/* hgcd.c.

   THE FUNCTIONS IN THIS FILE ARE INTERNAL WITH MUTABLE INTERFACES.  IT IS ONLY

   SAFE TO REACH THEM THROUGH DOCUMENTED INTERFACES.  IN FACT, IT IS ALMOST

   GUARANTEED THAT THEY'LL CHANGE OR DISAPPEAR IN A FUTURE GNU MP RELEASE.

Copyright 2003-2005, 2008, 2011, 2012 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.h"

#include "gmp-impl.h"

#include "longlong.h"

/* Size analysis for hgcd:

   For the recursive calls, we have n1 <= ceil(n / 2). Then the

   storage need is determined by the storage for the recursive call

   computing M1, and hgcd_matrix_adjust and hgcd_matrix_mul calls that use M1

   (after this, the storage needed for M1 can be recycled).

   Let S(r) denote the required storage. For M1 we need 4 * (ceil(n1/2) + 1)

   = 4 * (ceil(n/4) + 1), for the hgcd_matrix_adjust call, we need n + 2,

   and for the hgcd_matrix_mul, we may need 3 ceil(n/2) + 8. In total,

   4 * ceil(n/4) + 3 ceil(n/2) + 12 <= 10 ceil(n/4) + 12.

   For the recursive call, we need S(n1) = S(ceil(n/2)).

   S(n) <= 10*ceil(n/4) + 12 + S(ceil(n/2))

	<= 10*(ceil(n/4) + ... + ceil(n/2^(1+k))) + 12k + S(ceil(n/2^k))

	<= 10*(2 ceil(n/4) + k) + 12k + S(ceil(n/2^k))

	<= 20 ceil(n/4) + 22k + S(ceil(n/2^k))

*/

mp_size_t

mpn_hgcd_itch (mp_size_t n)

{

  unsigned k;

  int count;

  mp_size_t nscaled;

  if (BELOW_THRESHOLD (n, HGCD_THRESHOLD))

    return n;

  /* Get the recursion depth. */

  nscaled = (n - 1) / (HGCD_THRESHOLD - 1);

  count_leading_zeros (count, nscaled);

  k = GMP_LIMB_BITS - count;

  return 20 * ((n+3) / 4) + 22 * k + HGCD_THRESHOLD;

}

/* Reduces a,b until |a-b| fits in n/2 + 1 limbs. Constructs matrix M

   with elements of size at most (n+1)/2 - 1. Returns new size of a,

   b, or zero if no reduction is possible. */

mp_size_t

mpn_hgcd (mp_ptr ap, mp_ptr bp, mp_size_t n,

	  struct hgcd_matrix *M, mp_ptr tp)

{

  mp_size_t s = n/2 + 1;

  mp_size_t nn;

  int success = 0;

  if (n <= s)

    /* Happens when n <= 2, a fairly uninteresting case but exercised

       by the random inputs of the testsuite. */

    return 0;

  ASSERT ((ap[n-1] | bp[n-1]) > 0);

  ASSERT ((n+1)/2 - 1 < M->alloc);

  if (ABOVE_THRESHOLD (n, HGCD_THRESHOLD))

    {

      mp_size_t n2 = (3*n)/4 + 1;

      mp_size_t p = n/2;

      nn = mpn_hgcd_reduce (M, ap, bp, n, p, tp);

      if (nn)

	{

	  n = nn;

	  success = 1;

	}

      /* NOTE: It appears this loop never runs more than once (at

	 least when not recursing to hgcd_appr). */

      while (n > n2)

	{

	  /* Needs n + 1 storage */

	  nn = mpn_hgcd_step (n, ap, bp, s, M, tp);

	  if (!nn)

	    return success ? n : 0;

	  n = nn;

	  success = 1;

	}

      if (n > s + 2)

	{

	  struct hgcd_matrix M1;

	  mp_size_t scratch;

	  p = 2*s - n + 1;

	  scratch = MPN_HGCD_MATRIX_INIT_ITCH (n-p);

	  mpn_hgcd_matrix_init(&M1, n - p, tp);

	  /* FIXME: Should use hgcd_reduce, but that may require more

	     scratch space, which requires review. */

	  nn = mpn_hgcd (ap + p, bp + p, n - p, &M1, tp + scratch);

	  if (nn > 0)

	    {

	      /* We always have max(M) > 2^{-(GMP_NUMB_BITS + 1)} max(M1) */

	      ASSERT (M->n + 2 >= M1.n);

	      /* Furthermore, assume M ends with a quotient (1, q; 0, 1),

		 then either q or q + 1 is a correct quotient, and M1 will

		 start with either (1, 0; 1, 1) or (2, 1; 1, 1). This

		 rules out the case that the size of M * M1 is much

		 smaller than the expected M->n + M1->n. */

	      ASSERT (M->n + M1.n < M->alloc);

	      /* Needs 2 (p + M->n) <= 2 (2*s - n2 + 1 + n2 - s - 1)

		 = 2*s <= 2*(floor(n/2) + 1) <= n + 2. */

	      n = mpn_hgcd_matrix_adjust (&M1, p + nn, ap, bp, p, tp + scratch);

	      /* We need a bound for of M->n + M1.n. Let n be the original

		 input size. Then

		 ceil(n/2) - 1 >= size of product >= M.n + M1.n - 2

		 and it follows that

		 M.n + M1.n <= ceil(n/2) + 1

		 Then 3*(M.n + M1.n) + 5 <= 3 * ceil(n/2) + 8 is the

		 amount of needed scratch space. */

	      mpn_hgcd_matrix_mul (M, &M1, tp + scratch);

	      success = 1;

	    }

	}

    }

  for (;;)

    {

      /* Needs s+3 < n */

      nn = mpn_hgcd_step (n, ap, bp, s, M, tp);

      if (!nn)

	return success ? n : 0;

      n = nn;

      success = 1;

    }

}