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Factorization of Multivariate Polynomials with Coefficients in F_p

The ring of polynomials in X, X1,?,Xm are denoted by FpX, X1,?,Xm] in Fp, that is the field of integers defined modulo p. In the usual factorization algorithm defined by Wang, the given polynomial P is first factorized modulo ?n, where ?n is an ideal. This algorithm uses a generalization of Hensel's Lemma. If there are no extraneous factors, then the factors defined modulo ?n correspond to the factors of P in FpX, X1,?,Xm], or else the defined factors must be regrouped to find the factors of P in FpX, X1,?,Xm]. In this paper, a mapping that transforms the product of the factors into a sum is defined. A theorem that determines whether a subproduct of the factors of P corresponds to a factor of P in FpX, X1,?,Xm] is given. Therefore the regrouping of the factors of P reduces to solving a system of linear equations, as in the univariate case, with Berlekamp's algorithm.