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Nonvanishing $k$-flats of Boolean and vectorial functions

Christian Kaspers

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.

Nonvanishing $k$-flats of Boolean and vectorial functions

Abstract

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 th-order sum-free -function of algebraic degree gives rise to an th-order sum-free -function of algebraic degree . This implies the existence of millions of th-order sum-free functions.

Paper Structure

This paper contains 5 sections, 24 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nonvanishing $2$-flats and the Walsh spectrum
  4. A new technique to determine the nonvanishing $k$-flats of Boolean functions
  5. Applications to Boolean and vectorial functions

Key Result

Proposition 1

Let $k \in [n]$, $U \in \mathcal{U}_{n,k}$, and let $F$ be an $(n,m)$-function. If the algebraic degree of $F$ is $k$, then $\sum_{x \in U} F(x) = \sum_{x \in A} F(x)$ for all $A \in \mathcal{C}_U$.

Theorems & Definitions (40)

  • Proposition 1
  • proof
  • Corollary 1
  • Corollary 2: carlet2025generalizations
  • Corollary 3
  • Definition 1
  • Corollary 4
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • ...and 30 more