On lower bounds for the distances between APN functions
Whether two distinct APN functions can have a Hamming distance of $1$ remains an open problem. In 2020, L. Budaghyan et al. introduced a new CCZ-invariant $\Pi_F$ which can be used to provide lower bounds on the Hamming distance between a given APN function $F \colon \mathbb{F}_2^n \to \mathbb{F}_2^n$ and other APN functions. Lower bounds on the distance from an APN function $F$ to any other APN function $G$ are known when $F$ is an almost bent (AB) function or when $F$ is a $3$-to-$1$ quadratic function with $n$ even. In this paper, we reinterpret $\Pi_F$ in terms of the multiplicities of the 3-sums of the graph $\mathcal{G}_F=\{(x, F(x)) : x \in \mathbb{F}_2^n\}$ of $F$ as a Sidon set, which we call exclude multiplicities. For even $n$, we establish lower bounds on the distance between $F$ and any other APN function $G$ when $F$ is plateaued APN, and we generalize a previously known lower bound for quadratic $3$-to-$1$ functions to the case where $F$ is plateaued $3$-to-$1$ (e.g., when $F$ is a Kasami function). For odd $n$, we derive new lower bounds when $F$ is the APN inverse function over $\mathbb{F}_{2^n}$. We also study how the exclude multiplicities of $\mathcal{G}_F$ are directly connected to the existence of linear structures of $\gamma_F$ when $F$ is plateaued APN and to the ortho-derivative when $F$ is a quadratic APN function. In particular, we prove that $\gamma_F$ has no nontrivial linear structures when $F$ is plateaued APN. We also use the CCZ-invariance of exclude multiplicities to prove that the Brinkmann-Leander-Edel-Pott function is not CCZ-equivalent to a plateaued function.