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arXiv:1004.4619 · Physical Review A

Quantum secret sharing with qudit graph states

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

Published Dec 14, 2010• 164 citations
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Abstract

We present a unified formalism for threshold quantum secret sharing using graph states of systems with prime dimension. We construct protocols for three varieties of secret sharing: with classical and quantum secrets shared between parties over both classical and quantum channels.
DOI
10.1103/physreva.82.062315

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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.
Quantum secret sharing (QSS) is a cryptographic protocol in which a quantum secret is distributed among a number of parties where some subsets of the parties are able to recover the secret while some subsets are unable to recover the secret. In the standard ((k, n)) quantum threshold secret sharing scheme, any subset of k or more parties out of the total n parties can recover the secret while other subsets have no information about the secret. But recovery of the secret incurs a communication cost of at least k qudits for every qudit in the secret. Recently, a class of communication efficient QSS schemes were proposed which can improve this communication cost to dd−k+1\frac{d}{{d - k + 1}}d−k+1d​ by contacting d ≥ k parties where d is fixed prior to the distribution of shares. In this paper, we propose a more general class of ((k, n)) quantum secret sharing schemes with low communication complexity. In these schemes the combiner can contact any d parties at the time of recovery where k ≤ d ≤ n. This is the first such class of universal communication efficient quantum threshold schemes.
(ttt, nnn) Threshold Quantum Secret Sharing based on a Single ddd-level Quantum System
A (ttt, nnn) threshold quantum secret sharing (QSS) is proposed based on a single ddd-level quantum system. It enables the (ttt, nnn) threshold structure based on Shamir's secret sharing and simply requires sequential communication in the ddd-level quantum system to recover secret. Besides, the scheme employs an additional qudit to detect cheats and eavesdropping during secret reconstruction, and allows a participant to use the share repeatedly. Analyses show that the proposed scheme is resistant to typical attacks. The scheme is scalable in the number of participants and easier to realize compared to related schemes. More generally, our scheme also presents a generic method to construct new (ttt, nnn) threshold QSS schemes based on ddd-level quantum system with other classical threshold secret sharing.
Quantum secret sharing (QSS) allows a dealer to distribute a secret quantum state among a set of parties so that certain subsets can reconstruct the secret, while unauthorized subsets obtain no information. While QSS was introduced over twenty years ago, previous works focused only on existence of perfectly secure schemes, and the share size of the known schemes is exponential even for access structures computed by polynomial size monotone circuits. This stands in contrast to the classical case, where efficient computationally-secure schemes have been long known for all access structures in monotone P\mathsf{monotone~P}monotone P, and one can even obtain shares which are much shorter than the secret which is impossible with perfect security. In this work, we initiate the study of computationally-secure QSS and show that computational assumptions help significantly in building QSS schemes. We present a simple compiler and use it to obtain a large variety results: We construct polynomial-time QSS schemes under standard assumptions for a rich class of access structures. This includes many access structures for which previous results in QSS required exponential share size. We also construct QSS schemes for which the size of the shares is significantly smaller than the size of the secret. As in the classical case, this is impossible with perfect security. We also use our compiler to obtain results beyond computational QSS. In the information-theoretic setting, we improve the share size of perfect QSS schemes for a large class of access structures to 1.5n+o(n)1.5^{n+o(n)}1.5n+o(n), improving upon best known schemes and matching the best known result for general access structures in the classical case. Finally, we show construct efficient schemes for all access structures in P\mathsf{P}P and NP\mathsf{NP}NP when the quantum secret sharing scheme is given multiple of copies of the secret.