# 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 ofKnown 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 9
ID ${\displaystyle \Pi _{F}^{0}}$ ${\displaystyle m_{F}}$ lower bound
1 255511, 512 255 86
2 255511, 512 255 86
3 255511, 512 255 86
4 255511, 512 255 86
5 255511, 512 255 86
6 255511, 512 255 86
7 2313, 23745, 24027, 24336, 24654, 24936, 25236, 25537, 25827, 26145, 26454, 26745, 27036, 2739, 27618, 2793, 512 231 78
8 255511, 512 255 86
9 255511, 512 255 86
10 255511, 512 255 86
11 255511, 512 255 86
DIMENSION 10
ID ${\displaystyle \Pi _{F}^{0}}$ ${\displaystyle m_{F}}$ lower bound
1 495682, 543341, 1024 495 166
2 495682, 543341, 1024 495 166
3 495682, 543341, 1024 495 166
4 47740, 48330, 48960, 49557, 501220, 50770, 513160, 519105, 525150, 53140, 53750, 5431, 54940, 1024 477 160
5 495682, 543341, 1024 495 166
6 495682, 543341, 1024 495 166
7 495682, 543341, 1024 495 166
8 495682, 543341, 1024 495 166
1. Budaghyan L, Carlet C, Helleseth T, Kaleyski N. Changing Points in APN Functions. IACR Cryptology ePrint Archive. 2018;2018:1217.