A new method of choosing primitive elements for Brezing–Weng families of pairing-friendly elliptic curves
Kisoon Yoon · Journal of Mathematical Cryptology · 2014
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 ) ζ k $(a+b\sqrt{-D})\zeta _k$ for rational numbers a , b and a square-free positive integer D , where ζ k $\zeta _k$ 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.