/**************************************************************************\ 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 #include 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]. // 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 thread_local double ZZ_pXFileThresh // which can be set by the user. If the sizes of the tables // exceeds ZZ_pXFileThresh KB, external files are used. // Initial value is NTL_FILE_THRESH (defined in tools.h). 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. // NOTE: In most situations, you should use the CanZass factoring // routine, rather than Berlekamp: it is faster and uses less space. 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]