In this article we propose a study of the modified Tate pairing in characteristics two and three. Starting from the $\eta_T$ pairing introduced by Barreto {\em et al.}, we detail various algorithmic improvements in the case of characteristic two. As far as characteristic three is concerned, we refer to the survey by Beuchat {\em et al.}. We then show how to get back to the modified Tate pairing at almost no extra cost. Finally, we explore the trade-offs involved in the hardware implementation of this pairing for both characteristics two and three. From our experiments, characteristic three appears to have a slight advantage over characteristic two.