# 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 6.3 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 ${\displaystyle x^{3}+a(x^{2^{i}+1})^{2^{k}}+bx^{3\cdot 2^{m}}+c(x^{2^{i+m}+2^{m}})^{2^{k}}}$ ${\displaystyle n=2m=10,(a,b,c)=(\beta ,0,0),i=3,k=2,\beta {\text{ primitive in }}\mathbb {F} _{2^{2}}}$ [9]
${\displaystyle n=2m,m\ odd,3\nmid m,(a,b,c)=(\beta ,\beta ^{2},1),\beta {\text{ primitive in }}\mathbb {F} _{2^{2}}}$, ${\displaystyle i\in \{m-2,m,2m-1,(m-2)^{-1}\mod n\}}$
1. L. Budaghyan, C. Carlet, G. Leander. Two classes of quadratic APN binomials inequivalent to power functions. IEEE Trans. Inform. Theory, 54(9), pp. 4218-4229, 2008. https://doi.org/10.1109/TIT.2008.928275
2. L. Budaghyan and C. Carlet. Classes of Quadratic APN Trinomials and Hexanomials and Related Structures. IEEE Trans. Inform. Theory, vol. 54, no. 5, pp. 2354-2357, May 2008. https://doi.org/10.1109/TIT.2008.920246
3. L. Budaghyan, C. Carlet, G. Leander. Constructing new APN functions from known ones. Finite Fields and Their Applications, v. 15, issue 2, pp. 150-159, April 2009. https://doi.org/10.1016/j.ffa.2008.10.001
4. L. Budaghyan, C. Carlet, G. Leander. On a construction of quadratic APN functions. Proceedings of IEEE Information Theory Workshop, ITW’09, pp. 374-378, Taormina, Sicily, Oct. 2009. https://doi.org/10.1109/ITW.2009.5351383
5. C. Bracken, E. Byrne, N. Markin, G. McGuire. A few more quadratic APN functions. Cryptography and Communications 3, pp. 45-53, 2008. https://doi.org/10.1007/s12095-010-0038-7
6. Y. Zhou, A. Pott. A new family of semifields with 2 parameters. Advances in Mathematics, v. 234, pp. 43-60, 2013. https://doi.org/10.1016/j.aim.2012.10.014
7. L. Budaghyan, M. Calderini, C. Carlet, R. S. Coulter, I. Villa. Constructing APN Functions through Isotopic Shifts. IEEE Trans. Inform. Theory, early access article. https://doi.org/10.1109/TIT.2020.2974471
8. H. Taniguchi. On some quadratic APN functions. Designs, Codes and Cryptography 87, pp. 1973-1983, 2019. https://doi.org/10.1007/s10623-018-00598-2
9. L. Budaghyan, T. Helleseth, N. Kaleyski. A new family of APN quadrinomials. IEEE Trans. Inf. Theory, early access article. https://doi.org/10.1109/TIT.2020.3007513