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 ∏ᵢ₌₁ᵐ zᵢ⁻¹ cᵢ zᵢ = 1 (and its variants) over the class of finite metabelian groups Gp,n=Zₚⁿ Zₚ^, where 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).