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

[math]\displaystyle{ N^\circ }[/math]	Functions	Conditions	References
C1-C2	[math]\displaystyle{ x^{2^s+1}+u^{2^k-1}x^{2^{ik}+2^{mk+s}} }[/math]	[math]\displaystyle{ n = pk, \gcd(k,3) = \gcd(s,3k) = 1, p \in \{3,4\} }[/math], [math]\displaystyle{ i = sk\bmod p, m = p -i, n \ge 12, u \text{ primitive in } \mathbb{F}_{2^n}^* }[/math]
C3	[math]\displaystyle{ sx^{q+1}+x^{2^i+1}+x^{q(2^i+1)}+cx^{2^iq+1}+c^qx^{2^i+q} }[/math]	[math]\displaystyle{ 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} }[/math], [math]\displaystyle{ X^{2^i+1}+cX^{2^i}+c^{q}X+1 \text{ has no solution } x \text{ s.t. }x^{q+1}=1 }[/math]
C4	[math]\displaystyle{ x^3+a^{-1} \mathrm {Tr}_n (a^3x^9) }[/math]	[math]\displaystyle{ a\neq 0 }[/math]
C5	[math]\displaystyle{ x^3+a^{-1} \mathrm {Tr}_n^3 (a^3x^9+a^6x^{18}) }[/math]	[math]\displaystyle{ 3|n }[/math], [math]\displaystyle{ a\ne0 }[/math]
C6	[math]\displaystyle{ x^3+a^{-1} \mathrm{Tr}_n^3(a^6x^{18}+a^{12}x^{36}) }[/math]	[math]\displaystyle{ 3|n, a \ne 0 }[/math]
C7-C9	[math]\displaystyle{ 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}} }[/math]	[math]\displaystyle{ n=3k, \gcd(k,3)=\gcd(s,3k)=1, v, w\in\mathbb{F}_{2^k} }[/math], [math]\displaystyle{ vw \ne 1, 3|(k+s), u \text{ primitive in } \mathbb{F}_{2^n}^* }[/math]
C10	[math]\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}}) }[/math]	[math]\displaystyle{ n=2m, m\geqslant 2 }[/math] even, [math]\displaystyle{ \gcd(k, m)=1 }[/math] and [math]\displaystyle{ i \geqslant 2 }[/math] even, [math]\displaystyle{ u\text{ primitive in } \mathbb{F}_{2^n}^*, u' \in \mathbb{F}_{2^m} \text{ not a cube } }[/math]
C11	[math]\displaystyle{ L(x)^{2^i}x+L(x)x^{2^i} }[/math]	[math]\displaystyle{ n=km, m\gt 1, \gcd(n,i)=1, L(x)=\sum_{j=0}^{k-1}a_jx^{2^{jm}} }[/math] satisfies the conditions in Theorem 3.6 of [7]
C12	[math]\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} }[/math]	[math]\displaystyle{ n=2m, q=2^m, \gcd(m,i)=1, t(x)=u^qx+x^qu }[/math], [math]\displaystyle{ X^{2^i+1}+aX+b \mbox{ has no solution over }\mathbb{F}_{2^m} }[/math]
C13	[math]\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} }[/math]	[math]\displaystyle{ n = 2m = 10, (a,b,c) = (\beta,0,0), i = 3, k = 2, \beta \text{ primitive in } \mathbb{F}_{2^2} }[/math]
		[math]\displaystyle{ n = 2m, m\ odd, 3 \nmid m, (a,b,c) = (\beta, \beta^2, 1), \beta \text{ primitive in } \mathbb{F}_{2^2} }[/math], [math]\displaystyle{ i \in \{ m-2, m, 2m-1, (m-2)^{-1} \mod n \} }[/math]