论文标题
自由基异基因和模块化曲线
Radical isogenies and modular curves
论文作者
论文摘要
本文探讨了自由基同基因与模块化曲线之间的联系。激进的ISEGENES是公式的公式，用于计算Castryck，deptu和vercauteren在2020年在Asiacrypt中引入的固定小度$ n $的链链。与其他类似目的的激进等级配方相比，激进等级配方的一个重要优势是，它们消除了生成$ n $ n $ n $ n $ n Is kerne kerney的订单的需求。虽然最初使用以正常形式的椭圆形曲线开发了自由基的降血压配方，但Onuki和Moriya提出了蒙哥马利曲线上的激进的ISEGEN级$ 3 $和$ 4 $的激进等级等级公式，并试图使用增强的椭圆形和模块化曲线来获得更简单的自由基isenies。在本文中，我们将以正常形式的自由基异基因的原始设置转化为模块化曲线的语言。此外，我们解决了Onuki和Moriya引入的一个开放问题，内容涉及$ x_0（n）上的激进等级公式。
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 $X_0(N).$
