On near Prime-Order Elliptic curves with small embedding degrees

Abstract

In this paper, we extend the method of Scott and Barreto and present an explicit and simple algorithm to generate families of generalized MNT elliptic curves. Our algorithm allows us to obtain all families of generalized MNT curves with any given cofactor. Then, we analyze the complex multiplication equations of these families of curves and transform them into generalized Pell equations. As an example, we describe a way to generate Edwards curves with embedding degree 6, that is, elliptic curves having cofactor h = 4.