Lattices in Euclidean spaces are important research objects in geometric number theory, and they have important applications in many areas, such as cryptology. The shortest vector problem (SVP) and the closest vector problem (CVP) are two famous computational problems about lattices. In this paper, we consider p-adic lattices in local fields, and define the p-adic analogues of SVP and CVP in local fields. The authors find that, in contrast with lattices in Euclidean spaces, the situation is different and interesting. The SVP in Euclidean spaces corresponds to the Longest Vector Problem (LVP) in local fields. The authors develop relevant algorithms, indicating that these problems are computable.

欧几里得空间中的格是几何数论的重要研究对象，在密码学等许多领域都有重要的应用。最短向量问题（SVP）和最接近向量问题（CVP）是关于格的两个著名的计算问题。在本文中，我们考虑了局部场中的p- adic 格，并定义了局部场中 SVP 和 CVP的p- adic 类似物。作者发现，与欧几里得空间中的格子相比，情况不同且有趣。欧几里得空间中的 SVP 对应于局部场中的最长向量问题 (LVP)。作者开发了相关算法，表明这些问题是可计算的。