Mind the Gap? Not for SVP Hardness under ETH!

2025-04-03

AI Summary

This paper establishes tight hardness lower bounds for lattice problems under the Exponential Time Hypothesis (ETH). For the ℓₚ-norm (p > 2), it addresses the approximate Closest Vector Problem (CVPₚ), Shortest Vector Problem (SVPₚ), and Bounded Distance Decoding (BDDₚ). Methodologically, it introduces the first compact polynomial-time reductions from 3SAT via MAXLIN to these lattice problems—resolving a long-standing gap challenge. It delivers the first deterministic ETH-hardness proof for CVPₚ; leverages novel geometric properties of ℤⁿ under ℓₚ to achieve a randomized reduction from CVPₚ to SVPₚ; and substantially improves the ETH-based hardness threshold for BDDₚ to αₚ†. Consequently, unless ETH fails, no 2ᵒ⁽ⁿ⁾-time algorithms exist for CVPₚ,γ, SVPₚ,γ (p > 2), or BDDₚ,α (α > αₚ†), and γ-Minimum Distance Problem (γ-MDP) is also shown to be ETH-hard.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Learning with Manifolds

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Economics, Online Markets and Human Computation: Humans versus LLMs for data annotation and labeling

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 $gamma>1$ such that $mathsf{CVP}_{p,gamma}$ (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 $mathsf{CVP}_{p,gamma}$. Next, we prove a randomized ETH-hardness result for $mathsf{SVP}_{p,gamma}$ (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 $mathsf{CVP}_{p,gamma}$ to $mathsf{SVP}_{p,gamma'}$. Finally, we improve over prior reductions from $mathsf{3SAT}$ to $mathsf{BDD}_{p, alpha}$ (the Bounded Distance Decoding problem), yielding better ETH-hardness results for $mathsf{BDD}_{p, alpha}$ for any $p in [1, infty)$ and $alpha>alpha_p^{ddagger}$, where $alpha_p^{ddagger}$ is an explicit threshold depending on $p$. We additionally observe that prior work implies ETH hardness for the gap minimum distance problem ($gamma$-$mathsf{MDP}$) in codes.

Problem

• Proves ETH-hardness for approximate Closest Vector Problem (CVP).
• Establishes randomized ETH-hardness for approximate Shortest Vector Problem (SVP).
• Improves ETH-hardness results for Bounded Distance Decoding (BDD) problem.

Innovation

• Deterministic reduction from MAXLIN to CVP
• Randomized reduction from CVP to SVP
• Improved reductions from 3SAT to BDD