Smooth skew morphisms of dihedral groups

A skew morphism phi of a finite group A is a permutation on A fixing the identity element of A and for which there exists an integer-valued function pi on A such that phi(ab) = phi(a)phi(pi(a))(b) for all a, b is an element of A. In the case where pi(phi(a)) = pi(a), for all a is an element of A, the skew morphism is smooth. The concept of smooth skew morphism is a generalization of that of t-balanced skew morphism. The aim of this paper is to develop a general theory of smooth skew morphisms. As an application we classify smooth skew morphisms of dihedral groups.