While examining some $NP$-complete problems relating to sets of integers, a question flashed through my mind: whether the $NP$-completeness of these problems is retained when integer arithmetic is replaced by modular arithmetic?
A concrete example is the subset product problem, which was known to be $NP$-complete. I wonder if we consider the problem over ring $\mathbb{Z}_n$, as formulated below, it is still $NP$-complete?
Given $t, n \in \mathbb{N}$ and a finite set of integers $S$, determine whether there exists $S' \subseteq S$ such that $\prod_{s \in S'}s \equiv t \pmod n$.
My principal difficulty in approaching the problem is that I cannot figure out how the (polynomial) reduction works, supposing the $NP$-completeness is retained, due to the "cyclic" behavior of modular arithmetic. I've tried to search for variants of the problem but no luck. Thank you for any explanations, suggestions or references.