  Let $\mathcal{M}=CM(D_n,X,p)$ be a regular Cayley map on the dihedral group
$D_n$ of order $2n, n \ge 2,$ and let $\pi$ be the power function associated
with $\mathcal{M}$. In this paper it is shown that the kernel Ker$(\pi)$ of the
power function $\pi$ is a dihedral subgroup of $D_n$ and if $n \ne 3,$ then the
kernel Ker$(\pi)$ is of order at least $4$. Moreover, all $\mathcal{M}$ are
classified for which Ker$(\pi)$ is of order $4$. In particular, besides $4$
sporadic maps on $4,4,8$ and $12$ vertices respectively, two infinite families
of non-$t$-balanced Cayley maps on $D_n$ are obtained.
