Abstract

Unique shortest vector problem (uSVP) plays an important role in lattice based cryptography. Many cryptographic schemes based their security on it. For the cofidence of those applications, it is essential to clarify the complexity of uSVP with different parameters. However, proving the NP-hardness of uSVP appears quite hard. To the state of the art, we are even not able to prove the NP-hardness of uSVP with constant parameters. In this work, we gave a lower bound for the hardness of uSVP with constant parameters, i.e. we proved that uSVP is at least as hard as gap shortest vector problem (GapSVP) with gap of \(O(\sqrt{n/\log (n)})\), which is in \(NP \cap coAM\). Unlike previous works, our reduction works for paramters in a bigger range, especially when the constant hidden by the big-O in GapSVP is smaller than 1.

Graphical abstract

中文翻译：

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改进了唯一最短向量问题的复杂性下界

摘要

独特的最短向量问题（uSVP）在基于格的​​密码学中发挥着重要作用。许多加密方案的安全性都基于它。为了保证这些应用的一致性，有必要澄清具有不同参数的 uSVP 的复杂性。然而，证明 uSVP 的 NP 难度似乎相当困难。就现有技术而言，我们甚至无法证明具有恒定参数的 uSVP 的 NP 硬度。在这项工作中，我们给出了具有常数参数的 uSVP 的难度下界，即我们证明了 uSVP 至少与间隙为 \(O(\sqrt{n/\ log (n)})\)，位于\(NP \cap coAM\)中。与之前的工作不同，我们的归约适用于更大范围内的参数，特别是当 GapSVP 中的大O隐藏的常数小于 1 时。

图形概要