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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\cTable 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>}

Revision as of 13:33, 18 January 2019

Table 1: Classification of Quadratic APN Trinomials (CCZ-inequivalent to infinite monomial families) in Small Dimensions with Coefficients in [math]\displaystyle{ \mathbb{F}_2 }[/math]

[math]\displaystyle{ n }[/math] [math]\displaystyle{ N^\circ }[/math] Functions Families from tables 5 Relation to [6]
[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{ 7.2 }[/math]

[math]\displaystyle{ x^{20} + x^6 + x^3 }[/math]

[math]\displaystyle{ x^{34} + x^{28} + x^5 }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ Table 7: N^\circ8.1 }[/math]

[math]\displaystyle{ 9:N^\circ1.4 }[/math]

[math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 8.1 }[/math]

[math]\displaystyle{ 8.2 }[/math]

[math]\displaystyle{ x^{72} + x^6 + x^3 }[/math]

[math]\displaystyle{ x^{72} + x^{36} + x^3 }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ Table 7: N^\circ8.1 }[/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]


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

[math]\displaystyle{ n }[/math] [math]\displaystyle{ N^\circ }[/math] Functions Families from tables 5 Relation to [6]
[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{ 7.2 }[/math]

[math]\displaystyle{ 7.3 }[/math]

[math]\displaystyle{ 7.4 }[/math]

[math]\displaystyle{ 7.5 }[/math]

[math]\displaystyle{ 7.6 }[/math]

[math]\displaystyle{ x^{72} + x^{40} + x^{12} + x^3 }[/math]

[math]\displaystyle{ x^{33} + x^{17} + x^{12} + x^3 }[/math]

[math]\displaystyle{ x^{34} + x^{33} + x^{10} + x^3 }[/math]

[math]\displaystyle{ x^{66} + x^{34} + x^{20} + x^3 }[/math]

[math]\displaystyle{ x^{68} + x^{18} + x^5 + x^3 }[/math]

[math]\displaystyle{ x^{66} + x^{18} + x^9 + x^3 }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ Table 7: N^\circ12.1 }[/math]

[math]\displaystyle{ 7:N^\circ2.2 }[/math]

[math]\displaystyle{ 7:N^\circ10.1 }[/math]

[math]\displaystyle{ 7:N^\circ11.1 }[/math]

[math]\displaystyle{ 7:N^\circ8.1 }[/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]


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

[math]\displaystyle{ n }[/math] [math]\displaystyle{ N^\circ }[/math] Functions Families from tables 5 Relation to [6]
[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{ 7.2 }[/math]

[math]\displaystyle{ 7.3 }[/math]

[math]\displaystyle{ 7.4 }[/math]

[math]\displaystyle{ 7.5 }[/math]

[math]\displaystyle{ 7.6 }[/math]

[math]\displaystyle{ 7.7 }[/math]

[math]\displaystyle{ 7.8 }[/math]

[math]\displaystyle{ 7.9 }[/math]

[math]\displaystyle{ 7.10 }[/math]
[math]\displaystyle{ x^{68} + x^{40} + x^{24} + x^6 + x^3 }[/math]

[math]\displaystyle{ x^{65} + x^{20} + x^{18} + x^6 + x^3 }[/math]

[math]\displaystyle{ x^{40} + x^{34} + x^{18} + x^{10} + x^3 }[/math]

[math]\displaystyle{ x^{48} + x^{40} + x^{10} + x^9 + x^3 }[/math]

[math]\displaystyle{ x^{33} + x^9 + x^6 + x^5 + x^3 }[/math]

[math]\displaystyle{ x^{40} + x^{36} + x^{34} + x^{24} + x^3 }[/math]

[math]\displaystyle{ x^{24} + x^{10} + x^9 + x^6 + x^3 }[/math]

[math]\displaystyle{ x^{65} + x^{36} + x^{20} + x^{17} + x^3 }[/math]

[math]\displaystyle{ x^{40} + x^{33} + x^{17} + x^5 + x^3 }[/math]

[math]\displaystyle{ x^{36} + x^{33} + x^{18} + x^9 + x^5 }[/math]
[math]\displaystyle{ - }[/math]

[math]\displaystyle{ N^\circ5 }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]
[math]\displaystyle{ Table 7: N^\circ13.1 }[/math]

[math]\displaystyle{ 7:N^\circ1.2 }[/math]

[math]\displaystyle{ 7:N^\circ12.1 }[/math]

[math]\displaystyle{ 7:N^\circ2.1 }[/math]

[math]\displaystyle{ 7:N^\circ11.1 }[/math]

[math]\displaystyle{ 7:N^\circ10.1 }[/math]

[math]\displaystyle{ 7:N^\circ2.1 }[/math]

[math]\displaystyle{ 7:N^\circ14.1 }[/math]

[math]\displaystyle{ 7:N^\circ8.1 }[/math]

[math]\displaystyle{ 7:N^\circ10.1 }[/math]
[math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 8.1 }[/math]

[math]\displaystyle{ 8.2 }[/math]

[math]\displaystyle{ 8.3 }[/math]

[math]\displaystyle{ 8.4 }[/math]
[math]\displaystyle{ x^{36} + x^{33} + x^9 + x^6 + x^3 }[/math]

[math]\displaystyle{ x^{72} + x^{66} + x^{12} + x^6 + x^3 }[/math]

[math]\displaystyle{ x^{130} + x^{66} + x^{40} + x^{12} + x^3 }[/math]

[math]\displaystyle{ x^{66} + x^{40} + x^{18} + x^5 + x^3 }[/math]
[math]\displaystyle{ - }[/math]

[math]\displaystyle{ N^\circ5 }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]
[math]\displaystyle{ Table 9: N^\circ1.4 }[/math]

[math]\displaystyle{ 9:N^\circ1.3 }[/math]

[math]\displaystyle{ 9:N^\circ6.1 }[/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{ 11.2 }[/math]

[math]\displaystyle{ 11.3 }[/math]

[math]\displaystyle{ 11.4 }[/math]

[math]\displaystyle{ 11.5 }[/math]
[math]\displaystyle{ x^{12} + x^{10} + x^9 + x^5 + x^3 }[/math]

[math]\displaystyle{ x^{258} + x^{257} + x^{18} + x^{17} + x^3 }[/math]

[math]\displaystyle{ x^{96} + x^{66} + x^{34} + x^{33} + x^3 }[/math]

[math]\displaystyle{ x^{80} + x^{68} + x^{65} + x^{17} + x^5 }[/math]

[math]\displaystyle{ x^{260} + x^{257} + x^{36} + x^{33} + x^5 }[/math]
[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]
[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]


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

[math]\displaystyle{ n }[/math] [math]\displaystyle{ N^\circ }[/math] Functions Families from tables 5 Relation to [6]
[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{ 7.2 }[/math]

[math]\displaystyle{ 7.3 }[/math]

[math]\displaystyle{ 7.4 }[/math]

[math]\displaystyle{ 7.5 }[/math]

[math]\displaystyle{ 7.6 }[/math]

[math]\displaystyle{ 7.7 }[/math]

[math]\displaystyle{ 7.8 }[/math]

[math]\displaystyle{ 7.9 }[/math]

[math]\displaystyle{ 7.10 }[/math]

[math]\displaystyle{ 7.11 }[/math]

[math]\displaystyle{ 7.12 }[/math]
[math]\displaystyle{ x^{34} + x^{33} + x^{12} + x^6 + x^5 + x^3 }[/math]

[math]\displaystyle{ x^{40} + x^{24} + x^{20} + x^9 + x^5 + x^3 }[/math]

[math]\displaystyle{ x^{33} + x^{24} + x^{20} + x^{18} + x^{12} + x^3 }[/math]

[math]\displaystyle{ x^{24} + x^{17} + x^{12} + x^{10} + x^6 + x^3 }[/math]

[math]\displaystyle{ x^{40} + x^{34} + x^{18} + x^{17} + x^5 + x^3 }[/math]

[math]\displaystyle{ x^{48} + x^{40} + x^{18} + x^{10} + x^5 + x^3 }[/math]

[math]\displaystyle{ x^{40} + x^{12} + x^{10} + x^9 + x^5 + x^3 }[/math]

[math]\displaystyle{ x^{34} + x^{24} + x^{10} + x^9 + x^6 + x^3 }[/math]

[math]\displaystyle{ x^{34} + x^{33} + x^{20} + x^{17} + x^{10} + x^3 }[/math]

[math]\displaystyle{ x^{36} + x^{33} + x^{24} + x^9 + x^6 + x^3 }[/math]

[math]\displaystyle{ x^{40} + x^{36} + x^{20} + x^{10} + x^5 + x^3 }[/math]

[math]\displaystyle{ x^{36} + x^{34} + x^{20} + x^{10} + x^9 + x^3 }[/math]
[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ N^\circ5 }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]
[math]\displaystyle{ Table 7: N^\circ14.2 }[/math]

[math]\displaystyle{ 7:N^\circ14.1 }[/math]

[math]\displaystyle{ 7:N^\circ12.1 }[/math]

[math]\displaystyle{ 7:N^\circ2.1 }[/math]

[math]\displaystyle{ 7:N^\circ1.2 }[/math]

[math]\displaystyle{ 7:N^\circ11.1 }[/math]

[math]\displaystyle{ 7:N^\circ2.2 }[/math]

[math]\displaystyle{ 7:N^\circ9.1 }[/math]

[math]\displaystyle{ 7:N^\circ13.1 }[/math]

[math]\displaystyle{ 7:N^\circ10.1 }[/math]

[math]\displaystyle{ 7:N^\circ10.2 }[/math]

[math]\displaystyle{ 7:N^\circ8.1 }[/math]
[math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 8.1 }[/math]

[math]\displaystyle{ 8.2 }[/math]

[math]\displaystyle{ 8.3 }[/math]
[math]\displaystyle{ x^{68} + x^{34} + x^{17} + x^{12} + x^9 + x^3 }[/math]

[math]\displaystyle{ x^{72} + x^{40} + x^{34} + x^{20} + x^{12} + x^3 }[/math]

[math]\displaystyle{ x^{72} + x^{66} + x^{34} + x^{18} + x^{10} + x^5 }[/math]
[math]\displaystyle{ - }[/math]

[math]\displaystyle{ N^\circ5 }[/math]

[math]\displaystyle{ - }[/math]

[math]\displaystyle{ - }[/math]
[math]\displaystyle{ Table 9: N^\circ5.1 }[/math]

[math]\displaystyle{ 9:N^\circ6.1 }[/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]


\cTable 5: Known classes of quadratic APN polynomials CCZ-inequivalent to APN monomials on [math]\displaystyle{ \mathbb{F}_{2^n} }[/math] [math]\displaystyle{ u }[/math] is primitive in [math]\displaystyle{ \mathbb{F}_{2^n}^* }[/math]}