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 ({ \
176 asm volatile( \
177 "divl %4" \
178 : "=a"(q), "=d"(rem) \
179 : "a"(n1), "d"(n0), "r"(d0) \
180 : "cc"); \
181 q; \
182 })
183 #define REMAINDER_IS_ALREADY_CALCULATED
184 #elif defined(__x86_64) && defined(SIXTY_FOUR_BIT_LONG)
185 /*
186 * Same story here, but it's 128-bit by 64-bit division. Wow!
187 */
188 #undef bn_div_words
189 #define bn_div_words(n0, n1, d0) \
190 ({ \
191 asm volatile( \
192 "divq %4" \
193 : "=a"(q), "=d"(rem) \
194 : "a"(n1), "d"(n0), "r"(d0) \
195 : "cc"); \
196 q; \
197 })
198 #define REMAINDER_IS_ALREADY_CALCULATED
199 #endif /* __<cpu> */
200 #endif /* __GNUC__ */
201 #endif /* OPENSSL_NO_ASM */
202
203 /*-
204 * BN_div computes dv := num / divisor, rounding towards
205 * zero, and sets up rm such that dv*divisor + rm = num holds.
206 * Thus:
207 * dv->neg == num->neg ^ divisor->neg (unless the result is zero)
208 * rm->neg == num->neg (unless the remainder is zero)
209 * If 'dv' or 'rm' is NULL, the respective value is not returned.
210 */
BN_div(BIGNUM * dv,BIGNUM * rm,const BIGNUM * num,const BIGNUM * divisor,BN_CTX * ctx)211 int BN_div(BIGNUM *dv, BIGNUM *rm, const BIGNUM *num, const BIGNUM *divisor,
212 BN_CTX *ctx)
213 {
214 int ret;
215
216 if (BN_is_zero(divisor)) {
217 ERR_raise(ERR_LIB_BN, BN_R_DIV_BY_ZERO);
218 return 0;
219 }
220
221 /*
222 * Invalid zero-padding would have particularly bad consequences so don't
223 * just rely on bn_check_top() here (bn_check_top() works only for
224 * BN_DEBUG builds)
225 */
226 if (divisor->d[divisor->top - 1] == 0) {
227 ERR_raise(ERR_LIB_BN, BN_R_NOT_INITIALIZED);
228 return 0;
229 }
230
231 ret = bn_div_fixed_top(dv, rm, num, divisor, ctx);
232
233 if (ret) {
234 if (dv != NULL)
235 bn_correct_top(dv);
236 if (rm != NULL)
237 bn_correct_top(rm);
238 }
239
240 return ret;
241 }
242
243 /*
244 * It's argued that *length* of *significant* part of divisor is public.
245 * Even if it's private modulus that is. Again, *length* is assumed
246 * public, but not *value*. Former is likely to be pre-defined by
247 * algorithm with bit granularity, though below subroutine is invariant
248 * of limb length. Thanks to this assumption we can require that |divisor|
249 * may not be zero-padded, yet claim this subroutine "constant-time"(*).
250 * This is because zero-padded dividend, |num|, is tolerated, so that
251 * caller can pass dividend of public length(*), but with smaller amount
252 * of significant limbs. This naturally means that quotient, |dv|, would
253 * contain correspongly less significant limbs as well, and will be zero-
254 * padded accordingly. Returned remainder, |rm|, will have same bit length
255 * as divisor, also zero-padded if needed. These actually leave sign bits
256 * in ambiguous state. In sense that we try to avoid negative zeros, while
257 * zero-padded zeros would retain sign.
258 *
259 * (*) "Constant-time-ness" has two pre-conditions:
260 *
261 * - availability of constant-time bn_div_3_words;
262 * - dividend is at least as "wide" as divisor, limb-wise, zero-padded
263 * if so required, which shouldn't be a privacy problem, because
264 * divisor's length is considered public;
265 */
bn_div_fixed_top(BIGNUM * dv,BIGNUM * rm,const BIGNUM * num,const BIGNUM * divisor,BN_CTX * ctx)266 int bn_div_fixed_top(BIGNUM *dv, BIGNUM *rm, const BIGNUM *num,
267 const BIGNUM *divisor, BN_CTX *ctx)
268 {
269 int norm_shift, i, j, loop;
270 BIGNUM *tmp, *snum, *sdiv, *res;
271 BN_ULONG *resp, *wnum, *wnumtop;
272 BN_ULONG d0, d1;
273 int num_n, div_n, num_neg;
274
275 assert(divisor->top > 0 && divisor->d[divisor->top - 1] != 0);
276
277 bn_check_top(num);
278 bn_check_top(divisor);
279 bn_check_top(dv);
280 bn_check_top(rm);
281
282 BN_CTX_start(ctx);
283 res = (dv == NULL) ? BN_CTX_get(ctx) : dv;
284 tmp = BN_CTX_get(ctx);
285 snum = BN_CTX_get(ctx);
286 sdiv = BN_CTX_get(ctx);
287 if (sdiv == NULL)
288 goto err;
289
290 /* First we normalise the numbers */
291 if (!BN_copy(sdiv, divisor))
292 goto err;
293 norm_shift = bn_left_align(sdiv);
294 sdiv->neg = 0;
295 /*
296 * Note that bn_lshift_fixed_top's output is always one limb longer
297 * than input, even when norm_shift is zero. This means that amount of
298 * inner loop iterations is invariant of dividend value, and that one
299 * doesn't need to compare dividend and divisor if they were originally
300 * of the same bit length.
301 */
302 if (!(bn_lshift_fixed_top(snum, num, norm_shift)))
303 goto err;
304
305 div_n = sdiv->top;
306 num_n = snum->top;
307
308 if (num_n <= div_n) {
309 /* caller didn't pad dividend -> no constant-time guarantee... */
310 if (bn_wexpand(snum, div_n + 1) == NULL)
311 goto err;
312 memset(&(snum->d[num_n]), 0, (div_n - num_n + 1) * sizeof(BN_ULONG));
313 snum->top = num_n = div_n + 1;
314 }
315
316 loop = num_n - div_n;
317 /*
318 * Lets setup a 'window' into snum This is the part that corresponds to
319 * the current 'area' being divided
320 */
321 wnum = &(snum->d[loop]);
322 wnumtop = &(snum->d[num_n - 1]);
323
324 /* Get the top 2 words of sdiv */
325 d0 = sdiv->d[div_n - 1];
326 d1 = (div_n == 1) ? 0 : sdiv->d[div_n - 2];
327
328 /* Setup quotient */
329 if (!bn_wexpand(res, loop))
330 goto err;
331 num_neg = num->neg;
332 res->neg = (num_neg ^ divisor->neg);
333 res->top = loop;
334 res->flags |= BN_FLG_FIXED_TOP;
335 resp = &(res->d[loop]);
336
337 /* space for temp */
338 if (!bn_wexpand(tmp, (div_n + 1)))
339 goto err;
340
341 for (i = 0; i < loop; i++, wnumtop--) {
342 BN_ULONG q, l0;
343 /*
344 * the first part of the loop uses the top two words of snum and sdiv
345 * to calculate a BN_ULONG q such that | wnum - sdiv * q | < sdiv
346 */
347 #if defined(BN_DIV3W)
348 q = bn_div_3_words(wnumtop, d1, d0);
349 #else
350 BN_ULONG n0, n1, rem = 0;
351
352 n0 = wnumtop[0];
353 n1 = wnumtop[-1];
354 if (n0 == d0)
355 q = BN_MASK2;
356 else { /* n0 < d0 */
357 BN_ULONG n2 = (wnumtop == wnum) ? 0 : wnumtop[-2];
358 #ifdef BN_LLONG
359 BN_ULLONG t2;
360
361 #if defined(BN_LLONG) && defined(BN_DIV2W) && !defined(bn_div_words)
362 q = (BN_ULONG)(((((BN_ULLONG)n0) << BN_BITS2) | n1) / d0);
363 #else
364 q = bn_div_words(n0, n1, d0);
365 #endif
366
367 #ifndef REMAINDER_IS_ALREADY_CALCULATED
368 /*
369 * rem doesn't have to be BN_ULLONG. The least we
370 * know it's less that d0, isn't it?
371 */
372 rem = (n1 - q * d0) & BN_MASK2;
373 #endif
374 t2 = (BN_ULLONG)d1 * q;
375
376 for (;;) {
377 if (t2 <= ((((BN_ULLONG)rem) << BN_BITS2) | n2))
378 break;
379 q--;
380 rem += d0;
381 if (rem < d0)
382 break; /* don't let rem overflow */
383 t2 -= d1;
384 }
385 #else /* !BN_LLONG */
386 BN_ULONG t2l, t2h;
387
388 q = bn_div_words(n0, n1, d0);
389 #ifndef REMAINDER_IS_ALREADY_CALCULATED
390 rem = (n1 - q * d0) & BN_MASK2;
391 #endif
392
393 #if defined(BN_UMULT_LOHI)
394 BN_UMULT_LOHI(t2l, t2h, d1, q);
395 #elif defined(BN_UMULT_HIGH)
396 t2l = d1 * q;
397 t2h = BN_UMULT_HIGH(d1, q);
398 #else
399 {
400 BN_ULONG ql, qh;
401 t2l = LBITS(d1);
402 t2h = HBITS(d1);
403 ql = LBITS(q);
404 qh = HBITS(q);
405 mul64(t2l, t2h, ql, qh); /* t2=(BN_ULLONG)d1*q; */
406 }
407 #endif
408
409 for (;;) {
410 if ((t2h < rem) || ((t2h == rem) && (t2l <= n2)))
411 break;
412 q--;
413 rem += d0;
414 if (rem < d0)
415 break; /* don't let rem overflow */
416 if (t2l < d1)
417 t2h--;
418 t2l -= d1;
419 }
420 #endif /* !BN_LLONG */
421 }
422 #endif /* !BN_DIV3W */
423
424 l0 = bn_mul_words(tmp->d, sdiv->d, div_n, q);
425 tmp->d[div_n] = l0;
426 wnum--;
427 /*
428 * ignore top values of the bignums just sub the two BN_ULONG arrays
429 * with bn_sub_words
430 */
431 l0 = bn_sub_words(wnum, wnum, tmp->d, div_n + 1);
432 q -= l0;
433 /*
434 * Note: As we have considered only the leading two BN_ULONGs in
435 * the calculation of q, sdiv * q might be greater than wnum (but
436 * then (q-1) * sdiv is less or equal than wnum)
437 */
438 for (l0 = 0 - l0, j = 0; j < div_n; j++)
439 tmp->d[j] = sdiv->d[j] & l0;
440 l0 = bn_add_words(wnum, wnum, tmp->d, div_n);
441 (*wnumtop) += l0;
442 assert((*wnumtop) == 0);
443
444 /* store part of the result */
445 *--resp = q;
446 }
447 /* snum holds remainder, it's as wide as divisor */
448 snum->neg = num_neg;
449 snum->top = div_n;
450 snum->flags |= BN_FLG_FIXED_TOP;
451
452 if (rm != NULL && bn_rshift_fixed_top(rm, snum, norm_shift) == 0)
453 goto err;
454
455 BN_CTX_end(ctx);
456 return 1;
457 err:
458 bn_check_top(rm);
459 BN_CTX_end(ctx);
460 return 0;
461 }
462 #endif
463