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

Revision as of 12:36, 26 July 2019

CCZ-inequivalent APN Functions over $\mathbb {F} _{2^{n}}$ from the Known APN Classes for $6\leqslant n\leqslant 11$

Dimension	$N^{\circ }$	Functions	Equivalent to	Walsh spectrum	Γ-rank	Δ-rank
$6$	$6.1$	$x^{3}$	$Gold$	Gold	1102	94
	$6.2$	$x^{24}+ax^{17}+a^{8}x^{10}+ax^{9}+x^{3}$	$C3$	Gold	1146	94
	$6.3$	$ax^{3}+x^{17}+a^{4}x^{24}$	$C7-C9$	Gold	1166	96
$7$	$7.1$	$x^{3}$	$Gold$	Gold	3610	198
	$7.2$	$x^{5}$	$Gold$	Gold	3708	198
	$7.3$	$x^{9}$	$Gold$	Gold	3610	198
	$7.4$	$x^{13}$	$Kasami$	Gold	4270	338
	$7.5$	$x^{57}$	$Kasami$	Gold	4704	436
	$7.6$	$x^{63}$	$Inverse$	Inverse	8128	4928
	$7.7$	$x^{3}+Tr_{7}(x^{9})$	$C4$	Gold	4026	212
$8$	$8.1$	$x^{3}$	$Gold$	Gold	11818	420
	$8.2$	$x^{9}$	$Gold$	Gold	12370	420
	$8.3$	$x^{57}$	$Kasami$	Gold	15358	960
	$8.4$	$x^{3}+x^{17}+p^{48}x^{18}+p^{3}x^{33}+px^{34}+x^{48}$	$C3$	Gold	13200	414
	$8.5$	$x^{3}+Tr_{8}(x^{9})$	$C4$	Gold	13200	432
	$8.6$	$x^{3}+a^{-1}Tr_{8}(a^{3}x^{9})$	$C4$	Gold	13842	436
	$8.7$	$a(x+x^{16})(ax+a^{16}x^{16})+a^{17}(ax+a^{16}x^{16})^{12}$	$C10$	Gold	13642	436
$9$	$9.1$	$x^{3}$	$Gold$	Gold	38470	872
	$9.2$	$x^{5}$	$Gold$	Gold	41494	872
	$9.3$	$x^{17}$	$Gold$	Gold	38470	872
	$9.4$	$x^{13}$	$Kasami$	Gold	58676	3086
	$9.5$	$x^{241}$	$Kasami$	Gold	61726	3482
	$9.6$	$x^{19}$	$Welch$	Gold	60894	3956
	$9.7$	$x^{255}$	$Inverse$	Inverse	130816	93024
	$9.8$	$x^{3}+Tr_{9}(x^{9})$	$C4$	Gold	47890	920
	$9.9$	$x^{3}+Tr_{9}^{3}(x^{9}+x^{18})$	$C5$	Gold	48428	930
	$9.10$	$x^{3}+Tr_{9}^{3}(x^{18}+x^{36})$	$C6$	Gold	48460	944
	$9.11$	$x^{3}+a^{246}x^{10}+a^{47}x^{17}+a^{181}x^{66}+a^{428}x^{129}$	$C11$	Gold	48596	944
$10$	$10.1$	$x^{3}$	$Gold$	Gold	125042
	$10.2$	$x^{9}$	$Gold$	Gold	136492
	$10.3$	$x^{57}$	$Kasami$	Gold	186416
	$10.4$	$x^{339}$	$Dobbertin$	Dobbertin	280604
	$10.5$	$x^{6}+x^{33}+p^{31}x^{192}$	$C3$	Gold	151216
	$10.6$	$x^{33}+x^{72}+p^{31}x^{258}$	$C3$	Gold	153896
	$10.7$	$x^{3}+Tr_{10}(x^{9})$	$C4$	Gold	164034
	$10.8$	$x^{3}+a^{-1}Tr_{10}(a^{3}x^{9})$	$C4$	Gold	164098
$11$	$11.1$	$x^{3}$	$Gold$	Gold
	$11.2$	$x^{5}$	$Gold$	Gold
	$11.3$	$x^{9}$	$Gold$	Gold
	$11.4$	$x^{17}$	$Gold$	Gold
	$11.5$	$x^{33}$	$Gold$	Gold
	$11.6$	$x^{13}$	$Kasami$	Gold
	$11.7$	$x^{57}$	$Kasami$	Gold
	$11.8$	$x^{241}$	$Kasami$	Gold
	$11.9$	$x^{993}$	$Kasami$	Gold
	$11.10$	$x^{35}$	$Welch$	Gold
	$11.11$	$x^{287}$	$Niho$	Gold
	$11.12$	$x^{123}$	$Inverse$	Inverse
	$11.13$	$x^{3}+Tr_{11}(x^{9})$	$C4$	Gold

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

Revision as of 12:36, 26 July 2019

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@@ Line 7: / Line 7: @@
 <th>Functions</th>
 <th>Equivalent to</th>
+<th>Walsh spectrum</th>
+<th>Γ-rank</th>
+<th>Δ-rank</th>
 </tr>
@@ Line 14: / Line 17: @@
 <td><math>x^3</math></td>
 <td><math>Gold</math></td>
+<td>Gold</td>
+<td>1102</td>
+<td>94</td>
 </tr>
@@ Line 21: / Line 27: @@
 <td><math>x^{24}+ax^{17}+a^8x^{10}+ax^9+x^3</math></td>
 <td><math>C3</math></td>
+<td>Gold</td>
+<td>1146</td>
+<td>94</td>
 </tr>
@@ Line 27: / Line 36: @@
 <td><math>ax^3+x^{17}+a^4x^{24}</math></td>
 <td><math>C7-C9</math></td>
+<td>Gold</td>
+<td>1166</td>
+<td>96</td>
 </tr>
@@ Line 35: / Line 47: @@
 <td><math>x^3</math></td>
 <td><math>Gold</math></td>
+<td>Gold</td>
+<td>3610</td>
+<td>198</td>
 </tr>
@@ Line 42: / Line 57: @@
 <td><math>x^5</math></td>
 <td><math>Gold</math></td>
+<td>Gold</td>
+<td>3708</td>
+<td>198</td>
 </tr>
@@ Line 49: / Line 67: @@
 <td><math>x^9</math></td>
 <td><math>Gold</math></td>
+<td>Gold</td>
+<td>3610</td>
+<td>198</td>
 </tr>
@@ Line 56: / Line 77: @@
 <td><math>x^{13}</math></td>
 <td><math>Kasami</math></td>
+<td>Gold</td>
+<td>4270</td>
+<td>338</td>
 </tr>
@@ Line 62: / Line 86: @@
 <td><math>x^{57}</math></td>
 <td><math>Kasami</math></td>
+<td>Gold</td>
+<td>4704</td>
+<td>436</td>
 </tr>
@@ Line 69: / Line 96: @@
 <td><math>x^{63}</math></td>
 <td><math>Inverse</math></td>
+<td>Inverse</td>
+<td>8128</td>
+<td>4928</td>
 </tr>
@@ Line 77: / Line 107: @@
 <td><math>x^3+Tr_7(x^9)</math></td>
 <td><math>C4</math></td>
+<td>Gold</td>
+<td>4026</td>
+<td>212</td>
 </tr>
@@ Line 85: / Line 118: @@
 <td><math>x^3</math></td>
 <td><math>Gold</math></td>
+<td>Gold</td>
+<td>11818</td>
+<td>420</td>
 </tr>
@@ Line 91: / Line 127: @@
 <td><math>x^9</math></td>
 <td><math>Gold</math></td>
+<td>Gold</td>
+<td>12370</td>
+<td>420</td>
 </tr>
@@ Line 98: / Line 137: @@
 <td><math>x^{57}</math></td>
 <td><math>Kasami</math></td>
+<td>Gold</td>
+<td>15358</td>
+<td>960</td>
 </tr>
@@ Line 104: / Line 146: @@
 <td><math>x^3+x^{17}+p^{48}x^{18}+p^3x^{33}+px^{34}+x^{48}</math></td>
 <td><math>C3</math></td>
+<td>Gold</td>
+<td>13200</td>
+<td>414</td>
 </tr>
@@ Line 110: / Line 155: @@
 <td><math>x^3+Tr_8(x^9)</math></td>
 <td><math>C4</math></td>
+<td>Gold</td>
+<td>13200</td>
+<td>432</td>
 </tr>
@@ Line 116: / Line 164: @@
 <td><math>x^3+a^{-1}Tr_8(a^3x^9)</math></td>
 <td><math>C4</math></td>
+<td>Gold</td>
+<td>13842</td>
+<td>436</td>
 </tr>
@@ Line 122: / Line 173: @@
 <td><math>a(x+x^{16})(ax+a^{16}x^{16})+a^{17}(ax+a^{16}x^{16})^{12}</math></td>
 <td><math>C10</math></td>
+<td>Gold</td>
+<td>13642</td>
+<td>436</td>
 </tr>
@@ Line 129: / Line 183: @@
 <td><math>x^3</math></td>
 <td><math>Gold</math></td>
+<td>Gold</td>
+<td>38470</td>
+<td>872</td>
 </tr>
@@ Line 136: / Line 193: @@
 <td><math>x^5</math></td>
 <td><math>Gold</math></td>
+<td>Gold</td>
+<td>41494</td>
+<td>872</td>
 </tr>
@@ Line 142: / Line 202: @@
 <td><math>x^{17}</math></td>
 <td><math>Gold</math></td>
+<td>Gold</td>
+<td>38470</td>
+<td>872</td>
 </tr>
@@ Line 148: / Line 211: @@
 <td><math>x^{13}</math></td>
 <td><math>Kasami</math></td>
+<td>Gold</td>
+<td>58676</td>
+<td>3086</td>
 </tr>
@@ Line 154: / Line 220: @@
 <td><math>x^{241}</math></td>
 <td><math>Kasami</math></td>
+<td>Gold</td>
+<td>61726</td>
+<td>3482</td>
 </tr>
@@ Line 160: / Line 229: @@
 <td><math>x^{19}</math></td>
 <td><math>Welch</math></td>
+<td>Gold</td>
+<td>60894</td>
+<td>3956</td>
 </tr>
@@ Line 166: / Line 238: @@
 <td><math>x^{255}</math></td>
 <td><math>Inverse</math></td>
+<td>Inverse</td>
+<td>130816</td>
+<td>93024</td>
 </tr>
@@ Line 172: / Line 247: @@
 <td><math>x^3+Tr_9(x^9)</math></td>
 <td><math>C4</math></td>
+<td>Gold</td>
+<td>47890</td>
+<td>920</td>
 </tr>
@@ Line 178: / Line 256: @@
 <td><math>x^3+Tr^3_9(x^9+x^{18})</math></td>
 <td><math>C5</math></td>
+<td>Gold</td>
+<td>48428</td>
+<td>930</td>
 </tr>
@@ Line 184: / Line 265: @@
 <td><math>x^3+Tr^3_9(x^{18}+x^{36})</math></td>
 <td><math>C6</math></td>
+<td>Gold</td>
+<td>48460</td>
+<td>944</td>
 </tr>
@@ Line 190: / Line 274: @@
 <td><math>x^3+a^{246}x^{10}+a^{47}x^{17}+a^{181}x^{66}+a^{428}x^{129}</math></td>
 <td><math>C11</math></td>
+<td>Gold</td>
+<td>48596</td>
+<td>944</td>
 </tr>
@@ Line 197: / Line 284: @@
 <td><math>x^3</math></td>
 <td><math>Gold</math></td>
+<td>Gold</td>
+<td>125042</td>
+<td></td>
 </tr>
@@ Line 203: / Line 293: @@
 <td><math>x^9</math></td>
 <td><math>Gold</math></td>
+<td>Gold</td>
+<td>136492</td>
+<td></td>
 </tr>
@@ Line 209: / Line 302: @@
 <td><math>x^{57}</math></td>
 <td><math>Kasami</math></td>
+<td>Gold</td>
+<td>186416</td>
+<td></td>
 </tr>
@@ Line 215: / Line 311: @@
 <td><math>x^{339}</math></td>
 <td><math>Dobbertin</math></td>
+<td>Dobbertin</td>
+<td>280604</td>
+<td></td>
 </tr>
@@ Line 221: / Line 320: @@
 <td><math>x^6+x^{33}+p^{31}x^{192}</math></td>
 <td><math>C3</math></td>
+<td>Gold</td>
+<td>151216</td>
+<td></td>
 </tr>
@@ Line 227: / Line 329: @@
 <td><math>x^{33}+x^{72}+p^{31}x^{258}</math></td>
 <td><math>C3</math></td>
+<td>Gold</td>
+<td>153896</td>
+<td></td>
 </tr>
@@ Line 233: / Line 338: @@
 <td><math>x^3+Tr_{10}(x^9)</math></td>
 <td><math>C4</math></td>
+<td>Gold</td>
+<td>164034</td>
+<td></td>
 </tr>
@@ Line 239: / Line 347: @@
 <td><math>x^3+a^{-1}Tr_{10}(a^3x^9)</math></td>
 <td><math>C4</math></td>
+<td>Gold</td>
+<td>164098</td>
+<td></td>
 </tr>
@@ Line 246: / Line 357: @@
 <td><math>x^3</math></td>
 <td><math>Gold</math></td>
+<td>Gold</td>
+<td></td>
+<td></td>
 </tr>
@@ Line 252: / Line 366: @@
 <td><math>x^5</math></td>
 <td><math>Gold</math></td>
+<td>Gold</td>
+<td></td>
+<td></td>
 </tr>
@@ Line 258: / Line 375: @@
 <td><math>x^9</math></td>
 <td><math>Gold</math></td>
+<td>Gold</td>
+<td></td>
+<td></td>
 </tr>
@@ Line 264: / Line 384: @@
 <td><math>x^{17}</math></td>
 <td><math>Gold</math></td>
+<td>Gold</td>
+<td></td>
+<td></td>
 </tr>
@@ Line 270: / Line 393: @@
 <td><math>x^{33}</math></td>
 <td><math>Gold</math></td>
+<td>Gold</td>
+<td></td>
+<td></td>
 </tr>
@@ Line 276: / Line 402: @@
 <td><math>x^{13}</math></td>
 <td><math>Kasami</math></td>
+<td>Gold</td>
+<td></td>
+<td></td>
 </tr>
@@ Line 282: / Line 411: @@
 <td><math>x^{57}</math></td>
 <td><math>Kasami</math></td>
+<td>Gold</td>
+<td></td>
+<td></td>
 </tr>
@@ Line 287: / Line 419: @@
 <td><math>x^{241}</math></td>
 <td><math>Kasami</math></td>
+<td>Gold</td>
+<td></td>
+<td></td>
 </tr>
@@ Line 292: / Line 427: @@
 <td><math>x^{993}</math></td>
 <td><math>Kasami</math></td>
+<td>Gold</td>
+<td></td>
+<td></td>
 </tr>
@@ Line 297: / Line 435: @@
 <td><math>x^{35}</math></td>
 <td><math>Welch</math></td>
+<td>Gold</td>
+<td></td>
+<td></td>
 </tr>
@@ Line 302: / Line 443: @@
 <td><math>x^{287}</math></td>
 <td><math>Niho</math></td>
+<td>Gold</td>
+<td></td>
+<td></td>
 </tr>
@@ Line 307: / Line 451: @@
 <td><math>x^{123}</math></td>
 <td><math>Inverse</math></td>
+<td>Inverse</td>
+<td></td>
+<td></td>
 </tr>
@@ Line 312: / Line 459: @@
 <td><math>x^3+Tr_{11}(x^9)</math></td>
 <td><math>C4</math></td>
+<td>Gold</td>
+<td></td>
+<td></td>
 </tr>
 </table>