The Construction Of Three Classes Of Bent Functions

The study of cryptographic functions has a significance role in cryptography.A cryptographic function with appropriate and good cryptographic properties can greatly improve the security of the cryptographic system.Cryptographic functions are divided into boolean functions and vector-valued functions,among which boolean functions play an important role in designing stream cipher,block cipher and hash function.The security criteria of cryptographic functions include algebraic degree,correlation immunity,balancedness,difference uniformity,nonlinearity,algebraic immunity and so on.The nonlinearity is used to measure the ability of cryptographic functions to resist linear attack,and the bent function are these cryptographic functions with the best nonlinearity.Therefore,it is of great significance to study and construct bent functions.It is difficult to classify bent functions completely,and it is also challenging to construct and generalize bent functions.The construction of bent functions has always been the focus of scholars' research.In this paper,we investigate a class of p-ary functions over the finite field IFpn with the form f(x)=g(x)+?i=1 k Tr(uix).Firstly,we give the concrete expression of Walsh transform of f(x)based on that of g(x),where p is an arbitrary prime and k is a positive integer.Further,for k=2,we construct new bent functions by choosing appropriate g(x):When g(x)is a monomial bent function of Dillon type,we give a necessary and sufficient condition such that f(x)is bent,and we conjecture that f(x)cannot be bent in this case based on our computer experiments;when g(x)is a monomial trace function of Gold type,through in-depth discussions on the Walsh spectral values of g(x),three new bent functions are obtained.