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arXiv:1204.1072 · Physical Review A

Generalized semiquantum secret-sharing schemes

Vlad Gheorghiu

Published May 15, 2012• 32 citations
arXivHTMLPDF

Abstract

We investigate quantum secret-sharing schemes constructed from ${[[n,k,\ensuremath{\delta}]]}_{D}$ nonbinary stabilizer quantum error-correcting codes with carrier qudits of prime dimension $D$. We provide a systematic way of determining the access structure, which completely determines the forbidden and intermediate structures. We then show that the information available to the intermediate structure can be fully described and quantified by what we call the information group, a subgroup of the Pauli group of $k$ qudits, and we employ this group structure to construct a method for hiding the information from the intermediate structure via twirling of the information group and sharing of classical bits between the dealer and the players. Our scheme allows for the transformation of a ramp (intermediate) quantum secret-sharing scheme into a semiquantum perfect secret-sharing scheme with the same access structure as the ramp one but without any intermediate subsets, and is optimal in the amount of classical bits the dealer has to distribute.

DOI
10.1103/physreva.85.052309

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We investigate quantum secret-sharing schemes constructed from [[n,k,\ensuremathδ]]D{[[n,k,\ensuremath{\delta}]]}_{D}[[n,k,\ensuremathδ]]D​ nonbinary stabilizer quantum error-correcting codes with carrier qudits of prime dimension DDD. We provide a systematic way of determining the access structure, which completely determines the forbidden and intermediate structures. We then show that the information available to the intermediate structure can be fully described and quantified by what we call the information group, a subgroup of the Pauli group of kkk qudits, and we employ this group structure to construct a method for hiding the information from the intermediate structure via twirling of the information group and sharing of classical bits between the dealer and the players. Our scheme allows for the transformation of a ramp (intermediate) quantum secret-sharing scheme into a semiquantum perfect secret-sharing scheme with the same access structure as the ramp one but without any intermediate subsets, and is optimal in the amount of classical bits the dealer has to distribute.
We introduce a multiparty cryptographic protocol, which we call entanglement sharing schemes, wherein a dealer retains half of a maximally entangled bipartite state and encodes the other half into a multipartite state that is distributed among multiple players. In a close analog to quantum secret sharing, some subsets of players can recover maximal entanglement with the dealer whereas other subsets can recover no entanglement (though they may retain classical correlations with the dealer). We find a lower bound on the share size for such schemes and construct two nontrivial examples based on Shor's [[9,1,3]][[9,1,3]][[9,1,3]] and the [[4,2,2]][[4,2,2]][[4,2,2]] stabilizer code; we further demonstrate how other examples may be obtained from quantum error correcting codes through classical encryption. Finally, we demonstrate that entanglement sharing schemes can be applied to characterize leaked information in quantum ramp secret sharing.
Abstract In this paper, we consider to use the quantum stabilizer codes as secret sharing schemes for classical secrets. We give necessary and sufficient conditions for qualified and forbidden sets in terms of quantum stabilizers. Then, we give a Gilbert–Varshamov-type sufficient condition for existence of secret sharing schemes with given parameters, and by using that sufficient condition, we show that roughly 19% of participants can be made forbidden independently of the size of classical secret, in particular when an n -bit classical secret is shared among n participants having 1-qubit share each. We also consider how much information is obtained by an intermediate set and express that amount of information in terms of quantum stabilizers. All the results are stated in terms of linear spaces over finite fields associated with the quantum stabilizers.
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.
We demonstrate a new construction for perfect quantum secret sharing (QSS) schemes based on imperfect “ramp” secret sharing combined with classical encryption, in which the individual parties' shares are split into quantum and classical components, allowing the former to be of lower dimension than the secret itself. We show that such schemes can be performed with smaller quantum components and lower overall quantum communication than required for existing methods. We further demonstrate that one may combine both imperfect quantum and imperfect classical secret sharing to produce an overall perfect QSS scheme, and that examples of such schemes (which we construct) can have the smallest quantum and classical share components possible for their access structures, something provably not achievable using perfect underlying schemes. Our construction has significant potential for being adapted to other QSS schemes based on stabilizer codes.
Helper quantum secret sharing is a form of secret sharing defined by its unique access structure. One special fixed party, called the helper, can work together with any other party to fully decode the secret. A blind helper is one who can provide this assistance while not holding any local information about the encoded secret. In this work, we analyze the general structure of QSS helper codes and present new code constructions. We fully characterize the structure of blind helper stabilizer codes and show that for the encoding of a single qubit, recovery is always possible using one-way local operations and classical communication (LOCC) from the helper to the targeted party. Furthermore, we demonstrate how such codes also allow for the helper to target larger subsets of parties by one-way LOCC, enabling them to be authorized to recover the secret. Finally, when each party only holds a qubit, we identify the general form of all helper codes (including non-stabilizer codes) and find that LOCC recovery is only possible in special cases.
Quantum secret sharing (QSS) allows a dealer to distribute a secret quantum state among a set of parties in such a way that certain authorized subsets can reconstruct the secret, while unauthorized subsets obtain no information about it. Previous works on QSS for general access structures focused solely on the existence of perfectly secure schemes, and the share size of the known schemes is necessarily exponential even in cases where the access structure is computed by polynomial size monotone circuits. This stands in stark contrast to the classical setting, where polynomial-time computationally-secure secret sharing schemes have been long known for all access structures computed by polynomial-size monotone circuits under standard hardness assumptions, and one can even obtain shares which are much shorter than the secret (which is impossible with perfect security). While QSS was introduced over twenty years ago, previous works only considered information-theoretic privacy. In this work, we initiate the study of computationally-secure QSS and show that computational assumptions help significantly in building QSS schemes, just as in the classical case. We present a simple compiler and use it to obtain a large variety results: We construct polynomial-time computationally-secure QSS schemes under standard hardness assumptions for a rich class of access structures. This includes many access structures for which previous results in QSS necessarily required exponential share size. In fact, we can go even further: We construct QSS schemes for which the size of the quantum shares is significantly smaller than the size of the secret. As in the classical setting, this is impossible with perfect security. We also apply 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 n-party access structures to 1.5^{n+o(n)}, improving upon best known schemes and matching the best known result for general access structures in the classical setting. Finally, among other things, we study the class of access structures which can be efficiently implemented when the quantum secret sharing scheme has access to a given number of copies of the secret, including all such functions in 𝖯 and NP.