source: git/ntl/src/GF2EXFactoring.c @ de6a29

#include <NTL/GF2EXFactoring.h>
#include <NTL/vec_GF2XVec.h>
#include <NTL/fileio.h>
#include <NTL/FacVec.h>

#include <stdio.h>

#include <NTL/new.h>

NTL_START_IMPL




static
void IterSqr(GF2E& c, const GF2E& a, long n)
{
   GF2E res;

   long i;

   res = a;

   for (i = 0; i < n; i++)
      sqr(res, res);

   c = res;
}
   


void SquareFreeDecomp(vec_pair_GF2EX_long& u, const GF2EX& ff)
{
   GF2EX f = ff;

   if (!IsOne(LeadCoeff(f)))
      Error("SquareFreeDecomp: bad args");

   GF2EX r, t, v, tmp1;
   long m, j, finished, done;

   u.SetLength(0);

   if (deg(f) == 0)
      return;

   m = 1;
   finished = 0;

   do {
      j = 1;
      diff(tmp1, f);
      GCD(r, f, tmp1);
      div(t, f, r);

      if (deg(t) > 0) {
         done = 0;
         do {
            GCD(v, r, t);
            div(tmp1, t, v);
            if (deg(tmp1) > 0) append(u, cons(tmp1, j*m));
            if (deg(v) > 0) {
               div(r, r, v);
               t = v;
               j++;
            }
            else
               done = 1;
         } while (!done);
         if (deg(r) == 0) finished = 1;
      }

      if (!finished) {
         /* r is a square */

         long k, d;
         d = deg(r)/2;
         f.rep.SetLength(d+1);
         for (k = 0; k <= d; k++) 
            IterSqr(f.rep[k], r.rep[k*2], GF2E::degree()-1);
         m = m*2;
      }
   } while (!finished);
}
         


static
void NullSpace(long& r, vec_long& D, vec_GF2XVec& M, long verbose)
{
   long k, l, n;
   long i, j;
   long pos;
   GF2X t1, t2;
   GF2X *x, *y;

   const GF2XModulus& p = GF2E::modulus();

   n = M.length();

   D.SetLength(n);
   for (j = 0; j < n; j++) D[j] = -1;

   r = 0;

   l = 0;
   for (k = 0; k < n; k++) {

      pos = -1;
      for (i = l; i < n; i++) {
         rem(t1, M[i][k], p);
         M[i][k] = t1;
         if (pos == -1 && !IsZero(t1))
            pos = i;
      }

      if (pos != -1) {
         swap(M[pos], M[l]);

         // make M[l, k] == -1 mod p, and make row l reduced

         InvMod(t1, M[l][k], p);
         for (j = k+1; j < n; j++) {
            rem(t2, M[l][j], p);
            MulMod(M[l][j], t2, t1, p);
         }

         for (i = l+1; i < n; i++) {
            // M[i] = M[i] + M[l]*M[i,k]

            t1 = M[i][k];   // this is already reduced

            x = M[i].elts() + (k+1);
            y = M[l].elts() + (k+1);

            for (j = k+1; j < n; j++, x++, y++) {
               // *x = *x + (*y)*t1

               mul(t2, *y, t1);
               add(*x, *x, t2);
            }
         }

         D[k] = l;   // variable k is defined by row l
         l++;

      }
      else {
         r++;
      }
   }
}



static
void BuildMatrix(vec_GF2XVec& M, long n, const GF2EX& g, const GF2EXModulus& F,
                 long verbose)
{
   long i, j, m;
   GF2EX h;


   M.SetLength(n);
   for (i = 0; i < n; i++)
      M[i].SetSize(n, 2*GF2E::WordLength());

   set(h);
   for (j = 0; j < n; j++) {

      m = deg(h);
      for (i = 0; i < n; i++) {
         if (i <= m)
            M[i][j] = rep(h.rep[i]);
         else
            clear(M[i][j]);
      }

      if (j < n-1)
         MulMod(h, h, g, F);
   }

   for (i = 0; i < n; i++)
      add(M[i][i], M[i][i], 1);

}


static
void TraceMap(GF2EX& h, const GF2EX& a, const GF2EXModulus& F)

// one could consider making a version based on modular composition,
// as in ComposeFrobeniusMap...

{
   GF2EX res, tmp;

   res = a;
   tmp = a;

   long i;
   for (i = 0; i < GF2E::degree()-1; i++) {
      SqrMod(tmp, tmp, F);
      add(res, res, tmp);
   }

   h = res;
}

void PlainFrobeniusMap(GF2EX& h, const GF2EXModulus& F)
{
   GF2EX res;

   SetX(res);
   long i;
   for (i = 0; i < GF2E::degree(); i++) 
      SqrMod(res, res, F);

   h = res;
}

long UseComposeFrobenius(long d, long n)
{
   long i;
   i = 1;
   while (i <= d) i = i << 1;
   i = i >> 1;

   i = i >> 1;
   long m = 1;

   long dz;

   if (n == 2) {
      dz = 1;
   }
   else {
      while (i) {
         long m1 = 2*m;
         if (i & d) m1++;
   
         if (m1 >= NTL_BITS_PER_LONG-1 || (1L << m1) >= n) break;
   
         m = m1;
         i = i >> 1;
      }

      dz = 1L << m;
   }

   long rootn = SqrRoot(n);
   long cnt = 0;

   if (i) {
      cnt += SqrRoot(dz+1);
      i = i >> 1;
   }

   while (i) {
      cnt += rootn;
      i = i >> 1;
   }

   return 4*cnt <= d;
}

void ComposeFrobeniusMap(GF2EX& y, const GF2EXModulus& F)
{
   long d = GF2E::degree();
   long n = deg(F);

   long i;
   i = 1;
   while (i <= d) i = i << 1;
   i = i >> 1;

   GF2EX z(INIT_SIZE, n), z1(INIT_SIZE, n);

   i = i >> 1;
   long m = 1;

   if (n == 2) {
      SetX(z);
      SqrMod(z, z, F);
   }
   else {
      while (i) {
         long m1 = 2*m;
         if (i & d) m1++;
   
         if (m1 >= NTL_BITS_PER_LONG-1 || (1L << m1) >= n) break;
   
         m = m1;
         i = i >> 1;
      }

      clear(z);
      SetCoeff(z, 1L << m);
   }


   while (i) {
      z1 = z;

      long j, k, dz;
      dz = deg(z);

      for (j = 0; j <= dz; j++)
         for (k = 0; k < m; k++)
            sqr(z1.rep[j], z1.rep[j]);

      CompMod(z, z1, z, F);
      m = 2*m;

      if (d & i) {
         SqrMod(z, z, F);
         m++;
      }

      i = i >> 1;
   }

   y = z;
}

void FrobeniusMap(GF2EX& h, const GF2EXModulus& F)
{
   long n = deg(F);
   long d = GF2E::degree();

   if (n == 1) {
      h = ConstTerm(F);
      return;
   }

   if (UseComposeFrobenius(d, n))
      ComposeFrobeniusMap(h, F);
   else
      PlainFrobeniusMap(h, F);
}



   



static
void RecFindRoots(vec_GF2E& x, const GF2EX& f)
{
   if (deg(f) == 0) return;

   if (deg(f) == 1) {
      long k = x.length();
      x.SetLength(k+1);
      x[k] = ConstTerm(f);
      return;
   }
      
   GF2EX h;

   GF2E r;

   
   {
      GF2EXModulus F;
      build(F, f);

      do {
         random(r);
         clear(h);
         SetCoeff(h, 1, r);
         TraceMap(h, h, F);
         GCD(h, h, f);
      } while (deg(h) <= 0 || deg(h) == deg(f));
   }

   RecFindRoots(x, h);
   div(h, f, h); 
   RecFindRoots(x, h);
}

void FindRoots(vec_GF2E& x, const GF2EX& ff)
{
   GF2EX f = ff;

   if (!IsOne(LeadCoeff(f)))
      Error("FindRoots: bad args");

   x.SetMaxLength(deg(f));
   x.SetLength(0);
   RecFindRoots(x, f);
}


static
void RandomBasisElt(GF2EX& g, const vec_long& D, const vec_GF2XVec& M)
{
   static GF2X t1, t2;

   long n = D.length();

   long i, j, s;

   g.rep.SetLength(n);

   vec_GF2E& v = g.rep;

   for (j = n-1; j >= 0; j--) {
      if (D[j] == -1)
         random(v[j]);
      else {
         i = D[j];

         // v[j] = sum_{s=j+1}^{n-1} v[s]*M[i,s]

         clear(t1);

         for (s = j+1; s < n; s++) {
            mul(t2, rep(v[s]), M[i][s]);
            add(t1, t1, t2);
         }

         conv(v[j], t1);
      }
   }

   g.normalize();
}



static
void split(GF2EX& f1, GF2EX& g1, GF2EX& f2, GF2EX& g2,
           const GF2EX& f, const GF2EX& g, 
           const vec_GF2E& roots, long lo, long mid)
{
   long r = mid-lo+1;

   GF2EXModulus F;
   build(F, f);

   vec_GF2E lroots(INIT_SIZE, r);
   long i;

   for (i = 0; i < r; i++)
      lroots[i] = roots[lo+i];


   GF2EX h, a, d;
   BuildFromRoots(h, lroots);
   CompMod(a, h, g, F);


   GCD(f1, a, f);
   
   div(f2, f, f1);

   rem(g1, g, f1);
   rem(g2, g, f2);
}

static
void RecFindFactors(vec_GF2EX& factors, const GF2EX& f, const GF2EX& g,
                    const vec_GF2E& roots, long lo, long hi)
{
   long r = hi-lo+1;

   if (r == 0) return;

   if (r == 1) {
      append(factors, f);
      return;
   }

   GF2EX f1, g1, f2, g2;

   long mid = (lo+hi)/2;

   split(f1, g1, f2, g2, f, g, roots, lo, mid);

   RecFindFactors(factors, f1, g1, roots, lo, mid);
   RecFindFactors(factors, f2, g2, roots, mid+1, hi);
}


static
void FindFactors(vec_GF2EX& factors, const GF2EX& f, const GF2EX& g,
                 const vec_GF2E& roots)
{
   long r = roots.length();

   factors.SetMaxLength(r);
   factors.SetLength(0);

   RecFindFactors(factors, f, g, roots, 0, r-1);
}

#if 0

static
void IterFindFactors(vec_GF2EX& factors, const GF2EX& f,
                     const GF2EX& g, const vec_GF2E& roots)
{
   long r = roots.length();
   long i;
   GF2EX h;

   factors.SetLength(r);

   for (i = 0; i < r; i++) {
      add(h, g, roots[i]);
      GCD(factors[i], f, h);
   }
}

#endif


   

void SFBerlekamp(vec_GF2EX& factors, const GF2EX& ff, long verbose)
{
   GF2EX f = ff;

   if (!IsOne(LeadCoeff(f)))
      Error("SFBerlekamp: bad args");

   if (deg(f) == 0) {
      factors.SetLength(0);
      return;
   }

   if (deg(f) == 1) {
      factors.SetLength(1);
      factors[0] = f;
      return;
   }

   double t;

   long n = deg(f);

   GF2EXModulus F;

   build(F, f);

   GF2EX g, h;

   FrobeniusMap(g, F);

   vec_long D;
   long r;

   vec_GF2XVec M;

   BuildMatrix(M, n, g, F, verbose);

   NullSpace(r, D, M, verbose);



   if (r == 1) {
      factors.SetLength(1);
      factors[0] = f;
      return;
   }


   vec_GF2E roots;

   RandomBasisElt(g, D, M);
   MinPolyMod(h, g, F, r);
   if (deg(h) == r) M.kill();
   FindRoots(roots, h);
   FindFactors(factors, f, g, roots);

   GF2EX g1;
   vec_GF2EX S, S1;
   long i;

   while (factors.length() < r) {
      RandomBasisElt(g, D, M);
      S.kill();
      for (i = 0; i < factors.length(); i++) {
         const GF2EX& f = factors[i];
         if (deg(f) == 1) {
            append(S, f);
            continue;
         }
         build(F, f);
         rem(g1, g, F);
         if (deg(g1) <= 0) {
            append(S, f);
            continue;
         }
         MinPolyMod(h, g1, F, min(deg(f), r-factors.length()+1));
         FindRoots(roots, h);
         S1.kill();
         FindFactors(S1, f, g1, roots);
         append(S, S1);
      }
      swap(factors, S);
   }


}


void berlekamp(vec_pair_GF2EX_long& factors, const GF2EX& f, long verbose)
{
   double t;
   vec_pair_GF2EX_long sfd;
   vec_GF2EX x;

   if (!IsOne(LeadCoeff(f)))
      Error("berlekamp: bad args");

   
   SquareFreeDecomp(sfd, f);

   factors.SetLength(0);

   long i, j;

   for (i = 0; i < sfd.length(); i++) {

      SFBerlekamp(x, sfd[i].a, verbose);

      for (j = 0; j < x.length(); j++)
         append(factors, cons(x[j], sfd[i].b));
   }
}



static
void AddFactor(vec_pair_GF2EX_long& factors, const GF2EX& g, long d, long verbose)
{
   append(factors, cons(g, d));
}

static
void ProcessTable(GF2EX& f, vec_pair_GF2EX_long& factors, 
                  const GF2EXModulus& F, long limit, const vec_GF2EX& tbl,
                  long d, long verbose)

{
   if (limit == 0) return;

   GF2EX t1;

   if (limit == 1) {
      GCD(t1, f, tbl[0]);
      if (deg(t1) > 0) {
         AddFactor(factors, t1, d, verbose);
         div(f, f, t1);
      }

      return;
   }

   long i;

   t1 = tbl[0];
   for (i = 1; i < limit; i++)
      MulMod(t1, t1, tbl[i], F);

   GCD(t1, f, t1);

   if (deg(t1) == 0) return;

   div(f, f, t1);

   GF2EX t2;

   i = 0;
   d = d - limit + 1;

   while (2*d <= deg(t1)) {
      GCD(t2, tbl[i], t1); 
      if (deg(t2) > 0) {
         AddFactor(factors, t2, d, verbose);
         div(t1, t1, t2);
      }

      i++;
      d++;
   }

   if (deg(t1) > 0)
      AddFactor(factors, t1, deg(t1), verbose);
}


void TraceMap(GF2EX& w, const GF2EX& a, long d, const GF2EXModulus& F, 
              const GF2EX& b)

{
   if (d < 0) Error("TraceMap: bad args");

   GF2EX y, z, t;

   z = b;
   y = a;
   clear(w);

   while (d) {
      if (d == 1) {
         if (IsZero(w)) 
            w = y;
         else {
            CompMod(w, w, z, F);
            add(w, w, y);
         }
      }
      else if ((d & 1) == 0) {
         Comp2Mod(z, t, z, y, z, F);
         add(y, t, y);
      }
      else if (IsZero(w)) {
         w = y;
         Comp2Mod(z, t, z, y, z, F);
         add(y, t, y);
      }
      else {
         Comp3Mod(z, t, w, z, y, w, z, F);
         add(w, w, y);
         add(y, t, y);
      }

      d = d >> 1;
   }
}


void PowerCompose(GF2EX& y, const GF2EX& h, long q, const GF2EXModulus& F)
{
   if (q < 0) Error("powerCompose: bad args");

   GF2EX z(INIT_SIZE, F.n);
   long sw;

   z = h;
   SetX(y);

   while (q) {
      sw = 0;

      if (q > 1) sw = 2;
      if (q & 1) {
         if (IsX(y))
            y = z;
         else
            sw = sw | 1;
      }

      switch (sw) {
      case 0:
         break;

      case 1:
         CompMod(y, y, z, F);
         break;

      case 2:
         CompMod(z, z, z, F);
         break;

      case 3:
         Comp2Mod(y, z, y, z, z, F);
         break;
      }

      q = q >> 1;
   }
}


long ProbIrredTest(const GF2EX& f, long iter)
{
   long n = deg(f);

   if (n <= 0) return 0;
   if (n == 1) return 1;

   GF2EXModulus F;

   build(F, f);

   GF2EX b, r, s;

   FrobeniusMap(b, F);

   long all_zero = 1;

   long i;

   for (i = 0; i < iter; i++) {
      random(r, n);
      TraceMap(s, r, n, F, b);

      all_zero = all_zero && IsZero(s);

      if (deg(s) > 0) return 0;
   }

   if (!all_zero || (n & 1)) return 1;

   PowerCompose(s, b, n/2, F);
   return !IsX(s);
}


long GF2EX_BlockingFactor = 10;

void DDF(vec_pair_GF2EX_long& factors, const GF2EX& ff, const GF2EX& hh, 
         long verbose)
{
   GF2EX f = ff;
   GF2EX h = hh;

   if (!IsOne(LeadCoeff(f)))
      Error("DDF: bad args");

   factors.SetLength(0);

   if (deg(f) == 0)
      return;

   if (deg(f) == 1) {
      AddFactor(factors, f, 1, verbose);
      return;
   }

   long CompTableSize = 2*SqrRoot(deg(f)); 

   long GCDTableSize = GF2EX_BlockingFactor;

   GF2EXModulus F;
   build(F, f);

   GF2EXArgument H;

   build(H, h, F, min(CompTableSize, deg(f)));

   long i, d, limit, old_n;
   GF2EX g, X;


   vec_GF2EX tbl(INIT_SIZE, GCDTableSize);

   SetX(X);

   i = 0;
   g = h;
   d = 1;
   limit = GCDTableSize;


   while (2*d <= deg(f)) {

      old_n = deg(f);
      add(tbl[i], g, X);
      i++;
      if (i == limit) {
         ProcessTable(f, factors, F, i, tbl, d, verbose);
         i = 0;
      }

      d = d + 1;
      if (2*d <= deg(f)) {
         // we need to go further

         if (deg(f) < old_n) {
            // f has changed 

            build(F, f);
            rem(h, h, f);
            rem(g, g, f);
            build(H, h, F, min(CompTableSize, deg(f)));
         }

         CompMod(g, g, H, F);
      }
   }

   ProcessTable(f, factors, F, i, tbl, d-1, verbose);

   if (!IsOne(f)) AddFactor(factors, f, deg(f), verbose);
}



void RootEDF(vec_GF2EX& factors, const GF2EX& f, long verbose)
{
   vec_GF2E roots;
   double t;

   FindRoots(roots, f);

   long r = roots.length();
   factors.SetLength(r);
   for (long j = 0; j < r; j++) {
      SetX(factors[j]);
      add(factors[j], factors[j], roots[j]);
   }
}

static
void EDFSplit(vec_GF2EX& v, const GF2EX& f, const GF2EX& b, long d)
{
   GF2EX a, g, h;
   GF2EXModulus F;
   vec_GF2E roots;
   
   build(F, f);
   long n = F.n;
   long r = n/d;
   random(a, n);
   TraceMap(g, a, d, F, b);
   MinPolyMod(h, g, F, r);
   FindRoots(roots, h);
   FindFactors(v, f, g, roots);
}

static
void RecEDF(vec_GF2EX& factors, const GF2EX& f, const GF2EX& b, long d,
            long verbose)
{
   vec_GF2EX v;
   long i;
   GF2EX bb;

   EDFSplit(v, f, b, d);
   for (i = 0; i < v.length(); i++) {
      if (deg(v[i]) == d) {
         append(factors, v[i]);
      }
      else {
         GF2EX bb;
         rem(bb, b, v[i]);
         RecEDF(factors, v[i], bb, d, verbose);
      }
   }
}
         

void EDF(vec_GF2EX& factors, const GF2EX& ff, const GF2EX& bb,
         long d, long verbose)

{
   GF2EX f = ff;
   GF2EX b = bb;

   if (!IsOne(LeadCoeff(f)))
      Error("EDF: bad args");

   long n = deg(f);
   long r = n/d;

   if (r == 0) {
      factors.SetLength(0);
      return;
   }

   if (r == 1) {
      factors.SetLength(1);
      factors[0] = f;
      return;
   }

   if (d == 1) {
      RootEDF(factors, f, verbose);
      return;
   }

   
   factors.SetLength(0);

   RecEDF(factors, f, b, d, verbose);

}


void SFCanZass(vec_GF2EX& factors, const GF2EX& ff, long verbose)
{
   GF2EX f = ff;

   if (!IsOne(LeadCoeff(f)))
      Error("SFCanZass: bad args");

   if (deg(f) == 0) {
      factors.SetLength(0);
      return;
   }

   if (deg(f) == 1) {
      factors.SetLength(1);
      factors[0] = f;
      return;
   }

   factors.SetLength(0);

   double t;

   
   GF2EXModulus F;
   build(F, f);

   GF2EX h;

   FrobeniusMap(h, F);

   vec_pair_GF2EX_long u;
   NewDDF(u, f, h, verbose);

   GF2EX hh;
   vec_GF2EX v;

   long i;
   for (i = 0; i < u.length(); i++) {
      const GF2EX& g = u[i].a;
      long d = u[i].b;
      long r = deg(g)/d;

      if (r == 1) {
         // g is already irreducible

         append(factors, g);
      }
      else {
         // must perform EDF

         if (d == 1) {
            // root finding
            RootEDF(v, g, verbose);
            append(factors, v);
         }
         else {
            // general case
            rem(hh, h, g);
            EDF(v, g, hh, d, verbose);
            append(factors, v);
         }
      }
   }
}
   
void CanZass(vec_pair_GF2EX_long& factors, const GF2EX& f, long verbose)
{
   if (!IsOne(LeadCoeff(f)))
      Error("CanZass: bad args");

   double t;
   vec_pair_GF2EX_long sfd;
   vec_GF2EX x;

   
   SquareFreeDecomp(sfd, f);

   factors.SetLength(0);

   long i, j;

   for (i = 0; i < sfd.length(); i++) {

      SFCanZass(x, sfd[i].a, verbose);

      for (j = 0; j < x.length(); j++)
         append(factors, cons(x[j], sfd[i].b));
   }
}

void mul(GF2EX& f, const vec_pair_GF2EX_long& v)
{
   long i, j, n;

   n = 0;
   for (i = 0; i < v.length(); i++)
      n += v[i].b*deg(v[i].a);

   GF2EX g(INIT_SIZE, n+1);

   set(g);
   for (i = 0; i < v.length(); i++)
      for (j = 0; j < v[i].b; j++) {
         mul(g, g, v[i].a);
      }

   f = g;
}




static
long BaseCase(const GF2EX& h, long q, long a, const GF2EXModulus& F)
{
   long b, e;
   GF2EX lh(INIT_SIZE, F.n);

   lh = h;
   b = 1;
   e = 0;
   while (e < a-1 && !IsX(lh)) {
      e++;
      b *= q;
      PowerCompose(lh, lh, q, F);
   }

   if (!IsX(lh)) b *= q;

   return b;
}



static
void TandemPowerCompose(GF2EX& y1, GF2EX& y2, const GF2EX& h, 
                        long q1, long q2, const GF2EXModulus& F)
{
   GF2EX z(INIT_SIZE, F.n);
   long sw;

   z = h;
   SetX(y1);
   SetX(y2);

   while (q1 || q2) {
      sw = 0;

      if (q1 > 1 || q2 > 1) sw = 4;

      if (q1 & 1) {
         if (IsX(y1))
            y1 = z;
         else
            sw = sw | 2;
      }

      if (q2 & 1) {
         if (IsX(y2))
            y2 = z;
         else
            sw = sw | 1;
      }

      switch (sw) {
      case 0:
         break;

      case 1:
         CompMod(y2, y2, z, F);
         break;

      case 2:
         CompMod(y1, y1, z, F);
         break;

      case 3:
         Comp2Mod(y1, y2, y1, y2, z, F);
         break;

      case 4:
         CompMod(z, z, z, F);
         break;

      case 5:
         Comp2Mod(z, y2, z, y2, z, F);
         break;

      case 6:
         Comp2Mod(z, y1, z, y1, z, F);
         break;

      case 7:
         Comp3Mod(z, y1, y2, z, y1, y2, z, F);
         break;
      }

      q1 = q1 >> 1;
      q2 = q2 >> 1;
   }
}



static
long RecComputeDegree(long u, const GF2EX& h, const GF2EXModulus& F,
                      FacVec& fvec)
{
   if (IsX(h)) return 1;

   if (fvec[u].link == -1) return BaseCase(h, fvec[u].q, fvec[u].a, F);

   GF2EX h1, h2;
   long q1, q2, r1, r2;

   q1 = fvec[fvec[u].link].val; 
   q2 = fvec[fvec[u].link+1].val;

   TandemPowerCompose(h1, h2, h, q1, q2, F);
   r1 = RecComputeDegree(fvec[u].link, h2, F, fvec);
   r2 = RecComputeDegree(fvec[u].link+1, h1, F, fvec);
   return r1*r2;
}

   


long RecComputeDegree(const GF2EX& h, const GF2EXModulus& F)
   // f = F.f is assumed to be an "equal degree" polynomial
   // h = X^p mod f
   // the common degree of the irreducible factors of f is computed
{
   if (F.n == 1 || IsX(h)) 
      return 1;

   FacVec fvec;

   FactorInt(fvec, F.n);

   return RecComputeDegree(fvec.length()-1, h, F, fvec);
}


void FindRoot(GF2E& root, const GF2EX& ff)
// finds a root of ff.
// assumes that ff is monic and splits into distinct linear factors

{
   GF2EXModulus F;
   GF2EX h, h1, f;
   GF2E r;

   f = ff;
   
   if (!IsOne(LeadCoeff(f)))
      Error("FindRoot: bad args");

   if (deg(f) == 0)
      Error("FindRoot: bad args");


   while (deg(f) > 1) {
      build(F, f);
      random(r);
      clear(h);
      SetCoeff(h, 1, r);
      TraceMap(h, h, F);
      GCD(h, h, f);
      if (deg(h) > 0 && deg(h) < deg(f)) {
         if (deg(h) > deg(f)/2)
            div(f, f, h);
         else
            f = h;
      }
   }
 
   root = ConstTerm(f);
}


static
long power(long a, long e)
{
   long i, res;

   res = 1;
   for (i = 1; i <= e; i++)
      res = res * a;

   return res;
}


static
long IrredBaseCase(const GF2EX& h, long q, long a, const GF2EXModulus& F)
{
   long e;
   GF2EX X, s, d;

   e = power(q, a-1);
   PowerCompose(s, h, e, F);
   SetX(X);
   add(s, s, X);
   GCD(d, F.f, s);
   return IsOne(d);
}


static
long RecIrredTest(long u, const GF2EX& h, const GF2EXModulus& F,
                 const FacVec& fvec)
{
   long  q1, q2;
   GF2EX h1, h2;

   if (IsX(h)) return 0;

   if (fvec[u].link == -1) {
      return IrredBaseCase(h, fvec[u].q, fvec[u].a, F);
   }


   q1 = fvec[fvec[u].link].val; 
   q2 = fvec[fvec[u].link+1].val;

   TandemPowerCompose(h1, h2, h, q1, q2, F);
   return RecIrredTest(fvec[u].link, h2, F, fvec) 
          && RecIrredTest(fvec[u].link+1, h1, F, fvec);
}

long DetIrredTest(const GF2EX& f)
{
   if (deg(f) <= 0) return 0;
   if (deg(f) == 1) return 1;

   GF2EXModulus F;

   build(F, f);
   
   GF2EX h;

   FrobeniusMap(h, F);

   GF2EX s;
   PowerCompose(s, h, F.n, F);
   if (!IsX(s)) return 0;

   FacVec fvec;

   FactorInt(fvec, F.n);

   return RecIrredTest(fvec.length()-1, h, F, fvec);
}



long IterIrredTest(const GF2EX& f)
{
   if (deg(f) <= 0) return 0;
   if (deg(f) == 1) return 1;

   GF2EXModulus F;

   build(F, f);
   
   GF2EX h;

   FrobeniusMap(h, F);

   long CompTableSize = 2*SqrRoot(deg(f));

   GF2EXArgument H;

   build(H, h, F, CompTableSize);

   long i, d, limit, limit_sqr;
   GF2EX g, X, t, prod;


   SetX(X);

   i = 0;
   g = h;
   d = 1;
   limit = 2;
   limit_sqr = limit*limit;

   set(prod);


   while (2*d <= deg(f)) {
      add(t, g, X);
      MulMod(prod, prod, t, F);
      i++;
      if (i == limit_sqr) {
         GCD(t, f, prod);
         if (!IsOne(t)) return 0;

         set(prod);
         limit++;
         limit_sqr = limit*limit;
         i = 0;
      }

      d = d + 1;
      if (2*d <= deg(f)) {
         CompMod(g, g, H, F);
      }
   }

   if (i > 0) {
      GCD(t, f, prod);
      if (!IsOne(t)) return 0;
   }

   return 1;
}

static
void MulByXPlusY(vec_GF2EX& h, const GF2EX& f, const GF2EX& g)
// h represents the bivariate polynomial h[0] + h[1]*Y + ... + h[n-1]*Y^k,
// where the h[i]'s are polynomials in X, each of degree < deg(f),
// and k < deg(g).
// h is replaced by the bivariate polynomial h*(X+Y) (mod f(X), g(Y)).

{
   long n = deg(g);
   long k = h.length()-1;

   if (k < 0) return;

   if (k < n-1) {
      h.SetLength(k+2);
      h[k+1] = h[k];
      for (long i = k; i >= 1; i--) {
         MulByXMod(h[i], h[i], f);
         add(h[i], h[i], h[i-1]);
      }
      MulByXMod(h[0], h[0], f);
   }
   else {
      GF2EX b, t;

      b = h[n-1];
      for (long i = n-1; i >= 1; i--) {
         mul(t, b, g.rep[i]);
         MulByXMod(h[i], h[i], f);
         add(h[i], h[i], h[i-1]);
         add(h[i], h[i], t);
      }
      mul(t, b, g.rep[0]);
      MulByXMod(h[0], h[0], f);
      add(h[0], h[0], t);
   }

   // normalize

   k = h.length()-1;
   while (k >= 0 && IsZero(h[k])) k--;
   h.SetLength(k+1);
}


static
void IrredCombine(GF2EX& x, const GF2EX& f, const GF2EX& g)
{
   if (deg(f) < deg(g)) {
      IrredCombine(x, g, f);
      return;
   }

   // deg(f) >= deg(g)...not necessary, but maybe a little more
   //                    time & space efficient

   long df = deg(f);
   long dg = deg(g);
   long m = df*dg;

   vec_GF2EX h(INIT_SIZE, dg);

   long i;
   for (i = 0; i < dg; i++) h[i].SetMaxLength(df);

   h.SetLength(1);
   set(h[0]);

   vec_GF2E a;

   a.SetLength(2*m);

   for (i = 0; i < 2*m; i++) {
      a[i] = ConstTerm(h[0]);
      if (i < 2*m-1)
         MulByXPlusY(h, f, g);
   }

   MinPolySeq(x, a, m);
}


static
void BuildPrimePowerIrred(GF2EX& f, long q, long e)
{
   long n = power(q, e);

   do {
      random(f, n);
      SetCoeff(f, n);
   } while (!IterIrredTest(f));
}

static
void RecBuildIrred(GF2EX& f, long u, const FacVec& fvec)
{
   if (fvec[u].link == -1)
      BuildPrimePowerIrred(f, fvec[u].q, fvec[u].a);
   else {
      GF2EX g, h;
      RecBuildIrred(g, fvec[u].link, fvec);
      RecBuildIrred(h, fvec[u].link+1, fvec);
      IrredCombine(f, g, h);
   }
}


void BuildIrred(GF2EX& f, long n)
{
   if (n <= 0)
      Error("BuildIrred: n must be positive");

   if (NTL_OVERFLOW(n, 1, 0))
      Error("overflow in BuildIrred");

   if (n == 1) {
      SetX(f);
      return;
   }

   FacVec fvec;

   FactorInt(fvec, n);

   RecBuildIrred(f, fvec.length()-1, fvec);
}



#if 0
void BuildIrred(GF2EX& f, long n)
{
   if (n <= 0)
      Error("BuildIrred: n must be positive");

   if (NTL_OVERFLOW(n, 1, 0))
      Error("overflow in BuildIrred");

   if (n == 1) {
      SetX(f);
      return;
   }

   GF2EX g;

   do {
      random(g, n);
      SetCoeff(g, n);
   } while (!IterIrredTest(g));

   f = g;

}
#endif



void BuildRandomIrred(GF2EX& f, const GF2EX& g)
{
   GF2EXModulus G;
   GF2EX h, ff;

   build(G, g);
   do {
      random(h, deg(g));
      IrredPolyMod(ff, h, G);
   } while (deg(ff) < deg(g));

   f = ff;
}


/************* NEW DDF ****************/

long GF2EX_GCDTableSize = 4;
char GF2EX_stem[256] = "";

double GF2EXFileThresh = 256;

static vec_GF2EX BabyStepFile;
static vec_GF2EX GiantStepFile;

static
double CalcTableSize(long n, long k)
{
   double sz = GF2E::storage();
   sz = sz * n;
   sz = sz + NTL_VECTOR_HEADER_SIZE + sizeof(vec_GF2E);
   sz = sz * k;
   sz = sz/1024;
   return sz;
}



static
void GenerateBabySteps(GF2EX& h1, const GF2EX& f, const GF2EX& h, long k,
                       long verbose)

{
   double t;

   GF2EXModulus F;
   build(F, f);

   GF2EXArgument H;

#if 0
   double n2 = sqrt(double(F.n));
   double n4 = sqrt(n2);
   double n34 = n2*n4;
   long sz = long(ceil(n34/sqrt(sqrt(2.0))));
#else
   long sz = 2*SqrRoot(F.n);
#endif

   build(H, h, F, sz);


   h1 = h;

   long i;

   long HexOutput = GF2X::HexOutput;
   GF2X::HexOutput = 1;

      BabyStepFile.kill();
      BabyStepFile.SetLength(k-1);

   for (i = 1; i <= k-1; i++) {
         BabyStepFile(i) = h1;

      CompMod(h1, h1, H, F);
   }

   GF2X::HexOutput = HexOutput;
}


static
void GenerateGiantSteps(const GF2EX& f, const GF2EX& h, long l, long verbose)
{

   double t;


   GF2EXModulus F;
   build(F, f);

   GF2EXArgument H;

#if 0
   double n2 = sqrt(double(F.n));
   double n4 = sqrt(n2);
   double n34 = n2*n4;
   long sz = long(ceil(n34/sqrt(sqrt(2.0))));
#else
   long sz = 2*SqrRoot(F.n);
#endif

   build(H, h, F, sz);

   GF2EX h1;

   h1 = h;

   long i;

   long HexOutput = GF2X::HexOutput; 
   GF2X::HexOutput = 1;

      GiantStepFile.kill();
      GiantStepFile.SetLength(l);

   for (i = 1; i <= l-1; i++) {
         GiantStepFile(i) = h1;

      CompMod(h1, h1, H, F);
   }

      GiantStepFile(i) = h1;

   GF2X::HexOutput = HexOutput;
}

static
void FileCleanup(long k, long l)
{
      BabyStepFile.kill();
      GiantStepFile.kill();
}


static
void NewAddFactor(vec_pair_GF2EX_long& u, const GF2EX& g, long m, long verbose)
{
   long len = u.length();

   u.SetLength(len+1);
   u[len].a = g;
   u[len].b = m;

}

   


static
void NewProcessTable(vec_pair_GF2EX_long& u, GF2EX& f, const GF2EXModulus& F,
                     vec_GF2EX& buf, long size, long StartInterval,
                     long IntervalLength, long verbose)

{
   if (size == 0) return;

   GF2EX& g = buf[size-1];

   long i;

   for (i = 0; i < size-1; i++)
      MulMod(g, g, buf[i], F);

   GCD(g, f, g);

   if (deg(g) == 0) return;

   div(f, f, g);

   long d = (StartInterval-1)*IntervalLength + 1;
   i = 0;
   long interval = StartInterval;

   while (i < size-1 && 2*d <= deg(g)) {
      GCD(buf[i], buf[i], g);
      if (deg(buf[i]) > 0) {
         NewAddFactor(u, buf[i], interval, verbose);
         div(g, g, buf[i]);
      }

      i++;
      interval++;
      d += IntervalLength;
   }

   if (deg(g) > 0) {
      if (i == size-1)
         NewAddFactor(u, g, interval, verbose);
      else
         NewAddFactor(u, g, (deg(g)+IntervalLength-1)/IntervalLength, verbose);
   }
}


static
void FetchGiantStep(GF2EX& g, long gs, const GF2EXModulus& F)
{
      g = GiantStepFile(gs);

   rem(g, g, F);
}


static
void FetchBabySteps(vec_GF2EX& v, long k)
{
   v.SetLength(k);

   SetX(v[0]);

   long i;
   for (i = 1; i <= k-1; i++) {
         v[i] = BabyStepFile(i);
   }
}
      


static
void GiantRefine(vec_pair_GF2EX_long& u, const GF2EX& ff, long k, long l,
                 long verbose)

{
   u.SetLength(0);

   vec_GF2EX BabyStep;

   FetchBabySteps(BabyStep, k);

   vec_GF2EX buf(INIT_SIZE, GF2EX_GCDTableSize);

   GF2EX f;
   f = ff;

   GF2EXModulus F;
   build(F, f);

   GF2EX g;
   GF2EX h;

   long size = 0;

   long first_gs;

   long d = 1;

   while (2*d <= deg(f)) {

      long old_n = deg(f);

      long gs = (d+k-1)/k;
      long bs = gs*k - d;

      if (bs == k-1) {
         size++;
         if (size == 1) first_gs = gs;
         FetchGiantStep(g, gs, F);
         add(buf[size-1], g, BabyStep[bs]);
      }
      else {
         add(h, g, BabyStep[bs]);
         MulMod(buf[size-1], buf[size-1], h, F);
      }

      if (size == GF2EX_GCDTableSize && bs == 0) {
         NewProcessTable(u, f, F, buf, size, first_gs, k, verbose);
         size = 0;
      }

      d++;

      if (2*d <= deg(f) && deg(f) < old_n) {
         build(F, f);

         long i;
         for (i = 1; i <= k-1; i++) 
            rem(BabyStep[i], BabyStep[i], F);
      }
   }

   if (size > 0) {
      NewProcessTable(u, f, F, buf, size, first_gs, k, verbose);
   }

   if (deg(f) > 0) 
      NewAddFactor(u, f, 0, verbose);

}


static
void IntervalRefine(vec_pair_GF2EX_long& factors, const GF2EX& ff,
                    long k, long gs, const vec_GF2EX& BabyStep, long verbose)

{
   vec_GF2EX buf(INIT_SIZE, GF2EX_GCDTableSize);

   GF2EX f;
   f = ff;

   GF2EXModulus F;
   build(F, f);

   GF2EX g;

   FetchGiantStep(g, gs, F);

   long size = 0;

   long first_d;

   long d = (gs-1)*k + 1;
   long bs = k-1;

   while (bs >= 0 && 2*d <= deg(f)) {

      long old_n = deg(f);

      if (size == 0) first_d = d;
      rem(buf[size], BabyStep[bs], F);
      add(buf[size], buf[size], g);
      size++;

      if (size == GF2EX_GCDTableSize) {
         NewProcessTable(factors, f, F, buf, size, first_d, 1, verbose);
         size = 0;
      }

      d++;
      bs--;

      if (bs >= 0 && 2*d <= deg(f) && deg(f) < old_n) {
         build(F, f);
         rem(g, g, F);
      }
   }

   NewProcessTable(factors, f, F, buf, size, first_d, 1, verbose);

   if (deg(f) > 0) 
      NewAddFactor(factors, f, deg(f), verbose);
}
   



static
void BabyRefine(vec_pair_GF2EX_long& factors, const vec_pair_GF2EX_long& u,
                long k, long l, long verbose)

{

   factors.SetLength(0);

   vec_GF2EX BabyStep;

   long i;
   for (i = 0; i < u.length(); i++) {
      const GF2EX& g = u[i].a;
      long gs = u[i].b;

      if (gs == 0 || 2*((gs-1)*k+1) > deg(g))
         NewAddFactor(factors, g, deg(g), verbose);
      else {
         if (BabyStep.length() == 0)
            FetchBabySteps(BabyStep, k);
         IntervalRefine(factors, g, k, gs, BabyStep, verbose);
      }
   }

}

      
      

      

void NewDDF(vec_pair_GF2EX_long& factors,
            const GF2EX& f,
            const GF2EX& h,
            long verbose)

{
   if (!IsOne(LeadCoeff(f)))
      Error("NewDDF: bad args");

   if (deg(f) == 0) {
      factors.SetLength(0);
      return;
   }

   if (deg(f) == 1) {
      factors.SetLength(0);
      append(factors, cons(f, 1));
      return;
   }

   if (!GF2EX_stem[0])
      sprintf(GF2EX_stem, "ddf-%ld", RandomBnd(10000));
      
   long B = deg(f)/2;
   long k = SqrRoot(B);
   long l = (B+k-1)/k;

   GF2EX h1;

   GenerateBabySteps(h1, f, h, k, verbose);

   GenerateGiantSteps(f, h1, l, verbose);

   vec_pair_GF2EX_long u;
   GiantRefine(u, f, k, l, verbose);
   BabyRefine(factors, u, k, l, verbose);

   FileCleanup(k, l);
}

long IterComputeDegree(const GF2EX& h, const GF2EXModulus& F)
{
   long n = deg(F);

   if (n == 1 || IsX(h)) return 1;

   long B = n/2;
   long k = SqrRoot(B);
   long l = (B+k-1)/k;


   GF2EXArgument H;

#if 0
   double n2 = sqrt(double(n));
   double n4 = sqrt(n2);
   double n34 = n2*n4;
   long sz = long(ceil(n34/sqrt(sqrt(2.0))));
#else
   long sz = 2*SqrRoot(F.n);
#endif

   build(H, h, F, sz);

   GF2EX h1;
   h1 = h;

   vec_GF2EX baby;
   baby.SetLength(k);

   SetX(baby[0]);

   long i;

   for (i = 1; i <= k-1; i++) {
      baby[i] = h1;
      CompMod(h1, h1, H, F);
      if (IsX(h1)) return i+1;
   }

   build(H, h1, F, sz);

   long j;

   for (j = 2; j <= l; j++) {
      CompMod(h1, h1, H, F);

      for (i = k-1; i >= 0; i--) {
         if (h1 == baby[i])
            return j*k-i;
      }
   }

   return n;
}

NTL_END_IMPL