lzz_pXFactoring.txt

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

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 "zz_pX.h"
#include "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].

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.


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]