A classification of elliptic curves with respect to the GHS attack in odd characteristic

The GHS attack is known to solve discrete logarithm problems (DLP) in the Jacobian of a curve C0 defined over the d degree extension field kd of k := Fq by mapping it to the DLP in the Jacobian of a covering curve C of C0 over k. Recently, classifications for all elliptic curves and hyperelliptic curves C0/kd of genus 2,3 which possess (2, ..., 2)-covering C/k of P were shown under an isogeny condition (i.e. when g(C) = d · g(C0)). This paper presents a systematic classification procedure for hyperelliptic curves in the odd characteristic case. In particular, we show a complete classification of elliptic curves C0 over kd which have (2, ..., 2)-covering C/k of P for d = 2, 3, 5, 7. It has been reported by Diem[6] that the GHS attack fails for elliptic curves C0 over odd characteristic definition field kd with prime extension degree d greater than or equal to 11 since g(C) become very large. Therefore, for elliptic curves over kd with prime extension degree d, it is sufficient to analyze cases of d = 2, 3, 5, 7. As a result, a complete list of all elliptic curves C0/k which possess (2, ..., 2)-covering C/k of P thus are subjected to the GHS attack with odd characteristic and prime extension degree d is obtained.