Worst--Case to Average--Case Reductions for SIS over integers
This work provides a worst-case to average-case reduction for a non-modular SIS problem, which is significant for the cryptographic community and lattice-based cryptography.
This paper investigates a non-modular Short Integer Solution (SIS) problem over integers, aiming to find a short non-zero vector x such that Ax=0. The authors demonstrate that an algorithm solving random instances of this problem can be used to approximate the Shortest Independent Vector Problem (SIVP) within a factor of O(n^(3/2)) in the worst case for any n-dimensional integer lattice.
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.