On the non-idealness of cyclotomic families of pairing-friendly elliptic curves

TL;DR

This paper establishes lower bounds on the rho-values of cyclotomic families of pairing-friendly elliptic curves, demonstrating that these families cannot be ideal, thus highlighting limitations in their efficiency for cryptographic applications.

Contribution

It provides the first theoretical lower bounds for rho-values in cyclotomic families, showing their non-ideality and guiding future curve construction efforts.

Findings

Lower bounds on rho-values for cyclotomic families

Proof that these families are not ideal

Implications for cryptographic curve selection

Abstract

Let $k=2^mp^n$ for an odd prime $p$ and integers $m\ge 0$ and $n\ge 0$. We obtain lower bounds for the $\rho$-values of cyclotomic families of pairing-friendly elliptic curves with embedding degree $k$ and $r(x)=\Phi_k(x)$. Our bounds imply that none of these families are ideal.