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On isogeny classes of Edwards curves over finite fields (1103.3381v1)
Published 17 Mar 2011 in math.NT and cs.CR
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.