The Number Field Sieve in the Medium Prime Case
Résumé
In this paper, we study several variations of the number field sieve
to compute discrete logarithms in finite fields of the form
$\GF{p^n}$, with $p$ a medium to large prime. We show that when $n$
is not too large, this yields a $L_{p^n}(1/3)$ algorithm with
efficiency similar to that of the regular number field sieve over
prime fields. This approach complements the recent results of Joux
and Lercier on the function field sieve. Combining both results, we
deduce that computing discrete logarithms have heuristic complexity
$L_{p^n}(1/3)$ in all finite fields. To illustrate the efficiency
of our algorithm, we computed discrete logarithms in a 120-digit
finite field $\F_{p^3}$.