Research On Some Topics Of Boolean Functions

The studies of many problems of symmectric cipher system, such as the S-boxes in block cipher and the filter function or combining function in stream cipher based on LFSR, are equivalent to the studies of Boolean functions. And Boolean function also plays an important role in coding theory, combinatorial design and sequence design and has wide applications. Some topics of Boolean functions are studied in this thesis, including the properties and constructions of almost resilient functions, trace representation of bent functions and its applications in the constructions of generalized binary bent sequences, trace representation of binary bent sequences, properties of spectral of multi-valued logical functions over finite fields and its applications, relations between generalized bent functions and perfect nonlinear functions over finite fields etc. Main constributions of this work are as follows:1. Relations between almost resilient functions and its component functions are characterized and some concrete constructions of almost resilient functions are presented. A spectral characterization of almost (n, 1, k) resilient functions are obtained, which enable us to determined almost (n,1,k) resilient functions by computing its Walsh spectral.2. Trace representations of bent functions are studied in detail. We summarize the known results systemlly. Furthermore, we present trace representations of some bent functions of degree two and construct new classes of generalized binary bent sequences by using these bent functions.3. Linear onto mappings play important roles in the construction of binary bent sequences. We characrterize these linear onto mappings completely and analyze the relation between generalized binary bent sequences and binary bent sequences.4. We propose a new definition of linear spectrum of multi-valued...