Relevant Properties Of Derivatives Of Boolean Functions

The derivatives of Boolean function play an important role in the study of various cryptographic properties of Boolean function.In this paper,firstly,we study the relationship between the algebraic normal form and the derivatives of Boolean function,and obtain a necessary and sufficient condition of the coefficients any quadratic Boolean function for its derivative to be a constant or of function degree one,respectively.Secondly,based on the properties of the linear kernel of Boolean function,we give the concrete weight of any quadratic Boolean function and the dimension of its linear kernel,and extend to Boolean function degree k.Then we use the properties of Walsh transformation of Boolean function to determine the Walsh spectrum of a special kind of Boolean function.Finally,we generalize a result of Carlet to characterize a class of Bent function by using the derivatives of Boolean function.