# Difference between revisions of "Sigma multiplicities for APN functions in dimensions up to 10"

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− | For all known APN functions over GF(2^n) with n up to 10, the following tables list the partitions of GF(2^n) induced by the multiplicities of the Sigma sets <math>\Sigma_F^k(0)</ | + | For all known APN functions over GF(2^n) with n up to 10, the following tables list the partitions of GF(2^n) induced by the multiplicities of the Sigma sets <math>\Sigma_F^k(0)</math> for k = 4, as explained in the [https://seta-2020.org/assets/files/program/papers/paper-44.pdf SETA paper on testing EA-equivalences]. |

Due to the large number of functions, the results for dimension 8 are listed on a [[Sigma multiplicities for APN functions in dimension 8|separate page]]. | Due to the large number of functions, the results for dimension 8 are listed on a [[Sigma multiplicities for APN functions in dimension 8|separate page]]. | ||

+ | |||

+ | == Dimension 6 == | ||

+ | |||

+ | Tne observed sizes of the partition for the known switching classes are given in the following table; the functions are index according to [[Known switching classes of APN functions over GF(2^n) for n = 5,6,7,8]]. A Magma script listing these sizes is available [https://people.uib.no/nka041/sigma_6_sizes.m here], and one listing the partition themselves is available [https://people.uib.no/nka041/sigma_6_magma.m here]. | ||

+ | |||

+ | <table> | ||

+ | <tr><th>ID</th><th>Partition sizes</th></tr> | ||

+ | <tr><td>1</td><td> 1, 21, 42 </td></tr> | ||

+ | <tr><td>2</td><td> 1, 21, 42 </td></tr> | ||

+ | <tr><td>3</td><td> 1, 1, 1, 5, 10, 20, 26 </td></tr> | ||

+ | <tr><td>4</td><td> 1, 21, 42 </td></tr> | ||

+ | <tr><td>5</td><td> 1, 21, 42 </td></tr> | ||

+ | <tr><td>6</td><td> 1, 1, 1, 5, 10, 20, 26 </td></tr> | ||

+ | <tr><td>7</td><td> 1, 6, 6, 10, 10, 15, 16 </td></tr> | ||

+ | <tr><td>8</td><td> 1, 1, 1, 5, 10, 20, 26 </td></tr> | ||

+ | <tr><td>9</td><td> 1, 1, 1, 5, 10, 20, 26 </td></tr> | ||

+ | <tr><td>10</td><td> 1, 1, 1, 5, 10, 20, 26 </td></tr> | ||

+ | <tr><td>11</td><td> 1, 2, 6, 9, 22, 24 </td></tr> | ||

+ | <tr><td>12</td><td> 1, 2, 6, 9, 22, 24 </td></tr> | ||

+ | <tr><td>13</td><td> 1, 1, 1, 5, 10, 20, 26 </td></tr> | ||

+ | <tr><td>14</td><td> 1, 1, 3, 4, 6, 7, 10, 32 </td></tr> | ||

+ | </table> | ||

+ | |||

+ | |||

+ | == Dimension 7 == | ||

+ | |||

+ | In dimension 7, all tested functions induce the trivial partition of the field into zero and non-zero elements. This includes the inverse APN function. | ||

+ | |||

+ | == Dimension 8 == | ||

+ | |||

+ | The results for dimension 8 are given on a [[Sigma multiplicities for APN functions in dimension 8|separate page]] due to the large number of functions and distinct partitions. | ||

== Dimension 9 == | == Dimension 9 == | ||

− | + | In dimension 9, all tested functions (including the inverse function) from the list of [[CCZ-inequivalent_representatives_from_the_known_APN_families_for_dimensions_up_to_11]], as well as those obtained via self-equivalences, have the same trivial partition into zero and non-zero elements. | |

== Dimension 10 == | == Dimension 10 == | ||

− | Functions 1-10 are indexed according to [[ | + | Functions 1-10 are indexed according to [[CCZ-inequivalent_representatives_from_the_known_APN_families_for_dimensions_up_to_11]]. The remaining functions correspond to the ones found via self-equivalences, and are indexed in the same way as in the [https://zenodo.org/record/4235166#.X8ewfy2ZN27 provided dataset]. A Magma script containing the partitions size is available [https://people.uib.no/nka041/sigma_10_sizes.m here], and a script containing the actual partitions is available [https://people.uib.no/nka041/sigma_10_magma.m here]. |

<table> | <table> | ||

Line 23: | Line 54: | ||

<tr><td>9</td><td> 1, 341, 682 </td></tr> | <tr><td>9</td><td> 1, 341, 682 </td></tr> | ||

<tr><td>10</td><td> 1, 341, 682 </td></tr> | <tr><td>10</td><td> 1, 341, 682 </td></tr> | ||

− | <tr><td>11</td><td> 1, | + | <tr><td>11</td><td> 1, 341, 682 </td></tr> |

<tr><td>12</td><td> 1, 341, 682 </td></tr> | <tr><td>12</td><td> 1, 341, 682 </td></tr> | ||

− | <tr><td>13</td><td> 1, | + | <tr><td>13</td><td> 1, 341, 682 </td></tr> |

<tr><td>14</td><td> 1, 341, 682 </td></tr> | <tr><td>14</td><td> 1, 341, 682 </td></tr> | ||

− | <tr><td>15</td><td> 1, 11, 11, 165, 330, 506 </td></tr> | + | <tr><td>15</td><td> 1, 341, 682 </td></tr> |

+ | <tr><td>16</td><td> 1, 341, 682 </td></tr> | ||

+ | <tr><td>17</td><td> 1, 31, 155, 155, 155, 217, 310 </td></tr> | ||

+ | <tr><td>18</td><td> 1, 341, 682 </td></tr> | ||

+ | <tr><td>19</td><td> 1, 124, 217, 310, 372 </td></tr> | ||

+ | <tr><td>20</td><td> 1, 341, 682 </td></tr> | ||

+ | <tr><td>21</td><td> 1, 11, 11, 165, 330, 506 </td></tr> | ||

</table> | </table> |

## Latest revision as of 02:14, 3 December 2020

For all known APN functions over GF(2^n) with n up to 10, the following tables list the partitions of GF(2^n) induced by the multiplicities of the Sigma sets for k = 4, as explained in the SETA paper on testing EA-equivalences.

Due to the large number of functions, the results for dimension 8 are listed on a separate page.

## Dimension 6

Tne observed sizes of the partition for the known switching classes are given in the following table; the functions are index according to Known switching classes of APN functions over GF(2^n) for n = 5,6,7,8. A Magma script listing these sizes is available here, and one listing the partition themselves is available here.

ID | Partition sizes |
---|---|

1 | 1, 21, 42 |

2 | 1, 21, 42 |

3 | 1, 1, 1, 5, 10, 20, 26 |

4 | 1, 21, 42 |

5 | 1, 21, 42 |

6 | 1, 1, 1, 5, 10, 20, 26 |

7 | 1, 6, 6, 10, 10, 15, 16 |

8 | 1, 1, 1, 5, 10, 20, 26 |

9 | 1, 1, 1, 5, 10, 20, 26 |

10 | 1, 1, 1, 5, 10, 20, 26 |

11 | 1, 2, 6, 9, 22, 24 |

12 | 1, 2, 6, 9, 22, 24 |

13 | 1, 1, 1, 5, 10, 20, 26 |

14 | 1, 1, 3, 4, 6, 7, 10, 32 |

## Dimension 7

In dimension 7, all tested functions induce the trivial partition of the field into zero and non-zero elements. This includes the inverse APN function.

## Dimension 8

The results for dimension 8 are given on a separate page due to the large number of functions and distinct partitions.

## Dimension 9

In dimension 9, all tested functions (including the inverse function) from the list of CCZ-inequivalent_representatives_from_the_known_APN_families_for_dimensions_up_to_11, as well as those obtained via self-equivalences, have the same trivial partition into zero and non-zero elements.

## Dimension 10

Functions 1-10 are indexed according to CCZ-inequivalent_representatives_from_the_known_APN_families_for_dimensions_up_to_11. The remaining functions correspond to the ones found via self-equivalences, and are indexed in the same way as in the provided dataset. A Magma script containing the partitions size is available here, and a script containing the actual partitions is available here.

ID | Partition sizes |
---|---|

1 | 1, 341, 682 |

2 | 1, 341, 682 |

3 | 1, 341, 682 |

4 | 1, 341, 682 |

5 | 1, 341, 682 |

6 | 1, 341, 682 |

7 | 1, 341, 682 |

8 | 1, 341, 682 |

9 | 1, 341, 682 |

10 | 1, 341, 682 |

11 | 1, 341, 682 |

12 | 1, 341, 682 |

13 | 1, 341, 682 |

14 | 1, 341, 682 |

15 | 1, 341, 682 |

16 | 1, 341, 682 |

17 | 1, 31, 155, 155, 155, 217, 310 |

18 | 1, 341, 682 |

19 | 1, 124, 217, 310, 372 |

20 | 1, 341, 682 |

21 | 1, 11, 11, 165, 330, 506 |