Constructions Of Highly Nonlinear Functions And Their Applications In Sequences And Codes

Optimal nonlinear functions, i.e., Bent functions and perfect nonlinear functions, appose optimal resistances to linear cryptanalysis and difference cryptanalysis respectively, thus they play an important role in cryptography. Moreover, optimal nonlinear functions also have significant applications in coding theory, sequence design, combinatorial theory and so on.The first main topic of this thesis is the construction of Bent functions. From the ap-proach of linearized equations, several new classes of quadratic generalized Boolean Bent functions are obtained based on the theory of Z4-valued quadratic forms. This together with the link between Boolean Bent functions and generalized Boolean Bent functions leads to new Boolean Bent functions, and new p-ary Bent functions are also presented by using the same method in the finite fields with odd characteristic p. For the Bent functions with high degree, this thesis mainly focuses on the Bent functions of Dillon type and Niho type which have the optimal algebraic degree. Through the discussions on some partial exponential sums over finite fields, several new classes of Dillon type Boolean Bent and p-ary Bent functions are characterized, which generalize some known results. By adopting the same techniques, this thesis generalizes the well-known Leander-Kholosha’s class of Niho type Boolean Bent functions and a much simpler proof regarding the bentness is also given. Meanwhile, the investigated function of Niho type in the finite fields of odd characteristic is proven to be four-valued Walsh spectra and the spectra distribution is also determined.The second main topic of this thesis is the construction of optimal cyclic codes from perfect nonlinear and almost perfect nonlinear functions. By using the canonical factoriza-tions of polynomials with low degrees over finite fields, and the relation between the degree of an irreducible polynomial and its roots in finite fields, an open problem proposed by D-ing and Helleseth about optimal single-error correcting ternary cyclic codes is successfully solved, and four new classes of such optimal codes with parameters [3m-1,3m-1-2m,4] are also obtained by virtue of the quadratic character over finite fields, where m is a positive integer. Furthermore, two new classes of optimal two-error correcting ternary cyclic codes with parameters [3m-1,3m-2-2m,5] are also presented by making use of the properties of perfect nonlinear functions. In addition, the covering radiuses of the constructed optimal codes and the weight distributions of their dual codes are considered as well. However, only few results are obtained and there still some problems remain open. The third main topic of this thesis is the investigation of optimal or almost optimal quaternary sequence families by using generalized Boolean Bent functions and the highly nonlinear Gold functions. According to the theory of Z4-valued quadratic forms and the properties of generalized Boolean Bent functions, a class of exponential sums over Galois rings is investigated and then the exact correlation distributions of two classes of optimal sequences are determined. Moreover, again by the theory of Z4-valued quadratic forms, a class of known optimal quaternary sequences and a new class of quaternary sequences with low correlation are obtained by a unique method. On the other hand, based on the properties of the Gold functions over Galois rings, the correlation distribution of the quaternary Gold sequences is determined. From the constructed quaternary Gold sequences, the maximal nontrivial correlation value of its MSB sequences and the exact correlation distribution of its Gray sequences are also determined in virtue of the relation between quaternary sequences and binary sequences.