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

Abstract

Let $k=2^m{p}^n$ 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.