On the discrete logarithm problem in finite fields of fixed characteristic

Abstract

For  q q a prime power, the discrete logarithm problem (DLP) in  F q \mathbb {F}_{q} consists of finding, for any g ∈ F q × g \in \mathbb {F}_{q}^{\times } and h ∈ ⟨ g ⟩ h \in \langle g \rangle , an integer  x x such that g x = h g^x = h . We present an algorithm for computing discrete logarithms with which we prove that for each prime  p p there exist infinitely many explicit extension fields  F p n \mathbb {F}_{p^n} in which the DLP can be solved in expected quasi-polynomial time. Furthermore, subject to a conjecture on the existence of irreducible polynomials of a certain form, the algorithm solves the DLP in all extensions  F p n \mathbb {F}_{p^n} in expected quasi-polynomial time.