xref: /openssl/crypto/ec/curve25519.c (revision 7ed6de99)
1 /*
2  * Copyright 2016-2024 The OpenSSL Project Authors. All Rights Reserved.
3  *
4  * Licensed under the Apache License 2.0 (the "License").  You may not use
5  * this file except in compliance with the License.  You can obtain a copy
6  * in the file LICENSE in the source distribution or at
7  * https://www.openssl.org/source/license.html
8  */
9 
10 /*
11  * ECDSA low level APIs are deprecated for public use, but still ok for
12  * internal use.
13  */
14 #include "internal/deprecated.h"
15 
16 #include <string.h>
17 #include "crypto/ecx.h"
18 #include "ec_local.h"
19 #include <openssl/evp.h>
20 #include <openssl/sha.h>
21 
22 #include "internal/numbers.h"
23 
24 #if defined(X25519_ASM) && (defined(__x86_64) || defined(__x86_64__) || \
25                             defined(_M_AMD64) || defined(_M_X64))
26 
27 # define BASE_2_64_IMPLEMENTED
28 
29 typedef uint64_t fe64[4];
30 
31 int x25519_fe64_eligible(void);
32 
33 /*
34  * Following subroutines perform corresponding operations modulo
35  * 2^256-38, i.e. double the curve modulus. However, inputs and
36  * outputs are permitted to be partially reduced, i.e. to remain
37  * in [0..2^256) range. It's all tied up in final fe64_tobytes
38  * that performs full reduction modulo 2^255-19.
39  *
40  * There are no reference C implementations for these.
41  */
42 void x25519_fe64_mul(fe64 h, const fe64 f, const fe64 g);
43 void x25519_fe64_sqr(fe64 h, const fe64 f);
44 void x25519_fe64_mul121666(fe64 h, fe64 f);
45 void x25519_fe64_add(fe64 h, const fe64 f, const fe64 g);
46 void x25519_fe64_sub(fe64 h, const fe64 f, const fe64 g);
47 void x25519_fe64_tobytes(uint8_t *s, const fe64 f);
48 # define fe64_mul x25519_fe64_mul
49 # define fe64_sqr x25519_fe64_sqr
50 # define fe64_mul121666 x25519_fe64_mul121666
51 # define fe64_add x25519_fe64_add
52 # define fe64_sub x25519_fe64_sub
53 # define fe64_tobytes x25519_fe64_tobytes
54 
load_8(const uint8_t * in)55 static uint64_t load_8(const uint8_t *in)
56 {
57     uint64_t result;
58 
59     result = in[0];
60     result |= ((uint64_t)in[1]) << 8;
61     result |= ((uint64_t)in[2]) << 16;
62     result |= ((uint64_t)in[3]) << 24;
63     result |= ((uint64_t)in[4]) << 32;
64     result |= ((uint64_t)in[5]) << 40;
65     result |= ((uint64_t)in[6]) << 48;
66     result |= ((uint64_t)in[7]) << 56;
67 
68     return result;
69 }
70 
fe64_frombytes(fe64 h,const uint8_t * s)71 static void fe64_frombytes(fe64 h, const uint8_t *s)
72 {
73     h[0] = load_8(s);
74     h[1] = load_8(s + 8);
75     h[2] = load_8(s + 16);
76     h[3] = load_8(s + 24) & 0x7fffffffffffffff;
77 }
78 
fe64_0(fe64 h)79 static void fe64_0(fe64 h)
80 {
81     h[0] = 0;
82     h[1] = 0;
83     h[2] = 0;
84     h[3] = 0;
85 }
86 
fe64_1(fe64 h)87 static void fe64_1(fe64 h)
88 {
89     h[0] = 1;
90     h[1] = 0;
91     h[2] = 0;
92     h[3] = 0;
93 }
94 
fe64_copy(fe64 h,const fe64 f)95 static void fe64_copy(fe64 h, const fe64 f)
96 {
97     h[0] = f[0];
98     h[1] = f[1];
99     h[2] = f[2];
100     h[3] = f[3];
101 }
102 
fe64_cswap(fe64 f,fe64 g,unsigned int b)103 static void fe64_cswap(fe64 f, fe64 g, unsigned int b)
104 {
105     int i;
106     uint64_t mask = 0 - (uint64_t)b;
107 
108     for (i = 0; i < 4; i++) {
109         uint64_t x = f[i] ^ g[i];
110         x &= mask;
111         f[i] ^= x;
112         g[i] ^= x;
113     }
114 }
115 
fe64_invert(fe64 out,const fe64 z)116 static void fe64_invert(fe64 out, const fe64 z)
117 {
118     fe64 t0;
119     fe64 t1;
120     fe64 t2;
121     fe64 t3;
122     int i;
123 
124     /*
125      * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as
126      * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11.
127      */
128 
129     /* t0 = z ** 2 */
130     fe64_sqr(t0, z);
131 
132     /* t1 = t0 ** (2 ** 2) = z ** 8 */
133     fe64_sqr(t1, t0);
134     fe64_sqr(t1, t1);
135 
136     /* t1 = z * t1 = z ** 9 */
137     fe64_mul(t1, z, t1);
138     /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */
139     fe64_mul(t0, t0, t1);
140 
141     /* t2 = t0 ** 2 = z ** 22 */
142     fe64_sqr(t2, t0);
143 
144     /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */
145     fe64_mul(t1, t1, t2);
146 
147     /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */
148     fe64_sqr(t2, t1);
149     for (i = 1; i < 5; ++i)
150         fe64_sqr(t2, t2);
151 
152     /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */
153     fe64_mul(t1, t2, t1);
154 
155     /* Continuing similarly... */
156 
157     /* t2 = z ** (2 ** 20 - 1) */
158     fe64_sqr(t2, t1);
159     for (i = 1; i < 10; ++i)
160         fe64_sqr(t2, t2);
161 
162     fe64_mul(t2, t2, t1);
163 
164     /* t2 = z ** (2 ** 40 - 1) */
165     fe64_sqr(t3, t2);
166     for (i = 1; i < 20; ++i)
167         fe64_sqr(t3, t3);
168 
169     fe64_mul(t2, t3, t2);
170 
171     /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */
172     for (i = 0; i < 10; ++i)
173         fe64_sqr(t2, t2);
174 
175     /* t1 = z ** (2 ** 50 - 1) */
176     fe64_mul(t1, t2, t1);
177 
178     /* t2 = z ** (2 ** 100 - 1) */
179     fe64_sqr(t2, t1);
180     for (i = 1; i < 50; ++i)
181         fe64_sqr(t2, t2);
182 
183     fe64_mul(t2, t2, t1);
184 
185     /* t2 = z ** (2 ** 200 - 1) */
186     fe64_sqr(t3, t2);
187     for (i = 1; i < 100; ++i)
188         fe64_sqr(t3, t3);
189 
190     fe64_mul(t2, t3, t2);
191 
192     /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */
193     for (i = 0; i < 50; ++i)
194         fe64_sqr(t2, t2);
195 
196     /* t1 = z ** (2 ** 250 - 1) */
197     fe64_mul(t1, t2, t1);
198 
199     /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */
200     for (i = 0; i < 5; ++i)
201         fe64_sqr(t1, t1);
202 
203     /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */
204     fe64_mul(out, t1, t0);
205 }
206 
207 /*
208  * Duplicate of original x25519_scalar_mult_generic, but using
209  * fe64_* subroutines.
210  */
x25519_scalar_mulx(uint8_t out[32],const uint8_t scalar[32],const uint8_t point[32])211 static void x25519_scalar_mulx(uint8_t out[32], const uint8_t scalar[32],
212                                const uint8_t point[32])
213 {
214     fe64 x1, x2, z2, x3, z3, tmp0, tmp1;
215     uint8_t e[32];
216     unsigned swap = 0;
217     int pos;
218 
219     memcpy(e, scalar, 32);
220     e[0]  &= 0xf8;
221     e[31] &= 0x7f;
222     e[31] |= 0x40;
223     fe64_frombytes(x1, point);
224     fe64_1(x2);
225     fe64_0(z2);
226     fe64_copy(x3, x1);
227     fe64_1(z3);
228 
229     for (pos = 254; pos >= 0; --pos) {
230         unsigned int b = 1 & (e[pos / 8] >> (pos & 7));
231 
232         swap ^= b;
233         fe64_cswap(x2, x3, swap);
234         fe64_cswap(z2, z3, swap);
235         swap = b;
236         fe64_sub(tmp0, x3, z3);
237         fe64_sub(tmp1, x2, z2);
238         fe64_add(x2, x2, z2);
239         fe64_add(z2, x3, z3);
240         fe64_mul(z3, x2, tmp0);
241         fe64_mul(z2, z2, tmp1);
242         fe64_sqr(tmp0, tmp1);
243         fe64_sqr(tmp1, x2);
244         fe64_add(x3, z3, z2);
245         fe64_sub(z2, z3, z2);
246         fe64_mul(x2, tmp1, tmp0);
247         fe64_sub(tmp1, tmp1, tmp0);
248         fe64_sqr(z2, z2);
249         fe64_mul121666(z3, tmp1);
250         fe64_sqr(x3, x3);
251         fe64_add(tmp0, tmp0, z3);
252         fe64_mul(z3, x1, z2);
253         fe64_mul(z2, tmp1, tmp0);
254     }
255 
256     fe64_invert(z2, z2);
257     fe64_mul(x2, x2, z2);
258     fe64_tobytes(out, x2);
259 
260     OPENSSL_cleanse(e, sizeof(e));
261 }
262 #endif
263 
264 #if defined(X25519_ASM) \
265     || ( defined(INT128_MAX) \
266          && !defined(__sparc__) \
267          && (!defined(__SIZEOF_LONG__) || (__SIZEOF_LONG__ == 8)) \
268          && !(defined(__ANDROID__) && !defined(__clang__)) )
269 /*
270  * Base 2^51 implementation. It's virtually no different from reference
271  * base 2^25.5 implementation in respect to lax boundary conditions for
272  * intermediate values and even individual limbs. So that whatever you
273  * know about the reference, applies even here...
274  */
275 # define BASE_2_51_IMPLEMENTED
276 
277 typedef uint64_t fe51[5];
278 
279 static const uint64_t MASK51 = 0x7ffffffffffff;
280 
load_7(const uint8_t * in)281 static uint64_t load_7(const uint8_t *in)
282 {
283     uint64_t result;
284 
285     result = in[0];
286     result |= ((uint64_t)in[1]) << 8;
287     result |= ((uint64_t)in[2]) << 16;
288     result |= ((uint64_t)in[3]) << 24;
289     result |= ((uint64_t)in[4]) << 32;
290     result |= ((uint64_t)in[5]) << 40;
291     result |= ((uint64_t)in[6]) << 48;
292 
293     return result;
294 }
295 
load_6(const uint8_t * in)296 static uint64_t load_6(const uint8_t *in)
297 {
298     uint64_t result;
299 
300     result = in[0];
301     result |= ((uint64_t)in[1]) << 8;
302     result |= ((uint64_t)in[2]) << 16;
303     result |= ((uint64_t)in[3]) << 24;
304     result |= ((uint64_t)in[4]) << 32;
305     result |= ((uint64_t)in[5]) << 40;
306 
307     return result;
308 }
309 
fe51_frombytes(fe51 h,const uint8_t * s)310 static void fe51_frombytes(fe51 h, const uint8_t *s)
311 {
312     uint64_t h0 = load_7(s);                                /* 56 bits */
313     uint64_t h1 = load_6(s + 7) << 5;                       /* 53 bits */
314     uint64_t h2 = load_7(s + 13) << 2;                      /* 58 bits */
315     uint64_t h3 = load_6(s + 20) << 7;                      /* 55 bits */
316     uint64_t h4 = (load_6(s + 26) & 0x7fffffffffff) << 4;   /* 51 bits */
317 
318     h1 |= h0 >> 51; h0 &= MASK51;
319     h2 |= h1 >> 51; h1 &= MASK51;
320     h3 |= h2 >> 51; h2 &= MASK51;
321     h4 |= h3 >> 51; h3 &= MASK51;
322 
323     h[0] = h0;
324     h[1] = h1;
325     h[2] = h2;
326     h[3] = h3;
327     h[4] = h4;
328 }
329 
fe51_tobytes(uint8_t * s,const fe51 h)330 static void fe51_tobytes(uint8_t *s, const fe51 h)
331 {
332     uint64_t h0 = h[0];
333     uint64_t h1 = h[1];
334     uint64_t h2 = h[2];
335     uint64_t h3 = h[3];
336     uint64_t h4 = h[4];
337     uint64_t q;
338 
339     /* compare to modulus */
340     q = (h0 + 19) >> 51;
341     q = (h1 + q) >> 51;
342     q = (h2 + q) >> 51;
343     q = (h3 + q) >> 51;
344     q = (h4 + q) >> 51;
345 
346     /* full reduce */
347     h0 += 19 * q;
348     h1 += h0 >> 51; h0 &= MASK51;
349     h2 += h1 >> 51; h1 &= MASK51;
350     h3 += h2 >> 51; h2 &= MASK51;
351     h4 += h3 >> 51; h3 &= MASK51;
352                     h4 &= MASK51;
353 
354     /* smash */
355     s[0] = (uint8_t)(h0 >> 0);
356     s[1] = (uint8_t)(h0 >> 8);
357     s[2] = (uint8_t)(h0 >> 16);
358     s[3] = (uint8_t)(h0 >> 24);
359     s[4] = (uint8_t)(h0 >> 32);
360     s[5] = (uint8_t)(h0 >> 40);
361     s[6] = (uint8_t)((h0 >> 48) | ((uint32_t)h1 << 3));
362     s[7] = (uint8_t)(h1 >> 5);
363     s[8] = (uint8_t)(h1 >> 13);
364     s[9] = (uint8_t)(h1 >> 21);
365     s[10] = (uint8_t)(h1 >> 29);
366     s[11] = (uint8_t)(h1 >> 37);
367     s[12] = (uint8_t)((h1 >> 45) | ((uint32_t)h2 << 6));
368     s[13] = (uint8_t)(h2 >> 2);
369     s[14] = (uint8_t)(h2 >> 10);
370     s[15] = (uint8_t)(h2 >> 18);
371     s[16] = (uint8_t)(h2 >> 26);
372     s[17] = (uint8_t)(h2 >> 34);
373     s[18] = (uint8_t)(h2 >> 42);
374     s[19] = (uint8_t)((h2 >> 50) | ((uint32_t)h3 << 1));
375     s[20] = (uint8_t)(h3 >> 7);
376     s[21] = (uint8_t)(h3 >> 15);
377     s[22] = (uint8_t)(h3 >> 23);
378     s[23] = (uint8_t)(h3 >> 31);
379     s[24] = (uint8_t)(h3 >> 39);
380     s[25] = (uint8_t)((h3 >> 47) | ((uint32_t)h4 << 4));
381     s[26] = (uint8_t)(h4 >> 4);
382     s[27] = (uint8_t)(h4 >> 12);
383     s[28] = (uint8_t)(h4 >> 20);
384     s[29] = (uint8_t)(h4 >> 28);
385     s[30] = (uint8_t)(h4 >> 36);
386     s[31] = (uint8_t)(h4 >> 44);
387 }
388 
389 # if defined(X25519_ASM)
390 void x25519_fe51_mul(fe51 h, const fe51 f, const fe51 g);
391 void x25519_fe51_sqr(fe51 h, const fe51 f);
392 void x25519_fe51_mul121666(fe51 h, fe51 f);
393 #  define fe51_mul x25519_fe51_mul
394 #  define fe51_sq  x25519_fe51_sqr
395 #  define fe51_mul121666 x25519_fe51_mul121666
396 # else
397 
398 typedef uint128_t u128;
399 
fe51_mul(fe51 h,const fe51 f,const fe51 g)400 static void fe51_mul(fe51 h, const fe51 f, const fe51 g)
401 {
402     u128 h0, h1, h2, h3, h4;
403     uint64_t f_i, g0, g1, g2, g3, g4;
404 
405     f_i = f[0];
406     h0 = (u128)f_i * (g0 = g[0]);
407     h1 = (u128)f_i * (g1 = g[1]);
408     h2 = (u128)f_i * (g2 = g[2]);
409     h3 = (u128)f_i * (g3 = g[3]);
410     h4 = (u128)f_i * (g4 = g[4]);
411 
412     f_i = f[1];
413     h0 += (u128)f_i * (g4 *= 19);
414     h1 += (u128)f_i * g0;
415     h2 += (u128)f_i * g1;
416     h3 += (u128)f_i * g2;
417     h4 += (u128)f_i * g3;
418 
419     f_i = f[2];
420     h0 += (u128)f_i * (g3 *= 19);
421     h1 += (u128)f_i * g4;
422     h2 += (u128)f_i * g0;
423     h3 += (u128)f_i * g1;
424     h4 += (u128)f_i * g2;
425 
426     f_i = f[3];
427     h0 += (u128)f_i * (g2 *= 19);
428     h1 += (u128)f_i * g3;
429     h2 += (u128)f_i * g4;
430     h3 += (u128)f_i * g0;
431     h4 += (u128)f_i * g1;
432 
433     f_i = f[4];
434     h0 += (u128)f_i * (g1 *= 19);
435     h1 += (u128)f_i * g2;
436     h2 += (u128)f_i * g3;
437     h3 += (u128)f_i * g4;
438     h4 += (u128)f_i * g0;
439 
440     /* partial [lazy] reduction */
441     h3 += (uint64_t)(h2 >> 51); g2 = (uint64_t)h2 & MASK51;
442     h1 += (uint64_t)(h0 >> 51); g0 = (uint64_t)h0 & MASK51;
443 
444     h4 += (uint64_t)(h3 >> 51); g3 = (uint64_t)h3 & MASK51;
445     g2 += (uint64_t)(h1 >> 51); g1 = (uint64_t)h1 & MASK51;
446 
447     g0 += (uint64_t)(h4 >> 51) * 19; g4 = (uint64_t)h4 & MASK51;
448     g3 += g2 >> 51; g2 &= MASK51;
449     g1 += g0 >> 51; g0 &= MASK51;
450 
451     h[0] = g0;
452     h[1] = g1;
453     h[2] = g2;
454     h[3] = g3;
455     h[4] = g4;
456 }
457 
fe51_sq(fe51 h,const fe51 f)458 static void fe51_sq(fe51 h, const fe51 f)
459 {
460 #  if defined(OPENSSL_SMALL_FOOTPRINT)
461     fe51_mul(h, f, f);
462 #  else
463     /* dedicated squaring gives 16-25% overall improvement */
464     uint64_t g0 = f[0];
465     uint64_t g1 = f[1];
466     uint64_t g2 = f[2];
467     uint64_t g3 = f[3];
468     uint64_t g4 = f[4];
469     u128 h0, h1, h2, h3, h4;
470 
471     h0 = (u128)g0 * g0;     g0 *= 2;
472     h1 = (u128)g0 * g1;
473     h2 = (u128)g0 * g2;
474     h3 = (u128)g0 * g3;
475     h4 = (u128)g0 * g4;
476 
477     g0 = g4;                /* borrow g0 */
478     h3 += (u128)g0 * (g4 *= 19);
479 
480     h2 += (u128)g1 * g1;    g1 *= 2;
481     h3 += (u128)g1 * g2;
482     h4 += (u128)g1 * g3;
483     h0 += (u128)g1 * g4;
484 
485     g0 = g3;                /* borrow g0 */
486     h1 += (u128)g0 * (g3 *= 19);
487     h2 += (u128)(g0 * 2) * g4;
488 
489     h4 += (u128)g2 * g2;    g2 *= 2;
490     h0 += (u128)g2 * g3;
491     h1 += (u128)g2 * g4;
492 
493     /* partial [lazy] reduction */
494     h3 += (uint64_t)(h2 >> 51); g2 = (uint64_t)h2 & MASK51;
495     h1 += (uint64_t)(h0 >> 51); g0 = (uint64_t)h0 & MASK51;
496 
497     h4 += (uint64_t)(h3 >> 51); g3 = (uint64_t)h3 & MASK51;
498     g2 += (uint64_t)(h1 >> 51); g1 = (uint64_t)h1 & MASK51;
499 
500     g0 += (uint64_t)(h4 >> 51) * 19; g4 = (uint64_t)h4 & MASK51;
501     g3 += g2 >> 51; g2 &= MASK51;
502     g1 += g0 >> 51; g0 &= MASK51;
503 
504     h[0] = g0;
505     h[1] = g1;
506     h[2] = g2;
507     h[3] = g3;
508     h[4] = g4;
509 #  endif
510 }
511 
fe51_mul121666(fe51 h,fe51 f)512 static void fe51_mul121666(fe51 h, fe51 f)
513 {
514     u128 h0 = f[0] * (u128)121666;
515     u128 h1 = f[1] * (u128)121666;
516     u128 h2 = f[2] * (u128)121666;
517     u128 h3 = f[3] * (u128)121666;
518     u128 h4 = f[4] * (u128)121666;
519     uint64_t g0, g1, g2, g3, g4;
520 
521     h3 += (uint64_t)(h2 >> 51); g2 = (uint64_t)h2 & MASK51;
522     h1 += (uint64_t)(h0 >> 51); g0 = (uint64_t)h0 & MASK51;
523 
524     h4 += (uint64_t)(h3 >> 51); g3 = (uint64_t)h3 & MASK51;
525     g2 += (uint64_t)(h1 >> 51); g1 = (uint64_t)h1 & MASK51;
526 
527     g0 += (uint64_t)(h4 >> 51) * 19; g4 = (uint64_t)h4 & MASK51;
528     g3 += g2 >> 51; g2 &= MASK51;
529     g1 += g0 >> 51; g0 &= MASK51;
530 
531     h[0] = g0;
532     h[1] = g1;
533     h[2] = g2;
534     h[3] = g3;
535     h[4] = g4;
536 }
537 # endif
538 
fe51_add(fe51 h,const fe51 f,const fe51 g)539 static void fe51_add(fe51 h, const fe51 f, const fe51 g)
540 {
541     h[0] = f[0] + g[0];
542     h[1] = f[1] + g[1];
543     h[2] = f[2] + g[2];
544     h[3] = f[3] + g[3];
545     h[4] = f[4] + g[4];
546 }
547 
fe51_sub(fe51 h,const fe51 f,const fe51 g)548 static void fe51_sub(fe51 h, const fe51 f, const fe51 g)
549 {
550     /*
551      * Add 2*modulus to ensure that result remains positive
552      * even if subtrahend is partially reduced.
553      */
554     h[0] = (f[0] + 0xfffffffffffda) - g[0];
555     h[1] = (f[1] + 0xffffffffffffe) - g[1];
556     h[2] = (f[2] + 0xffffffffffffe) - g[2];
557     h[3] = (f[3] + 0xffffffffffffe) - g[3];
558     h[4] = (f[4] + 0xffffffffffffe) - g[4];
559 }
560 
fe51_0(fe51 h)561 static void fe51_0(fe51 h)
562 {
563     h[0] = 0;
564     h[1] = 0;
565     h[2] = 0;
566     h[3] = 0;
567     h[4] = 0;
568 }
569 
fe51_1(fe51 h)570 static void fe51_1(fe51 h)
571 {
572     h[0] = 1;
573     h[1] = 0;
574     h[2] = 0;
575     h[3] = 0;
576     h[4] = 0;
577 }
578 
fe51_copy(fe51 h,const fe51 f)579 static void fe51_copy(fe51 h, const fe51 f)
580 {
581     h[0] = f[0];
582     h[1] = f[1];
583     h[2] = f[2];
584     h[3] = f[3];
585     h[4] = f[4];
586 }
587 
fe51_cswap(fe51 f,fe51 g,unsigned int b)588 static void fe51_cswap(fe51 f, fe51 g, unsigned int b)
589 {
590     int i;
591     uint64_t mask = 0 - (uint64_t)b;
592 
593     for (i = 0; i < 5; i++) {
594         int64_t x = f[i] ^ g[i];
595         x &= mask;
596         f[i] ^= x;
597         g[i] ^= x;
598     }
599 }
600 
fe51_invert(fe51 out,const fe51 z)601 static void fe51_invert(fe51 out, const fe51 z)
602 {
603     fe51 t0;
604     fe51 t1;
605     fe51 t2;
606     fe51 t3;
607     int i;
608 
609     /*
610      * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as
611      * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11.
612      */
613 
614     /* t0 = z ** 2 */
615     fe51_sq(t0, z);
616 
617     /* t1 = t0 ** (2 ** 2) = z ** 8 */
618     fe51_sq(t1, t0);
619     fe51_sq(t1, t1);
620 
621     /* t1 = z * t1 = z ** 9 */
622     fe51_mul(t1, z, t1);
623     /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */
624     fe51_mul(t0, t0, t1);
625 
626     /* t2 = t0 ** 2 = z ** 22 */
627     fe51_sq(t2, t0);
628 
629     /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */
630     fe51_mul(t1, t1, t2);
631 
632     /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */
633     fe51_sq(t2, t1);
634     for (i = 1; i < 5; ++i)
635         fe51_sq(t2, t2);
636 
637     /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */
638     fe51_mul(t1, t2, t1);
639 
640     /* Continuing similarly... */
641 
642     /* t2 = z ** (2 ** 20 - 1) */
643     fe51_sq(t2, t1);
644     for (i = 1; i < 10; ++i)
645         fe51_sq(t2, t2);
646 
647     fe51_mul(t2, t2, t1);
648 
649     /* t2 = z ** (2 ** 40 - 1) */
650     fe51_sq(t3, t2);
651     for (i = 1; i < 20; ++i)
652         fe51_sq(t3, t3);
653 
654     fe51_mul(t2, t3, t2);
655 
656     /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */
657     for (i = 0; i < 10; ++i)
658         fe51_sq(t2, t2);
659 
660     /* t1 = z ** (2 ** 50 - 1) */
661     fe51_mul(t1, t2, t1);
662 
663     /* t2 = z ** (2 ** 100 - 1) */
664     fe51_sq(t2, t1);
665     for (i = 1; i < 50; ++i)
666         fe51_sq(t2, t2);
667 
668     fe51_mul(t2, t2, t1);
669 
670     /* t2 = z ** (2 ** 200 - 1) */
671     fe51_sq(t3, t2);
672     for (i = 1; i < 100; ++i)
673         fe51_sq(t3, t3);
674 
675     fe51_mul(t2, t3, t2);
676 
677     /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */
678     for (i = 0; i < 50; ++i)
679         fe51_sq(t2, t2);
680 
681     /* t1 = z ** (2 ** 250 - 1) */
682     fe51_mul(t1, t2, t1);
683 
684     /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */
685     for (i = 0; i < 5; ++i)
686         fe51_sq(t1, t1);
687 
688     /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */
689     fe51_mul(out, t1, t0);
690 }
691 
692 /*
693  * Duplicate of original x25519_scalar_mult_generic, but using
694  * fe51_* subroutines.
695  */
x25519_scalar_mult(uint8_t out[32],const uint8_t scalar[32],const uint8_t point[32])696 static void x25519_scalar_mult(uint8_t out[32], const uint8_t scalar[32],
697                                const uint8_t point[32])
698 {
699     fe51 x1, x2, z2, x3, z3, tmp0, tmp1;
700     uint8_t e[32];
701     unsigned swap = 0;
702     int pos;
703 
704 # ifdef BASE_2_64_IMPLEMENTED
705     if (x25519_fe64_eligible()) {
706         x25519_scalar_mulx(out, scalar, point);
707         return;
708     }
709 # endif
710 
711     memcpy(e, scalar, 32);
712     e[0]  &= 0xf8;
713     e[31] &= 0x7f;
714     e[31] |= 0x40;
715     fe51_frombytes(x1, point);
716     fe51_1(x2);
717     fe51_0(z2);
718     fe51_copy(x3, x1);
719     fe51_1(z3);
720 
721     for (pos = 254; pos >= 0; --pos) {
722         unsigned int b = 1 & (e[pos / 8] >> (pos & 7));
723 
724         swap ^= b;
725         fe51_cswap(x2, x3, swap);
726         fe51_cswap(z2, z3, swap);
727         swap = b;
728         fe51_sub(tmp0, x3, z3);
729         fe51_sub(tmp1, x2, z2);
730         fe51_add(x2, x2, z2);
731         fe51_add(z2, x3, z3);
732         fe51_mul(z3, tmp0, x2);
733         fe51_mul(z2, z2, tmp1);
734         fe51_sq(tmp0, tmp1);
735         fe51_sq(tmp1, x2);
736         fe51_add(x3, z3, z2);
737         fe51_sub(z2, z3, z2);
738         fe51_mul(x2, tmp1, tmp0);
739         fe51_sub(tmp1, tmp1, tmp0);
740         fe51_sq(z2, z2);
741         fe51_mul121666(z3, tmp1);
742         fe51_sq(x3, x3);
743         fe51_add(tmp0, tmp0, z3);
744         fe51_mul(z3, x1, z2);
745         fe51_mul(z2, tmp1, tmp0);
746     }
747 
748     fe51_invert(z2, z2);
749     fe51_mul(x2, x2, z2);
750     fe51_tobytes(out, x2);
751 
752     OPENSSL_cleanse(e, sizeof(e));
753 }
754 #endif
755 
756 /*
757  * Reference base 2^25.5 implementation.
758  *
759  * This code is mostly taken from the ref10 version of Ed25519 in SUPERCOP
760  * 20141124 (http://bench.cr.yp.to/supercop.html).
761  *
762  * The field functions are shared by Ed25519 and X25519 where possible.
763  */
764 
765 /*
766  * fe means field element. Here the field is \Z/(2^255-19). An element t,
767  * entries t[0]...t[9], represents the integer t[0]+2^26 t[1]+2^51 t[2]+2^77
768  * t[3]+2^102 t[4]+...+2^230 t[9]. Bounds on each t[i] vary depending on
769  * context.
770  */
771 typedef int32_t fe[10];
772 
773 static const int64_t kBottom21Bits =  0x1fffffLL;
774 static const int64_t kBottom25Bits = 0x1ffffffLL;
775 static const int64_t kBottom26Bits = 0x3ffffffLL;
776 static const int64_t kTop39Bits = 0xfffffffffe000000LL;
777 static const int64_t kTop38Bits = 0xfffffffffc000000LL;
778 
load_3(const uint8_t * in)779 static uint64_t load_3(const uint8_t *in)
780 {
781     uint64_t result;
782 
783     result  = ((uint64_t)in[0]);
784     result |= ((uint64_t)in[1]) << 8;
785     result |= ((uint64_t)in[2]) << 16;
786     return result;
787 }
788 
load_4(const uint8_t * in)789 static uint64_t load_4(const uint8_t *in)
790 {
791     uint64_t result;
792 
793     result  = ((uint64_t)in[0]);
794     result |= ((uint64_t)in[1]) << 8;
795     result |= ((uint64_t)in[2]) << 16;
796     result |= ((uint64_t)in[3]) << 24;
797     return result;
798 }
799 
fe_frombytes(fe h,const uint8_t * s)800 static void fe_frombytes(fe h, const uint8_t *s)
801 {
802     /* Ignores top bit of h. */
803     int64_t h0 =  load_4(s);
804     int64_t h1 =  load_3(s +  4) << 6;
805     int64_t h2 =  load_3(s +  7) << 5;
806     int64_t h3 =  load_3(s + 10) << 3;
807     int64_t h4 =  load_3(s + 13) << 2;
808     int64_t h5 =  load_4(s + 16);
809     int64_t h6 =  load_3(s + 20) << 7;
810     int64_t h7 =  load_3(s + 23) << 5;
811     int64_t h8 =  load_3(s + 26) << 4;
812     int64_t h9 = (load_3(s + 29) & 0x7fffff) << 2;
813     int64_t carry0;
814     int64_t carry1;
815     int64_t carry2;
816     int64_t carry3;
817     int64_t carry4;
818     int64_t carry5;
819     int64_t carry6;
820     int64_t carry7;
821     int64_t carry8;
822     int64_t carry9;
823 
824     carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
825     carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
826     carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
827     carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
828     carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;
829 
830     carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
831     carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
832     carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
833     carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
834     carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;
835 
836     h[0] = (int32_t)h0;
837     h[1] = (int32_t)h1;
838     h[2] = (int32_t)h2;
839     h[3] = (int32_t)h3;
840     h[4] = (int32_t)h4;
841     h[5] = (int32_t)h5;
842     h[6] = (int32_t)h6;
843     h[7] = (int32_t)h7;
844     h[8] = (int32_t)h8;
845     h[9] = (int32_t)h9;
846 }
847 
848 /*
849  * Preconditions:
850  *   |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
851  *
852  * Write p=2^255-19; q=floor(h/p).
853  * Basic claim: q = floor(2^(-255)(h + 19 2^(-25)h9 + 2^(-1))).
854  *
855  * Proof:
856  *   Have |h|<=p so |q|<=1 so |19^2 2^(-255) q|<1/4.
857  *   Also have |h-2^230 h9|<2^231 so |19 2^(-255)(h-2^230 h9)|<1/4.
858  *
859  *   Write y=2^(-1)-19^2 2^(-255)q-19 2^(-255)(h-2^230 h9).
860  *   Then 0<y<1.
861  *
862  *   Write r=h-pq.
863  *   Have 0<=r<=p-1=2^255-20.
864  *   Thus 0<=r+19(2^-255)r<r+19(2^-255)2^255<=2^255-1.
865  *
866  *   Write x=r+19(2^-255)r+y.
867  *   Then 0<x<2^255 so floor(2^(-255)x) = 0 so floor(q+2^(-255)x) = q.
868  *
869  *   Have q+2^(-255)x = 2^(-255)(h + 19 2^(-25) h9 + 2^(-1))
870  *   so floor(2^(-255)(h + 19 2^(-25) h9 + 2^(-1))) = q.
871  */
fe_tobytes(uint8_t * s,const fe h)872 static void fe_tobytes(uint8_t *s, const fe h)
873 {
874     int32_t h0 = h[0];
875     int32_t h1 = h[1];
876     int32_t h2 = h[2];
877     int32_t h3 = h[3];
878     int32_t h4 = h[4];
879     int32_t h5 = h[5];
880     int32_t h6 = h[6];
881     int32_t h7 = h[7];
882     int32_t h8 = h[8];
883     int32_t h9 = h[9];
884     int32_t q;
885 
886     q = (19 * h9 + (((int32_t) 1) << 24)) >> 25;
887     q = (h0 + q) >> 26;
888     q = (h1 + q) >> 25;
889     q = (h2 + q) >> 26;
890     q = (h3 + q) >> 25;
891     q = (h4 + q) >> 26;
892     q = (h5 + q) >> 25;
893     q = (h6 + q) >> 26;
894     q = (h7 + q) >> 25;
895     q = (h8 + q) >> 26;
896     q = (h9 + q) >> 25;
897 
898     /* Goal: Output h-(2^255-19)q, which is between 0 and 2^255-20. */
899     h0 += 19 * q;
900     /* Goal: Output h-2^255 q, which is between 0 and 2^255-20. */
901 
902     h1 += h0 >> 26; h0 &= kBottom26Bits;
903     h2 += h1 >> 25; h1 &= kBottom25Bits;
904     h3 += h2 >> 26; h2 &= kBottom26Bits;
905     h4 += h3 >> 25; h3 &= kBottom25Bits;
906     h5 += h4 >> 26; h4 &= kBottom26Bits;
907     h6 += h5 >> 25; h5 &= kBottom25Bits;
908     h7 += h6 >> 26; h6 &= kBottom26Bits;
909     h8 += h7 >> 25; h7 &= kBottom25Bits;
910     h9 += h8 >> 26; h8 &= kBottom26Bits;
911                     h9 &= kBottom25Bits;
912     /* h10 = carry9 */
913 
914     /*
915      * Goal: Output h0+...+2^255 h10-2^255 q, which is between 0 and 2^255-20.
916      * Have h0+...+2^230 h9 between 0 and 2^255-1;
917      * evidently 2^255 h10-2^255 q = 0.
918      * Goal: Output h0+...+2^230 h9.
919      */
920     s[ 0] = (uint8_t) (h0 >>  0);
921     s[ 1] = (uint8_t) (h0 >>  8);
922     s[ 2] = (uint8_t) (h0 >> 16);
923     s[ 3] = (uint8_t)((h0 >> 24) | ((uint32_t)(h1) << 2));
924     s[ 4] = (uint8_t) (h1 >>  6);
925     s[ 5] = (uint8_t) (h1 >> 14);
926     s[ 6] = (uint8_t)((h1 >> 22) | ((uint32_t)(h2) << 3));
927     s[ 7] = (uint8_t) (h2 >>  5);
928     s[ 8] = (uint8_t) (h2 >> 13);
929     s[ 9] = (uint8_t)((h2 >> 21) | ((uint32_t)(h3) << 5));
930     s[10] = (uint8_t) (h3 >>  3);
931     s[11] = (uint8_t) (h3 >> 11);
932     s[12] = (uint8_t)((h3 >> 19) | ((uint32_t)(h4) << 6));
933     s[13] = (uint8_t) (h4 >>  2);
934     s[14] = (uint8_t) (h4 >> 10);
935     s[15] = (uint8_t) (h4 >> 18);
936     s[16] = (uint8_t) (h5 >>  0);
937     s[17] = (uint8_t) (h5 >>  8);
938     s[18] = (uint8_t) (h5 >> 16);
939     s[19] = (uint8_t)((h5 >> 24) | ((uint32_t)(h6) << 1));
940     s[20] = (uint8_t) (h6 >>  7);
941     s[21] = (uint8_t) (h6 >> 15);
942     s[22] = (uint8_t)((h6 >> 23) | ((uint32_t)(h7) << 3));
943     s[23] = (uint8_t) (h7 >>  5);
944     s[24] = (uint8_t) (h7 >> 13);
945     s[25] = (uint8_t)((h7 >> 21) | ((uint32_t)(h8) << 4));
946     s[26] = (uint8_t) (h8 >>  4);
947     s[27] = (uint8_t) (h8 >> 12);
948     s[28] = (uint8_t)((h8 >> 20) | ((uint32_t)(h9) << 6));
949     s[29] = (uint8_t) (h9 >>  2);
950     s[30] = (uint8_t) (h9 >> 10);
951     s[31] = (uint8_t) (h9 >> 18);
952 }
953 
954 /* h = f */
fe_copy(fe h,const fe f)955 static void fe_copy(fe h, const fe f)
956 {
957     memmove(h, f, sizeof(int32_t) * 10);
958 }
959 
960 /* h = 0 */
fe_0(fe h)961 static void fe_0(fe h)
962 {
963     memset(h, 0, sizeof(int32_t) * 10);
964 }
965 
966 /* h = 1 */
fe_1(fe h)967 static void fe_1(fe h)
968 {
969     memset(h, 0, sizeof(int32_t) * 10);
970     h[0] = 1;
971 }
972 
973 /*
974  * h = f + g
975  *
976  * Can overlap h with f or g.
977  *
978  * Preconditions:
979  *    |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
980  *    |g| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
981  *
982  * Postconditions:
983  *    |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
984  */
fe_add(fe h,const fe f,const fe g)985 static void fe_add(fe h, const fe f, const fe g)
986 {
987     unsigned i;
988 
989     for (i = 0; i < 10; i++) {
990         h[i] = f[i] + g[i];
991     }
992 }
993 
994 /*
995  * h = f - g
996  *
997  * Can overlap h with f or g.
998  *
999  * Preconditions:
1000  *    |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
1001  *    |g| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
1002  *
1003  * Postconditions:
1004  *    |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
1005  */
fe_sub(fe h,const fe f,const fe g)1006 static void fe_sub(fe h, const fe f, const fe g)
1007 {
1008     unsigned i;
1009 
1010     for (i = 0; i < 10; i++) {
1011         h[i] = f[i] - g[i];
1012     }
1013 }
1014 
1015 /*
1016  * h = f * g
1017  *
1018  * Can overlap h with f or g.
1019  *
1020  * Preconditions:
1021  *    |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
1022  *    |g| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
1023  *
1024  * Postconditions:
1025  *    |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
1026  *
1027  * Notes on implementation strategy:
1028  *
1029  * Using schoolbook multiplication.
1030  * Karatsuba would save a little in some cost models.
1031  *
1032  * Most multiplications by 2 and 19 are 32-bit precomputations;
1033  * cheaper than 64-bit postcomputations.
1034  *
1035  * There is one remaining multiplication by 19 in the carry chain;
1036  * one *19 precomputation can be merged into this,
1037  * but the resulting data flow is considerably less clean.
1038  *
1039  * There are 12 carries below.
1040  * 10 of them are 2-way parallelizable and vectorizable.
1041  * Can get away with 11 carries, but then data flow is much deeper.
1042  *
1043  * With tighter constraints on inputs can squeeze carries into int32.
1044  */
fe_mul(fe h,const fe f,const fe g)1045 static void fe_mul(fe h, const fe f, const fe g)
1046 {
1047     int32_t f0 = f[0];
1048     int32_t f1 = f[1];
1049     int32_t f2 = f[2];
1050     int32_t f3 = f[3];
1051     int32_t f4 = f[4];
1052     int32_t f5 = f[5];
1053     int32_t f6 = f[6];
1054     int32_t f7 = f[7];
1055     int32_t f8 = f[8];
1056     int32_t f9 = f[9];
1057     int32_t g0 = g[0];
1058     int32_t g1 = g[1];
1059     int32_t g2 = g[2];
1060     int32_t g3 = g[3];
1061     int32_t g4 = g[4];
1062     int32_t g5 = g[5];
1063     int32_t g6 = g[6];
1064     int32_t g7 = g[7];
1065     int32_t g8 = g[8];
1066     int32_t g9 = g[9];
1067     int32_t g1_19 = 19 * g1; /* 1.959375*2^29 */
1068     int32_t g2_19 = 19 * g2; /* 1.959375*2^30; still ok */
1069     int32_t g3_19 = 19 * g3;
1070     int32_t g4_19 = 19 * g4;
1071     int32_t g5_19 = 19 * g5;
1072     int32_t g6_19 = 19 * g6;
1073     int32_t g7_19 = 19 * g7;
1074     int32_t g8_19 = 19 * g8;
1075     int32_t g9_19 = 19 * g9;
1076     int32_t f1_2 = 2 * f1;
1077     int32_t f3_2 = 2 * f3;
1078     int32_t f5_2 = 2 * f5;
1079     int32_t f7_2 = 2 * f7;
1080     int32_t f9_2 = 2 * f9;
1081     int64_t f0g0    = f0   * (int64_t) g0;
1082     int64_t f0g1    = f0   * (int64_t) g1;
1083     int64_t f0g2    = f0   * (int64_t) g2;
1084     int64_t f0g3    = f0   * (int64_t) g3;
1085     int64_t f0g4    = f0   * (int64_t) g4;
1086     int64_t f0g5    = f0   * (int64_t) g5;
1087     int64_t f0g6    = f0   * (int64_t) g6;
1088     int64_t f0g7    = f0   * (int64_t) g7;
1089     int64_t f0g8    = f0   * (int64_t) g8;
1090     int64_t f0g9    = f0   * (int64_t) g9;
1091     int64_t f1g0    = f1   * (int64_t) g0;
1092     int64_t f1g1_2  = f1_2 * (int64_t) g1;
1093     int64_t f1g2    = f1   * (int64_t) g2;
1094     int64_t f1g3_2  = f1_2 * (int64_t) g3;
1095     int64_t f1g4    = f1   * (int64_t) g4;
1096     int64_t f1g5_2  = f1_2 * (int64_t) g5;
1097     int64_t f1g6    = f1   * (int64_t) g6;
1098     int64_t f1g7_2  = f1_2 * (int64_t) g7;
1099     int64_t f1g8    = f1   * (int64_t) g8;
1100     int64_t f1g9_38 = f1_2 * (int64_t) g9_19;
1101     int64_t f2g0    = f2   * (int64_t) g0;
1102     int64_t f2g1    = f2   * (int64_t) g1;
1103     int64_t f2g2    = f2   * (int64_t) g2;
1104     int64_t f2g3    = f2   * (int64_t) g3;
1105     int64_t f2g4    = f2   * (int64_t) g4;
1106     int64_t f2g5    = f2   * (int64_t) g5;
1107     int64_t f2g6    = f2   * (int64_t) g6;
1108     int64_t f2g7    = f2   * (int64_t) g7;
1109     int64_t f2g8_19 = f2   * (int64_t) g8_19;
1110     int64_t f2g9_19 = f2   * (int64_t) g9_19;
1111     int64_t f3g0    = f3   * (int64_t) g0;
1112     int64_t f3g1_2  = f3_2 * (int64_t) g1;
1113     int64_t f3g2    = f3   * (int64_t) g2;
1114     int64_t f3g3_2  = f3_2 * (int64_t) g3;
1115     int64_t f3g4    = f3   * (int64_t) g4;
1116     int64_t f3g5_2  = f3_2 * (int64_t) g5;
1117     int64_t f3g6    = f3   * (int64_t) g6;
1118     int64_t f3g7_38 = f3_2 * (int64_t) g7_19;
1119     int64_t f3g8_19 = f3   * (int64_t) g8_19;
1120     int64_t f3g9_38 = f3_2 * (int64_t) g9_19;
1121     int64_t f4g0    = f4   * (int64_t) g0;
1122     int64_t f4g1    = f4   * (int64_t) g1;
1123     int64_t f4g2    = f4   * (int64_t) g2;
1124     int64_t f4g3    = f4   * (int64_t) g3;
1125     int64_t f4g4    = f4   * (int64_t) g4;
1126     int64_t f4g5    = f4   * (int64_t) g5;
1127     int64_t f4g6_19 = f4   * (int64_t) g6_19;
1128     int64_t f4g7_19 = f4   * (int64_t) g7_19;
1129     int64_t f4g8_19 = f4   * (int64_t) g8_19;
1130     int64_t f4g9_19 = f4   * (int64_t) g9_19;
1131     int64_t f5g0    = f5   * (int64_t) g0;
1132     int64_t f5g1_2  = f5_2 * (int64_t) g1;
1133     int64_t f5g2    = f5   * (int64_t) g2;
1134     int64_t f5g3_2  = f5_2 * (int64_t) g3;
1135     int64_t f5g4    = f5   * (int64_t) g4;
1136     int64_t f5g5_38 = f5_2 * (int64_t) g5_19;
1137     int64_t f5g6_19 = f5   * (int64_t) g6_19;
1138     int64_t f5g7_38 = f5_2 * (int64_t) g7_19;
1139     int64_t f5g8_19 = f5   * (int64_t) g8_19;
1140     int64_t f5g9_38 = f5_2 * (int64_t) g9_19;
1141     int64_t f6g0    = f6   * (int64_t) g0;
1142     int64_t f6g1    = f6   * (int64_t) g1;
1143     int64_t f6g2    = f6   * (int64_t) g2;
1144     int64_t f6g3    = f6   * (int64_t) g3;
1145     int64_t f6g4_19 = f6   * (int64_t) g4_19;
1146     int64_t f6g5_19 = f6   * (int64_t) g5_19;
1147     int64_t f6g6_19 = f6   * (int64_t) g6_19;
1148     int64_t f6g7_19 = f6   * (int64_t) g7_19;
1149     int64_t f6g8_19 = f6   * (int64_t) g8_19;
1150     int64_t f6g9_19 = f6   * (int64_t) g9_19;
1151     int64_t f7g0    = f7   * (int64_t) g0;
1152     int64_t f7g1_2  = f7_2 * (int64_t) g1;
1153     int64_t f7g2    = f7   * (int64_t) g2;
1154     int64_t f7g3_38 = f7_2 * (int64_t) g3_19;
1155     int64_t f7g4_19 = f7   * (int64_t) g4_19;
1156     int64_t f7g5_38 = f7_2 * (int64_t) g5_19;
1157     int64_t f7g6_19 = f7   * (int64_t) g6_19;
1158     int64_t f7g7_38 = f7_2 * (int64_t) g7_19;
1159     int64_t f7g8_19 = f7   * (int64_t) g8_19;
1160     int64_t f7g9_38 = f7_2 * (int64_t) g9_19;
1161     int64_t f8g0    = f8   * (int64_t) g0;
1162     int64_t f8g1    = f8   * (int64_t) g1;
1163     int64_t f8g2_19 = f8   * (int64_t) g2_19;
1164     int64_t f8g3_19 = f8   * (int64_t) g3_19;
1165     int64_t f8g4_19 = f8   * (int64_t) g4_19;
1166     int64_t f8g5_19 = f8   * (int64_t) g5_19;
1167     int64_t f8g6_19 = f8   * (int64_t) g6_19;
1168     int64_t f8g7_19 = f8   * (int64_t) g7_19;
1169     int64_t f8g8_19 = f8   * (int64_t) g8_19;
1170     int64_t f8g9_19 = f8   * (int64_t) g9_19;
1171     int64_t f9g0    = f9   * (int64_t) g0;
1172     int64_t f9g1_38 = f9_2 * (int64_t) g1_19;
1173     int64_t f9g2_19 = f9   * (int64_t) g2_19;
1174     int64_t f9g3_38 = f9_2 * (int64_t) g3_19;
1175     int64_t f9g4_19 = f9   * (int64_t) g4_19;
1176     int64_t f9g5_38 = f9_2 * (int64_t) g5_19;
1177     int64_t f9g6_19 = f9   * (int64_t) g6_19;
1178     int64_t f9g7_38 = f9_2 * (int64_t) g7_19;
1179     int64_t f9g8_19 = f9   * (int64_t) g8_19;
1180     int64_t f9g9_38 = f9_2 * (int64_t) g9_19;
1181     int64_t h0 = f0g0 + f1g9_38 + f2g8_19 + f3g7_38 + f4g6_19 + f5g5_38 + f6g4_19 + f7g3_38 + f8g2_19 + f9g1_38;
1182     int64_t h1 = f0g1 + f1g0    + f2g9_19 + f3g8_19 + f4g7_19 + f5g6_19 + f6g5_19 + f7g4_19 + f8g3_19 + f9g2_19;
1183     int64_t h2 = f0g2 + f1g1_2  + f2g0    + f3g9_38 + f4g8_19 + f5g7_38 + f6g6_19 + f7g5_38 + f8g4_19 + f9g3_38;
1184     int64_t h3 = f0g3 + f1g2    + f2g1    + f3g0    + f4g9_19 + f5g8_19 + f6g7_19 + f7g6_19 + f8g5_19 + f9g4_19;
1185     int64_t h4 = f0g4 + f1g3_2  + f2g2    + f3g1_2  + f4g0    + f5g9_38 + f6g8_19 + f7g7_38 + f8g6_19 + f9g5_38;
1186     int64_t h5 = f0g5 + f1g4    + f2g3    + f3g2    + f4g1    + f5g0    + f6g9_19 + f7g8_19 + f8g7_19 + f9g6_19;
1187     int64_t h6 = f0g6 + f1g5_2  + f2g4    + f3g3_2  + f4g2    + f5g1_2  + f6g0    + f7g9_38 + f8g8_19 + f9g7_38;
1188     int64_t h7 = f0g7 + f1g6    + f2g5    + f3g4    + f4g3    + f5g2    + f6g1    + f7g0    + f8g9_19 + f9g8_19;
1189     int64_t h8 = f0g8 + f1g7_2  + f2g6    + f3g5_2  + f4g4    + f5g3_2  + f6g2    + f7g1_2  + f8g0    + f9g9_38;
1190     int64_t h9 = f0g9 + f1g8    + f2g7    + f3g6    + f4g5    + f5g4    + f6g3    + f7g2    + f8g1    + f9g0   ;
1191     int64_t carry0;
1192     int64_t carry1;
1193     int64_t carry2;
1194     int64_t carry3;
1195     int64_t carry4;
1196     int64_t carry5;
1197     int64_t carry6;
1198     int64_t carry7;
1199     int64_t carry8;
1200     int64_t carry9;
1201 
1202     /* |h0| <= (1.65*1.65*2^52*(1+19+19+19+19)+1.65*1.65*2^50*(38+38+38+38+38))
1203      *   i.e. |h0| <= 1.4*2^60; narrower ranges for h2, h4, h6, h8
1204      * |h1| <= (1.65*1.65*2^51*(1+1+19+19+19+19+19+19+19+19))
1205      *   i.e. |h1| <= 1.7*2^59; narrower ranges for h3, h5, h7, h9 */
1206 
1207     carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
1208     carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
1209     /* |h0| <= 2^25 */
1210     /* |h4| <= 2^25 */
1211     /* |h1| <= 1.71*2^59 */
1212     /* |h5| <= 1.71*2^59 */
1213 
1214     carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
1215     carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
1216     /* |h1| <= 2^24; from now on fits into int32 */
1217     /* |h5| <= 2^24; from now on fits into int32 */
1218     /* |h2| <= 1.41*2^60 */
1219     /* |h6| <= 1.41*2^60 */
1220 
1221     carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
1222     carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
1223     /* |h2| <= 2^25; from now on fits into int32 unchanged */
1224     /* |h6| <= 2^25; from now on fits into int32 unchanged */
1225     /* |h3| <= 1.71*2^59 */
1226     /* |h7| <= 1.71*2^59 */
1227 
1228     carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
1229     carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;
1230     /* |h3| <= 2^24; from now on fits into int32 unchanged */
1231     /* |h7| <= 2^24; from now on fits into int32 unchanged */
1232     /* |h4| <= 1.72*2^34 */
1233     /* |h8| <= 1.41*2^60 */
1234 
1235     carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
1236     carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;
1237     /* |h4| <= 2^25; from now on fits into int32 unchanged */
1238     /* |h8| <= 2^25; from now on fits into int32 unchanged */
1239     /* |h5| <= 1.01*2^24 */
1240     /* |h9| <= 1.71*2^59 */
1241 
1242     carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
1243     /* |h9| <= 2^24; from now on fits into int32 unchanged */
1244     /* |h0| <= 1.1*2^39 */
1245 
1246     carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
1247     /* |h0| <= 2^25; from now on fits into int32 unchanged */
1248     /* |h1| <= 1.01*2^24 */
1249 
1250     h[0] = (int32_t)h0;
1251     h[1] = (int32_t)h1;
1252     h[2] = (int32_t)h2;
1253     h[3] = (int32_t)h3;
1254     h[4] = (int32_t)h4;
1255     h[5] = (int32_t)h5;
1256     h[6] = (int32_t)h6;
1257     h[7] = (int32_t)h7;
1258     h[8] = (int32_t)h8;
1259     h[9] = (int32_t)h9;
1260 }
1261 
1262 /*
1263  * h = f * f
1264  *
1265  * Can overlap h with f.
1266  *
1267  * Preconditions:
1268  *    |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
1269  *
1270  * Postconditions:
1271  *    |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
1272  *
1273  * See fe_mul.c for discussion of implementation strategy.
1274  */
fe_sq(fe h,const fe f)1275 static void fe_sq(fe h, const fe f)
1276 {
1277     int32_t f0 = f[0];
1278     int32_t f1 = f[1];
1279     int32_t f2 = f[2];
1280     int32_t f3 = f[3];
1281     int32_t f4 = f[4];
1282     int32_t f5 = f[5];
1283     int32_t f6 = f[6];
1284     int32_t f7 = f[7];
1285     int32_t f8 = f[8];
1286     int32_t f9 = f[9];
1287     int32_t f0_2 = 2 * f0;
1288     int32_t f1_2 = 2 * f1;
1289     int32_t f2_2 = 2 * f2;
1290     int32_t f3_2 = 2 * f3;
1291     int32_t f4_2 = 2 * f4;
1292     int32_t f5_2 = 2 * f5;
1293     int32_t f6_2 = 2 * f6;
1294     int32_t f7_2 = 2 * f7;
1295     int32_t f5_38 = 38 * f5; /* 1.959375*2^30 */
1296     int32_t f6_19 = 19 * f6; /* 1.959375*2^30 */
1297     int32_t f7_38 = 38 * f7; /* 1.959375*2^30 */
1298     int32_t f8_19 = 19 * f8; /* 1.959375*2^30 */
1299     int32_t f9_38 = 38 * f9; /* 1.959375*2^30 */
1300     int64_t f0f0    = f0   * (int64_t) f0;
1301     int64_t f0f1_2  = f0_2 * (int64_t) f1;
1302     int64_t f0f2_2  = f0_2 * (int64_t) f2;
1303     int64_t f0f3_2  = f0_2 * (int64_t) f3;
1304     int64_t f0f4_2  = f0_2 * (int64_t) f4;
1305     int64_t f0f5_2  = f0_2 * (int64_t) f5;
1306     int64_t f0f6_2  = f0_2 * (int64_t) f6;
1307     int64_t f0f7_2  = f0_2 * (int64_t) f7;
1308     int64_t f0f8_2  = f0_2 * (int64_t) f8;
1309     int64_t f0f9_2  = f0_2 * (int64_t) f9;
1310     int64_t f1f1_2  = f1_2 * (int64_t) f1;
1311     int64_t f1f2_2  = f1_2 * (int64_t) f2;
1312     int64_t f1f3_4  = f1_2 * (int64_t) f3_2;
1313     int64_t f1f4_2  = f1_2 * (int64_t) f4;
1314     int64_t f1f5_4  = f1_2 * (int64_t) f5_2;
1315     int64_t f1f6_2  = f1_2 * (int64_t) f6;
1316     int64_t f1f7_4  = f1_2 * (int64_t) f7_2;
1317     int64_t f1f8_2  = f1_2 * (int64_t) f8;
1318     int64_t f1f9_76 = f1_2 * (int64_t) f9_38;
1319     int64_t f2f2    = f2   * (int64_t) f2;
1320     int64_t f2f3_2  = f2_2 * (int64_t) f3;
1321     int64_t f2f4_2  = f2_2 * (int64_t) f4;
1322     int64_t f2f5_2  = f2_2 * (int64_t) f5;
1323     int64_t f2f6_2  = f2_2 * (int64_t) f6;
1324     int64_t f2f7_2  = f2_2 * (int64_t) f7;
1325     int64_t f2f8_38 = f2_2 * (int64_t) f8_19;
1326     int64_t f2f9_38 = f2   * (int64_t) f9_38;
1327     int64_t f3f3_2  = f3_2 * (int64_t) f3;
1328     int64_t f3f4_2  = f3_2 * (int64_t) f4;
1329     int64_t f3f5_4  = f3_2 * (int64_t) f5_2;
1330     int64_t f3f6_2  = f3_2 * (int64_t) f6;
1331     int64_t f3f7_76 = f3_2 * (int64_t) f7_38;
1332     int64_t f3f8_38 = f3_2 * (int64_t) f8_19;
1333     int64_t f3f9_76 = f3_2 * (int64_t) f9_38;
1334     int64_t f4f4    = f4   * (int64_t) f4;
1335     int64_t f4f5_2  = f4_2 * (int64_t) f5;
1336     int64_t f4f6_38 = f4_2 * (int64_t) f6_19;
1337     int64_t f4f7_38 = f4   * (int64_t) f7_38;
1338     int64_t f4f8_38 = f4_2 * (int64_t) f8_19;
1339     int64_t f4f9_38 = f4   * (int64_t) f9_38;
1340     int64_t f5f5_38 = f5   * (int64_t) f5_38;
1341     int64_t f5f6_38 = f5_2 * (int64_t) f6_19;
1342     int64_t f5f7_76 = f5_2 * (int64_t) f7_38;
1343     int64_t f5f8_38 = f5_2 * (int64_t) f8_19;
1344     int64_t f5f9_76 = f5_2 * (int64_t) f9_38;
1345     int64_t f6f6_19 = f6   * (int64_t) f6_19;
1346     int64_t f6f7_38 = f6   * (int64_t) f7_38;
1347     int64_t f6f8_38 = f6_2 * (int64_t) f8_19;
1348     int64_t f6f9_38 = f6   * (int64_t) f9_38;
1349     int64_t f7f7_38 = f7   * (int64_t) f7_38;
1350     int64_t f7f8_38 = f7_2 * (int64_t) f8_19;
1351     int64_t f7f9_76 = f7_2 * (int64_t) f9_38;
1352     int64_t f8f8_19 = f8   * (int64_t) f8_19;
1353     int64_t f8f9_38 = f8   * (int64_t) f9_38;
1354     int64_t f9f9_38 = f9   * (int64_t) f9_38;
1355     int64_t h0 = f0f0   + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38;
1356     int64_t h1 = f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38;
1357     int64_t h2 = f0f2_2 + f1f1_2  + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19;
1358     int64_t h3 = f0f3_2 + f1f2_2  + f4f9_38 + f5f8_38 + f6f7_38;
1359     int64_t h4 = f0f4_2 + f1f3_4  + f2f2    + f5f9_76 + f6f8_38 + f7f7_38;
1360     int64_t h5 = f0f5_2 + f1f4_2  + f2f3_2  + f6f9_38 + f7f8_38;
1361     int64_t h6 = f0f6_2 + f1f5_4  + f2f4_2  + f3f3_2  + f7f9_76 + f8f8_19;
1362     int64_t h7 = f0f7_2 + f1f6_2  + f2f5_2  + f3f4_2  + f8f9_38;
1363     int64_t h8 = f0f8_2 + f1f7_4  + f2f6_2  + f3f5_4  + f4f4    + f9f9_38;
1364     int64_t h9 = f0f9_2 + f1f8_2  + f2f7_2  + f3f6_2  + f4f5_2;
1365     int64_t carry0;
1366     int64_t carry1;
1367     int64_t carry2;
1368     int64_t carry3;
1369     int64_t carry4;
1370     int64_t carry5;
1371     int64_t carry6;
1372     int64_t carry7;
1373     int64_t carry8;
1374     int64_t carry9;
1375 
1376     carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
1377     carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
1378 
1379     carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
1380     carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
1381 
1382     carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
1383     carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
1384 
1385     carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
1386     carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;
1387 
1388     carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
1389     carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;
1390 
1391     carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
1392 
1393     carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
1394 
1395     h[0] = (int32_t)h0;
1396     h[1] = (int32_t)h1;
1397     h[2] = (int32_t)h2;
1398     h[3] = (int32_t)h3;
1399     h[4] = (int32_t)h4;
1400     h[5] = (int32_t)h5;
1401     h[6] = (int32_t)h6;
1402     h[7] = (int32_t)h7;
1403     h[8] = (int32_t)h8;
1404     h[9] = (int32_t)h9;
1405 }
1406 
fe_invert(fe out,const fe z)1407 static void fe_invert(fe out, const fe z)
1408 {
1409     fe t0;
1410     fe t1;
1411     fe t2;
1412     fe t3;
1413     int i;
1414 
1415     /*
1416      * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as
1417      * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11.
1418      */
1419 
1420     /* t0 = z ** 2 */
1421     fe_sq(t0, z);
1422 
1423     /* t1 = t0 ** (2 ** 2) = z ** 8 */
1424     fe_sq(t1, t0);
1425     fe_sq(t1, t1);
1426 
1427     /* t1 = z * t1 = z ** 9 */
1428     fe_mul(t1, z, t1);
1429     /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */
1430     fe_mul(t0, t0, t1);
1431 
1432     /* t2 = t0 ** 2 = z ** 22 */
1433     fe_sq(t2, t0);
1434 
1435     /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */
1436     fe_mul(t1, t1, t2);
1437 
1438     /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */
1439     fe_sq(t2, t1);
1440     for (i = 1; i < 5; ++i) {
1441         fe_sq(t2, t2);
1442     }
1443 
1444     /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */
1445     fe_mul(t1, t2, t1);
1446 
1447     /* Continuing similarly... */
1448 
1449     /* t2 = z ** (2 ** 20 - 1) */
1450     fe_sq(t2, t1);
1451     for (i = 1; i < 10; ++i) {
1452         fe_sq(t2, t2);
1453     }
1454     fe_mul(t2, t2, t1);
1455 
1456     /* t2 = z ** (2 ** 40 - 1) */
1457     fe_sq(t3, t2);
1458     for (i = 1; i < 20; ++i) {
1459         fe_sq(t3, t3);
1460     }
1461     fe_mul(t2, t3, t2);
1462 
1463     /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */
1464     for (i = 0; i < 10; ++i) {
1465         fe_sq(t2, t2);
1466     }
1467     /* t1 = z ** (2 ** 50 - 1) */
1468     fe_mul(t1, t2, t1);
1469 
1470     /* t2 = z ** (2 ** 100 - 1) */
1471     fe_sq(t2, t1);
1472     for (i = 1; i < 50; ++i) {
1473         fe_sq(t2, t2);
1474     }
1475     fe_mul(t2, t2, t1);
1476 
1477     /* t2 = z ** (2 ** 200 - 1) */
1478     fe_sq(t3, t2);
1479     for (i = 1; i < 100; ++i) {
1480         fe_sq(t3, t3);
1481     }
1482     fe_mul(t2, t3, t2);
1483 
1484     /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */
1485     fe_sq(t2, t2);
1486     for (i = 1; i < 50; ++i) {
1487         fe_sq(t2, t2);
1488     }
1489 
1490     /* t1 = z ** (2 ** 250 - 1) */
1491     fe_mul(t1, t2, t1);
1492 
1493     /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */
1494     fe_sq(t1, t1);
1495     for (i = 1; i < 5; ++i) {
1496         fe_sq(t1, t1);
1497     }
1498 
1499     /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */
1500     fe_mul(out, t1, t0);
1501 }
1502 
1503 /*
1504  * h = -f
1505  *
1506  * Preconditions:
1507  *    |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
1508  *
1509  * Postconditions:
1510  *    |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
1511  */
fe_neg(fe h,const fe f)1512 static void fe_neg(fe h, const fe f)
1513 {
1514     unsigned i;
1515 
1516     for (i = 0; i < 10; i++) {
1517         h[i] = -f[i];
1518     }
1519 }
1520 
1521 /*
1522  * Replace (f,g) with (g,g) if b == 1;
1523  * replace (f,g) with (f,g) if b == 0.
1524  *
1525  * Preconditions: b in {0,1}.
1526  */
fe_cmov(fe f,const fe g,unsigned b)1527 static void fe_cmov(fe f, const fe g, unsigned b)
1528 {
1529     size_t i;
1530 
1531     b = 0-b;
1532     for (i = 0; i < 10; i++) {
1533         int32_t x = f[i] ^ g[i];
1534         x &= b;
1535         f[i] ^= x;
1536     }
1537 }
1538 
1539 /*
1540  * return 0 if f == 0
1541  * return 1 if f != 0
1542  *
1543  * Preconditions:
1544  *    |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
1545  */
fe_isnonzero(const fe f)1546 static int fe_isnonzero(const fe f)
1547 {
1548     uint8_t s[32];
1549     static const uint8_t zero[32] = {0};
1550 
1551     fe_tobytes(s, f);
1552 
1553     return CRYPTO_memcmp(s, zero, sizeof(zero)) != 0;
1554 }
1555 
1556 /*
1557  * return 1 if f is in {1,3,5,...,q-2}
1558  * return 0 if f is in {0,2,4,...,q-1}
1559  *
1560  * Preconditions:
1561  *    |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
1562  */
fe_isnegative(const fe f)1563 static int fe_isnegative(const fe f)
1564 {
1565     uint8_t s[32];
1566 
1567     fe_tobytes(s, f);
1568     return s[0] & 1;
1569 }
1570 
1571 /*
1572  * h = 2 * f * f
1573  *
1574  * Can overlap h with f.
1575  *
1576  * Preconditions:
1577  *    |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
1578  *
1579  * Postconditions:
1580  *    |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
1581  *
1582  * See fe_mul.c for discussion of implementation strategy.
1583  */
fe_sq2(fe h,const fe f)1584 static void fe_sq2(fe h, const fe f)
1585 {
1586     int32_t f0 = f[0];
1587     int32_t f1 = f[1];
1588     int32_t f2 = f[2];
1589     int32_t f3 = f[3];
1590     int32_t f4 = f[4];
1591     int32_t f5 = f[5];
1592     int32_t f6 = f[6];
1593     int32_t f7 = f[7];
1594     int32_t f8 = f[8];
1595     int32_t f9 = f[9];
1596     int32_t f0_2 = 2 * f0;
1597     int32_t f1_2 = 2 * f1;
1598     int32_t f2_2 = 2 * f2;
1599     int32_t f3_2 = 2 * f3;
1600     int32_t f4_2 = 2 * f4;
1601     int32_t f5_2 = 2 * f5;
1602     int32_t f6_2 = 2 * f6;
1603     int32_t f7_2 = 2 * f7;
1604     int32_t f5_38 = 38 * f5; /* 1.959375*2^30 */
1605     int32_t f6_19 = 19 * f6; /* 1.959375*2^30 */
1606     int32_t f7_38 = 38 * f7; /* 1.959375*2^30 */
1607     int32_t f8_19 = 19 * f8; /* 1.959375*2^30 */
1608     int32_t f9_38 = 38 * f9; /* 1.959375*2^30 */
1609     int64_t f0f0    = f0   * (int64_t) f0;
1610     int64_t f0f1_2  = f0_2 * (int64_t) f1;
1611     int64_t f0f2_2  = f0_2 * (int64_t) f2;
1612     int64_t f0f3_2  = f0_2 * (int64_t) f3;
1613     int64_t f0f4_2  = f0_2 * (int64_t) f4;
1614     int64_t f0f5_2  = f0_2 * (int64_t) f5;
1615     int64_t f0f6_2  = f0_2 * (int64_t) f6;
1616     int64_t f0f7_2  = f0_2 * (int64_t) f7;
1617     int64_t f0f8_2  = f0_2 * (int64_t) f8;
1618     int64_t f0f9_2  = f0_2 * (int64_t) f9;
1619     int64_t f1f1_2  = f1_2 * (int64_t) f1;
1620     int64_t f1f2_2  = f1_2 * (int64_t) f2;
1621     int64_t f1f3_4  = f1_2 * (int64_t) f3_2;
1622     int64_t f1f4_2  = f1_2 * (int64_t) f4;
1623     int64_t f1f5_4  = f1_2 * (int64_t) f5_2;
1624     int64_t f1f6_2  = f1_2 * (int64_t) f6;
1625     int64_t f1f7_4  = f1_2 * (int64_t) f7_2;
1626     int64_t f1f8_2  = f1_2 * (int64_t) f8;
1627     int64_t f1f9_76 = f1_2 * (int64_t) f9_38;
1628     int64_t f2f2    = f2   * (int64_t) f2;
1629     int64_t f2f3_2  = f2_2 * (int64_t) f3;
1630     int64_t f2f4_2  = f2_2 * (int64_t) f4;
1631     int64_t f2f5_2  = f2_2 * (int64_t) f5;
1632     int64_t f2f6_2  = f2_2 * (int64_t) f6;
1633     int64_t f2f7_2  = f2_2 * (int64_t) f7;
1634     int64_t f2f8_38 = f2_2 * (int64_t) f8_19;
1635     int64_t f2f9_38 = f2   * (int64_t) f9_38;
1636     int64_t f3f3_2  = f3_2 * (int64_t) f3;
1637     int64_t f3f4_2  = f3_2 * (int64_t) f4;
1638     int64_t f3f5_4  = f3_2 * (int64_t) f5_2;
1639     int64_t f3f6_2  = f3_2 * (int64_t) f6;
1640     int64_t f3f7_76 = f3_2 * (int64_t) f7_38;
1641     int64_t f3f8_38 = f3_2 * (int64_t) f8_19;
1642     int64_t f3f9_76 = f3_2 * (int64_t) f9_38;
1643     int64_t f4f4    = f4   * (int64_t) f4;
1644     int64_t f4f5_2  = f4_2 * (int64_t) f5;
1645     int64_t f4f6_38 = f4_2 * (int64_t) f6_19;
1646     int64_t f4f7_38 = f4   * (int64_t) f7_38;
1647     int64_t f4f8_38 = f4_2 * (int64_t) f8_19;
1648     int64_t f4f9_38 = f4   * (int64_t) f9_38;
1649     int64_t f5f5_38 = f5   * (int64_t) f5_38;
1650     int64_t f5f6_38 = f5_2 * (int64_t) f6_19;
1651     int64_t f5f7_76 = f5_2 * (int64_t) f7_38;
1652     int64_t f5f8_38 = f5_2 * (int64_t) f8_19;
1653     int64_t f5f9_76 = f5_2 * (int64_t) f9_38;
1654     int64_t f6f6_19 = f6   * (int64_t) f6_19;
1655     int64_t f6f7_38 = f6   * (int64_t) f7_38;
1656     int64_t f6f8_38 = f6_2 * (int64_t) f8_19;
1657     int64_t f6f9_38 = f6   * (int64_t) f9_38;
1658     int64_t f7f7_38 = f7   * (int64_t) f7_38;
1659     int64_t f7f8_38 = f7_2 * (int64_t) f8_19;
1660     int64_t f7f9_76 = f7_2 * (int64_t) f9_38;
1661     int64_t f8f8_19 = f8   * (int64_t) f8_19;
1662     int64_t f8f9_38 = f8   * (int64_t) f9_38;
1663     int64_t f9f9_38 = f9   * (int64_t) f9_38;
1664     int64_t h0 = f0f0   + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38;
1665     int64_t h1 = f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38;
1666     int64_t h2 = f0f2_2 + f1f1_2  + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19;
1667     int64_t h3 = f0f3_2 + f1f2_2  + f4f9_38 + f5f8_38 + f6f7_38;
1668     int64_t h4 = f0f4_2 + f1f3_4  + f2f2    + f5f9_76 + f6f8_38 + f7f7_38;
1669     int64_t h5 = f0f5_2 + f1f4_2  + f2f3_2  + f6f9_38 + f7f8_38;
1670     int64_t h6 = f0f6_2 + f1f5_4  + f2f4_2  + f3f3_2  + f7f9_76 + f8f8_19;
1671     int64_t h7 = f0f7_2 + f1f6_2  + f2f5_2  + f3f4_2  + f8f9_38;
1672     int64_t h8 = f0f8_2 + f1f7_4  + f2f6_2  + f3f5_4  + f4f4    + f9f9_38;
1673     int64_t h9 = f0f9_2 + f1f8_2  + f2f7_2  + f3f6_2  + f4f5_2;
1674     int64_t carry0;
1675     int64_t carry1;
1676     int64_t carry2;
1677     int64_t carry3;
1678     int64_t carry4;
1679     int64_t carry5;
1680     int64_t carry6;
1681     int64_t carry7;
1682     int64_t carry8;
1683     int64_t carry9;
1684 
1685     h0 += h0;
1686     h1 += h1;
1687     h2 += h2;
1688     h3 += h3;
1689     h4 += h4;
1690     h5 += h5;
1691     h6 += h6;
1692     h7 += h7;
1693     h8 += h8;
1694     h9 += h9;
1695 
1696     carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
1697     carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
1698 
1699     carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
1700     carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
1701 
1702     carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
1703     carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
1704 
1705     carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
1706     carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;
1707 
1708     carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
1709     carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;
1710 
1711     carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
1712 
1713     carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
1714 
1715     h[0] = (int32_t)h0;
1716     h[1] = (int32_t)h1;
1717     h[2] = (int32_t)h2;
1718     h[3] = (int32_t)h3;
1719     h[4] = (int32_t)h4;
1720     h[5] = (int32_t)h5;
1721     h[6] = (int32_t)h6;
1722     h[7] = (int32_t)h7;
1723     h[8] = (int32_t)h8;
1724     h[9] = (int32_t)h9;
1725 }
1726 
fe_pow22523(fe out,const fe z)1727 static void fe_pow22523(fe out, const fe z)
1728 {
1729     fe t0;
1730     fe t1;
1731     fe t2;
1732     int i;
1733 
1734     fe_sq(t0, z);
1735     fe_sq(t1, t0);
1736     for (i = 1; i < 2; ++i) {
1737         fe_sq(t1, t1);
1738     }
1739     fe_mul(t1, z, t1);
1740     fe_mul(t0, t0, t1);
1741     fe_sq(t0, t0);
1742     fe_mul(t0, t1, t0);
1743     fe_sq(t1, t0);
1744     for (i = 1; i < 5; ++i) {
1745         fe_sq(t1, t1);
1746     }
1747     fe_mul(t0, t1, t0);
1748     fe_sq(t1, t0);
1749     for (i = 1; i < 10; ++i) {
1750         fe_sq(t1, t1);
1751     }
1752     fe_mul(t1, t1, t0);
1753     fe_sq(t2, t1);
1754     for (i = 1; i < 20; ++i) {
1755         fe_sq(t2, t2);
1756     }
1757     fe_mul(t1, t2, t1);
1758     fe_sq(t1, t1);
1759     for (i = 1; i < 10; ++i) {
1760         fe_sq(t1, t1);
1761     }
1762     fe_mul(t0, t1, t0);
1763     fe_sq(t1, t0);
1764     for (i = 1; i < 50; ++i) {
1765         fe_sq(t1, t1);
1766     }
1767     fe_mul(t1, t1, t0);
1768     fe_sq(t2, t1);
1769     for (i = 1; i < 100; ++i) {
1770         fe_sq(t2, t2);
1771     }
1772     fe_mul(t1, t2, t1);
1773     fe_sq(t1, t1);
1774     for (i = 1; i < 50; ++i) {
1775         fe_sq(t1, t1);
1776     }
1777     fe_mul(t0, t1, t0);
1778     fe_sq(t0, t0);
1779     for (i = 1; i < 2; ++i) {
1780         fe_sq(t0, t0);
1781     }
1782     fe_mul(out, t0, z);
1783 }
1784 
1785 /*
1786  * ge means group element.
1787  *
1788  * Here the group is the set of pairs (x,y) of field elements (see fe.h)
1789  * satisfying -x^2 + y^2 = 1 + d x^2y^2
1790  * where d = -121665/121666.
1791  *
1792  * Representations:
1793  *   ge_p2 (projective): (X:Y:Z) satisfying x=X/Z, y=Y/Z
1794  *   ge_p3 (extended): (X:Y:Z:T) satisfying x=X/Z, y=Y/Z, XY=ZT
1795  *   ge_p1p1 (completed): ((X:Z),(Y:T)) satisfying x=X/Z, y=Y/T
1796  *   ge_precomp (Duif): (y+x,y-x,2dxy)
1797  */
1798 typedef struct {
1799     fe X;
1800     fe Y;
1801     fe Z;
1802 } ge_p2;
1803 
1804 typedef struct {
1805     fe X;
1806     fe Y;
1807     fe Z;
1808     fe T;
1809 } ge_p3;
1810 
1811 typedef struct {
1812     fe X;
1813     fe Y;
1814     fe Z;
1815     fe T;
1816 } ge_p1p1;
1817 
1818 typedef struct {
1819     fe yplusx;
1820     fe yminusx;
1821     fe xy2d;
1822 } ge_precomp;
1823 
1824 typedef struct {
1825     fe YplusX;
1826     fe YminusX;
1827     fe Z;
1828     fe T2d;
1829 } ge_cached;
1830 
ge_tobytes(uint8_t * s,const ge_p2 * h)1831 static void ge_tobytes(uint8_t *s, const ge_p2 *h)
1832 {
1833     fe recip;
1834     fe x;
1835     fe y;
1836 
1837     fe_invert(recip, h->Z);
1838     fe_mul(x, h->X, recip);
1839     fe_mul(y, h->Y, recip);
1840     fe_tobytes(s, y);
1841     s[31] ^= fe_isnegative(x) << 7;
1842 }
1843 
ge_p3_tobytes(uint8_t * s,const ge_p3 * h)1844 static void ge_p3_tobytes(uint8_t *s, const ge_p3 *h)
1845 {
1846     fe recip;
1847     fe x;
1848     fe y;
1849 
1850     fe_invert(recip, h->Z);
1851     fe_mul(x, h->X, recip);
1852     fe_mul(y, h->Y, recip);
1853     fe_tobytes(s, y);
1854     s[31] ^= fe_isnegative(x) << 7;
1855 }
1856 
1857 static const fe d = {
1858     -10913610, 13857413, -15372611, 6949391,   114729,
1859     -8787816,  -6275908, -3247719,  -18696448, -12055116
1860 };
1861 
1862 static const fe sqrtm1 = {
1863     -32595792, -7943725,  9377950,  3500415, 12389472,
1864     -272473,   -25146209, -2005654, 326686,  11406482
1865 };
1866 
ge_frombytes_vartime(ge_p3 * h,const uint8_t * s)1867 static int ge_frombytes_vartime(ge_p3 *h, const uint8_t *s)
1868 {
1869     fe u;
1870     fe v;
1871     fe w;
1872     fe vxx;
1873     fe check;
1874 
1875     fe_frombytes(h->Y, s);
1876     fe_1(h->Z);
1877     fe_sq(u, h->Y);
1878     fe_mul(v, u, d);
1879     fe_sub(u, u, h->Z); /* u = y^2-1 */
1880     fe_add(v, v, h->Z); /* v = dy^2+1 */
1881 
1882     fe_mul(w, u, v); /* w = u*v */
1883 
1884     fe_pow22523(h->X, w); /* x = w^((q-5)/8) */
1885     fe_mul(h->X, h->X, u); /* x = u * w^((q-5)/8) */
1886 
1887     fe_sq(vxx, h->X);
1888     fe_mul(vxx, vxx, v);
1889     fe_sub(check, vxx, u); /* vx^2-u */
1890     if (fe_isnonzero(check)) {
1891         fe_add(check, vxx, u); /* vx^2+u */
1892         if (fe_isnonzero(check)) {
1893             return -1;
1894         }
1895         fe_mul(h->X, h->X, sqrtm1);
1896     }
1897 
1898     if (fe_isnegative(h->X) != (s[31] >> 7)) {
1899         fe_neg(h->X, h->X);
1900     }
1901 
1902     fe_mul(h->T, h->X, h->Y);
1903     return 0;
1904 }
1905 
ge_p2_0(ge_p2 * h)1906 static void ge_p2_0(ge_p2 *h)
1907 {
1908     fe_0(h->X);
1909     fe_1(h->Y);
1910     fe_1(h->Z);
1911 }
1912 
ge_p3_0(ge_p3 * h)1913 static void ge_p3_0(ge_p3 *h)
1914 {
1915     fe_0(h->X);
1916     fe_1(h->Y);
1917     fe_1(h->Z);
1918     fe_0(h->T);
1919 }
1920 
ge_precomp_0(ge_precomp * h)1921 static void ge_precomp_0(ge_precomp *h)
1922 {
1923     fe_1(h->yplusx);
1924     fe_1(h->yminusx);
1925     fe_0(h->xy2d);
1926 }
1927 
1928 /* r = p */
ge_p3_to_p2(ge_p2 * r,const ge_p3 * p)1929 static void ge_p3_to_p2(ge_p2 *r, const ge_p3 *p)
1930 {
1931     fe_copy(r->X, p->X);
1932     fe_copy(r->Y, p->Y);
1933     fe_copy(r->Z, p->Z);
1934 }
1935 
1936 static const fe d2 = {
1937     -21827239, -5839606,  -30745221, 13898782, 229458,
1938     15978800,  -12551817, -6495438,  29715968, 9444199
1939 };
1940 
1941 /* r = p */
ge_p3_to_cached(ge_cached * r,const ge_p3 * p)1942 static void ge_p3_to_cached(ge_cached *r, const ge_p3 *p)
1943 {
1944     fe_add(r->YplusX, p->Y, p->X);
1945     fe_sub(r->YminusX, p->Y, p->X);
1946     fe_copy(r->Z, p->Z);
1947     fe_mul(r->T2d, p->T, d2);
1948 }
1949 
1950 /* r = p */
ge_p1p1_to_p2(ge_p2 * r,const ge_p1p1 * p)1951 static void ge_p1p1_to_p2(ge_p2 *r, const ge_p1p1 *p)
1952 {
1953     fe_mul(r->X, p->X, p->T);
1954     fe_mul(r->Y, p->Y, p->Z);
1955     fe_mul(r->Z, p->Z, p->T);
1956 }
1957 
1958 /* r = p */
ge_p1p1_to_p3(ge_p3 * r,const ge_p1p1 * p)1959 static void ge_p1p1_to_p3(ge_p3 *r, const ge_p1p1 *p)
1960 {
1961     fe_mul(r->X, p->X, p->T);
1962     fe_mul(r->Y, p->Y, p->Z);
1963     fe_mul(r->Z, p->Z, p->T);
1964     fe_mul(r->T, p->X, p->Y);
1965 }
1966 
1967 /* r = 2 * p */
ge_p2_dbl(ge_p1p1 * r,const ge_p2 * p)1968 static void ge_p2_dbl(ge_p1p1 *r, const ge_p2 *p)
1969 {
1970     fe t0;
1971 
1972     fe_sq(r->X, p->X);
1973     fe_sq(r->Z, p->Y);
1974     fe_sq2(r->T, p->Z);
1975     fe_add(r->Y, p->X, p->Y);
1976     fe_sq(t0, r->Y);
1977     fe_add(r->Y, r->Z, r->X);
1978     fe_sub(r->Z, r->Z, r->X);
1979     fe_sub(r->X, t0, r->Y);
1980     fe_sub(r->T, r->T, r->Z);
1981 }
1982 
1983 /* r = 2 * p */
ge_p3_dbl(ge_p1p1 * r,const ge_p3 * p)1984 static void ge_p3_dbl(ge_p1p1 *r, const ge_p3 *p)
1985 {
1986     ge_p2 q;
1987     ge_p3_to_p2(&q, p);
1988     ge_p2_dbl(r, &q);
1989 }
1990 
1991 /* r = p + q */
ge_madd(ge_p1p1 * r,const ge_p3 * p,const ge_precomp * q)1992 static void ge_madd(ge_p1p1 *r, const ge_p3 *p, const ge_precomp *q)
1993 {
1994     fe t0;
1995 
1996     fe_add(r->X, p->Y, p->X);
1997     fe_sub(r->Y, p->Y, p->X);
1998     fe_mul(r->Z, r->X, q->yplusx);
1999     fe_mul(r->Y, r->Y, q->yminusx);
2000     fe_mul(r->T, q->xy2d, p->T);
2001     fe_add(t0, p->Z, p->Z);
2002     fe_sub(r->X, r->Z, r->Y);
2003     fe_add(r->Y, r->Z, r->Y);
2004     fe_add(r->Z, t0, r->T);
2005     fe_sub(r->T, t0, r->T);
2006 }
2007 
2008 /* r = p - q */
ge_msub(ge_p1p1 * r,const ge_p3 * p,const ge_precomp * q)2009 static void ge_msub(ge_p1p1 *r, const ge_p3 *p, const ge_precomp *q)
2010 {
2011     fe t0;
2012 
2013     fe_add(r->X, p->Y, p->X);
2014     fe_sub(r->Y, p->Y, p->X);
2015     fe_mul(r->Z, r->X, q->yminusx);
2016     fe_mul(r->Y, r->Y, q->yplusx);
2017     fe_mul(r->T, q->xy2d, p->T);
2018     fe_add(t0, p->Z, p->Z);
2019     fe_sub(r->X, r->Z, r->Y);
2020     fe_add(r->Y, r->Z, r->Y);
2021     fe_sub(r->Z, t0, r->T);
2022     fe_add(r->T, t0, r->T);
2023 }
2024 
2025 /* r = p + q */
ge_add(ge_p1p1 * r,const ge_p3 * p,const ge_cached * q)2026 static void ge_add(ge_p1p1 *r, const ge_p3 *p, const ge_cached *q)
2027 {
2028     fe t0;
2029 
2030     fe_add(r->X, p->Y, p->X);
2031     fe_sub(r->Y, p->Y, p->X);
2032     fe_mul(r->Z, r->X, q->YplusX);
2033     fe_mul(r->Y, r->Y, q->YminusX);
2034     fe_mul(r->T, q->T2d, p->T);
2035     fe_mul(r->X, p->Z, q->Z);
2036     fe_add(t0, r->X, r->X);
2037     fe_sub(r->X, r->Z, r->Y);
2038     fe_add(r->Y, r->Z, r->Y);
2039     fe_add(r->Z, t0, r->T);
2040     fe_sub(r->T, t0, r->T);
2041 }
2042 
2043 /* r = p - q */
ge_sub(ge_p1p1 * r,const ge_p3 * p,const ge_cached * q)2044 static void ge_sub(ge_p1p1 *r, const ge_p3 *p, const ge_cached *q)
2045 {
2046     fe t0;
2047 
2048     fe_add(r->X, p->Y, p->X);
2049     fe_sub(r->Y, p->Y, p->X);
2050     fe_mul(r->Z, r->X, q->YminusX);
2051     fe_mul(r->Y, r->Y, q->YplusX);
2052     fe_mul(r->T, q->T2d, p->T);
2053     fe_mul(r->X, p->Z, q->Z);
2054     fe_add(t0, r->X, r->X);
2055     fe_sub(r->X, r->Z, r->Y);
2056     fe_add(r->Y, r->Z, r->Y);
2057     fe_sub(r->Z, t0, r->T);
2058     fe_add(r->T, t0, r->T);
2059 }
2060 
equal(signed char b,signed char c)2061 static uint8_t equal(signed char b, signed char c)
2062 {
2063     uint8_t ub = b;
2064     uint8_t uc = c;
2065     uint8_t x = ub ^ uc; /* 0: yes; 1..255: no */
2066     uint32_t y = x;      /* 0: yes; 1..255: no */
2067     y -= 1;              /* 4294967295: yes; 0..254: no */
2068     y >>= 31;            /* 1: yes; 0: no */
2069     return y;
2070 }
2071 
cmov(ge_precomp * t,const ge_precomp * u,uint8_t b)2072 static void cmov(ge_precomp *t, const ge_precomp *u, uint8_t b)
2073 {
2074     fe_cmov(t->yplusx, u->yplusx, b);
2075     fe_cmov(t->yminusx, u->yminusx, b);
2076     fe_cmov(t->xy2d, u->xy2d, b);
2077 }
2078 
2079 /* k25519Precomp[i][j] = (j+1)*256^i*B */
2080 static const ge_precomp k25519Precomp[32][8] = {
2081     {
2082         {
2083             {25967493, -14356035, 29566456, 3660896, -12694345, 4014787,
2084              27544626, -11754271, -6079156, 2047605},
2085             {-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692,
2086              5043384, 19500929, -15469378},
2087             {-8738181, 4489570, 9688441, -14785194, 10184609, -12363380,
2088              29287919, 11864899, -24514362, -4438546},
2089         },
2090         {
2091             {-12815894, -12976347, -21581243, 11784320, -25355658, -2750717,
2092              -11717903, -3814571, -358445, -10211303},
2093             {-21703237, 6903825, 27185491, 6451973, -29577724, -9554005,
2094              -15616551, 11189268, -26829678, -5319081},
2095             {26966642, 11152617, 32442495, 15396054, 14353839, -12752335,
2096              -3128826, -9541118, -15472047, -4166697},
2097         },
2098         {
2099             {15636291, -9688557, 24204773, -7912398, 616977, -16685262,
2100              27787600, -14772189, 28944400, -1550024},
2101             {16568933, 4717097, -11556148, -1102322, 15682896, -11807043,
2102              16354577, -11775962, 7689662, 11199574},
2103             {30464156, -5976125, -11779434, -15670865, 23220365, 15915852,
2104              7512774, 10017326, -17749093, -9920357},
2105         },
2106         {
2107             {-17036878, 13921892, 10945806, -6033431, 27105052, -16084379,
2108              -28926210, 15006023, 3284568, -6276540},
2109             {23599295, -8306047, -11193664, -7687416, 13236774, 10506355,
2110              7464579, 9656445, 13059162, 10374397},
2111             {7798556, 16710257, 3033922, 2874086, 28997861, 2835604, 32406664,
2112              -3839045, -641708, -101325},
2113         },
2114         {
2115             {10861363, 11473154, 27284546, 1981175, -30064349, 12577861,
2116              32867885, 14515107, -15438304, 10819380},
2117             {4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668,
2118              12483688, -12668491, 5581306},
2119             {19563160, 16186464, -29386857, 4097519, 10237984, -4348115,
2120              28542350, 13850243, -23678021, -15815942},
2121         },
2122         {
2123             {-15371964, -12862754, 32573250, 4720197, -26436522, 5875511,
2124              -19188627, -15224819, -9818940, -12085777},
2125             {-8549212, 109983, 15149363, 2178705, 22900618, 4543417, 3044240,
2126              -15689887, 1762328, 14866737},
2127             {-18199695, -15951423, -10473290, 1707278, -17185920, 3916101,
2128              -28236412, 3959421, 27914454, 4383652},
2129         },
2130         {
2131             {5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852,
2132              5230134, -23952439, -15175766},
2133             {-30269007, -3463509, 7665486, 10083793, 28475525, 1649722,
2134              20654025, 16520125, 30598449, 7715701},
2135             {28881845, 14381568, 9657904, 3680757, -20181635, 7843316,
2136              -31400660, 1370708, 29794553, -1409300},
2137         },
2138         {
2139             {14499471, -2729599, -33191113, -4254652, 28494862, 14271267,
2140              30290735, 10876454, -33154098, 2381726},
2141             {-7195431, -2655363, -14730155, 462251, -27724326, 3941372,
2142              -6236617, 3696005, -32300832, 15351955},
2143             {27431194, 8222322, 16448760, -3907995, -18707002, 11938355,
2144              -32961401, -2970515, 29551813, 10109425},
2145         },
2146     },
2147     {
2148         {
2149             {-13657040, -13155431, -31283750, 11777098, 21447386, 6519384,
2150              -2378284, -1627556, 10092783, -4764171},
2151             {27939166, 14210322, 4677035, 16277044, -22964462, -12398139,
2152              -32508754, 12005538, -17810127, 12803510},
2153             {17228999, -15661624, -1233527, 300140, -1224870, -11714777,
2154              30364213, -9038194, 18016357, 4397660},
2155         },
2156         {
2157             {-10958843, -7690207, 4776341, -14954238, 27850028, -15602212,
2158              -26619106, 14544525, -17477504, 982639},
2159             {29253598, 15796703, -2863982, -9908884, 10057023, 3163536, 7332899,
2160              -4120128, -21047696, 9934963},
2161             {5793303, 16271923, -24131614, -10116404, 29188560, 1206517,
2162              -14747930, 4559895, -30123922, -10897950},
2163         },
2164         {
2165             {-27643952, -11493006, 16282657, -11036493, 28414021, -15012264,
2166              24191034, 4541697, -13338309, 5500568},
2167             {12650548, -1497113, 9052871, 11355358, -17680037, -8400164,
2168              -17430592, 12264343, 10874051, 13524335},
2169             {25556948, -3045990, 714651, 2510400, 23394682, -10415330, 33119038,
2170              5080568, -22528059, 5376628},
2171         },
2172         {
2173             {-26088264, -4011052, -17013699, -3537628, -6726793, 1920897,
2174              -22321305, -9447443, 4535768, 1569007},
2175             {-2255422, 14606630, -21692440, -8039818, 28430649, 8775819,
2176              -30494562, 3044290, 31848280, 12543772},
2177             {-22028579, 2943893, -31857513, 6777306, 13784462, -4292203,
2178              -27377195, -2062731, 7718482, 14474653},
2179         },
2180         {
2181             {2385315, 2454213, -22631320, 46603, -4437935, -15680415, 656965,
2182              -7236665, 24316168, -5253567},
2183             {13741529, 10911568, -33233417, -8603737, -20177830, -1033297,
2184              33040651, -13424532, -20729456, 8321686},
2185             {21060490, -2212744, 15712757, -4336099, 1639040, 10656336,
2186              23845965, -11874838, -9984458, 608372},
2187         },
2188         {
2189             {-13672732, -15087586, -10889693, -7557059, -6036909, 11305547,
2190              1123968, -6780577, 27229399, 23887},
2191             {-23244140, -294205, -11744728, 14712571, -29465699, -2029617,
2192              12797024, -6440308, -1633405, 16678954},
2193             {-29500620, 4770662, -16054387, 14001338, 7830047, 9564805,
2194              -1508144, -4795045, -17169265, 4904953},
2195         },
2196         {
2197             {24059557, 14617003, 19037157, -15039908, 19766093, -14906429,
2198              5169211, 16191880, 2128236, -4326833},
2199             {-16981152, 4124966, -8540610, -10653797, 30336522, -14105247,
2200              -29806336, 916033, -6882542, -2986532},
2201             {-22630907, 12419372, -7134229, -7473371, -16478904, 16739175,
2202              285431, 2763829, 15736322, 4143876},
2203         },
2204         {
2205             {2379352, 11839345, -4110402, -5988665, 11274298, 794957, 212801,
2206              -14594663, 23527084, -16458268},
2207             {33431127, -11130478, -17838966, -15626900, 8909499, 8376530,
2208              -32625340, 4087881, -15188911, -14416214},
2209             {1767683, 7197987, -13205226, -2022635, -13091350, 448826, 5799055,
2210              4357868, -4774191, -16323038},
2211         },
2212     },
2213     {
2214         {
2215             {6721966, 13833823, -23523388, -1551314, 26354293, -11863321,
2216              23365147, -3949732, 7390890, 2759800},
2217             {4409041, 2052381, 23373853, 10530217, 7676779, -12885954, 21302353,
2218              -4264057, 1244380, -12919645},
2219             {-4421239, 7169619, 4982368, -2957590, 30256825, -2777540, 14086413,
2220              9208236, 15886429, 16489664},
2221         },
2222         {
2223             {1996075, 10375649, 14346367, 13311202, -6874135, -16438411,
2224              -13693198, 398369, -30606455, -712933},
2225             {-25307465, 9795880, -2777414, 14878809, -33531835, 14780363,
2226              13348553, 12076947, -30836462, 5113182},
2227             {-17770784, 11797796, 31950843, 13929123, -25888302, 12288344,
2228              -30341101, -7336386, 13847711, 5387222},
2229         },
2230         {
2231             {-18582163, -3416217, 17824843, -2340966, 22744343, -10442611,
2232              8763061, 3617786, -19600662, 10370991},
2233             {20246567, -14369378, 22358229, -543712, 18507283, -10413996,
2234              14554437, -8746092, 32232924, 16763880},
2235             {9648505, 10094563, 26416693, 14745928, -30374318, -6472621,
2236              11094161, 15689506, 3140038, -16510092},
2237         },
2238         {
2239             {-16160072, 5472695, 31895588, 4744994, 8823515, 10365685,
2240              -27224800, 9448613, -28774454, 366295},
2241             {19153450, 11523972, -11096490, -6503142, -24647631, 5420647,
2242              28344573, 8041113, 719605, 11671788},
2243             {8678025, 2694440, -6808014, 2517372, 4964326, 11152271, -15432916,
2244              -15266516, 27000813, -10195553},
2245         },
2246         {
2247             {-15157904, 7134312, 8639287, -2814877, -7235688, 10421742, 564065,
2248              5336097, 6750977, -14521026},
2249             {11836410, -3979488, 26297894, 16080799, 23455045, 15735944,
2250              1695823, -8819122, 8169720, 16220347},
2251             {-18115838, 8653647, 17578566, -6092619, -8025777, -16012763,
2252              -11144307, -2627664, -5990708, -14166033},
2253         },
2254         {
2255             {-23308498, -10968312, 15213228, -10081214, -30853605, -11050004,
2256              27884329, 2847284, 2655861, 1738395},
2257             {-27537433, -14253021, -25336301, -8002780, -9370762, 8129821,
2258              21651608, -3239336, -19087449, -11005278},
2259             {1533110, 3437855, 23735889, 459276, 29970501, 11335377, 26030092,
2260              5821408, 10478196, 8544890},
2261         },
2262         {
2263             {32173121, -16129311, 24896207, 3921497, 22579056, -3410854,
2264              19270449, 12217473, 17789017, -3395995},
2265             {-30552961, -2228401, -15578829, -10147201, 13243889, 517024,
2266              15479401, -3853233, 30460520, 1052596},
2267             {-11614875, 13323618, 32618793, 8175907, -15230173, 12596687,
2268              27491595, -4612359, 3179268, -9478891},
2269         },
2270         {
2271             {31947069, -14366651, -4640583, -15339921, -15125977, -6039709,
2272              -14756777, -16411740, 19072640, -9511060},
2273             {11685058, 11822410, 3158003, -13952594, 33402194, -4165066,
2274              5977896, -5215017, 473099, 5040608},
2275             {-20290863, 8198642, -27410132, 11602123, 1290375, -2799760,
2276              28326862, 1721092, -19558642, -3131606},
2277         },
2278     },
2279     {
2280         {
2281             {7881532, 10687937, 7578723, 7738378, -18951012, -2553952, 21820786,
2282              8076149, -27868496, 11538389},
2283             {-19935666, 3899861, 18283497, -6801568, -15728660, -11249211,
2284              8754525, 7446702, -5676054, 5797016},
2285             {-11295600, -3793569, -15782110, -7964573, 12708869, -8456199,
2286              2014099, -9050574, -2369172, -5877341},
2287         },
2288         {
2289             {-22472376, -11568741, -27682020, 1146375, 18956691, 16640559,
2290              1192730, -3714199, 15123619, 10811505},
2291             {14352098, -3419715, -18942044, 10822655, 32750596, 4699007, -70363,
2292              15776356, -28886779, -11974553},
2293             {-28241164, -8072475, -4978962, -5315317, 29416931, 1847569,
2294              -20654173, -16484855, 4714547, -9600655},
2295         },
2296         {
2297             {15200332, 8368572, 19679101, 15970074, -31872674, 1959451,
2298              24611599, -4543832, -11745876, 12340220},
2299             {12876937, -10480056, 33134381, 6590940, -6307776, 14872440,
2300              9613953, 8241152, 15370987, 9608631},
2301             {-4143277, -12014408, 8446281, -391603, 4407738, 13629032, -7724868,
2302              15866074, -28210621, -8814099},
2303         },
2304         {
2305             {26660628, -15677655, 8393734, 358047, -7401291, 992988, -23904233,
2306              858697, 20571223, 8420556},
2307             {14620715, 13067227, -15447274, 8264467, 14106269, 15080814,
2308              33531827, 12516406, -21574435, -12476749},
2309             {236881, 10476226, 57258, -14677024, 6472998, 2466984, 17258519,
2310              7256740, 8791136, 15069930},
2311         },
2312         {
2313             {1276410, -9371918, 22949635, -16322807, -23493039, -5702186,
2314              14711875, 4874229, -30663140, -2331391},
2315             {5855666, 4990204, -13711848, 7294284, -7804282, 1924647, -1423175,
2316              -7912378, -33069337, 9234253},
2317             {20590503, -9018988, 31529744, -7352666, -2706834, 10650548,
2318              31559055, -11609587, 18979186, 13396066},
2319         },
2320         {
2321             {24474287, 4968103, 22267082, 4407354, 24063882, -8325180,
2322              -18816887, 13594782, 33514650, 7021958},
2323             {-11566906, -6565505, -21365085, 15928892, -26158305, 4315421,
2324              -25948728, -3916677, -21480480, 12868082},
2325             {-28635013, 13504661, 19988037, -2132761, 21078225, 6443208,
2326              -21446107, 2244500, -12455797, -8089383},
2327         },
2328         {
2329             {-30595528, 13793479, -5852820, 319136, -25723172, -6263899,
2330              33086546, 8957937, -15233648, 5540521},
2331             {-11630176, -11503902, -8119500, -7643073, 2620056, 1022908,
2332              -23710744, -1568984, -16128528, -14962807},
2333             {23152971, 775386, 27395463, 14006635, -9701118, 4649512, 1689819,
2334              892185, -11513277, -15205948},
2335         },
2336         {
2337             {9770129, 9586738, 26496094, 4324120, 1556511, -3550024, 27453819,
2338              4763127, -19179614, 5867134},
2339             {-32765025, 1927590, 31726409, -4753295, 23962434, -16019500,
2340              27846559, 5931263, -29749703, -16108455},
2341             {27461885, -2977536, 22380810, 1815854, -23033753, -3031938,
2342              7283490, -15148073, -19526700, 7734629},
2343         },
2344     },
2345     {
2346         {
2347             {-8010264, -9590817, -11120403, 6196038, 29344158, -13430885,
2348              7585295, -3176626, 18549497, 15302069},
2349             {-32658337, -6171222, -7672793, -11051681, 6258878, 13504381,
2350              10458790, -6418461, -8872242, 8424746},
2351             {24687205, 8613276, -30667046, -3233545, 1863892, -1830544,
2352              19206234, 7134917, -11284482, -828919},
2353         },
2354         {
2355             {11334899, -9218022, 8025293, 12707519, 17523892, -10476071,
2356              10243738, -14685461, -5066034, 16498837},
2357             {8911542, 6887158, -9584260, -6958590, 11145641, -9543680, 17303925,
2358              -14124238, 6536641, 10543906},
2359             {-28946384, 15479763, -17466835, 568876, -1497683, 11223454,
2360              -2669190, -16625574, -27235709, 8876771},
2361         },
2362         {
2363             {-25742899, -12566864, -15649966, -846607, -33026686, -796288,
2364              -33481822, 15824474, -604426, -9039817},
2365             {10330056, 70051, 7957388, -9002667, 9764902, 15609756, 27698697,
2366              -4890037, 1657394, 3084098},
2367             {10477963, -7470260, 12119566, -13250805, 29016247, -5365589,
2368              31280319, 14396151, -30233575, 15272409},
2369         },
2370         {
2371             {-12288309, 3169463, 28813183, 16658753, 25116432, -5630466,
2372              -25173957, -12636138, -25014757, 1950504},
2373             {-26180358, 9489187, 11053416, -14746161, -31053720, 5825630,
2374              -8384306, -8767532, 15341279, 8373727},
2375             {28685821, 7759505, -14378516, -12002860, -31971820, 4079242,
2376              298136, -10232602, -2878207, 15190420},
2377         },
2378         {
2379             {-32932876, 13806336, -14337485, -15794431, -24004620, 10940928,
2380              8669718, 2742393, -26033313, -6875003},
2381             {-1580388, -11729417, -25979658, -11445023, -17411874, -10912854,
2382              9291594, -16247779, -12154742, 6048605},
2383             {-30305315, 14843444, 1539301, 11864366, 20201677, 1900163,
2384              13934231, 5128323, 11213262, 9168384},
2385         },
2386         {
2387             {-26280513, 11007847, 19408960, -940758, -18592965, -4328580,
2388              -5088060, -11105150, 20470157, -16398701},
2389             {-23136053, 9282192, 14855179, -15390078, -7362815, -14408560,
2390              -22783952, 14461608, 14042978, 5230683},
2391             {29969567, -2741594, -16711867, -8552442, 9175486, -2468974,
2392              21556951, 3506042, -5933891, -12449708},
2393         },
2394         {
2395             {-3144746, 8744661, 19704003, 4581278, -20430686, 6830683,
2396              -21284170, 8971513, -28539189, 15326563},
2397             {-19464629, 10110288, -17262528, -3503892, -23500387, 1355669,
2398              -15523050, 15300988, -20514118, 9168260},
2399             {-5353335, 4488613, -23803248, 16314347, 7780487, -15638939,
2400              -28948358, 9601605, 33087103, -9011387},
2401         },
2402         {
2403             {-19443170, -15512900, -20797467, -12445323, -29824447, 10229461,
2404              -27444329, -15000531, -5996870, 15664672},
2405             {23294591, -16632613, -22650781, -8470978, 27844204, 11461195,
2406              13099750, -2460356, 18151676, 13417686},
2407             {-24722913, -4176517, -31150679, 5988919, -26858785, 6685065,
2408              1661597, -12551441, 15271676, -15452665},
2409         },
2410     },
2411     {
2412         {
2413             {11433042, -13228665, 8239631, -5279517, -1985436, -725718,
2414              -18698764, 2167544, -6921301, -13440182},
2415             {-31436171, 15575146, 30436815, 12192228, -22463353, 9395379,
2416              -9917708, -8638997, 12215110, 12028277},
2417             {14098400, 6555944, 23007258, 5757252, -15427832, -12950502,
2418              30123440, 4617780, -16900089, -655628},
2419         },
2420         {
2421             {-4026201, -15240835, 11893168, 13718664, -14809462, 1847385,
2422              -15819999, 10154009, 23973261, -12684474},
2423             {-26531820, -3695990, -1908898, 2534301, -31870557, -16550355,
2424              18341390, -11419951, 32013174, -10103539},
2425             {-25479301, 10876443, -11771086, -14625140, -12369567, 1838104,
2426              21911214, 6354752, 4425632, -837822},
2427         },
2428         {
2429             {-10433389, -14612966, 22229858, -3091047, -13191166, 776729,
2430              -17415375, -12020462, 4725005, 14044970},
2431             {19268650, -7304421, 1555349, 8692754, -21474059, -9910664, 6347390,
2432              -1411784, -19522291, -16109756},
2433             {-24864089, 12986008, -10898878, -5558584, -11312371, -148526,
2434              19541418, 8180106, 9282262, 10282508},
2435         },
2436         {
2437             {-26205082, 4428547, -8661196, -13194263, 4098402, -14165257,
2438              15522535, 8372215, 5542595, -10702683},
2439             {-10562541, 14895633, 26814552, -16673850, -17480754, -2489360,
2440              -2781891, 6993761, -18093885, 10114655},
2441             {-20107055, -929418, 31422704, 10427861, -7110749, 6150669,
2442              -29091755, -11529146, 25953725, -106158},
2443         },
2444         {
2445             {-4234397, -8039292, -9119125, 3046000, 2101609, -12607294,
2446              19390020, 6094296, -3315279, 12831125},
2447             {-15998678, 7578152, 5310217, 14408357, -33548620, -224739,
2448              31575954, 6326196, 7381791, -2421839},
2449             {-20902779, 3296811, 24736065, -16328389, 18374254, 7318640,
2450              6295303, 8082724, -15362489, 12339664},
2451         },
2452         {
2453             {27724736, 2291157, 6088201, -14184798, 1792727, 5857634, 13848414,
2454              15768922, 25091167, 14856294},
2455             {-18866652, 8331043, 24373479, 8541013, -701998, -9269457, 12927300,
2456              -12695493, -22182473, -9012899},
2457             {-11423429, -5421590, 11632845, 3405020, 30536730, -11674039,
2458              -27260765, 13866390, 30146206, 9142070},
2459         },
2460         {
2461             {3924129, -15307516, -13817122, -10054960, 12291820, -668366,
2462              -27702774, 9326384, -8237858, 4171294},
2463             {-15921940, 16037937, 6713787, 16606682, -21612135, 2790944,
2464              26396185, 3731949, 345228, -5462949},
2465             {-21327538, 13448259, 25284571, 1143661, 20614966, -8849387,
2466              2031539, -12391231, -16253183, -13582083},
2467         },
2468         {
2469             {31016211, -16722429, 26371392, -14451233, -5027349, 14854137,
2470              17477601, 3842657, 28012650, -16405420},
2471             {-5075835, 9368966, -8562079, -4600902, -15249953, 6970560,
2472              -9189873, 16292057, -8867157, 3507940},
2473             {29439664, 3537914, 23333589, 6997794, -17555561, -11018068,
2474              -15209202, -15051267, -9164929, 6580396},
2475         },
2476     },
2477     {
2478         {
2479             {-12185861, -7679788, 16438269, 10826160, -8696817, -6235611,
2480              17860444, -9273846, -2095802, 9304567},
2481             {20714564, -4336911, 29088195, 7406487, 11426967, -5095705,
2482              14792667, -14608617, 5289421, -477127},
2483             {-16665533, -10650790, -6160345, -13305760, 9192020, -1802462,
2484              17271490, 12349094, 26939669, -3752294},
2485         },
2486         {
2487             {-12889898, 9373458, 31595848, 16374215, 21471720, 13221525,
2488              -27283495, -12348559, -3698806, 117887},
2489             {22263325, -6560050, 3984570, -11174646, -15114008, -566785,
2490              28311253, 5358056, -23319780, 541964},
2491             {16259219, 3261970, 2309254, -15534474, -16885711, -4581916,
2492              24134070, -16705829, -13337066, -13552195},
2493         },
2494         {
2495             {9378160, -13140186, -22845982, -12745264, 28198281, -7244098,
2496              -2399684, -717351, 690426, 14876244},
2497             {24977353, -314384, -8223969, -13465086, 28432343, -1176353,
2498              -13068804, -12297348, -22380984, 6618999},
2499             {-1538174, 11685646, 12944378, 13682314, -24389511, -14413193,
2500              8044829, -13817328, 32239829, -5652762},
2501         },
2502         {
2503             {-18603066, 4762990, -926250, 8885304, -28412480, -3187315, 9781647,
2504              -10350059, 32779359, 5095274},
2505             {-33008130, -5214506, -32264887, -3685216, 9460461, -9327423,
2506              -24601656, 14506724, 21639561, -2630236},
2507             {-16400943, -13112215, 25239338, 15531969, 3987758, -4499318,
2508              -1289502, -6863535, 17874574, 558605},
2509         },
2510         {
2511             {-13600129, 10240081, 9171883, 16131053, -20869254, 9599700,
2512              33499487, 5080151, 2085892, 5119761},
2513             {-22205145, -2519528, -16381601, 414691, -25019550, 2170430,
2514              30634760, -8363614, -31999993, -5759884},
2515             {-6845704, 15791202, 8550074, -1312654, 29928809, -12092256,
2516              27534430, -7192145, -22351378, 12961482},
2517         },
2518         {
2519             {-24492060, -9570771, 10368194, 11582341, -23397293, -2245287,
2520              16533930, 8206996, -30194652, -5159638},
2521             {-11121496, -3382234, 2307366, 6362031, -135455, 8868177, -16835630,
2522              7031275, 7589640, 8945490},
2523             {-32152748, 8917967, 6661220, -11677616, -1192060, -15793393,
2524              7251489, -11182180, 24099109, -14456170},
2525         },
2526         {
2527             {5019558, -7907470, 4244127, -14714356, -26933272, 6453165,
2528              -19118182, -13289025, -6231896, -10280736},
2529             {10853594, 10721687, 26480089, 5861829, -22995819, 1972175,
2530              -1866647, -10557898, -3363451, -6441124},
2531             {-17002408, 5906790, 221599, -6563147, 7828208, -13248918, 24362661,
2532              -2008168, -13866408, 7421392},
2533         },
2534         {
2535             {8139927, -6546497, 32257646, -5890546, 30375719, 1886181,
2536              -21175108, 15441252, 28826358, -4123029},
2537             {6267086, 9695052, 7709135, -16603597, -32869068, -1886135,
2538              14795160, -7840124, 13746021, -1742048},
2539             {28584902, 7787108, -6732942, -15050729, 22846041, -7571236,
2540              -3181936, -363524, 4771362, -8419958},
2541         },
2542     },
2543     {
2544         {
2545             {24949256, 6376279, -27466481, -8174608, -18646154, -9930606,
2546              33543569, -12141695, 3569627, 11342593},
2547             {26514989, 4740088, 27912651, 3697550, 19331575, -11472339, 6809886,
2548              4608608, 7325975, -14801071},
2549             {-11618399, -14554430, -24321212, 7655128, -1369274, 5214312,
2550              -27400540, 10258390, -17646694, -8186692},
2551         },
2552         {
2553             {11431204, 15823007, 26570245, 14329124, 18029990, 4796082,
2554              -31446179, 15580664, 9280358, -3973687},
2555             {-160783, -10326257, -22855316, -4304997, -20861367, -13621002,
2556              -32810901, -11181622, -15545091, 4387441},
2557             {-20799378, 12194512, 3937617, -5805892, -27154820, 9340370,
2558              -24513992, 8548137, 20617071, -7482001},
2559         },
2560         {
2561             {-938825, -3930586, -8714311, 16124718, 24603125, -6225393,
2562              -13775352, -11875822, 24345683, 10325460},
2563             {-19855277, -1568885, -22202708, 8714034, 14007766, 6928528,
2564              16318175, -1010689, 4766743, 3552007},
2565             {-21751364, -16730916, 1351763, -803421, -4009670, 3950935, 3217514,
2566              14481909, 10988822, -3994762},
2567         },
2568         {
2569             {15564307, -14311570, 3101243, 5684148, 30446780, -8051356,
2570              12677127, -6505343, -8295852, 13296005},
2571             {-9442290, 6624296, -30298964, -11913677, -4670981, -2057379,
2572              31521204, 9614054, -30000824, 12074674},
2573             {4771191, -135239, 14290749, -13089852, 27992298, 14998318,
2574              -1413936, -1556716, 29832613, -16391035},
2575         },
2576         {
2577             {7064884, -7541174, -19161962, -5067537, -18891269, -2912736,
2578              25825242, 5293297, -27122660, 13101590},
2579             {-2298563, 2439670, -7466610, 1719965, -27267541, -16328445,
2580              32512469, -5317593, -30356070, -4190957},
2581             {-30006540, 10162316, -33180176, 3981723, -16482138, -13070044,
2582              14413974, 9515896, 19568978, 9628812},
2583         },
2584         {
2585             {33053803, 199357, 15894591, 1583059, 27380243, -4580435, -17838894,
2586              -6106839, -6291786, 3437740},
2587             {-18978877, 3884493, 19469877, 12726490, 15913552, 13614290,
2588              -22961733, 70104, 7463304, 4176122},
2589             {-27124001, 10659917, 11482427, -16070381, 12771467, -6635117,
2590              -32719404, -5322751, 24216882, 5944158},
2591         },
2592         {
2593             {8894125, 7450974, -2664149, -9765752, -28080517, -12389115,
2594              19345746, 14680796, 11632993, 5847885},
2595             {26942781, -2315317, 9129564, -4906607, 26024105, 11769399,
2596              -11518837, 6367194, -9727230, 4782140},
2597             {19916461, -4828410, -22910704, -11414391, 25606324, -5972441,
2598              33253853, 8220911, 6358847, -1873857},
2599         },
2600         {
2601             {801428, -2081702, 16569428, 11065167, 29875704, 96627, 7908388,
2602              -4480480, -13538503, 1387155},
2603             {19646058, 5720633, -11416706, 12814209, 11607948, 12749789,
2604              14147075, 15156355, -21866831, 11835260},
2605             {19299512, 1155910, 28703737, 14890794, 2925026, 7269399, 26121523,
2606              15467869, -26560550, 5052483},
2607         },
2608     },
2609     {
2610         {
2611             {-3017432, 10058206, 1980837, 3964243, 22160966, 12322533, -6431123,
2612              -12618185, 12228557, -7003677},
2613             {32944382, 14922211, -22844894, 5188528, 21913450, -8719943,
2614              4001465, 13238564, -6114803, 8653815},
2615             {22865569, -4652735, 27603668, -12545395, 14348958, 8234005,
2616              24808405, 5719875, 28483275, 2841751},
2617         },
2618         {
2619             {-16420968, -1113305, -327719, -12107856, 21886282, -15552774,
2620              -1887966, -315658, 19932058, -12739203},
2621             {-11656086, 10087521, -8864888, -5536143, -19278573, -3055912,
2622              3999228, 13239134, -4777469, -13910208},
2623             {1382174, -11694719, 17266790, 9194690, -13324356, 9720081,
2624              20403944, 11284705, -14013818, 3093230},
2625         },
2626         {
2627             {16650921, -11037932, -1064178, 1570629, -8329746, 7352753, -302424,
2628              16271225, -24049421, -6691850},
2629             {-21911077, -5927941, -4611316, -5560156, -31744103, -10785293,
2630              24123614, 15193618, -21652117, -16739389},
2631             {-9935934, -4289447, -25279823, 4372842, 2087473, 10399484,
2632              31870908, 14690798, 17361620, 11864968},
2633         },
2634         {
2635             {-11307610, 6210372, 13206574, 5806320, -29017692, -13967200,
2636              -12331205, -7486601, -25578460, -16240689},
2637             {14668462, -12270235, 26039039, 15305210, 25515617, 4542480,
2638              10453892, 6577524, 9145645, -6443880},
2639             {5974874, 3053895, -9433049, -10385191, -31865124, 3225009,
2640              -7972642, 3936128, -5652273, -3050304},
2641         },
2642         {
2643             {30625386, -4729400, -25555961, -12792866, -20484575, 7695099,
2644              17097188, -16303496, -27999779, 1803632},
2645             {-3553091, 9865099, -5228566, 4272701, -5673832, -16689700,
2646              14911344, 12196514, -21405489, 7047412},
2647             {20093277, 9920966, -11138194, -5343857, 13161587, 12044805,
2648              -32856851, 4124601, -32343828, -10257566},
2649         },
2650         {
2651             {-20788824, 14084654, -13531713, 7842147, 19119038, -13822605,
2652              4752377, -8714640, -21679658, 2288038},
2653             {-26819236, -3283715, 29965059, 3039786, -14473765, 2540457,
2654              29457502, 14625692, -24819617, 12570232},
2655             {-1063558, -11551823, 16920318, 12494842, 1278292, -5869109,
2656              -21159943, -3498680, -11974704, 4724943},
2657         },
2658         {
2659             {17960970, -11775534, -4140968, -9702530, -8876562, -1410617,
2660              -12907383, -8659932, -29576300, 1903856},
2661             {23134274, -14279132, -10681997, -1611936, 20684485, 15770816,
2662              -12989750, 3190296, 26955097, 14109738},
2663             {15308788, 5320727, -30113809, -14318877, 22902008, 7767164,
2664              29425325, -11277562, 31960942, 11934971},
2665         },
2666         {
2667             {-27395711, 8435796, 4109644, 12222639, -24627868, 14818669,
2668              20638173, 4875028, 10491392, 1379718},
2669             {-13159415, 9197841, 3875503, -8936108, -1383712, -5879801,
2670              33518459, 16176658, 21432314, 12180697},
2671             {-11787308, 11500838, 13787581, -13832590, -22430679, 10140205,
2672              1465425, 12689540, -10301319, -13872883},
2673         },
2674     },
2675     {
2676         {
2677             {5414091, -15386041, -21007664, 9643570, 12834970, 1186149,
2678              -2622916, -1342231, 26128231, 6032912},
2679             {-26337395, -13766162, 32496025, -13653919, 17847801, -12669156,
2680              3604025, 8316894, -25875034, -10437358},
2681             {3296484, 6223048, 24680646, -12246460, -23052020, 5903205,
2682              -8862297, -4639164, 12376617, 3188849},
2683         },
2684         {
2685             {29190488, -14659046, 27549113, -1183516, 3520066, -10697301,
2686              32049515, -7309113, -16109234, -9852307},
2687             {-14744486, -9309156, 735818, -598978, -20407687, -5057904,
2688              25246078, -15795669, 18640741, -960977},
2689             {-6928835, -16430795, 10361374, 5642961, 4910474, 12345252,
2690              -31638386, -494430, 10530747, 1053335},
2691         },
2692         {
2693             {-29265967, -14186805, -13538216, -12117373, -19457059, -10655384,
2694              -31462369, -2948985, 24018831, 15026644},
2695             {-22592535, -3145277, -2289276, 5953843, -13440189, 9425631,
2696              25310643, 13003497, -2314791, -15145616},
2697             {-27419985, -603321, -8043984, -1669117, -26092265, 13987819,
2698              -27297622, 187899, -23166419, -2531735},
2699         },
2700         {
2701             {-21744398, -13810475, 1844840, 5021428, -10434399, -15911473,
2702              9716667, 16266922, -5070217, 726099},
2703             {29370922, -6053998, 7334071, -15342259, 9385287, 2247707,
2704              -13661962, -4839461, 30007388, -15823341},
2705             {-936379, 16086691, 23751945, -543318, -1167538, -5189036, 9137109,
2706              730663, 9835848, 4555336},
2707         },
2708         {
2709             {-23376435, 1410446, -22253753, -12899614, 30867635, 15826977,
2710              17693930, 544696, -11985298, 12422646},
2711             {31117226, -12215734, -13502838, 6561947, -9876867, -12757670,
2712              -5118685, -4096706, 29120153, 13924425},
2713             {-17400879, -14233209, 19675799, -2734756, -11006962, -5858820,
2714              -9383939, -11317700, 7240931, -237388},
2715         },
2716         {
2717             {-31361739, -11346780, -15007447, -5856218, -22453340, -12152771,
2718              1222336, 4389483, 3293637, -15551743},
2719             {-16684801, -14444245, 11038544, 11054958, -13801175, -3338533,
2720              -24319580, 7733547, 12796905, -6335822},
2721             {-8759414, -10817836, -25418864, 10783769, -30615557, -9746811,
2722              -28253339, 3647836, 3222231, -11160462},
2723         },
2724         {
2725             {18606113, 1693100, -25448386, -15170272, 4112353, 10045021,
2726              23603893, -2048234, -7550776, 2484985},
2727             {9255317, -3131197, -12156162, -1004256, 13098013, -9214866,
2728              16377220, -2102812, -19802075, -3034702},
2729             {-22729289, 7496160, -5742199, 11329249, 19991973, -3347502,
2730              -31718148, 9936966, -30097688, -10618797},
2731         },
2732         {
2733             {21878590, -5001297, 4338336, 13643897, -3036865, 13160960,
2734              19708896, 5415497, -7360503, -4109293},
2735             {27736861, 10103576, 12500508, 8502413, -3413016, -9633558,
2736              10436918, -1550276, -23659143, -8132100},
2737             {19492550, -12104365, -29681976, -852630, -3208171, 12403437,
2738              30066266, 8367329, 13243957, 8709688},
2739         },
2740     },
2741     {
2742         {
2743             {12015105, 2801261, 28198131, 10151021, 24818120, -4743133,
2744              -11194191, -5645734, 5150968, 7274186},
2745             {2831366, -12492146, 1478975, 6122054, 23825128, -12733586,
2746              31097299, 6083058, 31021603, -9793610},
2747             {-2529932, -2229646, 445613, 10720828, -13849527, -11505937,
2748              -23507731, 16354465, 15067285, -14147707},
2749         },
2750         {
2751             {7840942, 14037873, -33364863, 15934016, -728213, -3642706,
2752              21403988, 1057586, -19379462, -12403220},
2753             {915865, -16469274, 15608285, -8789130, -24357026, 6060030,
2754              -17371319, 8410997, -7220461, 16527025},
2755             {32922597, -556987, 20336074, -16184568, 10903705, -5384487,
2756              16957574, 52992, 23834301, 6588044},
2757         },
2758         {
2759             {32752030, 11232950, 3381995, -8714866, 22652988, -10744103,
2760              17159699, 16689107, -20314580, -1305992},
2761             {-4689649, 9166776, -25710296, -10847306, 11576752, 12733943,
2762              7924251, -2752281, 1976123, -7249027},
2763             {21251222, 16309901, -2983015, -6783122, 30810597, 12967303, 156041,
2764              -3371252, 12331345, -8237197},
2765         },
2766         {
2767             {8651614, -4477032, -16085636, -4996994, 13002507, 2950805,
2768              29054427, -5106970, 10008136, -4667901},
2769             {31486080, 15114593, -14261250, 12951354, 14369431, -7387845,
2770              16347321, -13662089, 8684155, -10532952},
2771             {19443825, 11385320, 24468943, -9659068, -23919258, 2187569,
2772              -26263207, -6086921, 31316348, 14219878},
2773         },
2774         {
2775             {-28594490, 1193785, 32245219, 11392485, 31092169, 15722801,
2776              27146014, 6992409, 29126555, 9207390},
2777             {32382935, 1110093, 18477781, 11028262, -27411763, -7548111,
2778              -4980517, 10843782, -7957600, -14435730},
2779             {2814918, 7836403, 27519878, -7868156, -20894015, -11553689,
2780              -21494559, 8550130, 28346258, 1994730},
2781         },
2782         {
2783             {-19578299, 8085545, -14000519, -3948622, 2785838, -16231307,
2784              -19516951, 7174894, 22628102, 8115180},
2785             {-30405132, 955511, -11133838, -15078069, -32447087, -13278079,
2786              -25651578, 3317160, -9943017, 930272},
2787             {-15303681, -6833769, 28856490, 1357446, 23421993, 1057177,
2788              24091212, -1388970, -22765376, -10650715},
2789         },
2790         {
2791             {-22751231, -5303997, -12907607, -12768866, -15811511, -7797053,
2792              -14839018, -16554220, -1867018, 8398970},
2793             {-31969310, 2106403, -4736360, 1362501, 12813763, 16200670,
2794              22981545, -6291273, 18009408, -15772772},
2795             {-17220923, -9545221, -27784654, 14166835, 29815394, 7444469,
2796              29551787, -3727419, 19288549, 1325865},
2797         },
2798         {
2799             {15100157, -15835752, -23923978, -1005098, -26450192, 15509408,
2800              12376730, -3479146, 33166107, -8042750},
2801             {20909231, 13023121, -9209752, 16251778, -5778415, -8094914,
2802              12412151, 10018715, 2213263, -13878373},
2803             {32529814, -11074689, 30361439, -16689753, -9135940, 1513226,
2804              22922121, 6382134, -5766928, 8371348},
2805         },
2806     },
2807     {
2808         {
2809             {9923462, 11271500, 12616794, 3544722, -29998368, -1721626,
2810              12891687, -8193132, -26442943, 10486144},
2811             {-22597207, -7012665, 8587003, -8257861, 4084309, -12970062, 361726,
2812              2610596, -23921530, -11455195},
2813             {5408411, -1136691, -4969122, 10561668, 24145918, 14240566,
2814              31319731, -4235541, 19985175, -3436086},
2815         },
2816         {
2817             {-13994457, 16616821, 14549246, 3341099, 32155958, 13648976,
2818              -17577068, 8849297, 65030, 8370684},
2819             {-8320926, -12049626, 31204563, 5839400, -20627288, -1057277,
2820              -19442942, 6922164, 12743482, -9800518},
2821             {-2361371, 12678785, 28815050, 4759974, -23893047, 4884717,
2822              23783145, 11038569, 18800704, 255233},
2823         },
2824         {
2825             {-5269658, -1773886, 13957886, 7990715, 23132995, 728773, 13393847,
2826              9066957, 19258688, -14753793},
2827             {-2936654, -10827535, -10432089, 14516793, -3640786, 4372541,
2828              -31934921, 2209390, -1524053, 2055794},
2829             {580882, 16705327, 5468415, -2683018, -30926419, -14696000,
2830              -7203346, -8994389, -30021019, 7394435},
2831         },
2832         {
2833             {23838809, 1822728, -15738443, 15242727, 8318092, -3733104,
2834              -21672180, -3492205, -4821741, 14799921},
2835             {13345610, 9759151, 3371034, -16137791, 16353039, 8577942, 31129804,
2836              13496856, -9056018, 7402518},
2837             {2286874, -4435931, -20042458, -2008336, -13696227, 5038122,
2838              11006906, -15760352, 8205061, 1607563},
2839         },
2840         {
2841             {14414086, -8002132, 3331830, -3208217, 22249151, -5594188,
2842              18364661, -2906958, 30019587, -9029278},
2843             {-27688051, 1585953, -10775053, 931069, -29120221, -11002319,
2844              -14410829, 12029093, 9944378, 8024},
2845             {4368715, -3709630, 29874200, -15022983, -20230386, -11410704,
2846              -16114594, -999085, -8142388, 5640030},
2847         },
2848         {
2849             {10299610, 13746483, 11661824, 16234854, 7630238, 5998374, 9809887,
2850              -16694564, 15219798, -14327783},
2851             {27425505, -5719081, 3055006, 10660664, 23458024, 595578, -15398605,
2852              -1173195, -18342183, 9742717},
2853             {6744077, 2427284, 26042789, 2720740, -847906, 1118974, 32324614,
2854              7406442, 12420155, 1994844},
2855         },
2856         {
2857             {14012521, -5024720, -18384453, -9578469, -26485342, -3936439,
2858              -13033478, -10909803, 24319929, -6446333},
2859             {16412690, -4507367, 10772641, 15929391, -17068788, -4658621,
2860              10555945, -10484049, -30102368, -4739048},
2861             {22397382, -7767684, -9293161, -12792868, 17166287, -9755136,
2862              -27333065, 6199366, 21880021, -12250760},
2863         },
2864         {
2865             {-4283307, 5368523, -31117018, 8163389, -30323063, 3209128,
2866              16557151, 8890729, 8840445, 4957760},
2867             {-15447727, 709327, -6919446, -10870178, -29777922, 6522332,
2868              -21720181, 12130072, -14796503, 5005757},
2869             {-2114751, -14308128, 23019042, 15765735, -25269683, 6002752,
2870              10183197, -13239326, -16395286, -2176112},
2871         },
2872     },
2873     {
2874         {
2875             {-19025756, 1632005, 13466291, -7995100, -23640451, 16573537,
2876              -32013908, -3057104, 22208662, 2000468},
2877             {3065073, -1412761, -25598674, -361432, -17683065, -5703415,
2878              -8164212, 11248527, -3691214, -7414184},
2879             {10379208, -6045554, 8877319, 1473647, -29291284, -12507580,
2880              16690915, 2553332, -3132688, 16400289},
2881         },
2882         {
2883             {15716668, 1254266, -18472690, 7446274, -8448918, 6344164,
2884              -22097271, -7285580, 26894937, 9132066},
2885             {24158887, 12938817, 11085297, -8177598, -28063478, -4457083,
2886              -30576463, 64452, -6817084, -2692882},
2887             {13488534, 7794716, 22236231, 5989356, 25426474, -12578208, 2350710,
2888              -3418511, -4688006, 2364226},
2889         },
2890         {
2891             {16335052, 9132434, 25640582, 6678888, 1725628, 8517937, -11807024,
2892              -11697457, 15445875, -7798101},
2893             {29004207, -7867081, 28661402, -640412, -12794003, -7943086,
2894              31863255, -4135540, -278050, -15759279},
2895             {-6122061, -14866665, -28614905, 14569919, -10857999, -3591829,
2896              10343412, -6976290, -29828287, -10815811},
2897         },
2898         {
2899             {27081650, 3463984, 14099042, -4517604, 1616303, -6205604, 29542636,
2900              15372179, 17293797, 960709},
2901             {20263915, 11434237, -5765435, 11236810, 13505955, -10857102,
2902              -16111345, 6493122, -19384511, 7639714},
2903             {-2830798, -14839232, 25403038, -8215196, -8317012, -16173699,
2904              18006287, -16043750, 29994677, -15808121},
2905         },
2906         {
2907             {9769828, 5202651, -24157398, -13631392, -28051003, -11561624,
2908              -24613141, -13860782, -31184575, 709464},
2909             {12286395, 13076066, -21775189, -1176622, -25003198, 4057652,
2910              -32018128, -8890874, 16102007, 13205847},
2911             {13733362, 5599946, 10557076, 3195751, -5557991, 8536970, -25540170,
2912              8525972, 10151379, 10394400},
2913         },
2914         {
2915             {4024660, -16137551, 22436262, 12276534, -9099015, -2686099,
2916              19698229, 11743039, -33302334, 8934414},
2917             {-15879800, -4525240, -8580747, -2934061, 14634845, -698278,
2918              -9449077, 3137094, -11536886, 11721158},
2919             {17555939, -5013938, 8268606, 2331751, -22738815, 9761013, 9319229,
2920              8835153, -9205489, -1280045},
2921         },
2922         {
2923             {-461409, -7830014, 20614118, 16688288, -7514766, -4807119,
2924              22300304, 505429, 6108462, -6183415},
2925             {-5070281, 12367917, -30663534, 3234473, 32617080, -8422642,
2926              29880583, -13483331, -26898490, -7867459},
2927             {-31975283, 5726539, 26934134, 10237677, -3173717, -605053,
2928              24199304, 3795095, 7592688, -14992079},
2929         },
2930         {
2931             {21594432, -14964228, 17466408, -4077222, 32537084, 2739898,
2932              6407723, 12018833, -28256052, 4298412},
2933             {-20650503, -11961496, -27236275, 570498, 3767144, -1717540,
2934              13891942, -1569194, 13717174, 10805743},
2935             {-14676630, -15644296, 15287174, 11927123, 24177847, -8175568,
2936              -796431, 14860609, -26938930, -5863836},
2937         },
2938     },
2939     {
2940         {
2941             {12962541, 5311799, -10060768, 11658280, 18855286, -7954201,
2942              13286263, -12808704, -4381056, 9882022},
2943             {18512079, 11319350, -20123124, 15090309, 18818594, 5271736,
2944              -22727904, 3666879, -23967430, -3299429},
2945             {-6789020, -3146043, 16192429, 13241070, 15898607, -14206114,
2946              -10084880, -6661110, -2403099, 5276065},
2947         },
2948         {
2949             {30169808, -5317648, 26306206, -11750859, 27814964, 7069267,
2950              7152851, 3684982, 1449224, 13082861},
2951             {10342826, 3098505, 2119311, 193222, 25702612, 12233820, 23697382,
2952              15056736, -21016438, -8202000},
2953             {-33150110, 3261608, 22745853, 7948688, 19370557, -15177665,
2954              -26171976, 6482814, -10300080, -11060101},
2955         },
2956         {
2957             {32869458, -5408545, 25609743, 15678670, -10687769, -15471071,
2958              26112421, 2521008, -22664288, 6904815},
2959             {29506923, 4457497, 3377935, -9796444, -30510046, 12935080, 1561737,
2960              3841096, -29003639, -6657642},
2961             {10340844, -6630377, -18656632, -2278430, 12621151, -13339055,
2962              30878497, -11824370, -25584551, 5181966},
2963         },
2964         {
2965             {25940115, -12658025, 17324188, -10307374, -8671468, 15029094,
2966              24396252, -16450922, -2322852, -12388574},
2967             {-21765684, 9916823, -1300409, 4079498, -1028346, 11909559, 1782390,
2968              12641087, 20603771, -6561742},
2969             {-18882287, -11673380, 24849422, 11501709, 13161720, -4768874,
2970              1925523, 11914390, 4662781, 7820689},
2971         },
2972         {
2973             {12241050, -425982, 8132691, 9393934, 32846760, -1599620, 29749456,
2974              12172924, 16136752, 15264020},
2975             {-10349955, -14680563, -8211979, 2330220, -17662549, -14545780,
2976              10658213, 6671822, 19012087, 3772772},
2977             {3753511, -3421066, 10617074, 2028709, 14841030, -6721664, 28718732,
2978              -15762884, 20527771, 12988982},
2979         },
2980         {
2981             {-14822485, -5797269, -3707987, 12689773, -898983, -10914866,
2982              -24183046, -10564943, 3299665, -12424953},
2983             {-16777703, -15253301, -9642417, 4978983, 3308785, 8755439, 6943197,
2984              6461331, -25583147, 8991218},
2985             {-17226263, 1816362, -1673288, -6086439, 31783888, -8175991,
2986              -32948145, 7417950, -30242287, 1507265},
2987         },
2988         {
2989             {29692663, 6829891, -10498800, 4334896, 20945975, -11906496,
2990              -28887608, 8209391, 14606362, -10647073},
2991             {-3481570, 8707081, 32188102, 5672294, 22096700, 1711240, -33020695,
2992              9761487, 4170404, -2085325},
2993             {-11587470, 14855945, -4127778, -1531857, -26649089, 15084046,
2994              22186522, 16002000, -14276837, -8400798},
2995         },
2996         {
2997             {-4811456, 13761029, -31703877, -2483919, -3312471, 7869047,
2998              -7113572, -9620092, 13240845, 10965870},
2999             {-7742563, -8256762, -14768334, -13656260, -23232383, 12387166,
3000              4498947, 14147411, 29514390, 4302863},
3001             {-13413405, -12407859, 20757302, -13801832, 14785143, 8976368,
3002              -5061276, -2144373, 17846988, -13971927},
3003         },
3004     },
3005     {
3006         {
3007             {-2244452, -754728, -4597030, -1066309, -6247172, 1455299,
3008              -21647728, -9214789, -5222701, 12650267},
3009             {-9906797, -16070310, 21134160, 12198166, -27064575, 708126, 387813,
3010              13770293, -19134326, 10958663},
3011             {22470984, 12369526, 23446014, -5441109, -21520802, -9698723,
3012              -11772496, -11574455, -25083830, 4271862},
3013         },
3014         {
3015             {-25169565, -10053642, -19909332, 15361595, -5984358, 2159192,
3016              75375, -4278529, -32526221, 8469673},
3017             {15854970, 4148314, -8893890, 7259002, 11666551, 13824734,
3018              -30531198, 2697372, 24154791, -9460943},
3019             {15446137, -15806644, 29759747, 14019369, 30811221, -9610191,
3020              -31582008, 12840104, 24913809, 9815020},
3021         },
3022         {
3023             {-4709286, -5614269, -31841498, -12288893, -14443537, 10799414,
3024              -9103676, 13438769, 18735128, 9466238},
3025             {11933045, 9281483, 5081055, -5183824, -2628162, -4905629, -7727821,
3026              -10896103, -22728655, 16199064},
3027             {14576810, 379472, -26786533, -8317236, -29426508, -10812974,
3028              -102766, 1876699, 30801119, 2164795},
3029         },
3030         {
3031             {15995086, 3199873, 13672555, 13712240, -19378835, -4647646,
3032              -13081610, -15496269, -13492807, 1268052},
3033             {-10290614, -3659039, -3286592, 10948818, 23037027, 3794475,
3034              -3470338, -12600221, -17055369, 3565904},
3035             {29210088, -9419337, -5919792, -4952785, 10834811, -13327726,
3036              -16512102, -10820713, -27162222, -14030531},
3037         },
3038         {
3039             {-13161890, 15508588, 16663704, -8156150, -28349942, 9019123,
3040              -29183421, -3769423, 2244111, -14001979},
3041             {-5152875, -3800936, -9306475, -6071583, 16243069, 14684434,
3042              -25673088, -16180800, 13491506, 4641841},
3043             {10813417, 643330, -19188515, -728916, 30292062, -16600078,
3044              27548447, -7721242, 14476989, -12767431},
3045         },
3046         {
3047             {10292079, 9984945, 6481436, 8279905, -7251514, 7032743, 27282937,
3048              -1644259, -27912810, 12651324},
3049             {-31185513, -813383, 22271204, 11835308, 10201545, 15351028,
3050              17099662, 3988035, 21721536, -3148940},
3051             {10202177, -6545839, -31373232, -9574638, -32150642, -8119683,
3052              -12906320, 3852694, 13216206, 14842320},
3053         },
3054         {
3055             {-15815640, -10601066, -6538952, -7258995, -6984659, -6581778,
3056              -31500847, 13765824, -27434397, 9900184},
3057             {14465505, -13833331, -32133984, -14738873, -27443187, 12990492,
3058              33046193, 15796406, -7051866, -8040114},
3059             {30924417, -8279620, 6359016, -12816335, 16508377, 9071735,
3060              -25488601, 15413635, 9524356, -7018878},
3061         },
3062         {
3063             {12274201, -13175547, 32627641, -1785326, 6736625, 13267305,
3064              5237659, -5109483, 15663516, 4035784},
3065             {-2951309, 8903985, 17349946, 601635, -16432815, -4612556,
3066              -13732739, -15889334, -22258478, 4659091},
3067             {-16916263, -4952973, -30393711, -15158821, 20774812, 15897498,
3068              5736189, 15026997, -2178256, -13455585},
3069         },
3070     },
3071     {
3072         {
3073             {-8858980, -2219056, 28571666, -10155518, -474467, -10105698,
3074              -3801496, 278095, 23440562, -290208},
3075             {10226241, -5928702, 15139956, 120818, -14867693, 5218603, 32937275,
3076              11551483, -16571960, -7442864},
3077             {17932739, -12437276, -24039557, 10749060, 11316803, 7535897,
3078              22503767, 5561594, -3646624, 3898661},
3079         },
3080         {
3081             {7749907, -969567, -16339731, -16464, -25018111, 15122143, -1573531,
3082              7152530, 21831162, 1245233},
3083             {26958459, -14658026, 4314586, 8346991, -5677764, 11960072,
3084              -32589295, -620035, -30402091, -16716212},
3085             {-12165896, 9166947, 33491384, 13673479, 29787085, 13096535,
3086              6280834, 14587357, -22338025, 13987525},
3087         },
3088         {
3089             {-24349909, 7778775, 21116000, 15572597, -4833266, -5357778,
3090              -4300898, -5124639, -7469781, -2858068},
3091             {9681908, -6737123, -31951644, 13591838, -6883821, 386950, 31622781,
3092              6439245, -14581012, 4091397},
3093             {-8426427, 1470727, -28109679, -1596990, 3978627, -5123623,
3094              -19622683, 12092163, 29077877, -14741988},
3095         },
3096         {
3097             {5269168, -6859726, -13230211, -8020715, 25932563, 1763552,
3098              -5606110, -5505881, -20017847, 2357889},
3099             {32264008, -15407652, -5387735, -1160093, -2091322, -3946900,
3100              23104804, -12869908, 5727338, 189038},
3101             {14609123, -8954470, -6000566, -16622781, -14577387, -7743898,
3102              -26745169, 10942115, -25888931, -14884697},
3103         },
3104         {
3105             {20513500, 5557931, -15604613, 7829531, 26413943, -2019404,
3106              -21378968, 7471781, 13913677, -5137875},
3107             {-25574376, 11967826, 29233242, 12948236, -6754465, 4713227,
3108              -8940970, 14059180, 12878652, 8511905},
3109             {-25656801, 3393631, -2955415, -7075526, -2250709, 9366908,
3110              -30223418, 6812974, 5568676, -3127656},
3111         },
3112         {
3113             {11630004, 12144454, 2116339, 13606037, 27378885, 15676917,
3114              -17408753, -13504373, -14395196, 8070818},
3115             {27117696, -10007378, -31282771, -5570088, 1127282, 12772488,
3116              -29845906, 10483306, -11552749, -1028714},
3117             {10637467, -5688064, 5674781, 1072708, -26343588, -6982302,
3118              -1683975, 9177853, -27493162, 15431203},
3119         },
3120         {
3121             {20525145, 10892566, -12742472, 12779443, -29493034, 16150075,
3122              -28240519, 14943142, -15056790, -7935931},
3123             {-30024462, 5626926, -551567, -9981087, 753598, 11981191, 25244767,
3124              -3239766, -3356550, 9594024},
3125             {-23752644, 2636870, -5163910, -10103818, 585134, 7877383, 11345683,
3126              -6492290, 13352335, -10977084},
3127         },
3128         {
3129             {-1931799, -5407458, 3304649, -12884869, 17015806, -4877091,
3130              -29783850, -7752482, -13215537, -319204},
3131             {20239939, 6607058, 6203985, 3483793, -18386976, -779229, -20723742,
3132              15077870, -22750759, 14523817},
3133             {27406042, -6041657, 27423596, -4497394, 4996214, 10002360,
3134              -28842031, -4545494, -30172742, -4805667},
3135         },
3136     },
3137     {
3138         {
3139             {11374242, 12660715, 17861383, -12540833, 10935568, 1099227,
3140              -13886076, -9091740, -27727044, 11358504},
3141             {-12730809, 10311867, 1510375, 10778093, -2119455, -9145702,
3142              32676003, 11149336, -26123651, 4985768},
3143             {-19096303, 341147, -6197485, -239033, 15756973, -8796662, -983043,
3144              13794114, -19414307, -15621255},
3145         },
3146         {
3147             {6490081, 11940286, 25495923, -7726360, 8668373, -8751316, 3367603,
3148              6970005, -1691065, -9004790},
3149             {1656497, 13457317, 15370807, 6364910, 13605745, 8362338, -19174622,
3150              -5475723, -16796596, -5031438},
3151             {-22273315, -13524424, -64685, -4334223, -18605636, -10921968,
3152              -20571065, -7007978, -99853, -10237333},
3153         },
3154         {
3155             {17747465, 10039260, 19368299, -4050591, -20630635, -16041286,
3156              31992683, -15857976, -29260363, -5511971},
3157             {31932027, -4986141, -19612382, 16366580, 22023614, 88450, 11371999,
3158              -3744247, 4882242, -10626905},
3159             {29796507, 37186, 19818052, 10115756, -11829032, 3352736, 18551198,
3160              3272828, -5190932, -4162409},
3161         },
3162         {
3163             {12501286, 4044383, -8612957, -13392385, -32430052, 5136599,
3164              -19230378, -3529697, 330070, -3659409},
3165             {6384877, 2899513, 17807477, 7663917, -2358888, 12363165, 25366522,
3166              -8573892, -271295, 12071499},
3167             {-8365515, -4042521, 25133448, -4517355, -6211027, 2265927,
3168              -32769618, 1936675, -5159697, 3829363},
3169         },
3170         {
3171             {28425966, -5835433, -577090, -4697198, -14217555, 6870930, 7921550,
3172              -6567787, 26333140, 14267664},
3173             {-11067219, 11871231, 27385719, -10559544, -4585914, -11189312,
3174              10004786, -8709488, -21761224, 8930324},
3175             {-21197785, -16396035, 25654216, -1725397, 12282012, 11008919,
3176              1541940, 4757911, -26491501, -16408940},
3177         },
3178         {
3179             {13537262, -7759490, -20604840, 10961927, -5922820, -13218065,
3180              -13156584, 6217254, -15943699, 13814990},
3181             {-17422573, 15157790, 18705543, 29619, 24409717, -260476, 27361681,
3182              9257833, -1956526, -1776914},
3183             {-25045300, -10191966, 15366585, 15166509, -13105086, 8423556,
3184              -29171540, 12361135, -18685978, 4578290},
3185         },
3186         {
3187             {24579768, 3711570, 1342322, -11180126, -27005135, 14124956,
3188              -22544529, 14074919, 21964432, 8235257},
3189             {-6528613, -2411497, 9442966, -5925588, 12025640, -1487420,
3190              -2981514, -1669206, 13006806, 2355433},
3191             {-16304899, -13605259, -6632427, -5142349, 16974359, -10911083,
3192              27202044, 1719366, 1141648, -12796236},
3193         },
3194         {
3195             {-12863944, -13219986, -8318266, -11018091, -6810145, -4843894,
3196              13475066, -3133972, 32674895, 13715045},
3197             {11423335, -5468059, 32344216, 8962751, 24989809, 9241752,
3198              -13265253, 16086212, -28740881, -15642093},
3199             {-1409668, 12530728, -6368726, 10847387, 19531186, -14132160,
3200              -11709148, 7791794, -27245943, 4383347},
3201         },
3202     },
3203     {
3204         {
3205             {-28970898, 5271447, -1266009, -9736989, -12455236, 16732599,
3206              -4862407, -4906449, 27193557, 6245191},
3207             {-15193956, 5362278, -1783893, 2695834, 4960227, 12840725, 23061898,
3208              3260492, 22510453, 8577507},
3209             {-12632451, 11257346, -32692994, 13548177, -721004, 10879011,
3210              31168030, 13952092, -29571492, -3635906},
3211         },
3212         {
3213             {3877321, -9572739, 32416692, 5405324, -11004407, -13656635,
3214              3759769, 11935320, 5611860, 8164018},
3215             {-16275802, 14667797, 15906460, 12155291, -22111149, -9039718,
3216              32003002, -8832289, 5773085, -8422109},
3217             {-23788118, -8254300, 1950875, 8937633, 18686727, 16459170, -905725,
3218              12376320, 31632953, 190926},
3219         },
3220         {
3221             {-24593607, -16138885, -8423991, 13378746, 14162407, 6901328,
3222              -8288749, 4508564, -25341555, -3627528},
3223             {8884438, -5884009, 6023974, 10104341, -6881569, -4941533, 18722941,
3224              -14786005, -1672488, 827625},
3225             {-32720583, -16289296, -32503547, 7101210, 13354605, 2659080,
3226              -1800575, -14108036, -24878478, 1541286},
3227         },
3228         {
3229             {2901347, -1117687, 3880376, -10059388, -17620940, -3612781,
3230              -21802117, -3567481, 20456845, -1885033},
3231             {27019610, 12299467, -13658288, -1603234, -12861660, -4861471,
3232              -19540150, -5016058, 29439641, 15138866},
3233             {21536104, -6626420, -32447818, -10690208, -22408077, 5175814,
3234              -5420040, -16361163, 7779328, 109896},
3235         },
3236         {
3237             {30279744, 14648750, -8044871, 6425558, 13639621, -743509, 28698390,
3238              12180118, 23177719, -554075},
3239             {26572847, 3405927, -31701700, 12890905, -19265668, 5335866,
3240              -6493768, 2378492, 4439158, -13279347},
3241             {-22716706, 3489070, -9225266, -332753, 18875722, -1140095,
3242              14819434, -12731527, -17717757, -5461437},
3243         },
3244         {
3245             {-5056483, 16566551, 15953661, 3767752, -10436499, 15627060,
3246              -820954, 2177225, 8550082, -15114165},
3247             {-18473302, 16596775, -381660, 15663611, 22860960, 15585581,
3248              -27844109, -3582739, -23260460, -8428588},
3249             {-32480551, 15707275, -8205912, -5652081, 29464558, 2713815,
3250              -22725137, 15860482, -21902570, 1494193},
3251         },
3252         {
3253             {-19562091, -14087393, -25583872, -9299552, 13127842, 759709,
3254              21923482, 16529112, 8742704, 12967017},
3255             {-28464899, 1553205, 32536856, -10473729, -24691605, -406174,
3256              -8914625, -2933896, -29903758, 15553883},
3257             {21877909, 3230008, 9881174, 10539357, -4797115, 2841332, 11543572,
3258              14513274, 19375923, -12647961},
3259         },
3260         {
3261             {8832269, -14495485, 13253511, 5137575, 5037871, 4078777, 24880818,
3262              -6222716, 2862653, 9455043},
3263             {29306751, 5123106, 20245049, -14149889, 9592566, 8447059, -2077124,
3264              -2990080, 15511449, 4789663},
3265             {-20679756, 7004547, 8824831, -9434977, -4045704, -3750736,
3266              -5754762, 108893, 23513200, 16652362},
3267         },
3268     },
3269     {
3270         {
3271             {-33256173, 4144782, -4476029, -6579123, 10770039, -7155542,
3272              -6650416, -12936300, -18319198, 10212860},
3273             {2756081, 8598110, 7383731, -6859892, 22312759, -1105012, 21179801,
3274              2600940, -9988298, -12506466},
3275             {-24645692, 13317462, -30449259, -15653928, 21365574, -10869657,
3276              11344424, 864440, -2499677, -16710063},
3277         },
3278         {
3279             {-26432803, 6148329, -17184412, -14474154, 18782929, -275997,
3280              -22561534, 211300, 2719757, 4940997},
3281             {-1323882, 3911313, -6948744, 14759765, -30027150, 7851207,
3282              21690126, 8518463, 26699843, 5276295},
3283             {-13149873, -6429067, 9396249, 365013, 24703301, -10488939, 1321586,
3284              149635, -15452774, 7159369},
3285         },
3286         {
3287             {9987780, -3404759, 17507962, 9505530, 9731535, -2165514, 22356009,
3288              8312176, 22477218, -8403385},
3289             {18155857, -16504990, 19744716, 9006923, 15154154, -10538976,
3290              24256460, -4864995, -22548173, 9334109},
3291             {2986088, -4911893, 10776628, -3473844, 10620590, -7083203,
3292              -21413845, 14253545, -22587149, 536906},
3293         },
3294         {
3295             {4377756, 8115836, 24567078, 15495314, 11625074, 13064599, 7390551,
3296              10589625, 10838060, -15420424},
3297             {-19342404, 867880, 9277171, -3218459, -14431572, -1986443,
3298              19295826, -15796950, 6378260, 699185},
3299             {7895026, 4057113, -7081772, -13077756, -17886831, -323126, -716039,
3300              15693155, -5045064, -13373962},
3301         },
3302         {
3303             {-7737563, -5869402, -14566319, -7406919, 11385654, 13201616,
3304              31730678, -10962840, -3918636, -9669325},
3305             {10188286, -15770834, -7336361, 13427543, 22223443, 14896287,
3306              30743455, 7116568, -21786507, 5427593},
3307             {696102, 13206899, 27047647, -10632082, 15285305, -9853179,
3308              10798490, -4578720, 19236243, 12477404},
3309         },
3310         {
3311             {-11229439, 11243796, -17054270, -8040865, -788228, -8167967,
3312              -3897669, 11180504, -23169516, 7733644},
3313             {17800790, -14036179, -27000429, -11766671, 23887827, 3149671,
3314              23466177, -10538171, 10322027, 15313801},
3315             {26246234, 11968874, 32263343, -5468728, 6830755, -13323031,
3316              -15794704, -101982, -24449242, 10890804},
3317         },
3318         {
3319             {-31365647, 10271363, -12660625, -6267268, 16690207, -13062544,
3320              -14982212, 16484931, 25180797, -5334884},
3321             {-586574, 10376444, -32586414, -11286356, 19801893, 10997610,
3322              2276632, 9482883, 316878, 13820577},
3323             {-9882808, -4510367, -2115506, 16457136, -11100081, 11674996,
3324              30756178, -7515054, 30696930, -3712849},
3325         },
3326         {
3327             {32988917, -9603412, 12499366, 7910787, -10617257, -11931514,
3328              -7342816, -9985397, -32349517, 7392473},
3329             {-8855661, 15927861, 9866406, -3649411, -2396914, -16655781,
3330              -30409476, -9134995, 25112947, -2926644},
3331             {-2504044, -436966, 25621774, -5678772, 15085042, -5479877,
3332              -24884878, -13526194, 5537438, -13914319},
3333         },
3334     },
3335     {
3336         {
3337             {-11225584, 2320285, -9584280, 10149187, -33444663, 5808648,
3338              -14876251, -1729667, 31234590, 6090599},
3339             {-9633316, 116426, 26083934, 2897444, -6364437, -2688086, 609721,
3340              15878753, -6970405, -9034768},
3341             {-27757857, 247744, -15194774, -9002551, 23288161, -10011936,
3342              -23869595, 6503646, 20650474, 1804084},
3343         },
3344         {
3345             {-27589786, 15456424, 8972517, 8469608, 15640622, 4439847, 3121995,
3346              -10329713, 27842616, -202328},
3347             {-15306973, 2839644, 22530074, 10026331, 4602058, 5048462, 28248656,
3348              5031932, -11375082, 12714369},
3349             {20807691, -7270825, 29286141, 11421711, -27876523, -13868230,
3350              -21227475, 1035546, -19733229, 12796920},
3351         },
3352         {
3353             {12076899, -14301286, -8785001, -11848922, -25012791, 16400684,
3354              -17591495, -12899438, 3480665, -15182815},
3355             {-32361549, 5457597, 28548107, 7833186, 7303070, -11953545,
3356              -24363064, -15921875, -33374054, 2771025},
3357             {-21389266, 421932, 26597266, 6860826, 22486084, -6737172,
3358              -17137485, -4210226, -24552282, 15673397},
3359         },
3360         {
3361             {-20184622, 2338216, 19788685, -9620956, -4001265, -8740893,
3362              -20271184, 4733254, 3727144, -12934448},
3363             {6120119, 814863, -11794402, -622716, 6812205, -15747771, 2019594,
3364              7975683, 31123697, -10958981},
3365             {30069250, -11435332, 30434654, 2958439, 18399564, -976289,
3366              12296869, 9204260, -16432438, 9648165},
3367         },
3368         {
3369             {32705432, -1550977, 30705658, 7451065, -11805606, 9631813, 3305266,
3370              5248604, -26008332, -11377501},
3371             {17219865, 2375039, -31570947, -5575615, -19459679, 9219903, 294711,
3372              15298639, 2662509, -16297073},
3373             {-1172927, -7558695, -4366770, -4287744, -21346413, -8434326,
3374              32087529, -1222777, 32247248, -14389861},
3375         },
3376         {
3377             {14312628, 1221556, 17395390, -8700143, -4945741, -8684635,
3378              -28197744, -9637817, -16027623, -13378845},
3379             {-1428825, -9678990, -9235681, 6549687, -7383069, -468664, 23046502,
3380              9803137, 17597934, 2346211},
3381             {18510800, 15337574, 26171504, 981392, -22241552, 7827556,
3382              -23491134, -11323352, 3059833, -11782870},
3383         },
3384         {
3385             {10141598, 6082907, 17829293, -1947643, 9830092, 13613136,
3386              -25556636, -5544586, -33502212, 3592096},
3387             {33114168, -15889352, -26525686, -13343397, 33076705, 8716171,
3388              1151462, 1521897, -982665, -6837803},
3389             {-32939165, -4255815, 23947181, -324178, -33072974, -12305637,
3390              -16637686, 3891704, 26353178, 693168},
3391         },
3392         {
3393             {30374239, 1595580, -16884039, 13186931, 4600344, 406904, 9585294,
3394              -400668, 31375464, 14369965},
3395             {-14370654, -7772529, 1510301, 6434173, -18784789, -6262728,
3396              32732230, -13108839, 17901441, 16011505},
3397             {18171223, -11934626, -12500402, 15197122, -11038147, -15230035,
3398              -19172240, -16046376, 8764035, 12309598},
3399         },
3400     },
3401     {
3402         {
3403             {5975908, -5243188, -19459362, -9681747, -11541277, 14015782,
3404              -23665757, 1228319, 17544096, -10593782},
3405             {5811932, -1715293, 3442887, -2269310, -18367348, -8359541,
3406              -18044043, -15410127, -5565381, 12348900},
3407             {-31399660, 11407555, 25755363, 6891399, -3256938, 14872274,
3408              -24849353, 8141295, -10632534, -585479},
3409         },
3410         {
3411             {-12675304, 694026, -5076145, 13300344, 14015258, -14451394,
3412              -9698672, -11329050, 30944593, 1130208},
3413             {8247766, -6710942, -26562381, -7709309, -14401939, -14648910,
3414              4652152, 2488540, 23550156, -271232},
3415             {17294316, -3788438, 7026748, 15626851, 22990044, 113481, 2267737,
3416              -5908146, -408818, -137719},
3417         },
3418         {
3419             {16091085, -16253926, 18599252, 7340678, 2137637, -1221657,
3420              -3364161, 14550936, 3260525, -7166271},
3421             {-4910104, -13332887, 18550887, 10864893, -16459325, -7291596,
3422              -23028869, -13204905, -12748722, 2701326},
3423             {-8574695, 16099415, 4629974, -16340524, -20786213, -6005432,
3424              -10018363, 9276971, 11329923, 1862132},
3425         },
3426         {
3427             {14763076, -15903608, -30918270, 3689867, 3511892, 10313526,
3428              -21951088, 12219231, -9037963, -940300},
3429             {8894987, -3446094, 6150753, 3013931, 301220, 15693451, -31981216,
3430              -2909717, -15438168, 11595570},
3431             {15214962, 3537601, -26238722, -14058872, 4418657, -15230761,
3432              13947276, 10730794, -13489462, -4363670},
3433         },
3434         {
3435             {-2538306, 7682793, 32759013, 263109, -29984731, -7955452,
3436              -22332124, -10188635, 977108, 699994},
3437             {-12466472, 4195084, -9211532, 550904, -15565337, 12917920,
3438              19118110, -439841, -30534533, -14337913},
3439             {31788461, -14507657, 4799989, 7372237, 8808585, -14747943, 9408237,
3440              -10051775, 12493932, -5409317},
3441         },
3442         {
3443             {-25680606, 5260744, -19235809, -6284470, -3695942, 16566087,
3444              27218280, 2607121, 29375955, 6024730},
3445             {842132, -2794693, -4763381, -8722815, 26332018, -12405641,
3446              11831880, 6985184, -9940361, 2854096},
3447             {-4847262, -7969331, 2516242, -5847713, 9695691, -7221186, 16512645,
3448              960770, 12121869, 16648078},
3449         },
3450         {
3451             {-15218652, 14667096, -13336229, 2013717, 30598287, -464137,
3452              -31504922, -7882064, 20237806, 2838411},
3453             {-19288047, 4453152, 15298546, -16178388, 22115043, -15972604,
3454              12544294, -13470457, 1068881, -12499905},
3455             {-9558883, -16518835, 33238498, 13506958, 30505848, -1114596,
3456              -8486907, -2630053, 12521378, 4845654},
3457         },
3458         {
3459             {-28198521, 10744108, -2958380, 10199664, 7759311, -13088600,
3460              3409348, -873400, -6482306, -12885870},
3461             {-23561822, 6230156, -20382013, 10655314, -24040585, -11621172,
3462              10477734, -1240216, -3113227, 13974498},
3463             {12966261, 15550616, -32038948, -1615346, 21025980, -629444,
3464              5642325, 7188737, 18895762, 12629579},
3465         },
3466     },
3467     {
3468         {
3469             {14741879, -14946887, 22177208, -11721237, 1279741, 8058600,
3470              11758140, 789443, 32195181, 3895677},
3471             {10758205, 15755439, -4509950, 9243698, -4879422, 6879879, -2204575,
3472              -3566119, -8982069, 4429647},
3473             {-2453894, 15725973, -20436342, -10410672, -5803908, -11040220,
3474              -7135870, -11642895, 18047436, -15281743},
3475         },
3476         {
3477             {-25173001, -11307165, 29759956, 11776784, -22262383, -15820455,
3478              10993114, -12850837, -17620701, -9408468},
3479             {21987233, 700364, -24505048, 14972008, -7774265, -5718395,
3480              32155026, 2581431, -29958985, 8773375},
3481             {-25568350, 454463, -13211935, 16126715, 25240068, 8594567,
3482              20656846, 12017935, -7874389, -13920155},
3483         },
3484         {
3485             {6028182, 6263078, -31011806, -11301710, -818919, 2461772,
3486              -31841174, -5468042, -1721788, -2776725},
3487             {-12278994, 16624277, 987579, -5922598, 32908203, 1248608, 7719845,
3488              -4166698, 28408820, 6816612},
3489             {-10358094, -8237829, 19549651, -12169222, 22082623, 16147817,
3490              20613181, 13982702, -10339570, 5067943},
3491         },
3492         {
3493             {-30505967, -3821767, 12074681, 13582412, -19877972, 2443951,
3494              -19719286, 12746132, 5331210, -10105944},
3495             {30528811, 3601899, -1957090, 4619785, -27361822, -15436388,
3496              24180793, -12570394, 27679908, -1648928},
3497             {9402404, -13957065, 32834043, 10838634, -26580150, -13237195,
3498              26653274, -8685565, 22611444, -12715406},
3499         },
3500         {
3501             {22190590, 1118029, 22736441, 15130463, -30460692, -5991321,
3502              19189625, -4648942, 4854859, 6622139},
3503             {-8310738, -2953450, -8262579, -3388049, -10401731, -271929,
3504              13424426, -3567227, 26404409, 13001963},
3505             {-31241838, -15415700, -2994250, 8939346, 11562230, -12840670,
3506              -26064365, -11621720, -15405155, 11020693},
3507         },
3508         {
3509             {1866042, -7949489, -7898649, -10301010, 12483315, 13477547,
3510              3175636, -12424163, 28761762, 1406734},
3511             {-448555, -1777666, 13018551, 3194501, -9580420, -11161737,
3512              24760585, -4347088, 25577411, -13378680},
3513             {-24290378, 4759345, -690653, -1852816, 2066747, 10693769,
3514              -29595790, 9884936, -9368926, 4745410},
3515         },
3516         {
3517             {-9141284, 6049714, -19531061, -4341411, -31260798, 9944276,
3518              -15462008, -11311852, 10931924, -11931931},
3519             {-16561513, 14112680, -8012645, 4817318, -8040464, -11414606,
3520              -22853429, 10856641, -20470770, 13434654},
3521             {22759489, -10073434, -16766264, -1871422, 13637442, -10168091,
3522              1765144, -12654326, 28445307, -5364710},
3523         },
3524         {
3525             {29875063, 12493613, 2795536, -3786330, 1710620, 15181182,
3526              -10195717, -8788675, 9074234, 1167180},
3527             {-26205683, 11014233, -9842651, -2635485, -26908120, 7532294,
3528              -18716888, -9535498, 3843903, 9367684},
3529             {-10969595, -6403711, 9591134, 9582310, 11349256, 108879, 16235123,
3530              8601684, -139197, 4242895},
3531         },
3532     },
3533     {
3534         {
3535             {22092954, -13191123, -2042793, -11968512, 32186753, -11517388,
3536              -6574341, 2470660, -27417366, 16625501},
3537             {-11057722, 3042016, 13770083, -9257922, 584236, -544855, -7770857,
3538              2602725, -27351616, 14247413},
3539             {6314175, -10264892, -32772502, 15957557, -10157730, 168750,
3540              -8618807, 14290061, 27108877, -1180880},
3541         },
3542         {
3543             {-8586597, -7170966, 13241782, 10960156, -32991015, -13794596,
3544              33547976, -11058889, -27148451, 981874},
3545             {22833440, 9293594, -32649448, -13618667, -9136966, 14756819,
3546              -22928859, -13970780, -10479804, -16197962},
3547             {-7768587, 3326786, -28111797, 10783824, 19178761, 14905060,
3548              22680049, 13906969, -15933690, 3797899},
3549         },
3550         {
3551             {21721356, -4212746, -12206123, 9310182, -3882239, -13653110,
3552              23740224, -2709232, 20491983, -8042152},
3553             {9209270, -15135055, -13256557, -6167798, -731016, 15289673,
3554              25947805, 15286587, 30997318, -6703063},
3555             {7392032, 16618386, 23946583, -8039892, -13265164, -1533858,
3556              -14197445, -2321576, 17649998, -250080},
3557         },
3558         {
3559             {-9301088, -14193827, 30609526, -3049543, -25175069, -1283752,
3560              -15241566, -9525724, -2233253, 7662146},
3561             {-17558673, 1763594, -33114336, 15908610, -30040870, -12174295,
3562              7335080, -8472199, -3174674, 3440183},
3563             {-19889700, -5977008, -24111293, -9688870, 10799743, -16571957,
3564              40450, -4431835, 4862400, 1133},
3565         },
3566         {
3567             {-32856209, -7873957, -5422389, 14860950, -16319031, 7956142,
3568              7258061, 311861, -30594991, -7379421},
3569             {-3773428, -1565936, 28985340, 7499440, 24445838, 9325937, 29727763,
3570              16527196, 18278453, 15405622},
3571             {-4381906, 8508652, -19898366, -3674424, -5984453, 15149970,
3572              -13313598, 843523, -21875062, 13626197},
3573         },
3574         {
3575             {2281448, -13487055, -10915418, -2609910, 1879358, 16164207,
3576              -10783882, 3953792, 13340839, 15928663},
3577             {31727126, -7179855, -18437503, -8283652, 2875793, -16390330,
3578              -25269894, -7014826, -23452306, 5964753},
3579             {4100420, -5959452, -17179337, 6017714, -18705837, 12227141,
3580              -26684835, 11344144, 2538215, -7570755},
3581         },
3582         {
3583             {-9433605, 6123113, 11159803, -2156608, 30016280, 14966241,
3584              -20474983, 1485421, -629256, -15958862},
3585             {-26804558, 4260919, 11851389, 9658551, -32017107, 16367492,
3586              -20205425, -13191288, 11659922, -11115118},
3587             {26180396, 10015009, -30844224, -8581293, 5418197, 9480663, 2231568,
3588              -10170080, 33100372, -1306171},
3589         },
3590         {
3591             {15121113, -5201871, -10389905, 15427821, -27509937, -15992507,
3592              21670947, 4486675, -5931810, -14466380},
3593             {16166486, -9483733, -11104130, 6023908, -31926798, -1364923,
3594              2340060, -16254968, -10735770, -10039824},
3595             {28042865, -3557089, -12126526, 12259706, -3717498, -6945899,
3596              6766453, -8689599, 18036436, 5803270},
3597         },
3598     },
3599     {
3600         {
3601             {-817581, 6763912, 11803561, 1585585, 10958447, -2671165, 23855391,
3602              4598332, -6159431, -14117438},
3603             {-31031306, -14256194, 17332029, -2383520, 31312682, -5967183,
3604              696309, 50292, -20095739, 11763584},
3605             {-594563, -2514283, -32234153, 12643980, 12650761, 14811489, 665117,
3606              -12613632, -19773211, -10713562},
3607         },
3608         {
3609             {30464590, -11262872, -4127476, -12734478, 19835327, -7105613,
3610              -24396175, 2075773, -17020157, 992471},
3611             {18357185, -6994433, 7766382, 16342475, -29324918, 411174, 14578841,
3612              8080033, -11574335, -10601610},
3613             {19598397, 10334610, 12555054, 2555664, 18821899, -10339780,
3614              21873263, 16014234, 26224780, 16452269},
3615         },
3616         {
3617             {-30223925, 5145196, 5944548, 16385966, 3976735, 2009897, -11377804,
3618              -7618186, -20533829, 3698650},
3619             {14187449, 3448569, -10636236, -10810935, -22663880, -3433596,
3620              7268410, -10890444, 27394301, 12015369},
3621             {19695761, 16087646, 28032085, 12999827, 6817792, 11427614,
3622              20244189, -1312777, -13259127, -3402461},
3623         },
3624         {
3625             {30860103, 12735208, -1888245, -4699734, -16974906, 2256940,
3626              -8166013, 12298312, -8550524, -10393462},
3627             {-5719826, -11245325, -1910649, 15569035, 26642876, -7587760,
3628              -5789354, -15118654, -4976164, 12651793},
3629             {-2848395, 9953421, 11531313, -5282879, 26895123, -12697089,
3630              -13118820, -16517902, 9768698, -2533218},
3631         },
3632         {
3633             {-24719459, 1894651, -287698, -4704085, 15348719, -8156530,
3634              32767513, 12765450, 4940095, 10678226},
3635             {18860224, 15980149, -18987240, -1562570, -26233012, -11071856,
3636              -7843882, 13944024, -24372348, 16582019},
3637             {-15504260, 4970268, -29893044, 4175593, -20993212, -2199756,
3638              -11704054, 15444560, -11003761, 7989037},
3639         },
3640         {
3641             {31490452, 5568061, -2412803, 2182383, -32336847, 4531686,
3642              -32078269, 6200206, -19686113, -14800171},
3643             {-17308668, -15879940, -31522777, -2831, -32887382, 16375549,
3644              8680158, -16371713, 28550068, -6857132},
3645             {-28126887, -5688091, 16837845, -1820458, -6850681, 12700016,
3646              -30039981, 4364038, 1155602, 5988841},
3647         },
3648         {
3649             {21890435, -13272907, -12624011, 12154349, -7831873, 15300496,
3650              23148983, -4470481, 24618407, 8283181},
3651             {-33136107, -10512751, 9975416, 6841041, -31559793, 16356536,
3652              3070187, -7025928, 1466169, 10740210},
3653             {-1509399, -15488185, -13503385, -10655916, 32799044, 909394,
3654              -13938903, -5779719, -32164649, -15327040},
3655         },
3656         {
3657             {3960823, -14267803, -28026090, -15918051, -19404858, 13146868,
3658              15567327, 951507, -3260321, -573935},
3659             {24740841, 5052253, -30094131, 8961361, 25877428, 6165135,
3660              -24368180, 14397372, -7380369, -6144105},
3661             {-28888365, 3510803, -28103278, -1158478, -11238128, -10631454,
3662              -15441463, -14453128, -1625486, -6494814},
3663         },
3664     },
3665     {
3666         {
3667             {793299, -9230478, 8836302, -6235707, -27360908, -2369593, 33152843,
3668              -4885251, -9906200, -621852},
3669             {5666233, 525582, 20782575, -8038419, -24538499, 14657740, 16099374,
3670              1468826, -6171428, -15186581},
3671             {-4859255, -3779343, -2917758, -6748019, 7778750, 11688288,
3672              -30404353, -9871238, -1558923, -9863646},
3673         },
3674         {
3675             {10896332, -7719704, 824275, 472601, -19460308, 3009587, 25248958,
3676              14783338, -30581476, -15757844},
3677             {10566929, 12612572, -31944212, 11118703, -12633376, 12362879,
3678              21752402, 8822496, 24003793, 14264025},
3679             {27713862, -7355973, -11008240, 9227530, 27050101, 2504721,
3680              23886875, -13117525, 13958495, -5732453},
3681         },
3682         {
3683             {-23481610, 4867226, -27247128, 3900521, 29838369, -8212291,
3684              -31889399, -10041781, 7340521, -15410068},
3685             {4646514, -8011124, -22766023, -11532654, 23184553, 8566613,
3686              31366726, -1381061, -15066784, -10375192},
3687             {-17270517, 12723032, -16993061, 14878794, 21619651, -6197576,
3688              27584817, 3093888, -8843694, 3849921},
3689         },
3690         {
3691             {-9064912, 2103172, 25561640, -15125738, -5239824, 9582958,
3692              32477045, -9017955, 5002294, -15550259},
3693             {-12057553, -11177906, 21115585, -13365155, 8808712, -12030708,
3694              16489530, 13378448, -25845716, 12741426},
3695             {-5946367, 10645103, -30911586, 15390284, -3286982, -7118677,
3696              24306472, 15852464, 28834118, -7646072},
3697         },
3698         {
3699             {-17335748, -9107057, -24531279, 9434953, -8472084, -583362,
3700              -13090771, 455841, 20461858, 5491305},
3701             {13669248, -16095482, -12481974, -10203039, -14569770, -11893198,
3702              -24995986, 11293807, -28588204, -9421832},
3703             {28497928, 6272777, -33022994, 14470570, 8906179, -1225630,
3704              18504674, -14165166, 29867745, -8795943},
3705         },
3706         {
3707             {-16207023, 13517196, -27799630, -13697798, 24009064, -6373891,
3708              -6367600, -13175392, 22853429, -4012011},
3709             {24191378, 16712145, -13931797, 15217831, 14542237, 1646131,
3710              18603514, -11037887, 12876623, -2112447},
3711             {17902668, 4518229, -411702, -2829247, 26878217, 5258055, -12860753,
3712              608397, 16031844, 3723494},
3713         },
3714         {
3715             {-28632773, 12763728, -20446446, 7577504, 33001348, -13017745,
3716              17558842, -7872890, 23896954, -4314245},
3717             {-20005381, -12011952, 31520464, 605201, 2543521, 5991821, -2945064,
3718              7229064, -9919646, -8826859},
3719             {28816045, 298879, -28165016, -15920938, 19000928, -1665890,
3720              -12680833, -2949325, -18051778, -2082915},
3721         },
3722         {
3723             {16000882, -344896, 3493092, -11447198, -29504595, -13159789,
3724              12577740, 16041268, -19715240, 7847707},
3725             {10151868, 10572098, 27312476, 7922682, 14825339, 4723128,
3726              -32855931, -6519018, -10020567, 3852848},
3727             {-11430470, 15697596, -21121557, -4420647, 5386314, 15063598,
3728              16514493, -15932110, 29330899, -15076224},
3729         },
3730     },
3731     {
3732         {
3733             {-25499735, -4378794, -15222908, -6901211, 16615731, 2051784,
3734              3303702, 15490, -27548796, 12314391},
3735             {15683520, -6003043, 18109120, -9980648, 15337968, -5997823,
3736              -16717435, 15921866, 16103996, -3731215},
3737             {-23169824, -10781249, 13588192, -1628807, -3798557, -1074929,
3738              -19273607, 5402699, -29815713, -9841101},
3739         },
3740         {
3741             {23190676, 2384583, -32714340, 3462154, -29903655, -1529132,
3742              -11266856, 8911517, -25205859, 2739713},
3743             {21374101, -3554250, -33524649, 9874411, 15377179, 11831242,
3744              -33529904, 6134907, 4931255, 11987849},
3745             {-7732, -2978858, -16223486, 7277597, 105524, -322051, -31480539,
3746              13861388, -30076310, 10117930},
3747         },
3748         {
3749             {-29501170, -10744872, -26163768, 13051539, -25625564, 5089643,
3750              -6325503, 6704079, 12890019, 15728940},
3751             {-21972360, -11771379, -951059, -4418840, 14704840, 2695116, 903376,
3752              -10428139, 12885167, 8311031},
3753             {-17516482, 5352194, 10384213, -13811658, 7506451, 13453191,
3754              26423267, 4384730, 1888765, -5435404},
3755         },
3756         {
3757             {-25817338, -3107312, -13494599, -3182506, 30896459, -13921729,
3758              -32251644, -12707869, -19464434, -3340243},
3759             {-23607977, -2665774, -526091, 4651136, 5765089, 4618330, 6092245,
3760              14845197, 17151279, -9854116},
3761             {-24830458, -12733720, -15165978, 10367250, -29530908, -265356,
3762              22825805, -7087279, -16866484, 16176525},
3763         },
3764         {
3765             {-23583256, 6564961, 20063689, 3798228, -4740178, 7359225, 2006182,
3766              -10363426, -28746253, -10197509},
3767             {-10626600, -4486402, -13320562, -5125317, 3432136, -6393229,
3768              23632037, -1940610, 32808310, 1099883},
3769             {15030977, 5768825, -27451236, -2887299, -6427378, -15361371,
3770              -15277896, -6809350, 2051441, -15225865},
3771         },
3772         {
3773             {-3362323, -7239372, 7517890, 9824992, 23555850, 295369, 5148398,
3774              -14154188, -22686354, 16633660},
3775             {4577086, -16752288, 13249841, -15304328, 19958763, -14537274,
3776              18559670, -10759549, 8402478, -9864273},
3777             {-28406330, -1051581, -26790155, -907698, -17212414, -11030789,
3778              9453451, -14980072, 17983010, 9967138},
3779         },
3780         {
3781             {-25762494, 6524722, 26585488, 9969270, 24709298, 1220360, -1677990,
3782              7806337, 17507396, 3651560},
3783             {-10420457, -4118111, 14584639, 15971087, -15768321, 8861010,
3784              26556809, -5574557, -18553322, -11357135},
3785             {2839101, 14284142, 4029895, 3472686, 14402957, 12689363, -26642121,
3786              8459447, -5605463, -7621941},
3787         },
3788         {
3789             {-4839289, -3535444, 9744961, 2871048, 25113978, 3187018, -25110813,
3790              -849066, 17258084, -7977739},
3791             {18164541, -10595176, -17154882, -1542417, 19237078, -9745295,
3792              23357533, -15217008, 26908270, 12150756},
3793             {-30264870, -7647865, 5112249, -7036672, -1499807, -6974257, 43168,
3794              -5537701, -32302074, 16215819},
3795         },
3796     },
3797     {
3798         {
3799             {-6898905, 9824394, -12304779, -4401089, -31397141, -6276835,
3800              32574489, 12532905, -7503072, -8675347},
3801             {-27343522, -16515468, -27151524, -10722951, 946346, 16291093,
3802              254968, 7168080, 21676107, -1943028},
3803             {21260961, -8424752, -16831886, -11920822, -23677961, 3968121,
3804              -3651949, -6215466, -3556191, -7913075},
3805         },
3806         {
3807             {16544754, 13250366, -16804428, 15546242, -4583003, 12757258,
3808              -2462308, -8680336, -18907032, -9662799},
3809             {-2415239, -15577728, 18312303, 4964443, -15272530, -12653564,
3810              26820651, 16690659, 25459437, -4564609},
3811             {-25144690, 11425020, 28423002, -11020557, -6144921, -15826224,
3812              9142795, -2391602, -6432418, -1644817},
3813         },
3814         {
3815             {-23104652, 6253476, 16964147, -3768872, -25113972, -12296437,
3816              -27457225, -16344658, 6335692, 7249989},
3817             {-30333227, 13979675, 7503222, -12368314, -11956721, -4621693,
3818              -30272269, 2682242, 25993170, -12478523},
3819             {4364628, 5930691, 32304656, -10044554, -8054781, 15091131,
3820              22857016, -10598955, 31820368, 15075278},
3821         },
3822         {
3823             {31879134, -8918693, 17258761, 90626, -8041836, -4917709, 24162788,
3824              -9650886, -17970238, 12833045},
3825             {19073683, 14851414, -24403169, -11860168, 7625278, 11091125,
3826              -19619190, 2074449, -9413939, 14905377},
3827             {24483667, -11935567, -2518866, -11547418, -1553130, 15355506,
3828              -25282080, 9253129, 27628530, -7555480},
3829         },
3830         {
3831             {17597607, 8340603, 19355617, 552187, 26198470, -3176583, 4593324,
3832              -9157582, -14110875, 15297016},
3833             {510886, 14337390, -31785257, 16638632, 6328095, 2713355, -20217417,
3834              -11864220, 8683221, 2921426},
3835             {18606791, 11874196, 27155355, -5281482, -24031742, 6265446,
3836              -25178240, -1278924, 4674690, 13890525},
3837         },
3838         {
3839             {13609624, 13069022, -27372361, -13055908, 24360586, 9592974,
3840              14977157, 9835105, 4389687, 288396},
3841             {9922506, -519394, 13613107, 5883594, -18758345, -434263, -12304062,
3842              8317628, 23388070, 16052080},
3843             {12720016, 11937594, -31970060, -5028689, 26900120, 8561328,
3844              -20155687, -11632979, -14754271, -10812892},
3845         },
3846         {
3847             {15961858, 14150409, 26716931, -665832, -22794328, 13603569,
3848              11829573, 7467844, -28822128, 929275},
3849             {11038231, -11582396, -27310482, -7316562, -10498527, -16307831,
3850              -23479533, -9371869, -21393143, 2465074},
3851             {20017163, -4323226, 27915242, 1529148, 12396362, 15675764,
3852              13817261, -9658066, 2463391, -4622140},
3853         },
3854         {
3855             {-16358878, -12663911, -12065183, 4996454, -1256422, 1073572,
3856              9583558, 12851107, 4003896, 12673717},
3857             {-1731589, -15155870, -3262930, 16143082, 19294135, 13385325,
3858              14741514, -9103726, 7903886, 2348101},
3859             {24536016, -16515207, 12715592, -3862155, 1511293, 10047386,
3860              -3842346, -7129159, -28377538, 10048127},
3861         },
3862     },
3863     {
3864         {
3865             {-12622226, -6204820, 30718825, 2591312, -10617028, 12192840,
3866              18873298, -7297090, -32297756, 15221632},
3867             {-26478122, -11103864, 11546244, -1852483, 9180880, 7656409,
3868              -21343950, 2095755, 29769758, 6593415},
3869             {-31994208, -2907461, 4176912, 3264766, 12538965, -868111, 26312345,
3870              -6118678, 30958054, 8292160},
3871         },
3872         {
3873             {31429822, -13959116, 29173532, 15632448, 12174511, -2760094,
3874              32808831, 3977186, 26143136, -3148876},
3875             {22648901, 1402143, -22799984, 13746059, 7936347, 365344, -8668633,
3876              -1674433, -3758243, -2304625},
3877             {-15491917, 8012313, -2514730, -12702462, -23965846, -10254029,
3878              -1612713, -1535569, -16664475, 8194478},
3879         },
3880         {
3881             {27338066, -7507420, -7414224, 10140405, -19026427, -6589889,
3882              27277191, 8855376, 28572286, 3005164},
3883             {26287124, 4821776, 25476601, -4145903, -3764513, -15788984,
3884              -18008582, 1182479, -26094821, -13079595},
3885             {-7171154, 3178080, 23970071, 6201893, -17195577, -4489192,
3886              -21876275, -13982627, 32208683, -1198248},
3887         },
3888         {
3889             {-16657702, 2817643, -10286362, 14811298, 6024667, 13349505,
3890              -27315504, -10497842, -27672585, -11539858},
3891             {15941029, -9405932, -21367050, 8062055, 31876073, -238629,
3892              -15278393, -1444429, 15397331, -4130193},
3893             {8934485, -13485467, -23286397, -13423241, -32446090, 14047986,
3894              31170398, -1441021, -27505566, 15087184},
3895         },
3896         {
3897             {-18357243, -2156491, 24524913, -16677868, 15520427, -6360776,
3898              -15502406, 11461896, 16788528, -5868942},
3899             {-1947386, 16013773, 21750665, 3714552, -17401782, -16055433,
3900              -3770287, -10323320, 31322514, -11615635},
3901             {21426655, -5650218, -13648287, -5347537, -28812189, -4920970,
3902              -18275391, -14621414, 13040862, -12112948},
3903         },
3904         {
3905             {11293895, 12478086, -27136401, 15083750, -29307421, 14748872,
3906              14555558, -13417103, 1613711, 4896935},
3907             {-25894883, 15323294, -8489791, -8057900, 25967126, -13425460,
3908              2825960, -4897045, -23971776, -11267415},
3909             {-15924766, -5229880, -17443532, 6410664, 3622847, 10243618,
3910              20615400, 12405433, -23753030, -8436416},
3911         },
3912         {
3913             {-7091295, 12556208, -20191352, 9025187, -17072479, 4333801,
3914              4378436, 2432030, 23097949, -566018},
3915             {4565804, -16025654, 20084412, -7842817, 1724999, 189254, 24767264,
3916              10103221, -18512313, 2424778},
3917             {366633, -11976806, 8173090, -6890119, 30788634, 5745705, -7168678,
3918              1344109, -3642553, 12412659},
3919         },
3920         {
3921             {-24001791, 7690286, 14929416, -168257, -32210835, -13412986,
3922              24162697, -15326504, -3141501, 11179385},
3923             {18289522, -14724954, 8056945, 16430056, -21729724, 7842514,
3924              -6001441, -1486897, -18684645, -11443503},
3925             {476239, 6601091, -6152790, -9723375, 17503545, -4863900, 27672959,
3926              13403813, 11052904, 5219329},
3927         },
3928     },
3929     {
3930         {
3931             {20678546, -8375738, -32671898, 8849123, -5009758, 14574752,
3932              31186971, -3973730, 9014762, -8579056},
3933             {-13644050, -10350239, -15962508, 5075808, -1514661, -11534600,
3934              -33102500, 9160280, 8473550, -3256838},
3935             {24900749, 14435722, 17209120, -15292541, -22592275, 9878983,
3936              -7689309, -16335821, -24568481, 11788948},
3937         },
3938         {
3939             {-3118155, -11395194, -13802089, 14797441, 9652448, -6845904,
3940              -20037437, 10410733, -24568470, -1458691},
3941             {-15659161, 16736706, -22467150, 10215878, -9097177, 7563911,
3942              11871841, -12505194, -18513325, 8464118},
3943             {-23400612, 8348507, -14585951, -861714, -3950205, -6373419,
3944              14325289, 8628612, 33313881, -8370517},
3945         },
3946         {
3947             {-20186973, -4967935, 22367356, 5271547, -1097117, -4788838,
3948              -24805667, -10236854, -8940735, -5818269},
3949             {-6948785, -1795212, -32625683, -16021179, 32635414, -7374245,
3950              15989197, -12838188, 28358192, -4253904},
3951             {-23561781, -2799059, -32351682, -1661963, -9147719, 10429267,
3952              -16637684, 4072016, -5351664, 5596589},
3953         },
3954         {
3955             {-28236598, -3390048, 12312896, 6213178, 3117142, 16078565,
3956              29266239, 2557221, 1768301, 15373193},
3957             {-7243358, -3246960, -4593467, -7553353, -127927, -912245, -1090902,
3958              -4504991, -24660491, 3442910},
3959             {-30210571, 5124043, 14181784, 8197961, 18964734, -11939093,
3960              22597931, 7176455, -18585478, 13365930},
3961         },
3962         {
3963             {-7877390, -1499958, 8324673, 4690079, 6261860, 890446, 24538107,
3964              -8570186, -9689599, -3031667},
3965             {25008904, -10771599, -4305031, -9638010, 16265036, 15721635,
3966              683793, -11823784, 15723479, -15163481},
3967             {-9660625, 12374379, -27006999, -7026148, -7724114, -12314514,
3968              11879682, 5400171, 519526, -1235876},
3969         },
3970         {
3971             {22258397, -16332233, -7869817, 14613016, -22520255, -2950923,
3972              -20353881, 7315967, 16648397, 7605640},
3973             {-8081308, -8464597, -8223311, 9719710, 19259459, -15348212,
3974              23994942, -5281555, -9468848, 4763278},
3975             {-21699244, 9220969, -15730624, 1084137, -25476107, -2852390,
3976              31088447, -7764523, -11356529, 728112},
3977         },
3978         {
3979             {26047220, -11751471, -6900323, -16521798, 24092068, 9158119,
3980              -4273545, -12555558, -29365436, -5498272},
3981             {17510331, -322857, 5854289, 8403524, 17133918, -3112612, -28111007,
3982              12327945, 10750447, 10014012},
3983             {-10312768, 3936952, 9156313, -8897683, 16498692, -994647,
3984              -27481051, -666732, 3424691, 7540221},
3985         },
3986         {
3987             {30322361, -6964110, 11361005, -4143317, 7433304, 4989748, -7071422,
3988              -16317219, -9244265, 15258046},
3989             {13054562, -2779497, 19155474, 469045, -12482797, 4566042, 5631406,
3990              2711395, 1062915, -5136345},
3991             {-19240248, -11254599, -29509029, -7499965, -5835763, 13005411,
3992              -6066489, 12194497, 32960380, 1459310},
3993         },
3994     },
3995     {
3996         {
3997             {19852034, 7027924, 23669353, 10020366, 8586503, -6657907, 394197,
3998              -6101885, 18638003, -11174937},
3999             {31395534, 15098109, 26581030, 8030562, -16527914, -5007134,
4000              9012486, -7584354, -6643087, -5442636},
4001             {-9192165, -2347377, -1997099, 4529534, 25766844, 607986, -13222,
4002              9677543, -32294889, -6456008},
4003         },
4004         {
4005             {-2444496, -149937, 29348902, 8186665, 1873760, 12489863, -30934579,
4006              -7839692, -7852844, -8138429},
4007             {-15236356, -15433509, 7766470, 746860, 26346930, -10221762,
4008              -27333451, 10754588, -9431476, 5203576},
4009             {31834314, 14135496, -770007, 5159118, 20917671, -16768096,
4010              -7467973, -7337524, 31809243, 7347066},
4011         },
4012         {
4013             {-9606723, -11874240, 20414459, 13033986, 13716524, -11691881,
4014              19797970, -12211255, 15192876, -2087490},
4015             {-12663563, -2181719, 1168162, -3804809, 26747877, -14138091,
4016              10609330, 12694420, 33473243, -13382104},
4017             {33184999, 11180355, 15832085, -11385430, -1633671, 225884,
4018              15089336, -11023903, -6135662, 14480053},
4019         },
4020         {
4021             {31308717, -5619998, 31030840, -1897099, 15674547, -6582883,
4022              5496208, 13685227, 27595050, 8737275},
4023             {-20318852, -15150239, 10933843, -16178022, 8335352, -7546022,
4024              -31008351, -12610604, 26498114, 66511},
4025             {22644454, -8761729, -16671776, 4884562, -3105614, -13559366,
4026              30540766, -4286747, -13327787, -7515095},
4027         },
4028         {
4029             {-28017847, 9834845, 18617207, -2681312, -3401956, -13307506,
4030              8205540, 13585437, -17127465, 15115439},
4031             {23711543, -672915, 31206561, -8362711, 6164647, -9709987,
4032              -33535882, -1426096, 8236921, 16492939},
4033             {-23910559, -13515526, -26299483, -4503841, 25005590, -7687270,
4034              19574902, 10071562, 6708380, -6222424},
4035         },
4036         {
4037             {2101391, -4930054, 19702731, 2367575, -15427167, 1047675, 5301017,
4038              9328700, 29955601, -11678310},
4039             {3096359, 9271816, -21620864, -15521844, -14847996, -7592937,
4040              -25892142, -12635595, -9917575, 6216608},
4041             {-32615849, 338663, -25195611, 2510422, -29213566, -13820213,
4042              24822830, -6146567, -26767480, 7525079},
4043         },
4044         {
4045             {-23066649, -13985623, 16133487, -7896178, -3389565, 778788,
4046              -910336, -2782495, -19386633, 11994101},
4047             {21691500, -13624626, -641331, -14367021, 3285881, -3483596,
4048              -25064666, 9718258, -7477437, 13381418},
4049             {18445390, -4202236, 14979846, 11622458, -1727110, -3582980,
4050              23111648, -6375247, 28535282, 15779576},
4051         },
4052         {
4053             {30098053, 3089662, -9234387, 16662135, -21306940, 11308411,
4054              -14068454, 12021730, 9955285, -16303356},
4055             {9734894, -14576830, -7473633, -9138735, 2060392, 11313496,
4056              -18426029, 9924399, 20194861, 13380996},
4057             {-26378102, -7965207, -22167821, 15789297, -18055342, -6168792,
4058              -1984914, 15707771, 26342023, 10146099},
4059         },
4060     },
4061     {
4062         {
4063             {-26016874, -219943, 21339191, -41388, 19745256, -2878700,
4064              -29637280, 2227040, 21612326, -545728},
4065             {-13077387, 1184228, 23562814, -5970442, -20351244, -6348714,
4066              25764461, 12243797, -20856566, 11649658},
4067             {-10031494, 11262626, 27384172, 2271902, 26947504, -15997771, 39944,
4068              6114064, 33514190, 2333242},
4069         },
4070         {
4071             {-21433588, -12421821, 8119782, 7219913, -21830522, -9016134,
4072              -6679750, -12670638, 24350578, -13450001},
4073             {-4116307, -11271533, -23886186, 4843615, -30088339, 690623,
4074              -31536088, -10406836, 8317860, 12352766},
4075             {18200138, -14475911, -33087759, -2696619, -23702521, -9102511,
4076              -23552096, -2287550, 20712163, 6719373},
4077         },
4078         {
4079             {26656208, 6075253, -7858556, 1886072, -28344043, 4262326, 11117530,
4080              -3763210, 26224235, -3297458},
4081             {-17168938, -14854097, -3395676, -16369877, -19954045, 14050420,
4082              21728352, 9493610, 18620611, -16428628},
4083             {-13323321, 13325349, 11432106, 5964811, 18609221, 6062965,
4084              -5269471, -9725556, -30701573, -16479657},
4085         },
4086         {
4087             {-23860538, -11233159, 26961357, 1640861, -32413112, -16737940,
4088              12248509, -5240639, 13735342, 1934062},
4089             {25089769, 6742589, 17081145, -13406266, 21909293, -16067981,
4090              -15136294, -3765346, -21277997, 5473616},
4091             {31883677, -7961101, 1083432, -11572403, 22828471, 13290673,
4092              -7125085, 12469656, 29111212, -5451014},
4093         },
4094         {
4095             {24244947, -15050407, -26262976, 2791540, -14997599, 16666678,
4096              24367466, 6388839, -10295587, 452383},
4097             {-25640782, -3417841, 5217916, 16224624, 19987036, -4082269,
4098              -24236251, -5915248, 15766062, 8407814},
4099             {-20406999, 13990231, 15495425, 16395525, 5377168, 15166495,
4100              -8917023, -4388953, -8067909, 2276718},
4101         },
4102         {
4103             {30157918, 12924066, -17712050, 9245753, 19895028, 3368142,
4104              -23827587, 5096219, 22740376, -7303417},
4105             {2041139, -14256350, 7783687, 13876377, -25946985, -13352459,
4106              24051124, 13742383, -15637599, 13295222},
4107             {33338237, -8505733, 12532113, 7977527, 9106186, -1715251,
4108              -17720195, -4612972, -4451357, -14669444},
4109         },
4110         {
4111             {-20045281, 5454097, -14346548, 6447146, 28862071, 1883651,
4112              -2469266, -4141880, 7770569, 9620597},
4113             {23208068, 7979712, 33071466, 8149229, 1758231, -10834995, 30945528,
4114              -1694323, -33502340, -14767970},
4115             {1439958, -16270480, -1079989, -793782, 4625402, 10647766, -5043801,
4116              1220118, 30494170, -11440799},
4117         },
4118         {
4119             {-5037580, -13028295, -2970559, -3061767, 15640974, -6701666,
4120              -26739026, 926050, -1684339, -13333647},
4121             {13908495, -3549272, 30919928, -6273825, -21521863, 7989039,
4122              9021034, 9078865, 3353509, 4033511},
4123             {-29663431, -15113610, 32259991, -344482, 24295849, -12912123,
4124              23161163, 8839127, 27485041, 7356032},
4125         },
4126     },
4127     {
4128         {
4129             {9661027, 705443, 11980065, -5370154, -1628543, 14661173, -6346142,
4130              2625015, 28431036, -16771834},
4131             {-23839233, -8311415, -25945511, 7480958, -17681669, -8354183,
4132              -22545972, 14150565, 15970762, 4099461},
4133             {29262576, 16756590, 26350592, -8793563, 8529671, -11208050,
4134              13617293, -9937143, 11465739, 8317062},
4135         },
4136         {
4137             {-25493081, -6962928, 32500200, -9419051, -23038724, -2302222,
4138              14898637, 3848455, 20969334, -5157516},
4139             {-20384450, -14347713, -18336405, 13884722, -33039454, 2842114,
4140              -21610826, -3649888, 11177095, 14989547},
4141             {-24496721, -11716016, 16959896, 2278463, 12066309, 10137771,
4142              13515641, 2581286, -28487508, 9930240},
4143         },
4144         {
4145             {-17751622, -2097826, 16544300, -13009300, -15914807, -14949081,
4146              18345767, -13403753, 16291481, -5314038},
4147             {-33229194, 2553288, 32678213, 9875984, 8534129, 6889387, -9676774,
4148              6957617, 4368891, 9788741},
4149             {16660756, 7281060, -10830758, 12911820, 20108584, -8101676,
4150              -21722536, -8613148, 16250552, -11111103},
4151         },
4152         {
4153             {-19765507, 2390526, -16551031, 14161980, 1905286, 6414907, 4689584,
4154              10604807, -30190403, 4782747},
4155             {-1354539, 14736941, -7367442, -13292886, 7710542, -14155590,
4156              -9981571, 4383045, 22546403, 437323},
4157             {31665577, -12180464, -16186830, 1491339, -18368625, 3294682,
4158              27343084, 2786261, -30633590, -14097016},
4159         },
4160         {
4161             {-14467279, -683715, -33374107, 7448552, 19294360, 14334329,
4162              -19690631, 2355319, -19284671, -6114373},
4163             {15121312, -15796162, 6377020, -6031361, -10798111, -12957845,
4164              18952177, 15496498, -29380133, 11754228},
4165             {-2637277, -13483075, 8488727, -14303896, 12728761, -1622493,
4166              7141596, 11724556, 22761615, -10134141},
4167         },
4168         {
4169             {16918416, 11729663, -18083579, 3022987, -31015732, -13339659,
4170              -28741185, -12227393, 32851222, 11717399},
4171             {11166634, 7338049, -6722523, 4531520, -29468672, -7302055,
4172              31474879, 3483633, -1193175, -4030831},
4173             {-185635, 9921305, 31456609, -13536438, -12013818, 13348923,
4174              33142652, 6546660, -19985279, -3948376},
4175         },
4176         {
4177             {-32460596, 11266712, -11197107, -7899103, 31703694, 3855903,
4178              -8537131, -12833048, -30772034, -15486313},
4179             {-18006477, 12709068, 3991746, -6479188, -21491523, -10550425,
4180              -31135347, -16049879, 10928917, 3011958},
4181             {-6957757, -15594337, 31696059, 334240, 29576716, 14796075,
4182              -30831056, -12805180, 18008031, 10258577},
4183         },
4184         {
4185             {-22448644, 15655569, 7018479, -4410003, -30314266, -1201591,
4186              -1853465, 1367120, 25127874, 6671743},
4187             {29701166, -14373934, -10878120, 9279288, -17568, 13127210,
4188              21382910, 11042292, 25838796, 4642684},
4189             {-20430234, 14955537, -24126347, 8124619, -5369288, -5990470,
4190              30468147, -13900640, 18423289, 4177476},
4191         },
4192     },
4193 };
4194 
negative(signed char b)4195 static uint8_t negative(signed char b)
4196 {
4197     uint32_t x = b;
4198 
4199     x >>= 31; /* 1: yes; 0: no */
4200     return x;
4201 }
4202 
table_select(ge_precomp * t,int pos,signed char b)4203 static void table_select(ge_precomp *t, int pos, signed char b)
4204 {
4205     ge_precomp minust;
4206     uint8_t bnegative = negative(b);
4207     uint8_t babs = b - ((uint8_t)((-bnegative) & b) << 1);
4208 
4209     ge_precomp_0(t);
4210     cmov(t, &k25519Precomp[pos][0], equal(babs, 1));
4211     cmov(t, &k25519Precomp[pos][1], equal(babs, 2));
4212     cmov(t, &k25519Precomp[pos][2], equal(babs, 3));
4213     cmov(t, &k25519Precomp[pos][3], equal(babs, 4));
4214     cmov(t, &k25519Precomp[pos][4], equal(babs, 5));
4215     cmov(t, &k25519Precomp[pos][5], equal(babs, 6));
4216     cmov(t, &k25519Precomp[pos][6], equal(babs, 7));
4217     cmov(t, &k25519Precomp[pos][7], equal(babs, 8));
4218     fe_copy(minust.yplusx, t->yminusx);
4219     fe_copy(minust.yminusx, t->yplusx);
4220     fe_neg(minust.xy2d, t->xy2d);
4221     cmov(t, &minust, bnegative);
4222 }
4223 
4224 /*
4225  * h = a * B
4226  *
4227  * where a = a[0]+256*a[1]+...+256^31 a[31]
4228  * B is the Ed25519 base point (x,4/5) with x positive.
4229  *
4230  * Preconditions:
4231  *   a[31] <= 127
4232  */
ge_scalarmult_base(ge_p3 * h,const uint8_t * a)4233 static void ge_scalarmult_base(ge_p3 *h, const uint8_t *a)
4234 {
4235     signed char e[64];
4236     signed char carry;
4237     ge_p1p1 r;
4238     ge_p2 s;
4239     ge_precomp t;
4240     int i;
4241 
4242     for (i = 0; i < 32; ++i) {
4243         e[2 * i + 0] = (a[i] >> 0) & 15;
4244         e[2 * i + 1] = (a[i] >> 4) & 15;
4245     }
4246     /* each e[i] is between 0 and 15 */
4247     /* e[63] is between 0 and 7 */
4248 
4249     carry = 0;
4250     for (i = 0; i < 63; ++i) {
4251         e[i] += carry;
4252         carry = e[i] + 8;
4253         carry >>= 4;
4254         e[i] -= carry << 4;
4255     }
4256     e[63] += carry;
4257     /* each e[i] is between -8 and 8 */
4258 
4259     ge_p3_0(h);
4260     for (i = 1; i < 64; i += 2) {
4261         table_select(&t, i / 2, e[i]);
4262         ge_madd(&r, h, &t);
4263         ge_p1p1_to_p3(h, &r);
4264     }
4265 
4266     ge_p3_dbl(&r, h);
4267     ge_p1p1_to_p2(&s, &r);
4268     ge_p2_dbl(&r, &s);
4269     ge_p1p1_to_p2(&s, &r);
4270     ge_p2_dbl(&r, &s);
4271     ge_p1p1_to_p2(&s, &r);
4272     ge_p2_dbl(&r, &s);
4273     ge_p1p1_to_p3(h, &r);
4274 
4275     for (i = 0; i < 64; i += 2) {
4276         table_select(&t, i / 2, e[i]);
4277         ge_madd(&r, h, &t);
4278         ge_p1p1_to_p3(h, &r);
4279     }
4280 
4281     OPENSSL_cleanse(e, sizeof(e));
4282 }
4283 
4284 #if !defined(BASE_2_51_IMPLEMENTED)
4285 /*
4286  * Replace (f,g) with (g,f) if b == 1;
4287  * replace (f,g) with (f,g) if b == 0.
4288  *
4289  * Preconditions: b in {0,1}.
4290  */
fe_cswap(fe f,fe g,unsigned int b)4291 static void fe_cswap(fe f, fe g, unsigned int b)
4292 {
4293     size_t i;
4294 
4295     b = 0-b;
4296     for (i = 0; i < 10; i++) {
4297         int32_t x = f[i] ^ g[i];
4298         x &= b;
4299         f[i] ^= x;
4300         g[i] ^= x;
4301     }
4302 }
4303 
4304 /*
4305  * h = f * 121666
4306  *
4307  * Can overlap h with f.
4308  *
4309  * Preconditions:
4310  *    |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
4311  *
4312  * Postconditions:
4313  *    |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
4314  */
fe_mul121666(fe h,fe f)4315 static void fe_mul121666(fe h, fe f)
4316 {
4317     int32_t f0 = f[0];
4318     int32_t f1 = f[1];
4319     int32_t f2 = f[2];
4320     int32_t f3 = f[3];
4321     int32_t f4 = f[4];
4322     int32_t f5 = f[5];
4323     int32_t f6 = f[6];
4324     int32_t f7 = f[7];
4325     int32_t f8 = f[8];
4326     int32_t f9 = f[9];
4327     int64_t h0 = f0 * (int64_t) 121666;
4328     int64_t h1 = f1 * (int64_t) 121666;
4329     int64_t h2 = f2 * (int64_t) 121666;
4330     int64_t h3 = f3 * (int64_t) 121666;
4331     int64_t h4 = f4 * (int64_t) 121666;
4332     int64_t h5 = f5 * (int64_t) 121666;
4333     int64_t h6 = f6 * (int64_t) 121666;
4334     int64_t h7 = f7 * (int64_t) 121666;
4335     int64_t h8 = f8 * (int64_t) 121666;
4336     int64_t h9 = f9 * (int64_t) 121666;
4337     int64_t carry0;
4338     int64_t carry1;
4339     int64_t carry2;
4340     int64_t carry3;
4341     int64_t carry4;
4342     int64_t carry5;
4343     int64_t carry6;
4344     int64_t carry7;
4345     int64_t carry8;
4346     int64_t carry9;
4347 
4348     carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
4349     carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
4350     carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
4351     carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
4352     carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;
4353 
4354     carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
4355     carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
4356     carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
4357     carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
4358     carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;
4359 
4360     h[0] = (int32_t)h0;
4361     h[1] = (int32_t)h1;
4362     h[2] = (int32_t)h2;
4363     h[3] = (int32_t)h3;
4364     h[4] = (int32_t)h4;
4365     h[5] = (int32_t)h5;
4366     h[6] = (int32_t)h6;
4367     h[7] = (int32_t)h7;
4368     h[8] = (int32_t)h8;
4369     h[9] = (int32_t)h9;
4370 }
4371 
x25519_scalar_mult_generic(uint8_t out[32],const uint8_t scalar[32],const uint8_t point[32])4372 static void x25519_scalar_mult_generic(uint8_t out[32],
4373                                        const uint8_t scalar[32],
4374                                        const uint8_t point[32]) {
4375     fe x1, x2, z2, x3, z3, tmp0, tmp1;
4376     uint8_t e[32];
4377     unsigned swap = 0;
4378     int pos;
4379 
4380     memcpy(e, scalar, 32);
4381     e[0] &= 248;
4382     e[31] &= 127;
4383     e[31] |= 64;
4384     fe_frombytes(x1, point);
4385     fe_1(x2);
4386     fe_0(z2);
4387     fe_copy(x3, x1);
4388     fe_1(z3);
4389 
4390     for (pos = 254; pos >= 0; --pos) {
4391         unsigned b = 1 & (e[pos / 8] >> (pos & 7));
4392         swap ^= b;
4393         fe_cswap(x2, x3, swap);
4394         fe_cswap(z2, z3, swap);
4395         swap = b;
4396         fe_sub(tmp0, x3, z3);
4397         fe_sub(tmp1, x2, z2);
4398         fe_add(x2, x2, z2);
4399         fe_add(z2, x3, z3);
4400         fe_mul(z3, tmp0, x2);
4401         fe_mul(z2, z2, tmp1);
4402         fe_sq(tmp0, tmp1);
4403         fe_sq(tmp1, x2);
4404         fe_add(x3, z3, z2);
4405         fe_sub(z2, z3, z2);
4406         fe_mul(x2, tmp1, tmp0);
4407         fe_sub(tmp1, tmp1, tmp0);
4408         fe_sq(z2, z2);
4409         fe_mul121666(z3, tmp1);
4410         fe_sq(x3, x3);
4411         fe_add(tmp0, tmp0, z3);
4412         fe_mul(z3, x1, z2);
4413         fe_mul(z2, tmp1, tmp0);
4414     }
4415 
4416     fe_invert(z2, z2);
4417     fe_mul(x2, x2, z2);
4418     fe_tobytes(out, x2);
4419 
4420     OPENSSL_cleanse(e, sizeof(e));
4421 }
4422 
x25519_scalar_mult(uint8_t out[32],const uint8_t scalar[32],const uint8_t point[32])4423 static void x25519_scalar_mult(uint8_t out[32], const uint8_t scalar[32],
4424                                const uint8_t point[32]) {
4425     x25519_scalar_mult_generic(out, scalar, point);
4426 }
4427 #endif
4428 
slide(signed char * r,const uint8_t * a)4429 static void slide(signed char *r, const uint8_t *a)
4430 {
4431     int i;
4432     int b;
4433     int k;
4434 
4435     for (i = 0; i < 256; ++i) {
4436         r[i] = 1 & (a[i >> 3] >> (i & 7));
4437     }
4438 
4439     for (i = 0; i < 256; ++i) {
4440         if (r[i]) {
4441             for (b = 1; b <= 6 && i + b < 256; ++b) {
4442                 if (r[i + b]) {
4443                     if (r[i] + (r[i + b] << b) <= 15) {
4444                         r[i] += r[i + b] << b;
4445                         r[i + b] = 0;
4446                     } else if (r[i] - (r[i + b] << b) >= -15) {
4447                         r[i] -= r[i + b] << b;
4448                         for (k = i + b; k < 256; ++k) {
4449                             if (!r[k]) {
4450                                 r[k] = 1;
4451                                 break;
4452                             }
4453                             r[k] = 0;
4454                         }
4455                     } else {
4456                         break;
4457                     }
4458                 }
4459             }
4460         }
4461     }
4462 }
4463 
4464 static const ge_precomp Bi[8] = {
4465     {
4466         {25967493, -14356035, 29566456, 3660896, -12694345, 4014787, 27544626,
4467          -11754271, -6079156, 2047605},
4468         {-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692,
4469          5043384, 19500929, -15469378},
4470         {-8738181, 4489570, 9688441, -14785194, 10184609, -12363380, 29287919,
4471          11864899, -24514362, -4438546},
4472     },
4473     {
4474         {15636291, -9688557, 24204773, -7912398, 616977, -16685262, 27787600,
4475          -14772189, 28944400, -1550024},
4476         {16568933, 4717097, -11556148, -1102322, 15682896, -11807043, 16354577,
4477          -11775962, 7689662, 11199574},
4478         {30464156, -5976125, -11779434, -15670865, 23220365, 15915852, 7512774,
4479          10017326, -17749093, -9920357},
4480     },
4481     {
4482         {10861363, 11473154, 27284546, 1981175, -30064349, 12577861, 32867885,
4483          14515107, -15438304, 10819380},
4484         {4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668,
4485          12483688, -12668491, 5581306},
4486         {19563160, 16186464, -29386857, 4097519, 10237984, -4348115, 28542350,
4487          13850243, -23678021, -15815942},
4488     },
4489     {
4490         {5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852,
4491          5230134, -23952439, -15175766},
4492         {-30269007, -3463509, 7665486, 10083793, 28475525, 1649722, 20654025,
4493          16520125, 30598449, 7715701},
4494         {28881845, 14381568, 9657904, 3680757, -20181635, 7843316, -31400660,
4495          1370708, 29794553, -1409300},
4496     },
4497     {
4498         {-22518993, -6692182, 14201702, -8745502, -23510406, 8844726, 18474211,
4499          -1361450, -13062696, 13821877},
4500         {-6455177, -7839871, 3374702, -4740862, -27098617, -10571707, 31655028,
4501          -7212327, 18853322, -14220951},
4502         {4566830, -12963868, -28974889, -12240689, -7602672, -2830569, -8514358,
4503          -10431137, 2207753, -3209784},
4504     },
4505     {
4506         {-25154831, -4185821, 29681144, 7868801, -6854661, -9423865, -12437364,
4507          -663000, -31111463, -16132436},
4508         {25576264, -2703214, 7349804, -11814844, 16472782, 9300885, 3844789,
4509          15725684, 171356, 6466918},
4510         {23103977, 13316479, 9739013, -16149481, 817875, -15038942, 8965339,
4511          -14088058, -30714912, 16193877},
4512     },
4513     {
4514         {-33521811, 3180713, -2394130, 14003687, -16903474, -16270840, 17238398,
4515          4729455, -18074513, 9256800},
4516         {-25182317, -4174131, 32336398, 5036987, -21236817, 11360617, 22616405,
4517          9761698, -19827198, 630305},
4518         {-13720693, 2639453, -24237460, -7406481, 9494427, -5774029, -6554551,
4519          -15960994, -2449256, -14291300},
4520     },
4521     {
4522         {-3151181, -5046075, 9282714, 6866145, -31907062, -863023, -18940575,
4523          15033784, 25105118, -7894876},
4524         {-24326370, 15950226, -31801215, -14592823, -11662737, -5090925,
4525          1573892, -2625887, 2198790, -15804619},
4526         {-3099351, 10324967, -2241613, 7453183, -5446979, -2735503, -13812022,
4527          -16236442, -32461234, -12290683},
4528     },
4529 };
4530 
4531 /*
4532  * r = a * A + b * B
4533  *
4534  * where a = a[0]+256*a[1]+...+256^31 a[31].
4535  * and b = b[0]+256*b[1]+...+256^31 b[31].
4536  * B is the Ed25519 base point (x,4/5) with x positive.
4537  */
ge_double_scalarmult_vartime(ge_p2 * r,const uint8_t * a,const ge_p3 * A,const uint8_t * b)4538 static void ge_double_scalarmult_vartime(ge_p2 *r, const uint8_t *a,
4539                                          const ge_p3 *A, const uint8_t *b)
4540 {
4541     signed char aslide[256];
4542     signed char bslide[256];
4543     ge_cached Ai[8]; /* A,3A,5A,7A,9A,11A,13A,15A */
4544     ge_p1p1 t;
4545     ge_p3 u;
4546     ge_p3 A2;
4547     int i;
4548 
4549     slide(aslide, a);
4550     slide(bslide, b);
4551 
4552     ge_p3_to_cached(&Ai[0], A);
4553     ge_p3_dbl(&t, A);
4554     ge_p1p1_to_p3(&A2, &t);
4555     ge_add(&t, &A2, &Ai[0]);
4556     ge_p1p1_to_p3(&u, &t);
4557     ge_p3_to_cached(&Ai[1], &u);
4558     ge_add(&t, &A2, &Ai[1]);
4559     ge_p1p1_to_p3(&u, &t);
4560     ge_p3_to_cached(&Ai[2], &u);
4561     ge_add(&t, &A2, &Ai[2]);
4562     ge_p1p1_to_p3(&u, &t);
4563     ge_p3_to_cached(&Ai[3], &u);
4564     ge_add(&t, &A2, &Ai[3]);
4565     ge_p1p1_to_p3(&u, &t);
4566     ge_p3_to_cached(&Ai[4], &u);
4567     ge_add(&t, &A2, &Ai[4]);
4568     ge_p1p1_to_p3(&u, &t);
4569     ge_p3_to_cached(&Ai[5], &u);
4570     ge_add(&t, &A2, &Ai[5]);
4571     ge_p1p1_to_p3(&u, &t);
4572     ge_p3_to_cached(&Ai[6], &u);
4573     ge_add(&t, &A2, &Ai[6]);
4574     ge_p1p1_to_p3(&u, &t);
4575     ge_p3_to_cached(&Ai[7], &u);
4576 
4577     ge_p2_0(r);
4578 
4579     for (i = 255; i >= 0; --i) {
4580         if (aslide[i] || bslide[i]) {
4581             break;
4582         }
4583     }
4584 
4585     for (; i >= 0; --i) {
4586         ge_p2_dbl(&t, r);
4587 
4588         if (aslide[i] > 0) {
4589             ge_p1p1_to_p3(&u, &t);
4590             ge_add(&t, &u, &Ai[aslide[i] / 2]);
4591         } else if (aslide[i] < 0) {
4592             ge_p1p1_to_p3(&u, &t);
4593             ge_sub(&t, &u, &Ai[(-aslide[i]) / 2]);
4594         }
4595 
4596         if (bslide[i] > 0) {
4597             ge_p1p1_to_p3(&u, &t);
4598             ge_madd(&t, &u, &Bi[bslide[i] / 2]);
4599         } else if (bslide[i] < 0) {
4600             ge_p1p1_to_p3(&u, &t);
4601             ge_msub(&t, &u, &Bi[(-bslide[i]) / 2]);
4602         }
4603 
4604         ge_p1p1_to_p2(r, &t);
4605     }
4606 }
4607 
4608 /*
4609  * The set of scalars is \Z/l
4610  * where l = 2^252 + 27742317777372353535851937790883648493.
4611  *
4612  * Input:
4613  *   s[0]+256*s[1]+...+256^63*s[63] = s
4614  *
4615  * Output:
4616  *   s[0]+256*s[1]+...+256^31*s[31] = s mod l
4617  *   where l = 2^252 + 27742317777372353535851937790883648493.
4618  *   Overwrites s in place.
4619 */
x25519_sc_reduce(uint8_t * s)4620 static void x25519_sc_reduce(uint8_t *s)
4621 {
4622     int64_t s0  = kBottom21Bits &  load_3(s);
4623     int64_t s1  = kBottom21Bits & (load_4(s +  2) >> 5);
4624     int64_t s2  = kBottom21Bits & (load_3(s +  5) >> 2);
4625     int64_t s3  = kBottom21Bits & (load_4(s +  7) >> 7);
4626     int64_t s4  = kBottom21Bits & (load_4(s + 10) >> 4);
4627     int64_t s5  = kBottom21Bits & (load_3(s + 13) >> 1);
4628     int64_t s6  = kBottom21Bits & (load_4(s + 15) >> 6);
4629     int64_t s7  = kBottom21Bits & (load_3(s + 18) >> 3);
4630     int64_t s8  = kBottom21Bits &  load_3(s + 21);
4631     int64_t s9  = kBottom21Bits & (load_4(s + 23) >> 5);
4632     int64_t s10 = kBottom21Bits & (load_3(s + 26) >> 2);
4633     int64_t s11 = kBottom21Bits & (load_4(s + 28) >> 7);
4634     int64_t s12 = kBottom21Bits & (load_4(s + 31) >> 4);
4635     int64_t s13 = kBottom21Bits & (load_3(s + 34) >> 1);
4636     int64_t s14 = kBottom21Bits & (load_4(s + 36) >> 6);
4637     int64_t s15 = kBottom21Bits & (load_3(s + 39) >> 3);
4638     int64_t s16 = kBottom21Bits &  load_3(s + 42);
4639     int64_t s17 = kBottom21Bits & (load_4(s + 44) >> 5);
4640     int64_t s18 = kBottom21Bits & (load_3(s + 47) >> 2);
4641     int64_t s19 = kBottom21Bits & (load_4(s + 49) >> 7);
4642     int64_t s20 = kBottom21Bits & (load_4(s + 52) >> 4);
4643     int64_t s21 = kBottom21Bits & (load_3(s + 55) >> 1);
4644     int64_t s22 = kBottom21Bits & (load_4(s + 57) >> 6);
4645     int64_t s23 =                 (load_4(s + 60) >> 3);
4646     int64_t carry0;
4647     int64_t carry1;
4648     int64_t carry2;
4649     int64_t carry3;
4650     int64_t carry4;
4651     int64_t carry5;
4652     int64_t carry6;
4653     int64_t carry7;
4654     int64_t carry8;
4655     int64_t carry9;
4656     int64_t carry10;
4657     int64_t carry11;
4658     int64_t carry12;
4659     int64_t carry13;
4660     int64_t carry14;
4661     int64_t carry15;
4662     int64_t carry16;
4663 
4664     s11 += s23 * 666643;
4665     s12 += s23 * 470296;
4666     s13 += s23 * 654183;
4667     s14 -= s23 * 997805;
4668     s15 += s23 * 136657;
4669     s16 -= s23 * 683901;
4670     s23  = 0;
4671 
4672     s10 += s22 * 666643;
4673     s11 += s22 * 470296;
4674     s12 += s22 * 654183;
4675     s13 -= s22 * 997805;
4676     s14 += s22 * 136657;
4677     s15 -= s22 * 683901;
4678     s22  = 0;
4679 
4680     s9  += s21 * 666643;
4681     s10 += s21 * 470296;
4682     s11 += s21 * 654183;
4683     s12 -= s21 * 997805;
4684     s13 += s21 * 136657;
4685     s14 -= s21 * 683901;
4686     s21  = 0;
4687 
4688     s8  += s20 * 666643;
4689     s9  += s20 * 470296;
4690     s10 += s20 * 654183;
4691     s11 -= s20 * 997805;
4692     s12 += s20 * 136657;
4693     s13 -= s20 * 683901;
4694     s20  = 0;
4695 
4696     s7  += s19 * 666643;
4697     s8  += s19 * 470296;
4698     s9  += s19 * 654183;
4699     s10 -= s19 * 997805;
4700     s11 += s19 * 136657;
4701     s12 -= s19 * 683901;
4702     s19  = 0;
4703 
4704     s6  += s18 * 666643;
4705     s7  += s18 * 470296;
4706     s8  += s18 * 654183;
4707     s9  -= s18 * 997805;
4708     s10 += s18 * 136657;
4709     s11 -= s18 * 683901;
4710     s18  = 0;
4711 
4712     carry6 = (s6 + (1 << 20)) >> 21;
4713     s7  += carry6;
4714     s6  -= carry6 * (1 << 21);
4715     carry8 = (s8 + (1 << 20)) >> 21;
4716     s9  += carry8;
4717     s8  -= carry8 * (1 << 21);
4718     carry10 = (s10 + (1 << 20)) >> 21;
4719     s11 += carry10;
4720     s10 -= carry10 * (1 << 21);
4721     carry12 = (s12 + (1 << 20)) >> 21;
4722     s13 += carry12;
4723     s12 -= carry12 * (1 << 21);
4724     carry14 = (s14 + (1 << 20)) >> 21;
4725     s15 += carry14;
4726     s14 -= carry14 * (1 << 21);
4727     carry16 = (s16 + (1 << 20)) >> 21;
4728     s17 += carry16;
4729     s16 -= carry16 * (1 << 21);
4730 
4731     carry7 = (s7 + (1 << 20)) >> 21;
4732     s8  += carry7;
4733     s7  -= carry7 * (1 << 21);
4734     carry9 = (s9 + (1 << 20)) >> 21;
4735     s10 += carry9;
4736     s9  -= carry9 * (1 << 21);
4737     carry11 = (s11 + (1 << 20)) >> 21;
4738     s12 += carry11;
4739     s11 -= carry11 * (1 << 21);
4740     carry13 = (s13 + (1 << 20)) >> 21;
4741     s14 += carry13;
4742     s13 -= carry13 * (1 << 21);
4743     carry15 = (s15 + (1 << 20)) >> 21;
4744     s16 += carry15;
4745     s15 -= carry15 * (1 << 21);
4746 
4747     s5  += s17 * 666643;
4748     s6  += s17 * 470296;
4749     s7  += s17 * 654183;
4750     s8  -= s17 * 997805;
4751     s9  += s17 * 136657;
4752     s10 -= s17 * 683901;
4753     s17  = 0;
4754 
4755     s4  += s16 * 666643;
4756     s5  += s16 * 470296;
4757     s6  += s16 * 654183;
4758     s7  -= s16 * 997805;
4759     s8  += s16 * 136657;
4760     s9  -= s16 * 683901;
4761     s16  = 0;
4762 
4763     s3  += s15 * 666643;
4764     s4  += s15 * 470296;
4765     s5  += s15 * 654183;
4766     s6  -= s15 * 997805;
4767     s7  += s15 * 136657;
4768     s8  -= s15 * 683901;
4769     s15  = 0;
4770 
4771     s2  += s14 * 666643;
4772     s3  += s14 * 470296;
4773     s4  += s14 * 654183;
4774     s5  -= s14 * 997805;
4775     s6  += s14 * 136657;
4776     s7  -= s14 * 683901;
4777     s14  = 0;
4778 
4779     s1  += s13 * 666643;
4780     s2  += s13 * 470296;
4781     s3  += s13 * 654183;
4782     s4  -= s13 * 997805;
4783     s5  += s13 * 136657;
4784     s6  -= s13 * 683901;
4785     s13  = 0;
4786 
4787     s0  += s12 * 666643;
4788     s1  += s12 * 470296;
4789     s2  += s12 * 654183;
4790     s3  -= s12 * 997805;
4791     s4  += s12 * 136657;
4792     s5  -= s12 * 683901;
4793     s12  = 0;
4794 
4795     carry0 = (s0 + (1 << 20)) >> 21;
4796     s1  += carry0;
4797     s0  -= carry0 * (1 << 21);
4798     carry2 = (s2 + (1 << 20)) >> 21;
4799     s3  += carry2;
4800     s2  -= carry2 * (1 << 21);
4801     carry4 = (s4 + (1 << 20)) >> 21;
4802     s5  += carry4;
4803     s4  -= carry4 * (1 << 21);
4804     carry6 = (s6 + (1 << 20)) >> 21;
4805     s7 += carry6;
4806     s6 -= carry6 * (1 << 21);
4807     carry8 = (s8 + (1 << 20)) >> 21;
4808     s9  += carry8;
4809     s8  -= carry8 * (1 << 21);
4810     carry10 = (s10 + (1 << 20)) >> 21;
4811     s11 += carry10;
4812     s10 -= carry10 * (1 << 21);
4813 
4814     carry1 = (s1 + (1 << 20)) >> 21;
4815     s2  += carry1;
4816     s1  -= carry1 * (1 << 21);
4817     carry3 = (s3 + (1 << 20)) >> 21;
4818     s4  += carry3;
4819     s3  -= carry3 * (1 << 21);
4820     carry5 = (s5 + (1 << 20)) >> 21;
4821     s6  += carry5;
4822     s5  -= carry5 * (1 << 21);
4823     carry7 = (s7 + (1 << 20)) >> 21;
4824     s8  += carry7;
4825     s7  -= carry7 * (1 << 21);
4826     carry9 = (s9 + (1 << 20)) >> 21;
4827     s10 += carry9;
4828     s9  -= carry9 * (1 << 21);
4829     carry11 = (s11 + (1 << 20)) >> 21;
4830     s12 += carry11;
4831     s11 -= carry11 * (1 << 21);
4832 
4833     s0  += s12 * 666643;
4834     s1  += s12 * 470296;
4835     s2  += s12 * 654183;
4836     s3  -= s12 * 997805;
4837     s4  += s12 * 136657;
4838     s5  -= s12 * 683901;
4839     s12  = 0;
4840 
4841     carry0 = s0 >> 21;
4842     s1  += carry0;
4843     s0  -= carry0 * (1 << 21);
4844     carry1 = s1 >> 21;
4845     s2  += carry1;
4846     s1  -= carry1 * (1 << 21);
4847     carry2 = s2 >> 21;
4848     s3  += carry2;
4849     s2  -= carry2 * (1 << 21);
4850     carry3 = s3 >> 21;
4851     s4  += carry3;
4852     s3  -= carry3 * (1 << 21);
4853     carry4 = s4 >> 21;
4854     s5  += carry4;
4855     s4  -= carry4 * (1 << 21);
4856     carry5 = s5 >> 21;
4857     s6  += carry5;
4858     s5  -= carry5 * (1 << 21);
4859     carry6 = s6 >> 21;
4860     s7  += carry6;
4861     s6  -= carry6 * (1 << 21);
4862     carry7 = s7 >> 21;
4863     s8  += carry7;
4864     s7  -= carry7 * (1 << 21);
4865     carry8 = s8 >> 21;
4866     s9  += carry8;
4867     s8  -= carry8 * (1 << 21);
4868     carry9 = s9 >> 21;
4869     s10 += carry9;
4870     s9  -= carry9 * (1 << 21);
4871     carry10 = s10 >> 21;
4872     s11 += carry10;
4873     s10 -= carry10 * (1 << 21);
4874     carry11 = s11 >> 21;
4875     s12 += carry11;
4876     s11 -= carry11 * (1 << 21);
4877 
4878     s0  += s12 * 666643;
4879     s1  += s12 * 470296;
4880     s2  += s12 * 654183;
4881     s3  -= s12 * 997805;
4882     s4  += s12 * 136657;
4883     s5  -= s12 * 683901;
4884     s12  = 0;
4885 
4886     carry0 = s0 >> 21;
4887     s1  += carry0;
4888     s0  -= carry0 * (1 << 21);
4889     carry1 = s1 >> 21;
4890     s2  += carry1;
4891     s1  -= carry1 * (1 << 21);
4892     carry2 = s2 >> 21;
4893     s3  += carry2;
4894     s2  -= carry2 * (1 << 21);
4895     carry3 = s3 >> 21;
4896     s4  += carry3;
4897     s3  -= carry3 * (1 << 21);
4898     carry4 = s4 >> 21;
4899     s5  += carry4;
4900     s4  -= carry4 * (1 << 21);
4901     carry5 = s5 >> 21;
4902     s6  += carry5;
4903     s5  -= carry5 * (1 << 21);
4904     carry6 = s6 >> 21;
4905     s7  += carry6;
4906     s6  -= carry6 * (1 << 21);
4907     carry7 = s7 >> 21;
4908     s8  += carry7;
4909     s7  -= carry7 * (1 << 21);
4910     carry8 = s8 >> 21;
4911     s9  += carry8;
4912     s8  -= carry8 * (1 << 21);
4913     carry9 = s9 >> 21;
4914     s10 += carry9;
4915     s9  -= carry9 * (1 << 21);
4916     carry10 = s10 >> 21;
4917     s11 += carry10;
4918     s10 -= carry10 * (1 << 21);
4919 
4920     s[ 0] = (uint8_t) (s0  >>  0);
4921     s[ 1] = (uint8_t) (s0  >>  8);
4922     s[ 2] = (uint8_t)((s0  >> 16) | (s1  <<  5));
4923     s[ 3] = (uint8_t) (s1  >>  3);
4924     s[ 4] = (uint8_t) (s1  >> 11);
4925     s[ 5] = (uint8_t)((s1  >> 19) | (s2  <<  2));
4926     s[ 6] = (uint8_t) (s2  >>  6);
4927     s[ 7] = (uint8_t)((s2  >> 14) | (s3  <<  7));
4928     s[ 8] = (uint8_t) (s3  >>  1);
4929     s[ 9] = (uint8_t) (s3  >>  9);
4930     s[10] = (uint8_t)((s3  >> 17) | (s4  <<  4));
4931     s[11] = (uint8_t) (s4  >>  4);
4932     s[12] = (uint8_t) (s4  >> 12);
4933     s[13] = (uint8_t)((s4  >> 20) | (s5  <<  1));
4934     s[14] = (uint8_t) (s5  >>  7);
4935     s[15] = (uint8_t)((s5  >> 15) | (s6  <<  6));
4936     s[16] = (uint8_t) (s6  >>  2);
4937     s[17] = (uint8_t) (s6  >> 10);
4938     s[18] = (uint8_t)((s6  >> 18) | (s7  <<  3));
4939     s[19] = (uint8_t) (s7  >>  5);
4940     s[20] = (uint8_t) (s7  >> 13);
4941     s[21] = (uint8_t) (s8  >>  0);
4942     s[22] = (uint8_t) (s8  >>  8);
4943     s[23] = (uint8_t)((s8  >> 16) | (s9  <<  5));
4944     s[24] = (uint8_t) (s9  >>  3);
4945     s[25] = (uint8_t) (s9  >> 11);
4946     s[26] = (uint8_t)((s9  >> 19) | (s10 <<  2));
4947     s[27] = (uint8_t) (s10 >>  6);
4948     s[28] = (uint8_t)((s10 >> 14) | (s11 <<  7));
4949     s[29] = (uint8_t) (s11 >>  1);
4950     s[30] = (uint8_t) (s11 >>  9);
4951     s[31] = (uint8_t) (s11 >> 17);
4952 }
4953 
4954 /*
4955  * Input:
4956  *   a[0]+256*a[1]+...+256^31*a[31] = a
4957  *   b[0]+256*b[1]+...+256^31*b[31] = b
4958  *   c[0]+256*c[1]+...+256^31*c[31] = c
4959  *
4960  * Output:
4961  *   s[0]+256*s[1]+...+256^31*s[31] = (ab+c) mod l
4962  *   where l = 2^252 + 27742317777372353535851937790883648493.
4963  */
sc_muladd(uint8_t * s,const uint8_t * a,const uint8_t * b,const uint8_t * c)4964 static void sc_muladd(uint8_t *s, const uint8_t *a, const uint8_t *b,
4965                       const uint8_t *c)
4966 {
4967     int64_t a0  = kBottom21Bits &  load_3(a);
4968     int64_t a1  = kBottom21Bits & (load_4(a +  2) >> 5);
4969     int64_t a2  = kBottom21Bits & (load_3(a +  5) >> 2);
4970     int64_t a3  = kBottom21Bits & (load_4(a +  7) >> 7);
4971     int64_t a4  = kBottom21Bits & (load_4(a + 10) >> 4);
4972     int64_t a5  = kBottom21Bits & (load_3(a + 13) >> 1);
4973     int64_t a6  = kBottom21Bits & (load_4(a + 15) >> 6);
4974     int64_t a7  = kBottom21Bits & (load_3(a + 18) >> 3);
4975     int64_t a8  = kBottom21Bits &  load_3(a + 21);
4976     int64_t a9  = kBottom21Bits & (load_4(a + 23) >> 5);
4977     int64_t a10 = kBottom21Bits & (load_3(a + 26) >> 2);
4978     int64_t a11 =                 (load_4(a + 28) >> 7);
4979     int64_t b0  = kBottom21Bits &  load_3(b);
4980     int64_t b1  = kBottom21Bits & (load_4(b +  2) >> 5);
4981     int64_t b2  = kBottom21Bits & (load_3(b +  5) >> 2);
4982     int64_t b3  = kBottom21Bits & (load_4(b +  7) >> 7);
4983     int64_t b4  = kBottom21Bits & (load_4(b + 10) >> 4);
4984     int64_t b5  = kBottom21Bits & (load_3(b + 13) >> 1);
4985     int64_t b6  = kBottom21Bits & (load_4(b + 15) >> 6);
4986     int64_t b7  = kBottom21Bits & (load_3(b + 18) >> 3);
4987     int64_t b8  = kBottom21Bits &  load_3(b + 21);
4988     int64_t b9  = kBottom21Bits & (load_4(b + 23) >> 5);
4989     int64_t b10 = kBottom21Bits & (load_3(b + 26) >> 2);
4990     int64_t b11 =                 (load_4(b + 28) >> 7);
4991     int64_t c0  = kBottom21Bits &  load_3(c);
4992     int64_t c1  = kBottom21Bits & (load_4(c +  2) >> 5);
4993     int64_t c2  = kBottom21Bits & (load_3(c +  5) >> 2);
4994     int64_t c3  = kBottom21Bits & (load_4(c +  7) >> 7);
4995     int64_t c4  = kBottom21Bits & (load_4(c + 10) >> 4);
4996     int64_t c5  = kBottom21Bits & (load_3(c + 13) >> 1);
4997     int64_t c6  = kBottom21Bits & (load_4(c + 15) >> 6);
4998     int64_t c7  = kBottom21Bits & (load_3(c + 18) >> 3);
4999     int64_t c8  = kBottom21Bits &  load_3(c + 21);
5000     int64_t c9  = kBottom21Bits & (load_4(c + 23) >> 5);
5001     int64_t c10 = kBottom21Bits & (load_3(c + 26) >> 2);
5002     int64_t c11 =                 (load_4(c + 28) >> 7);
5003     int64_t s0;
5004     int64_t s1;
5005     int64_t s2;
5006     int64_t s3;
5007     int64_t s4;
5008     int64_t s5;
5009     int64_t s6;
5010     int64_t s7;
5011     int64_t s8;
5012     int64_t s9;
5013     int64_t s10;
5014     int64_t s11;
5015     int64_t s12;
5016     int64_t s13;
5017     int64_t s14;
5018     int64_t s15;
5019     int64_t s16;
5020     int64_t s17;
5021     int64_t s18;
5022     int64_t s19;
5023     int64_t s20;
5024     int64_t s21;
5025     int64_t s22;
5026     int64_t s23;
5027     int64_t carry0;
5028     int64_t carry1;
5029     int64_t carry2;
5030     int64_t carry3;
5031     int64_t carry4;
5032     int64_t carry5;
5033     int64_t carry6;
5034     int64_t carry7;
5035     int64_t carry8;
5036     int64_t carry9;
5037     int64_t carry10;
5038     int64_t carry11;
5039     int64_t carry12;
5040     int64_t carry13;
5041     int64_t carry14;
5042     int64_t carry15;
5043     int64_t carry16;
5044     int64_t carry17;
5045     int64_t carry18;
5046     int64_t carry19;
5047     int64_t carry20;
5048     int64_t carry21;
5049     int64_t carry22;
5050 
5051     s0  = c0   +   a0 * b0;
5052     s1  = c1   +   a0 * b1   +   a1 * b0;
5053     s2  = c2   +   a0 * b2   +   a1 * b1   +   a2 * b0;
5054     s3  = c3   +   a0 * b3   +   a1 * b2   +   a2 * b1  +   a3 * b0;
5055     s4  = c4   +   a0 * b4   +   a1 * b3   +   a2 * b2  +   a3 * b1  +   a4 * b0;
5056     s5  = c5   +   a0 * b5   +   a1 * b4   +   a2 * b3  +   a3 * b2  +   a4 * b1  +   a5 * b0;
5057     s6  = c6   +   a0 * b6   +   a1 * b5   +   a2 * b4  +   a3 * b3  +   a4 * b2  +   a5 * b1 +   a6 * b0;
5058     s7  = c7   +   a0 * b7   +   a1 * b6   +   a2 * b5  +   a3 * b4  +   a4 * b3  +   a5 * b2 +   a6 * b1   +   a7 * b0;
5059     s8  = c8   +   a0 * b8   +   a1 * b7   +   a2 * b6  +   a3 * b5  +   a4 * b4  +   a5 * b3 +   a6 * b2   +   a7 * b1   +   a8 * b0;
5060     s9  = c9   +   a0 * b9   +   a1 * b8   +   a2 * b7  +   a3 * b6  +   a4 * b5  +   a5 * b4 +   a6 * b3   +   a7 * b2   +   a8 * b1  +   a9 * b0;
5061     s10 = c10  +   a0 * b10  +   a1 * b9   +   a2 * b8  +   a3 * b7  +   a4 * b6  +   a5 * b5 +   a6 * b4   +   a7 * b3   +   a8 * b2  +   a9 * b1  +  a10 * b0;
5062     s11 = c11  +   a0 * b11  +   a1 * b10  +   a2 * b9  +   a3 * b8  +   a4 * b7  +   a5 * b6 +   a6 * b5   +   a7 * b4   +   a8 * b3  +   a9 * b2  +  a10 * b1  +  a11 * b0;
5063     s12 =          a1 * b11  +   a2 * b10  +   a3 * b9  +   a4 * b8  +   a5 * b7  +   a6 * b6 +   a7 * b5   +   a8 * b4   +   a9 * b3  +  a10 * b2  +  a11 * b1;
5064     s13 =          a2 * b11  +   a3 * b10  +   a4 * b9  +   a5 * b8  +   a6 * b7  +   a7 * b6 +   a8 * b5   +   a9 * b4   +  a10 * b3  +  a11 * b2;
5065     s14 =          a3 * b11  +   a4 * b10  +   a5 * b9  +   a6 * b8  +   a7 * b7  +   a8 * b6 +   a9 * b5   +  a10 * b4   +  a11 * b3;
5066     s15 =          a4 * b11  +   a5 * b10  +   a6 * b9  +   a7 * b8  +   a8 * b7  +   a9 * b6 +  a10 * b5   +  a11 * b4;
5067     s16 =          a5 * b11  +   a6 * b10  +   a7 * b9  +   a8 * b8  +   a9 * b7  +  a10 * b6 +  a11 * b5;
5068     s17 =          a6 * b11  +   a7 * b10  +   a8 * b9  +   a9 * b8  +  a10 * b7  +  a11 * b6;
5069     s18 =          a7 * b11  +   a8 * b10  +   a9 * b9  +  a10 * b8  +  a11 * b7;
5070     s19 =          a8 * b11  +   a9 * b10  +  a10 * b9  +  a11 * b8;
5071     s20 =          a9 * b11  +  a10 * b10  +  a11 * b9;
5072     s21 =         a10 * b11  +  a11 * b10;
5073     s22 =         a11 * b11;
5074     s23 =         0;
5075 
5076     carry0 = (s0 + (1 << 20)) >> 21;
5077     s1  += carry0;
5078     s0  -= carry0 * (1 << 21);
5079     carry2 = (s2 + (1 << 20)) >> 21;
5080     s3  += carry2;
5081     s2  -= carry2 * (1 << 21);
5082     carry4 = (s4 + (1 << 20)) >> 21;
5083     s5  += carry4;
5084     s4  -= carry4 * (1 << 21);
5085     carry6 = (s6 + (1 << 20)) >> 21;
5086     s7  += carry6;
5087     s6  -= carry6 * (1 << 21);
5088     carry8 = (s8 + (1 << 20)) >> 21;
5089     s9  += carry8;
5090     s8  -= carry8 * (1 << 21);
5091     carry10 = (s10 + (1 << 20)) >> 21;
5092     s11 += carry10;
5093     s10 -= carry10 * (1 << 21);
5094     carry12 = (s12 + (1 << 20)) >> 21;
5095     s13 += carry12;
5096     s12 -= carry12 * (1 << 21);
5097     carry14 = (s14 + (1 << 20)) >> 21;
5098     s15 += carry14;
5099     s14 -= carry14 * (1 << 21);
5100     carry16 = (s16 + (1 << 20)) >> 21;
5101     s17 += carry16;
5102     s16 -= carry16 * (1 << 21);
5103     carry18 = (s18 + (1 << 20)) >> 21;
5104     s19 += carry18;
5105     s18 -= carry18 * (1 << 21);
5106     carry20 = (s20 + (1 << 20)) >> 21;
5107     s21 += carry20;
5108     s20 -= carry20 * (1 << 21);
5109     carry22 = (s22 + (1 << 20)) >> 21;
5110     s23 += carry22;
5111     s22 -= carry22 * (1 << 21);
5112 
5113     carry1 = (s1 + (1 << 20)) >> 21;
5114     s2  += carry1;
5115     s1  -= carry1 * (1 << 21);
5116     carry3 = (s3 + (1 << 20)) >> 21;
5117     s4  += carry3;
5118     s3  -= carry3 * (1 << 21);
5119     carry5 = (s5 + (1 << 20)) >> 21;
5120     s6  += carry5;
5121     s5  -= carry5 * (1 << 21);
5122     carry7 = (s7 + (1 << 20)) >> 21;
5123     s8  += carry7;
5124     s7  -= carry7 * (1 << 21);
5125     carry9 = (s9 + (1 << 20)) >> 21;
5126     s10 += carry9;
5127     s9  -= carry9 * (1 << 21);
5128     carry11 = (s11 + (1 << 20)) >> 21;
5129     s12 += carry11;
5130     s11 -= carry11 * (1 << 21);
5131     carry13 = (s13 + (1 << 20)) >> 21;
5132     s14 += carry13;
5133     s13 -= carry13 * (1 << 21);
5134     carry15 = (s15 + (1 << 20)) >> 21;
5135     s16 += carry15;
5136     s15 -= carry15 * (1 << 21);
5137     carry17 = (s17 + (1 << 20)) >> 21;
5138     s18 += carry17;
5139     s17 -= carry17 * (1 << 21);
5140     carry19 = (s19 + (1 << 20)) >> 21;
5141     s20 += carry19;
5142     s19 -= carry19 * (1 << 21);
5143     carry21 = (s21 + (1 << 20)) >> 21;
5144     s22 += carry21;
5145     s21 -= carry21 * (1 << 21);
5146 
5147     s11 += s23 * 666643;
5148     s12 += s23 * 470296;
5149     s13 += s23 * 654183;
5150     s14 -= s23 * 997805;
5151     s15 += s23 * 136657;
5152     s16 -= s23 * 683901;
5153     s23  = 0;
5154 
5155     s10 += s22 * 666643;
5156     s11 += s22 * 470296;
5157     s12 += s22 * 654183;
5158     s13 -= s22 * 997805;
5159     s14 += s22 * 136657;
5160     s15 -= s22 * 683901;
5161     s22  = 0;
5162 
5163     s9  += s21 * 666643;
5164     s10 += s21 * 470296;
5165     s11 += s21 * 654183;
5166     s12 -= s21 * 997805;
5167     s13 += s21 * 136657;
5168     s14 -= s21 * 683901;
5169     s21  = 0;
5170 
5171     s8  += s20 * 666643;
5172     s9  += s20 * 470296;
5173     s10 += s20 * 654183;
5174     s11 -= s20 * 997805;
5175     s12 += s20 * 136657;
5176     s13 -= s20 * 683901;
5177     s20  = 0;
5178 
5179     s7  += s19 * 666643;
5180     s8  += s19 * 470296;
5181     s9  += s19 * 654183;
5182     s10 -= s19 * 997805;
5183     s11 += s19 * 136657;
5184     s12 -= s19 * 683901;
5185     s19  = 0;
5186 
5187     s6  += s18 * 666643;
5188     s7  += s18 * 470296;
5189     s8  += s18 * 654183;
5190     s9  -= s18 * 997805;
5191     s10 += s18 * 136657;
5192     s11 -= s18 * 683901;
5193     s18  = 0;
5194 
5195     carry6 = (s6 + (1 << 20)) >> 21;
5196     s7  += carry6;
5197     s6  -= carry6 * (1 << 21);
5198     carry8 = (s8 + (1 << 20)) >> 21;
5199     s9  += carry8;
5200     s8  -= carry8 * (1 << 21);
5201     carry10 = (s10 + (1 << 20)) >> 21;
5202     s11 += carry10;
5203     s10 -= carry10 * (1 << 21);
5204     carry12 = (s12 + (1 << 20)) >> 21;
5205     s13 += carry12;
5206     s12 -= carry12 * (1 << 21);
5207     carry14 = (s14 + (1 << 20)) >> 21;
5208     s15 += carry14;
5209     s14 -= carry14 * (1 << 21);
5210     carry16 = (s16 + (1 << 20)) >> 21;
5211     s17 += carry16;
5212     s16 -= carry16 * (1 << 21);
5213 
5214     carry7 = (s7 + (1 << 20)) >> 21;
5215     s8  += carry7;
5216     s7  -= carry7 * (1 << 21);
5217     carry9 = (s9 + (1 << 20)) >> 21;
5218     s10 += carry9;
5219     s9  -= carry9 * (1 << 21);
5220     carry11 = (s11 + (1 << 20)) >> 21;
5221     s12 += carry11;
5222     s11 -= carry11 * (1 << 21);
5223     carry13 = (s13 + (1 << 20)) >> 21;
5224     s14 += carry13;
5225     s13 -= carry13 * (1 << 21);
5226     carry15 = (s15 + (1 << 20)) >> 21;
5227     s16 += carry15;
5228     s15 -= carry15 * (1 << 21);
5229 
5230     s5  += s17 * 666643;
5231     s6  += s17 * 470296;
5232     s7  += s17 * 654183;
5233     s8  -= s17 * 997805;
5234     s9  += s17 * 136657;
5235     s10 -= s17 * 683901;
5236     s17  = 0;
5237 
5238     s4  += s16 * 666643;
5239     s5  += s16 * 470296;
5240     s6  += s16 * 654183;
5241     s7  -= s16 * 997805;
5242     s8  += s16 * 136657;
5243     s9  -= s16 * 683901;
5244     s16  = 0;
5245 
5246     s3  += s15 * 666643;
5247     s4  += s15 * 470296;
5248     s5  += s15 * 654183;
5249     s6  -= s15 * 997805;
5250     s7  += s15 * 136657;
5251     s8  -= s15 * 683901;
5252     s15  = 0;
5253 
5254     s2  += s14 * 666643;
5255     s3  += s14 * 470296;
5256     s4  += s14 * 654183;
5257     s5  -= s14 * 997805;
5258     s6  += s14 * 136657;
5259     s7  -= s14 * 683901;
5260     s14  = 0;
5261 
5262     s1  += s13 * 666643;
5263     s2  += s13 * 470296;
5264     s3  += s13 * 654183;
5265     s4  -= s13 * 997805;
5266     s5  += s13 * 136657;
5267     s6  -= s13 * 683901;
5268     s13  = 0;
5269 
5270     s0  += s12 * 666643;
5271     s1  += s12 * 470296;
5272     s2  += s12 * 654183;
5273     s3  -= s12 * 997805;
5274     s4  += s12 * 136657;
5275     s5  -= s12 * 683901;
5276     s12 = 0;
5277 
5278     carry0 = (s0 + (1 << 20)) >> 21;
5279     s1  += carry0;
5280     s0  -= carry0 * (1 << 21);
5281     carry2 = (s2 + (1 << 20)) >> 21;
5282     s3  += carry2;
5283     s2  -= carry2 * (1 << 21);
5284     carry4 = (s4 + (1 << 20)) >> 21;
5285     s5  += carry4;
5286     s4  -= carry4 * (1 << 21);
5287     carry6 = (s6 + (1 << 20)) >> 21;
5288     s7  += carry6;
5289     s6  -= carry6 * (1 << 21);
5290     carry8 = (s8 + (1 << 20)) >> 21;
5291     s9  += carry8;
5292     s8  -= carry8 * (1 << 21);
5293     carry10 = (s10 + (1 << 20)) >> 21;
5294     s11 += carry10;
5295     s10 -= carry10 * (1 << 21);
5296 
5297     carry1 = (s1 + (1 << 20)) >> 21;
5298     s2  += carry1;
5299     s1  -= carry1 * (1 << 21);
5300     carry3 = (s3 + (1 << 20)) >> 21;
5301     s4  += carry3;
5302     s3  -= carry3 * (1 << 21);
5303     carry5 = (s5 + (1 << 20)) >> 21;
5304     s6  += carry5;
5305     s5  -= carry5 * (1 << 21);
5306     carry7 = (s7 + (1 << 20)) >> 21;
5307     s8  += carry7;
5308     s7  -= carry7 * (1 << 21);
5309     carry9 = (s9 + (1 << 20)) >> 21;
5310     s10 += carry9;
5311     s9  -= carry9 * (1 << 21);
5312     carry11 = (s11 + (1 << 20)) >> 21;
5313     s12 += carry11;
5314     s11 -= carry11 * (1 << 21);
5315 
5316     s0  += s12 * 666643;
5317     s1  += s12 * 470296;
5318     s2  += s12 * 654183;
5319     s3  -= s12 * 997805;
5320     s4  += s12 * 136657;
5321     s5  -= s12 * 683901;
5322     s12  = 0;
5323 
5324     carry0 = s0 >> 21;
5325     s1  += carry0;
5326     s0  -= carry0 * (1 << 21);
5327     carry1 = s1 >> 21;
5328     s2  += carry1;
5329     s1  -= carry1 * (1 << 21);
5330     carry2 = s2 >> 21;
5331     s3  += carry2;
5332     s2  -= carry2 * (1 << 21);
5333     carry3 = s3 >> 21;
5334     s4  += carry3;
5335     s3  -= carry3 * (1 << 21);
5336     carry4 = s4 >> 21;
5337     s5  += carry4;
5338     s4  -= carry4 * (1 << 21);
5339     carry5 = s5 >> 21;
5340     s6  += carry5;
5341     s5  -= carry5 * (1 << 21);
5342     carry6 = s6 >> 21;
5343     s7  += carry6;
5344     s6  -= carry6 * (1 << 21);
5345     carry7 = s7 >> 21;
5346     s8  += carry7;
5347     s7  -= carry7 * (1 << 21);
5348     carry8 = s8 >> 21;
5349     s9  += carry8;
5350     s8  -= carry8 * (1 << 21);
5351     carry9 = s9 >> 21;
5352     s10 += carry9;
5353     s9  -= carry9 * (1 << 21);
5354     carry10 = s10 >> 21;
5355     s11 += carry10;
5356     s10 -= carry10 * (1 << 21);
5357     carry11 = s11 >> 21;
5358     s12 += carry11;
5359     s11 -= carry11 * (1 << 21);
5360 
5361     s0  += s12 * 666643;
5362     s1  += s12 * 470296;
5363     s2  += s12 * 654183;
5364     s3  -= s12 * 997805;
5365     s4  += s12 * 136657;
5366     s5  -= s12 * 683901;
5367     s12  = 0;
5368 
5369     carry0 = s0 >> 21;
5370     s1  += carry0;
5371     s0  -= carry0 * (1 << 21);
5372     carry1 = s1 >> 21;
5373     s2  += carry1;
5374     s1  -= carry1 * (1 << 21);
5375     carry2 = s2 >> 21;
5376     s3  += carry2;
5377     s2  -= carry2 * (1 << 21);
5378     carry3 = s3 >> 21;
5379     s4  += carry3;
5380     s3  -= carry3 * (1 << 21);
5381     carry4 = s4 >> 21;
5382     s5  += carry4;
5383     s4  -= carry4 * (1 << 21);
5384     carry5 = s5 >> 21;
5385     s6  += carry5;
5386     s5  -= carry5 * (1 << 21);
5387     carry6 = s6 >> 21;
5388     s7  += carry6;
5389     s6  -= carry6 * (1 << 21);
5390     carry7 = s7 >> 21;
5391     s8  += carry7;
5392     s7  -= carry7 * (1 << 21);
5393     carry8 = s8 >> 21;
5394     s9  += carry8;
5395     s8  -= carry8 * (1 << 21);
5396     carry9 = s9 >> 21;
5397     s10 += carry9;
5398     s9  -= carry9 * (1 << 21);
5399     carry10 = s10 >> 21;
5400     s11 += carry10;
5401     s10 -= carry10 * (1 << 21);
5402 
5403     s[ 0] = (uint8_t) (s0  >>  0);
5404     s[ 1] = (uint8_t) (s0  >>  8);
5405     s[ 2] = (uint8_t)((s0  >> 16) | (s1 << 5));
5406     s[ 3] = (uint8_t) (s1  >>  3);
5407     s[ 4] = (uint8_t) (s1  >> 11);
5408     s[ 5] = (uint8_t)((s1  >> 19) | (s2 << 2));
5409     s[ 6] = (uint8_t) (s2  >>  6);
5410     s[ 7] = (uint8_t)((s2  >> 14) | (s3 << 7));
5411     s[ 8] = (uint8_t) (s3  >>  1);
5412     s[ 9] = (uint8_t) (s3  >>  9);
5413     s[10] = (uint8_t)((s3  >> 17) | (s4 << 4));
5414     s[11] = (uint8_t) (s4  >>  4);
5415     s[12] = (uint8_t) (s4  >> 12);
5416     s[13] = (uint8_t)((s4  >> 20) | (s5 << 1));
5417     s[14] = (uint8_t) (s5  >>  7);
5418     s[15] = (uint8_t)((s5  >> 15) | (s6 << 6));
5419     s[16] = (uint8_t) (s6  >>  2);
5420     s[17] = (uint8_t) (s6  >> 10);
5421     s[18] = (uint8_t)((s6  >> 18) | (s7 << 3));
5422     s[19] = (uint8_t) (s7  >>  5);
5423     s[20] = (uint8_t) (s7  >> 13);
5424     s[21] = (uint8_t) (s8  >>  0);
5425     s[22] = (uint8_t) (s8  >>  8);
5426     s[23] = (uint8_t)((s8  >> 16) | (s9 << 5));
5427     s[24] = (uint8_t) (s9  >>  3);
5428     s[25] = (uint8_t) (s9  >> 11);
5429     s[26] = (uint8_t)((s9  >> 19) | (s10 << 2));
5430     s[27] = (uint8_t) (s10 >>  6);
5431     s[28] = (uint8_t)((s10 >> 14) | (s11 << 7));
5432     s[29] = (uint8_t) (s11 >>  1);
5433     s[30] = (uint8_t) (s11 >>  9);
5434     s[31] = (uint8_t) (s11 >> 17);
5435 }
5436 
hash_init_with_dom(EVP_MD_CTX * hash_ctx,EVP_MD * sha512,const uint8_t dom2flag,const uint8_t phflag,const uint8_t * context,const size_t context_len)5437 static int hash_init_with_dom(EVP_MD_CTX *hash_ctx,
5438                               EVP_MD *sha512,
5439                               const uint8_t dom2flag,
5440                               const uint8_t phflag,
5441                               const uint8_t *context,
5442                               const size_t context_len)
5443 {
5444     /* ASCII: "SigEd25519 no Ed25519 collisions", in hex for EBCDIC compatibility */
5445     const char dom_s[] =
5446             "\x53\x69\x67\x45\x64\x32\x35\x35\x31\x39\x20\x6e"
5447             "\x6f\x20\x45\x64\x32\x35\x35\x31\x39\x20\x63\x6f"
5448             "\x6c\x6c\x69\x73\x69\x6f\x6e\x73";
5449     uint8_t dom[2];
5450 
5451     if (!EVP_DigestInit_ex(hash_ctx, sha512, NULL))
5452         return 0;
5453 
5454     /* return early if dom2flag is not set */
5455     if (!dom2flag)
5456         return 1;
5457 
5458     if (context_len > UINT8_MAX)
5459         return 0;
5460 
5461     dom[0] = (uint8_t)(phflag >= 1 ? 1 : 0);
5462     dom[1] = (uint8_t)context_len;
5463 
5464     if (!EVP_DigestUpdate(hash_ctx, dom_s, sizeof(dom_s)-1)
5465         || !EVP_DigestUpdate(hash_ctx, dom, sizeof(dom))
5466         || !EVP_DigestUpdate(hash_ctx, context, context_len)) {
5467         return 0;
5468     }
5469 
5470     return 1;
5471 }
5472 
5473 int
ossl_ed25519_sign(uint8_t * out_sig,const uint8_t * tbs,size_t tbs_len,const uint8_t public_key[32],const uint8_t private_key[32],const uint8_t dom2flag,const uint8_t phflag,const uint8_t csflag,const uint8_t * context,size_t context_len,OSSL_LIB_CTX * libctx,const char * propq)5474 ossl_ed25519_sign(uint8_t *out_sig, const uint8_t *tbs, size_t tbs_len,
5475                   const uint8_t public_key[32], const uint8_t private_key[32],
5476                   const uint8_t dom2flag, const uint8_t phflag, const uint8_t csflag,
5477                   const uint8_t *context, size_t context_len,
5478                   OSSL_LIB_CTX *libctx, const char *propq)
5479 {
5480     uint8_t az[SHA512_DIGEST_LENGTH];
5481     uint8_t nonce[SHA512_DIGEST_LENGTH];
5482     ge_p3 R;
5483     uint8_t hram[SHA512_DIGEST_LENGTH];
5484     EVP_MD *sha512 = EVP_MD_fetch(libctx, SN_sha512, propq);
5485     EVP_MD_CTX *hash_ctx = EVP_MD_CTX_new();
5486     unsigned int sz;
5487     int res = 0;
5488 
5489     if (context == NULL)
5490         context_len = 0;
5491 
5492     /* if csflag is set, then a non-empty context-string is required */
5493     if (csflag && context_len == 0)
5494         goto err;
5495 
5496     /* if dom2flag is not set, then an empty context-string is required */
5497     if (!dom2flag && context_len > 0)
5498         goto err;
5499 
5500     if (sha512 == NULL || hash_ctx == NULL)
5501         goto err;
5502 
5503     if (!EVP_DigestInit_ex(hash_ctx, sha512, NULL)
5504         || !EVP_DigestUpdate(hash_ctx, private_key, 32)
5505         || !EVP_DigestFinal_ex(hash_ctx, az, &sz))
5506         goto err;
5507 
5508     az[0] &= 248;
5509     az[31] &= 63;
5510     az[31] |= 64;
5511 
5512     if (!hash_init_with_dom(hash_ctx, sha512, dom2flag, phflag, context, context_len)
5513         || !EVP_DigestUpdate(hash_ctx, az + 32, 32)
5514         || !EVP_DigestUpdate(hash_ctx, tbs, tbs_len)
5515         || !EVP_DigestFinal_ex(hash_ctx, nonce, &sz))
5516         goto err;
5517 
5518     x25519_sc_reduce(nonce);
5519     ge_scalarmult_base(&R, nonce);
5520     ge_p3_tobytes(out_sig, &R);
5521 
5522     if (!hash_init_with_dom(hash_ctx, sha512, dom2flag, phflag, context, context_len)
5523         || !EVP_DigestUpdate(hash_ctx, out_sig, 32)
5524         || !EVP_DigestUpdate(hash_ctx, public_key, 32)
5525         || !EVP_DigestUpdate(hash_ctx, tbs, tbs_len)
5526         || !EVP_DigestFinal_ex(hash_ctx, hram, &sz))
5527         goto err;
5528 
5529     x25519_sc_reduce(hram);
5530     sc_muladd(out_sig + 32, hram, az, nonce);
5531 
5532     res = 1;
5533 err:
5534     OPENSSL_cleanse(nonce, sizeof(nonce));
5535     OPENSSL_cleanse(az, sizeof(az));
5536     EVP_MD_free(sha512);
5537     EVP_MD_CTX_free(hash_ctx);
5538     return res;
5539 }
5540 
5541 /*
5542  * This function should not be necessary since ossl_ed25519_verify() already
5543  * does this check internally.
5544  * For some reason the FIPS ACVP requires a EDDSA KeyVer test.
5545  */
5546 int
ossl_ed25519_pubkey_verify(const uint8_t * pub,size_t pub_len)5547 ossl_ed25519_pubkey_verify(const uint8_t *pub, size_t pub_len)
5548 {
5549     ge_p3 A;
5550 
5551     if (pub_len != ED25519_KEYLEN)
5552         return 0;
5553     return (ge_frombytes_vartime(&A, pub) == 0);
5554 }
5555 
5556 static const char allzeroes[15];
5557 
5558 int
ossl_ed25519_verify(const uint8_t * tbs,size_t tbs_len,const uint8_t signature[64],const uint8_t public_key[32],const uint8_t dom2flag,const uint8_t phflag,const uint8_t csflag,const uint8_t * context,size_t context_len,OSSL_LIB_CTX * libctx,const char * propq)5559 ossl_ed25519_verify(const uint8_t *tbs, size_t tbs_len,
5560                     const uint8_t signature[64], const uint8_t public_key[32],
5561                     const uint8_t dom2flag, const uint8_t phflag, const uint8_t csflag,
5562                     const uint8_t *context, size_t context_len,
5563                     OSSL_LIB_CTX *libctx, const char *propq)
5564 {
5565     int i;
5566     ge_p3 A;
5567     const uint8_t *r, *s;
5568     EVP_MD *sha512;
5569     EVP_MD_CTX *hash_ctx = NULL;
5570     unsigned int sz;
5571     int res = 0;
5572     ge_p2 R;
5573     uint8_t rcheck[32];
5574     uint8_t h[SHA512_DIGEST_LENGTH];
5575     /* 27742317777372353535851937790883648493 in little endian format */
5576     const uint8_t l_low[16] = {
5577         0xED, 0xD3, 0xF5, 0x5C, 0x1A, 0x63, 0x12, 0x58, 0xD6, 0x9C, 0xF7, 0xA2,
5578         0xDE, 0xF9, 0xDE, 0x14
5579     };
5580 
5581     if (context == NULL)
5582         context_len = 0;
5583 
5584     /* if csflag is set, then a non-empty context-string is required */
5585     if (csflag && context_len == 0)
5586         return 0;
5587 
5588     /* if dom2flag is not set, then an empty context-string is required */
5589     if (!dom2flag && context_len > 0)
5590         return 0;
5591 
5592     r = signature;
5593     s = signature + 32;
5594 
5595     /*
5596      * Check 0 <= s < L where L = 2^252 + 27742317777372353535851937790883648493
5597      *
5598      * If not the signature is publicly invalid. Since it's public we can do the
5599      * check in variable time.
5600      *
5601      * First check the most significant byte
5602      */
5603     if (s[31] > 0x10)
5604         return 0;
5605     if (s[31] == 0x10) {
5606         /*
5607          * Most significant byte indicates a value close to 2^252 so check the
5608          * rest
5609          */
5610         if (memcmp(s + 16, allzeroes, sizeof(allzeroes)) != 0)
5611             return 0;
5612         for (i = 15; i >= 0; i--) {
5613             if (s[i] < l_low[i])
5614                 break;
5615             if (s[i] > l_low[i])
5616                 return 0;
5617         }
5618         if (i < 0)
5619             return 0;
5620     }
5621 
5622     if (ge_frombytes_vartime(&A, public_key) != 0) {
5623         return 0;
5624     }
5625 
5626     fe_neg(A.X, A.X);
5627     fe_neg(A.T, A.T);
5628 
5629     sha512 = EVP_MD_fetch(libctx, SN_sha512, propq);
5630     if (sha512 == NULL)
5631         return 0;
5632     hash_ctx = EVP_MD_CTX_new();
5633     if (hash_ctx == NULL)
5634         goto err;
5635 
5636     if (!hash_init_with_dom(hash_ctx, sha512, dom2flag, phflag, context, context_len)
5637         || !EVP_DigestUpdate(hash_ctx, r, 32)
5638         || !EVP_DigestUpdate(hash_ctx, public_key, 32)
5639         || !EVP_DigestUpdate(hash_ctx, tbs, tbs_len)
5640         || !EVP_DigestFinal_ex(hash_ctx, h, &sz))
5641         goto err;
5642 
5643     x25519_sc_reduce(h);
5644 
5645     ge_double_scalarmult_vartime(&R, h, &A, s);
5646 
5647     ge_tobytes(rcheck, &R);
5648 
5649     res = CRYPTO_memcmp(rcheck, r, sizeof(rcheck)) == 0;
5650 
5651     /* note that we have used the strict verification equation here.
5652      * we checked that  ENC( [h](-A) + [s]B ) == r
5653      * B is the base point.
5654      *
5655      * the less strict verification equation uses the curve cofactor:
5656      *          [h*8](-A) + [s*8]B == [8]R
5657      */
5658 err:
5659     EVP_MD_free(sha512);
5660     EVP_MD_CTX_free(hash_ctx);
5661     return res;
5662 }
5663 
5664 int
ossl_ed25519_public_from_private(OSSL_LIB_CTX * ctx,uint8_t out_public_key[32],const uint8_t private_key[32],const char * propq)5665 ossl_ed25519_public_from_private(OSSL_LIB_CTX *ctx, uint8_t out_public_key[32],
5666                                  const uint8_t private_key[32],
5667                                  const char *propq)
5668 {
5669     uint8_t az[SHA512_DIGEST_LENGTH];
5670     ge_p3 A;
5671     int r;
5672     EVP_MD *sha512 = NULL;
5673 
5674     sha512 = EVP_MD_fetch(ctx, SN_sha512, propq);
5675     if (sha512 == NULL)
5676         return 0;
5677     r = EVP_Digest(private_key, 32, az, NULL, sha512, NULL);
5678     EVP_MD_free(sha512);
5679     if (!r) {
5680         OPENSSL_cleanse(az, sizeof(az));
5681         return 0;
5682     }
5683 
5684     az[0] &= 248;
5685     az[31] &= 63;
5686     az[31] |= 64;
5687 
5688     ge_scalarmult_base(&A, az);
5689     ge_p3_tobytes(out_public_key, &A);
5690 
5691     OPENSSL_cleanse(az, sizeof(az));
5692     return 1;
5693 }
5694 
5695 int
ossl_x25519(uint8_t out_shared_key[32],const uint8_t private_key[32],const uint8_t peer_public_value[32])5696 ossl_x25519(uint8_t out_shared_key[32], const uint8_t private_key[32],
5697             const uint8_t peer_public_value[32])
5698 {
5699     static const uint8_t kZeros[32] = {0};
5700     x25519_scalar_mult(out_shared_key, private_key, peer_public_value);
5701     /* The all-zero output results when the input is a point of small order. */
5702     return CRYPTO_memcmp(kZeros, out_shared_key, 32) != 0;
5703 }
5704 
5705 void
ossl_x25519_public_from_private(uint8_t out_public_value[32],const uint8_t private_key[32])5706 ossl_x25519_public_from_private(uint8_t out_public_value[32],
5707                                 const uint8_t private_key[32])
5708 {
5709     uint8_t e[32];
5710     ge_p3 A;
5711     fe zplusy, zminusy, zminusy_inv;
5712 
5713     memcpy(e, private_key, 32);
5714     e[0] &= 248;
5715     e[31] &= 127;
5716     e[31] |= 64;
5717 
5718     ge_scalarmult_base(&A, e);
5719 
5720     /*
5721      * We only need the u-coordinate of the curve25519 point.
5722      * The map is u=(y+1)/(1-y). Since y=Y/Z, this gives
5723      * u=(Z+Y)/(Z-Y).
5724      */
5725     fe_add(zplusy, A.Z, A.Y);
5726     fe_sub(zminusy, A.Z, A.Y);
5727     fe_invert(zminusy_inv, zminusy);
5728     fe_mul(zplusy, zplusy, zminusy_inv);
5729     fe_tobytes(out_public_value, zplusy);
5730 
5731     OPENSSL_cleanse(e, sizeof(e));
5732 }
5733