Abstract
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 <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$a$</tex></formula> modulo 64, modulo 128, and modulo 256 in terms of coefficients of the characteristic polynomial of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$a$</tex> </formula> .
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