Differentially 4-uniform permutations

Functions	Conditions	References
$x^{2^{i}+1}$	$gcd(i,n)=2,n=2t$ and t is odd	^[1]^[2]
$x^{2^{2i}-2^{i}+1}$	$gcd(i,n)=2,n=2t$ and t is odd	^[3]
$x^{2^{n}-2}$	$n=2t$ (inverse)	^[2]^[4]
$x^{2^{2t}-2^{t}+1}$	$n=4t$ and t is odd	^[5]
$\alpha x^{2^{s}+1}+\alpha ^{2^{t}}x^{{2-t}+2^{t+s}}$	$n=3t,t/2$ is odd, $gcd(n,s)=2,3\|t+s$ 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]
$L_{u}(F^{-1}(x))\|_{H_{u}}$	$n=2t,F(x)$ is a quadratic APN permutation on ${\mathbb {F} }_{2^{n+1}},u\in {\mathbb {F} ^{*}}_{2^{n+1}}$	^[8]
$\displaystyle \sum _{i=0}^{2^{n}-3}x^{i}$	$n=2t,$ t is odd	^[9]
$x^{-1}+t(x^{2^{s}}+x)^{2^{sn}-1}$	$s$ is even $t\in {\mathbb {F} }_{2^{s}}^{},$ or $s,n$ are odd, $t\in {\mathbb {F} }_{2^{s}}^{}$	^[10]
$x^{2^{k}+1}+t(x^{2^{s}}+x)^{2^{sn}-1}$	$n,s$ are odd, $t\in {\mathbb {F} }_{2^{s}}^{*},gcd(k,sn)=1$	^[11]
$(x,x_{n})\to$ $((1+x_{n})x^{-1}+x_{n}\alpha x^{-1},f(x,x_{n}))$	$n$ is even $x,\alpha \in {\mathbb {F} }_{2^{n-1}},x_{n}\in {\mathbb {F} }_{2},\mathrm {Tr} _{1}^{n-1}(\alpha )=\mathrm {Tr} _{1}^{n-1}({\frac {1}{\alpha }})=1,$ $f(x,x_{n})$ is $(n,1)-$ function	^[12]

↑ 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 over ${\mathbb {F} }_{2^{2m}}$ from quadratic APN permutations over ${\mathbb {F} }_{2^{2m+1}}$ . Designs, Codes and Cryptography. 2014 Aug 1;72(2):249-64.
↑ Yu Y, Wang M, Li Y. Constructing low differential uniformity functions from known ones. Chinese Journal of Electronics. 2013;22(3):495-9.
↑ 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.
↑ 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.
↑ 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.

[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 over ${\mathbb {F} }_{2^{2m}}$ from quadratic APN permutations over ${\mathbb {F} }_{2^{2m+1}}$ . 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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Differentially 4-uniform permutations

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