# Lower bounds on APN-distance for all known APN functions

The following tables list a lower bound on the Hamming distance between all known CCZ-inequivalent APN representatives up to dimension 11 using the methods described used in [1]. Note that the lower bound is a CCZ-invariant (unlike the exact minimum distance itself) and it can be calculated via the formula ${\displaystyle l(F)=\lceil {\frac {m_{F}}{3}}\rceil +1}$, where ${\displaystyle l(F)}$ is the lower bound on the Hamming distance between an ${\displaystyle (n,n)}$-function ${\displaystyle F}$ and the closest APN function, and ${\displaystyle m_{F}}$ is defined as ${\displaystyle m_{F}=\min _{b,\beta \in \mathbb {F} _{2^{n}}}|\{a\in \mathbb {F} _{2^{n}}:(\exists x\in \mathbb {F} _{2^{n}})(F(x)+F(a+x)+F(a+\beta )=b)\}|}$. The values of ${\displaystyle m_{F}}$ for the CCZ-inequivalent representatives are provided in the tables for convenience. The representatives for dimensions 7 and 8 are taken from the list of Known quadratic APN polynomial functions over GF(2^7) and Known quadratic APN polynomial functions over GF(2^8), respectively, while the rest are taken from the table of CCZ-inequivalent APN functions from the known APN classes over GF(2^n) (for n between 6 and 11).

The tables for dimensions 7 and 8 can be found under Lower bounds on APN-distance for all known APN functions in dimension 7 and Lower bounds on APN-distance for all known APN functions in dimension 8, respectively, due to their large size.

Dimension F ${\displaystyle m_{F}}$ Lower bound
4 x3 3 2
5 x3 15 6
5 x5 15 6
5 x15 9 4
6 1.1 27 10
6 1.2 27 10
6 2.1 15 6
6 2.2 27 10
6 2.3 27 10
6 2.4 15 6
6 2.5 15 6
6 2.6 15 6
6 2.7 15 6
6 2.8 15 6
6 2.9 21 8
6 2.10 21 8
6 2.11 15 6
6 2.12 15 6
7 7.1 54 19
7 all others 63 22
8 1.1 - 1.13 111 38
8 1.14 99 34
8 1.15 - 1.17 111 38
8 2.1 111 38
8 3.1 111 38
8 4.1 99 34
8 5.1 105 36
8 6.1 105 36
8 7.1 111 38
1. Budaghyan L, Carlet C, Helleseth T, Kaleyski N. Changing Points in APN Functions. IACR Cryptology ePrint Archive. 2018;2018:1217.