Connecting Legendre with Kummer and Edwards
type: Item
instance of: scholarly article
title: Connecting Legendre with Kummer and Edwards
zbMATH Open document ID: 1409.14057
author: Sabyasachi Karati
author: Palash Sarkar
published in: Advances in Mathematics of Communications
publication date: 2019-02-15T00:00:00Z
review text: The paper under review deals with the scalar multiplication over Legendre form elliptic curves which is a basic operation in many public key cryptographic schemes. The bulk of the computation can be performed either on the associated Kummer line or on an appropriate twisted Edwards form elliptic curve. The first approach requires a method for recovering the $y$-coordinate of the result from the Kummer line computation. In this paper, a detailed explicit formula for doing this is provided. Furthermore three conversion methods from Legendre form curves to appropriate twisted Edwards form curves are given. The two methods are constructions of birational equivalences while the third a 2-isogeny. Furthermore, these methods are applied in order to obtain new concrete twisted Edwards curves from Legendre curves which correspond to known Kummer lines at the 128-bit security level. These Kummer lines provide very fast scalar multiplication on modern architectures supporting SIMD operations.
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 11T71
Mathematics Subject Classification ID: 14G50
Mathematics Subject Classification ID: 94A60
zbMATH DE Number: 7024180
zbMATH Keywords: elliptic curves
zbMATH Keywords: Legendre form
zbMATH Keywords: Kummer line
zbMATH Keywords: Edwards form
reviewed by: Dimitros Poulakis
describes a project that uses: KummerLineV02
describes a project that uses: EFD
describes a project that uses: Curve25519
describes a project that uses: GitHub
MaRDI profile type: Publication
full work available at URL: amc.2019003
OpenAlex ID: W2794919770
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DOI: 10.3934/AMC.2019003
Wikidata QID: Q128709370
description: scientific article; zbMATH DE number 7024180