A Class Of Permutation Polynomials With Good Cryptographic Properties

Permutation polynomials are widely used in combinatorics,cryptography,cod-ing theory.For example,the core component of symmetric cryptosystem,S-box,is usually designed by permutation polynomials over finite fields with even characteris-tic.Because the cryptographic properties of S-box are related to the security of the whole cryptosystem,it is necessary to consider its resistance to various cryptographic attacks when selecting permutation polynomials.The ability of cryptographic func-tions to resist known attacks is measured by their corresponding cryptographic indices,such as differential uniformity,nonlinearity and so on.Lower differential uniformity makes cryptographic functions well resistant to differential analysis;higher nonlinear-ity makes cryptographic functions effectively resistant to linear attacks or fast corre-lation attacks.Therefore,it is very important to study the difference uniformity and nonlinearity of permutation polynomials.In this paper,we mainly discuss the difference uniformity and nonlinearity of polynomial x3+ax2x+bxx2+cx3 over F22m.Firstly,we transform the problem of difference uniformity into the roots of equations over finite field by polar coordinate representation.By discussing coefficients and roots of equations,we obtain a class of polynomials with difference uniformity of 4.Secondly,by using the properties of trace function and unit circle,the problem of nonlinearity is also transformed into solutions of equations.The values of the nonlinearity of the above polynomials are determined.And necessary and sufficient condition of each value are given.Finally,a class of permutation polynomials with low difference and high nonlinearity are obtained.