Research On Properties And Constructions Of Bent Functions And Their Generalizations

Cryptography provides key technologies for the security of cyberspace,and cryptographic function,as one of the most important cryptographic primitives,is widely used in the fields of symmetric cryptography algorithms,coding theory,sequence,and combinatorial design.In particular,as a class of cryptographic functions with optimum nonlinearity,bent functions have attracted wide attention from many scholars.In this thesis,we concentrate our works on the properties and constructions of bent functions and their generalizations,including p-ary bent functions,generalized bent functions,and p-bent functions,et al.The main contributions are listed as follows:Firstly,we present the secondary construction of（non-）weakly regular and vectorial dual p-ary bent functions from generalized semi-direct sum and Carlet methods.This can be used to construct bent functions with different regularities by employing suitable initial vectorial bent functions and their combinations.More precisely,we present some classes of（weakly-）regular p-ary bent functions by using vectorial Maiorana-McFarland（M-M）and Partial Spread（PS）bent functions,respectively,and the explicit expressions for the dual of these functions are obtained.Based on the perfect nonlinear（PN）functions,the infinite families of non-weakly regular p-ary bent functions can also be produced.Secondly,we provide more new characterizations of generalized bent functions over Z_q with flexible coefficients by using new spectral decomposition techniques.Some sufficient（or also necessary）conditions and the dual of generalized bent functions under binomial and trinomial linear combinations are analyzed.Furthermore,the equivalent characterizations of generalized bent functions over different Z_q are derived.Based on these results,we give a class of generalized bent functions without increasing variables by generalized indirect sum,which suggests that it contains some known methods.Some generalized bent functions are also obtained by using the new concatenation and composite constructions.Next,we establish the connection among the sum-of-squares,the signal-to-noise ratio,and the Gowers U₂ norm of generalized Boolean functions.The tight bounds on the Gowers U₂ norm and the general properties of the signal-to-noise ratio are given.The expressions of the Gowers U₂ norm for two classes of generalized Boolean functions are obtained.We study the Gowers U_l（l∈ Z+）norm of q-ary functions.For l=2,the link between the Gowers U₂ norm and the nonlinearity of Lee distance measurement for functions over Z₄ is analyzed.The Gowers U₂ norm of three classes of（balanced）q-ary functions are calculated.For l=3,we analyze and calculate the Gowers U₃ norm of q-ary M-M functions.Finally,we investigate the characteristics and properties of generalized Boolean functions based on the p-Walsh spectrum and correlation function,as well as a class of generalized Boolean functions in terms Boolean functions under the p-Walsh spectrum.We derive the correlation functions for a class of concatenation generalized Boolean functions in n+2-variables,which proves that a class of symmetric functions about two variables are not p-bent functions.We provide the equivalent characterization and subspace decomposition of generalized Boolean functions to ρ-bent functions.The existence of p-bent functions with concrete functional forms when p takes some special values is discussed.