A Note on Vectorial Boolean Functions as Embeddings

Let $F$ be a vectorial Boolean function from ${\mathbb{F}}_2^n$ to ${\mathbb{F}}_2^m$, with $m\ge n$. 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.