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0bcb8184 MW |
1 | /* |
2 | * f25519.c: arithmetic modulo 2^255 - 19 | |
3 | */ | |
4 | /* | |
5 | * This file is Free Software. It has been modified to as part of its | |
6 | * incorporation into secnet. | |
7 | * | |
8 | * Copyright 2017 Mark Wooding | |
9 | * | |
10 | * You may redistribute this file and/or modify it under the terms of | |
11 | * the permissive licence shown below. | |
12 | * | |
13 | * You may redistribute secnet as a whole and/or modify it under the | |
14 | * terms of the GNU General Public License as published by the Free | |
15 | * Software Foundation; either version 3, or (at your option) any | |
16 | * later version. | |
17 | * | |
18 | * This program is distributed in the hope that it will be useful, | |
19 | * but WITHOUT ANY WARRANTY; without even the implied warranty of | |
20 | * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the | |
21 | * GNU General Public License for more details. | |
22 | * | |
23 | * You should have received a copy of the GNU General Public License | |
24 | * along with this program; if not, see | |
25 | * https://www.gnu.org/licenses/gpl.html. | |
26 | */ | |
27 | /* | |
28 | * Imported from Catacomb, and modified for Secnet (2017-04-30): | |
29 | * | |
30 | * * Use `fake-mLib-bits.h' in place of the real <mLib/bits.h>. | |
31 | * | |
32 | * * Remove the 16/32-bit implementation, since C99 always has 64-bit | |
33 | * arithmetic. | |
34 | * | |
35 | * * Remove the test rig code: a replacement is in a separate source file. | |
36 | * | |
37 | * * Disable some of the operations which aren't needed for X25519. | |
38 | * (They're used for Ed25519, which we don't need.) | |
39 | * | |
40 | * The file's original comment headers are preserved below. | |
41 | */ | |
b7a5ecfc MW |
42 | /* -*-c-*- |
43 | * | |
44 | * Arithmetic modulo 2^255 - 19 | |
45 | * | |
46 | * (c) 2017 Straylight/Edgeware | |
47 | */ | |
48 | ||
49 | /*----- Licensing notice --------------------------------------------------* | |
50 | * | |
51 | * This file is part of Catacomb. | |
52 | * | |
53 | * Catacomb is free software; you can redistribute it and/or modify | |
54 | * it under the terms of the GNU Library General Public License as | |
55 | * published by the Free Software Foundation; either version 2 of the | |
56 | * License, or (at your option) any later version. | |
57 | * | |
58 | * Catacomb is distributed in the hope that it will be useful, | |
59 | * but WITHOUT ANY WARRANTY; without even the implied warranty of | |
60 | * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the | |
61 | * GNU Library General Public License for more details. | |
62 | * | |
63 | * You should have received a copy of the GNU Library General Public | |
64 | * License along with Catacomb; if not, write to the Free | |
65 | * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, | |
66 | * MA 02111-1307, USA. | |
67 | */ | |
68 | ||
69 | /*----- Header files ------------------------------------------------------*/ | |
70 | ||
b7a5ecfc MW |
71 | #include "f25519.h" |
72 | ||
73 | /*----- Basic setup -------------------------------------------------------*/ | |
74 | ||
b7a5ecfc MW |
75 | /* Elements x of GF(2^255 - 19) are represented by ten signed integers x_i: x |
76 | * = SUM_{0<=i<10} x_i 2^ceil(51i/2), mostly following Bernstein's original | |
77 | * paper. | |
78 | */ | |
79 | ||
80 | typedef int32 piece; typedef int64 dblpiece; | |
81 | typedef uint32 upiece; typedef uint64 udblpiece; | |
82 | #define P p26 | |
83 | #define PIECEWD(i) ((i)%2 ? 25 : 26) | |
84 | #define NPIECE 10 | |
85 | ||
86 | #define M26 0x03ffffffu | |
87 | #define M25 0x01ffffffu | |
88 | #define B26 0x04000000u | |
89 | #define B25 0x02000000u | |
90 | #define B24 0x01000000u | |
91 | ||
92 | #define PIECES(v) v##0, v##1, v##2, v##3, v##4, v##5, v##6, v##7, v##8, v##9 | |
93 | #define FETCH(v, w) do { \ | |
94 | v##0 = (w)->P[0]; v##1 = (w)->P[1]; \ | |
95 | v##2 = (w)->P[2]; v##3 = (w)->P[3]; \ | |
96 | v##4 = (w)->P[4]; v##5 = (w)->P[5]; \ | |
97 | v##6 = (w)->P[6]; v##7 = (w)->P[7]; \ | |
98 | v##8 = (w)->P[8]; v##9 = (w)->P[9]; \ | |
99 | } while (0) | |
100 | #define STASH(w, v) do { \ | |
101 | (w)->P[0] = v##0; (w)->P[1] = v##1; \ | |
102 | (w)->P[2] = v##2; (w)->P[3] = v##3; \ | |
103 | (w)->P[4] = v##4; (w)->P[5] = v##5; \ | |
104 | (w)->P[6] = v##6; (w)->P[7] = v##7; \ | |
105 | (w)->P[8] = v##8; (w)->P[9] = v##9; \ | |
106 | } while (0) | |
107 | ||
b7a5ecfc MW |
108 | /*----- Debugging machinery -----------------------------------------------*/ |
109 | ||
0bcb8184 | 110 | #if defined(F25519_DEBUG) |
b7a5ecfc MW |
111 | |
112 | #include <stdio.h> | |
113 | ||
114 | #include "mp.h" | |
115 | #include "mptext.h" | |
116 | ||
117 | static mp *get_2p255m91(void) | |
118 | { | |
119 | mpw w19 = 19; | |
120 | mp *p = MP_NEW, m19; | |
121 | ||
122 | p = mp_setbit(p, MP_ZERO, 255); | |
123 | mp_build(&m19, &w19, &w19 + 1); | |
124 | p = mp_sub(p, p, &m19); | |
125 | return (p); | |
126 | } | |
127 | ||
128 | DEF_FDUMP(fdump, piece, PIECEWD, NPIECE, 32, get_2p255m91()) | |
129 | ||
130 | #endif | |
131 | ||
132 | /*----- Loading and storing -----------------------------------------------*/ | |
133 | ||
134 | /* --- @f25519_load@ --- * | |
135 | * | |
136 | * Arguments: @f25519 *z@ = where to store the result | |
137 | * @const octet xv[32]@ = source to read | |
138 | * | |
139 | * Returns: --- | |
140 | * | |
141 | * Use: Reads an element of %$\gf{2^{255} - 19}$% in external | |
142 | * representation from @xv@ and stores it in @z@. | |
143 | * | |
144 | * External representation is little-endian base-256. Some | |
145 | * elements have multiple encodings, which are not produced by | |
146 | * correct software; use of noncanonical encodings is not an | |
147 | * error, and toleration of them is considered a performance | |
148 | * feature. | |
149 | */ | |
150 | ||
151 | void f25519_load(f25519 *z, const octet xv[32]) | |
152 | { | |
b7a5ecfc MW |
153 | uint32 xw0 = LOAD32_L(xv + 0), xw1 = LOAD32_L(xv + 4), |
154 | xw2 = LOAD32_L(xv + 8), xw3 = LOAD32_L(xv + 12), | |
155 | xw4 = LOAD32_L(xv + 16), xw5 = LOAD32_L(xv + 20), | |
156 | xw6 = LOAD32_L(xv + 24), xw7 = LOAD32_L(xv + 28); | |
157 | piece PIECES(x), b, c; | |
158 | ||
159 | /* First, split the 32-bit words into the irregularly-sized pieces we need | |
160 | * for the field representation. These pieces are all positive. We'll do | |
161 | * the sign correction afterwards. | |
162 | * | |
163 | * It may be that the top bit of the input is set. Avoid trouble by | |
164 | * folding that back round into the bottom piece of the representation. | |
165 | * | |
166 | * Here, we briefly have 0 <= x_0 < 2^26 + 19, but will resolve this later. | |
167 | * Otherwise, we have 0 <= x_{2i} < 2^26, and 0 <= x_{2i+1} < 2^25. | |
168 | */ | |
169 | x0 = ((xw0 << 0)&0x03ffffff) + 19*((xw7 >> 31)&0x00000001); | |
170 | x1 = ((xw1 << 6)&0x01ffffc0) | ((xw0 >> 26)&0x0000003f); | |
171 | x2 = ((xw2 << 13)&0x03ffe000) | ((xw1 >> 19)&0x00001fff); | |
172 | x3 = ((xw3 << 19)&0x01f80000) | ((xw2 >> 13)&0x0007ffff); | |
173 | x4 = ((xw3 >> 6)&0x03ffffff); | |
174 | x5 = (xw4 << 0)&0x01ffffff; | |
175 | x6 = ((xw5 << 7)&0x03ffff80) | ((xw4 >> 25)&0x0000007f); | |
176 | x7 = ((xw6 << 13)&0x01ffe000) | ((xw5 >> 19)&0x00001fff); | |
177 | x8 = ((xw7 << 20)&0x03f00000) | ((xw6 >> 12)&0x000fffff); | |
178 | x9 = ((xw7 >> 6)&0x01ffffff); | |
179 | ||
180 | /* Next, we convert these pieces into a roughly balanced signed | |
181 | * representation. For each piece, check to see if its top bit is set. If | |
182 | * it is, then lend a bit to the next piece over. For x_9, this needs to | |
183 | * be carried around, which is a little fiddly. In particular, we delay | |
184 | * the carry until after all of the pieces have been balanced. If we don't | |
185 | * do this, then we have to do a more expensive test for nonzeroness to | |
186 | * decide whether to lend a bit leftwards rather than just testing a single | |
187 | * bit. | |
188 | * | |
189 | * This fixes up the anomalous size of x_0: the loan of a bit becomes an | |
190 | * actual carry if x_0 >= 2^26. By the end, then, we have: | |
191 | * | |
192 | * { 2^25 if i even | |
193 | * |x_i| <= { | |
194 | * { 2^24 if i odd | |
195 | * | |
196 | * Note that we don't try for a canonical representation here: both upper | |
197 | * and lower bounds are achievable. | |
198 | * | |
199 | * All of the x_i at this point are positive, so we don't need to do | |
200 | * anything wierd when masking them. | |
201 | */ | |
202 | b = x9&B24; c = 19&((b >> 19) - (b >> 24)); x9 -= b << 1; | |
203 | b = x8&B25; x9 += b >> 25; x8 -= b << 1; | |
204 | b = x7&B24; x8 += b >> 24; x7 -= b << 1; | |
205 | b = x6&B25; x7 += b >> 25; x6 -= b << 1; | |
206 | b = x5&B24; x6 += b >> 24; x5 -= b << 1; | |
207 | b = x4&B25; x5 += b >> 25; x4 -= b << 1; | |
208 | b = x3&B24; x4 += b >> 24; x3 -= b << 1; | |
209 | b = x2&B25; x3 += b >> 25; x2 -= b << 1; | |
210 | b = x1&B24; x2 += b >> 24; x1 -= b << 1; | |
211 | b = x0&B25; x1 += (b >> 25) + (x0 >> 26); x0 = (x0&M26) - (b << 1); | |
212 | x0 += c; | |
213 | ||
214 | /* And with that, we're done. */ | |
215 | STASH(z, x); | |
b7a5ecfc MW |
216 | } |
217 | ||
218 | /* --- @f25519_store@ --- * | |
219 | * | |
220 | * Arguments: @octet zv[32]@ = where to write the result | |
221 | * @const f25519 *x@ = the field element to write | |
222 | * | |
223 | * Returns: --- | |
224 | * | |
225 | * Use: Stores a field element in the given octet vector in external | |
226 | * representation. A canonical encoding is always stored, so, | |
227 | * in particular, the top bit of @xv[31]@ is always left clear. | |
228 | */ | |
229 | ||
230 | void f25519_store(octet zv[32], const f25519 *x) | |
231 | { | |
b7a5ecfc MW |
232 | piece PIECES(x), PIECES(y), c, d; |
233 | uint32 zw0, zw1, zw2, zw3, zw4, zw5, zw6, zw7; | |
234 | mask32 m; | |
235 | ||
236 | FETCH(x, x); | |
237 | ||
238 | /* First, propagate the carries throughout the pieces. By the end of this, | |
239 | * we'll have all of the pieces canonically sized and positive, and maybe | |
240 | * there'll be (signed) carry out. The carry c is in { -1, 0, +1 }, and | |
241 | * the remaining value will be in the half-open interval [0, 2^255). The | |
242 | * whole represented value is then x + 2^255 c. | |
243 | * | |
244 | * It's worth paying careful attention to the bounds. We assume that we | |
245 | * start out with |x_i| <= 2^30. We start by cutting off and reducing the | |
246 | * carry c_9 from the topmost piece, x_9. This leaves 0 <= x_9 < 2^25; and | |
247 | * we'll have |c_9| <= 2^5. We multiply this by 19 and we'll add it onto | |
248 | * x_0 and propagate the carries: but what bounds can we calculate on x | |
249 | * before this? | |
250 | * | |
251 | * Let o_i = floor(51 i/2). We have X_i = SUM_{0<=j<i} x_j 2^{o_i}, so | |
252 | * x = X_10. We see, inductively, that |X_i| < 2^{31+o_{i-1}}: X_0 = 0; | |
253 | * |x_i| <= 2^30; and |X_{i+1}| = |X_i + x_i 2^{o_i}| <= |X_i| + 2^{30+o_i} | |
254 | * < 2^{31+o_i}. Then x = X_9 + 2^230 x_9, and we have better bounds for | |
255 | * x_9, so | |
256 | * | |
257 | * -2^235 < x + 19 c_9 < 2^255 + 2^235 | |
258 | * | |
259 | * Here, the x_i are signed, so we must be cautious about bithacking them. | |
260 | */ | |
261 | c = ASR(piece, x9, 25); x9 = (upiece)x9&M25; | |
262 | x0 += 19*c; c = ASR(piece, x0, 26); x0 = (upiece)x0&M26; | |
263 | x1 += c; c = ASR(piece, x1, 25); x1 = (upiece)x1&M25; | |
264 | x2 += c; c = ASR(piece, x2, 26); x2 = (upiece)x2&M26; | |
265 | x3 += c; c = ASR(piece, x3, 25); x3 = (upiece)x3&M25; | |
266 | x4 += c; c = ASR(piece, x4, 26); x4 = (upiece)x4&M26; | |
267 | x5 += c; c = ASR(piece, x5, 25); x5 = (upiece)x5&M25; | |
268 | x6 += c; c = ASR(piece, x6, 26); x6 = (upiece)x6&M26; | |
269 | x7 += c; c = ASR(piece, x7, 25); x7 = (upiece)x7&M25; | |
270 | x8 += c; c = ASR(piece, x8, 26); x8 = (upiece)x8&M26; | |
271 | x9 += c; c = ASR(piece, x9, 25); x9 = (upiece)x9&M25; | |
272 | ||
273 | /* Now we have a slightly fiddly job to do. If c = +1, or if c = 0 and | |
274 | * x >= 2^255 - 19, then we should subtract 2^255 - 19 from the whole | |
275 | * value; if c = -1 then we should add 2^255 - 19; and otherwise we should | |
276 | * do nothing. | |
277 | * | |
278 | * But conditional behaviour is bad, m'kay. So here's what we do instead. | |
279 | * | |
280 | * The first job is to sort out what we wanted to do. If c = -1 then we | |
281 | * want to (a) invert the constant addend and (b) feed in a carry-in; | |
282 | * otherwise, we don't. | |
283 | */ | |
284 | m = SIGN(c); | |
285 | d = m&1; | |
286 | ||
287 | /* Now do the addition/subtraction. Remember that all of the x_i are | |
288 | * nonnegative, so shifting and masking are safe and easy. | |
289 | */ | |
290 | d += x0 + (19 ^ (M26&m)); y0 = d&M26; d >>= 26; | |
291 | d += x1 + (M25&m); y1 = d&M25; d >>= 25; | |
292 | d += x2 + (M26&m); y2 = d&M26; d >>= 26; | |
293 | d += x3 + (M25&m); y3 = d&M25; d >>= 25; | |
294 | d += x4 + (M26&m); y4 = d&M26; d >>= 26; | |
295 | d += x5 + (M25&m); y5 = d&M25; d >>= 25; | |
296 | d += x6 + (M26&m); y6 = d&M26; d >>= 26; | |
297 | d += x7 + (M25&m); y7 = d&M25; d >>= 25; | |
298 | d += x8 + (M26&m); y8 = d&M26; d >>= 26; | |
299 | d += x9 + (M25&m); y9 = d&M25; d >>= 25; | |
300 | ||
301 | /* The final carry-out is in d; since we only did addition, and the x_i are | |
302 | * nonnegative, then d is in { 0, 1 }. We want to keep y, rather than x, | |
303 | * if (a) c /= 0 (in which case we know that the old value was | |
304 | * unsatisfactory), or (b) if d = 1 (in which case, if c = 0, we know that | |
305 | * the subtraction didn't cause a borrow, so we must be in the case where | |
306 | * 2^255 - 19 <= x < 2^255). | |
307 | */ | |
308 | m = NONZEROP(c) | ~NONZEROP(d - 1); | |
309 | x0 = (y0&m) | (x0&~m); x1 = (y1&m) | (x1&~m); | |
310 | x2 = (y2&m) | (x2&~m); x3 = (y3&m) | (x3&~m); | |
311 | x4 = (y4&m) | (x4&~m); x5 = (y5&m) | (x5&~m); | |
312 | x6 = (y6&m) | (x6&~m); x7 = (y7&m) | (x7&~m); | |
313 | x8 = (y8&m) | (x8&~m); x9 = (y9&m) | (x9&~m); | |
314 | ||
315 | /* Extract 32-bit words from the value. */ | |
316 | zw0 = ((x0 >> 0)&0x03ffffff) | (((uint32)x1 << 26)&0xfc000000); | |
317 | zw1 = ((x1 >> 6)&0x0007ffff) | (((uint32)x2 << 19)&0xfff80000); | |
318 | zw2 = ((x2 >> 13)&0x00001fff) | (((uint32)x3 << 13)&0xffffe000); | |
319 | zw3 = ((x3 >> 19)&0x0000003f) | (((uint32)x4 << 6)&0xffffffc0); | |
320 | zw4 = ((x5 >> 0)&0x01ffffff) | (((uint32)x6 << 25)&0xfe000000); | |
321 | zw5 = ((x6 >> 7)&0x0007ffff) | (((uint32)x7 << 19)&0xfff80000); | |
322 | zw6 = ((x7 >> 13)&0x00000fff) | (((uint32)x8 << 12)&0xfffff000); | |
323 | zw7 = ((x8 >> 20)&0x0000003f) | (((uint32)x9 << 6)&0x7fffffc0); | |
324 | ||
325 | /* Store the result as an octet string. */ | |
326 | STORE32_L(zv + 0, zw0); STORE32_L(zv + 4, zw1); | |
327 | STORE32_L(zv + 8, zw2); STORE32_L(zv + 12, zw3); | |
328 | STORE32_L(zv + 16, zw4); STORE32_L(zv + 20, zw5); | |
329 | STORE32_L(zv + 24, zw6); STORE32_L(zv + 28, zw7); | |
b7a5ecfc MW |
330 | } |
331 | ||
332 | /* --- @f25519_set@ --- * | |
333 | * | |
334 | * Arguments: @f25519 *z@ = where to write the result | |
335 | * @int a@ = a small-ish constant | |
336 | * | |
337 | * Returns: --- | |
338 | * | |
339 | * Use: Sets @z@ to equal @a@. | |
340 | */ | |
341 | ||
342 | void f25519_set(f25519 *x, int a) | |
343 | { | |
344 | unsigned i; | |
345 | ||
346 | x->P[0] = a; | |
347 | for (i = 1; i < NPIECE; i++) x->P[i] = 0; | |
348 | } | |
349 | ||
350 | /*----- Basic arithmetic --------------------------------------------------*/ | |
351 | ||
352 | /* --- @f25519_add@ --- * | |
353 | * | |
354 | * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@) | |
355 | * @const f25519 *x, *y@ = two operands | |
356 | * | |
357 | * Returns: --- | |
358 | * | |
359 | * Use: Set @z@ to the sum %$x + y$%. | |
360 | */ | |
361 | ||
362 | void f25519_add(f25519 *z, const f25519 *x, const f25519 *y) | |
363 | { | |
b7a5ecfc MW |
364 | z->P[0] = x->P[0] + y->P[0]; z->P[1] = x->P[1] + y->P[1]; |
365 | z->P[2] = x->P[2] + y->P[2]; z->P[3] = x->P[3] + y->P[3]; | |
366 | z->P[4] = x->P[4] + y->P[4]; z->P[5] = x->P[5] + y->P[5]; | |
367 | z->P[6] = x->P[6] + y->P[6]; z->P[7] = x->P[7] + y->P[7]; | |
368 | z->P[8] = x->P[8] + y->P[8]; z->P[9] = x->P[9] + y->P[9]; | |
b7a5ecfc MW |
369 | } |
370 | ||
371 | /* --- @f25519_sub@ --- * | |
372 | * | |
373 | * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@) | |
374 | * @const f25519 *x, *y@ = two operands | |
375 | * | |
376 | * Returns: --- | |
377 | * | |
378 | * Use: Set @z@ to the difference %$x - y$%. | |
379 | */ | |
380 | ||
381 | void f25519_sub(f25519 *z, const f25519 *x, const f25519 *y) | |
382 | { | |
b7a5ecfc MW |
383 | z->P[0] = x->P[0] - y->P[0]; z->P[1] = x->P[1] - y->P[1]; |
384 | z->P[2] = x->P[2] - y->P[2]; z->P[3] = x->P[3] - y->P[3]; | |
385 | z->P[4] = x->P[4] - y->P[4]; z->P[5] = x->P[5] - y->P[5]; | |
386 | z->P[6] = x->P[6] - y->P[6]; z->P[7] = x->P[7] - y->P[7]; | |
387 | z->P[8] = x->P[8] - y->P[8]; z->P[9] = x->P[9] - y->P[9]; | |
b7a5ecfc MW |
388 | } |
389 | ||
0bcb8184 MW |
390 | #ifndef F25519_TRIM_X25519 |
391 | ||
b7a5ecfc MW |
392 | /* --- @f25519_neg@ --- * |
393 | * | |
394 | * Arguments: @f25519 *z@ = where to put the result (may alias @x@) | |
395 | * @const f25519 *x@ = an operand | |
396 | * | |
397 | * Returns: --- | |
398 | * | |
399 | * Use: Set @z = -x@. | |
400 | */ | |
401 | ||
402 | void f25519_neg(f25519 *z, const f25519 *x) | |
403 | { | |
b7a5ecfc MW |
404 | z->P[0] = -x->P[0]; z->P[1] = -x->P[1]; |
405 | z->P[2] = -x->P[2]; z->P[3] = -x->P[3]; | |
406 | z->P[4] = -x->P[4]; z->P[5] = -x->P[5]; | |
407 | z->P[6] = -x->P[6]; z->P[7] = -x->P[7]; | |
408 | z->P[8] = -x->P[8]; z->P[9] = -x->P[9]; | |
b7a5ecfc MW |
409 | } |
410 | ||
0bcb8184 MW |
411 | #endif |
412 | ||
b7a5ecfc MW |
413 | /*----- Constant-time utilities -------------------------------------------*/ |
414 | ||
0bcb8184 MW |
415 | #ifndef F25519_TRIM_X25519 |
416 | ||
b7a5ecfc MW |
417 | /* --- @f25519_pick2@ --- * |
418 | * | |
419 | * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@) | |
420 | * @const f25519 *x, *y@ = two operands | |
421 | * @uint32 m@ = a mask | |
422 | * | |
423 | * Returns: --- | |
424 | * | |
425 | * Use: If @m@ is zero, set @z = y@; if @m@ is all-bits-set, then set | |
426 | * @z = x@. If @m@ has some other value, then scramble @z@ in | |
427 | * an unhelpful way. | |
428 | */ | |
429 | ||
430 | void f25519_pick2(f25519 *z, const f25519 *x, const f25519 *y, uint32 m) | |
431 | { | |
432 | mask32 mm = FIX_MASK32(m); | |
433 | ||
b7a5ecfc MW |
434 | z->P[0] = PICK2(x->P[0], y->P[0], mm); |
435 | z->P[1] = PICK2(x->P[1], y->P[1], mm); | |
436 | z->P[2] = PICK2(x->P[2], y->P[2], mm); | |
437 | z->P[3] = PICK2(x->P[3], y->P[3], mm); | |
438 | z->P[4] = PICK2(x->P[4], y->P[4], mm); | |
439 | z->P[5] = PICK2(x->P[5], y->P[5], mm); | |
440 | z->P[6] = PICK2(x->P[6], y->P[6], mm); | |
441 | z->P[7] = PICK2(x->P[7], y->P[7], mm); | |
442 | z->P[8] = PICK2(x->P[8], y->P[8], mm); | |
443 | z->P[9] = PICK2(x->P[9], y->P[9], mm); | |
b7a5ecfc MW |
444 | } |
445 | ||
446 | /* --- @f25519_pickn@ --- * | |
447 | * | |
448 | * Arguments: @f25519 *z@ = where to put the result | |
449 | * @const f25519 *v@ = a table of entries | |
450 | * @size_t n@ = the number of entries in @v@ | |
451 | * @size_t i@ = an index | |
452 | * | |
453 | * Returns: --- | |
454 | * | |
455 | * Use: If @0 <= i < n < 32@ then set @z = v[i]@. If @n >= 32@ then | |
456 | * do something unhelpful; otherwise, if @i >= n@ then set @z@ | |
457 | * to zero. | |
458 | */ | |
459 | ||
460 | void f25519_pickn(f25519 *z, const f25519 *v, size_t n, size_t i) | |
461 | { | |
462 | uint32 b = (uint32)1 << (31 - i); | |
463 | mask32 m; | |
464 | ||
b7a5ecfc MW |
465 | z->P[0] = z->P[1] = z->P[2] = z->P[3] = z->P[4] = |
466 | z->P[5] = z->P[6] = z->P[7] = z->P[8] = z->P[9] = 0; | |
467 | while (n--) { | |
468 | m = SIGN(b); | |
469 | CONDPICK(z->P[0], v->P[0], m); | |
470 | CONDPICK(z->P[1], v->P[1], m); | |
471 | CONDPICK(z->P[2], v->P[2], m); | |
472 | CONDPICK(z->P[3], v->P[3], m); | |
473 | CONDPICK(z->P[4], v->P[4], m); | |
474 | CONDPICK(z->P[5], v->P[5], m); | |
475 | CONDPICK(z->P[6], v->P[6], m); | |
476 | CONDPICK(z->P[7], v->P[7], m); | |
477 | CONDPICK(z->P[8], v->P[8], m); | |
478 | CONDPICK(z->P[9], v->P[9], m); | |
479 | v++; b <<= 1; | |
480 | } | |
0bcb8184 | 481 | } |
b7a5ecfc | 482 | |
b7a5ecfc | 483 | #endif |
b7a5ecfc MW |
484 | |
485 | /* --- @f25519_condswap@ --- * | |
486 | * | |
487 | * Arguments: @f25519 *x, *y@ = two operands | |
488 | * @uint32 m@ = a mask | |
489 | * | |
490 | * Returns: --- | |
491 | * | |
492 | * Use: If @m@ is zero, do nothing; if @m@ is all-bits-set, then | |
493 | * exchange @x@ and @y@. If @m@ has some other value, then | |
494 | * scramble @x@ and @y@ in an unhelpful way. | |
495 | */ | |
496 | ||
497 | void f25519_condswap(f25519 *x, f25519 *y, uint32 m) | |
498 | { | |
499 | mask32 mm = FIX_MASK32(m); | |
500 | ||
b7a5ecfc MW |
501 | CONDSWAP(x->P[0], y->P[0], mm); |
502 | CONDSWAP(x->P[1], y->P[1], mm); | |
503 | CONDSWAP(x->P[2], y->P[2], mm); | |
504 | CONDSWAP(x->P[3], y->P[3], mm); | |
505 | CONDSWAP(x->P[4], y->P[4], mm); | |
506 | CONDSWAP(x->P[5], y->P[5], mm); | |
507 | CONDSWAP(x->P[6], y->P[6], mm); | |
508 | CONDSWAP(x->P[7], y->P[7], mm); | |
509 | CONDSWAP(x->P[8], y->P[8], mm); | |
510 | CONDSWAP(x->P[9], y->P[9], mm); | |
b7a5ecfc MW |
511 | } |
512 | ||
0bcb8184 MW |
513 | #ifndef F25519_TRIM_X25519 |
514 | ||
b7a5ecfc MW |
515 | /* --- @f25519_condneg@ --- * |
516 | * | |
517 | * Arguments: @f25519 *z@ = where to put the result (may alias @x@) | |
518 | * @const f25519 *x@ = an operand | |
519 | * @uint32 m@ = a mask | |
520 | * | |
521 | * Returns: --- | |
522 | * | |
523 | * Use: If @m@ is zero, set @z = x@; if @m@ is all-bits-set, then set | |
524 | * @z = -x@. If @m@ has some other value then scramble @z@ in | |
525 | * an unhelpful way. | |
526 | */ | |
527 | ||
528 | void f25519_condneg(f25519 *z, const f25519 *x, uint32 m) | |
529 | { | |
b7a5ecfc MW |
530 | mask32 m_xor = FIX_MASK32(m); |
531 | piece m_add = m&1; | |
532 | # define CONDNEG(x) (((x) ^ m_xor) + m_add) | |
b7a5ecfc | 533 | |
b7a5ecfc MW |
534 | z->P[0] = CONDNEG(x->P[0]); |
535 | z->P[1] = CONDNEG(x->P[1]); | |
536 | z->P[2] = CONDNEG(x->P[2]); | |
537 | z->P[3] = CONDNEG(x->P[3]); | |
538 | z->P[4] = CONDNEG(x->P[4]); | |
539 | z->P[5] = CONDNEG(x->P[5]); | |
540 | z->P[6] = CONDNEG(x->P[6]); | |
541 | z->P[7] = CONDNEG(x->P[7]); | |
542 | z->P[8] = CONDNEG(x->P[8]); | |
543 | z->P[9] = CONDNEG(x->P[9]); | |
b7a5ecfc MW |
544 | |
545 | #undef CONDNEG | |
546 | } | |
547 | ||
0bcb8184 | 548 | #endif |
b7a5ecfc | 549 | |
0bcb8184 | 550 | /*----- Multiplication ----------------------------------------------------*/ |
b7a5ecfc MW |
551 | |
552 | /* Let B = 2^63 - 1 be the largest value such that +B and -B can be | |
553 | * represented in a double-precision piece. On entry, it must be the case | |
554 | * that |X_i| <= M <= B - 2^25 for some M. If this is the case, then, on | |
555 | * exit, we will have |Z_i| <= 2^25 + 19 M/2^25. | |
556 | */ | |
557 | #define CARRYSTEP(z, x, m, b, f, xx, n) do { \ | |
558 | (z) = (dblpiece)((udblpiece)(x)&(m)) - (b) + \ | |
559 | (f)*ASR(dblpiece, (xx), (n)); \ | |
560 | } while (0) | |
561 | #define CARRY_REDUCE(z, x) do { \ | |
562 | dblpiece PIECES(_t); \ | |
563 | \ | |
564 | /* Bias the input pieces. This keeps the carries and so on centred \ | |
565 | * around zero rather than biased positive. \ | |
566 | */ \ | |
567 | _t0 = (x##0) + B25; _t1 = (x##1) + B24; \ | |
568 | _t2 = (x##2) + B25; _t3 = (x##3) + B24; \ | |
569 | _t4 = (x##4) + B25; _t5 = (x##5) + B24; \ | |
570 | _t6 = (x##6) + B25; _t7 = (x##7) + B24; \ | |
571 | _t8 = (x##8) + B25; _t9 = (x##9) + B24; \ | |
572 | \ | |
573 | /* Calculate the reduced pieces. Careful with the bithacking. */ \ | |
574 | CARRYSTEP(z##0, _t0, M26, B25, 19, _t9, 25); \ | |
575 | CARRYSTEP(z##1, _t1, M25, B24, 1, _t0, 26); \ | |
576 | CARRYSTEP(z##2, _t2, M26, B25, 1, _t1, 25); \ | |
577 | CARRYSTEP(z##3, _t3, M25, B24, 1, _t2, 26); \ | |
578 | CARRYSTEP(z##4, _t4, M26, B25, 1, _t3, 25); \ | |
579 | CARRYSTEP(z##5, _t5, M25, B24, 1, _t4, 26); \ | |
580 | CARRYSTEP(z##6, _t6, M26, B25, 1, _t5, 25); \ | |
581 | CARRYSTEP(z##7, _t7, M25, B24, 1, _t6, 26); \ | |
582 | CARRYSTEP(z##8, _t8, M26, B25, 1, _t7, 25); \ | |
583 | CARRYSTEP(z##9, _t9, M25, B24, 1, _t8, 26); \ | |
584 | } while (0) | |
585 | ||
b7a5ecfc MW |
586 | /* --- @f25519_mulconst@ --- * |
587 | * | |
588 | * Arguments: @f25519 *z@ = where to put the result (may alias @x@) | |
589 | * @const f25519 *x@ = an operand | |
590 | * @long a@ = a small-ish constant; %$|a| < 2^{20}$%. | |
591 | * | |
592 | * Returns: --- | |
593 | * | |
594 | * Use: Set @z@ to the product %$a x$%. | |
595 | */ | |
596 | ||
597 | void f25519_mulconst(f25519 *z, const f25519 *x, long a) | |
598 | { | |
b7a5ecfc MW |
599 | piece PIECES(x); |
600 | dblpiece PIECES(z), aa = a; | |
601 | ||
602 | FETCH(x, x); | |
603 | ||
604 | /* Suppose that |x_i| <= 2^27, and |a| <= 2^23. Then we'll have | |
605 | * |z_i| <= 2^50. | |
606 | */ | |
607 | z0 = aa*x0; z1 = aa*x1; z2 = aa*x2; z3 = aa*x3; z4 = aa*x4; | |
608 | z5 = aa*x5; z6 = aa*x6; z7 = aa*x7; z8 = aa*x8; z9 = aa*x9; | |
609 | ||
610 | /* Following `CARRY_REDUCE', we'll have |z_i| <= 2^26. */ | |
611 | CARRY_REDUCE(z, z); | |
612 | STASH(z, z); | |
b7a5ecfc MW |
613 | } |
614 | ||
615 | /* --- @f25519_mul@ --- * | |
616 | * | |
617 | * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@) | |
618 | * @const f25519 *x, *y@ = two operands | |
619 | * | |
620 | * Returns: --- | |
621 | * | |
622 | * Use: Set @z@ to the product %$x y$%. | |
623 | */ | |
624 | ||
625 | void f25519_mul(f25519 *z, const f25519 *x, const f25519 *y) | |
626 | { | |
b7a5ecfc MW |
627 | piece PIECES(x), PIECES(y); |
628 | dblpiece PIECES(z); | |
629 | unsigned i; | |
630 | ||
631 | FETCH(x, x); FETCH(y, y); | |
632 | ||
633 | /* Suppose that |x_i|, |y_i| <= 2^27. Then we'll have | |
634 | * | |
635 | * |z_0| <= 267*2^54 | |
636 | * |z_1| <= 154*2^54 | |
637 | * |z_2| <= 213*2^54 | |
638 | * |z_3| <= 118*2^54 | |
639 | * |z_4| <= 159*2^54 | |
640 | * |z_5| <= 82*2^54 | |
641 | * |z_6| <= 105*2^54 | |
642 | * |z_7| <= 46*2^54 | |
643 | * |z_8| <= 51*2^54 | |
644 | * |z_9| <= 10*2^54 | |
645 | * | |
646 | * all of which are less than 2^63 - 2^25. | |
647 | */ | |
648 | ||
649 | #define M(a, b) ((dblpiece)(a)*(b)) | |
650 | z0 = M(x0, y0) + | |
651 | 19*(M(x2, y8) + M(x4, y6) + M(x6, y4) + M(x8, y2)) + | |
652 | 38*(M(x1, y9) + M(x3, y7) + M(x5, y5) + M(x7, y3) + M(x9, y1)); | |
653 | z1 = M(x0, y1) + M(x1, y0) + | |
654 | 19*(M(x2, y9) + M(x3, y8) + M(x4, y7) + M(x5, y6) + | |
655 | M(x6, y5) + M(x7, y4) + M(x8, y3) + M(x9, y2)); | |
656 | z2 = M(x0, y2) + M(x2, y0) + | |
657 | 2* M(x1, y1) + | |
658 | 19*(M(x4, y8) + M(x6, y6) + M(x8, y4)) + | |
659 | 38*(M(x3, y9) + M(x5, y7) + M(x7, y5) + M(x9, y3)); | |
660 | z3 = M(x0, y3) + M(x1, y2) + M(x2, y1) + M(x3, y0) + | |
661 | 19*(M(x4, y9) + M(x5, y8) + M(x6, y7) + | |
662 | M(x7, y6) + M(x8, y5) + M(x9, y4)); | |
663 | z4 = M(x0, y4) + M(x2, y2) + M(x4, y0) + | |
664 | 2*(M(x1, y3) + M(x3, y1)) + | |
665 | 19*(M(x6, y8) + M(x8, y6)) + | |
666 | 38*(M(x5, y9) + M(x7, y7) + M(x9, y5)); | |
667 | z5 = M(x0, y5) + M(x1, y4) + M(x2, y3) + | |
668 | M(x3, y2) + M(x4, y1) + M(x5, y0) + | |
669 | 19*(M(x6, y9) + M(x7, y8) + M(x8, y7) + M(x9, y6)); | |
670 | z6 = M(x0, y6) + M(x2, y4) + M(x4, y2) + M(x6, y0) + | |
671 | 2*(M(x1, y5) + M(x3, y3) + M(x5, y1)) + | |
672 | 19* M(x8, y8) + | |
673 | 38*(M(x7, y9) + M(x9, y7)); | |
674 | z7 = M(x0, y7) + M(x1, y6) + M(x2, y5) + M(x3, y4) + | |
675 | M(x4, y3) + M(x5, y2) + M(x6, y1) + M(x7, y0) + | |
676 | 19*(M(x8, y9) + M(x9, y8)); | |
677 | z8 = M(x0, y8) + M(x2, y6) + M(x4, y4) + M(x6, y2) + M(x8, y0) + | |
678 | 2*(M(x1, y7) + M(x3, y5) + M(x5, y3) + M(x7, y1)) + | |
679 | 38* M(x9, y9); | |
680 | z9 = M(x0, y9) + M(x1, y8) + M(x2, y7) + M(x3, y6) + M(x4, y5) + | |
681 | M(x5, y4) + M(x6, y3) + M(x7, y2) + M(x8, y1) + M(x9, y0); | |
682 | #undef M | |
683 | ||
684 | /* From above, we have |z_i| <= 2^63 - 2^25. A pass of `CARRY_REDUCE' will | |
685 | * leave |z_i| <= 2^38 + 2^25; and a second pass will leave |z_i| <= 2^25 + | |
686 | * 2^13, which is comfortable for an addition prior to the next | |
687 | * multiplication. | |
688 | */ | |
689 | for (i = 0; i < 2; i++) CARRY_REDUCE(z, z); | |
690 | STASH(z, z); | |
b7a5ecfc MW |
691 | } |
692 | ||
693 | /* --- @f25519_sqr@ --- * | |
694 | * | |
695 | * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@) | |
696 | * @const f25519 *x@ = an operand | |
697 | * | |
698 | * Returns: --- | |
699 | * | |
700 | * Use: Set @z@ to the square %$x^2$%. | |
701 | */ | |
702 | ||
703 | void f25519_sqr(f25519 *z, const f25519 *x) | |
704 | { | |
b7a5ecfc MW |
705 | piece PIECES(x); |
706 | dblpiece PIECES(z); | |
707 | unsigned i; | |
708 | ||
709 | FETCH(x, x); | |
710 | ||
711 | /* See `f25519_mul' for bounds. */ | |
712 | ||
713 | #define M(a, b) ((dblpiece)(a)*(b)) | |
714 | z0 = M(x0, x0) + | |
715 | 38*(M(x2, x8) + M(x4, x6) + M(x5, x5)) + | |
716 | 76*(M(x1, x9) + M(x3, x7)); | |
717 | z1 = 2* M(x0, x1) + | |
718 | 38*(M(x2, x9) + M(x3, x8) + M(x4, x7) + M(x5, x6)); | |
719 | z2 = 2*(M(x0, x2) + M(x1, x1)) + | |
720 | 19* M(x6, x6) + | |
721 | 38* M(x4, x8) + | |
722 | 76*(M(x3, x9) + M(x5, x7)); | |
723 | z3 = 2*(M(x0, x3) + M(x1, x2)) + | |
724 | 38*(M(x4, x9) + M(x5, x8) + M(x6, x7)); | |
725 | z4 = M(x2, x2) + | |
726 | 2* M(x0, x4) + | |
727 | 4* M(x1, x3) + | |
728 | 38*(M(x6, x8) + M(x7, x7)) + | |
729 | 76* M(x5, x9); | |
730 | z5 = 2*(M(x0, x5) + M(x1, x4) + M(x2, x3)) + | |
731 | 38*(M(x6, x9) + M(x7, x8)); | |
732 | z6 = 2*(M(x0, x6) + M(x2, x4) + M(x3, x3)) + | |
733 | 4* M(x1, x5) + | |
734 | 19* M(x8, x8) + | |
735 | 76* M(x7, x9); | |
736 | z7 = 2*(M(x0, x7) + M(x1, x6) + M(x2, x5) + M(x3, x4)) + | |
737 | 38* M(x8, x9); | |
738 | z8 = M(x4, x4) + | |
739 | 2*(M(x0, x8) + M(x2, x6)) + | |
740 | 4*(M(x1, x7) + M(x3, x5)) + | |
741 | 38* M(x9, x9); | |
742 | z9 = 2*(M(x0, x9) + M(x1, x8) + M(x2, x7) + M(x3, x6) + M(x4, x5)); | |
743 | #undef M | |
744 | ||
745 | /* See `f25519_mul' for details. */ | |
746 | for (i = 0; i < 2; i++) CARRY_REDUCE(z, z); | |
747 | STASH(z, z); | |
b7a5ecfc MW |
748 | } |
749 | ||
750 | /*----- More complicated things -------------------------------------------*/ | |
751 | ||
752 | /* --- @f25519_inv@ --- * | |
753 | * | |
754 | * Arguments: @f25519 *z@ = where to put the result (may alias @x@) | |
755 | * @const f25519 *x@ = an operand | |
756 | * | |
757 | * Returns: --- | |
758 | * | |
759 | * Use: Stores in @z@ the multiplicative inverse %$x^{-1}$%. If | |
760 | * %$x = 0$% then @z@ is set to zero. This is considered a | |
761 | * feature. | |
762 | */ | |
763 | ||
764 | void f25519_inv(f25519 *z, const f25519 *x) | |
765 | { | |
766 | f25519 t, u, t2, t11, t2p10m1, t2p50m1; | |
767 | unsigned i; | |
768 | ||
769 | #define SQRN(z, x, n) do { \ | |
770 | f25519_sqr((z), (x)); \ | |
771 | for (i = 1; i < (n); i++) f25519_sqr((z), (z)); \ | |
772 | } while (0) | |
773 | ||
774 | /* Calculate x^-1 = x^(p - 2) = x^(2^255 - 21), which also handles x = 0 as | |
775 | * intended. The addition chain here is from Bernstein's implementation; I | |
776 | * couldn't find a better one. | |
777 | */ /* step | value */ | |
778 | f25519_sqr(&t2, x); /* 1 | 2 */ | |
779 | SQRN(&u, &t2, 2); /* 3 | 8 */ | |
780 | f25519_mul(&t, &u, x); /* 4 | 9 */ | |
781 | f25519_mul(&t11, &t, &t2); /* 5 | 11 = 2^5 - 21 */ | |
782 | f25519_sqr(&u, &t11); /* 6 | 22 */ | |
783 | f25519_mul(&t, &t, &u); /* 7 | 31 = 2^5 - 1 */ | |
784 | SQRN(&u, &t, 5); /* 12 | 2^10 - 2^5 */ | |
785 | f25519_mul(&t2p10m1, &t, &u); /* 13 | 2^10 - 1 */ | |
786 | SQRN(&u, &t2p10m1, 10); /* 23 | 2^20 - 2^10 */ | |
787 | f25519_mul(&t, &t2p10m1, &u); /* 24 | 2^20 - 1 */ | |
788 | SQRN(&u, &t, 20); /* 44 | 2^40 - 2^20 */ | |
789 | f25519_mul(&t, &t, &u); /* 45 | 2^40 - 1 */ | |
790 | SQRN(&u, &t, 10); /* 55 | 2^50 - 2^10 */ | |
791 | f25519_mul(&t2p50m1, &t2p10m1, &u); /* 56 | 2^50 - 1 */ | |
792 | SQRN(&u, &t2p50m1, 50); /* 106 | 2^100 - 2^50 */ | |
793 | f25519_mul(&t, &t2p50m1, &u); /* 107 | 2^100 - 1 */ | |
794 | SQRN(&u, &t, 100); /* 207 | 2^200 - 2^100 */ | |
795 | f25519_mul(&t, &t, &u); /* 208 | 2^200 - 1 */ | |
796 | SQRN(&u, &t, 50); /* 258 | 2^250 - 2^50 */ | |
797 | f25519_mul(&t, &t2p50m1, &u); /* 259 | 2^250 - 1 */ | |
798 | SQRN(&u, &t, 5); /* 264 | 2^255 - 2^5 */ | |
799 | f25519_mul(z, &u, &t11); /* 265 | 2^255 - 21 */ | |
800 | ||
801 | #undef SQRN | |
802 | } | |
803 | ||
0bcb8184 MW |
804 | #ifndef F25519_TRIM_X25519 |
805 | ||
b7a5ecfc MW |
806 | /* --- @f25519_quosqrt@ --- * |
807 | * | |
808 | * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@) | |
809 | * @const f25519 *x, *y@ = two operands | |
810 | * | |
811 | * Returns: Zero if successful, @-1@ if %$x/y$% is not a square. | |
812 | * | |
813 | * Use: Stores in @z@ the one of the square roots %$\pm\sqrt{x/y}$%. | |
814 | * If %$x = y = 0% then the result is zero; if %$y = 0$% but %$x | |
815 | * \ne 0$% then the operation fails. If you wanted a specific | |
816 | * square root then you'll have to pick it yourself. | |
817 | */ | |
818 | ||
819 | static const piece sqrtm1_pieces[NPIECE] = { | |
820 | #if F25519_IMPL == 26 | |
821 | -32595792, -7943725, 9377950, 3500415, 12389472, | |
822 | -272473, -25146209, -2005654, 326686, 11406482 | |
823 | #elif F25519_IMPL == 10 | |
824 | 176, -88, 161, 157, -485, -196, -231, -220, -416, | |
825 | -169, -255, 50, 189, -89, -266, -32, 202, -511, | |
826 | 423, 357, 248, -249, 80, 288, 50, 174 | |
827 | #endif | |
828 | }; | |
829 | #define SQRTM1 ((const f25519 *)sqrtm1_pieces) | |
830 | ||
831 | int f25519_quosqrt(f25519 *z, const f25519 *x, const f25519 *y) | |
832 | { | |
833 | f25519 t, u, w, beta, xy3, t2p50m1; | |
834 | octet xb[32], b0[32], b1[32]; | |
835 | int32 rc = -1; | |
836 | mask32 m; | |
837 | unsigned i; | |
838 | ||
839 | #define SQRN(z, x, n) do { \ | |
840 | f25519_sqr((z), (x)); \ | |
841 | for (i = 1; i < (n); i++) f25519_sqr((z), (z)); \ | |
842 | } while (0) | |
843 | ||
844 | /* This is a bit tricky; the algorithm is from Bernstein, Duif, Lange, | |
845 | * Schwabe, and Yang, `High-speed high-security signatures', 2011-09-26, | |
846 | * https://ed25519.cr.yp.to/ed25519-20110926.pdf. | |
847 | * | |
848 | * First of all, a complicated exponentation. The addition chain here is | |
849 | * mine. We start with some preliminary values. | |
850 | */ /* step | value */ | |
851 | SQRN(&u, y, 1); /* 1 | 0, 2 */ | |
852 | f25519_mul(&t, &u, y); /* 2 | 0, 3 */ | |
853 | f25519_mul(&xy3, &t, x); /* 3 | 1, 3 */ | |
854 | SQRN(&u, &u, 1); /* 4 | 0, 4 */ | |
855 | f25519_mul(&w, &u, &xy3); /* 5 | 1, 7 */ | |
856 | ||
857 | /* And now we calculate w^((p - 5)/8) = w^(252 - 3). */ | |
858 | SQRN(&u, &w, 1); /* 6 | 2 */ | |
859 | f25519_mul(&t, &w, &u); /* 7 | 3 */ | |
860 | SQRN(&u, &t, 1); /* 8 | 6 */ | |
861 | f25519_mul(&t, &u, &w); /* 9 | 7 */ | |
862 | SQRN(&u, &t, 3); /* 12 | 56 */ | |
863 | f25519_mul(&t, &t, &u); /* 13 | 63 = 2^6 - 1 */ | |
864 | SQRN(&u, &t, 6); /* 19 | 2^12 - 2^6 */ | |
865 | f25519_mul(&t, &t, &u); /* 20 | 2^12 - 1 */ | |
866 | SQRN(&u, &t, 12); /* 32 | 2^24 - 2^12 */ | |
867 | f25519_mul(&t, &t, &u); /* 33 | 2^24 - 1 */ | |
868 | SQRN(&u, &t, 1); /* 34 | 2^25 - 2 */ | |
869 | f25519_mul(&t, &u, &w); /* 35 | 2^25 - 1 */ | |
870 | SQRN(&u, &t, 25); /* 60 | 2^50 - 2^25 */ | |
871 | f25519_mul(&t2p50m1, &t, &u); /* 61 | 2^50 - 1 */ | |
872 | SQRN(&u, &t2p50m1, 50); /* 111 | 2^100 - 2^50 */ | |
873 | f25519_mul(&t, &t2p50m1, &u); /* 112 | 2^100 - 1 */ | |
874 | SQRN(&u, &t, 100); /* 212 | 2^200 - 2^100 */ | |
875 | f25519_mul(&t, &t, &u); /* 213 | 2^200 - 1 */ | |
876 | SQRN(&u, &t, 50); /* 263 | 2^250 - 2^50 */ | |
877 | f25519_mul(&t, &t2p50m1, &u); /* 264 | 2^250 - 1 */ | |
878 | SQRN(&u, &t, 2); /* 266 | 2^252 - 4 */ | |
879 | f25519_mul(&t, &u, &w); /* 267 | 2^252 - 3 */ | |
880 | ||
881 | /* And finally... */ | |
882 | f25519_mul(&beta, &t, &xy3); /* 268 | ... */ | |
883 | ||
884 | /* Now we have beta = (x y^3) (x y^7)^((p - 5)/8) = (x/y)^((p + 3)/8), and | |
885 | * we're ready to finish the computation. Suppose that alpha^2 = u/w. | |
886 | * Then beta^4 = (x/y)^((p + 3)/2) = alpha^(p + 3) = alpha^4 = (x/y)^2, so | |
887 | * we have beta^2 = ±x/y. If y beta^2 = x then beta is the one we wanted; | |
888 | * if -y beta^2 = x, then we want beta sqrt(-1), which we already know. Of | |
889 | * course, it might not match either, in which case we fail. | |
890 | * | |
891 | * The easiest way to compare is to encode. This isn't as wasteful as it | |
892 | * sounds: the hard part is normalizing the representations, which we have | |
893 | * to do anyway. | |
894 | */ | |
895 | f25519_sqr(&t, &beta); | |
896 | f25519_mul(&t, &t, y); | |
897 | f25519_neg(&u, &t); | |
898 | f25519_store(xb, x); | |
899 | f25519_store(b0, &t); | |
900 | f25519_store(b1, &u); | |
901 | f25519_mul(&u, &beta, SQRTM1); | |
902 | ||
903 | m = -ct_memeq(b0, xb, 32); | |
904 | rc = PICK2(0, rc, m); | |
905 | f25519_pick2(z, &beta, &u, m); | |
906 | m = -ct_memeq(b1, xb, 32); | |
907 | rc = PICK2(0, rc, m); | |
908 | ||
909 | /* And we're done. */ | |
910 | return (rc); | |
911 | } | |
912 | ||
b7a5ecfc MW |
913 | #endif |
914 | ||
915 | /*----- That's all, folks -------------------------------------------------*/ |