An Introduction to Torsors in Mathematics with a View Toward $\Sigma$-Protocols in Cryptography

TL;DR

This paper introduces torsors, focusing on their mathematical properties and how they relate to group actions and cocycle conditions, with an eye toward applications in cryptography, especially $\\Sigma$-protocols.

Contribution

It provides a focused exposition on torsors, highlighting their structure, examples, and connection to sheaf theory, to support cryptographic applications.

Findings

Torsors are characterized by free transitive group actions.

They naturally arise from gluing local trivial pieces with cocycle conditions.

The paper links torsors to concepts useful in cryptography, like $\\Sigma$-protocols.

Abstract

This paper provides a preparatory introduction to torsors, written with a view toward later applications in the author's work. Rather than aiming at a comprehensive survey, the exposition focuses on those aspects of torsors that are most useful for understanding torsor-based reasoning: group actions, orbits, free transitive actions, the absence of a canonically chosen origin, and the interpretation of group elements as transports between points. After developing the basic definition and several elementary examples, we emphasize a central theme: torsors are not only characterized abstractly by free transitive group actions, but also arise naturally as objects obtained by gluing local trivial pieces by means of transition data satisfying cocycle conditions. A brief optional section indicates a sheaf- and topos-theoretic perspective. In the final part, we explain how these ideas prepare the ground for later conceptual applications, including aspects of $\Sigma$-protocols.