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 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.

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