View source for APN polynomials over GF(2^n) CCZ-inequivalent to quadratic functions and monomials

= n = 6 =

The polynomial

<div>
<span class="htmlMath">x<sup>3</sup> + α<sup>17</sup>(x<sup>17</sup> + x<sup>18</sup> + x<sup>20</sup> + x<sup>24</sup>) + α<sup>14</sup>( <span class="htmlMathText">Tr</span>( α<sup>52</sup>x<sup>3</sup> + α<sup>6</sup>x<sup>5</sup> + α<sup>19</sup>x<sup>7</sup> + α<sup>28</sup>x<sup>11</sup> + α<sup>2</sup>x<sup>13</sup>) + (α<sup>2</sup>x)<sup>9</sup> + (α<sup>2</sup>x)<sup>18</sup> + (α<sup>2</sup>x)<sup>36</sup> + x<sup>21</sup> + x<sup>42</sup>)</span>
</div>

where is α is primitive in GF(2^6), is the only known example of an APN function CCZ-inequivalent to a monomial or quadratic function <ref>M. Brinkmann, G. Leander. On the classification of APN functions up to dimension five. Designs, Codes and Cryptography 49, pp. 273-288, 2008. https://doi.org/10.1007/s10623-008-9194-6</ref><ref>Y. Edel, A. Pott. A new almost perfect nonlinear function which is not quadratic.  Advances in Mathematics of Communications 3 (1), pp. 59-81, 2009. https://doi.org/10.3934/amc.2009.3.59</ref>. A [[:File:Cubic.txt|Magma implementation of the polynomial]] is available.