On isogeny classes of Edwards curves over finite fields

2011/03/17 by Omran Ahmadi, Ahmadi, Omran, Robert Granger +1
Computer Science · Mathematics · #14G17 #14H52 #14K02 #Algebraic Geometry and Number Theory #Coding theory and cryptography #Cryptography and Residue Arithmetic #Cryptography and Security (cs.CR) #FOS: Computer and information sciences #FOS: Mathematics #Number Theory (math.NT) #cs.CR #math.NT #msc:14G17 #msc:14H52 #msc:14K02

paper · pdf · doi:10.48550/arxiv.1103.3381

27 pages

arxiv created 2011/03/17 · openalex publication_date 2011/03/17 · arxiv updated 2011/03/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ≡ -1 \pmod4, and 16 if q ≡ 1 \pmod4. Furthermore, we give formulae for the proportion of d ∈ \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.

Citations

• Edwards curves and CM curves

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