A remark on the MOV algorithm for non-supersingular elliptic curves

In recent years, the study of the security of Elliptic Curve Cryptosystems (ECCs) have been received much at attention. The MOV algorithm, which reduc es the elliptic curve discrete log problem (ECDLP) to the discrete log prob lem in finite fields with the Well pairing, is a representative attack on E CCs. Recently Kanayama et ai. observed a realization of the MOV algorithm f or non-supersingular elliptic curves under the weakest condition. Shikata e t al. independently considered a realization of the MOV algorithm for non-s upersingular elliptic curves and proposed a generalization of the MOV algor ithm. This short note explicitly shows that, under a usual cryptographical condition, me can apply the MOV algorithm to non-supersingular elliptic cur ves by using the multiplication by constant maps as in the case of supersin gular. Namely, it is explicitly showed that we don't need such a generaliza tion in order to realize the MOV algorithm for non-supersingular elliptic c urves under a usual cryptographical condition.