Orthogonal Arrays And Multiple Output Resilient Functions

Orthogonal arrays on statistics are mainly used for experimental design,this means that the orthogonal arrays in the areas of human study play a very important role,such as industry,agriculture,medicine,quality control and product improvements.Orthogonal arrays are so popular in experimental design,since it can be used for main effect of orthog-onal design.Moreover,asymmetrical orthogonal arrays have additional appeal because they have greater flexibility,allowing for factors with different numbers of levels.Resilient functions are a kind of important Boolean functions,which are widely used in information security domains such as fault-tolerant distributed computing,quantum-cryptographic key distribution,random sequence generation of stream ciphers and so on.However,the con-struction of the resilient functions is a very challenging problem,because the total number of Boolean functions with eight variables is approxiiately 1.15 × 1077,larger than the estimated number of atoms in the universe while resilient functions in the proportion of Boolean functions are very small.Orthogonal arrays,which are indispensable in statistics and combination design,are also applied to cryptography,coding and computer science due to their simple definition and greater flexibility.Therefore,the deep studies on the equivalence between resilient functions and ort,hogonal arrays are of great significance for constructing Boolean functions with good cryptographic properties.Chapter 1 introduces the development and the current research status of orthogonal arrays and resilient function,and contains basic concepts and main lemmas.Chapter 2 firstly puts forward the definition of the output matrix,proves respectively the equivalence between the uniform distribution of resilient function and orthogonal array and output matrix of the multiple output resilient function is equivalent to an orthogonal array.Secondly,by using the relation between output matrix and the orthogonal array,we establish the relationship among the number of input variables,the number of output variables and the order of resilience for multiple output resilient functions.Thirdly,we prove that the support matrix of resilient function is also equivalent to an orthogonal array and present an easier method to prove that a resilient function has the same probability for its every output value.Finally,we find the relationship between multiple output resilient functions and nonzero linear combination of single output resilient functions by making use of the properties of Hadamard matrix.Chapter 3 proves that the orthogonal arrays can be used to construct some resilient functions through the support matrices.Conversely,resilient function provides a tool for constructing orthogonal arrays.By using an example we illustrate our construction methods for resilient function from the orthogonal partitions of orthogonal arrays.Our results provide some good ideas for constructing resilient function with good cryptographic properties.Chapter 4 concludes the main content of this paper and puts forward some suggestions.