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
5 1.1 x3 330 42 4960 4960
1.2 x5 330 42 4960 158720
2.1 x-1 496 232 310 310
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
7 1.1

x3

3610 198 113792 113792
1.2

x3 + Tr(x9)

4026 212 896 896
2.1

x34 + x18 + x5

4034 210 896 896
2.2

x3 + x17 + x33 + x34

4040 212 896 896
3.1

x5

3708 198 113792 113792
4.1

x9

3610 198 113792 14565376
5.1

x13

4270 338 889 889
6.1

x57

4704 436 889 889
7.1

x-1

8128 4928 1778 1778
8.1

x65 + x10 + x3

4038 212 896 896
9.1

x3 + x9 + x18 + x66

4044 212 896 896
10.1

x3 + x12 + x17 + x33

4048 210 896 896
10.2

x3 + x17 + x20 + x34 + x66

4040 210 896 896
11.1

x3 + x20 + x34 + x66

4048 210 896 896
12.1

x3 + x12 + x40 + x72

4048 210 896 896
13.1

x3 + x5 + x10 + x33 + x34

4040 212 896 896
14.1

x3 + x6 + x34 + x40 + x72

4048 212 896 896
14.2

x3 + x5 + x6 + x12 + x33 + x34

4050 210 896 896
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
8 1.1

x3

11818 420
1.2

x9

12370 420
1.3

x3+Tr(x9)

13800 432
1.4

x9+Tr(x3)

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

x57

15358 960