/**************************************************************************\MODULE: ZZ_pEXFactoringSUMMARY:Routines are provided for factorization of polynomials over ZZ_pE, aswell as routines for related problems such as testing irreducibilityand constructing irreducible polynomials of given degree.\**************************************************************************/#include <NTL/ZZ_pEX.h>#include <NTL/pair_ZZ_pEX_long.h>void SquareFreeDecomp(vec_pair_ZZ_pEX_long& u, const ZZ_pEX& f);vec_pair_ZZ_pEX_long SquareFreeDecomp(const ZZ_pEX& f);// Performs square-free decomposition.  f must be monic.  If f =// prod_i g_i^i, then u is set to a list of pairs (g_i, i).  The list// is is increasing order of i, with trivial terms (i.e., g_i = 1)// deleted.void FindRoots(vec_ZZ_pE& x, const ZZ_pEX& f);vec_ZZ_pE FindRoots(const ZZ_pEX& f);// f is monic, and has deg(f) distinct roots.  returns the list of// rootsvoid FindRoot(ZZ_pE& root, const ZZ_pEX& f);ZZ_pE FindRoot(const ZZ_pEX& f);// finds a single root of f.  assumes that f is monic and splits into// distinct linear factorsvoid NewDDF(vec_pair_ZZ_pEX_long& factors, const ZZ_pEX& f,             const ZZ_pEX& h, long verbose=0);vec_pair_ZZ_pEX_long NewDDF(const ZZ_pEX& f, const ZZ_pEX& h,         long verbose=0);// This computes a distinct-degree factorization.  The input must be// monic and square-free.  factors is set to a list of pairs (g, d),// where g is the product of all irreducible factors of f of degree d.// Only nontrivial pairs (i.e., g != 1) are included.  The polynomial// h is assumed to be equal to X^{ZZ_pE::cardinality()} mod f.  // This routine implements the baby step/giant step algorithm// of [Kaltofen and Shoup, STOC 1995].// further described in [Shoup, J. Symbolic Comp. 20:363-397, 1995].// NOTE: When factoring "large" polynomials,// this routine uses external files to store some intermediate// results, which are removed if the routine terminates normally.// These files are stored in the current directory under names of the// form ddf-*-baby-* and ddf-*-giant-*.// The definition of "large" is controlled by the variable      extern double ZZ_pEXFileThresh// which can be set by the user.  If the sizes of the tables// exceeds ZZ_pEXFileThresh KB, external files are used.// Initial value is 256.void EDF(vec_ZZ_pEX& factors, const ZZ_pEX& f, const ZZ_pEX& h,         long d, long verbose=0);vec_ZZ_pEX EDF(const ZZ_pEX& f, const ZZ_pEX& h,         long d, long verbose=0);// Performs equal-degree factorization.  f is monic, square-free, and// all irreducible factors have same degree.  h = X^{ZZ_pE::cardinality()} mod// f.  d = degree of irreducible factors of f.  This routine// implements the algorithm of [von zur Gathen and Shoup,// Computational Complexity 2:187-224, 1992]void RootEDF(vec_ZZ_pEX& factors, const ZZ_pEX& f, long verbose=0);vec_ZZ_pEX RootEDF(const ZZ_pEX& f, long verbose=0);// EDF for d==1void SFCanZass(vec_ZZ_pEX& factors, const ZZ_pEX& f, long verbose=0);vec_ZZ_pEX SFCanZass(const ZZ_pEX& f, long verbose=0);// Assumes f is monic and square-free.  returns list of factors of f.// Uses "Cantor/Zassenhaus" approach, using the routines NewDDF and// EDF above.void CanZass(vec_pair_ZZ_pEX_long& factors, const ZZ_pEX& f,              long verbose=0);vec_pair_ZZ_pEX_long CanZass(const ZZ_pEX& f, long verbose=0);// returns a list of factors, with multiplicities.  f must be monic.// Calls SquareFreeDecomp and SFCanZass.// NOTE: these routines use modular composition.  The space// used for the required tables can be controlled by the variable// ZZ_pEXArgBound (see ZZ_pEX.txt).void mul(ZZ_pEX& f, const vec_pair_ZZ_pEX_long& v);ZZ_pEX mul(const vec_pair_ZZ_pEX_long& v);// multiplies polynomials, with multiplicities/**************************************************************************\                            Irreducible Polynomials\**************************************************************************/long ProbIrredTest(const ZZ_pEX& f, long iter=1);// performs a fast, probabilistic irreduciblity test.  The test can// err only if f is reducible, and the error probability is bounded by// ZZ_pE::cardinality()^{-iter}.  This implements an algorithm from [Shoup,// J. Symbolic Comp. 17:371-391, 1994].long DetIrredTest(const ZZ_pEX& f);// performs a recursive deterministic irreducibility test.  Fast in// the worst-case (when input is irreducible).  This implements an// algorithm from [Shoup, J. Symbolic Comp. 17:371-391, 1994].long IterIrredTest(const ZZ_pEX& f);// performs an iterative deterministic irreducibility test, based on// DDF.  Fast on average (when f has a small factor).void BuildIrred(ZZ_pEX& f, long n);ZZ_pEX BuildIrred_ZZ_pEX(long n);// Build a monic irreducible poly of degree n. void BuildRandomIrred(ZZ_pEX& f, const ZZ_pEX& g);ZZ_pEX BuildRandomIrred(const ZZ_pEX& g);// g is a monic irreducible polynomial.  Constructs a random monic// irreducible polynomial f of the same degree.long IterComputeDegree(const ZZ_pEX& h, const ZZ_pEXModulus& F);// f is assumed to be an "equal degree" polynomial, and h =// X^{ZZ_pE::cardinality()} mod f.  The common degree of the irreducible // factors of f is computed.  Uses a "baby step/giant step" algorithm, similar// to NewDDF.  Although asymptotocally slower than RecComputeDegree// (below), it is faster for reasonably sized inputs.long RecComputeDegree(const ZZ_pEX& h, const ZZ_pEXModulus& F);// f is assumed to be an "equal degree" polynomial, // h = X^{ZZ_pE::cardinality()} mod f.  // The common degree of the irreducible factors of f is// computed Uses a recursive algorithm similar to DetIrredTest.void TraceMap(ZZ_pEX& w, const ZZ_pEX& a, long d, const ZZ_pEXModulus& F,              const ZZ_pEX& h);ZZ_pEX TraceMap(const ZZ_pEX& a, long d, const ZZ_pEXModulus& F,              const ZZ_pEX& h);// Computes w = a+a^q+...+^{q^{d-1}} mod f; it is assumed that d >= 0,// and h = X^q mod f, q a power of ZZ_pE::cardinality().  This routine// implements an algorithm from [von zur Gathen and Shoup,// Computational Complexity 2:187-224, 1992]void PowerCompose(ZZ_pEX& w, const ZZ_pEX& h, long d, const ZZ_pEXModulus& F);ZZ_pEX PowerCompose(const ZZ_pEX& h, long d, const ZZ_pEXModulus& F);// Computes w = X^{q^d} mod f; it is assumed that d >= 0, and h = X^q// mod f, q a power of ZZ_pE::cardinality().  This routine implements an// algorithm from [von zur Gathen and Shoup, Computational Complexity// 2:187-224, 1992]