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

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CCZ-inequivalent APN Functions over <math>\mathbb{F}_{2^n}</math> from the Known APN Classes for <math>6\leqslant n \leqslant 11</math>
+
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.
 
 
The function families are indexed according to the table of [[Known infinite families of APN power functions over GF(2^n)]] in the case of monomials, and according to the table of [[Known infinite families of quadratic APN polynomials over GF(2^n)]] in the case of polynomial families.
 
  
 
<table class="borderless">
 
<table class="borderless">
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<tr class="divider">
 
<tr class="divider">
<td rowspan="8"><span class="htmlMath">8</span></td>
+
<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>
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<td><span class="htmlMath">C4</span></td>
 
<td><span class="htmlMath">C4</span></td>
 
<td>Gold</td>
 
<td>Gold</td>
<td>13200</td>
+
<td>13800</td>
 
<td>432</td>
 
<td>432</td>
 
<td>6144 = 2<sup>11</sup> * 3</td>
 
<td>6144 = 2<sup>11</sup> * 3</td>
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<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>
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<td>434</td>
 
<td>434</td>
 
<td>6144 = 2<sup>11</sup> * 3</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>
  
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<tr class="divider">
 
<tr class="divider">
<td rowspan="10"><span class="htmlMath">10</span></td>
+
<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>
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<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">C12</span></td>
+
<td><span class="htmlMath">C13</span></td>
 
<td>Gold</td>
 
<td>Gold</td>
 
<td>166068</td>
 
<td>166068</td>
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<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">C12</span></td>
+
<td><span class="htmlMath">C13</span></td>
 
<td>Gold</td>
 
<td>Gold</td>
 
<td>166168</td>
 
<td>166168</td>
 
<td></td>
 
<td></td>
 
<td>476160 = 2<sup>10</sup> * 3 * 5 * 31</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>Gold</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>168960</td>
 
</tr>
 
</tr>
  
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</table>
 
</table>
  
Note.
+
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.
In dimension 10 not all instances of family C11 have been checked.
 

Latest revision as of 09:23, 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.