Constrained inhomogeneous spherical equations: average-case hardness

TL;DR

This paper investigates the average-case computational hardness of solving constrained spherical equations over specific finite metabelian groups, linking their difficulty to the hardness of certain lattice approximation problems.

Contribution

It establishes the average-case hardness of solving constrained spherical equations in finite metabelian groups, connecting it to lattice approximation problem complexity.

Findings

Proves average-case hardness of solving certain spherical equations

Links the problem's difficulty to lattice approximation complexity

Analyzes equations over specific finite metabelian groups

Abstract

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 $G_{p,n}=\mathbb{Z}_p^n \rtimes \mathbb{Z}_p^\ast$, 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).