# 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 ${\displaystyle x^{3}+{\mathrm {T} r}_{n}(x^{9})}$ and CCZ-inequivalent to the Gold functions over ${\displaystyle \mathbb {F} _{2^{n}}}$[1].

${\displaystyle N^{\circ }}$ Functions Conditions ${\displaystyle d^{\circ }}$
${\displaystyle 1}$ ${\displaystyle x^{3}+{\mathrm {T} r}_{n}(x^{9})+(x^{2}+x){\mathrm {T} r}_{n}(x^{3}+x^{9})}$ ${\displaystyle n\geqslant 5}$ odd, ${\displaystyle \gcd(i,n)=1}$ ${\displaystyle 3}$
${\displaystyle 2}$ ${\displaystyle x^{3}+{\mathrm {T} r}_{n}(x^{9})+(x^{2}+x+1){\mathrm {T} r}_{n}(x^{3})}$ ${\displaystyle n\geqslant 4}$ even, ${\displaystyle \gcd(i,n)=1}$ ${\displaystyle 3}$
${\displaystyle 3}$ ${\displaystyle {\Big (}x+{\mathrm {T} r}_{n}^{3}(x^{6}+x^{12})+{\mathrm {T} r}_{n}(x){\mathrm {T} r}_{n}^{3}(x^{3}+x^{12}){\Big )}^{3}+}$ ${\displaystyle {\mathrm {T} r}_{n}{\Big (}\left(x+{\mathrm {T} r}_{n}^{3}(x^{6}+x^{12})+{\mathrm {T} r}_{n}(x){\mathrm {T} r}_{n}^{3}(x^{3}+x^{12})\right)^{9}{\Big )}}$ ${\displaystyle 6|n}$, ${\displaystyle \gcd(i,n)=1}$ ${\displaystyle 4}$
${\displaystyle 4}$ ${\displaystyle \left(x^{\frac {1}{3}}+{\mathrm {T} r}_{n}^{3}(x+x^{4})\right)^{-1}+{\mathrm {T} r}_{n}\left(\left(\left(x^{\frac {1}{3}}+{\mathrm {T} r}_{n}^{3}(x+x^{4})\right)^{-1}\right)^{9}\right)}$ ${\displaystyle 3|n}$, ${\displaystyle n}$ odd ${\displaystyle 4}$
1. 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