Analysis of affine equivalent boolean functions for cryptography

Boolean functions are an important area of study for cryptography. These functions, consisting merely of one's and zero's, are the heart of numerous cryptographic systems and their ability to provide secure communication. Boolean functions have application in a variety of such systems, including block ciphers, stream ciphers and hash functions. The continued study of Boolean functions for cryptography is therefore fundamental to the provision of secure communication in the future. This thesis presents an investigation into the analysis of Boolean functions and in particular, analysis of affine transformations with respect to both the design and application of Boolean functions for cryptography. Past research has often been limited by the difficulties arising from the magnitude of the search space. The research presented in this thesis will be shown to provide an important step towards overcoming such restrictions and hence forms the basis for a new analysis methodology. The new perspective allows a reduced view of the Boolean space in which all Boolean functions are grouped into connected equivalence classes so that only one function from each class need be established. This approach is a significant development in Boolean function research with many applications, including class distinguishing, class structures, self mapping analysis and finite field based s-box analysis. The thesis will begin with a brief overview of Boolean function theory; including an introduction to the main theme of the research, namely the affine transformation. This will be followed by the presentation of a fundamental new theorem describing the connectivity that exists between equivalence classes. The theorem of connectivity will form the foundation for the remainder of the research presented in this thesis. A discussion of efficient algorithms for the manipulation of Boolean functions will then be presented. The ability of Boolean function research to achieve new levels of analysis and understanding is centered on the availability of computer based programs that can perform various manipulations. The development and optimisation of efficient algorithms specifically for execution on a computer will be shown to have a considerable advantage compared to those constructed using a more traditional approach to algorithm optimisation. The theorem of connectivety will be shown to be fundamental in the provision many avenues of new analysis and application. These applications include the first non-exhaustive test for determining equivalent Boolean functions, a visual representation of the connected equivalence class structure to aid in the understanding of the Boolean space and a self mapping constant that enables enumeration of the functions in each equivalence class. A detailed survey of the classes with six inputs is also presented, providing valuable insight into their range and structure. This theme is then continued in the application Boolean function construction. Two important new methodologies are presented; the first to yield bent functions and the second to yield the best currently known balanced functions of eight inputs with respect to nonlinearity. The implementation of these constructions is extremely efficient. The first construction yields bent functions of a variety of algebraic order and inputs sizes. The second construction provides better results than previously proposed heuristic techniques. Each construction is then analysed with respect to its ability to produce functions from a variety of equivalence classes. Finally, in a further application of affine equivalence analysis, the impact to both s-box design and construction will be considered. The effect of linear redundancy in finite field based s-boxes will be examined and in particular it will be shown that the AES s-box possesses complete linear redundancy. The effect of such analysis will be discussed and an alternative construction to s-box design that ensures removal of all linear redundancy will be presented in addition to the best known example of such an s-box.