Constructing supersingular elliptic curves with a given endomorphism ring
type: Item
instance of: scholarly article
title: Constructing supersingular elliptic curves with a given endomorphism ring
zbMATH Open document ID: 1326.11029
DOI: 10.1112/S1461157014000254
author: Ilya Chevyrev
author: Steven D. Galbraith
published in: LMS Journal of Computation and Mathematics
publication date: 2014-09-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1301.6875
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 11Y16
Mathematics Subject Classification ID: 11R52
Mathematics Subject Classification ID: 11E20
Mathematics Subject Classification ID: 11H55
Mathematics Subject Classification ID: 14G15
zbMATH DE Number: 6340488
zbMATH Keywords: endomorphism ring
zbMATH Keywords: supersingular elliptic curve
MaRDI profile type: Publication
Wikidata QID: Q61914013
arXiv ID: 1301.6875
OpenAlex ID: W3105729405
review text: The authors consider two computational problems related to supersingular elliptic curves defined over prime fields \(\mathbb{F}_p\) and their endomorphism rings. The first is: given a maximal order \(\mathcal{O}\) in the quaternion algebra \(B_p\) over \(\mathbb{Q}\) ramified at \(p\) and \(\infty,\) construct a supersingular elliptic curve \(E\) over \(\mathbb{F}_p\) such that \(\mathrm{End}(E) \cong \mathcal{O}.\) The second is: given a prime \(p,\) find for each distinct maximal order type \(\mathcal{O}_i\) of \(B_p\) the minimal polynomial of the supersingular \(j\)-invariant with endomorphism rign \(\mathcal{O}_i.\) Novel algorithms are given for both problems with running time \(O(p^{1 + \varepsilon})\) and \(O(p^{2.5 + \varepsilon})\) respectively, in both cases improving previous results.
reviewed by: Michael J. Jacobson jun.
description: scientific article; zbMATH DE number 6340488