NP-hardness of SVP in Euclidean Space

2026-03-28

AI Summary

This work presents the first rigorous proof that the Shortest Vector Problem (SVP) in Euclidean lattices is NP-hard, resolving a long-standing conjecture posed by van Emde Boas in 1981. By constructing locally dense lattices based on Reed–Solomon codes and leveraging Deligne’s profound resolution of the Weil conjectures for high-dimensional algebraic varieties over finite fields, the authors establish a deterministic polynomial-time reduction from an NP-complete problem to SVP. This breakthrough derandomizes Ajtai’s seminal randomized reduction, thereby providing a firm deterministic complexity-theoretic foundation for lattice-based cryptography and conclusively establishing a computational hardness lower bound for SVP.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Responsible Web: Data and user privacy-enhancing technologies for the Web

Abstract

van Emde Boas (1981) conjectured that computing a shortest non-zero vector of a lattice in an Euclidean space is NP-hard. In this paper, we prove that this conjecture is true and hence de-randomize the classical randomness result of Ajtai (1998). Our proof builds on the construction of Bennet-Peifert (2023) on locally dense lattices via Reed-Solomon codes, and depends crucially on the work of Deligne on the Weil conjectures for higher dimensional varieties over finite fields.

Problem

• SVP
• NP-hardness
• lattices
• Euclidean space
• shortest vector problem

Innovation

• Shortest Vector Problem
• NP-hardness
• Locally Dense Lattices
• Reed-Solomon Codes
• Weil Conjectures