# On the normality of $p$-ary bent functions

## TL;DR

This paper investigates the normality properties of $p$-ary bent functions, develops an algorithm to test their normality, and finds examples that do not reach the theoretical bounds, extending results to infinitely many dimensions.

## Contribution

It introduces a new algorithm for testing normality of $p$-ary functions and demonstrates that some bent functions do not attain the known normality bounds, with implications for infinite dimensions.

## Key findings

- Some $p$-ary bent functions are not $n/2$-normal.
- The developed algorithm effectively tests normality.
- Existence of bent functions with non-bound normality in infinitely many dimensions.

## Abstract

Depending on the parity of $n$ and the regularity of a bent function $f$ from $\mathbb F_p^n$ to $\mathbb F_p$, $f$ 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 $\mathbb F_p^n$ to $\mathbb F_p$. 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.