(Gap/S)ETH Hardness of SVP
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We prove the following quantitative hardness results for the Shortest Vector Problem in the ℓ_p norm (_p), where n is the rank of the input lattice. ∙ For "almost all" p > p_0 ≈ 2.1397, there no 2^n/C_p-time algorithm for _p for some explicit constant C_p > 0 unless the (randomized) Strong Exponential Time Hypothesis (SETH) is false. ∙ For any p > 2, there is no 2^o(n)-time algorithm for _p unless the (randomized) Gap-Exponential Time Hypothesis (Gap-ETH) is false. Furthermore, for each p > 2, there exists a constant γ_p > 1 such that the same result holds even for γ_p-approximate _p. ∙ There is no 2^o(n)-time algorithm for _p for any 1 ≤ p ≤ 2 unless either (1) (non-uniform) Gap-ETH is false; or (2) there is no family of lattices with exponential kissing number in the ℓ_2 norm. Furthermore, for each 1 ≤ p ≤ 2, there exists a constant γ_p > 1 such that the same result holds even for γ_p-approximate _p.
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