NP-hardness of SVP in Euclidean Space

TL;DR

This paper proves NP-hardness of finding the shortest non-zero vector in a lattice in Euclidean space, confirming a long-standing conjecture and advancing understanding of lattice problems.

Contribution

It establishes the NP-hardness of SVP in Euclidean space, using novel techniques based on locally dense lattices and algebraic geometry.

Findings

NP-hardness of SVP confirmed in Euclidean space

De-randomization of a classical randomized result

New proof techniques involving algebraic geometry and Reed-Solomon codes

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.