1 /*
2 * Copyright 1995-2022 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
17 int 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 ERR_raise(ERR_LIB_BN, 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)
91 BN_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 */
bn_div_3_words(const BN_ULONG * m,BN_ULONG d1,BN_ULONG d0)115 static 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
bn_left_align(BIGNUM * num)141 static 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 */
BN_div(BIGNUM * dv,BIGNUM * rm,const BIGNUM * num,const BIGNUM * divisor,BN_CTX * ctx)209 int 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 ERR_raise(ERR_LIB_BN, 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 ERR_raise(ERR_LIB_BN, 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 */
bn_div_fixed_top(BIGNUM * dv,BIGNUM * rm,const BIGNUM * num,const BIGNUM * divisor,BN_CTX * ctx)264 int 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, num_neg;
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 num_neg = num->neg;
330 res->neg = (num_neg ^ divisor->neg);
331 res->top = loop;
332 res->flags |= BN_FLG_FIXED_TOP;
333 resp = &(res->d[loop]);
334
335 /* space for temp */
336 if (!bn_wexpand(tmp, (div_n + 1)))
337 goto err;
338
339 for (i = 0; i < loop; i++, wnumtop--) {
340 BN_ULONG q, l0;
341 /*
342 * the first part of the loop uses the top two words of snum and sdiv
343 * to calculate a BN_ULONG q such that | wnum - sdiv * q | < sdiv
344 */
345 # if defined(BN_DIV3W)
346 q = bn_div_3_words(wnumtop, d1, d0);
347 # else
348 BN_ULONG n0, n1, rem = 0;
349
350 n0 = wnumtop[0];
351 n1 = wnumtop[-1];
352 if (n0 == d0)
353 q = BN_MASK2;
354 else { /* n0 < d0 */
355 BN_ULONG n2 = (wnumtop == wnum) ? 0 : wnumtop[-2];
356 # ifdef BN_LLONG
357 BN_ULLONG t2;
358
359 # if defined(BN_LLONG) && defined(BN_DIV2W) && !defined(bn_div_words)
360 q = (BN_ULONG)(((((BN_ULLONG) n0) << BN_BITS2) | n1) / d0);
361 # else
362 q = bn_div_words(n0, n1, d0);
363 # endif
364
365 # ifndef REMAINDER_IS_ALREADY_CALCULATED
366 /*
367 * rem doesn't have to be BN_ULLONG. The least we
368 * know it's less that d0, isn't it?
369 */
370 rem = (n1 - q * d0) & BN_MASK2;
371 # endif
372 t2 = (BN_ULLONG) d1 *q;
373
374 for (;;) {
375 if (t2 <= ((((BN_ULLONG) rem) << BN_BITS2) | n2))
376 break;
377 q--;
378 rem += d0;
379 if (rem < d0)
380 break; /* don't let rem overflow */
381 t2 -= d1;
382 }
383 # else /* !BN_LLONG */
384 BN_ULONG t2l, t2h;
385
386 q = bn_div_words(n0, n1, d0);
387 # ifndef REMAINDER_IS_ALREADY_CALCULATED
388 rem = (n1 - q * d0) & BN_MASK2;
389 # endif
390
391 # if defined(BN_UMULT_LOHI)
392 BN_UMULT_LOHI(t2l, t2h, d1, q);
393 # elif defined(BN_UMULT_HIGH)
394 t2l = d1 * q;
395 t2h = BN_UMULT_HIGH(d1, q);
396 # else
397 {
398 BN_ULONG ql, qh;
399 t2l = LBITS(d1);
400 t2h = HBITS(d1);
401 ql = LBITS(q);
402 qh = HBITS(q);
403 mul64(t2l, t2h, ql, qh); /* t2=(BN_ULLONG)d1*q; */
404 }
405 # endif
406
407 for (;;) {
408 if ((t2h < rem) || ((t2h == rem) && (t2l <= n2)))
409 break;
410 q--;
411 rem += d0;
412 if (rem < d0)
413 break; /* don't let rem overflow */
414 if (t2l < d1)
415 t2h--;
416 t2l -= d1;
417 }
418 # endif /* !BN_LLONG */
419 }
420 # endif /* !BN_DIV3W */
421
422 l0 = bn_mul_words(tmp->d, sdiv->d, div_n, q);
423 tmp->d[div_n] = l0;
424 wnum--;
425 /*
426 * ignore top values of the bignums just sub the two BN_ULONG arrays
427 * with bn_sub_words
428 */
429 l0 = bn_sub_words(wnum, wnum, tmp->d, div_n + 1);
430 q -= l0;
431 /*
432 * Note: As we have considered only the leading two BN_ULONGs in
433 * the calculation of q, sdiv * q might be greater than wnum (but
434 * then (q-1) * sdiv is less or equal than wnum)
435 */
436 for (l0 = 0 - l0, j = 0; j < div_n; j++)
437 tmp->d[j] = sdiv->d[j] & l0;
438 l0 = bn_add_words(wnum, wnum, tmp->d, div_n);
439 (*wnumtop) += l0;
440 assert((*wnumtop) == 0);
441
442 /* store part of the result */
443 *--resp = q;
444 }
445 /* snum holds remainder, it's as wide as divisor */
446 snum->neg = num_neg;
447 snum->top = div_n;
448 snum->flags |= BN_FLG_FIXED_TOP;
449
450 if (rm != NULL && bn_rshift_fixed_top(rm, snum, norm_shift) == 0)
451 goto err;
452
453 BN_CTX_end(ctx);
454 return 1;
455 err:
456 bn_check_top(rm);
457 BN_CTX_end(ctx);
458 return 0;
459 }
460 #endif
461
