F. Gologlu, P. Lisonek, G. McGuire, R. Moloney,
Binary Kloosterman sums modulo 256 and coefficients of the characteristic polynomial.
IEEE Transactions on Information Theory 58 (2012), 2516-2523.

Kloosterman sums are exponential sums on finite fields that have important applications in cryptography and coding theory. We use Stickelberger's theorem and the Gross-Koblitz formula to determine the value of the binary Kloosterman sum at $a$ modulo 64, modulo 128 and modulo 256 in terms of coefficients of the characteristic polynomial of $a$.

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