Differentially 4-uniform permutations
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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][2] |
| [math]\displaystyle{ x^{2^{2i}-2^i+1} }[/math] | [math]\displaystyle{ gcd(i,n) = 2, n = 2t }[/math] and t is odd | [3] |
| [math]\displaystyle{ x^{2^n-2} }[/math] | [math]\displaystyle{ n = 2t }[/math] (inverse) | [2][4] |
| [math]\displaystyle{ x^{2^{2t}+2^t+1} }[/math] | [math]\displaystyle{ n = 4t }[/math] and t is odd | [5] |
| [math]\displaystyle{ \alpha x^{2^s+1}+\alpha^{2^t}x^{{2-t}+2^{t+s}} }[/math] | [math]\displaystyle{ n = 3t, t/2 }[/math] is odd, [math]\displaystyle{ gcd(n,s) = 2, 3|t + s }[/math] and [math]\displaystyle{ \alpha }[/math] is a primitive element in [math]\displaystyle{ \mathbb{F}_{2^n} }[/math] | [6] |
| [math]\displaystyle{ x^{-1} + \mathrm {Tr}_{n}^{1}(x+ (x^{-1}+1)^{-1}) }[/math] | [math]\displaystyle{ n=2t }[/math] is even | [7] |
| [math]\displaystyle{ x^{-1} + \mathrm {Tr}_{n}^{1}(x^{-3(2^{k}+1)}+ (x^{-1}+1)^{3(2^{k}+1)}) }[/math] | [math]\displaystyle{ n=2t }[/math] and [math]\displaystyle{ 2\leq k \leq t-1 }[/math] | [7] |
| [math]\displaystyle{ L_u(F^{-1}(x))|_{H_u} }[/math] | [math]\displaystyle{ n=2t,F(x) }[/math] is a quadratic APN permutation on [math]\displaystyle{ {\mathbb F} _ {2^{n+1}}, u \in {\mathbb F}^{*}_{2^{n+1}}, }[/math] [math]\displaystyle{ L_u(x)= F(x)+F(x+u)+F(u), }[/math] [math]\displaystyle{ H_u = \{L_u(x)|x \in {\mathbb F} _ {2^{n+1}}\} }[/math] |
[8] |
| [math]\displaystyle{ \displaystyle\sum_{i=0}^{2^{n}-3} x^{i} }[/math] | [math]\displaystyle{ n=2t, }[/math] t is odd | [9] |
| [math]\displaystyle{ x^{-1} + t(x^{2^{s}}+x)^{2^{sn}-1} }[/math] | [math]\displaystyle{ s }[/math] is even [math]\displaystyle{ , t \in {\mathbb F}^{*} _ {2^{s}}, }[/math] or [math]\displaystyle{ s, n }[/math] are odd, [math]\displaystyle{ t \in {\mathbb F}^{*} _ {2^{s}} }[/math] | [10] |
| [math]\displaystyle{ x^{2^{k}+1} + t(x^{2^{s}}+x)^{2^{sn}-1} }[/math] | [math]\displaystyle{ n, s }[/math] are odd, [math]\displaystyle{ t \in {\mathbb F}^{*} _ {2^{s}}, gcd(k, sn) = 1 }[/math] | [11] |
| [math]\displaystyle{ (x, x_n) \mapsto }[/math] [math]\displaystyle{ ((1+x_{n})x^{-1}+x_{n}\alpha x^{-1}, f(x, x_{n})) }[/math] |
[math]\displaystyle{ n }[/math] is even [math]\displaystyle{ x, \alpha \in {\mathbb F} _ {2^{n-1}}, x_n \in {\mathbb F} _ {2}, \mathrm{Tr}_{n-1}(\alpha) = \mathrm{Tr}_{n-1}\left(\frac{1}{\alpha}\right) = 1, }[/math] [math]\displaystyle{ f(x, x_n) }[/math] is [math]\displaystyle{ (n, 1)- }[/math]function |
[12] |
- ↑ R. Gold. Maximal recursive sequences with 3-valued recursive cross-correlation functions. IEEE Trans. Inf. Theory, 14, pp. 154-156, 1968. https://doi.org/10.1109/TIT.1968.1054106
- ↑ 2.0 2.1 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
- ↑ 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
- ↑ 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
- ↑ 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
- ↑ 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
- ↑ 7.0 7.1 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
- ↑ 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
- ↑ 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.
- ↑ 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
- ↑ 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.
- ↑ 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