On Normality Of Boolean Functions

Cryptographic Boolean functions must satisfy several criteria simultaneously to sat-isfy the theory of confusion and diffusion proposed by Shannon. Normality is one of thecriteria on which the dissertation focus. The main results are as follows.(1). Plateaued function plays well in cryptography, and can achieve the trade o? amongseveral criteria. Based on the theory of Walsh transformation and of function de-composing, the decomposing of Plateaued functions on subspaces of large dimensionare discussed. We show that the functions keep high nonlinear after been decom-posed. Further, the theory of normality can be obtained by generalizing the theoryof decomposing. We characterize the normality of Plateaued functions by Walshtransformation, and then propose a simpler algorithm for checking the normalityof this kind of functions. Finally, the relation between the normality of a Booleanfunction and of its decomposing function is provided.(2). The normality of the duality function of a given Plateaued function is introduced,and its properties are discussed. We show that there is a strong relation betweenthe normality of a Plateaued function and of its dual function.(3). The normality of Plateaued functions that constructed by the known constructionmethods are analyzed. We show that, mostly the constructed functions are normal.Then we present several ways to construct nonnormal Plateaued functions.(4). The theory of generalized normality of Boolean function is presented as a general-ization of normality. We analyze the relations between generalized normality andnormality, generalized algebraic immunity. Finally, we propose to check the normal-ity of Boolean functions from its algebraic normal form(ANF).(5). The relations between normality and other cryptographic criteria such as nonlin-earity, linear structures, algebraic immunity are discussed, and then the balanced,nonnormal functions with high algebraic degree are proved to be exist by calculation.(6). Bent functions can achieve the highest nonlinearity and have the best propagationproperty, meanwhile the known Bent functions are nearly normal. So we want toconstruct Boolean functions which can satisfy several criteria simultaneously basedon normal Bent functions. We show that Boolean functions with high nonlinearity,balanced, correlation immunity, and SAC can be obtained by modifying, decompos-ing and concatenation of normal Bent functions.