Sigma multiplicities for APN functions in dimensions up to 10: Difference between revisions
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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 == | |||
== 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> |
Revision as of 23:58, 2 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 [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
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 |