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

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<td rowspan="19"><math>7</math></td>
 
<td rowspan="19"><math>7</math></td>
 
<td>1.1</td>
 
<td>1.1</td>
<td><math>
+
<td>
x^{3}
+
<span class="htmlMath">x<sup>3</sup></span>
</math></td>
+
</td>
 
<td>3610</td>
 
<td>3610</td>
 
<td>198</td>
 
<td>198</td>
Line 165: Line 165:
 
<tr>
 
<tr>
 
<td>1.2</td>
 
<td>1.2</td>
<td><math>
+
<td>
x^{3} + {\rm Tr}(x^{9})
+
<span class="htmlMath">x<sup>3</sup> + Tr(x<sup>9</sup>)</span>
</math></td>
+
</td>
 
<td>4026</td>
 
<td>4026</td>
 
<td>212</td>
 
<td>212</td>
Line 175: Line 175:
 
<tr>
 
<tr>
 
<td>2.1</td>
 
<td>2.1</td>
<td><math>
+
<td>
x^{34} + x^{18} + x^{5}
+
<span class="htmlMath">x<sup>34</sup> + x<sup>18</sup> + x<sup>5</sup></span>
</math></td>
+
</td>
 
<td>4034</td>
 
<td>4034</td>
 
<td>210</td>
 
<td>210</td>
Line 185: Line 185:
 
<tr>
 
<tr>
 
<td>2.2</td>
 
<td>2.2</td>
<td><math>
+
<td>
x^{3} + x^{17} + x^{33} + x^{34}
+
<span class="htmlMath">x<sup>3</sup> + x<sup>17</sup> + x<sup>33</sup> + x<sup>34</sup></span>
</math></td>
+
</td>
 
<td>4040</td>
 
<td>4040</td>
 
<td>212</td>
 
<td>212</td>
Line 195: Line 195:
 
<tr>
 
<tr>
 
<td>3.1</td>
 
<td>3.1</td>
<td><math>
+
<td>
x^{5}
+
<span class="htmlMath">x<sup>5</sup></span>
</math></td>
+
</td>
 
<td>3708</td>
 
<td>3708</td>
 
<td>198</td>
 
<td>198</td>
Line 205: Line 205:
 
<tr>
 
<tr>
 
<td>4.1</td>
 
<td>4.1</td>
<td><math>
+
<td>
x^{9}
+
<span class="htmlMath">x<sup>9</sup></span>
</math></td>
+
</td>
 
<td>3610</td>
 
<td>3610</td>
 
<td>198</td>
 
<td>198</td>
Line 215: Line 215:
 
<tr>
 
<tr>
 
<td>5.1</td>
 
<td>5.1</td>
<td><math>
+
<td>
x^{13}
+
<span class="htmlMath">x<sup>13</sup></span>
</math></td>
+
</td>
 
<td>4270</td>
 
<td>4270</td>
 
<td>338</td>
 
<td>338</td>
Line 225: Line 225:
 
<tr>
 
<tr>
 
<td>6.1</td>
 
<td>6.1</td>
<td><math>
+
<td>
x^{57}
+
<span class="htmlMath">x<sup>57</sup></span>
</math></td>
+
</td>
 
<td>4704</td>
 
<td>4704</td>
 
<td>436</td>
 
<td>436</td>
Line 235: Line 235:
 
<tr>
 
<tr>
 
<td>7.1</td>
 
<td>7.1</td>
<td><math>
+
<td>
x^{-1}
+
<span class="htmlMath">x<sup>-1</sup></span>
</math></td>
+
</td>
 
<td>8128</td>
 
<td>8128</td>
 
<td>4928</td>
 
<td>4928</td>
Line 245: Line 245:
 
<tr>
 
<tr>
 
<td>8.1</td>
 
<td>8.1</td>
<td><math>
+
<td>
x^{65} + x^{10} + x^{3}
+
<span class="htmlMath">x<sup>65</sup> + x<sup>10</sup> + x<sup>3</sup></span>
</math></td>
+
</td>
 
<td>4038</td>
 
<td>4038</td>
 
<td>212</td>
 
<td>212</td>
Line 255: Line 255:
 
<tr>
 
<tr>
 
<td>9.1</td>
 
<td>9.1</td>
<td><math>
+
<td>
x^{3} + x^{9} + x^{18} + x^{66}
+
<span class="htmlMath">x<sup>3</sup> + x<sup>9</sup> + x<sup>18</sup> + x<sup>66</sup></span>
</math></td>
+
</td>
 
<td>4044</td>
 
<td>4044</td>
 
<td>212</td>
 
<td>212</td>
Line 265: Line 265:
 
<tr>
 
<tr>
 
<td>10.1</td>
 
<td>10.1</td>
<td><math>
+
<td>
x^{3} + x^{12} + x^{17} + x^{33}
+
<span class="htmlMath">x<sup>3</sup> + x<sup>12</sup> + x<sup>17</sup> + x<sup>33</sup></span>
</math></td>
+
</td>
 
<td>4048</td>
 
<td>4048</td>
 
<td>210</td>
 
<td>210</td>
Line 275: Line 275:
 
<tr>
 
<tr>
 
<td>10.2</td>
 
<td>10.2</td>
<td><math>
+
<td>
x^{3} + x^{17} + x^{20} + x^{34} + x^{66}
+
<span class="htmlMath">x<sup>3</sup> + x<sup>17</sup> + x<sup>20</sup> + x<sup>34</sup> + x<sup>66</sup></span>
</math></td>
+
</td>
 
<td>4040</td>
 
<td>4040</td>
 
<td>210</td>
 
<td>210</td>
Line 285: Line 285:
 
<tr>
 
<tr>
 
<td>11.1</td>
 
<td>11.1</td>
<td><math>
+
<td>
x^{3} + x^{20} + x^{34} + x^{66}
+
<span class="htmlMath">x<sup>3</sup> + x<sup>20</sup> + x<sup>34</sup> + x<sup>66</sup></span>
</math></td>
+
</td>
 
<td>4048</td>
 
<td>4048</td>
 
<td>210</td>
 
<td>210</td>
Line 295: Line 295:
 
<tr>
 
<tr>
 
<td>12.1</td>
 
<td>12.1</td>
<td><math>
+
<td>
x^{3} + x^{12} + x^{40} + x^{72}
+
<span class="htmlMath">x<sup>3</sup> + x<sup>12</sup> + x<sup>40</sup> + x<sup>72</sup></span>
</math></td>
+
</td>
 
<td>4048</td>
 
<td>4048</td>
 
<td>210</td>
 
<td>210</td>
Line 305: Line 305:
 
<tr>
 
<tr>
 
<td>13.1</td>
 
<td>13.1</td>
<td><math>
+
<td>
x^{3} + x^{5} + x^{10} + x^{33} + x^{34}
+
<span class="htmlMath">x<sup>3</sup> + x<sup>5</sup> + x<sup>10</sup> + x<sup>33</sup> + x<sup>34</sup></span>
</math></td>
+
</td>
 
<td>4040</td>
 
<td>4040</td>
 
<td>212</td>
 
<td>212</td>
Line 315: Line 315:
 
<tr>
 
<tr>
 
<td>14.1</td>
 
<td>14.1</td>
<td><math>
+
<td>
x^{3} + x^{6} + x^{34} + x^{40} + x^{72}
+
<span class="htmlMath">x<sup>3</sup> + x<sup>6</sup> + x<sup>34</sup> + x<sup>40</sup> + x<sup>72</sup></span>
</math></td>
+
</td>
 
<td>4048</td>
 
<td>4048</td>
 
<td>212</td>
 
<td>212</td>
Line 325: Line 325:
 
<tr>
 
<tr>
 
<td>14.2</td>
 
<td>14.2</td>
<td><math>
+
<td>
x^{3} + x^{5} + x^{6} + x^{12} + x^{33} + x^{34}
+
<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>
</math></td>
+
</td>
 
<td>4050</td>
 
<td>4050</td>
 
<td>210</td>
 
<td>210</td>
Line 335: Line 335:
 
<tr>
 
<tr>
 
<td>14.3</td>
 
<td>14.3</td>
<td><math>
+
<td>
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}
+
<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>
</math></td>
+
</td>
 
<td>4046</td>
 
<td>4046</td>
 
<td>212</td>
 
<td>212</td>

Revision as of 19:29, 11 July 2020

Known switching classes of APN functions over , , and .

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

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

x3

3610 198 113792 113792
1.2

x3 + Tr(x9)

4026 212 896 896
2.1

x34 + x18 + x5

4034 210 896 896
2.2

x3 + x17 + x33 + x34

4040 212 896 896
3.1

x5

3708 198 113792 113792
4.1

x9

3610 198 113792 14565376
5.1

x13

4270 338 889 889
6.1

x57

4704 436 889 889
7.1

x-1

8128 4928 1778 1778
8.1

x65 + x10 + x3

4038 212 896 896
9.1

x3 + x9 + x18 + x66

4044 212 896 896
10.1

x3 + x12 + x17 + x33

4048 210 896 896
10.2

x3 + x17 + x20 + x34 + x66

4040 210 896 896
11.1

x3 + x20 + x34 + x66

4048 210 896 896
12.1

x3 + x12 + x40 + x72

4048 210 896 896
13.1

x3 + x5 + x10 + x33 + x34

4040 212 896 896
14.1

x3 + x6 + x34 + x40 + x72

4048 212 896 896
14.2

x3 + x5 + x6 + x12 + x33 + x34

4050 210 896 896
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
1.1 11818 420
1.2 12370 420
1.3 13800 432
1.4 13804 434
1.5 13842 436
1.6 13848 438
1.7 14034 438
1.8 14032 438
1.9 14036 438
1.10 14036 438
1.11 14032 438
1.12 14034 438
1.13 14030 438
1.14 14046 454
1.15 14036 454
1.16 14032 438
1.17 14028 438
2.1 13200 414
3.1 14024 438
4.1 14040 454
5.1 14044 446
6.1 14046 438
7.1 15358 960