On the discrete logarithm problem in finite fields of fixed characteristic

For $q$ a prime power, the discrete logarithm problem (DLP) in ${\mathbb{F}}_q$ consists in finding, for any $g\in {\mathbb{F}}_q^{\times }$ and $h\in ⟨g⟩$, an integer $x$ such that $g^x=h$. We present an algorithm for computing discrete logarithms with which we prove that for each prime $p$ there exist infinitely many explicit extension fields ${\mathbb{F}}_{p^n}$ 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 ${\mathbb{F}}_{p^n}$ in expected quasi-polynomial time.