Attacking a binary GLS elliptic curve with Magma

Abstract

In this paper we present a complete Magma implementation for solving the discrete logarithm problem (DLP) on a binary GLS curve defined over the field (formula presented). For this purpose, we constructed a curve vulnerable against the gGHS Weil descent attack and adapted the algorithm proposed by Enge and Gaudry to solve the DLP on the Jacobian of a genus-32 hyperelliptic curve. Furthermore, we describe a mechanism to check whether a randomly selected binary GLS curve is vulnerable against the gGHS attack. Such method works with all curves defined over binary fields and can be applied to each element of the isogeny class.