1 | |
---|
2 | |
---|
3 | #include <NTL/FFT.h> |
---|
4 | |
---|
5 | #include <NTL/new.h> |
---|
6 | |
---|
7 | NTL_START_IMPL |
---|
8 | |
---|
9 | long NumFFTPrimes = 0; |
---|
10 | |
---|
11 | long *FFTPrime = 0; |
---|
12 | long **RootTable = 0; |
---|
13 | long **RootInvTable = 0; |
---|
14 | long **TwoInvTable = 0; |
---|
15 | double *FFTPrimeInv = 0; |
---|
16 | |
---|
17 | |
---|
18 | static |
---|
19 | long IsFFTPrime(long n, long& w) |
---|
20 | { |
---|
21 | long m, x, y, z; |
---|
22 | long j, k; |
---|
23 | |
---|
24 | if (n % 3 == 0) return 0; |
---|
25 | |
---|
26 | if (n % 5 == 0) return 0; |
---|
27 | |
---|
28 | if (n % 7 == 0) return 0; |
---|
29 | |
---|
30 | m = n - 1; |
---|
31 | k = 0; |
---|
32 | while ((m & 1) == 0) { |
---|
33 | m = m >> 1; |
---|
34 | k++; |
---|
35 | } |
---|
36 | |
---|
37 | for (;;) { |
---|
38 | x = RandomBnd(n); |
---|
39 | |
---|
40 | if (x == 0) continue; |
---|
41 | z = PowerMod(x, m, n); |
---|
42 | if (z == 1) continue; |
---|
43 | |
---|
44 | x = z; |
---|
45 | j = 0; |
---|
46 | do { |
---|
47 | y = z; |
---|
48 | z = MulMod(y, y, n); |
---|
49 | j++; |
---|
50 | } while (j != k && z != 1); |
---|
51 | |
---|
52 | if (z != 1 || y != n-1) return 0; |
---|
53 | |
---|
54 | if (j == k) |
---|
55 | break; |
---|
56 | } |
---|
57 | |
---|
58 | /* x^{2^k} = 1 mod n, x^{2^{k-1}} = -1 mod n */ |
---|
59 | |
---|
60 | long TrialBound; |
---|
61 | |
---|
62 | TrialBound = m >> k; |
---|
63 | if (TrialBound > 0) { |
---|
64 | if (!ProbPrime(n, 5)) return 0; |
---|
65 | |
---|
66 | /* we have to do trial division by special numbers */ |
---|
67 | |
---|
68 | TrialBound = SqrRoot(TrialBound); |
---|
69 | |
---|
70 | long a, b; |
---|
71 | |
---|
72 | for (a = 1; a <= TrialBound; a++) { |
---|
73 | b = (a << k) + 1; |
---|
74 | if (n % b == 0) return 0; |
---|
75 | } |
---|
76 | } |
---|
77 | |
---|
78 | /* n is an FFT prime */ |
---|
79 | |
---|
80 | for (j = NTL_FFTMaxRoot; j < k; j++) |
---|
81 | x = MulMod(x, x, n); |
---|
82 | |
---|
83 | w = x; |
---|
84 | return 1; |
---|
85 | } |
---|
86 | |
---|
87 | |
---|
88 | static |
---|
89 | void NextFFTPrime(long& q, long& w) |
---|
90 | { |
---|
91 | static long m = NTL_FFTMaxRootBnd + 1; |
---|
92 | static long k = 0; |
---|
93 | |
---|
94 | long t, cand; |
---|
95 | |
---|
96 | for (;;) { |
---|
97 | if (k == 0) { |
---|
98 | m--; |
---|
99 | if (m < 5) Error("ran out of FFT primes"); |
---|
100 | k = 1L << (NTL_SP_NBITS-m-2); |
---|
101 | } |
---|
102 | |
---|
103 | k--; |
---|
104 | |
---|
105 | cand = (1L << (NTL_SP_NBITS-1)) + (k << (m+1)) + (1L << m) + 1; |
---|
106 | |
---|
107 | if (!IsFFTPrime(cand, t)) continue; |
---|
108 | q = cand; |
---|
109 | w = t; |
---|
110 | return; |
---|
111 | } |
---|
112 | } |
---|
113 | |
---|
114 | |
---|
115 | long CalcMaxRoot(long p) |
---|
116 | { |
---|
117 | p = p-1; |
---|
118 | long k = 0; |
---|
119 | while ((p & 1) == 0) { |
---|
120 | p = p >> 1; |
---|
121 | k++; |
---|
122 | } |
---|
123 | |
---|
124 | if (k > NTL_FFTMaxRoot) |
---|
125 | return NTL_FFTMaxRoot; |
---|
126 | else |
---|
127 | return k; |
---|
128 | } |
---|
129 | |
---|
130 | |
---|
131 | void UseFFTPrime(long index) |
---|
132 | { |
---|
133 | if (index < 0 || index > NumFFTPrimes) |
---|
134 | Error("invalid FFT prime index"); |
---|
135 | |
---|
136 | if (index < NumFFTPrimes) return; |
---|
137 | |
---|
138 | long q, w; |
---|
139 | |
---|
140 | NextFFTPrime(q, w); |
---|
141 | |
---|
142 | long mr = CalcMaxRoot(q); |
---|
143 | |
---|
144 | // tables are allocated in increments of 100 |
---|
145 | |
---|
146 | if (index == 0) { |
---|
147 | FFTPrime = (long *) NTL_MALLOC(100, sizeof(long), 0); |
---|
148 | RootTable = (long **) NTL_MALLOC(100, sizeof(long *), 0); |
---|
149 | RootInvTable = (long **) NTL_MALLOC(100, sizeof(long *), 0); |
---|
150 | TwoInvTable = (long **) NTL_MALLOC(100, sizeof(long *), 0); |
---|
151 | FFTPrimeInv = (double *) NTL_MALLOC(100, sizeof(double), 0); |
---|
152 | } |
---|
153 | else if ((index % 100) == 0) { |
---|
154 | FFTPrime = (long *) NTL_REALLOC(FFTPrime, index+100, sizeof(long), 0); |
---|
155 | RootTable = (long **) |
---|
156 | NTL_REALLOC(RootTable, index+100, sizeof(long *), 0); |
---|
157 | RootInvTable = (long **) |
---|
158 | NTL_REALLOC(RootInvTable, index+100, sizeof(long *), 0); |
---|
159 | TwoInvTable = (long **) |
---|
160 | NTL_REALLOC(TwoInvTable, index+100, sizeof(long *), 0); |
---|
161 | FFTPrimeInv = (double *) |
---|
162 | NTL_REALLOC(FFTPrimeInv, index+100, sizeof(double), 0); |
---|
163 | } |
---|
164 | |
---|
165 | if (!FFTPrime || !RootTable || !RootInvTable || !TwoInvTable || |
---|
166 | !FFTPrimeInv) |
---|
167 | Error("out of space"); |
---|
168 | |
---|
169 | FFTPrime[index] = q; |
---|
170 | |
---|
171 | long *rt, *rit, *tit; |
---|
172 | |
---|
173 | if (!(rt = RootTable[index] = (long*) NTL_MALLOC(mr+1, sizeof(long), 0))) |
---|
174 | Error("out of space"); |
---|
175 | if (!(rit = RootInvTable[index] = (long*) NTL_MALLOC(mr+1, sizeof(long), 0))) |
---|
176 | Error("out of space"); |
---|
177 | if (!(tit = TwoInvTable[index] = (long*) NTL_MALLOC(mr+1, sizeof(long), 0))) |
---|
178 | Error("out of space"); |
---|
179 | |
---|
180 | long j; |
---|
181 | long t; |
---|
182 | |
---|
183 | rt[mr] = w; |
---|
184 | for (j = mr-1; j >= 0; j--) |
---|
185 | rt[j] = MulMod(rt[j+1], rt[j+1], q); |
---|
186 | |
---|
187 | rit[mr] = InvMod(w, q); |
---|
188 | for (j = mr-1; j >= 0; j--) |
---|
189 | rit[j] = MulMod(rit[j+1], rit[j+1], q); |
---|
190 | |
---|
191 | t = InvMod(2, q); |
---|
192 | tit[0] = 1; |
---|
193 | for (j = 1; j <= mr; j++) |
---|
194 | tit[j] = MulMod(tit[j-1], t, q); |
---|
195 | |
---|
196 | FFTPrimeInv[index] = 1/double(q); |
---|
197 | |
---|
198 | NumFFTPrimes++; |
---|
199 | } |
---|
200 | |
---|
201 | |
---|
202 | static |
---|
203 | long RevInc(long a, long k) |
---|
204 | { |
---|
205 | long j, m; |
---|
206 | |
---|
207 | j = k; |
---|
208 | m = 1L << (k-1); |
---|
209 | |
---|
210 | while (j && (m & a)) { |
---|
211 | a ^= m; |
---|
212 | m >>= 1; |
---|
213 | j--; |
---|
214 | } |
---|
215 | if (j) a ^= m; |
---|
216 | return a; |
---|
217 | } |
---|
218 | |
---|
219 | static |
---|
220 | void BitReverseCopy(long *A, const long *a, long k) |
---|
221 | { |
---|
222 | static long* mem[NTL_FFTMaxRoot+1]; |
---|
223 | |
---|
224 | long n = 1L << k; |
---|
225 | long* rev; |
---|
226 | long i, j; |
---|
227 | |
---|
228 | rev = mem[k]; |
---|
229 | if (!rev) { |
---|
230 | rev = mem[k] = (long *) NTL_MALLOC(n, sizeof(long), 0); |
---|
231 | if (!rev) Error("out of memory in BitReverseCopy"); |
---|
232 | for (i = 0, j = 0; i < n; i++, j = RevInc(j, k)) |
---|
233 | rev[i] = j; |
---|
234 | } |
---|
235 | |
---|
236 | for (i = 0; i < n; i++) |
---|
237 | A[rev[i]] = a[i]; |
---|
238 | } |
---|
239 | |
---|
240 | |
---|
241 | #ifdef NTL_FFT_PIPELINE |
---|
242 | |
---|
243 | /***************************************************** |
---|
244 | |
---|
245 | This version of the FFT is written with an explicit |
---|
246 | "software pipeline", which sometimes speeds things up. |
---|
247 | |
---|
248 | *******************************************************/ |
---|
249 | |
---|
250 | void FFT(long* A, const long* a, long k, long q, const long* root) |
---|
251 | |
---|
252 | // performs a 2^k-point convolution modulo q |
---|
253 | |
---|
254 | { |
---|
255 | if (k == 0) { |
---|
256 | A[0] = a[0]; |
---|
257 | return; |
---|
258 | } |
---|
259 | |
---|
260 | if (k == 1) { |
---|
261 | A[0] = AddMod(a[0], a[1], q); |
---|
262 | A[1] = SubMod(a[0], a[1], q); |
---|
263 | return; |
---|
264 | } |
---|
265 | |
---|
266 | // assume k > 1 |
---|
267 | |
---|
268 | long n = 1L << k; |
---|
269 | long s, m, m2, j; |
---|
270 | long t, u, v, w, z, tt; |
---|
271 | long *p1, *p, *ub, *ub1; |
---|
272 | double qinv = ((double) 1)/((double) q); |
---|
273 | double wqinv, zqinv; |
---|
274 | |
---|
275 | BitReverseCopy(A, a, k); |
---|
276 | |
---|
277 | ub = A+n; |
---|
278 | |
---|
279 | p = A; |
---|
280 | while (p < ub) { |
---|
281 | u = *p; |
---|
282 | v = *(p+1); |
---|
283 | *p = AddMod(u, v, q); |
---|
284 | *(p+1) = SubMod(u, v, q); |
---|
285 | p += 2; |
---|
286 | } |
---|
287 | |
---|
288 | for (s = 2; s < k; s++) { |
---|
289 | m = 1L << s; |
---|
290 | m2 = m >> 1; |
---|
291 | |
---|
292 | p = A; |
---|
293 | while (p < ub) { |
---|
294 | u = *p; |
---|
295 | v = *(p+m2); |
---|
296 | *p = AddMod(u, v, q); |
---|
297 | *(p+m2) = SubMod(u, v, q); |
---|
298 | p += m; |
---|
299 | } |
---|
300 | |
---|
301 | z = root[s]; |
---|
302 | w = z; |
---|
303 | for (j = 1; j < m2; j++) { |
---|
304 | wqinv = ((double) w)*qinv; |
---|
305 | p = A + j; |
---|
306 | p1 = p + m2; |
---|
307 | |
---|
308 | ub1 = ub-m; |
---|
309 | |
---|
310 | u = *p; |
---|
311 | t = MulMod2(*p1, w, q, wqinv); |
---|
312 | |
---|
313 | while (p < ub1) { |
---|
314 | tt = MulMod2(*(p1+m), w, q, wqinv); |
---|
315 | *p = AddMod(u, t, q); |
---|
316 | *p1 = SubMod(u, t, q); |
---|
317 | p1 += m; |
---|
318 | p += m; |
---|
319 | u = *p; |
---|
320 | t = tt; |
---|
321 | } |
---|
322 | *p = AddMod(u, t, q); |
---|
323 | *p1 = SubMod(u, t, q); |
---|
324 | |
---|
325 | w = MulMod2(z, w, q, wqinv); |
---|
326 | } |
---|
327 | } |
---|
328 | |
---|
329 | m2 = n >> 1; |
---|
330 | z = root[k]; |
---|
331 | zqinv = ((double) z)*qinv; |
---|
332 | w = 1; |
---|
333 | p = A; |
---|
334 | p1 = A + m2; |
---|
335 | m2--; |
---|
336 | u = *p; |
---|
337 | t = *p1; |
---|
338 | while (m2) { |
---|
339 | w = MulMod2(w, z, q, zqinv); |
---|
340 | tt = MulMod(*(p1+1), w, q, qinv); |
---|
341 | *p = AddMod(u, t, q); |
---|
342 | *p1 = SubMod(u, t, q); |
---|
343 | p++; |
---|
344 | p1++; |
---|
345 | u = *p; |
---|
346 | t = tt; |
---|
347 | m2--; |
---|
348 | } |
---|
349 | *p = AddMod(u, t, q); |
---|
350 | *p1 = SubMod(u, t, q); |
---|
351 | } |
---|
352 | |
---|
353 | |
---|
354 | |
---|
355 | #else |
---|
356 | |
---|
357 | |
---|
358 | /***************************************************** |
---|
359 | |
---|
360 | This version of the FFT has no "software pipeline". |
---|
361 | |
---|
362 | ******************************************************/ |
---|
363 | |
---|
364 | |
---|
365 | |
---|
366 | void FFT(long* A, const long* a, long k, long q, const long* root) |
---|
367 | |
---|
368 | // performs a 2^k-point convolution modulo q |
---|
369 | |
---|
370 | { |
---|
371 | if (k == 0) { |
---|
372 | A[0] = a[0]; |
---|
373 | return; |
---|
374 | } |
---|
375 | |
---|
376 | if (k == 1) { |
---|
377 | A[0] = AddMod(a[0], a[1], q); |
---|
378 | A[1] = SubMod(a[0], a[1], q); |
---|
379 | return; |
---|
380 | } |
---|
381 | |
---|
382 | // assume k > 1 |
---|
383 | |
---|
384 | long n = 1L << k; |
---|
385 | long s, m, m2, j; |
---|
386 | long t, u, v, w, z; |
---|
387 | long *p, *ub, *p1, *ub1; |
---|
388 | double qinv = ((double) 1)/((double) q); |
---|
389 | double wqinv, zqinv; |
---|
390 | |
---|
391 | BitReverseCopy(A, a, k); |
---|
392 | |
---|
393 | ub = A+n; |
---|
394 | |
---|
395 | p = A; |
---|
396 | while (p < ub) { |
---|
397 | u = *p; |
---|
398 | v = *(p+1); |
---|
399 | *p = AddMod(u, v, q); |
---|
400 | *(p+1) = SubMod(u, v, q); |
---|
401 | p += 2; |
---|
402 | } |
---|
403 | |
---|
404 | for (s = 2; s < k; s++) { |
---|
405 | m = 1L << s; |
---|
406 | m2 = m >> 1; |
---|
407 | |
---|
408 | p = A; |
---|
409 | while (p < ub) { |
---|
410 | u = *p; |
---|
411 | v = *(p+m2); |
---|
412 | *p = AddMod(u, v, q); |
---|
413 | *(p+m2) = SubMod(u, v, q); |
---|
414 | p += m; |
---|
415 | } |
---|
416 | |
---|
417 | z = root[s]; |
---|
418 | w = z; |
---|
419 | for (j = 1; j < m2; j++) { |
---|
420 | wqinv = ((double) w)*qinv; |
---|
421 | p = A + j; |
---|
422 | p1 = p + m2; |
---|
423 | ub1 = ub-m; |
---|
424 | |
---|
425 | while (p < ub1) { |
---|
426 | u = *p; |
---|
427 | v = *p1; |
---|
428 | t = MulMod2(v, w, q, wqinv); |
---|
429 | *p = AddMod(u, t, q); |
---|
430 | *p1 = SubMod(u, t, q); |
---|
431 | p += m; |
---|
432 | p1 += m; |
---|
433 | } |
---|
434 | |
---|
435 | u = *p; |
---|
436 | v = *p1; |
---|
437 | t = MulMod2(v, w, q, wqinv); |
---|
438 | *p = AddMod(u, t, q); |
---|
439 | *p1 = SubMod(u, t, q); |
---|
440 | |
---|
441 | w = MulMod2(z, w, q, wqinv); |
---|
442 | } |
---|
443 | } |
---|
444 | |
---|
445 | m2 = n >> 1; |
---|
446 | z = root[k]; |
---|
447 | zqinv = ((double) z)*qinv; |
---|
448 | w = 1; |
---|
449 | p = A; |
---|
450 | for (j = 0; j < m2; j++) { |
---|
451 | u = *p; |
---|
452 | v = *(p+m2); |
---|
453 | t = MulMod(v, w, q, qinv); |
---|
454 | *p = AddMod(u, t, q); |
---|
455 | *(p+m2) = SubMod(u, t, q); |
---|
456 | w = MulMod2(w, z, q, zqinv); |
---|
457 | p++; |
---|
458 | } |
---|
459 | } |
---|
460 | |
---|
461 | |
---|
462 | #endif |
---|
463 | |
---|
464 | NTL_END_IMPL |
---|