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Quantum Information Theoretical Analysis of Various Constructions for Quantum Secret Sharing. (arXiv:quant-ph/0502009v1 CROSS LISTED)

Karin Rietjens, Berry Schoenmakers, Pim Tuyls

Published Feb 12, 2018• 0 citations
OpenAlex

Abstract

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.

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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, 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
In this paper, we investigate the notion of quantum secret sharing (QSS) schemes based on quantum operations. We present the information theoretical model for QSS schemes using reversible operations and erase operations at first. We show that in pure state schemes, the recoverability requirement and the secrecy requirement are equivalent, which is regarded as a variant of nocloning theorem. By establishing a relation between the quantum information quantity--coherent information and QSS schemes, the upper bound of quantum information rate is given. Finally, we propose a pure state threshold scheme with the form of quantum operations. The operations formalism of quantum secret sharing generalizes the theorey of QSS, and provides a unifying framework for the study of these schemes.
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.
In this paper we introduce a quantum information theoretical model for quantum secret sharing schemes. We show that quantum information theory provides a unifying framework for the study of these schemes. We prove that the information theoretical requirements for a class of quantum secret sharing schemes reduce to only one requirement (the recoverability condition) as a consequence of the no-cloning principle. We give also a shorter proof of the fact that the size of the shares in a quantum secret sharing scheme must be at least as large as the secret itself.
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.