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Quantum Information Processing

A rational hierarchical (t,n)-threshold quantum secret sharing scheme

Fulin Li, Zhuo Liu, Li Liu, Shixin Zhu

Published Feb 12, 2024• 15 citations
Publisher
DOI
10.1007/s11128-024-04269-1

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arXiv:1501.04212•Physical Review A•Aug 4, 2015•55 citations
Hierarchical secret sharing is among the most natural generalizations of threshold secret sharing, and it has attracted a lot of attention since the invention of secret sharing until nowadays. Several constructions of ideal hierarchical secret sharing schemes have been proposed, but it was not known what access structures admit such a scheme. We solve this problem by providing a natural definition for the family of the hierarchical access structures and, more importantly, by presenting a complete characterization of the ideal hierarchical access structures, that is, the ones admitting an ideal secret sharing scheme. Our characterization is based on the well-known connection between ideal secret sharing schemes and matroids and, more specifically, on the connection between ideal multipartite secret sharing schemes and integer polymatroids. In particular, we prove that every hierarchical matroid port admits an ideal linear secret sharing scheme over every large enough finite field. Finally, we use our results to present a new proof for the existing characterization of the ideal weighted threshold access structures.
ABSTRACT Quantum secret sharing (QSS) holds a significant place in quantum cryptography. Unlike traditional QSS schemes, which often simplistically model participants as either completely honest or fully malicious, rational QSS introduces a more realistic framework by modeling participants as rational entities who strategically act to maximize their own utility. In this paper, for the first time, a rational quantum multi‐secret sharing scheme designed to share a classical bit string by configuring multiple effective reconstruction rounds is proposed. Further, a novel method is proposed to facilitate equilibrium achievement—deception deterrence. In terms of efficiency, the scheme achieves, for the first time, an effective reconstruction round ratio and a communication efficiency are independent of the total number of reconstruction rounds configured in the scheme, overcoming the communication efficiency limitation inherent in existing schemes. In terms of equilibrium achievement, the proposed new method not only returns to the essence of strategic interaction among rational participants themselves but also avoids the substantial verification cost.
A rational secret sharing scheme is a game in which each party responsible for reconstructing a secret tries to maximize his or her utility by obtaining the secret alone. Quantum secret sharing schemes, derived either from quantum teleportation or from quantum error correcting code, do not succeed when we assume rational participants. This is because all existing quantum secret sharing schemes consider that the secret is reconstructed by a party chosen by the dealer. In this paper, for the first time, we propose a quantum secret sharing scheme which is resistant to rational parties. The proposed scheme is fair (everyone gets the secret), is correct, and achieves strict Nash equilibrium.