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97 lines
4 KiB
C++
97 lines
4 KiB
C++
/*++
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Copyright (c) 2011 Microsoft Corporation
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Module Name:
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upolynomial_factorization.h
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Abstract:
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Methods for factoring polynomials.
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Author:
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Dejan (t-dejanj) 2011-11-29
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Notes:
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[1] Elwyn Ralph Berlekamp. Factoring Polynomials over Finite Fields. Bell System Technical Journal,
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46(8-10):1853–1859, 1967.
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[2] Donald Ervin Knuth. The Art of Computer Programming, volume 2: Seminumerical Algorithms. Addison Wesley, third
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edition, 1997.
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[3] Henri Cohen. A Course in Computational Algebraic Number Theory. Springer Verlag, 1993.
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--*/
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#ifndef _UPOLYNOMIAL_FACTORIZATION_H_
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#define _UPOLYNOMIAL_FACTORIZATION_H_
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#include"upolynomial.h"
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#include"polynomial.h"
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#include"bit_vector.h"
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#include"z3_exception.h"
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namespace upolynomial {
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typedef manager::scoped_numeral scoped_numeral;
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/**
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\breif Factor f into f = f_1^k_1 * ... * p_n^k_n, such that p_i are square-free and coprime.
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*/
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void zp_square_free_factor(zp_manager & zp_upm, numeral_vector const & f, zp_factors & sq_free_factors);
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/**
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\brief Factor the monic square-free polynomial f from Z_p[x]. Returns true if factorization was sucesseful, or false
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if f is an irreducible square-free polynomial in Z_p[x].
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*/
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bool zp_factor_square_free(zp_manager & zp_upm, numeral_vector const & f, zp_factors & factors);
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inline bool zp_factor_square_free(zp_manager & zp_upm, numeral_vector const & f, zp_factors & factors, factor_params const & params) {
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return zp_factor_square_free(zp_upm, f, factors);
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}
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/**
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\brief Factor the monic square-free polynomial f from Z_p[x] using the Berlekamp algorithm. If randomized is true
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the factor splitting is done randomly [3], otherwise it is done as in the original Berlekamp [1].
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*/
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bool zp_factor_square_free_berlekamp(zp_manager & zp_upm, numeral_vector const & f, zp_factors & factors, bool randomized = true);
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/**
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\brief Factor the polynomial f from Z_p[x]. Returns true if factorization was sucesseful, or false if f is
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an irreducible polynomial in Z_p[x]
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*/
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bool zp_factor(zp_manager & zp_upm, numeral_vector const & f, zp_factors & factors);
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/**
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\brief Performs a Hensel lift of A and B in Z_a to Z_b, where p is prime and and a = p^{a_k}, b = p^{b_k},
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r = (a, b), with the following assumptions:
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* UA + VB = 1 (mod a)
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* C = AB (mod b)
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* (l(A), r) = 1 (importand in order to divide by A, i.e. to invert l(A))
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the output of is two polynomials A1, B1 (replacing A and B) such that A1 = A (mod b), B1 = B (mod b),
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l(A1) = l(A), deg(A1) = deg(A), deg(B1) = deg(B) and C = A1 B1 (mod b*r). Such A1, B1 are unique if
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r is prime. See [3] p. 138.
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The method will also change the zp_manager's module from b to b*r
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*/
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void hensel_lift(z_manager & upm, numeral const & a, numeral const & b, numeral const & r,
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numeral_vector const & U, numeral_vector const & A, numeral_vector const & V, numeral_vector const & B,
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numeral_vector const & C, numeral_vector & A_lifted, numeral_vector & B_lifted);
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/**
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\brief Performs the Hensel lift for the (monic!) factors_p of f in Z_p to Z_{p^e}.
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*/
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void hensel_lift(z_manager & upm, numeral_vector const & f, zp_factors const & factors_p, unsigned e, zp_factors & factors_pe);
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/**
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\brief Factor the square-free polynomial f from Z[x]. Returns true if factorization was sucesseful, or false if
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f is an irreducible polynomial in Z[x]. The vector of factors is cleared.
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*/
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bool factor_square_free(z_manager & upm, numeral_vector const & f, factors & fs, factor_params const & ps = factor_params());
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/**
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Similar to factor_square_free, but it is used to factor the k-th component f^k of a polynomial.
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That is, the factors of f are inserted as factors of degree k into fs.
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*/
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bool factor_square_free(z_manager & upm, numeral_vector const & f, factors & fs, unsigned k, factor_params const & ps = factor_params());
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};
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#endif
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