$2^{\log^{1-\eps} n}$ Hardness for Closest Vector Problem with Preprocessing

TL;DR

This paper establishes that, under certain complexity assumptions, the preprocessing versions of the closest vector and nearest codeword problems are extremely hard to approximate within a factor of roughly $2^{ ext{log}^{1- ext{eps}} n}$, significantly improving previous hardness bounds.

Contribution

It proves a stronger hardness of approximation for the preprocessing versions of CVP and NCP, assuming NP not in quasi-polynomial time, advancing the understanding of their computational difficulty.

Findings

Hardness factor of $2^{ ext{log}^{1- ext{eps}} n}$ for CVP and NCP

Improves previous hardness bounds from polylogarithmic to quasi-exponential

Conditional on a complexity assumption related to NP and quasi-polynomial time

Abstract

We prove that for an arbitrarily small constant $\eps>0,$ assuming NP$\not \subseteq$DTIME$(2^{{\log^{O(1/\eps)} n}})$, the preprocessing versions of the closest vector problem and the nearest codeword problem are hard to approximate within a factor better than $2^{\log ^{1-\eps}n}.$ This improves upon the previous hardness factor of $(\log n)^\delta$ for some $\delta > 0$ due to \cite{AKKV05}.