lll_xd.cpp

数值算法库for Windows

#include <NTL/LLL.h>
#include <NTL/fileio.h>
#include <NTL/vec_xdouble.h>
#include <NTL/vec_double.h>

#include <NTL/new.h>

NTL_START_IMPL


static xdouble InnerProduct(xdouble *a, xdouble *b, long n)
{
   xdouble s;
   long i;

   s = 0;
   for (i = 1; i <= n; i++) 
      MulAdd(s, s, a[i], b[i]);

   return s;
}


static void RowTransform(vec_ZZ& A, vec_ZZ& B, const ZZ& MU1)
// x = x - y*MU
{
   static ZZ T, MU;
   long k;

   long n = A.length();
   long i;

   MU = MU1;

   if (MU == 1) {
      for (i = 1; i <= n; i++)
         sub(A(i), A(i), B(i));

      return;
   }

   if (MU == -1) {
      for (i = 1; i <= n; i++)
         add(A(i), A(i), B(i));

      return;
   }

   if (MU == 0) return;

   if (NumTwos(MU) >= NTL_ZZ_NBITS) 
      k = MakeOdd(MU);
   else
      k = 0;


   if (MU.WideSinglePrecision()) {
      long mu1;
      conv(mu1, MU);

      for (i = 1; i <= n; i++) {
         mul(T, B(i), mu1);
         if (k > 0) LeftShift(T, T, k);
         sub(A(i), A(i), T);
      }
   }
   else {
      for (i = 1; i <= n; i++) {
         mul(T, B(i), MU);
         if (k > 0) LeftShift(T, T, k);
         sub(A(i), A(i), T);
      }
   }
}

static void RowTransform2(vec_ZZ& A, vec_ZZ& B, const ZZ& MU1)
// x = x + y*MU
{
   static ZZ T, MU;
   long k;

   long n = A.length();
   long i;

   MU = MU1;

   if (MU == 1) {
      for (i = 1; i <= n; i++)
         add(A(i), A(i), B(i));

      return;
   }

   if (MU == -1) {
      for (i = 1; i <= n; i++)
         sub(A(i), A(i), B(i));

      return;
   }

   if (MU == 0) return;

   if (NumTwos(MU) >= NTL_ZZ_NBITS) 
      k = MakeOdd(MU);
   else
      k = 0;

   if (MU.WideSinglePrecision()) {
      long mu1;
      conv(mu1, MU);

      for (i = 1; i <= n; i++) {
         mul(T, B(i), mu1);
         if (k > 0) LeftShift(T, T, k);
         add(A(i), A(i), T);
      }
   }
   else {
      for (i = 1; i <= n; i++) {
         mul(T, B(i), MU);
         if (k > 0) LeftShift(T, T, k);
         add(A(i), A(i), T);
      }
   }
}

static
void ComputeGS(mat_ZZ& B, xdouble **B1, xdouble **mu, xdouble *b, 
               xdouble *c, long k, xdouble bound, long st, xdouble *buf)
{
   long n = B.NumCols();
   long i, j;
   xdouble s, t1, y, t;
   ZZ T1;

   xdouble *mu_k = mu[k];

   if (st < k) {
      for (i = 1; i < st; i++)
         buf[i] = mu_k[i]*c[i];
   }

   for (j = st; j <= k-1; j++) {
      if (b[k]*b[j] < NTL_FDOUBLE_PRECISION*NTL_FDOUBLE_PRECISION) {
         double z = 0;
         xdouble *B1_k = B1[k];
         xdouble *B1_j = B1[j];

         for (i = 1; i <= n; i++)
            z += B1_k[i].x * B1_j[i].x;

         s = z;
      }
      else {
         s = InnerProduct(B1[k], B1[j], n);
   
         if (s*s <= b[k]*b[j]/bound) {
            InnerProduct(T1, B(k), B(j));
            conv(s, T1);
         }
      }

      xdouble *mu_j = mu[j];

      t1 = 0;
      for (i = 1; i <= j-1; i++)
         MulAdd(t1, t1, mu_j[i], buf[i]);

      mu_k[j] = (buf[j] = (s - t1))/c[j];
   }

   s = 0;
   for (j = 1; j <= k-1; j++)
      MulAdd(s, s, mu_k[j], buf[j]);

   c[k] = b[k] - s;
}

static xdouble red_fudge = to_xdouble(0);
static long log_red = 0;

static void init_red_fudge()
{
   long i;

   log_red = long(0.50*NTL_DOUBLE_PRECISION);
   red_fudge = 1;

   for (i = log_red; i > 0; i--)
      red_fudge = red_fudge*0.5;
}

static void inc_red_fudge()
{

   red_fudge = red_fudge * 2;
   log_red--;

   cerr << "LLL_XD: warning--relaxing reduction (" << log_red << ")\n";

   if (log_red < 4)
      Error("LLL_XD: can not continue...sorry");
}



static long verbose = 0;

static unsigned long NumSwaps = 0;
static double StartTime = 0;
static double LastTime = 0;



static void LLLStatus(long max_k, double t, long m, const mat_ZZ& B)
{
   cerr << "---- LLL_XD status ----\n";
   cerr << "elapsed time: ";
   PrintTime(cerr, t-StartTime);
   cerr << ", stage: " << max_k;
   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 = t;
   
}


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

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

   xdouble t1;
   ZZ T1;
   xdouble *tp;


   static xdouble bound = to_xdouble(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;
      }
   }


   xdouble half = to_xdouble(0.5);
   xdouble 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;

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

   long rst;
   long counter;

   long trigger_index;
   long small_trigger;
   long cnt;

   long max_k = 0;

   double tt;


   while (k <= m) {

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

      if (verbose) {
         tt = GetTime();

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


      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);
      st[k] = k;

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

      do {
         // size reduction

         counter++;
         if (counter > 10000) {
            cerr << "LLL_XD: warning--possible infinite loop\n";
            counter = 0;
         }


         Fc1 = 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 > 10) {
                        inc_red_fudge();
                        half_plus_fudge = 0.5 + red_fudge;
                        cnt = 0;
                     }
                  }

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


               Fc1 = 1;
   
               mu1 = mu[k][j];
               if (mu1 >= 0)
                  mu1 = ceil(mu1-half);
               else
                  mu1 = floor(mu1+half);
   
   
               xdouble *mu_k = mu[k];
               xdouble *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++)
                     MulSub(mu_k[i], mu_k[i], mu1, mu_j[i]);
               }
  
               mu_k[j] -= mu1;

               conv(MU, mu1);

               // cout << j << " " << MU << "\n";
   
               RowTransform(B(k), B(j), MU);
               if (U) RowTransform((*U)(k), (*U)(j), MU);
            }
         }

         if (Fc1) {
            for (i = 1; i <= n; i++)
               conv(B1[k][i], B(k, i));
   
            b[k] = InnerProduct(B1[k], B1[k], n);
            ComputeGS(B, B1, mu, b, c, k, bound, 1, buf);
         }
      } while (Fc1);

      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;
            if (U) swap((*U)(i), (*U)(i+1));
         }

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

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

      if (quit) break;

      if (deep > 0) {
         // deep insertions
   
         xdouble 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;
               if (U) swap((*U)(i), (*U)(i-1));
            }
   
            k = l;
            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;
         if (U) swap((*U)(k), (*U)(k-1));

         k--;
         NumSwaps++;

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

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


   delete [] buf;

   return m;
}




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

   long i, j;
   long new_m, dep, quit;
   xdouble s;
   ZZ MU;
   xdouble mu1;

   xdouble t1;
   ZZ T1;

   init_red_fudge();

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

   xdouble **B1;  // approximates B

   typedef xdouble *xdoubleptr;

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

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

   xdouble **mu;
   mu = NTL_NEW_OP xdoubleptr[m+1];
   if (!mu) Error("LLL_XD: out of memory");

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

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

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

   xdouble *b; // squared lengths of basis vectors

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



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


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


   new_m = ll_LLL_XD(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; i++) {
      delete [] B1[i];
   }

   delete [] B1;

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

   delete [] mu;

   delete [] c;

   delete [] b;

   return m;
}

         

long LLL_XD(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("LLL_XD: bad delta");
   if (deep < 0) Error("LLL_XD: bad deep");
   return LLL_XD(B, 0, to_xdouble(delta), deep, check);
}

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