# 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 ^{[1]}. 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 .

Dimension | F | Lower bound | |
---|---|---|---|

4 | x^{3} |
3 | 2 |

5 | x^{3} |
15 | 6 |

5 | x^{5} |
15 | 6 |

5 | x^{15} |
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 |

9 | 9.7 | 231 | 78 |

9 | all others | 255 | 86 |

10 | 10.4 | 477 | 160 |

10 | all others | 495 | 166 |

11 | 11.12 | 978 | 327 |

11 | all others | 1023 | 342 |

- ↑ 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