A subexponential algorithm for discrete logarithms over all finite fields

Abstract

There are numerous subexponential algorithms for computing discrete logarithms over certain classes of finite fields. However, there appears to be no published subexponential algorithm for computing discrete logarithms over all finite fields. We present such an algorithm and a heuristic argument that there exists a c ∈ ℜ > 0 c \in {\Re _{ > 0}} such that for all sufficiently large prime powers p n {p^n} , the algorithm computes discrete logarithms over GF ( p n ) {\text {GF}}({p^n}) within expected time: e c ( log ⁡ ( p n ) log ⁡ log ⁡ ( p n ) ) 1 / 2 {e^{c{{(\log ({p^n})\log \log ({p^n}))}^{1/2}}}} .