Effective Polynomial Families for Generating More Pairing-Friendly Elliptic Curves

Finding suitable non-supersingular elliptic curves becomes an important issue for the growing area of pairing-based cryptosystems. For this purpose, many methods have been proposed when embedding degree k and cofactor h are taken different values. In this paper we propose a new method to find pairing-friendly elliptic curves without restrictions on embedding degree k and cofactor h. We propose the idea of effective polynomial families for finding the curves through different kinds of Pell equations or special forms of D(x)V(x). In addition, we discover some efficient families which can be used to build pairing-friendly elliptic curves over extension fields, e.g. Fp and Fp.