Paper 2008/504 Original: eprint.iacr.org/2008/504 ↗

The $n^c$-Unique Shortest Vector Problem is Hard

Abstract

The unique Shortest Vector Problem (uSVP) gained prominence because it was the problem upon which the first provably-secure lattice-based cryptosystems were built. But it was an open problem as to whether uSVP was as hard as the standard, more general, version of the shortest vector problem. We show that there is a reduction from the approximate decision version of the shortest vector problem (GapSVP) to the unique shortest vector problem. In particular, we show that for any $\gamma>6\sqrt{n}$, there is a reduction from GapSVP$_\gamma$ to $\frac{\gamma}{6\sqrt{n}}$-uSVP. This implies that the Ajtai-Dwork and the Regev cryptosystems are based on the hardness of the worst-case GapSVP$_{O(n^{2.5})}$ and GapSVP$_{O(n^{2})}$, respectively. Our reduction is quite elementary, but it does use a clever, yet surprisingly simple (in retrospect!), idea of Peikert that was recently used by him to construct a cryptosystem based on the worst-case hardness of GapSVP$_{O(n^3)}$.