I wrote earlier:
Thanks for the reference. The paper gives a running time of exp(c(n log n)^(1/2)) for discrete log in GF(p) and exp(c*n^(1/3)*(log n)^(2/3)) for discrete log in GF(2^n). However, this paper was published in 1985. There is now an algorithm to calculate discrete logs in GF(p) in exp(c*n^(1/3)*(log n)^(2/3)) (see prime.discrete.logs.ps.Z in the same directory), so perhaps GF(2^n) isn't so bad after all.
To clarify my earlier post, although both of the latter two algorithms have a runtime of the form exp(c*n^(1/3)*(log n)^(2/3)), for GF(p) c=1.922+o(1), for GF(2^n) c=1.405+o(1). This seems to imply that if GF(2^n) is to be used, n needs to be 2.56*log p to achieve a comparable level of security to using GF(p). (2.56=1.922^3/1.405^3) Wei Dai