2025/09/02 by Mihaila, Maria, Thornburgh, Darrion · 1 citation
#11B13. Secondary: 94A60 #Combinatorics (math.CO) #FOS: Mathematics #Primary: 94D10
paper · doi:10.48550/arxiv.2509.02280
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 ΠF which can be used to provide lower bounds on the Hamming distance between a given APN function F \colon \mathbbF2n → \mathbbF2n and other APN functions. Lower bounds on the distance from an APN function F to any other are known for almost bent (AB) functions and when F is a 3-to-1 quadratic function with n even. In this paper, we reinterpret ΠF in terms of the exclude multiplicities of the graph GF=\(x, F(x)) : x ∈ \mathbbF2n\ of F as a Sidon set. We establish lower bounds on the distance for even n when F is plateaued APN, generalize the known lower bounds for quadratic 3-to-1 function to all 3-to-1 plateaued functions (e.g. Kasami functions), and derive new lower bounds for when F is the APN inverse function over \mathbbF2n for n odd. We also study how the exclude multiplicities of GF are directly connected to the existence of linear structures of γF when F is plateaued and APN and the ortho-derivative when F is a quadratic APN function. 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.