On Relevance of Derivative, E-derivative, and Correlation Immunity of Sum and Product for Boolean Functions

Abstract

In this paper, we discuss the relationship among derivative, e-derivative, and correlation immunity of the sum and the product for Boolean functions. We obtain the sufficient and necessary condition, which the sum of two elastic Boolean functions represented by the derivative and e-derivative are elastic Boolean functions. And we get the sufficient condition, which the sum and product of the correlation immune Boolean functions represented by the derivative are correlation immune. We also get the sufficient and necessary condition of the sum of two functions which is m-order correlation immunity, which is the product of two functions with m-order correlation immunity. At the same time, we obtain the sufficient conditions for correlation immunity of the derivatives and the ederivatives of the sum of two functions and so on. The correlation immunity of Boolean function is closely related to the derivative and e-derivative of functions.