An Attack on $p$-adic Lattice Public-key Cryptosystems and Signature Schemes

TL;DR

This paper presents a polynomial-time attack on $p$-adic lattice-based cryptosystems by improving the LVP algorithm, enabling forgery of signatures and decryption, and discusses potential modifications to secure these schemes.

Contribution

It introduces a deterministic polynomial-time algorithm for LVP in certain $p$-adic lattices and demonstrates its application in breaking existing cryptographic schemes based on these lattices.

Findings

The attack can forge signatures and decrypt ciphertexts.

The improved LVP algorithm is polynomial-time in specific cases.

Proposed modifications could prevent the attack.

Abstract

Lattices have many significant applications in cryptography. In 2021, the $p$-adic signature scheme and public-key encryption cryptosystem were introduced. They are based on the Longest Vector Problem (LVP) and the Closest Vector Problem (CVP) in $p$-adic lattices. These problems are considered to be challenging and there are no known deterministic polynomial time algorithms to solve them. In this paper, we improve the LVP algorithm in local fields. The modified LVP algorithm is a deterministic polynomial time algorithm when the field is totally ramified and $p$ is a polynomial in the rank of the input lattice. We utilize this algorithm to attack the above schemes so that we are able to forge a valid signature of any message and decrypt any ciphertext. Although these schemes are broken, this work does not mean that $p$-adic lattices are not suitable in constructing cryptographic primitives. We propose some possible modifications to avoid our attack at the end of this paper.