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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Graph state secret sharing in higher-dimensional systems

Ben Fortescue, Adrian Keet, Damian Markham, Barry C. Sanders

Published Jul 29, 2010• 1 citation
Publisher

Abstract

We present a formalism within which, using entangled graph states of prime-dimensional systems, a variety of different secret-sharing schemes (involving both quantum and classical secrets and quantum and classical channels shared between parties) may be unified. We review the explicit protocols we have found for three varieties of secret sharing within this formalism, including some for which the analogous formalism using qubit graph states is not sufficient.

DOI
10.1117/12.860711

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We present a formalism within which, using entangled graph states of prime-dimensional systems, a variety of different secret-sharing schemes (involving both quantum and classical secrets and quantum and classical channels shared between parties) may be unified. We review the explicit protocols we have found for three varieties of secret sharing within this formalism, including some for which the analogous formalism using qubit graph states is not sufficient.
We consider three broad classes of quantum secret sharing with and without eavesdropping and show how a graph state formalism unifies otherwise disparate quantum secret sharing models. In addition to the elegant unification provided by graph states, our approach provides a generalization of threshold classical secret sharing via insecure quantum channels beyond the current requirement of 100% collaboration by players to just a simple majority in the case of five players. Another innovation here is the introduction of embedded protocols within a larger graph state that serves as a one-way quantum-information processing system.
In a recent work, Markham and Sanders proposed a framework to study quantum secret-sharing (QSS) schemes using graph states. This framework unified three classes of QSS protocols, namely, sharing classical secrets over private and public channels, and sharing quantum secrets. However, previous work on graph-state secret sharing mostly focused on threshold schemes. In this paper, we focus on general access structures. We show how to realize a large class of arbitrary access structures using the graph-state formalism. We show an equivalence between [[n,1]][[n,1]][[n,1]] binary quantum codes and graph-state secret-sharing schemes sharing one bit. We also establish a similar (but restricted) equivalence between a class of [[n,1]][[n,1]][[n,1]] Calderbank-Shor-Steane codes and graph-state QSS schemes sharing one qubit. With these results we are able to construct a large class of graph-state quantum secret-sharing schemes with arbitrary access structures.
We give graphical characterisation of the access structure to both classical and quantum information encoded onto a multigraph defined for prime dimension qqq, as well as explicit decoding operations for quantum secret sharing based on graph state protocols. We give a lower bound on kkk for the existence of a ((k,n))q((k,n))_q((k,n))q​ scheme and prove, using probabilistic methods, that there exists ααα such that a random multigraph has an accessing parameter k≤αnk\leq αnk≤αn with high probability.
We give graphical characterisation of the access structure to both classical and quantum information encoded onto a multigraph defined for prime dimension q, as well as explicit decoding operations for quantum secret sharing based on graph state protocols. We give a lower bound on kkk for the existence of a ((k,n))_q scheme and prove, using probabilistic methods, that there exists alpha such that a random multigraph has an accessing parameter k => alpha*n with high probability.
We give an example of a wide class of problems for which quantum-information protocols based on multisystem entanglement can be mapped into much simpler ones involving one system. Secret sharing is a cryptographic primitive which plays a central role in various secure multiparty computation tasks and management of keys in cryptography. In secret sharing protocols, a classical message is divided into shares given to recipient parties in such a way that some number of parties need to collaborate in order to reconstruct the message. Quantum protocols for the task commonly rely on multipartite GHZ entanglement. We present a multiparty secret sharing protocol which requires only sequential communication of a single quantum ddd-level system (for any prime ddd). It has huge advantages in scalability and can be realized with state-of-the-art technology.