Paper 2013/487

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

Tsutomu Iijima and Fumiyuki Momose and Jinhui Chao

Abstract

The GHS attack is known as a method to map the discrete logarithm problem(DLP) in the Jacobian of a curve C_{0} defined over the d degree extension k_{d} of a finite field k to the DLP in the Jacobian of a new curve C over k which is a covering curve of C_{0}. Such curves C_{0}/k_{d} can be attacked by the GHS attack and index calculus algorithms. In this paper, we will classify all elliptic curves and hyperelliptic curves C_{0}/k_{d} of genus 2, 3 which possess (2,...,2) covering C/k of \Bbb{P}^1 under the isogeny condition (i.e. g(C)=d \cdot g(C_{0})) in odd characteristic case. Our main approach is analysis of ramification points and representation of the extension of Gal(k_{d}/k) acting on the covering group cov(C/\Bbb{P}^1). Consequently, all explicit defining equations of such curves C_0/k_d and existential conditions of a model of C over k are provided.