Constructing cryptographic schemes with tight or almost-tight security has long been one of the central problems in theoretical cryptography. At ASIACRYPT 2016, Boyen and Li posed an open problem: whether it is possible to construct a homomorphic signature scheme with tight or almost-tight security under the Short Integer Solution (SIS) assumption in the standard model. In 2024, Chen achieved the first construction with almost-tight security under a weaker security model. To further achieve tight security in the standard model, this paper introduces a new security model whose security requirements are weaker than those of the standard adaptive model but stronger than the model adopted by Chen. Under this model, we construct a linearly homomorphic signature scheme with tight security.

翻译：构建具有紧或几乎紧安全性的密码方案一直是理论密码学的核心问题之一。在ASIACRYPT 2016会议上，Boyen和Li提出一个公开问题：是否有可能在标准模型下，基于短整数解（SIS）假设，构造具有紧或几乎紧安全性的同态签名方案？2024年，Chen在较弱安全模型下实现了首个具有几乎紧安全性的构造。为进一步在标准模型下实现紧安全性，本文引入一种新安全模型，其安全要求弱于标准自适应模型但强于Chen所采用的模型。在该模型下，我们构造了一种具有紧安全性的线性同态签名方案。