🔐Quantum Cryptography Unit 8 – Quantum Secret Sharing & Multi-Party Computation

Key Concepts

• Quantum secret sharing (QSS) enables secure distribution of a secret among multiple parties
• Multi-party computation (MPC) allows joint computation on private inputs without revealing them
• Quantum entanglement, superposition, and measurement form the basis for QSS and quantum MPC
• Shamir's secret sharing scheme is a classical method for dividing a secret into shares
• Quantum key distribution (QKD) protocols, such as BB84, are used in conjunction with QSS
• Verifiable secret sharing (VSS) ensures honesty of participants and correctness of shares
• Quantum threshold schemes require a minimum number of parties to reconstruct the secret
• Quantum homomorphic encryption enables computation on encrypted data without decryption

Quantum Foundations

• Quantum bits (qubits) are the fundamental units of quantum information
  • Qubits exist in a superposition of states, allowing them to represent multiple values simultaneously
• Quantum entanglement occurs when two or more qubits are correlated, even when separated by large distances
  • Entangled states cannot be described independently and exhibit non-classical correlations
• Quantum measurement collapses the superposition of a qubit, revealing its state
  • Measurement outcomes are probabilistic and can only be predicted statistically
• No-cloning theorem states that an unknown quantum state cannot be perfectly copied
  • This property is crucial for the security of quantum cryptographic protocols
• Quantum teleportation enables the transfer of quantum information using entanglement and classical communication
• Quantum error correction codes protect quantum information from noise and decoherence

Secret Sharing Basics

• Secret sharing divides a secret into multiple shares distributed among participants
• Shamir's secret sharing scheme is based on polynomial interpolation
  • The secret is encoded as the constant term of a polynomial, and shares are points on the polynomial
• Blakley's secret sharing scheme uses hyperplane geometry for share distribution
• Verifiable secret sharing (VSS) ensures that the dealer distributes valid shares
  • VSS prevents cheating by the dealer and allows participants to verify the consistency of their shares
• Access structures define the subsets of participants authorized to reconstruct the secret
  • Threshold access structures require a minimum number of shares for secret reconstruction
• Secret sharing schemes can be perfect, meaning unauthorized subsets gain no information about the secret
• Visual secret sharing allows the secret to be reconstructed by stacking transparencies of the shares

Quantum Secret Sharing Protocols

• Hillery, Bužek, and Berthiaume (HBB) protocol is a pioneering QSS scheme using GHZ states
  • The secret is encoded in the phase of the GHZ state and shared among three parties
• Cleve, Gottesman, and Lo (CGL) protocol extends the HBB protocol to an arbitrary number of parties
  • CGL protocol uses a quantum error-correcting code to distribute shares of the secret
• Zhang, Li, and Guo (ZLG) protocol employs entanglement swapping for efficient share distribution
• Quantum verifiable secret sharing (QVSS) protocols, such as the Kogias-Xiang protocol, ensure the validity of quantum shares
• Quantum secret sharing with continuous variables uses squeezed states and homodyne detection
• Quantum secret sharing with graph states exploits the properties of graph-based entanglement structures
• Quantum secret sharing with multiple secrets allows the distribution of several secrets simultaneously

Multi-Party Computation Overview

• Multi-party computation (MPC) enables joint computation on private inputs without revealing them
  • Participants compute a function on their inputs while keeping the inputs secret
• Yao's garbled circuits protocol is a foundational technique for secure two-party computation
  • The function is represented as a garbled circuit, and inputs are encoded as garbled wire labels
• Oblivious transfer is a cryptographic primitive used in many MPC protocols
  • It allows a sender to transfer one of several messages to a receiver without knowing which message was chosen
• Secret sharing-based MPC protocols, such as the BGW protocol, use secret sharing to distribute computation
• Homomorphic encryption allows computation on encrypted data without decryption
  • Fully homomorphic encryption (FHE) supports arbitrary computations on encrypted data
• Secure multi-party computation (SMPC) ensures security against malicious adversaries
  • SMPC protocols, such as the SPDZ protocol, provide security even when some parties deviate from the protocol
• Verifiable computation enables the delegation of computations to untrusted parties while verifying the results

Quantum MPC Techniques

• Quantum multi-party computation (QMPC) combines quantum information processing with MPC
• Quantum homomorphic encryption (QHE) allows computation on encrypted quantum states
  • QHE schemes, such as the Broadbent-Jeffery scheme, support a limited set of quantum operations
• Quantum garbled circuits extend Yao's garbled circuits to the quantum domain
  • Quantum garbled circuits enable secure two-party quantum computation
• Quantum oblivious transfer (QOT) protocols, such as the Bennett-Brassard-Crépeau protocol, enable secure quantum message transfer
• Quantum secret sharing-based MPC uses QSS to distribute quantum computation among parties
• Quantum verifiable multi-party computation (QVMPC) ensures the correctness of quantum computations performed by untrusted parties
• Quantum fully homomorphic encryption (QFHE) is an active area of research, aiming to support arbitrary quantum computations on encrypted states

Real-World Applications

• Secure voting systems can be implemented using QSS and quantum MPC
  • Quantum protocols ensure the privacy and integrity of votes while preventing double voting
• Quantum auctions and marketplaces can be designed using quantum MPC techniques
  • Quantum protocols enable secure bidding and trading without revealing sensitive information
• Privacy-preserving machine learning can benefit from quantum MPC
  • Quantum techniques allow multiple parties to jointly train models on private data without compromising privacy
• Quantum anonymous communication can be achieved using quantum MPC and QSS
  • Quantum protocols enable secure and anonymous message transmission and routing
• Quantum secure multi-party computation has applications in finance, healthcare, and government
  • Quantum MPC can be used for secure data aggregation, private set intersection, and collaborative analysis
• Quantum secret sharing can enhance the security of distributed systems and cloud computing
  • QSS can be used to protect sensitive data and ensure its availability and integrity

Challenges and Future Directions

• Scalability of quantum secret sharing and MPC protocols is a significant challenge
  • Efficient quantum protocols are needed to handle large-scale secrets and computations
• Fault-tolerance and error correction are crucial for practical quantum cryptographic systems
  • Robust quantum error correction codes and fault-tolerant quantum computation are active research areas
• Integration with classical cryptographic primitives and protocols is necessary for hybrid quantum-classical systems
• Development of quantum-resistant classical cryptographic algorithms is essential for post-quantum security
• Standardization efforts, such as the NIST post-quantum cryptography standardization process, aim to establish quantum-safe cryptographic standards
• Experimental demonstrations and implementations of quantum secret sharing and MPC protocols are ongoing
  • Photonic quantum systems and superconducting qubits are promising platforms for quantum cryptography
• Theoretical foundations of quantum cryptography, including security proofs and composability frameworks, require further development