Euclidean SVP is deterministically NP-hard to approximate within any constant factor

2026-08-12

AI Summary

This work investigates the deterministic computational hardness of the Euclidean Shortest Vector Problem (SVP) under arbitrary constant approximation factors. By constructing a deterministic polynomial-time many-one reduction, it establishes for the first time that SVP is NP-hard for any constant approximation factor, substantially generalizing prior results that were limited to factors less than √2. The paper not only provides a deterministic counterpart to Khot’s randomized reduction but also simultaneously handles two classical dimension-dependent regimes, yielding deterministic NP-hardness for approximation factors of $2^{(\log n)^{1-\varepsilon}}$ and $n^{c/\log\log n}$. These advances significantly extend the known boundaries of complexity theory for lattice problems.

Abstract

We prove that, for every constant $ρ>1$, the Euclidean shortest vector problem is NP-hard to approximate within any constant factor $ρ$ under a deterministic polynomial-time many-one reduction. This extends our previous deterministic NP-hardness result from $ρ<\sqrt 2$ to arbitrary constants and gives a deterministic version of Khot's randomized arbitrary-constant theorem. Our proof also gives deterministic counterparts of the two classical dimension-dependent regimes of Haviv and Regev: $2^{(\log n)^{1-\varepsilon}}$ under quasipolynomial-time reductions and $n^{c/\log\log n}$ under subexponential-time reductions.

Problem

• Shortest Vector Problem
• NP-hardness
• Approximation
• Lattice Problems
• Computational Complexity

Innovation

• Shortest Vector Problem
• NP-hardness
• Deterministic Reduction
• Lattice-based Cryptography
• Approximation Hardness