Known switching classes of APN functions over GF(2^n) for n = 5,6,7,8: Difference between revisions
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| Line 8: | Line 8: | ||
<th><math>N^\circ</math></th> | <th><math>N^\circ</math></th> | ||
<th><math>F(x)</math></th> | <th><math>F(x)</math></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> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 13: | Line 17: | ||
<td>1.1</td> | <td>1.1</td> | ||
<td><math>x^3</math></td> | <td><math>x^3</math></td> | ||
<td>330</td> | |||
<td>42</td> | |||
<td>4960</td> | |||
<td>4960</td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>1.2</td> | <td>1.2</td> | ||
<td><math>x^5</math></td> | <td><math>x^5</math></td> | ||
<td>330</td> | |||
<td>42</td> | |||
<td>4960</td> | |||
<td>158720</td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>2.1</td> | <td>2.1</td> | ||
<td><math>x^{-1}</math></td> | <td><math>x^{-1}</math></td> | ||
<td>496</td> | |||
<td>232</td> | |||
<td>310</td> | |||
<td>310</td> | |||
</tr> | </tr> | ||
</tr> | </tr> | ||
| Line 27: | Line 43: | ||
<td>1.1</td> | <td>1.1</td> | ||
<td><math>x^{3}</math></td> | <td><math>x^{3}</math></td> | ||
<td>1102</td> | |||
<td>94</td> | |||
<td>24192</td> | |||
<td>48384</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><math>x^{3} + u^{11}x^{6} + ux^{9}</math></td> | ||
<td>1146</td> | |||
<td>94</td> | |||
<td>4032</td> | |||
<td>8064</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><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>1158</td> | |||
<td>96</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><math>u^{7}x^{3} + x^{5} + u^{3}x^{9} + u^{4}x^{10} + x^{17} + u^{6}x^{18}</math></td> | ||
<td>1166</td> | |||
<td>94</td> | |||
<td>448</td> | |||
<td>896</td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>2.3</td> | <td>2.3</td> | ||
<td><math>x^{3} + ux^{24} + x^{10}</math></td> | <td><math>x^{3} + ux^{24} + x^{10}</math></td> | ||
<td>1166</td> | |||
<td>96</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><math>x^{3} + u^{17}(x^{17} + x^{18} + x^{20} + x^{24})</math></td> | ||
<td>1168</td> | |||
<td>96</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><math>x^{3} + u^{11}x^{5} + u^{13}x^{9} + x^{17} + u^{11}x^{33} + x^{48}</math></td> | ||
<td>1170</td> | |||
<td>96</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><math>u^{25}x^{5} + x^{9} + u^{38}x^{12} + u^{25}x^{18} + u^{25}x^{36}</math></td> | ||
<td>1170</td> | |||
<td>96</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><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>1170</td> | |||
<td>96</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><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>1170</td> | |||
<td>96</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><math>x^{9} + u^{4}(x^{10} + x^{18}) + u^{9}(x^{12} + x^{20} + x^{40})</math></td> | ||
<td>1172</td> | |||
<td>96</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><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>1172</td> | |||
<td>96</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><math>u(x^{6} + x^{10} + x^{24} + x^{33}) + x^{9} + u^{4}x^{17}</math></td> | ||
<td>1174</td> | |||
<td>96</td> | |||
<td>64</td> | |||
<td>64</td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>2.12</td> | <td>2.12</td> | ||
<td><math>x^{3} + </math> <math>u^{17}(x^{17} + </math> <math>x^{18} + </math> <math>x^{20} + </math> <math>x^{24}) + </math> <math>u^{14}((u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})+ (u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})^{2} + </math> <math>(u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})^{4}+ (u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})^{8}+ (u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})^{16}+ (u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})^{32}+ (u^{2}x)^{9} +(u^{2}x)^{19} +(u^{2}x)^{36} + </math> <math>x^{21}+x^{42}</math></td> | <td><math>x^{3} + </math> <math>u^{17}(x^{17} + </math> <math>x^{18} + </math> <math>x^{20} + </math> <math>x^{24}) + </math> <math>u^{14}((u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})+ (u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})^{2} + </math> <math>(u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})^{4}+ (u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})^{8}+ (u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})^{16}+ (u^{52}x^{3} + </math> <math>u^{6}x^{5} + </math> <math>u^{19}x^{7} + </math> <math>u^{28}x^{11} + </math> <math>u^{2}x^{13})^{32}+ (u^{2}x)^{9} +(u^{2}x)^{19} +(u^{2}x)^{36} + </math> <math>x^{21}+x^{42}</math></td> | ||
<td>1300</td> | |||
<td>152</td> | |||
<td>8</td> | |||
<td>8</td> | |||
</tr> | </tr> | ||
<tr class="strongDivider"> | <tr class="strongDivider"> | ||
| Line 86: | Line 158: | ||
x^{3} | x^{3} | ||
</math></td> | </math></td> | ||
<td>3610</td> | |||
<td>198</td> | |||
<td>113792</td> | |||
<td>113792</td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 92: | Line 168: | ||
x^{3} + {\rm Tr}(x^{9}) | x^{3} + {\rm Tr}(x^{9}) | ||
</math></td> | </math></td> | ||
<td>4026</td> | |||
<td>212</td> | |||
<td>896</td> | |||
<td>896</td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 98: | Line 178: | ||
x^{34} + x^{18} + x^{5} | x^{34} + x^{18} + x^{5} | ||
</math></td> | </math></td> | ||
<td>4034</td> | |||
<td>210</td> | |||
<td>896</td> | |||
<td>896</td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 104: | Line 188: | ||
x^{3} + x^{17} + x^{33} + x^{34} | x^{3} + x^{17} + x^{33} + x^{34} | ||
</math></td> | </math></td> | ||
<td>4040</td> | |||
<td>212</td> | |||
<td>896</td> | |||
<td>896</td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 110: | Line 198: | ||
x^{5} | x^{5} | ||
</math></td> | </math></td> | ||
<td>3708</td> | |||
<td>198</td> | |||
<td>113792</td> | |||
<td>113792</td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 116: | Line 208: | ||
x^{9} | x^{9} | ||
</math></td> | </math></td> | ||
<td>3610</td> | |||
<td>198</td> | |||
<td>113792</td> | |||
<td>14565376</td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 122: | Line 218: | ||
x^{13} | x^{13} | ||
</math></td> | </math></td> | ||
<td>4270</td> | |||
<td>338</td> | |||
<td>889</td> | |||
<td>889</td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 128: | Line 228: | ||
x^{57} | x^{57} | ||
</math></td> | </math></td> | ||
<td>4704</td> | |||
<td>436</td> | |||
<td>889</td> | |||
<td>889</td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 134: | Line 238: | ||
x^{-1} | x^{-1} | ||
</math></td> | </math></td> | ||
<td>8128</td> | |||
<td>4928</td> | |||
<td>1778</td> | |||
<td>1778</td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 140: | Line 248: | ||
x^{65} + x^{10} + x^{3} | x^{65} + x^{10} + x^{3} | ||
</math></td> | </math></td> | ||
<td>4038</td> | |||
<td>212</td> | |||
<td>896</td> | |||
<td>896</td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 146: | Line 258: | ||
x^{3} + x^{9} + x^{18} + x^{66} | x^{3} + x^{9} + x^{18} + x^{66} | ||
</math></td> | </math></td> | ||
<td>4044</td> | |||
<td>212</td> | |||
<td>896</td> | |||
<td>896</td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 152: | Line 268: | ||
x^{3} + x^{12} + x^{17} + x^{33} | x^{3} + x^{12} + x^{17} + x^{33} | ||
</math></td> | </math></td> | ||
<td>4048</td> | |||
<td>210</td> | |||
<td>896</td> | |||
<td>896</td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 158: | Line 278: | ||
x^{3} + x^{17} + x^{20} + x^{34} + x^{66} | x^{3} + x^{17} + x^{20} + x^{34} + x^{66} | ||
</math></td> | </math></td> | ||
<td>4040</td> | |||
<td>210</td> | |||
<td>896</td> | |||
<td>896</td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 164: | Line 288: | ||
x^{3} + x^{20} + x^{34} + x^{66} | x^{3} + x^{20} + x^{34} + x^{66} | ||
</math></td> | </math></td> | ||
<td>4048</td> | |||
<td>210</td> | |||
<td>896</td> | |||
<td>896</td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 170: | Line 298: | ||
x^{3} + x^{12} + x^{40} + x^{72} | x^{3} + x^{12} + x^{40} + x^{72} | ||
</math></td> | </math></td> | ||
<td>4048</td> | |||
<td>210</td> | |||
<td>896</td> | |||
<td>896</td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 176: | Line 308: | ||
x^{3} + x^{5} + x^{10} + x^{33} + x^{34} | x^{3} + x^{5} + x^{10} + x^{33} + x^{34} | ||
</math></td> | </math></td> | ||
<td>4040</td> | |||
<td>212</td> | |||
<td>896</td> | |||
<td>896</td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 182: | Line 318: | ||
x^{3} + x^{6} + x^{34} + x^{40} + x^{72} | x^{3} + x^{6} + x^{34} + x^{40} + x^{72} | ||
</math></td> | </math></td> | ||
<td>4048</td> | |||
<td>212</td> | |||
<td>896</td> | |||
<td>896</td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 188: | Line 328: | ||
x^{3} + x^{5} + x^{6} + x^{12} + x^{33} + x^{34} | x^{3} + x^{5} + x^{6} + x^{12} + x^{33} + x^{34} | ||
</math></td> | </math></td> | ||
<td>4050</td> | |||
<td>210</td> | |||
<td>896</td> | |||
<td>896</td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 194: | Line 338: | ||
u^{2}x^{96} + </math> <math>u^{78}x^{80} + </math> <math>u^{121}x^{72} + </math> <math>u^{49}x^{68} + </math> <math>u^{77}x^{66} + </math> <math>u^{29}x^{65} + </math> <math>u^{119}x^{48} + </math> <math>u^{117}x^{40} + </math> <math>u^{28}x^{36} + </math> <math>u^{107}x^{34} +u^{62}x^{33} +u^{125}x^{24} +u^{76}x^{20} +u^{84}x^{18} +u^{110}x^{17} +u^{49}x^{12} +u^{102}x^{10} +u^{69}x^{9} + </math> <math>u^{14}x^{6} + </math> <math>x^{5} + </math> <math>x^{3} | u^{2}x^{96} + </math> <math>u^{78}x^{80} + </math> <math>u^{121}x^{72} + </math> <math>u^{49}x^{68} + </math> <math>u^{77}x^{66} + </math> <math>u^{29}x^{65} + </math> <math>u^{119}x^{48} + </math> <math>u^{117}x^{40} + </math> <math>u^{28}x^{36} + </math> <math>u^{107}x^{34} +u^{62}x^{33} +u^{125}x^{24} +u^{76}x^{20} +u^{84}x^{18} +u^{110}x^{17} +u^{49}x^{12} +u^{102}x^{10} +u^{69}x^{9} + </math> <math>u^{14}x^{6} + </math> <math>x^{5} + </math> <math>x^{3} | ||
</math></td> | </math></td> | ||
<td>4046</td> | |||
<td>212</td> | |||
<td>128</td> | |||
<td>128</td> | |||
</tr> | </tr> | ||
<tr class="strongDivider"> | <tr class="strongDivider"> | ||
| Line 201: | Line 349: | ||
x^{3} | x^{3} | ||
</math></td> | </math></td> | ||
<td>11818</td> | |||
<td>420</td> | |||
<td></td> | |||
<td></td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 207: | Line 359: | ||
x^{9} | x^{9} | ||
</math></td> | </math></td> | ||
<td>12370</td> | |||
<td>420</td> | |||
<td></td> | |||
<td></td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 213: | Line 369: | ||
x^{3}+{\rm Tr}(x^{9}) | x^{3}+{\rm Tr}(x^{9}) | ||
</math></td> | </math></td> | ||
<td>13800</td> | |||
<td>432</td> | |||
<td></td> | |||
<td></td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 219: | Line 379: | ||
x^{9}+{\rm Tr}(x^{3}) | x^{9}+{\rm Tr}(x^{3}) | ||
</math></td> | </math></td> | ||
<td>13804</td> | |||
<td>434</td> | |||
<td></td> | |||
<td></td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 225: | Line 389: | ||
x^{3}+u^{245}x^{33}+u^{183}x^{66}+u^{21}x^{144} | x^{3}+u^{245}x^{33}+u^{183}x^{66}+u^{21}x^{144} | ||
</math></td> | </math></td> | ||
<td>13842</td> | |||
<td>436</td> | |||
<td></td> | |||
<td></td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 231: | Line 399: | ||
x^{3} + u^{65}x^{18}+u^{120}x^{66}+u^{135}x^{144} | x^{3} + u^{65}x^{18}+u^{120}x^{66}+u^{135}x^{144} | ||
</math></td> | </math></td> | ||
<td>13848</td> | |||
<td>438</td> | |||
<td></td> | |||
<td></td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 237: | Line 409: | ||
u^{188}x^{192} + </math> <math>u^{129}x^{144} + </math> <math>u^{172}x^{132} + </math> <math> u^{138}x^{129} + </math> <math>u^{74}x^{96} + </math> <math>u^{244}x^{72} + </math> <math>u^{22}x^{66} + </math> <math> u^{178}x^{48} + </math> <math>u^{150}x^{36} + </math> <math>u^{146}x^{33} + </math> <math>u^{6}x^{24} + </math> <math> u^{60}x^{18} + </math> <math>u^{80}x^{12} + </math> <math>u^{140}x^{9} + </math> <math>u^{221}x^{6} + </math> <math>u^{19}x^{3} | u^{188}x^{192} + </math> <math>u^{129}x^{144} + </math> <math>u^{172}x^{132} + </math> <math> u^{138}x^{129} + </math> <math>u^{74}x^{96} + </math> <math>u^{244}x^{72} + </math> <math>u^{22}x^{66} + </math> <math> u^{178}x^{48} + </math> <math>u^{150}x^{36} + </math> <math>u^{146}x^{33} + </math> <math>u^{6}x^{24} + </math> <math> u^{60}x^{18} + </math> <math>u^{80}x^{12} + </math> <math>u^{140}x^{9} + </math> <math>u^{221}x^{6} + </math> <math>u^{19}x^{3} | ||
</math></td> | </math></td> | ||
<td>14034</td> | |||
<td>438</td> | |||
<td></td> | |||
<td></td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 243: | Line 419: | ||
u^{37}x^{192} + </math> <math>u^{110}x^{144} + </math> <math>u^{40}x^{132} + </math> <math>u^{53}x^{129} + </math> <math>u^{239}x^{96} + </math> <math>u^{235}x^{72} + </math> <math>u^{126}x^{66} + </math> <math>u^{215}x^{48} + </math> <math> u^{96}x^{36} + </math> <math>u^{29}x^{33} + </math> <math>u^{19}x^{24} + </math> <math>u^{14}x^{18} + </math> <math> u^{139}x^{12} + </math> <math>u^{230}x^{9} + </math> <math>u^{234}x^{6} + </math> <math>u^{228}x^{3} | u^{37}x^{192} + </math> <math>u^{110}x^{144} + </math> <math>u^{40}x^{132} + </math> <math>u^{53}x^{129} + </math> <math>u^{239}x^{96} + </math> <math>u^{235}x^{72} + </math> <math>u^{126}x^{66} + </math> <math>u^{215}x^{48} + </math> <math> u^{96}x^{36} + </math> <math>u^{29}x^{33} + </math> <math>u^{19}x^{24} + </math> <math>u^{14}x^{18} + </math> <math> u^{139}x^{12} + </math> <math>u^{230}x^{9} + </math> <math>u^{234}x^{6} + </math> <math>u^{228}x^{3} | ||
</math></td> | </math></td> | ||
<td>14032</td> | |||
<td>438</td> | |||
<td></td> | |||
<td></td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 249: | Line 429: | ||
u^{242}x^{192} + </math> <math>u^{100}x^{144} + </math> <math>u^{66}x^{132} + </math> <math>u^{230}x^{129} + </math> <math> u^{202}x^{96} + </math> <math>u^{156}x^{72} + </math> <math>u^{254}x^{66} + </math> <math>u^{18}x^{48} + </math> <math> u^{44}x^{36} + </math> <math>u^{95}x^{33} + </math> <math>u^{100}x^{24} + </math> <math>u^{245}x^{18} + </math> <math> u^{174}x^{12} + </math> <math>u^{175}x^{9} + </math> <math>u^{247}x^{6} + </math> <math>u^{166}x^{3} | u^{242}x^{192} + </math> <math>u^{100}x^{144} + </math> <math>u^{66}x^{132} + </math> <math>u^{230}x^{129} + </math> <math> u^{202}x^{96} + </math> <math>u^{156}x^{72} + </math> <math>u^{254}x^{66} + </math> <math>u^{18}x^{48} + </math> <math> u^{44}x^{36} + </math> <math>u^{95}x^{33} + </math> <math>u^{100}x^{24} + </math> <math>u^{245}x^{18} + </math> <math> u^{174}x^{12} + </math> <math>u^{175}x^{9} + </math> <math>u^{247}x^{6} + </math> <math>u^{166}x^{3} | ||
</math></td> | </math></td> | ||
<td>14036</td> | |||
<td>438</td> | |||
<td></td> | |||
<td></td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 256: | Line 440: | ||
u^{46}x^{66}+ u^{55}x^{48}+ u^{224}x^{36}+ u^{180}x^{33}+ u^{179}x^{24}+u^{226}x^{18}+ u^{54}x^{12}+ u^{168}x^{9}+ u^{89}x^{6}+ u^{56}x^{3} | u^{46}x^{66}+ u^{55}x^{48}+ u^{224}x^{36}+ u^{180}x^{33}+ u^{179}x^{24}+u^{226}x^{18}+ u^{54}x^{12}+ u^{168}x^{9}+ u^{89}x^{6}+ u^{56}x^{3} | ||
</math></td> | </math></td> | ||
<td>14036</td> | |||
<td>438</td> | |||
<td></td> | |||
<td></td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 262: | Line 450: | ||
u^{77}x^{192} + </math> <math> u^{133}x^{144} + </math> <math> u^{47}x^{132} + </math> <math> u^{229}x^{129} + </math> <math> u^{23}x^{96} + </math> <math> u^{242}x^{72} + </math> <math> u^{242}x^{66} + </math> <math> u^{245}x^{48} + </math> <math> u^{212}x^{36} + </math> <math> u^{231}x^{33} + </math> <math> u^{174}x^{24} + </math> <math> u^{216}x^{18} + </math> <math> u^{96}x^{12} + </math> <math> u^{253}x^{9} + </math> <math> u^{154}x^{6} + </math> <math> u^{71}x^{3} | u^{77}x^{192} + </math> <math> u^{133}x^{144} + </math> <math> u^{47}x^{132} + </math> <math> u^{229}x^{129} + </math> <math> u^{23}x^{96} + </math> <math> u^{242}x^{72} + </math> <math> u^{242}x^{66} + </math> <math> u^{245}x^{48} + </math> <math> u^{212}x^{36} + </math> <math> u^{231}x^{33} + </math> <math> u^{174}x^{24} + </math> <math> u^{216}x^{18} + </math> <math> u^{96}x^{12} + </math> <math> u^{253}x^{9} + </math> <math> u^{154}x^{6} + </math> <math> u^{71}x^{3} | ||
</math></td> | </math></td> | ||
<td>14032</td> | |||
<td>438</td> | |||
<td></td> | |||
<td></td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 268: | Line 460: | ||
u^{220}x^{192} + </math> <math> u^{94}x^{144} + </math> <math> u^{70}x^{132} + </math> <math> u^{159}x^{129} + </math> <math> u^{145}x^{96} + </math> <math>u^{160}x^{72} + </math> <math> u^{74}x^{66} + </math> <math> u^{184}x^{48} + </math> <math> u^{119}x^{36} + </math> <math> u^{106}x^{33} + </math> <math>u^{253}x^{24} + </math> <math> wx^{18} + </math> <math> u^{90}x^{12} + </math> <math> u^{169}x^{9} + </math> <math> u^{118}x^{6} + </math> <math> u^{187}x^{3} | u^{220}x^{192} + </math> <math> u^{94}x^{144} + </math> <math> u^{70}x^{132} + </math> <math> u^{159}x^{129} + </math> <math> u^{145}x^{96} + </math> <math>u^{160}x^{72} + </math> <math> u^{74}x^{66} + </math> <math> u^{184}x^{48} + </math> <math> u^{119}x^{36} + </math> <math> u^{106}x^{33} + </math> <math>u^{253}x^{24} + </math> <math> wx^{18} + </math> <math> u^{90}x^{12} + </math> <math> u^{169}x^{9} + </math> <math> u^{118}x^{6} + </math> <math> u^{187}x^{3} | ||
</math></td> | </math></td> | ||
<td>14034</td> | |||
<td>438</td> | |||
<td></td> | |||
<td></td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 274: | Line 470: | ||
u^{98}x^{192} + </math> <math> u^{225}x^{144} + </math> <math> u^{111}x^{132} + </math> <math> u^{238}x^{129} + </math> <math> u^{182}x^{96} + </math> <math> u^{125}x^{72} + </math> <math> u^{196}x^{66} + </math> <math> u^{219}x^{48} + </math> <math> u^{189}x^{36} + </math> <math> u^{199}x^{33} + </math> <math> u^{181}x^{24} + </math> <math> u^{110}x^{18} + </math> <math> u^{19}x^{12} + </math> <math> u^{175}x^{9} + </math> <math> u^{133}x^{6} + </math> <math> u^{47}x^{3} | u^{98}x^{192} + </math> <math> u^{225}x^{144} + </math> <math> u^{111}x^{132} + </math> <math> u^{238}x^{129} + </math> <math> u^{182}x^{96} + </math> <math> u^{125}x^{72} + </math> <math> u^{196}x^{66} + </math> <math> u^{219}x^{48} + </math> <math> u^{189}x^{36} + </math> <math> u^{199}x^{33} + </math> <math> u^{181}x^{24} + </math> <math> u^{110}x^{18} + </math> <math> u^{19}x^{12} + </math> <math> u^{175}x^{9} + </math> <math> u^{133}x^{6} + </math> <math> u^{47}x^{3} | ||
</math></td> | </math></td> | ||
<td>14030</td> | |||
<td>438</td> | |||
<td></td> | |||
<td></td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 280: | Line 480: | ||
u^{236}x^{192} + </math> <math> u^{212}x^{160} + </math> <math> u^{153}x^{144} + </math> <math> u^{185}x^{136} + </math> <math> u^{3}x^{132} + </math> <math>u^{89}x^{130} + </math> <math> u^{189}x^{129} + </math> <math> u^{182}x^{96} + </math> <math> u^{105}x^{80} + </math> <math> u^{232}x^{72} + </math> <math>u^{219}x^{68} + </math> <math> u^{145}x^{66} + </math> <math> u^{171}x^{65} + </math> <math> u^{107}x^{48} + </math> <math> u^{179}x^{40} + </math> <math> u^{227}x^{36} + </math> <math> u^{236}x^{34} + </math> <math> u^{189}x^{33} + </math> <math> u^{162}x^{24} + </math> <math> u^{216}x^{20} + </math> <math>u^{162}x^{18} + </math> <math> u^{117}x^{17} + </math> <math> u^{56}x^{12} + </math> <math> u^{107}x^{10} + </math> <math> u^{236}x^{9} + </math> <math>u^{253}x^{6} + </math> <math> u^{180}x^{5} + </math> <math> u^{18}x^{3} | u^{236}x^{192} + </math> <math> u^{212}x^{160} + </math> <math> u^{153}x^{144} + </math> <math> u^{185}x^{136} + </math> <math> u^{3}x^{132} + </math> <math>u^{89}x^{130} + </math> <math> u^{189}x^{129} + </math> <math> u^{182}x^{96} + </math> <math> u^{105}x^{80} + </math> <math> u^{232}x^{72} + </math> <math>u^{219}x^{68} + </math> <math> u^{145}x^{66} + </math> <math> u^{171}x^{65} + </math> <math> u^{107}x^{48} + </math> <math> u^{179}x^{40} + </math> <math> u^{227}x^{36} + </math> <math> u^{236}x^{34} + </math> <math> u^{189}x^{33} + </math> <math> u^{162}x^{24} + </math> <math> u^{216}x^{20} + </math> <math>u^{162}x^{18} + </math> <math> u^{117}x^{17} + </math> <math> u^{56}x^{12} + </math> <math> u^{107}x^{10} + </math> <math> u^{236}x^{9} + </math> <math>u^{253}x^{6} + </math> <math> u^{180}x^{5} + </math> <math> u^{18}x^{3} | ||
</math></td> | </math></td> | ||
<td>14046</td> | |||
<td>454</td> | |||
<td></td> | |||
<td></td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 286: | Line 490: | ||
u^{27}x^{192} + </math> <math> u^{167}x^{144} + </math> <math> u^{26}x^{132} + </math> <math>u^{231}x^{129} + </math> <math> u^{139}x^{96} + </math> <math>u^{30}x^{72} + </math> <math> u^{139}x^{66} + </math> <math> u^{203}x^{48} + </math> <math> u^{36}x^{36} + </math> <math> u^{210}x^{33} + </math> <math>u^{195}x^{24} + </math> <math> u^{12}x^{18} + </math> <math> u^{43}x^{12} + </math> <math> u^{97}x^{9} + </math> <math> u^{61}x^{6} + </math> <math>u^{39}x^{3} | u^{27}x^{192} + </math> <math> u^{167}x^{144} + </math> <math> u^{26}x^{132} + </math> <math>u^{231}x^{129} + </math> <math> u^{139}x^{96} + </math> <math>u^{30}x^{72} + </math> <math> u^{139}x^{66} + </math> <math> u^{203}x^{48} + </math> <math> u^{36}x^{36} + </math> <math> u^{210}x^{33} + </math> <math>u^{195}x^{24} + </math> <math> u^{12}x^{18} + </math> <math> u^{43}x^{12} + </math> <math> u^{97}x^{9} + </math> <math> u^{61}x^{6} + </math> <math>u^{39}x^{3} | ||
</math></td> | </math></td> | ||
<td>14036</td> | |||
<td>454</td> | |||
<td></td> | |||
<td></td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 292: | Line 500: | ||
u^{6}x^{192} + </math> <math> u^{85}x^{144} + </math> <math> u^{251}x^{132} + </math> <math> u^{215}x^{129} + </math> <math> u^{229}x^{96} + </math> <math> u^{195}x^{72} + </math> <math> u^{152}x^{66} + </math> <math> u^{173}x^{48} + </math> <math> u^{209}x^{36} + </math> <math> u^{165}x^{33} + </math> <math> u^{213}x^{24} + </math> <math> u^{214}x^{18} + </math> <math> u^{158}x^{12} + </math> <math> u^{146}x^{9} + </math> <math> x^{6} + </math> <math> u^{50}x^{3} | u^{6}x^{192} + </math> <math> u^{85}x^{144} + </math> <math> u^{251}x^{132} + </math> <math> u^{215}x^{129} + </math> <math> u^{229}x^{96} + </math> <math> u^{195}x^{72} + </math> <math> u^{152}x^{66} + </math> <math> u^{173}x^{48} + </math> <math> u^{209}x^{36} + </math> <math> u^{165}x^{33} + </math> <math> u^{213}x^{24} + </math> <math> u^{214}x^{18} + </math> <math> u^{158}x^{12} + </math> <math> u^{146}x^{9} + </math> <math> x^{6} + </math> <math> u^{50}x^{3} | ||
</math></td> | </math></td> | ||
<td>14032</td> | |||
<td>438</td> | |||
<td></td> | |||
<td></td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 298: | Line 510: | ||
u^{164}x^{192} + </math> <math> u^{224}x^{144} + </math> <math> u^{59}x^{132} + </math> <math> u^{124}x^{129} + </math> <math> u^{207}x^{96} + </math> <math> u^{211}x^{72} + </math> <math> u^{5}x^{66} + </math> <math> u^{26}x^{48} + </math> <math> u^{20}x^{36} + </math> <math> u^{101}x^{33} + </math> <math> u^{175}x^{24} + </math> <math> u^{241}x^{18} + </math> <math> x^{12} + </math> <math> u^{15}x^{9} + </math> <math> u^{217}x^{6} + </math> <math> u^{212}x^{3} | u^{164}x^{192} + </math> <math> u^{224}x^{144} + </math> <math> u^{59}x^{132} + </math> <math> u^{124}x^{129} + </math> <math> u^{207}x^{96} + </math> <math> u^{211}x^{72} + </math> <math> u^{5}x^{66} + </math> <math> u^{26}x^{48} + </math> <math> u^{20}x^{36} + </math> <math> u^{101}x^{33} + </math> <math> u^{175}x^{24} + </math> <math> u^{241}x^{18} + </math> <math> x^{12} + </math> <math> u^{15}x^{9} + </math> <math> u^{217}x^{6} + </math> <math> u^{212}x^{3} | ||
</math></td> | </math></td> | ||
<td>14028</td> | |||
<td>438</td> | |||
<td></td> | |||
<td></td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 304: | Line 520: | ||
x^{3}+ x^{17}+u^{16}(x^{18}+x^{33})+u^{15}x^{48} | x^{3}+ x^{17}+u^{16}(x^{18}+x^{33})+u^{15}x^{48} | ||
</math></td> | </math></td> | ||
<td>13200</td> | |||
<td>414</td> | |||
<td></td> | |||
<td></td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 310: | Line 530: | ||
x^{3}+ u^{24}x^{6}+u^{182}x^{132}+u^{67}x^{192} | x^{3}+ u^{24}x^{6}+u^{182}x^{132}+u^{67}x^{192} | ||
</math></td> | </math></td> | ||
<td>14024</td> | |||
<td>438</td> | |||
<td></td> | |||
<td></td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 316: | Line 540: | ||
x^{3}+x^{6}+x^{68}+x^{80}+x^{132}+x^{160} | x^{3}+x^{6}+x^{68}+x^{80}+x^{132}+x^{160} | ||
</math></td> | </math></td> | ||
<td>14040</td> | |||
<td>454</td> | |||
<td></td> | |||
<td></td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 322: | Line 550: | ||
x^{3}+x^{5}+x^{18}+x^{40}+x^{66} | x^{3}+x^{5}+x^{18}+x^{40}+x^{66} | ||
</math></td> | </math></td> | ||
<td>14044</td> | |||
<td>446</td> | |||
<td></td> | |||
<td></td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 328: | Line 560: | ||
x^{3}+x^{12}+x^{40}+x^{66}+x^{130} | x^{3}+x^{12}+x^{40}+x^{66}+x^{130} | ||
</math></td> | </math></td> | ||
<td>14046</td> | |||
<td>438</td> | |||
<td></td> | |||
<td></td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
| Line 334: | Line 570: | ||
x^{57} | x^{57} | ||
</math></td> | </math></td> | ||
<td>15358</td> | |||
<td>960</td> | |||
<td></td> | |||
<td></td> | |||
</tr> | </tr> | ||
</table> | </table> | ||
Revision as of 17:44, 8 October 2019
Known switching classes of APN functions over [math]\displaystyle{ \mathbb{F}_{2^5} }[/math], [math]\displaystyle{ \mathbb{F}_{2^6} }[/math], [math]\displaystyle{ \mathbb{F}_{2^7} }[/math] and [math]\displaystyle{ \mathbb{F}_{2^8} }[/math].
Also available is Magma code generating representatives from the switching classes.
| [math]\displaystyle{ n }[/math] | [math]\displaystyle{ N^\circ }[/math] | [math]\displaystyle{ F(x) }[/math] | Γ-rank | Δ-rank | Aut(dev(GF))/22n | Aut(dev(GF))/22n |
|---|---|---|---|---|---|---|
| [math]\displaystyle{ 5 }[/math] | 1.1 | [math]\displaystyle{ x^3 }[/math] | 330 | 42 | 4960 | 4960 |
| 1.2 | [math]\displaystyle{ x^5 }[/math] | 330 | 42 | 4960 | 158720 | |
| 2.1 | [math]\displaystyle{ x^{-1} }[/math] | 496 | 232 | 310 | 310 | |
| [math]\displaystyle{ 6 }[/math] | 1.1 | [math]\displaystyle{ x^{3} }[/math] | 1102 | 94 | 24192 | 48384 |
| 1.2 | [math]\displaystyle{ x^{3} + u^{11}x^{6} + ux^{9} }[/math] | 1146 | 94 | 4032 | 8064 | |
| 2.1 | [math]\displaystyle{ ux^{5} + x^{9} + u^{4}x^{17} + ux^{18} + u^{4}x^{20} + ux^{24} + u^{4}x^{34} + ux^{40} }[/math] | 1158 | 96 | 320 | 320 | |
| 2.2 | [math]\displaystyle{ u^{7}x^{3} + x^{5} + u^{3}x^{9} + u^{4}x^{10} + x^{17} + u^{6}x^{18} }[/math] | 1166 | 94 | 448 | 896 | |
| 2.3 | [math]\displaystyle{ x^{3} + ux^{24} + x^{10} }[/math] | 1166 | 96 | 896 | 896 | |
| 2.4 | [math]\displaystyle{ x^{3} + u^{17}(x^{17} + x^{18} + x^{20} + x^{24}) }[/math] | 1168 | 96 | 64 | 64 | |
| 2.5 | [math]\displaystyle{ x^{3} + u^{11}x^{5} + u^{13}x^{9} + x^{17} + u^{11}x^{33} + x^{48} }[/math] | 1170 | 96 | 320 | 320 | |
| 2.6 | [math]\displaystyle{ u^{25}x^{5} + x^{9} + u^{38}x^{12} + u^{25}x^{18} + u^{25}x^{36} }[/math] | 1170 | 96 | 64 | 64 | |
| 2.7 | [math]\displaystyle{ 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] | 1170 | 96 | 64 | 64 | |
| 2.8 | [math]\displaystyle{ 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] | 1170 | 96 | 64 | 64 | |
| 2.9 | [math]\displaystyle{ x^{9} + u^{4}(x^{10} + x^{18}) + u^{9}(x^{12} + x^{20} + x^{40}) }[/math] | 1172 | 96 | 64 | 64 | |
| 2.10 | [math]\displaystyle{ 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] | 1172 | 96 | 64 | 64 | |
| 2.11 | [math]\displaystyle{ u(x^{6} + x^{10} + x^{24} + x^{33}) + x^{9} + u^{4}x^{17} }[/math] | 1174 | 96 | 64 | 64 | |
| 2.12 | [math]\displaystyle{ x^{3} + }[/math] [math]\displaystyle{ u^{17}(x^{17} + }[/math] [math]\displaystyle{ x^{18} + }[/math] [math]\displaystyle{ x^{20} + }[/math] [math]\displaystyle{ x^{24}) + }[/math] [math]\displaystyle{ u^{14}((u^{52}x^{3} + }[/math] [math]\displaystyle{ u^{6}x^{5} + }[/math] [math]\displaystyle{ u^{19}x^{7} + }[/math] [math]\displaystyle{ u^{28}x^{11} + }[/math] [math]\displaystyle{ u^{2}x^{13})+ (u^{52}x^{3} + }[/math] [math]\displaystyle{ u^{6}x^{5} + }[/math] [math]\displaystyle{ u^{19}x^{7} + }[/math] [math]\displaystyle{ u^{28}x^{11} + }[/math] [math]\displaystyle{ u^{2}x^{13})^{2} + }[/math] [math]\displaystyle{ (u^{52}x^{3} + }[/math] [math]\displaystyle{ u^{6}x^{5} + }[/math] [math]\displaystyle{ u^{19}x^{7} + }[/math] [math]\displaystyle{ u^{28}x^{11} + }[/math] [math]\displaystyle{ u^{2}x^{13})^{4}+ (u^{52}x^{3} + }[/math] [math]\displaystyle{ u^{6}x^{5} + }[/math] [math]\displaystyle{ u^{19}x^{7} + }[/math] [math]\displaystyle{ u^{28}x^{11} + }[/math] [math]\displaystyle{ u^{2}x^{13})^{8}+ (u^{52}x^{3} + }[/math] [math]\displaystyle{ u^{6}x^{5} + }[/math] [math]\displaystyle{ u^{19}x^{7} + }[/math] [math]\displaystyle{ u^{28}x^{11} + }[/math] [math]\displaystyle{ u^{2}x^{13})^{16}+ (u^{52}x^{3} + }[/math] [math]\displaystyle{ u^{6}x^{5} + }[/math] [math]\displaystyle{ u^{19}x^{7} + }[/math] [math]\displaystyle{ u^{28}x^{11} + }[/math] [math]\displaystyle{ u^{2}x^{13})^{32}+ (u^{2}x)^{9} +(u^{2}x)^{19} +(u^{2}x)^{36} + }[/math] [math]\displaystyle{ x^{21}+x^{42} }[/math] | 1300 | 152 | 8 | 8 | |
| [math]\displaystyle{ 7 }[/math] | 1.1 | [math]\displaystyle{ x^{3} }[/math] | 3610 | 198 | 113792 | 113792 |
| 1.2 | [math]\displaystyle{ x^{3} + {\rm Tr}(x^{9}) }[/math] | 4026 | 212 | 896 | 896 | |
| 2.1 | [math]\displaystyle{ x^{34} + x^{18} + x^{5} }[/math] | 4034 | 210 | 896 | 896 | |
| 2.2 | [math]\displaystyle{ x^{3} + x^{17} + x^{33} + x^{34} }[/math] | 4040 | 212 | 896 | 896 | |
| 3.1 | [math]\displaystyle{ x^{5} }[/math] | 3708 | 198 | 113792 | 113792 | |
| 4.1 | [math]\displaystyle{ x^{9} }[/math] | 3610 | 198 | 113792 | 14565376 | |
| 5.1 | [math]\displaystyle{ x^{13} }[/math] | 4270 | 338 | 889 | 889 | |
| 6.1 | [math]\displaystyle{ x^{57} }[/math] | 4704 | 436 | 889 | 889 | |
| 7.1 | [math]\displaystyle{ x^{-1} }[/math] | 8128 | 4928 | 1778 | 1778 | |
| 8.1 | [math]\displaystyle{ x^{65} + x^{10} + x^{3} }[/math] | 4038 | 212 | 896 | 896 | |
| 9.1 | [math]\displaystyle{ x^{3} + x^{9} + x^{18} + x^{66} }[/math] | 4044 | 212 | 896 | 896 | |
| 10.1 | [math]\displaystyle{ x^{3} + x^{12} + x^{17} + x^{33} }[/math] | 4048 | 210 | 896 | 896 | |
| 10.2 | [math]\displaystyle{ x^{3} + x^{17} + x^{20} + x^{34} + x^{66} }[/math] | 4040 | 210 | 896 | 896 | |
| 11.1 | [math]\displaystyle{ x^{3} + x^{20} + x^{34} + x^{66} }[/math] | 4048 | 210 | 896 | 896 | |
| 12.1 | [math]\displaystyle{ x^{3} + x^{12} + x^{40} + x^{72} }[/math] | 4048 | 210 | 896 | 896 | |
| 13.1 | [math]\displaystyle{ x^{3} + x^{5} + x^{10} + x^{33} + x^{34} }[/math] | 4040 | 212 | 896 | 896 | |
| 14.1 | [math]\displaystyle{ x^{3} + x^{6} + x^{34} + x^{40} + x^{72} }[/math] | 4048 | 212 | 896 | 896 | |
| 14.2 | [math]\displaystyle{ x^{3} + x^{5} + x^{6} + x^{12} + x^{33} + x^{34} }[/math] | 4050 | 210 | 896 | 896 | |
| 14.3 | [math]\displaystyle{ u^{2}x^{96} + }[/math] [math]\displaystyle{ u^{78}x^{80} + }[/math] [math]\displaystyle{ u^{121}x^{72} + }[/math] [math]\displaystyle{ u^{49}x^{68} + }[/math] [math]\displaystyle{ u^{77}x^{66} + }[/math] [math]\displaystyle{ u^{29}x^{65} + }[/math] [math]\displaystyle{ u^{119}x^{48} + }[/math] [math]\displaystyle{ u^{117}x^{40} + }[/math] [math]\displaystyle{ u^{28}x^{36} + }[/math] [math]\displaystyle{ u^{107}x^{34} +u^{62}x^{33} +u^{125}x^{24} +u^{76}x^{20} +u^{84}x^{18} +u^{110}x^{17} +u^{49}x^{12} +u^{102}x^{10} +u^{69}x^{9} + }[/math] [math]\displaystyle{ u^{14}x^{6} + }[/math] [math]\displaystyle{ x^{5} + }[/math] [math]\displaystyle{ x^{3} }[/math] | 4046 | 212 | 128 | 128 | |
| [math]\displaystyle{ 8 }[/math] | 1.1 | [math]\displaystyle{ x^{3} }[/math] | 11818 | 420 | ||
| 1.2 | [math]\displaystyle{ x^{9} }[/math] | 12370 | 420 | |||
| 1.3 | [math]\displaystyle{ x^{3}+{\rm Tr}(x^{9}) }[/math] | 13800 | 432 | |||
| 1.4 | [math]\displaystyle{ x^{9}+{\rm Tr}(x^{3}) }[/math] | 13804 | 434 | |||
| 1.5 | [math]\displaystyle{ x^{3}+u^{245}x^{33}+u^{183}x^{66}+u^{21}x^{144} }[/math] | 13842 | 436 | |||
| 1.6 | [math]\displaystyle{ x^{3} + u^{65}x^{18}+u^{120}x^{66}+u^{135}x^{144} }[/math] | 13848 | 438 | |||
| 1.7 | [math]\displaystyle{ u^{188}x^{192} + }[/math] [math]\displaystyle{ u^{129}x^{144} + }[/math] [math]\displaystyle{ u^{172}x^{132} + }[/math] [math]\displaystyle{ u^{138}x^{129} + }[/math] [math]\displaystyle{ u^{74}x^{96} + }[/math] [math]\displaystyle{ u^{244}x^{72} + }[/math] [math]\displaystyle{ u^{22}x^{66} + }[/math] [math]\displaystyle{ u^{178}x^{48} + }[/math] [math]\displaystyle{ u^{150}x^{36} + }[/math] [math]\displaystyle{ u^{146}x^{33} + }[/math] [math]\displaystyle{ u^{6}x^{24} + }[/math] [math]\displaystyle{ u^{60}x^{18} + }[/math] [math]\displaystyle{ u^{80}x^{12} + }[/math] [math]\displaystyle{ u^{140}x^{9} + }[/math] [math]\displaystyle{ u^{221}x^{6} + }[/math] [math]\displaystyle{ u^{19}x^{3} }[/math] | 14034 | 438 | |||
| 1.8 | [math]\displaystyle{ u^{37}x^{192} + }[/math] [math]\displaystyle{ u^{110}x^{144} + }[/math] [math]\displaystyle{ u^{40}x^{132} + }[/math] [math]\displaystyle{ u^{53}x^{129} + }[/math] [math]\displaystyle{ u^{239}x^{96} + }[/math] [math]\displaystyle{ u^{235}x^{72} + }[/math] [math]\displaystyle{ u^{126}x^{66} + }[/math] [math]\displaystyle{ u^{215}x^{48} + }[/math] [math]\displaystyle{ u^{96}x^{36} + }[/math] [math]\displaystyle{ u^{29}x^{33} + }[/math] [math]\displaystyle{ u^{19}x^{24} + }[/math] [math]\displaystyle{ u^{14}x^{18} + }[/math] [math]\displaystyle{ u^{139}x^{12} + }[/math] [math]\displaystyle{ u^{230}x^{9} + }[/math] [math]\displaystyle{ u^{234}x^{6} + }[/math] [math]\displaystyle{ u^{228}x^{3} }[/math] | 14032 | 438 | |||
| 1.9 | [math]\displaystyle{ u^{242}x^{192} + }[/math] [math]\displaystyle{ u^{100}x^{144} + }[/math] [math]\displaystyle{ u^{66}x^{132} + }[/math] [math]\displaystyle{ u^{230}x^{129} + }[/math] [math]\displaystyle{ u^{202}x^{96} + }[/math] [math]\displaystyle{ u^{156}x^{72} + }[/math] [math]\displaystyle{ u^{254}x^{66} + }[/math] [math]\displaystyle{ u^{18}x^{48} + }[/math] [math]\displaystyle{ u^{44}x^{36} + }[/math] [math]\displaystyle{ u^{95}x^{33} + }[/math] [math]\displaystyle{ u^{100}x^{24} + }[/math] [math]\displaystyle{ u^{245}x^{18} + }[/math] [math]\displaystyle{ u^{174}x^{12} + }[/math] [math]\displaystyle{ u^{175}x^{9} + }[/math] [math]\displaystyle{ u^{247}x^{6} + }[/math] [math]\displaystyle{ u^{166}x^{3} }[/math] | 14036 | 438 | |||
| 1.10 | [math]\displaystyle{ u^{100}x^{192} + }[/math] [math]\displaystyle{ u^{83}x^{144} + }[/math] [math]\displaystyle{ u^{153}x^{132} + }[/math] [math]\displaystyle{ u^{65}x^{129} + }[/math] [math]\displaystyle{ u^{174}x^{96} + }[/math] [math]\displaystyle{ u^{136}x^{72} + }[/math] [math]\displaystyle{ u^{46}x^{66}+ u^{55}x^{48}+ u^{224}x^{36}+ u^{180}x^{33}+ u^{179}x^{24}+u^{226}x^{18}+ u^{54}x^{12}+ u^{168}x^{9}+ u^{89}x^{6}+ u^{56}x^{3} }[/math] | 14036 | 438 | |||
| 1.11 | [math]\displaystyle{ u^{77}x^{192} + }[/math] [math]\displaystyle{ u^{133}x^{144} + }[/math] [math]\displaystyle{ u^{47}x^{132} + }[/math] [math]\displaystyle{ u^{229}x^{129} + }[/math] [math]\displaystyle{ u^{23}x^{96} + }[/math] [math]\displaystyle{ u^{242}x^{72} + }[/math] [math]\displaystyle{ u^{242}x^{66} + }[/math] [math]\displaystyle{ u^{245}x^{48} + }[/math] [math]\displaystyle{ u^{212}x^{36} + }[/math] [math]\displaystyle{ u^{231}x^{33} + }[/math] [math]\displaystyle{ u^{174}x^{24} + }[/math] [math]\displaystyle{ u^{216}x^{18} + }[/math] [math]\displaystyle{ u^{96}x^{12} + }[/math] [math]\displaystyle{ u^{253}x^{9} + }[/math] [math]\displaystyle{ u^{154}x^{6} + }[/math] [math]\displaystyle{ u^{71}x^{3} }[/math] | 14032 | 438 | |||
| 1.12 | [math]\displaystyle{ u^{220}x^{192} + }[/math] [math]\displaystyle{ u^{94}x^{144} + }[/math] [math]\displaystyle{ u^{70}x^{132} + }[/math] [math]\displaystyle{ u^{159}x^{129} + }[/math] [math]\displaystyle{ u^{145}x^{96} + }[/math] [math]\displaystyle{ u^{160}x^{72} + }[/math] [math]\displaystyle{ u^{74}x^{66} + }[/math] [math]\displaystyle{ u^{184}x^{48} + }[/math] [math]\displaystyle{ u^{119}x^{36} + }[/math] [math]\displaystyle{ u^{106}x^{33} + }[/math] [math]\displaystyle{ u^{253}x^{24} + }[/math] [math]\displaystyle{ wx^{18} + }[/math] [math]\displaystyle{ u^{90}x^{12} + }[/math] [math]\displaystyle{ u^{169}x^{9} + }[/math] [math]\displaystyle{ u^{118}x^{6} + }[/math] [math]\displaystyle{ u^{187}x^{3} }[/math] | 14034 | 438 | |||
| 1.13 | [math]\displaystyle{ u^{98}x^{192} + }[/math] [math]\displaystyle{ u^{225}x^{144} + }[/math] [math]\displaystyle{ u^{111}x^{132} + }[/math] [math]\displaystyle{ u^{238}x^{129} + }[/math] [math]\displaystyle{ u^{182}x^{96} + }[/math] [math]\displaystyle{ u^{125}x^{72} + }[/math] [math]\displaystyle{ u^{196}x^{66} + }[/math] [math]\displaystyle{ u^{219}x^{48} + }[/math] [math]\displaystyle{ u^{189}x^{36} + }[/math] [math]\displaystyle{ u^{199}x^{33} + }[/math] [math]\displaystyle{ u^{181}x^{24} + }[/math] [math]\displaystyle{ u^{110}x^{18} + }[/math] [math]\displaystyle{ u^{19}x^{12} + }[/math] [math]\displaystyle{ u^{175}x^{9} + }[/math] [math]\displaystyle{ u^{133}x^{6} + }[/math] [math]\displaystyle{ u^{47}x^{3} }[/math] | 14030 | 438 | |||
| 1.14 | [math]\displaystyle{ u^{236}x^{192} + }[/math] [math]\displaystyle{ u^{212}x^{160} + }[/math] [math]\displaystyle{ u^{153}x^{144} + }[/math] [math]\displaystyle{ u^{185}x^{136} + }[/math] [math]\displaystyle{ u^{3}x^{132} + }[/math] [math]\displaystyle{ u^{89}x^{130} + }[/math] [math]\displaystyle{ u^{189}x^{129} + }[/math] [math]\displaystyle{ u^{182}x^{96} + }[/math] [math]\displaystyle{ u^{105}x^{80} + }[/math] [math]\displaystyle{ u^{232}x^{72} + }[/math] [math]\displaystyle{ u^{219}x^{68} + }[/math] [math]\displaystyle{ u^{145}x^{66} + }[/math] [math]\displaystyle{ u^{171}x^{65} + }[/math] [math]\displaystyle{ u^{107}x^{48} + }[/math] [math]\displaystyle{ u^{179}x^{40} + }[/math] [math]\displaystyle{ u^{227}x^{36} + }[/math] [math]\displaystyle{ u^{236}x^{34} + }[/math] [math]\displaystyle{ u^{189}x^{33} + }[/math] [math]\displaystyle{ u^{162}x^{24} + }[/math] [math]\displaystyle{ u^{216}x^{20} + }[/math] [math]\displaystyle{ u^{162}x^{18} + }[/math] [math]\displaystyle{ u^{117}x^{17} + }[/math] [math]\displaystyle{ u^{56}x^{12} + }[/math] [math]\displaystyle{ u^{107}x^{10} + }[/math] [math]\displaystyle{ u^{236}x^{9} + }[/math] [math]\displaystyle{ u^{253}x^{6} + }[/math] [math]\displaystyle{ u^{180}x^{5} + }[/math] [math]\displaystyle{ u^{18}x^{3} }[/math] | 14046 | 454 | |||
| 1.15 | [math]\displaystyle{ u^{27}x^{192} + }[/math] [math]\displaystyle{ u^{167}x^{144} + }[/math] [math]\displaystyle{ u^{26}x^{132} + }[/math] [math]\displaystyle{ u^{231}x^{129} + }[/math] [math]\displaystyle{ u^{139}x^{96} + }[/math] [math]\displaystyle{ u^{30}x^{72} + }[/math] [math]\displaystyle{ u^{139}x^{66} + }[/math] [math]\displaystyle{ u^{203}x^{48} + }[/math] [math]\displaystyle{ u^{36}x^{36} + }[/math] [math]\displaystyle{ u^{210}x^{33} + }[/math] [math]\displaystyle{ u^{195}x^{24} + }[/math] [math]\displaystyle{ u^{12}x^{18} + }[/math] [math]\displaystyle{ u^{43}x^{12} + }[/math] [math]\displaystyle{ u^{97}x^{9} + }[/math] [math]\displaystyle{ u^{61}x^{6} + }[/math] [math]\displaystyle{ u^{39}x^{3} }[/math] | 14036 | 454 | |||
| 1.16 | [math]\displaystyle{ u^{6}x^{192} + }[/math] [math]\displaystyle{ u^{85}x^{144} + }[/math] [math]\displaystyle{ u^{251}x^{132} + }[/math] [math]\displaystyle{ u^{215}x^{129} + }[/math] [math]\displaystyle{ u^{229}x^{96} + }[/math] [math]\displaystyle{ u^{195}x^{72} + }[/math] [math]\displaystyle{ u^{152}x^{66} + }[/math] [math]\displaystyle{ u^{173}x^{48} + }[/math] [math]\displaystyle{ u^{209}x^{36} + }[/math] [math]\displaystyle{ u^{165}x^{33} + }[/math] [math]\displaystyle{ u^{213}x^{24} + }[/math] [math]\displaystyle{ u^{214}x^{18} + }[/math] [math]\displaystyle{ u^{158}x^{12} + }[/math] [math]\displaystyle{ u^{146}x^{9} + }[/math] [math]\displaystyle{ x^{6} + }[/math] [math]\displaystyle{ u^{50}x^{3} }[/math] | 14032 | 438 | |||
| 1.17 | [math]\displaystyle{ u^{164}x^{192} + }[/math] [math]\displaystyle{ u^{224}x^{144} + }[/math] [math]\displaystyle{ u^{59}x^{132} + }[/math] [math]\displaystyle{ u^{124}x^{129} + }[/math] [math]\displaystyle{ u^{207}x^{96} + }[/math] [math]\displaystyle{ u^{211}x^{72} + }[/math] [math]\displaystyle{ u^{5}x^{66} + }[/math] [math]\displaystyle{ u^{26}x^{48} + }[/math] [math]\displaystyle{ u^{20}x^{36} + }[/math] [math]\displaystyle{ u^{101}x^{33} + }[/math] [math]\displaystyle{ u^{175}x^{24} + }[/math] [math]\displaystyle{ u^{241}x^{18} + }[/math] [math]\displaystyle{ x^{12} + }[/math] [math]\displaystyle{ u^{15}x^{9} + }[/math] [math]\displaystyle{ u^{217}x^{6} + }[/math] [math]\displaystyle{ u^{212}x^{3} }[/math] | 14028 | 438 | |||
| 2.1 | [math]\displaystyle{ x^{3}+ x^{17}+u^{16}(x^{18}+x^{33})+u^{15}x^{48} }[/math] | 13200 | 414 | |||
| 3.1 | [math]\displaystyle{ x^{3}+ u^{24}x^{6}+u^{182}x^{132}+u^{67}x^{192} }[/math] | 14024 | 438 | |||
| 4.1 | [math]\displaystyle{ x^{3}+x^{6}+x^{68}+x^{80}+x^{132}+x^{160} }[/math] | 14040 | 454 | |||
| 5.1 | [math]\displaystyle{ x^{3}+x^{5}+x^{18}+x^{40}+x^{66} }[/math] | 14044 | 446 | |||
| 6.1 | [math]\displaystyle{ x^{3}+x^{12}+x^{40}+x^{66}+x^{130} }[/math] | 14046 | 438 | |||
| 7.1 | [math]\displaystyle{ x^{57} }[/math] | 15358 | 960 |