CCZ-inequivalent representatives from the known APN families for dimensions up to 11: Difference between revisions
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CCZ-inequivalent APN | The following table presents CCZ-inequivalent representatives from all CCZ-equivalence classes corresponding to the [[known infinite families of APN power functions over GF(2^n)]] and the [[known infinite families of quadratic APN polynomials over GF(2^n)]] over GF(2<sup>n</sup>) for n from 6 to 11. Values of various invariants are given in the cases when this is possible; empty cells in the invariant columns indicate that their computation is infeasible using our current resources. | ||
<table class="borderless"> | <table class="borderless"> | ||
| Line 10: | Line 10: | ||
<th>Γ-rank</th> | <th>Γ-rank</th> | ||
<th>Δ-rank</th> | <th>Δ-rank</th> | ||
<th>Multiplier group</th> | |||
</tr> | </tr> | ||
| Line 20: | Line 21: | ||
<td>1102</td> | <td>1102</td> | ||
<td>94</td> | <td>94</td> | ||
<td>24192 = 2<sup>7</sup> * 3<sup>3</sup> * 7</td> | |||
</tr> | </tr> | ||
| Line 30: | Line 32: | ||
<td>1146</td> | <td>1146</td> | ||
<td>94</td> | <td>94</td> | ||
<td>4032 = 2<sup>6</sup> * 3<sup>2</sup> * 7</td> | |||
</tr> | </tr> | ||
| Line 39: | Line 42: | ||
<td>1166</td> | <td>1166</td> | ||
<td>96</td> | <td>96</td> | ||
<td>896 = 2<sup>7</sup> * 7</td> | |||
</tr> | </tr> | ||
| Line 50: | Line 54: | ||
<td>3610</td> | <td>3610</td> | ||
<td>198</td> | <td>198</td> | ||
<td>113792 = 2<sup>7</sup> * 7 * 127 </td> | |||
</tr> | </tr> | ||
| Line 60: | Line 65: | ||
<td>3708</td> | <td>3708</td> | ||
<td>198</td> | <td>198</td> | ||
<td>113792 = 2<sup>7</sup> * 7 * 127 </td> | |||
</tr> | </tr> | ||
| Line 70: | Line 76: | ||
<td>3610</td> | <td>3610</td> | ||
<td>198</td> | <td>198</td> | ||
<td>113792 = 2<sup>7</sup> * 7 * 127 </td> | |||
</tr> | </tr> | ||
| Line 80: | Line 87: | ||
<td>4270</td> | <td>4270</td> | ||
<td>338</td> | <td>338</td> | ||
<td>889 = 7 * 127</td> | |||
</tr> | </tr> | ||
| Line 89: | Line 97: | ||
<td>4704</td> | <td>4704</td> | ||
<td>436</td> | <td>436</td> | ||
<td>889 = 7 * 127</td> | |||
</tr> | </tr> | ||
| Line 99: | Line 108: | ||
<td>8128</td> | <td>8128</td> | ||
<td>4928</td> | <td>4928</td> | ||
<td>1778 = 2 * 7 * 127</td> | |||
</tr> | </tr> | ||
| Line 110: | Line 120: | ||
<td>4026</td> | <td>4026</td> | ||
<td>212</td> | <td>212</td> | ||
<td>896 = 2<sup>7</sup> * 7</td> | |||
</tr> | </tr> | ||
<tr class="divider"> | <tr class="divider"> | ||
<td rowspan=" | <td rowspan="10"><span class="htmlMath">8</span></td> | ||
<td class="noborderbelow"><span class="htmlMath">8.1</span></td> | <td class="noborderbelow"><span class="htmlMath">8.1</span></td> | ||
<td><span class="htmlMath">x<sup>3</sup></span></td> | <td><span class="htmlMath">x<sup>3</sup></span></td> | ||
| Line 121: | Line 132: | ||
<td>11818</td> | <td>11818</td> | ||
<td>420</td> | <td>420</td> | ||
<td>522240 = 2<sup>11</sup> * 3 * 5 * 17</td> | |||
</tr> | </tr> | ||
| Line 130: | Line 142: | ||
<td>12370</td> | <td>12370</td> | ||
<td>420</td> | <td>420</td> | ||
<td>522240 = 2<sup>11</sup> * 3 * 5 * 17</td> | |||
</tr> | </tr> | ||
| Line 140: | Line 153: | ||
<td>15358</td> | <td>15358</td> | ||
<td>960</td> | <td>960</td> | ||
<td>2040 = 2<sup>3</sup> * 3 * 5 * 17</td> | |||
</tr> | </tr> | ||
| Line 149: | Line 163: | ||
<td>13200</td> | <td>13200</td> | ||
<td>414</td> | <td>414</td> | ||
<td>46080 = 2<sup>10</sup> * 3<sup>2</sup> * 5</td> | |||
</tr> | </tr> | ||
| Line 156: | Line 171: | ||
<td><span class="htmlMath">C4</span></td> | <td><span class="htmlMath">C4</span></td> | ||
<td>Gold</td> | <td>Gold</td> | ||
<td> | <td>13800</td> | ||
<td>432</td> | <td>432</td> | ||
<td>6144 = 2<sup>11</sup> * 3</td> | |||
</tr> | </tr> | ||
| Line 167: | Line 183: | ||
<td>13842</td> | <td>13842</td> | ||
<td>436</td> | <td>436</td> | ||
<td>3072 = 2<sup>10</sup> * 3</td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td class="noborderbelow"><span class="htmlMath">8.7</span></td> | <td class="noborderbelow"><span class="htmlMath">8.7</span></td> | ||
<td><span class="htmlMath">a(x+x<sup>16</sup>)(ax+a<sup>16</sup>x<sup>16</sup>)+a<sup>17</sup>(ax+a<sup>16</sup>x<sup>16</sup>)<sup>12</sup></span></td> | <td><span class="htmlMath">(x+x<sup>16</sup>)<sup>3</sup>+a(x+x<sup>16</sup>)(ax+a<sup>16</sup>x<sup>16</sup>)+a<sup>17</sup>(ax+a<sup>16</sup>x<sup>16</sup>)<sup>12</sup></span></td> | ||
<td><span class="htmlMath">C10</span></td> | <td><span class="htmlMath">C10</span></td> | ||
<td>Gold</td> | <td>Gold</td> | ||
<td>13642</td> | <td>13642</td> | ||
<td>436</td> | <td>436</td> | ||
<td>46080 = 2<sup>10</sup> * 3<sup>2</sup> * 5</td> | |||
</tr> | |||
<tr> | |||
<td class="noborderbelow"><span class="htmlMath">8.8</span></td> | |||
<td><span class="htmlMath">x<sup>9</sup>+<span class="latexCommand">Tr</span><sub>8</sub>(x<sup>3</sup>)</span></td> | |||
<td><span class="htmlMath">C11</span></td> | |||
<td>Gold</td> | |||
<td>13804</td> | |||
<td>434</td> | |||
<td>6144 = 2<sup>11</sup> * 3</td> | |||
</tr> | |||
<tr> | |||
<td class="noborderbelow"><span class="htmlMath">8.9</span></td> | |||
<td><span class="htmlMath">a(a<sup>16</sup>x+x<sup>16</sup>a)(x<sup>16</sup>+x) + (a<sup>16</sup>x+x<sup>16</sup>a)<sup>12</sup> + a<sup>17</sup>(a<sup>16</sup>x+x<sup>16</sup>a)<sup>4</sup>(x<sup>16</sup>+x)<sup>2</sup> + a<sup>17</sup>(x<sup>16</sup>+x)<sup>3</sup>, where a is a zero of X<sup>8</sup> + X<sup>4</sup> + X<sup>3</sup> + X<sup>2</sup> + 1</span></td> | |||
<td><span class="htmlMath">C12</span></td> | |||
<td>Gold</td> | |||
<td>13798</td> | |||
<td>438</td> | |||
<td>3840 = 2<sup>8</sup> * 3 * 5</td> | |||
</tr> | |||
<tr> | |||
<td class="noborderbelow"><span class="htmlMath">8.10</span></td> | |||
<td><span class="htmlMath">a(a<sup>16</sup>x+x<sup>16</sup>a)(x<sup>16</sup>+x) + (a<sup>16</sup>x+x<sup>16</sup>a)<sup>66</sup> + a<sup>17</sup>(a<sup>16</sup>x+x<sup>16</sup>a)<sup>64</sup>(x<sup>16</sup>+x)<sup>8</sup> + a<sup>51</sup>(x<sup>16</sup>+x)<sup>9</sup>, where a is a zero of X<sup>8</sup> + X<sup>4</sup> + X<sup>3</sup> + X<sup>2</sup> + 1</span></td> | |||
<td><span class="htmlMath">C12</span></td> | |||
<td>Gold</td> | |||
<td>13700</td> | |||
<td>438</td> | |||
<td>15360 = 2<sup>10</sup> * 3 * 5</td> | |||
</tr> | </tr> | ||
| Line 186: | Line 234: | ||
<td>38470</td> | <td>38470</td> | ||
<td>872</td> | <td>872</td> | ||
<td>2354688 = 2<sup>9</sup> * 3<sup>2</sup> * 7 * 73</td> | |||
</tr> | </tr> | ||
| Line 196: | Line 245: | ||
<td>41494</td> | <td>41494</td> | ||
<td>872</td> | <td>872</td> | ||
<td>2354688 = 2<sup>9</sup> * 3<sup>2</sup> * 7 * 73</td> | |||
</tr> | </tr> | ||
| Line 205: | Line 255: | ||
<td>38470</td> | <td>38470</td> | ||
<td>872</td> | <td>872</td> | ||
<td>2354688 = 2<sup>9</sup> * 3<sup>2</sup> * 7 * 73</td> | |||
</tr> | </tr> | ||
| Line 214: | Line 265: | ||
<td>58676</td> | <td>58676</td> | ||
<td>3086</td> | <td>3086</td> | ||
<td>4599 = 3<sup>2</sup> * 7 * 73</td> | |||
</tr> | </tr> | ||
| Line 223: | Line 275: | ||
<td>61726</td> | <td>61726</td> | ||
<td>3482</td> | <td>3482</td> | ||
<td>4599 = 3<sup>2</sup> * 7 * 73</td> | |||
</tr> | </tr> | ||
| Line 232: | Line 285: | ||
<td>60894</td> | <td>60894</td> | ||
<td>3956</td> | <td>3956</td> | ||
<td>4599 = 3<sup>2</sup> * 7 * 73</td> | |||
</tr> | </tr> | ||
| Line 241: | Line 295: | ||
<td>130816</td> | <td>130816</td> | ||
<td>93024</td> | <td>93024</td> | ||
<td>9198 = 2 * 3<sup>2</sup> * 7 * 73</td> | |||
</tr> | </tr> | ||
| Line 250: | Line 305: | ||
<td>47890</td> | <td>47890</td> | ||
<td>920</td> | <td>920</td> | ||
<td>4608 = 2<sup>9</sup> * 3<sup>2</sup></td> | |||
</tr> | </tr> | ||
| Line 259: | Line 315: | ||
<td>48428</td> | <td>48428</td> | ||
<td>930</td> | <td>930</td> | ||
<td>4608 = 2<sup>9</sup> * 3<sup>2</sup></td> | |||
</tr> | </tr> | ||
| Line 268: | Line 325: | ||
<td>48460</td> | <td>48460</td> | ||
<td>944</td> | <td>944</td> | ||
<td>4608 = 2<sup>9</sup> * 3<sup>2</sup></td> | |||
</tr> | </tr> | ||
| Line 277: | Line 335: | ||
<td>48596</td> | <td>48596</td> | ||
<td>944</td> | <td>944</td> | ||
<td>10752 = 2<sup>9</sup> * 3 * 7</td> | |||
</tr> | </tr> | ||
<tr class="divider"> | <tr class="divider"> | ||
<td rowspan=" | <td rowspan="17"><span class="htmlMath">10</span></td> | ||
<td class="noborderbelow"><span class="htmlMath">10.1</span></td> | <td class="noborderbelow"><span class="htmlMath">10.1</span></td> | ||
<td><span class="htmlMath">x<sup>3</sup></span></td> | <td><span class="htmlMath">x<sup>3</sup></span></td> | ||
| Line 287: | Line 346: | ||
<td>125042</td> | <td>125042</td> | ||
<td></td> | <td></td> | ||
<td>10475520 = 2<sup>11</sup> * 3 * 5 * 11 * 31</td> | |||
</tr> | </tr> | ||
| Line 296: | Line 356: | ||
<td>136492</td> | <td>136492</td> | ||
<td></td> | <td></td> | ||
<td>10475520 = 2<sup>11</sup> * 3 * 5 * 11 * 31</td> | |||
</tr> | </tr> | ||
| Line 305: | Line 366: | ||
<td>186416</td> | <td>186416</td> | ||
<td></td> | <td></td> | ||
<td>10230 = 2 * 3 * 5 * 11 * 31</td> | |||
</tr> | </tr> | ||
| Line 314: | Line 376: | ||
<td>280604</td> | <td>280604</td> | ||
<td></td> | <td></td> | ||
<td>10230 = 2 * 3 * 5 * 11 * 31</td> | |||
</tr> | </tr> | ||
| Line 323: | Line 386: | ||
<td>151216</td> | <td>151216</td> | ||
<td></td> | <td></td> | ||
<td>476160 = 2<sup>10</sup> * 3 * 5 * 31</td> | |||
</tr> | </tr> | ||
| Line 332: | Line 396: | ||
<td>153896</td> | <td>153896</td> | ||
<td></td> | <td></td> | ||
<td>476160 = 2<sup>10</sup> * 3 * 5 * 31</td> | |||
</tr> | </tr> | ||
| Line 341: | Line 406: | ||
<td>164034</td> | <td>164034</td> | ||
<td></td> | <td></td> | ||
<td>30720 = 2<sup>11</sup> * 3 * 5</td> | |||
</tr> | </tr> | ||
| Line 350: | Line 416: | ||
<td>164098</td> | <td>164098</td> | ||
<td></td> | <td></td> | ||
<td>15360 = 2<sup>10</sup> * 3 * 5</td> | |||
</tr> | </tr> | ||
| Line 355: | Line 422: | ||
<td class="noborderbelow"><span class="htmlMath">10.9</span></td> | <td class="noborderbelow"><span class="htmlMath">10.9</span></td> | ||
<td><span class="htmlMath">x<sup>3</sup> + p<sup>341</sup>x<sup>9</sup> + p<sup>682</sup>x<sup>96</sup> + x<sup>288</sup></span></td> | <td><span class="htmlMath">x<sup>3</sup> + p<sup>341</sup>x<sup>9</sup> + p<sup>682</sup>x<sup>96</sup> + x<sup>288</sup></span></td> | ||
<td><span class="htmlMath"> | <td><span class="htmlMath">C13</span></td> | ||
<td>Gold</td> | <td>Gold</td> | ||
<td>166068</td> | <td>166068</td> | ||
<td></td> | <td></td> | ||
<td>476160 = 2<sup>10</sup> * 3 * 5 * 31</td> | |||
</tr> | </tr> | ||
| Line 364: | Line 432: | ||
<td class="noborderbelow"><span class="htmlMath">10.10</span></td> | <td class="noborderbelow"><span class="htmlMath">10.10</span></td> | ||
<td><span class="htmlMath">x<sup>3</sup> + p<sup>341</sup>x<sup>129</sup> + p<sup>682</sup>x<sup>96</sup> + x<sup>36</sup></span></td> | <td><span class="htmlMath">x<sup>3</sup> + p<sup>341</sup>x<sup>129</sup> + p<sup>682</sup>x<sup>96</sup> + x<sup>36</sup></span></td> | ||
<td><span class="htmlMath">C13</span></td> | |||
<td>Gold</td> | |||
<td>166168</td> | |||
<td></td> | |||
<td>476160 = 2<sup>10</sup> * 3 * 5 * 31</td> | |||
</tr> | |||
<tr> | |||
<td class="noborderbelow"><span class="htmlMath">10.11</span></td> | |||
<td><span class="htmlMath">x<sup>3</sup> + a<sup>128</sup>x<sup>6</sup> + a<sup>384</sup>x<sup>12</sup> + a<sup>133</sup>x<sup>33</sup> + x<sup>34</sup> + a<sup>2</sup>x<sup>64</sup> + x<sup>65</sup> + a<sup>128</sup>x<sup>68</sup> + x<sup>96</sup> + a<sup>4</sup>x<sup>130</sup> + a<sup>260</sup>x<sup>136</sup> + a<sup>4</sup>x<sup>192</sup> + a<sup>136</sup>x<sup>260</sup> + a<sup>12</sup>x<sup>384</sup></span></td> | |||
<td><span class="htmlMath">C12</span></td> | |||
<td>Gold</td> | |||
<td>162550</td> | |||
<td></td> | |||
<td>158720</td> | |||
</tr> | |||
<tr> | |||
<td class="noborderbelow"><span class="htmlMath">10.12</span></td> | |||
<td><span class="htmlMath">x<sup>3</sup> + a<sup>920</sup>x<sup>6</sup> + a<sup>153</sup>x<sup>12</sup> + a<sup>925</sup>x<sup>33</sup> + x<sup>34</sup> + a<sup>794</sup>x<sup>64</sup> + x<sup>65</sup> + a<sup>920</sup>x<sup>68</sup> + x<sup>96</sup> + a<sup>796</sup>x<sup>130</sup> + a<sup>29</sup>x<sup>136</sup> + a<sup>796</sup>x<sup>192</sup> + a<sup>928</sup>x<sup>260</sup> + a<sup>804</sup>x<sup>384</sup> </span></td> | |||
<td><span class="htmlMath">C12</span></td> | <td><span class="htmlMath">C12</span></td> | ||
<td>Gold</td> | <td>Gold</td> | ||
<td> | <td>163400</td> | ||
<td></td> | |||
<td>31744</td> | |||
</tr> | |||
<tr> | |||
<td class="noborderbelow"><span class="htmlMath">10.13</span></td> | |||
<td><span class="htmlMath">x<sup>3</sup> + p<sup>788</sup>x<sup>6</sup> + p<sup>21</sup>x<sup>12</sup> + p<sup>793</sup>x<sup>33</sup> + x<sup>34</sup> + p<sup>662</sup>x<sup>64</sup> + x<sup>65</sup> + p<sup>788</sup>x<sup>68</sup> + x<sup>96</sup> + p<sup>664</sup>x<sup>130</sup> + p<sup>920</sup>x<sup>136</sup> + p<sup>664</sup>x<sup>192</sup> + p<sup>796</sup>x<sup>260</sup> + p<sup>672</sup>x<sup>384</sup></span></td> | |||
<td><span class="htmlMath">C12</span></td> | |||
<td>Gold</td> | |||
<td>163398</td> | |||
<td></td> | |||
<td>31744</td> | |||
</tr> | |||
<tr> | |||
<td class="noborderbelow"><span class="htmlMath">10.14</span></td> | |||
<td><span class="htmlMath">x<sup>5</sup> + p<sup>576</sup>x<sup>18</sup> + p<sup>512</sup>x<sup>20</sup> + p<sup>133</sup>x<sup>33</sup> + x<sup>36</sup> + p<sup>2</sup>x<sup>64</sup> + p<sup>514</sup>x<sup>80</sup> + x<sup>129</sup> + p<sup>512</sup>x<sup>144</sup> + x<sup>160</sup> + p<sup>80</sup>x<sup>514</sup> + p<sup>16</sup>x<sup>516</sup> + p<sup>18</sup>x<sup>576</sup> + p<sup>16</sup>x<sup>640</sup></span></td> | |||
<td><span class="htmlMath">C12</span></td> | |||
<td>Gold</td> | |||
<td>163308</td> | |||
<td></td> | |||
<td>158720</td> | |||
</tr> | |||
<tr> | |||
<td class="noborderbelow"><span class="htmlMath">10.15</span></td> | |||
<td><span class="htmlMath">x<sup>5</sup> + a<sup>477</sup>x<sup>18</sup> + a<sup>413</sup>x<sup>20</sup> + a<sup>34</sup>x<sup>33</sup> + x<sup>36</sup> + a<sup>926</sup>x<sup>64</sup> + a<sup>415</sup>x<sup>80</sup> + x<sup>129</sup> + a<sup>413</sup>x<sup>144</sup> + x<sup>160</sup> + a<sup>1004</sup>x<sup>514</sup> + a<sup>940</sup>x<sup>516</sup> + a<sup>942</sup>x<sup>576</sup> + a<sup>940</sup>x<sup>640</sup></span></td> | |||
<td><span class="htmlMath">C12</span></td> | |||
<td>Gold</td> | |||
<td>164026</td> | |||
<td></td> | |||
<td>31744</td> | |||
</tr> | |||
<tr> | |||
<td class="noborderbelow"><span class="htmlMath">10.16</span></td> | |||
<td><span class="htmlMath">x<sup>5</sup> + a<sup>81</sup>x<sup>18</sup> + a<sup>17</sup>x<sup>20</sup> + a<sup>661</sup>x<sup>33</sup> + x<sup>36</sup> + a<sup>530</sup>x<sup>64</sup> + a<sup>19</sup>x<sup>80</sup> + x<sup>129</sup> + a<sup>17</sup>x<sup>144</sup> + x<sup>160</sup> + a<sup>608</sup>x<sup>514</sup> + a<sup>544</sup>x<sup>516</sup> + a<sup>546</sup>x<sup>576</sup> + a<sup>544</sup>x<sup>640</sup></span></td> | |||
<td><span class="htmlMath">C12</span></td> | |||
<td>Gold</td> | |||
<td>164026</td> | |||
<td></td> | |||
<td>31744</td> | |||
</tr> | |||
<tr> | |||
<td class="noborderbelow"><span class="htmlMath">10.17</span></td> | |||
<td><span class="htmlMath">x<sup>3</sup> + p<sup>341</sup>x<sup>36</sup></span></td> | |||
<td><span class="htmlMath">C13</span></td> | |||
<td>Gold</td> | |||
<td>169984</td> | |||
<td></td> | <td></td> | ||
<td>168960</td> | |||
</tr> | </tr> | ||
| Line 376: | Line 515: | ||
<td><span class="htmlMath">Gold</span></td> | <td><span class="htmlMath">Gold</span></td> | ||
<td>Gold</td> | <td>Gold</td> | ||
<td></td> | |||
<td></td> | <td></td> | ||
<td></td> | <td></td> | ||
| Line 385: | Line 525: | ||
<td><span class="htmlMath">Gold</span></td> | <td><span class="htmlMath">Gold</span></td> | ||
<td>Gold</td> | <td>Gold</td> | ||
<td></td> | |||
<td></td> | <td></td> | ||
<td></td> | <td></td> | ||
| Line 394: | Line 535: | ||
<td><span class="htmlMath">Gold</span></td> | <td><span class="htmlMath">Gold</span></td> | ||
<td>Gold</td> | <td>Gold</td> | ||
<td></td> | |||
<td></td> | <td></td> | ||
<td></td> | <td></td> | ||
| Line 403: | Line 545: | ||
<td><span class="htmlMath">Gold</span></td> | <td><span class="htmlMath">Gold</span></td> | ||
<td>Gold</td> | <td>Gold</td> | ||
<td></td> | |||
<td></td> | <td></td> | ||
<td></td> | <td></td> | ||
| Line 412: | Line 555: | ||
<td><span class="htmlMath">Gold</span></td> | <td><span class="htmlMath">Gold</span></td> | ||
<td>Gold</td> | <td>Gold</td> | ||
<td></td> | |||
<td></td> | <td></td> | ||
<td></td> | <td></td> | ||
| Line 421: | Line 565: | ||
<td><span class="htmlMath">Kasami</span></td> | <td><span class="htmlMath">Kasami</span></td> | ||
<td>Gold</td> | <td>Gold</td> | ||
<td></td> | |||
<td></td> | <td></td> | ||
<td></td> | <td></td> | ||
| Line 430: | Line 575: | ||
<td><span class="htmlMath">Kasami</span></td> | <td><span class="htmlMath">Kasami</span></td> | ||
<td>Gold</td> | <td>Gold</td> | ||
<td></td> | |||
<td></td> | <td></td> | ||
<td></td> | <td></td> | ||
| Line 438: | Line 584: | ||
<td><span class="htmlMath">Kasami</span></td> | <td><span class="htmlMath">Kasami</span></td> | ||
<td>Gold</td> | <td>Gold</td> | ||
<td></td> | |||
<td></td> | <td></td> | ||
<td></td> | <td></td> | ||
| Line 446: | Line 593: | ||
<td><span class="htmlMath">Kasami</span></td> | <td><span class="htmlMath">Kasami</span></td> | ||
<td>Gold</td> | <td>Gold</td> | ||
<td></td> | |||
<td></td> | <td></td> | ||
<td></td> | <td></td> | ||
| Line 454: | Line 602: | ||
<td><span class="htmlMath">Welch</span></td> | <td><span class="htmlMath">Welch</span></td> | ||
<td>Gold</td> | <td>Gold</td> | ||
<td></td> | |||
<td></td> | <td></td> | ||
<td></td> | <td></td> | ||
| Line 462: | Line 611: | ||
<td><span class="htmlMath">Niho</span></td> | <td><span class="htmlMath">Niho</span></td> | ||
<td>Gold</td> | <td>Gold</td> | ||
<td></td> | |||
<td></td> | <td></td> | ||
<td></td> | <td></td> | ||
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<td><span class="htmlMath">Inverse</span></td> | <td><span class="htmlMath">Inverse</span></td> | ||
<td>Inverse</td> | <td>Inverse</td> | ||
<td></td> | |||
<td></td> | <td></td> | ||
<td></td> | <td></td> | ||
| Line 478: | Line 629: | ||
<td><span class="htmlMath">C4</span></td> | <td><span class="htmlMath">C4</span></td> | ||
<td>Gold</td> | <td>Gold</td> | ||
<td></td> | |||
<td></td> | <td></td> | ||
<td></td> | <td></td> | ||
| Line 484: | Line 636: | ||
</table> | </table> | ||
Note: In dimension 10, not all instances of family C11 have been checked. The ones that we have checked are CCZ-equivalent to the representatives currently given in the table above, but, at the moment, we cannot assert that the representatives from the table exhaust all possible CCZ-classes corresponding to C11 in dimension 10. | |||
Latest revision as of 08:23, 5 May 2021
The following table presents CCZ-inequivalent representatives from all CCZ-equivalence classes corresponding to the known infinite families of APN power functions over GF(2^n) and the known infinite families of quadratic APN polynomials over GF(2^n) over GF(2n) for n from 6 to 11. Values of various invariants are given in the cases when this is possible; empty cells in the invariant columns indicate that their computation is infeasible using our current resources.
| Dimension | N○ | Functions | Equivalent to | Walsh spectrum | Γ-rank | Δ-rank | Multiplier group |
|---|---|---|---|---|---|---|---|
| 6 | 6.1 | x3 | Gold | Gold | 1102 | 94 | 24192 = 27 * 33 * 7 |
| 6.2 | x24+ax17+a8x10+ax9+x3 | C3 | Gold | 1146 | 94 | 4032 = 26 * 32 * 7 | |
| 6.3 | ax3+x17+a4x24 | C7-C9 | Gold | 1166 | 96 | 896 = 27 * 7 | |
| 7 | 7.1 | x3 | Gold | Gold | 3610 | 198 | 113792 = 27 * 7 * 127 |
| 7.2 | x5 | Gold | Gold | 3708 | 198 | 113792 = 27 * 7 * 127 | |
| 7.3 | x9 | Gold | Gold | 3610 | 198 | 113792 = 27 * 7 * 127 | |
| 7.4 | x13 | Kasami | Gold | 4270 | 338 | 889 = 7 * 127 | |
| 7.5 | x57 | Kasami | Gold | 4704 | 436 | 889 = 7 * 127 | |
| 7.6 | x63 | Inverse | Inverse | 8128 | 4928 | 1778 = 2 * 7 * 127 | |
| 7.7 | x3+Tr7(x9) | C4 | Gold | 4026 | 212 | 896 = 27 * 7 | |
| 8 | 8.1 | x3 | Gold | Gold | 11818 | 420 | 522240 = 211 * 3 * 5 * 17 |
| 8.2 | x9 | Gold | Gold | 12370 | 420 | 522240 = 211 * 3 * 5 * 17 | |
| 8.3 | x57 | Kasami | Gold | 15358 | 960 | 2040 = 23 * 3 * 5 * 17 | |
| 8.4 | x3+x17+p48x18+p3x33+px34+x48 | C3 | Gold | 13200 | 414 | 46080 = 210 * 32 * 5 | |
| 8.5 | x3+Tr8(x9) | C4 | Gold | 13800 | 432 | 6144 = 211 * 3 | |
| 8.6 | x3+a-1Tr8(a3x9) | C4 | Gold | 13842 | 436 | 3072 = 210 * 3 | |
| 8.7 | (x+x16)3+a(x+x16)(ax+a16x16)+a17(ax+a16x16)12 | C10 | Gold | 13642 | 436 | 46080 = 210 * 32 * 5 | |
| 8.8 | x9+Tr8(x3) | C11 | Gold | 13804 | 434 | 6144 = 211 * 3 | |
| 8.9 | a(a16x+x16a)(x16+x) + (a16x+x16a)12 + a17(a16x+x16a)4(x16+x)2 + a17(x16+x)3, where a is a zero of X8 + X4 + X3 + X2 + 1 | C12 | Gold | 13798 | 438 | 3840 = 28 * 3 * 5 | |
| 8.10 | a(a16x+x16a)(x16+x) + (a16x+x16a)66 + a17(a16x+x16a)64(x16+x)8 + a51(x16+x)9, where a is a zero of X8 + X4 + X3 + X2 + 1 | C12 | Gold | 13700 | 438 | 15360 = 210 * 3 * 5 | |
| 9 | 9.1 | x3 | Gold | Gold | 38470 | 872 | 2354688 = 29 * 32 * 7 * 73 |
| 9.2 | x5 | Gold | Gold | 41494 | 872 | 2354688 = 29 * 32 * 7 * 73 | |
| 9.3 | x17 | Gold | Gold | 38470 | 872 | 2354688 = 29 * 32 * 7 * 73 | |
| 9.4 | x13 | Kasami | Gold | 58676 | 3086 | 4599 = 32 * 7 * 73 | |
| 9.5 | x241 | Kasami | Gold | 61726 | 3482 | 4599 = 32 * 7 * 73 | |
| 9.6 | x19 | Welch | Gold | 60894 | 3956 | 4599 = 32 * 7 * 73 | |
| 9.7 | x255 | Inverse | Inverse | 130816 | 93024 | 9198 = 2 * 32 * 7 * 73 | |
| 9.8 | x3+Tr9(x9) | C4 | Gold | 47890 | 920 | 4608 = 29 * 32 | |
| 9.9 | x3+Tr39(x9+x18) | C5 | Gold | 48428 | 930 | 4608 = 29 * 32 | |
| 9.10 | x3+Tr39(x18+x36) | C6 | Gold | 48460 | 944 | 4608 = 29 * 32 | |
| 9.11 | x3+a246x10+a47x17+a181x66+a428x129 | C11 | Gold | 48596 | 944 | 10752 = 29 * 3 * 7 | |
| 10 | 10.1 | x3 | Gold | Gold | 125042 | 10475520 = 211 * 3 * 5 * 11 * 31 | |
| 10.2 | x9 | Gold | Gold | 136492 | 10475520 = 211 * 3 * 5 * 11 * 31 | ||
| 10.3 | x57 | Kasami | Gold | 186416 | 10230 = 2 * 3 * 5 * 11 * 31 | ||
| 10.4 | x339 | Dobbertin | Dobbertin | 280604 | 10230 = 2 * 3 * 5 * 11 * 31 | ||
| 10.5 | x6+x33+p31x192 | C3 | Gold | 151216 | 476160 = 210 * 3 * 5 * 31 | ||
| 10.6 | x33+x72+p31x258 | C3 | Gold | 153896 | 476160 = 210 * 3 * 5 * 31 | ||
| 10.7 | x3+Tr10(x9) | C4 | Gold | 164034 | 30720 = 211 * 3 * 5 | ||
| 10.8 | x3+a-1Tr10(a3x9) | C4 | Gold | 164098 | 15360 = 210 * 3 * 5 | ||
| 10.9 | x3 + p341x9 + p682x96 + x288 | C13 | Gold | 166068 | 476160 = 210 * 3 * 5 * 31 | ||
| 10.10 | x3 + p341x129 + p682x96 + x36 | C13 | Gold | 166168 | 476160 = 210 * 3 * 5 * 31 | ||
| 10.11 | x3 + a128x6 + a384x12 + a133x33 + x34 + a2x64 + x65 + a128x68 + x96 + a4x130 + a260x136 + a4x192 + a136x260 + a12x384 | C12 | Gold | 162550 | 158720 | ||
| 10.12 | x3 + a920x6 + a153x12 + a925x33 + x34 + a794x64 + x65 + a920x68 + x96 + a796x130 + a29x136 + a796x192 + a928x260 + a804x384 | C12 | Gold | 163400 | 31744 | ||
| 10.13 | x3 + p788x6 + p21x12 + p793x33 + x34 + p662x64 + x65 + p788x68 + x96 + p664x130 + p920x136 + p664x192 + p796x260 + p672x384 | C12 | Gold | 163398 | 31744 | ||
| 10.14 | x5 + p576x18 + p512x20 + p133x33 + x36 + p2x64 + p514x80 + x129 + p512x144 + x160 + p80x514 + p16x516 + p18x576 + p16x640 | C12 | Gold | 163308 | 158720 | ||
| 10.15 | x5 + a477x18 + a413x20 + a34x33 + x36 + a926x64 + a415x80 + x129 + a413x144 + x160 + a1004x514 + a940x516 + a942x576 + a940x640 | C12 | Gold | 164026 | 31744 | ||
| 10.16 | x5 + a81x18 + a17x20 + a661x33 + x36 + a530x64 + a19x80 + x129 + a17x144 + x160 + a608x514 + a544x516 + a546x576 + a544x640 | C12 | Gold | 164026 | 31744 | ||
| 10.17 | x3 + p341x36 | C13 | Gold | 169984 | 168960 | ||
| 11 | 11.1 | x3 | Gold | Gold | |||
| 11.2 | x5 | Gold | Gold | ||||
| 11.3 | x9 | Gold | Gold | ||||
| 11.4 | x17 | Gold | Gold | ||||
| 11.5 | x33 | Gold | Gold | ||||
| 11.6 | x13 | Kasami | Gold | ||||
| 11.7 | x57 | Kasami | Gold | ||||
| 11.8 | x241 | Kasami | Gold | ||||
| 11.9 | x993 | Kasami | Gold | ||||
| 11.10 | x35 | Welch | Gold | ||||
| 11.11 | x287 | Niho | Gold | ||||
| 11.12 | x1023 | Inverse | Inverse | ||||
| 11.13 | x3+Tr11(x9) | C4 | Gold |
Note: In dimension 10, not all instances of family C11 have been checked. The ones that we have checked are CCZ-equivalent to the representatives currently given in the table above, but, at the moment, we cannot assert that the representatives from the table exhaust all possible CCZ-classes corresponding to C11 in dimension 10.