lzz_pxfactoring.cpp

数值算法库for Windows

   zz_pX h;

   PowerXMod(h, zz_p::modulus(), F);

   zz_pX 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 zz_pX& f)
{
   if (deg(f) <= 0) return 0;
   if (deg(f) == 1) return 1;

   zz_pXModulus F;

   build(F, f);
   
   zz_pX h;

   PowerXMod(h, zz_p::modulus(), F);

   long rootn = SqrRoot(deg(f));

   long CompTableSize = 2*rootn;

   zz_pXArgument H;

   long UseModComp = 1;

   if (NumBits(zz_p::modulus()) < rootn/2)
      UseModComp = 0;

   if (UseModComp) build(H, h, F, CompTableSize);

   long i, d, limit, limit_sqr;
   zz_pX 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)) {
      sub(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)) {
         if (UseModComp)
            CompMod(g, g, H, F);
         else
            PowerMod(g, g, zz_p::modulus(), F);
      }
   }

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

   return 1;
}


static
void MulByXPlusY(vec_zz_pX& h, const zz_pX& f, const zz_pX& 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 {
      zz_pX 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]);
         sub(h[i], h[i], t);
      }
      mul(t, b, g.rep[0]);
      MulByXMod(h[0], h[0], f);
      sub(h[0], h[0], t);
   }

   // normalize

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



static
void IrredCombine(zz_pX& x, const zz_pX& f, const zz_pX& 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_zz_pX h(INIT_SIZE, dg);

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

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

   vec_zz_p 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(zz_pX& f, long q, long e)
{
   long n = power(q, e);

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

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


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

   if (n >= (1L << (NTL_BITS_PER_LONG-4))) Error("overflow in BuildIrred");

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

   FacVec fvec;

   FactorInt(fvec, n);

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



void BuildRandomIrred(zz_pX& f, const zz_pX& g)
{
   zz_pXModulus G;
   zz_pX h, ff;

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

   f = ff;
}


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

long zz_pX_GCDTableSize = 4;

static vec_zz_pX *BabyStepFile = 0;
static vec_zz_pX *GiantStepFile = 0;
static zz_pXArgument *HHH = 0;
static long OldN = 0;


static
void GenerateBabySteps(zz_pX& h1, const zz_pX& f, const zz_pX& h, long k,
                       long verbose)
{
   double t;

   if (verbose) { cerr << "generating baby steps..."; t = GetTime(); }

   zz_pXModulus F;
   build(F, f);


   BabyStepFile = NTL_NEW_OP vec_zz_pX;
   (*BabyStepFile).SetLength(k-1);

   h1 = h;

   long i;

   long rootn = SqrRoot(F.n);

   if (NumBits(zz_p::modulus()) < rootn/2) {
      for (i = 1; i <= k-1; i++) {
         (*BabyStepFile)(i) = h1;

         PowerMod(h1, h1, zz_p::modulus(), F);
         if (verbose) cerr << "+";
      }
   }
   else {
      zz_pXArgument H;
      build(H, h, F, 2*rootn);
   
   
      for (i = 1; i <= k-1; i++) {
         (*BabyStepFile)(i) = h1; 
   
         CompMod(h1, h1, H, F);
         if (verbose) cerr << "+";
      }
   }
   
   if (verbose)
      cerr << (GetTime()-t) << "\n";
}



static
void GenerateGiantSteps(const zz_pX& f, const zz_pX& h, long l, long verbose)
{
   zz_pXModulus F;

   build(F, f);

   HHH = NTL_NEW_OP zz_pXArgument;
   build(*HHH, h, F, 2*SqrRoot(F.n));

   OldN = F.n;

   GiantStepFile = NTL_NEW_OP vec_zz_pX;
   (*GiantStepFile).SetLength(1);
   (*GiantStepFile)(1) = h;
}


static
void FileCleanup(long k, long l)
{
   delete BabyStepFile;
   delete GiantStepFile;
   delete HHH;
}


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

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

   if (verbose) {
      cerr << "split " << m << " " << deg(g) << "\n";
   }
}

   


static
void NewProcessTable(vec_pair_zz_pX_long& u, zz_pX& f, const zz_pXModulus& F,
                     vec_zz_pX& buf, long size, long StartInterval,
                     long IntervalLength, long verbose)

{
   if (size == 0) return;

   zz_pX& 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(zz_pX& g, long gs, const zz_pXModulus& F)
{
   long l = (*GiantStepFile).length();
   zz_pX last;

   if (gs > l+1)
      Error("bad arg to FetchGiantStep");

   if (gs == l+1) {
      last = (*GiantStepFile)(l);
      if (F.n < OldN) {
         rem(last, last, F);
         for (long i = 0; i < (*HHH).H.length(); i++)
            rem((*HHH).H[i], (*HHH).H[i], F);
         OldN = F.n;
      }

      (*GiantStepFile).SetLength(l+1);
      CompMod((*GiantStepFile)(l+1), last, *HHH, F);
      g = (*GiantStepFile)(l+1);
   }
   else if (deg((*GiantStepFile)(gs)) >= F.n)
      rem(g, (*GiantStepFile)(gs), F);
   else
      g = (*GiantStepFile)(gs);
}


static
void FetchBabySteps(vec_zz_pX& 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_zz_pX_long& u, const zz_pX& ff, long k, long l,
                 long verbose)

{
   double t;

   if (verbose) {
      cerr << "giant refine...";
      t = GetTime();
   }

   u.SetLength(0);

   vec_zz_pX BabyStep;

   FetchBabySteps(BabyStep, k);

   vec_zz_pX buf(INIT_SIZE, zz_pX_GCDTableSize);

   zz_pX f;
   f = ff;

   zz_pXModulus F;
   build(F, f);

   zz_pX g;
   zz_pX 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);
         sub(buf[size-1], g, BabyStep[bs]);
      }
      else {
         sub(h, g, BabyStep[bs]);
         MulMod(buf[size-1], buf[size-1], h, F);
      }

      if (verbose && bs == 0) cerr << "+";

      if (size == zz_pX_GCDTableSize && bs == 0) {
         NewProcessTable(u, f, F, buf, size, first_gs, k, verbose);
         if (verbose) cerr << "*";
         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 (verbose) cerr << "*";
   }

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

   if (verbose) {
      t = GetTime()-t;
      cerr << "giant refine time: " << t << "\n";
   }
}


static
void IntervalRefine(vec_pair_zz_pX_long& factors, const zz_pX& ff,
                    long k, long gs, const vec_zz_pX& BabyStep, long verbose)

{
   vec_zz_pX buf(INIT_SIZE, zz_pX_GCDTableSize);

   zz_pX f;
   f = ff;

   zz_pXModulus F;
   build(F, f);

   zz_pX 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);
      sub(buf[size], buf[size], g);
      size++;

      if (size == zz_pX_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_zz_pX_long& factors, const vec_pair_zz_pX_long& u,
                long k, long l, long verbose)

{
   double t;

   if (verbose) {
      cerr << "baby refine...";
      t = GetTime();
   }

   factors.SetLength(0);

   vec_zz_pX BabyStep;

   long i;
   for (i = 0; i < u.length(); i++) {
      const zz_pX& 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);
      }
   }

   if (verbose) {
      t = GetTime()-t;
      cerr << "baby refine time: " << t << "\n";
   }
}

      

      

void NewDDF(vec_pair_zz_pX_long& factors,
            const zz_pX& f,
            const zz_pX& 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;
   }

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

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

   GenerateGiantSteps(f, h1, l, verbose);

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

   FileCleanup(k, l);
}

NTL_END_IMPL