Generating Pairing-friendly curves with the cm equation of degree 1

Abstract

Refinements of the Brezing-Weng method have provided families of pairing-friendly curves with improved ρ-values by using non-cyclotomic polynomials that define cyclotomic fields. We revisit these methods via a change-of-basis matrix and completely classify a basis for a cyclotomic field to produce a family of pairing-friendly curves with a CM equation of degree 1. Using this classification, we propose a new algorithm to construct Brezing-Weng-like elliptic curves having the CM equation of degree 1, and we present new families of curves with larger discriminants. © 2009 Springer Berlin Heidelberg.