# Differentially 4-uniform permutations

Functions Conditions References
${\displaystyle x^{2^{i}+1}}$ ${\displaystyle gcd(i,n)=2,n=2t}$ and t is odd [1][2]
${\displaystyle x^{2^{2i}-2^{i}+1}}$ ${\displaystyle gcd(i,n)=2,n=2t}$ and t is odd [3]
${\displaystyle x^{2^{n}-2}}$ ${\displaystyle n=2t}$ (inverse) [2][4]
${\displaystyle x^{2^{2t}-2^{t}+1}}$ ${\displaystyle n=4t}$ and t is odd [5]
${\displaystyle \alpha x^{2^{s}+1}+\alpha ^{2^{t}}x^{{2-t}+2^{t+s}}}$ ${\displaystyle n=3t,t/2}$ is odd, ${\displaystyle gcd(n,s)=2,3|t+s}$ and ${\displaystyle \alpha }$ is a primitive element in ${\displaystyle \mathbb {F} _{2^{n}}}$ [6]
${\displaystyle x^{-1}+\mathrm {Tr} _{1}^{n}(x+(x^{-1}+1)^{-1})}$ ${\displaystyle n=2t}$ is even [7]
${\displaystyle x^{-1}+\mathrm {Tr} _{1}^{n}(x^{-3(2^{k}+1)}+(x^{-1}+1)^{3(2^{k}+1)})}$ ${\displaystyle n=2t}$ and ${\displaystyle 2\leq k\leq t-1}$ [7]
${\displaystyle L_{u}(F^{-1}(x))|_{H_{u}}}$ ${\displaystyle n=2t,F(x)}$ is a quadratic APN permutation on ${\displaystyle {\mathbb {F} }_{2^{n+1}},u\in {\mathbb {F} }_{2^{n+1}}^{*},}$
${\displaystyle L_{u}(x)=F(x)+F(x+u)+F(u),}$
${\displaystyle H_{u}=\{L_{u}(x)|x\in {\mathbb {F} }_{2^{n+1}}\}}$
[8]
${\displaystyle \displaystyle \sum _{i=0}^{2^{n}-3}x^{i}}$ ${\displaystyle n=2t,}$ t is odd [9]
${\displaystyle x^{-1}+t(x^{2^{s}}+x)^{2^{sn}-1}}$ ${\displaystyle s}$ is even ${\displaystyle ,t\in {\mathbb {F} }_{2^{s}}^{*},}$ or ${\displaystyle s,n}$ are odd, ${\displaystyle t\in {\mathbb {F} }_{2^{s}}^{*}}$ [10]
${\displaystyle x^{2^{k}+1}+t(x^{2^{s}}+x)^{2^{sn}-1}}$ ${\displaystyle n,s}$ are odd, ${\displaystyle t\in {\mathbb {F} }_{2^{s}}^{*},gcd(k,sn)=1}$ [11]
${\displaystyle (x,x_{n})\mapsto }$
${\displaystyle ((1+x_{n})x^{-1}+x_{n}\alpha x^{-1},f(x,x_{n}))}$
${\displaystyle n}$ is even ${\displaystyle x,\alpha \in {\mathbb {F} }_{2^{n-1}},x_{n}\in {\mathbb {F} }_{2},\mathrm {Tr} _{1}^{n-1}(\alpha )=\mathrm {Tr} _{1}^{n-1}\left({\frac {1}{\alpha }}\right)=1,}$
${\displaystyle f(x,x_{n})}$ is ${\displaystyle (n,1)-}$function
[12]
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2. K. Nyberg. Differentially uniform mappings for cryptography. Advances in Cryptography, EUROCRYPT’93, Lecture Notes in Computer Science 765, pp. 55-64, 1994. Lecture Notes in Computer Science, vol 765. Springer, Berlin, Heidelberg https://doi.org/10.1007/3-540-48285-7_6
3. T. Kasami. The weight enumerators for several classes of subcodes of the second order binary Reed-Muller codes. Inform. and Control, 18, pp. 369-394, 1971. https://doi.org/10.1016/S0019-9958(71)90473-6
4. G. Lachaud, J. Wolfmann. The weights of the orthogonals of the extended quadratic binary Goppa codes. IEEE Trans. Inf. Theory, vol. 36, no. 3, pp. 686-692, 1990. https://doi.org/10.1109/18.54892
5. C. Bracken, G. Leander. A highly nonlinear differentially 4 uniform power mapping that permutes fields of even degree. Finite Fields and Their Applications, vol. 16, no. 4, pp. 231-242, 2010. https://doi.org/10.1016/j.ffa.2010.03.001
6. C. Bracken, C. H. Tan, Y. Tan. Binomial differentially 4 uniform permutations with high nonlinearity. Finite Fields and Their Applications, vol. 18, no. 3, pp. 537-546, 2012. https://doi.org/10.1016/j.ffa.2011.11.006
7. Y. Tan, L. Qu, C. H. Tan, C. Li. New Families of Differentially 4-Uniform Permutations over F(22k). In: T. Helleseth, J. Jedwab (eds) Sequences and Their Applications - SETA 2012. SETA 2012. Lecture Notes in Computer Science, vol. 7280. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30615-0_3
8. Y. Li, M. Wang. Constructing differentially 4-uniform permutations over GF(22m) from quadratic APN permutations over GF(22m+1). Designs, Codes and Cryptography, vol. 72, pp. 249-264, 2014. https://doi.org/10.1007/s10623-012-9760-9
9. Y. Yu, M. Wang, Y. Li. Constructing Differentially 4 Uniform Permutations from Known Ones. Chinese Journal of Electronics, vol. 22, no. 3, pp. 495-499, 2013.
10. Z. Zha, L. Hu, S. Sun. Constructing new differentially 4-uniform permutations from the inverse function. Finite Fields and Their Applications, vol. 25, pp. 64-78, 2014. https://doi.org/10.1016/j.ffa.2013.08.003
11. Xu G, Cao X, Xu S. Constructing new differentially 4-uniform permutations and APN functions over finite fields. Cryptography and Communications-Discrete Structures, Boolean Functions and Sequences. Pre-print. 2014.
12. C. Carlet, D. Tang, X. Tang, Q. Liao. New Construction of Differentially 4-Uniform Bijections. In: D. Lin, S. Xu, M. Yung (eds) Information Security and Cryptology. Inscrypt 2013. Lecture Notes in Computer Science, vol. 8567, Springer, Cham. https://doi.org/10.1007/978-3-319-12087-4_2