~/ntl-11.4.2/doc/GF2XFactoring.cpp.html

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

    MODULE: GF2XFactoring

    SUMMARY:

    Routines are provided for factorization in F_2[X], as well as routines
    for related problems such as testing irreducibility and constructing
    irreducible polynomials of given degree.

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

    #include <NTL/GF2X.h>
    #include <NTL/pair_GF2X_long.h>

    void SquareFreeDecomp(vec_pair_GF2X_long& u, const GF2X& f);
    vec_pair_GF2X_long SquareFreeDecomp(const GF2X& 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 DDF(vec_pair_GF2X_long& factors, const GF2X& f, long verbose=0);
    vec_pair_GF2X_long DDF(const GF2X& f, 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.

    void EDF(vec_GF2X& factors, const GF2X& f, long d, long verbose=0);
    vec_GF2X EDF(const GF2X& f, long d, long verbose=0);

    // Performs equal-degree factorization.  f is monic, square-free, and
    // all irreducible factors have same degree.  d = degree of
    // irreducible factors of f

    void SFCanZass(vec_GF2X& factors, const GF2X& f, long verbose=0);
    vec_GF2X SFCanZass(const GF2X& f, long verbose=0);

    // Assumes f is monic and square-free.  returns list of factors of f.

    void CanZass(vec_pair_GF2X_long& factors, const GF2X& f, long verbose=0);
    vec_pair_GF2X_long CanZass(const GF2X& f, long verbose=0);

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

    void mul(GF2X& f, const vec_pair_GF2X_long& v);
    GF2X mul(const vec_pair_GF2X_long& v);

    // multiplies polynomials, with multiplicities

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

                                Irreducible Polynomials

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

    long IterIrredTest(const GF2X& f);

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

    void BuildSparseIrred(GF2X& f, long n);
    GF2X BuildSparseIrred_GF2X(long n);

    // Builds a monic irreducible polynomial of degree n.
    // If there is an irreducible trinomial X^n + X^k + 1,
    // then the one with minimal k is chosen.
    // Otherwise, if there is an irreducible pentanomial
    // X^n + X^k3 + X^k2 + x^k1 + 1, then the one with minimal
    // k3 is chosen (minimizing first k2 and then k1).
    // Otherwise, if there is niether an irreducible trinomial
    // or pentanomial, the routine result from BuildIrred (see below)
    // is chosen---this is probably only of academic interest,
    // since it a reasonable, but unproved, conjecture that they
    // exist for every n > 1.

    // For n <= 2048, the polynomial is constructed
    // by table lookup in a pre-computed table.

    // The GF2XModulus data structure and routines (and indirectly GF2E)
    // are optimized to deal with the output from BuildSparseIrred.

    void BuildIrred(GF2X& f, long n);
    GF2X BuildIrred_GF2X(long n);

    // Build a monic irreducible poly of degree n.  The polynomial
    // constructed is "canonical" in the sense that it is of the form
    // f=X^n + g, where the bits of g are the those of the smallest
    // non-negative integer that make f irreducible.

    // The GF2XModulus data structure and routines (and indirectly GF2E)
    // are optimized to deal with the output from BuildIrred.

    // Note that the output from BuildSparseIrred will generally yield
    // a "better" representation (in terms of efficiency) for
    // GF(2^n) than the output from BuildIrred.

    void BuildRandomIrred(GF2X& f, const GF2X& g);
    GF2X BuildRandomIrred(const GF2X& g);

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