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

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<math>N^\circ</math> Functions Conditions References
C1-C2 <math>x^{2^s+1}+u^{2^k-1}x^{2^{ik}+2^{mk+s}}</math> <math>n = pk, \gcd(k,3) = \gcd(s,3k) = 1, p \in \{3,4\}, i = sk\bmod p, m = p -i, n \ge 12, u \text{ primitive in } \mathbb{F}_{2^n}^*</math> [1]
C3 <math>sx^{q+1}+x^{2^i+1}+x^{q(2^i+1)}+cx^{2^iq+1}+c^qx^{2^i+q}</math> <math>q=2^m, n=2m,</math> <math>gcd(i,m)=1</math>, <math>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</math> s.t. <math>x^{q+1}=1</math> [2]
C4 <math>x^3+a^{-1} \mathrm {Tr}_n (a^3x^9)</math> <math>a\neq 0</math> [3]
C5 <math>x^3+a^{-1} \mathrm {Tr}_n^3 (a^3x^9+a^6x^{18})</math> <math>3|n </math>, <math>a\ne0</math> [4]
C6 <math>x^3+a^{-1} \mathrm{Tr}_n^3(a^6x^{18}+a^{12}x^{36})</math> <math>3|n, a \ne 0</math> [4]
C7-C9 <math>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>n=3k, \gcd(k,3)=\gcd(s,3k)=1, v, w\in\mathbb{F}_{2^k}, vw \ne 1, 3|(k+s), u \text{ primitive in } \mathbb{F}_{2^n}^* </math> [5]
C10 <math>(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>n=2m, m\geqslant 2</math> even, <math>\gcd(k, m)=1</math> and <math> i \geqslant 2</math> even, <math>u\text{ primitive in } \mathbb{F}_{2^n}^*, u' \in \mathbb{F}_{2^m} \text{ not a cube }</math> [6]
C11 <math>L(x)^{2^i}x+L(x)x^{2^i}</math> <math>n=km, \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] [7]
C12 <math>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>n=2m, q=2^m, \gcd(m,i)=1, t(x)=u^qx+x^qu, X^{2^i+1}+aX+b \mbox{ has no solution over }\mathbb{F}_{2^m}</math> [8]
C13 <math>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>n = 2m = 10, (a,b,c) = (\beta,1,0,0), i = 3, k = 2, \beta \text{ primitive in } \mathbb{F}_{2^2}</math> [9]
<math>n = 2m, m\ odd, 3 \nmid m, (a,b,c) = (\beta, \beta^2, 1), \beta \text{ primitive in } \mathbb{F}_{2^2}, i \in \{ m-2, m, 2m-1, (m-2)^{-1} \mod n \}</math>
  1. ↑ Budaghyan L, Carlet C, Leander G. Two classes of quadratic APN binomials inequivalent to power functions. IEEE Transactions on Information Theory. 2008 Sep;54(9):4218-29.
  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. ↑ 4.0 4.1 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. ↑ Villa I, Budaghyan L, Calderini M, Carlet C, Coulter R. 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
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