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

Abstract: Bl\"omer and Seifert 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 Bl\"omer and Seifert 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 showed that under Gap-ETH, there is no $2<sup>{o(n)}$-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 a reduction from $\mathsf{Gap3SAT}$ to the Gap Closest Vector Problem with Bounded Minima. This, together with the above mentioned reduction, implies that, under Gap-ETH, there is no $2<sup>{o(n)}$-time algorithm for approximating $\mathsf{SIVP}_p$ up to some constant factor $\gamma \geq 1$ for any $1 \leq p \leq \infty$.