Worst--Case to Average--Case Reductions for SIS over integers
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915842595225600 |
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| author | Draziotis, Konstantinos A. Gkogkou, Myrto Eleftheria |
| author_facet | Draziotis, Konstantinos A. Gkogkou, Myrto Eleftheria |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_07274 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Worst--Case to Average--Case Reductions for SIS over integers Draziotis, Konstantinos A. Gkogkou, Myrto Eleftheria Cryptography and Security 94A60 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. |
| title | Worst--Case to Average--Case Reductions for SIS over integers |
| topic | Cryptography and Security 94A60 |
| url | https://arxiv.org/abs/2603.07274 |