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]\displaystyle{ \Sigma_F^k(0) }[/math] 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 |