View source for Classification of Quadratic APN Trinomials, Quadrinomials, Pentanomials, Hexanomials (CCZ-inequivalent with infinite monomial families) in Small Dimensions with all Coefficients equal to 1

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>\mathbb{F}_{2}</math> over <math>\mathbb{F}_{2^n}</math> with <math>6 \le n \le 11</math> <ref>Sun B. On Classification and Some Properties of APN Functions.</ref>. The tables also list which equivalence class from <ref name="edelPott">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.</ref> the functions belong to. Only polynomials inequivalent to power functions are considered. If the polynomial is equivalent to a family from the [[Known infinite families of quadratic APN polynomials over GF(2^n)|table of infinite families]], this is also listed.

== Trinomials ==

<table class="borderless">
<tr>
<th>Dimension</th>
<th><math>N^\circ</math></th>
<th>Functions</th>
<th>Familiy</th>
<th>Relation to <ref name="edelPott"></ref></th>
</tr>

<tr>
<td><math>6</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
</tr>

<tr class="divider">
<td rowspan="2"><math>7</math></td>
<td class="noborderbelow"><math>7.1</math></td>
<td><math>x^{20} + x^6 + x^3</math></td>
<td><math>-</math></td>
<td>Table 7: № 8.1</td>
</tr>

<tr>
<td><math>7.2</math></td>
<td><math>x^{34} + x^{18} + x^5</math></td>
<td><math>-</math></td>
<td>Table 7: № 2.1</td>
</tr>


<tr class="divider">
<td rowspan="2"><math>8</math></td>
<td class="noborderbelow"><math>8.1</math></td>
<td><math>x^{72} + x^6 + x^3</math></td>
<td><math>N^\circ5</math></td>
<td>Table 9: № 1.3</td>
</tr>

<tr>
<td><math>8.2</math></td>
<td><math>x^{72} + x^{36} + x^3</math></td>
<td><math>-</math></td>
<td>Table 9: № 1.4</td>
</tr>

<tr class="divider">
<td><math>9</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
</tr>

<tr class="divider">
<td><math>10</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
</tr>

<tr class="divider">
<td><math>11</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
</tr>

</table>

== Quadrinomials ==

<table class="borderless">
<tr>
<th>Dimension</th>
<th><math>N^\circ</math></th>
<th>Functions</th>
<th>Families</th>
<th>Relation to <ref name="edelPott"></ref></th>
</tr>

<tr>
<td><math>6</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
</tr>

<tr class="divider">
<td rowspan="6"><math>7</math></td>
<td class="noborderbelow"><math>7.1</math></td>
<td><math>x^{72} + x^{40} + x^{12} + x^3</math></td>
<td><math>-</math></td>
<td>Table 7: № 12.1</td>
</tr>

<td><math>7.2</math></td>
<td><math>x^{33} + x^{17} + x^{12} + x^3</math></td>
<td><math>-</math></td>
<td>Table 7: № 2.2</td>
</tr>

<td><math>7.3</math></td>
<td><math>x^{34} + x^{33} + x^{10} + x^3</math></td>
<td><math>-</math></td>
<td>Table 7: № 10.1</td>
</tr>

<td><math>7.4</math></td>
<td><math>x^{66} + x^{34} + x^{20} + x^3</math></td>
<td><math>-</math></td>
<td>Table 7: № 11.1</td>
</tr>

<td><math>7.5</math></td>
<td><math>x^{68} + x^{18} + x^5 + x^3</math></td>
<td><math>-</math></td>
<td>Table 7: № 8.1</td>
</tr>

<td><math>7.6</math></td>
<td><math>x^{66} + x^{18} + x^9 + x^3</math></td>
<td><math>-</math></td>
<td>Table 7: № 9.1</td>
</tr>

<tr class="divider">
<td><math>8</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
</tr>

<tr class="divider">
<td><math>9</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
</tr>

<tr class="divider">
<td><math>10</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
</tr>

<tr class="divider">
<td><math>11</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
</tr>

</table>

== Pentanomials ==


<table class="borderless">
<tr>
<th>Dimension</th>
<th><math>N^\circ</math></th>
<th>Functions</th>
<th>Families</th>
<th>Relation to <ref name="edelPott"></ref></th>
</tr>

<tr>
<td><math>6</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
</tr>

<tr class="divider">
<td rowspan="10"><math>7</math></td>
<td class="noborderbelow"><math>7.1</math></td>
<td><math>x^{68} + x^{40} + x^{24} + x^6 + x^3</math></td>
<td><math>-</math></td>
<td>Table 7: № 13.1</td>
</tr>

<td><math>7.2</math></td>
<td><math>x^{65} + x^{20} + x^{18} + x^6 + x^3</math></td>
<td><math>N^\circ5</math></td>
<td>Table 7: № 1.2</td>
</tr>

<td><math>7.3</math></td>
<td><math>x^{40} + x^{34} + x^{18} + x^{10} + x^3</math></td>
<td><math>-</math></td>
<td>Table 7: № 12.1</td>
</tr>

<td><math>7.4</math></td>
<td><math>x^{48} + x^{40} + x^{10} + x^9 + x^3</math></td>
<td><math>-</math></td>
<td>Table 7: № 1.2</td>
</tr>

<td><math>7.5</math></td>
<td><math>x^{33} + x^9 + x^6 + x^5 + x^3</math></td>
<td><math>-</math></td>
<td>Table 7: № 11.1</td>
</tr>

<td><math>7.6</math></td>
<td><math>x^{40} + x^{36} + x^{34} + x^{24} + x^3</math></td>
<td><math>-</math></td>
<td>Table 7: № 10.1</td>
</tr>

<td><math>7.7</math></td>
<td><math>x^{24} + x^{10} + x^9 + x^6 + x^3</math></td>
<td><math>-</math></td>
<td>Table 7: № 2.1</td>
</tr>

<td><math>7.8</math></td>
<td><math>x^{65} + x^{36} + x^{20} + x^{17} + x^3</math></td>
<td><math>-</math></td>
<td>Table 7: № 14.1</td>
</tr>

<td><math>7.9</math></td>
<td><math>x^{40} + x^{33} + x^{17} + x^5 + x^3</math></td>
<td><math>-</math></td>
<td>Table 7: № 8.1</td>
</tr>

<td><math>7.10</math></td>
<td><math>x^{36} + x^{33} + x^{18} + x^9 + x^5</math></td>
<td><math>-</math></td>
<td>Table 7: № 10.1</td>
</tr>

<tr class="divider">
<td rowspan="4"><math>8</math></td>
<td class="noborderbelow"><math>8.1</math></td>
<td><math>x^{36} + x^{33} + x^9 + x^6 + x^3</math></td>
<td><math>-</math></td>
<td>Table 9: № 1.4</td>
</tr>

<td><math>8.2</math></td>
<td><math>x^{72} + x^{66} + x^{12} + x^6 + x^3</math></td>
<td><math>N^\circ5</math></td>
<td>Table 9: № 1.3</td>
</tr>

<td><math>8.3</math></td>
<td><math>x^{130} + x^{66} + x^{40} + x^{12} + x^3</math></td>
<td><math>-</math></td>
<td>Table 9: № 6.1</td>
</tr>

<td><math>8.4</math></td>
<td><math>x^{66} + x^{40} + x^{18} + x^5 + x^3</math></td>
<td><math>-</math></td>
<td>Table 9: № 5.1</td>
</tr>

<tr class="divider">
<td><math>9</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
</tr>

<tr class="divider">
<td><math>10</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
</tr>

<tr class="divider">
<td rowspan="5"><math>11</math></td>
<td class="noborderbelow"><math>11.1</math></td>
<td><math>x^{12} + x^{10} + x^9 + x^5 + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
</tr>

<td><math>11.2</math></td>
<td><math>x^{258} + x^{257} + x^{18} + x^{17} + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
</tr>

<td><math>11.3</math></td>
<td><math>x^{96} + x^{66} + x^{34} + x^{33} + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
</tr>

<td><math>11.4</math></td>
<td><math>x^{80} + x^{68} + x^{65} + x^{17} + x^5</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
</tr>

<td><math>11.5</math></td>
<td><math>x^{260} + x^{257} + x^{36} + x^{33} + x^5</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
</tr>

</table>

== Hexanomials ==

<table>
<table class="borderless">
<tr>
<th>Dimension</th>
<th><math>N^\circ</math></th>
<th>Functions</th>
<th>Families</th>
<th>Relation to <ref name="edelPott"></ref></th>
</tr>

<tr>
<td><math>6</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
</tr>

<tr class="divider">
<td rowspan="12"><math>7</math></td>
<td class="noborderbelow"><math>7.1</math></td>
<td><math>x^{34} + x^{33} + x^{12} + x^6 + x^5 + x^3</math></td>
<td><math>-</math></td>
<td>Table 7: № 14.2</td>
</tr>

<td><math>7.2</math></td>
<td><math>x^{40} + x^{24} + x^{20} + x^9 + x^5 + x^3</math></td>
<td><math>-</math></td>
<td>Table 7: № 14.1</td>
</tr>

<td><math>7.3</math></td>
<td><math>x^{33} + x^{24} + x^{20} + x^{18} + x^{12} + x^3</math></td>
<td><math>-</math></td>
<td>Table 7: № 12.1</td>
</tr>

<td><math>7.4</math></td>
<td><math>x^{24} + x^{17} + x^{12} + x^{10} + x^6 + x^3</math></td>
<td><math>-</math></td>
<td>Table 7: № 2.1</td>
</tr>

<td><math>7.5</math></td>
<td><math>x^{40} + x^{34} + x^{18} + x^{17} + x^5 + x^3</math></td>
<td><math>N^\circ5</math></td>
<td>Table 7: № 1.2</td>
</tr>

<td><math>7.6</math></td>
<td><math>x^{48} + x^{40} + x^{18} + x^{10} + x^5 + x^3</math></td>
<td><math>-</math></td>
<td>Table 7: № 11.1</td>
</tr>

<td><math>7.7</math></td>
<td><math>x^{40} + x^{12} + x^{10} + x^9 + x^5 + x^3</math></td>
<td><math>-</math></td>
<td>Table 7: № 2.2</td>
</tr>

<td><math>7.8</math></td>
<td><math>x^{34} + x^{24} + x^{10} + x^9 + x^6 + x^3</math></td>
<td><math>-</math></td>
<td>Table 7: № 9.1</td>
</tr>

<td><math>7.9</math></td>
<td><math>x^{34} + x^{33} + x^{20} + x^{17} + x^{10} + x^3</math></td>
<td><math>-</math></td>
<td>Table 7: № 13.1</td>
</tr>

<td><math>7.10</math></td>
<td><math>x^{36} + x^{33} + x^{24} + x^9 + x^6 + x^3</math>
<td><math>-</math></td>
<td>Table 7: № 10.1</td>
</tr>

<td><math>7.11</math></td>
<td><math>x^{40} + x^{36} + x^{20} + x^{10} + x^5 + x^3</math></td>
<td><math>-</math></td>
<td>Table 7: № 10.2</td>
</tr>

<td><math>7.12</math></td>
<td><math>x^{36} + x^{34} + x^{20} + x^{10} + x^9 + x^3</math></td>
<td><math>-</math></td>
<td>Table 7: № 8.1</td>
</tr>

<tr class="divider">
<td rowspan="3"><math>8</math></td>
<td class="noborderbelow"><math>8.1</math></td>
<td><math>x^{68} + x^{34} + x^{17} + x^{12} + x^9 + x^3</math></td>
<td><math>-</math></td>
<td>Table 9: № 5.1</td>
</tr>

<td><math>8.2</math></td>
<td><math>x^{72} + x^{40} + x^{34} + x^{20} + x^{12} + x^3</math></td>
<td><math>N^\circ5</math></td>
<td>Table 9: № 6.1</td>
</tr>

<td><math>8.3</math></td>
<td><math>x^{72} + x^{66} + x^{34} + x^{18} + x^{10} + x^5</math></td></td>
<td><math>-</math></td>
<td>Table 9: № 4.1</td>
</tr>

<tr class="divider">
<td><math>9</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
</tr>

<tr class="divider">
<td><math>10</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
</tr>

<tr class="divider">
<td><math>11</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
</tr>

</table>