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

The following tables list CCZ-inequivalent representatives found by systematically searching for APN functions among all trinomials, quadrinomials, pentanomials and hexanomials with coefficients in ${\displaystyle \mathbb {F} _{2}}$ over ${\displaystyle \mathbb {F} _{2^{n}}}$ with ${\displaystyle 6\leq n\leq 11}$ [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 ${\displaystyle N^{\circ }}$ Functions Familiy Relation to [2]
${\displaystyle 6}$ ${\displaystyle -}$ ${\displaystyle -}$ ${\displaystyle -}$ ${\displaystyle -}$
${\displaystyle 7}$ ${\displaystyle 7.1}$ ${\displaystyle x^{20}+x^{6}+x^{3}}$ ${\displaystyle -}$ Table 7: № 8.1
${\displaystyle 7.2}$ ${\displaystyle x^{34}+x^{18}+x^{5}}$ ${\displaystyle -}$ Table 7: № 2.1
${\displaystyle 8}$ ${\displaystyle 8.1}$ ${\displaystyle x^{72}+x^{6}+x^{3}}$ ${\displaystyle N^{\circ }5}$ Table 9: № 1.3
${\displaystyle 8.2}$ ${\displaystyle x^{72}+x^{36}+x^{3}}$ ${\displaystyle -}$ Table 9: № 1.4
${\displaystyle 9}$ ${\displaystyle -}$ ${\displaystyle -}$ ${\displaystyle -}$ ${\displaystyle -}$
${\displaystyle 10}$ ${\displaystyle -}$ ${\displaystyle -}$ ${\displaystyle -}$ ${\displaystyle -}$
${\displaystyle 11}$ ${\displaystyle -}$ ${\displaystyle -}$ ${\displaystyle -}$ ${\displaystyle -}$

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

## Pentanomials

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

## Hexanomials

Dimension ${\displaystyle N^{\circ }}$ Functions Families Relation to [2]
${\displaystyle 6}$ ${\displaystyle -}$ ${\displaystyle -}$ ${\displaystyle -}$ ${\displaystyle -}$
${\displaystyle 7}$ ${\displaystyle 7.1}$ ${\displaystyle x^{34}+x^{33}+x^{12}+x^{6}+x^{5}+x^{3}}$ ${\displaystyle -}$ Table 7: № 14.2
${\displaystyle 7.2}$ ${\displaystyle x^{40}+x^{24}+x^{20}+x^{9}+x^{5}+x^{3}}$ ${\displaystyle -}$ Table 7: № 14.1
${\displaystyle 7.3}$ ${\displaystyle x^{33}+x^{24}+x^{20}+x^{18}+x^{12}+x^{3}}$ ${\displaystyle -}$ Table 7: № 12.1
${\displaystyle 7.4}$ ${\displaystyle x^{24}+x^{17}+x^{12}+x^{10}+x^{6}+x^{3}}$ ${\displaystyle -}$ Table 7: № 2.1
${\displaystyle 7.5}$ ${\displaystyle x^{40}+x^{34}+x^{18}+x^{17}+x^{5}+x^{3}}$ ${\displaystyle N^{\circ }5}$ Table 7: № 1.2
${\displaystyle 7.6}$ ${\displaystyle x^{48}+x^{40}+x^{18}+x^{10}+x^{5}+x^{3}}$ ${\displaystyle -}$ Table 7: № 11.1
${\displaystyle 7.7}$ ${\displaystyle x^{40}+x^{12}+x^{10}+x^{9}+x^{5}+x^{3}}$ ${\displaystyle -}$ Table 7: № 2.2
${\displaystyle 7.8}$ ${\displaystyle x^{34}+x^{24}+x^{10}+x^{9}+x^{6}+x^{3}}$ ${\displaystyle -}$ Table 7: № 9.1
${\displaystyle 7.9}$ ${\displaystyle x^{34}+x^{33}+x^{20}+x^{17}+x^{10}+x^{3}}$ ${\displaystyle -}$ Table 7: № 13.1
${\displaystyle 7.10}$ ${\displaystyle x^{36}+x^{33}+x^{24}+x^{9}+x^{6}+x^{3}}$ ${\displaystyle -}$ Table 7: № 10.1
${\displaystyle 7.11}$ ${\displaystyle x^{40}+x^{36}+x^{20}+x^{10}+x^{5}+x^{3}}$ ${\displaystyle -}$ Table 7: № 10.2
${\displaystyle 7.12}$ ${\displaystyle x^{36}+x^{34}+x^{20}+x^{10}+x^{9}+x^{3}}$ ${\displaystyle -}$ Table 7: № 8.1
${\displaystyle 8}$ ${\displaystyle 8.1}$ ${\displaystyle x^{68}+x^{34}+x^{17}+x^{12}+x^{9}+x^{3}}$ ${\displaystyle -}$ Table 9: № 5.1
${\displaystyle 8.2}$ ${\displaystyle x^{72}+x^{40}+x^{34}+x^{20}+x^{12}+x^{3}}$ ${\displaystyle N^{\circ }5}$ Table 9: № 6.1
${\displaystyle 8.3}$ ${\displaystyle x^{72}+x^{66}+x^{34}+x^{18}+x^{10}+x^{5}}$ ${\displaystyle -}$ Table 9: № 4.1
${\displaystyle 9}$ ${\displaystyle -}$ ${\displaystyle -}$ ${\displaystyle -}$ ${\displaystyle -}$
${\displaystyle 10}$ ${\displaystyle -}$ ${\displaystyle -}$ ${\displaystyle -}$ ${\displaystyle -}$
${\displaystyle 11}$ ${\displaystyle -}$ ${\displaystyle -}$ ${\displaystyle -}$ ${\displaystyle -}$
1. Sun B. On Classification and Some Properties of APN Functions.
2. 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.