Monomial and Quadratic Bent Functions over the Finite Fields of Odd Characteristic †

Abstract

We consider p-ary bent functions of the form f(x) = Trn Ès i=0 aixd i¡. A new class of ternary monomial regular bent function with the Dillon exponent is discovered. The existence of Dillon bent functions in the general case is an open problem of deciding whether a certain Kloosterman sum can take on the value −1. Also described is the general Gold-like form of a bent function that covers all the previously known monomial quadratic cases. We also discuss the (weak) regularity of our new as well as of known monomial bent functions and give the first example of a not weakly regular bent function. Finally we prove some criteria for an arbitrary quadratic functions to be bent.