Using number fields to compute logarithms in finite fields

We describe an adaptation of the number field sieve to the problem of computing logarithms in a finite field. We conjecture that the running time of the algorithm, when restricted to finite fields of an arbitrary but fixed degree, is Lq[1/3; (64/9)1/3 + o(1)], where q is the cardinality of the field, Lq [s; c] = exp(c(log q)s(log log q)1−s), and the o(1) is for q →∞. The number field sieve factoring algorithm is conjectured to factor a number the size of q in the same amount of time.