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Designs, Codes and Cryptography

, Volume 59, Issue 1–3, pp 223–230 | Cite as

On zeros of Kloosterman sums

Article

Abstract

A Kloosterman zero is a non-zero element of \({{\mathbb F}_q}\) for which the Kloosterman sum on \({{\mathbb F}_q}\) attains the value 0. Kloosterman zeros can be used to construct monomial hyperbent (bent) functions in even (odd) characteristic, respectively. We give an elementary proof of the fact that for characteristic 2 and 3, no Kloosterman zero in \({{\mathbb F}_q}\) belongs to a proper subfield of \({{\mathbb F}_q}\) with one exception that occurs at q = 16. It was recently proved that no Kloosterman zero exists in a field of characteristic greater than 3. We also characterize those binary Kloosterman sums that are divisible by 16 as well as those ternary Kloosterman sums that are divisible by 9. Hence we provide necessary conditions that Kloosterman zeros must satisfy.

Keywords

Kloosterman sum Bent function Hyperbent function Division polynomial 

Mathematics Subject Classification (2000)

11L05 94A60 

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsSimon Fraser UniversityBurnabyCanada
  2. 2.Department of Mathematics and StatisticsUniversity of VaasaVaasaFinland

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