On the normality of p-ary bent functions

Abstract

Depending on the parity of n and the regularity of a bent function f from Fn p to Fp, f can be affine on a subspace of dimension at most n/2, (n−1)/2 or n/2−1. We point out that many p-ary bent functions take on this bound, and it seems not easy to find examples for which one can show a different behaviour. This resembles the situation for Boolean bent functions of which many are (weakly) n/2-normal, i.e. affine on a n/2-dimensional subspace. However applying an algorithm by Canteaut et.al., some Boolean bent functions were shown to be not n/2-normal. We develop an algorithm for testing normality for functions from Fn p to Fp. Applying the algorithm, for some bent functions in small dimension we show that they do not take on the bound on normality. Applying direct sum of functions this yields bent functions with this property in infinitely many dimensions.