Characterizations of Bent and Almost Bent Function on ℤ²_p

Bent and almost-bent functions on ℤ²_p are studied in this paper. By calculating certain exponential sum and using a technique due to Hou (Finite Fields Appl 10:566-582, 2004), we obtain a degree bound for quasi-bent functions, and prove that almost-bent functions on ℤ²_p are equivalent to a degenerate quadratic form. From the viewpoint of relative difference sets, we also characterize bent functions on ℤ²_p in two classes of M's and PS's, and show that the graph set corresponding to a bent function on ℤ²_p can be written as the sum of a graph set of M's type bent function and another group ring element. By using our characterization and some technique of permutation polynomial, we obtain the result: a bent function must be of M's type if its corresponding set contains more than (p - 3)/2 flats. A problem proposed by Ma and Pott (J Algebra 175:505-525, 1995) is therefore partially answered.