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Bl\"omer and Seifert showed that $\mathsf{SIVP}_2$ is NP-hard to approximateby giving a reduction from $\mathsf{CVP}_2$ to $\mathsf{SIVP}_2$ for constantapproximation factors as long as the $\mathsf{CVP}$ instance has a certainproperty. In order to formally define this requirement on the $\mathsf{CVP}$instance, we introduce a new computational problem called the Gap ClosestVector Problem with Bounded Minima. We adapt the proof of Bl\"omer and Seifertto show a reduction from the Gap Closest Vector Problem with Bounded Minima to$\mathsf{SIVP}$ for any $\ell_p$ norm for some constant approximation factorgreater than $1$.  In a recent result, Bennett, Golovnev and Stephens-Davidowitz showed thatunder Gap-ETH, there is no $2^{o(n)}$-time algorithm for approximating$\mathsf{CVP}_p$ up to some constant factor $\gamma \geq 1$ for any $1 \leq p\leq \infty$. We observe that the reduction in their paper can be viewed as areduction from $\mathsf{Gap3SAT}$ to the Gap Closest Vector Problem withBounded Minima. This, together with the above mentioned reduction, impliesthat, under Gap-ETH, there is no $2^{o(n)}$-time algorithm for approximating$\mathsf{SIVP}_p$ up to some constant factor $\gamma \geq 1$ for any $1 \leq p\leq \infty$.

A Note on the Concrete Hardness of the Shortest Independent Vectors Problem in Lattices