A highly nonlinear differentially 4 uniform power mapping that permutes fields of even degree

Functions with low differential uniformity can be used as the s-boxes of symmetric cryptosystems as they have good resistance to differential attacks. The AES (Advanced Encryption Standard) uses a differentially 4 uniform function called the inverse function. Any function used in a symmetric cryptosystem should be a permutation. Also, it is required that the function is highly nonlinear so that it is resistant to Matsui's linear attack. In this article we demonstrate that the highly nonlinear permutation f(x)=x^2^^^2^^^k^+^2^^^k^+^1 on the field F"2"^"4"^"k, discovered by Hans Dobbertin (1998) [1], has differential uniformity of four and hence, with respect to differential and linear cryptanalysis, is just as suitable for use in a symmetric cryptosystem as the inverse function. Its suitability with respect to other attacks remains to be seen.

[1]  Kaisa Nyberg,et al.  Differentially Uniform Mappings for Cryptography , 1994, EUROCRYPT.

[2]  Eimear Byrne,et al.  New families of quadratic almost perfect nonlinear trinomials and multinomials , 2008, Finite Fields Their Appl..

[3]  Robert Gold,et al.  Maximal recursive sequences with 3-valued recursive cross-correlation functions (Corresp.) , 1968, IEEE Trans. Inf. Theory.

[4]  Cunsheng Ding,et al.  On Almost Perfect Nonlinear Permutations , 1994, EUROCRYPT.

[5]  Eimear Byrne,et al.  A few more quadratic APN functions , 2008, Cryptography and Communications.

[6]  Claude Carlet,et al.  Two Classes of Quadratic APN Binomials Inequivalent to Power Functions , 2008, IEEE Transactions on Information Theory.

[7]  Hans Dobbertin,et al.  One-to-One Highly Nonlinear Power Functions on GF(2n) , 1998, Applicable Algebra in Engineering, Communication and Computing.

[8]  Pascale Charpin,et al.  Cubic Monomial Bent Functions: A Subclass of M , 2008, SIAM J. Discret. Math..

[9]  Tadao Kasami,et al.  The Weight Enumerators for Several Clauses of Subcodes of the 2nd Order Binary Reed-Muller Codes , 1971, Inf. Control..

[10]  Claude Carlet,et al.  Constructing new APN functions from known ones , 2009, Finite Fields Their Appl..

[11]  Claude Carlet,et al.  Another class of quadratic APN binomials over F2n: the case n divisible by 4 , 2006, IACR Cryptol. ePrint Arch..

[12]  Mitsuru Matsui,et al.  Linear Cryptanalysis Method for DES Cipher , 1994, EUROCRYPT.

[13]  Hans Dobbertin,et al.  New cyclic difference sets with Singer parameters , 2004, Finite Fields Their Appl..

[14]  Anne Canteaut,et al.  A new class of monomial bent functions , 2006, 2006 IEEE International Symposium on Information Theory.

[15]  Claude Carlet,et al.  An infinite class of quadratic APN functions which are not equivalent to power mappings , 2006, 2006 IEEE International Symposium on Information Theory.