/Volumes/Unix/unix-files.noindex/ntl-new/ntl-9.6.0/doc/GF2EXFactoring.cpp.html

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

    MODULE: GF2EXFactoring

    SUMMARY:

    Routines are provided for factorization of polynomials over GF2E, as
    well as routines for related problems such as testing irreducibility
    and constructing irreducible polynomials of given degree.

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

    #include <NTL/GF2EX.h>
    #include <NTL/pair_GF2EX_long.h>

    void SquareFreeDecomp(vec_pair_GF2EX_long& u, const GF2EX& f);
    vec_pair_GF2EX_long SquareFreeDecomp(const GF2EX& 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_GF2E& x, const GF2EX& f);
    vec_GF2E FindRoots(const GF2EX& f);

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

    void FindRoot(GF2E& root, const GF2EX& f);
    GF2E FindRoot(const GF2EX& f);

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

    void SFBerlekamp(vec_GF2EX& factors, const GF2EX& f, long verbose=0);
    vec_GF2EX  SFBerlekamp(const GF2EX& 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_GF2EX_long& factors, const GF2EX& f,
                   long verbose=0);

    vec_pair_GF2EX_long berlekamp(const GF2EX& f, long verbose=0);

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

    void NewDDF(vec_pair_GF2EX_long& factors, const GF2EX& f, const GF2EX& h,
             long verbose=0);

    vec_pair_GF2EX_long NewDDF(const GF2EX& f, const GF2EX& 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^{2^{GF2E::degree()}} mod f,
    // which can be computed efficiently using the function FrobeniusMap
    // (see below).
    // 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 double GF2EXFileThresh

    // which can be set by the user.  If the sizes of the tables
    // exceeds GF2EXFileThresh KB, external files are used.
    // Initial value is NTL_FILE_THRESH (defined in tools.h).

    void EDF(vec_GF2EX& factors, const GF2EX& f, const GF2EX& h,
             long d, long verbose=0);

    vec_GF2EX EDF(const GF2EX& f, const GF2EX& h,
             long d, long verbose=0);

    // Performs equal-degree factorization.  f is monic, square-free, and
    // all irreducible factors have same degree.
    // h = X^{2^{GF2E::degree()}} mod f,
    // which can be computed efficiently using the function FrobeniusMap
    // (see below).
    // 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_GF2EX& factors, const GF2EX& f, long verbose=0);
    vec_GF2EX RootEDF(const GF2EX& f, long verbose=0);

    // EDF for d==1

    void SFCanZass(vec_GF2EX& factors, const GF2EX& f, long verbose=0);
    vec_GF2EX SFCanZass(const GF2EX& 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_GF2EX_long& factors, const GF2EX& f,
                 long verbose=0);

    vec_pair_GF2EX_long CanZass(const GF2EX& 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
    // GF2EXArgBound (see GF2EX.txt).

    void mul(GF2EX& f, const vec_pair_GF2EX_long& v);
    GF2EX mul(const vec_pair_GF2EX_long& v);

    // multiplies polynomials, with multiplicities

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

                                Irreducible Polynomials

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

    long ProbIrredTest(const GF2EX& 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
    // 2^{-iter*GF2E::degree()}.  This implements an algorithm from [Shoup,
    // J. Symbolic Comp. 17:371-391, 1994].

    long DetIrredTest(const GF2EX& 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 GF2EX& f);

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

    void BuildIrred(GF2EX& f, long n);
    GF2EX BuildIrred_GF2EX(long n);

    // Build a monic irreducible poly of degree n.

    void BuildRandomIrred(GF2EX& f, const GF2EX& g);
    GF2EX BuildRandomIrred(const GF2EX& g);

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

    void FrobeniusMap(GF2EX& h, const GF2EXModulus& F);
    GF2EX FrobeniusMap(const GF2EXModulus& F);

    // Computes h = X^{2^{GF2E::degree()}} mod F,
    // by either iterated squaring or modular
    // composition.  The latter method is based on a technique developed
    // in Kaltofen & Shoup (Faster polynomial factorization over high
    // algebraic extensions of finite fields, ISSAC 1997).  This method is
    // faster than iterated squaring when deg(F) is large relative to
    // GF2E::degree().

    long IterComputeDegree(const GF2EX& h, const GF2EXModulus& F);

    // f is assumed to be an "equal degree" polynomial, and h =
    // X^{2^{GF2E::degree()}} mod f (see function FrobeniusMap above)
    // 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 GF2EX& h, const GF2EXModulus& F);

    // f is assumed to be an "equal degree" polynomial, h = X^{2^{GF2E::degree()}}
    // mod f (see function FrobeniusMap above).
    // The common degree of the irreducible factors of f is
    // computed. Uses a recursive algorithm similar to DetIrredTest.

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

    GF2EX TraceMap(const GF2EX& a, long d, const GF2EXModulus& F,
                  const GF2EX& 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 2^{GF2E::degree()}.  This routine
    // implements an algorithm from [von zur Gathen and Shoup,
    // Computational Complexity 2:187-224, 1992].
    // If q = 2^{GF2E::degree()}, then h can be computed most efficiently
    // by using the function FrobeniusMap above.

    void PowerCompose(GF2EX& w, const GF2EX& h, long d, const GF2EXModulus& F);

    GF2EX PowerCompose(const GF2EX& h, long d, const GF2EXModulus& F);

    // Computes w = X^{q^d} mod f; it is assumed that d >= 0, and h = X^q
    // mod f, q a power of 2^{GF2E::degree()}.  This routine implements an
    // algorithm from [von zur Gathen and Shoup, Computational Complexity
    // 2:187-224, 1992].
    // If q = 2^{GF2E::degree()}, then h can be computed most efficiently
    // by using the function FrobeniusMap above.