  Consider $f:\Omega^n_K\to\mathbf{C}$ a function from the $n$-fold product of
multiplicative cyclic groups of order $K$. Any such $f$ may be extended via its
Fourier expansion to an analytic polynomial on the polytorus $\mathbf{T}^n$,
and the set of such polynomials coincides with the set of all analytic
polynomials on $\mathbf{T}^n$ of individual degree at most $K-1$.
  In this setting it is natural to ask how the supremum norms of $f$ over
$\mathbf{T}^n$ and over $\Omega_K^n$ compare. We prove the following Remez-type
inequality: if $f$ has degree at most $d$ as an analytic polynomial, then
$\|f\|_{\mathbf{T}^n}\leq C(d,K)\|f\|_{\Omega_K^n}$ with $C(d,K)$ independent
of dimension $n$. As a consequence we also obtain a new proof of the
Bohnenblust--Hille inequality for functions on products of cyclic groups.
  Key to our argument is a special class of Fourier multipliers on $\Omega_K^n$
which are $L^\infty\to L^\infty$ bounded independent of dimension when
restricted to low-degree polynomials. This class includes projections onto the
$k$-homogeneous parts of low-degree polynomials as well as projections of much
finer granularity.
