Smoothness and Factoring Polynomials Over Finite Fields

Let p be a prime number, and F p the nite eld with p elements. Let S(m) be the \smoothness" function that for integers m is deened as the largest prime divisor of m. In this note, we prove the following theorem. Theorem. There is a deterministic algorithm for factoring polynomials over F p , which on poly-nomials over F p of degree n runs in time S(p ? 1) 1=2 (n log p) O(1) under the assumption of the Extended Riemann Hypothesis (ERH). The algorithm we describe is a reenement of algorithms given by von zur Gathen 19] and RR onyai 13]. Assuming the ERH, these algorithms run in time S(p ? 1)(n log p) O(1) , thus our algorithm represents an improvement of a factor of S(p ? 1) 1=2. If the ERH is true, then in terms of the dependence on p, the bound on the running time of our algorithm is better than the worst-case bounds on the running times of current algorithms in the literature for factoring arbitrary polynomials over F p. See 15] for an unconditional running time bound of p 1=2 (n log p) O(1) , and 8, 14] for running time bounds (assuming ERH) of (n log p) O(1) for polynomials of a special form. The algorithms of von zur Gathen and Ronyai essentially reduce the problem of factoring a polynomial of degree n over F p to the following two problems in time (n log p) O(1) : (1) computing the prime factorization of p ? 1; (2) computing all of the roots of X q ?a, where q is a prime divisor of p?1, and a is a q-th power residue in F p .