lzz_pexfactoring.cpp

数值算法库for Windows

#include <NTL/lzz_pEXFactoring.h>
#include <NTL/FacVec.h>
#include <NTL/fileio.h>

#include <stdio.h>

#include <NTL/new.h>

NTL_START_IMPL



static
void IterPower(zz_pE& c, const zz_pE& a, long n)
{
   zz_pE res;

   long i;

   res = a;

   for (i = 0; i < n; i++)
      power(res, res, zz_p::modulus());

   c = res;
}
   


void SquareFreeDecomp(vec_pair_zz_pEX_long& u, const zz_pEX& ff)
{
   zz_pEX f = ff;

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

   zz_pEX 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 p-th power */

         long k, d;
         long p = to_long(zz_p::modulus()); 

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


static
void AbsTraceMap(zz_pEX& h, const zz_pEX& a, const zz_pEXModulus& F)
{
   zz_pEX res, tmp;

   long k = NumBits(zz_pE::cardinality())-1;

   res = a;
   tmp = a;

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

   h = res;
}

void FrobeniusMap(zz_pEX& h, const zz_pEXModulus& F)
{
   PowerXMod(h, zz_pE::cardinality(), F);
}


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

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

   zz_pEX r;

   
   {
      zz_pEXModulus F;
      build(F, f);

      do {
         random(r, deg(F));
         if (IsOdd(zz_pE::cardinality())) {
            PowerMod(h, r, RightShift(zz_pE::cardinality(), 1), F);
            sub(h, h, 1);
         }
         else {
            AbsTraceMap(h, r, 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_zz_pE& x, const zz_pEX& ff)
{
   zz_pEX f = ff;

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

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

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

   zz_pEXModulus F;
   build(F, f);

   vec_zz_pE lroots(INIT_SIZE, r);
   long i;

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


   zz_pEX 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);
}

void RecFindFactors(vec_zz_pEX& factors, const zz_pEX& f, const zz_pEX& g,
                    const vec_zz_pE& roots, long lo, long hi)
{
   long r = hi-lo+1;

   if (r == 0) return;

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

   zz_pEX 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);
}


void FindFactors(vec_zz_pEX& factors, const zz_pEX& f, const zz_pEX& g,
                 const vec_zz_pE& roots)
{
   long r = roots.length();

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

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

void IterFindFactors(vec_zz_pEX& factors, const zz_pEX& f,
                     const zz_pEX& g, const vec_zz_pE& roots)
{
   long r = roots.length();
   long i;
   zz_pEX h;

   factors.SetLength(r);

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


void TraceMap(zz_pEX& w, const zz_pEX& a, long d, const zz_pEXModulus& F, 
              const zz_pEX& b)

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

   zz_pEX 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(zz_pEX& y, const zz_pEX& h, long q, const zz_pEXModulus& F)
{
   if (q < 0) Error("PowerCompose: bad args");

   zz_pEX 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 zz_pEX& f, long iter)
{
   long n = deg(f);

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

   zz_pEXModulus F;

   build(F, f);

   zz_pEX 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 zz_pEX_BlockingFactor = 10;




void RootEDF(vec_zz_pEX& factors, const zz_pEX& f, long verbose)
{
   vec_zz_pE roots;
   double t;

   if (verbose) { cerr << "finding roots..."; t = GetTime(); }
   FindRoots(roots, f);
   if (verbose) { cerr << (GetTime()-t) << "\n"; }

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

void EDFSplit(vec_zz_pEX& v, const zz_pEX& f, const zz_pEX& b, long d)
{
   zz_pEX a, g, h;
   zz_pEXModulus F;
   vec_zz_pE 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);
}

void RecEDF(vec_zz_pEX& factors, const zz_pEX& f, const zz_pEX& b, long d,
            long verbose)
{
   vec_zz_pEX v;
   long i;
   zz_pEX bb;

   if (verbose) cerr << "+";

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

void EDF(vec_zz_pEX& factors, const zz_pEX& ff, const zz_pEX& bb,
         long d, long verbose)

{
   zz_pEX f = ff;
   zz_pEX 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;
   }

   
   double t;
   if (verbose) { 
      cerr << "computing EDF(" << d << "," << r << ")..."; 
      t = GetTime(); 
   }

   factors.SetLength(0);

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

   if (verbose) cerr << (GetTime()-t) << "\n";
}


void SFCanZass(vec_zz_pEX& factors, const zz_pEX& ff, long verbose)
{
   zz_pEX 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;

   
   zz_pEXModulus F;
   build(F, f);

   zz_pEX h;

   if (verbose) { cerr << "computing X^p..."; t = GetTime(); }
   FrobeniusMap(h, F);
   if (verbose) { cerr << (GetTime()-t) << "\n"; }

   vec_pair_zz_pEX_long u;
   if (verbose) { cerr << "computing DDF..."; t = GetTime(); }
   NewDDF(u, f, h, verbose);
   if (verbose) { 
      t = GetTime()-t; 
      cerr << "DDF time: " << t << "\n";
   }

   zz_pEX hh;
   vec_zz_pEX v;

   long i;
   for (i = 0; i < u.length(); i++) {
      const zz_pEX& 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_zz_pEX_long& factors, const zz_pEX& f, long verbose)
{
   if (!IsOne(LeadCoeff(f)))
      Error("CanZass: bad args");

   double t;
   vec_pair_zz_pEX_long sfd;
   vec_zz_pEX x;

   
   if (verbose) { cerr << "square-free decomposition..."; t = GetTime(); }
   SquareFreeDecomp(sfd, f);
   if (verbose) cerr << (GetTime()-t) << "\n";

   factors.SetLength(0);

   long i, j;

   for (i = 0; i < sfd.length(); i++) {
      if (verbose) {
         cerr << "factoring multiplicity " << sfd[i].b 
              << ", deg = " << deg(sfd[i].a) << "\n";
      }

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

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

void mul(zz_pEX& f, const vec_pair_zz_pEX_long& v)
{
   long i, j, n;

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

   zz_pEX 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;
}


long BaseCase(const zz_pEX& h, long q, long a, const zz_pEXModulus& F)
{
   long b, e;
   zz_pEX 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;
}



void TandemPowerCompose(zz_pEX& y1, zz_pEX& y2, const zz_pEX& h, 
                        long q1, long q2, const zz_pEXModulus& F)
{
   zz_pEX 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;
   }
}


long RecComputeDegree(long u, const zz_pEX& h, const zz_pEXModulus& F,
                      FacVec& fvec)
{
   if (IsX(h)) return 1;

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

   zz_pEX 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 zz_pEX& h, const zz_pEXModulus& 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(zz_pE& root, const zz_pEX& ff)
// finds a root of ff.
// assumes that ff is monic and splits into distinct linear factors

{
   zz_pEXModulus F;
   zz_pEX h, h1, f;
   zz_pEX 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, deg(F));
      if (IsOdd(zz_pE::cardinality())) {
         PowerMod(h, r, RightShift(zz_pE::cardinality(), 1), F);
         sub(h, h, 1);
      }
      else {
         AbsTraceMap(h, r, 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;
      }
   }
 
   negate(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 zz_pEX& h, long q, long a, const zz_pEXModulus& F)
{
   long e;
   zz_pEX X, s, d;

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


static
long RecIrredTest(long u, const zz_pEX& h, const zz_pEXModulus& F,
                 const FacVec& fvec)
{
   long  q1, q2;
   zz_pEX 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 zz_pEX& f)
{
   if (deg(f) <= 0) return 0;