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

From Boolean
Revision as of 20:09, 10 July 2020 by Nikolay (talk | contribs)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

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