View source for Sigma multiplicities for APN functions in dimensions up to 10

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]].

== 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 ==

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 [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>
<tr><th>ID</th><th>Partition sizes</th></tr>
<tr><td>1</td><td> 1, 341, 682 </td></tr>
<tr><td>2</td><td> 1, 341, 682 </td></tr>
<tr><td>3</td><td> 1, 341, 682 </td></tr>
<tr><td>4</td><td> 1, 341, 682 </td></tr>
<tr><td>5</td><td> 1, 341, 682 </td></tr>
<tr><td>6</td><td> 1, 341, 682 </td></tr>
<tr><td>7</td><td> 1, 341, 682 </td></tr>
<tr><td>8</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>11</td><td> 1, 341, 682 </td></tr>
<tr><td>12</td><td> 1, 341, 682 </td></tr>
<tr><td>13</td><td> 1, 341, 682 </td></tr>
<tr><td>14</td><td> 1, 341, 682 </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>