Divisibility properties of classical binary Kloosterman sums

Let K(a)K(a) be the so-called classical Kloosterman sum over F2m. In this paper, we compute K(a)K(a) modulo 24 for even mm, completing our previous results for odd mm. We extensively study the links between K(a)K(a) and other exponential sums, especially the cubic sums. We point out (as we did for odd mm) that the values K(a)K(a) are involved in the computation of the weight distributions of cosets of primitive narrow sense extended BCH codes of length 2m2m and minimum distance 8. We also complete some recent results on K(a)−1K(a)−1 modulo 3.