Constructions And Analysis Of Cryptographic Boolean Functions With Permutation And High Nonlinearity Properties

Cryptographic functions occupy a vital position in symmetric cryptosystems,whose security depends on some cryptographic properties of cryptographic functions,such as permutation,balancedness,resiliency,nonlinearity,algebraic degree,algebraic immunity and differential uniformity,etc.Furthermore,these cryptographic properties are emerged to resist certain attacks,such as distinguishing attack,correlation attacks,fast correlation attacks and best affine approximation in stream ciphers,as well as linear attacks in block ciphers,Berlekmap-Massey attacks,algebraic attacks and differential attacks,etc.Since these cryptographic properties are mutually constrained,it is a hot research topic to construct and analyse cryp-tographic functions with permutation and high nonlinearity properties.In the dissertation,we mainly study the problem of the constructions and analysis of several permutation poly-nomials over finite fields with odd characteristic,several permutation polynomials with Ni-ho exponents,several permutation polynomials and complete permutation polynomials over Fpn,Bent functions,resilient Boolean functions and vectorial Boolean functions with high nonlinearity.The main results are as follows:1)Basing on the piecewise construction method,we propose six classes of permutation polynomials with the form as(axqqm-bx-+?)s+L(x)for an integer s satisfying s=(?).Next,by determining the number of solutions for some certain equations over finite fields,for s satisfying s(pm-1)=pm-1(mod pn-1)or(?)= pkm-1(mod pn-1),the permutation behaviour of three classes of polynomials of the form(armn(x)+?)s+L(x)is analysed.2)Through analysing a seventh-degree and a fifth-degree Dickson polynomial over the finite field F32m,two conjectures on permutation trinomials over F32m which were presented by Li.et al are partially confirmed,furthermore,several classes of permutation trinomials from Niho exponents over F32m are proposed.Next,a class of permutation trinomials from Niho exponents over F52m is given,which generalize the results from Wu.et al,and some other generalization results are presented.Apart from constructing the permutation trinomials over finite fields with odd characteristic,the permutation behaviour of a class of the trinomial over finite fields with even characteristic is studied.Finally,by studying the root distribution in the unit circle of certain quadratic and cubic equations,a class of permutation quadrinomials over F2n is constructed3)Depending on the AGW criterion and determination of the number of solutions to some equations over finite fields to investigate the permutation behaviour of polynomials,nine classes of permutation polynomials over Fpn of the form(xpm-x+?)s1+(xpm-x+?)s2x are constructed.Next,the complete permutation behaviour of five classes of polynomials over Fp2m of the form axpm+bx+h(xpm-x)is analysed.It is a further studies and supplements on some previous works of Li.et al.4)For the traditional M-M class construction of Bent function,substituting the product of multiple linear functions and three linear functions to the original arbitrary Boolean function with m variables,respectively,a new Bent function is obtained by computing the Walsh spectrum of the given functions.Next,through a binomial involution and a function with a linear translator,three different permutations which satisfied some conditions are given,based on this,a new Bent function is obtained and its dual function is calculated.Finally,using two linear translators of any two functions,a permutation which satisfies the traditional M-M class construction of Bent function is given to obtain a new Bent function5)Basing on the concatenation of different resilient functions and taking advantage of var-ious lower resilient functions to construct higher ones,we have obtained a large class of strictly almost optimal nonlinearity resilient functions with the optimal algebraic degree.Next,by introducing disjoint linear codes and the concatenation of different resilient vecto-rial functions,and some lower resilient vectorial functions are used to design higher resilient vectorial Boolean functions,a large class of strictly almost optimal nonlinearity(n,m,r)vectorial Boolean functions with high algebraic degree is achieved.