On certain values of Kloosterman sums

TL;DR

This paper proves the subfield conjecture for Kloosterman sums over finite fields of characteristic greater than 3, showing that certain sums do not equal -1 when the element belongs to a proper subfield, with implications for bent functions.

Contribution

The paper proves the subfield conjecture for Kloosterman sums in characteristic p>3 and demonstrates the irreducibility of a large class of Dickson polynomial translates.

Findings

Confirmed the subfield conjecture for p>3

Established irreducibility of certain Dickson polynomial translates

Extended previous results by Shparlinski, Moisio, and Lisonek

Abstract

Let $K_{q^n}(a)$ be a Kloosterman sum over the finite field $\F_{q^n}$ of characteristic $p$. In this note so called subfield conjecture is proved in case $p>3$: if $a\ne0$ belongs to the proper subfield $\F_q$ of $\F_{q^n}$, then $K_{q^n}(a)\ne-1$. This completes recent works on the subfield conjecture by Shparlinski, and Moisio and Lisonek. The problem is motivated by some applications to bent functions. Moreover, in the course of the proof a large class of translates of Dickson polynomials are shown to be irreducible.