Cryptology ePrint Archive: Report 2009/613

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 21 Nov 2013

Contact author: tiijima at jt3 so-net ne jp

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Version: 20131122:044935 (All versions of this report)

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