Cryptographic security is traditionally formulated using game-based or simulation-based definitions. In this paper, we propose a structural reformulation of cryptographic security based on Grothendieck topologies and sheaf theory. Our key idea is to model attacker observations as a Grothendieck site, where covering families represent admissible decompositions of partial information determined by efficient simulation. Within this framework, protocol transcripts naturally form sheaves, and security properties arise as geometric conditions. As a first step, we focus on $Σ$-protocols. We show that the transcript structure of any $Σ$-protocol defines a torsor in the associated topos of sheaves. Local triviality of this torsor corresponds to zero-knowledge, while the absence of global sections reflects soundness. A concrete analysis of the Schnorr $Σ$-protocol is provided to illustrate the construction. This sheaf-theoretic perspective offers a conceptual explanation of simulation-based security and suggests a geometric foundation for further cryptographic abstractions.

翻译：密码学安全传统上采用基于游戏或基于模拟的定义进行表述。本文提出一种基于格罗滕迪克拓扑与层论的密码学安全结构性重构。我们的核心思想是将攻击者观测建模为一个格罗滕迪克位点，其中覆盖族表示由高效模拟所确定的部分信息的可容许分解。在此框架内，协议记录自然构成层，而安全属性则呈现为几何条件。作为第一步，我们聚焦于$Σ$-协议。我们证明任何$Σ$-协议的记录结构在其关联的层意象中定义了一个主齐性空间。该主齐性空间的局部平凡性对应于零知识性，而全局截影的缺失则反映可靠性。本文通过对Schnorr $Σ$-协议的具体分析来阐释该构造。这一层论视角为基于模拟的安全性提供了概念性解释，并为进一步的密码学抽象提出了几何基础。