Models of curves from GHS attack in odd characteristic

Abstract

The idea behind the GHS attack is to transform the discrete logarithm problem(DLP) in the Jacobian of a (hyper-)elliptic curve over an extension field into DLPs in Jacobians of covering curves over the base field. Diem gives a condition under which explicit defining equations for some coverings are computed. In this paper, we show that his method works without that condition. We also give explicit map from the covering to the original curve if the covering is hyperelliptic. Our method is based on a formula for the embedding of rational subfield of the function field of (hyper)elliptic curve in that of the hyperelliptic covering.