~/ntl-11.4.2/doc/ZZ_pXFactoring.cpp.html

    /**************************************************************************\

    MODULE: ZZ_pXFactoring

    SUMMARY:

    Routines are provided for factorization of polynomials over ZZ_p, as
    well as routines for related problems such as testing irreducibility
    and constructing irreducible polynomials of given degree.

    \**************************************************************************/

    #include <NTL/ZZ_pX.h>
    #include <NTL/pair_ZZ_pX_long.h>

    void SquareFreeDecomp(vec_pair_ZZ_pX_long& u, const ZZ_pX& f);
    vec_pair_ZZ_pX_long SquareFreeDecomp(const ZZ_pX& f);

    // Performs square-free decomposition.  f must be monic.  If f =
    // prod_i g_i^i, then u is set to a lest 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_p& x, const ZZ_pX& f);
    vec_ZZ_p FindRoots(const ZZ_pX& f);

    // f is monic, and has deg(f) distinct roots.  returns the list of
    // roots

    void FindRoot(ZZ_p& root, const ZZ_pX& f);
    ZZ_p FindRoot(const ZZ_pX& f);

    // finds a single root of f.  assumes that f is monic and splits into
    // distinct linear factors

    void SFBerlekamp(vec_ZZ_pX& factors, const ZZ_pX& f, long verbose=0);
    vec_ZZ_pX  SFBerlekamp(const ZZ_pX& f, long verbose=0);

    // Assumes f is square-free and monic.  returns list of factors of f.
    // Uses "Berlekamp" approach, as described in detail in [Shoup,
    // J. Symbolic Comp. 20:363-397, 1995].

    void berlekamp(vec_pair_ZZ_pX_long& factors, const ZZ_pX& f,
                   long verbose=0);

    vec_pair_ZZ_pX_long berlekamp(const ZZ_pX& f, long verbose=0);

    // returns a list of factors, with multiplicities.  f must be monic.
    // Calls SFBerlekamp.

    void NewDDF(vec_pair_ZZ_pX_long& factors, const ZZ_pX& f, const ZZ_pX& h,
             long verbose=0);

    vec_pair_ZZ_pX_long NewDDF(const ZZ_pX& f, const ZZ_pX& 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^p 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 tmp-*.
    // The definition of "large" is controlled by the variable

          extern thread_local double ZZ_pXFileThresh

    // which can be set by the user.  If the sizes of the tables
    // exceeds ZZ_pXFileThresh KB, external files are used.
    // Initial value is NTL_FILE_THRESH (defined in tools.h).

    void EDF(vec_ZZ_pX& factors, const ZZ_pX& f, const ZZ_pX& h,
             long d, long verbose=0);

    vec_ZZ_pX EDF(const ZZ_pX& f, const ZZ_pX& h,
             long d, long verbose=0);

    // Performs equal-degree factorization.  f is monic, square-free, and
    // all irreducible factors have same degree.  h = X^p 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_pX& factors, const ZZ_pX& f, long verbose=0);
    vec_ZZ_pX RootEDF(const ZZ_pX& f, long verbose=0);

    // EDF for d==1

    void SFCanZass(vec_ZZ_pX& factors, const ZZ_pX& f, long verbose=0);
    vec_ZZ_pX SFCanZass(const ZZ_pX& 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_pX_long& factors, const ZZ_pX& f,
                 long verbose=0);

    vec_pair_ZZ_pX_long CanZass(const ZZ_pX& f, long verbose=0);

    // returns a list of factors, with multiplicities.  f must be monic.
    // Calls SquareFreeDecomp and SFCanZass.

    // NOTE: In most situations, you should use the CanZass factoring
    // routine, rather than Berlekamp: it is faster and uses less space.

    void mul(ZZ_pX& f, const vec_pair_ZZ_pX_long& v);
    ZZ_pX mul(const vec_pair_ZZ_pX_long& v);

    // multiplies polynomials, with multiplicities

    /**************************************************************************\

                                Irreducible Polynomials

    \**************************************************************************/

    long ProbIrredTest(const ZZ_pX& 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
    // p^{-iter}.  This implements an algorithm from [Shoup, J. Symbolic
    // Comp. 17:371-391, 1994].

    long DetIrredTest(const ZZ_pX& 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_pX& f);

    // performs an iterative deterministic irreducibility test, based on
    // DDF.  Fast on average (when f has a small factor).

    void BuildIrred(ZZ_pX& f, long n);
    ZZ_pX BuildIrred_ZZ_pX(long n);

    // Build a monic irreducible poly of degree n.

    void BuildRandomIrred(ZZ_pX& f, const ZZ_pX& g);
    ZZ_pX BuildRandomIrred(const ZZ_pX& g);

    // g is a monic irreducible polynomial.  Constructs a random monic
    // irreducible polynomial f of the same degree.

    long ComputeDegree(const ZZ_pX& h, const ZZ_pXModulus& 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 This
    // routine is useful in counting points on elliptic curves

    long ProbComputeDegree(const ZZ_pX& h, const ZZ_pXModulus& F);

    // Same as above, but uses a slightly faster probabilistic algorithm.
    // The return value may be 0 or may be too big, but for large p
    // (relative to n), this happens with very low probability.

    void TraceMap(ZZ_pX& w, const ZZ_pX& a, long d, const ZZ_pXModulus& F,
                  const ZZ_pX& h);

    ZZ_pX TraceMap(const ZZ_pX& a, long d, const ZZ_pXModulus& F,
                  const ZZ_pX& h);

    // 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 p.  This routine implements an algorithm
    // from [von zur Gathen and Shoup, Computational Complexity 2:187-224,
    // 1992].

    void PowerCompose(ZZ_pX& w, const ZZ_pX& h, long d, const ZZ_pXModulus& F);

    ZZ_pX PowerCompose(const ZZ_pX& h, long d, const ZZ_pXModulus& F);

    // w = X^{q^d} mod f; it is assumed that d >= 0, and h = X^q mod f, q
    // a power of p.  This routine implements an algorithm from [von zur
    // Gathen and Shoup, Computational Complexity 2:187-224, 1992]