Difference between revisions of "Known switching classes of APN functions over GF(2^n) for n = 5,6,7,8"

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<td><math>5</math></td>
 
<td><math>5</math></td>
<td><math>1,1</math>
+
<td><math>1.1</math>
<math>1,2</math>
+
<math>1.2</math>
<math>2,1</math></td>
+
<math>2.1</math></td>
 
<td><math>x^3</math>
 
<td><math>x^3</math>
 
<math>x^5</math>
 
<math>x^5</math>
 
<math>x^{-1}</math></td>
 
<math>x^{-1}</math></td>
 
</tr>
 
</tr>
 +
</table>
 +
  
 +
<table>
 +
<tr>
 +
<th><math>n</math></th>
 +
<th><math>N^\circ</math></th>
 +
<th><math>F(x)</math></th>
 +
</tr>
 +
<tr>
 +
<td rowspan="14"><math>6</math></td>
 +
<td>1.1</td>
 +
<td><math>x^{3}</math></td>
 +
</tr>
 +
<tr>
 +
<td>1.2</td>
 +
<td><math>x^{3} + u^{11}x^{6} + ux^{9}</math></td>
 +
</tr>
 +
<tr>
 +
<td>2.1</td>
 +
<td><math>ux^{5} + x^{9} + u^{4}x^{17} + ux^{18} + u^{4}x^{20} + ux^{24} + u^{4}x^{34} + ux^{40}</math></td>
 +
</tr>
 +
<tr>
 +
<td>2.2</td>
 +
<td><math>u^{7}x^{3} + x^{5} + u^{3}x^{9} + u^{4}x^{10} + x^{17} + u^{6}x^{18}</math></td>
 +
</tr>
 +
<tr>
 +
<td>2.3</td>
 +
<td><math>x^{3} + ux^{24} + x^{10}</math></td>
 +
</tr>
 +
<tr>
 +
<td>2.4</td>
 +
<td><math>x^{3} + u^{17}(x^{17} + x^{18} + x^{20} + x^{24})</math></td>
 +
</tr>
 +
<tr>
 +
<td>2.5</td>
 +
<td><math>x^{3} + u^{11}x^{5} + u^{13}x^{9} + x^{17} + u^{11}x^{33} + x^{48}</math></td>
 +
</tr>
 +
<tr>
 +
<td>2.6</td>
 +
<td><math>u^{25}x^{5} + x^{9} + u^{38}x^{12} + u^{25}x^{18} + u^{25}x^{36}</math></td>
 +
</tr>
 +
<tr>
 +
<td>2.7</td>
 +
<td><math>u^{40}x^{5} + u^{10}x^{6} + u^{62}x^{20} + u^{35}x^{33} + u^{15}x^{34} + u^{29}x^{48}</math></td>
 +
</tr>
 +
<tr>
 +
<td>2.8</td>
 +
<td><math>u^{34}x^{6} + u^{52}x^{9} + u^{48}x^{12} + u^{6}x^{20} + u^{9}x^{33} + u^{23}x^{34} + u^{25}x^{40}</math></td>
 +
</tr>
 +
<tr>
 +
<td>2.9</td>
 +
<td><math>x^{9} + u^{4}(x^{10} + x^{18}) + u^{9}(x^{12} + x^{20} + x^{40})</math></td>
 +
</tr>
 +
<tr>
 +
<td>2.10</td>
 +
<td><math>u^{52}x^{3} + u^{47}x^{5} + ux^{6} + u^{9}x^{9} + u^{44}x^{12} + u^{47}x^{33} + u^{10}x^{34} + u^{33}x^{40}</math></td>
 +
</tr>
 +
<tr>
 +
<td>2.11</td>
 +
<td><math>u(x^{6} + x^{10} + x^{24} + x^{33}) + x^{9} + u^{4}x^{17}</math></td>
 +
</tr>
 
<tr>
 
<tr>
<td><math>6</math></td>
+
<td>2.12</td>
<td><math>1,1</math>
+
<td><math>x^{3} + u^{17}(x^{17} + x^{18} + x^{20} + x^{24}) + u^{14}((u^{52}x^{3} + u^{6}x^{5} + u^{19}x^{7} + u^{28}x^{11} + u^{2}x^{13})+ (u^{52}x^{3} + u^{6}x^{5} + u^{19}x^{7} + u^{28}x^{11} + u^{2}x^{13})^{2} + (u^{52}x^{3} + u^{6}x^{5} + u^{19}x^{7} + u^{28}x^{11} + u^{2}x^{13})^{4}+ (u^{52}x^{3} + u^{6}x^{5} + u^{19}x^{7} + u^{28}x^{11} + u^{2}x^{13})^{8}+ (u^{52}x^{3} + u^{6}x^{5} + u^{19}x^{7} + u^{28}x^{11} + u^{2}x^{13})^{1}6+ (u^{52}x^{3} + u^{6}x^{5} + u^{19}x^{7} + u^{28}x^{11} + u^{2}x^{13})^{3}2+ (u^{2}x)^{9} +(u^{2}x)^{1}8 +(u^{2}x)^{3}6 + x^{21}+x^{42}</math></td>
<math>1,2</math>
 
<math>2.1</math></td>
 
<td><math>x^3</math>
 
<math>x^3+u^{11}x6+ux^9</math>
 
<math>ux^5+x^9+u^4x^{17}+ux^{18}+u^4x^{20}+ux^{24}+u^4x^{34}+ux^{40}</math></td>
 
 
</tr>
 
</tr>
 +
</table>

Revision as of 17:06, 22 January 2019

Known switching classes of APN functions over , , and


1.1
1.2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12