Heuristic counting of Kachisa-Schaefer-Scott curves

Abstract

Estimating the number of pairing-friendly elliptic curves is important for obtaining such a curve with a suitable security level and high efficiency. For 128-bit security level, M. Naehrig and J. Boxall estimated the number of Barreto-Naehrig (BN) curves. For future use, we extend their results to higher security levels, that is, to count Kachisa-Schaefer-Scott (KSS) curves with 192- and 224-bit security levels. Our efficient counting is based on a number-theoretic conjecture, called the Bateman-Horn conjecture. We verify the validity of using the conjecture and confirm that an enough amount of KSS curves can be obtained for practical use.