Constructing efficient, tightly secure short signature schemes based on lattices has remained a significant challenge in cryptographic theory. In their ASIACRYPT 2016 paper, Boyen and Li raised an open problem: how to construct a tightly or almost-tightly secure homomorphic signature scheme in the standard model under the Short Integer Solution (SIS) hardness assumption. Under the same assumption, this paper presents the first lattice-based short signature scheme that achieves adaptive security and provides a proof of its almost-tight security. Furthermore, the proposed scheme retains the potential to be further refined into a tightly secure signature scheme.

翻译：在密码学理论中，构建高效且具有紧致安全性的基于格的短签名方案一直是一个重要挑战。Boyen和Li在其ASIACRYPT 2016论文中提出了一个开放性问题：如何在标准模型下，基于短整数解（SIS）困难性假设，构造一个具有紧致或近似紧致安全性的同态签名方案。在相同假设下，本文提出了首个基于格的短签名方案，该方案实现了自适应安全性，并给出了其近似紧致安全性的证明。此外，所提出的方案保留了进一步优化为紧致安全签名方案的潜力。