Almost optimum $\ell$-covering of $\mathbb{Z}_n$

TL;DR

This paper establishes near-optimal bounds on the size of $ ext{ell}$-covering sets in $ ext{Z}_n$, providing constructions and lower bounds, with applications to modular subset sum algorithms.

Contribution

It introduces a construction of $ ext{ell}$-covering sets of size $O(rac{n}{ ext{ell}} ext{log} n)$ and proves lower bounds, using advanced sieve theory techniques.

Findings

Existence of $ ext{ell}$-covering sets of size $O(rac{n}{ ext{ell}} ext{log} n)$

Lower bounds of $ ext{ell}$-covering set size of $ ext{Omega}(rac{n}{ ext{ell}}rac{ ext{log} n}{ ext{log} ext{log} n})$

Application to simplifying modular subset sum algorithms

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 $\Omega(\frac{n}{\ell}\frac{\log n}{\log \log n})$. 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.