An analysis of product ciphers based on the properties of boolean functions

Any encryption function E can be implemented as a circuit, which permits E to be modeled as a system of boolean equations. With this representation it is then possible to evaluate the strength of E according to several design criteria based on the properties of boolean functions. These criteria have been established either formally or empirically as essential to the security of a cipher, the most basic of which for product ciphers are diffusion and confusion. More specific criteria include balance, nonlinearity, nondegeneracy, correlation immunity, and satisfying the strict avalanche criterion. These properties will collectively be referred to as cryptographic criteria. In this thesis we will provide an analysis of the expected characteristics of boolean functions which model product ciphers, and their relation to certain cryptographic criteria. Parts of the analysis will be general enough to be applied to any substitution cipher, but the principal focus will be product ciphers. The analysis is probabilistic in nature, and is concerned with the properties assumed by large uniformly distributed boolean functions, and also by the boolean functions that describe elements of the symmetric group. The thesis explicitly deals with the properties of nondegeneracy, nonlinearity, the presence of linear structures, affine approximation, differential cryptanalysis and the construction of S-boxes.