Parametric families of elliptic curves with cyclic \(\mathbb F_p\)-rational points groups
type: Item
instance of: scholarly article
title: Parametric families of elliptic curves with cyclic \(\mathbb F_p\)-rational points groups
zbMATH Open document ID: 1171.11316
author: Naoya Nakazawa
published in: Tokyo Journal of Mathematics
publication date: 2006-03-29T00:00:00Z
review text: The author constructs some families of elliptic curves without complex multiplication over \(\mathbb{Q}\) such that for prime numbers of the form \(p=2^\alpha 3^\beta 5^\gamma q^\delta +1\) (\(q\) an odd prime) the groups \(\tilde{E}(\mathbb{F}_p)\) of the reductions of \(E\) modulo \(p\) are cyclic. Such families can be useful for cryptographic purposes. The constructions and proofs are based on a result by \textit{R. Gupta} and \textit{M. R. Murty} [Invent. Math. 101, No. 1, 225--235 (1990; Zbl 0731.14011)] which asserts, in particular, that for an elliptic curve over \(\mathbb{Q}\) with good reduction at prime \(p\) the group \(\tilde{E}(\mathbb{F}_p)\) is cyclic if and only if for any prime \(l\) \(p\) does not split completely in the field \(K_l(E)\) generated over \(\mathbb{Q}\) by all the \(l\)-division points of \(E\). Many relevant examples are included in the paper.
reviewed by: Vasyl I. Andriychuk
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14H25
zbMATH DE Number: 5015935
zbMATH Keywords: elliptic curves over number and finite fields
zbMATH Keywords: elliptic cryptography
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1244208196
OpenAlex ID: W2049972889
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DOI: 10.3836/TJM/1244208196
description: scientific article; zbMATH DE number 5015935