# Differentially 4-uniform permutations

Functions Conditions References
$x^{2^{i}+1}$ $gcd(i,n)=2,n=2t$ and t is odd 
$x^{2^{2i}-2^{i}+1}$ $gcd(i,n)=2,n=2t$ and t is odd 
$x^{2^{n}-2}$ $n=2t$ (inverse) 
$x^{2^{2t}+2^{t}+1}$ $n=4t$ and t is odd 
$\alpha x^{2^{s}+1}+\alpha ^{2^{t}}x^{{2-t}+2^{t+s}}$ $n=3t,t/2$ is odd, $gcd(n,s)=2,3|t+s$ and $\alpha$ is a primitive element in $\mathbb {F} _{2^{n}}$ $x^{-1}+\mathrm {Tr} _{n}^{1}(x+(x^{-1}+1)^{-1})$ $n=2t$ is even 
$x^{-1}+\mathrm {Tr} _{n}^{1}(x^{-3(2^{k}+1)}+(x^{-1}+1)^{3(2^{k}+1)})$ $n=2t$ and $2\leq k\leq t-1$ $L_{u}(F^{-1}(x))|_{H_{u}}$ $n=2t,F(x)$ is a quadratic APN permutation on ${\mathbb {F} }_{2^{n+1}},u\in {\mathbb {F} }_{2^{n+1}}^{*},$ $L_{u}(x)=F(x)+F(x+u)+F(u),$ $H_{u}=\{L_{u}(x)|x\in {\mathbb {F} }_{2^{n+1}}\}$ $\displaystyle \sum _{i=0}^{2^{n}-3}x^{i}$ $n=2t,$ t is odd 
$x^{-1}+t(x^{2^{s}}+x)^{2^{sn}-1}$ $s$ is even $,t\in {\mathbb {F} }_{2^{s}}^{*},$ or $s,n$ are odd, $t\in {\mathbb {F} }_{2^{s}}^{*}$ $x^{2^{k}+1}+t(x^{2^{s}}+x)^{2^{sn}-1}$ $n,s$ are odd, $t\in {\mathbb {F} }_{2^{s}}^{*},gcd(k,sn)=1$ $(x,x_{n})\mapsto$ $((1+x_{n})x^{-1}+x_{n}\alpha x^{-1},f(x,x_{n}))$ $n$ is even $x,\alpha \in {\mathbb {F} }_{2^{n-1}},x_{n}\in {\mathbb {F} }_{2},\mathrm {Tr} _{n-1}(\alpha )=\mathrm {Tr} _{n-1}\left({\frac {1}{\alpha }}\right)=1,$ $f(x,x_{n})$ is $(n,1)-$ function

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