Estimates for Discrete Logarithm Computations in Finite Fields of Small Characteristic

We give estimates for the running-time of the function field sieve (FFS) to compute discrete logarithms in $\mathbb{F}_{p^n}^{\times}$ for small $p$. Specifically, we obtain sharp probability estimates that allow us to select optimal parameters in cases of cryptographic interest, without appealing to the heuristics commonly relied upon in an asymptotic analysis. We also give evidence that for any fixed field size some may be weaker than others of a different characteristic or field representation, and compare the relative difficulty of computing discrete logarithms via the FFS in such cases.