Quantum secret sharing (QSS) is a quantum cryptographic protocol allows a secret quantum state or classical information to be circulated among some parties, such that only a specific subset of them can reconstruct the secret. This is similar to classical secret sharing schemes, but it controls the principles of quantum mechanics to provide improved security and new functionalities.
In a QSS situation, a sender, referred to as Alice, wants to share a secret with a group of participants. The secret could be a quantum state, or classical information encoded in a quantum system. The goal is to divide this secret into shares, which are distributed among the participants. Importantly, individual participants or even smaller groups of participants should not be able to gain any information about the secret from their share alone. Only when a pre-defined threshold number of participants (or a specific authorized subset) combine their shares can the original secret be recovered.
Concepts of Quantum Secret Sharing
QSS depend on some values of quantum mechanics
- Superposition: A qubit can exist in a superposition of the basis states |0〉 and |1〉 simultaneously, represented as α|0〉 + β|1〉, where α and β are complex amplitudes. This allows for a encoding of information compared to classical bits.
- Entanglement: Quantum entanglement is a connection between two or more quantum systems, where their fortunes are linked unrelatedly of the distance separating them. Measuring the state of one entangled particle influences the state of the others. Entanglement is a critical resource for many QSS protocols.
- No-Cloning Theorem: This theorem states that it is impossible to create an identical copy of a random unknown quantum state. This prevents any eavesdropper, or even a dishonest participant, from simply copying shares to learn the secret without being detected.
- Measurement and Disturbance: Any attempt to measure a quantum state generally disturbs it. This property is broken in QSS to detect any unauthorized attempts to access the shared secret.
How Quantum Secret Sharing Works in Quantum Computing?
Hillery, Bužek, and Berthiaume (HBB) Protocol:One of the earliest and most well-known QSS protocols was proposed by Hillery, Bužek, and Berthiaume in 1999. This protocol reveals how Alice can share a secret qubit among three parties (Bob, Charlie, and a potential eavesdropper, Eve) using a three-particle Greenberger-Horne-Zeilinger (GHZ) entangled state. The GHZ state for three qubits is given by 1/√2 (|000〉 + |111〉).
How HBB protocol Works?
- Alice prepares three-particle GHZ states.
- For each GHZ state, she preserves one particle and sends the other two to Bob and Charlie.
- Alice encodes her secret qubit |φ〉 = α|0〉 + β|1〉 onto her particle by applying a controlled-NOT (CNOT) gate between her qubit (as the control) and share of the GHZ state (as the target), followed by a Hadamard (H) gate on her secret qubit.
- Alice sends her modified qubit to Bob.
- Now, Bob and Charlie each possess one particle of the original GHZ state, and Bob also receives the encoded secret qubit from Alice.
- To reconstruct the secret, Bob and Charlie need to cooperate. They each perform a measurement in a specific basis (e.g., the computational basis {|0〉, |1〉} or other chosen bases).
- Based on their measurement outcomes, they can jointly apply a specific quantum gate on Bob’s qubit to retrieve the original secret state |φ〉.
An important part of this protocol is any attempt by an eavesdropper like Eve to intercept or measure the particles will disturb the entanglement, which can be detected by Alice, Bob, and Charlie by performing certain joint measurements and checking for connections.
Multiparty Quantum Secret Sharing
QSS has been extended to involving more than three parties. These multiparty protocols aim to distribute a secret among n participants such that a number k of them (the threshold) are required to reconstruct the secret. Various entangled states, beyond the GHZ state, can be used to achieve this, depending on the desired threshold and the number of participants.
Entanglement-Based QSS
Some QSS protocols depend on directly to the distribution of entangled pairs of qubits. For instance, Alice can create several Bell pairs, such as 1/√2 (|00〉 + |11〉), and distribute one qubit of each pair to each participant. The secret can be encoded by Alice on her share of the entangled pairs, and the participants collaborate, potentially by performing joint measurements, to decode the secret.
QSS via Quantum Teleportation
Quantum teleportation, a process that allows the transfer of an unknown quantum state from one location to another without transmitting the state itself, can also be used for quantum secret sharing. In this approach:
- Alice shares a secret qubit |φ〉 with Bob and Charlie.
- She creates an entangled pair of qubits, giving one to Bob and another one to Charlie.
- Alice performs a Bell measurement on her secret qubit and proposes to “teleport” (which is part of another entangled pair shared with a third party or prepared specifically).
- The result of Alice’s Bell measurement provides two classical bits of information.
- Alice sends two classical bits to Bob.
- Based on the classical information received from Alice, Bob can apply specific quantum gates to his qubit to reconstruct a state that is identical to the original secret qubit |φ〉.
To extend this to secret sharing, Alice can involve multiple entangled pairs distributed among some parties. The reconstruction of the secret qubit would then require the collaboration of those parties who hold the entangled qubits used in the teleportation process.
Security of Quantum Secret Sharing
The security of QSS protocols is rooted in the principles of quantum mechanics. Any attempt by an unauthorized party to access the shares or the secret will unescapably cause a trouble that can be detected by the honest participants. Furthermore, the no-cloning theorem ensures that eavesdroppers cannot simply copy the quantum shares to gain information without introducing detectable errors.
QSS, like quantum key distribution (QKD), goals for information-theoretic security, meaning that the security is guaranteed by the laws of physics rather than depend on the hardness of mathematical problems, as in classical cryptography. It provides an advantage in the advancing computational power, including the threat from quantum computers to break classical encryption algorithms.
Relation to Other Quantum Cryptographic Protocols
QSS is closely related to other quantum cryptographic techniques:
- Quantum Key Distribution (QKD): While QKD attentions on establishing a secret key between two parties, QSS to share a secret directly among multiple parties. However, the secret shared in QSS could also be a cryptographic key that is subsequently used for secure communication.
- Quantum Teleportation: As said earlier, teleportation can be a component in certain QSS protocols, enabling the transfer of quantum information necessary for secret sharing.
Challenges and Future Directions
The practical implementation of QSS appearances some challenges:
- Decoherence: Quantum states are delicate and It can lose their quantum properties due to interaction with the environment. Maintaining entanglement and superposition over periods and distances is critical for successful QSS.
- Scalability: Implementing QSS protocols with a large number of participants and complex entangled states is technically difficult.
- Losses in Quantum Channels: Transmitting qubits over long distances, for example through optical fibers, suffers from photon loss, which can affect the efficiency and security of QSS. Quantum repeaters are being explored to overcome these limitations.
Ongoing research is focused on developing more healthy, efficient, and scalable QSS protocols. Exploring different types of entangled states, error correction techniques, and advancements in quantum hardware are critical for the future of quantum secret sharing.
Applications of Quantum Secret Sharing
QSS has possible applications in areas where secure multi-party collaboration is required:
- Secure Multiparty Computation: QSS can enable multiple parties to perform computations on shared secret data without revealing their individual inputs to each other.
- Quantum Key Management: QSS can be used to distribute cryptographic keys among multiple custodians, requiring their joint consent for key retrieval and usage.
- Secure Cloud Quantum Computing: QSS could play a role in allowing multiple users to securely access and utilize quantum computing resources in the cloud.
- Distributed Quantum Information Processing: QSS can be a fundamental building block for distributed quantum networks and quantum communication protocols.
Conclusion
Quantum secret sharing offers a powerful and fundamentally secure way to distribute secrets among multiple parties, leveraging the unique principles of quantum mechanics. By exploiting entanglement, the no-cloning theorem, and the disturbance caused by measurement, QSS provides information-theoretic security and functionalities beyond the capabilities of classical secret sharing schemes. While practical implementation still faces challenges, ongoing research and technological advancements are paving the way for QSS to become a crucial tool in future secure communication and distributed quantum information processing systems.