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

TL;DR

This paper provides a simple proof that the approximate Shortest Vector Problem is NP-hard under randomized reductions, using Reed-Solomon codes and locally dense lattices, and discusses the challenges of derandomization.

Contribution

It introduces a straightforward proof of NP-hardness for approximate SVP using Reed-Solomon codes and locally dense lattices, and explores derandomization challenges and decoding algorithms.

Findings

Proves NP-hardness of approximate SVP in ℓ_p norm for certain approximation factors.

Constructs locally dense lattices via Reed-Solomon codes with elementary arguments.

Provides a polynomial-time decoding algorithm for certain Construction A Reed-Solomon lattices.

Abstract

$\newcommand{\NP}{\mathsf{NP}}\newcommand{\GapSVP}{\textrm{GapSVP}}$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 $p \geq 1$ and any constant $\gamma < 2^{1/p}$, the $\gamma$-approximate problem in the $\ell_p$ norm ($\gamma$-$\GapSVP_p$) is not in $\mathsf{RP}$ unless $\NP \subseteq \mathsf{RP}$. Our proof follows an approach pioneered by Ajtai (STOC 1998), and strengthened by Micciancio (FOCS 1998 and SICOMP 2000), for showing hardness of $\gamma$-$\GapSVP_p$ 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 $p < \infty$, 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 Trans. Inf. 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 $O(\sqrt{\log n})$ factor of Minkowski's bound. This asymptotically matches the best known distance for decoding near Minkowski's bound, due to Mook and Peikert (IEEE Trans. Inf. Theory 2022), whose work we build on with a somewhat simpler construction and analysis.