Tree Proof-of-Position Algorithms

TL;DR

This work introduces Tree-Proof-of-Position (T-PoP), a decentralised and privacy-preserving protocol enabling agents to prove their location without revealing it, even under adversarial conditions. It combines a commitment-based Commit stage, a height-$h$ witness-tree construction with per-level witnesses, and a Checks-then-Verification process, achieving a worst-case quadratic runtime suitable for hardware-constrained IoT. The authors develop a rigorous mathematical model that characterises the probability of satisfying three core criteria, incorporates platoon-attack detection, and integrates agent-density effects via Poisson point processes and random geometric graphs; they validate the model with extensive agent-based simulations that align with theory. The practical contribution includes guidance on tuning operating conditions for varying densities, a notarisation path via DLT/IPFS, and an open-source codebase for implementation and replication.

Abstract

We present a novel class of proof-of-position algorithms: Tree-Proof-of-Position (T-PoP). This algorithm is decentralised, collaborative and can be computed in a privacy preserving manner, such that agents do not need to reveal their position publicly. We make no assumptions of honest behaviour in the system, and consider varying ways in which agents may misbehave. Our algorithm is therefore resilient to highly adversarial scenarios. This makes it suitable for a wide class of applications, namely those in which trust in a centralised infrastructure may not be assumed, or high security risk scenarios. Our algorithm has a worst case quadratic runtime, making it suitable for hardware constrained IoT applications. We also provide a mathematical model that summarises T-PoP's performance for varying operating conditions. We then simulate T-PoP's behaviour with a large number of agent-based simulations, which are in complete agreement with our mathematical model, thus demonstrating its validity. T-PoP can achieve high levels of reliability and security by tuning its operating conditions, both in high and low density environments. Finally, we also present a mathematical model to probabilistically detect platooning attacks.

Figures (23)

• Figure 1: High-level Overview of the Proof of Position protocol.
• Figure 2: Tree building examples. Agent $a$ commits their alleged position. The panel on the top right shows the construction of a tree for $h = 1$ and $w_1 = 4$, while the panel on the bottom right shows the construction of a tree for $h = 2, w_1 = 2, w_2 =2$.
• Figure 3: We start by evaluating the outer level of the tree and we evaluate the witnesses in $W_2$ (the circled set). Agents $a_5$ and $a_6$ do not confirm that they see agent $a_2$, even though $a_2$ is an honest agent. This leads to agent $a_2$ being eliminated from the tree.
• Figure 4: We go down one level, and now evaluate the witnesses in $W_1$ (the circled set). $a_2$ has been eliminated by the tree (shown in grey) and so only agent $a_1$ is left.
• Figure 5: Venn Diagram Representing Possible Agent State Combinations