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Frontiers in Physics

Efficient (k, n) threshold semi-quantum secret sharing protocol

Jie Cao, Jinchao Xu

Published Feb 26, 2025• 4 citations
Publisher

Abstract

Most (k, n) threshold quantum secret sharing protocols are fully quantum. The message receivers must be equipped with complex quantum devices so as to prepare various quantum resources and perform complex quantum operations, which may affect the practice of these protocols. On the other hand, the qubit efficiency of most (k, n) threshold quantum secret sharing protocols is not more than 1/2. To simplify the (k, n) threshold quantum secret sharing protocol and improve its practice and qubit efficiency, a new (k, n) threshold secret sharing protocol with semi-quantum properties is proposed. In this protocol, the dealer prepares decoy particles and sends them to the receivers. The receivers insert particles carrying secret information along with Z-basis decoy particles into the received particle sequence to generate mixed-particle sequences, which are returned to the dealer. The dealer measures the received particle sequences to check for eavesdropping and establishes shared secret keys with the receivers. With the shared secret keys, the dealer distributes the secret pieces among the receivers using Shamir’s secret sharing scheme. Multiple secret messages can be recovered by k or more receivers. The qubit efficiency of our protocol is k/n. For an (n, n) threshold protocol, the qubit efficiency would be 100%. The proposed scheme is based on single particles without using any entangled system. Therefore, its quantum resources are relatively easy to prepare. Receivers must only prepare simple Z-basis qubits. Its semi-quantum properties enhance practice implementation. The proposed protocol has robust security against various types of attacks, including eavesdropping, internal, and collusion attacks. Furthermore, it can resist the unitary attack, which is seldom analyzed in other protocols.

DOI
10.3389/fphy.2025.1542675

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Most (k, n) threshold quantum secret sharing protocols are fully quantum. The message receivers must be equipped with complex quantum devices so as to prepare various quantum resources and perform complex quantum operations, which may affect the practice of these protocols. On the other hand, the qubit efficiency of most (k, n) threshold quantum secret sharing protocols is not more than 1/2. To simplify the (k, n) threshold quantum secret sharing protocol and improve its practice and qubit efficiency, a new (k, n) threshold secret sharing protocol with semi-quantum properties is proposed. In this protocol, the dealer prepares decoy particles and sends them to the receivers. The receivers insert particles carrying secret information along with Z-basis decoy particles into the received particle sequence to generate mixed-particle sequences, which are returned to the dealer. The dealer measures the received particle sequences to check for eavesdropping and establishes shared secret keys with the receivers. With the shared secret keys, the dealer distributes the secret pieces among the receivers using Shamir’s secret sharing scheme. Multiple secret messages can be recovered by k or more receivers. The qubit efficiency of our protocol is k/n. For an (n, n) threshold protocol, the qubit efficiency would be 100%. The proposed scheme is based on single particles without using any entangled system. Therefore, its quantum resources are relatively easy to prepare. Receivers must only prepare simple Z-basis qubits. Its semi-quantum properties enhance practice implementation. The proposed protocol has robust security against various types of attacks, including eavesdropping, internal, and collusion attacks. Furthermore, it can resist the unitary attack, which is seldom analyzed in other protocols.
Semi-quantum secret sharing is an important research issue in quantum cryptography. In this paper, we propose a (t, n) threshold semi-quantum secret sharing protocol, which combines the practicality of semi-quantum secret sharing protocols and the flexibility of (t, n) threshold quantum secret sharing protocols. Participants prepare and transmit single particles in a circular way, and then any t out of n participants can recover the secret according to Shamir’s secret sharing scheme. As quantum resources, single particles are easy to prepare. Furthermore, classical participants only need to possess the capability to prepare and insert particles. The security analysis shows our protocol has security against most attacks. Except decoy particles, all particles are useful to carry the secret message, so the efficiency of the proposed protocol can achieve 100%.
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.
Seldom quantum secret sharing protocols with both the flexible (t, n) threshold and semi-quantum properties have been proposed. Recently, a novel idea of (t, n) threshold semiquantum secret sharing was proposed, which has both the (t, [Formula: see text] threshold and semi-quantum properties. Its idea can simplify the quantum secret sharing process such that many classical users with simple ability of quantum operations can realize the communication goal by cooperating with one quantum party. Furthermore, the protocol is very flexible since any t collaborators of the n classical users can reconstruct the full secret by Shamir’s threshold technology. Unfortunately, their protocol is vulnerable to the NOT-gate attack. This paper shows that an attacker can break the (t, n) threshold protocol by performing two rounds of NOT-gate attack. Then, an improved (t, n) threshold semiquantum secret sharing protocol is proposed. In the improvement scheme, each receiver can perform the eavesdropping check by measuring the Z-basis sample states. The improved protocol not only has semi-quantum properties but also can repair the security hole of the old version and has enhanced security against various quantum attacks.
Most semi-quantum secret sharing protocols are suitable for the application with three participants. Based on the n-level states, a multiparty semi-quantum secret sharing protocol is proposed. Compared with similar protocols, it has the merits as follows. (1) The dealer only needs to prepare the single photon sequence without using any product state or entangle state. (2) To share a secret, the classical parties need not have the ability of measurement. They only need to reorder the photon sequence and send it to the receiver. (3) The dealer knows nothing about any agent’s secret shadow. Therefore, it is infeasible for the dealer to impersonate the classical participant to pool the secret shadow during the secret recovering phase. (4) Theoretically, the qubit efficiency of the proposed protocol is asymptotically 100%. The protocol is secure against famous attacks such as intercept-resend attack, measurement-resend attack and entangle-measure attack.
Semi-quantum secret sharing (SQSS) is a branch of quantum cryptography which only requires the dealer to have quantum capabilities, reducing the difficulty of protocol implementation. However, the efficiency of the SQSS protocol still needs to be further studied. In this paper, we propose a semi-quantum secret sharing protocol, whose efficiency can approach 100% as the length of message increases. The protocol is based on single particles to reduce the difficulty of resource preparation. Particle reordering, a simple but effective operation, is used in the protocol to improve efficiency and ensure security. Furthermore, our protocol can share specific secrets while most SQSS protocols could not. We also prove that the protocol is secure against common attacks.
The security of a secret sharing protocol is compromised if one or more dishonest participants can reconstruct the secret by deviating from the prescribed protocol, particularly when such malicious behavior remains undetected. Semi-Quantum Secret Sharing (SQSS) is a variant of secret sharing in which the participants possess only classical capabilities, such as preparing and measuring qubits in the computational (ZZZ) basis, while a quantum-capable third party assists the dealer in generating and distributing quantum shares of a classical secret. The limited quantum capabilities of the participants in SQSS protocols may introduce vulnerabilities to collusion attacks, enabling an assisting quantum third party, in collaboration with a dishonest classical participant, to jointly recover the secret even without detection. In this work, we propose a novel SQSS protocol that eliminates this vulnerability and achieves information-theoretic security against collusion attacks. We further prove that the proposed protocol is secure against a broad class of external and internal attacks. Finally, a comparative analysis demonstrates that our construction advances existing SQSS protocols by simultaneously optimizing qubit efficiency, involves a classical dealer, and resilience against DCNA attacks using basic Bell states.