  We are interested in the long time behavior of solutions of the nonlinear
Schr{\"o}dinger equation on the $d$-dimensional torus in low regularity, i.e.
for small initial data in the Sobolev space $H^{s_0}(\mathbb T^d)$ with
$s_0>d/2$. We prove that, even in this context of low regularity, the
$H^s$-norms, $s\geq 0$, remain under control during times, $T_\varepsilon= \exp
\big(-\frac{|\log\varepsilon|^2}{4\log|\log\varepsilon|} \big)$, exponential
with respect to the initial size of the initial datum in $H^{s_0}$,
$\|u(0)\|_{H^{s_0}}=\varepsilon$. For this, we add to the linear part of the
equation a random Fourier multiplier in $\ell^\infty(\mathbb Z^d)$ and show our
stability result for almost any realization of this multiplier. In particular,
with such Fourier multipliers, we obtain the almost global well posedness of
the nonlinear Schr{\"o}dinger equation on $H^{s_0}(\mathbb T^d)$ for any
$s_0>d/2$ and any $d\geq1$.
