View source for Known infinite families of APN power functions over GF(2^n)

The following table provides a summary of all known infinite families of power APN functions of the form <span class="htmlMath">F(x) = x<sup>d</sup></span>.

<table>
<tr>
<th>Family</th>
<th>Exponent</th>
<th>Conditions</th>
<th><span class="htmlMath"><span class="latexCommand">deg</span>(x<sup>d</sup>)</span></th>
<th>Reference</th>
</tr>

<tr>
<td>Gold</td>
<td><span class="htmlMath">2<sup>i</sup> + 1</span></td>
<td><span class="htmlMath"><span class="latexCommand">gcd</span>(i,n) = 1</span></td>
<td><span class="htmlMath">2</span></td>
<td> <ref>Gold R. Maximal recursive sequences with 3-valued recursive cross-correlation functions. IEEE Trans. Inform. Theory, 14, pp. 154-156, 1968. https://doi.org/10.1109/TIT.1968.1054106</ref><ref name="kaisa">Nyberg K. Differentially uniform mappings for cryptography. Advances in Cryptography, EUROCRYPT’93, Lecture Notes in Computer Science 765, pp. 55-64, 1994. Lecture Notes in Computer Science, vol 765. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48285-7_6</ref>
 </td>
 </tr>
 <tr>

 <tr>
 <td>Kasami</td>
 <td><span class="htmlMath">2<sup>2i</sup> - 2<sup>i</sup> + 1</span></td>
 <td><span class="htmlMath"><span class="latexCommand">gcd</span>(i,n) = 1</span></td>
 <td><span class="htmlMath">i + 1</span></td>
 <td><ref>Janwa H., Wilson R.M. Hyperplane sections of Fermat varieties in P 3 in char. 2 and some applications to cyclic codes. Proceedings of AAECC-10, LNCS, vol. 673, Berlin, Springer-Verlag, pp. 180-194, 1993. https://doi.org/10.1007/3-540-56686-4_43</ref><ref>Kasami T. The weight enumerators for several classes of subcodes of the second order binary Reed-Muller codes. Inform. and Control, 18, pp. 369-394, 1971. https://doi.org/10.1016/S0019-9958(71)90473-6</ref></td>
 </tr>

 <tr>
 <td>Welch</td>
 <td><span class="htmlMath">2<sup>t</sup> + 3</span></td>
 <td><span class="htmlMath">n = 2t + 1</span></td>
 <td><span class="htmlMath">3</span></td>
 <td><ref>Dobbertin H. Almost perfect nonlinear power functions over GF (2n): the Welch case. IEEE Trans. Inform. Theory, 45, pp. 1271-1275, 1999. https://doi.org/10.1109/18.761283</ref></td>
 </tr>

 <tr>
 <td rowspan="2">Niho</td>
 <td><span class="htmlMath">2<sup>t</sup> + 2<sup>t/2</sup> - 1, t</span> even</td>
 <td rowspan="2"><span class="htmlMath">n = 2t + 1</span></td>
 <td><span class="htmlMath">(t+2)/2</span></td>
 <td rowspan="2"><ref>Dobbertin H. Almost perfect nonlinear power functions over GF (2n): the Niho case. Inform. and Comput., 151, pp. 57-72, 1999. https://doi.org/10.1006/inco.1998.2764</ref></td>
 </tr>

 <tr>
 <td><span class="htmlMath">2<sup>t</sup> + 2<sup>(3t+1)/2</sup> - 1, t</span> odd</td>
 <td><span class="htmlMath">t + 1</span></td>
 </tr>

 <tr>
 <td>Inverse</td>
 <td><span class="htmlMath">2<sup>2t</sup> - 1</span></td>
 <td><span class="htmlMath">n = 2t + 1</span></td>
 <td><span class="htmlMath">n-1</span></td>
 <td><ref name="kaisa" /><ref>Beth T., Ding C. On almost perfect nonlinear permutations. Advances in Cryptology-EUROCRYPT’93, Lecture Notes in Computer Science, 765, Springer-Verlag, New York, pp. 65-76, 1993. https://doi.org/10.1007/3-540-48285-7_7</ref>
 </tr>

 <tr>
 <td>Dobbertin</td>
 <td><span class="htmlMath">2<sup>4i</sup> + 2<sup>3i</sup> + 2<sup>2i</sup> + 2<sup>i</sup> - 1</span></td>
 <td><span class="htmlMath">n = 5i</span></td>
 <td><span class="htmlMath">i + 3</span></td>
 <td><ref>Dobbertin H. Almost perfect nonlinear power functions over GF (2n): a new case for n divisible by 5. Proceedings of Finite Fields and Applications FQ5, pp. 113-121, 2000. https://doi.org/10.1007/978-3-642-56755-1_11</ref></td>
 </tr>

 </table>