Some APN functions CCZ-equivalent to Gold functions and EA-inequivalent to power functions over GF(2^n)
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Some APN functions CCZ-equivalent to Gold functions and EA-inequivalent to power functions over[math]\displaystyle{ \mathbb{F}_{2^n} }[/math] (constructed in [1])
| Functions | Conditions | [math]\displaystyle{ d^\circ }[/math] |
|---|---|---|
| [math]\displaystyle{ x^{2^i+1}+(x^{2^i}+x+tr_n(1)+1)tr_n(x^{2^i+1}+x\ tr_n(1)) }[/math] | [math]\displaystyle{ n\geqslant4 }[/math], [math]\displaystyle{ \gcd(i,n)=1 }[/math] | [math]\displaystyle{ 3 }[/math] |
| [math]\displaystyle{ [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] | [math]\displaystyle{ 6|n\ , \gcd(i,n)=1 }[/math] | [math]\displaystyle{ 4 }[/math] |
| [math]\displaystyle{ 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]
[math]\displaystyle{ +[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]\displaystyle{ +[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] |
[math]\displaystyle{ \ m\ne n\ , n\ odd\ , m|n\ , \gcd(i,n)=1\ }[/math] | [math]\displaystyle{ \ m+2\ }[/math] |
- ↑ Budaghyan L, Carlet C, Pott A. New classes of almost bent and almost perfect nonlinear polynomials. IEEE Transactions on Information Theory. 2006 Mar;52(3):1141-52.