Abstract

Boolean functions are vital components of symmetric-key ciphers such as block ciphers, stream ciphers and hash functions. When used in cipher systems, Boolean functions should satisfy several cryptographic properties such as balance, high nonlinearity, resiliency and high algebraic degree. Bent functions achieve the maximum possible nonlinearity and hence they provide optimal resistance to several cryptographic attacks such as linear and differential cryptanalysis. We present some simple constructions for binary bent functions of length 2 2 k using a known bent function of length 2 2 k -2 . Adams and Tavares introduced two classes of bent functions: bent-based bent functions and linear-based bent functions. In this thesis we explore different bent-based constructions. In particular, we show that all nonlinear resilient functions with maximum order resiliency are either bent-based or linear-based. We provide an explicit count for the number of such resilient functions that belong to both classes. We also provide a simple proof that all symmetric functions that achieve the maximum possible nonlinearity are bent-based. In particular, for n even, we have 4 bent-based bent functions. For n odd, we also have 4 bent-based functions. We also prove that there is no bent-based homogeneous functions with algebraic degree >2. Almost all cryptographic properties of Boolean functions can be determined efficiently from its Walsh transform. In this thesis, we present some restrictions on the partial sum of the Walsh transform of binary functions. In several parts of the thesis, we extend the obtained results to functions defined over GF(p)