The divisibility modulo 24 of Kloosterman sums on GF(m2), m odd

In a previous paper, we studied the cosets of weight 4 of binary extended 3-error-correcting BCH codes of length m2 (where m is odd). We expressed the number of codewords of weight 4 in such cosets in terms of exponential sums of three types, including the Kloosterman sums K(a), a∈F∗. In this paper, we derive some congruences which link Kloosterman sums and cubic sums. This allows us to study the divisibility of Kloosterman sums modulo 24. More precisely, if we know the traces of a and of a1/3, we are able to evaluate K(a) modulo 24 and to compute the number of those a giving the same value of K(a) modulo 24.