  A subset $B$ of the ring $\mathbb{Z}_n$ is referred to as a $\ell$-covering
set if $\{ ab \pmod n | 0\leq a \leq \ell, b\in B\} = \mathbb{Z}_n$. We show
that there exists a $\ell$-covering set of $\mathbb{Z}_n$ of size
$O(\frac{n}{\ell}\log n)$ for all $n$ and $\ell$, and how to construct such a
set. We also provide examples where any $\ell$-covering set must have a size of
$\Omega(\frac{n}{\ell}\frac{\log n}{\log \log n})$. The proof employs a refined
bound for the relative totient function obtained through sieve theory and the
existence of a large divisor with a linear divisor sum. The result can be used
to simplify a modular subset sum algorithm.
