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Hardness of the (Approximate) Shortest Vector Problem: A Simple Proof via Reed-Solomon Codes

Authors Huck Bennett, Chris Peikert



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Author Details

Huck Bennett
  • Oregon State University, Corvallis, OR, USA
Chris Peikert
  • University of Michigan, Ann Arbor, MI, USA
  • Algorand, Inc., Boston, MA, USA

Acknowledgements

We thank Swastik Kopparty [Swastik Kopparty, 2020] for very helpful answers to several of our questions.

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Huck Bennett and Chris Peikert. Hardness of the (Approximate) Shortest Vector Problem: A Simple Proof via Reed-Solomon Codes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 37:1-37:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.37

Abstract

We give a simple proof that the (approximate, decisional) Shortest Vector Problem is NP-hard under a randomized reduction. Specifically, we show that for any p ≥ 1 and any constant γ < 2^{1/p}, the γ-approximate problem in the 𝓁_p norm (γ-GapSVP_p) is not in RP unless NP ⊆ RP. Our proof follows an approach pioneered by Ajtai (STOC 1998), and strengthened by Micciancio (FOCS 1998 and SICOMP 2000), for showing hardness of γ-GapSVP_p using locally dense lattices. We construct such lattices simply by applying "Construction A" to Reed-Solomon codes with suitable parameters, and prove their local density via an elementary argument originally used in the context of Craig lattices. As in all known NP-hardness results for GapSVP_p with p < ∞, our reduction uses randomness. Indeed, it is a notorious open problem to prove NP-hardness via a deterministic reduction. To this end, we additionally discuss potential directions and associated challenges for derandomizing our reduction. In particular, we show that a close deterministic analogue of our local density construction would improve on the state-of-the-art explicit Reed-Solomon list-decoding lower bounds of Guruswami and Rudra (STOC 2005 and IEEE Transactions on Information Theory 2006). As a related contribution of independent interest, we also give a polynomial-time algorithm for decoding n-dimensional "Construction A Reed-Solomon lattices" (with different parameters than those used in our hardness proof) to a distance within an O(√log n) factor of Minkowski’s bound. This asymptotically matches the best known distance for decoding near Minkowski’s bound, due to Mook and Peikert (IEEE Transactions on Information Theory 2022), whose work we build on with a somewhat simpler construction and analysis.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Error-correcting codes
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Lattices
  • Shortest Vector Problem
  • Reed-Solomon codes
  • NP-hardness
  • derandomization

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