(Gap/S)ETH hardness of SVP

Abstract

We prove the following quantitative hardness results for the Shortest Vector Problem in the ℓp norm (SVP_p), where n is the rank of the input lattice. For “almost all” p > p0 ≈ 2.1397, there is no 2n/Cp-time algorithm for SVP_p for some explicit (easily computable) constant Cp > 0 unless the (randomized) Strong Exponential Time Hypothesis (SETH) is false. (E.g., for p ≥ 3, Cp < 1 + (p+3) 2−p + 10 p2 2−2p.) For any 1 ≤ p ≤ ∞, there is no 2o(n)-time algorithm for SVP_p unless the non-uniform Gap-Exponential Time Hypothesis (Gap-ETH) is false. Furthermore, for each such p, there exists a constant γp > 1 such that the same result holds even for γp-approximate SVP_p. For p > 2, the above statement holds under the weaker assumption of randomized Gap-ETH. I.e., there is no 2o(n)-time algorithm for γp-approximate SVP_p unless randomized Gap-ETH is false. See http://arxiv.org/abs/1712.00942 for a complete exposition.