Decomposition And Normality Of Generalized Boolean Functions In Cryptography

Boolean functions play an important role in cryptography and error correcting coding theory. They are extensively used in designing stream ciphers and block ciphers. Boolean functions have many cryptographic properties which have been extensively studied. One of them is the question if there exists a space of dimension n 2 such that the restriction of a given function is constant (resp. affine) on this space. We call the functions for which such a space exists normal (resp. weakly normal).This dissertation is devoted to the decomposition of generalized Boolean functions and the normality of generalized Boolean functions.Firstly, the restrictions of generalized Boolean functions to affine subspaces are studied. The connection between the Fourier transform of the derivatives and the Fourier transform of generalized Boolean functions is pointed out. At the same time, the property of Fourier transform of generalized Boolean functions which satisfy strict avalanche criterion and propagation criterion is presented. For a generalized Bent function, the connection between the Fourier transform of its restriction and the decompositions of its dual is described.Secondly, as a logical extension of notion of normal Boolean functions, we focus on considering the normality of generalized Boolean functions. We present a method for constructing an infinite class of generalized normal Boolean functions and an algorithm checking if a given generalized Boolean function is normal.At last, we give some consequences for the behavior of generalized normal Bent functions which are not included in generalized normal Boolean functions and construct a new class of genenralized Bent functions by using generalized normal Bent functions.