(Gap/S)ETH hardness of SVP

We prove the following quantitative hardness results for the Shortest Vector Problem in the l p norm (SVP_p), where n is the rank of the input lattice. For “almost all” p u003e p 0 ≈ 2.1397, there is no 2 n / C p -time algorithm for SVP_p for some explicit (easily computable) constant C p u003e 0 unless the (randomized) Strong Exponential Time Hypothesis (SETH) is false. (E.g., for p ≥ 3, C p p +3) 2 − p + 10 p 2 2 −2 p .) For any 1 ≤ p ≤ ∞, there is no 2 o ( 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 γ p u003e 1 such that the same result holds even for γ p -approximate SVP_p. For p u003e 2, the above statement holds under the weaker assumption of randomized Gap-ETH. I.e., there is no 2 o ( n ) -time algorithm for γ p -approximate SVP_p unless randomized Gap-ETH is false. See http://arxiv.org/abs/1712.00942 for a complete exposition.