Deterministic primality tests based on tori and elliptic curves

Alexander Gurevich; Boris Kunyavskiǐ

doi:10.1016/j.ffa.2011.07.011

Deterministic primality tests based on tori and elliptic curves

Alexander Gurevich, Boris Kunyavskiǐ^*

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

We develop a general framework for producing deterministic primality tests based on commutative group schemes over rings of integers. Our focus is on the cases of algebraic tori and elliptic curves. The proposed general machinery provides several series of tests which include, as special cases, tests discovered by Gross and by Denomme and Savin for Mersenne and Fermat primes, primes of the form 2 ^2l+1-2 ^l+1, as well as some new ones.

Original language	English
Pages (from-to)	222-236
Number of pages	15
Journal	Finite Fields and Their Applications
Volume	18
Issue number	1
DOIs	https://doi.org/10.1016/j.ffa.2011.07.011
State	Published - Jan 2012

Bibliographical note

Funding Information:
Kunyavski˘ı was supported in part by the Minerva foundation through the Emmy Noether Research Institute. This work was finished when he was visiting the MPIM (Bonn) in September 2010. The support of these institutions is gratefully appreciated. We thank the referees for careful reading and thoughtful critical remarks.

Keywords

Algebraic torus
Elliptic curve
Group scheme
Primality test

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10.1016/j.ffa.2011.07.011

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abstract = "We develop a general framework for producing deterministic primality tests based on commutative group schemes over rings of integers. Our focus is on the cases of algebraic tori and elliptic curves. The proposed general machinery provides several series of tests which include, as special cases, tests discovered by Gross and by Denomme and Savin for Mersenne and Fermat primes, primes of the form 2 2l+1-2 l+1, as well as some new ones.",

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AB - We develop a general framework for producing deterministic primality tests based on commutative group schemes over rings of integers. Our focus is on the cases of algebraic tori and elliptic curves. The proposed general machinery provides several series of tests which include, as special cases, tests discovered by Gross and by Denomme and Savin for Mersenne and Fermat primes, primes of the form 2 2l+1-2 l+1, as well as some new ones.

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KW - Elliptic curve

KW - Group scheme

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Deterministic primality tests based on tori and elliptic curves

Abstract

Bibliographical note

Keywords

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