/**************************************************************************\
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]