# 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 ${\displaystyle \mathbb {F} _{2^{8}}}$.

Also available is Magma code generating representatives from the switching classes.

${\displaystyle n}$ ${\displaystyle N^{\circ }}$ ${\displaystyle F(x)}$ Γ-rank Δ-rank Aut(dev(GF))/22n Aut(dev(GF))/22n
${\displaystyle 5}$ 1.1 x3 330 42 4960 4960
1.2 x5 330 42 4960 158720
2.1 x-1 496 232 310 310
${\displaystyle 6}$ 1.1 x3 1102 94 24192 48384
1.2 x3 + u11x6 + ux9 1146 94 4032 8064
2.1 ux5 + x9 + u4x17 + ux18 + u4x20 + ux24 + u4x34 + ux40 1158 96 320 320
2.2 u7x3 + x5 + u3x9 + u4x10 + x17 + u6x18 1166 94 448 896
2.3 x3 + ux24 + x10 1166 96 896 896
2.4 x3 + u17(x17 + x18 + x20 + x24) 1168 96 64 64
2.5 x3 + u11x5 + u13x9 + x17 + u11x33 + x48 1170 96 320 320
2.6 u25x5 + x9 + u38x12 + u25x18 + u25x36 1170 96 64 64
2.7 u40x5 + u10x6 + u62x20 + u35x33 + u15x34 + u29x48 1170 96 64 64
2.8 u34x6 + u52x9 + u48x12 + u6x20 + u9x33 + u23x34 + u25x40 1170 96 64 64
2.9 x9 + u4(x10 + x18) + u9(x12 + x20 + x40) 1172 96 64 64
2.10 u52x3 + u47x5 + ux6 + u9x9 + u44x12 + u47x33 + u10x34 + u33x40 1172 96 64 64
2.11 u(x6 + x10 + x24 + x33) + x9 + u4x17 1174 96 64 64
2.12 x3 + u17(x17 + x18 + x20 + x24) + u14((u52x3 + u6x5 + u19x7 + u28x11 + u2x13)+ (u52x3 + u6x5 + u19x7 + u28x11 + u2x13)2 + (u52x3 + u6x5 + u19x7 + u28x11 + u2x13)4+ (u52x3 + u6x5 + u19x7 + u28x11 + u2x13)8+ (u52x3 + u6x5 + u19x7 + u28x11 + u2x13)16+ (u52x3 + u6x5 + u19x7 + u28x11 + u2x13)32+ (u2x)9 +(u2x)19 +(u2x)36 + x21+x42 1300 152 8 8
${\displaystyle 7}$ 1.1 ${\displaystyle x^{3}}$ 3610 198 113792 113792
1.2 ${\displaystyle x^{3}+{\rm {Tr}}(x^{9})}$ 4026 212 896 896
2.1 ${\displaystyle x^{34}+x^{18}+x^{5}}$ 4034 210 896 896
2.2 ${\displaystyle x^{3}+x^{17}+x^{33}+x^{34}}$ 4040 212 896 896
3.1 ${\displaystyle x^{5}}$ 3708 198 113792 113792
4.1 ${\displaystyle x^{9}}$ 3610 198 113792 14565376
5.1 ${\displaystyle x^{13}}$ 4270 338 889 889
6.1 ${\displaystyle x^{57}}$ 4704 436 889 889
7.1 ${\displaystyle x^{-1}}$ 8128 4928 1778 1778
8.1 ${\displaystyle x^{65}+x^{10}+x^{3}}$ 4038 212 896 896
9.1 ${\displaystyle x^{3}+x^{9}+x^{18}+x^{66}}$ 4044 212 896 896
10.1 ${\displaystyle x^{3}+x^{12}+x^{17}+x^{33}}$ 4048 210 896 896
10.2 ${\displaystyle x^{3}+x^{17}+x^{20}+x^{34}+x^{66}}$ 4040 210 896 896
11.1 ${\displaystyle x^{3}+x^{20}+x^{34}+x^{66}}$ 4048 210 896 896
12.1 ${\displaystyle x^{3}+x^{12}+x^{40}+x^{72}}$ 4048 210 896 896
13.1 ${\displaystyle x^{3}+x^{5}+x^{10}+x^{33}+x^{34}}$ 4040 212 896 896
14.1 ${\displaystyle x^{3}+x^{6}+x^{34}+x^{40}+x^{72}}$ 4048 212 896 896
14.2 ${\displaystyle x^{3}+x^{5}+x^{6}+x^{12}+x^{33}+x^{34}}$ 4050 210 896 896
14.3 ${\displaystyle u^{2}x^{96}+}$ ${\displaystyle u^{78}x^{80}+}$ ${\displaystyle u^{121}x^{72}+}$ ${\displaystyle u^{49}x^{68}+}$ ${\displaystyle u^{77}x^{66}+}$ ${\displaystyle u^{29}x^{65}+}$ ${\displaystyle u^{119}x^{48}+}$ ${\displaystyle u^{117}x^{40}+}$ ${\displaystyle u^{28}x^{36}+}$ ${\displaystyle u^{107}x^{34}+u^{62}x^{33}+u^{125}x^{24}+u^{76}x^{20}+u^{84}x^{18}+u^{110}x^{17}+u^{49}x^{12}+u^{102}x^{10}+u^{69}x^{9}+}$ ${\displaystyle u^{14}x^{6}+}$ ${\displaystyle x^{5}+}$ ${\displaystyle x^{3}}$ 4046 212 128 128
${\displaystyle 8}$ 1.1 ${\displaystyle x^{3}}$ 11818 420
1.2 ${\displaystyle x^{9}}$ 12370 420
1.3 ${\displaystyle x^{3}+{\rm {Tr}}(x^{9})}$ 13800 432
1.4 ${\displaystyle x^{9}+{\rm {Tr}}(x^{3})}$ 13804 434
1.5 ${\displaystyle x^{3}+u^{245}x^{33}+u^{183}x^{66}+u^{21}x^{144}}$ 13842 436
1.6 ${\displaystyle x^{3}+u^{65}x^{18}+u^{120}x^{66}+u^{135}x^{144}}$ 13848 438
1.7 ${\displaystyle u^{188}x^{192}+}$ ${\displaystyle u^{129}x^{144}+}$ ${\displaystyle u^{172}x^{132}+}$ ${\displaystyle u^{138}x^{129}+}$ ${\displaystyle u^{74}x^{96}+}$ ${\displaystyle u^{244}x^{72}+}$ ${\displaystyle u^{22}x^{66}+}$ ${\displaystyle u^{178}x^{48}+}$ ${\displaystyle u^{150}x^{36}+}$ ${\displaystyle u^{146}x^{33}+}$ ${\displaystyle u^{6}x^{24}+}$ ${\displaystyle u^{60}x^{18}+}$ ${\displaystyle u^{80}x^{12}+}$ ${\displaystyle u^{140}x^{9}+}$ ${\displaystyle u^{221}x^{6}+}$ ${\displaystyle u^{19}x^{3}}$ 14034 438
1.8 ${\displaystyle u^{37}x^{192}+}$ ${\displaystyle u^{110}x^{144}+}$ ${\displaystyle u^{40}x^{132}+}$ ${\displaystyle u^{53}x^{129}+}$ ${\displaystyle u^{239}x^{96}+}$ ${\displaystyle u^{235}x^{72}+}$ ${\displaystyle u^{126}x^{66}+}$ ${\displaystyle u^{215}x^{48}+}$ ${\displaystyle u^{96}x^{36}+}$ ${\displaystyle u^{29}x^{33}+}$ ${\displaystyle u^{19}x^{24}+}$ ${\displaystyle u^{14}x^{18}+}$ ${\displaystyle u^{139}x^{12}+}$ ${\displaystyle u^{230}x^{9}+}$ ${\displaystyle u^{234}x^{6}+}$ ${\displaystyle u^{228}x^{3}}$ 14032 438
1.9 ${\displaystyle u^{242}x^{192}+}$ ${\displaystyle u^{100}x^{144}+}$ ${\displaystyle u^{66}x^{132}+}$ ${\displaystyle u^{230}x^{129}+}$ ${\displaystyle u^{202}x^{96}+}$ ${\displaystyle u^{156}x^{72}+}$ ${\displaystyle u^{254}x^{66}+}$ ${\displaystyle u^{18}x^{48}+}$ ${\displaystyle u^{44}x^{36}+}$ ${\displaystyle u^{95}x^{33}+}$ ${\displaystyle u^{100}x^{24}+}$ ${\displaystyle u^{245}x^{18}+}$ ${\displaystyle u^{174}x^{12}+}$ ${\displaystyle u^{175}x^{9}+}$ ${\displaystyle u^{247}x^{6}+}$ ${\displaystyle u^{166}x^{3}}$ 14036 438
1.10 ${\displaystyle u^{100}x^{192}+}$ ${\displaystyle u^{83}x^{144}+}$ ${\displaystyle u^{153}x^{132}+}$ ${\displaystyle u^{65}x^{129}+}$ ${\displaystyle u^{174}x^{96}+}$ ${\displaystyle u^{136}x^{72}+}$ ${\displaystyle u^{46}x^{66}+u^{55}x^{48}+u^{224}x^{36}+u^{180}x^{33}+u^{179}x^{24}+u^{226}x^{18}+u^{54}x^{12}+u^{168}x^{9}+u^{89}x^{6}+u^{56}x^{3}}$ 14036 438
1.11 ${\displaystyle u^{77}x^{192}+}$ ${\displaystyle u^{133}x^{144}+}$ ${\displaystyle u^{47}x^{132}+}$ ${\displaystyle u^{229}x^{129}+}$ ${\displaystyle u^{23}x^{96}+}$ ${\displaystyle u^{242}x^{72}+}$ ${\displaystyle u^{242}x^{66}+}$ ${\displaystyle u^{245}x^{48}+}$ ${\displaystyle u^{212}x^{36}+}$ ${\displaystyle u^{231}x^{33}+}$ ${\displaystyle u^{174}x^{24}+}$ ${\displaystyle u^{216}x^{18}+}$ ${\displaystyle u^{96}x^{12}+}$ ${\displaystyle u^{253}x^{9}+}$ ${\displaystyle u^{154}x^{6}+}$ ${\displaystyle u^{71}x^{3}}$ 14032 438
1.12 ${\displaystyle u^{220}x^{192}+}$ ${\displaystyle u^{94}x^{144}+}$ ${\displaystyle u^{70}x^{132}+}$ ${\displaystyle u^{159}x^{129}+}$ ${\displaystyle u^{145}x^{96}+}$ ${\displaystyle u^{160}x^{72}+}$ ${\displaystyle u^{74}x^{66}+}$ ${\displaystyle u^{184}x^{48}+}$ ${\displaystyle u^{119}x^{36}+}$ ${\displaystyle u^{106}x^{33}+}$ ${\displaystyle u^{253}x^{24}+}$ ${\displaystyle wx^{18}+}$ ${\displaystyle u^{90}x^{12}+}$ ${\displaystyle u^{169}x^{9}+}$ ${\displaystyle u^{118}x^{6}+}$ ${\displaystyle u^{187}x^{3}}$ 14034 438
1.13 ${\displaystyle u^{98}x^{192}+}$ ${\displaystyle u^{225}x^{144}+}$ ${\displaystyle u^{111}x^{132}+}$ ${\displaystyle u^{238}x^{129}+}$ ${\displaystyle u^{182}x^{96}+}$ ${\displaystyle u^{125}x^{72}+}$ ${\displaystyle u^{196}x^{66}+}$ ${\displaystyle u^{219}x^{48}+}$ ${\displaystyle u^{189}x^{36}+}$ ${\displaystyle u^{199}x^{33}+}$ ${\displaystyle u^{181}x^{24}+}$ ${\displaystyle u^{110}x^{18}+}$ ${\displaystyle u^{19}x^{12}+}$ ${\displaystyle u^{175}x^{9}+}$ ${\displaystyle u^{133}x^{6}+}$ ${\displaystyle u^{47}x^{3}}$ 14030 438
1.14 ${\displaystyle u^{236}x^{192}+}$ ${\displaystyle u^{212}x^{160}+}$ ${\displaystyle u^{153}x^{144}+}$ ${\displaystyle u^{185}x^{136}+}$ ${\displaystyle u^{3}x^{132}+}$ ${\displaystyle u^{89}x^{130}+}$ ${\displaystyle u^{189}x^{129}+}$ ${\displaystyle u^{182}x^{96}+}$ ${\displaystyle u^{105}x^{80}+}$ ${\displaystyle u^{232}x^{72}+}$ ${\displaystyle u^{219}x^{68}+}$ ${\displaystyle u^{145}x^{66}+}$ ${\displaystyle u^{171}x^{65}+}$ ${\displaystyle u^{107}x^{48}+}$ ${\displaystyle u^{179}x^{40}+}$ ${\displaystyle u^{227}x^{36}+}$ ${\displaystyle u^{236}x^{34}+}$ ${\displaystyle u^{189}x^{33}+}$ ${\displaystyle u^{162}x^{24}+}$ ${\displaystyle u^{216}x^{20}+}$ ${\displaystyle u^{162}x^{18}+}$ ${\displaystyle u^{117}x^{17}+}$ ${\displaystyle u^{56}x^{12}+}$ ${\displaystyle u^{107}x^{10}+}$ ${\displaystyle u^{236}x^{9}+}$ ${\displaystyle u^{253}x^{6}+}$ ${\displaystyle u^{180}x^{5}+}$ ${\displaystyle u^{18}x^{3}}$ 14046 454
1.15 ${\displaystyle u^{27}x^{192}+}$ ${\displaystyle u^{167}x^{144}+}$ ${\displaystyle u^{26}x^{132}+}$ ${\displaystyle u^{231}x^{129}+}$ ${\displaystyle u^{139}x^{96}+}$ ${\displaystyle u^{30}x^{72}+}$ ${\displaystyle u^{139}x^{66}+}$ ${\displaystyle u^{203}x^{48}+}$ ${\displaystyle u^{36}x^{36}+}$ ${\displaystyle u^{210}x^{33}+}$ ${\displaystyle u^{195}x^{24}+}$ ${\displaystyle u^{12}x^{18}+}$ ${\displaystyle u^{43}x^{12}+}$ ${\displaystyle u^{97}x^{9}+}$ ${\displaystyle u^{61}x^{6}+}$ ${\displaystyle u^{39}x^{3}}$ 14036 454
1.16 ${\displaystyle u^{6}x^{192}+}$ ${\displaystyle u^{85}x^{144}+}$ ${\displaystyle u^{251}x^{132}+}$ ${\displaystyle u^{215}x^{129}+}$ ${\displaystyle u^{229}x^{96}+}$ ${\displaystyle u^{195}x^{72}+}$ ${\displaystyle u^{152}x^{66}+}$ ${\displaystyle u^{173}x^{48}+}$ ${\displaystyle u^{209}x^{36}+}$ ${\displaystyle u^{165}x^{33}+}$ ${\displaystyle u^{213}x^{24}+}$ ${\displaystyle u^{214}x^{18}+}$ ${\displaystyle u^{158}x^{12}+}$ ${\displaystyle u^{146}x^{9}+}$ ${\displaystyle x^{6}+}$ ${\displaystyle u^{50}x^{3}}$ 14032 438
1.17 ${\displaystyle u^{164}x^{192}+}$ ${\displaystyle u^{224}x^{144}+}$ ${\displaystyle u^{59}x^{132}+}$ ${\displaystyle u^{124}x^{129}+}$ ${\displaystyle u^{207}x^{96}+}$ ${\displaystyle u^{211}x^{72}+}$ ${\displaystyle u^{5}x^{66}+}$ ${\displaystyle u^{26}x^{48}+}$ ${\displaystyle u^{20}x^{36}+}$ ${\displaystyle u^{101}x^{33}+}$ ${\displaystyle u^{175}x^{24}+}$ ${\displaystyle u^{241}x^{18}+}$ ${\displaystyle x^{12}+}$ ${\displaystyle u^{15}x^{9}+}$ ${\displaystyle u^{217}x^{6}+}$ ${\displaystyle u^{212}x^{3}}$ 14028 438
2.1 ${\displaystyle x^{3}+x^{17}+u^{16}(x^{18}+x^{33})+u^{15}x^{48}}$ 13200 414
3.1 ${\displaystyle x^{3}+u^{24}x^{6}+u^{182}x^{132}+u^{67}x^{192}}$ 14024 438
4.1 ${\displaystyle x^{3}+x^{6}+x^{68}+x^{80}+x^{132}+x^{160}}$ 14040 454
5.1 ${\displaystyle x^{3}+x^{5}+x^{18}+x^{40}+x^{66}}$ 14044 446
6.1 ${\displaystyle x^{3}+x^{12}+x^{40}+x^{66}+x^{130}}$ 14046 438
7.1 ${\displaystyle x^{57}}$ 15358 960