Divisibility properties of classical binary Kloosterman sums

Let K(a) be the so-called classical Kloosterman sum over F(2)m. In this paper, We compute K(a) modulo 24 for even m, completing our previous results for odd in. We extensively study the links between K(a) and other exponential sums, especially the Cubic Sums. We point Out (as we did for odd m) that the values K(a) are involved in the computation of the weight distributions of cosets of primitive narrow sense extended BCH codes of length 2(m) and minimum distance 8. We also complete some recent results on K(a) - 1 modulo 3. (C) 2008 Elsevier B.V. All rights reserved.