Relations of the E-Derivative, Derivative, Algebraic Immunity, and Algebraic Degree of Balanced H Boolean Functions

Abstract

We examine the algebraic immunity and algebraic degree of balanced H Boolean functions and resilient H Boolean functions that exhibit propagation, balance, and correlation immunity by using a derivative of Boolean functions and e-derivative that we defined ourselves. We determine that e-derivatives decide the algebraic immunity of balanced H Boolean functions and resilient H Boolean functions. We also identify that derivatives partly decide the algebraic degree of functions. Meanwhile, we deduce that without changing the algebraic immunity order, the algebraic degree of balanced H Boolean functions can be improved through a change in the e-derivative, and as a result, algebraic complexity is also improved. Furthermore, a method to develop resilient H Boolean functions is developed with the use of generalized complement transformation and cascade operation.