Binary Kloosterman sums using Stickelberger's theorem and the Gross–Koblitz formula
Faruk Göloğlu, Gary McGuire, Richard Moloney · Acta Arithmetica · 2011
defined by K p n (a) := x∈F p n ζ Tr(x p n -2 +ax) , for any a ∈ F p n , where ζ is a primitive pth root of unity and Tr denotes the absolute trace map Tr : F p n → F p defined as usual as Tr(c) :Finding explicit zeros (explicit a's with K p n (a) = 0) of Kloosterman sums is considered difficult.Recent research on Kloosterman sums is generally concentrated on proving divisibility results and characterisation of Kloosterman sums modulo some integer (see [15,12,2,1,13]).It is easy to see that binary Kloosterman sums are divisible by 4 = 2 2 , i.e., for all a ∈ F 2 n , (1) K 2 n (a) ≡ 0 (mod 4).They also satisfy (see [8])and take every value which is congruent to 0 modulo 4 in that range.Helleseth and Zinoviev proved the following result which improved (1) one level higher, i.e., modulo 2 3 , in the sense of describing the a for which K 2 n (a) is 0 or 4 modulo 8. Theorem 1.1 ([5]).For a ∈ F 2 n , K 2 n (a) ≡ 0 (mod 8) if Tr(a) = 0, 4 (mod 8) if Tr(a) = 1.