The Differential Spectra And Nonlinearity Of Some Power Mappings With Low Uniformity

S-box is one of the core modules of block cipher algorithm.In order to resist differential attacks and linear attacks,the cryptographic functions used for constructing S-box are required to have low differential uniformity and high nonlinearity respectively.The study on cryptographic functions with low differential uniformity is a hot topic due to their wide applications in coding,cryptography combinatorial design and so on.The differential spectrum of a cryptographic function is an important notion for estimating its differential properties more precisely.In this thesis,we investigate the differential spectra of several power functions with low differential unformity,and the nonlinearity of certain function is also studied.The main results are as follows:Let p be a prime and d-pⁿ-2 for some positive integer n.Firstly,we calculate the differential spectrum of the power function f(x)=xd over Fpⁿ.The diggerential uniformity of this power function have been determined by Helleseth and Dobbertin et al.When p=2,this power function is the well known inverse function,which was used to construct the S-box in AES.By further investigating some equations over finite fields,we determined the differential spectrum of this function.Then,we study the differential uniformity and the corresponding differential spectrum of the power function f(x)=x^3n-3 over F_3n.This power function has been studied by Helleseth et al in 1999,and for even n,they only gave a bound for the differential uniformity.By studying the relationship between the coefficients and the number of solutions of a quartic equation,we determine the exact differential uniformity and the corresponding differential spectra of this function.This solves an open problem of Helleseth et al.Finally,for an odd p,we also investigate the differential spectrum of the power function x⁴ over Fpⁿ.The differential uniformity of this function was also determined by Dobbertin in 2003.Here,we further studied its differential properties.By utilizing the results about the cyclotmic numbers,we determined its differential spectrum.Moreover,we also investigate the Walsh transform of x⁴ It is shown that he upper bound of the module of its Walsh transform is(?),and thus the lower bound of its Nonlinearity is pⁿ-1-3pⁿ/2-1.The results show that the power function x⁴ has low differential uniformity and high nonlinearity.