Explicit criterions for p-ary functions being non-bent

There has been only limited information on the existence of p-ary bent functions. Recently there has been a result by the authors on finding necessary conditions for the existence of regular p  -ary bent functions (from Zpn to ZpZp), where p is a prime. The general case of p-ary bent functions is, however, an open question for finding necessary conditions for their existence. In this paper we complete this open case. We state our main result in more detail. We find an explicit family of non-bent functions. We also show that there is no p-ary bent function f in n   variables with w(Mf)>n2 if n   is even (w(Mf)>n+32 if n is odd, respectively), and for a given nonnegative integer k there is no p-ary bent function f in n   variables with w(Mf)=n2−k (w(Mf)=n+32−k, respectively) for an even n≥Np,kn≥Np,k (for an odd n≥Np,kn≥Np,k, respectively), where Np,kNp,k is some positive integer which is explicitly determined and w(Mf)w(Mf) is some explicit value related to the power of each monomial of f. We point out that if f is not a p-ary bent function in n variables and g is any p-ary function in n   variables such that L(Gf)=GgL(Gf)=Gg for some CCZ-transformation LL of f, then g cannot be bent either. This shows that our result produces a larger family of non-bent functions.