Evaluating logarithms in GF(2ⁿ)

We present a method for determining logarithms in GF(2n). Its asymptotic running time is O( exp (en1/3log2/3n)) for a small constant c, while, by comparison, Adleman's scheme runs in time O( exp (cn1/2log1/2n)). The ideas give a dramatic improvement even for moderate-sized fields such as GF(2127), and make (barely) possible computations in fields of size around 2400. The method is not applicable to GF(q) for a large prime q.