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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

On the number of isogeny classes of pairing-friendly elliptic curves and statistics of MNT curves


Authors: Jorge Jiménez Urroz, Florian Luca and Igor E. Shparlinski
Journal: Math. Comp. 81 (2012), 1093-1110
MSC (2010): Primary 11G07, 11T71, 14H52
Published electronically: September 29, 2011
MathSciNet review: 2869051
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Abstract: We give an upper bound on the number of finite fields over which elliptic curves of cryptographic interest with a given embedding degree and small complex multiplication discriminant may exist, and present some heuristic arguments which indicate that this bound is tight. We also refine some heuristic arguments on the total number of so-called MNT curves with prime cardinalities which have been recently presented by various authors.


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Additional Information

Jorge Jiménez Urroz
Affiliation: Departamento de Matemática Aplicada IV, Universidad Politecnica de Catalunya, Barcelona, 08034, España
Email: jjimenez@ma4.upc.edu

Florian Luca
Affiliation: Instituto de Matemáticas, Universidad Nacional Autonoma de México, C.P. 58089, Morelia, Michoacán, México
Email: fluca@matmor.unam.mx

Igor E. Shparlinski
Affiliation: Department of Computing, Macquarie University, Sydney, NSW 2109, Australia
Email: igor.shparlinski@mq.edu.au

DOI: http://dx.doi.org/10.1090/S0025-5718-2011-02543-3
PII: S 0025-5718(2011)02543-3
Keywords: Elliptic curves, pairing based cryptography, embedding degree, MNT curves
Received by editor(s): February 23, 2010
Received by editor(s) in revised form: February 11, 2011
Published electronically: September 29, 2011
Article copyright: © Copyright 2011 American Mathematical Society