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

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Let K(a) denote the Kloosterman sum on the finite field of order 2ⁿ. We give a simple characterization of K(a) modulo 16, in terms of the trace of a and one other function. We also give a characterization of 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.