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Title:
Classification of Elliptic/hyperelliptic Curves with Weak Coverings against the GHS attack under an Isogeny Condition
Authors: Tsutomu Iijima, Fumiyuki Momose, Jinhui Chao
Abstract:The GHS attack is known 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, then solve the DLP of curves C/k by variations of index calculus algorithms. It is therefore important to know which curve C_0/k_d is subjected to the GHS attack, especially those whose covering C/k have the smallest genus g(C)=dg(C_0), which we called satisfying the isogeny condition. Until now, 4 classes of such curves were found by Thériault and 6 classes by Diem. In this paper, we present a classification i.e. a complete list of 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. In particular, classification of the Galois representation of \Gal(k_{d}/k) acting on the covering group \cov(C/\Bbb{P}^1) is used together with analysis of ramification points of these coverings. Besides, a general existential condition of a model of C over k is also obtained. As the result, a complete list of all defining equations of curves C_0/k_d with covering C/k are provided explicitly. Besides the 10 classes of C_0/k_d already known, 17 classes are newly found.
ePrint: https://eprint.iacr.org/2013/487
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