 27544626,
4462          -11754271, -6079156, 2047605},
4463         {-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692,
4464          5043384, 19500929, -15469378},
4465         {-8738181, 4489570, 9688441, -14785194, 10184609, -12363380, 29287919,
4466          11864899, -24514362, -4438546},
4467     },
4468     {
4469         {15636291, -9688557, 24204773, -7912398, 616977, -16685262, 27787600,
4470          -14772189, 28944400, -1550024},
4471         {16568933, 4717097, -11556148, -1102322, 15682896, -11807043, 16354577,
4472          -11775962, 7689662, 11199574},
4473         {30464156, -5976125, -11779434, -15670865, 23220365, 15915852, 7512774,
4474          10017326, -17749093, -9920357},
4475     },
4476     {
4477         {10861363, 11473154, 27284546, 1981175, -30064349, 12577861, 32867885,
4478          14515107, -15438304, 10819380},
4479         {4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668,
4480          12483688, -12668491, 5581306},
4481         {19563160, 16186464, -29386857, 4097519, 10237984, -4348115, 28542350,
4482          13850243, -23678021, -15815942},
4483     },
4484     {
4485         {5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852,
4486          5230134, -23952439, -15175766},
4487         {-30269007, -3463509, 7665486, 10083793, 28475525, 1649722, 20654025,
4488          16520125, 30598449, 7715701},
4489         {28881845, 14381568, 9657904, 3680757, -20181635, 7843316, -31400660,
4490          1370708, 29794553, -1409300},
4491     },
4492     {
4493         {-22518993, -6692182, 14201702, -8745502, -23510406, 8844726, 18474211,
4494          -1361450, -13062696, 13821877},
4495         {-6455177, -7839871, 3374702, -4740862, -27098617, -10571707, 31655028,
4496          -7212327, 18853322, -14220951},
4497         {4566830, -12963868, -28974889, -12240689, -7602672, -2830569, -8514358,
4498          -10431137, 2207753, -3209784},
4499     },
4500     {
4501         {-25154831, -4185821, 29681144, 7868801, -6854661, -9423865, -12437364,
4502          -663000, -31111463, -16132436},
4503         {25576264, -2703214, 7349804, -11814844, 16472782, 9300885, 3844789,
4504          15725684, 171356, 6466918},
4505         {23103977, 13316479, 9739013, -16149481, 817875, -15038942, 8965339,
4506          -14088058, -30714912, 16193877},
4507     },
4508     {
4509         {-33521811, 3180713, -2394130, 14003687, -16903474, -16270840, 17238398,
4510          4729455, -18074513, 9256800},
4511         {-25182317, -4174131, 32336398, 5036987, -21236817, 11360617, 22616405,
4512          9761698, -19827198, 630305},
4513         {-13720693, 2639453, -24237460, -7406481, 9494427, -5774029, -6554551,
4514          -15960994, -2449256, -14291300},
4515     },
4516     {
4517         {-3151181, -5046075, 9282714, 6866145, -31907062, -863023, -18940575,
4518          15033784, 25105118, -7894876},
4519         {-24326370, 15950226, -31801215, -14592823, -11662737, -5090925,
4520          1573892, -2625887, 2198790, -15804619},
4521         {-3099351, 10324967, -2241613, 7453183, -5446979, -2735503, -13812022,
4522          -16236442, -32461234, -12290683},
4523     },
4524 };
4525 
4526 /*
4527  * r = a * A + b * B
4528  *
4529  * where a = a[0]+256*a[1]+...+256^31 a[31].
4530  * and b = b[0]+256*b[1]+...+256^31 b[31].
4531  * B is the Ed25519 base point (x,4/5) with x positive.
4532  */
ge_double_scalarmult_vartime(ge_p2 * r,const uint8_t * a,const ge_p3 * A,const uint8_t * b)4533 static void ge_double_scalarmult_vartime(ge_p2 *r, const uint8_t *a,
4534                                          const ge_p3 *A, const uint8_t *b)
4535 {
4536     signed char aslide[256];
4537     signed char bslide[256];
4538     ge_cached Ai[8]; /* A,3A,5A,7A,9A,11A,13A,15A */
4539     ge_p1p1 t;
4540     ge_p3 u;
4541     ge_p3 A2;
4542     int i;
4543 
4544     slide(aslide, a);
4545     slide(bslide, b);
4546 
4547     ge_p3_to_cached(&Ai[0], A);
4548     ge_p3_dbl(&t, A);
4549     ge_p1p1_to_p3(&A2, &t);
4550     ge_add(&t, &A2, &Ai[0]);
4551     ge_p1p1_to_p3(&u, &t);
4552     ge_p3_to_cached(&Ai[1], &u);
4553     ge_add(&t, &A2, &Ai[1]);
4554     ge_p1p1_to_p3(&u, &t);
4555     ge_p3_to_cached(&Ai[2], &u);
4556     ge_add(&t, &A2, &Ai[2]);
4557     ge_p1p1_to_p3(&u, &t);
4558     ge_p3_to_cached(&Ai[3], &u);
4559     ge_add(&t, &A2, &Ai[3]);
4560     ge_p1p1_to_p3(&u, &t);
4561     ge_p3_to_cached(&Ai[4], &u);
4562     ge_add(&t, &A2, &Ai[4]);
4563     ge_p1p1_to_p3(&u, &t);
4564     ge_p3_to_cached(&Ai[5], &u);
4565     ge_add(&t, &A2, &Ai[5]);
4566     ge_p1p1_to_p3(&u, &t);
4567     ge_p3_to_cached(&Ai[6], &u);
4568     ge_add(&t, &A2, &Ai[6]);
4569     ge_p1p1_to_p3(&u, &t);
4570     ge_p3_to_cached(&Ai[7], &u);
4571 
4572     ge_p2_0(r);
4573 
4574     for (i = 255; i >= 0; --i) {
4575         if (aslide[i] || bslide[i]) {
4576             break;
4577         }
4578     }
4579 
4580     for (; i >= 0; --i) {
4581         ge_p2_dbl(&t, r);
4582 
4583         if (aslide[i] > 0) {
4584             ge_p1p1_to_p3(&u, &t);
4585             ge_add(&t, &u, &Ai[aslide[i] / 2]);
4586         } else if (aslide[i] < 0) {
4587             ge_p1p1_to_p3(&u, &t);
4588             ge_sub(&t, &u, &Ai[(-aslide[i]) / 2]);
4589         }
4590 
4591         if (bslide[i] > 0) {
4592             ge_p1p1_to_p3(&u, &t);
4593             ge_madd(&t, &u, &Bi[bslide[i] / 2]);
4594         } else if (bslide[i] < 0) {
4595             ge_p1p1_to_p3(&u, &t);
4596             ge_msub(&t, &u, &Bi[(-bslide[i]) / 2]);
4597         }
4598 
4599         ge_p1p1_to_p2(r, &t);
4600     }
4601 }
4602 
4603 /*
4604  * The set of scalars is \Z/l
4605  * where l = 2^252 + 27742317777372353535851937790883648493.
4606  *
4607  * Input:
4608  *   s[0]+256*s[1]+...+256^63*s[63] = s
4609  *
4610  * Output:
4611  *   s[0]+256*s[1]+...+256^31*s[31] = s mod l
4612  *   where l = 2^252 + 27742317777372353535851937790883648493.
4613  *   Overwrites s in place.
4614 */
x25519_sc_reduce(uint8_t * s)4615 static void x25519_sc_reduce(uint8_t *s)
4616 {
4617     int64_t s0  = kBottom21Bits &  load_3(s);
4618     int64_t s1  = kBottom21Bits & (load_4(s +  2) >> 5);
4619     int64_t s2  = kBottom21Bits & (load_3(s +  5) >> 2);
4620     int64_t s3  = kBottom21Bits & (load_4(s +  7) >> 7);
4621     int64_t s4  = kBottom21Bits & (load_4(s + 10) >> 4);
4622     int64_t s5  = kBottom21Bits & (load_3(s + 13) >> 1);
4623     int64_t s6  = kBottom21Bits & (load_4(s + 15) >> 6);
4624     int64_t s7  = kBottom21Bits & (load_3(s + 18) >> 3);
4625     int64_t s8  = kBottom21Bits &  load_3(s + 21);
4626     int64_t s9  = kBottom21Bits & (load_4(s + 23) >> 5);
4627     int64_t s10 = kBottom21Bits & (load_3(s + 26) >> 2);
4628     int64_t s11 = kBottom21Bits & (load_4(s + 28) >> 7);
4629     int64_t s12 = kBottom21Bits & (load_4(s + 31) >> 4);
4630     int64_t s13 = kBottom21Bits & (load_3(s + 34) >> 1);
4631     int64_t s14 = kBottom21Bits & (load_4(s + 36) >> 6);
4632     int64_t s15 = kBottom21Bits & (load_3(s + 39) >> 3);
4633     int64_t s16 = kBottom21Bits &  load_3(s + 42);
4634     int64_t s17 = kBottom21Bits & (load_4(s + 44) >> 5);
4635     int64_t s18 = kBottom21Bits & (load_3(s + 47) >> 2);
4636     int64_t s19 = kBottom21Bits & (load_4(s + 49) >> 7);
4637     int64_t s20 = kBottom21Bits & (load_4(s + 52) >> 4);
4638     int64_t s21 = kBottom21Bits & (load_3(s + 55) >> 1);
4639     int64_t s22 = kBottom21Bits & (load_4(s + 57) >> 6);
4640     int64_t s23 =                 (load_4(s + 60) >> 3);
4641     int64_t carry0;
4642     int64_t carry1;
4643     int64_t carry2;
4644     int64_t carry3;
4645     int64_t carry4;
4646     int64_t carry5;
4647     int64_t carry6;
4648     int64_t carry7;
4649     int64_t carry8;
4650     int64_t carry9;
4651     int64_t carry10;
4652     int64_t carry11;
4653     int64_t carry12;
4654     int64_t carry13;
4655     int64_t carry14;
4656     int64_t carry15;
4657     int64_t carry16;
4658 
4659     s11 += s23 * 666643;
4660     s12 += s23 * 470296;
4661     s13 += s23 * 654183;
4662     s14 -= s23 * 997805;
4663     s15 += s23 * 136657;
4664     s16 -= s23 * 683901;
4665     s23  = 0;
4666 
4667     s10 += s22 * 666643;
4668     s11 += s22 * 470296;
4669     s12 += s22 * 654183;
4670     s13 -= s22 * 997805;
4671     s14 += s22 * 136657;
4672     s15 -= s22 * 683901;
4673     s22  = 0;
4674 
4675     s9  += s21 * 666643;
4676     s10 += s21 * 470296;
4677     s11 += s21 * 654183;
4678     s12 -= s21 * 997805;
4679     s13 += s21 * 136657;
4680     s14 -= s21 * 683901;
4681     s21  = 0;
4682 
4683     s8  += s20 * 666643;
4684     s9  += s20 * 470296;
4685     s10 += s20 * 654183;
4686     s11 -= s20 * 997805;
4687     s12 += s20 * 136657;
4688     s13 -= s20 * 683901;
4689     s20  = 0;
4690 
4691     s7  += s19 * 666643;
4692     s8  += s19 * 470296;
4693     s9  += s19 * 654183;
4694     s10 -= s19 * 997805;
4695     s11 += s19 * 136657;
4696     s12 -= s19 * 683901;
4697     s19  = 0;
4698 
4699     s6  += s18 * 666643;
4700     s7  += s18 * 470296;
4701     s8  += s18 * 654183;
4702     s9  -= s18 * 997805;
4703     s10 += s18 * 136657;
4704     s11 -= s18 * 683901;
4705     s18  = 0;
4706 
4707     carry6 = (s6 + (1 << 20)) >> 21;
4708     s7  += carry6;
4709     s6  -= carry6 * (1 << 21);
4710     carry8 = (s8 + (1 << 20)) >> 21;
4711     s9  += carry8;
4712     s8  -= carry8 * (1 << 21);
4713     carry10 = (s10 + (1 << 20)) >> 21;
4714     s11 += carry10;
4715     s10 -= carry10 * (1 << 21);
4716     carry12 = (s12 + (1 << 20)) >> 21;
4717     s13 += carry12;
4718     s12 -= carry12 * (1 << 21);
4719     carry14 = (s14 + (1 << 20)) >> 21;
4720     s15 += carry14;
4721     s14 -= carry14 * (1 << 21);
4722     carry16 = (s16 + (1 << 20)) >> 21;
4723     s17 += carry16;
4724     s16 -= carry16 * (1 << 21);
4725 
4726     carry7 = (s7 + (1 << 20)) >> 21;
4727     s8  += carry7;
4728     s7  -= carry7 * (1 << 21);
4729     carry9 = (s9 + (1 << 20)) >> 21;
4730     s10 += carry9;
4731     s9  -= carry9 * (1 << 21);
4732     carry11 = (s11 + (1 << 20)) >> 21;
4733     s12 += carry11;
4734     s11 -= carry11 * (1 << 21);
4735     carry13 = (s13 + (1 << 20)) >> 21;
4736     s14 += carry13;
4737     s13 -= carry13 * (1 << 21);
4738     carry15 = (s15 + (1 << 20)) >> 21;
4739     s16 += carry15;
4740     s15 -= carry15 * (1 << 21);
4741 
4742     s5  += s17 * 666643;
4743     s6  += s17 * 470296;
4744     s7  += s17 * 654183;
4745     s8  -= s17 * 997805;
4746     s9  += s17 * 136657;
4747     s10 -= s17 * 683901;
4748     s17  = 0;
4749 
4750     s4  += s16 * 666643;
4751     s5  += s16 * 470296;
4752     s6  += s16 * 654183;
4753     s7  -= s16 * 997805;
4754     s8  += s16 * 136657;
4755     s9  -= s16 * 683901;
4756     s16  = 0;
4757 
4758     s3  += s15 * 666643;
4759     s4  += s15 * 470296;
4760     s5  += s15 * 654183;
4761     s6  -= s15 * 997805;
4762     s7  += s15 * 136657;
4763     s8  -= s15 * 683901;
4764     s15  = 0;
4765 
4766     s2  += s14 * 666643;
4767     s3  += s14 * 470296;
4768     s4  += s14 * 654183;
4769     s5  -= s14 * 997805;
4770     s6  += s14 * 136657;
4771     s7  -= s14 * 683901;
4772     s14  = 0;
4773 
4774     s1  += s13 * 666643;
4775     s2  += s13 * 470296;
4776     s3  += s13 * 654183;
4777     s4  -= s13 * 997805;
4778     s5  += s13 * 136657;
4779     s6  -= s13 * 683901;
4780     s13  = 0;
4781 
4782     s0  += s12 * 666643;
4783     s1  += s12 * 470296;
4784     s2  += s12 * 654183;
4785     s3  -= s12 * 997805;
4786     s4  += s12 * 136657;
4787     s5  -= s12 * 683901;
4788     s12  = 0;
4789 
4790     carry0 = (s0 + (1 << 20)) >> 21;
4791     s1  += carry0;
4792     s0  -= carry0 * (1 << 21);
4793     carry2 = (s2 + (1 << 20)) >> 21;
4794     s3  += carry2;
4795     s2  -= carry2 * (1 << 21);
4796     carry4 = (s4 + (1 << 20)) >> 21;
4797     s5  += carry4;
4798     s4  -= carry4 * (1 << 21);
4799     carry6 = (s6 + (1 << 20)) >> 21;
4800     s7 += carry6;
4801     s6 -= carry6 * (1 << 21);
4802     carry8 = (s8 + (1 << 20)) >> 21;
4803     s9  += carry8;
4804     s8  -= carry8 * (1 << 21);
4805     carry10 = (s10 + (1 << 20)) >> 21;
4806     s11 += carry10;
4807     s10 -= carry10 * (1 << 21);
4808 
4809     carry1 = (s1 + (1 << 20)) >> 21;
4810     s2  += carry1;
4811     s1  -= carry1 * (1 << 21);
4812     carry3 = (s3 + (1 << 20)) >> 21;
4813     s4  += carry3;
4814     s3  -= carry3 * (1 << 21);
4815     carry5 = (s5 + (1 << 20)) >> 21;
4816     s6  += carry5;
4817     s5  -= carry5 * (1 << 21);
4818     carry7 = (s7 + (1 << 20)) >> 21;
4819     s8  += carry7;
4820     s7  -= carry7 * (1 << 21);
4821     carry9 = (s9 + (1 << 20)) >> 21;
4822     s10 += carry9;
4823     s9  -= carry9 * (1 << 21);
4824     carry11 = (s11 + (1 << 20)) >> 21;
4825     s12 += carry11;
4826     s11 -= carry11 * (1 << 21);
4827 
4828     s0  += s12 * 666643;
4829     s1  += s12 * 470296;
4830     s2  += s12 * 654183;
4831     s3  -= s12 * 997805;
4832     s4  += s12 * 136657;
4833     s5  -= s12 * 683901;
4834     s12  = 0;
4835 
4836     carry0 = s0 >> 21;
4837     s1  += carry0;
4838     s0  -= carry0 * (1 << 21);
4839     carry1 = s1 >> 21;
4840     s2  += carry1;
4841     s1  -= carry1 * (1 << 21);
4842     carry2 = s2 >> 21;
4843     s3  += carry2;
4844     s2  -= carry2 * (1 << 21);
4845     carry3 = s3 >> 21;
4846     s4  += carry3;
4847     s3  -= carry3 * (1 << 21);
4848     carry4 = s4 >> 21;
4849     s5  += carry4;
4850     s4  -= carry4 * (1 << 21);
4851     carry5 = s5 >> 21;
4852     s6  += carry5;
4853     s5  -= carry5 * (1 << 21);
4854     carry6 = s6 >> 21;
4855     s7  += carry6;
4856     s6  -= carry6 * (1 << 21);
4857     carry7 = s7 >> 21;
4858     s8  += carry7;
4859     s7  -= carry7 * (1 << 21);
4860     carry8 = s8 >> 21;
4861     s9  += carry8;
4862     s8  -= carry8 * (1 << 21);
4863     carry9 = s9 >> 21;
4864     s10 += carry9;
4865     s9  -= carry9 * (1 << 21);
4866     carry10 = s10 >> 21;
4867     s11 += carry10;
4868     s10 -= carry10 * (1 << 21);
4869     carry11 = s11 >> 21;
4870     s12 += carry11;
4871     s11 -= carry11 * (1 << 21);
4872 
4873     s0  += s12 * 666643;
4874     s1  += s12 * 470296;
4875     s2  += s12 * 654183;
4876     s3  -= s12 * 997805;
4877     s4  += s12 * 136657;
4878     s5  -= s12 * 683901;
4879     s12  = 0;
4880 
4881     carry0 = s0 >> 21;
4882     s1  += carry0;
4883     s0  -= carry0 * (1 << 21);
4884     carry1 = s1 >> 21;
4885     s2  += carry1;
4886     s1  -= carry1 * (1 << 21);
4887     carry2 = s2 >> 21;
4888     s3  += carry2;
4889     s2  -= carry2 * (1 << 21);
4890     carry3 = s3 >> 21;
4891     s4  += carry3;
4892     s3  -= carry3 * (1 << 21);
4893     carry4 = s4 >> 21;
4894     s5  += carry4;
4895     s4  -= carry4 * (1 << 21);
4896     carry5 = s5 >> 21;
4897     s6  += carry5;
4898     s5  -= carry5 * (1 << 21);
4899     carry6 = s6 >> 21;
4900     s7  += carry6;
4901     s6  -= carry6 * (1 << 21);
4902     carry7 = s7 >> 21;
4903     s8  += carry7;
4904     s7  -= carry7 * (1 << 21);
4905     carry8 = s8 >> 21;
4906     s9  += carry8;
4907     s8  -= carry8 * (1 << 21);
4908     carry9 = s9 >> 21;
4909     s10 += carry9;
4910     s9  -= carry9 * (1 << 21);
4911     carry10 = s10 >> 21;
4912     s11 += carry10;
4913     s10 -= carry10 * (1 << 21);
4914 
4915     s[ 0] = (uint8_t) (s0  >>  0);
4916     s[ 1] = (uint8_t) (s0  >>  8);
4917     s[ 2] = (uint8_t)((s0  >> 16) | (s1  <<  5));
4918     s[ 3] = (uint8_t) (s1  >>  3);
4919     s[ 4] = (uint8_t) (s1  >> 11);
4920     s[ 5] = (uint8_t)((s1  >> 19) | (s2  <<  2));
4921     s[ 6] = (uint8_t) (s2  >>  6);
4922     s[ 7] = (uint8_t)((s2  >> 14) | (s3  <<  7));
4923     s[ 8] = (uint8_t) (s3  >>  1);
4924     s[ 9] = (uint8_t) (s3  >>  9);
4925     s[10] = (uint8_t)((s3  >> 17) | (s4  <<  4));
4926     s[11] = (uint8_t) (s4  >>  4);
4927     s[12] = (uint8_t) (s4  >> 12);
4928     s[13] = (uint8_t)((s4  >> 20) | (s5  <<  1));
4929     s[14] = (uint8_t) (s5  >>  7);
4930     s[15] = (uint8_t)((s5  >> 15) | (s6  <<  6));
4931     s[16] = (uint8_t) (s6  >>  2);
4932     s[17] = (uint8_t) (s6  >> 10);
4933     s[18] = (uint8_t)((s6  >> 18) | (s7  <<  3));
4934     s[19] = (uint8_t) (s7  >>  5);
4935     s[20] = (uint8_t) (s7  >> 13);
4936     s[21] = (uint8_t) (s8  >>  0);
4937     s[22] = (uint8_t) (s8  >>  8);
4938     s[23] = (uint8_t)((s8  >> 16) | (s9  <<  5));
4939     s[24] = (uint8_t) (s9  >>  3);
4940     s[25] = (uint8_t) (s9  >> 11);
4941     s[26] = (uint8_t)((s9  >> 19) | (s10 <<  2));
4942     s[27] = (uint8_t) (s10 >>  6);
4943     s[28] = (uint8_t)((s10 >> 14) | (s11 <<  7));
4944     s[29] = (uint8_t) (s11 >>  1);
4945     s[30] = (uint8_t) (s11 >>  9);
4946     s[31] = (uint8_t) (s11 >> 17);
4947 }
4948 
4949 /*
4950  * Input:
4951  *   a[0]+256*a[1]+...+256^31*a[31] = a
4952  *   b[0]+256*b[1]+...+256^31*b[31] = b
4953  *   c[0]+256*c[1]+...+256^31*c[31] = c
4954  *
4955  * Output:
4956  *   s[0]+256*s[1]+...+256^31*s[31] = (ab+c) mod l
4957  *   where l = 2^252 + 27742317777372353535851937790883648493.
4958  */
sc_muladd(uint8_t * s,const uint8_t * a,const uint8_t * b,const uint8_t * c)4959 static void sc_muladd(uint8_t *s, const uint8_t *a, const uint8_t *b,
4960                       const uint8_t *c)
4961 {
4962     int64_t a0  = kBottom21Bits &  load_3(a);
4963     int64_t a1  = kBottom21Bits & (load_4(a +  2) >> 5);
4964     int64_t a2  = kBottom21Bits & (load_3(a +  5) >> 2);
4965     int64_t a3  = kBottom21Bits & (load_4(a +  7) >> 7);
4966     int64_t a4  = kBottom21Bits & (load_4(a + 10) >> 4);
4967     int64_t a5  = kBottom21Bits & (load_3(a + 13) >> 1);
4968     int64_t a6  = kBottom21Bits & (load_4(a + 15) >> 6);
4969     int64_t a7  = kBottom21Bits & (load_3(a + 18) >> 3);
4970     int64_t a8  = kBottom21Bits &  load_3(a + 21);
4971     int64_t a9  = kBottom21Bits & (load_4(a + 23) >> 5);
4972     int64_t a10 = kBottom21Bits & (load_3(a + 26) >> 2);
4973     int64_t a11 =                 (load_4(a + 28) >> 7);
4974     int64_t b0  = kBottom21Bits &  load_3(b);
4975     int64_t b1  = kBottom21Bits & (load_4(b +  2) >> 5);
4976     int64_t b2  = kBottom21Bits & (load_3(b +  5) >> 2);
4977     int64_t b3  = kBottom21Bits & (load_4(b +  7) >> 7);
4978     int64_t b4  = kBottom21Bits & (load_4(b + 10) >> 4);
4979     int64_t b5  = kBottom21Bits & (load_3(b + 13) >> 1);
4980     int64_t b6  = kBottom21Bits & (load_4(b + 15) >> 6);
4981     int64_t b7  = kBottom21Bits & (load_3(b + 18) >> 3);
4982     int64_t b8  = kBottom21Bits &  load_3(b + 21);
4983     int64_t b9  = kBottom21Bits & (load_4(b + 23) >> 5);
4984     int64_t b10 = kBottom21Bits & (load_3(b + 26) >> 2);
4985     int64_t b11 =                 (load_4(b + 28) >> 7);
4986     int64_t c0  = kBottom21Bits &  load_3(c);
4987     int64_t c1  = kBottom21Bits & (load_4(c +  2) >> 5);
4988     int64_t c2  = kBottom21Bits & (load_3(c +  5) >> 2);
4989     int64_t c3  = kBottom21Bits & (load_4(c +  7) >> 7);
4990     int64_t c4  = kBottom21Bits & (load_4(c + 10) >> 4);
4991     int64_t c5  = kBottom21Bits & (load_3(c + 13) >> 1);
4992     int64_t c6  = kBottom21Bits & (load_4(c + 15) >> 6);
4993     int64_t c7  = kBottom21Bits & (load_3(c + 18) >> 3);
4994     int64_t c8  = kBottom21Bits &  load_3(c + 21);
4995     int64_t c9  = kBottom21Bits & (load_4(c + 23) >> 5);
4996     int64_t c10 = kBottom21Bits & (load_3(c + 26) >> 2);
4997     int64_t c11 =                 (load_4(c + 28) >> 7);
4998     int64_t s0;
4999     int64_t s1;
5000     int64_t s2;
5001     int64_t s3;
5002     int64_t s4;
5003     int64_t s5;
5004     int64_t s6;
5005     int64_t s7;
5006     int64_t s8;
5007     int64_t s9;
5008     int64_t s10;
5009     int64_t s11;
5010     int64_t s12;
5011     int64_t s13;
5012     int64_t s14;
5013     int64_t s15;
5014     int64_t s16;
5015     int64_t s17;
5016     int64_t s18;
5017     int64_t s19;
5018     int64_t s20;
5019     int64_t s21;
5020     int64_t s22;
5021     int64_t s23;
5022     int64_t carry0;
5023     int64_t carry1;
5024     int64_t carry2;
5025     int64_t carry3;
5026     int64_t carry4;
5027     int64_t carry5;
5028     int64_t carry6;
5029     int64_t carry7;
5030     int64_t carry8;
5031     int64_t carry9;
5032     int64_t carry10;
5033     int64_t carry11;
5034     int64_t carry12;
5035     int64_t carry13;
5036     int64_t carry14;
5037     int64_t carry15;
5038     int64_t carry16;
5039     int64_t carry17;
5040     int64_t carry18;
5041     int64_t carry19;
5042     int64_t carry20;
5043     int64_t carry21;
5044     int64_t carry22;
5045 
5046     s0  = c0   +   a0 * b0;
5047     s1  = c1   +   a0 * b1   +   a1 * b0;
5048     s2  = c2   +   a0 * b2   +   a1 * b1   +   a2 * b0;
5049     s3  = c3   +   a0 * b3   +   a1 * b2   +   a2 * b1  +   a3 * b0;
5050     s4  = c4   +   a0 * b4   +   a1 * b3   +   a2 * b2  +   a3 * b1  +   a4 * b0;
5051     s5  = c5   +   a0 * b5   +   a1 * b4   +   a2 * b3  +   a3 * b2  +   a4 * b1  +   a5 * b0;
5052     s6  = c6   +   a0 * b6   +   a1 * b5   +   a2 * b4  +   a3 * b3  +   a4 * b2  +   a5 * b1 +   a6 * b0;
5053     s7  = c7   +   a0 * b7   +   a1 * b6   +   a2 * b5  +   a3 * b4  +   a4 * b3  +   a5 * b2 +   a6 * b1   +   a7 * b0;
5054     s8  = c8   +   a0 * b8   +   a1 * b7   +   a2 * b6  +   a3 * b5  +   a4 * b4  +   a5 * b3 +   a6 * b2   +   a7 * b1   +   a8 * b0;
5055     s9  = c9   +   a0 * b9   +   a1 * b8   +   a2 * b7  +   a3 * b6  +   a4 * b5  +   a5 * b4 +   a6 * b3   +   a7 * b2   +   a8 * b1  +   a9 * b0;
5056     s10 = c10  +   a0 * b10  +   a1 * b9   +   a2 * b8  +   a3 * b7  +   a4 * b6  +   a5 * b5 +   a6 * b4   +   a7 * b3   +   a8 * b2  +   a9 * b1  +  a10 * b0;
5057     s11 = c11  +   a0 * b11  +   a1 * b10  +   a2 * b9  +   a3 * b8  +   a4 * b7  +   a5 * b6 +   a6 * b5   +   a7 * b4   +   a8 * b3  +   a9 * b2  +  a10 * b1  +  a11 * b0;
5058     s12 =          a1 * b11  +   a2 * b10  +   a3 * b9  +   a4 * b8  +   a5 * b7  +   a6 * b6 +   a7 * b5   +   a8 * b4   +   a9 * b3  +  a10 * b2  +  a11 * b1;
5059     s13 =          a2 * b11  +   a3 * b10  +   a4 * b9  +   a5 * b8  +   a6 * b7  +   a7 * b6 +   a8 * b5   +   a9 * b4   +  a10 * b3  +  a11 * b2;
5060     s14 =          a3 * b11  +   a4 * b10  +   a5 * b9  +   a6 * b8  +   a7 * b7  +   a8 * b6 +   a9 * b5   +  a10 * b4   +  a11 * b3;
5061     s15 =          a4 * b11  +   a5 * b10  +   a6 * b9  +   a7 * b8  +   a8 * b7  +   a9 * b6 +  a10 * b5   +  a11 * b4;
5062     s16 =          a5 * b11  +   a6 * b10  +   a7 * b9  +   a8 * b8  +   a9 * b7  +  a10 * b6 +  a11 * b5;
5063     s17 =          a6 * b11  +   a7 * b10  +   a8 * b9  +   a9 * b8  +  a10 * b7  +  a11 * b6;
5064     s18 =          a7 * b11  +   a8 * b10  +   a9 * b9  +  a10 * b8  +  a11 * b7;
5065     s19 =          a8 * b11  +   a9 * b10  +  a10 * b9  +  a11 * b8;
5066     s20 =          a9 * b11  +  a10 * b10  +  a11 * b9;
5067     s21 =         a10 * b11  +  a11 * b10;
5068     s22 =         a11 * b11;
5069     s23 =         0;
5070 
5071     carry0 = (s0 + (1 << 20)) >> 21;
5072     s1  += carry0;
5073     s0  -= carry0 * (1 << 21);
5074     carry2 = (s2 + (1 << 20)) >> 21;
5075     s3  += carry2;
5076     s2  -= carry2 * (1 << 21);
5077     carry4 = (s4 + (1 << 20)) >> 21;
5078     s5  += carry4;
5079     s4  -= carry4 * (1 << 21);
5080     carry6 = (s6 + (1 << 20)) >> 21;
5081     s7  += carry6;
5082     s6  -= carry6 * (1 << 21);
5083     carry8 = (s8 + (1 << 20)) >> 21;
5084     s9  += carry8;
5085     s8  -= carry8 * (1 << 21);
5086     carry10 = (s10 + (1 << 20)) >> 21;
5087     s11 += carry10;
5088     s10 -= carry10 * (1 << 21);
5089     carry12 = (s12 + (1 << 20)) >> 21;
5090     s13 += carry12;
5091     s12 -= carry12 * (1 << 21);
5092     carry14 = (s14 + (1 << 20)) >> 21;
5093     s15 += carry14;
5094     s14 -= carry14 * (1 << 21);
5095     carry16 = (s16 + (1 << 20)) >> 21;
5096     s17 += carry16;
5097     s16 -= carry16 * (1 << 21);
5098     carry18 = (s18 + (1 << 20)) >> 21;
5099     s19 += carry18;
5100     s18 -= carry18 * (1 << 21);
5101     carry20 = (s20 + (1 << 20)) >> 21;
5102     s21 += carry20;
5103     s20 -= carry20 * (1 << 21);
5104     carry22 = (s22 + (1 << 20)) >> 21;
5105     s23 += carry22;
5106     s22 -= carry22 * (1 << 21);
5107 
5108     carry1 = (s1 + (1 << 20)) >> 21;
5109     s2  += carry1;
5110     s1  -= carry1 * (1 << 21);
5111     carry3 = (s3 + (1 << 20)) >> 21;
5112     s4  += carry3;
5113     s3  -= carry3 * (1 << 21);
5114     carry5 = (s5 + (1 << 20)) >> 21;
5115     s6  += carry5;
5116     s5  -= carry5 * (1 << 21);
5117     carry7 = (s7 + (1 << 20)) >> 21;
5118     s8  += carry7;
5119     s7  -= carry7 * (1 << 21);
5120     carry9 = (s9 + (1 << 20)) >> 21;
5121     s10 += carry9;
5122     s9  -= carry9 * (1 << 21);
5123     carry11 = (s11 + (1 << 20)) >> 21;
5124     s12 += carry11;
5125     s11 -= carry11 * (1 << 21);
5126     carry13 = (s13 + (1 << 20)) >> 21;
5127     s14 += carry13;
5128     s13 -= carry13 * (1 << 21);
5129     carry15 = (s15 + (1 << 20)) >> 21;
5130     s16 += carry15;
5131     s15 -= carry15 * (1 << 21);
5132     carry17 = (s17 + (1 << 20)) >> 21;
5133     s18 += carry17;
5134     s17 -= carry17 * (1 << 21);
5135     carry19 = (s19 + (1 << 20)) >> 21;
5136     s20 += carry19;
5137     s19 -= carry19 * (1 << 21);
5138     carry21 = (s21 + (1 << 20)) >> 21;
5139     s22 += carry21;
5140     s21 -= carry21 * (1 << 21);
5141 
5142     s11 += s23 * 666643;
5143     s12 += s23 * 470296;
5144     s13 += s23 * 654183;
5145     s14 -= s23 * 997805;
5146     s15 += s23 * 136657;
5147     s16 -= s23 * 683901;
5148     s23  = 0;
5149 
5150     s10 += s22 * 666643;
5151     s11 += s22 * 470296;
5152     s12 += s22 * 654183;
5153     s13 -= s22 * 997805;
5154     s14 += s22 * 136657;
5155     s15 -= s22 * 683901;
5156     s22  = 0;
5157 
5158     s9  += s21 * 666643;
5159     s10 += s21 * 470296;
5160     s11 += s21 * 654183;
5161     s12 -= s21 * 997805;
5162     s13 += s21 * 136657;
5163     s14 -= s21 * 683901;
5164     s21  = 0;
5165 
5166     s8  += s20 * 666643;
5167     s9  += s20 * 470296;
5168     s10 += s20 * 654183;
5169     s11 -= s20 * 997805;
5170     s12 += s20 * 136657;
5171     s13 -= s20 * 683901;
5172     s20  = 0;
5173 
5174     s7  += s19 * 666643;
5175     s8  += s19 * 470296;
5176     s9  += s19 * 654183;
5177     s10 -= s19 * 997805;
5178     s11 += s19 * 136657;
5179     s12 -= s19 * 683901;
5180     s19  = 0;
5181 
5182     s6  += s18 * 666643;
5183     s7  += s18 * 470296;
5184     s8  += s18 * 654183;
5185     s9  -= s18 * 997805;
5186     s10 += s18 * 136657;
5187     s11 -= s18 * 683901;
5188     s18  = 0;
5189 
5190     carry6 = (s6 + (1 << 20)) >> 21;
5191     s7  += carry6;
5192     s6  -= carry6 * (1 << 21);
5193     carry8 = (s8 + (1 << 20)) >> 21;
5194     s9  += carry8;
5195     s8  -= carry8 * (1 << 21);
5196     carry10 = (s10 + (1 << 20)) >> 21;
5197     s11 += carry10;
5198     s10 -= carry10 * (1 << 21);
5199     carry12 = (s12 + (1 << 20)) >> 21;
5200     s13 += carry12;
5201     s12 -= carry12 * (1 << 21);
5202     carry14 = (s14 + (1 << 20)) >> 21;
5203     s15 += carry14;
5204     s14 -= carry14 * (1 << 21);
5205     carry16 = (s16 + (1 << 20)) >> 21;
5206     s17 += carry16;
5207     s16 -= carry16 * (1 << 21);
5208 
5209     carry7 = (s7 + (1 << 20)) >> 21;
5210     s8  += carry7;
5211     s7  -= carry7 * (1 << 21);
5212     carry9 = (s9 + (1 << 20)) >> 21;
5213     s10 += carry9;
5214     s9  -= carry9 * (1 << 21);
5215     carry11 = (s11 + (1 << 20)) >> 21;
5216     s12 += carry11;
5217     s11 -= carry11 * (1 << 21);
5218     carry13 = (s13 + (1 << 20)) >> 21;
5219     s14 += carry13;
5220     s13 -= carry13 * (1 << 21);
5221     carry15 = (s15 + (1 << 20)) >> 21;
5222     s16 += carry15;
5223     s15 -= carry15 * (1 << 21);
5224 
5225     s5  += s17 * 666643;
5226     s6  += s17 * 470296;
5227     s7  += s17 * 654183;
5228     s8  -= s17 * 997805;
5229     s9  += s17 * 136657;
5230     s10 -= s17 * 683901;
5231     s17  = 0;
5232 
5233     s4  += s16 * 666643;
5234     s5  += s16 * 470296;
5235     s6  += s16 * 654183;
5236     s7  -= s16 * 997805;
5237     s8  += s16 * 136657;
5238     s9  -= s16 * 683901;
5239     s16  = 0;
5240 
5241     s3  += s15 * 666643;
5242     s4  += s15 * 470296;
5243     s5  += s15 * 654183;
5244     s6  -= s15 * 997805;
5245     s7  += s15 * 136657;
5246     s8  -= s15 * 683901;
5247     s15  = 0;
5248 
5249     s2  += s14 * 666643;
5250     s3  += s14 * 470296;
5251     s4  += s14 * 654183;
5252     s5  -= s14 * 997805;
5253     s6  += s14 * 136657;
5254     s7  -= s14 * 683901;
5255     s14  = 0;
5256 
5257     s1  += s13 * 666643;
5258     s2  += s13 * 470296;
5259     s3  += s13 * 654183;
5260     s4  -= s13 * 997805;
5261     s5  += s13 * 136657;
5262     s6  -= s13 * 683901;
5263     s13  = 0;
5264 
5265     s0  += s12 * 666643;
5266     s1  += s12 * 470296;
5267     s2  += s12 * 654183;
5268     s3  -= s12 * 997805;
5269     s4  += s12 * 136657;
5270     s5  -= s12 * 683901;
5271     s12 = 0;
5272 
5273     carry0 = (s0 + (1 << 20)) >> 21;
5274     s1  += carry0;
5275     s0  -= carry0 * (1 << 21);
5276     carry2 = (s2 + (1 << 20)) >> 21;
5277     s3  += carry2;
5278     s2  -= carry2 * (1 << 21);
5279     carry4 = (s4 + (1 << 20)) >> 21;
5280     s5  += carry4;
5281     s4  -= carry4 * (1 << 21);
5282     carry6 = (s6 + (1 << 20)) >> 21;
5283     s7  += carry6;
5284     s6  -= carry6 * (1 << 21);
5285     carry8 = (s8 + (1 << 20)) >> 21;
5286     s9  += carry8;
5287     s8  -= carry8 * (1 << 21);
5288     carry10 = (s10 + (1 << 20)) >> 21;
5289     s11 += carry10;
5290     s10 -= carry10 * (1 << 21);
5291 
5292     carry1 = (s1 + (1 << 20)) >> 21;
5293     s2  += carry1;
5294     s1  -= carry1 * (1 << 21);
5295     carry3 = (s3 + (1 << 20)) >> 21;
5296     s4  += carry3;
5297     s3  -= carry3 * (1 << 21);
5298     carry5 = (s5 + (1 << 20)) >> 21;
5299     s6  += carry5;
5300     s5  -= carry5 * (1 << 21);
5301     carry7 = (s7 + (1 << 20)) >> 21;
5302     s8  += carry7;
5303     s7  -= carry7 * (1 << 21);
5304     carry9 = (s9 + (1 << 20)) >> 21;
5305     s10 += carry9;
5306     s9  -= carry9 * (1 << 21);
5307     carry11 = (s11 + (1 << 20)) >> 21;
5308     s12 += carry11;
5309     s11 -= carry11 * (1 << 21);
5310 
5311     s0  += s12 * 666643;
5312     s1  += s12 * 470296;
5313     s2  += s12 * 654183;
5314     s3  -= s12 * 997805;
5315     s4  += s12 * 136657;
5316     s5  -= s12 * 683901;
5317     s12  = 0;
5318 
5319     carry0 = s0 >> 21;
5320     s1  += carry0;
5321     s0  -= carry0 * (1 << 21);
5322     carry1 = s1 >> 21;
5323     s2  += carry1;
5324     s1  -= carry1 * (1 << 21);
5325     carry2 = s2 >> 21;
5326     s3  += carry2;
5327     s2  -= carry2 * (1 << 21);
5328     carry3 = s3 >> 21;
5329     s4  += carry3;
5330     s3  -= carry3 * (1 << 21);
5331     carry4 = s4 >> 21;
5332     s5  += carry4;
5333     s4  -= carry4 * (1 << 21);
5334     carry5 = s5 >> 21;
5335     s6  += carry5;
5336     s5  -= carry5 * (1 << 21);
5337     carry6 = s6 >> 21;
5338     s7  += carry6;
5339     s6  -= carry6 * (1 << 21);
5340     carry7 = s7 >> 21;
5341     s8  += carry7;
5342     s7  -= carry7 * (1 << 21);
5343     carry8 = s8 >> 21;
5344     s9  += carry8;
5345     s8  -= carry8 * (1 << 21);
5346     carry9 = s9 >> 21;
5347     s10 += carry9;
5348     s9  -= carry9 * (1 << 21);
5349     carry10 = s10 >> 21;
5350     s11 += carry10;
5351     s10 -= carry10 * (1 << 21);
5352     carry11 = s11 >> 21;
5353     s12 += carry11;
5354     s11 -= carry11 * (1 << 21);
5355 
5356     s0  += s12 * 666643;
5357     s1  += s12 * 470296;
5358     s2  += s12 * 654183;
5359     s3  -= s12 * 997805;
5360     s4  += s12 * 136657;
5361     s5  -= s12 * 683901;
5362     s12  = 0;
5363 
5364     carry0 = s0 >> 21;
5365     s1  += carry0;
5366     s0  -= carry0 * (1 << 21);
5367     carry1 = s1 >> 21;
5368     s2  += carry1;
5369     s1  -= carry1 * (1 << 21);
5370     carry2 = s2 >> 21;
5371     s3  += carry2;
5372     s2  -= carry2 * (1 << 21);
5373     carry3 = s3 >> 21;
5374     s4  += carry3;
5375     s3  -= carry3 * (1 << 21);
5376     carry4 = s4 >> 21;
5377     s5  += carry4;
5378     s4  -= carry4 * (1 << 21);
5379     carry5 = s5 >> 21;
5380     s6  += carry5;
5381     s5  -= carry5 * (1 << 21);
5382     carry6 = s6 >> 21;
5383     s7  += carry6;
5384     s6  -= carry6 * (1 << 21);
5385     carry7 = s7 >> 21;
5386     s8  += carry7;
5387     s7  -= carry7 * (1 << 21);
5388     carry8 = s8 >> 21;
5389     s9  += carry8;
5390     s8  -= carry8 * (1 << 21);
5391     carry9 = s9 >> 21;
5392     s10 += carry9;
5393     s9  -= carry9 * (1 << 21);
5394     carry10 = s10 >> 21;
5395     s11 += carry10;
5396     s10 -= carry10 * (1 << 21);
5397 
5398     s[ 0] = (uint8_t) (s0  >>  0);
5399     s[ 1] = (uint8_t) (s0  >>  8);
5400     s[ 2] = (uint8_t)((s0  >> 16) | (s1 << 5));
5401     s[ 3] = (uint8_t) (s1  >>  3);
5402     s[ 4] = (uint8_t) (s1  >> 11);
5403     s[ 5] = (uint8_t)((s1  >> 19) | (s2 << 2));
5404     s[ 6] = (uint8_t) (s2  >>  6);
5405     s[ 7] = (uint8_t)((s2  >> 14) | (s3 << 7));
5406     s[ 8] = (uint8_t) (s3  >>  1);
5407     s[ 9] = (uint8_t) (s3  >>  9);
5408     s[10] = (uint8_t)((s3  >> 17) | (s4 << 4));
5409     s[11] = (uint8_t) (s4  >>  4);
5410     s[12] = (uint8_t) (s4  >> 12);
5411     s[13] = (uint8_t)((s4  >> 20) | (s5 << 1));
5412     s[14] = (uint8_t) (s5  >>  7);
5413     s[15] = (uint8_t)((s5  >> 15) | (s6 << 6));
5414     s[16] = (uint8_t) (s6  >>  2);
5415     s[17] = (uint8_t) (s6  >> 10);
5416     s[18] = (uint8_t)((s6  >> 18) | (s7 << 3));
5417     s[19] = (uint8_t) (s7  >>  5);
5418     s[20] = (uint8_t) (s7  >> 13);
5419     s[21] = (uint8_t) (s8  >>  0);
5420     s[22] = (uint8_t) (s8  >>  8);
5421     s[23] = (uint8_t)((s8  >> 16) | (s9 << 5));
5422     s[24] = (uint8_t) (s9  >>  3);
5423     s[25] = (uint8_t) (s9  >> 11);
5424     s[26] = (uint8_t)((s9  >> 19) | (s10 << 2));
5425     s[27] = (uint8_t) (s10 >>  6);
5426     s[28] = (uint8_t)((s10 >> 14) | (s11 << 7));
5427     s[29] = (uint8_t) (s11 >>  1);
5428     s[30] = (uint8_t) (s11 >>  9);
5429     s[31] = (uint8_t) (s11 >> 17);
5430 }
5431 
ED25519_sign(uint8_t * out_sig,const uint8_t * message,size_t message_len,const uint8_t public_key[32],const uint8_t private_key[32])5432 int ED25519_sign(uint8_t *out_sig, const uint8_t *message, size_t message_len,
5433                  const uint8_t public_key[32], const uint8_t private_key[32])
5434 {
5435     uint8_t az[SHA512_DIGEST_LENGTH];
5436     uint8_t nonce[SHA512_DIGEST_LENGTH];
5437     ge_p3 R;
5438     uint8_t hram[SHA512_DIGEST_LENGTH];
5439     SHA512_CTX hash_ctx;
5440 
5441     SHA512_Init(&hash_ctx);
5442     SHA512_Update(&hash_ctx, private_key, 32);
5443     SHA512_Final(az, &hash_ctx);
5444 
5445     az[0] &= 248;
5446     az[31] &= 63;
5447     az[31] |= 64;
5448 
5449     SHA512_Init(&hash_ctx);
5450     SHA512_Update(&hash_ctx, az + 32, 32);
5451     SHA512_Update(&hash_ctx, message, message_len);
5452     SHA512_Final(nonce, &hash_ctx);
5453 
5454     x25519_sc_reduce(nonce);
5455     ge_scalarmult_base(&R, nonce);
5456     ge_p3_tobytes(out_sig, &R);
5457 
5458     SHA512_Init(&hash_ctx);
5459     SHA512_Update(&hash_ctx, out_sig, 32);
5460     SHA512_Update(&hash_ctx, public_key, 32);
5461     SHA512_Update(&hash_ctx, message, message_len);
5462     SHA512_Final(hram, &hash_ctx);
5463 
5464     x25519_sc_reduce(hram);
5465     sc_muladd(out_sig + 32, hram, az, nonce);
5466 
5467     OPENSSL_cleanse(&hash_ctx, sizeof(hash_ctx));
5468     OPENSSL_cleanse(nonce, sizeof(nonce));
5469     OPENSSL_cleanse(az, sizeof(az));
5470 
5471     return 1;
5472 }
5473 
5474 static const char allzeroes[15];
5475 
ED25519_verify(const uint8_t * message,size_t message_len,const uint8_t signature[64],const uint8_t public_key[32])5476 int ED25519_verify(const uint8_t *message, size_t message_len,
5477                    const uint8_t signature[64], const uint8_t public_key[32])
5478 {
5479     int i;
5480     ge_p3 A;
5481     const uint8_t *r, *s;
5482     SHA512_CTX hash_ctx;
5483     ge_p2 R;
5484     uint8_t rcheck[32];
5485     uint8_t h[SHA512_DIGEST_LENGTH];
5486     /* 27742317777372353535851937790883648493 in little endian format */
5487     const uint8_t l_low[16] = {
5488         0xED, 0xD3, 0xF5, 0x5C, 0x1A, 0x63, 0x12, 0x58, 0xD6, 0x9C, 0xF7, 0xA2,
5489         0xDE, 0xF9, 0xDE, 0x14
5490     };
5491 
5492     r = signature;
5493     s = signature + 32;
5494 
5495     /*
5496      * Check 0 <= s < L where L = 2^252 + 27742317777372353535851937790883648493
5497      *
5498      * If not the signature is publicly invalid. Since it's public we can do the
5499      * check in variable time.
5500      *
5501      * First check the most significant byte
5502      */
5503     if (s[31] > 0x10)
5504         return 0;
5505     if (s[31] == 0x10) {
5506         /*
5507          * Most significant byte indicates a value close to 2^252 so check the
5508          * rest
5509          */
5510         if (memcmp(s + 16, allzeroes, sizeof(allzeroes)) != 0)
5511             return 0;
5512         for (i = 15; i >= 0; i--) {
5513             if (s[i] < l_low[i])
5514                 break;
5515             if (s[i] > l_low[i])
5516                 return 0;
5517         }
5518         if (i < 0)
5519             return 0;
5520     }
5521 
5522     if (ge_frombytes_vartime(&A, public_key) != 0) {
5523         return 0;
5524     }
5525 
5526     fe_neg(A.X, A.X);
5527     fe_neg(A.T, A.T);
5528 
5529     SHA512_Init(&hash_ctx);
5530     SHA512_Update(&hash_ctx, r, 32);
5531     SHA512_Update(&hash_ctx, public_key, 32);
5532     SHA512_Update(&hash_ctx, message, message_len);
5533     SHA512_Final(h, &hash_ctx);
5534 
5535     x25519_sc_reduce(h);
5536 
5537     ge_double_scalarmult_vartime(&R, h, &A, s);
5538 
5539     ge_tobytes(rcheck, &R);
5540 
5541     return CRYPTO_memcmp(rcheck, r, sizeof(rcheck)) == 0;
5542 }
5543 
ED25519_public_from_private(uint8_t out_public_key[32],const uint8_t private_key[32])5544 void ED25519_public_from_private(uint8_t out_public_key[32],
5545                                  const uint8_t private_key[32])
5546 {
5547     uint8_t az[SHA512_DIGEST_LENGTH];
5548     ge_p3 A;
5549 
5550     SHA512(private_key, 32, az);
5551 
5552     az[0] &= 248;
5553     az[31] &= 63;
5554     az[31] |= 64;
5555 
5556     ge_scalarmult_base(&A, az);
5557     ge_p3_tobytes(out_public_key, &A);
5558 
5559     OPENSSL_cleanse(az, sizeof(az));
5560 }
5561 
X25519(uint8_t out_shared_key[32],const uint8_t private_key[32],const uint8_t peer_public_value[32])5562 int X25519(uint8_t out_shared_key[32], const uint8_t private_key[32],
5563            const uint8_t peer_public_value[32])
5564 {
5565     static const uint8_t kZeros[32] = {0};
5566     x25519_scalar_mult(out_shared_key, private_key, peer_public_value);
5567     /* The all-zero output results when the input is a point of small order. */
5568     return CRYPTO_memcmp(kZeros, out_shared_key, 32) != 0;
5569 }
5570 
X25519_public_from_private(uint8_t out_public_value[32],const uint8_t private_key[32])5571 void X25519_public_from_private(uint8_t out_public_value[32],
5572                                 const uint8_t private_key[32])
5573 {
5574     uint8_t e[32];
5575     ge_p3 A;
5576     fe zplusy, zminusy, zminusy_inv;
5577 
5578     memcpy(e, private_key, 32);
5579     e[0] &= 248;
5580     e[31] &= 127;
5581     e[31] |= 64;
5582 
5583     ge_scalarmult_base(&A, e);
5584 
5585     /*
5586      * We only need the u-coordinate of the curve25519 point.
5587      * The map is u=(y+1)/(1-y). Since y=Y/Z, this gives
5588      * u=(Z+Y)/(Z-Y).
5589      */
5590     fe_add(zplusy, A.Z, A.Y);
5591     fe_sub(zminusy, A.Z, A.Y);
5592     fe_invert(zminusy_inv, zminusy);
5593     fe_mul(zplusy, zplusy, zminusy_inv);
5594     fe_tobytes(out_public_value, zplusy);
5595 
5596     OPENSSL_cleanse(e, sizeof(e));
5597 }
5598