Distorting the volcano
type: Item
instance of: scholarly article
title: Distorting the volcano
zbMATH Open document ID: 1387.14091
author: Mireille Fouquet
author: Javier Valera
author: Josep M. Miret Biosca
published in: Finite Fields and their Applications
publication date: 2017-11-21T00:00:00Z
full work available at URL: http://hdl.handle.net/10459.1/62669
review text: This paper deals with ordinary elliptic curves over a finite field \(\mathbb{F}_q\) of characteristic \(p\) whose \(j\)-invariants are different from \(0\) and \(1728\). For a prime \(\ell\) with \(\ell\nmid q\) the volcanos of \(\ell\)-isogenies of such curves are considered. For a given elliptic curve \(E\) of a volcano of \(\ell\)-isogenies the question is how to determine the non-descending \(\ell\)-isogenies of \(E\). \textit{S. Ionica} and \textit{A. Joux}, [Math. Comput. 82, No. 281, 581--603 (2013; Zbl 1278.11067)] answered this question for a regular volcano or up to a second level of stability of an arbitrary volcano. In the paper under review the authors give an answer which works beyond the second stability level and does not need computing paths of \(\ell\)-isogenies. Denoting by \(D\) the discriminant of the order being isomorphic to the endomorphism ring of a given elliptic curve \(E\), they suppose \(\ell\neq 2,3, E[\ell]\subseteq E(\mathbb{F}_q)\), and \((\frac{D}{m})=1\), with \(m\neq \ell\). Then they consider a volcano of \(m\)-isogenies where \(E\) belongs to the crater and construct an endomorphism \(\varphi\) of \(E\) consisting of a cycle of \(m\)-isogenies of the crater and an isomorphism. They show: If \(P\) is a point of order \(\ell\) of \(E(\mathbb{F}_q)\) and \(\varphi\) is a distortion map for the subgroup \(\langle P\rangle\), then the \(\ell\)-isogeny with kernel \(\langle P\rangle\) is non-descending (Theorem 4.4). As a corollary they show that \(\varphi\) is a distortion map either for all kernels of the descending \(\ell\)-isogenies or for none of them. Further, they extend a result of \textit{D. Moody} (Proposition 5 in [Appl. Math. Comput. 218, No. 9, 5249--5258 (2012; Zbl 1251.14021)]) by comparing the crater size of volcanos of \(m\)-isogenies of two elliptic curves which belong to a given volcano of \(\ell\)-isogenies (Proposition 5.1). In addition, they present two algorithms: Taking the assumptions of Theorem 4.4 and the re…
reviewed by: Thorsten Herrig
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14K02
zbMATH DE Number: 6810611
zbMATH Keywords: elliptic curve
zbMATH Keywords: finite field
zbMATH Keywords: isogeny
zbMATH Keywords: distortion map
zbMATH Keywords: volcano
MaRDI profile type: Publication
OpenAlex ID: W2758643397
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DOI: 10.1016/J.FFA.2017.09.006
description: scientific article; zbMATH DE number 6810611