Lower bounds on APN-distance for all known APN functions
The following table lists a lower bound on the Hamming distance between a representative from each known CCZ-equivalence class of APN functions up to dimension 11, and the closes APN function (in terms of Hamming distance). The lower bound between an (n,n)-function F and the closest APN function is a CCZ-invariant, and is calculated via the formula , where . 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 indexed 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 under Lower bounds on APN-distance for all known APN functions in dimension 8. Note that all known APN functions in dimension 7 from the known quadratic APN polynomial functions over GF(2^7) have the same value of the lower bound as e.g. over .
|8||1.1 - 1.13||111||38|
|8||1.15 - 1.17||111||38|
- L. Budaghyan, C. Carlet, T. Helleseth, N. Kaleyski. On the distance between APN functions. IEEE Trans. Inf. Theory, early access article. https://doi.org/10.1109/TIT.2020.2983684