Geometry of Banach spaces: a new route towards Position Based Cryptography

Abstract

In this work we initiate the study of Position Based Quantum Cryptography (PBQC) from the perspective of geometric functional analysis and its connections with quantum games. The main question we are interested in asks for the optimal amount of entanglement that a coalition of attackers have to share in order to compromise the security of any PBQC protocol. Known upper bounds for that quantity are exponential in the size of the quantum systems manipulated in the honest implementation of the protocol. However, known lower bounds are only linear. In order to deepen the understanding of this question, here we propose a Position Verification (PV) protocol and find lower bounds on the resources needed to break it. The main idea behind the proof of these bounds is the understanding of cheating strategies as vector valued assignments on the Boolean hypercube. Then, the bounds follow from the understanding of some geometric properties of particular Banach spaces, their type constants. Under some regularity assumptions on the former assignment, these bounds lead to exponential lower bounds on the quantum resources employed, clarifying the question in this restricted case. Known attacks indeed satisfy the assumption we make, although we do not know how universal this feature is. Furthermore, we show that the understanding of the type properties of some more involved Banach spaces would allow to drop out the assumptions and lead to unconditional lower bounds on the resources used to attack our protocol. Unfortunately, we were not able to estimate the relevant type constant. Despite that, we conjecture an upper bound for this quantity and show some evidence supporting it. A positive solution of the conjecture would lead to stronger security guarantees for the proposed PV protocol providing a better understanding of the question asked above.

Presentation substantially improved. A couple of sections were removed for the sake of conciseness and some clarifications added

References in corpus (2)

• Holographic scattering requires a connected entanglement wedge
• Random Series of Trace Class Operators

Cited by in corpus (7)

• Complexity and entanglement in non-local computation and holography
• A single-qubit position verification protocol that is secure against multi-qubit attacks
• The connected wedge theorem and its consequences
• Relating non-local quantum computation to information theoretic cryptography
• Making Existing Quantum Position Verification Protocols Secure Against Arbitrary Transmission Loss
• Security of quantum position-verification limits Hamiltonian simulation via holography
• Impossibility of perfect cheating for single-qubit position verification