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 $\|{\bf x}\|_\infty \le β,$ for a given bound $β.$ 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.

翻译：本文研究整数上短整数解问题的一个非模变体。给定一个随机矩阵 $A \in \mathbb{Z}^{n\times m}$，其元素 $a_{ij}$ 满足 $0\le a_{ij}< Q$（$Q>0$），目标是找到一个非零向量 ${\bf x}\in\mathbb{Z}^m$，使得 $A{\bf x}={\bf 0}$ 且 $\|{\bf x}\|_\infty \le β$，其中 $β$ 为给定界。我们证明，能以不可忽略概率求解该问题随机实例的算法，可转化为在最坏情况下以 $\widetilde{O}(n^{3/2})$ 因子（采用 $\ell_2$ 范数）近似任意 $n$ 维整数格上 $\mathrm{SIVP}$ 问题的多项式时间算法。