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