Classification of Quadratic APN Trinomials, Quadrinomials, Pentanomials, Hexanomials (CCZ-inequivalent with infinite monomial families) in Small Dimensions with all Coefficients equal to 1: Difference between revisions

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Table 1: Classification of Quadratic APN Trinomials (CCZ-inequivalent to infinite monomial
The following tables list CCZ-inequivalent representatives found by systematically searching for APN functions among all trinomials, quadrinomials, pentanomials and hexanomials with coefficients in <math>\mathbb{F}_{2}</math> over <math>\mathbb{F}_{2^n}</math> with <math>6 \le n \le 11</math> <ref>Sun B. On Classification and Some Properties of APN Functions.</ref>. The tables also list which equivalence class from <ref name="edelPott">Edel Y, Pott A. A new almost perfect nonlinear function which is not quadratic. Adv. in Math. of Comm.. 2009 Mar;3(1):59-81.</ref> the functions belong to. Only polynomials inequivalent to power functions are considered. If the polynomial is equivalent to a family from the [[Known infinite families of quadratic APN polynomials over GF(2^n)|table of infinite families]], this is also listed.
families) in Small Dimensions with Coefficients in <math>\mathbb{F}_2</math>
 
== Trinomials ==


<table class="borderless">
<table class="borderless">
<tr>
<tr>
<th><math>n</math></th>
<th>Dimension</th>
<th><math>N^\circ</math></th>
<th><math>N^\circ</math></th>
<th>Functions</th>
<th>Functions</th>
<th>Families from tables 5</th>
<th>Familiy</th>
<th>Relation to [6]</th>
<th>Relation to <ref name="edelPott"></ref></th>
</tr>
</tr>


Line 24: Line 25:
<td><math>x^{20} + x^6 + x^3</math></td>
<td><math>x^{20} + x^6 + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>Table 7: N^\circ 8.1</math></td>
<td>Table 7: № 8.1</td>
</tr>
</tr>


Line 31: Line 32:
<td><math>x^{34} + x^{18} + x^5</math></td>
<td><math>x^{34} + x^{18} + x^5</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>7: N^\circ 2.1</math></td>
<td>Table 7: № 2.1</td>
</tr>
</tr>


Line 40: Line 41:
<td><math>x^{72} + x^6 + x^3</math></td>
<td><math>x^{72} + x^6 + x^3</math></td>
<td><math>N^\circ5</math></td>
<td><math>N^\circ5</math></td>
<td><math>Table 9: N^\circ1.3</math></td>
<td>Table 9: № 1.3</td>
</tr>
</tr>


Line 47: Line 48:
<td><math>x^{72} + x^{36} + x^3</math></td>
<td><math>x^{72} + x^{36} + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>9:N^\circ1.4</math></td>
<td>Table 9: № 1.4</td>
</tr>
</tr>


Line 76: Line 77:
</table>
</table>


 
== Quadrinomials ==
Table 2: Classification of Quadratic APN Quadrinomials (CCZ-inequivalent with infinite monomial
families) in Small Dimensions with Coefficients as 1


<table class="borderless">
<table class="borderless">
<tr>
<tr>
<th><math>n</math></th>
<th>Dimension</th>
<th><math>N^\circ</math></th>
<th><math>N^\circ</math></th>
<th>Functions</th>
<th>Functions</th>
<th>Families from tables 5</th>
<th>Families</th>
<th>Relation to [6]</th>
<th>Relation to <ref name="edelPott"></ref></th>
</tr>
</tr>


Line 102: Line 101:
<td><math>x^{72} + x^{40} + x^{12} + x^3</math></td>
<td><math>x^{72} + x^{40} + x^{12} + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>Table 7: N^\circ12.1</math></td>
<td>Table 7: № 12.1</td>
</tr>
</tr>


Line 108: Line 107:
<td><math>x^{33} + x^{17} + x^{12} + x^3</math></td>
<td><math>x^{33} + x^{17} + x^{12} + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>7:N^\circ2.2</math></td>
<td>Table 7: № 2.2</td>
</tr>
</tr>


Line 114: Line 113:
<td><math>x^{34} + x^{33} + x^{10} + x^3</math></td>
<td><math>x^{34} + x^{33} + x^{10} + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>7:N^\circ10.1</math></td>
<td>Table 7: № 10.1</td>
</tr>
</tr>


Line 120: Line 119:
<td><math>x^{66} + x^{34} + x^{20} + x^3</math></td>
<td><math>x^{66} + x^{34} + x^{20} + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>7:N^\circ11.1</math></td>
<td>Table 7: № 11.1</td>
</tr>
</tr>


Line 126: Line 125:
<td><math>x^{68} + x^{18} + x^5 + x^3</math></td>
<td><math>x^{68} + x^{18} + x^5 + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>7:N^\circ8.1</math></td>
<td>Table 7: № 8.1</td>
</tr>
</tr>


Line 132: Line 131:
<td><math>x^{66} + x^{18} + x^9 + x^3</math></td>
<td><math>x^{66} + x^{18} + x^9 + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>7:N^\circ9.1</math></td>
<td>Table 7: № 9.1</td>
</tr>
</tr>


Line 169: Line 168:
</table>
</table>


 
== Pentanomials ==
Table 3: Classification of Quadratic APN Quadrinomials (CCZ-inequivalent with infinite monomial
families) in Small Dimensions with Coefficients as 1




<table class="borderless">
<table class="borderless">
<tr>
<tr>
<th><math>n</math></th>
<th>Dimension</th>
<th><math>N^\circ</math></th>
<th><math>N^\circ</math></th>
<th>Functions</th>
<th>Functions</th>
<th>Families from tables 5</th>
<th>Families</th>
<th>Relation to [6]</th>
<th>Relation to <ref name="edelPott"></ref></th>
</tr>
</tr>


Line 196: Line 193:
<td><math>x^{68} + x^{40} + x^{24} + x^6 + x^3</math></td>
<td><math>x^{68} + x^{40} + x^{24} + x^6 + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>Table 7: N^\circ13.1</math></td>
<td>Table 7: № 13.1</td>
</tr>
</tr>


Line 202: Line 199:
<td><math>x^{65} + x^{20} + x^{18} + x^6 + x^3</math></td>
<td><math>x^{65} + x^{20} + x^{18} + x^6 + x^3</math></td>
<td><math>N^\circ5</math></td>
<td><math>N^\circ5</math></td>
<td><math>7:N^\circ1.2</math></td>
<td>Table 7: № 1.2</td>
</tr>
</tr>


Line 208: Line 205:
<td><math>x^{40} + x^{34} + x^{18} + x^{10} + x^3</math></td>
<td><math>x^{40} + x^{34} + x^{18} + x^{10} + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>7:N^\circ12.1</math></td>
<td>Table 7: № 12.1</td>
</tr>
</tr>


Line 214: Line 211:
<td><math>x^{48} + x^{40} + x^{10} + x^9 + x^3</math></td>
<td><math>x^{48} + x^{40} + x^{10} + x^9 + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>7:N^\circ2.1</math></td>
<td>Table 7: № 1.2</td>
</tr>
</tr>


Line 220: Line 217:
<td><math>x^{33} + x^9 + x^6 + x^5 + x^3</math></td>
<td><math>x^{33} + x^9 + x^6 + x^5 + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>7:N^\circ11.1</math></td>
<td>Table 7: № 11.1</td>
</tr>
</tr>


Line 226: Line 223:
<td><math>x^{40} + x^{36} + x^{34} + x^{24} + x^3</math></td>
<td><math>x^{40} + x^{36} + x^{34} + x^{24} + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>7:N^\circ10.1</math></td>
<td>Table 7: № 10.1</td>
</tr>
</tr>


Line 232: Line 229:
<td><math>x^{24} + x^{10} + x^9 + x^6 + x^3</math></td>
<td><math>x^{24} + x^{10} + x^9 + x^6 + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>7:N^\circ2.1</math></td>
<td>Table 7: № 2.1</td>
</tr>
</tr>


Line 238: Line 235:
<td><math>x^{65} + x^{36} + x^{20} + x^{17} + x^3</math></td>
<td><math>x^{65} + x^{36} + x^{20} + x^{17} + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>7:N^\circ14.1</math></td>
<td>Table 7: № 14.1</td>
</tr>
</tr>


Line 244: Line 241:
<td><math>x^{40} + x^{33} + x^{17} + x^5 + x^3</math></td>
<td><math>x^{40} + x^{33} + x^{17} + x^5 + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>7:N^\circ8.1</math></td>
<td>Table 7: № 8.1</td>
</tr>
</tr>


Line 250: Line 247:
<td><math>x^{36} + x^{33} + x^{18} + x^9 + x^5</math></td>
<td><math>x^{36} + x^{33} + x^{18} + x^9 + x^5</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>7:N^\circ10.1</math></td>
<td>Table 7: № 10.1</td>
</tr>
</tr>


Line 258: Line 255:
<td><math>x^{36} + x^{33} + x^9 + x^6 + x^3</math></td>
<td><math>x^{36} + x^{33} + x^9 + x^6 + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>Table 9: N^\circ1.4</math></td>
<td>Table 9: № 1.4</td>
</tr>
</tr>


Line 264: Line 261:
<td><math>x^{72} + x^{66} + x^{12} + x^6 + x^3</math></td>
<td><math>x^{72} + x^{66} + x^{12} + x^6 + x^3</math></td>
<td><math>N^\circ5</math></td>
<td><math>N^\circ5</math></td>
<td><math>9:N^\circ1.3</math></td>
<td>Table 9: № 1.3</td>
</tr>
</tr>


Line 270: Line 267:
<td><math>x^{130} + x^{66} + x^{40} + x^{12} + x^3</math></td>
<td><math>x^{130} + x^{66} + x^{40} + x^{12} + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>9:N^\circ6.1</math></td>
<td>Table 9: № 6.1</td>
</tr>
</tr>


Line 276: Line 273:
<td><math>x^{66} + x^{40} + x^{18} + x^5 + x^3</math></td>
<td><math>x^{66} + x^{40} + x^{18} + x^5 + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>9:N^\circ5.1</math></td>
<td>Table 9: № 5.1</td>
</tr>
</tr>


Line 296: Line 293:


<tr class="divider">
<tr class="divider">
<td rowspan="5"><math>11</math></td>
<td><math>11</math></td>
<td class="noborderbelow"><math>11.1</math></td>
<td><math>x^{12} + x^{10} + x^9 + x^5 + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
</tr>
 
<td><math>11.2</math></td>
<td><math>x^{258} + x^{257} + x^{18} + x^{17} + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
</tr>
 
<td><math>11.3</math></td>
<td><math>x^{96} + x^{66} + x^{34} + x^{33} + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
</tr>
<td><math>11.4</math></td>
<td><math>x^{80} + x^{68} + x^{65} + x^{17} + x^5</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
</tr>
<td><math>11.5</math></td>
<td><math>x^{260} + x^{257} + x^{36} + x^{33} + x^5</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
Line 329: Line 302:
</table>
</table>


 
== Hexanomials ==
Table 4: Classification of Quadratic APN Hexanomial (CCZ-inequivalent with infinite monomial
families) in Small Dimensions with Coefficients as 1


<table>
<table>
<table class="borderless">
<table class="borderless">
<tr>
<tr>
<th><math>n</math></th>
<th>Dimension</th>
<th><math>N^\circ</math></th>
<th><math>N^\circ</math></th>
<th>Functions</th>
<th>Functions</th>
<th>Families from tables 5</th>
<th>Families</th>
<th>Relation to [6]</th>
<th>Relation to <ref name="edelPott"></ref></th>
</tr>
</tr>


Line 356: Line 327:
<td><math>x^{34} + x^{33} + x^{12} + x^6 + x^5 + x^3</math></td>
<td><math>x^{34} + x^{33} + x^{12} + x^6 + x^5 + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>Table 7: N^\circ14.2</math></td>
<td>Table 7: № 14.2</td>
</tr>
</tr>


Line 362: Line 333:
<td><math>x^{40} + x^{24} + x^{20} + x^9 + x^5 + x^3</math></td>
<td><math>x^{40} + x^{24} + x^{20} + x^9 + x^5 + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>7:N^\circ14.1</math></td>
<td>Table 7: № 14.1</td>
</tr>
</tr>


Line 368: Line 339:
<td><math>x^{33} + x^{24} + x^{20} + x^{18} + x^{12} + x^3</math></td>
<td><math>x^{33} + x^{24} + x^{20} + x^{18} + x^{12} + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>7:N^\circ12.1</math></td>
<td>Table 7: № 12.1</td>
</tr>
</tr>


Line 374: Line 345:
<td><math>x^{24} + x^{17} + x^{12} + x^{10} + x^6 + x^3</math></td>
<td><math>x^{24} + x^{17} + x^{12} + x^{10} + x^6 + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>7:N^\circ2.1</math></td>
<td>Table 7: № 2.1</td>
</tr>
</tr>


Line 380: Line 351:
<td><math>x^{40} + x^{34} + x^{18} + x^{17} + x^5 + x^3</math></td>
<td><math>x^{40} + x^{34} + x^{18} + x^{17} + x^5 + x^3</math></td>
<td><math>N^\circ5</math></td>
<td><math>N^\circ5</math></td>
<td><math>7:N^\circ1.2</math></td>
<td>Table 7: № 1.2</td>
</tr>
</tr>


Line 386: Line 357:
<td><math>x^{48} + x^{40} + x^{18} + x^{10} + x^5 + x^3</math></td>
<td><math>x^{48} + x^{40} + x^{18} + x^{10} + x^5 + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>7:N^\circ11.1</math></td>
<td>Table 7: № 11.1</td>
</tr>
</tr>


Line 392: Line 363:
<td><math>x^{40} + x^{12} + x^{10} + x^9 + x^5 + x^3</math></td>
<td><math>x^{40} + x^{12} + x^{10} + x^9 + x^5 + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>7:N^\circ2.2</math></td>
<td>Table 7: № 2.2</td>
</tr>
</tr>


Line 398: Line 369:
<td><math>x^{34} + x^{24} + x^{10} + x^9 + x^6 + x^3</math></td>
<td><math>x^{34} + x^{24} + x^{10} + x^9 + x^6 + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>7:N^\circ9.1</math></td>
<td>Table 7: № 9.1</td>
</tr>
</tr>


Line 404: Line 375:
<td><math>x^{34} + x^{33} + x^{20} + x^{17} + x^{10} + x^3</math></td>
<td><math>x^{34} + x^{33} + x^{20} + x^{17} + x^{10} + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>7:N^\circ13.1</math></td>
<td>Table 7: № 13.1</td>
</tr>
</tr>


Line 410: Line 381:
<td><math>x^{36} + x^{33} + x^{24} + x^9 + x^6 + x^3</math>
<td><math>x^{36} + x^{33} + x^{24} + x^9 + x^6 + x^3</math>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>7:N^\circ10.1</math></td>
<td>Table 7: № 10.1</td>
</tr>
</tr>


Line 416: Line 387:
<td><math>x^{40} + x^{36} + x^{20} + x^{10} + x^5 + x^3</math></td>
<td><math>x^{40} + x^{36} + x^{20} + x^{10} + x^5 + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>7:N^\circ10.2</math></td>
<td>Table 7: № 10.2</td>
</tr>
</tr>


Line 422: Line 393:
<td><math>x^{36} + x^{34} + x^{20} + x^{10} + x^9 + x^3</math></td>
<td><math>x^{36} + x^{34} + x^{20} + x^{10} + x^9 + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>7:N^\circ8.1</math></td>
<td>Table 7: № 8.1</td>
</tr>
</tr>


Line 430: Line 401:
<td><math>x^{68} + x^{34} + x^{17} + x^{12} + x^9 + x^3</math></td>
<td><math>x^{68} + x^{34} + x^{17} + x^{12} + x^9 + x^3</math></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>Table 9: N^\circ5.1</math></td>
<td>Table 9: № 5.1</td>
</tr>
</tr>


Line 436: Line 407:
<td><math>x^{72} + x^{40} + x^{34} + x^{20} + x^{12} + x^3</math></td>
<td><math>x^{72} + x^{40} + x^{34} + x^{20} + x^{12} + x^3</math></td>
<td><math>N^\circ5</math></td>
<td><math>N^\circ5</math></td>
<td><math>9:N^\circ6.1</math></td>
<td>Table 9: № 6.1</td>
</tr>
</tr>


Line 442: Line 413:
<td><math>x^{72} + x^{66} + x^{34} + x^{18} + x^{10} + x^5</math></td></td>
<td><math>x^{72} + x^{66} + x^{34} + x^{18} + x^{10} + x^5</math></td></td>
<td><math>-</math></td>
<td><math>-</math></td>
<td><math>9:N^\circ4.1</math></td>
<td>Table 9: № 4.1</td>
</tr>
</tr>


Line 470: Line 441:


</table>
</table>
Table 5: Known classes of quadratic APN polynomials CCZ-inequivalent to APN monomials on <math>\mathbb{F}_{2^n}</math> <math>u</math> is primitive in <math>\mathbb{F}_{2^n}^*</math>}
<table>
<tr>
<th><math>N^\circ</math></th>
<th>Fanctions</th>
<th>Conditions</th>
<th>Reference</th>
</tr>
<tr>
<td><math>1-2</math></td>
<td><math>x^{2^s+1}+u^{2^k-1}x^{2^{ik}+2^{mk+s}}</math></td>
<td><math>n=pk,  p\in \{3,4\},  \gcd(k,3)=\gcd(s,3k)=1,  i=sk  \bmod p,  m=p-i,  n\geqslant12</math></td>
<td><ref>Budaghyan L, Carlet C, Leander G. Two classes of quadratic APN binomials inequivalent to power functions. IEEE Transactions on Information Theory. 2008 Sep;54(9):4218-29.</ref></td>
</tr>
<tr>
<td><math>3</math></td>
<td><math>x^{2^{2i}+2^i}+bx^{q+1}+cx^{q(2^{2i}+2^i)}</math></td>
<td><math>q=2^{m}, n=2m, cb^{q}+b\neq 0, \gcd(i,m)=1,  \gcd(2^{i}+1,q+1)\neq 1, c \not\in \{\lambda^{(2^{i}+1)(q-1)}, \lambda\in \mathbb{F}_{2^{n}}\},  c^{q+1}=1</math></td> 
<td><ref>Budaghyan L, Carlet C. Classes of quadratic APN trinomials and hexanomials and related structures. IEEE Transactions on Information Theory. 2008 May;54(5):2354-7.</ref></td>
</tr>
<tr>
<td><math>4</math></td>
<td><math>x(x^{2^i}+x^q+cx^{2^iq})+x^{2^i}(c^qx^q+sx^{2^iq})+x^{(2^i+1)q}</math></td>
<td><math>q=2^{m},n=2m, \gcd(i,m)=1, c\in\mathbb{F}_{2^n},  s\in\mathbb{F}_{2^n}\backslash \mathbb{F}_q, X^{2^i+1}+cX^{2^i}+c^qX+1</math> is irreducible over <math>\mathbb{F}_{2^n}</math></td> 
<td><ref>Budaghyan L, Carlet C. Classes of quadratic APN trinomials and hexanomials and related structures. IEEE Transactions on Information Theory. 2008 May;54(5):2354-7.</ref></td>
</tr>
<tr>
<td><math>5</math></td>
<td><math>x^3+a^{-1}tr_n(a^3x^9)</math></td>
<td><math>a\ne0</math></td> 
<td><ref>Budaghyan L, Carlet C, Leander G. Constructing new APN functions from known ones. Finite Fields and Their Applications. 2009 Apr 1;15(2):150-9.</ref><ref name="workshope">Budaghyan L, Carlet C, Leander G. On a construction of quadratic APN functions. In2009 IEEE Information Theory Workshop 2009 Oct 11 (pp. 374-378). IEEE.</ref></td>
</tr>
<tr>
<td><math>6</math></td>
<td><math>x^3+a^{-1}tr_n^3(a^3x^9+a^6x^{18})</math></td>
<td><math>3|n, a\ne0</math></td> 
<td><ref name="workshope"/></td>
</tr>
<tr>
<td><math>7</math></td>
<td><math>x^3+a^{-1}tr_n^3(a^6x^{18}+a^{12}x^{36})</math></td>
<td><math>3|n, a\ne0</math></td> 
<td><ref name="workshope"/></td>
</tr>
<tr>
<td><math>8-10</math></td>
<td><math>ux^{2^s+1}+u^{2^k} x^{2^{-k}+2^{k+s}}+vx^{2^{-k}+1}+wu^{2^k+1}x^{2^{s}+2^{k+s}}</math></td>
<td><math>n=3k, 3|(k+s), \gcd(k,3)=\gcd(s,3k)=1,v, w\in\mathbb{F}_{2^k}, vw \ne 1</math></td> 
<td><ref>Bracken C, Byrne E, Markin N, Mcguire G. New families of quadratic almost perfect nonlinear trinomials and multinomials. Finite Fields and Their Applications. 2008 Jul 1;14(3):703-14.</ref></td>
</tr>
<tr>
<td><math>11</math></td>
<td><math>(x+x^{2^m})^{2^k+1}+u^{(2^n-1)/(2^m-1)} (ux+u^{2^m}x^{2^m})^{(2^k+1)2^i}+u(x+x^{2^m})(ux+u^{2^m}x^{2^m})</math></td>
<td><math>m\geqslant 2, 2|m,n=2m,\gcd(k, m)=1, i</math> is even</td> 
<td><ref>Zhou Y, Pott A. A new family of semifields with 2 parameters. Advances in Mathematics. 2013 Feb 15;234:43-60.</ref></td>
</tr>

Latest revision as of 13:57, 19 August 2019

The following tables list CCZ-inequivalent representatives found by systematically searching for APN functions among all trinomials, quadrinomials, pentanomials and hexanomials with coefficients in [math]\displaystyle{ \mathbb{F}_{2} }[/math] over [math]\displaystyle{ \mathbb{F}_{2^n} }[/math] with [math]\displaystyle{ 6 \le n \le 11 }[/math] [1]. The tables also list which equivalence class from [2] the functions belong to. Only polynomials inequivalent to power functions are considered. If the polynomial is equivalent to a family from the table of infinite families, this is also listed.

Trinomials

Dimension [math]\displaystyle{ N^\circ }[/math] Functions Familiy Relation to [2]
[math]\displaystyle{ 6 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 7 }[/math] [math]\displaystyle{ 7.1 }[/math] [math]\displaystyle{ x^{20} + x^6 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 8.1
[math]\displaystyle{ 7.2 }[/math] [math]\displaystyle{ x^{34} + x^{18} + x^5 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 2.1
[math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 8.1 }[/math] [math]\displaystyle{ x^{72} + x^6 + x^3 }[/math] [math]\displaystyle{ N^\circ5 }[/math] Table 9: № 1.3
[math]\displaystyle{ 8.2 }[/math] [math]\displaystyle{ x^{72} + x^{36} + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 9: № 1.4
[math]\displaystyle{ 9 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 10 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 11 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]

Quadrinomials

Dimension [math]\displaystyle{ N^\circ }[/math] Functions Families Relation to [2]
[math]\displaystyle{ 6 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 7 }[/math] [math]\displaystyle{ 7.1 }[/math] [math]\displaystyle{ x^{72} + x^{40} + x^{12} + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 12.1
[math]\displaystyle{ 7.2 }[/math] [math]\displaystyle{ x^{33} + x^{17} + x^{12} + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 2.2
[math]\displaystyle{ 7.3 }[/math] [math]\displaystyle{ x^{34} + x^{33} + x^{10} + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 10.1
[math]\displaystyle{ 7.4 }[/math] [math]\displaystyle{ x^{66} + x^{34} + x^{20} + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 11.1
[math]\displaystyle{ 7.5 }[/math] [math]\displaystyle{ x^{68} + x^{18} + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 8.1
[math]\displaystyle{ 7.6 }[/math] [math]\displaystyle{ x^{66} + x^{18} + x^9 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 9.1
[math]\displaystyle{ 8 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 9 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 10 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 11 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]

Pentanomials

Dimension [math]\displaystyle{ N^\circ }[/math] Functions Families Relation to [2]
[math]\displaystyle{ 6 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 7 }[/math] [math]\displaystyle{ 7.1 }[/math] [math]\displaystyle{ x^{68} + x^{40} + x^{24} + x^6 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 13.1
[math]\displaystyle{ 7.2 }[/math] [math]\displaystyle{ x^{65} + x^{20} + x^{18} + x^6 + x^3 }[/math] [math]\displaystyle{ N^\circ5 }[/math] Table 7: № 1.2
[math]\displaystyle{ 7.3 }[/math] [math]\displaystyle{ x^{40} + x^{34} + x^{18} + x^{10} + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 12.1
[math]\displaystyle{ 7.4 }[/math] [math]\displaystyle{ x^{48} + x^{40} + x^{10} + x^9 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 1.2
[math]\displaystyle{ 7.5 }[/math] [math]\displaystyle{ x^{33} + x^9 + x^6 + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 11.1
[math]\displaystyle{ 7.6 }[/math] [math]\displaystyle{ x^{40} + x^{36} + x^{34} + x^{24} + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 10.1
[math]\displaystyle{ 7.7 }[/math] [math]\displaystyle{ x^{24} + x^{10} + x^9 + x^6 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 2.1
[math]\displaystyle{ 7.8 }[/math] [math]\displaystyle{ x^{65} + x^{36} + x^{20} + x^{17} + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 14.1
[math]\displaystyle{ 7.9 }[/math] [math]\displaystyle{ x^{40} + x^{33} + x^{17} + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 8.1
[math]\displaystyle{ 7.10 }[/math] [math]\displaystyle{ x^{36} + x^{33} + x^{18} + x^9 + x^5 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 10.1
[math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 8.1 }[/math] [math]\displaystyle{ x^{36} + x^{33} + x^9 + x^6 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 9: № 1.4
[math]\displaystyle{ 8.2 }[/math] [math]\displaystyle{ x^{72} + x^{66} + x^{12} + x^6 + x^3 }[/math] [math]\displaystyle{ N^\circ5 }[/math] Table 9: № 1.3
[math]\displaystyle{ 8.3 }[/math] [math]\displaystyle{ x^{130} + x^{66} + x^{40} + x^{12} + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 9: № 6.1
[math]\displaystyle{ 8.4 }[/math] [math]\displaystyle{ x^{66} + x^{40} + x^{18} + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 9: № 5.1
[math]\displaystyle{ 9 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 10 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 11 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]

Hexanomials

Dimension [math]\displaystyle{ N^\circ }[/math] Functions Families Relation to [2]
[math]\displaystyle{ 6 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 7 }[/math] [math]\displaystyle{ 7.1 }[/math] [math]\displaystyle{ x^{34} + x^{33} + x^{12} + x^6 + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 14.2
[math]\displaystyle{ 7.2 }[/math] [math]\displaystyle{ x^{40} + x^{24} + x^{20} + x^9 + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 14.1
[math]\displaystyle{ 7.3 }[/math] [math]\displaystyle{ x^{33} + x^{24} + x^{20} + x^{18} + x^{12} + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 12.1
[math]\displaystyle{ 7.4 }[/math] [math]\displaystyle{ x^{24} + x^{17} + x^{12} + x^{10} + x^6 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 2.1
[math]\displaystyle{ 7.5 }[/math] [math]\displaystyle{ x^{40} + x^{34} + x^{18} + x^{17} + x^5 + x^3 }[/math] [math]\displaystyle{ N^\circ5 }[/math] Table 7: № 1.2
[math]\displaystyle{ 7.6 }[/math] [math]\displaystyle{ x^{48} + x^{40} + x^{18} + x^{10} + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 11.1
[math]\displaystyle{ 7.7 }[/math] [math]\displaystyle{ x^{40} + x^{12} + x^{10} + x^9 + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 2.2
[math]\displaystyle{ 7.8 }[/math] [math]\displaystyle{ x^{34} + x^{24} + x^{10} + x^9 + x^6 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 9.1
[math]\displaystyle{ 7.9 }[/math] [math]\displaystyle{ x^{34} + x^{33} + x^{20} + x^{17} + x^{10} + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 13.1
[math]\displaystyle{ 7.10 }[/math] [math]\displaystyle{ x^{36} + x^{33} + x^{24} + x^9 + x^6 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 10.1
[math]\displaystyle{ 7.11 }[/math] [math]\displaystyle{ x^{40} + x^{36} + x^{20} + x^{10} + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 10.2
[math]\displaystyle{ 7.12 }[/math] [math]\displaystyle{ x^{36} + x^{34} + x^{20} + x^{10} + x^9 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 8.1
[math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 8.1 }[/math] [math]\displaystyle{ x^{68} + x^{34} + x^{17} + x^{12} + x^9 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 9: № 5.1
[math]\displaystyle{ 8.2 }[/math] [math]\displaystyle{ x^{72} + x^{40} + x^{34} + x^{20} + x^{12} + x^3 }[/math] [math]\displaystyle{ N^\circ5 }[/math] Table 9: № 6.1
[math]\displaystyle{ 8.3 }[/math] [math]\displaystyle{ x^{72} + x^{66} + x^{34} + x^{18} + x^{10} + x^5 }[/math] [math]\displaystyle{ - }[/math] Table 9: № 4.1
[math]\displaystyle{ 9 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 10 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 11 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
  1. ↑ Sun B. On Classification and Some Properties of APN Functions.
  2. ↑ 2.0 2.1 2.2 2.3 2.4 Edel Y, Pott A. A new almost perfect nonlinear function which is not quadratic. Adv. in Math. of Comm.. 2009 Mar;3(1):59-81.
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