Position-Based Quantum Cryptography and the Garden-Hose Game

We study position-based cryptography in the quantum setting. We examine a class of protocols that only require the communication of a single qubit and 2n bits of classical information. To this end, we define a new model of communication complexity, the garden-hose model, which enables us to prove upper bounds on the number of EPR pairs needed to attack such schemes. This model furthermore opens up a way to link the security of position-based quantum cryptography to traditional complexity theory.

💡 Research Summary

The paper investigates the security of position‑based cryptography in the quantum setting, focusing on a class of protocols that require only a single qubit together with 2n classical bits of communication. The authors first describe the basic setting: a verifier sends classical challenges and a quantum state to a prover whose claimed geographic location must be authenticated. An adversary consists of two cooperating parties placed at different positions who share an arbitrary amount of pre‑distributed entanglement (EPR pairs) and attempt to simulate a prover at the claimed location.

To analyze such attacks, the authors introduce a novel communication‑complexity framework called the garden‑hose model. In this model the adversaries’ shared entanglement is represented by a network of “hoses” (pipes). Each hose corresponds to one EPR pair, and the way the hoses are connected and switched mimics the sequence of quantum measurements and classical messages exchanged during the protocol. The goal of the garden‑hose game is, given inputs x and y, to route the “water” (quantum information) through the network so that it exits at the correct output pipe, using as few hoses as possible. The minimum number of hoses required for a given Boolean function f is defined as its garden‑hose complexity.

The paper then establishes a tight connection between garden‑hose complexity and traditional communication‑complexity measures. For many functions, the garden‑hose complexity is asymptotically equivalent to the quantum‑classical (or “bit‑qubit”) communication complexity. The authors prove upper bounds by constructing explicit garden‑hose protocols: for simple functions such as XOR, a linear number O(n) of EPR pairs suffices, implying that an attack can be mounted with only a modest amount of entanglement. Conversely, they derive exponential lower bounds for functions with high circuit depth or those that are hard for classical circuits; in these cases the number of required EPR pairs grows as Ω(2^n).

These results have direct implications for the security of position‑based quantum cryptography. If the verification function f is chosen to be computationally easy (low circuit complexity), an adversary can break the scheme using a relatively small entangled resource. However, if f is taken from a class of functions that are P‑complete, NP‑complete, or otherwise hard in the classical sense, the required entanglement becomes infeasibly large, rendering the protocol secure under realistic assumptions. The authors therefore argue that the choice of f is a critical design parameter, linking cryptographic security to well‑studied complexity‑theoretic hardness.

Beyond the immediate cryptographic applications, the garden‑hose model opens a pathway to relate quantum position‑verification to broader complexity classes. The paper sketches how garden‑hose complexity could be used to embed PSPACE‑complete problems, and suggests possible connections to quantum complexity classes such as QMA. It also outlines several directions for future work: extending the model to multi‑qubit protocols, analyzing more realistic network topologies where hoses cannot be arbitrarily rearranged, and exploring tighter lower‑bound techniques using advanced tools from communication complexity.

In summary, the authors provide a rigorous framework that quantifies the entanglement resources needed to attack single‑qubit position‑based protocols, demonstrate that these resources are tightly linked to the classical complexity of the verification function, and propose the garden‑hose model as a versatile bridge between quantum cryptography and traditional complexity theory. This work both clarifies the security landscape of position‑based quantum cryptography and suggests concrete avenues for designing provably secure schemes.