Differentially 4-uniform permutations: Difference between revisions

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<td><math>x^{2^i+1}</math></td>
<td><math>x^{2^i+1}</math></td>
<td><math>gcd(i,n) = 2, n = 2t</math> and t is odd</td>
<td><math>gcd(i,n) = 2, n = 2t</math> and t is odd</td>
<td><ref>Gold R. Maximal recursive sequences with 3-valued recursive cross-correlation functions (Corresp.). IEEE transactions on Information Theory. 1968 Jan;14(1):154-6.</ref><ref name="kaisa_ref">Nyberg K. Differentially uniform mappings for cryptography. InWorkshop on the Theory and Application of of Cryptographic Techniques 1993 May 23 (pp. 55-64).</ref></td>
<td><ref>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</ref><ref name="kaisa_ref">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</ref></td>
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<td><math>x^{2^{2i}-2^i+1}</math></td>
<td><math>x^{2^{2i}-2^i+1}</math></td>
<td><math>gcd(i,n) = 2, n = 2t</math> and t is odd</td>
<td><math>gcd(i,n) = 2, n = 2t</math> and t is odd</td>
<td><ref>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.</ref></td>
<td><ref>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</ref></td>
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<td><math>x^{2^n-2}</math></td>
<td><math>x^{2^n-2}</math></td>
<td><math> n = 2t</math> (inverse)</td>
<td><math> n = 2t</math> (inverse)</td>
<td><ref name="kaisa_ref" /><ref>Lachaud G, Wolfmann J. The weights of the orthogonals of the extended quadratic binary Goppa codes. IEEE transactions on information theory. 1990 May;36(3):686-92.</ref></td>
<td><ref name="kaisa_ref" /><ref>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</ref></td>
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<td><math>x^{2^{2t}-2^t+1}</math></td>
<td><math>x^{2^{2t}+2^t+1}</math></td>
<td><math>n = 4t</math> and t is odd</td>
<td><math>n = 4t</math> and t is odd</td>
<td><ref>Bracken C, Leander G. A highly nonlinear differentially 4 uniform power mapping that permutes fields of even degree. Finite Fields and Their Applications. 2010 Jul 1;16(4):231-42.</ref></td>
<td><ref>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</ref></td>
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<td><math>\alpha x^{2^s+1}+\alpha^{2^t}x^{{2-t}+2^{t+s}}</math></td>
<td><math>\alpha x^{2^s+1}+\alpha^{2^t}x^{{2-t}+2^{t+s}}</math></td>
<td> <math>n = 3t, t/2</math> is odd, <math>gcd(n,s) = 2, 3|t + s</math> and <math>\alpha</math> is a primitive element in <math>\mathbb{F}_{2^n}</math></td>
<td> <math>n = 3t, t/2</math> is odd, <math>gcd(n,s) = 2, 3|t + s</math> and <math>\alpha</math> is a primitive element in <math>\mathbb{F}_{2^n}</math></td>
<td><ref>Bracken C, Tan CH, Tan Y. Binomial differentially 4 uniform permutations with high nonlinearity. Finite Fields and Their Applications. 2012 May 1;18(3):537-46.</ref></td>
<td><ref>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</ref></td>
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<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">Tan Y, Qu L, Tan CH, Li C. New Families of Differentially 4-Uniform Permutations over <math>{\mathbb F} _ {2^{2k}} </math>. InInternational Conference on Sequences and Their Applications 2012 Jun 4 (pp. 25-39). Springer, Berlin, Heidelberg.</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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<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>L_u(F^{-1}(x))|_{H_u}</math></td>
<td><math>L_u(F^{-1}(x))|_{H_u}</math></td>
<td><math>n=2t,F(x)</math> is a quadratic APN permutation on <math>{\mathbb F} _ {2^{n+1}}, u \in {\mathbb F}^{*}_{2^{n+1}},</math><br/><math>L_u(x)= F(x)+F(x+u)+F(u),</math><br/><math> H_u = \{L_u(x)|x \in {\mathbb F} _ {2^{n+1}}\}</math></td>
<td><math>n=2t,F(x)</math> is a quadratic APN permutation on <math>{\mathbb F} _ {2^{n+1}}, u \in {\mathbb F}^{*}_{2^{n+1}},</math><br/><math>L_u(x)= F(x)+F(x+u)+F(u),</math><br/><math> H_u = \{L_u(x)|x \in {\mathbb F} _ {2^{n+1}}\}</math></td>
<td><ref>Li Y, Wang M. Constructing differentially 4-uniform permutations over<math>{\mathbb F} _ {2^{2m}} </math> from quadratic APN permutations over <math>{\mathbb F} _ {2^{2m+1}}</math>. Designs, Codes and Cryptography. 2014 Aug 1;72(2):249-64.</ref></td>
<td><ref>Y. Li, M. Wang. Constructing differentially 4-uniform permutations over GF(2<sup>2m</sup>) from quadratic APN permutations over GF(2<sup>2m+1</sup>). Designs, Codes and Cryptography, vol. 72, pp. 249-264, 2014. https://doi.org/10.1007/s10623-012-9760-9</ref></td>
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<td><math>\displaystyle\sum_{i=0}^{2^{n}-3} x^{i}</math></td>
<td><math>\displaystyle\sum_{i=0}^{2^{n}-3} x^{i}</math></td>
<td><math>n=2t, </math> t is odd</td>
<td><math>n=2t, </math> t is odd</td>
<td><ref>Yu Y, Wang M, Li Y. Constructing low differential uniformity functions from known ones. Chinese Journal of Electronics. 2013;22(3):495-9.</ref></td>
<td><ref>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.</ref></td>
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<td><math>x^{-1} + t(x^{2^{s}}+x)^{2^{sn}-1}</math></td>
<td><math>x^{-1} + t(x^{2^{s}}+x)^{2^{sn}-1}</math></td>
<td> <math> s</math> is even <math>, t \in {\mathbb F}^{*} _ {2^{s}},</math> or <math>s, n</math> are odd, <math>t \in {\mathbb F}^{*} _ {2^{s}}</math> </td>
<td> <math> s</math> is even <math>, t \in {\mathbb F}^{*} _ {2^{s}},</math> or <math>s, n</math> are odd, <math>t \in {\mathbb F}^{*} _ {2^{s}}</math> </td>
<td><ref>Zha Z, Hu L, Sun S. Constructing new differentially 4-uniform permutations from the inverse function. Finite Fields and Their Applications. 2014 Jan 1;25:64-78.</ref></td>
<td><ref>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</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>




<td><ref>Carlet C, Tang D, Tang X, Liao Q. New construction of differentially 4-uniform bijections. InInternational Conference on Information Security and Cryptology 2013 Nov 27 (pp. 22-38). Springer, Cham.</ref></td>
<td><ref>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</ref></td>
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Latest revision as of 15:10, 7 March 2022


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]
  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
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