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 . Note that the lower bound is a CCZ-invariant (unlike the exact minimum distance itself) and it can be calculated via the formula , where is the lower bound on the Hamming distance between an -function and the closest APN function, and is defined as . The values of 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 table below shows the lower bounds computed for some of the known APN functions for dimensions between 4 and 11. The functions in dimensions between 5 and 8 are indexed according to the table of Known switching classes of APN functions over GF(2^n) for n = 5,6,7,8. The ones between 9 and 11 are index according to the table of CCZ-inequivalent APN functions from the known APN classes over GF(2^n) (for n between 6 and 11). A separate table listing these results for the (more than 8000) known APN functions in dimension 8 is available as Lower bounds on APN-distance for all known APN functions in dimension 8. Note that all known APN functions in dimension 7 from Known quadratic APN polynomial functions over GF(2^7) have the same value of the lower bound as e.g. over as given in the table below.
|8||1.1 - 1.13||111||38|
|8||1.15 - 1.17||111||38|
- Budaghyan L, Carlet C, Helleseth T, Kaleyski N. Changing Points in APN Functions. IACR Cryptology ePrint Archive. 2018;2018:1217.