  Cyclic lattices are sublattices of $\mathbb Z^N$ that are preserved under the
rotational shift operator. Cyclic lattices were introduced by D.~Micciancio and
their properties were studied in the recent years by several authors due to
their importance in cryptography. In particular, Peikert and Rosen showed that
on cyclic lattices in prime dimensions, the shortest independent vectors
problem SIVP reduces to the shortest vector problem SVP with a particularly
small loss in approximation factor, as compared to general lattices. In this
paper, we further investigate geometric properties of cyclic lattices. Our main
result is a counting estimate for the number of well-rounded cyclic lattices,
indicating that well-rounded lattices are more common among cyclic lattices
than generically. We also show that SVP is equivalent to SIVP on a positive
proportion of Minkowskian well-rounded cyclic lattices in every dimension. As
an example, we demonstrate an explicit construction of a family of such
lattices on which this equivalence holds. To conclude, we introduce a class of
sublattices of $\mathbb Z^N$ closed under the action of subgroups of the
permutation group $S_N$, which are a natural generalization of cyclic lattices,
and show that our results extend to all such lattices closed under the action
of any $N$-cycle.
