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

From Boolean
Jump to navigation Jump to search
No edit summary
No edit summary
Line 9: Line 9:


<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 15:06, 22 January 2019

Known switching classes of APN functions over [math]\displaystyle{ \mathbb{F}_{2^5} }[/math], [math]\displaystyle{ \mathbb{F}_{2^6} }[/math], [math]\displaystyle{ \mathbb{F}_{2^7} }[/math] and [math]\displaystyle{ \mathbb{F}_{2^8} }[/math]

[math]\displaystyle{ n }[/math] [math]\displaystyle{ N^\circ }[/math] [math]\displaystyle{ F(x) }[/math]
[math]\displaystyle{ 5 }[/math] [math]\displaystyle{ 1.1 }[/math]

[math]\displaystyle{ 1.2 }[/math]

[math]\displaystyle{ 2.1 }[/math]
[math]\displaystyle{ x^3 }[/math]

[math]\displaystyle{ x^5 }[/math]

[math]\displaystyle{ x^{-1} }[/math]


[math]\displaystyle{ n }[/math] [math]\displaystyle{ N^\circ }[/math] [math]\displaystyle{ F(x) }[/math]
[math]\displaystyle{ 6 }[/math] 1.1 [math]\displaystyle{ x^{3} }[/math]
1.2 [math]\displaystyle{ x^{3} + u^{11}x^{6} + ux^{9} }[/math]
2.1 [math]\displaystyle{ ux^{5} + x^{9} + u^{4}x^{17} + ux^{18} + u^{4}x^{20} + ux^{24} + u^{4}x^{34} + ux^{40} }[/math]
2.2 [math]\displaystyle{ u^{7}x^{3} + x^{5} + u^{3}x^{9} + u^{4}x^{10} + x^{17} + u^{6}x^{18} }[/math]
2.3 [math]\displaystyle{ x^{3} + ux^{24} + x^{10} }[/math]
2.4 [math]\displaystyle{ x^{3} + u^{17}(x^{17} + x^{18} + x^{20} + x^{24}) }[/math]
2.5 [math]\displaystyle{ x^{3} + u^{11}x^{5} + u^{13}x^{9} + x^{17} + u^{11}x^{33} + x^{48} }[/math]
2.6 [math]\displaystyle{ u^{25}x^{5} + x^{9} + u^{38}x^{12} + u^{25}x^{18} + u^{25}x^{36} }[/math]
2.7 [math]\displaystyle{ 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]
2.8 [math]\displaystyle{ 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]
2.9 [math]\displaystyle{ x^{9} + u^{4}(x^{10} + x^{18}) + u^{9}(x^{12} + x^{20} + x^{40}) }[/math]
2.10 [math]\displaystyle{ 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]
2.11 [math]\displaystyle{ u(x^{6} + x^{10} + x^{24} + x^{33}) + x^{9} + u^{4}x^{17} }[/math]
2.12 [math]\displaystyle{ 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]