Binary Kloosterman Sums Modulo 256 and Coefficients of the Characteristic Polynomial

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
$a$
.