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NP-hardness of SVP in Euclidean Space

Daqing Wan

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.

NP-hardness of SVP in Euclidean Space

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.

Paper Structure

This paper contains 7 sections, 21 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Locally dense lattices
  3. Reed-Solomon lattices
  4. geometry of the variety $X_{k, h, u}$
  5. Rational points on the variety ${X}_{k,h, u}$
  6. Inclusion-exclusion sieving
  7. Linear Projections

Key Result

Theorem 1.2

SVP is NP-hard.

Theorems & Definitions (38)

  • Conjecture 1.1: van Emde Boas vEB81
  • Theorem 1.2
  • Definition 1.3
  • Conjecture 1.5
  • Theorem 1.6: Ajtai Ajt98
  • Theorem 1.7
  • Corollary 1.8
  • Remark 1.9
  • Definition 2.1: locally dense lattice
  • Theorem 2.2
  • ...and 28 more