Classification of Elliptic/Hyperelliptic Curves with Weak Coverings against the GHS Attack under an Isogeny Condition

The GHS attack is known to map the discrete logarithm problem(DLP) in the Jacobian of a curve C0 defined over the d degree extension kd of a finite field k to the DLP in the Jacobian of a new curve C over k which is a covering curve of C0, then solve the DLP of curves C/k by variations of index calculus algorithms. It is therefore important to know which curve C0/kd is subjected to the GHS attack, especially those whose covering C/k have the smallest genus g(C) = dg(C0), which we called satisfying the isogeny condition. Until now, 4 classes of such curves were found by Thériault in [35] and 6 classes by Diem in [3][5]. In this paper, we present a classification i.e. a complete list of all elliptic curves and hyperelliptic curves C0/kd of genus 2, 3 which possess (2, ..., 2) covering C/k of P under the isogeny condition (i.e. g(C) = d · g(C0)) in odd characteristic case. In particular, classification of the Galois representation of Gal(kd/k) acting on the covering group cov(C/P) is used together with analysis of ramification points of these coverings. Besides, a general existential condition of a model of C over k is also obtained. As the result, a complete list of all defining equations of curves C0/kd with covering C/k are provided explicitly. Besides the 10 classes of C0/kd already known, 17 classes are newly found.