Constructions Of Balanced Cryptographic Functions With High Nonlinearity

Cryptographic functions play an important role in the design of stream ciphers and block ciphers.Nonlinearity is an important indicator to measure the security of cryptographic functions.The functions with high nonlinearity can effectively resist linear attacks and best affine approximation attacks.Bent functions are Boolean functions with the highest nonlinearity,but they are not balanced and cannot be directly applied to the cryptosystem.Therefore,the constructions of balanced cryptographic functions with high nonlinearity are the most important research topics in the design of stream ciphers.It is still an open problem to determine the highest nonlinearity of the balanced functions.In addition,the multi-output Boolean functions are more efficient in cryptosystem.However,there are mutual restraints between the nonlinearity and the output dimension,it is a hard task to find a way to achieve the optimization in the design of cryptographic functions.In this thesis,disjoint linear codes are used to construct balanced cryptographic functions with high nonlinearity,which in-cludes almost optimal Boolean functions,resilient almost optimal Boolean functions and multi-output Boolean functions.The following results are obtained:1)The first construction method of balanced Boolean functions with high nonlinearity is proposed.Let input dimension be n ,when n is odd,new semi-bent functions are constructed.When n is even,new almost optimal functions are constructed.Let n =2k+1,F₂ⁿ can be divided into 2^k+1[ n ,k]linear codes and one[ n ,k+1]linear code,and these 2^k+1+1 linear codes form a disjoint codes set;let n =2k+2,F₂ⁿ can be divided into 2^k+2[ n ,k]linear codes and one[ n ,k+2]linear code,and these 2^k+2+1 linear codes form a disjoint codes set.Then,the Boolean functions are constructed by selecting the appropriate disjoint linear code as the support set of the functions,and the conclusion proved by calculating the Walsh spectra of the functions.These functions are derived from the perspective of vector spaces,which are simple and fast to implement.A large number of different functions can be constructed by this method.2)The designing method of 1-resilient almost optimal functions are given by modifying that in the first construction.In addition,by using disjoint linear codes and high non-linearity permutations,a class of? n , n -k?balanced multi-output Boolean functions with high nonlinearity are given,where n/3<k<n/2.The output dimension of the multi-output Boolean functions constructed by this method is greater than half of their input dimension.