Almost optimum $\ell$-covering of $\mathbb{Z}_n$
Ke Shi, Chao Xu
TL;DR
The paper tackles the problem of constructing small $\ell$-covering sets in the ring $\mathbb{Z}_n$, quantified by $f(n,\ell)$, and provides a constructive upper bound $f(n,\ell)=O\left(\frac{n}{\ell}\log n\right)$ for all $n,\ell$, along with a randomized variant achieving $\tilde{O}\left(\frac{n}{\ell}\right)$-type time. The authors also prove a near-matching lower bound in certain regimes and show how these coverings can significantly simplify modular subset-sum algorithms. Their approach refines the relative totient bound $\phi(n,\ell)$ via Brun’s sieve and leverages a large divisor with a linear divisor sum to lift coverings from the divisor semigroup $\mathbb{D}_n$ to $\mathbb{Z}_n$, while providing auxiliary number-theoretic results. The work combines semigroup-to-group reductions, greedy set-cover bounds, and constructive layering to achieve tight or near-tight bounds and yields practical implications for related modular arithmetic tasks. Overall, the results advance understanding of covering problems in semigroups and offer practical tools for efficient modular computations, including simpler modular subset-sum procedures.
Abstract
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 $Ω(\frac{n}{\ell}\frac{\log n}{\log \log n})$. 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.