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**2011/135**

**Title:**

On isogeny classes of Edwards curves over finite fields

**Authors:**
Omran Ahmadi, 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.

**ePrint:**
https://eprint.iacr.org/2011/135

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