# On the selection of polynomials for the DLP quasi-polynomial time algorithm in small characteristic

## TL;DR

This paper characterizes specific polynomials over finite fields that enable progress in quasi-polynomial algorithms for the discrete logarithm problem in small characteristic, providing a constructive approach to select such polynomials.

## Contribution

It offers a characterization and construction method for polynomials that facilitate the removal of heuristics in DLP algorithms in small characteristic.

## Key findings

- Characterization of polynomials with specific factorization properties over finite fields
- Construction method for polynomials satisfying the characterization
- Progress towards removing heuristics in DLP quasi-polynomial algorithms

## 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. The method is general and can be used to tackle similar problems which involve factorization patterns of polynomials over finite fields.