lll_qp.c

密码大家Shoup写的数论算法c语言实现

            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;

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

      if (quit) break;

      if (deep > 0) {
         // deep insertions
   
         quad_float 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++;
            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++;
         // cout << "- " << k << "\n";
      }
      else {
         k++;
         // cout << "+ " << k << "\n";
      }
   }

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


   delete [] buf;
   delete [] max_b;

   return m;
}

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

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

   quad_float t1;
   ZZ T1;

   init_red_fudge();

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

   quad_float **B1;  // approximates B

   typedef quad_float *quad_floatptr;

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

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

   quad_float **mu;
   mu = NTL_NEW_OP quad_floatptr[m+1];
   if (!mu) Error("LLL_QP: out of memory");

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

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

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

   quad_float *b; // squared lengths of basis vectors

   b = NTL_NEW_OP quad_float[m+1];
   if (!b) Error("LLL_QP: 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);
      if (!IsFinite(&b[i]))
         Error("LLL_QP: numbers too big...use LLL_XD");
   }


   new_m = ll_LLL_QP(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_QP(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_QP: bad delta");
   if (deep < 0) Error("LLL_QP: bad deep");
   return LLL_QP(B, 0, to_quad_float(delta), deep, check);
}

long LLL_QP(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("LLL_QP: bad delta");
   if (deep < 0) Error("LLL_QP: bad deep");
   return LLL_QP(B, &U, to_quad_float(delta), deep, check);
}



static vec_quad_float BKZConstant;

static
void ComputeBKZConstant(long beta, long p)
{
   const quad_float c_PI = 
      to_quad_float("3.141592653589793238462643383279502884197");
   const quad_float LogPI = 
      to_quad_float("1.144729885849400174143427351353058711647");

   BKZConstant.SetLength(beta-1);

   vec_quad_float Log;
   Log.SetLength(beta);


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

   for (j = 1; j <= beta; j++)
      Log(j) = log(to_quad_float(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/to_quad_float(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/to_quad_float(i));

         x = exp(x);
      }

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

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

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

static vec_quad_float BKZThresh;

static 
void ComputeBKZThresh(quad_float *c, long beta)
{
   BKZThresh.SetLength(beta-1);

   long i;
   quad_float x;

   x = 0;

   for (i = 1; i <= beta-1; i++) {
      x += log(c[i-1]);
      BKZThresh(i) = exp(x/to_quad_float(i))*BKZConstant(i);
      if (!IsFinite(&BKZThresh(i))) BKZThresh(i) = 0;
   }
}


static 
void BKZStatus(double tt, double enum_time, long NumIterations, 
               long NumTrivial, long NumNonTrivial, long NumNoOps, long m, 
               const mat_ZZ& B)
{
   cerr << "---- BKZ_QP 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 BKZ_QP(mat_ZZ& BB, mat_ZZ* UU, quad_float delta, 
         long beta, long prune, LLLCheckFct check)
{

   long m = BB.NumRows();
   long n = BB.NumCols();
   long m_orig = m;
   
   long i, j;
   ZZ MU;

   quad_float t1;
   ZZ T1;
   quad_float *tp;

   init_red_fudge();

   mat_ZZ B;
   B = BB;

   B.SetDims(m+1, n);


   quad_float **B1;  // approximates B

   typedef quad_float *quad_floatptr;

   B1 = NTL_NEW_OP quad_floatptr[m+2];
   if (!B1) Error("BKZ_QP: out of memory");

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

   quad_float **mu;
   mu = NTL_NEW_OP quad_floatptr[m+2];
   if (!mu) Error("BKZ_QP: out of memory");

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

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

   c = NTL_NEW_OP quad_float[m+2];
   if (!c) Error("BKZ_QP: out of memory");

   quad_float *b; // squared lengths of basis vectors

   b = NTL_NEW_OP quad_float[m+2];
   if (!b) Error("BKZ_QP: out of memory");

   quad_float cbar;

   quad_float *ctilda;
   ctilda = NTL_NEW_OP quad_float[m+2];
   if (!ctilda) Error("BKZ_QP: out of memory");

   quad_float *vvec;
   vvec = NTL_NEW_OP quad_float[m+2];
   if (!vvec) Error("BKZ_QP: out of memory");

   quad_float *yvec;
   yvec = NTL_NEW_OP quad_float[m+2];
   if (!yvec) Error("BKZ_QP: out of memory");

   quad_float *uvec;
   uvec = NTL_NEW_OP quad_float[m+2];
   if (!uvec) Error("BKZ_QP: out of memory");

   quad_float *utildavec;
   utildavec = NTL_NEW_OP quad_float[m+2];
   if (!utildavec) Error("BKZ_QP: out of memory");


   long *Deltavec;
   Deltavec = NTL_NEW_OP long[m+2];
   if (!Deltavec) Error("BKZ_QP: out of memory");

   long *deltavec;
   deltavec = NTL_NEW_OP long[m+2];
   if (!deltavec) Error("BKZ_QP: out of memory");

   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;
   quad_float eta;


   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);
      if (!IsFinite(&b[i]))
         Error("BKZ_FD: numbers too big...use BKZ_XD");
   }

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


   m = ll_LLL_QP(B, U, delta, 0, check, B1, mu, b, c, m, 1, quit);

   double tt;

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

   long verb = verbose;

   verbose = 0;

   long clean = 1;

   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));
      }
   }

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

      if (prune > 0)
         ComputeBKZConstant(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)
               BKZStatus(tt, enum_time, NumIterations, NumTrivial,
                         NumNonTrivial, NumNoOps, m, B);
         }



   
         // ENUM

         double tt1;

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


         if (prune > 0)
            ComputeBKZThresh(&c[jj], kk-jj+1);

   
         cbar = c[jj];
         utildavec[jj] = uvec[jj] = 1;
   
         yvec[jj] = vvec[jj] = 0;
         Deltavec[jj] = 0;
   
   
         s = t = jj;
         deltavec[jj] = 1;
   
         for (i = jj+1; i <= kk+1; i++) {
            ctilda[i] = uvec[i] = utildavec[i] = yvec[i] = 0;
            Deltavec[i] = 0;
            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;
                     BKZStatus(tt, enum_time, NumIterations, NumTrivial,
                               NumNonTrivial, NumNoOps, m, B);
                  }
               }
            }

            ctilda[t] = ctilda[t+1] + 
               (yvec[t]+utildavec[t])*(yvec[t]+utildavec[t])*c[t];
   
            if (prune > 0 && t > jj) {
               eta = BKZThresh(t-jj);
            }
            else
               eta = 0;

            if (ctilda[t] < cbar - eta) {
               if (t > jj) {
                  t--;
                  t1 = 0;
                  for (i = t+1; i <= s; i++) {
                     t1 += utildavec[i]*mu[i][t];
                  }


                  yvec[t] = t1;
                  t1 = -t1;
                  if (t1 >= 0)
                     t1 = ceil(t1-0.5);
                  else
                     t1 = floor(t1+0.5);

                  utildavec[t] = vvec[t] = t1;
                  Deltavec[t] = 0;
                  if (utildavec[t] > -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];
               utildavec[t] = vvec[t] + Deltavec[t];
            }
         }

         if (verb) {
            tt1 = GetTime() - tt1;
            enum_time += tt1;
         }

         NumIterations++;

         h = min(kk+1, m);
   
         if ((delta-8*red_fudge)*c[jj] > 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("BKZ_QP: internal error");
   
            if (s > 0) {
               // special case

               NumTrivial++;
   
               for (i = s; i > jj; i--) {
                  // swap i, i-1
                  swap(B(i-1), B(i));
                  if (U) swap((*U)(i-1), (*U)(i));
                  tp = B1[i-1]; B1[i-1] = B1[i]; B1[i] = tp;
                  t1 = b[i-1]; b[i-1] = b[i]; b[i] = t1;
               }
   
               // cerr << "special case\n";
               new_m = ll_LLL_QP(B, U, delta, 0, check, 
                                B1, mu, b, c, h, jj, quit);
               if (new_m != h) Error("BKZ_QP: 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);
               }