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