A simplified approach to rigorous degree 2 elimination in discrete logarithm algorithms

In this paper, we revisit the ZigZag strategy of Granger, Kleinjung, and Zumbrägel. In particular, we provide a new algorithm and proof for the so-called degree 2 elimination step. This allows us to provide a stronger theorem concerning discrete logarithm computations in small characteristic fields $\mathbb{F}_{q^{k_0k}}$ with $k$ close to $q$ and $k_0$ a small integer. As in the aforementioned paper, we rely on the existence of two polynomials $h_0$ and $h_1$ of degree $2$ providing a convenient representation of the finite field  $\mathbb{F}_{q^{k_0k}}$.