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arXiv · 2603.07274

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

Konstantinos A. Draziotis·Myrto Eleftheria Gkogkou
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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.

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Konstantinos A. Draziotis, Myrto Eleftheria Gkogkou. 2026-03-07. Worst--Case to Average--Case Reductions for SIS over integers. https://arxiv.org/abs/2603.07274

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