Differentially 4-uniform permutations: Difference between revisions

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Created page with "<table> <tr> <th>Functions</th> <th>Conditions</th> <th>References</th> </tr> <tr> <td><math>x^{2^i+1}</math></td> <td><math>gcd(i,n) = 2, n = 2t</math> and t is odd</td> <td..."
 
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<td><math>x^{-1} + \mathrm {Tr}(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">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>
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<td><math>x^{-1} + \mathrm {Tr}(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></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>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>
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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>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>
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<td><math>(x, x_n) \to</math>
<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>
<math>((1+x_{n})x^{-1}+x_{n}\alpha x^{-1}, f(x, x_{n}))</math>
 
<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(\frac{1}{\alpha}) = 1,</math>
<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>
<math> f(x, x_n)</math> is <math>(n, 1)-</math>function
 


<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>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>
</tr>
</tr>
</table>
</table>

Revision as of 14:24, 13 June 2019


Functions Conditions References
x2i+1 gcd(i,n)=2,n=2t and t is odd [1][2]
x22i−2i+1 gcd(i,n)=2,n=2t and t is odd [3]
x2n−2 n=2t (inverse) [2][4]
x22t−2t+1 n=4t and t is odd [5]
αx2s+1+α2tx2−t+2t+s n=3t,t/2 is odd, gcd(n,s)=2,3|t+s and α is a primitive element in 𝔽2n [6]
x−1+Tr1n(x+(x−1+1)−1) n=2t is even [7]
x−1+Tr1n(x−3(2k+1)+(x−1+1)3(2k+1)) n=2t and 2≤k≤t−1 [7]
Lu(F−1(x))|Hu n=2t,F(x) is a quadratic APN permutation on 𝔽2n+1,u∈𝔽2n+1∗,
Lu(x)=F(x)+F(x+u)+F(u),
Hu={Lu(x)|x∈𝔽2n+1}		 [8]
∑i=02n−3xi n=2t, t is odd [9]
x−1+t(x2s+x)2sn−1 s is even ,t∈𝔽2s∗, or s,n are odd, t∈𝔽2s∗ [10]
x2k+1+t(x2s+x)2sn−1 n,s are odd, t∈𝔽2s∗,gcd(k,sn)=1 [11]
(x,xn)↦
((1+xn)x−1+xnαx−1,f(x,xn))		 n is even x,α∈𝔽2n−1,xn∈𝔽2,Tr1n−1(α)=Tr1n−1(1α)=1,
f(x,xn) is (n,1)−function		 [12]
  1. ↑ Gold R. Maximal recursive sequences with 3-valued recursive cross-correlation functions (Corresp.). IEEE transactions on Information Theory. 1968 Jan;14(1):154-6.
  2. ↑ 2.0 2.1 Nyberg K. Differentially uniform mappings for cryptography. InWorkshop on the Theory and Application of of Cryptographic Techniques 1993 May 23 (pp. 55-64).
  3. ↑ 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.
  4. ↑ 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.
  5. ↑ 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.
  6. ↑ 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.
  7. ↑ 7.0 7.1 Tan Y, Qu L, Tan CH, Li C. New Families of Differentially 4-Uniform Permutations over 𝔽22k. InInternational Conference on Sequences and Their Applications 2012 Jun 4 (pp. 25-39). Springer, Berlin, Heidelberg.
  8. ↑ Li Y, Wang M. Constructing differentially 4-uniform permutations over𝔽22m from quadratic APN permutations over 𝔽22m+1. Designs, Codes and Cryptography. 2014 Aug 1;72(2):249-64.
  9. ↑ Yu Y, Wang M, Li Y. Constructing low differential uniformity functions from known ones. Chinese Journal of Electronics. 2013;22(3):495-9.
  10. ↑ 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.
  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. ↑ 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.
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