  The Number Field Sieve (NFS) algorithm is the best known method to compute
discrete logarithms (DL) in finite fields $\mathbb{F}\_{p^n}$, with $p$ medium
to large and $n \geq 1$ small. This algorithm comprises four steps: polynomial
selection, relation collection, linear algebra and finally, individual
logarithm computation. The first step outputs two polynomials defining two
number fields, and a map from the polynomial ring over the integers modulo each
of these polynomials to $\mathbb{F}\_{p^n}$. After the relation collection and
linear algebra phases, the (virtual) logarithm of a subset of elements in each
number field is known. Given the target element in $\mathbb{F}\_{p^n}$, the
fourth step computes a preimage in one number field. If one can write the
target preimage as a product of elements of known (virtual) logarithm, then one
can deduce the discrete logarithm of the target. As recently shown by the
Logjam attack, this final step can be critical when it can be computed very
quickly. But we realized that computing an individual DL is much slower in
medium-and large-characteristic non-prime fields $\mathbb{F}\_{p^n}$ with $n
\geq 3$, compared to prime fields and quadratic fields $\mathbb{F}\_{p^2}$. We
optimize the first part of individual DL: the \emph{booting step}, by reducing
dramatically the size of the preimage norm. Its smoothness probability is
higher, hence the running-time of the booting step is much improved. Our method
is very efficient for small extension fields with $2 \leq n \leq 6$ and applies
to any $n \textgreater{} 1$, in medium and large characteristic.
