| Boolean functions are an important part of cryptographic algorithms.At Eurocrypt 2016 proposed a new family of stream ciphers,called FLIP,which is intended to be combined with a homomorphic encryption scheme to create an acceptable system of fully homomorphic encryption.The filter function in the FLIP cipher is a Boolean function with input limiting weight.In order to resist various attacks,the design of the filter function needs to have good cryptographic properties.Since the most important property of cryptography is balance,it is of great significance to construct a weightwise perfectly balanced Boolean function with good cryptographic properties.This paper describes the design of several classes of Boolean functions that weightwise(almost)perfectly balanced Boolean functions by weight and analyzes their cryptographic properties.The main results are as follows:Firstly,we first present two classes of odd-weightwise perfectly balanced Boolean functions on 2m variables,where m is a positive integer.And then,we give two concrete constructions of weightwise perfectly balanced Boolean functions on 2m variables by modifying the support of the odd-weightwise perfectly balanced Boolean functions respectively.The weightwise nonlinearities and the algebraic immunities of the newly constructed weightwise perfectly balanced Boolean functions are discussed at the end of this paper.Secondly,we first introduce a class of quartic Boolean functions on 8m variables,where m is a positive integer.Then introduce a class of Boolean functions on 1≤n≤7 and sum them directly with the quartic Boolean function of the 8m variables to obtain a class of quartic Boolean functions with arbitrary arguments.And then,we give the construction method of weightwise almost perfectly balanced Boolean functions on n variables by modifying the support of the quartic Boolean functions.The weightwise nonlinearities of the newly constructed weightwise almost perfectly balanced Boolean functions are discussed at the end of this paper.Lastly,we present a systematic method of constructing weightwise almost perfectly balanced Boolean functions on 2p1+2p2,which equal the direct sum of two known weightwise perfectly balanced Boolean functions,where p1 and p2 are different from each other Nonnegative integer.Then,generalizing this conclusion to any weightwise perfectly balanced Boolean function can also obtain a weightwise almost perfectly balanced Boolean function on 2p1+2p2 +…+2pm after doing straight-sum operations,where pm is a non-negative integer that is not the same as each other,m≥3.This paper gives a general method for constructing a weightwise almost perfectly balanced Boolean function for the first time.At the same time,we show two concrete constructions of weightwise almost perfectly balanced Boolean functions,whose k-weight nonlinearities are discussed at the end of this paper. |
