Two Classes Of Combinatorial Configurations In Cryptograph And Bioinformatics

Combinatorial design theory has extensive interactions with numerous areas of mathematics: group theory, graph theory, number theory and finite fields, and finite geometry. It also has applications ranging from routine to complex in numerous disciplines: experimental design, coding theory, cryptography, computer science, and bioinformatics, among others. The use of combinatorial structures offers a powerful tool for these modern sciences.In this thesis, we focus on investigating the existence of (generalized) mix functions and grid-block designs, which are applied to cryptography and bioinformatics.To enrich the message space of a cipher and guarantee security, Ristenpart and Rogaway defined mix functions on two sets of equal size. To mix inputs from two sets of different sizes, Stinson generalized the definition of mix functions (called generalized mix functions), and established an existence result for generalized mix functions with 10 undetermined pairs of input sizes. Using direct construction and recursive construction, we present a complete solution to the existence problem for generalized mix functions.Combinatorial group testing is a basic tool in conducting experiments of tests which can be applied to molecular biology. For example, in DNA library screening, group testing is used to select positive clones efficiently. Grid-block design is derived from it and becomes an important tool in DNA library screening. The notion of grid-block design was first introduced by Fu, Hwang, Jimbo, Mutoh and Shiue.Recently more and more researchers are interested in it and extensive researches have been done on it. In this thesis, we list some known results for the existence of grid-block designs and focus on the existence of resolvable D_3?(K_s(9))s. The necessary condition for the existence of resolvable D_3?(K_s(9))s is s = 1 (mod 4). Applying difference method, we directly construct some useful frame group divisible grid-block designs. Using these designs, we establish an existence result of D_3?(K_s(9))s with s = 8n + 1, except possibly n = 2,3,6. For the remaining case s = 8n + 5, we construct some FD_3?(K_s(t))s, which are crucial to the complete solution for the whole problem.