(Gap/S)ETH Hardness of SVP

We prove the following quantitative hardness results for the Shortest Vector Problem in the `p norm (SVPp), where n is the rank of the input lattice. 1. For “almost all” p > p0 ≈ 2.1397, there no 2p -time algorithm for SVPp for some explicit constant Cp > 0 unless the (randomized) Strong Exponential Time Hypothesis (SETH) is false. 2. For any p > 2, there is no 2-time algorithm for SVPp unless the (randomized) GapExponential Time Hypothesis (Gap-ETH) is false. Furthermore, for each p > 2, there exists a constant γp > 1 such that the same result holds even for γp-approximate SVPp. 3. There is no 2-time algorithm for SVPp 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 `2 norm. Furthermore, for each 1 ≤ p ≤ 2, there exists a constant γp > 1 such that the same result holds even for γp-approximate SVPp. ∗Supported by the Singapore Ministry of Education and the National Research Foundation, also through the Tier 3 Grant “Random numbers from quantum processes” MOE2012-T3-1-009. †Supported by the Simons Collaboration on Algorithms and Geometry. ar X iv :1 71 2. 00 94 2v 1 [ cs .C C ] 4 D ec 2 01 7