1 /* mpihelp-div.c - MPI helper functions
2 * Copyright (C) 1994, 1996 Free Software Foundation, Inc.
3 * Copyright (C) 1998, 1999 Free Software Foundation, Inc.
4 *
5 * This file is part of GnuPG.
6 *
7 * GnuPG is free software; you can redistribute it and/or modify
8 * it under the terms of the GNU General Public License as published by
9 * the Free Software Foundation; either version 2 of the License, or
10 * (at your option) any later version.
11 *
12 * GnuPG is distributed in the hope that it will be useful,
13 * but WITHOUT ANY WARRANTY; without even the implied warranty of
14 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
15 * GNU General Public License for more details.
16 *
17 * You should have received a copy of the GNU General Public License
18 * along with this program; if not, write to the Free Software
19 * Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA
20 *
21 * Note: This code is heavily based on the GNU MP Library.
22 * Actually it's the same code with only minor changes in the
23 * way the data is stored; this is to support the abstraction
24 * of an optional secure memory allocation which may be used
25 * to avoid revealing of sensitive data due to paging etc.
26 * The GNU MP Library itself is published under the LGPL;
27 * however I decided to publish this code under the plain GPL.
28 */
29
30 #include "mpi-internal.h"
31 #include "longlong.h"
32
33 #ifndef UMUL_TIME
34 #define UMUL_TIME 1
35 #endif
36 #ifndef UDIV_TIME
37 #define UDIV_TIME UMUL_TIME
38 #endif
39
40 /* FIXME: We should be using invert_limb (or invert_normalized_limb)
41 * here (not udiv_qrnnd).
42 */
43
44 mpi_limb_t
mpihelp_mod_1(mpi_ptr_t dividend_ptr,mpi_size_t dividend_size,mpi_limb_t divisor_limb)45 mpihelp_mod_1(mpi_ptr_t dividend_ptr, mpi_size_t dividend_size,
46 mpi_limb_t divisor_limb)
47 {
48 mpi_size_t i;
49 mpi_limb_t n1, n0, r;
50 int dummy;
51
52 /* Botch: Should this be handled at all? Rely on callers? */
53 if (!dividend_size)
54 return 0;
55
56 /* If multiplication is much faster than division, and the
57 * dividend is large, pre-invert the divisor, and use
58 * only multiplications in the inner loop.
59 *
60 * This test should be read:
61 * Does it ever help to use udiv_qrnnd_preinv?
62 * && Does what we save compensate for the inversion overhead?
63 */
64 if (UDIV_TIME > (2 * UMUL_TIME + 6)
65 && (UDIV_TIME - (2 * UMUL_TIME + 6)) * dividend_size > UDIV_TIME) {
66 int normalization_steps;
67
68 count_leading_zeros(normalization_steps, divisor_limb);
69 if (normalization_steps) {
70 mpi_limb_t divisor_limb_inverted;
71
72 divisor_limb <<= normalization_steps;
73
74 /* Compute (2**2N - 2**N * DIVISOR_LIMB) / DIVISOR_LIMB. The
75 * result is a (N+1)-bit approximation to 1/DIVISOR_LIMB, with the
76 * most significant bit (with weight 2**N) implicit.
77 *
78 * Special case for DIVISOR_LIMB == 100...000.
79 */
80 if (!(divisor_limb << 1))
81 divisor_limb_inverted = ~(mpi_limb_t) 0;
82 else
83 udiv_qrnnd(divisor_limb_inverted, dummy,
84 -divisor_limb, 0, divisor_limb);
85
86 n1 = dividend_ptr[dividend_size - 1];
87 r = n1 >> (BITS_PER_MPI_LIMB - normalization_steps);
88
89 /* Possible optimization:
90 * if (r == 0
91 * && divisor_limb > ((n1 << normalization_steps)
92 * | (dividend_ptr[dividend_size - 2] >> ...)))
93 * ...one division less...
94 */
95 for (i = dividend_size - 2; i >= 0; i--) {
96 n0 = dividend_ptr[i];
97 UDIV_QRNND_PREINV(dummy, r, r,
98 ((n1 << normalization_steps)
99 | (n0 >>
100 (BITS_PER_MPI_LIMB -
101 normalization_steps))),
102 divisor_limb,
103 divisor_limb_inverted);
104 n1 = n0;
105 }
106 UDIV_QRNND_PREINV(dummy, r, r,
107 n1 << normalization_steps,
108 divisor_limb, divisor_limb_inverted);
109 return r >> normalization_steps;
110 } else {
111 mpi_limb_t divisor_limb_inverted;
112
113 /* Compute (2**2N - 2**N * DIVISOR_LIMB) / DIVISOR_LIMB. The
114 * result is a (N+1)-bit approximation to 1/DIVISOR_LIMB, with the
115 * most significant bit (with weight 2**N) implicit.
116 *
117 * Special case for DIVISOR_LIMB == 100...000.
118 */
119 if (!(divisor_limb << 1))
120 divisor_limb_inverted = ~(mpi_limb_t) 0;
121 else
122 udiv_qrnnd(divisor_limb_inverted, dummy,
123 -divisor_limb, 0, divisor_limb);
124
125 i = dividend_size - 1;
126 r = dividend_ptr[i];
127
128 if (r >= divisor_limb)
129 r = 0;
130 else
131 i--;
132
133 for (; i >= 0; i--) {
134 n0 = dividend_ptr[i];
135 UDIV_QRNND_PREINV(dummy, r, r,
136 n0, divisor_limb,
137 divisor_limb_inverted);
138 }
139 return r;
140 }
141 } else {
142 if (UDIV_NEEDS_NORMALIZATION) {
143 int normalization_steps;
144
145 count_leading_zeros(normalization_steps, divisor_limb);
146 if (normalization_steps) {
147 divisor_limb <<= normalization_steps;
148
149 n1 = dividend_ptr[dividend_size - 1];
150 r = n1 >> (BITS_PER_MPI_LIMB -
151 normalization_steps);
152
153 /* Possible optimization:
154 * if (r == 0
155 * && divisor_limb > ((n1 << normalization_steps)
156 * | (dividend_ptr[dividend_size - 2] >> ...)))
157 * ...one division less...
158 */
159 for (i = dividend_size - 2; i >= 0; i--) {
160 n0 = dividend_ptr[i];
161 udiv_qrnnd(dummy, r, r,
162 ((n1 << normalization_steps)
163 | (n0 >>
164 (BITS_PER_MPI_LIMB -
165 normalization_steps))),
166 divisor_limb);
167 n1 = n0;
168 }
169 udiv_qrnnd(dummy, r, r,
170 n1 << normalization_steps,
171 divisor_limb);
172 return r >> normalization_steps;
173 }
174 }
175 /* No normalization needed, either because udiv_qrnnd doesn't require
176 * it, or because DIVISOR_LIMB is already normalized. */
177 i = dividend_size - 1;
178 r = dividend_ptr[i];
179
180 if (r >= divisor_limb)
181 r = 0;
182 else
183 i--;
184
185 for (; i >= 0; i--) {
186 n0 = dividend_ptr[i];
187 udiv_qrnnd(dummy, r, r, n0, divisor_limb);
188 }
189 return r;
190 }
191 }
192
193 /* Divide num (NP/NSIZE) by den (DP/DSIZE) and write
194 * the NSIZE-DSIZE least significant quotient limbs at QP
195 * and the DSIZE long remainder at NP. If QEXTRA_LIMBS is
196 * non-zero, generate that many fraction bits and append them after the
197 * other quotient limbs.
198 * Return the most significant limb of the quotient, this is always 0 or 1.
199 *
200 * Preconditions:
201 * 0. NSIZE >= DSIZE.
202 * 1. The most significant bit of the divisor must be set.
203 * 2. QP must either not overlap with the input operands at all, or
204 * QP + DSIZE >= NP must hold true. (This means that it's
205 * possible to put the quotient in the high part of NUM, right after the
206 * remainder in NUM.
207 * 3. NSIZE >= DSIZE, even if QEXTRA_LIMBS is non-zero.
208 */
209
210 mpi_limb_t
mpihelp_divrem(mpi_ptr_t qp,mpi_size_t qextra_limbs,mpi_ptr_t np,mpi_size_t nsize,mpi_ptr_t dp,mpi_size_t dsize)211 mpihelp_divrem(mpi_ptr_t qp, mpi_size_t qextra_limbs,
212 mpi_ptr_t np, mpi_size_t nsize, mpi_ptr_t dp, mpi_size_t dsize)
213 {
214 mpi_limb_t most_significant_q_limb = 0;
215
216 switch (dsize) {
217 case 0:
218 /* We are asked to divide by zero, so go ahead and do it! (To make
219 the compiler not remove this statement, return the value.) */
220 /*
221 * existing clients of this function have been modified
222 * not to call it with dsize == 0, so this should not happen
223 */
224 return 1 / dsize;
225
226 case 1:
227 {
228 mpi_size_t i;
229 mpi_limb_t n1;
230 mpi_limb_t d;
231
232 d = dp[0];
233 n1 = np[nsize - 1];
234
235 if (n1 >= d) {
236 n1 -= d;
237 most_significant_q_limb = 1;
238 }
239
240 qp += qextra_limbs;
241 for (i = nsize - 2; i >= 0; i--)
242 udiv_qrnnd(qp[i], n1, n1, np[i], d);
243 qp -= qextra_limbs;
244
245 for (i = qextra_limbs - 1; i >= 0; i--)
246 udiv_qrnnd(qp[i], n1, n1, 0, d);
247
248 np[0] = n1;
249 }
250 break;
251
252 case 2:
253 {
254 mpi_size_t i;
255 mpi_limb_t n1, n0, n2;
256 mpi_limb_t d1, d0;
257
258 np += nsize - 2;
259 d1 = dp[1];
260 d0 = dp[0];
261 n1 = np[1];
262 n0 = np[0];
263
264 if (n1 >= d1 && (n1 > d1 || n0 >= d0)) {
265 sub_ddmmss(n1, n0, n1, n0, d1, d0);
266 most_significant_q_limb = 1;
267 }
268
269 for (i = qextra_limbs + nsize - 2 - 1; i >= 0; i--) {
270 mpi_limb_t q;
271 mpi_limb_t r;
272
273 if (i >= qextra_limbs)
274 np--;
275 else
276 np[0] = 0;
277
278 if (n1 == d1) {
279 /* Q should be either 111..111 or 111..110. Need special
280 * treatment of this rare case as normal division would
281 * give overflow. */
282 q = ~(mpi_limb_t) 0;
283
284 r = n0 + d1;
285 if (r < d1) { /* Carry in the addition? */
286 add_ssaaaa(n1, n0, r - d0,
287 np[0], 0, d0);
288 qp[i] = q;
289 continue;
290 }
291 n1 = d0 - (d0 != 0 ? 1 : 0);
292 n0 = -d0;
293 } else {
294 udiv_qrnnd(q, r, n1, n0, d1);
295 umul_ppmm(n1, n0, d0, q);
296 }
297
298 n2 = np[0];
299 q_test:
300 if (n1 > r || (n1 == r && n0 > n2)) {
301 /* The estimated Q was too large. */
302 q--;
303 sub_ddmmss(n1, n0, n1, n0, 0, d0);
304 r += d1;
305 if (r >= d1) /* If not carry, test Q again. */
306 goto q_test;
307 }
308
309 qp[i] = q;
310 sub_ddmmss(n1, n0, r, n2, n1, n0);
311 }
312 np[1] = n1;
313 np[0] = n0;
314 }
315 break;
316
317 default:
318 {
319 mpi_size_t i;
320 mpi_limb_t dX, d1, n0;
321
322 np += nsize - dsize;
323 dX = dp[dsize - 1];
324 d1 = dp[dsize - 2];
325 n0 = np[dsize - 1];
326
327 if (n0 >= dX) {
328 if (n0 > dX
329 || mpihelp_cmp(np, dp, dsize - 1) >= 0) {
330 mpihelp_sub_n(np, np, dp, dsize);
331 n0 = np[dsize - 1];
332 most_significant_q_limb = 1;
333 }
334 }
335
336 for (i = qextra_limbs + nsize - dsize - 1; i >= 0; i--) {
337 mpi_limb_t q;
338 mpi_limb_t n1, n2;
339 mpi_limb_t cy_limb;
340
341 if (i >= qextra_limbs) {
342 np--;
343 n2 = np[dsize];
344 } else {
345 n2 = np[dsize - 1];
346 MPN_COPY_DECR(np + 1, np, dsize - 1);
347 np[0] = 0;
348 }
349
350 if (n0 == dX) {
351 /* This might over-estimate q, but it's probably not worth
352 * the extra code here to find out. */
353 q = ~(mpi_limb_t) 0;
354 } else {
355 mpi_limb_t r;
356
357 udiv_qrnnd(q, r, n0, np[dsize - 1], dX);
358 umul_ppmm(n1, n0, d1, q);
359
360 while (n1 > r
361 || (n1 == r
362 && n0 > np[dsize - 2])) {
363 q--;
364 r += dX;
365 if (r < dX) /* I.e. "carry in previous addition?" */
366 break;
367 n1 -= n0 < d1;
368 n0 -= d1;
369 }
370 }
371
372 /* Possible optimization: We already have (q * n0) and (1 * n1)
373 * after the calculation of q. Taking advantage of that, we
374 * could make this loop make two iterations less. */
375 cy_limb = mpihelp_submul_1(np, dp, dsize, q);
376
377 if (n2 != cy_limb) {
378 mpihelp_add_n(np, np, dp, dsize);
379 q--;
380 }
381
382 qp[i] = q;
383 n0 = np[dsize - 1];
384 }
385 }
386 }
387
388 return most_significant_q_limb;
389 }
390
391 /****************
392 * Divide (DIVIDEND_PTR,,DIVIDEND_SIZE) by DIVISOR_LIMB.
393 * Write DIVIDEND_SIZE limbs of quotient at QUOT_PTR.
394 * Return the single-limb remainder.
395 * There are no constraints on the value of the divisor.
396 *
397 * QUOT_PTR and DIVIDEND_PTR might point to the same limb.
398 */
399
400 mpi_limb_t
mpihelp_divmod_1(mpi_ptr_t quot_ptr,mpi_ptr_t dividend_ptr,mpi_size_t dividend_size,mpi_limb_t divisor_limb)401 mpihelp_divmod_1(mpi_ptr_t quot_ptr,
402 mpi_ptr_t dividend_ptr, mpi_size_t dividend_size,
403 mpi_limb_t divisor_limb)
404 {
405 mpi_size_t i;
406 mpi_limb_t n1, n0, r;
407 int dummy;
408
409 if (!dividend_size)
410 return 0;
411
412 /* If multiplication is much faster than division, and the
413 * dividend is large, pre-invert the divisor, and use
414 * only multiplications in the inner loop.
415 *
416 * This test should be read:
417 * Does it ever help to use udiv_qrnnd_preinv?
418 * && Does what we save compensate for the inversion overhead?
419 */
420 if (UDIV_TIME > (2 * UMUL_TIME + 6)
421 && (UDIV_TIME - (2 * UMUL_TIME + 6)) * dividend_size > UDIV_TIME) {
422 int normalization_steps;
423
424 count_leading_zeros(normalization_steps, divisor_limb);
425 if (normalization_steps) {
426 mpi_limb_t divisor_limb_inverted;
427
428 divisor_limb <<= normalization_steps;
429
430 /* Compute (2**2N - 2**N * DIVISOR_LIMB) / DIVISOR_LIMB. The
431 * result is a (N+1)-bit approximation to 1/DIVISOR_LIMB, with the
432 * most significant bit (with weight 2**N) implicit.
433 */
434 /* Special case for DIVISOR_LIMB == 100...000. */
435 if (!(divisor_limb << 1))
436 divisor_limb_inverted = ~(mpi_limb_t) 0;
437 else
438 udiv_qrnnd(divisor_limb_inverted, dummy,
439 -divisor_limb, 0, divisor_limb);
440
441 n1 = dividend_ptr[dividend_size - 1];
442 r = n1 >> (BITS_PER_MPI_LIMB - normalization_steps);
443
444 /* Possible optimization:
445 * if (r == 0
446 * && divisor_limb > ((n1 << normalization_steps)
447 * | (dividend_ptr[dividend_size - 2] >> ...)))
448 * ...one division less...
449 */
450 for (i = dividend_size - 2; i >= 0; i--) {
451 n0 = dividend_ptr[i];
452 UDIV_QRNND_PREINV(quot_ptr[i + 1], r, r,
453 ((n1 << normalization_steps)
454 | (n0 >>
455 (BITS_PER_MPI_LIMB -
456 normalization_steps))),
457 divisor_limb,
458 divisor_limb_inverted);
459 n1 = n0;
460 }
461 UDIV_QRNND_PREINV(quot_ptr[0], r, r,
462 n1 << normalization_steps,
463 divisor_limb, divisor_limb_inverted);
464 return r >> normalization_steps;
465 } else {
466 mpi_limb_t divisor_limb_inverted;
467
468 /* Compute (2**2N - 2**N * DIVISOR_LIMB) / DIVISOR_LIMB. The
469 * result is a (N+1)-bit approximation to 1/DIVISOR_LIMB, with the
470 * most significant bit (with weight 2**N) implicit.
471 */
472 /* Special case for DIVISOR_LIMB == 100...000. */
473 if (!(divisor_limb << 1))
474 divisor_limb_inverted = ~(mpi_limb_t) 0;
475 else
476 udiv_qrnnd(divisor_limb_inverted, dummy,
477 -divisor_limb, 0, divisor_limb);
478
479 i = dividend_size - 1;
480 r = dividend_ptr[i];
481
482 if (r >= divisor_limb)
483 r = 0;
484 else
485 quot_ptr[i--] = 0;
486
487 for (; i >= 0; i--) {
488 n0 = dividend_ptr[i];
489 UDIV_QRNND_PREINV(quot_ptr[i], r, r,
490 n0, divisor_limb,
491 divisor_limb_inverted);
492 }
493 return r;
494 }
495 } else {
496 if (UDIV_NEEDS_NORMALIZATION) {
497 int normalization_steps;
498
499 count_leading_zeros(normalization_steps, divisor_limb);
500 if (normalization_steps) {
501 divisor_limb <<= normalization_steps;
502
503 n1 = dividend_ptr[dividend_size - 1];
504 r = n1 >> (BITS_PER_MPI_LIMB -
505 normalization_steps);
506
507 /* Possible optimization:
508 * if (r == 0
509 * && divisor_limb > ((n1 << normalization_steps)
510 * | (dividend_ptr[dividend_size - 2] >> ...)))
511 * ...one division less...
512 */
513 for (i = dividend_size - 2; i >= 0; i--) {
514 n0 = dividend_ptr[i];
515 udiv_qrnnd(quot_ptr[i + 1], r, r,
516 ((n1 << normalization_steps)
517 | (n0 >>
518 (BITS_PER_MPI_LIMB -
519 normalization_steps))),
520 divisor_limb);
521 n1 = n0;
522 }
523 udiv_qrnnd(quot_ptr[0], r, r,
524 n1 << normalization_steps,
525 divisor_limb);
526 return r >> normalization_steps;
527 }
528 }
529 /* No normalization needed, either because udiv_qrnnd doesn't require
530 * it, or because DIVISOR_LIMB is already normalized. */
531 i = dividend_size - 1;
532 r = dividend_ptr[i];
533
534 if (r >= divisor_limb)
535 r = 0;
536 else
537 quot_ptr[i--] = 0;
538
539 for (; i >= 0; i--) {
540 n0 = dividend_ptr[i];
541 udiv_qrnnd(quot_ptr[i], r, r, n0, divisor_limb);
542 }
543 return r;
544 }
545 }
546