The question of the complexity of the DLP in characteristic 2 has been answered multiple times here:
How robust is discrete logarithm in GF(2^n)?
Complexity of Discrete Logarithms in binary fields without special exponents or special structure
Is there any open implementation of Joux's algorithm (mentioned in the first link, with complexity L(1/4)) or of the method by Barbulescu et al. (mentioned in the last link, with quasi-polynomial complexity)?
A related question: what is currently used in SAGE, and can better implementations be found elsewhere?
