Factoring Polynomials over Special Finite Fields

We exhibit a deterministic algorithm for factoring polynomials in one variable over finite fields. It is efficient only if a positive integer k is known for which Φk(p) is built up from small prime factors; here Φk denotes the kth cyclotomic polynomial, and p is the characteristic of the field. In the case k=1, when Φk(p)=p−1, such an algorithm was known, and its analysis required the generalized Riemann hypothesis. Our algorithm depends on a similar, but weaker, assumption; specifically, the algorithm requires the availability of an irreducible polynomial of degree r over Z/pZ for each prime number r for which Φk(p) has a prime factor l with l≡1 mod r. An auxiliary procedure is devoted to the construction of roots of unity by means of Gauss sums. We do not claim that our algorithm has any practical value.