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.

翻译：van Emde Boas（1981）曾猜想，在欧几里得空间中计算一个格的最短非零向量是NP困难的。本文证明了该猜想成立，从而去随机化了Ajtai（1998）中经典随机性结果。我们的证明基于Bennet-Peifert（2023）通过Reed-Solomon码构造局部密集格的方法，并关键依赖于Deligne关于有限域上高维簇的Weil猜想的工作。