CCZ-inequivalent representatives from the known APN families for dimensions up to 11: Difference between revisions
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<td class="noborderbelow"><math>11.12</math></td> | <td class="noborderbelow"><math>11.12</math></td> | ||
<td><math>x^{ | <td><math>x^{1023}</math></td> | ||
<td><math>Inverse</math></td> | <td><math>Inverse</math></td> | ||
<td>Inverse</td> | <td>Inverse</td> |
Revision as of 11:15, 21 August 2019
CCZ-inequivalent APN Functions over [math]\displaystyle{ \mathbb{F}_{2^n} }[/math] from the Known APN Classes for [math]\displaystyle{ 6\leqslant n \leqslant 11 }[/math]
Dimension | [math]\displaystyle{ N^\circ }[/math] | Functions | Equivalent to | Walsh spectrum | Γ-rank | Δ-rank |
---|---|---|---|---|---|---|
[math]\displaystyle{ 6 }[/math] | [math]\displaystyle{ 6.1 }[/math] | [math]\displaystyle{ x^3 }[/math] | [math]\displaystyle{ Gold }[/math] | Gold | 1102 | 94 |
[math]\displaystyle{ 6.2 }[/math] | [math]\displaystyle{ x^{24}+ax^{17}+a^8x^{10}+ax^9+x^3 }[/math] | [math]\displaystyle{ C3 }[/math] | Gold | 1146 | 94 | |
[math]\displaystyle{ 6.3 }[/math] | [math]\displaystyle{ ax^3+x^{17}+a^4x^{24} }[/math] | [math]\displaystyle{ C7-C9 }[/math] | Gold | 1166 | 96 | |
[math]\displaystyle{ 7 }[/math] | [math]\displaystyle{ 7.1 }[/math] | [math]\displaystyle{ x^3 }[/math] | [math]\displaystyle{ Gold }[/math] | Gold | 3610 | 198 |
[math]\displaystyle{ 7.2 }[/math] | [math]\displaystyle{ x^5 }[/math] | [math]\displaystyle{ Gold }[/math] | Gold | 3708 | 198 | |
[math]\displaystyle{ 7.3 }[/math] | [math]\displaystyle{ x^9 }[/math] | [math]\displaystyle{ Gold }[/math] | Gold | 3610 | 198 | |
[math]\displaystyle{ 7.4 }[/math] | [math]\displaystyle{ x^{13} }[/math] | [math]\displaystyle{ Kasami }[/math] | Gold | 4270 | 338 | |
[math]\displaystyle{ 7.5 }[/math] | [math]\displaystyle{ x^{57} }[/math] | [math]\displaystyle{ Kasami }[/math] | Gold | 4704 | 436 | |
[math]\displaystyle{ 7.6 }[/math] | [math]\displaystyle{ x^{63} }[/math] | [math]\displaystyle{ Inverse }[/math] | Inverse | 8128 | 4928 | |
[math]\displaystyle{ 7.7 }[/math] | [math]\displaystyle{ x^3+Tr_7(x^9) }[/math] | [math]\displaystyle{ C4 }[/math] | Gold | 4026 | 212 | |
[math]\displaystyle{ 8 }[/math] | [math]\displaystyle{ 8.1 }[/math] | [math]\displaystyle{ x^3 }[/math] | [math]\displaystyle{ Gold }[/math] | Gold | 11818 | 420 |
[math]\displaystyle{ 8.2 }[/math] | [math]\displaystyle{ x^9 }[/math] | [math]\displaystyle{ Gold }[/math] | Gold | 12370 | 420 | |
[math]\displaystyle{ 8.3 }[/math] | [math]\displaystyle{ x^{57} }[/math] | [math]\displaystyle{ Kasami }[/math] | Gold | 15358 | 960 | |
[math]\displaystyle{ 8.4 }[/math] | [math]\displaystyle{ x^3+x^{17}+p^{48}x^{18}+p^3x^{33}+px^{34}+x^{48} }[/math] | [math]\displaystyle{ C3 }[/math] | Gold | 13200 | 414 | |
[math]\displaystyle{ 8.5 }[/math] | [math]\displaystyle{ x^3+Tr_8(x^9) }[/math] | [math]\displaystyle{ C4 }[/math] | Gold | 13200 | 432 | |
[math]\displaystyle{ 8.6 }[/math] | [math]\displaystyle{ x^3+a^{-1}Tr_8(a^3x^9) }[/math] | [math]\displaystyle{ C4 }[/math] | Gold | 13842 | 436 | |
[math]\displaystyle{ 8.7 }[/math] | [math]\displaystyle{ a(x+x^{16})(ax+a^{16}x^{16})+a^{17}(ax+a^{16}x^{16})^{12} }[/math] | [math]\displaystyle{ C10 }[/math] | Gold | 13642 | 436 | |
[math]\displaystyle{ 9 }[/math] | [math]\displaystyle{ 9.1 }[/math] | [math]\displaystyle{ x^3 }[/math] | [math]\displaystyle{ Gold }[/math] | Gold | 38470 | 872 |
[math]\displaystyle{ 9.2 }[/math] | [math]\displaystyle{ x^5 }[/math] | [math]\displaystyle{ Gold }[/math] | Gold | 41494 | 872 | |
[math]\displaystyle{ 9.3 }[/math] | [math]\displaystyle{ x^{17} }[/math] | [math]\displaystyle{ Gold }[/math] | Gold | 38470 | 872 | |
[math]\displaystyle{ 9.4 }[/math] | [math]\displaystyle{ x^{13} }[/math] | [math]\displaystyle{ Kasami }[/math] | Gold | 58676 | 3086 | |
[math]\displaystyle{ 9.5 }[/math] | [math]\displaystyle{ x^{241} }[/math] | [math]\displaystyle{ Kasami }[/math] | Gold | 61726 | 3482 | |
[math]\displaystyle{ 9.6 }[/math] | [math]\displaystyle{ x^{19} }[/math] | [math]\displaystyle{ Welch }[/math] | Gold | 60894 | 3956 | |
[math]\displaystyle{ 9.7 }[/math] | [math]\displaystyle{ x^{255} }[/math] | [math]\displaystyle{ Inverse }[/math] | Inverse | 130816 | 93024 | |
[math]\displaystyle{ 9.8 }[/math] | [math]\displaystyle{ x^3+Tr_9(x^9) }[/math] | [math]\displaystyle{ C4 }[/math] | Gold | 47890 | 920 | |
[math]\displaystyle{ 9.9 }[/math] | [math]\displaystyle{ x^3+Tr^3_9(x^9+x^{18}) }[/math] | [math]\displaystyle{ C5 }[/math] | Gold | 48428 | 930 | |
[math]\displaystyle{ 9.10 }[/math] | [math]\displaystyle{ x^3+Tr^3_9(x^{18}+x^{36}) }[/math] | [math]\displaystyle{ C6 }[/math] | Gold | 48460 | 944 | |
[math]\displaystyle{ 9.11 }[/math] | [math]\displaystyle{ x^3+a^{246}x^{10}+a^{47}x^{17}+a^{181}x^{66}+a^{428}x^{129} }[/math] | [math]\displaystyle{ C11 }[/math] | Gold | 48596 | 944 | |
[math]\displaystyle{ 10 }[/math] | [math]\displaystyle{ 10.1 }[/math] | [math]\displaystyle{ x^3 }[/math] | [math]\displaystyle{ Gold }[/math] | Gold | 125042 | |
[math]\displaystyle{ 10.2 }[/math] | [math]\displaystyle{ x^9 }[/math] | [math]\displaystyle{ Gold }[/math] | Gold | 136492 | ||
[math]\displaystyle{ 10.3 }[/math] | [math]\displaystyle{ x^{57} }[/math] | [math]\displaystyle{ Kasami }[/math] | Gold | 186416 | ||
[math]\displaystyle{ 10.4 }[/math] | [math]\displaystyle{ x^{339} }[/math] | [math]\displaystyle{ Dobbertin }[/math] | Dobbertin | 280604 | ||
[math]\displaystyle{ 10.5 }[/math] | [math]\displaystyle{ x^6+x^{33}+p^{31}x^{192} }[/math] | [math]\displaystyle{ C3 }[/math] | Gold | 151216 | ||
[math]\displaystyle{ 10.6 }[/math] | [math]\displaystyle{ x^{33}+x^{72}+p^{31}x^{258} }[/math] | [math]\displaystyle{ C3 }[/math] | Gold | 153896 | ||
[math]\displaystyle{ 10.7 }[/math] | [math]\displaystyle{ x^3+Tr_{10}(x^9) }[/math] | [math]\displaystyle{ C4 }[/math] | Gold | 164034 | ||
[math]\displaystyle{ 10.8 }[/math] | [math]\displaystyle{ x^3+a^{-1}Tr_{10}(a^3x^9) }[/math] | [math]\displaystyle{ C4 }[/math] | Gold | 164098 | ||
[math]\displaystyle{ 11 }[/math] | [math]\displaystyle{ 11.1 }[/math] | [math]\displaystyle{ x^3 }[/math] | [math]\displaystyle{ Gold }[/math] | Gold | ||
[math]\displaystyle{ 11.2 }[/math] | [math]\displaystyle{ x^5 }[/math] | [math]\displaystyle{ Gold }[/math] | Gold | |||
[math]\displaystyle{ 11.3 }[/math] | [math]\displaystyle{ x^9 }[/math] | [math]\displaystyle{ Gold }[/math] | Gold | |||
[math]\displaystyle{ 11.4 }[/math] | [math]\displaystyle{ x^{17} }[/math] | [math]\displaystyle{ Gold }[/math] | Gold | |||
[math]\displaystyle{ 11.5 }[/math] | [math]\displaystyle{ x^{33} }[/math] | [math]\displaystyle{ Gold }[/math] | Gold | |||
[math]\displaystyle{ 11.6 }[/math] | [math]\displaystyle{ x^{13} }[/math] | [math]\displaystyle{ Kasami }[/math] | Gold | |||
[math]\displaystyle{ 11.7 }[/math] | [math]\displaystyle{ x^{57} }[/math] | [math]\displaystyle{ Kasami }[/math] | Gold | |||
[math]\displaystyle{ 11.8 }[/math] | [math]\displaystyle{ x^{241} }[/math] | [math]\displaystyle{ Kasami }[/math] | Gold | |||
[math]\displaystyle{ 11.9 }[/math] | [math]\displaystyle{ x^{993} }[/math] | [math]\displaystyle{ Kasami }[/math] | Gold | |||
[math]\displaystyle{ 11.10 }[/math] | [math]\displaystyle{ x^{35} }[/math] | [math]\displaystyle{ Welch }[/math] | Gold | |||
[math]\displaystyle{ 11.11 }[/math] | [math]\displaystyle{ x^{287} }[/math] | [math]\displaystyle{ Niho }[/math] | Gold | |||
[math]\displaystyle{ 11.12 }[/math] | [math]\displaystyle{ x^{1023} }[/math] | [math]\displaystyle{ Inverse }[/math] | Inverse | |||
[math]\displaystyle{ 11.13 }[/math] | [math]\displaystyle{ x^3+Tr_{11}(x^9) }[/math] | [math]\displaystyle{ C4 }[/math] | Gold |