Secure Multi-Party Computation (SMPC)

The foundational property of SMPC is that no party learns anything about another party's private input beyond what can be logically inferred from the output of the computation itself.

• Uses secret sharing to split inputs into random fragments distributed across participants
• Each fragment individually reveals nothing about the original data
• Computation proceeds on these encrypted shares without ever reconstructing the original inputs
• Provides provable security against semi-honest and malicious adversaries

Example: Three hospitals can compute the average patient recovery rate across their combined datasets without any hospital revealing its individual patient records to the others.

SMPC ensures that the final computed result is identical to what would have been obtained if a trusted third party had collected all private inputs and performed the computation in the clear.

• Guarantees mathematical accuracy despite inputs never being centralized
• Protects against malicious participants attempting to corrupt the computation
• Uses verifiable secret sharing and zero-knowledge proofs to detect cheating
• Maintains integrity even when a subset of parties are compromised

This property is critical for financial applications like sealed-bid auctions, where the winning bid must be correctly determined without revealing any losing bids.

A foundational SMPC technique where a function is represented as a Boolean circuit and evaluated gate-by-gate on encrypted inputs.

• One party (the garbler) encrypts the circuit by assigning two random cryptographic labels to each possible wire value (0 and 1)
• The other party (the evaluator) receives labels corresponding to its input via oblivious transfer
• The evaluator decrypts the circuit gate-by-gate, learning only the final output labels
• Provides constant-round complexity, making it efficient for low-latency settings

Best suited for functions with small circuit depth, such as private set intersection and simple comparison operations.

Secret sharing is the mechanism that distributes private inputs across parties so that no individual share reveals any information about the original secret.

• Additive sharing: A value x is split into random shares that sum to x; each party holds one share
• Shamir's Secret Sharing: Uses polynomial interpolation; any t+1 out of n shares can reconstruct the secret, but t or fewer reveal nothing
• Replicated secret sharing: Each party holds a unique subset of shares, optimized for three-party computation with extremely fast local operations
• Enables information-theoretic security when combined with honest-majority assumptions

Secret sharing forms the backbone of most modern SMPC frameworks, including SPDZ and Sharemind.

SMPC protocols are categorized by their adversarial model, which determines the security guarantees and performance trade-offs.

• Honest-majority: Assumes more than half of parties follow the protocol correctly. Enables information-theoretic security and extremely fast computation using secret sharing alone
• Dishonest-majority: Assumes an adversary may corrupt all but one party. Requires computationally expensive cryptographic primitives like homomorphic encryption and zero-knowledge proofs
• Semi-honest model: Corrupted parties follow the protocol but attempt to learn additional information from the transcript
• Malicious model: Corrupted parties may arbitrarily deviate from the protocol; requires robust cheating detection

Choosing the right model involves balancing security requirements against computational overhead.

SMPC enables collaborative machine learning on sensitive data without centralizing or exposing private datasets.

• Private inference: Multiple parties jointly evaluate a model on their combined private inputs without revealing the inputs or the model
• Secure aggregation: Used in federated learning to sum model updates from clients while hiding individual contributions
• Privacy-preserving analytics: Enables cross-institutional statistical analysis on healthcare, financial, or census data
• Secure key management: Distributes cryptographic signing keys across multiple parties so no single party can sign alone

Real-world deployments include the UN's privacy-preserving poverty estimation across multiple national statistical offices and private contact discovery in secure messaging applications.