Differentially 4-uniform permutation

Functions	Conditions	References
$x^{2^i+1}$	$gcd(i,n)=2,n=2t$ and t is odd
$s{x}^{q+1}+x^{2^i+1}+x^{q(2^i+1)}+c{x}^{2^{i}q+1}+c^q{x}^{2^i+q}$	$q=2^m,n=2m,$ $gcd(i,m)=1$, $c\in {𝔽}_{2^n},s\in {𝔽}_{2^n}\setminus {𝔽}_q,X^{2^i+1}+c{X}^{2^i}+c^{q}X+1\text{ has no solution }x$ s.t. $x^{q+1}=1$
$x^3+a^{-1}{\mathrm{T}\mathrm{r}}_n(a^3{x}^9)$	$a\ne 0$
$x^3+a^{-1}{\mathrm{T}\mathrm{r}}_n^3(a^3{x}^9+a^6{x}^{18})$	$3|n$, $a\ne 0$
$x^3+a^{-1}{\mathrm{T}\mathrm{r}}_n^3(a^6{x}^{18}+a^{12}{x}^{36})$	$3|n,a\ne 0$
$u{x}^{2^s+1}+u^{2^k}{x}^{2^{-k}+2^{k+s}}+v{x}^{2^{-k}+1}+w{u}^{2^k+1}{x}^{2^s+2^{k+s}}$	$n=3k,gcd(k,3)=gcd(s,3k)=1,v,w\in {𝔽}_{2^k},vw\ne 1,3|(k+s),u\text{ primitive in }{𝔽}_{2^n}^{\ast }$
$(x+x^{2{}^m}{)}^{2^k+1}+u^{\prime }(ux+u^{2^m}{x}^{2^m}{)}^{(2^k+1)2^i}+u(x+x^{2^m})(ux+u^{2^m}{x}^{2^m})$	$n=2m,m⩾2$ even, $gcd(k,m)=1$ and $i⩾2$ even, $u\text{ primitive in }{𝔽}_{2^n}^{\ast },u^{\prime }\in {𝔽}_{2^m}\text{ not a cube }$
$a^2{x}^{2^{2m+1}+1}+b^2{x}^{2^{m+1}+1}+a{x}^{2^{2m}+2}+b{x}^{2^m+2}+(c^2+c)x^3$	$n=3m,m\text{odd},L(x)=a{x}^{2^{2m}}+b{x}^{2^m}+cx$ satisfies the conditions in Lemma 8 of [7]