conv(buf[i], rr_buf[i]);
   }


   RR::SetPrecision(old_p);

   tt = GetTime()-tt;
   RR_GS_time += tt;
   cerr << tt << " (" << RR_GS_time << ")\n";
}

void ComputeGS(const mat_ZZ& B, mat_RR& mu, vec_RR& c)
{
   long n = B.NumCols();
   long k = B.NumRows();

   mat_RR B1;
   vec_RR b;

   B1.SetDims(k, n);
   mu.SetDims(k, k);
   b.SetLength(k);
   c.SetLength(k);

   vec_RR buf;
   buf.SetLength(k);

   long i, j;

   for (i = 1; i <= k; i++)
      for (j = 1; j <= n; j++)
         conv(B1(i, j), B(i, j));

   for (i = 1; i <= k; i++)
      InnerProduct(b(i), B1(i), B1(i));

   

   RR bound;
   power2(bound, 2*long(0.15*RR::precision()));

   RR bound2;
   power2(bound2, 2*RR::precision());


   for (i = 1; i <= k; i++)
      ComputeGS(B, B1, mu, b, c, i, bound, 1, buf, bound2);

}





static
long ll_LLL_FP(mat_ZZ& B, mat_ZZ* U, double delta, long deep, 
           LLLCheckFct check, double **B1, double **mu, 
           double *b, double *c,
           long m, long init_k, long &quit)
{
   long n = B.NumCols();

   long i, j, k, Fc1;
   ZZ MU;
   double mu1;

   double t1;
   ZZ T1;
   double *tp;


   static double bound = 0;

   if (bound == 0) {
      // we tolerate a 15% loss of precision in computing
      // inner products in ComputeGS.

      bound = 1;
      for (i = 2*long(0.15*NTL_DOUBLE_PRECISION); i > 0; i--)
         bound = bound * 2;
   }

   double half_plus_fudge = 0.5 + red_fudge;

   quit = 0;
   k = init_k;


   vec_long st_mem;
   st_mem.SetLength(m+2);
   long *st = st_mem.elts();

   for (i = 1; i < k; i++)
      st[i] = i;

   for (i = k; i <= m+1; i++)
      st[i] = 1;

   double *buf;
   buf = NTL_NEW_OP double [m+1];
   if (!buf) Error("out of memory in lll_LLL_FP");

   vec_long in_vec_mem;
   in_vec_mem.SetLength(n+1);
   long *in_vec = in_vec_mem.elts();

   double *max_b;
   max_b = NTL_NEW_OP double [m+1];
   if (!max_b) Error("out of memory in lll_LLL_FP");

   for (i = 1; i <= m; i++)
      max_b[i] = max_abs(B1[i], n);

   long in_float;

   long rst;
   long counter;
   long start_over;

   long trigger_index;
   long small_trigger;
   long cnt;

   mat_RR rr_B1;
   mat_RR rr_mu;
   vec_RR rr_c;
   vec_RR rr_b;

   long m_orig = m;

   long rr_st = 1;

   long max_k = 0;

   long prec = RR::precision();

   double tt;

   long swap_cnt = 0;


   while (k <= m) {

      if (k > max_k) {
         max_k = k;
         swap_cnt = 0;
      }

      if (verbose) {
         tt = GetTime();

         if (tt > LastTime + LLLStatusInterval)
            LLLStatus(max_k, tt, m, B);
      }

      if (k < rr_st) rr_st = k;

      if (st[k] == k)
         rst = 1;
      else
         rst = k;

      if (st[k] < st[k+1]) st[k+1] = st[k];
      ComputeGS(B, B1, mu, b, c, k, bound, st[k], buf);
      CheckFinite(&c[k]);
      st[k] = k;

      if (swap_cnt > 200000) {
         cerr << "LLL_FP: swap loop?\n";
         RR_GS(B, B1, mu, b, c, buf, prec,
               rr_st, k, m_orig, rr_B1, rr_mu, rr_b, rr_c);
         if (rr_st < st[k+1]) st[k+1] = rr_st;
         rr_st = k+1;
         rst = k;
         swap_cnt = 0;
      }

      counter = 0;
      trigger_index = k;
      small_trigger = 0;
      cnt = 0;

      long thresh = 10;
      long sz=0, new_sz;

      long did_rr_gs = 0;


      do {
         // size reduction

         counter++;
         if ((counter & 127) == 0) {

            new_sz = 0;
            for (j = 1; j <= n; j++)
               new_sz += NumBits(B(k,j));

            if ((counter >> 7) == 1 || new_sz < sz) {
               sz = new_sz;
            }
            else {
               cerr << "LLL_FP: warning--infinite loop?\n";
            }
         }

         Fc1 = 0;
         start_over = 0;
   
         for (j = rst-1; j >= 1; j--) {
            t1 = fabs(mu[k][j]);
            if (t1 > half_plus_fudge) { 


               if (!Fc1) {
                  if (j > trigger_index || 
                      (j == trigger_index && small_trigger)) {

                     cnt++;

                     if (cnt > thresh) {
                        if (log_red <= 15) { 

                           while (log_red > 10)
                              inc_red_fudge();

                           half_plus_fudge = 0.5 + red_fudge;

                           if (!did_rr_gs) {
                              RR_GS(B, B1, mu, b, c, buf, prec,
                                    rr_st, k, m_orig, rr_B1, rr_mu, rr_b, rr_c);
                              if (rr_st < st[k+1]) st[k+1] = rr_st;
                              rr_st = k+1;
                              did_rr_gs = 1;
                              rst = k;
                              trigger_index = k;
                              small_trigger = 0;
                              start_over = 1;
                              break;
                           }
                        }
                        else {
                           inc_red_fudge();
                           half_plus_fudge = 0.5 + red_fudge;
                           cnt = 0;
                        }
                     }
                  }

                  trigger_index = j;
                  small_trigger = (t1 < 4);

                  Fc1 = 1;
                  if (k < rr_st) rr_st = k;
                  RowTransformStart(B1[k], in_vec, in_float, n);
               }
                  

               mu1 = mu[k][j];
               if (mu1 >= 0)
                  mu1 = ceil(mu1-0.5);
               else
                  mu1 = floor(mu1+0.5);
   
               double *mu_k = mu[k];
               double *mu_j = mu[j];
   
               if (mu1 == 1) {
                  for (i = 1; i <= j-1; i++)
                     mu_k[i] -= mu_j[i];
               }
               else if (mu1 == -1) {
                  for (i = 1; i <= j-1; i++)
                     mu_k[i] += mu_j[i];
               }
               else {
                  for (i = 1; i <= j-1; i++)
                     mu_k[i] -= mu1*mu_j[i];
               }
   
               mu_k[j] -= mu1;
   
               conv(MU, mu1);

               RowTransform(B(k), B(j), MU, B1[k], B1[j], in_vec,
                            max_b[k], max_b[j], in_float);
               if (U) RowTransform((*U)(k), (*U)(j), MU);
            }
         }


         if (Fc1) {
            RowTransformFinish(B(k), B1[k], in_vec);
            max_b[k] = max_abs(B1[k], n);

            if (!did_rr_gs) {
               b[k] = InnerProduct(B1[k], B1[k], n);
               CheckFinite(&b[k]);

               ComputeGS(B, B1, mu, b, c, k, bound, 1, buf);
               CheckFinite(&c[k]);
            }
            else {
               RR_GS(B, B1, mu, b, c, buf, prec,
                     rr_st, k, m_orig, rr_B1, rr_mu, rr_b, rr_c);
               rr_st = k+1;
            }

            rst = k;
         }
      } while (Fc1 || start_over);

      if (check && (*check)(B(k))) 
         quit = 1;

      if (b[k] == 0) {
         for (i = k; i < m; i++) {
            // swap i, i+1
            swap(B(i), B(i+1));
            tp = B1[i]; B1[i] = B1[i+1]; B1[i+1] = tp;
            t1 = b[i]; b[i] = b[i+1]; b[i+1] = t1;
            t1 = max_b[i]; max_b[i] = max_b[i+1]; max_b[i+1] = t1;
            if (U) swap((*U)(i), (*U)(i+1));
         }

         for (i = k; i <= m+1; i++) st[i] = 1;
         if (k < rr_st) rr_st = k;

         m--;
         if (quit) break;
         continue;
      }

      if (quit) break;

      if (deep > 0) {
         // deep insertions

         double cc = b[k];
         long l = 1;
         while (l <= k-1 && delta*c[l] <= cc) {
            cc = cc - mu[k][l]*mu[k][l]*c[l];
            l++;
         }
   
         if (l <= k-1 && (l <= deep || k-l <= deep)) {
            // deep insertion at position l
   
            for (i = k; i > l; i--) {
               // swap rows i, i-1
               swap(B(i), B(i-1));
               tp = B1[i]; B1[i] = B1[i-1]; B1[i-1] = tp;
               tp = mu[i]; mu[i] = mu[i-1]; mu[i-1] = tp;
               t1 = b[i]; b[i] = b[i-1]; b[i-1] = t1;
               t1 = max_b[i]; max_b[i] = max_b[i-1]; max_b[i-1] = t1;
               if (U) swap((*U)(i), (*U)(i-1));
            }
   
            k = l;
            NumSwaps++;
            swap_cnt++;
            continue;
         }
      } // end deep insertions

      // test LLL reduction condition

      if (k > 1 && delta*c[k-1] > c[k] + mu[k][k-1]*mu[k][k-1]*c[k-1]) {
         // swap rows k, k-1
         swap(B(k), B(k-1));
         tp = B1[k]; B1[k] = B1[k-1]; B1[k-1] = tp;
         tp = mu[k]; mu[k] = mu[k-1]; mu[k-1] = tp;
         t1 = b[k]; b[k] = b[k-1]; b[k-1] = t1;
         t1 = max_b[k]; max_b[k] = max_b[k-1]; max_b[k-1] = t1;
         if (U) swap((*U)(k), (*U)(k-1));

         k--;
         NumSwaps++;
         swap_cnt++;
         // cout << "-\n";
      }
      else {

         k++;
         // cout << "+\n";
      }

   }

   if (verbose) {
      LLLStatus(m+1, GetTime(), m, B);
   }


   delete [] buf;
   delete [] max_b;

   return m;
}





static
long LLL_FP(mat_ZZ& B, mat_ZZ* U, double delta, long deep, 
           LLLCheckFct check)
{
   long m = B.NumRows();
   long n = B.NumCols();

   long i, j;
   long new_m, dep, quit;
   ZZ MU;

   ZZ T1;

   init_red_fudge();

   if (U) ident(*U, m);

   double **B1;  // approximates B

   typedef double *doubleptr;

   B1 = NTL_NEW_OP doubleptr[m+1];
   if (!B1) Error("LLL_FP: out of memory");

   for (i = 1; i <= m; i++) {
      B1[i] = NTL_NEW_OP double[n+1];
      if (!B1[i]) Error("LLL_FP: out of memory");
   }

   double **mu;
   mu = NTL_NEW_OP doubleptr[m+1];
   if (!mu) Error("LLL_FP: out of memory");

   for (i = 1; i <= m; i++) {
      mu[i] = NTL_NEW_OP double[m+1];
      if (!mu[i]) Error("LLL_FP: out of memory");
   }

   double *c; // squared lengths of Gramm-Schmidt basis vectors

   c = NTL_NEW_OP double[m+1];
   if (!c) Error("LLL_FP: out of memory");

   double *b; // squared lengths of basis vectors

   b = NTL_NEW_OP double[m+1];
   if (!b) Error("LLL_FP: out of memory");


   for (i = 1; i <=m; i++)
      for (j = 1; j <= n; j++) {
         conv(B1[i][j], B(i, j));
         CheckFinite(&B1[i][j]);
      }

         
   for (i = 1; i <= m; i++) {
      b[i] = InnerProduct(B1[i], B1[i], n);
      CheckFinite(&b[i]);
   }

   new_m = ll_LLL_FP(B, U, delta, deep, check, B1, mu, b, c, m, 1, quit);
   dep = m - new_m;
   m = new_m;

   if (dep > 0) {
      // for consistency, we move all of the zero rows to the front

      for (i = 0; i < m; i++) {
         swap(B(m+dep-i), B(m-i));
         if (U) swap((*U)(m+dep-i), (*U)(m-i));
      }
   }


   // clean-up

   for (i = 1; i <= m+dep; i++) {
      delete [] B1[i];
   }

   delete [] B1;

   for (i = 1; i <= m+dep; i++) {
      delete [] mu[i];
   }

   delete [] mu;

   delete [] c;

   delete [] b;

   return m;
}

         

long LLL_FP(mat_ZZ& B, double delta, long deep, LLLCheckFct check, 
           long verb)
{
   verbose = verb;
   RR_GS_time = 0;
   NumSwaps = 0;
   if (verbose) {
      StartTime = GetTime();
      LastTime = StartTime;
   }

   if (delta < 0.50 || delta >= 1) Error("LLL_FP: bad delta");
   if (deep < 0) Error("LLL_FP: bad deep");
   return LLL_FP(B, 0, delta, deep, check);
}

long LLL_FP(mat_ZZ& B, mat_ZZ& U, double delta, long deep, 
           LLLCheckFct check, long verb)
{
   verbose = verb;
   RR_GS_time = 0;
   NumSwaps = 0;
   if (verbose) {
      StartTime = GetTime();
      LastTime = StartTime;
   }

   if (delta < 0.50 || delta >= 1) Error("LLL_FP: bad delta");
   if (deep < 0) Error("LLL_FP: bad deep");
   return LLL_FP(B, &U, delta, deep, check);
}



static vec_double BKZConstant;

static
void ComputeBKZConstant(long beta, long p)
{
   const double c_PI = 3.14159265358979323846264338328;
   const double LogPI = 1.14472988584940017414342735135;

   BKZConstant.SetLength(beta-1);

   vec_double Log;
   Log.SetLength(beta);


   long i, j, k;
   double x, y;

   for (j = 1; j <= beta; j++)
      Log(j) = log(double(j));

   for (i = 1; i <= beta-1; i++) {
      // First, we compute x = gamma(i/2)^{2/i}

      k = i/2;

      if ((i & 1) == 0) { // i even
         x = 0;
         for (j = 1; j <= k; j++)
            x = x + Log(j);
          
         x = x * (1/double(k));

         x = exp(x);
      }
      else { // i odd
         x = 0;
         for (j = k + 2; j <= 2*k + 2; j++)
            x = x + Log(j);

         x = 0.5*LogPI + x - 2*(k+1)*Log(2);

         x = x * (2.0/double(i));

         x = exp(x);
      }

      // Second, we compute y = 2^{2*p/i}

      y = -(2*p/double(i))*Log(2);
      y = exp(y);