On Computing Logarithms Over Finite Fields

The problem of computing logarithms over finite fields has proved to be of interest in different fields [4]. Subexponential time algorithms for computing logarithms over the special cases GF(p), GF(p 2) and GF(p m) for a fixed p and m → ∞ have been obtained. In this paper, we present some results for obtaining a subexponential time algorithms for the remaining cases GF(p m) for p → ∞ and fixed m ≠ 1, 2. The algorithm depends on mapping the field GF(p m) into a suitable cyclotomic extension of the integers (or rationals). Once an isomorphism between GF(p m) and a subset of the cyclotomic field Q(θ q) is obtained, the algorithms becomes similar to the previous algorithms for m = 1.2.