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8a7654a6 MW |
1 | /* -*-c-*- |
2 | * | |
3 | * Arithmetic modulo 2^255 - 19 | |
4 | * | |
5 | * (c) 2017 Straylight/Edgeware | |
6 | */ | |
7 | ||
8 | /*----- Licensing notice --------------------------------------------------* | |
9 | * | |
10 | * This file is part of secnet. | |
11 | * See README for full list of copyright holders. | |
12 | * | |
13 | * secnet is free software; you can redistribute it and/or modify it | |
14 | * under the terms of the GNU General Public License as published by | |
15 | * the Free Software Foundation; either version d of the License, or | |
16 | * (at your option) any later version. | |
17 | * | |
18 | * secnet is distributed in the hope that it will be useful, but | |
19 | * WITHOUT ANY WARRANTY; without even the implied warranty of | |
20 | * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
21 | * General Public License for more details. | |
22 | * | |
23 | * You should have received a copy of the GNU General Public License | |
24 | * version 3 along with secnet; if not, see | |
25 | * https://www.gnu.org/licenses/gpl.html. | |
26 | * | |
27 | * This file was originally part of Catacomb, but has been automatically | |
28 | * modified for incorporation into secnet: see `import-catacomb-crypto' | |
29 | * for details. | |
30 | * | |
31 | * Catacomb is free software; you can redistribute it and/or modify | |
32 | * it under the terms of the GNU Library General Public License as | |
33 | * published by the Free Software Foundation; either version 2 of the | |
34 | * License, or (at your option) any later version. | |
35 | * | |
36 | * Catacomb is distributed in the hope that it will be useful, | |
37 | * but WITHOUT ANY WARRANTY; without even the implied warranty of | |
38 | * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the | |
39 | * GNU Library General Public License for more details. | |
40 | * | |
41 | * You should have received a copy of the GNU Library General Public | |
42 | * License along with Catacomb; if not, write to the Free | |
43 | * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, | |
44 | * MA 02111-1307, USA. | |
45 | */ | |
46 | ||
47 | /*----- Header files ------------------------------------------------------*/ | |
48 | ||
49 | #include "f25519.h" | |
50 | ||
51 | /*----- Basic setup -------------------------------------------------------*/ | |
52 | ||
a1a6042e MW |
53 | typedef f25519_piece piece; |
54 | ||
8a7654a6 MW |
55 | /* Elements x of GF(2^255 - 19) are represented by ten signed integers x_i: x |
56 | * = SUM_{0<=i<10} x_i 2^ceil(51i/2), mostly following Bernstein's original | |
57 | * paper. | |
58 | */ | |
59 | ||
a1a6042e | 60 | typedef int64 dblpiece; |
8a7654a6 MW |
61 | typedef uint32 upiece; typedef uint64 udblpiece; |
62 | #define P p26 | |
63 | #define PIECEWD(i) ((i)%2 ? 25 : 26) | |
64 | #define NPIECE 10 | |
65 | ||
66 | #define M26 0x03ffffffu | |
67 | #define M25 0x01ffffffu | |
8a7654a6 MW |
68 | #define B25 0x02000000u |
69 | #define B24 0x01000000u | |
70 | ||
71 | #define PIECES(v) v##0, v##1, v##2, v##3, v##4, v##5, v##6, v##7, v##8, v##9 | |
72 | #define FETCH(v, w) do { \ | |
73 | v##0 = (w)->P[0]; v##1 = (w)->P[1]; \ | |
74 | v##2 = (w)->P[2]; v##3 = (w)->P[3]; \ | |
75 | v##4 = (w)->P[4]; v##5 = (w)->P[5]; \ | |
76 | v##6 = (w)->P[6]; v##7 = (w)->P[7]; \ | |
77 | v##8 = (w)->P[8]; v##9 = (w)->P[9]; \ | |
78 | } while (0) | |
79 | #define STASH(w, v) do { \ | |
80 | (w)->P[0] = v##0; (w)->P[1] = v##1; \ | |
81 | (w)->P[2] = v##2; (w)->P[3] = v##3; \ | |
82 | (w)->P[4] = v##4; (w)->P[5] = v##5; \ | |
83 | (w)->P[6] = v##6; (w)->P[7] = v##7; \ | |
84 | (w)->P[8] = v##8; (w)->P[9] = v##9; \ | |
85 | } while (0) | |
86 | ||
87 | /*----- Debugging machinery -----------------------------------------------*/ | |
88 | ||
89 | #if defined(F25519_DEBUG) | |
90 | ||
91 | #include <stdio.h> | |
92 | ||
93 | #include "mp.h" | |
94 | #include "mptext.h" | |
95 | ||
96 | static mp *get_2p255m91(void) | |
97 | { | |
98 | mpw w19 = 19; | |
99 | mp *p = MP_NEW, m19; | |
100 | ||
101 | p = mp_setbit(p, MP_ZERO, 255); | |
102 | mp_build(&m19, &w19, &w19 + 1); | |
103 | p = mp_sub(p, p, &m19); | |
104 | return (p); | |
105 | } | |
106 | ||
107 | DEF_FDUMP(fdump, piece, PIECEWD, NPIECE, 32, get_2p255m91()) | |
108 | ||
109 | #endif | |
110 | ||
111 | /*----- Loading and storing -----------------------------------------------*/ | |
112 | ||
113 | /* --- @f25519_load@ --- * | |
114 | * | |
115 | * Arguments: @f25519 *z@ = where to store the result | |
116 | * @const octet xv[32]@ = source to read | |
117 | * | |
118 | * Returns: --- | |
119 | * | |
120 | * Use: Reads an element of %$\gf{2^{255} - 19}$% in external | |
121 | * representation from @xv@ and stores it in @z@. | |
122 | * | |
123 | * External representation is little-endian base-256. Some | |
124 | * elements have multiple encodings, which are not produced by | |
125 | * correct software; use of noncanonical encodings is not an | |
126 | * error, and toleration of them is considered a performance | |
127 | * feature. | |
128 | */ | |
129 | ||
130 | void f25519_load(f25519 *z, const octet xv[32]) | |
131 | { | |
132 | ||
133 | uint32 xw0 = LOAD32_L(xv + 0), xw1 = LOAD32_L(xv + 4), | |
134 | xw2 = LOAD32_L(xv + 8), xw3 = LOAD32_L(xv + 12), | |
135 | xw4 = LOAD32_L(xv + 16), xw5 = LOAD32_L(xv + 20), | |
136 | xw6 = LOAD32_L(xv + 24), xw7 = LOAD32_L(xv + 28); | |
137 | piece PIECES(x), b, c; | |
138 | ||
139 | /* First, split the 32-bit words into the irregularly-sized pieces we need | |
140 | * for the field representation. These pieces are all positive. We'll do | |
141 | * the sign correction afterwards. | |
142 | * | |
143 | * It may be that the top bit of the input is set. Avoid trouble by | |
144 | * folding that back round into the bottom piece of the representation. | |
145 | * | |
146 | * Here, we briefly have 0 <= x_0 < 2^26 + 19, but will resolve this later. | |
147 | * Otherwise, we have 0 <= x_{2i} < 2^26, and 0 <= x_{2i+1} < 2^25. | |
148 | */ | |
149 | x0 = ((xw0 << 0)&0x03ffffff) + 19*((xw7 >> 31)&0x00000001); | |
150 | x1 = ((xw1 << 6)&0x01ffffc0) | ((xw0 >> 26)&0x0000003f); | |
151 | x2 = ((xw2 << 13)&0x03ffe000) | ((xw1 >> 19)&0x00001fff); | |
152 | x3 = ((xw3 << 19)&0x01f80000) | ((xw2 >> 13)&0x0007ffff); | |
153 | x4 = ((xw3 >> 6)&0x03ffffff); | |
154 | x5 = (xw4 << 0)&0x01ffffff; | |
155 | x6 = ((xw5 << 7)&0x03ffff80) | ((xw4 >> 25)&0x0000007f); | |
156 | x7 = ((xw6 << 13)&0x01ffe000) | ((xw5 >> 19)&0x00001fff); | |
157 | x8 = ((xw7 << 20)&0x03f00000) | ((xw6 >> 12)&0x000fffff); | |
158 | x9 = ((xw7 >> 6)&0x01ffffff); | |
159 | ||
160 | /* Next, we convert these pieces into a roughly balanced signed | |
161 | * representation. For each piece, check to see if its top bit is set. If | |
162 | * it is, then lend a bit to the next piece over. For x_9, this needs to | |
163 | * be carried around, which is a little fiddly. In particular, we delay | |
164 | * the carry until after all of the pieces have been balanced. If we don't | |
165 | * do this, then we have to do a more expensive test for nonzeroness to | |
166 | * decide whether to lend a bit leftwards rather than just testing a single | |
167 | * bit. | |
168 | * | |
169 | * This fixes up the anomalous size of x_0: the loan of a bit becomes an | |
170 | * actual carry if x_0 >= 2^26. By the end, then, we have: | |
171 | * | |
172 | * { 2^25 if i even | |
173 | * |x_i| <= { | |
174 | * { 2^24 if i odd | |
175 | * | |
176 | * Note that we don't try for a canonical representation here: both upper | |
177 | * and lower bounds are achievable. | |
178 | * | |
179 | * All of the x_i at this point are positive, so we don't need to do | |
180 | * anything wierd when masking them. | |
181 | */ | |
182 | b = x9&B24; c = 19&((b >> 19) - (b >> 24)); x9 -= b << 1; | |
183 | b = x8&B25; x9 += b >> 25; x8 -= b << 1; | |
184 | b = x7&B24; x8 += b >> 24; x7 -= b << 1; | |
185 | b = x6&B25; x7 += b >> 25; x6 -= b << 1; | |
186 | b = x5&B24; x6 += b >> 24; x5 -= b << 1; | |
187 | b = x4&B25; x5 += b >> 25; x4 -= b << 1; | |
188 | b = x3&B24; x4 += b >> 24; x3 -= b << 1; | |
189 | b = x2&B25; x3 += b >> 25; x2 -= b << 1; | |
190 | b = x1&B24; x2 += b >> 24; x1 -= b << 1; | |
191 | b = x0&B25; x1 += (b >> 25) + (x0 >> 26); x0 = (x0&M26) - (b << 1); | |
192 | x0 += c; | |
193 | ||
194 | /* And with that, we're done. */ | |
195 | STASH(z, x); | |
196 | } | |
197 | ||
198 | /* --- @f25519_store@ --- * | |
199 | * | |
200 | * Arguments: @octet zv[32]@ = where to write the result | |
201 | * @const f25519 *x@ = the field element to write | |
202 | * | |
203 | * Returns: --- | |
204 | * | |
205 | * Use: Stores a field element in the given octet vector in external | |
206 | * representation. A canonical encoding is always stored, so, | |
207 | * in particular, the top bit of @xv[31]@ is always left clear. | |
208 | */ | |
209 | ||
210 | void f25519_store(octet zv[32], const f25519 *x) | |
211 | { | |
212 | ||
213 | piece PIECES(x), PIECES(y), c, d; | |
214 | uint32 zw0, zw1, zw2, zw3, zw4, zw5, zw6, zw7; | |
215 | mask32 m; | |
216 | ||
217 | FETCH(x, x); | |
218 | ||
219 | /* First, propagate the carries throughout the pieces. By the end of this, | |
220 | * we'll have all of the pieces canonically sized and positive, and maybe | |
221 | * there'll be (signed) carry out. The carry c is in { -1, 0, +1 }, and | |
222 | * the remaining value will be in the half-open interval [0, 2^255). The | |
223 | * whole represented value is then x + 2^255 c. | |
224 | * | |
225 | * It's worth paying careful attention to the bounds. We assume that we | |
226 | * start out with |x_i| <= 2^30. We start by cutting off and reducing the | |
227 | * carry c_9 from the topmost piece, x_9. This leaves 0 <= x_9 < 2^25; and | |
228 | * we'll have |c_9| <= 2^5. We multiply this by 19 and we'll add it onto | |
229 | * x_0 and propagate the carries: but what bounds can we calculate on x | |
230 | * before this? | |
231 | * | |
232 | * Let o_i = floor(51 i/2). We have X_i = SUM_{0<=j<i} x_j 2^{o_i}, so | |
233 | * x = X_10. We see, inductively, that |X_i| < 2^{31+o_{i-1}}: X_0 = 0; | |
234 | * |x_i| <= 2^30; and |X_{i+1}| = |X_i + x_i 2^{o_i}| <= |X_i| + 2^{30+o_i} | |
235 | * < 2^{31+o_i}. Then x = X_9 + 2^230 x_9, and we have better bounds for | |
236 | * x_9, so | |
237 | * | |
238 | * -2^235 < x + 19 c_9 < 2^255 + 2^235 | |
239 | * | |
240 | * Here, the x_i are signed, so we must be cautious about bithacking them. | |
241 | */ | |
242 | c = ASR(piece, x9, 25); x9 = (upiece)x9&M25; | |
243 | x0 += 19*c; c = ASR(piece, x0, 26); x0 = (upiece)x0&M26; | |
244 | x1 += c; c = ASR(piece, x1, 25); x1 = (upiece)x1&M25; | |
245 | x2 += c; c = ASR(piece, x2, 26); x2 = (upiece)x2&M26; | |
246 | x3 += c; c = ASR(piece, x3, 25); x3 = (upiece)x3&M25; | |
247 | x4 += c; c = ASR(piece, x4, 26); x4 = (upiece)x4&M26; | |
248 | x5 += c; c = ASR(piece, x5, 25); x5 = (upiece)x5&M25; | |
249 | x6 += c; c = ASR(piece, x6, 26); x6 = (upiece)x6&M26; | |
250 | x7 += c; c = ASR(piece, x7, 25); x7 = (upiece)x7&M25; | |
251 | x8 += c; c = ASR(piece, x8, 26); x8 = (upiece)x8&M26; | |
252 | x9 += c; c = ASR(piece, x9, 25); x9 = (upiece)x9&M25; | |
253 | ||
254 | /* Now we have a slightly fiddly job to do. If c = +1, or if c = 0 and | |
255 | * x >= 2^255 - 19, then we should subtract 2^255 - 19 from the whole | |
256 | * value; if c = -1 then we should add 2^255 - 19; and otherwise we should | |
257 | * do nothing. | |
258 | * | |
259 | * But conditional behaviour is bad, m'kay. So here's what we do instead. | |
260 | * | |
261 | * The first job is to sort out what we wanted to do. If c = -1 then we | |
262 | * want to (a) invert the constant addend and (b) feed in a carry-in; | |
263 | * otherwise, we don't. | |
264 | */ | |
265 | m = SIGN(c); | |
266 | d = m&1; | |
267 | ||
268 | /* Now do the addition/subtraction. Remember that all of the x_i are | |
269 | * nonnegative, so shifting and masking are safe and easy. | |
270 | */ | |
271 | d += x0 + (19 ^ (M26&m)); y0 = d&M26; d >>= 26; | |
272 | d += x1 + (M25&m); y1 = d&M25; d >>= 25; | |
273 | d += x2 + (M26&m); y2 = d&M26; d >>= 26; | |
274 | d += x3 + (M25&m); y3 = d&M25; d >>= 25; | |
275 | d += x4 + (M26&m); y4 = d&M26; d >>= 26; | |
276 | d += x5 + (M25&m); y5 = d&M25; d >>= 25; | |
277 | d += x6 + (M26&m); y6 = d&M26; d >>= 26; | |
278 | d += x7 + (M25&m); y7 = d&M25; d >>= 25; | |
279 | d += x8 + (M26&m); y8 = d&M26; d >>= 26; | |
280 | d += x9 + (M25&m); y9 = d&M25; d >>= 25; | |
281 | ||
282 | /* The final carry-out is in d; since we only did addition, and the x_i are | |
283 | * nonnegative, then d is in { 0, 1 }. We want to keep y, rather than x, | |
284 | * if (a) c /= 0 (in which case we know that the old value was | |
285 | * unsatisfactory), or (b) if d = 1 (in which case, if c = 0, we know that | |
286 | * the subtraction didn't cause a borrow, so we must be in the case where | |
287 | * 2^255 - 19 <= x < 2^255). | |
288 | */ | |
289 | m = NONZEROP(c) | ~NONZEROP(d - 1); | |
290 | x0 = (y0&m) | (x0&~m); x1 = (y1&m) | (x1&~m); | |
291 | x2 = (y2&m) | (x2&~m); x3 = (y3&m) | (x3&~m); | |
292 | x4 = (y4&m) | (x4&~m); x5 = (y5&m) | (x5&~m); | |
293 | x6 = (y6&m) | (x6&~m); x7 = (y7&m) | (x7&~m); | |
294 | x8 = (y8&m) | (x8&~m); x9 = (y9&m) | (x9&~m); | |
295 | ||
296 | /* Extract 32-bit words from the value. */ | |
297 | zw0 = ((x0 >> 0)&0x03ffffff) | (((uint32)x1 << 26)&0xfc000000); | |
298 | zw1 = ((x1 >> 6)&0x0007ffff) | (((uint32)x2 << 19)&0xfff80000); | |
299 | zw2 = ((x2 >> 13)&0x00001fff) | (((uint32)x3 << 13)&0xffffe000); | |
300 | zw3 = ((x3 >> 19)&0x0000003f) | (((uint32)x4 << 6)&0xffffffc0); | |
301 | zw4 = ((x5 >> 0)&0x01ffffff) | (((uint32)x6 << 25)&0xfe000000); | |
302 | zw5 = ((x6 >> 7)&0x0007ffff) | (((uint32)x7 << 19)&0xfff80000); | |
303 | zw6 = ((x7 >> 13)&0x00000fff) | (((uint32)x8 << 12)&0xfffff000); | |
304 | zw7 = ((x8 >> 20)&0x0000003f) | (((uint32)x9 << 6)&0x7fffffc0); | |
305 | ||
306 | /* Store the result as an octet string. */ | |
307 | STORE32_L(zv + 0, zw0); STORE32_L(zv + 4, zw1); | |
308 | STORE32_L(zv + 8, zw2); STORE32_L(zv + 12, zw3); | |
309 | STORE32_L(zv + 16, zw4); STORE32_L(zv + 20, zw5); | |
310 | STORE32_L(zv + 24, zw6); STORE32_L(zv + 28, zw7); | |
311 | } | |
312 | ||
313 | /* --- @f25519_set@ --- * | |
314 | * | |
315 | * Arguments: @f25519 *z@ = where to write the result | |
316 | * @int a@ = a small-ish constant | |
317 | * | |
318 | * Returns: --- | |
319 | * | |
320 | * Use: Sets @z@ to equal @a@. | |
321 | */ | |
322 | ||
323 | void f25519_set(f25519 *x, int a) | |
324 | { | |
325 | unsigned i; | |
326 | ||
327 | x->P[0] = a; | |
328 | for (i = 1; i < NPIECE; i++) x->P[i] = 0; | |
329 | } | |
330 | ||
331 | /*----- Basic arithmetic --------------------------------------------------*/ | |
332 | ||
333 | /* --- @f25519_add@ --- * | |
334 | * | |
335 | * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@) | |
336 | * @const f25519 *x, *y@ = two operands | |
337 | * | |
338 | * Returns: --- | |
339 | * | |
340 | * Use: Set @z@ to the sum %$x + y$%. | |
341 | */ | |
342 | ||
343 | void f25519_add(f25519 *z, const f25519 *x, const f25519 *y) | |
344 | { | |
345 | z->P[0] = x->P[0] + y->P[0]; z->P[1] = x->P[1] + y->P[1]; | |
346 | z->P[2] = x->P[2] + y->P[2]; z->P[3] = x->P[3] + y->P[3]; | |
347 | z->P[4] = x->P[4] + y->P[4]; z->P[5] = x->P[5] + y->P[5]; | |
348 | z->P[6] = x->P[6] + y->P[6]; z->P[7] = x->P[7] + y->P[7]; | |
349 | z->P[8] = x->P[8] + y->P[8]; z->P[9] = x->P[9] + y->P[9]; | |
350 | } | |
351 | ||
352 | /* --- @f25519_sub@ --- * | |
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 difference %$x - y$%. | |
360 | */ | |
361 | ||
362 | void f25519_sub(f25519 *z, const f25519 *x, const f25519 *y) | |
363 | { | |
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]; | |
369 | } | |
370 | ||
371 | /* --- @f25519_neg@ --- * | |
372 | * | |
373 | * Arguments: @f25519 *z@ = where to put the result (may alias @x@) | |
374 | * @const f25519 *x@ = an operand | |
375 | * | |
376 | * Returns: --- | |
377 | * | |
378 | * Use: Set @z = -x@. | |
379 | */ | |
380 | ||
381 | void f25519_neg(f25519 *z, const f25519 *x) | |
382 | { | |
383 | z->P[0] = -x->P[0]; z->P[1] = -x->P[1]; | |
384 | z->P[2] = -x->P[2]; z->P[3] = -x->P[3]; | |
385 | z->P[4] = -x->P[4]; z->P[5] = -x->P[5]; | |
386 | z->P[6] = -x->P[6]; z->P[7] = -x->P[7]; | |
387 | z->P[8] = -x->P[8]; z->P[9] = -x->P[9]; | |
388 | } | |
389 | ||
390 | /*----- Constant-time utilities -------------------------------------------*/ | |
391 | ||
392 | /* --- @f25519_pick2@ --- * | |
393 | * | |
394 | * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@) | |
395 | * @const f25519 *x, *y@ = two operands | |
396 | * @uint32 m@ = a mask | |
397 | * | |
398 | * Returns: --- | |
399 | * | |
400 | * Use: If @m@ is zero, set @z = y@; if @m@ is all-bits-set, then set | |
401 | * @z = x@. If @m@ has some other value, then scramble @z@ in | |
402 | * an unhelpful way. | |
403 | */ | |
404 | ||
405 | void f25519_pick2(f25519 *z, const f25519 *x, const f25519 *y, uint32 m) | |
406 | { | |
407 | mask32 mm = FIX_MASK32(m); | |
408 | ||
409 | z->P[0] = PICK2(x->P[0], y->P[0], mm); | |
410 | z->P[1] = PICK2(x->P[1], y->P[1], mm); | |
411 | z->P[2] = PICK2(x->P[2], y->P[2], mm); | |
412 | z->P[3] = PICK2(x->P[3], y->P[3], mm); | |
413 | z->P[4] = PICK2(x->P[4], y->P[4], mm); | |
414 | z->P[5] = PICK2(x->P[5], y->P[5], mm); | |
415 | z->P[6] = PICK2(x->P[6], y->P[6], mm); | |
416 | z->P[7] = PICK2(x->P[7], y->P[7], mm); | |
417 | z->P[8] = PICK2(x->P[8], y->P[8], mm); | |
418 | z->P[9] = PICK2(x->P[9], y->P[9], mm); | |
419 | } | |
420 | ||
421 | /* --- @f25519_pickn@ --- * | |
422 | * | |
423 | * Arguments: @f25519 *z@ = where to put the result | |
424 | * @const f25519 *v@ = a table of entries | |
425 | * @size_t n@ = the number of entries in @v@ | |
426 | * @size_t i@ = an index | |
427 | * | |
428 | * Returns: --- | |
429 | * | |
430 | * Use: If @0 <= i < n < 32@ then set @z = v[i]@. If @n >= 32@ then | |
431 | * do something unhelpful; otherwise, if @i >= n@ then set @z@ | |
432 | * to zero. | |
433 | */ | |
434 | ||
435 | void f25519_pickn(f25519 *z, const f25519 *v, size_t n, size_t i) | |
436 | { | |
437 | uint32 b = (uint32)1 << (31 - i); | |
438 | mask32 m; | |
439 | ||
440 | z->P[0] = z->P[1] = z->P[2] = z->P[3] = z->P[4] = | |
441 | z->P[5] = z->P[6] = z->P[7] = z->P[8] = z->P[9] = 0; | |
442 | while (n--) { | |
443 | m = SIGN(b); | |
444 | CONDPICK(z->P[0], v->P[0], m); | |
445 | CONDPICK(z->P[1], v->P[1], m); | |
446 | CONDPICK(z->P[2], v->P[2], m); | |
447 | CONDPICK(z->P[3], v->P[3], m); | |
448 | CONDPICK(z->P[4], v->P[4], m); | |
449 | CONDPICK(z->P[5], v->P[5], m); | |
450 | CONDPICK(z->P[6], v->P[6], m); | |
451 | CONDPICK(z->P[7], v->P[7], m); | |
452 | CONDPICK(z->P[8], v->P[8], m); | |
453 | CONDPICK(z->P[9], v->P[9], m); | |
454 | v++; b <<= 1; | |
455 | } | |
456 | } | |
457 | ||
458 | /* --- @f25519_condswap@ --- * | |
459 | * | |
460 | * Arguments: @f25519 *x, *y@ = two operands | |
461 | * @uint32 m@ = a mask | |
462 | * | |
463 | * Returns: --- | |
464 | * | |
465 | * Use: If @m@ is zero, do nothing; if @m@ is all-bits-set, then | |
466 | * exchange @x@ and @y@. If @m@ has some other value, then | |
467 | * scramble @x@ and @y@ in an unhelpful way. | |
468 | */ | |
469 | ||
470 | void f25519_condswap(f25519 *x, f25519 *y, uint32 m) | |
471 | { | |
472 | mask32 mm = FIX_MASK32(m); | |
473 | ||
474 | CONDSWAP(x->P[0], y->P[0], mm); | |
475 | CONDSWAP(x->P[1], y->P[1], mm); | |
476 | CONDSWAP(x->P[2], y->P[2], mm); | |
477 | CONDSWAP(x->P[3], y->P[3], mm); | |
478 | CONDSWAP(x->P[4], y->P[4], mm); | |
479 | CONDSWAP(x->P[5], y->P[5], mm); | |
480 | CONDSWAP(x->P[6], y->P[6], mm); | |
481 | CONDSWAP(x->P[7], y->P[7], mm); | |
482 | CONDSWAP(x->P[8], y->P[8], mm); | |
483 | CONDSWAP(x->P[9], y->P[9], mm); | |
484 | } | |
485 | ||
486 | /* --- @f25519_condneg@ --- * | |
487 | * | |
488 | * Arguments: @f25519 *z@ = where to put the result (may alias @x@) | |
489 | * @const f25519 *x@ = an operand | |
490 | * @uint32 m@ = a mask | |
491 | * | |
492 | * Returns: --- | |
493 | * | |
494 | * Use: If @m@ is zero, set @z = x@; if @m@ is all-bits-set, then set | |
495 | * @z = -x@. If @m@ has some other value then scramble @z@ in | |
496 | * an unhelpful way. | |
497 | */ | |
498 | ||
499 | void f25519_condneg(f25519 *z, const f25519 *x, uint32 m) | |
500 | { | |
501 | mask32 m_xor = FIX_MASK32(m); | |
502 | piece m_add = m&1; | |
503 | # define CONDNEG(x) (((x) ^ m_xor) + m_add) | |
504 | ||
505 | z->P[0] = CONDNEG(x->P[0]); | |
506 | z->P[1] = CONDNEG(x->P[1]); | |
507 | z->P[2] = CONDNEG(x->P[2]); | |
508 | z->P[3] = CONDNEG(x->P[3]); | |
509 | z->P[4] = CONDNEG(x->P[4]); | |
510 | z->P[5] = CONDNEG(x->P[5]); | |
511 | z->P[6] = CONDNEG(x->P[6]); | |
512 | z->P[7] = CONDNEG(x->P[7]); | |
513 | z->P[8] = CONDNEG(x->P[8]); | |
514 | z->P[9] = CONDNEG(x->P[9]); | |
515 | ||
516 | #undef CONDNEG | |
517 | } | |
518 | ||
519 | /*----- Multiplication ----------------------------------------------------*/ | |
520 | ||
521 | /* Let B = 2^63 - 1 be the largest value such that +B and -B can be | |
522 | * represented in a double-precision piece. On entry, it must be the case | |
523 | * that |X_i| <= M <= B - 2^25 for some M. If this is the case, then, on | |
524 | * exit, we will have |Z_i| <= 2^25 + 19 M/2^25. | |
525 | */ | |
526 | #define CARRYSTEP(z, x, m, b, f, xx, n) do { \ | |
527 | (z) = (dblpiece)((udblpiece)(x)&(m)) - (b) + \ | |
528 | (f)*ASR(dblpiece, (xx), (n)); \ | |
529 | } while (0) | |
530 | #define CARRY_REDUCE(z, x) do { \ | |
531 | dblpiece PIECES(_t); \ | |
532 | \ | |
533 | /* Bias the input pieces. This keeps the carries and so on centred \ | |
534 | * around zero rather than biased positive. \ | |
535 | */ \ | |
536 | _t0 = (x##0) + B25; _t1 = (x##1) + B24; \ | |
537 | _t2 = (x##2) + B25; _t3 = (x##3) + B24; \ | |
538 | _t4 = (x##4) + B25; _t5 = (x##5) + B24; \ | |
539 | _t6 = (x##6) + B25; _t7 = (x##7) + B24; \ | |
540 | _t8 = (x##8) + B25; _t9 = (x##9) + B24; \ | |
541 | \ | |
542 | /* Calculate the reduced pieces. Careful with the bithacking. */ \ | |
543 | CARRYSTEP(z##0, _t0, M26, B25, 19, _t9, 25); \ | |
544 | CARRYSTEP(z##1, _t1, M25, B24, 1, _t0, 26); \ | |
545 | CARRYSTEP(z##2, _t2, M26, B25, 1, _t1, 25); \ | |
546 | CARRYSTEP(z##3, _t3, M25, B24, 1, _t2, 26); \ | |
547 | CARRYSTEP(z##4, _t4, M26, B25, 1, _t3, 25); \ | |
548 | CARRYSTEP(z##5, _t5, M25, B24, 1, _t4, 26); \ | |
549 | CARRYSTEP(z##6, _t6, M26, B25, 1, _t5, 25); \ | |
550 | CARRYSTEP(z##7, _t7, M25, B24, 1, _t6, 26); \ | |
551 | CARRYSTEP(z##8, _t8, M26, B25, 1, _t7, 25); \ | |
552 | CARRYSTEP(z##9, _t9, M25, B24, 1, _t8, 26); \ | |
553 | } while (0) | |
554 | ||
555 | /* --- @f25519_mulconst@ --- * | |
556 | * | |
557 | * Arguments: @f25519 *z@ = where to put the result (may alias @x@) | |
558 | * @const f25519 *x@ = an operand | |
559 | * @long a@ = a small-ish constant; %$|a| < 2^{20}$%. | |
560 | * | |
561 | * Returns: --- | |
562 | * | |
563 | * Use: Set @z@ to the product %$a x$%. | |
564 | */ | |
565 | ||
566 | void f25519_mulconst(f25519 *z, const f25519 *x, long a) | |
567 | { | |
568 | ||
569 | piece PIECES(x); | |
570 | dblpiece PIECES(z), aa = a; | |
571 | ||
572 | FETCH(x, x); | |
573 | ||
574 | /* Suppose that |x_i| <= 2^27, and |a| <= 2^23. Then we'll have | |
575 | * |z_i| <= 2^50. | |
576 | */ | |
577 | z0 = aa*x0; z1 = aa*x1; z2 = aa*x2; z3 = aa*x3; z4 = aa*x4; | |
578 | z5 = aa*x5; z6 = aa*x6; z7 = aa*x7; z8 = aa*x8; z9 = aa*x9; | |
579 | ||
580 | /* Following `CARRY_REDUCE', we'll have |z_i| <= 2^26. */ | |
581 | CARRY_REDUCE(z, z); | |
582 | STASH(z, z); | |
583 | } | |
584 | ||
585 | /* --- @f25519_mul@ --- * | |
586 | * | |
587 | * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@) | |
588 | * @const f25519 *x, *y@ = two operands | |
589 | * | |
590 | * Returns: --- | |
591 | * | |
592 | * Use: Set @z@ to the product %$x y$%. | |
593 | */ | |
594 | ||
595 | void f25519_mul(f25519 *z, const f25519 *x, const f25519 *y) | |
596 | { | |
597 | ||
598 | piece PIECES(x), PIECES(y); | |
599 | dblpiece PIECES(z); | |
600 | unsigned i; | |
601 | ||
602 | FETCH(x, x); FETCH(y, y); | |
603 | ||
604 | /* Suppose that |x_i|, |y_i| <= 2^27. Then we'll have | |
605 | * | |
606 | * |z_0| <= 267*2^54 | |
607 | * |z_1| <= 154*2^54 | |
608 | * |z_2| <= 213*2^54 | |
609 | * |z_3| <= 118*2^54 | |
610 | * |z_4| <= 159*2^54 | |
611 | * |z_5| <= 82*2^54 | |
612 | * |z_6| <= 105*2^54 | |
613 | * |z_7| <= 46*2^54 | |
614 | * |z_8| <= 51*2^54 | |
615 | * |z_9| <= 10*2^54 | |
616 | * | |
617 | * all of which are less than 2^63 - 2^25. | |
618 | */ | |
619 | ||
620 | #define M(a, b) ((dblpiece)(a)*(b)) | |
621 | z0 = M(x0, y0) + | |
622 | 19*(M(x2, y8) + M(x4, y6) + M(x6, y4) + M(x8, y2)) + | |
623 | 38*(M(x1, y9) + M(x3, y7) + M(x5, y5) + M(x7, y3) + M(x9, y1)); | |
624 | z1 = M(x0, y1) + M(x1, y0) + | |
625 | 19*(M(x2, y9) + M(x3, y8) + M(x4, y7) + M(x5, y6) + | |
626 | M(x6, y5) + M(x7, y4) + M(x8, y3) + M(x9, y2)); | |
627 | z2 = M(x0, y2) + M(x2, y0) + | |
628 | 2* M(x1, y1) + | |
629 | 19*(M(x4, y8) + M(x6, y6) + M(x8, y4)) + | |
630 | 38*(M(x3, y9) + M(x5, y7) + M(x7, y5) + M(x9, y3)); | |
631 | z3 = M(x0, y3) + M(x1, y2) + M(x2, y1) + M(x3, y0) + | |
632 | 19*(M(x4, y9) + M(x5, y8) + M(x6, y7) + | |
633 | M(x7, y6) + M(x8, y5) + M(x9, y4)); | |
634 | z4 = M(x0, y4) + M(x2, y2) + M(x4, y0) + | |
635 | 2*(M(x1, y3) + M(x3, y1)) + | |
636 | 19*(M(x6, y8) + M(x8, y6)) + | |
637 | 38*(M(x5, y9) + M(x7, y7) + M(x9, y5)); | |
638 | z5 = M(x0, y5) + M(x1, y4) + M(x2, y3) + | |
639 | M(x3, y2) + M(x4, y1) + M(x5, y0) + | |
640 | 19*(M(x6, y9) + M(x7, y8) + M(x8, y7) + M(x9, y6)); | |
641 | z6 = M(x0, y6) + M(x2, y4) + M(x4, y2) + M(x6, y0) + | |
642 | 2*(M(x1, y5) + M(x3, y3) + M(x5, y1)) + | |
643 | 19* M(x8, y8) + | |
644 | 38*(M(x7, y9) + M(x9, y7)); | |
645 | z7 = M(x0, y7) + M(x1, y6) + M(x2, y5) + M(x3, y4) + | |
646 | M(x4, y3) + M(x5, y2) + M(x6, y1) + M(x7, y0) + | |
647 | 19*(M(x8, y9) + M(x9, y8)); | |
648 | z8 = M(x0, y8) + M(x2, y6) + M(x4, y4) + M(x6, y2) + M(x8, y0) + | |
649 | 2*(M(x1, y7) + M(x3, y5) + M(x5, y3) + M(x7, y1)) + | |
650 | 38* M(x9, y9); | |
651 | z9 = M(x0, y9) + M(x1, y8) + M(x2, y7) + M(x3, y6) + M(x4, y5) + | |
652 | M(x5, y4) + M(x6, y3) + M(x7, y2) + M(x8, y1) + M(x9, y0); | |
653 | #undef M | |
654 | ||
655 | /* From above, we have |z_i| <= 2^63 - 2^25. A pass of `CARRY_REDUCE' will | |
656 | * leave |z_i| <= 2^38 + 2^25; and a second pass will leave |z_i| <= 2^25 + | |
657 | * 2^13, which is comfortable for an addition prior to the next | |
658 | * multiplication. | |
659 | */ | |
660 | for (i = 0; i < 2; i++) CARRY_REDUCE(z, z); | |
661 | STASH(z, z); | |
662 | } | |
663 | ||
664 | /* --- @f25519_sqr@ --- * | |
665 | * | |
666 | * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@) | |
667 | * @const f25519 *x@ = an operand | |
668 | * | |
669 | * Returns: --- | |
670 | * | |
671 | * Use: Set @z@ to the square %$x^2$%. | |
672 | */ | |
673 | ||
674 | void f25519_sqr(f25519 *z, const f25519 *x) | |
675 | { | |
676 | ||
677 | piece PIECES(x); | |
678 | dblpiece PIECES(z); | |
679 | unsigned i; | |
680 | ||
681 | FETCH(x, x); | |
682 | ||
683 | /* See `f25519_mul' for bounds. */ | |
684 | ||
685 | #define M(a, b) ((dblpiece)(a)*(b)) | |
686 | z0 = M(x0, x0) + | |
687 | 38*(M(x2, x8) + M(x4, x6) + M(x5, x5)) + | |
688 | 76*(M(x1, x9) + M(x3, x7)); | |
689 | z1 = 2* M(x0, x1) + | |
690 | 38*(M(x2, x9) + M(x3, x8) + M(x4, x7) + M(x5, x6)); | |
691 | z2 = 2*(M(x0, x2) + M(x1, x1)) + | |
692 | 19* M(x6, x6) + | |
693 | 38* M(x4, x8) + | |
694 | 76*(M(x3, x9) + M(x5, x7)); | |
695 | z3 = 2*(M(x0, x3) + M(x1, x2)) + | |
696 | 38*(M(x4, x9) + M(x5, x8) + M(x6, x7)); | |
697 | z4 = M(x2, x2) + | |
698 | 2* M(x0, x4) + | |
699 | 4* M(x1, x3) + | |
700 | 38*(M(x6, x8) + M(x7, x7)) + | |
701 | 76* M(x5, x9); | |
702 | z5 = 2*(M(x0, x5) + M(x1, x4) + M(x2, x3)) + | |
703 | 38*(M(x6, x9) + M(x7, x8)); | |
704 | z6 = 2*(M(x0, x6) + M(x2, x4) + M(x3, x3)) + | |
705 | 4* M(x1, x5) + | |
706 | 19* M(x8, x8) + | |
707 | 76* M(x7, x9); | |
708 | z7 = 2*(M(x0, x7) + M(x1, x6) + M(x2, x5) + M(x3, x4)) + | |
709 | 38* M(x8, x9); | |
710 | z8 = M(x4, x4) + | |
711 | 2*(M(x0, x8) + M(x2, x6)) + | |
712 | 4*(M(x1, x7) + M(x3, x5)) + | |
713 | 38* M(x9, x9); | |
714 | z9 = 2*(M(x0, x9) + M(x1, x8) + M(x2, x7) + M(x3, x6) + M(x4, x5)); | |
715 | #undef M | |
716 | ||
717 | /* See `f25519_mul' for details. */ | |
718 | for (i = 0; i < 2; i++) CARRY_REDUCE(z, z); | |
719 | STASH(z, z); | |
720 | } | |
721 | ||
722 | /*----- More complicated things -------------------------------------------*/ | |
723 | ||
724 | /* --- @f25519_inv@ --- * | |
725 | * | |
726 | * Arguments: @f25519 *z@ = where to put the result (may alias @x@) | |
727 | * @const f25519 *x@ = an operand | |
728 | * | |
729 | * Returns: --- | |
730 | * | |
731 | * Use: Stores in @z@ the multiplicative inverse %$x^{-1}$%. If | |
732 | * %$x = 0$% then @z@ is set to zero. This is considered a | |
733 | * feature. | |
734 | */ | |
735 | ||
736 | void f25519_inv(f25519 *z, const f25519 *x) | |
737 | { | |
738 | f25519 t, u, t2, t11, t2p10m1, t2p50m1; | |
739 | unsigned i; | |
740 | ||
741 | #define SQRN(z, x, n) do { \ | |
742 | f25519_sqr((z), (x)); \ | |
743 | for (i = 1; i < (n); i++) f25519_sqr((z), (z)); \ | |
744 | } while (0) | |
745 | ||
746 | /* Calculate x^-1 = x^(p - 2) = x^(2^255 - 21), which also handles x = 0 as | |
747 | * intended. The addition chain here is from Bernstein's implementation; I | |
748 | * couldn't find a better one. | |
749 | */ /* step | value */ | |
750 | f25519_sqr(&t2, x); /* 1 | 2 */ | |
751 | SQRN(&u, &t2, 2); /* 3 | 8 */ | |
752 | f25519_mul(&t, &u, x); /* 4 | 9 */ | |
753 | f25519_mul(&t11, &t, &t2); /* 5 | 11 = 2^5 - 21 */ | |
754 | f25519_sqr(&u, &t11); /* 6 | 22 */ | |
755 | f25519_mul(&t, &t, &u); /* 7 | 31 = 2^5 - 1 */ | |
756 | SQRN(&u, &t, 5); /* 12 | 2^10 - 2^5 */ | |
757 | f25519_mul(&t2p10m1, &t, &u); /* 13 | 2^10 - 1 */ | |
758 | SQRN(&u, &t2p10m1, 10); /* 23 | 2^20 - 2^10 */ | |
759 | f25519_mul(&t, &t2p10m1, &u); /* 24 | 2^20 - 1 */ | |
760 | SQRN(&u, &t, 20); /* 44 | 2^40 - 2^20 */ | |
761 | f25519_mul(&t, &t, &u); /* 45 | 2^40 - 1 */ | |
762 | SQRN(&u, &t, 10); /* 55 | 2^50 - 2^10 */ | |
763 | f25519_mul(&t2p50m1, &t2p10m1, &u); /* 56 | 2^50 - 1 */ | |
764 | SQRN(&u, &t2p50m1, 50); /* 106 | 2^100 - 2^50 */ | |
765 | f25519_mul(&t, &t2p50m1, &u); /* 107 | 2^100 - 1 */ | |
766 | SQRN(&u, &t, 100); /* 207 | 2^200 - 2^100 */ | |
767 | f25519_mul(&t, &t, &u); /* 208 | 2^200 - 1 */ | |
768 | SQRN(&u, &t, 50); /* 258 | 2^250 - 2^50 */ | |
769 | f25519_mul(&t, &t2p50m1, &u); /* 259 | 2^250 - 1 */ | |
770 | SQRN(&u, &t, 5); /* 264 | 2^255 - 2^5 */ | |
771 | f25519_mul(z, &u, &t11); /* 265 | 2^255 - 21 */ | |
772 | ||
773 | #undef SQRN | |
774 | } | |
775 | ||
776 | /* --- @f25519_quosqrt@ --- * | |
777 | * | |
778 | * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@) | |
779 | * @const f25519 *x, *y@ = two operands | |
780 | * | |
781 | * Returns: Zero if successful, @-1@ if %$x/y$% is not a square. | |
782 | * | |
783 | * Use: Stores in @z@ the one of the square roots %$\pm\sqrt{x/y}$%. | |
784 | * If %$x = y = 0% then the result is zero; if %$y = 0$% but %$x | |
785 | * \ne 0$% then the operation fails. If you wanted a specific | |
786 | * square root then you'll have to pick it yourself. | |
787 | */ | |
788 | ||
789 | static const piece sqrtm1_pieces[NPIECE] = { | |
790 | -32595792, -7943725, 9377950, 3500415, 12389472, | |
791 | -272473, -25146209, -2005654, 326686, 11406482 | |
792 | }; | |
793 | #define SQRTM1 ((const f25519 *)sqrtm1_pieces) | |
794 | ||
795 | int f25519_quosqrt(f25519 *z, const f25519 *x, const f25519 *y) | |
796 | { | |
a1a6042e | 797 | f25519 t, u, v, w, t15; |
8a7654a6 MW |
798 | octet xb[32], b0[32], b1[32]; |
799 | int32 rc = -1; | |
800 | mask32 m; | |
801 | unsigned i; | |
802 | ||
803 | #define SQRN(z, x, n) do { \ | |
804 | f25519_sqr((z), (x)); \ | |
805 | for (i = 1; i < (n); i++) f25519_sqr((z), (z)); \ | |
806 | } while (0) | |
807 | ||
a1a6042e MW |
808 | /* This is a bit tricky; the algorithm is loosely based on Bernstein, Duif, |
809 | * Lange, Schwabe, and Yang, `High-speed high-security signatures', | |
810 | * 2011-09-26, https://ed25519.cr.yp.to/ed25519-20110926.pdf. | |
811 | */ | |
812 | f25519_mul(&v, x, y); | |
813 | ||
814 | /* Now for an addition chain. */ /* step | value */ | |
815 | f25519_sqr(&u, &v); /* 1 | 2 */ | |
816 | f25519_mul(&t, &u, &v); /* 2 | 3 */ | |
817 | SQRN(&u, &t, 2); /* 4 | 12 */ | |
818 | f25519_mul(&t15, &u, &t); /* 5 | 15 */ | |
819 | f25519_sqr(&u, &t15); /* 6 | 30 */ | |
820 | f25519_mul(&t, &u, &v); /* 7 | 31 = 2^5 - 1 */ | |
821 | SQRN(&u, &t, 5); /* 12 | 2^10 - 2^5 */ | |
822 | f25519_mul(&t, &u, &t); /* 13 | 2^10 - 1 */ | |
823 | SQRN(&u, &t, 10); /* 23 | 2^20 - 2^10 */ | |
824 | f25519_mul(&u, &u, &t); /* 24 | 2^20 - 1 */ | |
825 | SQRN(&u, &u, 10); /* 34 | 2^30 - 2^10 */ | |
826 | f25519_mul(&t, &u, &t); /* 35 | 2^30 - 1 */ | |
827 | f25519_sqr(&u, &t); /* 36 | 2^31 - 2 */ | |
828 | f25519_mul(&t, &u, &v); /* 37 | 2^31 - 1 */ | |
829 | SQRN(&u, &t, 31); /* 68 | 2^62 - 2^31 */ | |
830 | f25519_mul(&t, &u, &t); /* 69 | 2^62 - 1 */ | |
831 | SQRN(&u, &t, 62); /* 131 | 2^124 - 2^62 */ | |
832 | f25519_mul(&t, &u, &t); /* 132 | 2^124 - 1 */ | |
833 | SQRN(&u, &t, 124); /* 256 | 2^248 - 2^124 */ | |
834 | f25519_mul(&t, &u, &t); /* 257 | 2^248 - 1 */ | |
835 | f25519_sqr(&u, &t); /* 258 | 2^249 - 2 */ | |
836 | f25519_mul(&t, &u, &v); /* 259 | 2^249 - 1 */ | |
837 | SQRN(&t, &t, 3); /* 262 | 2^252 - 8 */ | |
838 | f25519_sqr(&u, &t); /* 263 | 2^253 - 16 */ | |
839 | f25519_mul(&t, &u, &t); /* 264 | 3*2^252 - 24 */ | |
840 | f25519_mul(&t, &t, &t15); /* 265 | 3*2^252 - 9 */ | |
841 | f25519_mul(&w, &t, &v); /* 266 | 3*2^252 - 8 */ | |
842 | ||
843 | /* Awesome. Now let me explain. Let v be a square in GF(p), and let w = | |
844 | * v^(3*2^252 - 8). In particular, let's consider | |
8a7654a6 | 845 | * |
a1a6042e MW |
846 | * v^2 w^4 = v^2 v^{3*2^254 - 32} = (v^{2^254 - 10})^3 |
847 | * | |
848 | * But 2^254 - 10 = ((2^255 - 19) - 1)/2 = (p - 1)/2. Since v is a square, | |
849 | * it has order dividing (p - 1)/2, and therefore v^2 w^4 = 1 and | |
850 | * | |
851 | * w^4 = 1/v^2 | |
852 | * | |
853 | * That in turn implies that w^2 = ±1/v. Now, recall that v = x y, and let | |
854 | * w' = w x. Then w'^2 = ±x^2/v = ±x/y. If y w'^2 = x then we set | |
855 | * z = w', since we have z^2 = x/y; otherwise let z = i w', where i^2 = -1, | |
856 | * so z^2 = -w^2 = x/y, and we're done. | |
8a7654a6 MW |
857 | * |
858 | * The easiest way to compare is to encode. This isn't as wasteful as it | |
859 | * sounds: the hard part is normalizing the representations, which we have | |
860 | * to do anyway. | |
861 | */ | |
a1a6042e MW |
862 | f25519_mul(&w, &w, x); |
863 | f25519_sqr(&t, &w); | |
8a7654a6 MW |
864 | f25519_mul(&t, &t, y); |
865 | f25519_neg(&u, &t); | |
866 | f25519_store(xb, x); | |
867 | f25519_store(b0, &t); | |
868 | f25519_store(b1, &u); | |
a1a6042e | 869 | f25519_mul(&u, &w, SQRTM1); |
8a7654a6 MW |
870 | |
871 | m = -consttime_memeq(b0, xb, 32); | |
872 | rc = PICK2(0, rc, m); | |
a1a6042e | 873 | f25519_pick2(z, &w, &u, m); |
8a7654a6 MW |
874 | m = -consttime_memeq(b1, xb, 32); |
875 | rc = PICK2(0, rc, m); | |
876 | ||
877 | /* And we're done. */ | |
878 | return (rc); | |
879 | } | |
880 | ||
881 | /*----- That's all, folks -------------------------------------------------*/ |