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.
Categories: cs.CR