Ternary Kloosterman sums modulo 4

Garaschuk and Lisoněk (2008) in [3] characterised ternary Kloosterman sums modulo 4, leaving the cases and as open problems. In this paper we complete the characterisation using well-known theorems on Gauss sums and Kloosterman sums. We also give the number of elements satisfying these congruences.