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1 | /* -*-c-*- |
2 | * |
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3 | * $Id: exp.h,v 1.3 2004/03/22 02:19:10 mdw Exp $ |
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4 | * |
5 | * Generalized exponentiation |
6 | * |
7 | * (c) 2001 Straylight/Edgeware |
8 | */ |
9 | |
10 | /*----- Licensing notice --------------------------------------------------* |
11 | * |
12 | * This file is part of Catacomb. |
13 | * |
14 | * Catacomb is free software; you can redistribute it and/or modify |
15 | * it under the terms of the GNU Library General Public License as |
16 | * published by the Free Software Foundation; either version 2 of the |
17 | * License, or (at your option) any later version. |
18 | * |
19 | * Catacomb is distributed in the hope that it will be useful, |
20 | * but WITHOUT ANY WARRANTY; without even the implied warranty of |
21 | * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
22 | * GNU Library General Public License for more details. |
23 | * |
24 | * You should have received a copy of the GNU Library General Public |
25 | * License along with Catacomb; if not, write to the Free |
26 | * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, |
27 | * MA 02111-1307, USA. |
28 | */ |
29 | |
30 | /*----- Revision history --------------------------------------------------* |
31 | * |
32 | * $Log: exp.h,v $ |
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33 | * Revision 1.3 2004/03/22 02:19:10 mdw |
34 | * Rationalise the sliding-window threshold. Drop guarantee that right |
35 | * arguments to EC @add@ are canonical, and fix up projective implementations |
36 | * to cope. |
37 | * |
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38 | * Revision 1.2 2004/03/21 22:52:06 mdw |
39 | * Merge and close elliptic curve branch. |
40 | * |
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41 | * Revision 1.1.4.1 2004/03/20 00:13:31 mdw |
42 | * Projective coordinates for prime curves |
43 | * |
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44 | * Revision 1.1 2001/06/16 13:00:59 mdw |
45 | * New generic exponentation code. Includes sliding-window simultaneous |
46 | * exponentiation. |
47 | * |
48 | */ |
49 | |
50 | #ifdef CATACOMB_EXP_H |
51 | # error "Multiple inclusion of <catacomb/exp.h>" |
52 | #endif |
53 | |
54 | #define CATACOMB_EXP_H |
55 | |
56 | #ifdef __cplusplus |
57 | extern "C" { |
58 | #endif |
59 | |
60 | /*----- Header files ------------------------------------------------------*/ |
61 | |
62 | #include <stddef.h> |
63 | |
64 | #include <mLib/alloc.h> |
65 | |
66 | #ifndef CATACOMB_MP_H |
67 | # include "mp.h" |
68 | #endif |
69 | |
70 | /*----- Data structures ---------------------------------------------------*/ |
71 | |
72 | typedef struct exp_simulscan { |
73 | mpw w; |
74 | size_t len; |
75 | const mpw *v; |
76 | } exp_simulscan; |
77 | |
78 | typedef struct exp_simul { |
79 | unsigned b; |
80 | size_t o, n; |
81 | exp_simulscan *s; |
82 | } exp_simul; |
83 | |
84 | /*----- Macros provided ---------------------------------------------------*/ |
85 | |
86 | /* --- Parameters --- */ |
87 | |
88 | #ifndef EXP_WINSZ /* Sliding window size */ |
89 | # define EXP_WINSZ 4 /* Predefine if you need to */ |
90 | #endif |
91 | |
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92 | /* --- These are determined from the window size --- * |
93 | * |
94 | * Given a %$k$%-bit exponent, I expect to do %$k/2$% multiplies if I use the |
95 | * simple way. If I use an n-bit sliding window, then I do %$2^n$% |
96 | * multiplies up front, but I only do %$(2^n - 1)/2^n k/n$% multiplies for |
97 | * the exponentiation. This is a win when |
98 | * |
99 | * %$k \ge \frac{n 2^{n+1}}{n - 2}$% |
100 | */ |
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101 | |
102 | #define EXP_TABSZ (1 << EXP_WINSZ) |
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103 | #define EXP_THRESH \ |
104 | ((EXP_WINSZ * (2 << EXP_WINSZ))/((EXP_WINSZ - 2) * MPW_BITS)) |
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105 | |
106 | /* --- Required operations --- * |
107 | * |
108 | * The macros here are independent of the underlying group elements. You |
109 | * must provide the necessary group operations and other definitions. The |
110 | * group operation is assumed to be written multiplicatively. |
111 | * |
112 | * @EXP_TYPE@ The type of a group element, e.g., @mp *@. |
113 | * |
114 | * @EXP_COPY(d, x)@ Makes @d@ be a copy of @x@. |
115 | * |
116 | * @EXP_DROP(x)@ Discards the element @x@, reclaiming any |
117 | * memory it used. |
118 | * |
119 | * @EXP_MUL(a, x)@ Multiplies @a@ by @x@ (writing the result |
120 | * back to @a@). |
121 | * |
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122 | * @EXP_FIX(x)@ Makes @x@ be a canonical representation of |
123 | * its value. All multiplications have the |
124 | * right argument canonical. |
125 | * |
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126 | * @EXP_SQR(a)@ Multiplies @a@ by itself. |
127 | * |
128 | * @EXP_SETMUL(d, x, y)@ Sets @d@ to be the product of @x@ and @y@. |
129 | * The value @d@ has not been initialized. |
130 | * |
131 | * @EXP_SETSQR(d, x)@ Sets @d@ to be the square of @x@. |
132 | * |
133 | * Only @EXP_TYPE@, @EXP_MUL@ and @EXP_SQR@ are required for simple |
134 | * exponentation. Sliding window and simultaneous exponentation require all |
135 | * of the operations. |
136 | */ |
137 | |
138 | #ifndef EXP_TYPE |
139 | # error "EXP_TYPE not defined for <catacomb/exp.h>" |
140 | #endif |
141 | |
142 | /* --- @EXP_SIMPLE@ --- * |
143 | * |
144 | * Arguments: @a@ = the result object, initially a multiplicative identity |
145 | * @g@ = the object to exponentiate |
146 | * @x@ = the exponent, as a multiprecision integer |
147 | * |
148 | * Use: Performs a simple left-to-right exponentiation. At the end |
149 | * of the code, the answer is left in @a@; @g@ and @x@ are |
150 | * unchanged. |
151 | */ |
152 | |
153 | #define EXP_SIMPLE(a, g, x) do { \ |
154 | mpscan sc; \ |
155 | unsigned sq = 0; \ |
156 | \ |
157 | /* --- Begin scanning --- */ \ |
158 | \ |
159 | mp_rscan(&sc, x); \ |
160 | if (!MP_RSTEP(&sc)) \ |
161 | goto exp_simple_exit; \ |
162 | while (!MP_RBIT(&sc)) \ |
163 | MP_RSTEP(&sc); \ |
164 | \ |
165 | /* --- Do the main body of the work --- */ \ |
166 | \ |
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167 | EXP_FIX(g); \ |
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168 | for (;;) { \ |
169 | EXP_MUL(a, g); \ |
170 | sq = 0; \ |
171 | for (;;) { \ |
172 | if (!MP_RSTEP(&sc)) \ |
173 | goto exp_simple_done; \ |
174 | sq++; \ |
175 | if (MP_RBIT(&sc)) \ |
176 | break; \ |
177 | } \ |
178 | while (sq--) EXP_SQR(a); \ |
179 | } \ |
180 | \ |
181 | /* --- Do a final round of squaring --- */ \ |
182 | \ |
183 | exp_simple_done: \ |
184 | while (sq--) EXP_SQR(a); \ |
185 | exp_simple_exit:; \ |
186 | } while (0) |
187 | |
188 | /* --- @EXP_WINDOW@ --- * |
189 | * |
190 | * Arguments: @a@ = the result object, initially a multiplicative identity |
191 | * @g@ = the object to exponentiate |
192 | * @x@ = the exponent, as a multiprecision integer |
193 | * |
194 | * Use: Performs a sliding-window exponentiation. At the end of the |
195 | * code, the answer is left in @a@; @g@ and @x@ are unchanged. |
196 | */ |
197 | |
198 | #define EXP_WINDOW(a, g, x) do { \ |
199 | EXP_TYPE *v; \ |
200 | EXP_TYPE g2; \ |
201 | unsigned i, sq = 0; \ |
202 | mpscan sc; \ |
203 | \ |
204 | /* --- Get going --- */ \ |
205 | \ |
206 | mp_rscan(&sc, x); \ |
207 | if (!MP_RSTEP(&sc)) \ |
208 | goto exp_window_exit; \ |
209 | \ |
210 | /* --- Do the precomputation --- */ \ |
211 | \ |
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212 | EXP_FIX(g); \ |
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213 | EXP_SETSQR(g2, g); \ |
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214 | EXP_FIX(g2); \ |
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215 | v = xmalloc(EXP_TABSZ * sizeof(EXP_TYPE)); \ |
216 | EXP_COPY(v[0], g); \ |
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217 | for (i = 1; i < EXP_TABSZ; i++) { \ |
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218 | EXP_SETMUL(v[i], v[i - 1], g2); \ |
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219 | EXP_FIX(v[i]); \ |
220 | } \ |
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221 | EXP_DROP(g2); \ |
222 | \ |
223 | /* --- Skip top-end zero bits --- * \ |
224 | * \ |
225 | * If the initial step worked, there must be a set bit somewhere, so \ |
226 | * keep stepping until I find it. \ |
227 | */ \ |
228 | \ |
229 | while (!MP_RBIT(&sc)) \ |
230 | MP_RSTEP(&sc); \ |
231 | \ |
232 | /* --- Now for the main work --- */ \ |
233 | \ |
234 | for (;;) { \ |
235 | unsigned l = 1; \ |
236 | unsigned z = 0; \ |
237 | \ |
238 | /* --- The next bit is set, so read a window index --- * \ |
239 | * \ |
240 | * Reset @i@ to zero and increment @sq@. Then, until either I read \ |
241 | * @WINSZ@ bits or I run out of bits, scan in a bit: if it's clear, \ |
242 | * bump the @z@ counter; if it's set, push a set bit into @i@, \ |
243 | * shift it over by @z@ bits, bump @sq@ by @z + 1@ and clear @z@. \ |
244 | * By the end of this palaver, @i@ is an index to the precomputed \ |
245 | * value in @v@. \ |
246 | */ \ |
247 | \ |
248 | i = 0; \ |
249 | sq++; \ |
250 | while (l < EXP_WINSZ && MP_RSTEP(&sc)) { \ |
251 | l++; \ |
252 | if (!MP_RBIT(&sc)) \ |
253 | z++; \ |
254 | else { \ |
255 | i = ((i << 1) | 1) << z; \ |
256 | sq += z + 1; \ |
257 | z = 0; \ |
258 | } \ |
259 | } \ |
260 | \ |
261 | /* --- Do the squaring --- * \ |
262 | * \ |
263 | * Remember that @sq@ carries over from the zero-skipping stuff \ |
264 | * below. \ |
265 | */ \ |
266 | \ |
267 | while (sq--) EXP_SQR(a); \ |
268 | \ |
269 | /* --- Do the multiply --- */ \ |
270 | \ |
271 | EXP_MUL(a, v[i]); \ |
272 | \ |
273 | /* --- Now grind along through the rest of the bits --- */ \ |
274 | \ |
275 | sq = z; \ |
276 | for (;;) { \ |
277 | if (!MP_RSTEP(&sc)) \ |
278 | goto exp_window_done; \ |
279 | if (MP_RBIT(&sc)) \ |
280 | break; \ |
281 | sq++; \ |
282 | } \ |
283 | } \ |
284 | \ |
285 | /* --- Do a final round of squaring --- */ \ |
286 | \ |
287 | exp_window_done: \ |
288 | while (sq--) EXP_SQR(a); \ |
289 | for (i = 0; i < EXP_TABSZ; i++) \ |
290 | EXP_DROP(v[i]); \ |
291 | xfree(v); \ |
292 | exp_window_exit:; \ |
293 | } while (0) |
294 | |
295 | /* --- @EXP_SIMUL@ --- * |
296 | * |
297 | * Arguments: @a@ = the result object, initially a multiplicative identity |
298 | * @f@ = pointer to a vector of base/exp pairs |
299 | * @n@ = the number of base/exp pairs |
300 | * |
301 | * Use: Performs a simultaneous sliding-window exponentiation. The |
302 | * @f@ table is an array of structures containing members @base@ |
303 | * of type @EXP_TYPE@, and @exp@ of type @mp *@. |
304 | */ |
305 | |
306 | #define EXP_SIMUL(a, f, n) do { \ |
307 | size_t i, j, jj, k; \ |
308 | size_t vn = 1 << (EXP_WINSZ * n), m = (1 << n) - 1; \ |
309 | EXP_TYPE *v = xmalloc(vn * sizeof(EXP_TYPE)); \ |
310 | exp_simul e; \ |
311 | unsigned sq = 0; \ |
312 | \ |
313 | /* --- Fill in the precomputed table --- */ \ |
314 | \ |
315 | j = 1; \ |
316 | for (i = 0; i < n; i++) { \ |
317 | EXP_COPY(v[j], f[n - 1 - i].base); \ |
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318 | EXP_FIX(v[j]); \ |
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319 | j <<= 1; \ |
320 | } \ |
321 | k = n * EXP_WINSZ; \ |
322 | jj = 1; \ |
323 | for (; i < k; i++) { \ |
324 | EXP_SETSQR(v[j], v[jj]); \ |
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325 | EXP_FIX(v[j]); \ |
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326 | j <<= 1; jj <<= 1; \ |
327 | } \ |
328 | for (i = 1; i < vn; i <<= 1) { \ |
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329 | for (j = 1; j < i; j++) { \ |
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330 | EXP_SETMUL(v[j + i], v[j], v[i]); \ |
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331 | EXP_FIX(v[j + i]); \ |
332 | } \ |
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333 | } \ |
334 | \ |
335 | /* --- Set up the bitscanners --- * \ |
336 | * \ |
337 | * Got to use custom scanners, to keep them all in sync. \ |
338 | */ \ |
339 | \ |
340 | e.n = n; \ |
341 | e.b = 0; \ |
342 | e.s = xmalloc(n * sizeof(*e.s)); \ |
343 | e.o = 0; \ |
344 | for (i = 0; i < n; i++) { \ |
345 | MP_SHRINK(f[i].exp); \ |
346 | e.s[i].len = MP_LEN(f[i].exp); \ |
347 | e.s[i].v = f[i].exp->v; \ |
348 | if (e.s[i].len > e.o) \ |
349 | e.o = e.s[i].len; \ |
350 | } \ |
351 | \ |
352 | /* --- Skip as far as a nonzero column in the exponent matrix --- */ \ |
353 | \ |
354 | do { \ |
355 | if (!e.o && !e.b) \ |
356 | goto exp_simul_done; \ |
357 | i = exp_simulnext(&e, 0); \ |
358 | } while (!(i & m)); \ |
359 | \ |
360 | /* --- Now for the main work --- */ \ |
361 | \ |
362 | for (;;) { \ |
363 | unsigned l = 1; \ |
364 | unsigned z = 0; \ |
365 | \ |
366 | /* --- Just read a nonzero column, so read a window index --- * \ |
367 | * \ |
368 | * Clear high bits of @i@ and increment @sq@. Then, until either I \ |
369 | * read @WINSZ@ columns or I run out, scan in a column and append \ |
370 | * it to @i@. If it's zero, bump the @z@ counter; if it's nonzero, \ |
371 | * bump @sq@ by @z + 1@ and clear @z@. By the end of this palaver, \ |
372 | * @i@ is an index to the precomputed value in @v@, followed by \ |
373 | * @n * z@ zero bits. \ |
374 | */ \ |
375 | \ |
376 | sq++; \ |
377 | while (l < EXP_WINSZ && (e.o || e.b)) { \ |
378 | l++; \ |
379 | i = exp_simulnext(&e, i); \ |
380 | if (!(i & m)) \ |
381 | z++; \ |
382 | else { \ |
383 | sq += z + 1; \ |
384 | z = 0; \ |
385 | } \ |
386 | } \ |
387 | \ |
388 | /* --- Do the squaring --- * \ |
389 | * \ |
390 | * Remember that @sq@ carries over from the zero-skipping stuff \ |
391 | * below. \ |
392 | */ \ |
393 | \ |
394 | while (sq--) EXP_SQR(a); \ |
395 | \ |
396 | /* --- Do the multiply --- */ \ |
397 | \ |
398 | i >>= (z * n); \ |
399 | EXP_MUL(a, v[i]); \ |
400 | \ |
401 | /* --- Now grind along through the rest of the bits --- */ \ |
402 | \ |
403 | sq = z; \ |
404 | for (;;) { \ |
405 | if (!e.o && !e.b) \ |
406 | goto exp_simul_done; \ |
407 | if ((i = exp_simulnext(&e, 0)) != 0) \ |
408 | break; \ |
409 | sq++; \ |
410 | } \ |
411 | } \ |
412 | \ |
413 | /* --- Do a final round of squaring --- */ \ |
414 | \ |
415 | exp_simul_done: \ |
416 | while (sq--) EXP_SQR(a); \ |
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417 | for (i = 1; i < vn; i++) \ |
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418 | EXP_DROP(v[i]); \ |
419 | xfree(v); \ |
420 | } while (0) |
421 | |
422 | /*----- Functions provided ------------------------------------------------*/ |
423 | |
424 | /* --- @exp_simulnext@ --- * |
425 | * |
426 | * Arguments: @exp_simul *e@ = pointer to state structure |
427 | * @size_t x@ = a current accumulator |
428 | * |
429 | * Returns: The next column of bits. |
430 | * |
431 | * Use: Scans the next column of bits for a simultaneous |
432 | * exponentiation. |
433 | */ |
434 | |
435 | extern size_t exp_simulnext(exp_simul */*e*/, size_t /*x*/); |
436 | |
437 | /*----- That's all, folks -------------------------------------------------*/ |
438 | |
439 | #ifdef __cplusplus |
440 | } |
441 | #endif |