Several recent papers have established limits on the computational difficulty of lattice problems, focusing primarily on the l2 (Euclidean) norm. We demonstrate close analogues of these results in lp norms, for every 2 < p ≤ ∞. In particular, for lattices of dimension n: #R##N##R##N##R##N#Approximating the closest vector problem, the shortest vector problem, and other related problems to within \(O(\sqrt{n})\) factors (or \(O(\sqrt{n \log n})\) factors, for p = ∞) is in coNP.#R##N##R##N##R##N#Approximating the closest vector and bounded distance decoding problems with preprocessing to within \(O(\sqrt{n})\) factors can be accomplished in deterministic polynomial time.#R##N##R##N##R##N#Approximating several problems (such as the shortest independent vectors problem) to within O(n) factors in the worst case reduces to solving the average-case problems defined in prior works (Ajtai 2004; Micciancio & Regev 2007; Regev 2005).
['Chris Peikert']
Limits on the Hardness of Lattice Problems in lp Norms