In  this talk we investigate  the possibility of constructing bent functions over fields with odd characteristic. We show that the necessary and sufficient bent conditions for both the Boolean function of the form

f (x)=Tr_1^{2k} ( x^{ 2^k – 1} + a x^{r (2^k – 1)} ) and the associated mapping

F (x)=Tr_k^{2k} ( x^{2^k – 1} + a x^{ r (2^k – 1)} ), where F: GF( p^{2k} ) –> GF( p^k ), are very similar and can be expressed in terms of the image of a set V used in the direct sum decomposition of GF( p^{2k} ).

Furthermore, we observe that  multiple output bent functions are easily constructed using the Maiorana-McFarland method.