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

   mat_RR B1;  // approximates B
   B1.SetDims(m, n);


   mat_RR mu;
   mu.SetDims(m, n+1);

   mat_RR aux;
   aux.SetDims(m, n);


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

   GivensCache_RR cache(m, n);

   new_m = ll_G_LLL_RR(B, U, delta, deep, check, B1, mu, aux, m, 1, quit, cache);

   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));
      }
   }


   return m;
}

         

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

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

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

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



static vec_RR G_BKZConstant;

static
void ComputeG_BKZConstant(long beta, long p)
{
   RR c_PI;
   ComputePi(c_PI);

   RR LogPI = log(c_PI);

   G_BKZConstant.SetLength(beta-1);

   vec_RR Log;
   Log.SetLength(beta);


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

   for (j = 1; j <= beta; j++)
      Log(j) = log(to_RR(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 += Log(j);
          
         x = exp(x/k);

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

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

         x = exp(2*x/i);
      }

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

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

      G_BKZConstant(i) = x*y/c_PI;
   }

}

static vec_RR G_BKZThresh;

static 
void ComputeG_BKZThresh(RR *c, long beta)
{
   G_BKZThresh.SetLength(beta-1);

   long i;
   RR x;
   RR t1;

   x = 0;

   for (i = 1; i <= beta-1; i++) {
      log(t1, c[i-1]);
      add(x, x, t1);
      div(t1, x, i);
      exp(t1, t1);
      mul(G_BKZThresh(i), t1, G_BKZConstant(i));
   }
}




static 
void G_BKZStatus(double tt, double enum_time, unsigned long NumIterations, 
               unsigned long NumTrivial, unsigned long NumNonTrivial, 
               unsigned long NumNoOps, long m, 
               const mat_ZZ& B)
{
   cerr << "---- G_BKZ_RR status ----\n";
   cerr << "elapsed time: ";
   PrintTime(cerr, tt-StartTime);
   cerr << ", enum time: ";
   PrintTime(cerr, enum_time);
   cerr << ", iter: " << NumIterations << "\n";
   cerr << "triv: " << NumTrivial;
   cerr << ", nontriv: " << NumNonTrivial;
   cerr << ", no ops: " << NumNoOps;
   cerr << ", rank: " << m;
   cerr << ", swaps: " << NumSwaps << "\n";



   ZZ t1;
   long i;
   double prodlen = 0;

   for (i = 1; i <= m; i++) {
      InnerProduct(t1, B(i), B(i));
      if (!IsZero(t1))
         prodlen += log(t1);
   }

   cerr << "log of prod of lengths: " << prodlen/(2.0*log(2.0)) << "\n";


   if (LLLDumpFile) {
      cerr << "dumping to " << LLLDumpFile << "...";

      ofstream f;
      OpenWrite(f, LLLDumpFile);
      
      f << "[";
      for (i = 1; i <= m; i++) {
         f << B(i) << "\n";
      }
      f << "]\n";

      f.close();

      cerr << "\n";
   }

   LastTime = tt;
   
}




static
long G_BKZ_RR(mat_ZZ& BB, mat_ZZ* UU, const RR& delta, 
         long beta, long prune, LLLCheckFct check)
{
   long m = BB.NumRows();
   long n = BB.NumCols();
   long m_orig = m;
   
   long i, j;
   ZZ MU;

   RR t1, t2;
   ZZ T1;

   init_red_fudge();

   mat_ZZ B;
   B = BB;

   B.SetDims(m+1, n);


   mat_RR B1;
   B1.SetDims(m+1, n);

   mat_RR mu;
   mu.SetDims(m+1, n+1);

   mat_RR aux;
   aux.SetDims(m+1, n);

   vec_RR c;
   c.SetLength(m+1);

   RR cbar;

   vec_RR ctilda;
   ctilda.SetLength(m+1);

   vec_RR vvec;
   vvec.SetLength(m+1);

   vec_RR yvec;
   yvec.SetLength(m+1);

   vec_RR uvec;
   uvec.SetLength(m+1);

   vec_RR utildavec;
   utildavec.SetLength(m+1);

   vec_long Deltavec;
   Deltavec.SetLength(m+1);

   vec_long deltavec;
   deltavec.SetLength(m+1);

   mat_ZZ Ulocal;
   mat_ZZ *U;

   if (UU) {
      Ulocal.SetDims(m+1, m);
      for (i = 1; i <= m; i++)
         conv(Ulocal(i, i), 1);
      U = &Ulocal;
   }
   else
      U = 0;

   long quit;
   long new_m;
   long z, jj, kk;
   long s, t;
   long h;


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

   // cerr << "\n";
   // cerr << "first G_LLL\n";

   GivensCache_RR cache(m, n);

   m = ll_G_LLL_RR(B, U, delta, 0, check, B1, mu, aux, m, 1, quit, cache);


   double tt;

   double enum_time = 0;
   unsigned long NumIterations = 0;
   unsigned long NumTrivial = 0;
   unsigned long NumNonTrivial = 0;
   unsigned long NumNoOps = 0;

   long verb = verbose;

   verbose = 0;


   if (m < m_orig) {
      for (i = m_orig+1; i >= m+2; i--) {
         // swap i, i-1

         swap(B(i), B(i-1));
         if (U) swap((*U)(i), (*U)(i-1));
      }
   }

   long clean = 1;

   if (!quit && m > 1) {
      // cerr << "continuing\n";

      if (beta > m) beta = m;

      if (prune > 0)
         ComputeG_BKZConstant(beta, prune);

      z = 0;
      jj = 0;
   
      while (z < m-1) {
         jj++;
         kk = min(jj+beta-1, m);
   
         if (jj == m) {
            jj = 1;
            kk = beta;
            clean = 1;
         }

         if (verb) {
            tt = GetTime();
            if (tt > LastTime + LLLStatusInterval)
               G_BKZStatus(tt, enum_time, NumIterations, NumTrivial,
                         NumNonTrivial, NumNoOps, m, B);
         }

         // ENUM

         double tt1;

         if (verb) {
            tt1 = GetTime();
         }

         for (i = jj; i <= kk; i++)
            sqr(c(i), mu(i,i));


         if (prune > 0)
            ComputeG_BKZThresh(&c(jj), kk-jj+1);

         cbar = c(jj);
         conv(utildavec(jj), 1);
         conv(uvec(jj), 1);
   
         conv(yvec(jj), 0);
         conv(vvec(jj), 0);
         Deltavec(jj) = 0;
   
   
         s = t = jj;
         deltavec(jj) = 1;
   
         for (i = jj+1; i <= kk+1; i++) {
            conv(ctilda(i), 0);
            conv(uvec(i), 0);
            conv(utildavec(i), 0);
            conv(yvec(i), 0);
            Deltavec(i) = 0;
            conv(vvec(i), 0);
            deltavec(i) = 1;
         }

         long enum_cnt = 0;
   
         while (t <= kk) {
            if (verb) {
               enum_cnt++;
               if (enum_cnt > 100000) {
                  enum_cnt = 0;
                  tt = GetTime();
                  if (tt > LastTime + LLLStatusInterval) {
                     enum_time += tt - tt1;
                     tt1 = tt;
                     G_BKZStatus(tt, enum_time, NumIterations, NumTrivial,
                               NumNonTrivial, NumNoOps, m, B);
                  }
               }
            }


            add(t1, yvec(t), utildavec(t));
            sqr(t1, t1);
            mul(t1, t1, c(t));
            add(ctilda(t), ctilda(t+1), t1);

            if (prune > 0 && t > jj) 
               sub(t1, cbar, G_BKZThresh(t-jj));
            else
               t1 = cbar;

   
            if (ctilda(t) <t1) {
               if (t > jj) {
                  t--;
                  clear(t1);
                  for (i = t+1; i <= s; i++) {
                     mul(t2, utildavec(i), mu(i,t));
                     add(t1, t1, t2);
                  }

                  yvec(t) = t1;
                  negate(t1, t1);
                  if (sign(t1) >= 0) {
                     sub(t1, t1, 0.5);
                     ceil(t1, t1);
                  }
                  else {
                     add(t1, t1, 0.5);
                     floor(t1, t1);
                  }

                  utildavec(t) = t1;
                  vvec(t) = t1;
                  Deltavec(t) = 0;

                  negate(t1, t1);

                  if (t1 < yvec(t)) 
                     deltavec(t) = -1;
                  else
                     deltavec(t) = 1;
               }
               else {
                  cbar = ctilda(jj);
                  for (i = jj; i <= kk; i++) {
                     uvec(i) = utildavec(i);
                  }
               }
            }
            else {
               t++;
               s = max(s, t);
               if (t < s) Deltavec(t) = -Deltavec(t);
               if (Deltavec(t)*deltavec(t) >= 0) Deltavec(t) += deltavec(t);
               add(utildavec(t), vvec(t), Deltavec(t));
            }
         }
         
         if (verb) {
            tt1 = GetTime() - tt1;
            enum_time += tt1;
         }

         NumIterations++;
   
         h = min(kk+1, m);

         mul(t1, red_fudge, -8);
         add(t1, t1, delta);
         mul(t1, t1, c(jj));
   
         if (t1 > cbar) {
 
            clean = 0;

            // we treat the case that the new vector is b_s (jj < s <= kk)
            // as a special case that appears to occur most of the time.
   
            s = 0;
            for (i = jj+1; i <= kk; i++) {
               if (uvec(i) != 0) {
                  if (s == 0)
                     s = i;
                  else
                     s = -1;
               }
            }
   
            if (s == 0) Error("G_BKZ_RR: internal error");
   
            if (s > 0) {
               // special case
               // cerr << "special case\n";

               NumTrivial++;
   
               for (i = s; i > jj; i--) {
                  // swap i, i-1
                  swap(B(i-1), B(i));
                  swap(B1(i-1), B1(i));
                  if (U) swap((*U)(i-1), (*U)(i));
               }
   
               new_m = ll_G_LLL_RR(B, U, delta, 0, check,
                                B1, mu, aux, h, jj, quit, cache);
               if (new_m != h) Error("G_BKZ_RR: internal error");
               if (quit) break;
            }
            else {
               // the general case

               NumNonTrivial++;
   
               for (i = 1; i <= n; i++) conv(B(m+1, i), 0);

               if (U) {
                  for (i = 1; i <= m_orig; i++)
                     conv((*U)(m+1, i), 0);
               }

               for (i = jj; i <= kk; i++) {
                  if (uvec(i) == 0) continue;
                  conv(MU, uvec(i));
                  RowTransform2(B(m+1), B(i), MU);
                  if (U) RowTransform2((*U)(m+1), (*U)(i), MU);
               }
      
               for (i = m+1; i >= jj+1; i--) {
                  // swap i, i-1
                  swap(B(i-1), B(i));
                  swap(B1(i-1), B1(i));
                  if (U) swap((*U)(i-1), (*U)(i));
               }
      
               for (i = 1; i <= n; i++)
                  conv(B1(jj, i), B(jj, i));
      
               if (IsZero(B(jj))) Error("G_BKZ_RR: internal error"); 
      
               // remove linear dependencies
   
               // cerr << "general case\n";
               new_m = ll_G_LLL_RR(B, U, delta, 0, 0, B1, mu, aux,
                                  kk+1, jj, quit, cache);

              
               if (new_m != kk) Error("G_BKZ_RR: internal error"); 

               // remove zero vector
      
               for (i = kk+2; i <= m+1; i++) {
                  // swap i, i-1
                  swap(B(i-1), B(i));
                  swap(B1(i-1), B1(i));
                  if (U) swap((*U)(i-1), (*U)(i));
               }
      
               quit = 0;
               if (check) {
                  for (i = 1; i <= kk; i++)
                     if ((*check)(B(i))) {
                        quit = 1;
                        break;
                     }
               }

               if (quit) break;
   
               if (h > kk) {
                  // extend reduced basis
   
                  new_m = ll_G_LLL_RR(B, U, delta, 0, check,
                                   B1, mu, aux, h, h, quit, cache);
   
                  if (new_m != h) Error("G_BKZ_RR: internal error");
                  if (quit) break;
               }
            }
   
            z = 0;
         }
         else {
            // G_LLL_RR
            // cerr << "progress\n";

            NumNoOps++;

            if (!clean) {
               new_m = ll_G_LLL_RR(B, U, delta, 0, check, B1, mu, aux,
                                   h, h, quit, cache);
               if (new_m != h) Error("G_BKZ_RR: internal error");
               if (quit) break;
            }
   
            z++;
         }
      }
   }

   if (verb) {
      G_BKZStatus(GetTime(), enum_time, NumIterations, NumTrivial, NumNonTrivial,
                NumNoOps, m, B);
   }


   // clean up

   if (m_orig > m) {
      // for consistency, we move zero vectors to the front

      for (i = m+1; i <= m_orig; i++) {
         swap(B(i), B(i+1));
         if (U) swap((*U)(i), (*U)(i+1));
      }

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

   B.SetDims(m_orig, n);
   BB = B;

   if (U) {
      U->SetDims(m_orig, m_orig);
      *UU = *U;
   }

   return m;
}

long G_BKZ_RR(mat_ZZ& BB, mat_ZZ& UU, double delta, 
         long beta, long prune, LLLCheckFct check, long verb)
{
   verbose = verb;
   NumSwaps = 0;
   if (verbose) {
      StartTime = GetTime();
      LastTime = StartTime;
   }

   if (delta < 0.50 || delta >= 1) Error("G_BKZ_RR: bad delta");
   if (beta < 2) Error("G_BKZ_RR: bad block size");

   RR Delta;
   conv(Delta, delta);

   return G_BKZ_RR(BB, &UU, Delta, beta, prune, check);
}

long G_BKZ_RR(mat_ZZ& BB, double delta, 
         long beta, long prune, LLLCheckFct check, long verb)
{
   verbose = verb;
   NumSwaps = 0;
   if (verbose) {
      StartTime = GetTime();
      LastTime = StartTime;
   }

   if (delta < 0.50 || delta >= 1) Error("G_BKZ_RR: bad delta");
   if (beta < 2) Error("G_BKZ_RR: bad block size");

   RR Delta;
   conv(Delta, delta);

   return G_BKZ_RR(BB, 0, Delta, beta, prune, check);
}


NTL_END_IMPL