Generating Pairing-Friendly Curves with the CM Equation of Degree 1

Refinements of the Brezing-Weng method have provided families of pairing-friendly curves with improved rho-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.