On q-ary Bent and Plateaued Functions

TL;DR

This paper investigates the properties and distances of q-ary bent and plateaued functions, providing bounds, exact counts, and constructions, with implications for their nonlinearity and correlation immunity.

Contribution

It establishes minimal Hamming distances, counts of functions at specific distances, and bounds on nonlinearity for q-ary plateaued functions, extending known results to larger q.

Findings

Minimal Hamming distance between regular q-ary bent functions is q^n.

Exact count of q-ary regular bent functions at distance q^n from a quadratic bent function.

Lower bounds on Hamming distances for binary and ternary plateaued functions, proven to be tight.

Abstract

We obtain the following results. For any prime $q$ the minimal Hamming distance between distinct regular $q$-ary bent functions of $2n$ variables is equal to $q^n$. The number of $q$-ary regular bent functions at the distance $q^n$ from the quadratic bent function $Q_n=x_1x_2+\dots+x_{2n-1}x_{2n}$ is equal to $q^n(q^{n-1}+1)\cdots(q+1)(q-1)$ for $q>2$. The Hamming distance between distinct binary $s$-plateaued functions of $n$ variables is not less than $2^{\frac{s+n-2}{2}}$ and the Hamming distance between distinctternary $s$-plateaued functions of $n$ variables is not less than $3^{\frac{s+n-1}{2}}$. These bounds are tight. For $q=3$ we prove an upper bound on nonlinearity of ternary functions in terms of their correlation immunity. Moreover, functions reaching this bound are plateaued. For $q=2$ analogous result are well known but for large $q$ it seems impossible. Constructions and some properties of $q$-ary plateaued functions are discussed.