ISIS problem in the case of $m=n$

Question

The Inhomogeneous Short Integer Solution (ISIS) problem is as follows: given an integer $q$, a matrix $A\in \mathbb{Z}^{n\times m}_q$, a vector $b\in \mathbb{Z}^{n}_q$, and a real $\beta$, find an integer vector $e\in\mathbb{Z}^m$ such that $Ae=b\mod q$ and $0<\Vert e\Vert_2\leq\beta$.

if we assume that $n=m$ is this average-case problem is still hard for a well-chosen $(n,q,\beta)$?

because (I have tested many matrices and solved it) in that case a Gaussian Elimination can be performed given $(A,b)$.

Answer 1

Generally gaussian elimination is ruled out through choosing $\beta$ appropriately, as in general Gaussian elimination will find you a solution $e'$ such that $Ae'\equiv b\bmod q$, but this solution is generally not short. In particular $\beta > \sqrt{n\log q}$ suffices, and $\beta\geq q$ is trivial to find.

See for example this paper, though it is for SIS (rather than ISIS). I am under the (admittedly vague) impression that the problems have similar hardness.