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.

翻译：格在密码学中具有诸多重要应用。2021年，研究者提出了基于$p$进格中最长向量问题与最近向量问题的$p$进签名方案及公钥加密体制。这些问题被认为具有计算困难性，且目前不存在已知的确定性多项式时间算法予以求解。本文改进了局部域中的最长向量问题算法。当域为完全分歧且$p$为输入格秩的多项式函数时，改进后的最长向量问题算法为确定性多项式时间算法。我们利用该算法对上述密码方案进行攻击，实现了对任意消息的有效签名伪造及任意密文的解密。尽管上述方案已被攻破，本工作并不意味着$p$进格不适用于密码原语构造。文末我们提出了若干可能的改进方案以规避此类攻击。