Analysis of Optimum Pairing Products at High Security Levels

Abstract

In modern pairing implementations, considerable researches target at the optimum pairings at different security levels. However, in many cryptographic protocols, computing products or quotients of pairings is needed instead of computing single pairings. In this paper, we mainly analyze the computations of fast pairings on Kachisa-Schaefer-Scott curves with embedding degree 16 (KSS16) for the 192-bit security and Barreto-Lynn-Scott curves with embedding degree 27 (BLS27) for the 256-bit security, and then compare the cost estimations for implementing products and quotients of pairings at the 192 and 256-bit security levels. Being different from implementing single pairings, our results show that KSS16 curves could be most efficient for computing products or quotients of pairings for the 192-bit security; and for the 256-bit security, BLS27 curves might be more efficient for computing products of no less than 25 pairings, otherwise BLS24 curves are much more efficient. In addition, for the fast pairing computation on BLS27 curves, we propose faster Miller formulas in both affine and projective coordinates on curves with only cubic twist and embedding degree divisible by 3.