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/* hgcd.c.
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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'LL CHANGE OR DISAPPEAR IN A FUTURE GNU MP RELEASE.
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Copyright 2003-2005, 2008, 2011, 2012 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.h"
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#include "gmp-impl.h"
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#include "longlong.h"
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/* Size analysis for hgcd:
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For the recursive calls, we have n1 <= ceil(n / 2). Then the
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storage need is determined by the storage for the recursive call
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computing M1, and hgcd_matrix_adjust and hgcd_matrix_mul calls that use M1
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(after this, the storage needed for M1 can be recycled).
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Let S(r) denote the required storage. For M1 we need 4 * (ceil(n1/2) + 1)
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= 4 * (ceil(n/4) + 1), for the hgcd_matrix_adjust call, we need n + 2,
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and for the hgcd_matrix_mul, we may need 3 ceil(n/2) + 8. In total,
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4 * ceil(n/4) + 3 ceil(n/2) + 12 <= 10 ceil(n/4) + 12.
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For the recursive call, we need S(n1) = S(ceil(n/2)).
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S(n) <= 10*ceil(n/4) + 12 + S(ceil(n/2))
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<= 10*(ceil(n/4) + ... + ceil(n/2^(1+k))) + 12k + S(ceil(n/2^k))
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<= 10*(2 ceil(n/4) + k) + 12k + S(ceil(n/2^k))
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<= 20 ceil(n/4) + 22k + S(ceil(n/2^k))
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*/
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mp_size_t
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mpn_hgcd_itch (mp_size_t n)
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{
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unsigned k;
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int count;
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mp_size_t nscaled;
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if (BELOW_THRESHOLD (n, HGCD_THRESHOLD))
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return n;
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/* Get the recursion depth. */
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nscaled = (n - 1) / (HGCD_THRESHOLD - 1);
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count_leading_zeros (count, nscaled);
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k = GMP_LIMB_BITS - count;
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return 20 * ((n+3) / 4) + 22 * k + HGCD_THRESHOLD;
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}
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/* Reduces a,b until |a-b| fits in n/2 + 1 limbs. Constructs matrix M
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with elements of size at most (n+1)/2 - 1. Returns new size of a,
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b, or zero if no reduction is possible. */
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mp_size_t
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mpn_hgcd (mp_ptr ap, mp_ptr bp, mp_size_t n,
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struct hgcd_matrix *M, mp_ptr tp)
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{
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mp_size_t s = n/2 + 1;
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mp_size_t nn;
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int success = 0;
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if (n <= s)
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/* Happens when n <= 2, a fairly uninteresting case but exercised
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by the random inputs of the testsuite. */
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return 0;
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ASSERT ((ap[n-1] | bp[n-1]) > 0);
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ASSERT ((n+1)/2 - 1 < M->alloc);
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if (ABOVE_THRESHOLD (n, HGCD_THRESHOLD))
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{
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mp_size_t n2 = (3*n)/4 + 1;
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mp_size_t p = n/2;
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nn = mpn_hgcd_reduce (M, ap, bp, n, p, tp);
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if (nn)
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{
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n = nn;
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success = 1;
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}
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/* NOTE: It appears this loop never runs more than once (at
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least when not recursing to hgcd_appr). */
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while (n > n2)
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{
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/* Needs n + 1 storage */
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nn = mpn_hgcd_step (n, ap, bp, s, M, tp);
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if (!nn)
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return success ? n : 0;
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n = nn;
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success = 1;
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}
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if (n > s + 2)
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{
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struct hgcd_matrix M1;
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mp_size_t scratch;
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p = 2*s - n + 1;
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scratch = MPN_HGCD_MATRIX_INIT_ITCH (n-p);
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mpn_hgcd_matrix_init(&M1, n - p, tp);
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/* FIXME: Should use hgcd_reduce, but that may require more
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scratch space, which requires review. */
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nn = mpn_hgcd (ap + p, bp + p, n - p, &M1, tp + scratch);
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if (nn > 0)
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{
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/* We always have max(M) > 2^{-(GMP_NUMB_BITS + 1)} max(M1) */
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ASSERT (M->n + 2 >= M1.n);
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/* Furthermore, assume M ends with a quotient (1, q; 0, 1),
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then either q or q + 1 is a correct quotient, and M1 will
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start with either (1, 0; 1, 1) or (2, 1; 1, 1). This
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rules out the case that the size of M * M1 is much
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smaller than the expected M->n + M1->n. */
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ASSERT (M->n + M1.n < M->alloc);
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/* Needs 2 (p + M->n) <= 2 (2*s - n2 + 1 + n2 - s - 1)
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= 2*s <= 2*(floor(n/2) + 1) <= n + 2. */
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n = mpn_hgcd_matrix_adjust (&M1, p + nn, ap, bp, p, tp + scratch);
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/* We need a bound for of M->n + M1.n. Let n be the original
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input size. Then
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ceil(n/2) - 1 >= size of product >= M.n + M1.n - 2
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and it follows that
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M.n + M1.n <= ceil(n/2) + 1
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Then 3*(M.n + M1.n) + 5 <= 3 * ceil(n/2) + 8 is the
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amount of needed scratch space. */
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mpn_hgcd_matrix_mul (M, &M1, tp + scratch);
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success = 1;
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}
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}
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}
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for (;;)
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{
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/* Needs s+3 < n */
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nn = mpn_hgcd_step (n, ap, bp, s, M, tp);
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if (!nn)
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return success ? n : 0;
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n = nn;
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success = 1;
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}
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}
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