CCZ-inequivalent representatives from the known APN families for dimensions up to 11: Difference between revisions

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CCZ-inequivalent APN Functions over <math>\mathbb{F}_{2^n}</math> from the Known APN Classes for <math>6\leqslant n \leqslant 11</math>
CCZ-inequivalent APN Functions over <math>\mathbb{F}_{2^n}</math> from the Known APN Classes for <math>6\leqslant n \leqslant 11</math>


 
<table class="borderless">
<table>
<tr>
<tr>
<th><math>n</math></th>
<th><math>N^\circ</math></th>
<th><math>N^\circ</math></th>
<th>Fanctions</th>
<th>Functions</th>
<th>Equivalent to</th>
<th>Equivalent to</th>
</tr>
<tr class="divider">
<td rowspan="3"><math>6</math></td>
<td class="noborderbelow"><math>6.1</math></td>
<td><math>x^3</math></td>
<td><math>Gold</math></td>
</tr>
</tr>




<tr>
<tr>
<td><math>6.1</math>
<td class="noborderbelow"><math>6.2</math></td>
<math>6.2</math>
<td><math>x^{24}+ax^{17}+a^8x^{10}+ax^9+x^3</math></td>
<math>6.3</math></td>
<td><math>C3</math></td>
<td><math>x^3</math>
<math>x^{24}+ax^{17}+a^8x^{10}+ax^9+x^3</math>
<math>ax^3+x^{17}+a^4x^{24}</math></td>
<td><math>Gold</math>
<math>C3</math>
<math>C7-C9</math></td>
</tr>
</tr>


<tr>
<tr>
<td><math>7.1</math>
<td class="noborderbelow"><math>6.3</math></td>
<td><math>ax^3+x^{17}+a^4x^{24}</math></td>
<td><math>C7-C9</math></td>
</tr>


<math>7.2</math>


<math>7.3</math>
<tr class="divider">
<td rowspan="7"><math>7</math></td>
<td class="noborderbelow"><math>7.1</math></td>
<td><math>x^3</math></td>
<td><math>Gold</math></td>
</tr>


<math>7.4</math>


<math>7.5</math>
<tr>
<td class="noborderbelow"><math>7.2</math></td>
<td><math>x^5</math></td>
<td><math>Gold</math></td>
</tr>


<math>7.6</math>


<math>7.7</math></td>
<tr>
<td class="noborderbelow"><math>7.3</math></td>
<td><math>x^9</math></td>
<td><math>Gold</math></td>
</tr>


<td><math>x^3</math>


<math>x^5</math>
<tr>
<td class="noborderbelow"><math>7.4</math></td>
<td><math>x^{13}</math></td>
<td><math>Kasami</math></td>
</tr>


<math>x^9</math>
<tr>
<td class="noborderbelow"><math>7.5</math></td>
<td><math>x^{57}</math></td>
<td><math>Kasami</math></td>
</tr>


<math>x^{13}</math>


<math>x^{57}</math>
<tr>
<td class="noborderbelow"><math>7.6</math></td>
<td><math>x^{63}</math></td>
<td><math>Inverse</math></td>
</tr>


<math>x^{63}</math>


<math>x^3+Tr_7(x^9)</math></td>


<td><math>Gold</math>
<tr>
<td class="noborderbelow"><math>7.7</math></td>
<td><math>x^3+Tr_7(x^9)</math></td>
<td><math>C4</math></td>
</tr>


<math>Gold</math>


<math>Gold</math>
<tr class="divider">
<td rowspan="7"><math>8</math></td>
<td class="noborderbelow"><math>8.1</math></td>
<td><math>x^3</math></td>
<td><math>Gold</math></td>
</tr>


<math>Kasami</math>
<tr>
<td class="noborderbelow"><math>8.2</math></td>
<td><math>x^9</math></td>
<td><math>Gold</math></td>
</tr>


<math>Kasami</math>


<math>Inverse</math>
<tr>
<td class="noborderbelow"><math>8.3</math></td>
<td><math>x^{57}</math></td>
<td><math>Kasami</math></td>
</tr>


<math>C4</math></td>
<tr>
<td class="noborderbelow"><math>8.4</math></td>
<td><math>x^3+x^{17}+p^{48}x^{18}+p^3x^{33}+px^{34}+x^{48}</math></td>
<td><math>C3</math></td>
</tr>
</tr>


<tr>
<tr>
<td><math>8.1</math>
<td class="noborderbelow"><math>8.5</math></td>
<td><math>x^3+Tr_8(x^9)</math></td>
<td><math>C4</math></td>
</tr>


<math>8.2</math>
<tr>
<td class="noborderbelow"><math>8.6</math></td>
<td><math>x^3+a^{-1}Tr_8(a^3x^9)</math></td>
<td><math>C4</math></td>
</tr>


<math>8.3</math>
<tr>
<td class="noborderbelow"><math>8.7</math></td>
<td><math>a(x+x^{16})(ax+a^{16}x^{16})+a^{17}(ax+a^{16}x^{16})^{12}</math></td>
<td><math>C10</math></td>
</tr>


<math>8.4</math>
<tr class="divider">
<td rowspan="11"><math>9</math></td>
<td class="noborderbelow"><math>9.1</math></td>
<td><math>x^3</math></td>
<td><math>Gold</math></td>
</tr>


<math>8.5</math>


<math>8.6</math>
<tr>
<td class="noborderbelow"><math>9.2</math></td>
<td><math>x^5</math></td>
<td><math>Gold</math></td>
</tr>


<math>8.7</math></td>
<tr>
<td class="noborderbelow"><math>9.3</math></td>
<td><math>x^{17}</math></td>
<td><math>Gold</math></td>
</tr>


<td><math>x^3</math>
<tr>
<td class="noborderbelow"><math>9.4</math></td>
<td><math>x^{13}</math></td>
<td><math>Kasami</math></td>
</tr>


<math>x^9</math>
<tr>
<td class="noborderbelow"><math>9.5</math></td>
<td><math>x^{241}</math></td>
<td><math>Kasami</math></td>
</tr>


<math>x^{57}</math>
<tr>
<td class="noborderbelow"><math>9.6</math></td>
<td><math>x^{19}</math></td>
<td><math>Welch</math></td>
</tr>


<math>x^3+x^{17}+p^{48}x^{18}+p^3x^{33}+px^{34}+x^{48}</math>
<tr>
<td class="noborderbelow"><math>9.7</math></td>
<td><math>x^{255}</math></td>
<td><math>Inverse</math></td>
</tr>


<math>x^3+Tr_8(x^9)</math>
<tr>
<td class="noborderbelow"><math>9.8</math></td>
<td><math>x^3+Tr_9(x^9)</math></td>
<td><math>C4</math></td>
</tr>


<math>x^3+a^{-1}Tr_8(a^3x^9)</math>
<tr>
<td class="noborderbelow"><math>9.9</math></td>
<td><math>x^3+Tr^3_9(x^9+x^{18})</math></td>
<td><math>C5</math></td>
</tr>


<math>a(x+x^{16})(ax+a^{16}x^{16})+a^{17}(ax+a^{16}x^{16})^{12}</math></td>
<tr>
<td class="noborderbelow"><math>9.10</math></td>
<td><math>x^3+Tr^3_9(x^{18}+x^{36})</math></td>
<td><math>C6</math></td>
</tr>


<td><math>Gold</math>
<tr>
<td class="noborderbelow"><math>9.11</math></td>
<td><math>x^3+a^{246}x^{10}+a^{47}x^{17}+a^{181}x^{66}+a^{428}x^{129}</math></td>
<td><math>C11</math></td>
</tr>


<math>Gold</math>
<tr class="divider">
<td rowspan="8"><math>10</math></td>
<td class="noborderbelow"><math>10.1</math></td>
<td><math>x^3</math></td>
<td><math>Gold</math></td>
</tr>


<math>Kasami</math>
<tr>
<td class="noborderbelow"><math>10.2</math></td>
<td><math>x^9</math></td>
<td><math>Gold</math></td>
</tr>


<math>C3</math>
<tr>
<td class="noborderbelow"><math>10.3</math></td>
<td><math>x^{57}</math></td>
<td><math>Kasami</math></td>
</tr>


<math>C4</math>
<tr>
<td class="noborderbelow"><math>10.4</math></td>
<td><math>x^{339}</math></td>
<td><math>Dobbertin</math></td>
</tr>


<math>C4</math>
<tr>
<td class="noborderbelow"><math>10.5</math></td>
<td><math>x^6+x^{33}+p^{31}x^{192}</math></td>
<td><math>C3</math></td>
</tr>


<math>C10</math></td>
<tr>
<td class="noborderbelow"><math>10.6</math></td>
<td><math>x^{3}+x^{72}+p^{31}x^{258}</math></td>
<td><math>C3</math></td>
</tr>
</tr>


<tr>
<tr>
<td><math>9.1</math>
<td class="noborderbelow"><math>10.7</math></td>
<td><math>x^3+Tr_{10}(x^9)</math></td>
<td><math>C4</math></td>
</tr>


<math>9.2</math>
<tr>
<td class="noborderbelow"><math>10.8</math></td>
<td><math>x^3+a^{-1}Tr_{10}(a^3x^9)</math></td>
<td><math>C4</math></td>
</tr>


<math>9.3</math>
<tr class="divider">
<td rowspan="13"><math>11</math></td>
<td class="noborderbelow"><math>11.1</math></td>
<td><math>x^3</math></td>
<td><math>Gold</math></td>
</tr>


<math>9.4</math>
<tr>
<td class="noborderbelow"><math>11.2</math></td>
<td><math>x^5</math></td>
<td><math>Gold</math></td>
</tr>


<math>9.5</math>
<tr>
<td class="noborderbelow"><math>11.3</math></td>
<td><math>x^9</math></td>
<td><math>Gold</math></td>
</tr>


<math>9.6</math>
<tr>
<td class="noborderbelow"><math>11.4</math></td>
<td><math>x^{17}</math></td>
<td><math>Gold</math></td>
</tr>


<math>9.7</math>
<tr>
 
<td class="noborderbelow"><math>11.5</math></td>
<math>9.8</math>
<td><math>x^{33}</math></td>
 
<td><math>Gold</math></td>
<math>9.9</math>
 
<math>9.10</math>
 
<math>9.11</math></td>
 
<td><math>x^3</math>
 
<math>x^5</math>
 
<math>x^{17}</math>
 
<math>x^{13}</math>
 
<math>x^{241}</math>
 
<math>x^{19}</math>
 
<math>x^{255}</math>
 
<math>x^3+Tr_9(x^9)</math>
 
<math>x^3+Tr^3_9(x^9+x^{18})</math>
 
<math>x^3+Tr^3_9(x^{18}+x^{36})</math>
 
<math>x^3+a^{246}x^{10}+a^{47}x^{17}+a^{181}x^{66}+a^{428}x^{129}</math></td>
 
<td><math>Gold</math>
 
<math>Gold</math>
 
<math>Gold</math>
 
<math>Kasami</math>
 
<math>Kasami</math>
 
<math>Welch</math>
 
<math>Inverse</math>
 
<math>C4</math>
 
<math>C5</math>
 
<math>C6</math>
 
<math>C11</math></tD>
</tr>
</tr>


<tr>
<tr>
<td><math>10.1</math>
<td class="noborderbelow"><math>11.6</math></td>
 
<td><math>x^{13}</math></td>
<math>10.2</math>
<td><math>Kasami</math></td>
 
<math>10.3</math>
 
<math>10.4</math>
 
<math>10.5</math>
 
<math>10.6</math>
 
<math>10.7</math>
 
<math>10.8</math></td>
 
<td><math>10.1</math>
 
<math>x^3</math>
 
<math>x^9</math>
 
<math>x^{57}</math>
 
<math>x^{339}</math>
 
<math>x^6+x^{33}+p^{31}x^{192}</math>
 
<math>x^{3}+x^{72}+p^{31}x^{258}</math>
 
<math>x^3+Tr_{10}(x^9)</math>
 
<math>x^3+a^{-1}Tr_{10}(a^3x^9)</math></td>
 
<td><math>Gold</math>
 
<math>Gold</math>
 
<math>Kasami</math>
 
<math>Dobbertin</math>
 
<math>C3</math>
 
<math>C3</math>
 
<math>C4</math>
 
<math>C4</math></td>
 
</tr>
</tr>


<tr>
<tr>
<td><math>11.1</math>
<td class="noborderbelow"><math>11.7</math></td>
<td><math>x^{57}</math></td>
<td><math>Kasami</math></td>
</tr>


<math>11.2</math>
<td class="noborderbelow"><math>11.8</math></td>
<td><math>x^{241}</math></td>
<td><math>Kasami</math></td>
</tr>


<math>11.3</math>
<td class="noborderbelow"><math>11.9</math></td>
<td><math>x^{993}</math></td>
<td><math>Kasami</math></td>
</tr>


<math>11.4</math>
<td class="noborderbelow"><math>11.10</math></td>
<td><math>x^{35}</math></td>
<td><math>Welch</math></td>
</tr>


<math>11.5</math>
<td class="noborderbelow"><math>11.11</math></td>
<td><math>x^{287}</math></td>
<td><math>Niho</math></td>
</tr>


<math>11.6</math>
<td class="noborderbelow"><math>11.12</math></td>
<td><math>x^{123}</math></td>
<td><math>Inverse</math></td>
</tr>


<math>11.7</math>
<td class="noborderbelow"><math>11.13</math></td>
 
<td><math>x^3+Tr_{11}(x^9)</math></td>
<math>11.8</math>
<td><math>C4</math></td>
 
</tr>
<math>11.9</math>
 
<math>11.10</math>
 
<math>11.11</math>
 
<math>11.12</math>
 
<math>11.13</math></td>
 
<td><math>x^3</math>
 
<math>x^5</math>
 
<math>x^9</math>
 
<math>x^{17}</math>
 
<math>x^{33}</math>
 
<math>x^{13}</math>
 
<math>x^{57}</math>
 
<math>x^{241}</math>
 
<math>x^{993}</math>
 
<math>x^{35}</math>
 
<math>x^{287}</math>
 
<math>x^{123}</math>
 
<math>x^3+Tr_{11}(x^9)</math></td>
 
<td><math>Gold</math>
 
<math>Gold</math>
 
<math>Gold</math>
 
<math>Gold</math>
 
<math>Gold</math>
 
<math>Kasami</math>
 
<math>Kasami</math>
 
<math>Kasami</math>
 
<math>Kasami</math>


<math>Welch</math>
<math>Niho</math>
<math>Inverse</math>
<math>C4</math></td>
</tr>


</table>
</table>

Revision as of 11:54, 15 March 2019

CCZ-inequivalent APN Functions over [math]\displaystyle{ \mathbb{F}_{2^n} }[/math] from the Known APN Classes for [math]\displaystyle{ 6\leqslant n \leqslant 11 }[/math]

[math]\displaystyle{ n }[/math] [math]\displaystyle{ N^\circ }[/math] Functions Equivalent to
[math]\displaystyle{ 6 }[/math] [math]\displaystyle{ 6.1 }[/math] [math]\displaystyle{ x^3 }[/math] [math]\displaystyle{ Gold }[/math]
[math]\displaystyle{ 6.2 }[/math] [math]\displaystyle{ x^{24}+ax^{17}+a^8x^{10}+ax^9+x^3 }[/math] [math]\displaystyle{ C3 }[/math]
[math]\displaystyle{ 6.3 }[/math] [math]\displaystyle{ ax^3+x^{17}+a^4x^{24} }[/math] [math]\displaystyle{ C7-C9 }[/math]
[math]\displaystyle{ 7 }[/math] [math]\displaystyle{ 7.1 }[/math] [math]\displaystyle{ x^3 }[/math] [math]\displaystyle{ Gold }[/math]
[math]\displaystyle{ 7.2 }[/math] [math]\displaystyle{ x^5 }[/math] [math]\displaystyle{ Gold }[/math]
[math]\displaystyle{ 7.3 }[/math] [math]\displaystyle{ x^9 }[/math] [math]\displaystyle{ Gold }[/math]
[math]\displaystyle{ 7.4 }[/math] [math]\displaystyle{ x^{13} }[/math] [math]\displaystyle{ Kasami }[/math]
[math]\displaystyle{ 7.5 }[/math] [math]\displaystyle{ x^{57} }[/math] [math]\displaystyle{ Kasami }[/math]
[math]\displaystyle{ 7.6 }[/math] [math]\displaystyle{ x^{63} }[/math] [math]\displaystyle{ Inverse }[/math]
[math]\displaystyle{ 7.7 }[/math] [math]\displaystyle{ x^3+Tr_7(x^9) }[/math] [math]\displaystyle{ C4 }[/math]
[math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 8.1 }[/math] [math]\displaystyle{ x^3 }[/math] [math]\displaystyle{ Gold }[/math]
[math]\displaystyle{ 8.2 }[/math] [math]\displaystyle{ x^9 }[/math] [math]\displaystyle{ Gold }[/math]
[math]\displaystyle{ 8.3 }[/math] [math]\displaystyle{ x^{57} }[/math] [math]\displaystyle{ Kasami }[/math]
[math]\displaystyle{ 8.4 }[/math] [math]\displaystyle{ x^3+x^{17}+p^{48}x^{18}+p^3x^{33}+px^{34}+x^{48} }[/math] [math]\displaystyle{ C3 }[/math]
[math]\displaystyle{ 8.5 }[/math] [math]\displaystyle{ x^3+Tr_8(x^9) }[/math] [math]\displaystyle{ C4 }[/math]
[math]\displaystyle{ 8.6 }[/math] [math]\displaystyle{ x^3+a^{-1}Tr_8(a^3x^9) }[/math] [math]\displaystyle{ C4 }[/math]
[math]\displaystyle{ 8.7 }[/math] [math]\displaystyle{ a(x+x^{16})(ax+a^{16}x^{16})+a^{17}(ax+a^{16}x^{16})^{12} }[/math] [math]\displaystyle{ C10 }[/math]
[math]\displaystyle{ 9 }[/math] [math]\displaystyle{ 9.1 }[/math] [math]\displaystyle{ x^3 }[/math] [math]\displaystyle{ Gold }[/math]
[math]\displaystyle{ 9.2 }[/math] [math]\displaystyle{ x^5 }[/math] [math]\displaystyle{ Gold }[/math]
[math]\displaystyle{ 9.3 }[/math] [math]\displaystyle{ x^{17} }[/math] [math]\displaystyle{ Gold }[/math]
[math]\displaystyle{ 9.4 }[/math] [math]\displaystyle{ x^{13} }[/math] [math]\displaystyle{ Kasami }[/math]
[math]\displaystyle{ 9.5 }[/math] [math]\displaystyle{ x^{241} }[/math] [math]\displaystyle{ Kasami }[/math]
[math]\displaystyle{ 9.6 }[/math] [math]\displaystyle{ x^{19} }[/math] [math]\displaystyle{ Welch }[/math]
[math]\displaystyle{ 9.7 }[/math] [math]\displaystyle{ x^{255} }[/math] [math]\displaystyle{ Inverse }[/math]
[math]\displaystyle{ 9.8 }[/math] [math]\displaystyle{ x^3+Tr_9(x^9) }[/math] [math]\displaystyle{ C4 }[/math]
[math]\displaystyle{ 9.9 }[/math] [math]\displaystyle{ x^3+Tr^3_9(x^9+x^{18}) }[/math] [math]\displaystyle{ C5 }[/math]
[math]\displaystyle{ 9.10 }[/math] [math]\displaystyle{ x^3+Tr^3_9(x^{18}+x^{36}) }[/math] [math]\displaystyle{ C6 }[/math]
[math]\displaystyle{ 9.11 }[/math] [math]\displaystyle{ x^3+a^{246}x^{10}+a^{47}x^{17}+a^{181}x^{66}+a^{428}x^{129} }[/math] [math]\displaystyle{ C11 }[/math]
[math]\displaystyle{ 10 }[/math] [math]\displaystyle{ 10.1 }[/math] [math]\displaystyle{ x^3 }[/math] [math]\displaystyle{ Gold }[/math]
[math]\displaystyle{ 10.2 }[/math] [math]\displaystyle{ x^9 }[/math] [math]\displaystyle{ Gold }[/math]
[math]\displaystyle{ 10.3 }[/math] [math]\displaystyle{ x^{57} }[/math] [math]\displaystyle{ Kasami }[/math]
[math]\displaystyle{ 10.4 }[/math] [math]\displaystyle{ x^{339} }[/math] [math]\displaystyle{ Dobbertin }[/math]
[math]\displaystyle{ 10.5 }[/math] [math]\displaystyle{ x^6+x^{33}+p^{31}x^{192} }[/math] [math]\displaystyle{ C3 }[/math]
[math]\displaystyle{ 10.6 }[/math] [math]\displaystyle{ x^{3}+x^{72}+p^{31}x^{258} }[/math] [math]\displaystyle{ C3 }[/math]
[math]\displaystyle{ 10.7 }[/math] [math]\displaystyle{ x^3+Tr_{10}(x^9) }[/math] [math]\displaystyle{ C4 }[/math]
[math]\displaystyle{ 10.8 }[/math] [math]\displaystyle{ x^3+a^{-1}Tr_{10}(a^3x^9) }[/math] [math]\displaystyle{ C4 }[/math]
[math]\displaystyle{ 11 }[/math] [math]\displaystyle{ 11.1 }[/math] [math]\displaystyle{ x^3 }[/math] [math]\displaystyle{ Gold }[/math]
[math]\displaystyle{ 11.2 }[/math] [math]\displaystyle{ x^5 }[/math] [math]\displaystyle{ Gold }[/math]
[math]\displaystyle{ 11.3 }[/math] [math]\displaystyle{ x^9 }[/math] [math]\displaystyle{ Gold }[/math]
[math]\displaystyle{ 11.4 }[/math] [math]\displaystyle{ x^{17} }[/math] [math]\displaystyle{ Gold }[/math]
[math]\displaystyle{ 11.5 }[/math] [math]\displaystyle{ x^{33} }[/math] [math]\displaystyle{ Gold }[/math]
[math]\displaystyle{ 11.6 }[/math] [math]\displaystyle{ x^{13} }[/math] [math]\displaystyle{ Kasami }[/math]
[math]\displaystyle{ 11.7 }[/math] [math]\displaystyle{ x^{57} }[/math] [math]\displaystyle{ Kasami }[/math]
[math]\displaystyle{ 11.8 }[/math] [math]\displaystyle{ x^{241} }[/math] [math]\displaystyle{ Kasami }[/math]
[math]\displaystyle{ 11.9 }[/math] [math]\displaystyle{ x^{993} }[/math] [math]\displaystyle{ Kasami }[/math]
[math]\displaystyle{ 11.10 }[/math] [math]\displaystyle{ x^{35} }[/math] [math]\displaystyle{ Welch }[/math]
[math]\displaystyle{ 11.11 }[/math] [math]\displaystyle{ x^{287} }[/math] [math]\displaystyle{ Niho }[/math]
[math]\displaystyle{ 11.12 }[/math] [math]\displaystyle{ x^{123} }[/math] [math]\displaystyle{ Inverse }[/math]
[math]\displaystyle{ 11.13 }[/math] [math]\displaystyle{ x^3+Tr_{11}(x^9) }[/math] [math]\displaystyle{ C4 }[/math]
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