We present a new deterministic factorization algorithm for polynomials over a finite prime fieldFp. As in other factorization algorithms for polynomials over finite fields such as the Berlekamp algorithm, the key step is the “linearization” of the factorization problem, i.e., the reduction of the problem to a system of linear equations. The theoretical justification for our algorithm is based on a study of the differential equationy(p−1)+yp=0 of orderp−1 in the rational function fieldFp(x). In the casep=2 the new algorithm is more efficient than the Berlekamp algorithm since there is no set-up cost for the coefficient matrix of the system of linear equations.
Factorization of polynomials over finite fields Differential equations over rational function fields