Two kinds of division polynomials for twisted Edwards curves
type: Item
instance of: scholarly article
title: Two kinds of division polynomials for twisted Edwards curves
zbMATH Open document ID: 1264.11046
author: Richard Moloney
author: Gary McGuire
published in: Applicable Algebra in Engineering, Communication and Computing
publication date: 2012-06-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/0907.4347
review text: Division polynomials for elliptic curves are polynomials which have as roots the \(n\)-torsion points. The authors give two kinds of division polynomials for the twisted Edwards \(E_{a,d}\) curves defined as \[ ax^2+y^2=1+dx^2y^2. \] The first approach uses the birational transformation from elliptic curves in Weierstrass form to elliptic curves in Edwards form and the computations for the \(n\)-th power in Weierstrass form, in order to obtain an analogous computation for the \(n\)-th power of a point on an elliptic curve in Edwards form. From this computation the authors arrive at polynomials \(\tilde{\psi}\in \mathbb{Z}[a,d][y]\), which have as roots the \(n\)-th torsion points of the Edwards curve. Their degree is bounded by \(n^2/2\). In the second approach the authors give polynomials that can be used to give formulas for multiplication. The method is motivated by the work of Abel and Eisenstein on lemniscape sine functions. The \(n\)-th such polynomial is recursively expressed in terms of the \((n-1)\)th and \((n-2)\)th polynomial. Section 6 is devoted to the comparison of the \(E_{1,-1}\) curves, which were also studied by Gauss.
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14G05
zbMATH DE Number: 6048435
zbMATH Keywords: Elliptic curves
zbMATH Keywords: Edwards curves
zbMATH Keywords: division polynomials
reviewed by: Aristides Kontogeorgis
describes a project that uses: SageMath
MaRDI profile type: Publication
OpenAlex ID: W2137046350
arXiv ID: 0907.4347
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DOI: 10.1007/S00200-011-0153-5
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description: scientific article; zbMATH DE number 6048435