Classification of Quadratic APN Trinomials, Quadrinomials, Pentanomials, Hexanomials (CCZ-inequivalent with infinite monomial families) in Small Dimensions with all Coefficients equal to 1: Difference between revisions

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<td><math>x^{20} + x^6 + x^3</math></td>
<td><math>x^{20} + x^6 + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>Table 7: N^\circ 8.1</math></td>
<td><math>Table\ 7: N^\circ 8.1</math></td>
</tr>
</tr>


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<td><math>x^{34} + x^{18} + x^5</math></td>
<td><math>x^{34} + x^{18} + x^5</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>7: N^\circ 2.1</math></td>
<td><math>Table\ 7: N^\circ 2.1</math></td>
</tr>
</tr>


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<td><math>x^{72} + x^6 + x^3</math></td>
<td><math>x^{72} + x^6 + x^3</math></td>
<td><math>N^\circ5</math></td>
<td><math>N^\circ5</math></td>
<td><math>Table 9: N^\circ1.3</math></td>
<td><math>Table\ 9: N^\circ1.3</math></td>
</tr>
</tr>


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<td><math>x^{72} + x^{36} + x^3</math></td>
<td><math>x^{72} + x^{36} + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>9:N^\circ1.4</math></td>
<td><math>Table\ 9:N^\circ1.4</math></td>
</tr>
</tr>


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</table>
</table>


== Quadrinomials ==
== Quadrinomials ==

Revision as of 14:07, 26 February 2019

The following tables list CCZ-inequivalent representatives found by systematically searching for APN functions among all trinomials, quadrinomials, pentanomials and hexanomials with coefficients in [math]\displaystyle{ \mathbb{F}_{2} }[/math] over [math]\displaystyle{ \mathbb{F}_{2^n} }[/math] with [math]\displaystyle{ 6 \le n \le 11 }[/math] [1]. The tables also list which equivalence class from [2] the functions belong to. Only polynomials inequivalent to power functions are considered. If the polynomial is equivalent to a family from the table of infinite families, this is also listed.

Trinomials

[math]\displaystyle{ n }[/math] [math]\displaystyle{ N^\circ }[/math] Functions Familiy Relation to [2]
[math]\displaystyle{ 6 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 7 }[/math] [math]\displaystyle{ 7.1 }[/math] [math]\displaystyle{ x^{20} + x^6 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ Table\ 7: N^\circ 8.1 }[/math]
[math]\displaystyle{ 7.2 }[/math] [math]\displaystyle{ x^{34} + x^{18} + x^5 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ Table\ 7: N^\circ 2.1 }[/math]
[math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 8.1 }[/math] [math]\displaystyle{ x^{72} + x^6 + x^3 }[/math] [math]\displaystyle{ N^\circ5 }[/math] [math]\displaystyle{ Table\ 9: N^\circ1.3 }[/math]
[math]\displaystyle{ 8.2 }[/math] [math]\displaystyle{ x^{72} + x^{36} + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ Table\ 9:N^\circ1.4 }[/math]
[math]\displaystyle{ 9 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 10 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 11 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]

Quadrinomials

[math]\displaystyle{ n }[/math] [math]\displaystyle{ N^\circ }[/math] Functions Families Relation to [2]
[math]\displaystyle{ 6 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 7 }[/math] [math]\displaystyle{ 7.1 }[/math] [math]\displaystyle{ x^{72} + x^{40} + x^{12} + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ Table 7: N^\circ12.1 }[/math]
[math]\displaystyle{ 7.2 }[/math] [math]\displaystyle{ x^{33} + x^{17} + x^{12} + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ2.2 }[/math]
[math]\displaystyle{ 7.3 }[/math] [math]\displaystyle{ x^{34} + x^{33} + x^{10} + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ10.1 }[/math]
[math]\displaystyle{ 7.4 }[/math] [math]\displaystyle{ x^{66} + x^{34} + x^{20} + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ11.1 }[/math]
[math]\displaystyle{ 7.5 }[/math] [math]\displaystyle{ x^{68} + x^{18} + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ8.1 }[/math]
[math]\displaystyle{ 7.6 }[/math] [math]\displaystyle{ x^{66} + x^{18} + x^9 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ9.1 }[/math]
[math]\displaystyle{ 8 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 9 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 10 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 11 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]


Pentanomials

[math]\displaystyle{ n }[/math] [math]\displaystyle{ N^\circ }[/math] Functions Families Relation to [2]
[math]\displaystyle{ 6 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 7 }[/math] [math]\displaystyle{ 7.1 }[/math] [math]\displaystyle{ x^{68} + x^{40} + x^{24} + x^6 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ Table\ 7: N^\circ13.1 }[/math]
[math]\displaystyle{ 7.2 }[/math] [math]\displaystyle{ x^{65} + x^{20} + x^{18} + x^6 + x^3 }[/math] [math]\displaystyle{ N^\circ5 }[/math] [math]\displaystyle{ Table\ 7: N^\circ1.2 }[/math]
[math]\displaystyle{ 7.3 }[/math] [math]\displaystyle{ x^{40} + x^{34} + x^{18} + x^{10} + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ Table\ 7: N^\circ12.1 }[/math]
[math]\displaystyle{ 7.4 }[/math] [math]\displaystyle{ x^{48} + x^{40} + x^{10} + x^9 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ Table\ 7: N^\circ2.1 }[/math]
[math]\displaystyle{ 7.5 }[/math] [math]\displaystyle{ x^{33} + x^9 + x^6 + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ Table\ 7: N^\circ11.1 }[/math]
[math]\displaystyle{ 7.6 }[/math] [math]\displaystyle{ x^{40} + x^{36} + x^{34} + x^{24} + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ Table\ 7: N^\circ10.1 }[/math]
[math]\displaystyle{ 7.7 }[/math] [math]\displaystyle{ x^{24} + x^{10} + x^9 + x^6 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ Table\ 7: N^\circ2.1 }[/math]
[math]\displaystyle{ 7.8 }[/math] [math]\displaystyle{ x^{65} + x^{36} + x^{20} + x^{17} + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ Table\ 7: N^\circ14.1 }[/math]
[math]\displaystyle{ 7.9 }[/math] [math]\displaystyle{ x^{40} + x^{33} + x^{17} + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ Table\ 7: N^\circ8.1 }[/math]
[math]\displaystyle{ 7.10 }[/math] [math]\displaystyle{ x^{36} + x^{33} + x^{18} + x^9 + x^5 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ Table\ 7: N^\circ10.1 }[/math]
[math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 8.1 }[/math] [math]\displaystyle{ x^{36} + x^{33} + x^9 + x^6 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ Table\ 9: N^\circ1.4 }[/math]
[math]\displaystyle{ 8.2 }[/math] [math]\displaystyle{ x^{72} + x^{66} + x^{12} + x^6 + x^3 }[/math] [math]\displaystyle{ N^\circ5 }[/math] [math]\displaystyle{ Table\ 9:N^\circ1.3 }[/math]
[math]\displaystyle{ 8.3 }[/math] [math]\displaystyle{ x^{130} + x^{66} + x^{40} + x^{12} + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ Table\ 9:N^\circ6.1 }[/math]
[math]\displaystyle{ 8.4 }[/math] [math]\displaystyle{ x^{66} + x^{40} + x^{18} + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ Table\ 9:N^\circ5.1 }[/math]
[math]\displaystyle{ 9 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 10 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 11 }[/math] [math]\displaystyle{ 11.1 }[/math] [math]\displaystyle{ x^{12} + x^{10} + x^9 + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 11.2 }[/math] [math]\displaystyle{ x^{258} + x^{257} + x^{18} + x^{17} + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 11.3 }[/math] [math]\displaystyle{ x^{96} + x^{66} + x^{34} + x^{33} + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 11.4 }[/math] [math]\displaystyle{ x^{80} + x^{68} + x^{65} + x^{17} + x^5 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 11.5 }[/math] [math]\displaystyle{ x^{260} + x^{257} + x^{36} + x^{33} + x^5 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]

Hexanomials

[math]\displaystyle{ n }[/math] [math]\displaystyle{ N^\circ }[/math] Functions Families Relation to [2]
[math]\displaystyle{ 6 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 7 }[/math] [math]\displaystyle{ 7.1 }[/math] [math]\displaystyle{ x^{34} + x^{33} + x^{12} + x^6 + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ Table 7: N^\circ14.2 }[/math]
[math]\displaystyle{ 7.2 }[/math] [math]\displaystyle{ x^{40} + x^{24} + x^{20} + x^9 + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ14.1 }[/math]
[math]\displaystyle{ 7.3 }[/math] [math]\displaystyle{ x^{33} + x^{24} + x^{20} + x^{18} + x^{12} + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ12.1 }[/math]
[math]\displaystyle{ 7.4 }[/math] [math]\displaystyle{ x^{24} + x^{17} + x^{12} + x^{10} + x^6 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ2.1 }[/math]
[math]\displaystyle{ 7.5 }[/math] [math]\displaystyle{ x^{40} + x^{34} + x^{18} + x^{17} + x^5 + x^3 }[/math] [math]\displaystyle{ N^\circ5 }[/math] [math]\displaystyle{ 7:N^\circ1.2 }[/math]
[math]\displaystyle{ 7.6 }[/math] [math]\displaystyle{ x^{48} + x^{40} + x^{18} + x^{10} + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ11.1 }[/math]
[math]\displaystyle{ 7.7 }[/math] [math]\displaystyle{ x^{40} + x^{12} + x^{10} + x^9 + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ2.2 }[/math]
[math]\displaystyle{ 7.8 }[/math] [math]\displaystyle{ x^{34} + x^{24} + x^{10} + x^9 + x^6 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ9.1 }[/math]
[math]\displaystyle{ 7.9 }[/math] [math]\displaystyle{ x^{34} + x^{33} + x^{20} + x^{17} + x^{10} + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ13.1 }[/math]
[math]\displaystyle{ 7.10 }[/math] [math]\displaystyle{ x^{36} + x^{33} + x^{24} + x^9 + x^6 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ10.1 }[/math]
[math]\displaystyle{ 7.11 }[/math] [math]\displaystyle{ x^{40} + x^{36} + x^{20} + x^{10} + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ10.2 }[/math]
[math]\displaystyle{ 7.12 }[/math] [math]\displaystyle{ x^{36} + x^{34} + x^{20} + x^{10} + x^9 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ8.1 }[/math]
[math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 8.1 }[/math] [math]\displaystyle{ x^{68} + x^{34} + x^{17} + x^{12} + x^9 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ Table 9: N^\circ5.1 }[/math]
[math]\displaystyle{ 8.2 }[/math] [math]\displaystyle{ x^{72} + x^{40} + x^{34} + x^{20} + x^{12} + x^3 }[/math] [math]\displaystyle{ N^\circ5 }[/math] [math]\displaystyle{ 9:N^\circ6.1 }[/math]
[math]\displaystyle{ 8.3 }[/math] [math]\displaystyle{ x^{72} + x^{66} + x^{34} + x^{18} + x^{10} + x^5 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 9:N^\circ4.1 }[/math]
[math]\displaystyle{ 9 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 10 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 11 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
  1. Sun B. On Classification and Some Properties of APN Functions.
  2. 2.0 2.1 2.2 2.3 2.4 Edel Y, Pott A. A new almost perfect nonlinear function which is not quadratic. Adv. in Math. of Comm.. 2009 Mar;3(1):59-81.