1/*
2 * Copyright 1995-2018 The OpenSSL Project Authors. All Rights Reserved.
3 *
4 * Licensed under the Apache License 2.0 (the "License"). You may not use
5 * this file except in compliance with the License. You can obtain a copy
6 * in the file LICENSE in the source distribution or at
7 * https://www.openssl.org/source/license.html
8 */
9
10#include <assert.h>
11#include <openssl/bn.h>
12#include "internal/cryptlib.h"
13#include "bn_local.h"
14
15/* The old slow way */
16#if 0
17int BN_div(BIGNUM *dv, BIGNUM *rem, const BIGNUM *m, const BIGNUM *d,
18 BN_CTX *ctx)
19{
20 int i, nm, nd;
21 int ret = 0;
22 BIGNUM *D;
23
24 bn_check_top(m);
25 bn_check_top(d);
26 if (BN_is_zero(d)) {
27 BNerr(BN_F_BN_DIV, BN_R_DIV_BY_ZERO);
28 return 0;
29 }
30
31 if (BN_ucmp(m, d) < 0) {
32 if (rem != NULL) {
33 if (BN_copy(rem, m) == NULL)
34 return 0;
35 }
36 if (dv != NULL)
37 BN_zero(dv);
38 return 1;
39 }
40
41 BN_CTX_start(ctx);
42 D = BN_CTX_get(ctx);
43 if (dv == NULL)
44 dv = BN_CTX_get(ctx);
45 if (rem == NULL)
46 rem = BN_CTX_get(ctx);
47 if (D == NULL || dv == NULL || rem == NULL)
48 goto end;
49
50 nd = BN_num_bits(d);
51 nm = BN_num_bits(m);
52 if (BN_copy(D, d) == NULL)
53 goto end;
54 if (BN_copy(rem, m) == NULL)
55 goto end;
56
57 /*
58 * The next 2 are needed so we can do a dv->d[0]|=1 later since
59 * BN_lshift1 will only work once there is a value :-)
60 */
61 BN_zero(dv);
62 if (bn_wexpand(dv, 1) == NULL)
63 goto end;
64 dv->top = 1;
65
66 if (!BN_lshift(D, D, nm - nd))
67 goto end;
68 for (i = nm - nd; i >= 0; i--) {
69 if (!BN_lshift1(dv, dv))
70 goto end;
71 if (BN_ucmp(rem, D) >= 0) {
72 dv->d[0] |= 1;
73 if (!BN_usub(rem, rem, D))
74 goto end;
75 }
76/* CAN IMPROVE (and have now :=) */
77 if (!BN_rshift1(D, D))
78 goto end;
79 }
80 rem->neg = BN_is_zero(rem) ? 0 : m->neg;
81 dv->neg = m->neg ^ d->neg;
82 ret = 1;
83 end:
84 BN_CTX_end(ctx);
85 return ret;
86}
87
88#else
89
90# if defined(BN_DIV3W)
91BN_ULONG bn_div_3_words(const BN_ULONG *m, BN_ULONG d1, BN_ULONG d0);
92# elif 0
93/*
94 * This is #if-ed away, because it's a reference for assembly implementations,
95 * where it can and should be made constant-time. But if you want to test it,
96 * just replace 0 with 1.
97 */
98# if BN_BITS2 == 64 && defined(__SIZEOF_INT128__) && __SIZEOF_INT128__==16
99# undef BN_ULLONG
100# define BN_ULLONG __uint128_t
101# define BN_LLONG
102# endif
103
104# ifdef BN_LLONG
105# define BN_DIV3W
106/*
107 * Interface is somewhat quirky, |m| is pointer to most significant limb,
108 * and less significant limb is referred at |m[-1]|. This means that caller
109 * is responsible for ensuring that |m[-1]| is valid. Second condition that
110 * has to be met is that |d0|'s most significant bit has to be set. Or in
111 * other words divisor has to be "bit-aligned to the left." bn_div_fixed_top
112 * does all this. The subroutine considers four limbs, two of which are
113 * "overlapping," hence the name...
114 */
115static BN_ULONG bn_div_3_words(const BN_ULONG *m, BN_ULONG d1, BN_ULONG d0)
116{
117 BN_ULLONG R = ((BN_ULLONG)m[0] << BN_BITS2) | m[-1];
118 BN_ULLONG D = ((BN_ULLONG)d0 << BN_BITS2) | d1;
119 BN_ULONG Q = 0, mask;
120 int i;
121
122 for (i = 0; i < BN_BITS2; i++) {
123 Q <<= 1;
124 if (R >= D) {
125 Q |= 1;
126 R -= D;
127 }
128 D >>= 1;
129 }
130
131 mask = 0 - (Q >> (BN_BITS2 - 1)); /* does it overflow? */
132
133 Q <<= 1;
134 Q |= (R >= D);
135
136 return (Q | mask) & BN_MASK2;
137}
138# endif
139# endif
140
141static int bn_left_align(BIGNUM *num)
142{
143 BN_ULONG *d = num->d, n, m, rmask;
144 int top = num->top;
145 int rshift = BN_num_bits_word(d[top - 1]), lshift, i;
146
147 lshift = BN_BITS2 - rshift;
148 rshift %= BN_BITS2; /* say no to undefined behaviour */
149 rmask = (BN_ULONG)0 - rshift; /* rmask = 0 - (rshift != 0) */
150 rmask |= rmask >> 8;
151
152 for (i = 0, m = 0; i < top; i++) {
153 n = d[i];
154 d[i] = ((n << lshift) | m) & BN_MASK2;
155 m = (n >> rshift) & rmask;
156 }
157
158 return lshift;
159}
160
161# if !defined(OPENSSL_NO_ASM) && !defined(OPENSSL_NO_INLINE_ASM) \
162 && !defined(PEDANTIC) && !defined(BN_DIV3W)
163# if defined(__GNUC__) && __GNUC__>=2
164# if defined(__i386) || defined (__i386__)
165 /*-
166 * There were two reasons for implementing this template:
167 * - GNU C generates a call to a function (__udivdi3 to be exact)
168 * in reply to ((((BN_ULLONG)n0)<<BN_BITS2)|n1)/d0 (I fail to
169 * understand why...);
170 * - divl doesn't only calculate quotient, but also leaves
171 * remainder in %edx which we can definitely use here:-)
172 */
173# undef bn_div_words
174# define bn_div_words(n0,n1,d0) \
175 ({ asm volatile ( \
176 "divl %4" \
177 : "=a"(q), "=d"(rem) \
178 : "a"(n1), "d"(n0), "r"(d0) \
179 : "cc"); \
180 q; \
181 })
182# define REMAINDER_IS_ALREADY_CALCULATED
183# elif defined(__x86_64) && defined(SIXTY_FOUR_BIT_LONG)
184 /*
185 * Same story here, but it's 128-bit by 64-bit division. Wow!
186 */
187# undef bn_div_words
188# define bn_div_words(n0,n1,d0) \
189 ({ asm volatile ( \
190 "divq %4" \
191 : "=a"(q), "=d"(rem) \
192 : "a"(n1), "d"(n0), "r"(d0) \
193 : "cc"); \
194 q; \
195 })
196# define REMAINDER_IS_ALREADY_CALCULATED
197# endif /* __<cpu> */
198# endif /* __GNUC__ */
199# endif /* OPENSSL_NO_ASM */
200
201/*-
202 * BN_div computes dv := num / divisor, rounding towards
203 * zero, and sets up rm such that dv*divisor + rm = num holds.
204 * Thus:
205 * dv->neg == num->neg ^ divisor->neg (unless the result is zero)
206 * rm->neg == num->neg (unless the remainder is zero)
207 * If 'dv' or 'rm' is NULL, the respective value is not returned.
208 */
209int BN_div(BIGNUM *dv, BIGNUM *rm, const BIGNUM *num, const BIGNUM *divisor,
210 BN_CTX *ctx)
211{
212 int ret;
213
214 if (BN_is_zero(divisor)) {
215 BNerr(BN_F_BN_DIV, BN_R_DIV_BY_ZERO);
216 return 0;
217 }
218
219 /*
220 * Invalid zero-padding would have particularly bad consequences so don't
221 * just rely on bn_check_top() here (bn_check_top() works only for
222 * BN_DEBUG builds)
223 */
224 if (divisor->d[divisor->top - 1] == 0) {
225 BNerr(BN_F_BN_DIV, BN_R_NOT_INITIALIZED);
226 return 0;
227 }
228
229 ret = bn_div_fixed_top(dv, rm, num, divisor, ctx);
230
231 if (ret) {
232 if (dv != NULL)
233 bn_correct_top(dv);
234 if (rm != NULL)
235 bn_correct_top(rm);
236 }
237
238 return ret;
239}
240
241/*
242 * It's argued that *length* of *significant* part of divisor is public.
243 * Even if it's private modulus that is. Again, *length* is assumed
244 * public, but not *value*. Former is likely to be pre-defined by
245 * algorithm with bit granularity, though below subroutine is invariant
246 * of limb length. Thanks to this assumption we can require that |divisor|
247 * may not be zero-padded, yet claim this subroutine "constant-time"(*).
248 * This is because zero-padded dividend, |num|, is tolerated, so that
249 * caller can pass dividend of public length(*), but with smaller amount
250 * of significant limbs. This naturally means that quotient, |dv|, would
251 * contain correspongly less significant limbs as well, and will be zero-
252 * padded accordingly. Returned remainder, |rm|, will have same bit length
253 * as divisor, also zero-padded if needed. These actually leave sign bits
254 * in ambiguous state. In sense that we try to avoid negative zeros, while
255 * zero-padded zeros would retain sign.
256 *
257 * (*) "Constant-time-ness" has two pre-conditions:
258 *
259 * - availability of constant-time bn_div_3_words;
260 * - dividend is at least as "wide" as divisor, limb-wise, zero-padded
261 * if so required, which shouldn't be a privacy problem, because
262 * divisor's length is considered public;
263 */
264int bn_div_fixed_top(BIGNUM *dv, BIGNUM *rm, const BIGNUM *num,
265 const BIGNUM *divisor, BN_CTX *ctx)
266{
267 int norm_shift, i, j, loop;
268 BIGNUM *tmp, *snum, *sdiv, *res;
269 BN_ULONG *resp, *wnum, *wnumtop;
270 BN_ULONG d0, d1;
271 int num_n, div_n;
272
273 assert(divisor->top > 0 && divisor->d[divisor->top - 1] != 0);
274
275 bn_check_top(num);
276 bn_check_top(divisor);
277 bn_check_top(dv);
278 bn_check_top(rm);
279
280 BN_CTX_start(ctx);
281 res = (dv == NULL) ? BN_CTX_get(ctx) : dv;
282 tmp = BN_CTX_get(ctx);
283 snum = BN_CTX_get(ctx);
284 sdiv = BN_CTX_get(ctx);
285 if (sdiv == NULL)
286 goto err;
287
288 /* First we normalise the numbers */
289 if (!BN_copy(sdiv, divisor))
290 goto err;
291 norm_shift = bn_left_align(sdiv);
292 sdiv->neg = 0;
293 /*
294 * Note that bn_lshift_fixed_top's output is always one limb longer
295 * than input, even when norm_shift is zero. This means that amount of
296 * inner loop iterations is invariant of dividend value, and that one
297 * doesn't need to compare dividend and divisor if they were originally
298 * of the same bit length.
299 */
300 if (!(bn_lshift_fixed_top(snum, num, norm_shift)))
301 goto err;
302
303 div_n = sdiv->top;
304 num_n = snum->top;
305
306 if (num_n <= div_n) {
307 /* caller didn't pad dividend -> no constant-time guarantee... */
308 if (bn_wexpand(snum, div_n + 1) == NULL)
309 goto err;
310 memset(&(snum->d[num_n]), 0, (div_n - num_n + 1) * sizeof(BN_ULONG));
311 snum->top = num_n = div_n + 1;
312 }
313
314 loop = num_n - div_n;
315 /*
316 * Lets setup a 'window' into snum This is the part that corresponds to
317 * the current 'area' being divided
318 */
319 wnum = &(snum->d[loop]);
320 wnumtop = &(snum->d[num_n - 1]);
321
322 /* Get the top 2 words of sdiv */
323 d0 = sdiv->d[div_n - 1];
324 d1 = (div_n == 1) ? 0 : sdiv->d[div_n - 2];
325
326 /* Setup quotient */
327 if (!bn_wexpand(res, loop))
328 goto err;
329 res->neg = (num->neg ^ divisor->neg);
330 res->top = loop;
331 res->flags |= BN_FLG_FIXED_TOP;
332 resp = &(res->d[loop]);
333
334 /* space for temp */
335 if (!bn_wexpand(tmp, (div_n + 1)))
336 goto err;
337
338 for (i = 0; i < loop; i++, wnumtop--) {
339 BN_ULONG q, l0;
340 /*
341 * the first part of the loop uses the top two words of snum and sdiv
342 * to calculate a BN_ULONG q such that | wnum - sdiv * q | < sdiv
343 */
344# if defined(BN_DIV3W)
345 q = bn_div_3_words(wnumtop, d1, d0);
346# else
347 BN_ULONG n0, n1, rem = 0;
348
349 n0 = wnumtop[0];
350 n1 = wnumtop[-1];
351 if (n0 == d0)
352 q = BN_MASK2;
353 else { /* n0 < d0 */
354 BN_ULONG n2 = (wnumtop == wnum) ? 0 : wnumtop[-2];
355# ifdef BN_LLONG
356 BN_ULLONG t2;
357
358# if defined(BN_LLONG) && defined(BN_DIV2W) && !defined(bn_div_words)
359 q = (BN_ULONG)(((((BN_ULLONG) n0) << BN_BITS2) | n1) / d0);
360# else
361 q = bn_div_words(n0, n1, d0);
362# endif
363
364# ifndef REMAINDER_IS_ALREADY_CALCULATED
365 /*
366 * rem doesn't have to be BN_ULLONG. The least we
367 * know it's less that d0, isn't it?
368 */
369 rem = (n1 - q * d0) & BN_MASK2;
370# endif
371 t2 = (BN_ULLONG) d1 *q;
372
373 for (;;) {
374 if (t2 <= ((((BN_ULLONG) rem) << BN_BITS2) | n2))
375 break;
376 q--;
377 rem += d0;
378 if (rem < d0)
379 break; /* don't let rem overflow */
380 t2 -= d1;
381 }
382# else /* !BN_LLONG */
383 BN_ULONG t2l, t2h;
384
385 q = bn_div_words(n0, n1, d0);
386# ifndef REMAINDER_IS_ALREADY_CALCULATED
387 rem = (n1 - q * d0) & BN_MASK2;
388# endif
389
390# if defined(BN_UMULT_LOHI)
391 BN_UMULT_LOHI(t2l, t2h, d1, q);
392# elif defined(BN_UMULT_HIGH)
393 t2l = d1 * q;
394 t2h = BN_UMULT_HIGH(d1, q);
395# else
396 {
397 BN_ULONG ql, qh;
398 t2l = LBITS(d1);
399 t2h = HBITS(d1);
400 ql = LBITS(q);
401 qh = HBITS(q);
402 mul64(t2l, t2h, ql, qh); /* t2=(BN_ULLONG)d1*q; */
403 }
404# endif
405
406 for (;;) {
407 if ((t2h < rem) || ((t2h == rem) && (t2l <= n2)))
408 break;
409 q--;
410 rem += d0;
411 if (rem < d0)
412 break; /* don't let rem overflow */
413 if (t2l < d1)
414 t2h--;
415 t2l -= d1;
416 }
417# endif /* !BN_LLONG */
418 }
419# endif /* !BN_DIV3W */
420
421 l0 = bn_mul_words(tmp->d, sdiv->d, div_n, q);
422 tmp->d[div_n] = l0;
423 wnum--;
424 /*
425 * ignore top values of the bignums just sub the two BN_ULONG arrays
426 * with bn_sub_words
427 */
428 l0 = bn_sub_words(wnum, wnum, tmp->d, div_n + 1);
429 q -= l0;
430 /*
431 * Note: As we have considered only the leading two BN_ULONGs in
432 * the calculation of q, sdiv * q might be greater than wnum (but
433 * then (q-1) * sdiv is less or equal than wnum)
434 */
435 for (l0 = 0 - l0, j = 0; j < div_n; j++)
436 tmp->d[j] = sdiv->d[j] & l0;
437 l0 = bn_add_words(wnum, wnum, tmp->d, div_n);
438 (*wnumtop) += l0;
439 assert((*wnumtop) == 0);
440
441 /* store part of the result */
442 *--resp = q;
443 }
444 /* snum holds remainder, it's as wide as divisor */
445 snum->neg = num->neg;
446 snum->top = div_n;
447 snum->flags |= BN_FLG_FIXED_TOP;
448 if (rm != NULL)
449 bn_rshift_fixed_top(rm, snum, norm_shift);
450 BN_CTX_end(ctx);
451 return 1;
452 err:
453 bn_check_top(rm);
454 BN_CTX_end(ctx);
455 return 0;
456}
457#endif
458