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

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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

Dimension [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] Table 7: № 8.1
[math]\displaystyle{ 7.2 }[/math] [math]\displaystyle{ x^{34} + x^{18} + x^5 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 2.1
[math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 8.1 }[/math] [math]\displaystyle{ x^{72} + x^6 + x^3 }[/math] [math]\displaystyle{ N^\circ5 }[/math] Table 9: № 1.3
[math]\displaystyle{ 8.2 }[/math] [math]\displaystyle{ x^{72} + x^{36} + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 9: № 1.4
[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

Dimension [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] Table 7: № 12.1
[math]\displaystyle{ 7.2 }[/math] [math]\displaystyle{ x^{33} + x^{17} + x^{12} + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 2.2
[math]\displaystyle{ 7.3 }[/math] [math]\displaystyle{ x^{34} + x^{33} + x^{10} + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 10.1
[math]\displaystyle{ 7.4 }[/math] [math]\displaystyle{ x^{66} + x^{34} + x^{20} + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 11.1
[math]\displaystyle{ 7.5 }[/math] [math]\displaystyle{ x^{68} + x^{18} + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 8.1
[math]\displaystyle{ 7.6 }[/math] [math]\displaystyle{ x^{66} + x^{18} + x^9 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 9.1
[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

Dimension [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] Table 7: № 13.1
[math]\displaystyle{ 7.2 }[/math] [math]\displaystyle{ x^{65} + x^{20} + x^{18} + x^6 + x^3 }[/math] [math]\displaystyle{ N^\circ5 }[/math] Table 7: № 1.2
[math]\displaystyle{ 7.3 }[/math] [math]\displaystyle{ x^{40} + x^{34} + x^{18} + x^{10} + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 12.1
[math]\displaystyle{ 7.4 }[/math] [math]\displaystyle{ x^{48} + x^{40} + x^{10} + x^9 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 1.2
[math]\displaystyle{ 7.5 }[/math] [math]\displaystyle{ x^{33} + x^9 + x^6 + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 11.1
[math]\displaystyle{ 7.6 }[/math] [math]\displaystyle{ x^{40} + x^{36} + x^{34} + x^{24} + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 10.1
[math]\displaystyle{ 7.7 }[/math] [math]\displaystyle{ x^{24} + x^{10} + x^9 + x^6 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 2.1
[math]\displaystyle{ 7.8 }[/math] [math]\displaystyle{ x^{65} + x^{36} + x^{20} + x^{17} + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 14.1
[math]\displaystyle{ 7.9 }[/math] [math]\displaystyle{ x^{40} + x^{33} + x^{17} + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 8.1
[math]\displaystyle{ 7.10 }[/math] [math]\displaystyle{ x^{36} + x^{33} + x^{18} + x^9 + x^5 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 10.1
[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] Table 9: № 1.4
[math]\displaystyle{ 8.2 }[/math] [math]\displaystyle{ x^{72} + x^{66} + x^{12} + x^6 + x^3 }[/math] [math]\displaystyle{ N^\circ5 }[/math] Table 9: № 1.3
[math]\displaystyle{ 8.3 }[/math] [math]\displaystyle{ x^{130} + x^{66} + x^{40} + x^{12} + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 9: № 6.1
[math]\displaystyle{ 8.4 }[/math] [math]\displaystyle{ x^{66} + x^{40} + x^{18} + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 9: № 5.1
[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]

Hexanomials

Dimension [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] Table 7: № 14.2
[math]\displaystyle{ 7.2 }[/math] [math]\displaystyle{ x^{40} + x^{24} + x^{20} + x^9 + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 14.1
[math]\displaystyle{ 7.3 }[/math] [math]\displaystyle{ x^{33} + x^{24} + x^{20} + x^{18} + x^{12} + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 12.1
[math]\displaystyle{ 7.4 }[/math] [math]\displaystyle{ x^{24} + x^{17} + x^{12} + x^{10} + x^6 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 2.1
[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] Table 7: № 1.2
[math]\displaystyle{ 7.6 }[/math] [math]\displaystyle{ x^{48} + x^{40} + x^{18} + x^{10} + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 11.1
[math]\displaystyle{ 7.7 }[/math] [math]\displaystyle{ x^{40} + x^{12} + x^{10} + x^9 + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 2.2
[math]\displaystyle{ 7.8 }[/math] [math]\displaystyle{ x^{34} + x^{24} + x^{10} + x^9 + x^6 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 9.1
[math]\displaystyle{ 7.9 }[/math] [math]\displaystyle{ x^{34} + x^{33} + x^{20} + x^{17} + x^{10} + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 13.1
[math]\displaystyle{ 7.10 }[/math] [math]\displaystyle{ x^{36} + x^{33} + x^{24} + x^9 + x^6 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 10.1
[math]\displaystyle{ 7.11 }[/math] [math]\displaystyle{ x^{40} + x^{36} + x^{20} + x^{10} + x^5 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 10.2
[math]\displaystyle{ 7.12 }[/math] [math]\displaystyle{ x^{36} + x^{34} + x^{20} + x^{10} + x^9 + x^3 }[/math] [math]\displaystyle{ - }[/math] Table 7: № 8.1
[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] Table 9: № 5.1
[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] Table 9: № 6.1
[math]\displaystyle{ 8.3 }[/math] [math]\displaystyle{ x^{72} + x^{66} + x^{34} + x^{18} + x^{10} + x^5 }[/math] [math]\displaystyle{ - }[/math] Table 9: № 4.1
[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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