A new method of choosing primitive elements for Brezing–Weng families of pairing-friendly elliptic curves

Abstract

In this paper we present a new method of choosing primitive elements for Brezing–Weng families of pairing-friendly elliptic curves with small rho-values, and we improve on previously known best rho-values of families [J. Cryptology 23 (2010), 224–280], [Lecture Notes in Comput. Sci. 5209, Springer (2008), 126–135] for the embedding degrees k=16$k=16$, 22, 28, 40 and 46. We consider elements of the form (a+b-D)ζk$(a+b\sqrt{-D})\zeta _k$ for rational numbers a, b and a square-free positive integer D, where ζk$\zeta _k$ is a primitive k-th root of unity. We also investigate the conditions for an element of the proposed form to be suitable for our construction. Our construction uses fixed discriminants.

Keywords

• elliptic curves
• finite fields
• pairing-based cryptography
• complete families
• 14h52
• 11t71
• 11g20
• 94a60