The GHS attack is known as a method to map the discrete logarithm problem(DLP) in the Jacobian of a curve C0 defined over the d degree extension kd of a finite field k to the DLP in the Jacobian of a new curve C over k which is a covering curve of C0. Such curves C0/kd can be attacked by the GHS attack and index calculus algorithms. In this paper, we will classify all elliptic curves and hyperelliptic curves C0/kd of genus 2, 3 which possess (2,..., 2) covering C/k of P 1 under the isogeny condition (i.e. g(C) = d · g(C0)) in odd characteristic case. Our main approach is analysis of ramification points and representation of the extension of Gal(kd/k) acting on the covering group cov(C/P 1). Consequently, all explicit defining equations of such curve
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