Some APN functions CCZ-equivalent to Gold functions and EA-inequivalent to power functions over GF(2^n): Difference between revisions

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<td><math>x^{2^i+1}+(x^{2^i}+x+tr_n(1)+1)tr_n(x^{2^i+1}+x\ tr_n(1))</math></td>
<td><math>x^{2^i+1}+(x^{2^i}+x+{\mathrm Tr}_n(1)+1){\mathrm Tr}_n(x^{2^i+1}+x\ {\mathrm Tr}_n(1))</math></td>
<td><math>n\geqslant4</math>, <math>\gcd(i,n)=1</math></td>
<td><math>n\geqslant4</math>, <math>\gcd(i,n)=1</math></td>
<td><math>3</math></td>
<td><math>3</math></td>
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<td><math>[x+tr_{n/3}(x^{2(2^i+1)}+x^{4(2^i+1)})+tr_n(x)tr_{n/3}(x^{2^i+1}+x^{2^{2i}(2^i+1)})]^{2^i+1}</math></td>
<td><math>[x+{\mathrm Tr}_n^3(x^{2(2^i+1)}+x^{4(2^i+1)})+{\mathrm Tr}_n(x){\mathrm Tr}_n^3(x^{2^i+1}+x^{2^{2i}(2^i+1)})]^{2^i+1}</math></td>
<td><math>6|n\ , \gcd(i,n)=1</math></td>
<td><math>6|n\ , \gcd(i,n)=1</math></td>
<td><math>4</math></td>
<td><math>4</math></td>
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<td><math>x^{2^i+1}+tr_{n/m}(x^{2^i+1})+x^{2^i}tr_{n/m}(x)+x \ tr_{n/m}(x)^{2^i}</math>
<td><math>x^{2^i+1}+{\mathrm Tr}_n^m(x^{2^i+1})+x^{2^i}{\mathrm Tr}_n^m(x)+x \ {\mathrm Tr}_n^m(x)^{2^i}</math>
<math>+[tr_{n/m}(x)^{2^i+1}+tr_{n/m}(x^{2^i+1})+tr_{n/m}(x)]^{\frac{1}{2^i+1}}(x^{2^i}+tr_{n/m}(x)^{2^i}+1)</math>
<math>+[{\mathrm Tr}_n^m(x)^{2^i+1}+{\mathrm Tr}_n^m(x^{2^i+1})+{\mathrm Tr}_n^m(x)]^{\frac{1}{2^i+1}}(x^{2^i}+{\mathrm Tr}_n^m(x)^{2^i}+1)</math>
<math>+[tr_{n/m}(x)^{2^i+1}+tr_{n/m}(x^{2^i+1})+tr_{n/m}(x)]^{\frac{2^i}{2^i+1}}(x+tr_{n/m}(x))</math></td>
<math>+[{\mathrm Tr}_n^m(x)^{2^i+1}+{\mathrm Tr}_n^m(x^{2^i+1})+{\mathrm Tr}_n^m(x)]^{\frac{2^i}{2^i+1}}(x+{\mathrm Tr}_n^m(x))</math></td>
<td><math>\ m\ne n\ , n\ odd\ , m|n\ , \gcd(i,n)=1\ </math></td>
<td><math>\ m\ne n\ , n\ odd\ , m|n\ , \gcd(i,n)=1\ </math></td>
<td><math>\ m+2\ </math></td>
<td><math>\ m+2\ </math></td>
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Latest revision as of 20:09, 10 July 2020

Some APN functions CCZ-equivalent to Gold functions and EA-inequivalent to power functions over[math]\displaystyle{ \mathbb{F}_{2^n} }[/math][1].

Functions Conditions [math]\displaystyle{ d^\circ }[/math]
[math]\displaystyle{ x^{2^i+1}+(x^{2^i}+x+{\mathrm Tr}_n(1)+1){\mathrm Tr}_n(x^{2^i+1}+x\ {\mathrm Tr}_n(1)) }[/math] [math]\displaystyle{ n\geqslant4 }[/math], [math]\displaystyle{ \gcd(i,n)=1 }[/math] [math]\displaystyle{ 3 }[/math]
[math]\displaystyle{ [x+{\mathrm Tr}_n^3(x^{2(2^i+1)}+x^{4(2^i+1)})+{\mathrm Tr}_n(x){\mathrm Tr}_n^3(x^{2^i+1}+x^{2^{2i}(2^i+1)})]^{2^i+1} }[/math] [math]\displaystyle{ 6|n\ , \gcd(i,n)=1 }[/math] [math]\displaystyle{ 4 }[/math]
[math]\displaystyle{ x^{2^i+1}+{\mathrm Tr}_n^m(x^{2^i+1})+x^{2^i}{\mathrm Tr}_n^m(x)+x \ {\mathrm Tr}_n^m(x)^{2^i} }[/math]

[math]\displaystyle{ +[{\mathrm Tr}_n^m(x)^{2^i+1}+{\mathrm Tr}_n^m(x^{2^i+1})+{\mathrm Tr}_n^m(x)]^{\frac{1}{2^i+1}}(x^{2^i}+{\mathrm Tr}_n^m(x)^{2^i}+1) }[/math]

[math]\displaystyle{ +[{\mathrm Tr}_n^m(x)^{2^i+1}+{\mathrm Tr}_n^m(x^{2^i+1})+{\mathrm Tr}_n^m(x)]^{\frac{2^i}{2^i+1}}(x+{\mathrm Tr}_n^m(x)) }[/math]		 [math]\displaystyle{ \ m\ne n\ , n\ odd\ , m|n\ , \gcd(i,n)=1\ }[/math] [math]\displaystyle{ \ m+2\ }[/math]
  1. ↑ L. Budaghyan, C.Carlet, A. Pott. New classes of almost bent and almost perfect nonlinear polynomials. IEEE Trans. Inf. Theoery, vol. 52, no. 3, pp. 1141-1152, 2006. https://doi.org/10.1109/TIT.2005.864481
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