Construction Of Orthogonal Arrays And Their Applications In Resilient Functions

Ever since the seminal work of statistician Rao in 1947,Orthogonal arrays have become so prominent in the design literature as to form the backbone of designs for multifactor experiments.Many combinatorial mathematicians and statisticians devoted themselves to the research of orthogonal arrays,and obtained abundant achievements.In the vast majority of method,most of them focused on symmetric arrays,but relatively less work on the construction of asymmetric orthogonal arrays of strength greater than two is available.However,asymmetric orthogonal arrays of high strength have better properties which can be widely used in industrially mass manufacturing,or employed in theoretical studies in the favor of computer science,information science,coding theorists or algebraists.How to construct OAs of high strength needed in practice remains an open problem.In this paper,a new construction method orthogonal arrays of high strength is pro-posed by using orthogonal partitions and small OAs.This method can construct symmetric or asymmetric OAs of arbitrary strength,arbitrary level and the size of prime power or non-prime power.On the other hand,Boolean functions which play an important role in cryptography are also studied extensively in recent years and the support table of resilient function which as a subclass of Boolean functions is an orthogonal array.Moreover,many properties of Boolean function can be described by Walsh transform,so we further study the close relationship between Walsh transform of a Boolean function and the orthogonali-ty of some columns of its support table and find the orthogonality of orthogonal array can be presented and characterized by Walsh transform of a Boolean function.Moreover,we prove that the equivalence of two linear resilient functions constructed by the right cosets of a systematic code and by linear code.The thesis consists of four chapters and is organized as follows.Chapter 1 introduces the research background of the thesis,the related concepts and some existing results.Chapter 2 studies a new iterative construction method OAs of high strength from orthogonal partitions and small OAs by using the property of Kronecker product and permutation matrix.And some new infinite classes of orthogonal arrays can be obtained.Chapter 3 presents the application of OA and Walsh transform in resilient function.we not only study the close relationship between Walsh transform of a Boolean function and the orthogonality of some columns of its support table,but also prove that the equivalence of two construction methods of resilient functions.The summary of this paper is proposed in Chapter 4,we put forward some suggestions and the direction of efforts.