Differentially 4-uniform permutation

Functions	Conditions	References
$x^{2^{i}+1}$	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle gcd(i,n)=2,n=2t} and t is odd	^[1]^[2]
$x^{2^{2i}-2^{i}+1}$	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle gcd(i,n)=2,n=2t} and t is odd	^[3]
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x^{2^{n}-2}}	$n=2t$ (inverse)	^[2]^[4]
$x^{2^{2t}-2^{t}+1}$	$n=4t$ and t is odd	^[5]
$\alpha x^{2^{s}+1}+\alpha ^{2^{t}}x^{{2-t}+2^{t+s}}$	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle n=3t,t/2} is odd, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle gcd(n,s)=2,3\|t+s} and $\alpha$ is a primitive element in $\mathbb {F} _{2^{n}}$	^[6]
$x^{-1}+\mathrm {Tr} (x+(x^{-1}+1)^{-1})$	$n=2t$ is even	^[7]
$x^{-1}+\mathrm {Tr} (x^{-3(2^{k}+1)}+(x^{-1}+1)^{3(2^{k}+1)})$	$n=2t$ and $2\leq k\leq t-1$	^[7]
$L_{u}(F^{-1}(x))\|_{H_{u}}$	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle n=2t,F(x)} is a quadratic APN permutation on ${\mathbb {F} }_{2^{n+1}},u\in {\mathbb {F} ^{*}}_{2^{n+1}}$	^[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 ${\mathbb {F} }_{2^{2k}}$ . 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 overFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb F} _ {2^{2m}} } from quadratic APN permutations over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb F} _ {2^{2m+1}}} . Designs, Codes and Cryptography. 2014 Aug 1;72(2):249-64.

[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.

[kaisa_ref-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.

[kai_ref-7] 7.0 ^7.1 Tan Y, Qu L, Tan CH, Li C. New Families of Differentially 4-Uniform Permutations over ${\mathbb {F} }_{2^{2k}}$ . 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 overFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb F} _ {2^{2m}} } from quadratic APN permutations over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb F} _ {2^{2m+1}}} . Designs, Codes and Cryptography. 2014 Aug 1;72(2):249-64.

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