--- /dev/null
+/* mpz_jacobi, mpz_legendre, mpz_kronecker -- mpz/mpz Jacobi symbols.
+
+Copyright 2000, 2001, 2002, 2005 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 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.
+
+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 Lesser General Public License
+for more details.
+
+You should have received a copy of the GNU Lesser General Public License along
+with the GNU MP Library. If not, see http://www.gnu.org/licenses/. */
+
+#include <stdio.h>
+#include "gmp.h"
+#include "gmp-impl.h"
+#include "longlong.h"
+
+
+/* Change this to "#define TRACE(x) x" for some traces. */
+#define TRACE(x)
+
+
+#define MPN_RSHIFT_OR_COPY(dst,src,size,shift) \
+ do { \
+ if ((shift) != 0) \
+ { \
+ ASSERT_NOCARRY (mpn_rshift (dst, src, size, shift)); \
+ (size) -= ((dst)[(size)-1] == 0); \
+ } \
+ else \
+ MPN_COPY (dst, src, size); \
+ } while (0)
+
+
+/* This code does triple duty as mpz_jacobi, mpz_legendre and mpz_kronecker.
+
+ mpz_jacobi could assume b is odd, but the improvements from that seem
+ small compared to other operations, and anything significant should be
+ checked at run-time since we'd like odd b to go fast in mpz_kronecker
+ too.
+
+ mpz_legendre could assume b is an odd prime, but knowing this doesn't
+ present any obvious benefits. Result 0 wouldn't arise (unless "a" is a
+ multiple of b), but the checking for that takes little time compared to
+ other operations.
+
+ The main loop is just a simple binary GCD with the jacobi symbol result
+ tracked during the reduction.
+
+ The special cases for a or b fitting in one limb let mod_1 or modexact_1
+ get used, without any copying, and end up just as efficient as the mixed
+ precision mpz_kronecker_ui etc.
+
+ When tdiv_qr is called it's not necessary to make "a" odd or make a
+ working copy of it, but tdiv_qr is going to be pretty slow so it's not
+ worth bothering trying to save anything for that case.
+
+ Enhancements:
+
+ mpn_bdivmod could be used instead of mpn_tdiv_qr, like in mpn_gcd.
+ Currently tdiv_qr is preferred since it's sub-quadratic on big sizes,
+ although bdivmod might be a touch quicker on small sizes. This can be
+ revised when bdivmod becomes sub-quadratic too.
+
+ Some sort of multi-step algorithm should be used. The current subtract
+ and shift for every bit is very inefficient. Lehmer (per current gcdext)
+ would need some low bits included in its calculation to apply the sign
+ change for reciprocity. Binary Lehmer keeps low bits to strip twos
+ anyway, so might be better suited. Maybe the accelerated GCD style k-ary
+ reduction would work, if sign changes due to the extra factors it
+ introduces can be accounted for (or maybe they can be ignored). */
+
+
+int
+mpz_jacobi (mpz_srcptr a, mpz_srcptr b)
+{
+ mp_srcptr asrcp, bsrcp;
+ mp_size_t asize, bsize;
+ mp_ptr ap, bp;
+ mp_limb_t alow, blow, ahigh, bhigh, asecond, bsecond;
+ unsigned atwos, btwos;
+ int result_bit1;
+ TMP_DECL;
+
+ TRACE (printf ("start asize=%d bsize=%d\n", SIZ(a), SIZ(b));
+ mpz_trace (" a", a);
+ mpz_trace (" b", b));
+
+ asize = SIZ(a);
+ asrcp = PTR(a);
+ alow = asrcp[0];
+
+ bsize = SIZ(b);
+ if (bsize == 0)
+ return JACOBI_LS0 (alow, asize); /* (a/0) */
+
+ bsrcp = PTR(b);
+ blow = bsrcp[0];
+
+ if (asize == 0)
+ return JACOBI_0LS (blow, bsize); /* (0/b) */
+
+ /* (even/even)=0 */
+ if (((alow | blow) & 1) == 0)
+ return 0;
+
+ /* account for effect of sign of b, then ignore it */
+ result_bit1 = JACOBI_BSGN_SS_BIT1 (asize, bsize);
+ bsize = ABS (bsize);
+
+ /* low zero limbs on b can be discarded */
+ JACOBI_STRIP_LOW_ZEROS (result_bit1, alow, bsrcp, bsize, blow);
+
+ count_trailing_zeros (btwos, blow);
+ TRACE (printf ("b twos %u\n", btwos));
+
+ /* establish shifted blow */
+ blow >>= btwos;
+ if (bsize > 1)
+ {
+ bsecond = bsrcp[1];
+ if (btwos != 0)
+ blow |= (bsecond << (GMP_NUMB_BITS - btwos)) & GMP_NUMB_MASK;
+ }
+
+ /* account for effect of sign of a, then ignore it */
+ result_bit1 ^= JACOBI_ASGN_SU_BIT1 (asize, blow);
+ asize = ABS (asize);
+
+ if (bsize == 1 || (bsize == 2 && (bsecond >> btwos) == 0))
+ {
+ /* special case one limb b, use modexact and no copying */
+
+ /* (a/2)=(2/a) with a odd, and if b is even then a is odd here */
+ result_bit1 ^= JACOBI_TWOS_U_BIT1 (btwos, alow);
+
+ if (blow == 1) /* (a/1)=1 always */
+ return JACOBI_BIT1_TO_PN (result_bit1);
+
+ JACOBI_MOD_OR_MODEXACT_1_ODD (result_bit1, alow, asrcp, asize, blow);
+ TRACE (printf ("base (%lu/%lu) with %d\n",
+ alow, blow, JACOBI_BIT1_TO_PN (result_bit1)));
+ return mpn_jacobi_base (alow, blow, result_bit1);
+ }
+
+ /* Discard low zero limbs of a. Usually there won't be anything to
+ strip, hence not bothering with it for the bsize==1 case. */
+ JACOBI_STRIP_LOW_ZEROS (result_bit1, blow, asrcp, asize, alow);
+
+ count_trailing_zeros (atwos, alow);
+ TRACE (printf ("a twos %u\n", atwos));
+ result_bit1 ^= JACOBI_TWOS_U_BIT1 (atwos, blow);
+
+ /* establish shifted alow */
+ alow >>= atwos;
+ if (asize > 1)
+ {
+ asecond = asrcp[1];
+ if (atwos != 0)
+ alow |= (asecond << (GMP_NUMB_BITS - atwos)) & GMP_NUMB_MASK;
+ }
+
+ /* (a/2)=(2/a) with a odd */
+ result_bit1 ^= JACOBI_TWOS_U_BIT1 (btwos, alow);
+
+ if (asize == 1 || (asize == 2 && (asecond >> atwos) == 0))
+ {
+ /* another special case with modexact and no copying */
+
+ if (alow == 1) /* (1/b)=1 always */
+ return JACOBI_BIT1_TO_PN (result_bit1);
+
+ /* b still has its twos, so cancel out their effect */
+ result_bit1 ^= JACOBI_TWOS_U_BIT1 (btwos, alow);
+
+ result_bit1 ^= JACOBI_RECIP_UU_BIT1 (alow, blow); /* now (b/a) */
+ JACOBI_MOD_OR_MODEXACT_1_ODD (result_bit1, blow, bsrcp, bsize, alow);
+ TRACE (printf ("base (%lu/%lu) with %d\n",
+ blow, alow, JACOBI_BIT1_TO_PN (result_bit1)));
+ return mpn_jacobi_base (blow, alow, result_bit1);
+ }
+
+
+ TMP_MARK;
+ TMP_ALLOC_LIMBS_2 (ap, asize, bp, bsize);
+
+ MPN_RSHIFT_OR_COPY (ap, asrcp, asize, atwos);
+ ASSERT (alow == ap[0]);
+ TRACE (mpn_trace ("stripped a", ap, asize));
+
+ MPN_RSHIFT_OR_COPY (bp, bsrcp, bsize, btwos);
+ ASSERT (blow == bp[0]);
+ TRACE (mpn_trace ("stripped b", bp, bsize));
+
+ /* swap if necessary to make a longer than b */
+ if (asize < bsize)
+ {
+ TRACE (printf ("swap\n"));
+ MPN_PTR_SWAP (ap,asize, bp,bsize);
+ MP_LIMB_T_SWAP (alow, blow);
+ result_bit1 ^= JACOBI_RECIP_UU_BIT1 (alow, blow);
+ }
+
+ /* If a is bigger than b then reduce to a mod b.
+ Division is much faster than chipping away at "a" bit-by-bit. */
+ if (asize > bsize)
+ {
+ mp_ptr rp, qp;
+
+ TRACE (printf ("tdiv_qr asize=%ld bsize=%ld\n", asize, bsize));
+
+ TMP_ALLOC_LIMBS_2 (rp, bsize, qp, asize-bsize+1);
+ mpn_tdiv_qr (qp, rp, (mp_size_t) 0, ap, asize, bp, bsize);
+ ap = rp;
+ asize = bsize;
+ MPN_NORMALIZE (ap, asize);
+
+ TRACE (printf ("tdiv_qr asize=%ld bsize=%ld\n", asize, bsize);
+ mpn_trace (" a", ap, asize);
+ mpn_trace (" b", bp, bsize));
+
+ if (asize == 0) /* (0/b)=0 for b!=1 */
+ goto zero;
+
+ alow = ap[0];
+ goto strip_a;
+ }
+
+ for (;;)
+ {
+ ASSERT (asize >= 1); /* a,b non-empty */
+ ASSERT (bsize >= 1);
+ ASSERT (ap[asize-1] != 0); /* a,b normalized (and hence non-zero) */
+ ASSERT (bp[bsize-1] != 0);
+ ASSERT (alow == ap[0]); /* low limb copies should be correct */
+ ASSERT (blow == bp[0]);
+ ASSERT (alow & 1); /* a,b odd */
+ ASSERT (blow & 1);
+
+ TRACE (printf ("top asize=%ld bsize=%ld\n", asize, bsize);
+ mpn_trace (" a", ap, asize);
+ mpn_trace (" b", bp, bsize));
+
+ /* swap if necessary to make a>=b, applying reciprocity
+ high limbs are almost always enough to tell which is bigger */
+ if (asize < bsize
+ || (asize == bsize
+ && ((ahigh=ap[asize-1]) < (bhigh=bp[asize-1])
+ || (ahigh == bhigh
+ && mpn_cmp (ap, bp, asize-1) < 0))))
+ {
+ TRACE (printf ("swap\n"));
+ MPN_PTR_SWAP (ap,asize, bp,bsize);
+ MP_LIMB_T_SWAP (alow, blow);
+ result_bit1 ^= JACOBI_RECIP_UU_BIT1 (alow, blow);
+ }
+
+ if (asize == 1)
+ break;
+
+ /* a = a-b */
+ ASSERT (asize >= bsize);
+ ASSERT_NOCARRY (mpn_sub (ap, ap, asize, bp, bsize));
+ MPN_NORMALIZE (ap, asize);
+ alow = ap[0];
+
+ /* (0/b)=0 for b!=1. b!=1 when a==0 because otherwise would have had
+ a==1 which is asize==1 and would have exited above. */
+ if (asize == 0)
+ goto zero;
+
+ strip_a:
+ /* low zero limbs on a can be discarded */
+ JACOBI_STRIP_LOW_ZEROS (result_bit1, blow, ap, asize, alow);
+
+ if ((alow & 1) == 0)
+ {
+ /* factors of 2 from a */
+ unsigned twos;
+ count_trailing_zeros (twos, alow);
+ TRACE (printf ("twos %u\n", twos));
+ result_bit1 ^= JACOBI_TWOS_U_BIT1 (twos, blow);
+ ASSERT_NOCARRY (mpn_rshift (ap, ap, asize, twos));
+ asize -= (ap[asize-1] == 0);
+ alow = ap[0];
+ }
+ }
+
+ ASSERT (asize == 1 && bsize == 1); /* just alow and blow left */
+ TMP_FREE;
+
+ /* (1/b)=1 always (in this case have b==1 because a>=b) */
+ if (alow == 1)
+ return JACOBI_BIT1_TO_PN (result_bit1);
+
+ /* swap with reciprocity and do (b/a) */
+ result_bit1 ^= JACOBI_RECIP_UU_BIT1 (alow, blow);
+ TRACE (printf ("base (%lu/%lu) with %d\n",
+ blow, alow, JACOBI_BIT1_TO_PN (result_bit1)));
+ return mpn_jacobi_base (blow, alow, result_bit1);
+
+ zero:
+ TMP_FREE;
+ return 0;
+}
projects / msp430-gcc.git / blobdiff