Ternary Kloosterman sums modulo 4

Garaschuk and Lisoněk (2008) in 3] characterised ternary Kloosterman sums modulo 4, leaving the cases $K(a)\equiv 1(\mathrm{mod}4)$ and $K(a)\equiv 3(\mathrm{mod}4)$ 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.