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 define so-called p-adic lattices, and consider the p-adic analogues of SVP and CVP in local fields. We find that, in contrast with lattices in Euclidean spaces, the situation is completely different and interesting. We also develop relevant algorithms, indicating that these problems are computable

Public-key Cryptosystems and Signature Schemes from p-adic Lattices

In 2018, the longest vector problem and closest vector problem in local fields were introduced, as the p-adic analogues of the shortest vector problem and closest vector problem in lattices of Euclidean spaces. They are considered to be hard and useful in constructing cryptographic primitives, but no applications in cryptography were given. In this paper, we construct the first signature scheme and public-key encryption cryptosystem based on p-adic lattice by proposing a trapdoor function with the orthogonal basis of p-adic lattice. These cryptographic schemes have reasonable key size and efficiency, which shows that p-adic lattice can be a new alternative to construct cryptographic primitives and well worth studying