Weil descent attack for Kummer extensions

In this paper, we show how the Weil descent attack of Gaudry, Hess and Smart can be adapted to work for some hyperelliptic curves defined over fields of odd characteristic. This attack applies to a family of hyperelliptic and superelliptic curves over quadratic field extensions, as well as two families of hyperelliptic curves defined over cubic extensions. We also show that those are the only families of nonsingular curves defining Kummer extensions for which this method will work.