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