Factorization of Polynomials over Finite Fields and Characteristic Sequences

A new factorization algorithm for polynomials over finite fields was recently developed by the first author. For finite fields of characteristic 2, it is known from previous work that this algorithm has several advantages over the classical Berlekamp algorithm. In this paper we show that the linearization step in the new algorithm is feasible—in the sense that it can be carried out in polynomial time—for arbitrary finite fields, by using an approach based on decimation operators and characteristic linear recurring sequences. We also introduce a general principle for the linearization of the factorization problem for polynomials over finite fields.