On the Values of Kloosterman Sums

Given a prime p and a positive integer n, we show that the shifted Kloosterman sums Sigma<sub>xisinF</sub> <sub>p</sub> <sub>n</sub>Psi(x + alphax<sup>pn-2</sup>)=Sigma<sub>xisinF*</sub> <sub>p</sub> <sub>n</sub>Psi(x+alphax<sup>-1</sup>)+1, alphaisinF*<sub>p</sub>n where Psi is a nontrivial additive character of a finite field F<sub>p</sub>n of p<sup>n</sup> elements, do not vanish if alpha belongs to a small subfield F<sub>p</sub>m sube F<sub>p</sub>n. This complements recent results of P. Charpin and G. Gong which in turn were motivated by some applications to bent functions.

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