Fine-grained deterministic hardness of the shortest vector problem
Markus Hittmeir
TL;DR
This work studies the deterministic, fine-grained hardness of the shortest vector problem in $\ell_p$ norms, focusing on the GapSVP$_p$ decision problem. It proves, for infinitely many $p$, that the $(2-\varepsilon)$-approximation variant cannot be solved in time $O\left(2^{O(p)}\cdot n^{O(1)}\right)$ unless $\mathrm{P}=\mathrm{NP}$ and cannot be solved in time $O\left(2^{2^{o(p)}}\cdot 2^{o(n)}\right)$ unless $\mathsf{SETH}$ is false, via a fully deterministic polynomial-time reduction from a restricted subset-sum problem characterized by a $p$-dependent constraint. The core technique builds a lattice $\mathcal{L}_r$ from a subset-sum instance so that a solution to the restricted subset-sum problem corresponds to the shortest vector, enabling the hardness reductions without randomness. The results yield both polynomial and subexponential lower bounds and have implications for the fixed-parameter tractability of GapSVP$_p$, while leaving open questions about NP-hardness for fixed $p$ and suggesting directions for extending deterministic hardness to broader $p$-regimes.
Abstract
Let $γ$-$\mathsf{GapSVP}_p$ be the decision version of the shortest vector problem in the $\ell_p$-norm with approximation factor $γ$, let $n$ be the lattice dimension and $0<\varepsilon\leq 1$. We prove that the following statements hold for infinitely many values of $p$. $(2-\varepsilon)$-$\mathsf{GapSVP}_p$ is not in $O\left(2^{O(p)}\cdot n^{O(1)}\right)$-time, unless $\text{P}=\text{NP}$. $(2-\varepsilon)$-$\mathsf{GapSVP}_p$ is not in $O\left(2^{2^{o(p)}}\cdot 2^{o(n)}\right)$-time, unless the Strong Exponential Time Hypothesis is false. The proofs are based on a Karp reduction from a variant of the subset-sum problem that imposes restrictions on vectors orthogonal to the vector of its weights. While more extensive hardness results for the shortest vector problem in all $\ell_p$-norms have already been established under randomized reductions, the results in this paper are fully deterministic.