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 BrezingWeng families of pairing friendly elliptic curves with small rho-value, and we improve on previously-known best rho-values of families [10] for the cases k = 16, 22, 28 and 46. Our construction uses xed discriminants.

Generating pairing-friendly elliptic curve parameters using sparse families

“…Complete families are studied in [1,2,12,23,24] and they are attractive for applications due to their small CM discriminant. However, in [6] it is recommended to use curves with large discriminant to avoid various attacks on the DLP.…”

“…Finally, it sets t(x) and y(x) as the polynomials mapping to ζ k +1 and (ζ k −1)/ √ −D respectively and q(x) = (t(x) 2 +Dy(x) 2 )/4. For more examples of this type of families, see [10,12,23,24].…”

A generating method of CM parameters of pairing-friendly abelian surfaces using Brezing-Weng family

“…Brezing과 Weng은 수체의 다항식 표현을 이용하 여 페어링 친화 타원곡선의 CM 파라미터를 생성하기 위한 다항식족(family of polynomials)을 찾는 방 법을 제안했고, Freeman [3], Yoon [4] [1,2,5].…”

Note on families of pairing-friendly elliptic curves with small embedding degree

“…There is only one known example of a complete family with ρ(t, r, q) = 1; it was constructed by Barreto and Naehrig [2] (Remark 2.7). Various approaches have been used to search for the ideal case; see, for example, [9], [11], [13], [15], [18], and [19].…”