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

The Hebrew University of Jerusalem, Bar-Ilan University

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

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

Keywords

Algebraic torus
Elliptic curve
Group scheme
Primality test

All Science Journal Classification (ASJC) codes

General Engineering
Theoretical Computer Science
Applied Mathematics
Algebra and Number Theory

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10.1016/j.ffa.2011.07.011

Cite this

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title = "Deterministic primality tests based on tori and elliptic curves",

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.",

keywords = "Algebraic torus, Elliptic curve, Group scheme, Primality test",

author = "Alexander Gurevich and Boris Kunyavskiǐ",

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.",

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AU - Gurevich, Alexander

AU - Kunyavskiǐ, Boris

N1 - 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.

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N2 - 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.

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.

KW - Algebraic torus

KW - Elliptic curve

KW - Group scheme

KW - Primality test

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DO - 10.1016/j.ffa.2011.07.011

M3 - مقالة

SN - 1071-5797

VL - 18

SP - 222

EP - 236

JO - Finite Fields and their Applications

JF - Finite Fields and their Applications

IS - 1

ER -

Deterministic primality tests based on tori and elliptic curves

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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