Abstract
Elliptic curves over a finite field 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 , 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 need a fixed kth root of the unity in the underlying field. Moreover, when the elliptic curve admits distortion maps, an example is provided.
Acknowledgments
We thank the anonymous referee for the helpful comments and suggestions. An extended abstract including some results of this manuscript (only for elliptic curves with j-invariant 1728) was previously presented at the conference Recsi 2014, [Citation18].
Disclosure statement
No potential conflict of interest was reported by the authors.