/*****************************************************************************\ MODULE: ZZXFactoring SUMMARY: Routines are provided for factoring in ZZX. See IMPLEMENTATION DETAILS below for a discussion of the algorithms used, and of the flags available for selecting among these algorithms. \*****************************************************************************/ #include <NTL/ZZX.h> #include <NTL/pair_ZZX_long.h> void SquareFreeDecomp(vec_pair_ZZX_long& u, const ZZX& f); const vector(pair_ZZX_long SquareFreeDecomp(const ZZX& f); // input is primitive, with positive leading coefficient. Performs // square-free decomposition. 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 MultiLift(vec_ZZX& A, const vec_zz_pX& a, const ZZX& f, long e, long verbose=0); // Using current value p of zz_p::modulus(), this lifts the // square-free factorization a mod p of f to a factorization A mod p^e // of f. It is required that f and all the polynomials in a are // monic. void SFFactor(vec_ZZX& factors, const ZZX& f, long verbose=0, long bnd=0); vec_ZZX SFFactor(const ZZX& f, long verbose=0, long bnd=0); // input f is primitive and square-free, with positive leading // coefficient. bnd, if not zero, indicates that f divides a // polynomial h whose Euclidean norm is bounded by 2^{bnd} in absolute // value. This uses the routine SFCanZass in zz_pXFactoring and then // performs a MultiLift, followed by a brute-force search for the // factors. // A number of heuristics are used to speed up the factor-search step. // See "implementation details" below. void factor(ZZ& c, vec_pair_ZZX_long& factors, const ZZX& f, long verbose=0, long bnd=0); // input f is is an arbitrary polynomial. c is the content of f, and // factors is the facrorization of its primitive part. bnd is as in // SFFactor. The routine calls SquareFreeDecomp and SFFactor. void mul(ZZX& x, const vec_pair_ZZX_long& a); ZZX mul(const vec_pair_ZZX_long& a); // multiplies polynomials, with multiplcities. /*****************************************************************************\ IMPLEMENTATION DETAILS To factor a polynomial, first its content is extracted, and it is made squarefree. This is typically very fast. Second, a simple hack is performed: if the polynomial is of the form g(x^l), then an attempt is made to factor g(k^m), for divisors m of l, which can in some cases greatly simplify the factorization task. You can turn this "power hack" on/off by setting the following variable to 1/0: extern long ZZXFac_PowerHack; // initial value = 1 Third, the polynomial is factored modulo several small primes, and one small prime p is selected as the "best". You can choose the number of small primes that you want to use by setting the following variable: extern long ZZXFac_InitNumPrimes; // initial value = 7 Fourth, The factorization mod p is "lifted" to a factorization mod p^k for a sufficiently large k. This is done via quadratic Hensel lifting. Despite "folk wisdom" to the contrary, this is much more efficient than linear Hensel lifting, especially since NTL has very fast polynomial arithmetic. After the "lifting phase" comes the "factor recombination phase". The factorization mod p^k may be "finer" than the true factorization over the integers, hence we have to "combine" subsets of modular factors and test if these are factors over the integers. There are two basic strategies: the "Zassenhaus" method and the "van Hoeij" method. The van Hoeij method: The van Hoeij method is fairly new, but it is so much better than the older, Zassenhaus method, that it is now the default. For a description of the method, go to Mark van Hoeij's home page: http://www.openmath.org/~hoeij/ The van Hoeij method is not really a specific algorithm, but a general algorithmic approach: many parameters and strategies have to be selected to obtain a specific algorithm, and it is a challenge to make all of these choices so that the resulting algorithm works fairly well on all input polynomials. Set the following variable to 1 to enable the van Hoeij method, and to 0 to revert to the Zassenhaus method: extern long ZZXFac_van_Hoeij; // initial value = 1 Note that the "power hack" is still on by default when using van Hoeij's method, but we have arranged things so that the "power hack" strategy is abandoned if it appears to be too much a waste of time. Unlike with the Zassenhaus method, using the "power hack" method with van Hoeij can sometimes be a huge waste of time if one is not careful. The Zassenhaus method: The Zassenhaus method is essentially a brute-force search, but with a lot of fancy "pruning" techniques, as described in the paper [J. Abbott, V. Shoup, P. Zimmermann, "Factoring in Z[x]: the searching phase", ISSAC 2000]. These heuristics are fairly effective, and allow one to easily deal with up to around 30-40 modular factors, which is *much* more than other Zassenhaus-based factorizers can deal with; however, after this, the exponential behavior of the algorithm really starts to dominate. The behaviour of these heuristics can be fine tuned by setting the following global variables: extern long ZZXFac_MaxNumPrimes; // initial value = 50 // During the factor recombination phase, if not much progress // is being made, occasionally more "local" information is // collected by factoring f modulo another prime. // This "local" information is used to rule out degrees // of potential factors during recombination. // This value bounds the total number of primes modulo which f // is factored. extern long ZZXFac_MaxPrune; // initial value = 10 // A kind of "meet in the middle" strategy is used // to prune the search space during recombination. // For many (but not all) polynomials, this can greatly // reduce the running time. // When it does work, there is a time-space tradeoff: // If t = ZZXFac_MaxPrune, the running time will be reduced by a factor near // 2^t, but the table will take (at most) t*2^(t-1) bytes of storage. // Note that ZZXFac_MaxPrune is treated as an upper bound on t---the // factoring algorithm may decide to use a smaller value of t for // a number of reasons. \*****************************************************************************/