Limits on the Hardness of Lattice Problems in L(P) Norms

Abstract

Several recent papers have established limits on the computational difficulty of lattice problems, focusing primarily on the ℓ ₂ (Euclidean) norm. We demonstrate close analogues of these results in ℓ _p norms, for every 2 < p ≤ ∞. In particular, for lattices of dimension n:

• Approximating the closest vector problem, the shortest vector problem, and other related problems to within 𝑂(𝑛√)O(n) factors (or 𝑂(𝑛log𝑛‾‾‾‾‾‾√)O(nlog⁡n) factors, for p = ∞) is in coNP.
• Approximating the closest vector and bounded distance decoding problems with preprocessing to within 𝑂(𝑛√)O(n) factors can be accomplished in deterministic polynomial time.
• Approximating several problems (such as the shortest independent vectors problem) to within Õ(n) factors in the worst case reduces to solving the average–case problems defined in prior works (Ajtai 2004; Micciancio & Regev 2007; Regev 2005).

Our results improve prior approximation factors for ℓ _p norms by up to 𝑛√n factors. Taken all together, they complement recent reductions from the ℓ ₂ norm to ℓ _p norms (Regev & Rosen 2006), and provide some evidence that lattice problems in ℓ _p norms (for p > 2) may not be substantially harder than they are in the ℓ ₂ norm.

One of our main technical contributions is a very general analysis of Gaussian distributions over lattices, which may be of independent interest. Our proofs employ analytical techniques of Banaszczyk that, to our knowledge, have yet to be exploited in computer science.