  We say that $M$ and $S$ form a \textsl{splitting} of $G$ if every nonzero
element $g$ of $G$ has a unique representation of the form $g=ms$ with $m\in M$
and $s\in S$, while $0$ has no such representation. The splitting is called
{\it nonsingular} if $\gcd(|G|, a) = 1$ for any $a\in M$.
  In this paper, we focus our study on nonsingular splittings of cyclic groups.
We introduce a new notation --direct KM logarithm and we prove that if there is
a prime $q$ such that $M$ splits $\mathbb{Z}_q$, then there are infinitely many
primes $p$ such that $M$ splits $\mathbb{Z}_p$.
