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.

Non-local computation and the black hole interior

“…Fixed second rounds also appear naturally elsewhere in cryptography [52]. 15 We were not able to connect the constraints of [51] directly to holography and our setting, but this work does suggest that adding that the second round is fixed may allow for stronger constraints on bulk computation to be understood. Another comment is that all non-local computation protocols devised so far do involve fixed second rounds, and the observation here is just that protocols built from studying the holographic setting also share this property.…”

“…Non-local computation with a fixed second round has already appeared in the quantum information literature. In [51] the authors assumed a fixed second round and derived interesting new constraints on non-local computation, under some mathematical conjectures on type constants…”

Complexity and entanglement in non-local computation and holography

“…For example linear bounds with α > 1, or other explicit bounds with a super-linear behaviour. See for example [50] for work in this direction, where a 'smoothness' assumption is made on the form of the non-local computation protocol, and new interesting bounds can be obtained. 14 For bounds linear in ln d A 0 with α > 1, one would have to carefully understand if the subtraction term in the CEB, interpreted appropriately, suffices to enforce this constraint, or if new physics is needed.…”

“…In particular, this would require an exponential improvement to the known general purpose protocols[24]. Evidence this isn't possible is given in[50].Accepted in Quantum 2022-09-31, click title to verify. Published under CC-BY 4.0.…”