Known switching classes of APN functions over GF(2^n) for n = 5,6,7,8: Difference between revisions
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<td>2.12</td> | <td>2.12</td> | ||
<td><math>x^{3} + </math> <math>u^{17}(x^{17} + </math> <math>x^{18} + </math> <math>x^{20} + </math> <math>x^{24}) + </math> <math>u^{14}((u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})+ (u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})^{2} + </math> <math>(u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})^{4}+ (u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})^{8}+ (u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})^{16}+ (u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})^{32}+ (u^{2}x)^{9} +(u^{2}x)^{19} +(u^{2}x)^{36} + </math> <math>x^{21}+x^{42}</math></td> | <td><math>x^{3} + </math> <math>u^{17}(x^{17} + </math> <math>x^{18} + </math> <math>x^{20} + </math> <math>x^{24}) + </math> <math>u^{14}((u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})+ (u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})^{2} + </math> <math>(u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})^{4}+ (u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})^{8}+ (u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})^{16}+ (u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})^{32}+ (u^{2}x)^{9} +(u^{2}x)^{19} +(u^{2}x)^{36} + </math> <math>x^{21}+x^{42}</math></td> | ||
</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="19"><math>7</math></td> | |||
<td>1.1</td> | |||
<td><math> | |||
x^{3} | |||
</math></td> | |||
</tr> | |||
<tr> | |||
<td>1.2</td> | |||
<td><math> | |||
x^{3} + {\rm Tr}(x^{9}) | |||
</math></td> | |||
</tr> | |||
<tr> | |||
<td>2.1</td> | |||
<td><math> | |||
x^{34} + x^{18} + x^{5} | |||
</math></td> | |||
</tr> | |||
<tr> | |||
<td>2.2</td> | |||
<td><math> | |||
x^{3} + x^{17} + x^{33} + x^{34} | |||
</math></td> | |||
</tr> | |||
<tr> | |||
<td>3.1</td> | |||
<td><math> | |||
x^{5} | |||
</math></td> | |||
</tr> | |||
<tr> | |||
<td>4.1</td> | |||
<td><math> | |||
x^{9} | |||
</math></td> | |||
</tr> | |||
<tr> | |||
<td>5.1</td> | |||
<td><math> | |||
x^{13} | |||
</math></td> | |||
</tr> | |||
<tr> | |||
<td>6.1</td> | |||
<td><math> | |||
x^{57} | |||
</math></td> | |||
</tr> | |||
<tr> | |||
<td>7.1</td> | |||
<td><math> | |||
x^{-1} | |||
</math></td> | |||
</tr> | |||
<tr> | |||
<td>8.1</td> | |||
<td><math> | |||
x^{65} + x^{10} + x^{3} | |||
</math></td> | |||
</tr> | |||
<tr> | |||
<td>9.1</td> | |||
<td><math> | |||
x^{3} + x^{9} + x^{18} + x^{66} | |||
</math></td> | |||
</tr> | |||
<tr> | |||
<td>10.1</td> | |||
<td><math> | |||
x^{3} + x^{12} + x^{17} + x^{33} | |||
</math></td> | |||
</tr> | |||
<tr> | |||
<td>10.2</td> | |||
<td><math> | |||
x^{3} + x^{17} + x^{20} + x^{34} + x^{66} | |||
</math></td> | |||
</tr> | |||
<tr> | |||
<td>11.1</td> | |||
<td><math> | |||
x^{3} + x^{20} + x^{34} + x^{66} | |||
</math></td> | |||
</tr> | |||
<tr> | |||
<td>12.1</td> | |||
<td><math> | |||
x^{3} + x^{12} + x^{40} + x^{72} | |||
</math></td> | |||
</tr> | |||
<tr> | |||
<td>13.1</td> | |||
<td><math> | |||
x^{3} + x^{5} + x^{10} + x^{33} + x^{34} | |||
</math></td> | |||
</tr> | |||
<tr> | |||
<td>14.1</td> | |||
<td><math> | |||
x^{3} + x^{6} + x^{34} + x^{40} + x^{72} | |||
</math></td> | |||
</tr> | |||
<tr> | |||
<td>14.2</td> | |||
<td><math> | |||
x^{3} + x^{5} + x^{6} + x^{12} + x^{33} + x^{34} | |||
</math></td> | |||
</tr> | |||
<tr> | |||
<td>14.3</td> | |||
<td><math> | |||
u^{2}x^{96} + </math> <math>u^{78}x^{80} + </math> <math>u^{121}x^{72} + </math> <math>u^{49}x^{68} + </math> <math>u^{77}x^{66} + </math> <math>u^{29}x^{65} + </math> <math>u^{119}x^{48} + </math> <math>u^{117}x^{40} + </math> <math>u^{28}x^{36} + </math> <math>u^{107}x^{34} +u^{62}x^{33} +u^{125}x^{24} +u^{76}x^{20} +u^{84}x^{18} +u^{110}x^{17} +u^{49}x^{12} +u^{102}x^{10} +u^{69}x^{9} + </math> <math>u^{14}x^{6} + </math> <math>x^{5} + </math> <math>x^{3} | |||
</math></td> | |||
</tr> | </tr> | ||
</table> | </table> |
Revision as of 15:26, 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} + }[/math] [math]\displaystyle{ u^{17}(x^{17} + }[/math] [math]\displaystyle{ x^{18} + }[/math] [math]\displaystyle{ x^{20} + }[/math] [math]\displaystyle{ x^{24}) + }[/math] [math]\displaystyle{ u^{14}((u^{52}x^{3} + }[/math] [math]\displaystyle{ u^{6}x^{5} + }[/math] [math]\displaystyle{ u^{19}x^{7} + }[/math] [math]\displaystyle{ u^{28}x^{11} + }[/math] [math]\displaystyle{ u^{2}x^{13})+ (u^{52}x^{3} + }[/math] [math]\displaystyle{ u^{6}x^{5} + }[/math] [math]\displaystyle{ u^{19}x^{7} + }[/math] [math]\displaystyle{ u^{28}x^{11} + }[/math] [math]\displaystyle{ u^{2}x^{13})^{2} + }[/math] [math]\displaystyle{ (u^{52}x^{3} + }[/math] [math]\displaystyle{ u^{6}x^{5} + }[/math] [math]\displaystyle{ u^{19}x^{7} + }[/math] [math]\displaystyle{ u^{28}x^{11} + }[/math] [math]\displaystyle{ u^{2}x^{13})^{4}+ (u^{52}x^{3} + }[/math] [math]\displaystyle{ u^{6}x^{5} + }[/math] [math]\displaystyle{ u^{19}x^{7} + }[/math] [math]\displaystyle{ u^{28}x^{11} + }[/math] [math]\displaystyle{ u^{2}x^{13})^{8}+ (u^{52}x^{3} + }[/math] [math]\displaystyle{ u^{6}x^{5} + }[/math] [math]\displaystyle{ u^{19}x^{7} + }[/math] [math]\displaystyle{ u^{28}x^{11} + }[/math] [math]\displaystyle{ u^{2}x^{13})^{16}+ (u^{52}x^{3} + }[/math] [math]\displaystyle{ u^{6}x^{5} + }[/math] [math]\displaystyle{ u^{19}x^{7} + }[/math] [math]\displaystyle{ u^{28}x^{11} + }[/math] [math]\displaystyle{ u^{2}x^{13})^{32}+ (u^{2}x)^{9} +(u^{2}x)^{19} +(u^{2}x)^{36} + }[/math] [math]\displaystyle{ x^{21}+x^{42} }[/math] |
[math]\displaystyle{ n }[/math] | [math]\displaystyle{ N^\circ }[/math] | [math]\displaystyle{ F(x) }[/math] |
---|---|---|
[math]\displaystyle{ 7 }[/math] | 1.1 | [math]\displaystyle{ x^{3} }[/math] |
1.2 | [math]\displaystyle{ x^{3} + {\rm Tr}(x^{9}) }[/math] | |
2.1 | [math]\displaystyle{ x^{34} + x^{18} + x^{5} }[/math] | |
2.2 | [math]\displaystyle{ x^{3} + x^{17} + x^{33} + x^{34} }[/math] | |
3.1 | [math]\displaystyle{ x^{5} }[/math] | |
4.1 | [math]\displaystyle{ x^{9} }[/math] | |
5.1 | [math]\displaystyle{ x^{13} }[/math] | |
6.1 | [math]\displaystyle{ x^{57} }[/math] | |
7.1 | [math]\displaystyle{ x^{-1} }[/math] | |
8.1 | [math]\displaystyle{ x^{65} + x^{10} + x^{3} }[/math] | |
9.1 | [math]\displaystyle{ x^{3} + x^{9} + x^{18} + x^{66} }[/math] | |
10.1 | [math]\displaystyle{ x^{3} + x^{12} + x^{17} + x^{33} }[/math] | |
10.2 | [math]\displaystyle{ x^{3} + x^{17} + x^{20} + x^{34} + x^{66} }[/math] | |
11.1 | [math]\displaystyle{ x^{3} + x^{20} + x^{34} + x^{66} }[/math] | |
12.1 | [math]\displaystyle{ x^{3} + x^{12} + x^{40} + x^{72} }[/math] | |
13.1 | [math]\displaystyle{ x^{3} + x^{5} + x^{10} + x^{33} + x^{34} }[/math] | |
14.1 | [math]\displaystyle{ x^{3} + x^{6} + x^{34} + x^{40} + x^{72} }[/math] | |
14.2 | [math]\displaystyle{ x^{3} + x^{5} + x^{6} + x^{12} + x^{33} + x^{34} }[/math] | |
14.3 | [math]\displaystyle{ u^{2}x^{96} + }[/math] [math]\displaystyle{ u^{78}x^{80} + }[/math] [math]\displaystyle{ u^{121}x^{72} + }[/math] [math]\displaystyle{ u^{49}x^{68} + }[/math] [math]\displaystyle{ u^{77}x^{66} + }[/math] [math]\displaystyle{ u^{29}x^{65} + }[/math] [math]\displaystyle{ u^{119}x^{48} + }[/math] [math]\displaystyle{ u^{117}x^{40} + }[/math] [math]\displaystyle{ u^{28}x^{36} + }[/math] [math]\displaystyle{ u^{107}x^{34} +u^{62}x^{33} +u^{125}x^{24} +u^{76}x^{20} +u^{84}x^{18} +u^{110}x^{17} +u^{49}x^{12} +u^{102}x^{10} +u^{69}x^{9} + }[/math] [math]\displaystyle{ u^{14}x^{6} + }[/math] [math]\displaystyle{ x^{5} + }[/math] [math]\displaystyle{ x^{3} }[/math] |