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)</sigma> for k = 4, as explained in the [https://seta-2020.org/assets/files/program/papers/paper-44.pdf SETA paper on testing EA-equivalences].
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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)</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]].
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== Dimension 6 ==
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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].
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<table>
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<tr><th>ID</th><th>Partition sizes</th></tr>
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<tr><td>1</td><td> 1, 21, 42 </td></tr>
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<tr><td>2</td><td> 1, 21, 42 </td></tr>
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<tr><td>3</td><td> 1, 1, 1, 5, 10, 20, 26 </td></tr>
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<tr><td>4</td><td> 1, 21, 42 </td></tr>
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<tr><td>5</td><td> 1, 21, 42 </td></tr>
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<tr><td>6</td><td> 1, 1, 1, 5, 10, 20, 26 </td></tr>
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<tr><td>7</td><td> 1, 6, 6, 10, 10, 15, 16 </td></tr>
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<tr><td>8</td><td> 1, 1, 1, 5, 10, 20, 26 </td></tr>
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<tr><td>9</td><td> 1, 1, 1, 5, 10, 20, 26 </td></tr>
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<tr><td>10</td><td> 1, 1, 1, 5, 10, 20, 26 </td></tr>
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<tr><td>11</td><td> 1, 2, 6, 9, 22, 24 </td></tr>
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<tr><td>12</td><td> 1, 2, 6, 9, 22, 24 </td></tr>
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<tr><td>13</td><td> 1, 1, 1, 5, 10, 20, 26 </td></tr>
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<tr><td>14</td><td> 1, 1, 3, 4, 6, 7, 10, 32 </td></tr>
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</table>
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== Dimension 7 ==
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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.
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== Dimension 8 ==
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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 ==
  
 
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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 [[https://boolean.h.uib.no/mediawiki/index.php/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].  
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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>
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<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, 31, 155, 155, 155, 217, 310 </td></tr>
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<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, 124, 217, 310, 372 </td></tr>
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<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>
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<tr><td>15</td><td> 1, 341, 682 </td></tr>
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<tr><td>16</td><td> 1, 341, 682 </td></tr>
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<tr><td>17</td><td> 1, 31, 155, 155, 155, 217, 310 </td></tr>
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<tr><td>18</td><td> 1, 341, 682 </td></tr>
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<tr><td>19</td><td> 1, 124, 217, 310, 372 </td></tr>
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<tr><td>20</td><td> 1, 341, 682 </td></tr>
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<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.

IDPartition 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.

IDPartition 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