Constrained inhomogeneous spherical equations: average-case hardness (Q6601473)

Let \(p\) be a prime, \(n \in \mathbb{N}\) and let \(G_{p,n}=\mathbb{Z}^{n}_{p} \rtimes \mathbb{Z}^{\times}_{p}\), where the action of \(\mathbb{Z}^{\times}_{p}\) on \(\mathbb{Z}^{n}_{p}\) is given by \((x_{1}, \ldots, x_{n}) \mapsto (\alpha x_{1}, \ldots, \alpha x_{n})\), the multiplication by an element \(\alpha \in \mathbb{Z}^{\times}_{p}\).\N\NIn the paper under review, the author analyzes computational properties of the Diophantine problem (and its search variant) for spherical equations \(\prod_{i=1}^{m}z_{i}^{-1}c_{i}z_{i}=1\) over the class of finite metabelian groups \(G_{p,n}\). Assuming that some lattice approximation problem is hard in the worst case, he proves that the problem of finding solutions for certain constrained spherical equations is computationally hard on average.