On Some Computational Problems in Local Fields

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.