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

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Line 7: Line 7:
 
<th><math>F(x)</math></th>
 
<th><math>F(x)</math></th>
 
</tr>
 
</tr>
 
+
<tr>
<td><math>5</math></td>
+
<td rowspan="3"><math>5</math></td>
<td><math>1.1</math>
+
<td>1.1</td>
<math>1.2</math>
+
<td><math>x^3</math></td>
<math>2.1</math></td>
 
<td><math>x^3</math>
 
<math>x^5</math>
 
<math>x^{-1}</math></td>
 
 
</tr>
 
</tr>
</table>
 
 
 
<table>
 
 
<tr>
 
<tr>
<th><math>n</math></th>
+
<td>1.2</td>
<th><math>N^\circ</math></th>
+
<td><math>x^5</math></td>
<th><math>F(x)</math></th>
 
 
</tr>
 
</tr>
 
<tr>
 
<tr>
 +
<td>2.1</td>
 +
<td><math>x^{-1}</math></td>
 +
</tr>
 +
</tr>
 +
<tr class="strongDivider">
 
<td rowspan="14"><math>6</math></td>
 
<td rowspan="14"><math>6</math></td>
 
<td>1.1</td>
 
<td>1.1</td>
Line 82: Line 78:
 
<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>
 
</tr>
</table>
+
<tr class="strongDivider">
 
 
 
 
 
 
<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 rowspan="19"><math>7</math></td>
 
<td>1.1</td>
 
<td>1.1</td>
Line 205: Line 191:
 
<td><math>
 
<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}
 
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 class="strongDivider">
 +
<td rowspan="23">8</td>
 +
<td>1.1</td>
 +
<td><math>
 +
x^{3}
 +
</math></td>
 +
</tr>
 +
<tr>
 +
<td>1.2</td>
 +
<td><math>
 +
x^{9}
 +
</math></td>
 +
</tr>
 +
<tr>
 +
<td>1.3</td>
 +
<td><math>
 +
x^{3}+{\rm Tr}(x^{9})
 +
</math></td>
 +
</tr>
 +
<tr>
 +
<td>1.4</td>
 +
<td><math>
 +
x^{9}+{\rm Tr}(x^{3})
 +
</math></td>
 +
</tr>
 +
<tr>
 +
<td>1.5</td>
 +
<td><math>
 +
x^{3}+u^{245}x^{33}+u^{183}x^{66}+u^{21}x^{144}
 +
</math></td>
 +
</tr>
 +
<tr>
 +
<td>1.6</td>
 +
<td><math>
 +
x^{3} + u^{65}x^{18}+u^{120}x^{66}+u^{135}x^{144}
 +
</math></td>
 +
</tr>
 +
<tr>
 +
<td>1.7</td>
 +
<td><math>
 +
u^{188}x^{192} + </math> <math>u^{129}x^{144} + </math> <math>u^{172}x^{132} + </math> <math> u^{138}x^{129} + </math> <math>u^{74}x^{96} + </math> <math>u^{244}x^{72} + </math> <math>u^{22}x^{66} + </math> <math> u^{178}x^{48} + </math> <math>u^{150}x^{36} + </math> <math>u^{146}x^{33} + </math> <math>u^{6}x^{24} + </math> <math> u^{60}x^{18} + </math> <math>u^{80}x^{12} + </math> <math>u^{140}x^{9} + </math> <math>u^{221}x^{6} + </math> <math>u^{19}x^{3}
 +
</math></td>
 +
</tr>
 +
<tr>
 +
<td>1.8</td>
 +
<td><math>
 +
u^{37}x^{192} + </math> <math>u^{110}x^{144} + </math> <math>u^{40}x^{132} + </math> <math>u^{53}x^{129}  + </math> <math>u^{239}x^{96} + </math> <math>u^{235}x^{72} + </math> <math>u^{126}x^{66} + </math> <math>u^{215}x^{48} + </math> <math> u^{96}x^{36} + </math> <math>u^{29}x^{33} + </math> <math>u^{19}x^{24} + </math> <math>u^{14}x^{18} + </math> <math> u^{139}x^{12} + </math> <math>u^{230}x^{9} + </math> <math>u^{234}x^{6} + </math> <math>u^{228}x^{3}
 +
</math></td>
 +
</tr>
 +
<tr>
 +
<td>1.9</td>
 +
<td><math>
 +
u^{242}x^{192} + </math> <math>u^{100}x^{144} + </math> <math>u^{66}x^{132} + </math> <math>u^{230}x^{129} + </math> <math> u^{202}x^{96} + </math> <math>u^{156}x^{72} + </math> <math>u^{254}x^{66} + </math> <math>u^{18}x^{48} + </math> <math> u^{44}x^{36} + </math> <math>u^{95}x^{33} + </math> <math>u^{100}x^{24} + </math> <math>u^{245}x^{18} + </math> <math> u^{174}x^{12} + </math> <math>u^{175}x^{9} + </math> <math>u^{247}x^{6} + </math> <math>u^{166}x^{3}
 +
</math></td>
 +
</tr>
 +
<tr>
 +
<td>1.10</td>
 +
<td><math>
 +
u^{100}x^{192} + </math> <math> u^{83}x^{144} + </math> <math> u^{153}x^{132} + </math> <math> u^{65}x^{129} + </math> <math> u^{174}x^{96} + </math> <math> u^{136}x^{72} + </math> <math>
 +
u^{46}x^{66}+ u^{55}x^{48}+ u^{224}x^{36}+ u^{180}x^{33}+ u^{179}x^{24}+u^{226}x^{18}+ u^{54}x^{12}+ u^{168}x^{9}+ u^{89}x^{6}+ u^{56}x^{3}
 +
</math></td>
 +
</tr>
 +
<tr>
 +
<td>1.11</td>
 +
<td><math>
 +
u^{77}x^{192} + </math> <math> u^{133}x^{144} + </math> <math> u^{47}x^{132} + </math> <math> u^{229}x^{129} + </math> <math> u^{23}x^{96} + </math> <math> u^{242}x^{72} + </math> <math> u^{242}x^{66} + </math> <math> u^{245}x^{48} + </math> <math> u^{212}x^{36} + </math> <math> u^{231}x^{33} + </math> <math> u^{174}x^{24} + </math> <math> u^{216}x^{18} + </math> <math> u^{96}x^{12} + </math> <math> u^{253}x^{9} + </math> <math> u^{154}x^{6} + </math> <math> u^{71}x^{3}
 +
</math></td>
 +
</tr>
 +
<tr>
 +
<td>1.12</td>
 +
<td><math>
 +
u^{220}x^{192} + </math> <math> u^{94}x^{144} + </math> <math> u^{70}x^{132} + </math> <math> u^{159}x^{129} + </math> <math> u^{145}x^{96} + </math> <math>u^{160}x^{72} + </math> <math> u^{74}x^{66} + </math> <math> u^{184}x^{48} + </math> <math> u^{119}x^{36} + </math> <math> u^{106}x^{33} + </math> <math>u^{253}x^{24} + </math> <math> wx^{18} + </math> <math> u^{90}x^{12} + </math> <math> u^{169}x^{9} + </math> <math> u^{118}x^{6} + </math> <math> u^{187}x^{3}
 +
</math></td>
 +
</tr>
 +
<tr>
 +
<td>1.13</td>
 +
<td><math>
 +
u^{98}x^{192} + </math> <math> u^{225}x^{144} + </math> <math> u^{111}x^{132} + </math> <math> u^{238}x^{129} + </math> <math> u^{182}x^{96} + </math> <math> u^{125}x^{72} + </math> <math> u^{196}x^{66} + </math> <math> u^{219}x^{48} + </math> <math> u^{189}x^{36} + </math> <math> u^{199}x^{33} + </math> <math> u^{181}x^{24} + </math> <math> u^{110}x^{18} + </math> <math> u^{19}x^{12} + </math> <math> u^{175}x^{9} + </math> <math> u^{133}x^{6} + </math> <math> u^{47}x^{3}
 +
</math></td>
 +
</tr>
 +
<tr>
 +
<td>1.14</td>
 +
<td><math>
 +
u^{236}x^{192} + </math> <math> u^{212}x^{160} + </math> <math> u^{153}x^{144} + </math> <math> u^{185}x^{136} + </math> <math> u^{3}x^{132} + </math> <math>u^{89}x^{130} + </math> <math> u^{189}x^{129} + </math> <math> u^{182}x^{96} + </math> <math> u^{105}x^{80} + </math> <math> u^{232}x^{72} + </math> <math>u^{219}x^{68} + </math> <math> u^{145}x^{66} + </math> <math> u^{171}x^{65} + </math> <math> u^{107}x^{48} + </math> <math> u^{179}x^{40} + </math> <math> u^{227}x^{36} + </math> <math> u^{236}x^{34} + </math> <math> u^{189}x^{33} + </math> <math> u^{162}x^{24} + </math> <math> u^{216}x^{20} + </math> <math>u^{162}x^{18} + </math> <math> u^{117}x^{17} + </math> <math> u^{56}x^{12} + </math> <math> u^{107}x^{10} + </math> <math> u^{236}x^{9} + </math> <math>u^{253}x^{6} + </math> <math> u^{180}x^{5} + </math> <math> u^{18}x^{3}
 +
</math></td>
 +
</tr>
 +
<tr>
 +
<td>1.15</td>
 +
<td><math>
 +
u^{27}x^{192} + </math> <math> u^{167}x^{144} + </math> <math> u^{26}x^{132} + </math> <math>u^{231}x^{129} + </math> <math> u^{139}x^{96} + </math> <math>u^{30}x^{72} + </math> <math> u^{139}x^{66} + </math> <math> u^{203}x^{48} + </math> <math> u^{36}x^{36} + </math> <math> u^{210}x^{33} + </math> <math>u^{195}x^{24} + </math> <math> u^{12}x^{18} + </math> <math> u^{43}x^{12} + </math> <math> u^{97}x^{9} + </math> <math> u^{61}x^{6} + </math> <math>u^{39}x^{3}
 +
</math></td>
 +
</tr>
 +
<tr>
 +
<td>1.16</td>
 +
<td><math>
 +
u^{6}x^{192} + </math> <math> u^{85}x^{144} + </math> <math> u^{251}x^{132} + </math> <math> u^{215}x^{129} + </math> <math> u^{229}x^{96} + </math> <math> u^{195}x^{72} + </math> <math> u^{152}x^{66} + </math> <math> u^{173}x^{48} + </math> <math> u^{209}x^{36} + </math> <math> u^{165}x^{33} + </math> <math> u^{213}x^{24} + </math> <math> u^{214}x^{18} + </math> <math> u^{158}x^{12} + </math> <math> u^{146}x^{9} + </math> <math> x^{6} + </math> <math> u^{50}x^{3}
 +
</math></td>
 +
</tr>
 +
<tr>
 +
<td>1.17</td>
 +
<td><math>
 +
u^{164}x^{192} + </math> <math> u^{224}x^{144} + </math> <math> u^{59}x^{132} + </math> <math> u^{124}x^{129} + </math> <math> u^{207}x^{96} + </math> <math> u^{211}x^{72} + </math> <math> u^{5}x^{66} + </math> <math> u^{26}x^{48} + </math> <math> u^{20}x^{36} + </math> <math> u^{101}x^{33} + </math> <math> u^{175}x^{24} + </math> <math> u^{241}x^{18} + </math> <math> x^{12} + </math> <math> u^{15}x^{9} + </math> <math> u^{217}x^{6} + </math> <math> u^{212}x^{3}
 +
</math></td>
 +
</tr>
 +
<tr>
 +
<td>2.1</td>
 +
<td><math>
 +
x^{3}+ x^{17}+u^{16}(x^{18}+x^{33})+u^{15}x^{48}
 +
</math></td>
 +
</tr>
 +
<tr>
 +
<td>3.1</td>
 +
<td><math>
 +
x^{3}+ u^{24}x^{6}+u^{182}x^{132}+u^{67}x^{192}
 +
</math></td>
 +
</tr>
 +
<tr>
 +
<td>4.1</td>
 +
<td><math>
 +
x^{3}+x^{6}+x^{68}+x^{80}+x^{132}+x^{160}
 +
</math></td>
 +
</tr>
 +
<tr>
 +
<td>5.1</td>
 +
<td><math>
 +
x^{3}+x^{5}+x^{18}+x^{40}+x^{66}
 +
</math></td>
 +
</tr>
 +
<tr>
 +
<td>6.1</td>
 +
<td><math>
 +
x^{3}+x^{12}+x^{40}+x^{66}+x^{130}
 +
</math></td>
 +
</tr>
 +
<tr>
 +
<td>7.1</td>
 +
<td><math>
 +
x^{57}
 
</math></td>
 
</math></td>
 
</tr>
 
</tr>
 
</table>
 
</table>

Revision as of 17:43, 22 January 2019

Known switching classes of APN functions over , , and

1.1
1.2
2.1
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
1.1
1.2
2.1
2.2
3.1
4.1
5.1
6.1
7.1
8.1
9.1
10.1
10.2
11.1
12.1
13.1
14.1
14.2
14.3
8 1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
1.12
1.13
1.14
1.15
1.16
1.17
2.1
3.1
4.1
5.1
6.1
7.1