#ifndef NTL_ZZ_pXFactoring__H #define NTL_ZZ_pXFactoring__H #include #include #include #include NTL_OPEN_NNS /************************************************************ factorization routines ************************************************************/ void SquareFreeDecomp(vec_pair_ZZ_pX_long& u, const ZZ_pX& f); inline vec_pair_ZZ_pX_long SquareFreeDecomp(const ZZ_pX& f) { vec_pair_ZZ_pX_long x; SquareFreeDecomp(x, f); return x; } // 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); inline vec_ZZ_p FindRoots(const ZZ_pX& f) { vec_ZZ_p x; FindRoots(x, f); return x; } // f is monic, and has deg(f) distinct roots. // returns the list of roots void FindRoot(ZZ_p& root, const ZZ_pX& f); inline ZZ_p FindRoot(const ZZ_pX& f) { ZZ_p x; FindRoot(x, f); return x; } // 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); inline vec_ZZ_pX SFBerlekamp(const ZZ_pX& f, long verbose=0) { vec_ZZ_pX x; SFBerlekamp(x, f, verbose); return x; } // Assumes f is square-free and monic. // returns list of factors of f. // Uses "Berlekamp" appraoch. void berlekamp(vec_pair_ZZ_pX_long& factors, const ZZ_pX& f, long verbose=0); inline vec_pair_ZZ_pX_long berlekamp(const ZZ_pX& f, long verbose=0) { vec_pair_ZZ_pX_long x; berlekamp(x, f, verbose); return x; } // returns a list of factors, with multiplicities. // f must be monic. // Uses "Berlekamp" appraoch. extern long ZZ_pX_BlockingFactor; // Controls GCD blocking for DDF. void DDF(vec_pair_ZZ_pX_long& factors, const ZZ_pX& f, const ZZ_pX& h, long verbose=0); inline vec_pair_ZZ_pX_long DDF(const ZZ_pX& f, const ZZ_pX& h, long verbose=0) { vec_pair_ZZ_pX_long x; DDF(x, f, h, verbose); return x; } // Performs distinct-degree factorization. // Assumes f is monic and square-free, and h = X^p mod f // Obsolete: see NewDDF, below. extern long ZZ_pX_GCDTableSize; /* = 4 */ // Controls GCD blocking for NewDDF extern char ZZ_pX_stem[]; // Determines filename stem for external storage in NewDDF. extern double ZZ_pXFileThresh; /* = 128 */ // external files are used for baby/giant steps if size // of these tables exceeds ZZ_pXFileThresh KB. void NewDDF(vec_pair_ZZ_pX_long& factors, const ZZ_pX& f, const ZZ_pX& h, long verbose=0); inline vec_pair_ZZ_pX_long NewDDF(const ZZ_pX& f, const ZZ_pX& h, long verbose=0) { vec_pair_ZZ_pX_long x; NewDDF(x, f, h, verbose); return x; } // same as above, but uses baby-step/giant-step method void EDF(vec_ZZ_pX& factors, const ZZ_pX& f, const ZZ_pX& b, long d, long verbose=0); inline vec_ZZ_pX EDF(const ZZ_pX& f, const ZZ_pX& b, long d, long verbose=0) { vec_ZZ_pX x; EDF(x, f, b, d, verbose); return x; } // Performs equal-degree factorization. // f is monic, square-free, and all irreducible factors have same degree. // b = X^p mod f. // d = degree of irreducible factors of f // Space for the trace-map computation can be controlled via ComposeBound. void RootEDF(vec_ZZ_pX& factors, const ZZ_pX& f, long verbose=0); inline vec_ZZ_pX RootEDF(const ZZ_pX& f, long verbose=0) { vec_ZZ_pX x; RootEDF(x, f, verbose); return x; } // EDF for d==1 void SFCanZass(vec_ZZ_pX& factors, const ZZ_pX& f, long verbose=0); inline vec_ZZ_pX SFCanZass(const ZZ_pX& f, long verbose=0) { vec_ZZ_pX x; SFCanZass(x, f, verbose); return x; } // Assumes f is monic and square-free. // returns list of factors of f. // Uses "Cantor/Zassenhaus" approach. void CanZass(vec_pair_ZZ_pX_long& factors, const ZZ_pX& f, long verbose=0); inline vec_pair_ZZ_pX_long CanZass(const ZZ_pX& f, long verbose=0) { vec_pair_ZZ_pX_long x; CanZass(x, f, verbose); return x; } // returns a list of factors, with multiplicities. // f must be monic. // Uses "Cantor/Zassenhaus" approach. void mul(ZZ_pX& f, const vec_pair_ZZ_pX_long& v); inline ZZ_pX mul(const vec_pair_ZZ_pX_long& v) { ZZ_pX x; mul(x, v); return x; } // multiplies polynomials, with multiplicities /************************************************************* irreducible poly's: tests and constructions **************************************************************/ 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}. long DetIrredTest(const ZZ_pX& f); // performs a recursive deterministic irreducibility test // fast in the worst-case (when input is irreducible). 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); inline ZZ_pX BuildIrred_ZZ_pX(long n) { ZZ_pX x; BuildIrred(x, n); NTL_OPT_RETURN(ZZ_pX, x); } // Build a monic irreducible poly of degree n. void BuildRandomIrred(ZZ_pX& f, const ZZ_pX& g); inline ZZ_pX BuildRandomIrred(const ZZ_pX& g) { ZZ_pX x; BuildRandomIrred(x, g); NTL_OPT_RETURN(ZZ_pX, x); } // 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 = 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& b); inline ZZ_pX TraceMap(const ZZ_pX& a, long d, const ZZ_pXModulus& F, const ZZ_pX& b) { ZZ_pX x; TraceMap(x, a, d, F, b); return x; } // w = a+a^q+...+^{q^{d-1}} mod f; // it is assumed that d >= 0, and b = X^q mod f, q a power of p // Space allocation can be controlled via ComposeBound (see ) void PowerCompose(ZZ_pX& w, const ZZ_pX& a, long d, const ZZ_pXModulus& F); inline ZZ_pX PowerCompose(const ZZ_pX& a, long d, const ZZ_pXModulus& F) { ZZ_pX x; PowerCompose(x, a, d, F); return x; } // w = X^{q^d} mod f; // it is assumed that d >= 0, and b = X^q mod f, q a power of p // Space allocation can be controlled via ComposeBound (see ) NTL_CLOSE_NNS #endif