Dirichlet Product for Boolean Functions

Boolean functions play an important role in many symmetric cryptosystems and are crucial for their security. It is important to design boolean functions with reliable cryptographic properties such as balancedness and nonlinearity. Most of these properties are based on specific structures such as Möbius transform and Algebraic Normal Form. In this paper, we introduce the notion of Dirichlet product and use it to study the arithmetical properties of boolean functions. We show that, with the Dirichlet product, the set of boolean functions is an Abelian monoid with interesting algebraic structure. In addition, we apply the Dirichlet product to the sub-family of coincident functions and exhibit many properties satisfied by such functions.