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

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

${\displaystyle n}$ ${\displaystyle N^{\circ }}$ ${\displaystyle F(x)}$
${\displaystyle 5}$ 1.1 ${\displaystyle x^{3}}$
1.2 ${\displaystyle x^{5}}$
2.1 ${\displaystyle x^{-1}}$
${\displaystyle 6}$ 1.1 ${\displaystyle x^{3}}$
1.2 ${\displaystyle x^{3}+u^{11}x^{6}+ux^{9}}$
2.1 ${\displaystyle ux^{5}+x^{9}+u^{4}x^{17}+ux^{18}+u^{4}x^{20}+ux^{24}+u^{4}x^{34}+ux^{40}}$
2.2 ${\displaystyle u^{7}x^{3}+x^{5}+u^{3}x^{9}+u^{4}x^{10}+x^{17}+u^{6}x^{18}}$
2.3 ${\displaystyle x^{3}+ux^{24}+x^{10}}$
2.4 ${\displaystyle x^{3}+u^{17}(x^{17}+x^{18}+x^{20}+x^{24})}$
2.5 ${\displaystyle x^{3}+u^{11}x^{5}+u^{13}x^{9}+x^{17}+u^{11}x^{33}+x^{48}}$
2.6 ${\displaystyle u^{25}x^{5}+x^{9}+u^{38}x^{12}+u^{25}x^{18}+u^{25}x^{36}}$
2.7 ${\displaystyle u^{40}x^{5}+u^{10}x^{6}+u^{62}x^{20}+u^{35}x^{33}+u^{15}x^{34}+u^{29}x^{48}}$
2.8 ${\displaystyle u^{34}x^{6}+u^{52}x^{9}+u^{48}x^{12}+u^{6}x^{20}+u^{9}x^{33}+u^{23}x^{34}+u^{25}x^{40}}$
2.9 ${\displaystyle x^{9}+u^{4}(x^{10}+x^{18})+u^{9}(x^{12}+x^{20}+x^{40})}$
2.10 ${\displaystyle u^{52}x^{3}+u^{47}x^{5}+ux^{6}+u^{9}x^{9}+u^{44}x^{12}+u^{47}x^{33}+u^{10}x^{34}+u^{33}x^{40}}$
2.11 ${\displaystyle u(x^{6}+x^{10}+x^{24}+x^{33})+x^{9}+u^{4}x^{17}}$
2.12 ${\displaystyle x^{3}+}$ ${\displaystyle u^{17}(x^{17}+}$ ${\displaystyle x^{18}+}$ ${\displaystyle x^{20}+}$ ${\displaystyle x^{24})+}$ ${\displaystyle u^{14}((u^{52}x^{3}+}$ ${\displaystyle u^{6}x^{5}+}$ ${\displaystyle u^{19}x^{7}+}$ ${\displaystyle u^{28}x^{11}+}$ $\displaystyle u^{2}x^{13})+ (u^{52}x^{3} +$ $\displaystyle u^{6}x^{5} +$ $\displaystyle u^{19}x^{7} +$ $\displaystyle u^{28}x^{11} +$ $\displaystyle u^{2}x^{13})^{2} +$ $\displaystyle (u^{52}x^{3} +$ $\displaystyle u^{6}x^{5} +$ ${\displaystyle u^{19}x^{7}+}$ $\displaystyle u^{28}x^{11} +$ $\displaystyle u^{2}x^{13})^{4}+ (u^{52}x^{3} +$ $\displaystyle u^{6}x^{5} +$ $\displaystyle u^{19}x^{7} +$ $\displaystyle u^{28}x^{11} +$ $\displaystyle u^{2}x^{13})^{8}+ (u^{52}x^{3} +$ $\displaystyle u^{6}x^{5} +$ $\displaystyle u^{19}x^{7} +$ ${\displaystyle u^{28}x^{11}+}$ ${\displaystyle u^{2}x^{13})^{16}+(u^{52}x^{3}+}$ ${\displaystyle u^{6}x^{5}+}$ ${\displaystyle u^{19}x^{7}+}$ ${\displaystyle u^{28}x^{11}+}$ ${\displaystyle u^{2}x^{13})^{32}+(u^{2}x)^{9}+(u^{2}x)^{19}+(u^{2}x)^{36}+}$ ${\displaystyle x^{21}+x^{42}}$
${\displaystyle 7}$ 1.1 ${\displaystyle x^{3}}$
1.2 ${\displaystyle x^{3}+{\rm {Tr}}(x^{9})}$
2.1 ${\displaystyle x^{34}+x^{18}+x^{5}}$
2.2 ${\displaystyle x^{3}+x^{17}+x^{33}+x^{34}}$
3.1 ${\displaystyle x^{5}}$
4.1 ${\displaystyle x^{9}}$
5.1 ${\displaystyle x^{13}}$
6.1 ${\displaystyle x^{57}}$
7.1 ${\displaystyle x^{-1}}$
8.1 ${\displaystyle x^{65}+x^{10}+x^{3}}$
9.1 ${\displaystyle x^{3}+x^{9}+x^{18}+x^{66}}$
10.1 ${\displaystyle x^{3}+x^{12}+x^{17}+x^{33}}$
10.2 ${\displaystyle x^{3}+x^{17}+x^{20}+x^{34}+x^{66}}$
11.1 ${\displaystyle x^{3}+x^{20}+x^{34}+x^{66}}$
12.1 ${\displaystyle x^{3}+x^{12}+x^{40}+x^{72}}$
13.1 ${\displaystyle x^{3}+x^{5}+x^{10}+x^{33}+x^{34}}$
14.1 ${\displaystyle x^{3}+x^{6}+x^{34}+x^{40}+x^{72}}$
14.2 ${\displaystyle x^{3}+x^{5}+x^{6}+x^{12}+x^{33}+x^{34}}$
14.3 ${\displaystyle u^{2}x^{96}+}$ ${\displaystyle u^{78}x^{80}+}$ ${\displaystyle u^{121}x^{72}+}$ ${\displaystyle u^{49}x^{68}+}$ ${\displaystyle u^{77}x^{66}+}$ ${\displaystyle u^{29}x^{65}+}$ ${\displaystyle u^{119}x^{48}+}$ ${\displaystyle u^{117}x^{40}+}$ ${\displaystyle u^{28}x^{36}+}$ ${\displaystyle u^{107}x^{34}+u^{62}x^{33}+u^{125}x^{24}+u^{76}x^{20}+u^{84}x^{18}+u^{110}x^{17}+u^{49}x^{12}+u^{102}x^{10}+u^{69}x^{9}+}$ ${\displaystyle u^{14}x^{6}+}$ ${\displaystyle x^{5}+}$ ${\displaystyle x^{3}}$
8 1.1 ${\displaystyle x^{3}}$
1.2 ${\displaystyle x^{9}}$
1.3 ${\displaystyle x^{3}+{\rm {Tr}}(x^{9})}$
1.4 ${\displaystyle x^{9}+{\rm {Tr}}(x^{3})}$
1.5 ${\displaystyle x^{3}+u^{245}x^{33}+u^{183}x^{66}+u^{21}x^{144}}$
1.6 ${\displaystyle x^{3}+u^{65}x^{18}+u^{120}x^{66}+u^{135}x^{144}}$
1.7 ${\displaystyle u^{188}x^{192}+}$ ${\displaystyle u^{129}x^{144}+}$ ${\displaystyle u^{172}x^{132}+}$ ${\displaystyle u^{138}x^{129}+}$ ${\displaystyle u^{74}x^{96}+}$ ${\displaystyle u^{244}x^{72}+}$ ${\displaystyle u^{22}x^{66}+}$ ${\displaystyle u^{178}x^{48}+}$ ${\displaystyle u^{150}x^{36}+}$ ${\displaystyle u^{146}x^{33}+}$ ${\displaystyle u^{6}x^{24}+}$ ${\displaystyle u^{60}x^{18}+}$ ${\displaystyle u^{80}x^{12}+}$ ${\displaystyle u^{140}x^{9}+}$ ${\displaystyle u^{221}x^{6}+}$ ${\displaystyle u^{19}x^{3}}$
1.8 ${\displaystyle u^{37}x^{192}+}$ ${\displaystyle u^{110}x^{144}+}$ ${\displaystyle u^{40}x^{132}+}$ ${\displaystyle u^{53}x^{129}+}$ ${\displaystyle u^{239}x^{96}+}$ ${\displaystyle u^{235}x^{72}+}$ ${\displaystyle u^{126}x^{66}+}$ ${\displaystyle u^{215}x^{48}+}$ ${\displaystyle u^{96}x^{36}+}$ ${\displaystyle u^{29}x^{33}+}$ ${\displaystyle u^{19}x^{24}+}$ ${\displaystyle u^{14}x^{18}+}$ ${\displaystyle u^{139}x^{12}+}$ ${\displaystyle u^{230}x^{9}+}$ ${\displaystyle u^{234}x^{6}+}$ ${\displaystyle u^{228}x^{3}}$
1.9 ${\displaystyle u^{242}x^{192}+}$ ${\displaystyle u^{100}x^{144}+}$ ${\displaystyle u^{66}x^{132}+}$ ${\displaystyle u^{230}x^{129}+}$ ${\displaystyle u^{202}x^{96}+}$ ${\displaystyle u^{156}x^{72}+}$ ${\displaystyle u^{254}x^{66}+}$ ${\displaystyle u^{18}x^{48}+}$ ${\displaystyle u^{44}x^{36}+}$ ${\displaystyle u^{95}x^{33}+}$ ${\displaystyle u^{100}x^{24}+}$ ${\displaystyle u^{245}x^{18}+}$ ${\displaystyle u^{174}x^{12}+}$ ${\displaystyle u^{175}x^{9}+}$ ${\displaystyle u^{247}x^{6}+}$ ${\displaystyle u^{166}x^{3}}$
1.10 ${\displaystyle u^{100}x^{192}+}$ ${\displaystyle u^{83}x^{144}+}$ ${\displaystyle u^{153}x^{132}+}$ ${\displaystyle u^{65}x^{129}+}$ ${\displaystyle u^{174}x^{96}+}$ ${\displaystyle u^{136}x^{72}+}$ ${\displaystyle u^{46}x^{66}+u^{55}x^{48}+u^{224}x^{36}+u^{180}x^{33}+u^{179}x^{24}+u^{226}x^{18}+u^{54}x^{12}+u^{168}x^{9}+u^{89}x^{6}+u^{56}x^{3}}$
1.11 ${\displaystyle u^{77}x^{192}+}$ ${\displaystyle u^{133}x^{144}+}$ ${\displaystyle u^{47}x^{132}+}$ ${\displaystyle u^{229}x^{129}+}$ ${\displaystyle u^{23}x^{96}+}$ ${\displaystyle u^{242}x^{72}+}$ ${\displaystyle u^{242}x^{66}+}$ ${\displaystyle u^{245}x^{48}+}$ ${\displaystyle u^{212}x^{36}+}$ ${\displaystyle u^{231}x^{33}+}$ ${\displaystyle u^{174}x^{24}+}$ ${\displaystyle u^{216}x^{18}+}$ ${\displaystyle u^{96}x^{12}+}$ ${\displaystyle u^{253}x^{9}+}$ ${\displaystyle u^{154}x^{6}+}$ ${\displaystyle u^{71}x^{3}}$
1.12 ${\displaystyle u^{220}x^{192}+}$ ${\displaystyle u^{94}x^{144}+}$ ${\displaystyle u^{70}x^{132}+}$ ${\displaystyle u^{159}x^{129}+}$ ${\displaystyle u^{145}x^{96}+}$ ${\displaystyle u^{160}x^{72}+}$ ${\displaystyle u^{74}x^{66}+}$ ${\displaystyle u^{184}x^{48}+}$ ${\displaystyle u^{119}x^{36}+}$ ${\displaystyle u^{106}x^{33}+}$ ${\displaystyle u^{253}x^{24}+}$ ${\displaystyle wx^{18}+}$ ${\displaystyle u^{90}x^{12}+}$ ${\displaystyle u^{169}x^{9}+}$ ${\displaystyle u^{118}x^{6}+}$ ${\displaystyle u^{187}x^{3}}$
1.13 ${\displaystyle u^{98}x^{192}+}$ ${\displaystyle u^{225}x^{144}+}$ ${\displaystyle u^{111}x^{132}+}$ ${\displaystyle u^{238}x^{129}+}$ ${\displaystyle u^{182}x^{96}+}$ ${\displaystyle u^{125}x^{72}+}$ ${\displaystyle u^{196}x^{66}+}$ ${\displaystyle u^{219}x^{48}+}$ ${\displaystyle u^{189}x^{36}+}$ ${\displaystyle u^{199}x^{33}+}$ ${\displaystyle u^{181}x^{24}+}$ ${\displaystyle u^{110}x^{18}+}$ ${\displaystyle u^{19}x^{12}+}$ ${\displaystyle u^{175}x^{9}+}$ ${\displaystyle u^{133}x^{6}+}$ ${\displaystyle u^{47}x^{3}}$
1.14 ${\displaystyle u^{236}x^{192}+}$ ${\displaystyle u^{212}x^{160}+}$ ${\displaystyle u^{153}x^{144}+}$ ${\displaystyle u^{185}x^{136}+}$ ${\displaystyle u^{3}x^{132}+}$ ${\displaystyle u^{89}x^{130}+}$ ${\displaystyle u^{189}x^{129}+}$ ${\displaystyle u^{182}x^{96}+}$ ${\displaystyle u^{105}x^{80}+}$ ${\displaystyle u^{232}x^{72}+}$ ${\displaystyle u^{219}x^{68}+}$ ${\displaystyle u^{145}x^{66}+}$ ${\displaystyle u^{171}x^{65}+}$ ${\displaystyle u^{107}x^{48}+}$ ${\displaystyle u^{179}x^{40}+}$ ${\displaystyle u^{227}x^{36}+}$ ${\displaystyle u^{236}x^{34}+}$ ${\displaystyle u^{189}x^{33}+}$ ${\displaystyle u^{162}x^{24}+}$ ${\displaystyle u^{216}x^{20}+}$ ${\displaystyle u^{162}x^{18}+}$ ${\displaystyle u^{117}x^{17}+}$ ${\displaystyle u^{56}x^{12}+}$ ${\displaystyle u^{107}x^{10}+}$ ${\displaystyle u^{236}x^{9}+}$ ${\displaystyle u^{253}x^{6}+}$ ${\displaystyle u^{180}x^{5}+}$ ${\displaystyle u^{18}x^{3}}$
1.15 ${\displaystyle u^{27}x^{192}+}$ ${\displaystyle u^{167}x^{144}+}$ ${\displaystyle u^{26}x^{132}+}$ ${\displaystyle u^{231}x^{129}+}$ ${\displaystyle u^{139}x^{96}+}$ ${\displaystyle u^{30}x^{72}+}$ ${\displaystyle u^{139}x^{66}+}$ ${\displaystyle u^{203}x^{48}+}$ ${\displaystyle u^{36}x^{36}+}$ ${\displaystyle u^{210}x^{33}+}$ ${\displaystyle u^{195}x^{24}+}$ ${\displaystyle u^{12}x^{18}+}$ ${\displaystyle u^{43}x^{12}+}$ ${\displaystyle u^{97}x^{9}+}$ ${\displaystyle u^{61}x^{6}+}$ ${\displaystyle u^{39}x^{3}}$
1.16 ${\displaystyle u^{6}x^{192}+}$ ${\displaystyle u^{85}x^{144}+}$ ${\displaystyle u^{251}x^{132}+}$ ${\displaystyle u^{215}x^{129}+}$ ${\displaystyle u^{229}x^{96}+}$ ${\displaystyle u^{195}x^{72}+}$ ${\displaystyle u^{152}x^{66}+}$ ${\displaystyle u^{173}x^{48}+}$ ${\displaystyle u^{209}x^{36}+}$ ${\displaystyle u^{165}x^{33}+}$ ${\displaystyle u^{213}x^{24}+}$ ${\displaystyle u^{214}x^{18}+}$ ${\displaystyle u^{158}x^{12}+}$ ${\displaystyle u^{146}x^{9}+}$ ${\displaystyle x^{6}+}$ ${\displaystyle u^{50}x^{3}}$
1.17 ${\displaystyle u^{164}x^{192}+}$ ${\displaystyle u^{224}x^{144}+}$ ${\displaystyle u^{59}x^{132}+}$ ${\displaystyle u^{124}x^{129}+}$ ${\displaystyle u^{207}x^{96}+}$ ${\displaystyle u^{211}x^{72}+}$ ${\displaystyle u^{5}x^{66}+}$ ${\displaystyle u^{26}x^{48}+}$ ${\displaystyle u^{20}x^{36}+}$ ${\displaystyle u^{101}x^{33}+}$ ${\displaystyle u^{175}x^{24}+}$ ${\displaystyle u^{241}x^{18}+}$ ${\displaystyle x^{12}+}$ ${\displaystyle u^{15}x^{9}+}$ ${\displaystyle u^{217}x^{6}+}$ ${\displaystyle u^{212}x^{3}}$
2.1 ${\displaystyle x^{3}+x^{17}+u^{16}(x^{18}+x^{33})+u^{15}x^{48}}$
3.1 ${\displaystyle x^{3}+u^{24}x^{6}+u^{182}x^{132}+u^{67}x^{192}}$
4.1 ${\displaystyle x^{3}+x^{6}+x^{68}+x^{80}+x^{132}+x^{160}}$
5.1 ${\displaystyle x^{3}+x^{5}+x^{18}+x^{40}+x^{66}}$
6.1 ${\displaystyle x^{3}+x^{12}+x^{40}+x^{66}+x^{130}}$
7.1 ${\displaystyle x^{57}}$