Title: On the selection of polynomials for the DLP algorithm,
Abstract: In this paper we characterize the set of polynomials $f\in\mathbb F_q[X]$
satisfying the following property: there exists a positive integer $d$ such
that for any positive integer $\ell$ less or equal than the degree of $f$,
there exists $t_0$ in $\mathbb F_{q^d}$ such that the polynomial $f-t_0$ has an
irreducible factor of degree $\ell$ over $\mathbb F_{q^d}[X]$. This result is
then used to progress in the last step which is needed to remove the heuristic
from one of the quasi-polynomial time algorithms for discrete logarithm
problems (DLP) in small characteristic. Our characterization allows a
construction of polynomials satisfying the wanted property.