Difference between revisions of "Differentially 4-uniform permutations"

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<tr>
 
<tr>
<td><math>x^{-1} + \mathrm {Tr}^{n}_{1}(x+ (x^{-1}+1)^{-1})</math></td>
+
<td><math>x^{-1} + \mathrm {Tr}_{n}^{1}(x+ (x^{-1}+1)^{-1})</math></td>
 
<td><math>n=2t</math> is even</td>
 
<td><math>n=2t</math> is even</td>
 
<td><ref name="kai_ref">Y. Tan, L. Qu, C. H. Tan, C. Li. New Families of Differentially 4-Uniform Permutations over F(2<sup>2k</sup>). 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</ref></td>
 
<td><ref name="kai_ref">Y. Tan, L. Qu, C. H. Tan, C. Li. New Families of Differentially 4-Uniform Permutations over F(2<sup>2k</sup>). 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</ref></td>
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<tr>
 
<tr>
<td><math>x^{-1} + \mathrm {Tr}^{n}_{1}(x^{-3(2^{k}+1)}+ (x^{-1}+1)^{3(2^{k}+1)})</math></td>
+
<td><math>x^{-1} + \mathrm {Tr}_{n}^{1}(x^{-3(2^{k}+1)}+ (x^{-1}+1)^{3(2^{k}+1)})</math></td>
 
<td><math>n=2t</math> and <math>2\leq k \leq t-1</math></td>
 
<td><math>n=2t</math> and <math>2\leq k \leq t-1</math></td>
 
<td><ref name="kai_ref" /></td>
 
<td><ref name="kai_ref" /></td>
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<td><math>(x, x_n) \mapsto</math><br/><math>((1+x_{n})x^{-1}+x_{n}\alpha x^{-1}, f(x, x_{n}))</math></td>
 
<td><math>(x, x_n) \mapsto</math><br/><math>((1+x_{n})x^{-1}+x_{n}\alpha x^{-1}, f(x, x_{n}))</math></td>
  
<td> <math>n</math> is even <math> x, \alpha \in {\mathbb F} _ {2^{n-1}}, x_n \in {\mathbb F} _ {2}, \mathrm{Tr}^{n-1}_1(\alpha) = \mathrm{Tr}^{n-1}_1\left(\frac{1}{\alpha}\right) = 1,</math><br/><math> f(x, x_n)</math> is <math>(n, 1)-</math>function</td>
+
<td> <math>n</math> is even <math> 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><br/><math> f(x, x_n)</math> is <math>(n, 1)-</math>function</td>
  
  

Latest revision as of 22:12, 10 July 2020


Functions Conditions References
and t is odd [1][2]
and t is odd [3]
(inverse) [2][4]
and t is odd [5]
is odd, and is a primitive element in [6]
is even [7]
and [7]
is a quadratic APN permutation on

[8]
t is odd [9]
is even or are odd, [10]
are odd, [11]

is even
is function
[12]
  1. 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. 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
  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. 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
  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