Doctor of Philosophy
Department of Computer Science
Qu, Cheng-Xin, Boolean functions in cryptography, Doctor of Philosophy thesis, Department of Computer Science, University of Wollongong, 2000. https://ro.uow.edu.au/theses/1292
This thesis is about Boolean functions and their cryptographic properties. Two kinds of Boolean functions are discussed - balanced functions and bent functions. In addition to surveying recent activities of research into Boolean functions, a new representation of bent functions - degree-3 homogeneous bent functions are discovered. The complete set of degree-3 homogeneous bent functions on the lowest dimension Boolean spaces V6 is given. By using bent functions, some ways to construct highly nonlinear balanced Boolean functions are shown in this thesis, which yield a new property of bent functions. The structure of degree-3 highly nonlinear homogeneous balanced functions is also discussed. These results are based on computer searching. The theory of symmetric groups is applied in the research. In this study symmetric groups are applied to Boolean functions. Any Boolean function on Vn has its own symmetric properties associated with the symmetric group Sn. The relations between Boolean functions and symmetric groups are highlighted. This may lead to a new way to design good S-boxes by using an additive group of Boolean functions which is a subset of the function group generated by the symmetric group. Because good symmetric properties have the potential to be faster for implementation, the applications of homogeneous Boolean functions taken as rotation functions are discussed. Bent-like balanced functions are very good candidates of Boolean functions for good S-box design. In a degree-3 homogeneous bent or balanced Boolean function, each term is considered as a three variety block. Then it is found that the homogeneous Boolean function is tightly related with block designs BIBD and PBIBD. So in this thesis, the method of combinatorial block designs to discuss Boolean functions is also used. The connection of symmetric group theory with Boolean functions is established.