Blömer and Seifert [1] showed that ${\mathsf{SIVP}}_2$ is NP-hard to approximate by giving a reduction from ${\mathsf{CVP}}_2$ to ${\mathsf{SIVP}}_2$ for constant approximation factors as long as the $\mathsf{CVP}$ instance has a certain property. In order to formally define this requirement on the $\mathsf{CVP}$ instance, we introduce a new computational problem called the Gap Closest Vector Problem with Bounded Minima. We adapt the proof of [1] to show a reduction from the Gap Closest Vector Problem with Bounded Minima to $\mathsf{SIVP}$ for any ${\ell }_p$ norm for some constant approximation factor greater than 1.

In a recent result, Bennett, Golovnev and Stephens-Davidowitz [2] showed that under Gap-ETH, there is no $2^{o(n)}$-time algorithm for approximating ${\mathsf{CVP}}_p$ up to some constant factor $\gamma \ge 1$ for any $1\le p\le \infty$. We observe that the reduction in [2] can be viewed as a reduction from $\mathsf{Gap}\mathsf{\text{-}}\mathsf{3}\mathsf{\text{-}}\mathsf{SAT}$ to the Gap Closest Vector Problem with Bounded Minima. This, together with the above mentioned reduction, implies that, under Gap-ETH, there is no randomised $2^{o(n)}$-time algorithm for approximating ${\mathsf{SIVP}}_p$ up to some constant factor $\gamma \ge 1$ for any $1\le p\le \infty$.

Blömer和Seifert [1]表明 ${\mathsf{锡普}}_2$ NP很难通过减少 ${\mathsf{副总裁}}_2$ 至 ${\mathsf{锡普}}_2$ 对于恒定的近似因子，只要 $\mathsf{副总裁}$实例具有一定的属性。为了正式定义此要求$\mathsf{副总裁}$例如，我们引入了一个新的计算问题，即有界极小值的差距最近向量问题。我们采用[1]的证明来显示从有界极小值的差距最近向量问题到$\mathsf{锡普}$ 对于任何 ${\ell }_p$ 一些大于1的恒定近似因子的范数

Bennett，Golovnev和Stephens-Davidowitz [2]在最近的结果中表明，在Gap-ETH下，没有 $2^{Ø（ñ）}$近似的时间算法 ${\mathsf{副总裁}}_p$ 达到一定的常数 $\gamma \ge \mathrm{1个}$ 对于任何 $\mathrm{1个}\le p\le \infty$。我们观察到[2]的减少可以看作是从$\mathsf{间隙}\mathsf{\text{--}}\mathsf{3}\mathsf{\text{--}}\mathsf{SAT考试}$有界极小值的间隙最近向量问题。这与上述减少相结合，意味着在Gap-ETH下，没有随机分组$2^{Ø（ñ）}$近似的时间算法 ${\mathsf{锡普}}_p$ 达到一定的常数 $\gamma \ge \mathrm{1个}$ 对于任何 $\mathrm{1个}\le p\le \infty$。