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

 

A primality test for $ Kp^n+1$ numbers


Authors: José María Grau, Antonio M. Oller-Marcén and Daniel Sadornil
Journal: Math. Comp.
MSC (2010): Primary 11Y11, 11Y16, 11A51, 11B99
Published electronically: June 10, 2014
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Abstract: In this paper we generalize the classical Proth's theorem and the Miller-Rabin test for integers of the form $ N=Kp^n+1$. For these families, we present variations on the classical Pocklington's results and, in particular, a primality test whose computational complexity is $ \widetilde {O}(\log ^2 N)$ and, what is more important, that requires only one modular exponentiation modulo $ N$ similar to that of Fermat's test.


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Additional Information

José María Grau
Affiliation: Departamento de Matemáticas, Universidad de Oviedo, Avda. Calvo Sotelo, s/n, 33007 Oviedo, Spain
Email: grau@uniovi.es

Antonio M. Oller-Marcén
Affiliation: Centro Universitario de la Defensa de Zaragoza, Ctra. de Huesca, s/n, 50090 Zaragoza, Spain
Email: oller@unizar.es

Daniel Sadornil
Affiliation: Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, F. Ciencias, Avda de los Castros s/n, 39005 Santander, Spain
Email: sadornild@unican.es

DOI: http://dx.doi.org/10.1090/S0025-5718-2014-02849-4
PII: S 0025-5718(2014)02849-4
Received by editor(s): June 12, 2012
Received by editor(s) in revised form: May 13, 2013
Published electronically: June 10, 2014
Additional Notes: Daniel Sadornil was partially supported by the Spanish Government under projects MTM2010-21580-C02-02 and MTM2010-16051.
Article copyright: © Copyright 2014 American Mathematical Society