Existence and number of fixed points of Boolean transformations via the semi-tensor product method

Using the semi-tensor product method, this paper investigates the existence and number of fixed points of Boolean transformations. First, a Boolean function is expressed in an algebraic form via constructing its unique structural matrix. Second, based on the matrix expression, the existence of fixed points of Boolean transformations is converted into finding solutions to algebraic equations, and a set of new results is presented. Moreover, an effective algorithm is established to find all the fixed points of Boolean transformations. Finally, as an application, we study how to construct all the Boolean transformations with fixed number of fixed points.