  Polynomial representations of Boolean functions over various rings such as
$\mathbb{Z}$ and $\mathbb{Z}_m$ have been studied since Minsky and Papert
(1969). From then on, they have been employed in a large variety of fields
including communication complexity, circuit complexity, learning theory, coding
theory and so on. For any integer $m\ge2$, each Boolean function has a unique
multilinear polynomial representation over ring $\mathbb Z_m$. The degree of
such polynomial is called modulo-$m$ degree, denoted as
$\mathrm{deg}_m(\cdot)$.
  In this paper, we investigate the lower bound of modulo-$m$ degree of Boolean
functions. When $m=p^k$ ($k\ge 1$) for some prime $p$, we give a tight lower
bound that $\mathrm{deg}_m(f)\geq k(p-1)$ for any non-degenerated function
$f:\{0,1\}^n\to\{0,1\}$, provided that $n$ is sufficient large. When $m$
contains two different prime factors $p$ and $q$, we give a nearly optimal
lower bound for any symmetric function $f:\{0,1\}^n\to\{0,1\}$ that
$\mathrm{deg}_m(f) \geq \frac{n}{2+\frac{1}{p-1}+\frac{1}{q-1}}$.
