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

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Known switching classes of APN functions over GF(25), GF(26), GF(27) and GF(28).

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

n N F(x) Γ-rank Δ-rank Aut(dev(GF))/22n Aut(dev(GF))/22n Multiplier group
5 1.1 x3 330 42 4960 4960 4960
1.2 x5 330 42 4960 158720 4960
2.1 x-1 496 232 310 310 310
6 1.1 x3 1102 94 24192 48384 24192
1.2 x3 + u11x6 + ux9 1146 94 4032 8064 4032
2.1 ux5 + x9 + u4x17 + ux18 + u4x20 + ux24 + u4x34 + ux40 1158 96 320 320 320
2.2 u7x3 + x5 + u3x9 + u4x10 + x17 + u6x18 1166 94 448 896 448
2.3 x3 + ux24 + x10 1166 96 896 896 896
2.4 x3 + u17(x17 + x18 + x20 + x24) 1168 96 64 64 64
2.5 x3 + u11x5 + u13x9 + x17 + u11x33 + x48 1170 96 320 320 320
2.6 u25x5 + x9 + u38x12 + u25x18 + u25x36 1170 96 64 64 64
2.7 u40x5 + u10x6 + u62x20 + u35x33 + u15x34 + u29x48 1170 96 64 64 64
2.8 u34x6 + u52x9 + u48x12 + u6x20 + u9x33 + u23x34 + u25x40 1170 96 64 64 64
2.9 x9 + u4(x10 + x18) + u9(x12 + x20 + x40) 1172 96 64 64 64
2.10 u52x3 + u47x5 + ux6 + u9x9 + u44x12 + u47x33 + u10x34 + u33x40 1172 96 64 64 64
2.11 u(x6 + x10 + x24 + x33) + x9 + u4x17 1174 96 64 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 8
7 1.1

x3

3610 198 113792 113792 113792
1.2

x3 + Tr(x9)

4026 212 896 896 896
2.1

x34 + x18 + x5

4034 210 896 896 896
2.2

x3 + x17 + x33 + x34

4040 212 896 896 896
3.1

x5

3708 198 113792 113792 113792
4.1

x9

3610 198 113792 14565376 113792
5.1

x13

4270 338 889 889 889
6.1

x57

4704 436 889 889 889
7.1

x-1

8128 4928 1778 1778 1778
8.1

x65 + x10 + x3

4038 212 896 896 896
9.1

x3 + x9 + x18 + x66

4044 212 896 896 896
10.1

x3 + x12 + x17 + x33

4048 210 896 896 896
10.2

x3 + x17 + x20 + x34 + x66

4040 210 896 896 896
11.1

x3 + x20 + x34 + x66

4048 210 896 896 896
12.1

x3 + x12 + x40 + x72

4048 210 896 896 896
13.1

x3 + x5 + x10 + x33 + x34

4040 212 896 896 896
14.1

x3 + x6 + x34 + x40 + x72

4048 212 896 896 896
14.2

x3 + x5 + x6 + x12 + x33 + x34

4050 210 896 896 896
14.3

u2x96 + u78x80 + u121x72 + u49x68 + u77x66 + u29x65 + u119x48 + u117x40 + u28x36 + u107x34 +u62x33 +u125x24 +u76x20 +u84x18 +u110x17 +u49x12 +u102x10 +u69x9 + u14x6 + x5 + x3

4046 212 128 128 128
8 1.1

x3

11818 420 522240
1.2

x9

12370 420 522240
1.3

x3+Tr(x9)

13800 432 6144
1.4

x9+Tr(x3)

13804 434 6144
1.5 x3+u245x33+u183x66+u21x144 13842 436 3072
1.6 x3 + u65x18+u120x66+u135x144 13848 438 3072
1.7 u188x192 + u129x144 + u172x132 + u138x129 + u74x96 + u244x72 + u22x66 + u178x48 + u150x36 + u146x33 + u6x24 + u60x18 + u80x12 + u140x9 + u221x6 + u19x3 14034 438 768
1.8 u37x192 + u110x144 + u40x132 + u53x129 + u239x96 + u235x72 + u126x66 + u215x48 + u96x36 + u29x33 + u19x24 + u14x18 + u139x12 + u230x9 + u234x6 + u228x3 14032 438 3072
1.9 u242x192 + u100x144 + u66x132 + u230x129 + u202x96 + u156x72 + u254x66 + u18x48 + u44x36 + u95x33 + u100x24 + u245x18 + u174x12 + u175x9 + u247x6 + u166x3 14036 438 3072
1.10 u100x192 + u83x144 + u153x132 + u65x129 + u174x96 + u136x72 + u46x66+ u55x48+ u224x36+ u180x33+ u179x24+u226x18+ u54x12+ u168x9+ u89x6+ u56x3 14036 438 1536
1.11 u77x192 + u133x144 + u47x132 + u229x129 + u23x96 + u242x72 + u242x66 + u245x48 + u212x36 + u231x33 + u174x24 + u216x18 + u96x12 + u253x9 + u154x6 + u71x3 14032 438 3072
1.12 u220x192 + u94x144 + u70x132 + u159x129 + u145x96 + u160x72 + u74x66 + u184x48 + u119x36 + u106x33 + u253x24 + wx18 + u90x12 + u169x9 + u118x6 + u187x3 14034 438 3072
1.13 u98x192 + u225x144 + u111x132 + u238x129 + u182x96 + u125x72 + u196x66 + u219x48 + u189x36 + u199x33 + u181x24 + u110x18 + u19x12 + u175x9 + u133x6 + u47x3 14030 438 1536
1.14 u236x192 + u212x160 + u153x144 + u185x136 + u3x132 + u89x130 + u189x129 + u182x96 + u105x80 + u232x72 + u219x68 + u145x66 + u171x65 + u107x48 + u179x40 + u227x36 + u236x34 + u189x33 + u162x24 + u216x20 + u162x18 + u117x17 + u56x12 + u107x10 + u236x9 + u253x6 + u180x5 + u18x3 14046 454 512
1.15 u27x192 + u167x144 + u26x132 + u231x129 + u139x96 + u30x72 + u139x66 + u203x48 + u36x36 + u210x33 + u195x24 + u12x18 + u43x12 + u97x9 + u61x6 + u39x3 14036 454 768
1.16 u6x192 + u85x144 + u251x132 + u215x129 + u229x96 + u195x72 + u152x66 + u173x48 + u209x36 + u165x33 + u213x24 + u214x18 + u158x12 + u146x9 + x6 + u50x3 14032 438 1536
1.17 u164x192 + u224x144 + u59x132 + u124x129 + u207x96 + u211x72 + u5x66 + u26x48 + u20x36 + u101x33 + u175x24 + u241x18 + x12 + u15x9 + u217x6 + u212x3 14028 438 1536
2.1 x3+ x17+u16(x18+x33)+u15x48 13200 414 46080
3.1 x3+ u24x6+u182x132+u67x192 14024 438 3072
4.1 x3+x6+x68+x80+x132+x160 14040 454 2048
5.1 x3+x5+x18+x40+x66 14044 446 2048
6.1 x3+x12+x40+x66+x130 14046 438 2048
7.1

x57

15358 960 2040