**Classification of Elliptic/hyperelliptic Curves with Weak Coverings against GHS Attack without 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}.
Recently, classification and density analysis were shown for all elliptic and hyperelliptic curves C_{0}/k_d of genus 2, 3 which possess (2, \ldots ,2) covering C/k of {\mathbb{P}^{1}} under the isogeny condition (i.e. when g(C)=d \cdot g(C_{0})).
In this paper, we show a complete classification of small genus hyperelliptic curves C_0/k_d which possesses (2,..,2) covering C over k without the isogeny condition. Our main approach is to use representation of the extension of Gal(k_{d}/k) acting on cov(C/\mathbb{P}^{1}). Explicit defining equations of such curves and the existence of a model of C over k are also presented.

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

**Date: **received 10 Dec 2009, last revised 4 Dec 2011

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

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

**Version: **20111204:123821 (All versions of this report)

**Short URL: **ia.cr/2009/613

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