Elliptic curves with j = 0, 1728 and low embedding degree. Issue 12 (1st December 2016)

• Record Type: Journal Article
• Authors: Miret, J.M.; Sadornil, D.; Tena, J.
• Is Part Of: International journal of computer mathematics. Volume 93:Issue 12(2016)
• Page Start: 2042
• Page End: 2053
• Publication Date: 2016-12-01
• Subjects: elliptic curves -- embedding degree -- distortion maps -- pairing-based Cryptography -- Bateman-Horn's conjecture; 14H52 -- 94A60; Computers -- Periodicals; Numerical analysis -- Periodicals; Automation -- Periodicals; 004.0151
• DOI: 10.1080/00207160.2015.1083556
• Languages: English
• ISSNs: 0020-7160

Abstract

Abstract : Elliptic curves over a finite field F q with j -invariant 0 or 1728, both supersingular and ordinary, whose embedding degree k is low are studied. In the ordinary case we give conditions characterizing such elliptic curves with fixed embedding degree with respect to a subgroup of prime order ℓ. For k = 1, 2, these conditions give parameterizations of q in terms of ℓ and two integers m, n . We show several examples of families with infinitely many curves. Similar parameterizations for k ≥ 3 need a fixed k th root of the unity in the underlying field. Moreover, when the elliptic curve admits distortion maps, an example is provided.