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

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Line 12: Line 12:
 
<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>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>
 
<tr>
Line 19: Line 20:
 
<td>330</td>
 
<td>330</td>
 
<td>42</td>
 
<td>42</td>
 +
<td>4960</td>
 
<td>4960</td>
 
<td>4960</td>
 
<td>4960</td>
 
<td>4960</td>
Line 29: Line 31:
 
<td>4960</td>
 
<td>4960</td>
 
<td>158720</td>
 
<td>158720</td>
 +
<td>4960</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 35: Line 38:
 
<td>496</td>
 
<td>496</td>
 
<td>232</td>
 
<td>232</td>
 +
<td>310</td>
 
<td>310</td>
 
<td>310</td>
 
<td>310</td>
 
<td>310</td>
Line 47: Line 51:
 
<td>24192</td>
 
<td>24192</td>
 
<td>48384</td>
 
<td>48384</td>
 +
<td>24192</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 55: Line 60:
 
<td>4032</td>
 
<td>4032</td>
 
<td>8064</td>
 
<td>8064</td>
 +
<td>4032</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 61: Line 67:
 
<td>1158</td>
 
<td>1158</td>
 
<td>96</td>
 
<td>96</td>
 +
<td>320</td>
 
<td>320</td>
 
<td>320</td>
 
<td>320</td>
 
<td>320</td>
Line 71: Line 78:
 
<td>448</td>
 
<td>448</td>
 
<td>896</td>
 
<td>896</td>
 +
<td>448</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 77: Line 85:
 
<td>1166</td>
 
<td>1166</td>
 
<td>96</td>
 
<td>96</td>
 +
<td>896</td>
 
<td>896</td>
 
<td>896</td>
 
<td>896</td>
 
<td>896</td>
Line 85: Line 94:
 
<td>1168</td>
 
<td>1168</td>
 
<td>96</td>
 
<td>96</td>
 +
<td>64</td>
 
<td>64</td>
 
<td>64</td>
 
<td>64</td>
 
<td>64</td>
Line 93: Line 103:
 
<td>1170</td>
 
<td>1170</td>
 
<td>96</td>
 
<td>96</td>
 +
<td>320</td>
 
<td>320</td>
 
<td>320</td>
 
<td>320</td>
 
<td>320</td>
Line 101: Line 112:
 
<td>1170</td>
 
<td>1170</td>
 
<td>96</td>
 
<td>96</td>
 +
<td>64</td>
 
<td>64</td>
 
<td>64</td>
 
<td>64</td>
 
<td>64</td>
Line 109: Line 121:
 
<td>1170</td>
 
<td>1170</td>
 
<td>96</td>
 
<td>96</td>
 +
<td>64</td>
 
<td>64</td>
 
<td>64</td>
 
<td>64</td>
 
<td>64</td>
Line 117: Line 130:
 
<td>1170</td>
 
<td>1170</td>
 
<td>96</td>
 
<td>96</td>
 +
<td>64</td>
 
<td>64</td>
 
<td>64</td>
 
<td>64</td>
 
<td>64</td>
Line 125: Line 139:
 
<td>1172</td>
 
<td>1172</td>
 
<td>96</td>
 
<td>96</td>
 +
<td>64</td>
 
<td>64</td>
 
<td>64</td>
 
<td>64</td>
 
<td>64</td>
Line 133: Line 148:
 
<td>1172</td>
 
<td>1172</td>
 
<td>96</td>
 
<td>96</td>
 +
<td>64</td>
 
<td>64</td>
 
<td>64</td>
 
<td>64</td>
 
<td>64</td>
Line 141: Line 157:
 
<td>1174</td>
 
<td>1174</td>
 
<td>96</td>
 
<td>96</td>
 +
<td>64</td>
 
<td>64</td>
 
<td>64</td>
 
<td>64</td>
 
<td>64</td>
Line 149: Line 166:
 
<td>1300</td>
 
<td>1300</td>
 
<td>152</td>
 
<td>152</td>
 +
<td>8</td>
 
<td>8</td>
 
<td>8</td>
 
<td>8</td>
 
<td>8</td>
Line 160: Line 178:
 
<td>3610</td>
 
<td>3610</td>
 
<td>198</td>
 
<td>198</td>
 +
<td>113792</td>
 
<td>113792</td>
 
<td>113792</td>
 
<td>113792</td>
 
<td>113792</td>
Line 170: Line 189:
 
<td>4026</td>
 
<td>4026</td>
 
<td>212</td>
 
<td>212</td>
 +
<td>896</td>
 
<td>896</td>
 
<td>896</td>
 
<td>896</td>
 
<td>896</td>
Line 180: Line 200:
 
<td>4034</td>
 
<td>4034</td>
 
<td>210</td>
 
<td>210</td>
 +
<td>896</td>
 
<td>896</td>
 
<td>896</td>
 
<td>896</td>
 
<td>896</td>
Line 190: Line 211:
 
<td>4040</td>
 
<td>4040</td>
 
<td>212</td>
 
<td>212</td>
 +
<td>896</td>
 
<td>896</td>
 
<td>896</td>
 
<td>896</td>
 
<td>896</td>
Line 200: Line 222:
 
<td>3708</td>
 
<td>3708</td>
 
<td>198</td>
 
<td>198</td>
 +
<td>113792</td>
 
<td>113792</td>
 
<td>113792</td>
 
<td>113792</td>
 
<td>113792</td>
Line 212: Line 235:
 
<td>113792</td>
 
<td>113792</td>
 
<td>14565376</td>
 
<td>14565376</td>
 +
<td>113792</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 220: Line 244:
 
<td>4270</td>
 
<td>4270</td>
 
<td>338</td>
 
<td>338</td>
 +
<td>889</td>
 
<td>889</td>
 
<td>889</td>
 
<td>889</td>
 
<td>889</td>
Line 230: Line 255:
 
<td>4704</td>
 
<td>4704</td>
 
<td>436</td>
 
<td>436</td>
 +
<td>889</td>
 
<td>889</td>
 
<td>889</td>
 
<td>889</td>
 
<td>889</td>
Line 240: Line 266:
 
<td>8128</td>
 
<td>8128</td>
 
<td>4928</td>
 
<td>4928</td>
 +
<td>1778</td>
 
<td>1778</td>
 
<td>1778</td>
 
<td>1778</td>
 
<td>1778</td>
Line 250: Line 277:
 
<td>4038</td>
 
<td>4038</td>
 
<td>212</td>
 
<td>212</td>
 +
<td>896</td>
 
<td>896</td>
 
<td>896</td>
 
<td>896</td>
 
<td>896</td>
Line 260: Line 288:
 
<td>4044</td>
 
<td>4044</td>
 
<td>212</td>
 
<td>212</td>
 +
<td>896</td>
 
<td>896</td>
 
<td>896</td>
 
<td>896</td>
 
<td>896</td>
Line 270: Line 299:
 
<td>4048</td>
 
<td>4048</td>
 
<td>210</td>
 
<td>210</td>
 +
<td>896</td>
 
<td>896</td>
 
<td>896</td>
 
<td>896</td>
 
<td>896</td>
Line 280: Line 310:
 
<td>4040</td>
 
<td>4040</td>
 
<td>210</td>
 
<td>210</td>
 +
<td>896</td>
 
<td>896</td>
 
<td>896</td>
 
<td>896</td>
 
<td>896</td>
Line 290: Line 321:
 
<td>4048</td>
 
<td>4048</td>
 
<td>210</td>
 
<td>210</td>
 +
<td>896</td>
 
<td>896</td>
 
<td>896</td>
 
<td>896</td>
 
<td>896</td>
Line 300: Line 332:
 
<td>4048</td>
 
<td>4048</td>
 
<td>210</td>
 
<td>210</td>
 +
<td>896</td>
 
<td>896</td>
 
<td>896</td>
 
<td>896</td>
 
<td>896</td>
Line 310: Line 343:
 
<td>4040</td>
 
<td>4040</td>
 
<td>212</td>
 
<td>212</td>
 +
<td>896</td>
 
<td>896</td>
 
<td>896</td>
 
<td>896</td>
 
<td>896</td>
Line 320: Line 354:
 
<td>4048</td>
 
<td>4048</td>
 
<td>212</td>
 
<td>212</td>
 +
<td>896</td>
 
<td>896</td>
 
<td>896</td>
 
<td>896</td>
 
<td>896</td>
Line 330: Line 365:
 
<td>4050</td>
 
<td>4050</td>
 
<td>210</td>
 
<td>210</td>
 +
<td>896</td>
 
<td>896</td>
 
<td>896</td>
 
<td>896</td>
 
<td>896</td>
Line 340: Line 376:
 
<td>4046</td>
 
<td>4046</td>
 
<td>212</td>
 
<td>212</td>
 +
<td>128</td>
 
<td>128</td>
 
<td>128</td>
 
<td>128</td>
 
<td>128</td>
Line 353: Line 390:
 
<td></td>
 
<td></td>
 
<td></td>
 
<td></td>
 +
<td>522240</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 363: Line 401:
 
<td></td>
 
<td></td>
 
<td></td>
 
<td></td>
 +
<td>522240</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 373: Line 412:
 
<td></td>
 
<td></td>
 
<td></td>
 
<td></td>
 +
<td>6144</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 383: Line 423:
 
<td></td>
 
<td></td>
 
<td></td>
 
<td></td>
 +
<td>6144</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 391: Line 432:
 
<td></td>
 
<td></td>
 
<td></td>
 
<td></td>
 +
<td>3072</td>
 +
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 399: Line 442:
 
<td></td>
 
<td></td>
 
<td></td>
 
<td></td>
 +
<td>3072</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 407: Line 451:
 
<td></td>
 
<td></td>
 
<td></td>
 
<td></td>
 +
<td>768</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 415: Line 460:
 
<td></td>
 
<td></td>
 
<td></td>
 
<td></td>
 +
<td>3072</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 423: Line 469:
 
<td></td>
 
<td></td>
 
<td></td>
 
<td></td>
 +
<td>3072</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 431: Line 478:
 
<td></td>
 
<td></td>
 
<td></td>
 
<td></td>
 +
<td>1536</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 439: Line 487:
 
<td></td>
 
<td></td>
 
<td></td>
 
<td></td>
 +
<td>3072</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 447: Line 496:
 
<td></td>
 
<td></td>
 
<td></td>
 
<td></td>
 +
<td>3072</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 455: Line 505:
 
<td></td>
 
<td></td>
 
<td></td>
 
<td></td>
 +
<td>1536</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 463: Line 514:
 
<td></td>
 
<td></td>
 
<td></td>
 
<td></td>
 +
<td>512</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 471: Line 523:
 
<td></td>
 
<td></td>
 
<td></td>
 
<td></td>
 +
<td>768</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 479: Line 532:
 
<td></td>
 
<td></td>
 
<td></td>
 
<td></td>
 +
<td>1536</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 487: Line 541:
 
<td></td>
 
<td></td>
 
<td></td>
 
<td></td>
 +
<td>1536</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 495: Line 550:
 
<td></td>
 
<td></td>
 
<td></td>
 
<td></td>
 +
<td>46080</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 503: Line 559:
 
<td></td>
 
<td></td>
 
<td></td>
 
<td></td>
 +
<td>3072</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 511: Line 568:
 
<td></td>
 
<td></td>
 
<td></td>
 
<td></td>
 +
<td>2048</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 519: Line 577:
 
<td></td>
 
<td></td>
 
<td></td>
 
<td></td>
 +
<td>2048</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 527: Line 586:
 
<td></td>
 
<td></td>
 
<td></td>
 
<td></td>
 +
<td>2048</td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 537: Line 597:
 
<td></td>
 
<td></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