Exploring Quantum Supremacy in Access Structures of Secret Sharing by Coding Theory
aa r X i v : . [ qu a n t - ph ] M a r Exploring Quantum Supremacy in AccessStructures of Secret Sharing by Coding Theory
Ryutaroh Matsumoto
Department of Information and Communication EngineeringNagoya University, Japan
Abstract —We consider secret sharing schemes with aclassical secret and quantum shares. One example of suchschemes was recently reported whose access structurecannot be realized by any secret sharing schemes withclassical shares. In this paper, we report further quantumsecret sharing schemes whose access structures cannot berealized by any classical secret sharing schemes.
I. I ntroduction
Secret sharing [13] is a cryptographic scheme toencode a secret into multiple pieces of information(called shares) so that only qualified sets of sharescan reconstruct the original secret. Secret sharinghas become even more important as its applicationto the cloud storage is spreading [1]. The securitycriterion of secret sharing is usually informationtheoretic one and thus cannot be broken even byquantum computers [14].Quantum supremacy [12] is the potential abilityof quantum computing devices to solve problemsthat classical computers practically cannot. Dis-covery of new quantum supremacy is importantin research of quantum information processing.Since majority of secret sharing schemes are se-cure against both classical and quantum computers,quantum supremacy cannot be found in that respect.On the other hand, the author recently reported newquantum supremacy in the access structure of secretsharing [8]. An access structure of a secret sharingschemes is a set of qualified share sets and forbiddenshare sets, where a share set is said to be forbidden(resp. qualified) if the set has no information aboutthe secret (resp. can reconstruct the secret) [11].Specifically, when we use the famous [[5 , , ff erentnecessary conditions on the existence of accessstructures realized by secret sharing schemes withclassical shares, and report 9 new quantum secretsharing schemes whose access structures cannot berealized by secret sharing schemes with classicalshares.II. Q uantum E rror -C orrecting C odes and S ecret S haring Quantum error-correcting codes have been usedfor constructing secret sharing schemes for quantumsecrets [5], [7], [9]. Since classical information canbe regarded as a special case of quantum informa-tion [10], it is easy to construct a secret sharingscheme for a classical secret from a quantum error-correcting code. Suppose that we have a k -bit string ~ s as a classical secret and we want to encode ~ s into n shares. For this goal, we select a binary [[ n , k , d ]]quantum error-correcting code Q , where [[ n , k , d ]]means that the code encodes k qubits into n qubitsand has the minimum distance d . We prepare a k -qubit quantum state | ~ s i and encode | ~ s i into n qubits | ~ x i by Q . Then each qubit in the quantum codeword | ~ x i is distributed to each of n participants.We say that a secret sharing scheme has t -privacyif any set of t shares has absolutely no informationabout the secret, and has r -reconstruction if anyset of r shares uniquely reconstruct the secret [3].For simplicity, r is assumed to be smallest possibleand t to be largest possible. For a secret sharingscheme to be useful, we must know r and t . We willrelate r and t in order to demonstrate the quantumsupremacy.II. Q uantum S upremacy in A ccess S tructures Suppose that one has n − d + d − n shares | ~ x i from avail-able shares [8], and the secret ~ s can be reconstructedfrom | ~ x i . This means that r ≤ n − d + t ≥ d − ff erence r − t is called the threshold gap.When we construct a secret sharing scheme froma binary [[ n , k , d ]] quantum error-correcting codes,we have r − t ≤ n + − d . (1)On the other hand, when we have a secret sharingscheme in which each classical share has log q bitsand the classical secret has k log q bits, we musthave [2] r − t ≥ r + q . (2)A secret sharing scheme with classical shares issaid to be linear if the reconstruction from sharesto secrets is a linear map [4]. Most of studied secretsharing schemes with classical shares are linear, asthey enable e ffi cient encoding and reconstructionby linear algebraic algorithms. When a scheme islinear, we must have [3] r − t ≥ q m − q m + − n + + q m + − q m q m + − k − m )(for all 0 ≤ m ≤ k − . (3)We consider the case that each share is one bit orone qubit, and search for an access structure that canbe realized by quantum shares but cannot be realizedby classical shares. If we have a binary [[ n , k , d ]]quantum code and we also have n + − d < n + − d , (4)then by Eqs. (1) and (2) the binary [[ n , k , d ]] quan-tum code realizes an access structure that cannot TABLE IP arameters of binary [[ n , k , d ]] quantum error - correcting codesthat exhibit quantum supremacy in the access structures ofassociated secret sharing schemes n k d Eq. (4) Eq. (5)6 1 3 true false11 1 5 true false12 1 5 true false17 1 7 true false18 1 7 true false27 3 9 false true with m =
228 3 9 false true with m =
229 1 11 true false30 1 11 true false be realized by secret sharing schemes with classical1-bit shares, thus it exhibits quantum supremacy inthe access structure.In addition, if we have a binary [[ n , k , d ]] quantumcode and we also have n + − d < q m − q m + − n + + q m + − q m q m + − k − m )(for some 0 ≤ m ≤ k − , (5)then by Eqs. (1) and (3) the binary [[ n , k , d ]] quan-tum code realizes an access structure that cannotbe realized by linear secret sharing schemes withclassical 1-bit shares, thus it also exhibits quantumsupremacy in the access structure.Grassl [6] maintains the table of best binaryquantum error-correcting codes. We searched forcodes with properties (4) or (5), and found the codesin Table I. IV. C onclusion As a continuation of the author’s recent paper [8],we searched quantum error-correcting codes thatgive secret sharing schemes whose access structurescannot be realized by classical information process-ing. We reported 9 new codes having access struc-tures impossible by classical information processingin Table I. However, it remains unknown whetheror not there exist infinitely many quantum error-correcting codes having access structures impossibleby classical information processing. It is a furtherresearch agenda. A cknowledgment
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