In 2010, Freeman, Scott, and Teske published a well-known taxonomy compiling the
best known families of pairing-friendly elliptic curves. Since then, the
research effort mostly shifted from the generation of pairing-friendly curves to
the improvement of algorithms or the assessment of security parameters to resist
the latest attacks on the discrete logarithm problem. Consequently, very few new
families were discovered. However, the need of pairing-friendly curves of prime
order in some new applications such as SNARKs has reignited the interest in the
generation of pairing-friendly curves, with hope of finding families similar to
the one discovered by Barreto and Naehrig.
Building on the work of Kachisa, Schaefer, and Scott, we show that some elements
of extensions of a cyclotomic field have a higher probability of generating a
family of pairing-friendly curves. We present a general framework which embraces
the KSS families and many of the other families in the taxonomy paper. We finally
introduce a new family with embedding degree k=20 which we estimate to provide
a faster Miller loop compared to KSS16 and KSS18 at the 192-bit security level.