[Paper Review] On isogeny classes of Edwards curves over finite fields

Omran Ahmadi, Robert Granger|arXiv (Cornell University)|Mar 17, 2011

Cryptography and Residue Arithmetic20 references4 citations

TL;DR

This paper fully classifies the isogeny classes of Edwards curves over finite fields of odd characteristic by establishing a 2-isogeny between Edwards curves and Legendre curves, enabling exact counting of isogeny classes and proving that every isogeny class contains a complete Edwards curve. Key results include criteria for isogeny to original Edwards curves based on group order divisibility and explicit formulae for the proportion of parameters yielding complete or original curves within each isogeny class.

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.

Motivation & Objective

• To resolve a recent open question on counting isogeny classes of Edwards curves over finite fields.
• To determine conditions under which an Edwards curve is isogenous to an original Edwards curve over $\mathbb{F}_q$.
• To compute the proportion of parameters $d \in \mathbb{F}_q \setminus \{0,1\}$ that yield complete or original Edwards curves within each isogeny class.
• To establish a structural link between Edwards curves and Legendre curves via explicit 2-isogenies for isogeny class enumeration.

Proposed method

• Establish a 2-isogeny between Edwards curves $E_d$ and Legendre curves $L_d$ over $\mathbb{F}_q$, leveraging character sum identities and Tate's theorem.
• Use Katz's recent classification of isogeny classes of Legendre curves to derive the number of isogeny classes of Edwards curves.
• Apply Deuring-style class number formulae to compute the number of complete Edwards curve parameters $d$ per isogeny class.
• Construct explicit bijections between parameter sets of different curve types (e.g., complete, original) to derive proportion formulae.
• Utilize the fact that $E_d$ is 4-isogenous to $E_{1-d}, E_{1/d}, E_{1-1/d}, E_{d/(d-1)}$ depending on quadratic characters in $\mathbb{F}_q$.
• Leverage known isomorphisms between Huff curves and Edwards curves to extend results to related curve families.

Experimental results

Research questions

• RQ1 How many isogeny classes of Edwards curves exist over a finite field $\mathbb{F}_q$ of odd characteristic?
• RQ2 Under what conditions is an Edwards curve isogenous to an original Edwards curve over $\mathbb{F}_q$?
• RQ3 What proportion of parameters $d \in \mathbb{F}_q \setminus \{0,1\}$ yield complete Edwards curves within a given isogeny class?
• RQ4 What proportion of parameters yield original Edwards curves, and how does this depend on $q \mod 4$?
• RQ5 Can the isogeny structure of Edwards curves be fully characterized via their connection to Legendre curves?

Key findings

• The number of isogeny classes of Edwards curves over $\mathbb{F}_q$ is determined by the isogeny class structure of Legendre curves, which is fully classified via Katz's results.
• Every isogeny class of Edwards curves contains at least one complete Edwards curve, and the number of such $d$ values per isogeny class can be computed using a Deuring-style class number formula.
• An Edwards curve $E_d$ over $\mathbb{F}_q$ is isogenous to an original Edwards curve if and only if its group order is divisible by 8 when $q \equiv -1 \pmod{4}$, and by 16 when $q \equiv 1 \pmod{4}$.
• For $q \equiv -1 \pmod{4}$, the proportion of $d$ yielding complete Edwards curves is determined by the same formula as for original curves; for $q \equiv 1 \pmod{4}$, the proportion is derived using Katz's ratio theorems.
• The proportion of $d \in \mathbb{F}_q \setminus \{0,1\}$ for which $E_d$ is complete is explicitly computable per isogeny class using the Deuring formula and the 2-isogeny to Legendre curves.
• The paper establishes that $E_d$ and $E_{1-d}$ are 4-isogenous over $\mathbb{F}_q$ when $q \equiv 1 \pmod{4}$, and more generally, $E_d$ is 4-isogenous to five other Edwards curves depending on the quadratic characters of $-1$, $d$, and $1-d$.