TY - JOUR

T1 - Geometry of Banach Spaces

T2 - A New Route Towards Position Based Cryptography

AU - Junge, Marius

AU - Kubicki, Aleksander M.

AU - Palazuelos, Carlos

AU - Pérez-García, David

N1 - Funding Information:
We thank Jop Briët for kindly sharing some personal notes on Pisier’s method for bounding the cotype-2 constant of the projective tensor product of type-2 spaces. We also thank Elisabeth Werner, Matthias Christandl and Alex May for their kind correspondence during the preparation of this work. We acknowledge financial support from MICINN (Grants MTM2017-88385-P and SEV-2015-0554), from Comunidad de Madrid (Grant QUITEMAD-CM, ref. S2018/TCS-4342), and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No 648913). A.M.K. also acknowledge support from Spanish MICINN Project MTM2014-57838-C2-2-P.
Publisher Copyright:
© 2022, The Author(s).

PY - 2022/9

Y1 - 2022/9

N2 - 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.

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.

UR - http://www.scopus.com/inward/record.url?scp=85131328539&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85131328539&partnerID=8YFLogxK

U2 - 10.1007/s00220-022-04407-9

DO - 10.1007/s00220-022-04407-9

M3 - Article

AN - SCOPUS:85131328539

VL - 394

SP - 625

EP - 678

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -