Nonvanishing $k$-flats of Boolean and vectorial functions

TL;DR

This paper introduces a new combinatorial method to analyze nonvanishing flats in Boolean functions, revealing the abundance of higher-order sum-free functions with specific algebraic degrees.

Contribution

It develops a novel technique for studying nonvanishing flats and demonstrates the existence of numerous higher-order sum-free Boolean functions with particular degrees.

Findings

Determined the number of nonvanishing flats for an infinite family of Boolean functions.

Showed that certain sum-free functions of degree k lead to functions of degree n-k.

Proved the existence of millions of (n-2)th-order sum-free functions.

Abstract

$k$th-order sum-free functions are a natural generalization of APN functions using the concept of (non)vanishing flats. In this paper, we introduce a new combinatorial technique to study the nonvanishing flats of Boolean functions. This approach allows us to determine the number of nonvanishing flats for an infinite family of Boolean functions. We moreover use it to show that any $k$th-order sum-free $(n,n)$-function of algebraic degree $k$ gives rise to an $(n-k)$th-order sum-free $(n,n)$-function of algebraic degree $n-k$. This implies the existence of millions of $(n-2)$th-order sum-free functions.