RT Journal Article
T1 Geometry of Banach Spaces: A New Route Towards Position Based Cryptography
A1 Junge, Marius
A1 Kubicki, Aleksander
A1 Palazuelos Cabezón, Carlos
A1 Pérez García, David
AB 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.
PB Springer Nature
SN 0010-3616
YR 2022
FD 2022-06-02
LK https://hdl.handle.net/20.500.14352/71650
UL https://hdl.handle.net/20.500.14352/71650
LA eng
NO Junge, M., Kubicki, A., Palazuelos Cabezón, C. & Pérez García, D. «Geometry of Banach Spaces: A New Route Towards Position Based Cryptography». Communications in Mathematical Physics, vol. 394, n.o 2, septiembre de 2022, pp. 625-78. DOI.org (Crossref), https://doi.org/10.1007/s00220-022-04407-9.
NO CRUE-CSIC (Acuerdos Transformativos 2022)
NO Unión Europea
NO Ministerio de Ciencia, Innovación y Universidades (España)
NO Comunidad de Madrid
DS Docta Complutense
RD 7 abr 2025