On The Cryptographic Properties Of Some 4-differentially Functions And Their Applications

An S-box is the only nonlinear component in the block cipher,and its strength directly determines the security of the whole cryptographic algorithm.In order to resist differential attacks and linear attacks effectively,the cryptographic functions that are used for constructing S-boxes are required to have lower differential uniformity and higher nonlinearity.To show the differential properties of a cryptographic function more precisely,one needs to calculate its differential spectrum.However,the differential spectra of cryptographic functions are not well studied.Up to now,only for a few number of power functions and some polynomials with low degrees,the differential spectra have been determined.There are still a lot of functions with low differential uniformities whose differential spectra have not yet been determined,especially including some differentially 4-uniform or 6-uniform functions.In addition,the differential spectra also have some applications in the areas of coding theory and combinatorial designs.In this paper,we investigate the differential spectra of several differentially 4-uniform nonlinear functions over the finite field F2n.We also characterize other cryptographic properties of these functions and show some applications of them in coding theory.The main contributions are as follows:1)The differential spectrum of the Welch permutation g(x)=x2m+1+1+x3+x is deduced over F22m+1,where m is a positive integer.An upper bound on the boomerang uniformity of g(x)is also presented.Based on the power sum method and the theory of quadratic forms,the Walsh spectrum of g(x)is calculated.Finally as an application,a conjecture about the weights of some linear codes constructed from g(x)is partly solved.2)The differential spectrum of the quadratic function f(x)=x22t+1+?x2t+1 is studied over F2n,where n and t are positive integers satisfying gcd(n,t)=1.By using the theory of linearized equations,we address the differential spectrum of f(x).In addition,the Walsh spectrum of f(x)is derived for odd n and the possible values of the Walsh spectrum of f(x)are given for even n.Finally,two distinct binary linear codes are constructed from f(x)and their weight distributions are also determined.