Note on families of pairing-friendly elliptic curves with small embedding degree

Abstract

The set of pairing-friendly elliptic curves that are generated by given polynomials forms a complete family. Although a complete family with a $\rho$-value of 1 is the ideal case, there is only one such example that is known. We prove that there are no ideal families with embedding degree 3, 4, or 6 and that many complete families with embedding degree 8 or 12 are nonideal, even if we choose noncyclotomic families.