A Note on Vectorial Boolean Functions as Embeddings

TL;DR

This paper investigates the properties of vectorial Boolean functions as embeddings, analyzing their component functions, and establishing bounds on balanced components, with implications for cryptographic functions.

Contribution

It provides new bounds on the number of balanced components in embeddings and explores properties of partially-bent embeddings, linking to APN functions.

Findings

At most 2^m - 2^{m-n} components of an embedding can be balanced.

Partially-bent embeddings have at least 2^n - 1 balanced components when n is even.

A relation between embeddings and APN functions is established.

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.