  In the recent breakthrough paper by Barbulescu, Gaudry, Joux and Thom{\'e}, a
quasi-polynomial time algorithm (QPA) is proposed for the discrete logarithm
problem over finite fields of small characteristic. The time complexity
analysis of the algorithm is based on several heuristics presented in their
paper. We show that some of the heuristics are problematic in their original
forms, in particular, when the field is not a Kummer extension. We believe that
the basic idea behind the new approach should still work, and propose a fix to
the algorithm in non-Kummer cases, without altering the quasi-polynomial time
complexity. The modified algorithm is also heuristic. Further study is required
in order to fully understand the effectiveness of the new approach.
