Grothendieck Topologies and Sheaf-Theoretic Foundations of Cryptographic Security: Attacker Models and $\Sigma$-Protocols as the First Step

TL;DR

This paper introduces a novel geometric framework for cryptographic security using Grothendieck topologies and sheaf theory, providing new insights into attacker models and protocol properties.

Contribution

It reformulates cryptographic security notions within a sheaf-theoretic framework, modeling attacker observations as Grothendieck sites and protocol transcripts as sheaves, with initial focus on $\Sigma$-protocols.

Findings

Sheaf-theoretic model captures security properties as geometric conditions.

Transcript structures of $\Sigma$-protocols form torsors in the associated topos.

Concrete analysis of Schnorr $\Sigma$-protocol demonstrates the framework's applicability.

Abstract

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 $\Sigma$-protocols. We show that the transcript structure of any $\Sigma$-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 $\Sigma$-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.