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

05/24/2020

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by   Divesh Aggarwal, et al.

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Blömer and Seifert showed that 𝖲𝖨𝖵𝖯_2 is NP-hard to approximate by giving a reduction from 𝖢𝖵𝖯_2 to 𝖲𝖨𝖵𝖯_2 for constant approximation factors as long as the 𝖢𝖵𝖯 instance has a certain property. In order to formally define this requirement on the 𝖢𝖵𝖯 instance, we introduce a new computational problem called the Gap Closest Vector Problem with Bounded Minima. We adapt the proof of Blömer and Seifert to show a reduction from the Gap Closest Vector Problem with Bounded Minima to 𝖲𝖨𝖵𝖯 for any ℓ_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^o(n)-time algorithm for approximating 𝖢𝖵𝖯_p up to some constant factor γ≥ 1 for any 1 ≤ p ≤∞. We observe that the reduction in their paper can be viewed as a reduction from 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^o(n)-time algorithm for approximating 𝖲𝖨𝖵𝖯_p up to some constant factor γ≥ 1 for any 1 ≤ p ≤∞.