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Dealer-Based (5,11) Threshold Quantum Secret Sharing Scheme: Design, Security, and Practical Considerations

Ajay Kumar, Mukesh Kumar Sharma

Published Oct 12, 2026• 0 citations

Abstract

We present DQ-(5,11), a dealer-based (5,11) threshold quantum secret sharing (QSS) scheme where any 5 of the 11 participants can find a dealer's secret and 4 or fewer get nothing about it. A key point is that, due to the Cleve-Gottesman-Lo no-cloning bound, there is only a (k,n) threshold scheme for any quantum secret if n < 2k. Since 11≥10, is a (5,11) quantity, the threshold scheme is impossible for quantum secrets, and in any (5,11) quantum secret sharing scheme there is a classical secret that is encrypted in a way that is protected by quantum communication. DQ-(5,11) is a combination of Shamir's polynomial sharing over GF13 and a single-qudit reconstruction in dimension d=13. Each of the five cooperating participants in DQ-(5,11) performs one phase operation with the share of the quantum secret and a zero-sum mask so that the intermediate states will not have any partial sum. Decoy qudits from two mutually unbiased bases detect eavesdropping, and a dealer-published commitment lets participants check the reconstructed value. We achieve correctness of all 462 subsets of the network, information-theoretic secrecy against coalitions of up to four insiders and a requirement for the success of intercept-resend attacks to be (d+1)/2d)m for m decoys (8.6×10-9) for m=30). We use numerical data to verify the deterministic reconstruction for all 2,310 runs and show the effect of depolarizing noise (79.1% success at 5% noise at each stage) and channel loss. Furthermore, we compare our orbital-angular-momentum and time-bin qudit model to Trojan-horse countermeasures and other schemes.

DOI: 10.20944/preprints202610.1060.v1

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We present DQ-(5,11), a dealer-based (5,11) threshold quantum secret sharing (QSS) scheme where any 5 of the 11 participants can find a dealer's secret and 4 or fewer get nothing about it. A key point is that, due to the Cleve-Gottesman-Lo no-cloning bound, there is only a (k,n) threshold scheme for any quantum secret if n < 2k. Since 11≥10, is a (5,11) quantity, the threshold scheme is impossible for quantum secrets, and in any (5,11) quantum secret sharing scheme there is a classical secret that is encrypted in a way that is protected by quantum communication. DQ-(5,11) is a combination of Shamir's polynomial sharing over GF13 and a single-qudit reconstruction in dimension d=13. Each of the five cooperating participants in DQ-(5,11) performs one phase operation with the share of the quantum secret and a zero-sum mask so that the intermediate states will not have any partial sum. Decoy qudits from two mutually unbiased bases detect eavesdropping, and a dealer-published commitment lets participants check the reconstructed value. We achieve correctness of all 462 subsets of the network, information-theoretic secrecy against coalitions of up to four insiders and a requirement for the success of intercept-resend attacks to be (d+1)/2d)m for m decoys (8.6×10-9) for m=30). We use numerical data to verify the deterministic reconstruction for all 2,310 runs and show the effect of depolarizing noise (79.1% success at 5% noise at each stage) and channel loss. Furthermore, we compare our orbital-angular-momentum and time-bin qudit model to Trojan-horse countermeasures and other schemes.

Abstract Threshold quantum secret sharing is a cryptography technique for dividing and reconstructing secret information. The dealer shares a secret based on quantum mechanics principles, and a subset of participants cooperate to recover that secret. For example, in k , n quantum threshold scheme and n , n quantum threshold scheme, the former can only reconstruct the secret when at least k shares are collected. The latter requires all n shares to be collected to reconstruct the secret. However, in real scenarios, the priority of participant identities may vary. In order to meet the actual needs of participants with different rights, this paper proposes a w , ω , n quantum secret sharing (QSS) scheme. When the sum of participants’ weights w is greater than or equal to a threshold ω, they can cooperate to reconstruct the secret. In order to achieve this, this paper uses the entangled state of single particles and constructs a verifiable QSS scheme based on the Chinese remainder theorem. Significantly, in case of a dishonest participant, the dealer can delete the participant before recovering the secret to ensure the security of the communication. This scheme is more flexible than other threshold protocols. At the same time, it reduces the communication cost of the participants. In addition, security analysis shows that the scheme resists typical external and internal attacks.

(

t

,

n

) Threshold Quantum Secret Sharing based on a Single

d

-level Quantum System

A (

t

,

n

) threshold quantum secret sharing (QSS) is proposed based on a single

d

-level quantum system. It enables the (

t

,

n

) threshold structure based on Shamir's secret sharing and simply requires sequential communication in the

d

-level quantum system to recover secret. Besides, the scheme employs an additional qudit to detect cheats and eavesdropping during secret reconstruction, and allows a participant to use the share repeatedly. Analyses show that the proposed scheme is resistant to typical attacks. The scheme is scalable in the number of participants and easier to realize compared to related schemes. More generally, our scheme also presents a generic method to construct new (

t

,

n

) threshold QSS schemes based on

d

-level quantum system with other classical threshold secret sharing.

Abstract Quantum secret sharing (QSS) can address the increasing threat of computing power. While semi-QSS protocols alleviate participants’ reliance on quantum devices, ensuring security and lowering the participation difficulty. In this paper, we propose a semi-QSS protocol based on four quantum entangled states within a four-dimensional quantum system. It generates a four-dimensional four-quantum entangled state through a dealer, requiring full quantum capability for the dealer and only partial quantum capability for the participants. Participants perform random operations on the received qubits, and eavesdropping detection is based on these random operations. Then the dealer can choose the correct measurement basis based on the receiver’s specific operations. The security and quantum efficiency of protocol depend on the ratio of information to decoy particles inserted by the dealer. Furthermore, the security analysis shows that the protocol is resistant to common quantum attacks.

In this paper, a standard (3, 5)-threshold quantum secret sharing scheme is presented, in which any three of five participants can resume cooperatively the classical secret from the dealer, but one or two shares contain absolutely no information about the secret. Our scheme can be fulfilled by using the singular properties of maximally entangled 6-qubit states found by Borras. We analyze the scheme's security by several ways, for example, intercept-and-resend attack, entangle-and-measure attack, and so on. Compared with the other standard threshold quantum secret sharing schemes, our scheme needs neither to use d-level multipartite entangled states, nor to produce shares by classical secret splitting techniques, so it is feasible to be realized.

Abstract Threshold quantum secret sharing is a typical method for quantum secret sharing (QSS) schemes. In this paper, we propose a verifiable t , n threshold QSS scheme based on the d -dimensional Bell state and asymmetric binary polynomial. In this scheme, the dealer encodes the secret using the asymmetric binary polynomial and generates the corresponding secret share for each participant. Then, the dealer prepares the d -dimensional Bell state, and the participants perform corresponding unitary operations on the particles in transmission to recover the secret. With the hash function and the session key pairs, not only the cheating behavior of the dishonest participants can be detected, but also the specific cheaters can be identified. Furthermore, we consider the case when no less than t participants cooperate to recover the secret, which makes the proposed scheme more practical. Analyses show that the scheme can resist forgery attack, collusion attack and other common attacks.

A verifiable (t, n) threshold quantum secret sharing scheme with sequential communication was proposed recently. In this work, we analyze its security and then give two new participant attacks. Using the first participant attack, the first participant can obtain the dealer's secrets by himself with nonzero probability without being detected. Using the second participant attack, a dishonest participant can gain access to the dealer's secrets by himself in the secret reconstruction phase while he can make the other participants recover false secrets instead of the real ones without being detected. Furthermore, we present an effective way to prevent these attacks.