Geodesic
The Divisibility Modulo 4 of Kloosterman Sums over Finite Fields of Characteristic 3
Serdica Journal of Computing, Tome 5 (2011) no. 1, pp. 1-14.

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Recently Garashuk and Lisonek evaluated Kloosterman sums K (a) modulo 4 over a finite field F3m in the case of even K (a). They posed it as an open problem to characterize elements a in F3m for which K (a) ≡ 1 (mod4) and K (a) ≡ 3 (mod4). In this paper, we will give an answer to this problem. The result allows us to count the number of elements a in F3m belonging to each of these two classes.
Keywords: Kloosterman Sums, Divisibility, Exponential Sum 
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Sin, Changhyon. The Divisibility Modulo 4 of Kloosterman Sums over Finite Fields of Characteristic 3. Serdica Journal of Computing, Tome 5 (2011) no. 1, pp. 1-14. http://geodesic.mathdoc.fr/item/SJC_2011_5_1_a0/