Known infinite families of quadratic APN polynomials over GF(2^n)

$N^{\circ }$	Functions	Conditions	References
C1-C2	$x^{2^{s}+1}+u^{2^{k}-1}x^{2^{ik}+2^{mk+s}}$	$n=pk,\gcd(k,3)=\gcd(s,3k)=1,p\in \{3,4\}$, $i=sk{\bmod {p}},m=p-i,n\geq 12,u{\text{ primitive in }}\mathbb {F} _{2^{n}}^{*}$
C3	$sx^{q+1}+x^{2^{i}+1}+x^{q(2^{i}+1)}+cx^{2^{i}q+1}+c^{q}x^{2^{i}+q}$	$q=2^{m},n=2m,gcd(i,m)=1,c\in \mathbb {F} _{2^{n}},s\in \mathbb {F} _{2^{n}}\setminus \mathbb {F} _{q}$, $X^{2^{i}+1}+cX^{2^{i}}+c^{q}X+1{\text{ has no solution }}x{\text{ s.t. }}x^{q+1}=1$
C4	$x^{3}+a^{-1}\mathrm {Tr} _{n}(a^{3}x^{9})$	$a\neq 0$
C5	$x^{3}+a^{-1}\mathrm {Tr} _{n}^{3}(a^{3}x^{9}+a^{6}x^{18})$	$3|n$, $a\neq 0$
C6	$x^{3}+a^{-1}\mathrm {Tr} _{n}^{3}(a^{6}x^{18}+a^{12}x^{36})$	$3|n,a\neq 0$
C7-C9	$ux^{2^{s}+1}+u^{2^{k}}x^{2^{-k}+2^{k+s}}+vx^{2^{-k}+1}+wu^{2^{k}+1}x^{2^{s}+2^{k+s}}$	$n=3k,\gcd(k,3)=\gcd(s,3k)=1,v,w\in \mathbb {F} _{2^{k}}$, $vw\neq 1,3|(k+s),u{\text{ primitive in }}\mathbb {F} _{2^{n}}^{*}$
C10	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x+x^{2{^m}})^{2^k+1}+u'(ux+u^{2^{m}} x^{2^{m}})^{(2^k+1)2^i}+u(x+x^{2^{m}})(ux+u^{2^{m}} x^{2^{m}})}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=2m, m\geqslant 2} even, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gcd(k, m)=1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i \geqslant 2} even, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u\text{ primitive in } \mathbb{F}_{2^n}^*, u' \in \mathbb{F}_{2^m} \text{ not a cube }}
C11	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L(x)^{2^i}x+L(x)x^{2^i}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=km, m>1, \gcd(n,i)=1, L(x)=\sum_{j=0}^{k-1}a_jx^{2^{jm}}} satisfies the conditions in Theorem 3.6 of [7]
C12	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ut(x)(x^q+x)+t(x)^{2^{2i}+2^{3i}}+at(x)^{2^{2i}}(x^q+x)^{2^i}+b(x^q+x)^{2^i+1}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=2m, q=2^m, \gcd(m,i)=1, t(x)=u^qx+x^qu} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X^{2^i+1}+aX+b \mbox{ has no solution over }\mathbb{F}_{2^m}}
C13	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^3 + a (x^{2^i + 1})^{2^k} + b x^{3 \cdot 2^m} + c (x^{2^{i+m} + 2^m})^{2^k}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = 2m = 10, (a,b,c) = (\beta,1,0,0), i = 3, k = 2, \beta \text{ primitive in } \mathbb{F}_{2^2}}
		Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = 2m, m\ odd, 3 \nmid m, (a,b,c) = (\beta, \beta^2, 1), \beta \text{ primitive in } \mathbb{F}_{2^2}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i \in \{ m-2, m, 2m-1, (m-2)^{-1} \mod n \}}