mpn/generic/mul_fft.c

/* Schoenhage's fast multiplication modulo 2^N+1.

   Contributed by Paul Zimmermann.

   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 WILL CHANGE OR DISAPPEAR IN A FUTURE GNU MP RELEASE.

Copyright 1998-2010, 2012, 2013, 2018, 2020, 2022 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/.  */


/* References:

   Schnelle Multiplikation grosser Zahlen, by Arnold Schoenhage and Volker
   Strassen, Computing 7, p. 281-292, 1971.

   Asymptotically fast algorithms for the numerical multiplication and division
   of polynomials with complex coefficients, by Arnold Schoenhage, Computer
   Algebra, EUROCAM'82, LNCS 144, p. 3-15, 1982.

   Tapes versus Pointers, a study in implementing fast algorithms, by Arnold
   Schoenhage, Bulletin of the EATCS, 30, p. 23-32, 1986.

   TODO:

   Implement some of the tricks published at ISSAC'2007 by Gaudry, Kruppa, and
   Zimmermann.

   It might be possible to avoid a small number of MPN_COPYs by using a
   rotating temporary or two.

   Cleanup and simplify the code!
*/

#ifdef TRACE
#undef TRACE
#define TRACE(x) x
#include <stdio.h>
#else
#define TRACE(x)
#endif

#include "gmp-impl.h"

#ifdef WANT_ADDSUB
#include "generic/add_n_sub_n.c"
#define HAVE_NATIVE_mpn_add_n_sub_n 1
#endif

static mp_limb_t mpn_mul_fft_internal (mp_ptr, mp_size_t, int, mp_ptr *,
				       mp_ptr *, mp_ptr, mp_ptr, mp_size_t,
				       mp_size_t, mp_size_t, int **, mp_ptr, int);
static void mpn_mul_fft_decompose (mp_ptr, mp_ptr *, mp_size_t, mp_size_t, mp_srcptr,
				   mp_size_t, mp_size_t, mp_size_t, mp_ptr);


/* Find the best k to use for a mod 2^(m*GMP_NUMB_BITS)+1 FFT for m >= n.
   We have sqr=0 if for a multiply, sqr=1 for a square.
   There are three generations of this code; we keep the old ones as long as
   some gmp-mparam.h is not updated.  */


/*****************************************************************************/

#if TUNE_PROGRAM_BUILD || (defined (MUL_FFT_TABLE3) && defined (SQR_FFT_TABLE3))

#ifndef FFT_TABLE3_SIZE		/* When tuning this is defined in gmp-impl.h */
#if defined (MUL_FFT_TABLE3_SIZE) && defined (SQR_FFT_TABLE3_SIZE)
#if MUL_FFT_TABLE3_SIZE > SQR_FFT_TABLE3_SIZE
#define FFT_TABLE3_SIZE MUL_FFT_TABLE3_SIZE
#else
#define FFT_TABLE3_SIZE SQR_FFT_TABLE3_SIZE
#endif
#endif
#endif

#ifndef FFT_TABLE3_SIZE
#define FFT_TABLE3_SIZE 200
#endif

FFT_TABLE_ATTRS struct fft_table_nk mpn_fft_table3[2][FFT_TABLE3_SIZE] =
{
  MUL_FFT_TABLE3,
  SQR_FFT_TABLE3
};

int
mpn_fft_best_k (mp_size_t n, int sqr)
{
  const struct fft_table_nk *fft_tab, *tab;
  mp_size_t tab_n, thres;
  int last_k;

  fft_tab = mpn_fft_table3[sqr];
  last_k = fft_tab->k;
  for (tab = fft_tab + 1; ; tab++)
    {
      tab_n = tab->n;
      thres = tab_n << last_k;
      if (n <= thres)
	break;
      last_k = tab->k;
    }
  return last_k;
}

#define MPN_FFT_BEST_READY 1
#endif

/*****************************************************************************/

#if ! defined (MPN_FFT_BEST_READY)
FFT_TABLE_ATTRS mp_size_t mpn_fft_table[2][MPN_FFT_TABLE_SIZE] =
{
  MUL_FFT_TABLE,
  SQR_FFT_TABLE
};

int
mpn_fft_best_k (mp_size_t n, int sqr)
{
  int i;

  for (i = 0; mpn_fft_table[sqr][i] != 0; i++)
    if (n < mpn_fft_table[sqr][i])
      return i + FFT_FIRST_K;

  /* treat 4*last as one further entry */
  if (i == 0 || n < 4 * mpn_fft_table[sqr][i - 1])
    return i + FFT_FIRST_K;
  else
    return i + FFT_FIRST_K + 1;
}
#endif

/*****************************************************************************/


/* Returns smallest possible number of limbs >= pl for a fft of size 2^k,
   i.e. smallest multiple of 2^k >= pl.

   Don't declare static: needed by tuneup.
*/

mp_size_t
mpn_fft_next_size (mp_size_t pl, int k)
{
  pl = 1 + ((pl - 1) >> k); /* ceil (pl/2^k) */
  return pl << k;
}


/* Initialize l[i][j] with bitrev(j) */
static void
mpn_fft_initl (int **l, int k)
{
  int i, j, K;
  int *li;

  l[0][0] = 0;
  for (i = 1, K = 1; i <= k; i++, K *= 2)
    {
      li = l[i];
      for (j = 0; j < K; j++)
	{
	  li[j] = 2 * l[i - 1][j];
	  li[K + j] = 1 + li[j];
	}
    }
}


/* r <- a*2^d mod 2^(n*GMP_NUMB_BITS)+1 with a = {a, n+1}
   Assumes a is semi-normalized, i.e. a[n] <= 1.
   r and a must have n+1 limbs, and not overlap.
*/
static void
mpn_fft_mul_2exp_modF (mp_ptr r, mp_srcptr a, mp_bitcnt_t d, mp_size_t n)
{
  unsigned int sh;
  mp_size_t m;
  mp_limb_t cc, rd;

  sh = d % GMP_NUMB_BITS;
  m = d / GMP_NUMB_BITS;

  if (m >= n)			/* negate */
    {
      /* r[0..m-1]  <-- lshift(a[n-m]..a[n-1], sh)
	 r[m..n-1]  <-- -lshift(a[0]..a[n-m-1],  sh) */

      m -= n;
      if (sh != 0)
	{
	  /* no out shift below since a[n] <= 1 */
	  mpn_lshift (r, a + n - m, m + 1, sh);
	  rd = r[m];
	  cc = mpn_lshiftc (r + m, a, n - m, sh);
	}
      else
	{
	  MPN_COPY (r, a + n - m, m);
	  rd = a[n];
	  mpn_com (r + m, a, n - m);
	  cc = 0;
	}

      /* add cc to r[0], and add rd to r[m] */

      /* now add 1 in r[m], subtract 1 in r[n], i.e. add 1 in r[0] */

      r[n] = 0;
      /* cc < 2^sh <= 2^(GMP_NUMB_BITS-1) thus no overflow here */
      ++cc;
      MPN_INCR_U (r, n + 1, cc);

      ++rd;
      /* rd might overflow when sh=GMP_NUMB_BITS-1 */
      cc = rd + (rd == 0);
      r = r + m + (rd == 0);
      MPN_INCR_U (r, n + 1 - m - (rd == 0), cc);
    }
  else
    {
      /* r[0..m-1]  <-- -lshift(a[n-m]..a[n-1], sh)
	 r[m..n-1]  <-- lshift(a[0]..a[n-m-1],  sh)  */
      if (sh != 0)
	{
	  /* no out bits below since a[n] <= 1 */
	  mpn_lshiftc (r, a + n - m, m + 1, sh);
	  rd = ~r[m];
	  /* {r, m+1} = {a+n-m, m+1} << sh */
	  cc = mpn_lshift (r + m, a, n - m, sh); /* {r+m, n-m} = {a, n-m}<<sh */
	}
      else
	{
	  /* r[m] is not used below, but we save a test for m=0 */
	  mpn_com (r, a + n - m, m + 1);
	  rd = a[n];
	  MPN_COPY (r + m, a, n - m);
	  cc = 0;
	}

      /* now complement {r, m}, subtract cc from r[0], subtract rd from r[m] */

      /* if m=0 we just have r[0]=a[n] << sh */
      if (m != 0)
	{
	  /* now add 1 in r[0], subtract 1 in r[m] */
	  if (cc-- == 0) /* then add 1 to r[0] */
	    cc = mpn_add_1 (r, r, n, CNST_LIMB(1));
	  cc = mpn_sub_1 (r, r, m, cc) + 1;
	  /* add 1 to cc instead of rd since rd might overflow */
	}

      /* now subtract cc and rd from r[m..n] */

      r[n] = 2; /* Add a value, to avoid borrow propagation */
      MPN_DECR_U (r + m, n - m + 1, cc);
      MPN_DECR_U (r + m, n - m + 1, rd);
      /* Remove the added value, and check for a possible borrow. */
      if (UNLIKELY ((r[n] -= 2) != 0))
	{
	  mp_limb_t cy = -r[n];
	  /* cy should always be 1, except in the very unlikely case
	     m=n-1, r[m]=0, cc+rd>GMP_NUMB_MAX+1. Never triggered.
	     Is it actually possible? */
	  r[n] = 0;
	  MPN_INCR_U (r, n + 1, cy);
	}
    }
}

#if HAVE_NATIVE_mpn_add_n_sub_n
static inline void
mpn_fft_add_sub_modF (mp_ptr A0, mp_ptr Ai, mp_srcptr tp, mp_size_t n)
{
  mp_limb_t cyas, c, x;

  cyas = mpn_add_n_sub_n (A0, Ai, A0, tp, n);

  c = A0[n] - tp[n] - (cyas & 1);
  x = (-c) & -((c & GMP_LIMB_HIGHBIT) != 0);
  Ai[n] = x + c;
  MPN_INCR_U (Ai, n + 1, x);

  c = A0[n] + tp[n] + (cyas >> 1);
  x = (c - 1) & -(c != 0);
  A0[n] = c - x;
  MPN_DECR_U (A0, n + 1, x);
}

#else /* ! HAVE_NATIVE_mpn_add_n_sub_n  */

/* r <- a+b mod 2^(n*GMP_NUMB_BITS)+1.
   Assumes a and b are semi-normalized.
*/
static inline void
mpn_fft_add_modF (mp_ptr r, mp_srcptr a, mp_srcptr b, mp_size_t n)
{
  mp_limb_t c, x;

  c = a[n] + b[n] + mpn_add_n (r, a, b, n);
  /* 0 <= c <= 3 */

#if 1
  /* GCC 4.1 outsmarts most expressions here, and generates a 50% branch.  The
     result is slower code, of course.  But the following outsmarts GCC.  */
  x = (c - 1) & -(c != 0);
  r[n] = c - x;
  MPN_DECR_U (r, n + 1, x);
#endif
#if 0
  if (c > 1)
    {
      r[n] = 1;                       /* r[n] - c = 1 */
      MPN_DECR_U (r, n + 1, c - 1);
    }
  else
    {
      r[n] = c;
    }
#endif
}

/* r <- a-b mod 2^(n*GMP_NUMB_BITS)+1.
   Assumes a and b are semi-normalized.
*/
static inline void
mpn_fft_sub_modF (mp_ptr r, mp_srcptr a, mp_srcptr b, mp_size_t n)
{
  mp_limb_t c, x;

  c = a[n] - b[n] - mpn_sub_n (r, a, b, n);
  /* -2 <= c <= 1 */

#if 1
  /* GCC 4.1 outsmarts most expressions here, and generates a 50% branch.  The
     result is slower code, of course.  But the following outsmarts GCC.  */
  x = (-c) & -((c & GMP_LIMB_HIGHBIT) != 0);
  r[n] = x + c;
  MPN_INCR_U (r, n + 1, x);
#endif
#if 0
  if ((c & GMP_LIMB_HIGHBIT) != 0)
    {
      r[n] = 0;
      MPN_INCR_U (r, n + 1, -c);
    }
  else
    {
      r[n] = c;
    }
#endif
}
#endif /* HAVE_NATIVE_mpn_add_n_sub_n */

/* input: A[0] ... A[inc*(K-1)] are residues mod 2^N+1 where
	  N=n*GMP_NUMB_BITS, and 2^omega is a primitive root mod 2^N+1
   output: A[inc*l[k][i]] <- \sum (2^omega)^(ij) A[inc*j] mod 2^N+1 */

static void
mpn_fft_fft (mp_ptr *Ap, mp_size_t K, int **ll,
	     mp_size_t omega, mp_size_t n, mp_size_t inc, mp_ptr tp)
{
  if (K == 2)
    {
      mp_limb_t cy;
#if HAVE_NATIVE_mpn_add_n_sub_n
      cy = mpn_add_n_sub_n (Ap[0], Ap[inc], Ap[0], Ap[inc], n + 1) & 1;
#else
      MPN_COPY (tp, Ap[0], n + 1);
      mpn_add_n (Ap[0], Ap[0], Ap[inc], n + 1);
      cy = mpn_sub_n (Ap[inc], tp, Ap[inc], n + 1);
#endif
      if (Ap[0][n] > 1) /* can be 2 or 3 */
	{ /* Ap[0][n] = 1 - mpn_sub_1 (Ap[0], Ap[0], n, Ap[0][n] - 1); */
	  mp_limb_t cc = Ap[0][n] - 1;
	  Ap[0][n] = 1;
	  MPN_DECR_U (Ap[0], n + 1, cc);
	}
      if (cy) /* Ap[inc][n] can be -1 or -2 */
	{ /* Ap[inc][n] = mpn_add_1 (Ap[inc], Ap[inc], n, ~Ap[inc][n] + 1); */
	  mp_limb_t cc = ~Ap[inc][n] + 1;
	  Ap[inc][n] = 0;
	  MPN_INCR_U (Ap[inc], n + 1, cc);
	}
    }
  else
    {
      mp_size_t j, K2 = K >> 1;
      int *lk = *ll;

      mpn_fft_fft (Ap,     K2, ll-1, 2 * omega, n, inc * 2, tp);
      mpn_fft_fft (Ap+inc, K2, ll-1, 2 * omega, n, inc * 2, tp);
      /* A[2*j*inc]   <- A[2*j*inc] + omega^l[k][2*j*inc] A[(2j+1)inc]
	 A[(2j+1)inc] <- A[2*j*inc] + omega^l[k][(2j+1)inc] A[(2j+1)inc] */
      for (j = 0; j < K2; j++, lk += 2, Ap += 2 * inc)
	{
	  /* Ap[inc] <- Ap[0] + Ap[inc] * 2^(lk[1] * omega)
	     Ap[0]   <- Ap[0] + Ap[inc] * 2^(lk[0] * omega) */
	  mpn_fft_mul_2exp_modF (tp, Ap[inc], lk[0] * omega, n);
#if HAVE_NATIVE_mpn_add_n_sub_n
	  mpn_fft_add_sub_modF (Ap[0], Ap[inc], tp, n);
#else
	  mpn_fft_sub_modF (Ap[inc], Ap[0], tp, n);
	  mpn_fft_add_modF (Ap[0],   Ap[0], tp, n);
#endif
	}
    }
}

/* input: A[0] ... A[inc*(K-1)] are residues mod 2^N+1 where
	  N=n*GMP_NUMB_BITS, and 2^omega is a primitive root mod 2^N+1
   output: A[inc*l[k][i]] <- \sum (2^omega)^(ij) A[inc*j] mod 2^N+1
   tp must have space for 2*(n+1) limbs.
*/


/* Given ap[0..n] with ap[n]<=1, reduce it modulo 2^(n*GMP_NUMB_BITS)+1,
   by subtracting that modulus if necessary.

   If ap[0..n] is exactly 2^(n*GMP_NUMB_BITS) then mpn_sub_1 produces a
   borrow and the limbs must be zeroed out again.  This will occur very
   infrequently.  */

static inline void
mpn_fft_normalize (mp_ptr ap, mp_size_t n)
{
  if (ap[n] != 0)
    {
      MPN_DECR_U (ap, n + 1, CNST_LIMB(1));
      if (ap[n] == 0)
	{
	  /* This happens with very low probability; we have yet to trigger it,
	     and thereby make sure this code is correct.  */
	  MPN_ZERO (ap, n);
	  ap[n] = 1;
	}
      else
	ap[n] = 0;
    }
}

/* a[i] <- a[i]*b[i] mod 2^(n*GMP_NUMB_BITS)+1 for 0 <= i < K */
static void
mpn_fft_mul_modF_K (mp_ptr *ap, mp_ptr *bp, mp_size_t n, mp_size_t K)
{
  int i;
  unsigned k;
  int sqr = (ap == bp);
  TMP_DECL;

  TMP_MARK;

  if (n >= (sqr ? SQR_FFT_MODF_THRESHOLD : MUL_FFT_MODF_THRESHOLD))
    {
      mp_size_t K2, nprime2, Nprime2, M2, maxLK, l, Mp2;
      int k;
      int **fft_l, *tmp;
      mp_ptr *Ap, *Bp, A, B, T;

      k = mpn_fft_best_k (n, sqr);
      K2 = (mp_size_t) 1 << k;
      ASSERT_ALWAYS((n & (K2 - 1)) == 0);
      maxLK = (K2 > GMP_NUMB_BITS) ? K2 : GMP_NUMB_BITS;
      M2 = n * GMP_NUMB_BITS >> k;
      l = n >> k;
      Nprime2 = ((2 * M2 + k + 2 + maxLK) / maxLK) * maxLK;
      /* Nprime2 = ceil((2*M2+k+3)/maxLK)*maxLK*/
      nprime2 = Nprime2 / GMP_NUMB_BITS;

      /* we should ensure that nprime2 is a multiple of the next K */
      if (nprime2 >= (sqr ? SQR_FFT_MODF_THRESHOLD : MUL_FFT_MODF_THRESHOLD))
	{
	  mp_size_t K3;
	  for (;;)
	    {
	      K3 = (mp_size_t) 1 << mpn_fft_best_k (nprime2, sqr);
	      if ((nprime2 & (K3 - 1)) == 0)
		break;
	      nprime2 = (nprime2 + K3 - 1) & -K3;
	      Nprime2 = nprime2 * GMP_LIMB_BITS;
	      /* warning: since nprime2 changed, K3 may change too! */
	    }
	}
      ASSERT_ALWAYS(nprime2 < n); /* otherwise we'll loop */

      Mp2 = Nprime2 >> k;

      Ap = TMP_BALLOC_MP_PTRS (K2);
      Bp = TMP_BALLOC_MP_PTRS (K2);
      A = TMP_BALLOC_LIMBS (2 * (nprime2 + 1) << k);
      T = TMP_BALLOC_LIMBS (2 * (nprime2 + 1));
      B = A + ((nprime2 + 1) << k);
      fft_l = TMP_BALLOC_TYPE (k + 1, int *);
      tmp = TMP_BALLOC_TYPE ((size_t) 2 << k, int);
      for (i = 0; i <= k; i++)
	{
	  fft_l[i] = tmp;
	  tmp += (mp_size_t) 1 << i;
	}

      mpn_fft_initl (fft_l, k);

      TRACE (printf ("recurse: %ldx%ld limbs -> %ld times %ldx%ld (%1.2f)\n", n,
		    n, K2, nprime2, nprime2, 2.0*(double)n/nprime2/K2));
      for (i = 0; i < K; i++, ap++, bp++)
	{
	  mp_limb_t cy;
	  mpn_fft_normalize (*ap, n);
	  if (!sqr)
	    mpn_fft_normalize (*bp, n);

	  mpn_mul_fft_decompose (A, Ap, K2, nprime2, *ap, (l << k) + 1, l, Mp2, T);
	  if (!sqr)
	    mpn_mul_fft_decompose (B, Bp, K2, nprime2, *bp, (l << k) + 1, l, Mp2, T);

	  cy = mpn_mul_fft_internal (*ap, n, k, Ap, Bp, A, B, nprime2,
				     l, Mp2, fft_l, T, sqr);
	  (*ap)[n] = cy;
	}
    }
#if ! TUNE_PROGRAM_BUILD
  else if (MPN_MULMOD_BKNP1_USABLE (n, k, MUL_FFT_MODF_THRESHOLD))
    {
      mp_ptr a;
      mp_size_t n_k = n / k;

      if (sqr)
       {
	 mp_ptr tp = TMP_SALLOC_LIMBS (mpn_sqrmod_bknp1_itch (n));
         for (i = 0; i < K; i++)
           {
             a = *ap++;
             mpn_sqrmod_bknp1 (a, a, n_k, k, tp);
           }
       }
      else
       {
	 mp_ptr b, tp = TMP_SALLOC_LIMBS (mpn_mulmod_bknp1_itch (n));
         for (i = 0; i < K; i++)
           {
             a = *ap++;
             b = *bp++;
             mpn_mulmod_bknp1 (a, a, b, n_k, k, tp);
           }
       }
    }
#endif
  else
    {
      mp_ptr a, b, tp, tpn;
      mp_limb_t cc;
      mp_size_t n2 = 2 * n;
      tp = TMP_BALLOC_LIMBS (n2);
      tpn = tp + n;
      TRACE (printf ("  mpn_mul_n %ld of %ld limbs\n", K, n));
      for (i = 0; i < K; i++)
	{
	  a = *ap++;
	  b = *bp++;
	  if (sqr)
	    mpn_sqr (tp, a, n);
	  else
	    mpn_mul_n (tp, b, a, n);
	  if (a[n] != 0)
	    cc = mpn_add_n (tpn, tpn, b, n);
	  else
	    cc = 0;
	  if (b[n] != 0)
	    cc += mpn_add_n (tpn, tpn, a, n) + a[n];
	  if (cc != 0)
	    {
	      cc = mpn_add_1 (tp, tp, n2, cc);
	      /* If mpn_add_1 give a carry (cc != 0),
		 the result (tp) is at most GMP_NUMB_MAX - 1,
		 so the following addition can't overflow.
	      */
	      tp[0] += cc;
	    }
	  cc = mpn_sub_n (a, tp, tpn, n);
	  a[n] = 0;
	  MPN_INCR_U (a, n + 1, cc);
	}
    }
  TMP_FREE;
}


/* input: A^[l[k][0]] A^[l[k][1]] ... A^[l[k][K-1]]
   output: K*A[0] K*A[K-1] ... K*A[1].
   Assumes the Ap[] are pseudo-normalized, i.e. 0 <= Ap[][n] <= 1.
   This condition is also fulfilled at exit.
*/
static void
mpn_fft_fftinv (mp_ptr *Ap, mp_size_t K, mp_size_t omega, mp_size_t n, mp_ptr tp)
{
  if (K == 2)
    {
      mp_limb_t cy;
#if HAVE_NATIVE_mpn_add_n_sub_n
      cy = mpn_add_n_sub_n (Ap[0], Ap[1], Ap[0], Ap[1], n + 1) & 1;
#else
      MPN_COPY (tp, Ap[0], n + 1);
      mpn_add_n (Ap[0], Ap[0], Ap[1], n + 1);
      cy = mpn_sub_n (Ap[1], tp, Ap[1], n + 1);
#endif
      if (Ap[0][n] > 1) /* can be 2 or 3 */
	{ /* Ap[0][n] = 1 - mpn_sub_1 (Ap[0], Ap[0], n, Ap[0][n] - 1); */
	  mp_limb_t cc = Ap[0][n] - 1;
	  Ap[0][n] = 1;
	  MPN_DECR_U (Ap[0], n + 1, cc);
	}
      if (cy) /* Ap[1][n] can be -1 or -2 */
	{ /* Ap[1][n] = mpn_add_1 (Ap[1], Ap[1], n, ~Ap[1][n] + 1); */
	  mp_limb_t cc = ~Ap[1][n] + 1;
	  Ap[1][n] = 0;
	  MPN_INCR_U (Ap[1], n + 1, cc);
	}
    }
  else
    {
      mp_size_t j, K2 = K >> 1;

      mpn_fft_fftinv (Ap,      K2, 2 * omega, n, tp);
      mpn_fft_fftinv (Ap + K2, K2, 2 * omega, n, tp);
      /* A[j]     <- A[j] + omega^j A[j+K/2]
	 A[j+K/2] <- A[j] + omega^(j+K/2) A[j+K/2] */
      for (j = 0; j < K2; j++, Ap++)
	{
	  /* Ap[K2] <- Ap[0] + Ap[K2] * 2^((j + K2) * omega)
	     Ap[0]  <- Ap[0] + Ap[K2] * 2^(j * omega) */
	  mpn_fft_mul_2exp_modF (tp, Ap[K2], j * omega, n);
#if HAVE_NATIVE_mpn_add_n_sub_n
	  mpn_fft_add_sub_modF (Ap[0], Ap[K2], tp, n);
#else
	  mpn_fft_sub_modF (Ap[K2], Ap[0], tp, n);
	  mpn_fft_add_modF (Ap[0],  Ap[0], tp, n);
#endif
	}
    }
}


/* R <- A/2^k mod 2^(n*GMP_NUMB_BITS)+1 */
static void
mpn_fft_div_2exp_modF (mp_ptr r, mp_srcptr a, mp_bitcnt_t k, mp_size_t n)
{
  mp_bitcnt_t i;

  ASSERT (r != a);
  i = (mp_bitcnt_t) 2 * n * GMP_NUMB_BITS - k;
  mpn_fft_mul_2exp_modF (r, a, i, n);
  /* 1/2^k = 2^(2nL-k) mod 2^(n*GMP_NUMB_BITS)+1 */
  /* normalize so that R < 2^(n*GMP_NUMB_BITS)+1 */
  mpn_fft_normalize (r, n);
}


/* {rp,n} <- {ap,an} mod 2^(n*GMP_NUMB_BITS)+1, n <= an <= 3*n.
   Returns carry out, i.e. 1 iff {ap,an} = -1 mod 2^(n*GMP_NUMB_BITS)+1,
   then {rp,n}=0.
*/
static mp_size_t
mpn_fft_norm_modF (mp_ptr rp, mp_size_t n, mp_ptr ap, mp_size_t an)
{
  mp_size_t l, m, rpn;
  mp_limb_t cc;

  ASSERT ((n <= an) && (an <= 3 * n));
  m = an - 2 * n;
  if (m > 0)
    {
      l = n;
      /* add {ap, m} and {ap+2n, m} in {rp, m} */
      cc = mpn_add_n (rp, ap, ap + 2 * n, m);
      /* copy {ap+m, n-m} to {rp+m, n-m} */
      rpn = mpn_add_1 (rp + m, ap + m, n - m, cc);
    }
  else
    {
      l = an - n; /* l <= n */
      MPN_COPY (rp, ap, n);
      rpn = 0;
    }

  /* remains to subtract {ap+n, l} from {rp, n+1} */
  rpn -= mpn_sub (rp, rp, n, ap + n, l);
  if (rpn < 0) /* necessarily rpn = -1 */
    rpn = mpn_add_1 (rp, rp, n, CNST_LIMB(1));
  return rpn;
}

/* store in A[0..nprime] the first M bits from {n, nl},
   in A[nprime+1..] the following M bits, ...
   Assumes M is a multiple of GMP_NUMB_BITS (M = l * GMP_NUMB_BITS).
   T must have space for at least (nprime + 1) limbs.
   We must have nl <= 2*K*l.
*/
static void
mpn_mul_fft_decompose (mp_ptr A, mp_ptr *Ap, mp_size_t K, mp_size_t nprime,
		       mp_srcptr n, mp_size_t nl, mp_size_t l, mp_size_t Mp,
		       mp_ptr T)
{
  mp_size_t i, j;
  mp_ptr tmp;
  mp_size_t Kl = K * l;
  TMP_DECL;
  TMP_MARK;

  if (nl > Kl) /* normalize {n, nl} mod 2^(Kl*GMP_NUMB_BITS)+1 */
    {
      mp_size_t dif = nl - Kl;

      tmp = TMP_BALLOC_LIMBS(Kl + 1);
      tmp[Kl] = 0;

#if ! WANT_OLD_FFT_FULL
      ASSERT_ALWAYS (dif <= Kl);
#else
      /* The comment "We must have nl <= 2*K*l." says that
	 ((dif = nl - Kl) > Kl) should never happen. */
      if (UNLIKELY (dif > Kl))
	{
	  mp_limb_signed_t cy;
	  int subp = 0;

	  cy = mpn_sub_n (tmp, n, n + Kl, Kl);
	  n += 2 * Kl;
	  dif -= Kl;

	  /* now dif > 0 */
	  while (dif > Kl)
	    {
	      if (subp)
		cy += mpn_sub_n (tmp, tmp, n, Kl);
	      else
		cy -= mpn_add_n (tmp, tmp, n, Kl);
	      subp ^= 1;
	      n += Kl;
	      dif -= Kl;
	    }
	  /* now dif <= Kl */
	  if (subp)
	    cy += mpn_sub (tmp, tmp, Kl, n, dif);
	  else
	    cy -= mpn_add (tmp, tmp, Kl, n, dif);
	  if (cy >= 0)
	    MPN_INCR_U (tmp, Kl + 1, cy);
	  else
	    {
	      tmp[Kl] = 1;
	      MPN_DECR_U (tmp, Kl + 1, -cy - 1);
	    }
	}
      else /* dif <= Kl, i.e. nl <= 2 * Kl */
#endif
	{
	  mp_limb_t cy;
	  cy = mpn_sub (tmp, n, Kl, n + Kl, dif);
	  MPN_INCR_U (tmp, Kl + 1, cy);
	}
      nl = Kl + 1;
      n = tmp;
    }
  for (i = 0; i < K; i++)
    {
      Ap[i] = A;
      /* store the next M bits of n into A[0..nprime] */
      if (nl > 0) /* nl is the number of remaining limbs */
	{
	  j = (l <= nl && i < K - 1) ? l : nl; /* store j next limbs */
	  nl -= j;
	  MPN_COPY (T, n, j);
	  MPN_ZERO (T + j, nprime + 1 - j);
	  n += l;
	  mpn_fft_mul_2exp_modF (A, T, i * Mp, nprime);
	}
      else
	MPN_ZERO (A, nprime + 1);
      A += nprime + 1;
    }
  ASSERT_ALWAYS (nl == 0);
  TMP_FREE;
}

/* op <- n*m mod 2^N+1 with fft of size 2^k where N=pl*GMP_NUMB_BITS
   op is pl limbs, its high bit is returned.
   One must have pl = mpn_fft_next_size (pl, k).
   T must have space for 2 * (nprime + 1) limbs.
*/

static mp_limb_t
mpn_mul_fft_internal (mp_ptr op, mp_size_t pl, int k,
		      mp_ptr *Ap, mp_ptr *Bp, mp_ptr unusedA, mp_ptr B,
		      mp_size_t nprime, mp_size_t l, mp_size_t Mp,
		      int **fft_l, mp_ptr T, int sqr)
{
  mp_size_t K, i, pla, lo, sh, j;
  mp_ptr p;
  mp_limb_t cc;

  K = (mp_size_t) 1 << k;

  /* direct fft's */
  mpn_fft_fft (Ap, K, fft_l + k, 2 * Mp, nprime, 1, T);
  if (!sqr)
    mpn_fft_fft (Bp, K, fft_l + k, 2 * Mp, nprime, 1, T);

  /* term to term multiplications */
  mpn_fft_mul_modF_K (Ap, sqr ? Ap : Bp, nprime, K);

  /* inverse fft's */
  mpn_fft_fftinv (Ap, K, 2 * Mp, nprime, T);

  /* division of terms after inverse fft */
  Bp[0] = T + nprime + 1;
  mpn_fft_div_2exp_modF (Bp[0], Ap[0], k, nprime);
  for (i = 1; i < K; i++)
    {
      Bp[i] = Ap[i - 1];
      mpn_fft_div_2exp_modF (Bp[i], Ap[i], k + (K - i) * Mp, nprime);
    }

  /* addition of terms in result p */
  MPN_ZERO (T, nprime + 1);
  pla = l * (K - 1) + nprime + 1; /* number of required limbs for p */
  p = B; /* B has K*(n' + 1) limbs, which is >= pla, i.e. enough */
  MPN_ZERO (p, pla);
  cc = 0; /* will accumulate the (signed) carry at p[pla] */
  for (i = K - 1, lo = l * i + nprime,sh = l * i; i >= 0; i--,lo -= l,sh -= l)
    {
      mp_ptr n = p + sh;

      j = (K - i) & (K - 1);

      cc += mpn_add (n, n, pla - sh, Bp[j], nprime + 1);
      T[2 * l] = i + 1; /* T = (i + 1)*2^(2*M) */
      if (mpn_cmp (Bp[j], T, nprime + 1) > 0)
	{ /* subtract 2^N'+1 */
	  cc -= mpn_sub_1 (n, n, pla - sh, CNST_LIMB(1));
	  cc -= mpn_sub_1 (p + lo, p + lo, pla - lo, CNST_LIMB(1));
	}
    }
  if (cc == -CNST_LIMB(1))
    {
      if ((cc = mpn_add_1 (p + pla - pl, p + pla - pl, pl, CNST_LIMB(1))))
	{
	  /* p[pla-pl]...p[pla-1] are all zero */
	  mpn_sub_1 (p + pla - pl - 1, p + pla - pl - 1, pl + 1, CNST_LIMB(1));
	  mpn_sub_1 (p + pla - 1, p + pla - 1, 1, CNST_LIMB(1));
	}
    }
  else if (cc == 1)
    {
      if (pla >= 2 * pl)
	{
	  while ((cc = mpn_add_1 (p + pla - 2 * pl, p + pla - 2 * pl, 2 * pl, cc)))
	    ;
	}
      else
	{
	  MPN_DECR_U (p + pla - pl, pl, cc);
	}
    }
  else
    ASSERT (cc == 0);

  /* here p < 2^(2M) [K 2^(M(K-1)) + (K-1) 2^(M(K-2)) + ... ]
     < K 2^(2M) [2^(M(K-1)) + 2^(M(K-2)) + ... ]
     < K 2^(2M) 2^(M(K-1))*2 = 2^(M*K+M+k+1) */
  return mpn_fft_norm_modF (op, pl, p, pla);
}

/* return the lcm of a and 2^k */
static mp_bitcnt_t
mpn_mul_fft_lcm (mp_bitcnt_t a, int k)
{
  mp_bitcnt_t l = k;

  while (a % 2 == 0 && k > 0)
    {
      a >>= 1;
      k --;
    }
  return a << l;
}


mp_limb_t
mpn_mul_fft (mp_ptr op, mp_size_t pl,
	     mp_srcptr n, mp_size_t nl,
	     mp_srcptr m, mp_size_t ml,
	     int k)
{
  int i;
  mp_size_t K, maxLK;
  mp_size_t N, Nprime, nprime, M, Mp, l;
  mp_ptr *Ap, *Bp, A, T, B;
  int **fft_l, *tmp;
  int sqr = (n == m && nl == ml);
  mp_limb_t h;
  TMP_DECL;

  TRACE (printf ("\nmpn_mul_fft pl=%ld nl=%ld ml=%ld k=%d\n", pl, nl, ml, k));
  ASSERT_ALWAYS (mpn_fft_next_size (pl, k) == pl);

  TMP_MARK;
  N = pl * GMP_NUMB_BITS;
  fft_l = TMP_BALLOC_TYPE (k + 1, int *);
  tmp = TMP_BALLOC_TYPE ((size_t) 2 << k, int);
  for (i = 0; i <= k; i++)
    {
      fft_l[i] = tmp;
      tmp += (mp_size_t) 1 << i;
    }

  mpn_fft_initl (fft_l, k);
  K = (mp_size_t) 1 << k;
  M = N >> k;	/* N = 2^k M */
  l = 1 + (M - 1) / GMP_NUMB_BITS;
  maxLK = mpn_mul_fft_lcm (GMP_NUMB_BITS, k); /* lcm (GMP_NUMB_BITS, 2^k) */

  Nprime = (1 + (2 * M + k + 2) / maxLK) * maxLK;
  /* Nprime = ceil((2*M+k+3)/maxLK)*maxLK; */
  nprime = Nprime / GMP_NUMB_BITS;
  TRACE (printf ("N=%ld K=%ld, M=%ld, l=%ld, maxLK=%ld, Np=%ld, np=%ld\n",
		 N, K, M, l, maxLK, Nprime, nprime));
  /* we should ensure that recursively, nprime is a multiple of the next K */
  if (nprime >= (sqr ? SQR_FFT_MODF_THRESHOLD : MUL_FFT_MODF_THRESHOLD))
    {
      mp_size_t K2;
      for (;;)
	{
	  K2 = (mp_size_t) 1 << mpn_fft_best_k (nprime, sqr);
	  if ((nprime & (K2 - 1)) == 0)
	    break;
	  nprime = (nprime + K2 - 1) & -K2;
	  Nprime = nprime * GMP_LIMB_BITS;
	  /* warning: since nprime changed, K2 may change too! */
	}
      TRACE (printf ("new maxLK=%ld, Np=%ld, np=%ld\n", maxLK, Nprime, nprime));
    }
  ASSERT_ALWAYS (nprime < pl); /* otherwise we'll loop */

  T = TMP_BALLOC_LIMBS (2 * (nprime + 1));
  Mp = Nprime >> k;

  TRACE (printf ("%ldx%ld limbs -> %ld times %ldx%ld limbs (%1.2f)\n",
		pl, pl, K, nprime, nprime, 2.0 * (double) N / Nprime / K);
	 printf ("   temp space %ld\n", 2 * K * (nprime + 1)));

  A = TMP_BALLOC_LIMBS (K * (nprime + 1));
  Ap = TMP_BALLOC_MP_PTRS (K);
  Bp = TMP_BALLOC_MP_PTRS (K);
  mpn_mul_fft_decompose (A, Ap, K, nprime, n, nl, l, Mp, T);
  if (sqr)
    {
      mp_size_t pla;
      pla = l * (K - 1) + nprime + 1; /* number of required limbs for p */
      B = TMP_BALLOC_LIMBS (pla);
    }
  else
    {
      B = TMP_BALLOC_LIMBS (K * (nprime + 1));
      mpn_mul_fft_decompose (B, Bp, K, nprime, m, ml, l, Mp, T);
    }
  h = mpn_mul_fft_internal (op, pl, k, Ap, Bp, A, B, nprime, l, Mp, fft_l, T, sqr);

  TMP_FREE;
  return h;
}

#if WANT_OLD_FFT_FULL
/* multiply {n, nl} by {m, ml}, and put the result in {op, nl+ml} */
void
mpn_mul_fft_full (mp_ptr op,
		  mp_srcptr n, mp_size_t nl,
		  mp_srcptr m, mp_size_t ml)
{
  mp_ptr pad_op;
  mp_size_t pl, pl2, pl3, l;
  mp_size_t cc, c2, oldcc;
  int k2, k3;
  int sqr = (n == m && nl == ml);

  pl = nl + ml; /* total number of limbs of the result */

  /* perform a fft mod 2^(2N)+1 and one mod 2^(3N)+1.
     We must have pl3 = 3/2 * pl2, with pl2 a multiple of 2^k2, and
     pl3 a multiple of 2^k3. Since k3 >= k2, both are multiples of 2^k2,
     and pl2 must be an even multiple of 2^k2. Thus (pl2,pl3) =
     (2*j*2^k2,3*j*2^k2), which works for 3*j <= pl/2^k2 <= 5*j.
     We need that consecutive intervals overlap, i.e. 5*j >= 3*(j+1),
     which requires j>=2. Thus this scheme requires pl >= 6 * 2^FFT_FIRST_K. */

  /*  ASSERT_ALWAYS(pl >= 6 * (1 << FFT_FIRST_K)); */

  pl2 = (2 * pl - 1) / 5; /* ceil (2pl/5) - 1 */
  do
    {
      pl2++;
      k2 = mpn_fft_best_k (pl2, sqr); /* best fft size for pl2 limbs */
      pl2 = mpn_fft_next_size (pl2, k2);
      pl3 = 3 * pl2 / 2; /* since k>=FFT_FIRST_K=4, pl2 is a multiple of 2^4,
			    thus pl2 / 2 is exact */
      k3 = mpn_fft_best_k (pl3, sqr);
    }
  while (mpn_fft_next_size (pl3, k3) != pl3);

  TRACE (printf ("mpn_mul_fft_full nl=%ld ml=%ld -> pl2=%ld pl3=%ld k=%d\n",
		 nl, ml, pl2, pl3, k2));

  ASSERT_ALWAYS(pl3 <= pl);
  cc = mpn_mul_fft (op, pl3, n, nl, m, ml, k3);     /* mu */
  ASSERT(cc == 0);
  pad_op = __GMP_ALLOCATE_FUNC_LIMBS (pl2);
  cc = mpn_mul_fft (pad_op, pl2, n, nl, m, ml, k2); /* lambda */
  cc = -cc + mpn_sub_n (pad_op, pad_op, op, pl2);    /* lambda - low(mu) */
  /* 0 <= cc <= 1 */
  ASSERT(0 <= cc && cc <= 1);
  l = pl3 - pl2; /* l = pl2 / 2 since pl3 = 3/2 * pl2 */
  c2 = mpn_add_n (pad_op, pad_op, op + pl2, l);
  cc = mpn_add_1 (pad_op + l, pad_op + l, l, (mp_limb_t) c2) - cc;
  ASSERT(-1 <= cc && cc <= 1);
  if (cc < 0)
    cc = mpn_add_1 (pad_op, pad_op, pl2, (mp_limb_t) -cc);
  ASSERT(0 <= cc && cc <= 1);
  /* now lambda-mu = {pad_op, pl2} - cc mod 2^(pl2*GMP_NUMB_BITS)+1 */
  oldcc = cc;
#if HAVE_NATIVE_mpn_add_n_sub_n
  c2 = mpn_add_n_sub_n (pad_op + l, pad_op, pad_op, pad_op + l, l);
  cc += c2 >> 1; /* carry out from high <- low + high */
  c2 = c2 & 1; /* borrow out from low <- low - high */
#else
  {
    mp_ptr tmp;
    TMP_DECL;

    TMP_MARK;
    tmp = TMP_BALLOC_LIMBS (l);
    MPN_COPY (tmp, pad_op, l);
    c2 = mpn_sub_n (pad_op,      pad_op, pad_op + l, l);
    cc += mpn_add_n (pad_op + l, tmp,    pad_op + l, l);
    TMP_FREE;
  }
#endif
  c2 += oldcc;
  /* first normalize {pad_op, pl2} before dividing by 2: c2 is the borrow
     at pad_op + l, cc is the carry at pad_op + pl2 */
  /* 0 <= cc <= 2 */
  cc -= mpn_sub_1 (pad_op + l, pad_op + l, l, (mp_limb_t) c2);
  /* -1 <= cc <= 2 */
  if (cc > 0)
    cc = -mpn_sub_1 (pad_op, pad_op, pl2, (mp_limb_t) cc);
  /* now -1 <= cc <= 0 */
  if (cc < 0)
    cc = mpn_add_1 (pad_op, pad_op, pl2, (mp_limb_t) -cc);
  /* now {pad_op, pl2} is normalized, with 0 <= cc <= 1 */
  if (pad_op[0] & 1) /* if odd, add 2^(pl2*GMP_NUMB_BITS)+1 */
    cc += 1 + mpn_add_1 (pad_op, pad_op, pl2, CNST_LIMB(1));
  /* now 0 <= cc <= 2, but cc=2 cannot occur since it would give a carry
     out below */
  mpn_rshift (pad_op, pad_op, pl2, 1); /* divide by two */
  if (cc) /* then cc=1 */
    pad_op [pl2 - 1] |= (mp_limb_t) 1 << (GMP_NUMB_BITS - 1);
  /* now {pad_op,pl2}-cc = (lambda-mu)/(1-2^(l*GMP_NUMB_BITS))
     mod 2^(pl2*GMP_NUMB_BITS) + 1 */
  c2 = mpn_add_n (op, op, pad_op, pl2); /* no need to add cc (is 0) */
  /* since pl2+pl3 >= pl, necessary the extra limbs (including cc) are zero */
  MPN_COPY (op + pl3, pad_op, pl - pl3);
  ASSERT_MPN_ZERO_P (pad_op + pl - pl3, pl2 + pl3 - pl);
  __GMP_FREE_FUNC_LIMBS (pad_op, pl2);
  /* since the final result has at most pl limbs, no carry out below */
  MPN_INCR_U (op + pl2, pl - pl2, (mp_limb_t) c2);
}
#endif