Construction Of Resilient Boolean Functions With Multi-dimensional Vectorial Outputs And High Nonlinearity

How to construct resilient Boolean functions with multi-dimensional vectorial outputs and high nonlinearity is one of the most important problems in the design and analysis of stream cipher.Generally,a good vector-valued function has to be satisfied with several criteria,such as high nonlinearity,resiliency,high algebraic degree and differential uniform.But there are mutual restraints among the cryptographic parameters.Finding a way to achieve the optimization is always regarded as a hard task.Especially,as the most important parameters of measuring the safety of Boolean functions,the researches of nonlinearity and resiliency are of great significance.1)A construction of resilient Semi-Bent Boolean functions with high dimensional vectorial outputs has been proposed.Set the input dimension to n.When n is odd,n=2k+1,a mapping from F₂^kto F₂^k+1can be built,and the component functions of vector-valued functions are constructed by using 2^klinear functions with k+1variables.Furthermore,if n is even,n=2k,a mapping from F₂^k-1to F₂^k+1can be built,and the component functions are constructed by using 2^k-1linear functions with k+1 variables.It has been proved that the nonlinearities of all the constructed vector-valued functions are almost optimal.Compared with the existing results,on the condition of remaining the same order of resiliency,the vector-valued functions have higher dimensional vectorial outputs.2)A construction of resilient Boolean functions with strictly almost optimal nonlinearity and multi-dimensional vectorial outputs has been proposed.By using a number of[n/2,m,t+1]disjoint codes and known high nonlinear?n/2+k,m,t?vector-valued functions,a number of new?n,m,t?vector-valued functions with strictly almost optimal nonlinearity have been constructed.Compared to other methods,the nonlinearity have been largely improved on the condition of remaining the same re-siliency order.Moreover,many of the vector-valued functions constructed by our method have the currently best known nonlinearity.