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Abstract

We show that unless NP ⊆ RTIME(2 poly(log n) ), there is no polynomial-time algorithm approximating the Shortest Vector Problem (SVP) on n-dimensional lattices in the p norm (1 ≤ p < ∞) to within a factor of 2 (log n) 1−ε for any ε > 0. This improves the previous best factor of 2 (log n) 1/2−ε under the same complexity assumption due to Khot (J. ACM, 2005). Under the stronger assumption NP RSUBEXP, we obtain a hardness factor of n c/ log log n for some c > 0.Our proof starts with Khot's SVP instances that are hard to approximate to within some constant. To boost the hardness factor we simply apply the standard tensor product of lattices. The main novelty is in the analysis, where we show that the lattices of Khot behave nicely under tensorization. At the heart of the analysis is a certain matrix inequality which was first used in the context of lattices by de Shalit and Parzanchevski (2006).

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“…• the hardness of SVP for subpolynomial factors n 1/O(log log n) under the assumption that NP is not in subexponential time, as in [14,12], thus matching the strongest known hardness results for SVP, but under probabilistic reductions with onesided error. We regard our results as a partial derandomization of the reductions [14,12] with two-sided error, and a step toward an NP-hardness proof for SVP under deterministic reductions.…”

“…Our results We present a new, simpler proof that SVP is NP-hard to approximate within any constant factor which goes back to the geometrically appealing approach of [20] and avoids the introduction of additional probabilistic techniques from [14,12]. In particular, we prove…”

“…We regard our results as a partial derandomization of the reductions [14,12] with two-sided error, and a step toward an NP-hardness proof for SVP under deterministic reductions. Randomness is used within our proof exclusively for the construction of a geometric gadget with properties similar to the one originally introduced by Micciancio in [20].…”

“…This suggests the use of tensor products of lattices to prove the NP-hardness of SVP within large constant factors, starting from the inapproximability result of [20] for factors below √ 2. In fact, using the tensor product is a common theme in the sequence of papers [20,14,12] proving hardness of approximation results for SVP. Unfortunately, while the minimum distance of a linear code gets squared when one takes the tensor product of the code with itself, the same is not always true for the length of the shortest vectors in a lattice.…”

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“…On the negative side, SVP in 2 is NP-hard (under randomized reductions) to solve exactly, or even to approximate to within any constant factor [1,9,34,29]. Many more hardness results are known for other p norms and under stronger complexity assumptions than P = NP [17,14,43,22,35]. CVP is NP-hard to approximate to within n c/ log log n factors for some constant c > 0 [4,15], where n is the dimension of the lattice.…”