/* mpn_brootinv, compute r such that r^k * y = 1 (mod 2^b).

   Contributed to the GNU project by Martin Boij (as part of perfpow.c).

Copyright 2009, 2010, 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 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 "gmp.h"
#include "gmp-impl.h"

/* Computes a^e (mod B). Uses right-to-left binary algorithm, since
   typical use will have e small. */
static mp_limb_t
powlimb (mp_limb_t a, mp_limb_t e)
{
  mp_limb_t r = 1;
  mp_limb_t s = a;

  for (r = 1, s = a; e > 0; e >>= 1, s *= s)
    if (e & 1)
      r *= s;

  return r;
}

/* Compute r such that r^k * y = 1 (mod B^n).

   Iterates
     r' <-- k^{-1} ((k+1) r - r^{k+1} y) (mod 2^b)
   using Hensel lifting, each time doubling the number of known bits in r.

   Works just for odd k.  Else the Hensel lifting degenerates.

   FIXME:

     (1) Make it work for k == GMP_LIMB_MAX (k+1 below overflows).

     (2) Rewrite iteration as
	   r' <-- r - k^{-1} r (r^k y - 1)
	 and take advantage of the zero low part of r^k y - 1.

     (3) Use wrap-around trick.

     (4) Use a small table to get starting value.

   Scratch need: 5*bn, where bn = ceil (bnb / GMP_NUMB_BITS).
*/

void
mpn_brootinv (mp_ptr rp, mp_srcptr yp, mp_size_t bn, mp_limb_t k, mp_ptr tp)
{
  mp_ptr tp2, tp3;
  mp_limb_t kinv, k2, r0, y0;
  mp_size_t order[GMP_LIMB_BITS + 1];
  int i, d;

  ASSERT (bn > 0);
  ASSERT ((k & 1) != 0);

  tp2 = tp + bn;
  tp3 = tp + 2 * bn;
  k2 = k + 1;

  binvert_limb (kinv, k);

  /* 4-bit initial approximation:

   y%16 | 1  3  5  7  9 11 13 15,
    k%4 +-----------------------------
     1  | 1 11 13  7  9  3  5 15
     3  | 1  3  5  7  9 11 13 15

  */
  y0 = yp[0];

  r0 = y0 ^ (((y0 << 1) ^ (y0 << 2)) & ~(k << 2) & 8);		/* 4 bits */
  r0 = kinv * (k2 * r0 - y0 * powlimb(r0, k2 & 0x7f));		/* 8 bits */
  r0 = kinv * (k2 * r0 - y0 * powlimb(r0, k2 & 0xffff));	/* 16 bits */
  r0 = kinv * (k2 * r0 - y0 * powlimb(r0, k2));			/* 32 bits */
#if GMP_NUMB_BITS > 32
  {
    unsigned prec = 32;
    do
      {
	r0 = kinv * (k2 * r0 - y0 * powlimb(r0, k2));
	prec *= 2;
      }
    while (prec < GMP_NUMB_BITS);
  }
#endif

  rp[0] = r0;
  if (bn == 1)
    return;

  /* This initialization doesn't matter for the result (any garbage is
     cancelled in the iteration), but proper initialization makes
     valgrind happier. */
  MPN_ZERO (rp+1, bn-1);

  d = 0;
  for (; bn > 1; bn = (bn + 1) >> 1)
    order[d++] = bn;

  for (i = d - 1; i >= 0; i--)
    {
      bn = order[i];

      mpn_mul_1 (tp, rp, bn, k2);

      mpn_powlo (tp2, rp, &k2, 1, bn, tp3);
      mpn_mullo_n (rp, yp, tp2, bn);

      mpn_sub_n (tp2, tp, rp, bn);
      mpn_pi1_bdiv_q_1 (rp, tp2, bn, k, kinv, 0);
    }
}
