Difference between revisions of "Known switching classes of APN functions over GF(2^n) for n = 5,6,7,8"

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Known switching classes of APN functions over <math>\mathbb{F}_{2^5}</math>, <math>\mathbb{F}_{2^6}</math>, <math>\mathbb{F}_{2^7}</math> and <math>\mathbb{F}_{2^8}</math>
+
Known switching classes of APN functions over <span class="htmlMath">GF(2<sup>5</sup>), GF(2<sup>6</sup>), GF(2<sup>7</sup>)</span> and <span class="htmlMath">GF(2<sup>8</sup>)</span>.
 +
 
 +
Also available is [[:File:switching_classes.txt|Magma code generating representatives from the switching classes]].
  
 
<table>
 
<table>
 
<tr>
 
<tr>
<th><math>n</math></th>
+
<th><span class="htmlMath">n</span></th>
<th><math>N^\circ</math></th>
+
<th><span class="htmlMath">N</span></th>
<th><math>F(x)</math></th>
+
<th><span class="htmlMath">F(x)</span></th>
 +
<th>Γ-rank</th>
 +
<th>Δ-rank</th>
 +
<th>Aut(dev(G<sub>F</sub>))/2<sup>2n</sup></th>
 +
<th>Aut(dev(G<sub>F</sub>))/2<sup>2n</sup></th>
 +
<th>Multiplier group</td>
 
</tr>
 
</tr>
 
+
<tr>
<td><math>5</math></td>
+
<td rowspan="3">5</td>
<td><math>1.1</math>
+
<td>1.1</td>
<math>1.2</math>
+
<td><span class="htmlMath">x<sup>3</sup></span></td>
<math>2.1</math></td>
+
<td>330</td>
<td><math>x^3</math>
+
<td>42</td>
<math>x^5</math>
+
<td>4960</td>
<math>x^{-1}</math></td>
+
<td>4960</td>
 +
<td>4960</td>
 
</tr>
 
</tr>
</table>
 
 
 
<table>
 
 
<tr>
 
<tr>
<th><math>n</math></th>
+
<td>1.2</td>
<th><math>N^\circ</math></th>
+
<td><span class="htmlMath">x<sup>5</sup></span></td>
<th><math>F(x)</math></th>
+
<td>330</td>
 +
<td>42</td>
 +
<td>4960</td>
 +
<td>158720</td>
 +
<td>4960</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
<td rowspan="14"><math>6</math></td>
+
<td>2.1</td>
 +
<td><span class="htmlMath">x<sup>-1</sup></span></td>
 +
<td>496</td>
 +
<td>232</td>
 +
<td>310</td>
 +
<td>310</td>
 +
<td>310</td>
 +
</tr>
 +
</tr>
 +
<tr class="strongDivider">
 +
<td rowspan="14">6</td>
 
<td>1.1</td>
 
<td>1.1</td>
<td><math>x^{3}</math></td>
+
<td><span class="htmlMath">x<sup>3</sup></span></td>
 +
<td>1102</td>
 +
<td>94</td>
 +
<td>24192</td>
 +
<td>48384</td>
 +
<td>24192</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
 
<td>1.2</td>
 
<td>1.2</td>
<td><math>x^{3} + u^{11}x^{6} + ux^{9}</math></td>
+
<td><span class="htmlMath">x<sup>3</sup> + u<sup>11</sup>x<sup>6</sup> + ux<sup>9</sup></span></td>
 +
<td>1146</td>
 +
<td>94</td>
 +
<td>4032</td>
 +
<td>8064</td>
 +
<td>4032</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
 
<td>2.1</td>
 
<td>2.1</td>
<td><math>ux^{5} + x^{9} + u^{4}x^{17} + ux^{18} + u^{4}x^{20} + ux^{24} + u^{4}x^{34} + ux^{40}</math></td>
+
<td><span class="htmlMath">ux<sup>5</sup> + x<sup>9</sup> + u<sup>4</sup>x<sup>17</sup> + ux<sup>18</sup> + u<sup>4</sup>x<sup>20</sup> + ux<sup>24</sup> + u<sup>4</sup>x<sup>34</sup> + ux<sup>40</sup></span></td>
 +
<td>1158</td>
 +
<td>96</td>
 +
<td>320</td>
 +
<td>320</td>
 +
<td>320</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
 
<td>2.2</td>
 
<td>2.2</td>
<td><math>u^{7}x^{3} + x^{5} + u^{3}x^{9} + u^{4}x^{10} + x^{17} + u^{6}x^{18}</math></td>
+
<td><span class="htmlMath">u<sup>7</sup>x<sup>3</sup> + x<sup>5</sup> + u<sup>3</sup>x<sup>9</sup> + u<sup>4</sup>x<sup>10</sup> + x<sup>17</sup> + u<sup>6</sup>x<sup>18</sup></span></td>
 +
<td>1166</td>
 +
<td>94</td>
 +
<td>448</td>
 +
<td>896</td>
 +
<td>448</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
 
<td>2.3</td>
 
<td>2.3</td>
<td><math>x^{3} + ux^{24} + x^{10}</math></td>
+
<td><span class="htmlMath">x<sup>3</sup> + ux<sup>24</sup> + x<sup>10</sup></span></td>
 +
<td>1166</td>
 +
<td>96</td>
 +
<td>896</td>
 +
<td>896</td>
 +
<td>896</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
 
<td>2.4</td>
 
<td>2.4</td>
<td><math>x^{3} + u^{17}(x^{17} + x^{18} + x^{20} + x^{24})</math></td>
+
<td><span class="htmlMath">x<sup>3</sup> + u<sup>17</sup>(x<sup>17</sup> + x<sup>18</sup> + x<sup>20</sup> + x<sup>24</sup>)</span></td>
 +
<td>1168</td>
 +
<td>96</td>
 +
<td>64</td>
 +
<td>64</td>
 +
<td>64</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
 
<td>2.5</td>
 
<td>2.5</td>
<td><math>x^{3} + u^{11}x^{5} + u^{13}x^{9} + x^{17} + u^{11}x^{33} + x^{48}</math></td>
+
<td><span class="htmlMath">x<sup>3</sup> + u<sup>11</sup>x<sup>5</sup> + u<sup>13</sup>x<sup>9</sup> + x<sup>17</sup> + u<sup>11</sup>x<sup>33</sup> + x<sup>48</sup></span></td>
 +
<td>1170</td>
 +
<td>96</td>
 +
<td>320</td>
 +
<td>320</td>
 +
<td>320</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
 
<td>2.6</td>
 
<td>2.6</td>
<td><math>u^{25}x^{5} + x^{9} + u^{38}x^{12} + u^{25}x^{18} + u^{25}x^{36}</math></td>
+
<td><span class="htmlMath">u<sup>25</sup>x<sup>5</sup> + x<sup>9</sup> + u<sup>38</sup>x<sup>12</sup> + u<sup>25</sup>x<sup>18</sup> + u<sup>25</sup>x<sup>36</sup></span></td>
 +
<td>1170</td>
 +
<td>96</td>
 +
<td>64</td>
 +
<td>64</td>
 +
<td>64</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
 
<td>2.7</td>
 
<td>2.7</td>
<td><math>u^{40}x^{5} + u^{10}x^{6} + u^{62}x^{20} + u^{35}x^{33} + u^{15}x^{34} + u^{29}x^{48}</math></td>
+
<td><span class="htmlMath">u<sup>40</sup>x<sup>5</sup> + u<sup>10</sup>x<sup>6</sup> + u<sup>62</sup>x<sup>20</sup> + u<sup>35</sup>x<sup>33</sup> + u<sup>15</sup>x<sup>34</sup> + u<sup>29</sup>x<sup>48</sup></span></td>
 +
<td>1170</td>
 +
<td>96</td>
 +
<td>64</td>
 +
<td>64</td>
 +
<td>64</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
 
<td>2.8</td>
 
<td>2.8</td>
<td><math>u^{34}x^{6} + u^{52}x^{9} + u^{48}x^{12} + u^{6}x^{20} + u^{9}x^{33} + u^{23}x^{34} + u^{25}x^{40}</math></td>
+
<td><span class="htmlMath">u<sup>34</sup>x<sup>6</sup> + u<sup>52</sup>x<sup>9</sup> + u<sup>48</sup>x<sup>12</sup> + u<sup>6</sup>x<sup>20</sup> + u<sup>9</sup>x<sup>33</sup> + u<sup>23</sup>x<sup>34</sup> + u<sup>25</sup>x<sup>40</sup></span></td>
 +
<td>1170</td>
 +
<td>96</td>
 +
<td>64</td>
 +
<td>64</td>
 +
<td>64</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
 
<td>2.9</td>
 
<td>2.9</td>
<td><math>x^{9} + u^{4}(x^{10} + x^{18}) + u^{9}(x^{12} + x^{20} + x^{40})</math></td>
+
<td><span class="htmlMath">x<sup>9</sup> + u<sup>4</sup>(x<sup>10</sup> + x<sup>18</sup>) + u<sup>9</sup>(x<sup>12</sup> + x<sup>20</sup> + x<sup>40</sup>)</span></td>
 +
<td>1172</td>
 +
<td>96</td>
 +
<td>64</td>
 +
<td>64</td>
 +
<td>64</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
 
<td>2.10</td>
 
<td>2.10</td>
<td><math>u^{52}x^{3} + u^{47}x^{5} + ux^{6} + u^{9}x^{9} + u^{44}x^{12} + u^{47}x^{33} + u^{10}x^{34} + u^{33}x^{40}</math></td>
+
<td><span class="htmlMath">u<sup>52</sup>x<sup>3</sup> + u<sup>47</sup>x<sup>5</sup> + ux<sup>6</sup> + u<sup>9</sup>x<sup>9</sup> + u<sup>44</sup>x<sup>12</sup> + u<sup>47</sup>x<sup>33</sup> + u<sup>10</sup>x<sup>34</sup> + u<sup>33</sup>x<sup>40</sup></span></td>
 +
<td>1172</td>
 +
<td>96</td>
 +
<td>64</td>
 +
<td>64</td>
 +
<td>64</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
 
<td>2.11</td>
 
<td>2.11</td>
<td><math>u(x^{6} + x^{10} + x^{24} + x^{33}) + x^{9} + u^{4}x^{17}</math></td>
+
<td><span class="htmlMath">u(x<sup>6</sup> + x<sup>10</sup> + x<sup>24</sup> + x<sup>33</sup>) + x<sup>9</sup> + u<sup>4</sup>x<sup>17</sup></span></td>
 +
<td>1174</td>
 +
<td>96</td>
 +
<td>64</td>
 +
<td>64</td>
 +
<td>64</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
 
<td>2.12</td>
 
<td>2.12</td>
<td><math>x^{3} + u^{17}(x^{17} + x^{18} + x^{20} + x^{24}) + u^{14}((u^{52}x^{3} + u^{6}x^{5} + u^{19}x^{7} + u^{28}x^{11} + u^{2}x^{13})+ (u^{52}x^{3} + u^{6}x^{5} + u^{19}x^{7} + u^{28}x^{11} + u^{2}x^{13})^{2} + (u^{52}x^{3} + u^{6}x^{5} + u^{19}x^{7} + u^{28}x^{11} + u^{2}x^{13})^{4}+ (u^{52}x^{3} + u^{6}x^{5} + u^{19}x^{7} + u^{28}x^{11} + u^{2}x^{13})^{8}+ (u^{52}x^{3} + u^{6}x^{5} + u^{19}x^{7} + u^{28}x^{11} + u^{2}x^{13})^{1}6+ (u^{52}x^{3} + u^{6}x^{5} + u^{19}x^{7} + u^{28}x^{11} + u^{2}x^{13})^{3}2+ (u^{2}x)^{9} +(u^{2}x)^{1}8 +(u^{2}x)^{3}6 + x^{21}+x^{42}</math></td>
+
<td><span class="htmlMath">x<sup>3</sup> + u<sup>17</sup>(x<sup>17</sup> + x<sup>18</sup> + x<sup>20</sup> + x<sup>24</sup>) + u<sup>14</sup>(u<sup>52</sup>x<sup>3</sup> + u<sup>6</sup>x<sup>5</sup> + u<sup>19</sup>x<sup>7</sup> + u<sup>28</sup>x<sup>11</sup> + u<sup>2</sup>x<sup>13</sup>)+ (u<sup>52</sup>x<sup>3</sup> + u<sup>6</sup>x<sup>5</sup> + u<sup>19</sup>x<sup>7</sup> + u<sup>28</sup>x<sup>11</sup> + u<sup>2</sup>x<sup>13</sup>)<sup>2</sup> + (u<sup>52</sup>x<sup>3</sup> + u<sup>6</sup>x<sup>5</sup> + u<sup>19</sup>x<sup>7</sup> + u<sup>28</sup>x<sup>11</sup> + u<sup>2</sup>x<sup>13</sup>)<sup>4</sup>+ (u<sup>52</sup>x<sup>3</sup> + u<sup>6</sup>x<sup>5</sup> + u<sup>19</sup>x<sup>7</sup> + u<sup>28</sup>x<sup>11</sup> + u<sup>2</sup>x<sup>13</sup>)<sup>8</sup>+ (u<sup>52</sup>x<sup>3</sup> + u<sup>6</sup>x<sup>5</sup> + u<sup>19</sup>x<sup>7</sup> + u<sup>28</sup>x<sup>11</sup> + u<sup>2</sup>x<sup>13</sup>)<sup>16</sup>+ (u<sup>52</sup>x<sup>3</sup> + u<sup>6</sup>x<sup>5</sup> + u<sup>19</sup>x<sup>7</sup> + u<sup>28</sup>x<sup>11</sup> + u<sup>2</sup>x<sup>13</sup>)<sup>32</sup>+ (u<sup>2</sup>x)<sup>9</sup> +(u<sup>2</sup>x)<sup>19</sup> +(u<sup>2</sup>x)<sup>36</sup> + x<sup>21</sup>+x<sup>42</sup></span></td>
 +
<td>1300</td>
 +
<td>152</td>
 +
<td>8</td>
 +
<td>8</td>
 +
<td>8</td>
 +
</tr>
 +
<tr class="strongDivider">
 +
<td rowspan="19">7</td>
 +
<td>1.1</td>
 +
<td>
 +
<span class="htmlMath">x<sup>3</sup></span>
 +
</td>
 +
<td>3610</td>
 +
<td>198</td>
 +
<td>113792</td>
 +
<td>113792</td>
 +
<td>113792</td>
 +
</tr>
 +
<tr>
 +
<td>1.2</td>
 +
<td>
 +
<span class="htmlMath">x<sup>3</sup> + Tr(x<sup>9</sup>)</span>
 +
</td>
 +
<td>4026</td>
 +
<td>212</td>
 +
<td>896</td>
 +
<td>896</td>
 +
<td>896</td>
 +
</tr>
 +
<tr>
 +
<td>2.1</td>
 +
<td>
 +
<span class="htmlMath">x<sup>34</sup> + x<sup>18</sup> + x<sup>5</sup></span>
 +
</td>
 +
<td>4034</td>
 +
<td>210</td>
 +
<td>896</td>
 +
<td>896</td>
 +
<td>896</td>
 +
</tr>
 +
<tr>
 +
<td>2.2</td>
 +
<td>
 +
<span class="htmlMath">x<sup>3</sup> + x<sup>17</sup> + x<sup>33</sup> + x<sup>34</sup></span>
 +
</td>
 +
<td>4040</td>
 +
<td>212</td>
 +
<td>896</td>
 +
<td>896</td>
 +
<td>896</td>
 +
</tr>
 +
<tr>
 +
<td>3.1</td>
 +
<td>
 +
<span class="htmlMath">x<sup>5</sup></span>
 +
</td>
 +
<td>3708</td>
 +
<td>198</td>
 +
<td>113792</td>
 +
<td>113792</td>
 +
<td>113792</td>
 +
</tr>
 +
<tr>
 +
<td>4.1</td>
 +
<td>
 +
<span class="htmlMath">x<sup>9</sup></span>
 +
</td>
 +
<td>3610</td>
 +
<td>198</td>
 +
<td>113792</td>
 +
<td>14565376</td>
 +
<td>113792</td>
 +
</tr>
 +
<tr>
 +
<td>5.1</td>
 +
<td>
 +
<span class="htmlMath">x<sup>13</sup></span>
 +
</td>
 +
<td>4270</td>
 +
<td>338</td>
 +
<td>889</td>
 +
<td>889</td>
 +
<td>889</td>
 +
</tr>
 +
<tr>
 +
<td>6.1</td>
 +
<td>
 +
<span class="htmlMath">x<sup>57</sup></span>
 +
</td>
 +
<td>4704</td>
 +
<td>436</td>
 +
<td>889</td>
 +
<td>889</td>
 +
<td>889</td>
 +
</tr>
 +
<tr>
 +
<td>7.1</td>
 +
<td>
 +
<span class="htmlMath">x<sup>-1</sup></span>
 +
</td>
 +
<td>8128</td>
 +
<td>4928</td>
 +
<td>1778</td>
 +
<td>1778</td>
 +
<td>1778</td>
 +
</tr>
 +
<tr>
 +
<td>8.1</td>
 +
<td>
 +
<span class="htmlMath">x<sup>65</sup> + x<sup>10</sup> + x<sup>3</sup></span>
 +
</td>
 +
<td>4038</td>
 +
<td>212</td>
 +
<td>896</td>
 +
<td>896</td>
 +
<td>896</td>
 +
</tr>
 +
<tr>
 +
<td>9.1</td>
 +
<td>
 +
<span class="htmlMath">x<sup>3</sup> + x<sup>9</sup> + x<sup>18</sup> + x<sup>66</sup></span>
 +
</td>
 +
<td>4044</td>
 +
<td>212</td>
 +
<td>896</td>
 +
<td>896</td>
 +
<td>896</td>
 +
</tr>
 +
<tr>
 +
<td>10.1</td>
 +
<td>
 +
<span class="htmlMath">x<sup>3</sup> + x<sup>12</sup> + x<sup>17</sup> + x<sup>33</sup></span>
 +
</td>
 +
<td>4048</td>
 +
<td>210</td>
 +
<td>896</td>
 +
<td>896</td>
 +
<td>896</td>
 +
</tr>
 +
<tr>
 +
<td>10.2</td>
 +
<td>
 +
<span class="htmlMath">x<sup>3</sup> + x<sup>17</sup> + x<sup>20</sup> + x<sup>34</sup> + x<sup>66</sup></span>
 +
</td>
 +
<td>4040</td>
 +
<td>210</td>
 +
<td>896</td>
 +
<td>896</td>
 +
<td>896</td>
 +
</tr>
 +
<tr>
 +
<td>11.1</td>
 +
<td>
 +
<span class="htmlMath">x<sup>3</sup> + x<sup>20</sup> + x<sup>34</sup> + x<sup>66</sup></span>
 +
</td>
 +
<td>4048</td>
 +
<td>210</td>
 +
<td>896</td>
 +
<td>896</td>
 +
<td>896</td>
 +
</tr>
 +
<tr>
 +
<td>12.1</td>
 +
<td>
 +
<span class="htmlMath">x<sup>3</sup> + x<sup>12</sup> + x<sup>40</sup> + x<sup>72</sup></span>
 +
</td>
 +
<td>4048</td>
 +
<td>210</td>
 +
<td>896</td>
 +
<td>896</td>
 +
<td>896</td>
 +
</tr>
 +
<tr>
 +
<td>13.1</td>
 +
<td>
 +
<span class="htmlMath">x<sup>3</sup> + x<sup>5</sup> + x<sup>10</sup> + x<sup>33</sup> + x<sup>34</sup></span>
 +
</td>
 +
<td>4040</td>
 +
<td>212</td>
 +
<td>896</td>
 +
<td>896</td>
 +
<td>896</td>
 +
</tr>
 +
<tr>
 +
<td>14.1</td>
 +
<td>
 +
<span class="htmlMath">x<sup>3</sup> + x<sup>6</sup> + x<sup>34</sup> + x<sup>40</sup> + x<sup>72</sup></span>
 +
</td>
 +
<td>4048</td>
 +
<td>212</td>
 +
<td>896</td>
 +
<td>896</td>
 +
<td>896</td>
 +
</tr>
 +
<tr>
 +
<td>14.2</td>
 +
<td>
 +
<span class="htmlMath">x<sup>3</sup> + x<sup>5</sup> + x<sup>6</sup> + x<sup>12</sup> + x<sup>33</sup> + x<sup>34</sup></span>
 +
</td>
 +
<td>4050</td>
 +
<td>210</td>
 +
<td>896</td>
 +
<td>896</td>
 +
<td>896</td>
 +
</tr>
 +
<tr>
 +
<td>14.3</td>
 +
<td>
 +
<span class="htmlMath">u<sup>2</sup>x<sup>96</sup> +  u<sup>78</sup>x<sup>80</sup> +  u<sup>121</sup>x<sup>72</sup> +  u<sup>49</sup>x<sup>68</sup> +  u<sup>77</sup>x<sup>66</sup> +  u<sup>29</sup>x<sup>65</sup> +  u<sup>119</sup>x<sup>48</sup> +  u<sup>117</sup>x<sup>40</sup> +  u<sup>28</sup>x<sup>36</sup> +  u<sup>107</sup>x<sup>34</sup> +u<sup>62</sup>x<sup>33</sup> +u<sup>125</sup>x<sup>24</sup> +u<sup>76</sup>x<sup>20</sup> +u<sup>84</sup>x<sup>18</sup> +u<sup>110</sup>x<sup>17</sup> +u<sup>49</sup>x<sup>12</sup> +u<sup>102</sup>x<sup>10</sup> +u<sup>69</sup>x<sup>9</sup> +  u<sup>14</sup>x<sup>6</sup> +  x<sup>5</sup> +  x<sup>3</sup></span>
 +
</td>
 +
<td>4046</td>
 +
<td>212</td>
 +
<td>128</td>
 +
<td>128</td>
 +
<td>128</td>
 +
</tr>
 +
<tr class="strongDivider">
 +
<td rowspan="23">8</td>
 +
<td>1.1</td>
 +
<td>
 +
<span class="htmlMath">x<sup>3</sup></span>
 +
</td>
 +
<td>11818</td>
 +
<td>420</td>
 +
<td></td>
 +
<td></td>
 +
<td>522240</td>
 +
</tr>
 +
<tr>
 +
<td>1.2</td>
 +
<td>
 +
<span class="htmlMath">x<sup>9</sup></span>
 +
</td>
 +
<td>12370</td>
 +
<td>420</td>
 +
<td></td>
 +
<td></td>
 +
<td>522240</td>
 +
</tr>
 +
<tr>
 +
<td>1.3</td>
 +
<td>
 +
<span class="htmlMath">x<sup>3</sup>+Tr(x<sup>9</sup>)</span>
 +
</td>
 +
<td>13800</td>
 +
<td>432</td>
 +
<td></td>
 +
<td></td>
 +
<td>6144</td>
 +
</tr>
 +
<tr>
 +
<td>1.4</td>
 +
<td>
 +
<span class="htmlMath">x<sup>9</sup>+Tr(x<sup>3</sup>)</span>
 +
</td>
 +
<td>13804</td>
 +
<td>434</td>
 +
<td></td>
 +
<td></td>
 +
<td>6144</td>
 +
</tr>
 +
<tr>
 +
<td>1.5</td>
 +
<td><span class="htmlMath">x<sup>3</sup>+u<sup>245</sup>x<sup>33</sup>+u<sup>183</sup>x<sup>66</sup>+u<sup>21</sup>x<sup>144</sup></span></td>
 +
<td>13842</td>
 +
<td>436</td>
 +
<td></td>
 +
<td></td>
 +
<td>3072</td>
 +
 
 +
</tr>
 +
<tr>
 +
<td>1.6</td>
 +
<td><span class="htmlMath">x<sup>3</sup> + u<sup>65</sup>x<sup>18</sup>+u<sup>120</sup>x<sup>66</sup>+u<sup>135</sup>x<sup>144</sup></span></td>
 +
<td>13848</td>
 +
<td>438</td>
 +
<td></td>
 +
<td></td>
 +
<td>3072</td>
 +
</tr>
 +
<tr>
 +
<td>1.7</td>
 +
<td><span class="htmlMath">u<sup>188</sup>x<sup>192</sup> + u<sup>129</sup>x<sup>144</sup> + u<sup>172</sup>x<sup>132</sup> + u<sup>138</sup>x<sup>129</sup> + u<sup>74</sup>x<sup>96</sup> + u<sup>244</sup>x<sup>72</sup> + u<sup>22</sup>x<sup>66</sup> + u<sup>178</sup>x<sup>48</sup> + u<sup>150</sup>x<sup>36</sup> + u<sup>146</sup>x<sup>33</sup> + u<sup>6</sup>x<sup>24</sup> + u<sup>60</sup>x<sup>18</sup> + u<sup>80</sup>x<sup>12</sup> + u<sup>140</sup>x<sup>9</sup> + u<sup>221</sup>x<sup>6</sup> + u<sup>19</sup>x<sup>3</sup></span></td>
 +
<td>14034</td>
 +
<td>438</td>
 +
<td></td>
 +
<td></td>
 +
<td>768</td>
 +
</tr>
 +
<tr>
 +
<td>1.8</td>
 +
<td><span class="htmlMath">u<sup>37</sup>x<sup>192</sup> + u<sup>110</sup>x<sup>144</sup> + u<sup>40</sup>x<sup>132</sup> + u<sup>53</sup>x<sup>129</sup> + u<sup>239</sup>x<sup>96</sup> + u<sup>235</sup>x<sup>72</sup> + u<sup>126</sup>x<sup>66</sup> + u<sup>215</sup>x<sup>48</sup> + u<sup>96</sup>x<sup>36</sup> + u<sup>29</sup>x<sup>33</sup> + u<sup>19</sup>x<sup>24</sup> + u<sup>14</sup>x<sup>18</sup> + u<sup>139</sup>x<sup>12</sup> + u<sup>230</sup>x<sup>9</sup> + u<sup>234</sup>x<sup>6</sup> + u<sup>228</sup>x<sup>3</sup></span></td>
 +
<td>14032</td>
 +
<td>438</td>
 +
<td></td>
 +
<td></td>
 +
<td>3072</td>
 +
</tr>
 +
<tr>
 +
<td>1.9</td>
 +
<td><span class="htmlMath">u<sup>242</sup>x<sup>192</sup> + u<sup>100</sup>x<sup>144</sup> + u<sup>66</sup>x<sup>132</sup> + u<sup>230</sup>x<sup>129</sup> + u<sup>202</sup>x<sup>96</sup> + u<sup>156</sup>x<sup>72</sup> + u<sup>254</sup>x<sup>66</sup> + u<sup>18</sup>x<sup>48</sup> + u<sup>44</sup>x<sup>36</sup> + u<sup>95</sup>x<sup>33</sup> + u<sup>100</sup>x<sup>24</sup> + u<sup>245</sup>x<sup>18</sup> + u<sup>174</sup>x<sup>12</sup> + u<sup>175</sup>x<sup>9</sup> + u<sup>247</sup>x<sup>6</sup> + u<sup>166</sup>x<sup>3</sup></span></td>
 +
<td>14036</td>
 +
<td>438</td>
 +
<td></td>
 +
<td></td>
 +
<td>3072</td>
 +
</tr>
 +
<tr>
 +
<td>1.10</td>
 +
<td><span class="htmlMath">u<sup>100</sup>x<sup>192</sup> + u<sup>83</sup>x<sup>144</sup> + u<sup>153</sup>x<sup>132</sup> + u<sup>65</sup>x<sup>129</sup> + u<sup>174</sup>x<sup>96</sup> + u<sup>136</sup>x<sup>72</sup> + u<sup>46</sup>x<sup>66</sup>+ u<sup>55</sup>x<sup>48</sup>+ u<sup>224</sup>x<sup>36</sup>+ u<sup>180</sup>x<sup>33</sup>+ u<sup>179</sup>x<sup>24</sup>+u<sup>226</sup>x<sup>18</sup>+ u<sup>54</sup>x<sup>12</sup>+ u<sup>168</sup>x<sup>9</sup>+ u<sup>89</sup>x<sup>6</sup>+ u<sup>56</sup>x<sup>3</sup></span></td>
 +
<td>14036</td>
 +
<td>438</td>
 +
<td></td>
 +
<td></td>
 +
<td>1536</td>
 +
</tr>
 +
<tr>
 +
<td>1.11</td>
 +
<td><span class="htmlMath">u<sup>77</sup>x<sup>192</sup> + u<sup>133</sup>x<sup>144</sup> + u<sup>47</sup>x<sup>132</sup> + u<sup>229</sup>x<sup>129</sup> + u<sup>23</sup>x<sup>96</sup> + u<sup>242</sup>x<sup>72</sup> + u<sup>242</sup>x<sup>66</sup> + u<sup>245</sup>x<sup>48</sup> + u<sup>212</sup>x<sup>36</sup> + u<sup>231</sup>x<sup>33</sup> + u<sup>174</sup>x<sup>24</sup> + u<sup>216</sup>x<sup>18</sup> + u<sup>96</sup>x<sup>12</sup> + u<sup>253</sup>x<sup>9</sup> + u<sup>154</sup>x<sup>6</sup> + u<sup>71</sup>x<sup>3</sup></span></td>
 +
<td>14032</td>
 +
<td>438</td>
 +
<td></td>
 +
<td></td>
 +
<td>3072</td>
 +
</tr>
 +
<tr>
 +
<td>1.12</td>
 +
<td><span class="htmlMath">u<sup>220</sup>x<sup>192</sup> + u<sup>94</sup>x<sup>144</sup> + u<sup>70</sup>x<sup>132</sup> + u<sup>159</sup>x<sup>129</sup> + u<sup>145</sup>x<sup>96</sup> + u<sup>160</sup>x<sup>72</sup> + u<sup>74</sup>x<sup>66</sup> + u<sup>184</sup>x<sup>48</sup> + u<sup>119</sup>x<sup>36</sup> + u<sup>106</sup>x<sup>33</sup> + u<sup>253</sup>x<sup>24</sup> + wx<sup>18</sup> + u<sup>90</sup>x<sup>12</sup> + u<sup>169</sup>x<sup>9</sup> + u<sup>118</sup>x<sup>6</sup> + u<sup>187</sup>x<sup>3</sup></span></td>
 +
<td>14034</td>
 +
<td>438</td>
 +
<td></td>
 +
<td></td>
 +
<td>3072</td>
 +
</tr>
 +
<tr>
 +
<td>1.13</td>
 +
<td><span class="htmlMath">u<sup>98</sup>x<sup>192</sup> + u<sup>225</sup>x<sup>144</sup> + u<sup>111</sup>x<sup>132</sup> + u<sup>238</sup>x<sup>129</sup> + u<sup>182</sup>x<sup>96</sup> + u<sup>125</sup>x<sup>72</sup> + u<sup>196</sup>x<sup>66</sup> + u<sup>219</sup>x<sup>48</sup> + u<sup>189</sup>x<sup>36</sup> + u<sup>199</sup>x<sup>33</sup> + u<sup>181</sup>x<sup>24</sup> + u<sup>110</sup>x<sup>18</sup> + u<sup>19</sup>x<sup>12</sup> + u<sup>175</sup>x<sup>9</sup> + u<sup>133</sup>x<sup>6</sup> + u<sup>47</sup>x<sup>3</sup></span></td>
 +
<td>14030</td>
 +
<td>438</td>
 +
<td></td>
 +
<td></td>
 +
<td>1536</td>
 +
</tr>
 +
<tr>
 +
<td>1.14</td>
 +
<td><span class="htmlMath">u<sup>236</sup>x<sup>192</sup> + u<sup>212</sup>x<sup>160</sup> + u<sup>153</sup>x<sup>144</sup> + u<sup>185</sup>x<sup>136</sup> + u<sup>3</sup>x<sup>132</sup> + u<sup>89</sup>x<sup>130</sup> + u<sup>189</sup>x<sup>129</sup> + u<sup>182</sup>x<sup>96</sup> + u<sup>105</sup>x<sup>80</sup> + u<sup>232</sup>x<sup>72</sup> + u<sup>219</sup>x<sup>68</sup> + u<sup>145</sup>x<sup>66</sup> + u<sup>171</sup>x<sup>65</sup> + u<sup>107</sup>x<sup>48</sup> + u<sup>179</sup>x<sup>40</sup> + u<sup>227</sup>x<sup>36</sup> + u<sup>236</sup>x<sup>34</sup> + u<sup>189</sup>x<sup>33</sup> + u<sup>162</sup>x<sup>24</sup> + u<sup>216</sup>x<sup>20</sup> + u<sup>162</sup>x<sup>18</sup> + u<sup>117</sup>x<sup>17</sup> + u<sup>56</sup>x<sup>12</sup> + u<sup>107</sup>x<sup>10</sup> + u<sup>236</sup>x<sup>9</sup> + u<sup>253</sup>x<sup>6</sup> + u<sup>180</sup>x<sup>5</sup> + u<sup>18</sup>x<sup>3</sup></span></td>
 +
<td>14046</td>
 +
<td>454</td>
 +
<td></td>
 +
<td></td>
 +
<td>512</td>
 +
</tr>
 +
<tr>
 +
<td>1.15</td>
 +
<td><span class="htmlMath">u<sup>27</sup>x<sup>192</sup> + u<sup>167</sup>x<sup>144</sup> + u<sup>26</sup>x<sup>132</sup> + u<sup>231</sup>x<sup>129</sup> + u<sup>139</sup>x<sup>96</sup> + u<sup>30</sup>x<sup>72</sup> + u<sup>139</sup>x<sup>66</sup> + u<sup>203</sup>x<sup>48</sup> + u<sup>36</sup>x<sup>36</sup> + u<sup>210</sup>x<sup>33</sup> + u<sup>195</sup>x<sup>24</sup> + u<sup>12</sup>x<sup>18</sup> + u<sup>43</sup>x<sup>12</sup> + u<sup>97</sup>x<sup>9</sup> + u<sup>61</sup>x<sup>6</sup> + u<sup>39</sup>x<sup>3</sup></span></td>
 +
<td>14036</td>
 +
<td>454</td>
 +
<td></td>
 +
<td></td>
 +
<td>768</td>
 +
</tr>
 +
<tr>
 +
<td>1.16</td>
 +
<td><span class="htmlMath">u<sup>6</sup>x<sup>192</sup> + u<sup>85</sup>x<sup>144</sup> + u<sup>251</sup>x<sup>132</sup> + u<sup>215</sup>x<sup>129</sup> + u<sup>229</sup>x<sup>96</sup> + u<sup>195</sup>x<sup>72</sup> + u<sup>152</sup>x<sup>66</sup> + u<sup>173</sup>x<sup>48</sup> + u<sup>209</sup>x<sup>36</sup> + u<sup>165</sup>x<sup>33</sup> + u<sup>213</sup>x<sup>24</sup> + u<sup>214</sup>x<sup>18</sup> + u<sup>158</sup>x<sup>12</sup> + u<sup>146</sup>x<sup>9</sup> + x<sup>6</sup> + u<sup>50</sup>x<sup>3</sup></span></td>
 +
<td>14032</td>
 +
<td>438</td>
 +
<td></td>
 +
<td></td>
 +
<td>1536</td>
 +
</tr>
 +
<tr>
 +
<td>1.17</td>
 +
<td><span class="htmlMath">u<sup>164</sup>x<sup>192</sup> + u<sup>224</sup>x<sup>144</sup> + u<sup>59</sup>x<sup>132</sup> + u<sup>124</sup>x<sup>129</sup> + u<sup>207</sup>x<sup>96</sup> + u<sup>211</sup>x<sup>72</sup> + u<sup>5</sup>x<sup>66</sup> + u<sup>26</sup>x<sup>48</sup> + u<sup>20</sup>x<sup>36</sup> + u<sup>101</sup>x<sup>33</sup> + u<sup>175</sup>x<sup>24</sup> + u<sup>241</sup>x<sup>18</sup> + x<sup>12</sup> + u<sup>15</sup>x<sup>9</sup> + u<sup>217</sup>x<sup>6</sup> + u<sup>212</sup>x<sup>3</sup></span></td>
 +
<td>14028</td>
 +
<td>438</td>
 +
<td></td>
 +
<td></td>
 +
<td>1536</td>
 +
</tr>
 +
<tr>
 +
<td>2.1</td>
 +
<td><span class="htmlMath">x<sup>3</sup>+ x<sup>17</sup>+u<sup>16</sup>(x<sup>18</sup>+x<sup>33</sup>)+u<sup>15</sup>x<sup>48</sup></span></td>
 +
<td>13200</td>
 +
<td>414</td>
 +
<td></td>
 +
<td></td>
 +
<td>46080</td>
 +
</tr>
 +
<tr>
 +
<td>3.1</td>
 +
<td><span class="htmlMath">x<sup>3</sup>+ u<sup>24</sup>x<sup>6</sup>+u<sup>182</sup>x<sup>132</sup>+u<sup>67</sup>x<sup>192</sup></span></td>
 +
<td>14024</td>
 +
<td>438</td>
 +
<td></td>
 +
<td></td>
 +
<td>3072</td>
 +
</tr>
 +
<tr>
 +
<td>4.1</td>
 +
<td><span class="htmlMath">x<sup>3</sup>+x<sup>6</sup>+x<sup>68</sup>+x<sup>80</sup>+x<sup>132</sup>+x<sup>160</sup></span></td>
 +
<td>14040</td>
 +
<td>454</td>
 +
<td></td>
 +
<td></td>
 +
<td>2048</td>
 +
</tr>
 +
<tr>
 +
<td>5.1</td>
 +
<td><span class="htmlMath">x<sup>3</sup>+x<sup>5</sup>+x<sup>18</sup>+x<sup>40</sup>+x<sup>66</sup></span></td>
 +
<td>14044</td>
 +
<td>446</td>
 +
<td></td>
 +
<td></td>
 +
<td>2048</td>
 +
</tr>
 +
<tr>
 +
<td>6.1</td>
 +
<td><span class="htmlMath">x<sup>3</sup>+x<sup>12</sup>+x<sup>40</sup>+x<sup>66</sup>+x<sup>130</sup></span></td>
 +
<td>14046</td>
 +
<td>438</td>
 +
<td></td>
 +
<td></td>
 +
<td>2048</td>
 +
</tr>
 +
<tr>
 +
<td>7.1</td>
 +
<td>
 +
<span class="htmlMath">x<sup>57</sup></span>
 +
</td>
 +
<td>15358</td>
 +
<td>960</td>
 +
<td></td>
 +
<td></td>
 +
<td>2040</td>
 
</tr>
 
</tr>
 
</table>
 
</table>

Latest revision as of 12:48, 15 July 2020

Known switching classes of APN functions over GF(25), GF(26), GF(27) and GF(28).

Also available is Magma code generating representatives from the switching classes.

n N F(x) Γ-rank Δ-rank Aut(dev(GF))/22n Aut(dev(GF))/22n Multiplier group
5 1.1 x3 330 42 4960 4960 4960
1.2 x5 330 42 4960 158720 4960
2.1 x-1 496 232 310 310 310
6 1.1 x3 1102 94 24192 48384 24192
1.2 x3 + u11x6 + ux9 1146 94 4032 8064 4032
2.1 ux5 + x9 + u4x17 + ux18 + u4x20 + ux24 + u4x34 + ux40 1158 96 320 320 320
2.2 u7x3 + x5 + u3x9 + u4x10 + x17 + u6x18 1166 94 448 896 448
2.3 x3 + ux24 + x10 1166 96 896 896 896
2.4 x3 + u17(x17 + x18 + x20 + x24) 1168 96 64 64 64
2.5 x3 + u11x5 + u13x9 + x17 + u11x33 + x48 1170 96 320 320 320
2.6 u25x5 + x9 + u38x12 + u25x18 + u25x36 1170 96 64 64 64
2.7 u40x5 + u10x6 + u62x20 + u35x33 + u15x34 + u29x48 1170 96 64 64 64
2.8 u34x6 + u52x9 + u48x12 + u6x20 + u9x33 + u23x34 + u25x40 1170 96 64 64 64
2.9 x9 + u4(x10 + x18) + u9(x12 + x20 + x40) 1172 96 64 64 64
2.10 u52x3 + u47x5 + ux6 + u9x9 + u44x12 + u47x33 + u10x34 + u33x40 1172 96 64 64 64
2.11 u(x6 + x10 + x24 + x33) + x9 + u4x17 1174 96 64 64 64
2.12 x3 + u17(x17 + x18 + x20 + x24) + u14(u52x3 + u6x5 + u19x7 + u28x11 + u2x13)+ (u52x3 + u6x5 + u19x7 + u28x11 + u2x13)2 + (u52x3 + u6x5 + u19x7 + u28x11 + u2x13)4+ (u52x3 + u6x5 + u19x7 + u28x11 + u2x13)8+ (u52x3 + u6x5 + u19x7 + u28x11 + u2x13)16+ (u52x3 + u6x5 + u19x7 + u28x11 + u2x13)32+ (u2x)9 +(u2x)19 +(u2x)36 + x21+x42 1300 152 8 8 8
7 1.1

x3

3610 198 113792 113792 113792
1.2

x3 + Tr(x9)

4026 212 896 896 896
2.1

x34 + x18 + x5

4034 210 896 896 896
2.2

x3 + x17 + x33 + x34

4040 212 896 896 896
3.1

x5

3708 198 113792 113792 113792
4.1

x9

3610 198 113792 14565376 113792
5.1

x13

4270 338 889 889 889
6.1

x57

4704 436 889 889 889
7.1

x-1

8128 4928 1778 1778 1778
8.1

x65 + x10 + x3

4038 212 896 896 896
9.1

x3 + x9 + x18 + x66

4044 212 896 896 896
10.1

x3 + x12 + x17 + x33

4048 210 896 896 896
10.2

x3 + x17 + x20 + x34 + x66

4040 210 896 896 896
11.1

x3 + x20 + x34 + x66

4048 210 896 896 896
12.1

x3 + x12 + x40 + x72

4048 210 896 896 896
13.1

x3 + x5 + x10 + x33 + x34

4040 212 896 896 896
14.1

x3 + x6 + x34 + x40 + x72

4048 212 896 896 896
14.2

x3 + x5 + x6 + x12 + x33 + x34

4050 210 896 896 896
14.3

u2x96 + u78x80 + u121x72 + u49x68 + u77x66 + u29x65 + u119x48 + u117x40 + u28x36 + u107x34 +u62x33 +u125x24 +u76x20 +u84x18 +u110x17 +u49x12 +u102x10 +u69x9 + u14x6 + x5 + x3

4046 212 128 128 128
8 1.1

x3

11818 420 522240
1.2

x9

12370 420 522240
1.3

x3+Tr(x9)

13800 432 6144
1.4

x9+Tr(x3)

13804 434 6144
1.5 x3+u245x33+u183x66+u21x144 13842 436 3072
1.6 x3 + u65x18+u120x66+u135x144 13848 438 3072
1.7 u188x192 + u129x144 + u172x132 + u138x129 + u74x96 + u244x72 + u22x66 + u178x48 + u150x36 + u146x33 + u6x24 + u60x18 + u80x12 + u140x9 + u221x6 + u19x3 14034 438 768
1.8 u37x192 + u110x144 + u40x132 + u53x129 + u239x96 + u235x72 + u126x66 + u215x48 + u96x36 + u29x33 + u19x24 + u14x18 + u139x12 + u230x9 + u234x6 + u228x3 14032 438 3072
1.9 u242x192 + u100x144 + u66x132 + u230x129 + u202x96 + u156x72 + u254x66 + u18x48 + u44x36 + u95x33 + u100x24 + u245x18 + u174x12 + u175x9 + u247x6 + u166x3 14036 438 3072
1.10 u100x192 + u83x144 + u153x132 + u65x129 + u174x96 + u136x72 + u46x66+ u55x48+ u224x36+ u180x33+ u179x24+u226x18+ u54x12+ u168x9+ u89x6+ u56x3 14036 438 1536
1.11 u77x192 + u133x144 + u47x132 + u229x129 + u23x96 + u242x72 + u242x66 + u245x48 + u212x36 + u231x33 + u174x24 + u216x18 + u96x12 + u253x9 + u154x6 + u71x3 14032 438 3072
1.12 u220x192 + u94x144 + u70x132 + u159x129 + u145x96 + u160x72 + u74x66 + u184x48 + u119x36 + u106x33 + u253x24 + wx18 + u90x12 + u169x9 + u118x6 + u187x3 14034 438 3072
1.13 u98x192 + u225x144 + u111x132 + u238x129 + u182x96 + u125x72 + u196x66 + u219x48 + u189x36 + u199x33 + u181x24 + u110x18 + u19x12 + u175x9 + u133x6 + u47x3 14030 438 1536
1.14 u236x192 + u212x160 + u153x144 + u185x136 + u3x132 + u89x130 + u189x129 + u182x96 + u105x80 + u232x72 + u219x68 + u145x66 + u171x65 + u107x48 + u179x40 + u227x36 + u236x34 + u189x33 + u162x24 + u216x20 + u162x18 + u117x17 + u56x12 + u107x10 + u236x9 + u253x6 + u180x5 + u18x3 14046 454 512
1.15 u27x192 + u167x144 + u26x132 + u231x129 + u139x96 + u30x72 + u139x66 + u203x48 + u36x36 + u210x33 + u195x24 + u12x18 + u43x12 + u97x9 + u61x6 + u39x3 14036 454 768
1.16 u6x192 + u85x144 + u251x132 + u215x129 + u229x96 + u195x72 + u152x66 + u173x48 + u209x36 + u165x33 + u213x24 + u214x18 + u158x12 + u146x9 + x6 + u50x3 14032 438 1536
1.17 u164x192 + u224x144 + u59x132 + u124x129 + u207x96 + u211x72 + u5x66 + u26x48 + u20x36 + u101x33 + u175x24 + u241x18 + x12 + u15x9 + u217x6 + u212x3 14028 438 1536
2.1 x3+ x17+u16(x18+x33)+u15x48 13200 414 46080
3.1 x3+ u24x6+u182x132+u67x192 14024 438 3072
4.1 x3+x6+x68+x80+x132+x160 14040 454 2048
5.1 x3+x5+x18+x40+x66 14044 446 2048
6.1 x3+x12+x40+x66+x130 14046 438 2048
7.1

x57

15358 960 2040