Constrained inhomogeneous spherical equations: average-case hardness

In this paper we analyze computational properties of the Diophantine problem (and its search variant) for spherical equations ${\prod }_{i=1}^m{z}_i^{-1}{c}_i{z}_i=1$ (and its variants) over the class of finite metabelian groups $G_{p,n}={\mathbb{Z}}_p^n⋊{\mathbb{Z}}_p^{\ast }$, where $n\in \mathbb{N}$ and $p$ is prime. We prove that the problem of finding solutions for certain constrained spherical equations is computationally hard on average (assuming that some lattice approximation problem is hard in the worst case).