/*
* Copyright 1995-2019 The OpenSSL Project Authors. All Rights Reserved.
*
* Licensed under the OpenSSL license (the "License"). You may not use
* this file except in compliance with the License. You can obtain a copy
* in the file LICENSE in the source distribution or at
* https://www.openssl.org/source/license.html
*/
#include <assert.h>
#include <openssl/bn.h>
#include "internal/cryptlib.h"
#include "bn_local.h"
/* The old slow way */
#if 0
int BN_div(BIGNUM *dv, BIGNUM *rem, const BIGNUM *m, const BIGNUM *d,
BN_CTX *ctx)
{
int i, nm, nd;
int ret = 0;
BIGNUM *D;
bn_check_top(m);
bn_check_top(d);
if (BN_is_zero(d)) {
BNerr(BN_F_BN_DIV, BN_R_DIV_BY_ZERO);
return 0;
}
if (BN_ucmp(m, d) < 0) {
if (rem != NULL) {
if (BN_copy(rem, m) == NULL)
return 0;
}
if (dv != NULL)
BN_zero(dv);
return 1;
}
BN_CTX_start(ctx);
D = BN_CTX_get(ctx);
if (dv == NULL)
dv = BN_CTX_get(ctx);
if (rem == NULL)
rem = BN_CTX_get(ctx);
if (D == NULL || dv == NULL || rem == NULL)
goto end;
nd = BN_num_bits(d);
nm = BN_num_bits(m);
if (BN_copy(D, d) == NULL)
goto end;
if (BN_copy(rem, m) == NULL)
goto end;
/*
* The next 2 are needed so we can do a dv->d[0]|=1 later since
* BN_lshift1 will only work once there is a value :-)
*/
BN_zero(dv);
if (bn_wexpand(dv, 1) == NULL)
goto end;
dv->top = 1;
if (!BN_lshift(D, D, nm - nd))
goto end;
for (i = nm - nd; i >= 0; i--) {
if (!BN_lshift1(dv, dv))
goto end;
if (BN_ucmp(rem, D) >= 0) {
dv->d[0] |= 1;
if (!BN_usub(rem, rem, D))
goto end;
}
/* CAN IMPROVE (and have now :=) */
if (!BN_rshift1(D, D))
goto end;
}
rem->neg = BN_is_zero(rem) ? 0 : m->neg;
dv->neg = m->neg ^ d->neg;
ret = 1;
end:
BN_CTX_end(ctx);
return ret;
}
#else
# if defined(BN_DIV3W)
BN_ULONG bn_div_3_words(const BN_ULONG *m, BN_ULONG d1, BN_ULONG d0);
# elif 0
/*
* This is #if-ed away, because it's a reference for assembly implementations,
* where it can and should be made constant-time. But if you want to test it,
* just replace 0 with 1.
*/
# if BN_BITS2 == 64 && defined(__SIZEOF_INT128__) && __SIZEOF_INT128__==16
# undef BN_ULLONG
# define BN_ULLONG __uint128_t
# define BN_LLONG
# endif
# ifdef BN_LLONG
# define BN_DIV3W
/*
* Interface is somewhat quirky, |m| is pointer to most significant limb,
* and less significant limb is referred at |m[-1]|. This means that caller
* is responsible for ensuring that |m[-1]| is valid. Second condition that
* has to be met is that |d0|'s most significant bit has to be set. Or in
* other words divisor has to be "bit-aligned to the left." bn_div_fixed_top
* does all this. The subroutine considers four limbs, two of which are
* "overlapping," hence the name...
*/
static BN_ULONG bn_div_3_words(const BN_ULONG *m, BN_ULONG d1, BN_ULONG d0)
{
BN_ULLONG R = ((BN_ULLONG)m[0] << BN_BITS2) | m[-1];
BN_ULLONG D = ((BN_ULLONG)d0 << BN_BITS2) | d1;
BN_ULONG Q = 0, mask;
int i;
for (i = 0; i < BN_BITS2; i++) {
Q <<= 1;
if (R >= D) {
Q |= 1;
R -= D;
}
D >>= 1;
}
mask = 0 - (Q >> (BN_BITS2 - 1)); /* does it overflow? */
Q <<= 1;
Q |= (R >= D);
return (Q | mask) & BN_MASK2;
}
# endif
# endif
static int bn_left_align(BIGNUM *num)
{
BN_ULONG *d = num->d, n, m, rmask;
int top = num->top;
int rshift = BN_num_bits_word(d[top - 1]), lshift, i;
lshift = BN_BITS2 - rshift;
rshift %= BN_BITS2; /* say no to undefined behaviour */
rmask = (BN_ULONG)0 - rshift; /* rmask = 0 - (rshift != 0) */
rmask |= rmask >> 8;
for (i = 0, m = 0; i < top; i++) {
n = d[i];
d[i] = ((n << lshift) | m) & BN_MASK2;
m = (n >> rshift) & rmask;
}
return lshift;
}
# if !defined(OPENSSL_NO_ASM) && !defined(OPENSSL_NO_INLINE_ASM) \
&& !defined(PEDANTIC) && !defined(BN_DIV3W)
# if defined(__GNUC__) && __GNUC__>=2
# if defined(__i386) || defined (__i386__)
/*-
* There were two reasons for implementing this template:
* - GNU C generates a call to a function (__udivdi3 to be exact)
* in reply to ((((BN_ULLONG)n0)<<BN_BITS2)|n1)/d0 (I fail to
* understand why...);
* - divl doesn't only calculate quotient, but also leaves
* remainder in %edx which we can definitely use here:-)
*/
# undef bn_div_words
# define bn_div_words(n0,n1,d0) \
({ asm volatile ( \
"divl %4" \
: "=a"(q), "=d"(rem) \
: "a"(n1), "d"(n0), "r"(d0) \
: "cc"); \
q; \
})
# define REMAINDER_IS_ALREADY_CALCULATED
# elif defined(__x86_64) && defined(SIXTY_FOUR_BIT_LONG)
/*
* Same story here, but it's 128-bit by 64-bit division. Wow!
*/
# undef bn_div_words
# define bn_div_words(n0,n1,d0) \
({ asm volatile ( \
"divq %4" \
: "=a"(q), "=d"(rem) \
: "a"(n1), "d"(n0), "r"(d0) \
: "cc"); \
q; \
})
# define REMAINDER_IS_ALREADY_CALCULATED
# endif /* __<cpu> */
# endif /* __GNUC__ */
# endif /* OPENSSL_NO_ASM */
/*-
* BN_div computes dv := num / divisor, rounding towards
* zero, and sets up rm such that dv*divisor + rm = num holds.
* Thus:
* dv->neg == num->neg ^ divisor->neg (unless the result is zero)
* rm->neg == num->neg (unless the remainder is zero)
* If 'dv' or 'rm' is NULL, the respective value is not returned.
*/
int BN_div(BIGNUM *dv, BIGNUM *rm, const BIGNUM *num, const BIGNUM *divisor,
BN_CTX *ctx)
{
int ret;
if (BN_is_zero(divisor)) {
BNerr(BN_F_BN_DIV, BN_R_DIV_BY_ZERO);
return 0;
}
/*
* Invalid zero-padding would have particularly bad consequences so don't
* just rely on bn_check_top() here (bn_check_top() works only for
* BN_DEBUG builds)
*/
if (divisor->d[divisor->top - 1] == 0) {
BNerr(BN_F_BN_DIV, BN_R_NOT_INITIALIZED);
return 0;
}
ret = bn_div_fixed_top(dv, rm, num, divisor, ctx);
if (ret) {
if (dv != NULL)
bn_correct_top(dv);
if (rm != NULL)
bn_correct_top(rm);
}
return ret;
}
/*
* It's argued that *length* of *significant* part of divisor is public.
* Even if it's private modulus that is. Again, *length* is assumed
* public, but not *value*. Former is likely to be pre-defined by
* algorithm with bit granularity, though below subroutine is invariant
* of limb length. Thanks to this assumption we can require that |divisor|
* may not be zero-padded, yet claim this subroutine "constant-time"(*).
* This is because zero-padded dividend, |num|, is tolerated, so that
* caller can pass dividend of public length(*), but with smaller amount
* of significant limbs. This naturally means that quotient, |dv|, would
* contain correspongly less significant limbs as well, and will be zero-
* padded accordingly. Returned remainder, |rm|, will have same bit length
* as divisor, also zero-padded if needed. These actually leave sign bits
* in ambiguous state. In sense that we try to avoid negative zeros, while
* zero-padded zeros would retain sign.
*
* (*) "Constant-time-ness" has two pre-conditions:
*
* - availability of constant-time bn_div_3_words;
* - dividend is at least as "wide" as divisor, limb-wise, zero-padded
* if so required, which shouldn't be a privacy problem, because
* divisor's length is considered public;
*/
int bn_div_fixed_top(BIGNUM *dv, BIGNUM *rm, const BIGNUM *num,
const BIGNUM *divisor, BN_CTX *ctx)
{
int norm_shift, i, j, loop;
BIGNUM *tmp, *snum, *sdiv, *res;
BN_ULONG *resp, *wnum, *wnumtop;
BN_ULONG d0, d1;
int num_n, div_n;
assert(divisor->top > 0 && divisor->d[divisor->top - 1] != 0);
bn_check_top(num);
bn_check_top(divisor);
bn_check_top(dv);
bn_check_top(rm);
BN_CTX_start(ctx);
res = (dv == NULL) ? BN_CTX_get(ctx) : dv;
tmp = BN_CTX_get(ctx);
snum = BN_CTX_get(ctx);
sdiv = BN_CTX_get(ctx);
if (sdiv == NULL)
goto err;
/* First we normalise the numbers */
if (!BN_copy(sdiv, divisor))
goto err;
norm_shift = bn_left_align(sdiv);
sdiv->neg = 0;
/*
* Note that bn_lshift_fixed_top's output is always one limb longer
* than input, even when norm_shift is zero. This means that amount of
* inner loop iterations is invariant of dividend value, and that one
* doesn't need to compare dividend and divisor if they were originally
* of the same bit length.
*/
if (!(bn_lshift_fixed_top(snum, num, norm_shift)))
goto err;
div_n = sdiv->top;
num_n = snum->top;
if (num_n <= div_n) {
/* caller didn't pad dividend -> no constant-time guarantee... */
if (bn_wexpand(snum, div_n + 1) == NULL)
goto err;
memset(&(snum->d[num_n]), 0, (div_n - num_n + 1) * sizeof(BN_ULONG));
snum->top = num_n = div_n + 1;
}
loop = num_n - div_n;
/*
* Lets setup a 'window' into snum This is the part that corresponds to
* the current 'area' being divided
*/
wnum = &(snum->d[loop]);
wnumtop = &(snum->d[num_n - 1]);
/* Get the top 2 words of sdiv */
d0 = sdiv->d[div_n - 1];
d1 = (div_n == 1) ? 0 : sdiv->d[div_n - 2];
/* Setup quotient */
if (!bn_wexpand(res, loop))
goto err;
res->neg = (num->neg ^ divisor->neg);
res->top = loop;
res->flags |= BN_FLG_FIXED_TOP;
resp = &(res->d[loop]);
/* space for temp */
if (!bn_wexpand(tmp, (div_n + 1)))
goto err;
for (i = 0; i < loop; i++, wnumtop--) {
BN_ULONG q, l0;
/*
* the first part of the loop uses the top two words of snum and sdiv
* to calculate a BN_ULONG q such that | wnum - sdiv * q | < sdiv
*/
# if defined(BN_DIV3W)
q = bn_div_3_words(wnumtop, d1, d0);
# else
BN_ULONG n0, n1, rem = 0;
n0 = wnumtop[0];
n1 = wnumtop[-1];
if (n0 == d0)
q = BN_MASK2;
else { /* n0 < d0 */
BN_ULONG n2 = (wnumtop == wnum) ? 0 : wnumtop[-2];
# ifdef BN_LLONG
BN_ULLONG t2;
# if defined(BN_LLONG) && defined(BN_DIV2W) && !defined(bn_div_words)
q = (BN_ULONG)(((((BN_ULLONG) n0) << BN_BITS2) | n1) / d0);
# else
q = bn_div_words(n0, n1, d0);
# endif
# ifndef REMAINDER_IS_ALREADY_CALCULATED
/*
* rem doesn't have to be BN_ULLONG. The least we
* know it's less that d0, isn't it?
*/
rem = (n1 - q * d0) & BN_MASK2;
# endif
t2 = (BN_ULLONG) d1 *q;
for (;;) {
if (t2 <= ((((BN_ULLONG) rem) << BN_BITS2) | n2))
break;
q--;
rem += d0;
if (rem < d0)
break; /* don't let rem overflow */
t2 -= d1;
}
# else /* !BN_LLONG */
BN_ULONG t2l, t2h;
q = bn_div_words(n0, n1, d0);
# ifndef REMAINDER_IS_ALREADY_CALCULATED
rem = (n1 - q * d0) & BN_MASK2;
# endif
# if defined(BN_UMULT_LOHI)
BN_UMULT_LOHI(t2l, t2h, d1, q);
# elif defined(BN_UMULT_HIGH)
t2l = d1 * q;
t2h = BN_UMULT_HIGH(d1, q);
# else
{
BN_ULONG ql, qh;
t2l = LBITS(d1);
t2h = HBITS(d1);
ql = LBITS(q);
qh = HBITS(q);
mul64(t2l, t2h, ql, qh); /* t2=(BN_ULLONG)d1*q; */
}
# endif
for (;;) {
if ((t2h < rem) || ((t2h == rem) && (t2l <= n2)))
break;
q--;
rem += d0;
if (rem < d0)
break; /* don't let rem overflow */
if (t2l < d1)
t2h--;
t2l -= d1;
}
# endif /* !BN_LLONG */
}
# endif /* !BN_DIV3W */
l0 = bn_mul_words(tmp->d, sdiv->d, div_n, q);
tmp->d[div_n] = l0;
wnum--;
/*
* ignore top values of the bignums just sub the two BN_ULONG arrays
* with bn_sub_words
*/
l0 = bn_sub_words(wnum, wnum, tmp->d, div_n + 1);
q -= l0;
/*
* Note: As we have considered only the leading two BN_ULONGs in
* the calculation of q, sdiv * q might be greater than wnum (but
* then (q-1) * sdiv is less or equal than wnum)
*/
for (l0 = 0 - l0, j = 0; j < div_n; j++)
tmp->d[j] = sdiv->d[j] & l0;
l0 = bn_add_words(wnum, wnum, tmp->d, div_n);
(*wnumtop) += l0;
assert((*wnumtop) == 0);
/* store part of the result */
*--resp = q;
}
/* snum holds remainder, it's as wide as divisor */
snum->neg = num->neg;
snum->top = div_n;
snum->flags |= BN_FLG_FIXED_TOP;
if (rm != NULL)
bn_rshift_fixed_top(rm, snum, norm_shift);
BN_CTX_end(ctx);
return 1;
err:
bn_check_top(rm);
BN_CTX_end(ctx);
return 0;
}
#endif