Factorization of Multivariate Polynomials with Coefficients in Fp

The ring of polynomials in X, X1,…,Xm are denoted by Fp[X, 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 Δn, where Δn is an ideal. This algorithm uses a generalization of Henselʹs Lemma. If there are no extraneous factors, then the factors defined modulo Δn correspond to the factors of P in Fp [X, X1,…,Xm], or else the defined factors must be regrouped to find the factors of P in Fp[X, 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 Fp[X, 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 Berlekampʹs algorithm.