Constructions Of Low Differential Uniformity Functions And Bent Fuctions And Their Applications In Coding Theory

The low differential uniformity functions and bent functions have applications in cryptography(design of stream ciphers),coding theory(Reed-Muller codes and two-weight codes),association schemes,sequences,graph theory(strong regular graphs),combinational design,etc.In this paper,we give a research on several topics:constructions of permutation polynomials,constructions of differentially 4-uniform permutations,constructions of bent and semi-bent functions and their applications in constructions of linear codes.Based on the methods used by Dobbertin,we present three new classes of monomial complete permutations over finite fields of odd characteristic,not corresponding to any known monomial complete permutation.Furthermore,we find that the complete permutation polynomials in the second class are related to Dickson polynomials.Inspired by the work of Wan,we obtain the third class of complete permutation monomials over finite fields of odd characteristic.Meanwhile,the compositional inverses of these polynomials are also investigated.We continue the work of Tu and construct more permutation polynomials of the form(xpm—x + ?)t(pm+1)+1 + L(x)over Fp2m.The resulted permutation polynomials have flexible parameters t.Based on the switching method,the constructions of differentially 4-uniform permutations over finite fields in even characteristic and almost perfect nonlinear functions over finite fields in odd characteristic are investigated.By using Gold monomial,we present two classes differentially 4-uniform permutations in even characteristic and two classes of almost perfect nonlinear functions in odd characteristic respectively.In addition,two classes of permutation monomials with low differential uniformity over finite fields in odd characteristic are also provided by determining the number of solutions of certain equations.By means of the second order derivative of the dual of known bent functions,Mesnager presented two new infinite families of bent functions by adding the product of two linear functions to some known bent functions.Using this method,we obtain several classes of bent,near-bent and semi-bent functions.The proofs of our main results are based on the study of the Walsh transform.As a generalization of the result[47],we obtain not only bent functions but also semi-bent functions from our constructions.A general bridge between PN functions,APN functions and optimal cyclic codes is established.We use perfect nonlinear monomials and the inverse function to construct several classes of optimal p-ary cyclic codes C(0,1,e)with parameters[pm-1,pm-2m-2,4]for p ? 5.We show that the minimum distance of quinary cyclic codesC(1,e)is equal to 2 or 3 depending on whether e is odd or e is even.To obtain optimal quinary cyclic codes,we investigate a class of subcodes of C(1,e)and employ some known almost perfect nonlinear monomials and other monomials to construct optimal quinary cyclic codes C(0.1.e)with parameters[5m-1,5m-2m-2,4].Finally,we construct the linear codes with three or four weights from the resulted weakly regular p-ary bent functions which don't belong to RF and determine the weight distribution of three-weightcode.Using non-qudratic function f(x)= Tr(axp2m-1/p+1),a class of p-ary linear codes with two weights is constructed by using the properties of cyclotomic classes of Fp2*.To investigate the complete weight enumerators of linear codes obtained in this paper,a crucial problem is to determine in which cyclotomic class of Fp2*the elements of Fp*are.