Public-Key Cryptosystems and Signature Schemes from -Adic Lattices

In 2018, the longest vector problem and closest vector problem in local fields were introduced, as the -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 -adic lattice by proposing a trapdoor function with the norm-orthogonal basis of -adic lattice. These cryptographic schemes have reasonable key size and the signature scheme is efficient, while the encryption scheme works only for short messages, which shows that -adic lattice can be a new alternative to construct cryptographic primitives and well worth studying.