  Self-dual cyclic codes form an important class of linear codes. It has been
shown that there exists a self-dual cyclic code of length $n$ over a finite
field if and only if $n$ and the field characteristic are even. The enumeration
of such codes has been given under both the Euclidean and Hermitian products.
However, in each case, the formula for self-dual cyclic codes of length $n$
over a finite field contains a characteristic function which is not easily
computed. In this paper, we focus on more efficient ways to enumerate self-dual
cyclic codes of lengths $2^\nu p^r$ and $2^\nu p^rq^s$, where $\nu$, $r$, and
$s$ are positive integers. Some number theoretical tools are established. Based
on these results, alternative formulas and efficient algorithms to determine
the number of self-dual cyclic codes of such lengths are provided.
