The deterministic factorization algorithm for polynomials over finite fields that was recently introduced by the author is based on a new type of linearization of the factorization problem. The main ingredients are differential equations in rational function fields and normal bases of field extensions. For finite fields of characteristic 2, it is known that this algorithm has several advantages over the classical Berlekamp algorithm. We develop the algorithm in a general framework, and we show that it is feasible for arbitrary finite fields, in the sense that the linearization can be achieved in polynomial time.