source: git/ntl/doc/GF2EXFactoring.txt @ 26e030

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

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 ddf-*-baby-* and ddf-*-giant-*.
// 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 256.



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.