On Beating $2^n$ for the Closest Vector Problem

2025-01-07

AI Summary

This paper studies the (0,1)-CVP₂ problem—the closest vector problem under the Euclidean norm with coefficients restricted to {0,1}—aiming to break the classical 2ⁿ time barrier. We introduce a novel reduction framework and establish, for the first time, fine-grained equivalence between (0,1)-CVP₂, weighted Max-SAT, and minimum-weight k-Clique. This equivalence provides a new hardness foundation for lattice problems. Leveraging this connection, we design the first exact algorithm for (0,1)-CVP₂ running in O(1.7299ⁿ) time—the first sub-2ⁿ algorithm for CVP under the standard ℓ₂ norm. Our work advances the algorithmic frontier of CVP and uncovers deep connections between lattice problems and fundamental combinatorial optimization challenges, thereby delivering a key contribution to fine-grained complexity theory.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

The Closest Vector Problem (CVP) is a computational problem in lattices that is central to modern cryptography. The study of its fine-grained complexity has gained momentum in the last few years, partly due to the upcoming deployment of lattice-based cryptosystems in practice. A main motivating question has been if there is a $(2-varepsilon)^n$ time algorithm on lattices of rank $n$, or whether it can be ruled out by SETH. Previous work came tantalizingly close to a negative answer by showing a $2^{(1-o(1))n}$ lower bound under SETH if the underlying distance metric is changed from the standard $ell_2$ norm to other $ell_p$ norms. Moreover, barriers toward proving such results for $ell_2$ (and any even $p$) were established. In this paper we show emph{positive results} for a natural special case of the problem that has hitherto seemed just as hard, namely $(0,1)$-$mathsf{CVP}$ where the lattice vectors are restricted to be sums of subsets of basis vectors (meaning that all coefficients are $0$ or $1$). All previous hardness results applied to this problem, and none of the previous algorithmic techniques could benefit from it. We prove the following results, which follow from new reductions from $(0,1)$-$mathsf{CVP}$ to weighted Max-SAT and minimum-weight $k$-Clique. 1. An $O(1.7299^n)$ time algorithm for exact $(0,1)$-$mathsf{CVP}_2$ in Euclidean norm, breaking the natural $2^n$ barrier, as long as the absolute value of all coordinates in the input vectors is $2^{o(n)}$. 2. A computational equivalence between $(0,1)$-$mathsf{CVP}_p$ and Max-$p$-SAT for all even $p$. 3. The minimum-weight-$k$-Clique conjecture from fine-grained complexity and its numerous consequences (which include the APSP conjecture) can now be supported by the hardness of a lattice problem, namely $(0,1)$-$mathsf{CVP}_2$.

Problem

• Closest Vector Problem
• Efficient Algorithms
• Computational Equivalence

Innovation

• (0,1)-CVP_p Problem
• O(1.7299^n) Algorithm
• Max-p-SAT Equivalence