Research On 2^k-bent And ?_p-bent Functions

Boolean function is an important component of designing sequence ciphers and block ciphers,the quality of its cryptographic properties is directly related to the security of the entire cryptographic system.Bent function,which has optimal nonlinearity,has important applications in relative difference set theory.Generalized Bent function has attracted the attention of many scholars.This dissertation stud-ies the properties and construction of 2^k-bent function and?_p-bent function.Main constributions of this work are listed as follows:(1)The properties of the 2^k-correlation function and the 2^k-Hadamard transform of Boolean function are studied.Firstly,the relationship among 2^k-crosscorrelation functions of arbitrary four Boolean functions is given.Based on this relation,the relationship between 2^k-crosscorrelation function and 2^k-autocorrelation function is proved.Secondly,some properties of 2^k-correlation function and 2^k-Hadamard trans-form are given.(2)Using secondary construction,a class of n+m variables Boolean function is constructed,and the necessary and sufficient conditions for it to be 2^k-bent function are given by analyzing the property of 2^k-Hadamard transform.(3)The construction of?_p-bent function is studied.Five classes of different?_p-Boolean functions are constructed by using secondary construction and concatenation methods.Through the study of?_p-Walsh-Hadamard transform,the spectral value formula of the function is given.Furthermore,we obtain the necessary and sufficient conditions for it to be?_p-bent function,and give a specific?_p-Boolean function to prove the necessary and sufficient conditions when it is a?_p-bent function.