Known inifinte families of quadratic APN polynomials over GF(2^n)
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| [math]\displaystyle{ N^\circ }[/math] | Functions | Conditions | References |
|---|---|---|---|
| C1-C2 | [math]\displaystyle{ x^{2^s+1}+u^{2^k-1}x^{2^{ik}+2^{mk+s}} }[/math] | [math]\displaystyle{ n = pk, \gcd(k,3) = \gcd(s,3k) = 1, p \in \{3,4\}, i = sk \mod p, m = p -i, n \ge 12, u \text{ primitive in } \mathbb{F}_{2^n}^* }[/math] | [1] |
| C3 | [math]\displaystyle{ sx^{q+1}+x^{2^i+1}+x^{q(2^i+1)}+cx^{2^iq+1}+c^qx^{2^i+q} }[/math] | [math]\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 }[/math] s.t. [math]\displaystyle{ x^{q+1}=1 }[/math] | [2] |
| C4 | [math]\displaystyle{ x^3+a^{-1} \mathrm {Tr}_n (a^3x^9) }[/math] | [math]\displaystyle{ a\neq 0 }[/math] | [3] |
| C5 | [math]\displaystyle{ x^3+a^{-1} \mathrm {Tr}_n^3 (a^3x^9+a^6x^{18}) }[/math] | [math]\displaystyle{ 3|n }[/math], [math]\displaystyle{ a\ne0 }[/math] | [4] |
| C6 | [math]\displaystyle{ x^3+a^{-1} \mathrm{Tr}_n^3(a^6x^{18}+a^{12}x^{36}) }[/math] | [math]\displaystyle{ 3|n, a \ne 0 }[/math] | [4] |
| C7-C9 | [math]\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}} }[/math] | [math]\displaystyle{ 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]\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}) }[/math] | [math]\displaystyle{ n=2m, m\geqslant 2 }[/math] even, [math]\displaystyle{ \gcd(k, m)=1, }[/math] and [math]\displaystyle{ i \geqslant 2 }[/math] even [math]\displaystyle{ u\text{ primitive in } \mathbb{F}_{2^n}^*, u' \in \mathbb{F}_{2^m} \text{ not a cube } }[/math] | [6] |
| C11 | [math]\displaystyle{ a^2x^{2^{2m+1}+1}+b^2x^{2^{m+1}+1}+ax^{2^{2m}+2}+bx^{2^{m}+2}+(c^2+c)x^3 }[/math] | [math]\displaystyle{ n=3m, m \ \text{odd}\ L(x)=ax^{2^{2m}}+bx^{2m}+cx \ \text{satisfies the conditions in lemma 8 of}\ [3] }[/math] | [7] |
- ↑ 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
- ↑ 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.
- ↑ 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.
- ↑ 4.0 4.1 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.
- ↑ Bracken, C., Byrne, E., Markin, N., & Mcguire, G. (2011). A few more quadratic APN functions. Cryptography and Communications, 3(1), 43-53.
- ↑ Göloğlu, Faruk. Almost perfect nonlinear trinomials and hexanomials. Finite Fields and Their Applications 33 (2015): 258-282.
- ↑ Villa, I., Budaghyan, L., Calderini, M., Carlet, C., & Coulter, R. On Isotopic Construction of APN Functions. SETA 2018