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

Computing Individual Discrete Logarithms Faster in GF$(p^n)$ with the NFS-DL Algorithm