version 1.1.1.2, 2000/09/09 14:12:54 |
version 1.1.1.3, 2003/08/25 16:06:33 |
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/* mpz_jacobi (op1, op2). |
/* mpz_jacobi, mpz_legendre, mpz_kronecker -- mpz/mpz Jacobi symbols. |
Contributed by Bennet Yee (bsy) at Carnegie-Mellon University |
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Copyright (C) 1991, 1996 Free Software Foundation, Inc. |
Copyright 2000, 2001, 2002 Free Software Foundation, Inc. |
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This file is part of the GNU MP Library. |
This file is part of the GNU MP Library. |
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The GNU MP Library is free software; you can redistribute it and/or modify |
The GNU MP Library is free software; you can redistribute it and/or modify |
it under the terms of the GNU Lesser General Public License as published by |
it under the terms of the GNU Library General Public License as published by |
the Free Software Foundation; either version 2.1 of the License, or (at your |
the Free Software Foundation; either version 2 of the License, or (at your |
option) any later version. |
option) any later version. |
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The GNU MP Library is distributed in the hope that it will be useful, but |
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 |
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY |
or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public |
or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Library General Public |
License for more details. |
License for more details. |
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You should have received a copy of the GNU Lesser General Public License |
You should have received a copy of the GNU Library General Public License |
along with the GNU MP Library; see the file COPYING.LIB. If not, write to |
along with the GNU MP Library; see the file COPYING.LIB. If not, write to |
the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, |
the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, |
MA 02111-1307, USA. */ |
MA 02111-1307, USA. */ |
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#include <stdio.h> |
#include "gmp.h" |
#include "gmp.h" |
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#include "gmp-impl.h" |
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#include "longlong.h" |
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/* Precondition: both p and q are positive */ |
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int |
/* Change this to "#define TRACE(x) x" for some traces. */ |
#if __STDC__ |
#define TRACE(x) |
mpz_jacobi (mpz_srcptr pi, mpz_srcptr qi) |
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#else |
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mpz_jacobi (pi, qi) |
#define MPN_RSHIFT_OR_COPY(dst,src,size,shift) \ |
mpz_srcptr pi, qi; |
do { \ |
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if ((shift) != 0) \ |
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{ \ |
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ASSERT_NOCARRY (mpn_rshift (dst, src, size, shift)); \ |
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(size) -= ((dst)[(size)-1] == 0); \ |
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} \ |
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else \ |
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MPN_COPY (dst, src, size); \ |
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} while (0) |
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/* This code does triple duty as mpz_jacobi, mpz_legendre and mpz_kronecker. |
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mpz_jacobi could assume b is odd, but the improvements from that seem |
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small compared to other operations, and anything significant should be |
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checked at run-time since we'd like odd b to go fast in mpz_kronecker |
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too. |
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mpz_legendre could assume b is an odd prime, but knowing this doesn't |
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present any obvious benefits. Result 0 wouldn't arise (unless "a" is a |
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multiple of b), but the checking for that takes little time compared to |
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other operations. |
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The main loop is just a simple binary GCD with the jacobi symbol result |
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tracked during the reduction. |
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The special cases for a or b fitting in one limb let mod_1 or modexact_1 |
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get used, without any copying, and end up just as efficient as the mixed |
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precision mpz_kronecker_ui etc. |
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When tdiv_qr is called it's not necessary to make "a" odd or make a |
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working copy of it, but tdiv_qr is going to be pretty slow so it's not |
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worth bothering trying to save anything for that case. |
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Enhancements: |
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mpn_bdivmod could be used instead of mpn_tdiv_qr, like in mpn_gcd. |
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Currently tdiv_qr is preferred since it's sub-quadratic on big sizes, |
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although bdivmod might be a touch quicker on small sizes. This can be |
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revised when bdivmod becomes sub-quadratic too. |
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Some sort of multi-step algorithm should be used. The current subtract |
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and shift for every bit is very inefficient. Lehmer (per current gcdext) |
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would need some low bits included in its calculation to apply the sign |
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change for reciprocity. Binary Lehmer keeps low bits to strip twos |
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anyway, so might be better suited. Maybe the accelerated GCD style k-ary |
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reduction would work, if sign changes due to the extra factors it |
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introduces can be accounted for (or maybe they can be ignored). */ |
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/* This implementation depends on BITS_PER_MP_LIMB being even, so that |
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(a/2)^BITS_PER_MP_LIMB = 1 and so there's no need to pay attention to how |
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many low zero limbs are stripped. */ |
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#if BITS_PER_MP_LIMB % 2 != 0 |
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Error, error, need BITS_PER_MP_LIMB even |
#endif |
#endif |
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int |
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mpz_jacobi (mpz_srcptr a, mpz_srcptr b) |
{ |
{ |
#if GCDCHECK |
mp_srcptr asrcp, bsrcp; |
int retval; |
mp_size_t asize, bsize; |
mpz_t gcdval; |
mp_ptr ap, bp; |
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mp_limb_t alow, blow, ahigh, bhigh, asecond, bsecond; |
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unsigned atwos, btwos; |
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int result_bit1; |
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TMP_DECL (marker); |
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mpz_init (gcdval); |
TRACE (printf ("start asize=%d bsize=%d\n", SIZ(a), SIZ(b)); |
mpz_gcd (gcdval, pi, qi); |
mpz_trace (" a", a); |
if (!mpz_cmp_ui (gcdval, 1L)) |
mpz_trace (" b", b)); |
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asize = SIZ(a); |
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asrcp = PTR(a); |
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alow = asrcp[0]; |
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bsize = SIZ(b); |
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if (bsize == 0) |
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return JACOBI_LS0 (alow, asize); /* (a/0) */ |
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bsrcp = PTR(b); |
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blow = bsrcp[0]; |
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if (asize == 0) |
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return JACOBI_0LS (blow, bsize); /* (0/b) */ |
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/* (even/even)=0 */ |
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if (((alow | blow) & 1) == 0) |
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return 0; |
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/* account for effect of sign of b, then ignore it */ |
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result_bit1 = JACOBI_BSGN_SS_BIT1 (asize, bsize); |
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bsize = ABS (bsize); |
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/* low zero limbs on b can be discarded */ |
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MPN_STRIP_LOW_ZEROS_NOT_ZERO (bsrcp, bsize, blow); |
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count_trailing_zeros (btwos, blow); |
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TRACE (printf ("b twos %u\n", btwos)); |
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/* establish shifted blow */ |
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blow >>= btwos; |
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if (bsize > 1) |
{ |
{ |
/* J(ab,cb) = J(ab,c)J(ab,b) = J(ab,c)J(0,b) = J(ab,c)*0 */ |
bsecond = bsrcp[1]; |
retval = 0; |
if (btwos != 0) |
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blow |= (bsecond << (GMP_NUMB_BITS - btwos)) & GMP_NUMB_MASK; |
} |
} |
else |
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retval = mpz_legendre (pi, qi); |
/* account for effect of sign of a, then ignore it */ |
mpz_clear (gcdval); |
result_bit1 ^= JACOBI_ASGN_SU_BIT1 (asize, blow); |
return retval; |
asize = ABS (asize); |
#else |
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return mpz_legendre (pi, qi); |
if (bsize == 1 || (bsize == 2 && (bsecond >> btwos) == 0)) |
#endif |
{ |
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/* special case one limb b, use modexact and no copying */ |
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/* (a/2)=(2/a) with a odd, and if b is even then a is odd here */ |
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result_bit1 ^= JACOBI_TWOS_U_BIT1 (btwos, alow); |
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if (blow == 1) /* (a/1)=1 always */ |
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return JACOBI_BIT1_TO_PN (result_bit1); |
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JACOBI_MOD_OR_MODEXACT_1_ODD (result_bit1, alow, asrcp, asize, blow); |
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TRACE (printf ("base (%lu/%lu) with %d\n", |
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alow, blow, JACOBI_BIT1_TO_PN (result_bit1))); |
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return mpn_jacobi_base (alow, blow, result_bit1); |
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} |
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/* Discard low zero limbs of a. Usually there won't be anything to |
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strip, hence not bothering with it for the bsize==1 case. */ |
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MPN_STRIP_LOW_ZEROS_NOT_ZERO (asrcp, asize, alow); |
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count_trailing_zeros (atwos, alow); |
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TRACE (printf ("a twos %u\n", atwos)); |
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result_bit1 ^= JACOBI_TWOS_U_BIT1 (atwos, blow); |
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/* establish shifted alow */ |
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alow >>= atwos; |
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if (asize > 1) |
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{ |
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asecond = asrcp[1]; |
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if (atwos != 0) |
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alow |= (asecond << (GMP_NUMB_BITS - atwos)) & GMP_NUMB_MASK; |
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} |
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/* (a/2)=(2/a) with a odd */ |
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result_bit1 ^= JACOBI_TWOS_U_BIT1 (btwos, alow); |
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if (asize == 1 || (asize == 2 && (asecond >> atwos) == 0)) |
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{ |
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/* another special case with modexact and no copying */ |
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if (alow == 1) /* (1/b)=1 always */ |
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return JACOBI_BIT1_TO_PN (result_bit1); |
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/* b still has its twos, so cancel out their effect */ |
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result_bit1 ^= JACOBI_TWOS_U_BIT1 (btwos, alow); |
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result_bit1 ^= JACOBI_RECIP_UU_BIT1 (alow, blow); /* now (b/a) */ |
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JACOBI_MOD_OR_MODEXACT_1_ODD (result_bit1, blow, bsrcp, bsize, alow); |
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TRACE (printf ("base (%lu/%lu) with %d\n", |
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blow, alow, JACOBI_BIT1_TO_PN (result_bit1))); |
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return mpn_jacobi_base (blow, alow, result_bit1); |
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} |
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TMP_MARK (marker); |
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TMP_ALLOC_LIMBS_2 (ap, asize, bp, bsize); |
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MPN_RSHIFT_OR_COPY (ap, asrcp, asize, atwos); |
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ASSERT (alow == ap[0]); |
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TRACE (mpn_trace ("stripped a", ap, asize)); |
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MPN_RSHIFT_OR_COPY (bp, bsrcp, bsize, btwos); |
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ASSERT (blow == bp[0]); |
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TRACE (mpn_trace ("stripped b", bp, bsize)); |
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/* swap if necessary to make a longer than b */ |
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if (asize < bsize) |
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{ |
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TRACE (printf ("swap\n")); |
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MPN_PTR_SWAP (ap,asize, bp,bsize); |
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MP_LIMB_T_SWAP (alow, blow); |
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result_bit1 ^= JACOBI_RECIP_UU_BIT1 (alow, blow); |
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} |
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/* If a is bigger than b then reduce to a mod b. |
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Division is much faster than chipping away at "a" bit-by-bit. */ |
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if (asize > bsize) |
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{ |
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mp_ptr rp, qp; |
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TRACE (printf ("tdiv_qr asize=%ld bsize=%ld\n", asize, bsize)); |
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TMP_ALLOC_LIMBS_2 (rp, bsize, qp, asize-bsize+1); |
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mpn_tdiv_qr (qp, rp, 0, ap, asize, bp, bsize); |
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ap = rp; |
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asize = bsize; |
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MPN_NORMALIZE (ap, asize); |
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TRACE (printf ("tdiv_qr asize=%ld bsize=%ld\n", asize, bsize); |
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mpn_trace (" a", ap, asize); |
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mpn_trace (" b", bp, bsize)); |
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if (asize == 0) /* (0/b)=0 for b!=1 */ |
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goto zero; |
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alow = ap[0]; |
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goto strip_a; |
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} |
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for (;;) |
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{ |
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ASSERT (asize >= 1); /* a,b non-empty */ |
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ASSERT (bsize >= 1); |
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ASSERT (ap[asize-1] != 0); /* a,b normalized (and hence non-zero) */ |
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ASSERT (bp[bsize-1] != 0); |
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ASSERT (alow == ap[0]); /* low limb copies should be correct */ |
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ASSERT (blow == bp[0]); |
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ASSERT (alow & 1); /* a,b odd */ |
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ASSERT (blow & 1); |
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TRACE (printf ("top asize=%ld bsize=%ld\n", asize, bsize); |
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mpn_trace (" a", ap, asize); |
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mpn_trace (" b", bp, bsize)); |
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/* swap if necessary to make a>=b, applying reciprocity |
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high limbs are almost always enough to tell which is bigger */ |
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if (asize < bsize |
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|| (asize == bsize |
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&& ((ahigh=ap[asize-1]) < (bhigh=bp[asize-1]) |
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|| (ahigh == bhigh |
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&& mpn_cmp (ap, bp, asize-1) < 0)))) |
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{ |
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TRACE (printf ("swap\n")); |
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MPN_PTR_SWAP (ap,asize, bp,bsize); |
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MP_LIMB_T_SWAP (alow, blow); |
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result_bit1 ^= JACOBI_RECIP_UU_BIT1 (alow, blow); |
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} |
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if (asize == 1) |
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break; |
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/* a = a-b */ |
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ASSERT (asize >= bsize); |
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ASSERT_NOCARRY (mpn_sub (ap, ap, asize, bp, bsize)); |
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MPN_NORMALIZE (ap, asize); |
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alow = ap[0]; |
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/* (0/b)=0 for b!=1. b!=1 when a==0 because otherwise would have had |
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a==1 which is asize==1 and would have exited above. */ |
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if (asize == 0) |
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goto zero; |
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strip_a: |
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/* low zero limbs on a can be discarded */ |
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MPN_STRIP_LOW_ZEROS_NOT_ZERO (ap, asize, alow); |
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if ((alow & 1) == 0) |
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{ |
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/* factors of 2 from a */ |
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unsigned twos; |
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count_trailing_zeros (twos, alow); |
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TRACE (printf ("twos %u\n", twos)); |
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result_bit1 ^= JACOBI_TWOS_U_BIT1 (twos, blow); |
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ASSERT_NOCARRY (mpn_rshift (ap, ap, asize, twos)); |
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asize -= (ap[asize-1] == 0); |
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alow = ap[0]; |
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} |
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} |
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ASSERT (asize == 1 && bsize == 1); /* just alow and blow left */ |
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TMP_FREE (marker); |
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/* (1/b)=1 always (in this case have b==1 because a>=b) */ |
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if (alow == 1) |
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return JACOBI_BIT1_TO_PN (result_bit1); |
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/* swap with reciprocity and do (b/a) */ |
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result_bit1 ^= JACOBI_RECIP_UU_BIT1 (alow, blow); |
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TRACE (printf ("base (%lu/%lu) with %d\n", |
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blow, alow, JACOBI_BIT1_TO_PN (result_bit1))); |
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return mpn_jacobi_base (blow, alow, result_bit1); |
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zero: |
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TMP_FREE (marker); |
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return 0; |
} |
} |