View source for Known inifinte families of quadratic APN polynomials over GF(2^n)

<table>
<tr>
<th><math>N^\circ</math></th>
<th>Functions</th>
<th>Conditions</th>
<th>References</th>
</tr>

<tr>
<td>C1-C2</td>
<td><math>x^{2^s+1}+u^{2^k-1}x^{2^{ik}+2^{mk+s}}</math></td>
<td><math>n = pk, \gcd(k,3) = \gcd(s,3k) = 1, p \in \{3,4\}, i = sk \mod p, m = p -i, n \ge 12, u \text{ primitive in } \mathbb{F}_{2^n}^*</math></td>
<td><ref>L. Budaghyan, C. Carlet, G. Leander, ''Two Classes of Quadratic APN Binomials Inequivalent to Power Functions'', IEEE Trans. Inform. Theory 54(9), 2008, pp. 4218-4229</ref></td>
</td>
</tr>

<tr>
<td>C3</th>
<td><math>sx^{q+1}+x^{2^i+1}+x^{q(2^i+1)}+cx^{2^iq+1}+c^qx^{2^i+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} \setminus \mathbb{F}_{q}, X^{2^i+1}+cX^{2^i}+c^{q}X+1  \text{ has no solution } x</math> s.t. <math>x^{q+1}=1</math></td>
<td><ref>L. Budaghyan and C. Carlet. ''Classes of Quadratic APN Trinomials and Hexanomials and Related Structures''. {\em IEEE Trans. Inform. Theory}, vol. 54, no. 5, pp. 2354-2357, 2008.</ref></td>
</td>
</tr>

<tr>
<td>C4</td>
<td><math>x^3+a^{-1} \mathrm {Tr}_n (a^3x^9)</math></td>
<td><math>a\neq 0</math></td>
<td><ref>L. Budaghyan, C. Carlet and G.Leander, ''Constructinig new APN functions from known ones, Finite Fields and their applications'', vol.15, issue 2, Apr. 2009, pp. 150-159.</ref></td>
</td>
</tr>

<tr>
<td>C5</td>
<td><math>x^3+a^{-1} \mathrm {Tr}_n^3 (a^3x^9+a^6x^{18})</math></td>
<td><math>3|n </math>, <math>a\ne0</math></td>
<td><ref name="2_ref">L. Budaghyan, C. Carlet and G.Leander, ''On a Construction of quadratic APN functions, Proceedings of IEEE information Theory workshop'' ITW'09, Oct. 2009, 374-378.</ref></td>
</tr>

<tr>
<td>C6</td>
<td><math>x^3+a^{-1} \mathrm{Tr}_n^3(a^6x^{18}+a^{12}x^{36})</math></td>
<td> <math>3|n, a \ne 0</math></td>
<td><ref name="2_ref" />
</tr>


<td>C7-C9</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, \gcd(k,3)=\gcd(s,3k)=1, v, w\in\mathbb{F}_{2^k}, vw \ne 1, 3|(k+s) u \text{ primitive in } \mathbb{F}_{2^n}^* </math></td>
<td><ref>Bracken, C., Byrne, E., Markin, N., & Mcguire, G. (2011). ''A few more quadratic APN functions. Cryptography and Communications'', 3(1), 43-53.</ref></td>
</tr>

<tr>
<td>C10</td>
<td><math>(x+x^{2m})^{2^k+1}+u'(ux+u^{2m} x^{2m})^{(2^k+1)2^i}+u(x+x^{2m})(ux+u^{2m} x^{2m})</math></td>
<td><math>n=2m, m\geqslant 2</math> even, <math>\gcd(k, m)=1,</math> and <math> i \geqslant 2</math> even <math>u\text{ primitive in } \mathbb{F}_{2^n}^*, u' \in \mathbb{F}_{2^m} \text{ not a cube }</math></td>
<td><ref>Göloğlu, Faruk. ''Almost perfect nonlinear trinomials and hexanomials.'' Finite Fields and Their Applications 33 (2015): 258-282.</ref></td>
</tr>

<tr>
<td>C11</td>
<td><math>a^2x^{2^{2m+1}+1}+b^2x^{2^{m+1}+1}+ax^{2^{2m}+2}+bx^{2^{m}+2}+(c^2+c)x^3</math></td>
<td><math>n=3m, m \ \text{odd}\  L(x)=ax^{2^{2m}}+bx^{2m}+cx \ \text{satisfies the conditions in lemma 8 of}\ [3]</math></td>
<td><ref>Villa, I., Budaghyan, L., Calderini, M., Carlet, C., & Coulter, R. ''On Isotopic Construction of APN Functions.'' SETA 2018</ref></td>
</tr>


</table>