Deterministic Hardness of Approximation For SVP in all Finite $\ell_p$ Norms

2026-04-01

AI Summary

This work investigates the deterministic approximability of the Shortest Vector Problem (SVP) in lattices under arbitrary finite $\ell_p$ norms. By devising a constructive deterministic reduction that integrates refined lattice structural analysis with norm-specific techniques, the authors establish—under standard complexity assumptions—that for every finite $p$, SVP cannot be efficiently approximated within any factor better than $2^{(\log n)^{1 - o(1)}}$. This result constitutes the first unified hardness guarantee for all finite $\ell_p$ norms within a deterministic reduction framework, overcoming prior limitations that either relied on randomized reductions or applied only to specific norms. Consequently, it substantially advances the theoretical foundations of lattice-based cryptography and computational complexity.

Abstract

We show that, assuming NP $\not\subseteq$ $\cap_{δ> 0}$DTIME$\left(\exp{n^δ}\right)$, the shortest vector problem for lattices of rank $n$ in any finite $\ell_p$ norm is hard to approximate within a factor of $2^{(\log n)^{1 - o(1)}}$, via a deterministic reduction. Previously, for the Euclidean case $p=2$, even hardness of the exact shortest vector problem was not known under a deterministic reduction.

Problem

• Shortest Vector Problem
• Lattice
• Hardness of Approximation
• Deterministic Reduction
• ℓ_p Norm

Innovation

• Shortest Vector Problem
• Deterministic Reduction
• Hardness of Approximation
• Lattice Problems
• ℓ_p Norms