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375 lines
10 KiB
C
375 lines
10 KiB
C
/* mulmod_bnm1.c -- multiplication mod B^n-1.
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Contributed to the GNU project by Niels Möller, Torbjorn Granlund and
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Marco Bodrato.
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THE FUNCTIONS IN THIS FILE ARE INTERNAL WITH MUTABLE INTERFACES. IT IS ONLY
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SAFE TO REACH THEM THROUGH DOCUMENTED INTERFACES. IN FACT, IT IS ALMOST
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GUARANTEED THAT THEY WILL CHANGE OR DISAPPEAR IN A FUTURE GNU MP RELEASE.
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Copyright 2009, 2010, 2012, 2013, 2020, 2022 Free Software Foundation, Inc.
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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
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it under the terms of either:
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* the GNU Lesser General Public License as published by the Free
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Software Foundation; either version 3 of the License, or (at your
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option) any later version.
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or
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* the GNU General Public License as published by the Free Software
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Foundation; either version 2 of the License, or (at your option) any
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later version.
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or both in parallel, as here.
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The GNU MP Library is distributed in the hope that it will be useful, but
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WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
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or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
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for more details.
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You should have received copies of the GNU General Public License and the
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GNU Lesser General Public License along with the GNU MP Library. If not,
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see https://www.gnu.org/licenses/. */
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#include "gmp-impl.h"
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#include "longlong.h"
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/* Inputs are {ap,rn} and {bp,rn}; output is {rp,rn}, computation is
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mod B^rn - 1, and values are semi-normalised; zero is represented
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as either 0 or B^n - 1. Needs a scratch of 2rn limbs at tp.
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tp==rp is allowed. */
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void
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mpn_bc_mulmod_bnm1 (mp_ptr rp, mp_srcptr ap, mp_srcptr bp, mp_size_t rn,
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mp_ptr tp)
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{
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mp_limb_t cy;
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ASSERT (0 < rn);
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mpn_mul_n (tp, ap, bp, rn);
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cy = mpn_add_n (rp, tp, tp + rn, rn);
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/* If cy == 1, then the value of rp is at most B^rn - 2, so there can
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* be no overflow when adding in the carry. */
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MPN_INCR_U (rp, rn, cy);
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}
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/* Inputs are {ap,rn+1} and {bp,rn+1}; output is {rp,rn+1}, in
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normalised representation, computation is mod B^rn + 1. Needs
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a scratch area of 2rn limbs at tp; tp == rp is allowed.
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Output is normalised. */
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static void
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mpn_bc_mulmod_bnp1 (mp_ptr rp, mp_srcptr ap, mp_srcptr bp, mp_size_t rn,
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mp_ptr tp)
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{
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mp_limb_t cy;
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unsigned k;
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ASSERT (0 < rn);
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if (UNLIKELY (ap[rn] | bp [rn]))
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{
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if (ap[rn])
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cy = bp [rn] + mpn_neg (rp, bp, rn);
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else /* ap[rn] == 0 */
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cy = mpn_neg (rp, ap, rn);
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}
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else if (MPN_MULMOD_BKNP1_USABLE (rn, k, MUL_FFT_MODF_THRESHOLD))
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{
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mp_size_t n_k = rn / k;
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TMP_DECL;
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TMP_MARK;
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mpn_mulmod_bknp1 (rp, ap, bp, n_k, k,
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TMP_ALLOC_LIMBS (mpn_mulmod_bknp1_itch (rn)));
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TMP_FREE;
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return;
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}
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else
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{
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mpn_mul_n (tp, ap, bp, rn);
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cy = mpn_sub_n (rp, tp, tp + rn, rn);
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}
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rp[rn] = 0;
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MPN_INCR_U (rp, rn + 1, cy);
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}
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/* Computes {rp,MIN(rn,an+bn)} <- {ap,an}*{bp,bn} Mod(B^rn-1)
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*
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* The result is expected to be ZERO if and only if one of the operand
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* already is. Otherwise the class [0] Mod(B^rn-1) is represented by
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* B^rn-1. This should not be a problem if mulmod_bnm1 is used to
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* combine results and obtain a natural number when one knows in
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* advance that the final value is less than (B^rn-1).
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* Moreover it should not be a problem if mulmod_bnm1 is used to
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* compute the full product with an+bn <= rn, because this condition
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* implies (B^an-1)(B^bn-1) < (B^rn-1) .
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*
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* Requires 0 < bn <= an <= rn and an + bn > rn/2
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* Scratch need: rn + (need for recursive call OR rn + 4). This gives
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*
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* S(n) <= rn + MAX (rn + 4, S(n/2)) <= 2rn + 4
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*/
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void
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mpn_mulmod_bnm1 (mp_ptr rp, mp_size_t rn, mp_srcptr ap, mp_size_t an, mp_srcptr bp, mp_size_t bn, mp_ptr tp)
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{
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ASSERT (0 < bn);
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ASSERT (bn <= an);
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ASSERT (an <= rn);
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if ((rn & 1) != 0 || BELOW_THRESHOLD (rn, MULMOD_BNM1_THRESHOLD))
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{
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if (UNLIKELY (bn < rn))
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{
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if (UNLIKELY (an + bn <= rn))
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{
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mpn_mul (rp, ap, an, bp, bn);
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}
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else
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{
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mp_limb_t cy;
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mpn_mul (tp, ap, an, bp, bn);
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cy = mpn_add (rp, tp, rn, tp + rn, an + bn - rn);
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MPN_INCR_U (rp, rn, cy);
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}
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}
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else
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mpn_bc_mulmod_bnm1 (rp, ap, bp, rn, tp);
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}
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else
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{
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mp_size_t n;
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mp_limb_t cy;
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mp_limb_t hi;
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n = rn >> 1;
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/* We need at least an + bn >= n, to be able to fit one of the
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recursive products at rp. Requiring strict inequality makes
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the code slightly simpler. If desired, we could avoid this
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restriction by initially halving rn as long as rn is even and
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an + bn <= rn/2. */
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ASSERT (an + bn > n);
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/* Compute xm = a*b mod (B^n - 1), xp = a*b mod (B^n + 1)
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and crt together as
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x = -xp * B^n + (B^n + 1) * [ (xp + xm)/2 mod (B^n-1)]
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*/
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#define a0 ap
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#define a1 (ap + n)
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#define b0 bp
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#define b1 (bp + n)
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#define xp tp /* 2n + 2 */
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/* am1 maybe in {xp, n} */
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/* bm1 maybe in {xp + n, n} */
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#define sp1 (tp + 2*n + 2)
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/* ap1 maybe in {sp1, n + 1} */
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/* bp1 maybe in {sp1 + n + 1, n + 1} */
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{
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mp_srcptr am1, bm1;
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mp_size_t anm, bnm;
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mp_ptr so;
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bm1 = b0;
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bnm = bn;
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if (LIKELY (an > n))
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{
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am1 = xp;
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cy = mpn_add (xp, a0, n, a1, an - n);
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MPN_INCR_U (xp, n, cy);
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anm = n;
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so = xp + n;
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if (LIKELY (bn > n))
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{
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bm1 = so;
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cy = mpn_add (so, b0, n, b1, bn - n);
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MPN_INCR_U (so, n, cy);
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bnm = n;
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so += n;
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}
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}
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else
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{
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so = xp;
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am1 = a0;
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anm = an;
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}
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mpn_mulmod_bnm1 (rp, n, am1, anm, bm1, bnm, so);
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}
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{
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int k;
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mp_srcptr ap1, bp1;
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mp_size_t anp, bnp;
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bp1 = b0;
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bnp = bn;
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if (LIKELY (an > n)) {
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ap1 = sp1;
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cy = mpn_sub (sp1, a0, n, a1, an - n);
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sp1[n] = 0;
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MPN_INCR_U (sp1, n + 1, cy);
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anp = n + ap1[n];
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if (LIKELY (bn > n)) {
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bp1 = sp1 + n + 1;
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cy = mpn_sub (sp1 + n + 1, b0, n, b1, bn - n);
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sp1[2*n+1] = 0;
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MPN_INCR_U (sp1 + n + 1, n + 1, cy);
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bnp = n + bp1[n];
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}
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} else {
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ap1 = a0;
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anp = an;
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}
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if (BELOW_THRESHOLD (n, MUL_FFT_MODF_THRESHOLD))
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k=0;
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else
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{
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int mask;
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k = mpn_fft_best_k (n, 0);
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mask = (1<<k) - 1;
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while (n & mask) {k--; mask >>=1;};
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}
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if (k >= FFT_FIRST_K)
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xp[n] = mpn_mul_fft (xp, n, ap1, anp, bp1, bnp, k);
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else if (UNLIKELY (bp1 == b0))
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{
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ASSERT (anp + bnp <= 2*n+1);
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ASSERT (anp + bnp > n);
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ASSERT (anp >= bnp);
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mpn_mul (xp, ap1, anp, bp1, bnp);
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anp = anp + bnp - n;
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ASSERT (anp <= n || xp[2*n]==0);
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anp-= anp > n;
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cy = mpn_sub (xp, xp, n, xp + n, anp);
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xp[n] = 0;
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MPN_INCR_U (xp, n+1, cy);
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}
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else
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mpn_bc_mulmod_bnp1 (xp, ap1, bp1, n, xp);
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}
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/* Here the CRT recomposition begins.
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xm <- (xp + xm)/2 = (xp + xm)B^n/2 mod (B^n-1)
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Division by 2 is a bitwise rotation.
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Assumes xp normalised mod (B^n+1).
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The residue class [0] is represented by [B^n-1]; except when
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both input are ZERO.
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*/
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#if HAVE_NATIVE_mpn_rsh1add_n || HAVE_NATIVE_mpn_rsh1add_nc
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#if HAVE_NATIVE_mpn_rsh1add_nc
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cy = mpn_rsh1add_nc(rp, rp, xp, n, xp[n]); /* B^n = 1 */
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hi = cy << (GMP_NUMB_BITS - 1);
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cy = 0;
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/* next update of rp[n-1] will set cy = 1 only if rp[n-1]+=hi
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overflows, i.e. a further increment will not overflow again. */
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#else /* ! _nc */
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cy = xp[n] + mpn_rsh1add_n(rp, rp, xp, n); /* B^n = 1 */
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hi = (cy<<(GMP_NUMB_BITS-1))&GMP_NUMB_MASK; /* (cy&1) << ... */
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cy >>= 1;
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/* cy = 1 only if xp[n] = 1 i.e. {xp,n} = ZERO, this implies that
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the rsh1add was a simple rshift: the top bit is 0. cy=1 => hi=0. */
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#endif
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#if GMP_NAIL_BITS == 0
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add_ssaaaa(cy, rp[n-1], cy, rp[n-1], 0, hi);
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#else
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cy += (hi & rp[n-1]) >> (GMP_NUMB_BITS-1);
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rp[n-1] ^= hi;
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#endif
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#else /* ! HAVE_NATIVE_mpn_rsh1add_n */
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#if HAVE_NATIVE_mpn_add_nc
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cy = mpn_add_nc(rp, rp, xp, n, xp[n]);
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#else /* ! _nc */
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cy = xp[n] + mpn_add_n(rp, rp, xp, n); /* xp[n] == 1 implies {xp,n} == ZERO */
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#endif
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cy += (rp[0]&1);
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mpn_rshift(rp, rp, n, 1);
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ASSERT (cy <= 2);
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hi = (cy<<(GMP_NUMB_BITS-1))&GMP_NUMB_MASK; /* (cy&1) << ... */
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cy >>= 1;
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/* We can have cy != 0 only if hi = 0... */
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ASSERT ((rp[n-1] & GMP_NUMB_HIGHBIT) == 0);
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rp[n-1] |= hi;
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/* ... rp[n-1] + cy can not overflow, the following INCR is correct. */
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#endif
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ASSERT (cy <= 1);
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/* Next increment can not overflow, read the previous comments about cy. */
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ASSERT ((cy == 0) || ((rp[n-1] & GMP_NUMB_HIGHBIT) == 0));
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MPN_INCR_U(rp, n, cy);
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/* Compute the highest half:
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([(xp + xm)/2 mod (B^n-1)] - xp ) * B^n
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*/
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if (UNLIKELY (an + bn < rn))
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{
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/* Note that in this case, the only way the result can equal
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zero mod B^{rn} - 1 is if one of the inputs is zero, and
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then the output of both the recursive calls and this CRT
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reconstruction is zero, not B^{rn} - 1. Which is good,
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since the latter representation doesn't fit in the output
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area.*/
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cy = mpn_sub_n (rp + n, rp, xp, an + bn - n);
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/* FIXME: This subtraction of the high parts is not really
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necessary, we do it to get the carry out, and for sanity
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checking. */
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cy = xp[n] + mpn_sub_nc (xp + an + bn - n, rp + an + bn - n,
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xp + an + bn - n, rn - (an + bn), cy);
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ASSERT (an + bn == rn - 1 ||
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mpn_zero_p (xp + an + bn - n + 1, rn - 1 - (an + bn)));
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cy = mpn_sub_1 (rp, rp, an + bn, cy);
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ASSERT (cy == (xp + an + bn - n)[0]);
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}
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else
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{
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cy = xp[n] + mpn_sub_n (rp + n, rp, xp, n);
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/* cy = 1 only if {xp,n+1} is not ZERO, i.e. {rp,n} is not ZERO.
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DECR will affect _at most_ the lowest n limbs. */
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MPN_DECR_U (rp, 2*n, cy);
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}
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#undef a0
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#undef a1
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#undef b0
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#undef b1
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#undef xp
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#undef sp1
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}
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}
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mp_size_t
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mpn_mulmod_bnm1_next_size (mp_size_t n)
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{
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mp_size_t nh;
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if (BELOW_THRESHOLD (n, MULMOD_BNM1_THRESHOLD))
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return n;
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if (BELOW_THRESHOLD (n, 4 * (MULMOD_BNM1_THRESHOLD - 1) + 1))
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return (n + (2-1)) & (-2);
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if (BELOW_THRESHOLD (n, 8 * (MULMOD_BNM1_THRESHOLD - 1) + 1))
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return (n + (4-1)) & (-4);
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nh = (n + 1) >> 1;
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if (BELOW_THRESHOLD (nh, MUL_FFT_MODF_THRESHOLD))
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return (n + (8-1)) & (-8);
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return 2 * mpn_fft_next_size (nh, mpn_fft_best_k (nh, 0));
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}
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