Some APN functions CCZ-equivalent to x^3 + tr n(x^9) and CCZ-inequivalent to the Gold functions over GF(2^n)
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Some APN functions CCZ-equivalent to [math]\displaystyle{ x^3+{\mathrm Tr}_{n}(x^9) }[/math] and CCZ-inequivalent to the Gold functions over [math]\displaystyle{ \mathbb{F}_{2^n} }[/math][1].
| [math]\displaystyle{ N^\circ }[/math] | Functions | Conditions | [math]\displaystyle{ d^\circ }[/math] |
|---|---|---|---|
| [math]\displaystyle{ 1 }[/math] | [math]\displaystyle{ x^3+{\mathrm Tr}_n(x^9)+(x^2+x){\mathrm Tr}_n(x^3+x^9) }[/math] | [math]\displaystyle{ n\geqslant5 }[/math] odd, [math]\displaystyle{ \gcd(i,n)=1 }[/math] | [math]\displaystyle{ 3 }[/math] |
| [math]\displaystyle{ 2 }[/math] | [math]\displaystyle{ x^3+{\mathrm Tr}_n(x^9)+(x^2+x+1){\mathrm Tr}_n(x^3) }[/math] | [math]\displaystyle{ n\geqslant4 }[/math] even, [math]\displaystyle{ \gcd(i,n)=1 }[/math] | [math]\displaystyle{ 3 }[/math] |
| [math]\displaystyle{ 3 }[/math] | [math]\displaystyle{ \Big(x+{\mathrm Tr}_n^3(x^6+x^{12})+{\mathrm Tr}_n(x){\mathrm Tr}_n^3(x^3+x^{12})\Big)^3+ }[/math] [math]\displaystyle{ {\mathrm Tr}_n\Big(\left(x+{\mathrm Tr}_n^3(x^6+x^{12})+{\mathrm Tr}_n(x){\mathrm Tr}_n^3(x^3+x^{12})\right)^9\Big) }[/math] | [math]\displaystyle{ 6|n }[/math], [math]\displaystyle{ \gcd(i,n)=1 }[/math] | [math]\displaystyle{ 4 }[/math] |
| [math]\displaystyle{ 4 }[/math] | [math]\displaystyle{ \left(x^{\frac{1}{3}}+{\mathrm Tr}_n^3(x+x^4)\right)^{-1}+{\mathrm Tr}_n\left(\left(\left(x^{\frac{1}{3}}+{\mathrm Tr}_n^3(x+x^4)\right)^{-1}\right)^{9}\right) }[/math] | [math]\displaystyle{ 3|n }[/math], [math]\displaystyle{ n }[/math] odd | [math]\displaystyle{ 4 }[/math] |
- ↑ L. Budaghyan, C. Carlet, G. Leander. Constructing new APN functions from known ones. Finite Fields and Their Applications, v. 15, issue 2, pp. 150-159, April 2009. https://doi.org/10.1016/j.ffa.2008.10.001