On isogeny classes of Edwards curves over finite fields

Omran Ahmadi; Robert Granger

Paper 2011/135

On isogeny classes of Edwards curves over finite fields

Omran Ahmadi and Robert Granger

Abstract

We count the number of isogeny classes of Edwards curves over finite fields, answering a question recently posed by Rezaeian and Shparlinski. We also show that each isogeny class contains a {\em complete} Edwards curve, and that an Edwards curve is isogenous to an {\em original} Edwards curve over $\F_{q}$ if and only if its group order is divisible by $8$ if $q \equiv - 1 (\mod 4)$ , and $16$ if $q \equiv 1 (\mod 4)$ . Furthermore, we give formulae for the proportion of $d \in \F_{q} ∖ {0, 1}$ for which the Edwards curve $E_{d}$ is complete or original, relative to the total number of $d$ in each isogeny class.

Note: Comments welcome.

Metadata

Available format(s): PDF
Category: Public-key cryptography
Publication info: Published elsewhere. preprint
Keywords: number theory
Contact author(s): rgranger @ computing dcu ie
History: 2011-03-21: received
Short URL: https://ia.cr/2011/135
License: CC BY

BibTeX

@misc{cryptoeprint:2011/135,
      author = {Omran Ahmadi and Robert Granger},
      title = {On isogeny classes of Edwards curves over finite fields},
      howpublished = {Cryptology {ePrint} Archive, Paper 2011/135},
      year = {2011},
      url = {https://eprint.iacr.org/2011/135}
}