**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 $q \equiv -1 \pmod{4}$, and $16$ if $q \equiv 1 \pmod{4}$.
Furthermore, we give formulae for the proportion of $d \in \F_q \setminus \{0,1\}$ for which the Edwards curve $E_d$ is complete or original, relative to the total number of $d$ in each isogeny class.

**Category / Keywords: **public-key cryptography / number theory

**Publication Info: **preprint

**Date: **received 16 Mar 2011, last revised 17 Mar 2011

**Contact author: **rgranger at computing dcu ie

**Available format(s): **PDF | BibTeX Citation

**Note: **Comments welcome.

**Version: **20110321:023954 (All versions of this report)

**Short URL: **ia.cr/2011/135

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