Radical isogenies and modular curves

TL;DR

This paper connects radical isogenies with modular curves, translating formulas into modular curve language and solving an open problem related to radical isogeny formulas on $X_0(N)$, advancing computational methods in elliptic curve isogenies.

Contribution

It translates radical isogeny formulas into the framework of modular curves and resolves an open problem on radical isogenies on $X_0(N)$, enhancing theoretical understanding.

Findings

Radical isogeny formulas are expressed using modular curves.

An open problem on radical isogenies on $X_0(N)$ is solved.

The approach simplifies the computation of isogeny chains.

Abstract

This article explores the connection between radical isogenies and modular curves. Radical isogenies are formulas designed for the computation of chains of isogenies of fixed small degree $N$, introduced by Castryck, Decru, and Vercauteren at Asiacrypt 2020. One significant advantage of radical isogeny formulas over other formulas with a similar purpose is that they eliminate the need to generate a point of order $N$ that generates the kernel of the isogeny. While radical isogeny formulas were originally developed using elliptic curves in Tate normal form, Onuki and Moriya have proposed radical isogeny formulas of degrees $3$ and $4$ on Montgomery curves and attempted to obtain a simpler form of radical isogenies using enhanced elliptic and modular curves. In this article, we translate the original setup of radical isogenies in Tate normal form into the language of modular curves. Additionally, we solve an open problem introduced by Onuki and Moriya regarding radical isogeny formulas on $X_0(N).$