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/* mpn_modexact_1c_odd -- mpn by limb exact division style remainder.
THE FUNCTIONS IN THIS FILE ARE FOR INTERNAL USE ONLY. THEY'RE ALMOST
CERTAIN TO BE SUBJECT TO INCOMPATIBLE CHANGES OR DISAPPEAR COMPLETELY IN
FUTURE GNU MP RELEASES.
Copyright 2000-2004 Free Software Foundation, Inc.
This file is part of the GNU MP Library.
The GNU MP Library is free software; you can redistribute it and/or modify
it under the terms of either:
* the GNU Lesser General Public License as published by the Free
Software Foundation; either version 3 of the License, or (at your
option) any later version.
or
* the GNU General Public License as published by the Free Software
Foundation; either version 2 of the License, or (at your option) any
later version.
or both in parallel, as here.
The GNU MP Library is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
for more details.
You should have received copies of the GNU General Public License and the
GNU Lesser General Public License along with the GNU MP Library. If not,
see https://www.gnu.org/licenses/. */
#include "gmp-impl.h"
#include "longlong.h"
/* Calculate an r satisfying
r*B^k + a - c == q*d
where B=2^GMP_LIMB_BITS, a is {src,size}, k is either size or size-1
(the caller won't know which), and q is the quotient (discarded). d must
be odd, c can be any limb value.
If c<d then r will be in the range 0<=r<d, or if c>=d then 0<=r<=d.
This slightly strange function suits the initial Nx1 reduction for GCDs
or Jacobi symbols since the factors of 2 in B^k can be ignored, leaving
-r == a mod d (by passing c=0). For a GCD the factor of -1 on r can be
ignored, or for the Jacobi symbol it can be accounted for. The function
also suits divisibility and congruence testing since if r=0 (or r=d) is
obtained then a==c mod d.
r is a bit like the remainder returned by mpn_divexact_by3c, and is the
sort of remainder mpn_divexact_1 might return. Like mpn_divexact_by3c, r
represents a borrow, since effectively quotient limbs are chosen so that
subtracting that multiple of d from src at each step will produce a zero
limb.
A long calculation can be done piece by piece from low to high by passing
the return value from one part as the carry parameter to the next part.
The effective final k becomes anything between size and size-n, if n
pieces are used.
A similar sort of routine could be constructed based on adding multiples
of d at each limb, much like redc in mpz_powm does. Subtracting however
has a small advantage that when subtracting to cancel out l there's never
a borrow into h, whereas using an addition would put a carry into h
depending whether l==0 or l!=0.
In terms of efficiency, this function is similar to a mul-by-inverse
mpn_mod_1. Both are essentially two multiplies and are best suited to
CPUs with low latency multipliers (in comparison to a divide instruction
at least.) But modexact has a few less supplementary operations, only
needs low part and high part multiplies, and has fewer working quantities
(helping CPUs with few registers).
In the main loop it will be noted that the new carry (call it r) is the
sum of the high product h and any borrow from l=s-c. If c<d then we will
have r<d too, for the following reasons. Let q=l*inverse be the quotient
limb, so that q*d = B*h + l, where B=2^GMP_NUMB_BITS. Now if h=d-1 then
l = q*d - B*(d-1) <= (B-1)*d - B*(d-1) = B-d
But if l=s-c produces a borrow when c<d, then l>=B-d+1 and hence will
never have h=d-1 and so r=h+borrow <= d-1.
When c>=d, on the other hand, h=d-1 can certainly occur together with a
borrow, thereby giving only r<=d, as per the function definition above.
As a design decision it's left to the caller to check for r=d if it might
be passing c>=d. Several applications have c<d initially so the extra
test is often unnecessary, for example the GCDs or a plain divisibility
d|a test will pass c=0.
The special case for size==1 is so that it can be assumed c<=d in the
high<=divisor test at the end. c<=d is only guaranteed after at least
one iteration of the main loop. There's also a decent chance one % is
faster than a binvert_limb, though that will depend on the processor.
A CPU specific implementation might want to omit the size==1 code or the
high<divisor test. mpn/x86/k6/mode1o.asm for instance finds neither
useful. */
mp_limb_t
mpn_modexact_1c_odd (mp_srcptr src, mp_size_t size, mp_limb_t d,
mp_limb_t orig_c)
{
mp_limb_t s, h, l, inverse, dummy, dmul, ret;
mp_limb_t c = orig_c;
mp_size_t i;
ASSERT (size >= 1);
ASSERT (d & 1);
ASSERT_MPN (src, size);
ASSERT_LIMB (d);
ASSERT_LIMB (c);
if (size == 1)
{
s = src[0];
if (s > c)
{
l = s-c;
h = l % d;
if (h != 0)
h = d - h;
}
else
{
l = c-s;
h = l % d;
}
return h;
}
binvert_limb (inverse, d);
dmul = d << GMP_NAIL_BITS;
i = 0;
do
{
s = src[i];
SUBC_LIMB (c, l, s, c);
l = (l * inverse) & GMP_NUMB_MASK;
umul_ppmm (h, dummy, l, dmul);
c += h;
}
while (++i < size-1);
s = src[i];
if (s <= d)
{
/* With high<=d the final step can be a subtract and addback. If c==0
then the addback will restore to l>=0. If c==d then will get l==d
if s==0, but that's ok per the function definition. */
l = c - s;
if (c < s)
l += d;
ret = l;
}
else
{
/* Can't skip a divide, just do the loop code once more. */
SUBC_LIMB (c, l, s, c);
l = (l * inverse) & GMP_NUMB_MASK;
umul_ppmm (h, dummy, l, dmul);
c += h;
ret = c;
}
ASSERT (orig_c < d ? ret < d : ret <= d);
return ret;
}
#if 0
/* The following is an alternate form that might shave one cycle on a
superscalar processor since it takes c+=h off the dependent chain,
leaving just a low product, high product, and a subtract.
This is for CPU specific implementations to consider. A special case for
high<divisor and/or size==1 can be added if desired.
Notice that c is only ever 0 or 1, since if s-c produces a borrow then
x=0xFF..FF and x-h cannot produce a borrow. The c=(x>s) could become
c=(x==0xFF..FF) too, if that helped. */
mp_limb_t
mpn_modexact_1c_odd (mp_srcptr src, mp_size_t size, mp_limb_t d, mp_limb_t h)
{
mp_limb_t s, x, y, inverse, dummy, dmul, c1, c2;
mp_limb_t c = 0;
mp_size_t i;
ASSERT (size >= 1);
ASSERT (d & 1);
binvert_limb (inverse, d);
dmul = d << GMP_NAIL_BITS;
for (i = 0; i < size; i++)
{
ASSERT (c==0 || c==1);
s = src[i];
SUBC_LIMB (c1, x, s, c);
SUBC_LIMB (c2, y, x, h);
c = c1 + c2;
y = (y * inverse) & GMP_NUMB_MASK;
umul_ppmm (h, dummy, y, dmul);
}
h += c;
return h;
}
#endif
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