Equivalences between elliptic curves and real quadratic congruence function fields
type: Item
instance of: scholarly article
title: Equivalences between elliptic curves and real quadratic congruence function fields
zbMATH Open document ID: 0899.11054
author: Andreas Stein
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1998-03-25T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93688
full work available at URL: jtnb9-1.html
review text: The Diffie-Hellman key exchange protocol has previously been considered in the principal ideal class of a real quadratic number field as well as real quadratic congruence function fields. This set does not possess a group structure but rather an infrastructure. The security of the scheme depends on that of the discrete logarithm problem in this setting. This article shows that the discrete logarithm problem in real quadratic congruence function fields of genus 1, called real elliptic congruence function fields, is equivalent to that for elliptic curves. The properties and arithmetic of the set of reduced principal ideals in elliptic congruence function fields are discussed. The connection between elliptic curves and real quadratic congruence function fields is made. The one-to-one correspondence between the set of reduced principal ideals of an elliptic congruence function field and the group \(\langle{\mathcal P}\rangle/\{ {\mathcal P} \}\) is made, where \({\mathcal P}\) denotes an \({\mathbb{F}}_q\)-rational point on the corresponding elliptic curve. Several questions of importance that arise in establishing this correspondence are also discussed.
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11T71
Mathematics Subject Classification ID: 94A60
Mathematics Subject Classification ID: 11Y16
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 1132889
zbMATH Keywords: Diffie-Hellman key exchange
zbMATH Keywords: discrete logarithm problem
zbMATH Keywords: real quadratic congruence function fields
zbMATH Keywords: elliptic curves
reviewed by: Ian Blake
MaRDI profile type: Publication
OpenAlex ID: W2062451877
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DOI: 10.5802/JTNB.191
description: scientific article; zbMATH DE number 1132889