1 | // SPDX-License-Identifier: GPL-2.0-or-later |
2 | /* mpihelp-div.c - MPI helper functions |
3 | * Copyright (C) 1994, 1996 Free Software Foundation, Inc. |
4 | * Copyright (C) 1998, 1999 Free Software Foundation, Inc. |
5 | * |
6 | * This file is part of GnuPG. |
7 | * |
8 | * Note: This code is heavily based on the GNU MP Library. |
9 | * Actually it's the same code with only minor changes in the |
10 | * way the data is stored; this is to support the abstraction |
11 | * of an optional secure memory allocation which may be used |
12 | * to avoid revealing of sensitive data due to paging etc. |
13 | * The GNU MP Library itself is published under the LGPL; |
14 | * however I decided to publish this code under the plain GPL. |
15 | */ |
16 | |
17 | #include "mpi-internal.h" |
18 | #include "longlong.h" |
19 | |
20 | #ifndef UMUL_TIME |
21 | #define UMUL_TIME 1 |
22 | #endif |
23 | #ifndef UDIV_TIME |
24 | #define UDIV_TIME UMUL_TIME |
25 | #endif |
26 | |
27 | |
28 | mpi_limb_t |
29 | mpihelp_mod_1(mpi_ptr_t dividend_ptr, mpi_size_t dividend_size, |
30 | mpi_limb_t divisor_limb) |
31 | { |
32 | mpi_size_t i; |
33 | mpi_limb_t n1, n0, r; |
34 | mpi_limb_t dummy __maybe_unused; |
35 | |
36 | /* Botch: Should this be handled at all? Rely on callers? */ |
37 | if (!dividend_size) |
38 | return 0; |
39 | |
40 | /* If multiplication is much faster than division, and the |
41 | * dividend is large, pre-invert the divisor, and use |
42 | * only multiplications in the inner loop. |
43 | * |
44 | * This test should be read: |
45 | * Does it ever help to use udiv_qrnnd_preinv? |
46 | * && Does what we save compensate for the inversion overhead? |
47 | */ |
48 | if (UDIV_TIME > (2 * UMUL_TIME + 6) |
49 | && (UDIV_TIME - (2 * UMUL_TIME + 6)) * dividend_size > UDIV_TIME) { |
50 | int normalization_steps; |
51 | |
52 | normalization_steps = count_leading_zeros(x: divisor_limb); |
53 | if (normalization_steps) { |
54 | mpi_limb_t divisor_limb_inverted; |
55 | |
56 | divisor_limb <<= normalization_steps; |
57 | |
58 | /* Compute (2**2N - 2**N * DIVISOR_LIMB) / DIVISOR_LIMB. The |
59 | * result is a (N+1)-bit approximation to 1/DIVISOR_LIMB, with the |
60 | * most significant bit (with weight 2**N) implicit. |
61 | * |
62 | * Special case for DIVISOR_LIMB == 100...000. |
63 | */ |
64 | if (!(divisor_limb << 1)) |
65 | divisor_limb_inverted = ~(mpi_limb_t)0; |
66 | else |
67 | udiv_qrnnd(divisor_limb_inverted, dummy, |
68 | -divisor_limb, 0, divisor_limb); |
69 | |
70 | n1 = dividend_ptr[dividend_size - 1]; |
71 | r = n1 >> (BITS_PER_MPI_LIMB - normalization_steps); |
72 | |
73 | /* Possible optimization: |
74 | * if (r == 0 |
75 | * && divisor_limb > ((n1 << normalization_steps) |
76 | * | (dividend_ptr[dividend_size - 2] >> ...))) |
77 | * ...one division less... |
78 | */ |
79 | for (i = dividend_size - 2; i >= 0; i--) { |
80 | n0 = dividend_ptr[i]; |
81 | UDIV_QRNND_PREINV(dummy, r, r, |
82 | ((n1 << normalization_steps) |
83 | | (n0 >> (BITS_PER_MPI_LIMB - normalization_steps))), |
84 | divisor_limb, divisor_limb_inverted); |
85 | n1 = n0; |
86 | } |
87 | UDIV_QRNND_PREINV(dummy, r, r, |
88 | n1 << normalization_steps, |
89 | divisor_limb, divisor_limb_inverted); |
90 | return r >> normalization_steps; |
91 | } else { |
92 | mpi_limb_t divisor_limb_inverted; |
93 | |
94 | /* Compute (2**2N - 2**N * DIVISOR_LIMB) / DIVISOR_LIMB. The |
95 | * result is a (N+1)-bit approximation to 1/DIVISOR_LIMB, with the |
96 | * most significant bit (with weight 2**N) implicit. |
97 | * |
98 | * Special case for DIVISOR_LIMB == 100...000. |
99 | */ |
100 | if (!(divisor_limb << 1)) |
101 | divisor_limb_inverted = ~(mpi_limb_t)0; |
102 | else |
103 | udiv_qrnnd(divisor_limb_inverted, dummy, |
104 | -divisor_limb, 0, divisor_limb); |
105 | |
106 | i = dividend_size - 1; |
107 | r = dividend_ptr[i]; |
108 | |
109 | if (r >= divisor_limb) |
110 | r = 0; |
111 | else |
112 | i--; |
113 | |
114 | for ( ; i >= 0; i--) { |
115 | n0 = dividend_ptr[i]; |
116 | UDIV_QRNND_PREINV(dummy, r, r, |
117 | n0, divisor_limb, divisor_limb_inverted); |
118 | } |
119 | return r; |
120 | } |
121 | } else { |
122 | if (UDIV_NEEDS_NORMALIZATION) { |
123 | int normalization_steps; |
124 | |
125 | normalization_steps = count_leading_zeros(x: divisor_limb); |
126 | if (normalization_steps) { |
127 | divisor_limb <<= normalization_steps; |
128 | |
129 | n1 = dividend_ptr[dividend_size - 1]; |
130 | r = n1 >> (BITS_PER_MPI_LIMB - normalization_steps); |
131 | |
132 | /* Possible optimization: |
133 | * if (r == 0 |
134 | * && divisor_limb > ((n1 << normalization_steps) |
135 | * | (dividend_ptr[dividend_size - 2] >> ...))) |
136 | * ...one division less... |
137 | */ |
138 | for (i = dividend_size - 2; i >= 0; i--) { |
139 | n0 = dividend_ptr[i]; |
140 | udiv_qrnnd(dummy, r, r, |
141 | ((n1 << normalization_steps) |
142 | | (n0 >> (BITS_PER_MPI_LIMB - normalization_steps))), |
143 | divisor_limb); |
144 | n1 = n0; |
145 | } |
146 | udiv_qrnnd(dummy, r, r, |
147 | n1 << normalization_steps, |
148 | divisor_limb); |
149 | return r >> normalization_steps; |
150 | } |
151 | } |
152 | /* No normalization needed, either because udiv_qrnnd doesn't require |
153 | * it, or because DIVISOR_LIMB is already normalized. |
154 | */ |
155 | i = dividend_size - 1; |
156 | r = dividend_ptr[i]; |
157 | |
158 | if (r >= divisor_limb) |
159 | r = 0; |
160 | else |
161 | i--; |
162 | |
163 | for (; i >= 0; i--) { |
164 | n0 = dividend_ptr[i]; |
165 | udiv_qrnnd(dummy, r, r, n0, divisor_limb); |
166 | } |
167 | return r; |
168 | } |
169 | } |
170 | |
171 | /* Divide num (NP/NSIZE) by den (DP/DSIZE) and write |
172 | * the NSIZE-DSIZE least significant quotient limbs at QP |
173 | * and the DSIZE long remainder at NP. If QEXTRA_LIMBS is |
174 | * non-zero, generate that many fraction bits and append them after the |
175 | * other quotient limbs. |
176 | * Return the most significant limb of the quotient, this is always 0 or 1. |
177 | * |
178 | * Preconditions: |
179 | * 0. NSIZE >= DSIZE. |
180 | * 1. The most significant bit of the divisor must be set. |
181 | * 2. QP must either not overlap with the input operands at all, or |
182 | * QP + DSIZE >= NP must hold true. (This means that it's |
183 | * possible to put the quotient in the high part of NUM, right after the |
184 | * remainder in NUM. |
185 | * 3. NSIZE >= DSIZE, even if QEXTRA_LIMBS is non-zero. |
186 | */ |
187 | |
188 | mpi_limb_t |
189 | mpihelp_divrem(mpi_ptr_t qp, mpi_size_t , |
190 | mpi_ptr_t np, mpi_size_t nsize, mpi_ptr_t dp, mpi_size_t dsize) |
191 | { |
192 | mpi_limb_t most_significant_q_limb = 0; |
193 | |
194 | switch (dsize) { |
195 | case 0: |
196 | /* We are asked to divide by zero, so go ahead and do it! (To make |
197 | the compiler not remove this statement, return the value.) */ |
198 | /* |
199 | * existing clients of this function have been modified |
200 | * not to call it with dsize == 0, so this should not happen |
201 | */ |
202 | return 1 / dsize; |
203 | |
204 | case 1: |
205 | { |
206 | mpi_size_t i; |
207 | mpi_limb_t n1; |
208 | mpi_limb_t d; |
209 | |
210 | d = dp[0]; |
211 | n1 = np[nsize - 1]; |
212 | |
213 | if (n1 >= d) { |
214 | n1 -= d; |
215 | most_significant_q_limb = 1; |
216 | } |
217 | |
218 | qp += qextra_limbs; |
219 | for (i = nsize - 2; i >= 0; i--) |
220 | udiv_qrnnd(qp[i], n1, n1, np[i], d); |
221 | qp -= qextra_limbs; |
222 | |
223 | for (i = qextra_limbs - 1; i >= 0; i--) |
224 | udiv_qrnnd(qp[i], n1, n1, 0, d); |
225 | |
226 | np[0] = n1; |
227 | } |
228 | break; |
229 | |
230 | case 2: |
231 | { |
232 | mpi_size_t i; |
233 | mpi_limb_t n1, n0, n2; |
234 | mpi_limb_t d1, d0; |
235 | |
236 | np += nsize - 2; |
237 | d1 = dp[1]; |
238 | d0 = dp[0]; |
239 | n1 = np[1]; |
240 | n0 = np[0]; |
241 | |
242 | if (n1 >= d1 && (n1 > d1 || n0 >= d0)) { |
243 | sub_ddmmss(n1, n0, n1, n0, d1, d0); |
244 | most_significant_q_limb = 1; |
245 | } |
246 | |
247 | for (i = qextra_limbs + nsize - 2 - 1; i >= 0; i--) { |
248 | mpi_limb_t q; |
249 | mpi_limb_t r; |
250 | |
251 | if (i >= qextra_limbs) |
252 | np--; |
253 | else |
254 | np[0] = 0; |
255 | |
256 | if (n1 == d1) { |
257 | /* Q should be either 111..111 or 111..110. Need special |
258 | * treatment of this rare case as normal division would |
259 | * give overflow. */ |
260 | q = ~(mpi_limb_t) 0; |
261 | |
262 | r = n0 + d1; |
263 | if (r < d1) { /* Carry in the addition? */ |
264 | add_ssaaaa(n1, n0, r - d0, |
265 | np[0], 0, d0); |
266 | qp[i] = q; |
267 | continue; |
268 | } |
269 | n1 = d0 - (d0 != 0 ? 1 : 0); |
270 | n0 = -d0; |
271 | } else { |
272 | udiv_qrnnd(q, r, n1, n0, d1); |
273 | umul_ppmm(n1, n0, d0, q); |
274 | } |
275 | |
276 | n2 = np[0]; |
277 | q_test: |
278 | if (n1 > r || (n1 == r && n0 > n2)) { |
279 | /* The estimated Q was too large. */ |
280 | q--; |
281 | sub_ddmmss(n1, n0, n1, n0, 0, d0); |
282 | r += d1; |
283 | if (r >= d1) /* If not carry, test Q again. */ |
284 | goto q_test; |
285 | } |
286 | |
287 | qp[i] = q; |
288 | sub_ddmmss(n1, n0, r, n2, n1, n0); |
289 | } |
290 | np[1] = n1; |
291 | np[0] = n0; |
292 | } |
293 | break; |
294 | |
295 | default: |
296 | { |
297 | mpi_size_t i; |
298 | mpi_limb_t dX, d1, n0; |
299 | |
300 | np += nsize - dsize; |
301 | dX = dp[dsize - 1]; |
302 | d1 = dp[dsize - 2]; |
303 | n0 = np[dsize - 1]; |
304 | |
305 | if (n0 >= dX) { |
306 | if (n0 > dX |
307 | || mpihelp_cmp(op1_ptr: np, op2_ptr: dp, size: dsize - 1) >= 0) { |
308 | mpihelp_sub_n(res_ptr: np, s1_ptr: np, s2_ptr: dp, size: dsize); |
309 | n0 = np[dsize - 1]; |
310 | most_significant_q_limb = 1; |
311 | } |
312 | } |
313 | |
314 | for (i = qextra_limbs + nsize - dsize - 1; i >= 0; i--) { |
315 | mpi_limb_t q; |
316 | mpi_limb_t n1, n2; |
317 | mpi_limb_t cy_limb; |
318 | |
319 | if (i >= qextra_limbs) { |
320 | np--; |
321 | n2 = np[dsize]; |
322 | } else { |
323 | n2 = np[dsize - 1]; |
324 | MPN_COPY_DECR(np + 1, np, dsize - 1); |
325 | np[0] = 0; |
326 | } |
327 | |
328 | if (n0 == dX) { |
329 | /* This might over-estimate q, but it's probably not worth |
330 | * the extra code here to find out. */ |
331 | q = ~(mpi_limb_t) 0; |
332 | } else { |
333 | mpi_limb_t r; |
334 | |
335 | udiv_qrnnd(q, r, n0, np[dsize - 1], dX); |
336 | umul_ppmm(n1, n0, d1, q); |
337 | |
338 | while (n1 > r |
339 | || (n1 == r |
340 | && n0 > np[dsize - 2])) { |
341 | q--; |
342 | r += dX; |
343 | if (r < dX) /* I.e. "carry in previous addition?" */ |
344 | break; |
345 | n1 -= n0 < d1; |
346 | n0 -= d1; |
347 | } |
348 | } |
349 | |
350 | /* Possible optimization: We already have (q * n0) and (1 * n1) |
351 | * after the calculation of q. Taking advantage of that, we |
352 | * could make this loop make two iterations less. */ |
353 | cy_limb = mpihelp_submul_1(res_ptr: np, s1_ptr: dp, s1_size: dsize, s2_limb: q); |
354 | |
355 | if (n2 != cy_limb) { |
356 | mpihelp_add_n(res_ptr: np, s1_ptr: np, s2_ptr: dp, size: dsize); |
357 | q--; |
358 | } |
359 | |
360 | qp[i] = q; |
361 | n0 = np[dsize - 1]; |
362 | } |
363 | } |
364 | } |
365 | |
366 | return most_significant_q_limb; |
367 | } |
368 | |
369 | /**************** |
370 | * Divide (DIVIDEND_PTR,,DIVIDEND_SIZE) by DIVISOR_LIMB. |
371 | * Write DIVIDEND_SIZE limbs of quotient at QUOT_PTR. |
372 | * Return the single-limb remainder. |
373 | * There are no constraints on the value of the divisor. |
374 | * |
375 | * QUOT_PTR and DIVIDEND_PTR might point to the same limb. |
376 | */ |
377 | |
378 | mpi_limb_t |
379 | mpihelp_divmod_1(mpi_ptr_t quot_ptr, |
380 | mpi_ptr_t dividend_ptr, mpi_size_t dividend_size, |
381 | mpi_limb_t divisor_limb) |
382 | { |
383 | mpi_size_t i; |
384 | mpi_limb_t n1, n0, r; |
385 | mpi_limb_t dummy __maybe_unused; |
386 | |
387 | if (!dividend_size) |
388 | return 0; |
389 | |
390 | /* If multiplication is much faster than division, and the |
391 | * dividend is large, pre-invert the divisor, and use |
392 | * only multiplications in the inner loop. |
393 | * |
394 | * This test should be read: |
395 | * Does it ever help to use udiv_qrnnd_preinv? |
396 | * && Does what we save compensate for the inversion overhead? |
397 | */ |
398 | if (UDIV_TIME > (2 * UMUL_TIME + 6) |
399 | && (UDIV_TIME - (2 * UMUL_TIME + 6)) * dividend_size > UDIV_TIME) { |
400 | int normalization_steps; |
401 | |
402 | normalization_steps = count_leading_zeros(x: divisor_limb); |
403 | if (normalization_steps) { |
404 | mpi_limb_t divisor_limb_inverted; |
405 | |
406 | divisor_limb <<= normalization_steps; |
407 | |
408 | /* Compute (2**2N - 2**N * DIVISOR_LIMB) / DIVISOR_LIMB. The |
409 | * result is a (N+1)-bit approximation to 1/DIVISOR_LIMB, with the |
410 | * most significant bit (with weight 2**N) implicit. |
411 | */ |
412 | /* Special case for DIVISOR_LIMB == 100...000. */ |
413 | if (!(divisor_limb << 1)) |
414 | divisor_limb_inverted = ~(mpi_limb_t)0; |
415 | else |
416 | udiv_qrnnd(divisor_limb_inverted, dummy, |
417 | -divisor_limb, 0, divisor_limb); |
418 | |
419 | n1 = dividend_ptr[dividend_size - 1]; |
420 | r = n1 >> (BITS_PER_MPI_LIMB - normalization_steps); |
421 | |
422 | /* Possible optimization: |
423 | * if (r == 0 |
424 | * && divisor_limb > ((n1 << normalization_steps) |
425 | * | (dividend_ptr[dividend_size - 2] >> ...))) |
426 | * ...one division less... |
427 | */ |
428 | for (i = dividend_size - 2; i >= 0; i--) { |
429 | n0 = dividend_ptr[i]; |
430 | UDIV_QRNND_PREINV(quot_ptr[i + 1], r, r, |
431 | ((n1 << normalization_steps) |
432 | | (n0 >> (BITS_PER_MPI_LIMB - normalization_steps))), |
433 | divisor_limb, divisor_limb_inverted); |
434 | n1 = n0; |
435 | } |
436 | UDIV_QRNND_PREINV(quot_ptr[0], r, r, |
437 | n1 << normalization_steps, |
438 | divisor_limb, divisor_limb_inverted); |
439 | return r >> normalization_steps; |
440 | } else { |
441 | mpi_limb_t divisor_limb_inverted; |
442 | |
443 | /* Compute (2**2N - 2**N * DIVISOR_LIMB) / DIVISOR_LIMB. The |
444 | * result is a (N+1)-bit approximation to 1/DIVISOR_LIMB, with the |
445 | * most significant bit (with weight 2**N) implicit. |
446 | */ |
447 | /* Special case for DIVISOR_LIMB == 100...000. */ |
448 | if (!(divisor_limb << 1)) |
449 | divisor_limb_inverted = ~(mpi_limb_t) 0; |
450 | else |
451 | udiv_qrnnd(divisor_limb_inverted, dummy, |
452 | -divisor_limb, 0, divisor_limb); |
453 | |
454 | i = dividend_size - 1; |
455 | r = dividend_ptr[i]; |
456 | |
457 | if (r >= divisor_limb) |
458 | r = 0; |
459 | else |
460 | quot_ptr[i--] = 0; |
461 | |
462 | for ( ; i >= 0; i--) { |
463 | n0 = dividend_ptr[i]; |
464 | UDIV_QRNND_PREINV(quot_ptr[i], r, r, |
465 | n0, divisor_limb, divisor_limb_inverted); |
466 | } |
467 | return r; |
468 | } |
469 | } else { |
470 | if (UDIV_NEEDS_NORMALIZATION) { |
471 | int normalization_steps; |
472 | |
473 | normalization_steps = count_leading_zeros(x: divisor_limb); |
474 | if (normalization_steps) { |
475 | divisor_limb <<= normalization_steps; |
476 | |
477 | n1 = dividend_ptr[dividend_size - 1]; |
478 | r = n1 >> (BITS_PER_MPI_LIMB - normalization_steps); |
479 | |
480 | /* Possible optimization: |
481 | * if (r == 0 |
482 | * && divisor_limb > ((n1 << normalization_steps) |
483 | * | (dividend_ptr[dividend_size - 2] >> ...))) |
484 | * ...one division less... |
485 | */ |
486 | for (i = dividend_size - 2; i >= 0; i--) { |
487 | n0 = dividend_ptr[i]; |
488 | udiv_qrnnd(quot_ptr[i + 1], r, r, |
489 | ((n1 << normalization_steps) |
490 | | (n0 >> (BITS_PER_MPI_LIMB - normalization_steps))), |
491 | divisor_limb); |
492 | n1 = n0; |
493 | } |
494 | udiv_qrnnd(quot_ptr[0], r, r, |
495 | n1 << normalization_steps, |
496 | divisor_limb); |
497 | return r >> normalization_steps; |
498 | } |
499 | } |
500 | /* No normalization needed, either because udiv_qrnnd doesn't require |
501 | * it, or because DIVISOR_LIMB is already normalized. |
502 | */ |
503 | i = dividend_size - 1; |
504 | r = dividend_ptr[i]; |
505 | |
506 | if (r >= divisor_limb) |
507 | r = 0; |
508 | else |
509 | quot_ptr[i--] = 0; |
510 | |
511 | for (; i >= 0; i--) { |
512 | n0 = dividend_ptr[i]; |
513 | udiv_qrnnd(quot_ptr[i], r, r, n0, divisor_limb); |
514 | } |
515 | return r; |
516 | } |
517 | } |
518 | |