  We propose various strategies for improving the computation of discrete
logarithms in non-prime fields of medium to large characteristic using the
Number Field Sieve. This includes new methods for selecting the polynomials;
the use of explicit automorphisms; explicit computations in the number fields;
and prediction that some units have a zero virtual logarithm. On the
theoretical side, we obtain a new complexity bound of
$L_{p^n}(1/3,\sqrt[3]{96/9})$ in the medium characteristic case. On the
practical side, we computed discrete logarithms in $F_{p^2}$ for a prime number
$p$ with $80$ decimal digits.Warning: This unpublished version contains some
inexact statements.
