Fast evaluation of logarithms in fields of characteristic two

A method for determining logarithms in GF
(2^{n})
is presented. Its asymptotic running time is
O(\exp (cn^{1/3} \log^{2/3} n))
for a small constant
c
, while, by comparison, Adleman's scheme runs in time
O(\exp (c^{'}n^{1/2} \log^{1/2} n ))
. The ideas give a dramatic improvement even for moderate-sized fields such as GF
(2^{127})
, and make (barely) possible computations in fields of size around
2^{400}
. The method is not applicable to GF
(q)
for a large prime
q
.