# Difference between revisions of "Lower bounds on APN-distance for all known APN functions"

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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. | 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. | ||

+ | |||

+ | <table> | ||

+ | |||

+ | <tr> | ||

+ | <th colspan="4">DIMENSION 9</th> | ||

+ | </tr> | ||

+ | |||

+ | <tr> | ||

+ | <th>ID</th> | ||

+ | <th><math>\Pi_F^0</math></th> | ||

+ | <th><math>m_F</math></th> | ||

+ | <th>lower bound</th> | ||

+ | </tr> | ||

+ | |||

+ | <tr> | ||

+ | <td>1</td> | ||

+ | <td>255<sup>511</sup>, 512</td> | ||

+ | <td>255</td> | ||

+ | <td>86</td> | ||

+ | </tr> | ||

+ | |||

+ | <tr> | ||

+ | <td>2</td> | ||

+ | <td>255<sup>511</sup>, 512</td> | ||

+ | <td>255</td> | ||

+ | <td>86</td> | ||

+ | </tr> | ||

+ | |||

+ | <tr> | ||

+ | <td>3</td> | ||

+ | <td>255<sup>511</sup>, 512</td> | ||

+ | <td>255</td> | ||

+ | <td>86</td> | ||

+ | </tr> | ||

+ | |||

+ | <tr> | ||

+ | <td>4</td> | ||

+ | <td>255<sup>511</sup>, 512</td> | ||

+ | <td>255</td> | ||

+ | <td>86</td> | ||

+ | </tr> | ||

+ | |||

+ | <tr> | ||

+ | <td>5</td> | ||

+ | <td>255<sup>511</sup>, 512</td> | ||

+ | <td>255</td> | ||

+ | <td>86</td> | ||

+ | </tr> | ||

+ | |||

+ | <tr> | ||

+ | <td>6</td> | ||

+ | <td>255<sup>511</sup>, 512</td> | ||

+ | <td>255</td> | ||

+ | <td>86</td> | ||

+ | </tr> | ||

+ | |||

+ | <tr> | ||

+ | <td>7</td> | ||

+ | <td>231<sup>3</sup>, 237<sup>45</sup>, 240<sup>27</sup>, 243<sup>36</sup>, 246<sup>54</sup>, 249<sup>36</sup>, 252<sup>36</sup>, 255<sup>37</sup>, 258<sup>27</sup>, 261<sup>45</sup>, 264<sup>54</sup>, 267<sup>45</sup>, 270<sup>36</sup>, 273<sup>9</sup>, 276<sup>18</sup>, 279<sup>3</sup>, 512</td> | ||

+ | <td>231</td> | ||

+ | <td>78</td> | ||

+ | </tr> | ||

+ | |||

+ | <tr> | ||

+ | <td>8</td> | ||

+ | <td>255<sup>511</sup>, 512</td> | ||

+ | <td>255</td> | ||

+ | <td>86</td> | ||

+ | </tr> | ||

+ | |||

+ | <tr> | ||

+ | <td>9</td> | ||

+ | <td>255<sup>511</sup>, 512</td> | ||

+ | <td>255</td> | ||

+ | <td>86</td> | ||

+ | </tr> | ||

+ | |||

+ | <tr> | ||

+ | <td>10</td> | ||

+ | <td>255<sup>511</sup>, 512</td> | ||

+ | <td>255</td> | ||

+ | <td>86</td> | ||

+ | </tr> | ||

+ | |||

+ | <tr> | ||

+ | <td>11</td> | ||

+ | <td>255<sup>511</sup>, 512</td> | ||

+ | <td>255</td> | ||

+ | <td>86</td> | ||

+ | </tr> | ||

+ | |||

+ | </table> |

## Revision as of 18:18, 19 August 2019

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 , 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 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 | lower bound | ||

1 | 255^{511}, 512 |
255 | 86 |

2 | 255^{511}, 512 |
255 | 86 |

3 | 255^{511}, 512 |
255 | 86 |

4 | 255^{511}, 512 |
255 | 86 |

5 | 255^{511}, 512 |
255 | 86 |

6 | 255^{511}, 512 |
255 | 86 |

7 | 231^{3}, 237^{45}, 240^{27}, 243^{36}, 246^{54}, 249^{36}, 252^{36}, 255^{37}, 258^{27}, 261^{45}, 264^{54}, 267^{45}, 270^{36}, 273^{9}, 276^{18}, 279^{3}, 512 |
231 | 78 |

8 | 255^{511}, 512 |
255 | 86 |

9 | 255^{511}, 512 |
255 | 86 |

10 | 255^{511}, 512 |
255 | 86 |

11 | 255^{511}, 512 |
255 | 86 |

- ↑ Budaghyan L, Carlet C, Helleseth T, Kaleyski N. Changing Points in APN Functions. IACR Cryptology ePrint Archive. 2018;2018:1217.