Abstract

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 norm-orthogonal basis of \(p\) -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 \(p\) -adic lattice can be a new alternative to construct cryptographic primitives and well worth studying.

中文翻译：

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来自 $$p$$ 的公钥密码系统和签名方案 -Adic Lattices

摘要

2018年，引入了局部域中的最长向量问题和最近向量问题，作为欧几里德空间格中最短向量问题和最近向量问题的\(p\) -adic类似物。它们被认为在构造密码原语方面是困难且有用的，但没有给出密码学中的应用。本文通过提出以\(p\) -adic 格为范数正交基的陷门函数，构造了第一个基于\(p\) -adic 格的签名方案和公钥加密密码系统。这些密码方案具有合理的密钥大小，签名方案高效，而加密方案仅适用于短消息，这表明\(p\) -adic 格可以作为构造密码原语的新替代方案，非常值得研究。