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

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 $Σ$-protocols.

Figures (8)

• Figure 1: The transporter from $x$ to $y$ is the unique group element sending $x$ to $y$.
• Figure 2: Composition of transporters: $\operatorname{tr}(y,z)\operatorname{tr}(x,y)=\operatorname{tr}(x,z)$.
• Figure 3: In a torsor, relative position is measured by the unique transporter between two points.
• Figure 4: Two local trivializations are compared on the overlap by transition data $g_{ij}$.
• Figure 5: On a triple overlap, the two ways of passing between local trivializations must agree; this is the cocycle condition.

Theorems & Definitions (76)

• Definition 2.1
• Remark 2.2
• Definition 2.3
• Definition 2.4
• Example 2.5
• Example 2.6
• Definition 3.1
• Remark 3.2
• Example 3.3
• Example 3.4