On Construction Of Weightwise Perfectly Balanced Boolean Functions With High Nonlinearity

With the development of information technology,the network has become an important means of information transmission in modern society,and the consequent security problems have become increasingly prominent,which have also driven the application and development of cryptography.As an important part of block and stream ciphers,Boolean functions play an important role in the study of cryptography.Boolean functions are usually studied in the whole vector space.However,Boolean functions restricted to a subset ofk₂ⁿ have attracted a lot of attention in the context of a new homomorphic-friendly stream cipher which is called FLIP.The study of such functions with good cryptographic properties has become one of the hot topics in cryptography.As with the classical Boolean functions,the nonlinearity and balancedness of the Boolean functions restricted to a subset is still an important cryptographic criterion.The difference is that when the functions’inputs are restricted to a set of vectors with Hamming weight k,the criterion is called k-weight nonlinearity.An n-variable Boolean function is said to be weightwise perfectly balanced if it always keeps balance in all setsE_n,k={x∈k₂ⁿ|wt（x）=k},where 1≤k≤n-1.In this paper,we focus on the construction of weightwise perfectly balanced Boolean functions with high k-weight nonlinearity,with the following main results:（1）A unified construction is given by extending the known algebraic construction of weightwise perfectly balanced Boolean functions.Firstly,a family of 2^m-variable Boolean functions with algebraic degree of2^d-1 are introduced,where m and d are positive integers and d is not greater than m.Secondly,a unified construction of 2^m-variable weightwise perfectly balanced Boolean functions is obtained by modifying the support of this family of functions.When d is fixed to any positive integer,it represents a class of weightwise perfectly balanced Boolean functions with 2^m variables.In particular,when d=1,2,3,m,four known classes of weightwise perfectly balanced functions are included as special cases,and when d takes other integer values,they are new functions that have never appeared before.In addition,some cryptographic criteria such as algebraic degree,k-weight nonlinearity and algebraic immunity of this family of weightwise perfectly balanced Boolean functions are discussed,and the results show that in the case of small variables,the newly constructed functions have high nonlinearity when k takes middle values.（2）A construction of a new class of weightwise perfectly balanced Boolean functions is given.First,a new class of quartic functions is defined with algebraic normal form,where m is a positive integer and m?3.The k-weight of the functions is theoretically computed.Then,by modifying the support of the quartic functions,a new class of2^m-variable weightwise perfectly balanced Boolean functions is constructed,where m is any positive integer,and the weightwise perfectly balancedness and algebraic degree of these functions is proved.This class of functions does not belong to the unified construction of weightwise perfectly balanced Boolean functions in（1）.At the same time,the k-weight nonlinearities of the newly constructed functions in 8 and 16 variables are calculated,and the result is improved compared to the construction of similar methods when k takes some values.