# Difference between revisions of "Known infinite families of quadratic APN polynomials over GF(2^n)"

${\displaystyle N^{\circ }}$ Functions Conditions References
C1-C2 ${\displaystyle x^{2^{s}+1}+u^{2^{k}-1}x^{2^{ik}+2^{mk+s}}}$ ${\displaystyle n=pk,\gcd(k,3)=\gcd(s,3k)=1,p\in \{3,4\},i=sk{\bmod {p}},m=p-i,n\geq 12,u{\text{ primitive in }}\mathbb {F} _{2^{n}}^{*}}$ [1]
C3 ${\displaystyle sx^{q+1}+x^{2^{i}+1}+x^{q(2^{i}+1)}+cx^{2^{i}q+1}+c^{q}x^{2^{i}+q}}$ ${\displaystyle q=2^{m},n=2m,}$ ${\displaystyle gcd(i,m)=1}$, ${\displaystyle c\in \mathbb {F} _{2^{n}},s\in \mathbb {F} _{2^{n}}\setminus \mathbb {F} _{q},X^{2^{i}+1}+cX^{2^{i}}+c^{q}X+1{\text{ has no solution }}x}$ 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 ${\displaystyle (x+x^{2{^{m}}})^{2^{k}+1}+u'(ux+u^{2^{m}}x^{2^{m}})^{(2^{k}+1)2^{i}}+u(x+x^{2^{m}})(ux+u^{2^{m}}x^{2^{m}})}$ ${\displaystyle n=2m,m\geqslant 2}$ even, ${\displaystyle \gcd(k,m)=1,}$, ${\displaystyle i\geqslant 2}$ even, ${\displaystyle u{\text{ primitive in }}\mathbb {F} _{2^{n}}^{*},u'\in \mathbb {F} _{2^{m}}{\text{ not a cube }}}$ [6]
C11 ${\displaystyle a^{2}x^{2^{2m+1}+1}+b^{2}x^{2^{m+1}+1}+ax^{2^{2m}+2}+bx^{2^{m}+2}+(c^{2}+c)x^{3}}$ ${\displaystyle n=3m,m\ {\text{odd}},L(x)=ax^{2^{m}}+bx^{2^{m}}+cx}$ satisfies the conditions in lemma 8 of [7] [7]
1. Budaghyan, L., Carlet, C. and Leander, G., 2008. Two classes of quadratic APN binomials inequivalent to power functions. IEEE Transactions on Information Theory, 54(9), pp.4218-4229.
2. Budaghyan, L. and Carlet, C., 2008. Classes of quadratic APN trinomials and hexanomials and related structures. IEEE Transactions on Information Theory, 54(5), pp.2354-2357.
3. Budaghyan, L., Carlet, C. and Leander, G., 2009. Constructing new APN functions from known ones. Finite Fields and Their Applications, 15(2), pp.150-159.
4. Budaghyan, L., Carlet, C. and Leander, G., 2009, October. On a construction of quadratic APN functions. In Information Theory Workshop, 2009. ITW 2009. IEEE (pp. 374-378). IEEE.
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, F., 2015. Almost perfect nonlinear trinomials and hexanomials. Finite Fields and Their Applications, 33, pp.258-282.
7. Villa, I., Budaghyan, L., Calderini, M., Carlet, C., & Coulter, R. On Isotopic Construction of APN Functions. SETA 2018