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

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${\displaystyle N^{\circ }}$ Functions Conditions References
C1-C2 ${\displaystyle x^{2^{s}+1}+u^{2^{k}-1}x^{2^{ik}+2^{mk+s}}}$ ${\displaystyle n=pk,\gcd(k,3)=\gcd(s,3k)=1,p\in \{3,4\},i=sk\mod p,m=p-i,n\geq 12,u{\text{ primitive in }}\mathbb {F} _{2^{n}}^{*}}$ [1]
C3 ${\displaystyle sx^{q+1}+x^{2^{i}+1}+x^{q(2^{i}+1)}+cx^{2^{i}q+1}+c^{q}x^{2^{i}+q}}$ ${\displaystyle q=2^{m},n=2m,gcd(i,m)=1,c\in \mathbb {F} _{2^{n}},s\in \mathbb {F} _{2^{n}}\setminus \mathbb {F} _{q},X^{2^{i}+1}+cX^{2^{i}}+c^{q}X+1{\text{ has no solution }}x}$ s.t. ${\displaystyle x^{q+1}=1}$ [2]
C4 ${\displaystyle x^{3}+a^{-1}\mathrm {Tr} _{n}(a^{3}x^{9})}$ ${\displaystyle a\neq 0}$ [3]
C5 ${\displaystyle x^{3}+a^{-1}\mathrm {Tr} _{n}^{3}(a^{3}x^{9}+a^{6}x^{18})}$ ${\displaystyle 3|n}$, ${\displaystyle a\neq 0}$ [4]
C6 ${\displaystyle x^{3}+a^{-1}\mathrm {Tr} _{n}^{3}(a^{6}x^{18}+a^{12}x^{36})}$ ${\displaystyle 3|n,a\neq 0}$ [4]
C7-C9 ${\displaystyle ux^{2^{s}+1}+u^{2^{k}}x^{2^{-k}+2^{k+s}}+vx^{2^{-k}+1}+wu^{2^{k}+1},x^{2^{s}+2^{k+s}}}$ ${\displaystyle n=3k,\gcd(k,3)=\gcd(s,3k)=1,v,w\in \mathbb {F} _{2^{k}},vw\neq 1,3|(k+s)u{\text{ primitive in }}\mathbb {F} _{2^{n}}^{*}}$ [5]
C10 ${\displaystyle (x+x^{2m})^{2^{k}+1}+u'(ux+u^{2m}x^{2m})^{(2^{k}+1)2^{i}}+u(x+x^{2m})(ux+u^{2m}x^{2m})}$ ${\displaystyle n=2m,m\geqslant 2}$ even, ${\displaystyle \gcd(k,m)=1,}$ and ${\displaystyle i\geqslant 2}$ even ${\displaystyle u{\text{ primitive in }}\mathbb {F} _{2^{n}}^{*},u'\in \mathbb {F} _{2^{m}}{\text{ not a cube }}}$ [6]
C11 ${\displaystyle a^{2}x^{2^{2m+1}+1}+b^{2}x^{2^{m+1}+1}+ax^{2^{2m}+2}+bx^{2^{m}+2}+(c^{2}+c)x^{3}}$ ${\displaystyle n=3m,m\ {\text{odd}}\ L(x)=ax^{2^{2m}}+bx^{2m}+cx\ {\text{satisfies the conditions in lemma 8 of}}\ [3]}$ [7]
1. L. Budaghyan, C. Carlet, G. Leander, Two Classes of Quadratic APN Binomials Inequivalent to Power Functions, IEEE Trans. Inform. Theory 54(9), 2008, pp. 4218-4229
2. L. Budaghyan and C. Carlet. Classes of Quadratic APN Trinomials and Hexanomials and Related Structures. {\em IEEE Trans. Inform. Theory}, vol. 54, no. 5, pp. 2354-2357, 2008.
3. L. Budaghyan, C. Carlet and G.Leander, Constructinig new APN functions from known ones, Finite Fields and their applications, vol.15, issue 2, Apr. 2009, pp. 150-159.
4. L. Budaghyan, C. Carlet and G.Leander, On a Construction of quadratic APN functions, Proceedings of IEEE information Theory workshop ITW'09, Oct. 2009, 374-378.
5. Bracken, C., Byrne, E., Markin, N., & Mcguire, G. (2011). A few more quadratic APN functions. Cryptography and Communications, 3(1), 43-53.
6. Göloğlu, Faruk. Almost perfect nonlinear trinomials and hexanomials. Finite Fields and Their Applications 33 (2015): 258-282.
7. Villa, I., Budaghyan, L., Calderini, M., Carlet, C., & Coulter, R. On Isotopic Construction of APN Functions. SETA 2018