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math/Makefile.am, symm/Makefile.am: Use `--no-install' on oddball tests.
[catacomb] / math / f25519.c
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ee39a683
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 Catacomb.
11 *
12 * Catacomb is free software; you can redistribute it and/or modify
13 * it under the terms of the GNU Library General Public License as
14 * published by the Free Software Foundation; either version 2 of the
15 * License, or (at your option) any later version.
16 *
17 * Catacomb is distributed in the hope that it will be useful,
18 * but WITHOUT ANY WARRANTY; without even the implied warranty of
19 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
20 * GNU Library General Public License for more details.
21 *
22 * You should have received a copy of the GNU Library General Public
23 * License along with Catacomb; if not, write to the Free
24 * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
25 * MA 02111-1307, USA.
26 */
27
28/*----- Header files ------------------------------------------------------*/
29
30#include "config.h"
31
25f67362 32#include "ct.h"
ee39a683
MW		 33#include "f25519.h"
34
35/*----- Basic setup -------------------------------------------------------*/
36
f521d4c7
MW		 37typedef f25519_piece piece;
38
ee39a683
MW		 39#if F25519_IMPL == 26
40/* Elements x of GF(2^255 - 19) are represented by ten signed integers x_i: x
41 * = SUM_{0<=i<10} x_i 2^ceil(51i/2), mostly following Bernstein's original
42 * paper.
43 */
44
f521d4c7 45 typedef int64 dblpiece;
ee39a683
MW		 46typedef uint32 upiece; typedef uint64 udblpiece;
47#define P p26
48#define PIECEWD(i) ((i)%2 ? 25 : 26)
49#define NPIECE 10
50
51#define M26 0x03ffffffu
52#define M25 0x01ffffffu
53#define B26 0x04000000u
54#define B25 0x02000000u
55#define B24 0x01000000u
56
57#define PIECES(v) v##0, v##1, v##2, v##3, v##4, v##5, v##6, v##7, v##8, v##9
58#define FETCH(v, w) do { \
59 v##0 = (w)->P[0]; v##1 = (w)->P[1]; \
60 v##2 = (w)->P[2]; v##3 = (w)->P[3]; \
61 v##4 = (w)->P[4]; v##5 = (w)->P[5]; \
62 v##6 = (w)->P[6]; v##7 = (w)->P[7]; \
63 v##8 = (w)->P[8]; v##9 = (w)->P[9]; \
64} while (0)
65#define STASH(w, v) do { \
66 (w)->P[0] = v##0; (w)->P[1] = v##1; \
67 (w)->P[2] = v##2; (w)->P[3] = v##3; \
68 (w)->P[4] = v##4; (w)->P[5] = v##5; \
69 (w)->P[6] = v##6; (w)->P[7] = v##7; \
70 (w)->P[8] = v##8; (w)->P[9] = v##9; \
71} while (0)
72
73#elif F25519_IMPL == 10
74/* Elements x of GF(2^255 - 19) are represented by 26 signed integers x_i: x
75 * = SUM_{0<=i<26} x_i 2^ceil(255i/26); i.e., most pieces are 10 bits wide,
76 * except for pieces 5, 10, 15, 20, and 25 which have 9 bits.
77 */
78
f521d4c7 79 typedef int32 dblpiece;
ee39a683
MW		 80typedef uint16 upiece; typedef uint32 udblpiece;
81#define P p10
82#define PIECEWD(i) \
83 ((i) == 5 || (i) == 10 || (i) == 15 || (i) == 20 || (i) == 25 ? 9 : 10)
84#define NPIECE 26
85
86#define B10 0x0400
87#define B9 0x200
88#define B8 0x100
89#define M10 0x3ff
90#define M9 0x1ff
91
92#endif
93
94/*----- Debugging machinery -----------------------------------------------*/
95
96#if defined(F25519_DEBUG) || defined(TEST_RIG)
97
98#include <stdio.h>
99
100#include "mp.h"
101#include "mptext.h"
102
103static mp *get_2p255m91(void)
104{
105 mpw w19 = 19;
106 mp *p = MP_NEW, m19;
107
108 p = mp_setbit(p, MP_ZERO, 255);
109 mp_build(&m19, &w19, &w19 + 1);
110 p = mp_sub(p, p, &m19);
111 return (p);
112}
113
114DEF_FDUMP(fdump, piece, PIECEWD, NPIECE, 32, get_2p255m91())
115
116#endif
117
118/*----- Loading and storing -----------------------------------------------*/
119
120/* --- @f25519_load@ --- *
121 *
122 * Arguments: @f25519 *z@ = where to store the result
123 * @const octet xv[32]@ = source to read
124 *
125 * Returns: ---
126 *
127 * Use: Reads an element of %$\gf{2^{255} - 19}$% in external
128 * representation from @xv@ and stores it in @z@.
129 *
130 * External representation is little-endian base-256. Some
131 * elements have multiple encodings, which are not produced by
132 * correct software; use of noncanonical encodings is not an
133 * error, and toleration of them is considered a performance
134 * feature.
135 */
136
137void f25519_load(f25519 *z, const octet xv[32])
138{
139#if F25519_IMPL == 26
140
141 uint32 xw0 = LOAD32_L(xv + 0), xw1 = LOAD32_L(xv + 4),
142 xw2 = LOAD32_L(xv + 8), xw3 = LOAD32_L(xv + 12),
143 xw4 = LOAD32_L(xv + 16), xw5 = LOAD32_L(xv + 20),
144 xw6 = LOAD32_L(xv + 24), xw7 = LOAD32_L(xv + 28);
145 piece PIECES(x), b, c;
146
147 /* First, split the 32-bit words into the irregularly-sized pieces we need
148 * for the field representation. These pieces are all positive. We'll do
149 * the sign correction afterwards.
150 *
151 * It may be that the top bit of the input is set. Avoid trouble by
152 * folding that back round into the bottom piece of the representation.
153 *
154 * Here, we briefly have 0 <= x_0 < 2^26 + 19, but will resolve this later.
155 * Otherwise, we have 0 <= x_{2i} < 2^26, and 0 <= x_{2i+1} < 2^25.
156 */
157 x0 = ((xw0 << 0)&0x03ffffff) + 19*((xw7 >> 31)&0x00000001);
158 x1 = ((xw1 << 6)&0x01ffffc0) | ((xw0 >> 26)&0x0000003f);
159 x2 = ((xw2 << 13)&0x03ffe000) | ((xw1 >> 19)&0x00001fff);
160 x3 = ((xw3 << 19)&0x01f80000) | ((xw2 >> 13)&0x0007ffff);
161 x4 = ((xw3 >> 6)&0x03ffffff);
162 x5 = (xw4 << 0)&0x01ffffff;
163 x6 = ((xw5 << 7)&0x03ffff80) | ((xw4 >> 25)&0x0000007f);
164 x7 = ((xw6 << 13)&0x01ffe000) | ((xw5 >> 19)&0x00001fff);
165 x8 = ((xw7 << 20)&0x03f00000) | ((xw6 >> 12)&0x000fffff);
166 x9 = ((xw7 >> 6)&0x01ffffff);
167
168 /* Next, we convert these pieces into a roughly balanced signed
169 * representation. For each piece, check to see if its top bit is set. If
170 * it is, then lend a bit to the next piece over. For x_9, this needs to
171 * be carried around, which is a little fiddly. In particular, we delay
172 * the carry until after all of the pieces have been balanced. If we don't
173 * do this, then we have to do a more expensive test for nonzeroness to
174 * decide whether to lend a bit leftwards rather than just testing a single
175 * bit.
176 *
177 * This fixes up the anomalous size of x_0: the loan of a bit becomes an
178 * actual carry if x_0 >= 2^26. By the end, then, we have:
179 *
180 * { 2^25 if i even
181 * |x_i| <= {
182 * { 2^24 if i odd
183 *
184 * Note that we don't try for a canonical representation here: both upper
185 * and lower bounds are achievable.
186 *
187 * All of the x_i at this point are positive, so we don't need to do
188 * anything wierd when masking them.
189 */
190 b = x9&B24; c = 19&((b >> 19) - (b >> 24)); x9 -= b << 1;
191 b = x8&B25; x9 += b >> 25; x8 -= b << 1;
192 b = x7&B24; x8 += b >> 24; x7 -= b << 1;
193 b = x6&B25; x7 += b >> 25; x6 -= b << 1;
194 b = x5&B24; x6 += b >> 24; x5 -= b << 1;
195 b = x4&B25; x5 += b >> 25; x4 -= b << 1;
196 b = x3&B24; x4 += b >> 24; x3 -= b << 1;
197 b = x2&B25; x3 += b >> 25; x2 -= b << 1;
198 b = x1&B24; x2 += b >> 24; x1 -= b << 1;
199 b = x0&B25; x1 += (b >> 25) + (x0 >> 26); x0 = (x0&M26) - (b << 1);
200 x0 += c;
201
202 /* And with that, we're done. */
203 STASH(z, x);
204
205#elif F25519_IMPL == 10
206
207 piece x[NPIECE];
208 unsigned i, j, n, wd;
209 uint32 a;
210 int b, c;
211
212 /* First, just get the content out of the buffer. */
213 for (i = j = a = n = 0, wd = 10; j < NPIECE; i++) {
214 a |= (uint32)xv[i] << n; n += 8;
215 if (n >= wd) {
216 x[j++] = a&MASK(wd);
217 a >>= wd; n -= wd;
218 wd = PIECEWD(j);
219 }
220 }
221
222 /* There's a little bit left over from the top byte. Carry it into the low
223 * piece.
224 */
225 x[0] += 19*(int)(a&MASK(n));
226
227 /* Next, convert the pieces into a roughly balanced signed representation.
228 * If a piece's top bit is set, lend a bit to the next piece over. For
229 * x_25, this needs to be carried around, which is a bit fiddly.
230 */
231 b = x[NPIECE - 1]&B8;
232 c = 19&((b >> 3) - (b >> 8));
233 x[NPIECE - 1] -= b << 1;
234 for (i = NPIECE - 2; i > 0; i--) {
235 wd = PIECEWD(i) - 1;
236 b = x[i]&BIT(wd);
237 x[i + 1] += b >> wd;
238 x[i] -= b << 1;
239 }
240 b = x[0]&B9;
241 x[1] += (b >> 9) + (x[0] >> 10);
242 x[0] = (x[0]&M10) - (b << 1) + c;
243
244 /* And we're done. */
245 for (i = 0; i < NPIECE; i++) z->P[i] = x[i];
246
247#endif
248}
249
250/* --- @f25519_store@ --- *
251 *
252 * Arguments: @octet zv[32]@ = where to write the result
253 * @const f25519 *x@ = the field element to write
254 *
255 * Returns: ---
256 *
257 * Use: Stores a field element in the given octet vector in external
258 * representation. A canonical encoding is always stored, so,
259 * in particular, the top bit of @xv[31]@ is always left clear.
260 */
261
262void f25519_store(octet zv[32], const f25519 *x)
263{
264#if F25519_IMPL == 26
265
266 piece PIECES(x), PIECES(y), c, d;
267 uint32 zw0, zw1, zw2, zw3, zw4, zw5, zw6, zw7;
268 mask32 m;
269
270 FETCH(x, x);
271
272 /* First, propagate the carries throughout the pieces. By the end of this,
273 * we'll have all of the pieces canonically sized and positive, and maybe
274 * there'll be (signed) carry out. The carry c is in { -1, 0, +1 }, and
275 * the remaining value will be in the half-open interval [0, 2^255). The
276 * whole represented value is then x + 2^255 c.
277 *
278 * It's worth paying careful attention to the bounds. We assume that we
279 * start out with |x_i| <= 2^30. We start by cutting off and reducing the
280 * carry c_9 from the topmost piece, x_9. This leaves 0 <= x_9 < 2^25; and
281 * we'll have |c_9| <= 2^5. We multiply this by 19 and we'll add it onto
282 * x_0 and propagate the carries: but what bounds can we calculate on x
283 * before this?
284 *
285 * Let o_i = floor(51 i/2). We have X_i = SUM_{0<=j<i} x_j 2^{o_i}, so
286 * x = X_10. We see, inductively, that |X_i| < 2^{31+o_{i-1}}: X_0 = 0;
287 * |x_i| <= 2^30; and |X_{i+1}| = |X_i + x_i 2^{o_i}| <= |X_i| + 2^{30+o_i}
288 * < 2^{31+o_i}. Then x = X_9 + 2^230 x_9, and we have better bounds for
289 * x_9, so
290 *
291 * -2^235 < x + 19 c_9 < 2^255 + 2^235
292 *
293 * Here, the x_i are signed, so we must be cautious about bithacking them.
294 */
295 c = ASR(piece, x9, 25); x9 = (upiece)x9&M25;
296 x0 += 19*c; c = ASR(piece, x0, 26); x0 = (upiece)x0&M26;
297 x1 += c; c = ASR(piece, x1, 25); x1 = (upiece)x1&M25;
298 x2 += c; c = ASR(piece, x2, 26); x2 = (upiece)x2&M26;
299 x3 += c; c = ASR(piece, x3, 25); x3 = (upiece)x3&M25;
300 x4 += c; c = ASR(piece, x4, 26); x4 = (upiece)x4&M26;
301 x5 += c; c = ASR(piece, x5, 25); x5 = (upiece)x5&M25;
302 x6 += c; c = ASR(piece, x6, 26); x6 = (upiece)x6&M26;
303 x7 += c; c = ASR(piece, x7, 25); x7 = (upiece)x7&M25;
304 x8 += c; c = ASR(piece, x8, 26); x8 = (upiece)x8&M26;
305 x9 += c; c = ASR(piece, x9, 25); x9 = (upiece)x9&M25;
306
307 /* Now we have a slightly fiddly job to do. If c = +1, or if c = 0 and
308 * x >= 2^255 - 19, then we should subtract 2^255 - 19 from the whole
309 * value; if c = -1 then we should add 2^255 - 19; and otherwise we should
310 * do nothing.
311 *
312 * But conditional behaviour is bad, m'kay. So here's what we do instead.
313 *
314 * The first job is to sort out what we wanted to do. If c = -1 then we
315 * want to (a) invert the constant addend and (b) feed in a carry-in;
316 * otherwise, we don't.
317 */
318 m = SIGN(c);
319 d = m&1;
320
321 /* Now do the addition/subtraction. Remember that all of the x_i are
322 * nonnegative, so shifting and masking are safe and easy.
323 */
324 d += x0 + (19 ^ (M26&m)); y0 = d&M26; d >>= 26;
325 d += x1 + (M25&m); y1 = d&M25; d >>= 25;
326 d += x2 + (M26&m); y2 = d&M26; d >>= 26;
327 d += x3 + (M25&m); y3 = d&M25; d >>= 25;
328 d += x4 + (M26&m); y4 = d&M26; d >>= 26;
329 d += x5 + (M25&m); y5 = d&M25; d >>= 25;
330 d += x6 + (M26&m); y6 = d&M26; d >>= 26;
331 d += x7 + (M25&m); y7 = d&M25; d >>= 25;
332 d += x8 + (M26&m); y8 = d&M26; d >>= 26;
333 d += x9 + (M25&m); y9 = d&M25; d >>= 25;
334
335 /* The final carry-out is in d; since we only did addition, and the x_i are
336 * nonnegative, then d is in { 0, 1 }. We want to keep y, rather than x,
337 * if (a) c /= 0 (in which case we know that the old value was
338 * unsatisfactory), or (b) if d = 1 (in which case, if c = 0, we know that
339 * the subtraction didn't cause a borrow, so we must be in the case where
340 * 2^255 - 19 <= x < 2^255).
341 */
342 m = NONZEROP(c) | ~NONZEROP(d - 1);
343 x0 = (y0&m) | (x0&~m); x1 = (y1&m) | (x1&~m);
344 x2 = (y2&m) | (x2&~m); x3 = (y3&m) | (x3&~m);
345 x4 = (y4&m) | (x4&~m); x5 = (y5&m) | (x5&~m);
346 x6 = (y6&m) | (x6&~m); x7 = (y7&m) | (x7&~m);
347 x8 = (y8&m) | (x8&~m); x9 = (y9&m) | (x9&~m);
348
349 /* Extract 32-bit words from the value. */
350 zw0 = ((x0 >> 0)&0x03ffffff) | (((uint32)x1 << 26)&0xfc000000);
351 zw1 = ((x1 >> 6)&0x0007ffff) | (((uint32)x2 << 19)&0xfff80000);
352 zw2 = ((x2 >> 13)&0x00001fff) | (((uint32)x3 << 13)&0xffffe000);
353 zw3 = ((x3 >> 19)&0x0000003f) | (((uint32)x4 << 6)&0xffffffc0);
354 zw4 = ((x5 >> 0)&0x01ffffff) | (((uint32)x6 << 25)&0xfe000000);
355 zw5 = ((x6 >> 7)&0x0007ffff) | (((uint32)x7 << 19)&0xfff80000);
356 zw6 = ((x7 >> 13)&0x00000fff) | (((uint32)x8 << 12)&0xfffff000);
357 zw7 = ((x8 >> 20)&0x0000003f) | (((uint32)x9 << 6)&0x7fffffc0);
358
359 /* Store the result as an octet string. */
360 STORE32_L(zv + 0, zw0); STORE32_L(zv + 4, zw1);
361 STORE32_L(zv + 8, zw2); STORE32_L(zv + 12, zw3);
362 STORE32_L(zv + 16, zw4); STORE32_L(zv + 20, zw5);
363 STORE32_L(zv + 24, zw6); STORE32_L(zv + 28, zw7);
364
365#elif F25519_IMPL == 10
366
367 piece y[NPIECE], yy[NPIECE], c, d;
368 unsigned i, j, n, wd;
369 uint32 m, a;
370
371 /* Before we do anything, copy the input so we can hack on it. */
372 for (i = 0; i < NPIECE; i++) y[i] = x->P[i];
373
374 /* First, propagate the carries throughout the pieces.
375 *
376 * It's worth paying careful attention to the bounds. We assume that we
377 * start out with |y_i| <= 2^14. We start by cutting off and reducing the
378 * carry c_25 from the topmost piece, y_25. This leaves 0 <= y_25 < 2^9;
379 * and we'll have |c_25| <= 2^5. We multiply this by 19 and we'll ad it
380 * onto y_0 and propagte the carries: but what bounds can we calculate on
381 * y before this?
382 *
383 * Let o_i = floor(255 i/26). We have Y_i = SUM_{0<=j<i} y_j 2^{o_i}, so
384 * y = Y_26. We see, inductively, that |Y_i| < 2^{31+o_{i-1}}: Y_0 = 0;
385 * |y_i| <= 2^14; and |Y_{i+1}| = |Y_i + y_i 2^{o_i}| <= |Y_i| + 2^{14+o_i}
386 * < 2^{15+o_i}. Then x = Y_25 + 2^246 y_25, and we have better bounds for
387 * y_25, so
388 *
389 * -2^251 < y + 19 c_25 < 2^255 + 2^251
390 *
391 * Here, the y_i are signed, so we must be cautious about bithacking them.
392 *
393 * (Rather closer than the 10-piece case above, but still doable in one
394 * pass.)
395 */
396 c = 19*ASR(piece, y[NPIECE - 1], 9);
397 y[NPIECE - 1] = (upiece)y[NPIECE - 1]&M9;
398 for (i = 0; i < NPIECE; i++) {
399 wd = PIECEWD(i);
400 y[i] += c;
401 c = ASR(piece, y[i], wd);
402 y[i] = (upiece)y[i]&MASK(wd);
403 }
404
405 /* Now the addition or subtraction. */
406 m = SIGN(c);
407 d = m&1;
408
409 d += y[0] + (19 ^ (M10&m));
410 yy[0] = d&M10;
411 d >>= 10;
412 for (i = 1; i < NPIECE; i++) {
413 wd = PIECEWD(i);
414 d += y[i] + (MASK(wd)&m);
415 yy[i] = d&MASK(wd);
416 d >>= wd;
417 }
418
419 /* Choose which value to keep. */
420 m = NONZEROP(c) | ~NONZEROP(d - 1);
421 for (i = 0; i < NPIECE; i++) y[i] = (yy[i]&m) | (y[i]&~m);
422
423 /* Store the result as an octet string. */
424 for (i = j = a = n = 0; i < NPIECE; i++) {
425 a |= (upiece)y[i] << n; n += PIECEWD(i);
426 while (n >= 8) {
427 zv[j++] = a&0xff;
428 a >>= 8; n -= 8;
429 }
430 }
431 zv[j++] = a;
432
433#endif
434}
435
436/* --- @f25519_set@ --- *
437 *
438 * Arguments: @f25519 *z@ = where to write the result
439 * @int a@ = a small-ish constant
440 *
441 * Returns: ---
442 *
443 * Use: Sets @z@ to equal @a@.
444 */
445
446void f25519_set(f25519 *x, int a)
447{
448 unsigned i;
449
450 x->P[0] = a;
451 for (i = 1; i < NPIECE; i++) x->P[i] = 0;
452}
453
454/*----- Basic arithmetic --------------------------------------------------*/
455
456/* --- @f25519_add@ --- *
457 *
458 * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@)
459 * @const f25519 *x, *y@ = two operands
460 *
461 * Returns: ---
462 *
463 * Use: Set @z@ to the sum %$x + y$%.
464 */
465
466void f25519_add(f25519 *z, const f25519 *x, const f25519 *y)
467{
468#if F25519_IMPL == 26
469 z->P[0] = x->P[0] + y->P[0]; z->P[1] = x->P[1] + y->P[1];
470 z->P[2] = x->P[2] + y->P[2]; z->P[3] = x->P[3] + y->P[3];
471 z->P[4] = x->P[4] + y->P[4]; z->P[5] = x->P[5] + y->P[5];
472 z->P[6] = x->P[6] + y->P[6]; z->P[7] = x->P[7] + y->P[7];
473 z->P[8] = x->P[8] + y->P[8]; z->P[9] = x->P[9] + y->P[9];
474#elif F25519_IMPL == 10
475 unsigned i;
476 for (i = 0; i < NPIECE; i++) z->P[i] = x->P[i] + y->P[i];
477#endif
478}
479
480/* --- @f25519_sub@ --- *
481 *
482 * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@)
483 * @const f25519 *x, *y@ = two operands
484 *
485 * Returns: ---
486 *
487 * Use: Set @z@ to the difference %$x - y$%.
488 */
489
490void f25519_sub(f25519 *z, const f25519 *x, const f25519 *y)
491{
492#if F25519_IMPL == 26
493 z->P[0] = x->P[0] - y->P[0]; z->P[1] = x->P[1] - y->P[1];
494 z->P[2] = x->P[2] - y->P[2]; z->P[3] = x->P[3] - y->P[3];
495 z->P[4] = x->P[4] - y->P[4]; z->P[5] = x->P[5] - y->P[5];
496 z->P[6] = x->P[6] - y->P[6]; z->P[7] = x->P[7] - y->P[7];
497 z->P[8] = x->P[8] - y->P[8]; z->P[9] = x->P[9] - y->P[9];
498#elif F25519_IMPL == 10
499 unsigned i;
500 for (i = 0; i < NPIECE; i++) z->P[i] = x->P[i] - y->P[i];
501#endif
502}
503
25f67362
MW		 504/* --- @f25519_neg@ --- *
505 *
506 * Arguments: @f25519 *z@ = where to put the result (may alias @x@)
507 * @const f25519 *x@ = an operand
508 *
509 * Returns: ---
510 *
511 * Use: Set @z = -x@.
512 */
513
514void f25519_neg(f25519 *z, const f25519 *x)
515{
516#if F25519_IMPL == 26
517 z->P[0] = -x->P[0]; z->P[1] = -x->P[1];
518 z->P[2] = -x->P[2]; z->P[3] = -x->P[3];
519 z->P[4] = -x->P[4]; z->P[5] = -x->P[5];
520 z->P[6] = -x->P[6]; z->P[7] = -x->P[7];
521 z->P[8] = -x->P[8]; z->P[9] = -x->P[9];
522#elif F25519_IMPL == 10
523 unsigned i;
524 for (i = 0; i < NPIECE; i++) z->P[i] = -x->P[i];
525#endif
526}
527
ee39a683
MW		 528/*----- Constant-time utilities -------------------------------------------*/
529
25f67362
MW		 530/* --- @f25519_pick2@ --- *
531 *
532 * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@)
533 * @const f25519 *x, *y@ = two operands
534 * @uint32 m@ = a mask
535 *
536 * Returns: ---
537 *
538 * Use: If @m@ is zero, set @z = y@; if @m@ is all-bits-set, then set
539 * @z = x@. If @m@ has some other value, then scramble @z@ in
540 * an unhelpful way.
541 */
542
543void f25519_pick2(f25519 *z, const f25519 *x, const f25519 *y, uint32 m)
544{
545 mask32 mm = FIX_MASK32(m);
546
547#if F25519_IMPL == 26
548 z->P[0] = PICK2(x->P[0], y->P[0], mm);
549 z->P[1] = PICK2(x->P[1], y->P[1], mm);
550 z->P[2] = PICK2(x->P[2], y->P[2], mm);
551 z->P[3] = PICK2(x->P[3], y->P[3], mm);
552 z->P[4] = PICK2(x->P[4], y->P[4], mm);
553 z->P[5] = PICK2(x->P[5], y->P[5], mm);
554 z->P[6] = PICK2(x->P[6], y->P[6], mm);
555 z->P[7] = PICK2(x->P[7], y->P[7], mm);
556 z->P[8] = PICK2(x->P[8], y->P[8], mm);
557 z->P[9] = PICK2(x->P[9], y->P[9], mm);
558#elif F25519_IMPL == 10
559 unsigned i;
560 for (i = 0; i < NPIECE; i++) z->P[i] = PICK2(x->P[i], y->P[i], mm);
561#endif
562}
563
564/* --- @f25519_pickn@ --- *
565 *
566 * Arguments: @f25519 *z@ = where to put the result
567 * @const f25519 *v@ = a table of entries
568 * @size_t n@ = the number of entries in @v@
569 * @size_t i@ = an index
570 *
571 * Returns: ---
572 *
573 * Use: If @0 <= i < n < 32@ then set @z = v[i]@. If @n >= 32@ then
574 * do something unhelpful; otherwise, if @i >= n@ then set @z@
575 * to zero.
576 */
577
578void f25519_pickn(f25519 *z, const f25519 *v, size_t n, size_t i)
579{
580 uint32 b = (uint32)1 << (31 - i);
581 mask32 m;
582
583#if F25519_IMPL == 26
584 z->P[0] = z->P[1] = z->P[2] = z->P[3] = z->P[4] =
585 z->P[5] = z->P[6] = z->P[7] = z->P[8] = z->P[9] = 0;
586 while (n--) {
587 m = SIGN(b);
588 CONDPICK(z->P[0], v->P[0], m);
589 CONDPICK(z->P[1], v->P[1], m);
590 CONDPICK(z->P[2], v->P[2], m);
591 CONDPICK(z->P[3], v->P[3], m);
592 CONDPICK(z->P[4], v->P[4], m);
593 CONDPICK(z->P[5], v->P[5], m);
594 CONDPICK(z->P[6], v->P[6], m);
595 CONDPICK(z->P[7], v->P[7], m);
596 CONDPICK(z->P[8], v->P[8], m);
597 CONDPICK(z->P[9], v->P[9], m);
598 v++; b <<= 1;
599 }
600#elif F25519_IMPL == 10
601 unsigned j;
602
603 for (j = 0; j < NPIECE; j++) z->P[j] = 0;
604 while (n--) {
605 m = SIGN(b);
606 for (j = 0; j < NPIECE; j++) CONDPICK(z->P[j], v->P[j], m);
607 v++; b <<= 1;
608 }
609#endif
610}
611
ee39a683
MW		 612/* --- @f25519_condswap@ --- *
613 *
614 * Arguments: @f25519 *x, *y@ = two operands
615 * @uint32 m@ = a mask
616 *
617 * Returns: ---
618 *
619 * Use: If @m@ is zero, do nothing; if @m@ is all-bits-set, then
620 * exchange @x@ and @y@. If @m@ has some other value, then
621 * scramble @x@ and @y@ in an unhelpful way.
622 */
623
624void f25519_condswap(f25519 *x, f25519 *y, uint32 m)
625{
626 mask32 mm = FIX_MASK32(m);
627
628#if F25519_IMPL == 26
629 CONDSWAP(x->P[0], y->P[0], mm);
630 CONDSWAP(x->P[1], y->P[1], mm);
631 CONDSWAP(x->P[2], y->P[2], mm);
632 CONDSWAP(x->P[3], y->P[3], mm);
633 CONDSWAP(x->P[4], y->P[4], mm);
634 CONDSWAP(x->P[5], y->P[5], mm);
635 CONDSWAP(x->P[6], y->P[6], mm);
636 CONDSWAP(x->P[7], y->P[7], mm);
637 CONDSWAP(x->P[8], y->P[8], mm);
638 CONDSWAP(x->P[9], y->P[9], mm);
639#elif F25519_IMPL == 10
640 unsigned i;
641 for (i = 0; i < NPIECE; i++) CONDSWAP(x->P[i], y->P[i], mm);
642#endif
643}
644
25f67362
MW		 645/* --- @f25519_condneg@ --- *
646 *
647 * Arguments: @f25519 *z@ = where to put the result (may alias @x@)
648 * @const f25519 *x@ = an operand
649 * @uint32 m@ = a mask
650 *
651 * Returns: ---
652 *
653 * Use: If @m@ is zero, set @z = x@; if @m@ is all-bits-set, then set
654 * @z = -x@. If @m@ has some other value then scramble @z@ in
655 * an unhelpful way.
656 */
657
658void f25519_condneg(f25519 *z, const f25519 *x, uint32 m)
659{
660#ifdef NEG_TWOC
661 mask32 m_xor = FIX_MASK32(m);
662 piece m_add = m&1;
663# define CONDNEG(x) (((x) ^ m_xor) + m_add)
664#else
665 int s = PICK2(-1, +1, m);
666# define CONDNEG(x) (s*(x))
667#endif
668
669#if F25519_IMPL == 26
670 z->P[0] = CONDNEG(x->P[0]);
671 z->P[1] = CONDNEG(x->P[1]);
672 z->P[2] = CONDNEG(x->P[2]);
673 z->P[3] = CONDNEG(x->P[3]);
674 z->P[4] = CONDNEG(x->P[4]);
675 z->P[5] = CONDNEG(x->P[5]);
676 z->P[6] = CONDNEG(x->P[6]);
677 z->P[7] = CONDNEG(x->P[7]);
678 z->P[8] = CONDNEG(x->P[8]);
679 z->P[9] = CONDNEG(x->P[9]);
680#elif F25519_IMPL == 10
681 unsigned i;
682 for (i = 0; i < NPIECE; i++) z->P[i] = CONDNEG(x->P[i]);
683#endif
684
685#undef CONDNEG
686}
687
ee39a683
MW		 688/*----- Multiplication ----------------------------------------------------*/
689
690#if F25519_IMPL == 26
691
692/* Let B = 2^63 - 1 be the largest value such that +B and -B can be
693 * represented in a double-precision piece. On entry, it must be the case
694 * that |X_i| <= M <= B - 2^25 for some M. If this is the case, then, on
695 * exit, we will have |Z_i| <= 2^25 + 19 M/2^25.
696 */
697#define CARRYSTEP(z, x, m, b, f, xx, n) do { \
698 (z) = (dblpiece)((udblpiece)(x)&(m)) - (b) + \
699 (f)*ASR(dblpiece, (xx), (n)); \
700} while (0)
701#define CARRY_REDUCE(z, x) do { \
702 dblpiece PIECES(_t); \
703 \
704 /* Bias the input pieces. This keeps the carries and so on centred \
705 * around zero rather than biased positive. \
706 */ \
707 _t0 = (x##0) + B25; _t1 = (x##1) + B24; \
708 _t2 = (x##2) + B25; _t3 = (x##3) + B24; \
709 _t4 = (x##4) + B25; _t5 = (x##5) + B24; \
710 _t6 = (x##6) + B25; _t7 = (x##7) + B24; \
711 _t8 = (x##8) + B25; _t9 = (x##9) + B24; \
712 \
713 /* Calculate the reduced pieces. Careful with the bithacking. */ \
714 CARRYSTEP(z##0, _t0, M26, B25, 19, _t9, 25); \
715 CARRYSTEP(z##1, _t1, M25, B24, 1, _t0, 26); \
716 CARRYSTEP(z##2, _t2, M26, B25, 1, _t1, 25); \
717 CARRYSTEP(z##3, _t3, M25, B24, 1, _t2, 26); \
718 CARRYSTEP(z##4, _t4, M26, B25, 1, _t3, 25); \
719 CARRYSTEP(z##5, _t5, M25, B24, 1, _t4, 26); \
720 CARRYSTEP(z##6, _t6, M26, B25, 1, _t5, 25); \
721 CARRYSTEP(z##7, _t7, M25, B24, 1, _t6, 26); \
722 CARRYSTEP(z##8, _t8, M26, B25, 1, _t7, 25); \
723 CARRYSTEP(z##9, _t9, M25, B24, 1, _t8, 26); \
724} while (0)
725
726#elif F25519_IMPL == 10
727
728/* Perform carry propagation on X. */
729static void carry_reduce(dblpiece x[NPIECE])
730{
731 /* Initial bounds: we assume |x_i| < 2^31 - 2^27. */
732
733 unsigned i, j;
734 dblpiece c;
735
736 /* The result is nearly canonical, because we do sequential carry
737 * propagation, because smaller processors are more likely to prefer the
738 * smaller working set than the instruction-level parallelism.
739 *
740 * Start at x_23; truncate it to 10 bits, and propagate the carry to x_24.
741 * Truncate x_24 to 10 bits, and add the carry onto x_25. Truncate x_25 to
742 * 9 bits, and add 19 times the carry onto x_0. And so on.
743 *
744 * Let c_i be the portion of x_i to be carried onto x_{i+1}. I claim that
745 * |c_i| <= 2^22. Then the carry /into/ any x_i has magnitude at most
746 * 19*2^22 < 2^27 (allowing for the reduction as we carry from x_25 to
747 * x_0), and x_i after carry is bounded above by 2^31. Hence, the carry
748 * out is at most 2^22, as claimed.
749 *
750 * Once we reach x_23 for the second time, we start with |x_23| <= 2^9.
751 * The carry into x_23 is at most 2^27 as calculated above; so the carry
752 * out into x_24 has magnitude at most 2^17. In turn, |x_24| <= 2^9 before
753 * the carry, so is now no more than 2^18 in magnitude, and the carry out
754 * into x_25 is at most 2^8. This leaves |x_25| < 2^9 after carry
755 * propagation.
756 *
757 * Be careful with the bit hacking because the quantities involved are
758 * signed.
759 */
760
761 /*For each piece, we bias it so that floor division (as done by an
762 * arithmetic right shift) and modulus (as done by bitwise-AND) does the
763 * right thing.
764 */
765#define CARRY(i, wd, b, m) do { \
766 x[i] += (b); \
767 c = ASR(dblpiece, x[i], (wd)); \
768 x[i] = (dblpiece)((udblpiece)x[i]&(m)) - (b); \
769} while (0)
770
771 { CARRY(23, 10, B9, M10); }
772 { x[24] += c; CARRY(24, 10, B9, M10); }
773 { x[25] += c; CARRY(25, 9, B8, M9); }
774 { x[0] += 19*c; CARRY( 0, 10, B9, M10); }
775 for (i = 1; i < 21; ) {
776 for (j = i + 4; i < j; ) { x[i] += c; CARRY( i, 10, B9, M10); i++; }
777 { x[i] += c; CARRY( i, 9, B8, M9); i++; }
778 }
779 while (i < 25) { x[i] += c; CARRY( i, 10, B9, M10); i++; }
780 x[25] += c;
781
782#undef CARRY
783}
784
785#endif
786
787/* --- @f25519_mulconst@ --- *
788 *
789 * Arguments: @f25519 *z@ = where to put the result (may alias @x@)
790 * @const f25519 *x@ = an operand
791 * @long a@ = a small-ish constant; %$|a| < 2^{20}$%.
792 *
793 * Returns: ---
794 *
795 * Use: Set @z@ to the product %$a x$%.
796 */
797
798void f25519_mulconst(f25519 *z, const f25519 *x, long a)
799{
800#if F25519_IMPL == 26
801
802 piece PIECES(x);
803 dblpiece PIECES(z), aa = a;
804
805 FETCH(x, x);
806
807 /* Suppose that |x_i| <= 2^27, and |a| <= 2^23. Then we'll have
808 * |z_i| <= 2^50.
809 */
810 z0 = aa*x0; z1 = aa*x1; z2 = aa*x2; z3 = aa*x3; z4 = aa*x4;
811 z5 = aa*x5; z6 = aa*x6; z7 = aa*x7; z8 = aa*x8; z9 = aa*x9;
812
813 /* Following `CARRY_REDUCE', we'll have |z_i| <= 2^26. */
814 CARRY_REDUCE(z, z);
815 STASH(z, z);
816
817#elif F25519_IMPL == 10
818
819 dblpiece y[NPIECE];
820 unsigned i;
821
822 for (i = 0; i < NPIECE; i++) y[i] = a*x->P[i];
823 carry_reduce(y);
824 for (i = 0; i < NPIECE; i++) z->P[i] = y[i];
825
826#endif
827}
828
829/* --- @f25519_mul@ --- *
830 *
831 * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@)
832 * @const f25519 *x, *y@ = two operands
833 *
834 * Returns: ---
835 *
836 * Use: Set @z@ to the product %$x y$%.
837 */
838
839void f25519_mul(f25519 *z, const f25519 *x, const f25519 *y)
840{
841#if F25519_IMPL == 26
842
843 piece PIECES(x), PIECES(y);
844 dblpiece PIECES(z);
845 unsigned i;
846
847 FETCH(x, x); FETCH(y, y);
848
849 /* Suppose that |x_i|, |y_i| <= 2^27. Then we'll have
850 *
851 * |z_0| <= 267*2^54
852 * |z_1| <= 154*2^54
853 * |z_2| <= 213*2^54
854 * |z_3| <= 118*2^54
855 * |z_4| <= 159*2^54
856 * |z_5| <= 82*2^54
857 * |z_6| <= 105*2^54
858 * |z_7| <= 46*2^54
859 * |z_8| <= 51*2^54
860 * |z_9| <= 10*2^54
861 *
862 * all of which are less than 2^63 - 2^25.
863 */
864
865#define M(a, b) ((dblpiece)(a)*(b))
866 z0 = M(x0, y0) +
867 19*(M(x2, y8) + M(x4, y6) + M(x6, y4) + M(x8, y2)) +
868 38*(M(x1, y9) + M(x3, y7) + M(x5, y5) + M(x7, y3) + M(x9, y1));
869 z1 = M(x0, y1) + M(x1, y0) +
870 19*(M(x2, y9) + M(x3, y8) + M(x4, y7) + M(x5, y6) +
871 M(x6, y5) + M(x7, y4) + M(x8, y3) + M(x9, y2));
872 z2 = M(x0, y2) + M(x2, y0) +
873 2* M(x1, y1) +
874 19*(M(x4, y8) + M(x6, y6) + M(x8, y4)) +
875 38*(M(x3, y9) + M(x5, y7) + M(x7, y5) + M(x9, y3));
876 z3 = M(x0, y3) + M(x1, y2) + M(x2, y1) + M(x3, y0) +
877 19*(M(x4, y9) + M(x5, y8) + M(x6, y7) +
878 M(x7, y6) + M(x8, y5) + M(x9, y4));
879 z4 = M(x0, y4) + M(x2, y2) + M(x4, y0) +
880 2*(M(x1, y3) + M(x3, y1)) +
881 19*(M(x6, y8) + M(x8, y6)) +
882 38*(M(x5, y9) + M(x7, y7) + M(x9, y5));
883 z5 = M(x0, y5) + M(x1, y4) + M(x2, y3) +
884 M(x3, y2) + M(x4, y1) + M(x5, y0) +
885 19*(M(x6, y9) + M(x7, y8) + M(x8, y7) + M(x9, y6));
886 z6 = M(x0, y6) + M(x2, y4) + M(x4, y2) + M(x6, y0) +
887 2*(M(x1, y5) + M(x3, y3) + M(x5, y1)) +
888 19* M(x8, y8) +
889 38*(M(x7, y9) + M(x9, y7));
890 z7 = M(x0, y7) + M(x1, y6) + M(x2, y5) + M(x3, y4) +
891 M(x4, y3) + M(x5, y2) + M(x6, y1) + M(x7, y0) +
892 19*(M(x8, y9) + M(x9, y8));
893 z8 = M(x0, y8) + M(x2, y6) + M(x4, y4) + M(x6, y2) + M(x8, y0) +
894 2*(M(x1, y7) + M(x3, y5) + M(x5, y3) + M(x7, y1)) +
895 38* M(x9, y9);
896 z9 = M(x0, y9) + M(x1, y8) + M(x2, y7) + M(x3, y6) + M(x4, y5) +
897 M(x5, y4) + M(x6, y3) + M(x7, y2) + M(x8, y1) + M(x9, y0);
898#undef M
899
900 /* From above, we have |z_i| <= 2^63 - 2^25. A pass of `CARRY_REDUCE' will
901 * leave |z_i| <= 2^38 + 2^25; and a second pass will leave |z_i| <= 2^25 +
902 * 2^13, which is comfortable for an addition prior to the next
903 * multiplication.
904 */
905 for (i = 0; i < 2; i++) CARRY_REDUCE(z, z);
906 STASH(z, z);
907
908#elif F25519_IMPL == 10
909
910 dblpiece u[NPIECE], t, tt, p;
911 unsigned i, j, k;
912
913 /* This is unpleasant. Honestly, this table seems to be the best way of
914 * doing it.
915 */
916 static const unsigned short off[NPIECE] = {
917 0, 10, 20, 30, 40, 50, 59, 69, 79, 89, 99, 108, 118,
918 128, 138, 148, 157, 167, 177, 187, 197, 206, 216, 226, 236, 246
919 };
920
921 /* First pass: things we must multiply by 19 or 38. */
922 for (i = 0; i < NPIECE - 1; i++) {
923 t = tt = 0;
924 for (j = i + 1; j < NPIECE; j++) {
925 k = NPIECE + i - j; p = (dblpiece)x->P[j]*y->P[k];
926 if (off[i] < off[j] + off[k] - 255) tt += p;
927 else t += p;
928 }
929 u[i] = 19*(t + 2*tt);
930 }
931 u[NPIECE - 1] = 0;
932
933 /* Second pass: things we must multiply by 1 or 2. */
934 for (i = 0; i < NPIECE; i++) {
935 t = tt = 0;
936 for (j = 0; j <= i; j++) {
937 k = i - j; p = (dblpiece)x->P[j]*y->P[k];
938 if (off[i] < off[j] + off[k]) tt += p;
939 else t += p;
940 }
941 u[i] += t + 2*tt;
942 }
943
944 /* And we're done. */
945 carry_reduce(u);
946 for (i = 0; i < NPIECE; i++) z->P[i] = u[i];
947
948#endif
949}
950
951/* --- @f25519_sqr@ --- *
952 *
953 * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@)
954 * @const f25519 *x@ = an operand
955 *
956 * Returns: ---
957 *
958 * Use: Set @z@ to the square %$x^2$%.
959 */
960
961void f25519_sqr(f25519 *z, const f25519 *x)
962{
963#if F25519_IMPL == 26
964
965 piece PIECES(x);
966 dblpiece PIECES(z);
967 unsigned i;
968
969 FETCH(x, x);
970
971 /* See `f25519_mul' for bounds. */
972
973#define M(a, b) ((dblpiece)(a)*(b))
974 z0 = M(x0, x0) +
975 38*(M(x2, x8) + M(x4, x6) + M(x5, x5)) +
976 76*(M(x1, x9) + M(x3, x7));
977 z1 = 2* M(x0, x1) +
978 38*(M(x2, x9) + M(x3, x8) + M(x4, x7) + M(x5, x6));
979 z2 = 2*(M(x0, x2) + M(x1, x1)) +
980 19* M(x6, x6) +
981 38* M(x4, x8) +
982 76*(M(x3, x9) + M(x5, x7));
983 z3 = 2*(M(x0, x3) + M(x1, x2)) +
984 38*(M(x4, x9) + M(x5, x8) + M(x6, x7));
985 z4 = M(x2, x2) +
986 2* M(x0, x4) +
987 4* M(x1, x3) +
988 38*(M(x6, x8) + M(x7, x7)) +
989 76* M(x5, x9);
990 z5 = 2*(M(x0, x5) + M(x1, x4) + M(x2, x3)) +
991 38*(M(x6, x9) + M(x7, x8));
992 z6 = 2*(M(x0, x6) + M(x2, x4) + M(x3, x3)) +
993 4* M(x1, x5) +
994 19* M(x8, x8) +
995 76* M(x7, x9);
996 z7 = 2*(M(x0, x7) + M(x1, x6) + M(x2, x5) + M(x3, x4)) +
997 38* M(x8, x9);
998 z8 = M(x4, x4) +
999 2*(M(x0, x8) + M(x2, x6)) +
1000 4*(M(x1, x7) + M(x3, x5)) +
1001 38* M(x9, x9);
1002 z9 = 2*(M(x0, x9) + M(x1, x8) + M(x2, x7) + M(x3, x6) + M(x4, x5));
1003#undef M
1004
1005 /* See `f25519_mul' for details. */
1006 for (i = 0; i < 2; i++) CARRY_REDUCE(z, z);
1007 STASH(z, z);
1008
1009#elif F25519_IMPL == 10
1010 f25519_mul(z, x, x);
1011#endif
1012}
1013
1014/*----- More complicated things -------------------------------------------*/
1015
1016/* --- @f25519_inv@ --- *
1017 *
1018 * Arguments: @f25519 *z@ = where to put the result (may alias @x@)
1019 * @const f25519 *x@ = an operand
1020 *
1021 * Returns: ---
1022 *
1023 * Use: Stores in @z@ the multiplicative inverse %$x^{-1}$%. If
1024 * %$x = 0$% then @z@ is set to zero. This is considered a
1025 * feature.
1026 */
1027
1028void f25519_inv(f25519 *z, const f25519 *x)
1029{
1030 f25519 t, u, t2, t11, t2p10m1, t2p50m1;
1031 unsigned i;
1032
1033#define SQRN(z, x, n) do { \
1034 f25519_sqr((z), (x)); \
1035 for (i = 1; i < (n); i++) f25519_sqr((z), (z)); \
1036} while (0)
1037
1038 /* Calculate x^-1 = x^(p - 2) = x^(2^255 - 21), which also handles x = 0 as
1039 * intended. The addition chain here is from Bernstein's implementation; I
1040 * couldn't find a better one.
1041 */ /* step | value */
1042 f25519_sqr(&t2, x); /* 1 | 2 */
1043 SQRN(&u, &t2, 2); /* 3 | 8 */
1044 f25519_mul(&t, &u, x); /* 4 | 9 */
1045 f25519_mul(&t11, &t, &t2); /* 5 | 11 = 2^5 - 21 */
1046 f25519_sqr(&u, &t11); /* 6 | 22 */
1047 f25519_mul(&t, &t, &u); /* 7 | 31 = 2^5 - 1 */
1048 SQRN(&u, &t, 5); /* 12 | 2^10 - 2^5 */
1049 f25519_mul(&t2p10m1, &t, &u); /* 13 | 2^10 - 1 */
1050 SQRN(&u, &t2p10m1, 10); /* 23 | 2^20 - 2^10 */
1051 f25519_mul(&t, &t2p10m1, &u); /* 24 | 2^20 - 1 */
1052 SQRN(&u, &t, 20); /* 44 | 2^40 - 2^20 */
1053 f25519_mul(&t, &t, &u); /* 45 | 2^40 - 1 */
1054 SQRN(&u, &t, 10); /* 55 | 2^50 - 2^10 */
1055 f25519_mul(&t2p50m1, &t2p10m1, &u); /* 56 | 2^50 - 1 */
1056 SQRN(&u, &t2p50m1, 50); /* 106 | 2^100 - 2^50 */
1057 f25519_mul(&t, &t2p50m1, &u); /* 107 | 2^100 - 1 */
1058 SQRN(&u, &t, 100); /* 207 | 2^200 - 2^100 */
1059 f25519_mul(&t, &t, &u); /* 208 | 2^200 - 1 */
1060 SQRN(&u, &t, 50); /* 258 | 2^250 - 2^50 */
1061 f25519_mul(&t, &t2p50m1, &u); /* 259 | 2^250 - 1 */
1062 SQRN(&u, &t, 5); /* 264 | 2^255 - 2^5 */
1063 f25519_mul(z, &u, &t11); /* 265 | 2^255 - 21 */
1064
1065#undef SQRN
1066}
1067
25f67362
MW		 1068/* --- @f25519_quosqrt@ --- *
1069 *
1070 * Arguments: @f25519 *z@ = where to put the result (may alias @x@ or @y@)
1071 * @const f25519 *x, *y@ = two operands
1072 *
1073 * Returns: Zero if successful, @-1@ if %$x/y$% is not a square.
1074 *
1075 * Use: Stores in @z@ the one of the square roots %$\pm\sqrt{x/y}$%.
1076 * If %$x = y = 0% then the result is zero; if %$y = 0$% but %$x
1077 * \ne 0$% then the operation fails. If you wanted a specific
1078 * square root then you'll have to pick it yourself.
1079 */
1080
1081static const piece sqrtm1_pieces[NPIECE] = {
1082#if F25519_IMPL == 26
1083 -32595792, -7943725, 9377950, 3500415, 12389472,
1084 -272473, -25146209, -2005654, 326686, 11406482
1085#elif F25519_IMPL == 10
1086 176, -88, 161, 157, -485, -196, -231, -220, -416,
1087 -169, -255, 50, 189, -89, -266, -32, 202, -511,
1088 423, 357, 248, -249, 80, 288, 50, 174
1089#endif
1090};
1091#define SQRTM1 ((const f25519 *)sqrtm1_pieces)
1092
1093int f25519_quosqrt(f25519 *z, const f25519 *x, const f25519 *y)
1094{
e830bb69 1095 f25519 t, u, v, w, t15;
25f67362
MW		 1096 octet xb[32], b0[32], b1[32];
1097 int32 rc = -1;
1098 mask32 m;
1099 unsigned i;
1100
1101#define SQRN(z, x, n) do { \
1102 f25519_sqr((z), (x)); \
1103 for (i = 1; i < (n); i++) f25519_sqr((z), (z)); \
1104} while (0)
1105
e830bb69
MW		 1106 /* This is a bit tricky; the algorithm is loosely based on Bernstein, Duif,
1107 * Lange, Schwabe, and Yang, `High-speed high-security signatures',
1108 * 2011-09-26, https://ed25519.cr.yp.to/ed25519-20110926.pdf.
1109 */
1110 f25519_mul(&v, x, y);
1111
1112 /* Now for an addition chain. */ /* step | value */
1113 f25519_sqr(&u, &v); /* 1 | 2 */
1114 f25519_mul(&t, &u, &v); /* 2 | 3 */
1115 SQRN(&u, &t, 2); /* 4 | 12 */
1116 f25519_mul(&t15, &u, &t); /* 5 | 15 */
1117 f25519_sqr(&u, &t15); /* 6 | 30 */
1118 f25519_mul(&t, &u, &v); /* 7 | 31 = 2^5 - 1 */
1119 SQRN(&u, &t, 5); /* 12 | 2^10 - 2^5 */
1120 f25519_mul(&t, &u, &t); /* 13 | 2^10 - 1 */
1121 SQRN(&u, &t, 10); /* 23 | 2^20 - 2^10 */
1122 f25519_mul(&u, &u, &t); /* 24 | 2^20 - 1 */
1123 SQRN(&u, &u, 10); /* 34 | 2^30 - 2^10 */
1124 f25519_mul(&t, &u, &t); /* 35 | 2^30 - 1 */
1125 f25519_sqr(&u, &t); /* 36 | 2^31 - 2 */
1126 f25519_mul(&t, &u, &v); /* 37 | 2^31 - 1 */
1127 SQRN(&u, &t, 31); /* 68 | 2^62 - 2^31 */
1128 f25519_mul(&t, &u, &t); /* 69 | 2^62 - 1 */
1129 SQRN(&u, &t, 62); /* 131 | 2^124 - 2^62 */
1130 f25519_mul(&t, &u, &t); /* 132 | 2^124 - 1 */
1131 SQRN(&u, &t, 124); /* 256 | 2^248 - 2^124 */
1132 f25519_mul(&t, &u, &t); /* 257 | 2^248 - 1 */
1133 f25519_sqr(&u, &t); /* 258 | 2^249 - 2 */
1134 f25519_mul(&t, &u, &v); /* 259 | 2^249 - 1 */
1135 SQRN(&t, &t, 3); /* 262 | 2^252 - 8 */
1136 f25519_sqr(&u, &t); /* 263 | 2^253 - 16 */
1137 f25519_mul(&t, &u, &t); /* 264 | 3*2^252 - 24 */
1138 f25519_mul(&t, &t, &t15); /* 265 | 3*2^252 - 9 */
1139 f25519_mul(&w, &t, &v); /* 266 | 3*2^252 - 8 */
1140
1141 /* Awesome. Now let me explain. Let v be a square in GF(p), and let w =
1142 * v^(3*2^252 - 8). In particular, let's consider
25f67362 1143 *
e830bb69
MW		 1144 * v^2 w^4 = v^2 v^{3*2^254 - 32} = (v^{2^254 - 10})^3
1145 *
1146 * But 2^254 - 10 = ((2^255 - 19) - 1)/2 = (p - 1)/2. Since v is a square,
1147 * it has order dividing (p - 1)/2, and therefore v^2 w^4 = 1 and
1148 *
1149 * w^4 = 1/v^2
1150 *
1151 * That in turn implies that w^2 = ±1/v. Now, recall that v = x y, and let
1152 * w' = w x. Then w'^2 = ±x^2/v = ±x/y. If y w'^2 = x then we set
1153 * z = w', since we have z^2 = x/y; otherwise let z = i w', where i^2 = -1,
1154 * so z^2 = -w^2 = x/y, and we're done.
25f67362
MW		 1155 *
1156 * The easiest way to compare is to encode. This isn't as wasteful as it
1157 * sounds: the hard part is normalizing the representations, which we have
1158 * to do anyway.
1159 */
e830bb69
MW		 1160 f25519_mul(&w, &w, x);
1161 f25519_sqr(&t, &w);
25f67362
MW		 1162 f25519_mul(&t, &t, y);
1163 f25519_neg(&u, &t);
1164 f25519_store(xb, x);
1165 f25519_store(b0, &t);
1166 f25519_store(b1, &u);
e830bb69 1167 f25519_mul(&u, &w, SQRTM1);
25f67362
MW		 1168
1169 m = -ct_memeq(b0, xb, 32);
1170 rc = PICK2(0, rc, m);
e830bb69 1171 f25519_pick2(z, &w, &u, m);
25f67362
MW		 1172 m = -ct_memeq(b1, xb, 32);
1173 rc = PICK2(0, rc, m);
1174
1175 /* And we're done. */
1176 return (rc);
1177}
1178
ee39a683
MW		 1179/*----- Test rig ----------------------------------------------------------*/
1180
1181#ifdef TEST_RIG
1182
1183#include <mLib/report.h>
25f67362 1184#include <mLib/str.h>
ee39a683
MW		 1185#include <mLib/testrig.h>
1186
1187static void fixdstr(dstr *d)
1188{
1189 if (d->len > 32)
1190 die(1, "invalid length for f25519");
1191 else if (d->len < 32) {
1192 dstr_ensure(d, 32);
1193 memset(d->buf + d->len, 0, 32 - d->len);
1194 d->len = 32;
1195 }
1196}
1197
1198static void cvt_f25519(const char *buf, dstr *d)
1199{
1200 dstr dd = DSTR_INIT;
1201
1202 type_hex.cvt(buf, &dd); fixdstr(&dd);
1203 dstr_ensure(d, sizeof(f25519)); d->len = sizeof(f25519);
1204 f25519_load((f25519 *)d->buf, (const octet *)dd.buf);
1205 dstr_destroy(&dd);
1206}
1207
1208static void dump_f25519(dstr *d, FILE *fp)
1209 { fdump(stderr, "???", (const piece *)d->buf); }
1210
1211static void cvt_f25519_ref(const char *buf, dstr *d)
1212 { type_hex.cvt(buf, d); fixdstr(d); }
1213
1214static void dump_f25519_ref(dstr *d, FILE *fp)
1215{
1216 f25519 x;
1217
1218 f25519_load(&x, (const octet *)d->buf);
1219 fdump(stderr, "???", x.P);
1220}
1221
1222static int eq(const f25519 *x, dstr *d)
1223 { octet b[32]; f25519_store(b, x); return (memcmp(b, d->buf, 32) == 0); }
1224
1225static const test_type
1226 type_f25519 = { cvt_f25519, dump_f25519 },
1227 type_f25519_ref = { cvt_f25519_ref, dump_f25519_ref };
1228
1229#define TEST_UNOP(op) \
1230 static int vrf_##op(dstr dv[]) \
1231 { \
1232 f25519 *x = (f25519 *)dv[0].buf; \
1233 f25519 z, zz; \
1234 int ok = 1; \
1235 \
1236 f25519_##op(&z, x); \
1237 if (!eq(&z, &dv[1])) { \
1238 ok = 0; \
1239 fprintf(stderr, "failed!\n"); \
1240 fdump(stderr, "x", x->P); \
1241 fdump(stderr, "calc", z.P); \
1242 f25519_load(&zz, (const octet *)dv[1].buf); \
1243 fdump(stderr, "z", zz.P); \
1244 } \
1245 \
1246 return (ok); \
1247 }
1248
25f67362 1249TEST_UNOP(neg)
ee39a683
MW		 1250TEST_UNOP(sqr)
1251TEST_UNOP(inv)
1252
1253#define TEST_BINOP(op) \
1254 static int vrf_##op(dstr dv[]) \
1255 { \
1256 f25519 *x = (f25519 *)dv[0].buf, *y = (f25519 *)dv[1].buf; \
1257 f25519 z, zz; \
1258 int ok = 1; \
1259 \
1260 f25519_##op(&z, x, y); \
1261 if (!eq(&z, &dv[2])) { \
1262 ok = 0; \
1263 fprintf(stderr, "failed!\n"); \
1264 fdump(stderr, "x", x->P); \
1265 fdump(stderr, "y", y->P); \
1266 fdump(stderr, "calc", z.P); \
1267 f25519_load(&zz, (const octet *)dv[2].buf); \
1268 fdump(stderr, "z", zz.P); \
1269 } \
1270 \
1271 return (ok); \
1272 }
1273
1274TEST_BINOP(add)
1275TEST_BINOP(sub)
1276TEST_BINOP(mul)
1277
1278static int vrf_mulc(dstr dv[])
1279{
1280 f25519 *x = (f25519 *)dv[0].buf;
1281 long a = *(const long *)dv[1].buf;
1282 f25519 z, zz;
1283 int ok = 1;
1284
1285 f25519_mulconst(&z, x, a);
1286 if (!eq(&z, &dv[2])) {
1287 ok = 0;
1288 fprintf(stderr, "failed!\n");
1289 fdump(stderr, "x", x->P);
1290 fprintf(stderr, "a = %ld\n", a);
1291 fdump(stderr, "calc", z.P);
1292 f25519_load(&zz, (const octet *)dv[2].buf);
1293 fdump(stderr, "z", zz.P);
1294 }
1295
1296 return (ok);
1297}
1298
25f67362
MW		 1299static int vrf_condneg(dstr dv[])
1300{
1301 f25519 *x = (f25519 *)dv[0].buf;
1302 uint32 m = *(uint32 *)dv[1].buf;
1303 f25519 z;
1304 int ok = 1;
1305
1306 f25519_condneg(&z, x, m);
1307 if (!eq(&z, &dv[2])) {
1308 ok = 0;
1309 fprintf(stderr, "failed!\n");
1310 fdump(stderr, "x", x->P);
1311 fprintf(stderr, "m = 0x%08lx\n", (unsigned long)m);
1312 fdump(stderr, "calc z", z.P);
1313 f25519_load(&z, (const octet *)dv[1].buf);
1314 fdump(stderr, "want z", z.P);
1315 }
1316
1317 return (ok);
1318}
1319
1320static int vrf_pick2(dstr dv[])
1321{
1322 f25519 *x = (f25519 *)dv[0].buf, *y = (f25519 *)dv[1].buf;
1323 uint32 m = *(uint32 *)dv[2].buf;
1324 f25519 z;
1325 int ok = 1;
1326
1327 f25519_pick2(&z, x, y, m);
1328 if (!eq(&z, &dv[3])) {
1329 ok = 0;
1330 fprintf(stderr, "failed!\n");
1331 fdump(stderr, "x", x->P);
1332 fdump(stderr, "y", y->P);
1333 fprintf(stderr, "m = 0x%08lx\n", (unsigned long)m);
1334 fdump(stderr, "calc z", z.P);
1335 f25519_load(&z, (const octet *)dv[3].buf);
1336 fdump(stderr, "want z", z.P);
1337 }
1338
1339 return (ok);
1340}
1341
1342static int vrf_pickn(dstr dv[])
1343{
1344 dstr d = DSTR_INIT;
1345 f25519 v[32], z;
1346 size_t i = *(uint32 *)dv[1].buf, j, n;
1347 const char *p;
1348 char *q;
1349 int ok = 1;
1350
1351 for (q = dv[0].buf, n = 0; (p = str_qword(&q, 0)) != 0; n++)
1352 { cvt_f25519(p, &d); v[n] = *(f25519 *)d.buf; }
1353
1354 f25519_pickn(&z, v, n, i);
1355 if (!eq(&z, &dv[2])) {
1356 ok = 0;
1357 fprintf(stderr, "failed!\n");
1358 for (j = 0; j < n; j++) {
1359 fprintf(stderr, "v[%2u]", (unsigned)j);
1360 fdump(stderr, "", v[j].P);
1361 }
1362 fprintf(stderr, "i = %u\n", (unsigned)i);
1363 fdump(stderr, "calc z", z.P);
1364 f25519_load(&z, (const octet *)dv[2].buf);
1365 fdump(stderr, "want z", z.P);
1366 }
1367
1368 dstr_destroy(&d);
1369 return (ok);
1370}
1371
ee39a683
MW		 1372static int vrf_condswap(dstr dv[])
1373{
1374 f25519 *x = (f25519 *)dv[0].buf, *y = (f25519 *)dv[1].buf;
1375 f25519 xx = *x, yy = *y;
1376 uint32 m = *(uint32 *)dv[2].buf;
1377 int ok = 1;
1378
1379 f25519_condswap(&xx, &yy, m);
1380 if (!eq(&xx, &dv[3]) || !eq(&yy, &dv[4])) {
1381 ok = 0;
1382 fprintf(stderr, "failed!\n");
1383 fdump(stderr, "x", x->P);
1384 fdump(stderr, "y", y->P);
1385 fprintf(stderr, "m = 0x%08lx\n", (unsigned long)m);
1386 fdump(stderr, "calc xx", xx.P);
1387 fdump(stderr, "calc yy", yy.P);
1388 f25519_load(&xx, (const octet *)dv[3].buf);
1389 f25519_load(&yy, (const octet *)dv[4].buf);
1390 fdump(stderr, "want xx", xx.P);
1391 fdump(stderr, "want yy", yy.P);
1392 }
1393
1394 return (ok);
1395}
1396
25f67362
MW		 1397static int vrf_quosqrt(dstr dv[])
1398{
1399 f25519 *x = (f25519 *)dv[0].buf, *y = (f25519 *)dv[1].buf;
1400 f25519 z, zz;
1401 int rc;
1402 int ok = 1;
1403
1404 if (dv[2].len) { fixdstr(&dv[2]); fixdstr(&dv[3]); }
1405 rc = f25519_quosqrt(&z, x, y);
1406 if (!dv[2].len ? !rc : (rc || (!eq(&z, &dv[2]) && !eq(&z, &dv[3])))) {
1407 ok = 0;
1408 fprintf(stderr, "failed!\n");
1409 fdump(stderr, "x", x->P);
1410 fdump(stderr, "y", y->P);
1411 if (rc) fprintf(stderr, "calc: FAIL\n");
1412 else fdump(stderr, "calc", z.P);
1413 if (!dv[2].len)
1414 fprintf(stderr, "exp: FAIL\n");
1415 else {
1416 f25519_load(&zz, (const octet *)dv[2].buf);
1417 fdump(stderr, "z", zz.P);
1418 f25519_load(&zz, (const octet *)dv[3].buf);
1419 fdump(stderr, "z'", zz.P);
1420 }
1421 }
1422
1423 return (ok);
1424}
1425
ee39a683
MW		 1426static int vrf_sub_mulc_add_sub_mul(dstr dv[])
1427{
1428 f25519 *u = (f25519 *)dv[0].buf, *v = (f25519 *)dv[1].buf,
1429 *w = (f25519 *)dv[3].buf, *x = (f25519 *)dv[4].buf,
1430 *y = (f25519 *)dv[5].buf;
1431 long a = *(const long *)dv[2].buf;
1432 f25519 umv, aumv, wpaumv, xmy, z, zz;
1433 int ok = 1;
1434
1435 f25519_sub(&umv, u, v);
1436 f25519_mulconst(&aumv, &umv, a);
1437 f25519_add(&wpaumv, w, &aumv);
1438 f25519_sub(&xmy, x, y);
1439 f25519_mul(&z, &wpaumv, &xmy);
1440
1441 if (!eq(&z, &dv[6])) {
1442 ok = 0;
1443 fprintf(stderr, "failed!\n");
1444 fdump(stderr, "u", u->P);
1445 fdump(stderr, "v", v->P);
1446 fdump(stderr, "u - v", umv.P);
1447 fprintf(stderr, "a = %ld\n", a);
1448 fdump(stderr, "a (u - v)", aumv.P);
1449 fdump(stderr, "w + a (u - v)", wpaumv.P);
1450 fdump(stderr, "x", x->P);
1451 fdump(stderr, "y", y->P);
1452 fdump(stderr, "x - y", xmy.P);
1453 fdump(stderr, "(x - y) (w + a (u - v))", z.P);
1454 f25519_load(&zz, (const octet *)dv[6].buf); fdump(stderr, "z", zz.P);
1455 }
1456
1457 return (ok);
1458}
1459
1460static test_chunk tests[] = {
1461 { "add", vrf_add, { &type_f25519, &type_f25519, &type_f25519_ref } },
1462 { "sub", vrf_sub, { &type_f25519, &type_f25519, &type_f25519_ref } },
25f67362
MW		 1463 { "neg", vrf_neg, { &type_f25519, &type_f25519_ref } },
1464 { "condneg", vrf_condneg,
1465 { &type_f25519, &type_uint32, &type_f25519_ref } },
ee39a683
MW		 1466 { "mul", vrf_mul, { &type_f25519, &type_f25519, &type_f25519_ref } },
1467 { "mulconst", vrf_mulc, { &type_f25519, &type_long, &type_f25519_ref } },
25f67362
MW		 1468 { "pick2", vrf_pick2,
1469 { &type_f25519, &type_f25519, &type_uint32, &type_f25519_ref } },
1470 { "pickn", vrf_pickn,
1471 { &type_string, &type_uint32, &type_f25519_ref } },
ee39a683
MW		 1472 { "condswap", vrf_condswap,
1473 { &type_f25519, &type_f25519, &type_uint32,
1474 &type_f25519_ref, &type_f25519_ref } },
1475 { "sqr", vrf_sqr, { &type_f25519, &type_f25519_ref } },
1476 { "inv", vrf_inv, { &type_f25519, &type_f25519_ref } },
25f67362
MW		 1477 { "quosqrt", vrf_quosqrt,
1478 { &type_f25519, &type_f25519, &type_hex, &type_hex } },
ee39a683
MW		 1479 { "sub-mulc-add-sub-mul", vrf_sub_mulc_add_sub_mul,
1480 { &type_f25519, &type_f25519, &type_long, &type_f25519,
1481 &type_f25519, &type_f25519, &type_f25519_ref } },
1482 { 0, 0, { 0 } }
1483};
1484
1485int main(int argc, char *argv[])
1486{
1487 test_run(argc, argv, tests, SRCDIR "/t/f25519");
1488 return (0);
1489}
1490
1491#endif
1492
1493/*----- That's all, folks -------------------------------------------------*/
Catacomb cryptographic library