**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.

**Category / Keywords: **public-key cryptography / Weil descent attack, GHS attack, Elliptic curve cryptosystems, Hyperelliptic curve cryptosystems, Index calculus, Galois representation

**Date: **received 10 Aug 2013, last revised 20 Sep 2013

**Contact author: **tiijima at jt3 so-net ne jp

**Available format(s): **PDF | BibTeX Citation

**Version: **20130920:115148 (All versions of this report)

**Short URL: **ia.cr/2013/487

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