A Note on Vectorial Boolean Functions as Embeddings

TL;DR

This work analyzes vectorial Boolean functions $F:\mathbb{F}_2^n\to\mathbb{F}_2^m$ with $m\ge n$, focusing on the distribution of balanced component functions. It establishes a tight bound: at most $2^m-2^{m-n}$ components can be balanced, with equality exactly for embeddings, leaving $2^{m-n}$ constant components. For quadratic or partially-bent embeddings, it proves high balancedness: at least $2^n-1$ balanced components when $n$ is even, or at least $2^{m-1}+2^{n-1}-1$ when $n$ is odd, with further refinements tied to the bent/semi-bent status of other components. The paper also connects these embedding properties to APN functions, showing that derivative embeddings arising in cubic APN cases yield strong lower bounds on balanced components, which may illuminate why certain APN permutations are scarce. Overall, the results combine Fourier-analytic techniques with derivative-based criteria to characterize when embeddings exhibit affine-like behavior and high balancedness, with concrete implications for APN theory and S-box design.

Abstract

Let $F$ be a vectorial Boolean function from $\mathbb{F}_2^n$ to $\mathbb{F}_2^m$, with $m \geq n$. We define $F$ as an embedding if $F$ is injective. In this paper, we examine the component functions of $F$, focusing on constant and balanced components. Our findings reveal that at most $2^m - 2^{m-n}$ components of $F$ can be balanced, and this maximum is achieved precisely when $F$ is an embedding, with the remaining $2^{m-n}$ components being constants. Additionally, for partially-bent embeddings, we demonstrate that there are always at least $2^n - 1$ balanced components when $n$ is even, and $2^{m-1} + 2^{n-1} - 1$ balanced components when $n$ is odd. A relation with APN functions is shown.

Theorems & Definitions (11)

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