xref: /openssl/crypto/bn/bn_div.c (revision fecb3aae)
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