Fine-grained deterministic hardness of the shortest vector problem

TL;DR

This paper establishes deterministic hardness results for the shortest vector problem in various $oldsymbol{ ext{l}_p}$-norms, showing no efficient algorithms exist under ETH for certain approximation factors, via novel deterministic reductions from subset-sum.

Contribution

It introduces the first deterministic reductions from subset-sum to $ ext{GapSVP}_p$, proving tight hardness results in a broad range of norms under ETH.

Findings

No algorithms solve $(2- ext{epsilon})$-$ ext{GapSVP}_p$ in subexponential time for all $p$

Deterministic reductions from subset-sum to $ ext{GapSVP}_p$ and $ ext{GapSVP}_ ext{infinity}$

Hardness results hold under the Exponential Time Hypothesis

Abstract

Let $\gamma$-$\mathsf{GapSVP}_p$ be the decision version of the shortest vector problem in the $\ell_p$-norm with approximation factor $\gamma$, let $n$ be the lattice rank and $0<\varepsilon\leq 1$. We prove that there is no algorithm that solves $(2-\varepsilon)$-$\mathsf{GapSVP}_p$ uniformly for all $p\in\mathbb{N}$ in time\[ 2^{2^{o(p)}}\cdot 2^{o(n)},\] unless the Exponential Time Hypothesis is false. The proof is based on a deterministic Karp reduction from a constrained variant of the subset-sum problem to $\mathsf{GapSVP}_p$ for fixed $p$. While most hardness results for the shortest vector problem in finite norms rely on randomized reductions, our method is entirely deterministic. As a consequence, we also obtain a deterministic Karp reduction from the standard subset-sum problem to $(2-\varepsilon)$-$\mathsf{GapSVP}_{\infty}$.