Constructions Of Permutation Polynomials And Functions With Low Differential Uniformity

In the design of block ciphers,the design of S-boxes is crutial,since their crypto-graphic properties are crucial to the security of the ciphers.On one hand,in SPN-structure ciphers,S-boxes should be permutations.In addition to cryptography,permutation poly-nomials and their compositional inverses also have wide applications in coding theory,combinatorial design,and etc.On the other hand,as for Feistel-structure ciphers,(n,m)-functions with low differential uniformity when n/2<m<n actually play an important role in Feistel-structure ciphers,preventing several cryptographic attacks.In this thesis,we devote to the problems of constructing involutory permutation poly-nomials,solving the compositional inverse of permutation polynomials over finite fields,and constructing(n,m)-functions with low differential uniformity.The main results are as follows:(1)We construct involutions over finite fields by proposing an involutory version of the AGW Criterion.We demonstrate this general construction method by considering polynomials of different forms.First,we provide three explicit classes of involutions of the form x^rh(x^q-1)over (?)_q².Second,we consider the further relationship between two classes of permutation polynomials discovered by Zheng et al in 2018.On one hand,we reveal the relationship of being involutory between them;on the other hand,the compositional inverses of permutation polynomials of the form g(x^qi-x+?)+cx are computed.In addition,a class of involutions of the form g(x^qi-x+?)+cx is constructed.Finally,we study the fixed points of constructed involutions.(2)We propose a general unified method to solve the compositional inverse of permutation polynomials constructing by the AGW criterion.We convert the inverse of a permutation into constructing related mappings,where half of them is obtained from the AGW criterion directly and the other half is constructed carefully and accordingly.As applications,we apply this method to four classes of explicit permutations based on the AGW criterion as examples,which are divided into three cases,i.e.,multiplicative,additive and combinatorial cases.First,as for the multiplicative case,we obtain the compositional inverses of permutation polynomials of the form x^rh(x^s),completely,which improves previous work of Li et al.in 2019.Second,the new method is applied into the additive case and we obtain the compositional inverses of p(x)=f(x)+g(?(x))by the new design.Third,for the combinatorial case,we obtain the compositional inverses of permutations xh(B(x))and x+?G(F(x))respectively.(3)We construct(m+k,m)-functions with low differential uniformity.In this paper,we improve the method by Carlet et al.in 2018 to construct explicit infinite families of(m+k,m)-functions with low differential uniformity by constructing special sets.We construct a kind of special sets with structure by linearized polynomials,and provide differentially (?)-uniform(m+k,m)-functions with (?)<2^k+1when k?m-2.Specifically when k=m-2,the cryptographic properties of our specific constructions are better than the function constructed by Carlet et al.The constructed functions provide more choices for the design of Feistel ciphers.