Complexity of Discrete Logarithms in binary fields without special exponents or special structure
Edit: My hunch that the best complexity is exponential was incorrect. It is actually quasi-polynomial (thanks to Samuel Neves for his comment).

Recall that  Coppersmith's algorithm has complexity
$$
L_{2^n}(1/3,c)
$$
for a small constant $c,$ while even for $n$ prime, which is the hardest case,  discrete logarithms over $GF(2^n)$ using Barbulescu et. al.'s method   have quasi polynomial complexity
$$
O(2^{c(\log n)^2})\asymp c' n^{c \log n}.
$$

Note that the authors state "The crossing point between the (best knoen before) L(1/4) algorithm and our quasi-polynomial one is not determined yet".