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">
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<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>


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


<table class="borderless">
<table class="borderless">
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<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>


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




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<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>


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


<table>
<table>
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<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>


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</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>

Revision as of 11:19, 26 February 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

[math]\displaystyle{ n }[/math] [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] [math]\displaystyle{ Table 7: N^\circ 8.1 }[/math]
[math]\displaystyle{ 7.2 }[/math] [math]\displaystyle{ x^{34} + x^{18} + x^5 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7: N^\circ 2.1 }[/math]
[math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 8.1 }[/math] [math]\displaystyle{ x^{72} + x^6 + x^3 }[/math] [math]\displaystyle{ N^\circ5 }[/math] [math]\displaystyle{ Table 9: N^\circ1.3 }[/math]
[math]\displaystyle{ 8.2 }[/math] [math]\displaystyle{ x^{72} + x^{36} + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 9:N^\circ1.4 }[/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]


Quadrinomials

[math]\displaystyle{ n }[/math] [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] [math]\displaystyle{ Table 7: N^\circ12.1 }[/math]
[math]\displaystyle{ 7.2 }[/math] [math]\displaystyle{ x^{33} + x^{17} + x^{12} + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ2.2 }[/math]
[math]\displaystyle{ 7.3 }[/math] [math]\displaystyle{ x^{34} + x^{33} + x^{10} + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ10.1 }[/math]
[math]\displaystyle{ 7.4 }[/math] [math]\displaystyle{ x^{66} + x^{34} + x^{20} + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ11.1 }[/math]
[math]\displaystyle{ 7.5 }[/math] [math]\displaystyle{ x^{68} + x^{18} + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ8.1 }[/math]
[math]\displaystyle{ 7.6 }[/math] [math]\displaystyle{ x^{66} + x^{18} + x^9 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ9.1 }[/math]
[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

[math]\displaystyle{ n }[/math] [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] [math]\displaystyle{ Table 7: N^\circ13.1 }[/math]
[math]\displaystyle{ 7.2 }[/math] [math]\displaystyle{ x^{65} + x^{20} + x^{18} + x^6 + x^3 }[/math] [math]\displaystyle{ N^\circ5 }[/math] [math]\displaystyle{ 7:N^\circ1.2 }[/math]
[math]\displaystyle{ 7.3 }[/math] [math]\displaystyle{ x^{40} + x^{34} + x^{18} + x^{10} + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ12.1 }[/math]
[math]\displaystyle{ 7.4 }[/math] [math]\displaystyle{ x^{48} + x^{40} + x^{10} + x^9 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ2.1 }[/math]
[math]\displaystyle{ 7.5 }[/math] [math]\displaystyle{ x^{33} + x^9 + x^6 + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ11.1 }[/math]
[math]\displaystyle{ 7.6 }[/math] [math]\displaystyle{ x^{40} + x^{36} + x^{34} + x^{24} + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ10.1 }[/math]
[math]\displaystyle{ 7.7 }[/math] [math]\displaystyle{ x^{24} + x^{10} + x^9 + x^6 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ2.1 }[/math]
[math]\displaystyle{ 7.8 }[/math] [math]\displaystyle{ x^{65} + x^{36} + x^{20} + x^{17} + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ14.1 }[/math]
[math]\displaystyle{ 7.9 }[/math] [math]\displaystyle{ x^{40} + x^{33} + x^{17} + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ8.1 }[/math]
[math]\displaystyle{ 7.10 }[/math] [math]\displaystyle{ x^{36} + x^{33} + x^{18} + x^9 + x^5 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ10.1 }[/math]
[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] [math]\displaystyle{ Table 9: N^\circ1.4 }[/math]
[math]\displaystyle{ 8.2 }[/math] [math]\displaystyle{ x^{72} + x^{66} + x^{12} + x^6 + x^3 }[/math] [math]\displaystyle{ N^\circ5 }[/math] [math]\displaystyle{ 9:N^\circ1.3 }[/math]
[math]\displaystyle{ 8.3 }[/math] [math]\displaystyle{ x^{130} + x^{66} + x^{40} + x^{12} + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 9:N^\circ6.1 }[/math]
[math]\displaystyle{ 8.4 }[/math] [math]\displaystyle{ x^{66} + x^{40} + x^{18} + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 9:N^\circ5.1 }[/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{ 11.1 }[/math] [math]\displaystyle{ x^{12} + x^{10} + x^9 + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 11.2 }[/math] [math]\displaystyle{ x^{258} + x^{257} + x^{18} + x^{17} + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 11.3 }[/math] [math]\displaystyle{ x^{96} + x^{66} + x^{34} + x^{33} + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 11.4 }[/math] [math]\displaystyle{ x^{80} + x^{68} + x^{65} + x^{17} + x^5 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]
[math]\displaystyle{ 11.5 }[/math] [math]\displaystyle{ x^{260} + x^{257} + x^{36} + x^{33} + x^5 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ - }[/math]


Hexanomials

[math]\displaystyle{ n }[/math] [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] [math]\displaystyle{ Table 7: N^\circ14.2 }[/math]
[math]\displaystyle{ 7.2 }[/math] [math]\displaystyle{ x^{40} + x^{24} + x^{20} + x^9 + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ14.1 }[/math]
[math]\displaystyle{ 7.3 }[/math] [math]\displaystyle{ x^{33} + x^{24} + x^{20} + x^{18} + x^{12} + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ12.1 }[/math]
[math]\displaystyle{ 7.4 }[/math] [math]\displaystyle{ x^{24} + x^{17} + x^{12} + x^{10} + x^6 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ2.1 }[/math]
[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] [math]\displaystyle{ 7:N^\circ1.2 }[/math]
[math]\displaystyle{ 7.6 }[/math] [math]\displaystyle{ x^{48} + x^{40} + x^{18} + x^{10} + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ11.1 }[/math]
[math]\displaystyle{ 7.7 }[/math] [math]\displaystyle{ x^{40} + x^{12} + x^{10} + x^9 + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ2.2 }[/math]
[math]\displaystyle{ 7.8 }[/math] [math]\displaystyle{ x^{34} + x^{24} + x^{10} + x^9 + x^6 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ9.1 }[/math]
[math]\displaystyle{ 7.9 }[/math] [math]\displaystyle{ x^{34} + x^{33} + x^{20} + x^{17} + x^{10} + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ13.1 }[/math]
[math]\displaystyle{ 7.10 }[/math] [math]\displaystyle{ x^{36} + x^{33} + x^{24} + x^9 + x^6 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ10.1 }[/math]
[math]\displaystyle{ 7.11 }[/math] [math]\displaystyle{ x^{40} + x^{36} + x^{20} + x^{10} + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ10.2 }[/math]
[math]\displaystyle{ 7.12 }[/math] [math]\displaystyle{ x^{36} + x^{34} + x^{20} + x^{10} + x^9 + x^3 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 7:N^\circ8.1 }[/math]
[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] [math]\displaystyle{ Table 9: N^\circ5.1 }[/math]
[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] [math]\displaystyle{ 9:N^\circ6.1 }[/math]
[math]\displaystyle{ 8.3 }[/math] [math]\displaystyle{ x^{72} + x^{66} + x^{34} + x^{18} + x^{10} + x^5 }[/math] [math]\displaystyle{ - }[/math] [math]\displaystyle{ 9:N^\circ4.1 }[/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]
  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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