  We consider the cubic Hyperbolic Schr\"odinger equation \eqref{eq:nls} on
torus $\T^2$. We prove that sharp $L^4$ Strichartz estimate, which implies that
\eqref{eq:nls} is analytic locally well-posed in in $H^s(\T^2)$ with $s>1/2$,
meanwhile, the ill-posedness in $H^s(\T^2)$ for $s<1/2$ is also obtained. The
main difficulty comes from estimating the number of representations of an
integer as a difference of squares.
