Differentially 4-uniform permutation: Difference between revisions
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<td><math>x^{-1} + \mathrm {Tr}(x+ (x^{-1}+1)^{-1})</math></td> | <td><math>x^{-1} + \mathrm {Tr}(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" | <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><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> | ||
</tr> | </tr> | ||
</table> | </table> | ||
Revision as of 11:09, 13 June 2019
| 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}(x+ (x^{-1}+1)^{-1}) }[/math] | [math]\displaystyle{ n=2t }[/math] is even | [7] |
| [math]\displaystyle{ x^{-1} + \mathrm {Tr}(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] | [8] |
- ↑ Gold R. Maximal recursive sequences with 3-valued recursive cross-correlation functions (Corresp.). IEEE transactions on Information Theory. 1968 Jan;14(1):154-6.
- ↑ 2.0 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).
- ↑ 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.
- ↑ 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.
- ↑ 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.
- ↑ 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.
- ↑ 7.0 7.1 Tan Y, Qu L, Tan CH, Li C. New Families of Differentially 4-Uniform Permutations over [math]\displaystyle{ {\mathbb F} _ {2^{2k}} }[/math]. InInternational Conference on Sequences and Their Applications 2012 Jun 4 (pp. 25-39). Springer, Berlin, Heidelberg.
- ↑ Li Y, Wang M. Constructing differentially 4-uniform permutations over[math]\displaystyle{ {\mathbb F} _ {2^{2m}} }[/math] from quadratic APN permutations over [math]\displaystyle{ {\mathbb F} _ {2^{2m+1}} }[/math]. Designs, Codes and Cryptography. 2014 Aug 1;72(2):249-64.