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

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Line 7: Line 7:
 
<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 12:36, 26 July 2019

CCZ-inequivalent APN Functions over from the Known APN Classes for

Dimension Functions Equivalent to Walsh spectrum Γ-rank Δ-rank
Gold 1102 94
Gold 1146 94
Gold 1166 96
Gold 3610 198
Gold 3708 198
Gold 3610 198
Gold 4270 338
Gold 4704 436
Inverse 8128 4928
Gold 4026 212
Gold 11818 420
Gold 12370 420
Gold 15358 960
Gold 13200 414
Gold 13200 432
Gold 13842 436
Gold 13642 436
Gold 38470 872
Gold 41494 872
Gold 38470 872
Gold 58676 3086
Gold 61726 3482
Gold 60894 3956
Inverse 130816 93024
Gold 47890 920
Gold 48428 930
Gold 48460 944
Gold 48596 944
Gold 125042
Gold 136492
Gold 186416
Dobbertin 280604
Gold 151216
Gold 153896
Gold 164034
Gold 164098
Gold
Gold
Gold
Gold
Gold
Gold
Gold
Gold
Gold
Gold
Gold
Inverse
Gold