Worst--Case to Average--Case Reductions for SIS over integers

TL;DR

This paper explores a non-modular variant of the SIS problem over integers, establishing that solving average-case instances efficiently can lead to polynomial-time approximation algorithms for the worst-case SIVP problem within a specific factor.

Contribution

It introduces a reduction from average-case SIS over integers to worst-case SIVP approximation, linking the complexity of these problems.

Findings

Efficient algorithms for average-case SIS imply polynomial-time SIVP approximations.

The reduction achieves an approximation factor of roughly n^{3/2} in the worst case.

The work connects average-case hardness to worst-case lattice problems.

Abstract

In the present paper we study a non-modular variant of the Short Integer Solution problem over the integers. Given a random matrix $A \in \mathbb{Z}^{n\times m}$ with entries $a_{ij}$ such that $0\le a_{ij}< Q,$ for some $Q>0,$ the goal is to find a nonzero vector ${\bf x}\in\mathbb{Z}^m$ such that $A{\bf x}={\bf 0}$ and $\|{\bf x}\|_\infty \le \beta,$ for a given bound $\beta.$ We show that an algorithm that solves random instances of this problem with non-negligible probability yields a polynomial-time algorithm for approximating $\mathrm{SIVP}$ within a factor $\widetilde{O}(n^{3/2})$ (with $\ell_2$ norm) in the worst case for any $n-$dimensional integer lattice.