On integer values of Kloosterman sums

Authors:
Keijo Petteri Kononen;Marko Juhani Rinta-Aho;Keijo O. Väänänen
Affiliations:
Department of Mathematical Sciences, University of Oulu, Oulun, Yliopisto, Finland;Department of Mathematical Sciences, University of Oulu, Oulun, Yliopisto, Finland;Department of Mathematical Sciences, University of Oulu, Oulun, Yliopisto, Finland
Venue:
IEEE Transactions on Information Theory
Year:
2010

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Abstract

This paper considers rational integer values of Kloosterman sums over finite fields of characteristic p 3. We shall prove two main results. The first one is a congruence relation satisfied by possible integer values. One consequence is that there are no Kloosterman zeroes in the case of characteristic p 3, which generalizes recent works by Shparlinski, Moisio, and Lisonek on this subject. This, in turn, implies that there are no Dillon type bent functions in the case p 3, thus answering a question posed recently by Helleseth and Kholosha. Our other main result states that the Kloosterman sum obtains an integer value at a point if and only if the same sum lifted to any extension field remains an integer.