Mind the Gap? Not for SVP Hardness under ETH!

TL;DR

This work establishes subexponential-time ETH-hardness for fundamental lattice problems across $\ell_p$ norms: it shows ETH-hardness of $\mathsf{CVP}_{p,\gamma}$ for any $p\in[1,\infty)$ with constant $\gamma>1$, derives randomized ETH-hardness for $\mathsf{SVP}_{p,\gamma}$ when $p>2$, and improves ETH-hardness results for $\mathsf{BDD}_{p,\alpha}$ for all $p$ and $\alpha>\alpha_p^{\ddagger}$. The authors build on the Gap-$\mathsf{MAXLIN}$ reductions of Bitansky et al. to translate $\mathsf{3SAT}$-style hardness into lattice problems via a deterministic MAXLIN$_{\varepsilon}$→$\mathsf{CVP}_{p,\gamma}$ reduction, and then deploy a sparsification-based pipeline and a locally dense integer gadget to transfer hardness to SVP and BDD. A key technical contribution is the construction of locally dense gadgets in the integer lattice that create exponentially more close lattice vectors near a carefully chosen target than near the origin, enabling subexponential hardness for $p>2$. The results extend to codes, yielding ETH-hardness for the gap minimum distance problem in certain regimes, and collectively strengthen the fine-grained landscape for lattice-based problems with broad implications for cryptography and subexponential algorithms.

Abstract

We prove new hardness results for fundamental lattice problems under the Exponential Time Hypothesis (ETH). Building on a recent breakthrough by Bitansky et al. [BHIRW24], who gave a polynomial-time reduction from $\mathsf{3SAT}$ to the (gap) $\mathsf{MAXLIN}$ problem-a class of CSPs with linear equations over finite fields-we derive ETH-hardness for several lattice problems. First, we show that for any $p \in [1, \infty)$, there exists an explicit constant $γ> 1$ such that $\mathsf{CVP}_{p,γ}$ (the $\ell_p$-norm approximate Closest Vector Problem) does not admit a $2^{o(n)}$-time algorithm unless ETH is false. Our reduction is deterministic and proceeds via a direct reduction from (gap) $\mathsf{MAXLIN}$ to $\mathsf{CVP}_{p,γ}$. Next, we prove a randomized ETH-hardness result for $\mathsf{SVP}_{p,γ}$ (the $\ell_p$-norm approximate Shortest Vector Problem) for all $p > 2$. This result relies on a novel property of the integer lattice $\mathbb{Z}^n$ in the $\ell_p$ norm and a randomized reduction from $\mathsf{CVP}_{p,γ}$ to $\mathsf{SVP}_{p,γ'}$. Finally, we improve over prior reductions from $\mathsf{3SAT}$ to $\mathsf{BDD}_{p, α}$ (the Bounded Distance Decoding problem), yielding better ETH-hardness results for $\mathsf{BDD}_{p, α}$ for any $p \in [1, \infty)$ and $α> α_p^{\ddagger}$, where $α_p^{\ddagger}$ is an explicit threshold depending on $p$. We additionally observe that prior work implies ETH hardness for the gap minimum distance problem ($γ$-$\mathsf{MDP}$) in codes.

Theorems & Definitions (55)

• Theorem 1.1: hardness of $\cvp_{p, \gamma}$
• Theorem 1.2: hardness of $\svp_{p, \gamma}$
• Theorem 1.3: $\cvp_{p, \gamma}$ reduces to $\bdd_{p, \alpha}$
• Theorem 1.4: hardness of $\bdd_{p, \alpha}$
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4: Shortest Vector Problem ($\mathsf{SVP} _{p, \gamma}$)
• Definition 2.5: Closest Vector Problem ($\mathsf{CVP} _{p, \gamma}$)
• Definition 2.6: Bounded Distance Decoding ($\mathsf{BDD} _{p, \alpha}$)