Approximate CVP_∞ in time 2^0.802 n

05/11/2020

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by   Friedrich Eisenbrand, et al.

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We show that a constant factor approximation of the shortest and closest lattice vector problem w.r.t. ℓ_∞ can be computed in time 2^(0.802 +ϵ) n. This is breaking the kissing number barrier of 3^n that is inherent in the previous best approaches tackling this problem. We obtain this improvement by incorporating a bound on the number of scaled hypercubes that are necessary to cover the ℓ_2-ball of radius √(n). The final procedure is then a modification of the list-sieve algorithm for ℓ_2. It is to pick the smallest pairwise difference w.r.t. ℓ_∞ of the generated lattice vectors.