Suppose $\mathbb{Z}/m \mathbb{Z}$ is a residue ring for some $m \in \mathbb{N}$. If $m=p$ is a prime number then every function $f:\mathbb{Z}/p \mathbb{Z} \rightarrow \mathbb{Z}/p \mathbb{Z}$ is a polynomial $f(x) \in (\mathbb{Z}/p \mathbb{Z})[x]$.

**Question:** Are there any simple criteria of polynomiality of $f$ if $m$ is composite (particularly, if $m$ is a power of a prime number or $m$ is a product of two *distinct* primes)?

**Update:** Recently I found the article On polynomial functions $\text{(mod $m$)}$ by D. Singmaster; there are also a series of papers by Z. Chen (1 and 2) about polynomial functions. One interesting result is the following:

Let $f$ be a polynomial function $\text{(mod $m$)}$. Then $f$ has a unique polynomial representation $$f=\sum_{k=0}^{n-1} b_k x^k\quad \text{with}\quad 0 \leq b_k \leq m/(k!,m)$$ and $n$ is the least integer s.t. $m | n!$.

International Committee for the Abolition of the Notation$\mathbb{Z}_p$for Denoting Other Than p-Adic Numbers) $\endgroup$ – Qfwfq Sep 27 '13 at 13:49