Differentially 4-uniform permutation
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| Functions | Conditions | References |
|---|---|---|
| [math]\displaystyle{ x^{2^i+1} }[/math] | [math]\displaystyle{ gcd(i,n) = 2, n = 2t }[/math] and t is odd | [1] |
| [math]\displaystyle{ x^{2^{2i}-2^i+1} }[/math] | [math]\displaystyle{ gcd(i,n) = 2, n = 2t }[/math] and t is odd | [2] |
| [math]\displaystyle{ x^3+a^{-1} \mathrm {Tr}_n (a^3x^9) }[/math] | [math]\displaystyle{ a\neq 0 }[/math] | [3] |
| [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] |
| [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] |
| [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] |
| [math]\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}}) }[/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] |
| [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^{2^{m}}+cx }[/math] satisfies the conditions in Lemma 8 of [7] | [7] |
- ↑ Gold R. Maximal recursive sequences with 3-valued recursive cross-correlation functions (Corresp.). IEEE transactions on Information Theory. 1968 Jan;14(1):154-6.
- ↑ Kasami T. The weight enumerators for several classes of subcodes of the 2nd order binary Reed-Muller codes. Information and Control. 1971 May 1;18(4):369-94.
- ↑ 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.
- ↑ 4.0 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.
- ↑ Bracken C, Byrne E, Markin N, Mcguire G. A few more quadratic APN functions. Cryptography and Communications. 2011 Mar 1;3(1):43-53.
- ↑ Zhou Y, Pott A. A new family of semifields with 2 parameters. Advances in Mathematics. 2013 Feb 15;234:43-60.
- ↑ Villa I, Budaghyan L, Calderini M, Carlet C, & Coulter R. On Isotopic Construction of APN Functions. SETA 2018