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

${\displaystyle N^{\circ }}$ Functions Conditions References
C1-C2 ${\displaystyle x^{2^{s}+1}+u^{2^{k}-1}x^{2^{ik}+2^{mk+s}}}$ ${\displaystyle n=pk,\gcd(k,3)=\gcd(s,3k)=1,p\in \{3,4\}}$, ${\displaystyle i=sk{\bmod {p}},m=p-i,n\geq 12,u{\text{ primitive in }}\mathbb {F} _{2^{n}}^{*}}$ [1]
C3 ${\displaystyle sx^{q+1}+x^{2^{i}+1}+x^{q(2^{i}+1)}+cx^{2^{i}q+1}+c^{q}x^{2^{i}+q}}$ ${\displaystyle q=2^{m},n=2m,gcd(i,m)=1,c\in \mathbb {F} _{2^{n}},s\in \mathbb {F} _{2^{n}}\setminus \mathbb {F} _{q}}$, ${\displaystyle X^{2^{i}+1}+cX^{2^{i}}+c^{q}X+1{\text{ has no solution }}x{\text{ s.t. }}x^{q+1}=1}$ [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 ${\displaystyle ux^{2^{s}+1}+u^{2^{k}}x^{2^{-k}+2^{k+s}}+vx^{2^{-k}+1}+wu^{2^{k}+1}x^{2^{s}+2^{k+s}}}$ ${\displaystyle n=3k,\gcd(k,3)=\gcd(s,3k)=1,v,w\in \mathbb {F} _{2^{k}}}$, ${\displaystyle vw\neq 1,3|(k+s),u{\text{ primitive in }}\mathbb {F} _{2^{n}}^{*}}$ [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, ${\displaystyle \gcd(k,m)=1}$ 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 ${\displaystyle L(x)^{2^{i}}x+L(x)x^{2^{i}}}$ ${\displaystyle n=km,m>1,\gcd(n,i)=1,L(x)=\sum _{j=0}^{k-1}a_{j}x^{2^{jm}}}$ satisfies the conditions in Theorem 3.6 of [7] [7]
C12 ${\displaystyle ut(x)(x^{q}+x)+t(x)^{2^{2i}+2^{3i}}+at(x)^{2^{2i}}(x^{q}+x)^{2^{i}}+b(x^{q}+x)^{2^{i}+1}}$ ${\displaystyle n=2m,q=2^{m},\gcd(m,i)=1,t(x)=u^{q}x+x^{q}u}$, ${\displaystyle X^{2^{i}+1}+aX+b{\mbox{ has no solution over }}\mathbb {F} _{2^{m}}}$ [8]
C13 ${\displaystyle x^{3}+a(x^{2^{i}+1})^{2^{k}}+bx^{3\cdot 2^{m}}+c(x^{2^{i+m}+2^{m}})^{2^{k}}}$ ${\displaystyle n=2m=10,(a,b,c)=(\beta ,1,0,0),i=3,k=2,\beta {\text{ primitive in }}\mathbb {F} _{2^{2}}}$ [9]
${\displaystyle n=2m,m\ odd,3\nmid m,(a,b,c)=(\beta ,\beta ^{2},1),\beta {\text{ primitive in }}\mathbb {F} _{2^{2}}}$, ${\displaystyle i\in \{m-2,m,2m-1,(m-2)^{-1}\mod n\}}$
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2. Budaghyan L, Carlet C. Classes of quadratic APN trinomials and hexanomials and related structures. IEEE Transactions on Information Theory. 2008 May;54(5):2354-7.
3. Budaghyan L, Carlet C, Leander G. Constructing new APN functions from known ones. Finite Fields and Their Applications. 2009 Apr 1;15(2):150-9.
4. Budaghyan L, Carlet C, Leander G. On a construction of quadratic APN functions. InInformation Theory Workshop, 2009. ITW 2009. IEEE 2009 Oct 11 (pp. 374-378). IEEE.
5. Bracken C, Byrne E, Markin N, Mcguire G. A few more quadratic APN functions. Cryptography and Communications. 2011 Mar 1;3(1):43-53.
6. Zhou Y, Pott A. A new family of semifields with 2 parameters. Advances in Mathematics. 2013 Feb 15;234:43-60.
7. Budaghyan L, Calderini M, Carlet C, Coulter R, Villa I. Constructing APN functions through isotopic shift. Cryptology ePrint Archive, Report 2018/769
8. Taniguchi H. On some quadratic APN functions. Des. Codes Cryptogr. 2019, https://doi.org/10.1007/s10623-018-00598-2
9. Budaghyan L, Helleseth T, Kaleyski N. A new family of APN quadrinomials. Cryptology ePrint Archive, Report 2019/994