Analysis Of The Properties Of Generalized Boolean Functions And Bent Functions

As important cryptographic functions,Boolean functions has been widely used in the design of sequence and block cipher.Specially,Bent functions,which has optimal nonlinearity,has important application in the areas of coding and combination theory.Generalized Boolean functions have attracted the attention of researcher.This dissertation studies the properties of generalized Boolean functions and Bent functions.Main constributions of this work are listed as follows:(1)Use correlation properties,the correlation functions and the sum of squares indicator of generalized Boolean functions are obtained.The relationship between generalized bent and bent is proven,and sufficient(also necessary)conditions of generalized bent is given from autocorrelation function.Moreover,we shows the sum of squares indicator of generalized Boolean functions.(2)The relationship and characterization between different fixed forms of generalized Boolean functions are described though decomposition.The decomposition demonstrate that the relationship between different -primitive roots in correlation and Walsh-Hadamard transform.Furthermore,the relationship of Walsh-Hadamard transform of composition generalized Boolean functions can be described.In addition,the auto-correlation function of a class of generalized Boolean functions is achieved,and some sufficient conditions for generalized bent functions are also obtained.(3)The related condition and construction of bent and generalized bent are investigated by using Walsh-Hadamard transform and concatenation.Firstly,a class of the generalized Boolean functions under a linear combination is analyzed,and general situation under certain restrictions is given.Second,we present the WalshHadamard transform of a class of generalized Boolean functions,and demonstra that some previous forms are special cases of the results.We also present the sufficient and necessary conditions of generalized bent functions.Last,we provide the construction of generalized bent and bent functions by concatenation method.