On the selection of polynomials for the DLP algorithm

In this paper we characterize the set of polynomials f ∈ Fq[X] satisfying the following property: there exists a positive integer d such that for any positive integer l ≤ deg(f) there exists t0 in Fqd such that the polynomial f − t0 has an irreducible factor of degree l over Fqd [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.