On the ρ-values of complete families of pairing-friendly elliptic curves

Abstract

The parameter ρ of a complete family of pairing-friendly elliptic curves represents how suitable some given elliptic curves are in pairing-based cryptographic schemes. The superiority of the curves depends on how close ρ is to 1. Especially, the situation is ideal. We study whether a complete family of pairing-friendly elliptic curves has a ρ-value 1 or not. In this paper, we consider the original idea of Brezing and Weng that we put a parameter for constructing families of elliptic curves as cyclotomic polynomials. We show that the ρ-values cannot be 1 in many cases.

Keywords

• elliptic curves
• pairing-based cryptography
• embedding degree