On constructing parameterized families of pairing-friendly elliptic curves with ρ = 1

Abstract

The problem of constructing pairing-friendly elliptic curves is the key ingredients for implementing pairing-based cryptographic systems. In this paper, we aim at constructing such curves with ρ = 1. By offering a more generalized concept “parameterized families”, we propose a method for constructing parameterized families of pairing-friendly elliptic curves which can naturally include many existent (and even more new) families of curves without exhaustive survey. We demonstrate the utility of the method by constructing concrete parameterized family in the cases of embedding degree 3, 4 and 6. An interesting result is proved that all the possible quadratic families of pairing-friendly elliptic curves of desired embedding degrees satisfying ρ = 1 have been covered in our parameterized families. As a by-product, we also revisit the supersingular elliptic curves from a new perspective.