CCZ-inequivalent representatives from the known APN families for dimensions up to 11: Difference between revisions

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<th>Functions</th>
<th>Functions</th>
<th>Equivalent to</th>
<th>Equivalent to</th>
<th>Walsh spectrum</th>
<th>Γ-rank</th>
<th>Δ-rank</th>
</tr>
</tr>


Line 14: Line 17:
<td><math>x^3</math></td>
<td><math>x^3</math></td>
<td><math>Gold</math></td>
<td><math>Gold</math></td>
<td>Gold</td>
<td>1102</td>
<td>94</td>
</tr>
</tr>


Line 21: Line 27:
<td><math>x^{24}+ax^{17}+a^8x^{10}+ax^9+x^3</math></td>
<td><math>x^{24}+ax^{17}+a^8x^{10}+ax^9+x^3</math></td>
<td><math>C3</math></td>
<td><math>C3</math></td>
<td>Gold</td>
<td>1146</td>
<td>94</td>
</tr>
</tr>


Line 27: Line 36:
<td><math>ax^3+x^{17}+a^4x^{24}</math></td>
<td><math>ax^3+x^{17}+a^4x^{24}</math></td>
<td><math>C7-C9</math></td>
<td><math>C7-C9</math></td>
<td>Gold</td>
<td>1166</td>
<td>96</td>
</tr>
</tr>


Line 35: Line 47:
<td><math>x^3</math></td>
<td><math>x^3</math></td>
<td><math>Gold</math></td>
<td><math>Gold</math></td>
<td>Gold</td>
<td>3610</td>
<td>198</td>
</tr>
</tr>


Line 42: Line 57:
<td><math>x^5</math></td>
<td><math>x^5</math></td>
<td><math>Gold</math></td>
<td><math>Gold</math></td>
<td>Gold</td>
<td>3708</td>
<td>198</td>
</tr>
</tr>


Line 49: Line 67:
<td><math>x^9</math></td>
<td><math>x^9</math></td>
<td><math>Gold</math></td>
<td><math>Gold</math></td>
<td>Gold</td>
<td>3610</td>
<td>198</td>
</tr>
</tr>


Line 56: Line 77:
<td><math>x^{13}</math></td>
<td><math>x^{13}</math></td>
<td><math>Kasami</math></td>
<td><math>Kasami</math></td>
<td>Gold</td>
<td>4270</td>
<td>338</td>
</tr>
</tr>


Line 62: Line 86:
<td><math>x^{57}</math></td>
<td><math>x^{57}</math></td>
<td><math>Kasami</math></td>
<td><math>Kasami</math></td>
<td>Gold</td>
<td>4704</td>
<td>436</td>
</tr>
</tr>


Line 69: Line 96:
<td><math>x^{63}</math></td>
<td><math>x^{63}</math></td>
<td><math>Inverse</math></td>
<td><math>Inverse</math></td>
<td>Inverse</td>
<td>8128</td>
<td>4928</td>
</tr>
</tr>


Line 77: Line 107:
<td><math>x^3+Tr_7(x^9)</math></td>
<td><math>x^3+Tr_7(x^9)</math></td>
<td><math>C4</math></td>
<td><math>C4</math></td>
<td>Gold</td>
<td>4026</td>
<td>212</td>
</tr>
</tr>


Line 85: Line 118:
<td><math>x^3</math></td>
<td><math>x^3</math></td>
<td><math>Gold</math></td>
<td><math>Gold</math></td>
<td>Gold</td>
<td>11818</td>
<td>420</td>
</tr>
</tr>


Line 91: Line 127:
<td><math>x^9</math></td>
<td><math>x^9</math></td>
<td><math>Gold</math></td>
<td><math>Gold</math></td>
<td>Gold</td>
<td>12370</td>
<td>420</td>
</tr>
</tr>


Line 98: Line 137:
<td><math>x^{57}</math></td>
<td><math>x^{57}</math></td>
<td><math>Kasami</math></td>
<td><math>Kasami</math></td>
<td>Gold</td>
<td>15358</td>
<td>960</td>
</tr>
</tr>


Line 104: Line 146:
<td><math>x^3+x^{17}+p^{48}x^{18}+p^3x^{33}+px^{34}+x^{48}</math></td>
<td><math>x^3+x^{17}+p^{48}x^{18}+p^3x^{33}+px^{34}+x^{48}</math></td>
<td><math>C3</math></td>
<td><math>C3</math></td>
<td>Gold</td>
<td>13200</td>
<td>414</td>
</tr>
</tr>


Line 110: Line 155:
<td><math>x^3+Tr_8(x^9)</math></td>
<td><math>x^3+Tr_8(x^9)</math></td>
<td><math>C4</math></td>
<td><math>C4</math></td>
<td>Gold</td>
<td>13200</td>
<td>432</td>
</tr>
</tr>


Line 116: Line 164:
<td><math>x^3+a^{-1}Tr_8(a^3x^9)</math></td>
<td><math>x^3+a^{-1}Tr_8(a^3x^9)</math></td>
<td><math>C4</math></td>
<td><math>C4</math></td>
<td>Gold</td>
<td>13842</td>
<td>436</td>
</tr>
</tr>


Line 122: Line 173:
<td><math>a(x+x^{16})(ax+a^{16}x^{16})+a^{17}(ax+a^{16}x^{16})^{12}</math></td>
<td><math>a(x+x^{16})(ax+a^{16}x^{16})+a^{17}(ax+a^{16}x^{16})^{12}</math></td>
<td><math>C10</math></td>
<td><math>C10</math></td>
<td>Gold</td>
<td>13642</td>
<td>436</td>
</tr>
</tr>


Line 129: Line 183:
<td><math>x^3</math></td>
<td><math>x^3</math></td>
<td><math>Gold</math></td>
<td><math>Gold</math></td>
<td>Gold</td>
<td>38470</td>
<td>872</td>
</tr>
</tr>


Line 136: Line 193:
<td><math>x^5</math></td>
<td><math>x^5</math></td>
<td><math>Gold</math></td>
<td><math>Gold</math></td>
<td>Gold</td>
<td>41494</td>
<td>872</td>
</tr>
</tr>


Line 142: Line 202:
<td><math>x^{17}</math></td>
<td><math>x^{17}</math></td>
<td><math>Gold</math></td>
<td><math>Gold</math></td>
<td>Gold</td>
<td>38470</td>
<td>872</td>
</tr>
</tr>


Line 148: Line 211:
<td><math>x^{13}</math></td>
<td><math>x^{13}</math></td>
<td><math>Kasami</math></td>
<td><math>Kasami</math></td>
<td>Gold</td>
<td>58676</td>
<td>3086</td>
</tr>
</tr>


Line 154: Line 220:
<td><math>x^{241}</math></td>
<td><math>x^{241}</math></td>
<td><math>Kasami</math></td>
<td><math>Kasami</math></td>
<td>Gold</td>
<td>61726</td>
<td>3482</td>
</tr>
</tr>


Line 160: Line 229:
<td><math>x^{19}</math></td>
<td><math>x^{19}</math></td>
<td><math>Welch</math></td>
<td><math>Welch</math></td>
<td>Gold</td>
<td>60894</td>
<td>3956</td>
</tr>
</tr>


Line 166: Line 238:
<td><math>x^{255}</math></td>
<td><math>x^{255}</math></td>
<td><math>Inverse</math></td>
<td><math>Inverse</math></td>
<td>Inverse</td>
<td>130816</td>
<td>93024</td>
</tr>
</tr>


Line 172: Line 247:
<td><math>x^3+Tr_9(x^9)</math></td>
<td><math>x^3+Tr_9(x^9)</math></td>
<td><math>C4</math></td>
<td><math>C4</math></td>
<td>Gold</td>
<td>47890</td>
<td>920</td>
</tr>
</tr>


Line 178: Line 256:
<td><math>x^3+Tr^3_9(x^9+x^{18})</math></td>
<td><math>x^3+Tr^3_9(x^9+x^{18})</math></td>
<td><math>C5</math></td>
<td><math>C5</math></td>
<td>Gold</td>
<td>48428</td>
<td>930</td>
</tr>
</tr>


Line 184: Line 265:
<td><math>x^3+Tr^3_9(x^{18}+x^{36})</math></td>
<td><math>x^3+Tr^3_9(x^{18}+x^{36})</math></td>
<td><math>C6</math></td>
<td><math>C6</math></td>
<td>Gold</td>
<td>48460</td>
<td>944</td>
</tr>
</tr>


Line 190: Line 274:
<td><math>x^3+a^{246}x^{10}+a^{47}x^{17}+a^{181}x^{66}+a^{428}x^{129}</math></td>
<td><math>x^3+a^{246}x^{10}+a^{47}x^{17}+a^{181}x^{66}+a^{428}x^{129}</math></td>
<td><math>C11</math></td>
<td><math>C11</math></td>
<td>Gold</td>
<td>48596</td>
<td>944</td>
</tr>
</tr>


Line 197: Line 284:
<td><math>x^3</math></td>
<td><math>x^3</math></td>
<td><math>Gold</math></td>
<td><math>Gold</math></td>
<td>Gold</td>
<td>125042</td>
<td></td>
</tr>
</tr>


Line 203: Line 293:
<td><math>x^9</math></td>
<td><math>x^9</math></td>
<td><math>Gold</math></td>
<td><math>Gold</math></td>
<td>Gold</td>
<td>136492</td>
<td></td>
</tr>
</tr>


Line 209: Line 302:
<td><math>x^{57}</math></td>
<td><math>x^{57}</math></td>
<td><math>Kasami</math></td>
<td><math>Kasami</math></td>
<td>Gold</td>
<td>186416</td>
<td></td>
</tr>
</tr>


Line 215: Line 311:
<td><math>x^{339}</math></td>
<td><math>x^{339}</math></td>
<td><math>Dobbertin</math></td>
<td><math>Dobbertin</math></td>
<td>Dobbertin</td>
<td>280604</td>
<td></td>
</tr>
</tr>


Line 221: Line 320:
<td><math>x^6+x^{33}+p^{31}x^{192}</math></td>
<td><math>x^6+x^{33}+p^{31}x^{192}</math></td>
<td><math>C3</math></td>
<td><math>C3</math></td>
<td>Gold</td>
<td>151216</td>
<td></td>
</tr>
</tr>


Line 227: Line 329:
<td><math>x^{33}+x^{72}+p^{31}x^{258}</math></td>
<td><math>x^{33}+x^{72}+p^{31}x^{258}</math></td>
<td><math>C3</math></td>
<td><math>C3</math></td>
<td>Gold</td>
<td>153896</td>
<td></td>
</tr>
</tr>


Line 233: Line 338:
<td><math>x^3+Tr_{10}(x^9)</math></td>
<td><math>x^3+Tr_{10}(x^9)</math></td>
<td><math>C4</math></td>
<td><math>C4</math></td>
<td>Gold</td>
<td>164034</td>
<td></td>
</tr>
</tr>


Line 239: Line 347:
<td><math>x^3+a^{-1}Tr_{10}(a^3x^9)</math></td>
<td><math>x^3+a^{-1}Tr_{10}(a^3x^9)</math></td>
<td><math>C4</math></td>
<td><math>C4</math></td>
<td>Gold</td>
<td>164098</td>
<td></td>
</tr>
</tr>


Line 246: Line 357:
<td><math>x^3</math></td>
<td><math>x^3</math></td>
<td><math>Gold</math></td>
<td><math>Gold</math></td>
<td>Gold</td>
<td></td>
<td></td>
</tr>
</tr>


Line 252: Line 366:
<td><math>x^5</math></td>
<td><math>x^5</math></td>
<td><math>Gold</math></td>
<td><math>Gold</math></td>
<td>Gold</td>
<td></td>
<td></td>
</tr>
</tr>


Line 258: Line 375:
<td><math>x^9</math></td>
<td><math>x^9</math></td>
<td><math>Gold</math></td>
<td><math>Gold</math></td>
<td>Gold</td>
<td></td>
<td></td>
</tr>
</tr>


Line 264: Line 384:
<td><math>x^{17}</math></td>
<td><math>x^{17}</math></td>
<td><math>Gold</math></td>
<td><math>Gold</math></td>
<td>Gold</td>
<td></td>
<td></td>
</tr>
</tr>


Line 270: Line 393:
<td><math>x^{33}</math></td>
<td><math>x^{33}</math></td>
<td><math>Gold</math></td>
<td><math>Gold</math></td>
<td>Gold</td>
<td></td>
<td></td>
</tr>
</tr>


Line 276: Line 402:
<td><math>x^{13}</math></td>
<td><math>x^{13}</math></td>
<td><math>Kasami</math></td>
<td><math>Kasami</math></td>
<td>Gold</td>
<td></td>
<td></td>
</tr>
</tr>


Line 282: Line 411:
<td><math>x^{57}</math></td>
<td><math>x^{57}</math></td>
<td><math>Kasami</math></td>
<td><math>Kasami</math></td>
<td>Gold</td>
<td></td>
<td></td>
</tr>
</tr>


Line 287: Line 419:
<td><math>x^{241}</math></td>
<td><math>x^{241}</math></td>
<td><math>Kasami</math></td>
<td><math>Kasami</math></td>
<td>Gold</td>
<td></td>
<td></td>
</tr>
</tr>


Line 292: Line 427:
<td><math>x^{993}</math></td>
<td><math>x^{993}</math></td>
<td><math>Kasami</math></td>
<td><math>Kasami</math></td>
<td>Gold</td>
<td></td>
<td></td>
</tr>
</tr>


Line 297: Line 435:
<td><math>x^{35}</math></td>
<td><math>x^{35}</math></td>
<td><math>Welch</math></td>
<td><math>Welch</math></td>
<td>Gold</td>
<td></td>
<td></td>
</tr>
</tr>


Line 302: Line 443:
<td><math>x^{287}</math></td>
<td><math>x^{287}</math></td>
<td><math>Niho</math></td>
<td><math>Niho</math></td>
<td>Gold</td>
<td></td>
<td></td>
</tr>
</tr>


Line 307: Line 451:
<td><math>x^{123}</math></td>
<td><math>x^{123}</math></td>
<td><math>Inverse</math></td>
<td><math>Inverse</math></td>
<td>Inverse</td>
<td></td>
<td></td>
</tr>
</tr>


Line 312: Line 459:
<td><math>x^3+Tr_{11}(x^9)</math></td>
<td><math>x^3+Tr_{11}(x^9)</math></td>
<td><math>C4</math></td>
<td><math>C4</math></td>
<td>Gold</td>
<td></td>
<td></td>
</tr>
</tr>




</table>
</table>

Revision as of 10:36, 26 July 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^{123} }[/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