Particularly Friendly Members of Family Trees

The last decade has witnessed many clever constructions of parameterized families of pairing-friendly elliptic curves that now enable implementors targeting a particular security level to gather suitable curves in bulk. However, choosing the best curves from a (usually very large) set of candidates belonging to any particular family involves juggling a number of efficiency issues, such as the nature of binomials used to construct extension fields, the hamming-weight of key pairing parameters and the existence of compact generators in the pairing groups. In light of these issues, two recent works considered the best families for k = 12 and k = 24 respectively, and detailed subfamilies that offer very efficient pairing instantiations. In this paper we closely investigate the other eight attractive families with 8 ≤ k < 50, and systematically sub-divide each family into its family tree, branching off until concrete subfamilies are highlighted that simultaneously provide highly-efficient solutions to all of the above computational issues.