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

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 $\mathbf{x}\in {\mathbb{Z}}^m$ such that $A\mathbf{x}=0$ and $\Vert \mathbf{x}{\Vert }_{\infty }\le \beta ,$ 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 $\tilde{O}(n^{3/2})$ (with ${\ell }_2$ norm) in the worst case for any $n-$dimensional integer lattice.