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Title:
Classification of Elliptic/hyperelliptic Curves with Weak Coverings against GHS Attack without Isogeny Condition
Authors: Tsutomu Iijima, Fumiyuki Momose, 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.
ePrint: https://eprint.iacr.org/2009/613
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