Bent Functions In Cryptography

Boolean functions play an important role in the design and analysis of crypto-systems. The main reason is that to some extent, the security of a cryptosystem is depend on the cryptographic performance of the Boolean functions.Bent functions are a kind of special Boolean functions with good cryptographic properties. Its concept was first presented by O.S.Rothaus in 1976. The nonlinearity combiner that constructed by bent functions can resist correlation attacks, best affine attacks and difference analysis attacks. Bent function has been successfully used by J. Olsen et al to design balance binary sequences (bent sequences) with optimum correlation properties in 1982. From then on, bent functions have attracted the researchers' attention. The study of bent functions has been among the important fields of coding and cryptology theory. Bent function has the highest nonlinearity and satisfies the highest degree propagation criterion, so it has very important applications in cryptography, coding theory, sequences and signal designing theory. The construction of bent function has been a hot spot of research in cryptography.Multi-output Boolean functions play an important role in designing block ciphers. For example, multi-output Boolean functions with many good cryptographic properties have often been adopted in designing S-box which is the core of block ciphers. So it is especially important that how to construct those multi-output Boolean functions with many good cryptographic properties.This dissertation studies the constructions and properties of cryptographic bent functions including the constructions of multi-output bent functions, the constructions and cryptographic properties of multi-output semi-bent functions, and the constructions of k-quasi bent functions over finite field.Firstly, the cryptographic properties of bent functions and its constructions are summarized. At the same time, some kinds of constructions are analyzed and their advantages or disadvantages are pointed out. Moreover, the concept of multi-output semi-bent functions is introduced. Based on the new concept, a method to construct...