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arXiv:1106.2619 · Lecture notes in computer science

Approximating the Closest Vector Problem Using an Approximate Shortest Vector Oracle

Chandan K. Dubey, Thomas Holenstein

Published Jan 1, 2011• 6 citations
arXivHTMLPDF
DOI
10.1007/978-3-642-22935-0_16

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In this article, we present a survey of known inapproximability results for computational problems on lattices, viz. the Shortest Vector Problem (SVP), the Closest Vector Problem (CVP), the Closest Vector Problem with Preprocessing (CVPP), the Covering Radius Problem (CRP), the Shortest Independent Vectors Problem (SIVP), and the Shortest Basis Problem (SBP).

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We show simple constant-round interactive proof systems for problems capturing the approximability, to within a factor of n , of optimization problems in integer lattices, specifically, the closest vector problem (CVP) and the shortest vector problem (SVP). These interactive proofs are for the coNP direction; that is, we give an interactive protocol showing that a vector is far from the lattice (for CVP) and an interactive protocol showing that the shortest-lattice-vector is long (for SVP). Furthermore, these interactive proof systems are honest-verifier perfect zero-knowledge. We conclude that approximating CVP (resp., SVP) within a factor of n is in N P ∩co A M . Thus, it seems unlikely that approximating these problems to within a n factor is NP-hard. Previously, for the CVP (resp., SVP) problem, Lagarias et al. (1990, Combinatorica 10 , 333–348), Håstad (1988, Combinatorica 8 , 75–81), and Banaszczyk (1993, Math. Annal. 296 , 625–635) showed that the gap problem corresponding to approximating CVP (resp., SVP) within n is in N P ∩co N P . On the other hand, Arora et al. (1997, J. Comput. System Sci. 54 , 317–331) showed that the gap problem corresponding to approximating CVP within 2 log 0.999 n is quasi-NP-hard.
Let p > 1 be any fixed real. We show that assuming NP ⊈ RP, there is no polynomial time algorithm that approximates the Shortest Vector Problem (SVP) in ℓ p norm within a constant factor. Under the stronger assumption NP ⊈ RTIME(2 poly (log n ) ), we show that there is no polynomial-time algorithm with approximation ratio 2 (log n ) 1/2−ϵ where n is the dimension of the lattice and ϵ > 0 is an arbitrarily small constant.We first give a new (randomized) reduction from Closest Vector Problem (CVP) to SVP that achieves some constant factor hardness. The reduction is based on BCH Codes. Its advantage is that the SVP instances produced by the reduction behave well under the augmented tensor product , a new variant of tensor product that we introduce. This enables us to boost the hardness factor to 2 (log n ) 1/2-ϵ .