The divisibility modulo 24 of Kloosterman sums on $GF(2^m)$, $m$ even

In a recent work by Charpin, Helleseth, and Zinoviev Kloosterman sums $K(a)$ over a finite field $\F_{2^m}$ were evaluated modulo 24 in the case $m$ odd, and the number of those $a$ giving the same value for $K(a)$ modulo 24 was given. In this paper the same is done in the case $m$ even. The key techniques used in this paper are different from those used in the aforementioned work. In particular, we exploit recent results on the number of irreducible polynomials with prescribed coefficients.