Lower bounds on APN-distance for all known APN functions: Difference between revisions
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<td>255</td> | <td>255</td> | ||
<td>86</td> | <td>86</td> | ||
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<tr> | |||
<th colspan="4">DIMENSION 10</th> | |||
</tr> | |||
<tr> | |||
<th>ID</th> | |||
<th><math>\Pi_F^0</math></th> | |||
<th><math>m_F</math></th> | |||
<th>lower bound</th> | |||
</tr> | |||
<tr> | |||
<td>1</td> | |||
<td>495<sup>682</sup>, 543<sup>341</sup>, 1024</td> | |||
<td>495</td> | |||
<td>166</td> | |||
</tr> | |||
<tr> | |||
<td>2</td> | |||
<td>495<sup>682</sup>, 543<sup>341</sup>, 1024</td> | |||
<td>495</td> | |||
<td>166</td> | |||
</tr> | |||
<tr> | |||
<td>3</td> | |||
<td>495<sup>682</sup>, 543<sup>341</sup>, 1024</td> | |||
<td>495</td> | |||
<td>166</td> | |||
</tr> | |||
<tr> | |||
<td>4</td> | |||
<td>477<sup>40</sup>, 483<sup>30</sup>, 489<sup>60</sup>, 495<sup>57</sup>, 501<sup>220</sup>, 507<sup>70</sup>, 513<sup>160</sup>, 519<sup>105</sup>, 525<sup>150</sup>, 531<sup>40</sup>, 537<sup>50</sup>, 543<sup>1</sup>, 549<sup>40</sup>, 1024</td> | |||
<td>477</td> | |||
<td>160</td> | |||
</tr> | |||
<tr> | |||
<td>5</td> | |||
<td>495<sup>682</sup>, 543<sup>341</sup>, 1024</td> | |||
<td>495</td> | |||
<td>166</td> | |||
</tr> | |||
<tr> | |||
<td>6</td> | |||
<td>495<sup>682</sup>, 543<sup>341</sup>, 1024</td> | |||
<td>495</td> | |||
<td>166</td> | |||
</tr> | |||
<tr> | |||
<td>7</td> | |||
<td>495<sup>682</sup>, 543<sup>341</sup>, 1024</td> | |||
<td>495</td> | |||
<td>166</td> | |||
</tr> | |||
<tr> | |||
<td>8</td> | |||
<td>495<sup>682</sup>, 543<sup>341</sup>, 1024</td> | |||
<td>495</td> | |||
<td>166</td> | |||
</tr> | </tr> | ||
</table> | </table> |
Revision as of 16:22, 19 August 2019
The following tables list a lower bound on the Hamming distance between all known CCZ-inequivalent APN representatives up to dimension 11 using the methods described used in [1]. Note that the lower bound is a CCZ-invariant (unlike the exact minimum distance itself) and it can be calculated via the formula [math]\displaystyle{ l(F) = \lceil \frac{m_F}{3} \rceil + 1 }[/math], where [math]\displaystyle{ l(F) }[/math] is the lower bound on the Hamming distance between an [math]\displaystyle{ (n,n) }[/math]-function [math]\displaystyle{ F }[/math] and the closest APN function, and [math]\displaystyle{ m_F }[/math] is defined as [math]\displaystyle{ m_F = \min_{b, \beta \in \mathbb{F}_{2^n}} | \{ a \in \mathbb{F}_{2^n} : (\exists x \in \mathbb{F}_{2^n})( F(x) + F(a+x) + F(a + \beta) = b ) \} | }[/math]. The values of [math]\displaystyle{ m_F }[/math] for the CCZ-inequivalent representatives are provided in the tables for convenience. The representatives for dimensions 7 and 8 are taken from the list ofKnown quadratic APN polynomial functions over GF(2^7) and Known quadratic APN polynomial functions over GF(2^8), respectively, while the rest are taken from the table of CCZ-inequivalent APN functions from the known APN classes over GF(2^n) (for n between 6 and 11).
The tables for dimensions 7 and 8 can be found under Lower bounds on APN-distance for all known APN functions in dimension 7 and Lower bounds on APN-distance for all known APN functions in dimension 8, respectively, due to their large size.
DIMENSION 9 | |||
---|---|---|---|
ID | [math]\displaystyle{ \Pi_F^0 }[/math] | [math]\displaystyle{ m_F }[/math] | lower bound |
1 | 255511, 512 | 255 | 86 |
2 | 255511, 512 | 255 | 86 |
3 | 255511, 512 | 255 | 86 |
4 | 255511, 512 | 255 | 86 |
5 | 255511, 512 | 255 | 86 |
6 | 255511, 512 | 255 | 86 |
7 | 2313, 23745, 24027, 24336, 24654, 24936, 25236, 25537, 25827, 26145, 26454, 26745, 27036, 2739, 27618, 2793, 512 | 231 | 78 |
8 | 255511, 512 | 255 | 86 |
9 | 255511, 512 | 255 | 86 |
10 | 255511, 512 | 255 | 86 |
11 | 255511, 512 | 255 | 86 |
DIMENSION 10 | |||
---|---|---|---|
ID | [math]\displaystyle{ \Pi_F^0 }[/math] | [math]\displaystyle{ m_F }[/math] | lower bound |
1 | 495682, 543341, 1024 | 495 | 166 |
2 | 495682, 543341, 1024 | 495 | 166 |
3 | 495682, 543341, 1024 | 495 | 166 |
4 | 47740, 48330, 48960, 49557, 501220, 50770, 513160, 519105, 525150, 53140, 53750, 5431, 54940, 1024 | 477 | 160 |
5 | 495682, 543341, 1024 | 495 | 166 |
6 | 495682, 543341, 1024 | 495 | 166 |
7 | 495682, 543341, 1024 | 495 | 166 |
8 | 495682, 543341, 1024 | 495 | 166 |
- ↑ Budaghyan L, Carlet C, Helleseth T, Kaleyski N. Changing Points in APN Functions. IACR Cryptology ePrint Archive. 2018;2018:1217.