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arXiv:2101.12419 · IEEE Transactions on Information Theory

Theory of Communication Efficient Quantum Secret Sharing

Kaushik Senthoor, Pradeep Kiran Sarvepalli

Published Jan 7, 2022• 40 citations
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Abstract

A$((k,n))$quantum threshold secret sharing (QTS) scheme is a quantum cryptographic protocol for sharing a quantum secret among$n$parties such that the secret can be recovered by any$k$or more parties while$k-1$or fewer parties have no information about the secret. Despite extensive research on these schemes, there has been very little study on optimizing the quantum communication cost during recovery. Recently, we initiated the study of communication efficient quantum threshold secret sharing (CE-QTS) schemes. These schemes reduce the communication complexity in QTS schemes by accessing$d>k$parties for recovery; here$d$is fixed ahead of encoding the secret. In contrast to the standard QTS schemes which require$k$qudits for recovering each qudit in the secret, these schemes have a lower communication cost of$\frac {d}{d-k+1}$. In this paper, we further develop the theory of communication efficient quantum threshold schemes. Here, we propose universal CE-QTS schemes which reduce the communication cost for all$d>k$simultaneously. We provide a framework based on ramp quantum secret sharing to construct CE-QTS and universal CE-QTS schemes. We give another construction for universal CE-QTS schemes based on Staircase codes. We derived a lower bound on communication complexity and show that our constructions are optimal. Finally, an information theoretic model is developed to analyse CE-QTS schemes and the lower bound on communication complexity is proved again using this model.

DOI
10.1109/tit.2021.3139839

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A((k,n))((k,n))((k,n))quantum threshold secret sharing (QTS) scheme is a quantum cryptographic protocol for sharing a quantum secret amongnnnparties such that the secret can be recovered by anykkkor more parties whilek−1k-1k−1or fewer parties have no information about the secret. Despite extensive research on these schemes, there has been very little study on optimizing the quantum communication cost during recovery. Recently, we initiated the study of communication efficient quantum threshold secret sharing (CE-QTS) schemes. These schemes reduce the communication complexity in QTS schemes by accessingd>kd>kd>kparties for recovery; heredddis fixed ahead of encoding the secret. In contrast to the standard QTS schemes which requirekkkqudits for recovering each qudit in the secret, these schemes have a lower communication cost ofdd−k+1\frac {d}{d-k+1}d−k+1d​. In this paper, we further develop the theory of communication efficient quantum threshold schemes. Here, we propose universal CE-QTS schemes which reduce the communication cost for alld>kd>kd>ksimultaneously. We provide a framework based on ramp quantum secret sharing to construct CE-QTS and universal CE-QTS schemes. We give another construction for universal CE-QTS schemes based on Staircase codes. We derived a lower bound on communication complexity and show that our constructions are optimal. Finally, an information theoretic model is developed to analyse CE-QTS schemes and the lower bound on communication complexity is proved again using this model.
Recently, a class of quantum secret sharing schemes called communication efficient quantum threshold secret sharing schemes (CE-QTS) was introduced. These schemes reduced the communication cost during secret recovery. In this paper, we introduce a general class of communication efficient quantum secret sharing schemes (CE-QSS) which include both threshold and non-threshold schemes. We propose a framework for constructing CE-QSS schemes to generalize the earlier construction of CE-QTS schemes which was based on the staircase codes. The main component in this framework is a class of quantum codes which we call the extended Calderbank-Shor-Steane codes. These extended CSS codes could have other applications. We derive a bound on communication cost for CE-QSS schemes. Finally, we provide a construction of CE-QSS schemes meeting this bound using the proposed framework.
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.
Quantum secret sharing is a fundamental technique in quantum cryptography. However, in practical quantum networks, it still faces several bottlenecks, such as high quantum communication cost and low transmission efficiency. To reduce the communication cost, ramp quantum secret sharing schemes have been proposed in existing studies. Nevertheless, intermediate sets in such schemes may leak partial information about the secret. To address this problem, this paper presents a method for detecting intermediate sets. Based on this method, we further propose a communication-efficient perfect quantum secret sharing scheme with eavesdropping detection capability.We analyze the communication cost under different numbers of participating parties and determine the range of participants that minimizes the reconstruction communication cost. The results verify the communication efficiency and security of the proposed scheme.
A quantum secret sharing scheme is a cryptographic protocol by which a dealer can share a secret among a group of nnn players so that only certain subsets of players can recover the secret by collaboration. In this paper we propose communication efficient quantum threshold secret sharing schemes. They minimize the amount of quantum communication required to reconstruct the secret when more than the necessary number of players collaborate. They are based on a class of staircase codes proposed by Bitar and El Rouayheb. In a standard ((k,n))((k,n))((k,n)) quantum threshold scheme, any subset of kkk or more players can recover the secret. The quantum communication cost for reconstruction in such schemes is kkk qudits for each secret qudit. Using the proposed construction, any subset of $d\ensuremath{\ge}k$ players can also collaborate to recover the secret with a communication cost of ddd qudits for $d\ensuremath{-}k+1$ secret qudits. In other words, for the proposed schemes the quantum communication cost is only $\frac{d}{d\ensuremath{-}k+1}$ qudits for every secret qudit. For d>kd>kd>k, proposed schemes are communication efficient with respect to standard schemes; and when $d=2k\ensuremath{-}1$, the quantum communication cost is reduced by a factor O(k)O(k)O(k). Further, when $n=2k\ensuremath{-}1$, the proposed schemes have optimal communication cost for secret reconstruction.
Recently, an information theoretical model for Quantum Secret Sharing (QSS) schemes was introduced. By using this model, we prove that pure state Quantum Threshold Schemes (QTS) can be constructed from quantum MDS codes and vice versa. In particular, we consider stabilizer codes and give a constructive proof of their relation with QTS. Furthermore, we reformulate the Monotone Span Program (MSP) construction according to the information theoretical model and check the recoverability and secrecy requirement. Finally, we consider QSS schemes which are based on quantum teleportation.
Recently, a class of quantum secret sharing schemes called communication efficient quantum threshold secret sharing schemes (CE-QTS) was introduced. These schemes reduced the quantum communication cost during secret recovery in quantum threshold secret sharing schemes. However, non-threshold schemes are known to have less storage cost compared to threshold schemes. This motivates us to extend the idea of communication efficiency to non-threshold schemes as well. In this paper, we introduce a general class of communication efficient quantum secret sharing schemes (CE-QSS) which include both threshold and non-threshold schemes. Specifically, we provide a framework for constructing CE-QSS schemes by concatenating a pair of CSS codes. The main component in this framework is a new class of Calderbank-Shor-Steane (CSS) codes called the extended CSS codes.