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

Abstract

Pairing-based cryptographic schemes require so-called pairing-friendly elliptic curves, which have special properties. The set of pairing-friendly elliptic curves that are generated by given polynomials form a complete family. Although a complete family with a ρ-value of 1 is the ideal case, there is only one such example that is known; this was given by Barreto and Naehrig (Lecture Notes in Computer Science, 3897, Springer, Berlin, 2006, pp. 319-331). 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 chose noncyclotomic families.