pp1.c

/* The 'P+1' algorithm.

Copyright 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012
Paul Zimmermann and Alexander Kruppa.

This file is part of the ECM Library.

The ECM 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 ECM 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 ECM Library; see the file COPYING.LIB.  If not, see
http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */

/* References:

A p+1 Method of Factoring, H. C. Williams, Mathematics of Computation,
volume 39, number 159, pages 225-234, 1982.

Evaluating recurrences of form X_{m+n} = f(X_m, X_n, X_{m-n}) via
Lucas chains, Peter L. Montgomery, December 1983, revised January 1992. */

#include <math.h>
#include <stdlib.h>
#include "ecm-impl.h"

#ifdef HAVE_LIMITS_H
# include <limits.h>
#else
# ifndef ULONG_MAX
#  define ULONG_MAX __GMP_ULONG_MAX
# endif
#endif


/******************************************************************************
*                                                                             *
*                                  Stage 1                                    *
*                                                                             *
******************************************************************************/

/* prime powers are accumulated up to about n^L1 */
#define L1 1

/* P1 <- V_e(P0), using P, Q as auxiliary variables,
   where V_{2k}(P0) = V_k(P0)^2 - 2
         V_{2k-1}(P0) = V_k(P0)*V_{k-1}(P0) - P0.
   (More generally V_{m+n} = V_m * V_n - V_{m-n}.)
   Warning: P1 and P0 may be equal.
*/
static void
pp1_mul (mpres_t P1, mpres_t P0, mpz_t e, mpmod_t n, mpres_t P, mpres_t Q)
{
  mp_size_t size_e;
  unsigned long i;
  int sign;

  sign = mpz_sgn (e);
  mpz_abs (e, e);
  if (sign == 0)
    {
      mpres_set_ui (P1, 2, n);
      goto unnegate;
    }

  if (mpz_cmp_ui (e, 1) == 0)
    {
      mpres_set (P1, P0, n);
      goto unnegate;
    }
  
  /* now e >= 2 */
  mpz_sub_ui (e, e, 1);
  mpres_sqr (P, P0, n);
  mpres_sub_ui (P, P, 2, n); /* P = V_2(P0) = P0^2-2 */
  mpres_set (Q, P0, n);      /* Q = V_1(P0) = P0 */

  /* invariant: (P, Q) = (V_{k+1}(P0), V_k(P0)), start with k=1 */
  size_e = mpz_sizeinbase (e, 2);
  for (i = size_e - 1; i > 0;)
    {
      if (mpz_tstbit (e, --i)) /* k -> 2k+1 */
        {
          if (i) /* Q is not needed for last iteration */
            {
              mpres_mul (Q, P, Q, n);
              mpres_sub (Q, Q, P0, n);
            }
          mpres_sqr (P, P, n);
          mpres_sub_ui (P, P, 2, n);
        }
      else /* k -> 2k */
        {
          mpres_mul (P, P, Q, n);
          mpres_sub (P, P, P0, n);
          if (i) /* Q is not needed for last iteration */
            {
              mpres_sqr (Q, Q, n);
              mpres_sub_ui (Q, Q, 2, n);
            }
        }
    }

  mpres_set (P1, P, n);
  mpz_add_ui (e, e, 1); /* recover original value of e */
unnegate:
  if (sign == -1)
    mpz_neg (e, e);
  
  return;
}

/* Input:  P0 is the initial point (sigma)
           n is the number to factor
           B1 is the stage 1 bound
	   B1done: stage 1 was already done up to that limit
	   go: if <> 1, group order to preload
   Output: a is the factor found, or the value at end of stage 1
	   B1done is set to B1 if stage 1 completed normally,
	   or to the largest prime processed if interrupted, but never
	   to a smaller value than B1done was upon function entry.
   Return value: non-zero iff a factor was found.
*/
static int
pp1_stage1 (mpz_t f, mpres_t P0, mpmod_t n, double B1, double *B1done, 
            mpz_t go, int (*stop_asap)(void), char *chkfilename)
{
  double B0, p, q, r, last_chkpnt_p;
  mpz_t g;
  mpres_t P, Q;
  mpres_t R, S, T;
  int youpi = ECM_NO_FACTOR_FOUND;
  unsigned int max_size, size_n;
  long last_chkpnt_time;

  mpz_init (g);
  mpres_init (P, n);
  mpres_init (Q, n);
  mpres_init (R, n);
  mpres_init (S, n);
  mpres_init (T, n);

  B0 = ceil (sqrt (B1));

  size_n = mpz_sizeinbase (n->orig_modulus, 2);
  max_size = L1 * size_n;

  if (mpz_cmp_ui (go, 1) > 0)
    pp1_mul (P0, P0, go, n, P, Q);

  /* suggestion from Peter Montgomery: start with exponent n^2-1,
     as factors of Lucas and Fibonacci number are either +/-1 (mod index),
     and so is n. Therefore, index will appear as a factor
     of n^2-1 and be included in stage 1.
     Do this only when n is composite, otherwise all tests with prime
     n factor of a Cunningham number will succeed in stage 1.

     As in P-1, for small overhead, use that trick only when lg(n) <= sqrt(B1).
  */
  if ((double) size_n <= B0 &&
      mpz_probab_prime_p (n->orig_modulus, PROBAB_PRIME_TESTS) == 0)
    {
      mpz_mul (g, n->orig_modulus, n->orig_modulus);
      mpz_sub_ui (g, g, 1);
      pp1_mul (P0, P0, g, n, P, Q);
    }

  mpz_set_ui (g, 1);

  last_chkpnt_p = 2.;
  last_chkpnt_time = cputime ();
  /* first loop through small primes <= sqrt(B1) */
  for (p = 2.0; p <= B0; p = getprime ())
    {
      for (q = 1, r = p; r <= B1; r *= p)
        if (r > *B1done) q *= p;
      mpz_mul_d (g, g, q, Q);
      if (mpz_sizeinbase (g, 2) >= max_size)
	{
	  pp1_mul (P0, P0, g, n, P, Q);
	  mpz_set_ui (g, 1);
          if (stop_asap != NULL && (*stop_asap) ())
            {
              outputf (OUTPUT_NORMAL, "Interrupted at prime %.0f\n", p);
	      if (p > *B1done)
		  *B1done = p;
              goto clear_and_exit;
            }
	}
    }

  pp1_mul (P0, P0, g, n, P, Q);

#if 1
  /* All primes sqrt(B1) < p <= B1 appear in exponent 1. All primes <= B1done
     are already included in exponent of at least 1, so it's save to skip 
     ahead to B1done+1 */
  
  if (*B1done > p)
    {
      getprime_seek ((*B1done) + 1.);
      p = getprime ();
    }
#endif

  /* then all primes > sqrt(B1) and taken with exponent 1 */
  for (; p <= B1; p = getprime ())
    {
      pp1_mul_prac (P0, (ecm_uint) p, n, P, Q, R, S, T);
  
      if (stop_asap != NULL && (*stop_asap) ())
        {
          outputf (OUTPUT_NORMAL, "Interrupted at prime %.0f\n", p);
	  if (p > *B1done)
	      *B1done = p;
          goto clear_and_exit;
        }
      if (chkfilename != NULL && p > last_chkpnt_p + 10000. &&
          elltime (last_chkpnt_time, cputime ()) > CHKPNT_PERIOD)
        {
          writechkfile (chkfilename, ECM_PP1, p, n, NULL, P0, NULL);
          last_chkpnt_p = p;
          last_chkpnt_time = cputime ();
        }
    }

  /* If stage 1 finished normally, p is the smallest prime >B1 here.
     In that case, set to B1 */
  if (p > B1)
    p = B1;
  
  if (p > *B1done)
    *B1done = p;
  
  mpres_sub_ui (P, P0, 2, n);
  mpres_gcd (f, P, n);
  youpi = mpz_cmp_ui (f, 1);

clear_and_exit:
  if (chkfilename != NULL)
    writechkfile (chkfilename, ECM_PP1, p, n, NULL, P0, NULL);
  getprime_clear (); /* free the prime tables, and reinitialize */
  mpres_clear (Q, n);
  mpres_clear (R, n);
  mpres_clear (S, n);
  mpres_clear (T, n);
  mpz_clear (g);
  mpres_clear (P, n);
  
  return youpi;
}

/* checks if the factor p was found by P+1 or P-1 (when prime).
   a is the initial seed.
*/
static void
pp1_check_factor (mpz_t a, mpz_t p)
{
  if (mpz_probab_prime_p (p, PROBAB_PRIME_TESTS))
    {
      mpz_mul (a, a, a);
      mpz_sub_ui (a, a, 4);
      if (mpz_jacobi (a, p) == 1)
        outputf (OUTPUT_NORMAL, "[factor found by P-1]\n");
    }
}

/******************************************************************************
*                                                                             *
*                                  Stage 2                                    *
*                                                                             *
******************************************************************************/

/* let alpha, beta be the roots of x^2-Px+1=0
   set a, b such that alpha^e = a*alpha+b (idem for beta),
   i.e. a*x+b = rem(x^e, x^2-Px+1).
   Since (x-alpha)*(x-beta) = x^2-Px+1, we have alpha*beta = 1
   and alpha+beta = P, i.e. 1/alpha = beta = -alpha + P.
   It seems that if x^e % (x^2-Px+1) = a*x+b, then
   x^{-e+1} % (x^2-Px+1) = b*x+a. Proof?
*/

static void
pp1_mul2 (mpres_t a, mpres_t b, mpres_t P, mpz_t e, mpmod_t n)
{
  unsigned long l;
  mpres_t t;
  mpz_t abs_e;
  const int positive_e = (mpz_sgn (e) > 0);

  if (mpz_cmp_ui (e, 0UL) == 0) /* x^0 = 1 */
    {
      mpres_set_ui (a, 0, n);
      mpres_set_ui (b, 1, n);
      return;
    }
  
  mpres_init (t, n);
  mpz_init (abs_e);
  mpz_abs (abs_e, e);

  if (positive_e)
    {
      mpres_set_ui (a, 1, n);
      mpres_set_ui (b, 0, n);
    }
  else
    {
      /* Set to -x+P */
      mpres_set_ui (a, 1, n);
      mpres_neg (a, a, n);
      mpres_set (b, P, n);
    }

  l = mpz_sizeinbase (abs_e, 2) - 1; /* number of bits of e (minus 1) */
  
  while (l--)
    {
      /* square: (ax+b)^2 = (a^2P+2ab) x + (b^2-a^2) */
      mpres_sqr (t, a, n);    /* a^2 */
      mpres_mul (a, a, b, n);
      mpres_add (a, a, a, n); /* 2ab */
      mpres_sqr (b, b, n);    /* b^2 */
      mpres_sub (b, b, t, n); /* b^2-a^2 */
      mpres_mul (t, t, P, n); /* a^2P */
      mpres_add (a, t, a, n); /* a^2P+2ab */
      
      if (mpz_tstbit (abs_e, l))
	{
	  if (positive_e)
	    {
	      /* multiply: (ax+b)*x = (aP+b) x - a */
	      mpres_mul (t, a, P, n);
	      mpres_add (t, t, b, n);
	      mpres_neg (b, a, n);
	      mpres_set (a, t, n);
	    }
	  else
	    {
	      /* multiply: (ax+b)*(-x+P) = 
                           -ax^2+(aP-b)x+b*P == 
			   -bx + (bP + a) (mod x^2-P*x+1) */
	      mpres_mul (t, b, P, n);
	      mpres_add (t, t, a, n);
	      mpres_neg (a, b, n);
	      mpres_set (b, t, n);
	    }
	}
    }

  mpz_clear (abs_e);
  mpres_clear (t, n);
}

/* 
   Performs the following:
   for (i=0;i<m;i++)
      for (j=0;j<n;j++)
        (x[j+(n+1)*i],y[j+(n+1)*i]) += (x[j+1+(n+1)*i],y[j+1+(n+1)*i])
 */
static void
addWnm (point *X, mpres_t P, mpmod_t modulus, unsigned int m,
        unsigned int n, unsigned long *tot_muls)
{
  unsigned long i, j, k;
  mpres_t t, u;

  mpres_init (t, modulus);
  mpres_init (u, modulus);
  for (i = 0; i < m; i++)
    for (j = 0; j < n; j++)
      { /* (a*x+b)*(c*x+d) = (Pac+ad+bc) x + (bd-ac) */
	k = (n + 1) * i + j; /* n is the polynomial degree, so each table of
				differences contains n+1 entries */
	mpres_add (t, X[k].x, X[k].y, modulus); /* a+b */
	mpres_add (u, X[k+1].x, X[k+1].y, modulus); /* c+d */
	mpres_mul (t, t, u, modulus); /* ad+bc+ac+bd */
	mpres_mul (X[k].y, X[k].y, X[k+1].y, modulus); /* bd */
	mpres_sub (t, t, X[k].y, modulus); /* ad+bc+ac */
	mpres_mul (u, X[k].x, X[k+1].x, modulus); /* ac */
	mpres_sub (X[k].y, X[k].y, u, modulus); /* bd-ac */
	mpres_sub (t, t, u, modulus); /* ad+bc */
	mpres_mul (u, u, P, modulus); /* Pac */
	mpres_add (X[k].x, t, u, modulus); /* ad+bc+Pac */
	*tot_muls += 4;
      }
  mpres_clear (t, modulus);
  mpres_clear (u, modulus);
}

/* puts in F[0..dF-1] the successive values of 
   V_f(j*d2)(P) for Williams P+1,
   where f(x) = x^S for positive S, Dickson_{S,a}(x) for negative S,
   and 0 < j = 1 mod 6 < d1, j and d1 coprime.

   Return non-zero iff a factor was found (always zero in fact).
*/
int
pp1_rootsF (listz_t F, root_params_t *root_params, unsigned long dF, 
            mpres_t *x, listz_t t, mpmod_t modulus)
{
  unsigned long i, j;
  unsigned long muls = 0;
  long st, st1;
  int youpi = ECM_NO_FACTOR_FOUND;
  mpres_t fd[3];
  mpres_t u, v; /* auxiliary variables */
  listz_t coeffs;
  ecm_roots_state_t state;
  progression_params_t *params = &state.params; /* for less typing */

  if (dF == 0)
    return youpi;

  st1 = st = cputime ();

  outputf (OUTPUT_DEVVERBOSE, 
	   "pp1_rootsF: d1 = %lu, d2 = %lu, S = %d, dF = %lu\n",
	   root_params->d1, root_params->d2, root_params->S, dF);

  mpres_init (u, modulus);
  mpres_init (v, modulus);

  if (ABS(root_params->S) == 1) /* special code with d1/6 muls */
    {
      mpres_init (fd[0], modulus);
      mpres_init (fd[1], modulus);
      mpres_init (fd[2], modulus);
  
      mpz_set_ui (*t, root_params->d2);
      pp1_mul (fd[2], *x, *t, modulus, u, v);
      mpres_get_z (F[0], fd[2], modulus);
  
      mpz_set_ui (*t, 7UL);
      pp1_mul (fd[0], fd[2], *t, modulus, u, v);

      mpz_set_ui (*t, 6UL);
      pp1_mul (fd[1], fd[2], *t, modulus, u, v);

      /* fd[0] = V_{7*d2}(P), fd[1] = V_{6*d2}(P), fd[2] = V_{d2}(P) */

      outputf (OUTPUT_VERBOSE,
	       "Initializing table of differences for F took %ldms\n",
	       elltime (st1, cputime ()));

      i = 1;
      j = 7;
      while (i < dF)
	{
	  if (gcd (j, root_params->d1) == 1) /* (d2,d1) == 1 ==> (j*d2,d1) == (j,d1) */
	    mpres_get_z (F[i++], fd[0], modulus);

	  /* V_{m+n} = V_m * V_n - V_{m-n} */
	  /* fd[0] = V_m, fd[1] = V_n, fd[2] = V_{m-n} */
	  mpres_swap (fd[0], fd[2], modulus);
	  /* fd[0] = V_{m-n}, fd[1] = V_n, fd[2] = V_m */
	  mpres_mul (u, fd[2], fd[1], modulus);
	  /* u = V_n * V_m */
	  mpres_sub (fd[0], u, fd[0], modulus);
	  /* fd[0] = V_n * V_m - V_{m-n} = V_{m+n}, hence */
	  /* fd[0] = V_{m+n}, fd[1] = V_n, fd[2] = V_m */
	  j += 6;
	  muls ++;
	}

      mpres_clear (fd[0], modulus);
      mpres_clear (fd[1], modulus);
      mpres_clear (fd[2], modulus);
    }
  else /* case |S| <> 1: this code works also for S=1, but is more
	  expensive, since it can use up to 4*(d1/6) muls */
    {
      init_roots_params (params, root_params->S, root_params->d1, 
                         root_params->d2, 1.0);
      mpz_set_ui (*t, 0UL);
      coeffs = init_progression_coeffs (*t, params->dsieve, root_params->d2, 1, 
                                        6, params->S, params->dickson_a);
      
      if (coeffs == NULL)
        return ECM_ERROR;

      state.fd = (point *) malloc (params->size_fd * sizeof (point));
      if (state.fd == NULL)
        {
	  clear_list (coeffs, params->size_fd);
	  return ECM_ERROR;
        }
      for (i = 0; i < params->size_fd; i++)
	{
          mpres_init (state.fd[i].x, modulus);
          mpres_init (state.fd[i].y, modulus);
	  /* if i = k*(S+1) + S for k>=1, we can copy x and y from i - (S+1) */
	  if (i > params->S && (i % (params->S + 1) == params->S))
	    {
	      mpres_set (state.fd[i].x, state.fd[params->S].x, modulus);
	      mpres_set (state.fd[i].y, state.fd[params->S].y, modulus);
	    }
	  else
	    pp1_mul2 (state.fd[i].x, state.fd[i].y, x[0], coeffs[i], modulus);
	}
      clear_list (coeffs, params->size_fd);

      outputf (OUTPUT_VERBOSE,
	       "Initializing table of differences for F took %ldms\n",
	       elltime (st1, cputime ()));

      /* Now for the actual calculation of the roots. */
      for (i = 0; i < dF && !youpi;)
	{
	  /* Is this a rsieve value where we computed Dickson(j * d2) * X? */
	  if (gcd (params->rsieve, params->dsieve) == 1)
	    {
	      /* Did we use every progression since the last update? */
	      if (params->next == params->nr)
		{
		  /* Yes, time to update again */
		  addWnm (state.fd, x[0], modulus, params->nr, params->S, &muls);
		  params->next = 0;
		}

	      /* Is this a j value where we want Dickson(j*d2)*X as a root? */
	      if (gcd (params->rsieve, root_params->d1) == 1)
		{
		  /* we have alpha^k = x * alpha + y
		     thus alpha^k + beta^k = x * P + 2 * y.
                     FIXME: can we avoid returning to the Lucas form?
                  */
		  mpres_mul (u, state.fd[params->next * (params->S + 1)].x, x[0],
			     modulus);
		  mpres_add (v, state.fd[params->next * (params->S + 1)].y,
                             state.fd[params->next * (params->S + 1)].y, modulus);
		  mpres_add (u, u, v, modulus);
		  mpres_get_z (F[i++], u, modulus);
		}

	      params->next ++;
	    }
	  params->rsieve += 6;
	}

      for (i = 0; i < params->size_fd; i++)
        {
          mpres_clear (state.fd[i].x, modulus);
          mpres_clear (state.fd[i].y, modulus);
        }
      free (state.fd);
    }

  mpres_clear (u, modulus);
  mpres_clear (v, modulus);

  outputf (OUTPUT_VERBOSE, "Computing roots of F took %ldms",
	   elltime (st, cputime ()));
  outputf (OUTPUT_DEVVERBOSE, " and %d muls", muls);
  outputf (OUTPUT_VERBOSE, "\n");
  
  return youpi;
}

/* return NULL if an error occurred */
pp1_roots_state_t *
pp1_rootsG_init (mpres_t *x, root_params_t *root_params, mpmod_t modulus)
{
  mpres_t P;
  pp1_roots_state_t *state;
  progression_params_t *params; /* for less typing */
  unsigned long i;

  ASSERT (gcd (root_params->d1, root_params->d2) == 1);

  state = (pp1_roots_state_t *) malloc (sizeof (pp1_roots_state_t));
  if (state == NULL)
    return NULL;
  params = &(state->params);

  /* we don't need the sign anymore after pp1_rootsG_init */
  params->S = ABS(root_params->S); 

  if (params->S == 1)
    {
      mpz_t t;
      mpz_init (t);
      mpres_init (P, modulus);
      for (i = 0; i < 4; i++)
        mpres_init (state->tmp[i], modulus);

      params->dsieve = root_params->d2; /* needed in pp1_rootsG */
      /* We want to skip values where gcd((i0 + i) * d1, d2) != 1.
	 We can test for gcd(i0 + i, d2) instead and let pp1_rootsG()
	 advance params->rsieve in steps of 1 */
      /* params->rsieve = i0 % d2 */
      params->rsieve = mpz_fdiv_ui (root_params->i0, root_params->d2);
      
      outputf (OUTPUT_DEVVERBOSE, "pp1_rootsG_init: i0 = %Zd, state: "
               "dsieve = %d, rsieve = %d, S = %d\n", root_params->i0, 
               params->dsieve, params->rsieve, params->S);
      
      mpz_set_ui (t, root_params->d1);
      pp1_mul (state->tmp[1], *x, t, modulus, state->tmp[3], P);
      pp1_mul (state->tmp[0], state->tmp[1], root_params->i0, modulus, 
               state->tmp[3], P);
      mpz_sub_ui (t, root_params->i0, 1);
      mpz_abs (t, t);
      pp1_mul (state->tmp[2], state->tmp[1], t, modulus, state->tmp[3], P);
      /* for P+1, tmp[0] = V_s(P), tmp[1] = V_d1(P), tmp[2] = V_{|s-d1|}(P) */

      mpres_clear (P, modulus);
      mpz_clear (t);
    }
  else
    {
      listz_t coeffs;
      
      params->dickson_a = (root_params->S < 0) ? -1 : 0;
      params->nr = (root_params->d2 > 1) ? root_params->d2 - 1 : 1;
      params->size_fd = params->nr * (params->S + 1);
      params->next = 0;
      params->dsieve = 1;
      params->rsieve = 1;

      state->fd = (point *) malloc (params->size_fd * sizeof (point));
      if (state->fd == NULL)
        {
          free (state);
          return NULL;
        }
      
      coeffs = init_progression_coeffs (root_params->i0, root_params->d2, 
                 root_params->d1, 1, 1, params->S, params->dickson_a);
      if (coeffs == NULL)
        {
          free (state->fd);
          free (state);
          return NULL;
        }

      for (i = 0; i < params->size_fd; i++) 
        {
          mpres_init (state->fd[i].x, modulus);
          mpres_init (state->fd[i].y, modulus);
          /* The S-th coeff of all progressions is identical */
          if (i > params->S && i % (params->S + 1) == params->S) 
            {
              /* Simply copy from the first progression */
              mpres_set (state->fd[i].x, state->fd[params->S].x, modulus); 
              mpres_set (state->fd[i].y, state->fd[params->S].y, modulus); 
            }
          else
            pp1_mul2 (state->fd[i].x, state->fd[i].y, x[0], coeffs[i], modulus);
        }

      clear_list (coeffs, params->size_fd);
    }

  return state;
}

void 
pp1_rootsG_clear (pp1_roots_state_t *state, ATTRIBUTE_UNUSED mpmod_t modulus)
{
  unsigned long i;

  if (state->params.S == 1)
    {
      for (i = 0; i < 4; i++)
        mpres_clear (state->tmp[i], modulus);
    }
  else
    {
      for (i = 0; i < state->params.size_fd; i++)
        {
          mpres_clear (state->fd[i].x, modulus);
          mpres_clear (state->fd[i].y, modulus);
        }
      free (state->fd);
    }

  free (state);
}

int
pp1_rootsG (listz_t G, unsigned long dF, pp1_roots_state_t *state, 
	    mpmod_t modulus, mpres_t *x)
{
  unsigned long i;
  unsigned long muls = 0;
  long st;
  progression_params_t *params = &(state->params); /* for less typing */

  st = cputime ();

  /* params->S is positive: we don't need the sign anymore, since the
     polynomial is defined by the table of differences */

  if (params->S == 1)
    {
      for (i = 0; i < dF;)
        {
          if (gcd (params->rsieve, params->dsieve) == 1)
            {
              outputf (OUTPUT_TRACE, "pp1_rootsG: Taking root G[%d], rsieve = %d\n", 
                       i, params->rsieve);
              mpres_get_z (G[i++], state->tmp[0], modulus);
            }
          else
            {
              outputf (OUTPUT_TRACE, "pp1_rootsG: NOT taking root, rsieve = %d, gcd = %d\n", 
                       params->rsieve, gcd (params->rsieve, params->dsieve));
            }

          mpres_swap (state->tmp[0], state->tmp[2], modulus);
          mpres_mul (state->tmp[3], state->tmp[2], state->tmp[1], modulus);
          mpres_sub (state->tmp[0], state->tmp[3], state->tmp[0], modulus);
          params->rsieve++;
        }
    }
  else
    {
      mpres_t u, v;

      mpres_init (u, modulus);
      mpres_init (v, modulus);
      for (i = 0; i < dF;)
        {
          /* Did we use every progression since the last update? */
          if (params->next == params->nr)
            {
              /* Yes, time to update again */
              addWnm (state->fd, x[0], modulus, params->nr, params->S, &muls);
              params->next = 0;
            }
      
          /* Is this a root we should skip? (Take only if gcd == 1) */
          if (gcd (params->rsieve, params->dsieve) == 1)
            {
              mpres_mul (u, state->fd[params->next * (params->S + 1)].x, x[0],
                         modulus);
              mpres_add (v, state->fd[params->next * (params->S + 1)].y,
                         state->fd[params->next * (params->S + 1)].y, modulus);
              mpres_add (u, u, v, modulus);
              mpres_get_z (G[i++], u, modulus);
            }
      
          params->next ++;
          params->rsieve ++;
        }
      mpres_clear (u, modulus);
      mpres_clear (v, modulus);
    }

  outputf (OUTPUT_VERBOSE, "Computing roots of G took %ldms",
	   elltime (st, cputime ()));
  outputf (OUTPUT_DEVVERBOSE, ", %lu muls", dF);
  outputf (OUTPUT_VERBOSE, "\n");
  
  return ECM_NO_FACTOR_FOUND;
}


/******************************************************************************
*                                                                             *
*                               Williams P+1                                  *
*                                                                             *
******************************************************************************/

/* Input: p is the initial generator (sigma), if 0 generate it at random.
          n is the number to factor
	  B1 is the stage 1 bound
	  B2 is the stage 2 bound
          k is the number of blocks for stage 2
          verbose is the verbosity level
   Output: p is the factor found
   Return value: non-zero iff a factor is found (1 for stage 1, 2 for stage 2)
*/
int
pp1 (mpz_t f, mpz_t p, mpz_t n, mpz_t go, double *B1done, double B1,
     mpz_t B2min_parm, mpz_t B2_parm, double B2scale, unsigned long k, 
     const int S, int verbose, int repr, int use_ntt, FILE *os, FILE *es, 
     char *chkfilename, char *TreeFilename, double maxmem, 
     gmp_randstate_t rng, int (*stop_asap)(void))
{
  int youpi = ECM_NO_FACTOR_FOUND;
  int po2 = 0;    /* Whether we should use power-of-2 poly degree */
  long st;
  mpres_t a;
  mpmod_t modulus;
  mpz_t B2min, B2; /* Local B2, B2min to avoid changing caller's values */
  unsigned long dF;
  root_params_t root_params;
  faststage2_param_t faststage2_params;
  const int stage2_variant = (S == 1 || S == ECM_DEFAULT_S);
  int twopass = 0;

  set_verbose (verbose);
  ECM_STDOUT = (os == NULL) ? stdout : os;
  ECM_STDERR = (es == NULL) ? stdout : es;

  /* if n is even, return 2 */
  if (mpz_divisible_2exp_p (n, 1))
    {
      mpz_set_ui (f, 2);
      return ECM_FACTOR_FOUND_STEP1;
    }

  st = cputime ();

  if (mpz_cmp_ui (p, 0) == 0)
    pm1_random_seed (p, n, rng);

  mpz_init_set (B2min, B2min_parm);
  mpz_init_set (B2, B2_parm);

  /* Set default B2. See ecm.c for comments */
  if (ECM_IS_DEFAULT_B2(B2))
    {
      if (stage2_variant == 0)
        mpz_set_d (B2, B2scale * pow (B1 * PP1_COST, DEFAULT_B2_EXPONENT));
      else
        mpz_set_d (B2, B2scale * pow (B1 * PP1FS2_COST, 
                   PM1FS2_DEFAULT_B2_EXPONENT));
    }

  /* set B2min */
  if (mpz_sgn (B2min) < 0)
    mpz_set_d (B2min, B1);

  mpmod_init (modulus, n, repr);

  if (use_ntt)
    po2 = 1;
  
  if (stage2_variant != 0)
    {
      long P;
      const unsigned long lmax = 1UL<<28; /* An upper bound */
      unsigned long lmax_NTT, lmax_noNTT;
      
      mpz_init (faststage2_params.m_1);
      faststage2_params.l = 0;
      
      /* Find out what the longest transform length is we can do at all.
	 If no maxmem is given, the non-NTT can theoretically do any length. */

      lmax_NTT = 0;
      if (use_ntt)
	{
	  unsigned long t, t2 = 0;
	  /* See what transform length that the NTT can handle (due to limited 
	     primes and limited memory) */
	  t = mpzspm_max_len (n);
	  lmax_NTT = MIN (lmax, t);
	  if (maxmem != 0.)
	    {
	      t = pp1fs2_maxlen (double_to_size (maxmem), n, use_ntt, 0);
	      t = MIN (t, lmax_NTT);
	      /* Maybe the two pass variant lets us use a longer transform */
	      t2 = pp1fs2_maxlen (double_to_size (maxmem), n, use_ntt, 1);
	      t2 = MIN (t2, lmax_NTT);
	      if (t2 > t)
		{
		  t = t2;
		  twopass = 1;
		}
	      lmax_NTT = t;
	    }
	  outputf (OUTPUT_DEVVERBOSE, "NTT can handle lmax <= %lu\n", lmax_NTT);
	}

      /* See what transform length that the non-NTT code can handle */
      lmax_noNTT = lmax;
      if (maxmem != 0.)
	{
	  unsigned long t;
	  t = pp1fs2_maxlen (double_to_size (maxmem), n, 0, 0);
	  lmax_noNTT = MIN (lmax_noNTT, t);
	  outputf (OUTPUT_DEVVERBOSE, "non-NTT can handle lmax <= %lu\n", 
		   lmax_noNTT);
	}

      P = choose_P (B2min, B2, MAX(lmax_noNTT, lmax_NTT), k, 
		    &faststage2_params, B2min, B2, use_ntt, ECM_PP1);
      if (P == ECM_ERROR)
	{
          outputf (OUTPUT_ERROR, 
                   "Error: cannot choose suitable P value for your stage 2 "
                   "parameters.\nTry a shorter B2min,B2 interval.\n");
	  mpz_clear (faststage2_params.m_1);
	  return ECM_ERROR;
	}

      /* See if the selected parameters let us use NTT or not */
      if (faststage2_params.l > lmax_NTT)
	use_ntt = 0;
      
      if (maxmem != 0.)
	{
	  unsigned long MB;
	  char *s;
	  if (!use_ntt)
	    s = "out";
	  else if (twopass)
	    s = " two pass";
	  else
	    s = " one pass";

	  MB = pp1fs2_memory_use (faststage2_params.l, n, use_ntt, twopass)
	    / 1048576;
	  outputf (OUTPUT_VERBOSE, "Using lmax = %lu with%s NTT which takes "
		   "about %luMB of memory\n", faststage2_params.l, s, MB);
	}
    }
  else 
    {
      mpz_init (root_params.i0);
      root_params.d2 = 0; /* Enable automatic choice of d2 */
      if (bestD (&root_params, &k, &dF, B2min, B2, po2, use_ntt, maxmem,
		 (TreeFilename != NULL), modulus) == ECM_ERROR)
	{
	  youpi = ECM_ERROR;
	  goto clear_and_exit;
	}
      
      /* Set default degree for Brent-Suyama extension */
      root_params.S = S;
      if (root_params.S == ECM_DEFAULT_S)
	{
	  if (modulus->repr == ECM_MOD_BASE2 && modulus->Fermat > 0)
	    {
	      /* For Fermat numbers, default is 1 (no Brent-Suyama) */
	      root_params.S = 1;
	    }
	  else
	    {
	      mpz_t t;
	      mpz_init (t);
	      mpz_sub (t, B2, B2min);
	      root_params.S = choose_S (t);
	      mpz_clear (t);
	    }
	}
    }

  /* Print B1, B2, polynomial and x0 */
  print_B1_B2_poly (OUTPUT_NORMAL, ECM_PP1, B1, *B1done, B2min_parm, B2min, 
		    B2, (stage2_variant == 0) ? root_params.S : 1, p, 0, NULL, 0, 0);

  /* If we do a stage 2, print its parameters */
  if (mpz_cmp (B2, B2min) >= 0)
    {
      if (stage2_variant != 0)
        outputf (OUTPUT_VERBOSE, "P = %lu, l = %lu, s_1 = %lu, k = s_2 = %lu, "
                 "m_1 = %Zd\n", faststage2_params.P, faststage2_params.l,
                 faststage2_params.s_1,faststage2_params.s_2,
                 faststage2_params.m_1);
      else
        outputf (OUTPUT_VERBOSE, "dF=%lu, k=%lu, d=%lu, d2=%lu, i0=%Zd\n", 
                 dF, k, root_params.d1, root_params.d2, 
                 S == 1 ? faststage2_params.m_1 : root_params.i0);
    }

  mpres_init (a, modulus);
  mpres_set_z (a, p, modulus);

  /* since pp1_mul_prac takes an ecm_uint, we have to check
     that B1 <= ECM_UINT_MAX */
  if (B1 > (double) ECM_UINT_MAX)
    {
      outputf (OUTPUT_ERROR, "Error, maximal step1 bound for P+1 is %lu\n", 
               ECM_UINT_MAX);
      youpi = ECM_ERROR;
      goto clear_and_exit;
    }

  if (B1 > *B1done)
    youpi = pp1_stage1 (f, a, modulus, B1, B1done, go, stop_asap, 
                        chkfilename);

  outputf (OUTPUT_NORMAL, "Step 1 took %ldms\n", elltime (st, cputime ()));
  if (test_verbose (OUTPUT_RESVERBOSE))
    {
      mpz_t t;
      
      mpz_init (t);
      mpres_get_z (t, a, modulus);
      outputf (OUTPUT_RESVERBOSE, "x=%Zd\n", t);
      mpz_clear (t);
    }

  mpres_get_z (p, a, modulus);

  if (stop_asap != NULL && (*stop_asap) ())
    goto clear_and_exit;
      
  if (youpi == ECM_NO_FACTOR_FOUND && mpz_cmp (B2, B2min) >= 0)
    {
      if (stage2_variant != 0)
        {
          if (use_ntt)
            youpi = pp1fs2_ntt (f, a, modulus, &faststage2_params, twopass);
          else 
            youpi = pp1fs2 (f, a, modulus, &faststage2_params);
        }
      else
	youpi = stage2 (f, &a, modulus, dF, k, &root_params, ECM_PP1, 
			use_ntt, TreeFilename, stop_asap);
    }

  if (youpi > 0 && test_verbose (OUTPUT_NORMAL))
    pp1_check_factor (p, f); /* tell user if factor was found by P-1 */

 clear_and_exit:
  mpres_clear (a, modulus);
  mpmod_clear (modulus);
  if (stage2_variant != 0)
    mpz_clear (faststage2_params.m_1);
  else
    mpz_clear (root_params.i0);
  mpz_clear (B2);
  mpz_clear (B2min);

  return youpi;
}