Character Sums and Deterministic Polynomial Root Finding in Finite Fields

Let Fq be a finite field of q elements of characteristic p. The classical algorithm of Berlekamp [1] reduces the problem of factoring polynomials of degree n over Fq to the problem of factoring squarefree polynomials of degree n over Fp that fully split in Fp, see also [8, Chapter 14]. Shoup [15, Theorem 3.1] has given a deterministic algorithm that fully factors any polynomial of degree n over Fp in O(n p(log p)) arithmetic operations over Fp; in particular it runs in time n p. Furthermore, Shoup [15, Remark 3.5] has also announced an algorithm of complexity O(np(log p)) for factoring arbitrary univariate polynomials of degree n over Fp. We remark, that although the efficiency of deterministic polynomial factorisation algorithms falls far behind the fastest probabilistic algorithms, see, for example, [9, 11, 12], the question is of great theoretic interest. Here we address a special case of the polynomial factorisation problem when the polynomial f fully splits over Fp (as we have noticed there is a polynomial time reduction between factoring general polynomials and polynomials that split over Fp). That is, here we deal with the root finding problem. We also note that in order to find a root (or all roots) of a polynomial f ∈ Fp[X ], it is enough to do the same for the polynomial gcd (f(X), X − 1) which is squarefree fully splits over Fp. We consider two variants of the root finding problem: