Z3 sources

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>

lib/upolynomial_factorization.h

@@ -0,0 +1,96 @@
+/*++
+Copyright (c) 2011 Microsoft Corporation
+
+Module Name:
+
+    upolynomial_factorization.h
+
+Abstract:
+
+    Methods for factoring polynomials.
+
+Author:
+
+    Dejan (t-dejanj) 2011-11-29
+
+Notes:
+
+   [1] Elwyn Ralph Berlekamp. Factoring Polynomials over Finite Fields. Bell System Technical Journal, 
+       46(8-10):1853–1859, 1967.
+   [2] Donald Ervin Knuth. The Art of Computer Programming, volume 2: Seminumerical Algorithms. Addison Wesley, third 
+       edition, 1997.
+   [3] Henri Cohen. A Course in Computational Algebraic Number Theory. Springer Verlag, 1993.
+
+--*/
+#ifndef _UPOLYNOMIAL_FACTORIZATION_H_
+#define _UPOLYNOMIAL_FACTORIZATION_H_
+
+#include"upolynomial.h"
+#include"polynomial.h"
+#include"bit_vector.h"
+#include"z3_exception.h"
+
+namespace upolynomial {
+    typedef manager::scoped_numeral scoped_numeral;
+
+    /**
+       \breif Factor f into f = f_1^k_1 * ... * p_n^k_n, such that p_i are square-free and coprime.
+    */
+    void zp_square_free_factor(zp_manager & zp_upm, numeral_vector const & f, zp_factors & sq_free_factors);
+
+    /**
+       \brief Factor the monic square-free polynomial f from Z_p[x]. Returns true if factorization was sucesseful, or false 
+       if f is an irreducible square-free polynomial in Z_p[x].
+    */
+    bool zp_factor_square_free(zp_manager & zp_upm, numeral_vector const & f, zp_factors & factors);
+
+    inline bool zp_factor_square_free(zp_manager & zp_upm, numeral_vector const & f, zp_factors & factors, factor_params const & params) {
+        return zp_factor_square_free(zp_upm, f, factors);
+    }
+
+    /**
+       \brief Factor the monic square-free polynomial f from Z_p[x] using the Berlekamp algorithm. If randomized is true
+       the factor splitting is done randomly [3], otherwise it is done as in the original Berlekamp [1].
+    */
+    bool zp_factor_square_free_berlekamp(zp_manager & zp_upm, numeral_vector const & f, zp_factors & factors, bool randomized = true);
+
+    /**
+       \brief Factor the polynomial f from Z_p[x]. Returns true if factorization was sucesseful, or false if f is
+       an irreducible polynomial in Z_p[x]
+    */
+    bool zp_factor(zp_manager & zp_upm, numeral_vector const & f, zp_factors & factors);
+
+    /**
+        \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},
+        r = (a, b), with the following assumptions:
+         * UA + VB = 1 (mod a) 
+         * C = AB (mod b)
+         * (l(A), r) = 1 (importand in order to divide by A, i.e. to invert l(A))
+        the output of is two polynomials A1, B1 (replacing A and B) such that A1 = A (mod b), B1 = B (mod b), 
+        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 
+        r is prime. See [3] p. 138.
+
+        The method will also change the zp_manager's module from b to b*r
+    */
+    void hensel_lift(z_manager & upm, numeral const & a, numeral const & b, numeral const & r,
+        numeral_vector const & U, numeral_vector const & A, numeral_vector const & V, numeral_vector const & B,
+        numeral_vector const & C, numeral_vector & A_lifted, numeral_vector & B_lifted);
+
+    /**
+       \brief Performs the Hensel lift for the (monic!) factors_p of f in Z_p to Z_{p^e}.
+    */
+    void hensel_lift(z_manager & upm, numeral_vector const & f, zp_factors const & factors_p, unsigned e, zp_factors & factors_pe);
+
+    /**
+       \brief Factor the square-free polynomial f from Z[x]. Returns true if factorization was sucesseful, or false if 
+       f is an irreducible polynomial in Z[x]. The vector of factors is cleared.
+    */
+    bool factor_square_free(z_manager & upm, numeral_vector const & f, factors & fs, factor_params const & ps = factor_params());
+    /**
+       Similar to factor_square_free, but it is used to factor the k-th component f^k of a polynomial.
+       That is, the factors of f are inserted as factors of degree k into fs.
+    */
+    bool factor_square_free(z_manager & upm, numeral_vector const & f, factors & fs, unsigned k, factor_params const & ps = factor_params());
+};
+
+#endif