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8a7654a6 MW |
1 | /* -*-c-*- |
2 | * | |
3 | * Arithmetic in the Goldilocks field GF(2^448 - 2^224 - 1) | |
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 "fgoldi.h" | |
50 | ||
51 | /*----- Basic setup -------------------------------------------------------* | |
52 | * | |
53 | * Let φ = 2^224; then p = φ^2 - φ - 1, and, in GF(p), we have φ^2 = φ + 1 | |
54 | * (hence the name). | |
55 | */ | |
56 | ||
a1a6042e MW |
57 | typedef fgoldi_piece piece; |
58 | ||
8a7654a6 MW |
59 | /* We represent an element of GF(p) as 16 28-bit signed integer pieces x_i: |
60 | * x = SUM_{0<=i<16} x_i 2^(28i). | |
61 | */ | |
62 | ||
a1a6042e | 63 | typedef int64 dblpiece; |
8a7654a6 MW |
64 | typedef uint32 upiece; typedef uint64 udblpiece; |
65 | #define PIECEWD(i) 28 | |
66 | #define NPIECE 16 | |
67 | #define P p28 | |
68 | ||
8a7654a6 MW |
69 | #define B27 0x08000000u |
70 | #define M28 0x0fffffffu | |
8a7654a6 MW |
71 | #define M32 0xffffffffu |
72 | ||
73 | /*----- Debugging machinery -----------------------------------------------*/ | |
74 | ||
75 | #if defined(FGOLDI_DEBUG) | |
76 | ||
77 | #include <stdio.h> | |
78 | ||
79 | #include "mp.h" | |
80 | #include "mptext.h" | |
81 | ||
82 | static mp *get_pgoldi(void) | |
83 | { | |
84 | mp *p = MP_NEW, *t = MP_NEW; | |
85 | ||
86 | p = mp_setbit(p, MP_ZERO, 448); | |
87 | t = mp_setbit(t, MP_ZERO, 224); | |
88 | p = mp_sub(p, p, t); | |
89 | p = mp_sub(p, p, MP_ONE); | |
90 | mp_drop(t); | |
91 | return (p); | |
92 | } | |
93 | ||
94 | DEF_FDUMP(fdump, piece, PIECEWD, NPIECE, 56, get_pgoldi()) | |
95 | ||
96 | #endif | |
97 | ||
98 | /*----- Loading and storing -----------------------------------------------*/ | |
99 | ||
100 | /* --- @fgoldi_load@ --- * | |
101 | * | |
102 | * Arguments: @fgoldi *z@ = where to store the result | |
103 | * @const octet xv[56]@ = source to read | |
104 | * | |
105 | * Returns: --- | |
106 | * | |
107 | * Use: Reads an element of %$\gf{2^{448} - 2^{224} - 19}$% in | |
108 | * external representation from @xv@ and stores it in @z@. | |
109 | * | |
110 | * External representation is little-endian base-256. Some | |
111 | * elements have multiple encodings, which are not produced by | |
112 | * correct software; use of noncanonical encodings is not an | |
113 | * error, and toleration of them is considered a performance | |
114 | * feature. | |
115 | */ | |
116 | ||
117 | void fgoldi_load(fgoldi *z, const octet xv[56]) | |
118 | { | |
119 | ||
120 | unsigned i; | |
121 | uint32 xw[14]; | |
122 | piece b, c; | |
123 | ||
124 | /* First, read the input value as words. */ | |
125 | for (i = 0; i < 14; i++) xw[i] = LOAD32_L(xv + 4*i); | |
126 | ||
127 | /* Extract unsigned 28-bit pieces from the words. */ | |
128 | z->P[ 0] = (xw[ 0] >> 0)&M28; | |
129 | z->P[ 7] = (xw[ 6] >> 4)&M28; | |
130 | z->P[ 8] = (xw[ 7] >> 0)&M28; | |
131 | z->P[15] = (xw[13] >> 4)&M28; | |
132 | for (i = 1; i < 7; i++) { | |
133 | z->P[i + 0] = ((xw[i + 0] << (4*i)) | (xw[i - 1] >> (32 - 4*i)))&M28; | |
134 | z->P[i + 8] = ((xw[i + 7] << (4*i)) | (xw[i + 6] >> (32 - 4*i)))&M28; | |
135 | } | |
136 | ||
137 | /* Convert the nonnegative pieces into a balanced signed representation, so | |
138 | * each piece ends up in the interval |z_i| <= 2^27. For each piece, if | |
139 | * its top bit is set, lend a bit leftwards; in the case of z_15, reduce | |
140 | * this bit by adding it onto z_0 and z_8, since this is the φ^2 bit, and | |
141 | * φ^2 = φ + 1. We delay this carry until after all of the pieces have | |
142 | * been balanced. If we don't do this, then we have to do a more expensive | |
143 | * test for nonzeroness to decide whether to lend a bit leftwards rather | |
144 | * than just testing a single bit. | |
145 | * | |
146 | * Note that we don't try for a canonical representation here: both upper | |
147 | * and lower bounds are achievable. | |
148 | */ | |
149 | b = z->P[15]&B27; z->P[15] -= b << 1; c = b >> 27; | |
150 | for (i = NPIECE - 1; i--; ) | |
151 | { b = z->P[i]&B27; z->P[i] -= b << 1; z->P[i + 1] += b >> 27; } | |
152 | z->P[0] += c; z->P[8] += c; | |
153 | } | |
154 | ||
155 | /* --- @fgoldi_store@ --- * | |
156 | * | |
157 | * Arguments: @octet zv[56]@ = where to write the result | |
158 | * @const fgoldi *x@ = the field element to write | |
159 | * | |
160 | * Returns: --- | |
161 | * | |
162 | * Use: Stores a field element in the given octet vector in external | |
163 | * representation. A canonical encoding is always stored. | |
164 | */ | |
165 | ||
166 | void fgoldi_store(octet zv[56], const fgoldi *x) | |
167 | { | |
168 | ||
169 | piece y[NPIECE], yy[NPIECE], c, d; | |
170 | uint32 u, v; | |
171 | mask32 m; | |
172 | unsigned i; | |
173 | ||
174 | for (i = 0; i < NPIECE; i++) y[i] = x->P[i]; | |
175 | ||
176 | /* First, propagate the carries. By the end of this, we'll have all of the | |
177 | * the pieces canonically sized and positive, and maybe there'll be | |
178 | * (signed) carry out. The carry c is in { -1, 0, +1 }, and the remaining | |
179 | * value will be in the half-open interval [0, φ^2). The whole represented | |
180 | * value is then y + φ^2 c. | |
181 | * | |
182 | * Assume that we start out with |y_i| <= 2^30. We start off by cutting | |
183 | * off and reducing the carry c_15 from the topmost piece, y_15. This | |
184 | * leaves 0 <= y_15 < 2^28; and we'll have |c_15| <= 4. We'll add this | |
185 | * onto y_0 and y_8, and propagate the carries. It's very clear that we'll | |
186 | * end up with |y + (φ + 1) c_15 - φ^2/2| << φ^2. | |
187 | * | |
188 | * Here, the y_i are signed, so we must be cautious about bithacking them. | |
189 | */ | |
190 | c = ASR(piece, y[15], 28); y[15] = (upiece)y[15]&M28; y[8] += c; | |
191 | for (i = 0; i < NPIECE; i++) | |
192 | { y[i] += c; c = ASR(piece, y[i], 28); y[i] = (upiece)y[i]&M28; } | |
193 | ||
194 | /* Now we have a slightly fiddly job to do. If c = +1, or if c = 0 and | |
195 | * y >= p, then we should subtract p from the whole value; if c = -1 then | |
196 | * we should add p; and otherwise we should do nothing. | |
197 | * | |
198 | * But conditional behaviour is bad, m'kay. So here's what we do instead. | |
199 | * | |
200 | * The first job is to sort out what we wanted to do. If c = -1 then we | |
201 | * want to (a) invert the constant addend and (b) feed in a carry-in; | |
202 | * otherwise, we don't. | |
203 | */ | |
204 | m = SIGN(c)&M28; | |
205 | d = m&1; | |
206 | ||
207 | /* Now do the addition/subtraction. Remember that all of the y_i are | |
208 | * nonnegative, so shifting and masking are safe and easy. | |
209 | */ | |
210 | d += y[0] + (1 ^ m); yy[0] = d&M28; d >>= 28; | |
211 | for (i = 1; i < 8; i++) | |
212 | { d += y[i] + m; yy[i] = d&M28; d >>= 28; } | |
213 | d += y[8] + (1 ^ m); yy[8] = d&M28; d >>= 28; | |
214 | for (i = 9; i < 16; i++) | |
215 | { d += y[i] + m; yy[i] = d&M28; d >>= 28; } | |
216 | ||
217 | /* The final carry-out is in d; since we only did addition, and the y_i are | |
218 | * nonnegative, then d is in { 0, 1 }. We want to keep y', rather than y, | |
219 | * if (a) c /= 0 (in which case we know that the old value was | |
220 | * unsatisfactory), or (b) if d = 1 (in which case, if c = 0, we know that | |
221 | * the subtraction didn't cause a borrow, so we must be in the case where | |
222 | * p <= y < φ^2. | |
223 | */ | |
224 | m = NONZEROP(c) | ~NONZEROP(d - 1); | |
225 | for (i = 0; i < NPIECE; i++) y[i] = (yy[i]&m) | (y[i]&~m); | |
226 | ||
227 | /* Extract 32-bit words from the value. */ | |
228 | for (i = 0; i < 7; i++) { | |
229 | u = ((y[i + 0] >> (4*i)) | ((uint32)y[i + 1] << (28 - 4*i)))&M32; | |
230 | v = ((y[i + 8] >> (4*i)) | ((uint32)y[i + 9] << (28 - 4*i)))&M32; | |
231 | STORE32_L(zv + 4*i, u); | |
232 | STORE32_L(zv + 4*i + 28, v); | |
233 | } | |
234 | } | |
235 | ||
236 | /* --- @fgoldi_set@ --- * | |
237 | * | |
238 | * Arguments: @fgoldi *z@ = where to write the result | |
239 | * @int a@ = a small-ish constant | |
240 | * | |
241 | * Returns: --- | |
242 | * | |
243 | * Use: Sets @z@ to equal @a@. | |
244 | */ | |
245 | ||
246 | void fgoldi_set(fgoldi *x, int a) | |
247 | { | |
248 | unsigned i; | |
249 | ||
250 | x->P[0] = a; | |
251 | for (i = 1; i < NPIECE; i++) x->P[i] = 0; | |
252 | } | |
253 | ||
254 | /*----- Basic arithmetic --------------------------------------------------*/ | |
255 | ||
256 | /* --- @fgoldi_add@ --- * | |
257 | * | |
258 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@) | |
259 | * @const fgoldi *x, *y@ = two operands | |
260 | * | |
261 | * Returns: --- | |
262 | * | |
263 | * Use: Set @z@ to the sum %$x + y$%. | |
264 | */ | |
265 | ||
266 | void fgoldi_add(fgoldi *z, const fgoldi *x, const fgoldi *y) | |
267 | { | |
268 | unsigned i; | |
269 | for (i = 0; i < NPIECE; i++) z->P[i] = x->P[i] + y->P[i]; | |
270 | } | |
271 | ||
272 | /* --- @fgoldi_sub@ --- * | |
273 | * | |
274 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@) | |
275 | * @const fgoldi *x, *y@ = two operands | |
276 | * | |
277 | * Returns: --- | |
278 | * | |
279 | * Use: Set @z@ to the difference %$x - y$%. | |
280 | */ | |
281 | ||
282 | void fgoldi_sub(fgoldi *z, const fgoldi *x, const fgoldi *y) | |
283 | { | |
284 | unsigned i; | |
285 | for (i = 0; i < NPIECE; i++) z->P[i] = x->P[i] - y->P[i]; | |
286 | } | |
287 | ||
a1a6042e MW |
288 | /* --- @fgoldi_neg@ --- * |
289 | * | |
290 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@) | |
291 | * @const fgoldi *x@ = an operand | |
292 | * | |
293 | * Returns: --- | |
294 | * | |
295 | * Use: Set @z = -x@. | |
296 | */ | |
297 | ||
298 | void fgoldi_neg(fgoldi *z, const fgoldi *x) | |
299 | { | |
300 | unsigned i; | |
301 | for (i = 0; i < NPIECE; i++) z->P[i] = -x->P[i]; | |
302 | } | |
303 | ||
8a7654a6 MW |
304 | /*----- Constant-time utilities -------------------------------------------*/ |
305 | ||
a1a6042e MW |
306 | /* --- @fgoldi_pick2@ --- * |
307 | * | |
308 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@) | |
309 | * @const fgoldi *x, *y@ = two operands | |
310 | * @uint32 m@ = a mask | |
311 | * | |
312 | * Returns: --- | |
313 | * | |
314 | * Use: If @m@ is zero, set @z = y@; if @m@ is all-bits-set, then set | |
315 | * @z = x@. If @m@ has some other value, then scramble @z@ in | |
316 | * an unhelpful way. | |
317 | */ | |
318 | ||
319 | void fgoldi_pick2(fgoldi *z, const fgoldi *x, const fgoldi *y, uint32 m) | |
320 | { | |
321 | mask32 mm = FIX_MASK32(m); | |
322 | unsigned i; | |
323 | for (i = 0; i < NPIECE; i++) z->P[i] = PICK2(x->P[i], y->P[i], mm); | |
324 | } | |
325 | ||
326 | /* --- @fgoldi_pickn@ --- * | |
327 | * | |
328 | * Arguments: @fgoldi *z@ = where to put the result | |
329 | * @const fgoldi *v@ = a table of entries | |
330 | * @size_t n@ = the number of entries in @v@ | |
331 | * @size_t i@ = an index | |
332 | * | |
333 | * Returns: --- | |
334 | * | |
335 | * Use: If @0 <= i < n < 32@ then set @z = v[i]@. If @n >= 32@ then | |
336 | * do something unhelpful; otherwise, if @i >= n@ then set @z@ | |
337 | * to zero. | |
338 | */ | |
339 | ||
340 | void fgoldi_pickn(fgoldi *z, const fgoldi *v, size_t n, size_t i) | |
341 | { | |
342 | uint32 b = (uint32)1 << (31 - i); | |
343 | mask32 m; | |
344 | unsigned j; | |
345 | ||
346 | for (j = 0; j < NPIECE; j++) z->P[j] = 0; | |
347 | while (n--) { | |
348 | m = SIGN(b); | |
349 | for (j = 0; j < NPIECE; j++) CONDPICK(z->P[j], v->P[j], m); | |
350 | v++; b <<= 1; | |
351 | } | |
352 | } | |
353 | ||
8a7654a6 MW |
354 | /* --- @fgoldi_condswap@ --- * |
355 | * | |
356 | * Arguments: @fgoldi *x, *y@ = two operands | |
357 | * @uint32 m@ = a mask | |
358 | * | |
359 | * Returns: --- | |
360 | * | |
361 | * Use: If @m@ is zero, do nothing; if @m@ is all-bits-set, then | |
362 | * exchange @x@ and @y@. If @m@ has some other value, then | |
363 | * scramble @x@ and @y@ in an unhelpful way. | |
364 | */ | |
365 | ||
366 | void fgoldi_condswap(fgoldi *x, fgoldi *y, uint32 m) | |
367 | { | |
368 | unsigned i; | |
369 | mask32 mm = FIX_MASK32(m); | |
370 | ||
371 | for (i = 0; i < NPIECE; i++) CONDSWAP(x->P[i], y->P[i], mm); | |
372 | } | |
373 | ||
a1a6042e MW |
374 | /* --- @fgoldi_condneg@ --- * |
375 | * | |
376 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@) | |
377 | * @const fgoldi *x@ = an operand | |
378 | * @uint32 m@ = a mask | |
379 | * | |
380 | * Returns: --- | |
381 | * | |
382 | * Use: If @m@ is zero, set @z = x@; if @m@ is all-bits-set, then set | |
383 | * @z = -x@. If @m@ has some other value then scramble @z@ in | |
384 | * an unhelpful way. | |
385 | */ | |
386 | ||
387 | void fgoldi_condneg(fgoldi *z, const fgoldi *x, uint32 m) | |
388 | { | |
389 | mask32 m_xor = FIX_MASK32(m); | |
390 | piece m_add = m&1; | |
391 | # define CONDNEG(x) (((x) ^ m_xor) + m_add) | |
392 | ||
393 | unsigned i; | |
394 | for (i = 0; i < NPIECE; i++) z->P[i] = CONDNEG(x->P[i]); | |
395 | ||
396 | #undef CONDNEG | |
397 | } | |
398 | ||
8a7654a6 MW |
399 | /*----- Multiplication ----------------------------------------------------*/ |
400 | ||
401 | /* Let B = 2^63 - 1 be the largest value such that +B and -B can be | |
402 | * represented in a double-precision piece. On entry, it must be the case | |
403 | * that |X_i| <= M <= B - 2^27 for some M. If this is the case, then, on | |
404 | * exit, we will have |Z_i| <= 2^27 + M/2^27. | |
405 | */ | |
406 | #define CARRY_REDUCE(z, x) do { \ | |
407 | dblpiece _t[NPIECE], _c; \ | |
408 | unsigned _i; \ | |
409 | \ | |
410 | /* Bias the input pieces. This keeps the carries and so on centred \ | |
411 | * around zero rather than biased positive. \ | |
412 | */ \ | |
413 | for (_i = 0; _i < NPIECE; _i++) _t[_i] = (x)[_i] + B27; \ | |
414 | \ | |
415 | /* Calculate the reduced pieces. Careful with the bithacking. */ \ | |
416 | _c = ASR(dblpiece, _t[15], 28); \ | |
417 | (z)[0] = (dblpiece)((udblpiece)_t[0]&M28) - B27 + _c; \ | |
418 | for (_i = 1; _i < NPIECE; _i++) { \ | |
419 | (z)[_i] = (dblpiece)((udblpiece)_t[_i]&M28) - B27 + \ | |
420 | ASR(dblpiece, _t[_i - 1], 28); \ | |
421 | } \ | |
422 | (z)[8] += _c; \ | |
423 | } while (0) | |
424 | ||
425 | /* --- @fgoldi_mulconst@ --- * | |
426 | * | |
427 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@) | |
428 | * @const fgoldi *x@ = an operand | |
429 | * @long a@ = a small-ish constant; %$|a| < 2^{20}$%. | |
430 | * | |
431 | * Returns: --- | |
432 | * | |
433 | * Use: Set @z@ to the product %$a x$%. | |
434 | */ | |
435 | ||
436 | void fgoldi_mulconst(fgoldi *z, const fgoldi *x, long a) | |
437 | { | |
438 | unsigned i; | |
439 | dblpiece zz[NPIECE], aa = a; | |
440 | ||
441 | for (i = 0; i < NPIECE; i++) zz[i] = aa*x->P[i]; | |
442 | CARRY_REDUCE(z->P, zz); | |
443 | } | |
444 | ||
445 | /* --- @fgoldi_mul@ --- * | |
446 | * | |
447 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@) | |
448 | * @const fgoldi *x, *y@ = two operands | |
449 | * | |
450 | * Returns: --- | |
451 | * | |
452 | * Use: Set @z@ to the product %$x y$%. | |
453 | */ | |
454 | ||
455 | void fgoldi_mul(fgoldi *z, const fgoldi *x, const fgoldi *y) | |
456 | { | |
457 | dblpiece zz[NPIECE], u[NPIECE]; | |
458 | piece ab[NPIECE/2], cd[NPIECE/2]; | |
459 | const piece | |
460 | *a = x->P + NPIECE/2, *b = x->P, | |
461 | *c = y->P + NPIECE/2, *d = y->P; | |
462 | unsigned i, j; | |
463 | ||
464 | # define M(x,i, y,j) ((dblpiece)(x)[i]*(y)[j]) | |
465 | ||
466 | /* Behold the magic. | |
467 | * | |
468 | * Write x = a φ + b, and y = c φ + d. Then x y = a c φ^2 + | |
469 | * (a d + b c) φ + b d. Karatsuba and Ofman observed that a d + b c = | |
470 | * (a + b) (c + d) - a c - b d, saving a multiplication, and Hamburg chose | |
471 | * the prime p so that φ^2 = φ + 1. So | |
472 | * | |
473 | * x y = ((a + b) (c + d) - b d) φ + a c + b d | |
474 | */ | |
475 | ||
476 | for (i = 0; i < NPIECE; i++) zz[i] = 0; | |
477 | ||
478 | /* Our first job will be to calculate (1 - φ) b d, and write the result | |
479 | * into z. As we do this, an interesting thing will happen. Write | |
480 | * b d = u φ + v; then (1 - φ) b d = u φ + v - u φ^2 - v φ = (1 - φ) v - u. | |
481 | * So, what we do is to write the product end-swapped and negated, and then | |
482 | * we'll subtract the (negated, remember) high half from the low half. | |
483 | */ | |
484 | for (i = 0; i < NPIECE/2; i++) { | |
485 | for (j = 0; j < NPIECE/2 - i; j++) | |
486 | zz[i + j + NPIECE/2] -= M(b,i, d,j); | |
487 | for (; j < NPIECE/2; j++) | |
488 | zz[i + j - NPIECE/2] -= M(b,i, d,j); | |
489 | } | |
490 | for (i = 0; i < NPIECE/2; i++) | |
491 | zz[i] -= zz[i + NPIECE/2]; | |
492 | ||
493 | /* Next, we add on a c. There are no surprises here. */ | |
494 | for (i = 0; i < NPIECE/2; i++) | |
495 | for (j = 0; j < NPIECE/2; j++) | |
496 | zz[i + j] += M(a,i, c,j); | |
497 | ||
498 | /* Now, calculate a + b and c + d. */ | |
499 | for (i = 0; i < NPIECE/2; i++) | |
500 | { ab[i] = a[i] + b[i]; cd[i] = c[i] + d[i]; } | |
501 | ||
502 | /* Finally (for the multiplication) we must add on (a + b) (c + d) φ. | |
503 | * Write (a + b) (c + d) as u φ + v; then we actually want u φ^2 + v φ = | |
504 | * v φ + (1 + φ) u. We'll store u in a temporary place and add it on | |
505 | * twice. | |
506 | */ | |
507 | for (i = 0; i < NPIECE; i++) u[i] = 0; | |
508 | for (i = 0; i < NPIECE/2; i++) { | |
509 | for (j = 0; j < NPIECE/2 - i; j++) | |
510 | zz[i + j + NPIECE/2] += M(ab,i, cd,j); | |
511 | for (; j < NPIECE/2; j++) | |
512 | u[i + j - NPIECE/2] += M(ab,i, cd,j); | |
513 | } | |
514 | for (i = 0; i < NPIECE/2; i++) | |
515 | { zz[i] += u[i]; zz[i + NPIECE/2] += u[i]; } | |
516 | ||
517 | #undef M | |
518 | ||
519 | /* That wraps it up for the multiplication. Let's figure out some bounds. | |
520 | * Fortunately, Karatsuba is a polynomial identity, so all of the pieces | |
521 | * end up the way they'd be if we'd done the thing the easy way, which | |
522 | * simplifies the analysis. Suppose we started with |x_i|, |y_i| <= 9/5 | |
523 | * 2^28. The overheads in the result are given by the coefficients of | |
524 | * | |
525 | * ((u^16 - 1)/(u - 1))^2 mod u^16 - u^8 - 1 | |
526 | * | |
527 | * the greatest of which is 38. So |z_i| <= 38*81/25*2^56 < 2^63. | |
528 | * | |
529 | * Anyway, a round of `CARRY_REDUCE' will leave us with |z_i| < 2^27 + | |
530 | * 2^36; and a second round will leave us with |z_i| < 2^27 + 512. | |
531 | */ | |
532 | for (i = 0; i < 2; i++) CARRY_REDUCE(zz, zz); | |
533 | for (i = 0; i < NPIECE; i++) z->P[i] = zz[i]; | |
534 | } | |
535 | ||
536 | /* --- @fgoldi_sqr@ --- * | |
537 | * | |
538 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@) | |
539 | * @const fgoldi *x@ = an operand | |
540 | * | |
541 | * Returns: --- | |
542 | * | |
543 | * Use: Set @z@ to the square %$x^2$%. | |
544 | */ | |
545 | ||
546 | void fgoldi_sqr(fgoldi *z, const fgoldi *x) | |
547 | { | |
548 | ||
549 | dblpiece zz[NPIECE], u[NPIECE]; | |
550 | piece ab[NPIECE]; | |
551 | const piece *a = x->P + NPIECE/2, *b = x->P; | |
552 | unsigned i, j; | |
553 | ||
554 | # define M(x,i, y,j) ((dblpiece)(x)[i]*(y)[j]) | |
555 | ||
556 | /* The magic is basically the same as `fgoldi_mul' above. We write | |
557 | * x = a φ + b and use Karatsuba and the special prime shape. This time, | |
558 | * we have | |
559 | * | |
560 | * x^2 = ((a + b)^2 - b^2) φ + a^2 + b^2 | |
561 | */ | |
562 | ||
563 | for (i = 0; i < NPIECE; i++) zz[i] = 0; | |
564 | ||
565 | /* Our first job will be to calculate (1 - φ) b^2, and write the result | |
566 | * into z. Again, this interacts pleasantly with the prime shape. | |
567 | */ | |
568 | for (i = 0; i < NPIECE/4; i++) { | |
569 | zz[2*i + NPIECE/2] -= M(b,i, b,i); | |
570 | for (j = i + 1; j < NPIECE/2 - i; j++) | |
571 | zz[i + j + NPIECE/2] -= 2*M(b,i, b,j); | |
572 | for (; j < NPIECE/2; j++) | |
573 | zz[i + j - NPIECE/2] -= 2*M(b,i, b,j); | |
574 | } | |
575 | for (; i < NPIECE/2; i++) { | |
576 | zz[2*i - NPIECE/2] -= M(b,i, b,i); | |
577 | for (j = i + 1; j < NPIECE/2; j++) | |
578 | zz[i + j - NPIECE/2] -= 2*M(b,i, b,j); | |
579 | } | |
580 | for (i = 0; i < NPIECE/2; i++) | |
581 | zz[i] -= zz[i + NPIECE/2]; | |
582 | ||
583 | /* Next, we add on a^2. There are no surprises here. */ | |
584 | for (i = 0; i < NPIECE/2; i++) { | |
585 | zz[2*i] += M(a,i, a,i); | |
586 | for (j = i + 1; j < NPIECE/2; j++) | |
587 | zz[i + j] += 2*M(a,i, a,j); | |
588 | } | |
589 | ||
590 | /* Now, calculate a + b. */ | |
591 | for (i = 0; i < NPIECE/2; i++) | |
592 | ab[i] = a[i] + b[i]; | |
593 | ||
594 | /* Finally (for the multiplication) we must add on (a + b)^2 φ. | |
595 | * Write (a + b)^2 as u φ + v; then we actually want (u + v) φ + u. We'll | |
596 | * store u in a temporary place and add it on twice. | |
597 | */ | |
598 | for (i = 0; i < NPIECE; i++) u[i] = 0; | |
599 | for (i = 0; i < NPIECE/4; i++) { | |
600 | zz[2*i + NPIECE/2] += M(ab,i, ab,i); | |
601 | for (j = i + 1; j < NPIECE/2 - i; j++) | |
602 | zz[i + j + NPIECE/2] += 2*M(ab,i, ab,j); | |
603 | for (; j < NPIECE/2; j++) | |
604 | u[i + j - NPIECE/2] += 2*M(ab,i, ab,j); | |
605 | } | |
606 | for (; i < NPIECE/2; i++) { | |
607 | u[2*i - NPIECE/2] += M(ab,i, ab,i); | |
608 | for (j = i + 1; j < NPIECE/2; j++) | |
609 | u[i + j - NPIECE/2] += 2*M(ab,i, ab,j); | |
610 | } | |
611 | for (i = 0; i < NPIECE/2; i++) | |
612 | { zz[i] += u[i]; zz[i + NPIECE/2] += u[i]; } | |
613 | ||
614 | #undef M | |
615 | ||
616 | /* Finally, carrying. */ | |
617 | for (i = 0; i < 2; i++) CARRY_REDUCE(zz, zz); | |
618 | for (i = 0; i < NPIECE; i++) z->P[i] = zz[i]; | |
619 | } | |
620 | ||
621 | /*----- More advanced operations ------------------------------------------*/ | |
622 | ||
623 | /* --- @fgoldi_inv@ --- * | |
624 | * | |
625 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@) | |
626 | * @const fgoldi *x@ = an operand | |
627 | * | |
628 | * Returns: --- | |
629 | * | |
630 | * Use: Stores in @z@ the multiplicative inverse %$x^{-1}$%. If | |
631 | * %$x = 0$% then @z@ is set to zero. This is considered a | |
632 | * feature. | |
633 | */ | |
634 | ||
635 | void fgoldi_inv(fgoldi *z, const fgoldi *x) | |
636 | { | |
637 | fgoldi t, u; | |
638 | unsigned i; | |
639 | ||
640 | #define SQRN(z, x, n) do { \ | |
641 | fgoldi_sqr((z), (x)); \ | |
642 | for (i = 1; i < (n); i++) fgoldi_sqr((z), (z)); \ | |
643 | } while (0) | |
644 | ||
645 | /* Calculate x^-1 = x^(p - 2) = x^(2^448 - 2^224 - 3), which also handles | |
646 | * x = 0 as intended. The addition chain is home-made. | |
647 | */ /* step | value */ | |
648 | fgoldi_sqr(&u, x); /* 1 | 2 */ | |
649 | fgoldi_mul(&t, &u, x); /* 2 | 3 */ | |
650 | SQRN(&u, &t, 2); /* 4 | 12 */ | |
651 | fgoldi_mul(&t, &u, &t); /* 5 | 15 */ | |
652 | SQRN(&u, &t, 4); /* 9 | 240 */ | |
653 | fgoldi_mul(&u, &u, &t); /* 10 | 255 = 2^8 - 1 */ | |
654 | SQRN(&u, &u, 4); /* 14 | 2^12 - 16 */ | |
655 | fgoldi_mul(&t, &u, &t); /* 15 | 2^12 - 1 */ | |
656 | SQRN(&u, &t, 12); /* 27 | 2^24 - 2^12 */ | |
657 | fgoldi_mul(&u, &u, &t); /* 28 | 2^24 - 1 */ | |
658 | SQRN(&u, &u, 12); /* 40 | 2^36 - 2^12 */ | |
659 | fgoldi_mul(&t, &u, &t); /* 41 | 2^36 - 1 */ | |
660 | fgoldi_sqr(&t, &t); /* 42 | 2^37 - 2 */ | |
661 | fgoldi_mul(&t, &t, x); /* 43 | 2^37 - 1 */ | |
662 | SQRN(&u, &t, 37); /* 80 | 2^74 - 2^37 */ | |
663 | fgoldi_mul(&u, &u, &t); /* 81 | 2^74 - 1 */ | |
664 | SQRN(&u, &u, 37); /* 118 | 2^111 - 2^37 */ | |
665 | fgoldi_mul(&t, &u, &t); /* 119 | 2^111 - 1 */ | |
666 | SQRN(&u, &t, 111); /* 230 | 2^222 - 2^111 */ | |
667 | fgoldi_mul(&t, &u, &t); /* 231 | 2^222 - 1 */ | |
668 | fgoldi_sqr(&u, &t); /* 232 | 2^223 - 2 */ | |
669 | fgoldi_mul(&u, &u, x); /* 233 | 2^223 - 1 */ | |
670 | SQRN(&u, &u, 223); /* 456 | 2^446 - 2^223 */ | |
671 | fgoldi_mul(&t, &u, &t); /* 457 | 2^446 - 2^222 - 1 */ | |
672 | SQRN(&t, &t, 2); /* 459 | 2^448 - 2^224 - 4 */ | |
673 | fgoldi_mul(z, &t, x); /* 460 | 2^448 - 2^224 - 3 */ | |
674 | ||
675 | #undef SQRN | |
676 | } | |
677 | ||
a1a6042e MW |
678 | /* --- @fgoldi_quosqrt@ --- * |
679 | * | |
680 | * Arguments: @fgoldi *z@ = where to put the result (may alias @x@ or @y@) | |
681 | * @const fgoldi *x, *y@ = two operands | |
682 | * | |
683 | * Returns: Zero if successful, @-1@ if %$x/y$% is not a square. | |
684 | * | |
685 | * Use: Stores in @z@ the one of the square roots %$\pm\sqrt{x/y}$%. | |
686 | * If %$x = y = 0% then the result is zero; if %$y = 0$% but %$x | |
687 | * \ne 0$% then the operation fails. If you wanted a specific | |
688 | * square root then you'll have to pick it yourself. | |
689 | */ | |
690 | ||
691 | int fgoldi_quosqrt(fgoldi *z, const fgoldi *x, const fgoldi *y) | |
692 | { | |
693 | fgoldi t, u, v; | |
694 | octet xb[56], b0[56]; | |
695 | int32 rc = -1; | |
696 | mask32 m; | |
697 | unsigned i; | |
698 | ||
699 | #define SQRN(z, x, n) do { \ | |
700 | fgoldi_sqr((z), (x)); \ | |
701 | for (i = 1; i < (n); i++) fgoldi_sqr((z), (z)); \ | |
702 | } while (0) | |
703 | ||
704 | /* This is, fortunately, significantly easier than the equivalent problem | |
705 | * in GF(2^255 - 19), since p == 3 (mod 4). | |
706 | * | |
707 | * If x/y is square, then so is v = y^2 x/y = x y, and therefore u has | |
708 | * order r = (p - 1)/2. Let w = v^{(p-3)/4}. Then w^2 = v^{(p-3)/2} = | |
709 | * u^{r-1} = 1/v = 1/x y. Clearly, then, (x w)^2 = x^2/x y = x/y, so x w | |
710 | * is one of the square roots we seek. | |
711 | * | |
712 | * The addition chain, then, is a prefix of the previous one. | |
713 | */ | |
714 | fgoldi_mul(&v, x, y); | |
715 | ||
716 | fgoldi_sqr(&u, &v); /* 1 | 2 */ | |
717 | fgoldi_mul(&t, &u, &v); /* 2 | 3 */ | |
718 | SQRN(&u, &t, 2); /* 4 | 12 */ | |
719 | fgoldi_mul(&t, &u, &t); /* 5 | 15 */ | |
720 | SQRN(&u, &t, 4); /* 9 | 240 */ | |
721 | fgoldi_mul(&u, &u, &t); /* 10 | 255 = 2^8 - 1 */ | |
722 | SQRN(&u, &u, 4); /* 14 | 2^12 - 16 */ | |
723 | fgoldi_mul(&t, &u, &t); /* 15 | 2^12 - 1 */ | |
724 | SQRN(&u, &t, 12); /* 27 | 2^24 - 2^12 */ | |
725 | fgoldi_mul(&u, &u, &t); /* 28 | 2^24 - 1 */ | |
726 | SQRN(&u, &u, 12); /* 40 | 2^36 - 2^12 */ | |
727 | fgoldi_mul(&t, &u, &t); /* 41 | 2^36 - 1 */ | |
728 | fgoldi_sqr(&t, &t); /* 42 | 2^37 - 2 */ | |
729 | fgoldi_mul(&t, &t, &v); /* 43 | 2^37 - 1 */ | |
730 | SQRN(&u, &t, 37); /* 80 | 2^74 - 2^37 */ | |
731 | fgoldi_mul(&u, &u, &t); /* 81 | 2^74 - 1 */ | |
732 | SQRN(&u, &u, 37); /* 118 | 2^111 - 2^37 */ | |
733 | fgoldi_mul(&t, &u, &t); /* 119 | 2^111 - 1 */ | |
734 | SQRN(&u, &t, 111); /* 230 | 2^222 - 2^111 */ | |
735 | fgoldi_mul(&t, &u, &t); /* 231 | 2^222 - 1 */ | |
736 | fgoldi_sqr(&u, &t); /* 232 | 2^223 - 2 */ | |
737 | fgoldi_mul(&u, &u, &v); /* 233 | 2^223 - 1 */ | |
738 | SQRN(&u, &u, 223); /* 456 | 2^446 - 2^223 */ | |
739 | fgoldi_mul(&t, &u, &t); /* 457 | 2^446 - 2^222 - 1 */ | |
740 | ||
741 | #undef SQRN | |
742 | ||
743 | /* Now we must decide whether the answer was right. We should have z^2 = | |
744 | * x/y, so y z^2 = x. | |
745 | * | |
746 | * The easiest way to compare is to encode. This isn't as wasteful as it | |
747 | * sounds: the hard part is normalizing the representations, which we have | |
748 | * to do anyway. | |
749 | */ | |
750 | fgoldi_mul(z, x, &t); | |
751 | fgoldi_sqr(&t, z); | |
752 | fgoldi_mul(&t, &t, y); | |
753 | fgoldi_store(xb, x); | |
754 | fgoldi_store(b0, &t); | |
755 | m = -consttime_memeq(xb, b0, 56); | |
756 | rc = PICK2(0, rc, m); | |
757 | return (rc); | |
758 | } | |
759 | ||
8a7654a6 | 760 | /*----- That's all, folks -------------------------------------------------*/ |