Binomial differentially 4 uniform permutations with high nonlinearity

Abstract Differentially 4 uniform permutations with high nonlinearity on fields of even degree are crucial to the design of S-boxes in many symmetric cryptographic algorithms. Until now, there are not many known such functions and all functions known are power functions. In this paper, we construct the first class of binomial differentially 4 uniform permutations with high nonlinearity on F 2 6 m , where m is an odd integer. This result gives a positive answer to an open problem proposed in Bracken and Leander (2010) [7] .

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

[2]  Claude Carlet,et al.  Vectorial Boolean Functions for Cryptography , 2006 .

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

[4]  Carl Bracken,et al.  A highly nonlinear differentially 4 uniform power mapping that permutes fields of even degree , 2009, Finite Fields Their Appl..

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

[6]  Eimear Byrne,et al.  Determining the Nonlinearity of a New Family of APN Functions , 2007, AAECC.

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

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

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

[10]  Eli Biham,et al.  Differential cryptanalysis of DES-like cryptosystems , 1990, Journal of Cryptology.

[11]  Jürgen Bierbrauer.  New semifields, PN and APN functions , 2010, Des. Codes Cryptogr..

[12]  Claude Carlet,et al.  New classes of almost bent and almost perfect nonlinear polynomials , 2006, IEEE Transactions on Information Theory.

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

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

[15]  Pulak Mishra,et al.  Mergers, Acquisitions and Export Competitive- ness: Experience of Indian Manufacturing Sector , 2012 .

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