On lower bounds for the distances between APN functions

Abstract 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 $$F :\mathbb {F}_2^n \rightarrow \mathbb {F}_2^n$$ 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.