On generalized bent functions with Dillonʼs exponents

Abstract

In this paper we investigate the possibility of constructing bent functions over fields with odd characteristic. While in the binary case, and for n=2k, the bent property of monomials of the form Tr1n(axr(2k−1)) and binomials Tr1n(x2k−1+axr(2k−1)) were investigated in several papers, generalized bent functions f:GF(pn)→GF(p) of the form Tr1n(∑i=1taixri(pk−1)), p being an odd prime and n=2k, were not analyzed previously. In particular, the construction of vectorial (generalized) bent functions has not been addressed. It is shown that the necessary and sufficient bent conditions for both the single output function of the form f(x)=Tr1n(∑i=1taixri(pk−1)) and the associated mapping F(x)=Trk2k(∑i=1taixri(pk−1)), where F:GF(p2k)→GF(pk), are very similar and can be expressed in terms of the image of a set V used in the direct sum decomposition of GF(p2k). Furthermore, it is observed that vectorial bent functions are easily constructed using the Maiorana–McFarland method.