Binary Kloosterman sums using Stickelberger's theorem and the Gross-Koblitz formula

TL;DR

This paper characterizes binary Kloosterman sums modulo 16 and 64 using trace functions, Fourier analysis, Stickelberger's theorem, and the Gross-Koblitz formula, providing explicit residue class descriptions.

Contribution

It offers new explicit formulas for Kloosterman sums modulo 16 and 64 based on trace functions, enhancing understanding of their arithmetic properties.

Findings

Characterization of Kloosterman sums modulo 16 using trace and other functions.

Extension of the characterization to modulo 64 with the lifted trace.

Application of Fourier analysis, Stickelberger's theorem, and Gross-Koblitz formula in proofs.

Abstract

Let $\mathcal{K}(a)$ denote the Kloosterman sum on the finite field of order $2^n$. We give a simple characterization of $\mathcal{K}(a)$ modulo 16, in terms of the trace of $a$ and one other function. We also give a characterization of $\mathcal{K}(a)$ modulo 64 in terms of a different function, which we call the lifted trace. Our proofs use Fourier analysis, Stickelberger's theorem and the Gross-Koblitz formula.