Secure multi-party computation lets multiple parties work together on data without revealing their individual inputs. It's like having a trusted friend do math for you and your pals, but without anyone seeing each other's numbers.

This topic builds on cryptographic concepts we've learned, showing how they can be combined to solve real-world privacy problems. From secret sharing to homomorphic encryption, we'll see how these tools enable secure collaboration in various applications.

Secure Multi-Party Computation

Fundamentals and Objectives

• Secure multi-party computation (MPC) enables multiple parties to jointly compute a function over private inputs without revealing those inputs
• MPC achieves privacy-preserving distributed computation while maintaining confidentiality of individual data
• Simulates a trusted third party receiving all inputs, performing computation, and returning results
• Ensures input privacy, correctness of computation, and fairness in result distribution
• Applications include privacy-preserving data analysis, secure auctions, and collaborative machine learning
• Introduced by Andrew Yao in 1982 through the "millionaires' problem" (comparing net worth without revealing actual values)
• Designed to resist various adversaries (semi-honest and malicious)

Security Properties and Adversary Models

• Input privacy protects confidentiality of each party's individual data
• Correctness guarantees accurate computation results
• Fairness ensures all parties receive results simultaneously
• Semi-honest (honest-but-curious) adversaries follow protocol but attempt to learn additional information
• Malicious adversaries may deviate from protocol to compromise security
• Covert adversaries behave maliciously but fear detection
• Information-theoretic security provides unconditional protection against computationally unbounded adversaries
• Computational security relies on hardness assumptions of certain mathematical problems

Cryptographic Techniques in Secure Multi-Party Computation

Foundational Cryptographic Primitives

• Secret sharing schemes (Shamir's Secret Sharing) split data among participants
  • Example: Distributing encryption key shares among multiple parties
• Homomorphic encryption enables computations on encrypted data
  • Example: Performing statistical analysis on encrypted medical records
• Oblivious Transfer (OT) protocols transfer one piece of information without revealing which was selected
  • Example: Private information retrieval from a database
• Garbled circuits represent functions as encrypted boolean circuits for secure two-party computation
  • Example: Secure evaluation of decision trees in machine learning
• Zero-knowledge proofs verify computation correctness without revealing additional information
  • Example: Proving knowledge of a solution to a puzzle without revealing the solution

Advanced Cryptographic Techniques

• Threshold cryptography distributes trust and enhances security
  • Example: Distributed key generation for a cryptocurrency wallet
• Commitment schemes ensure parties cannot change inputs or intermediate results
  • Example: Committing to a bid in a sealed-bid auction
• Secure multiparty machine learning protects privacy in collaborative model training
  • Example: Privacy-preserving federated learning across multiple organizations
• Privacy-preserving set intersection finds common elements without revealing full sets
  • Example: Identifying common contacts between two users without sharing entire contact lists
• Secure sum computation aggregates values without revealing individual inputs
  • Example: Calculating average salary across companies without disclosing individual salaries

Secure Multi-Party Computation Schemes

Classic MPC Protocols

• Millionaires' problem using Yao's garbled circuits for secure two-party computation
  • Compares two values without revealing them
• Threshold secret sharing distributes sensitive information among multiple parties
  • Example: $t$-out-of-$n$ scheme requires $t$ shares to reconstruct the secret
• Secure sum protocols for distributed data aggregation
  • Example: Computing total votes in an election without revealing individual votes
• Privacy-preserving set intersection using oblivious polynomial evaluation
  • Finds common elements between two sets without revealing the sets themselves

Advanced MPC Applications

• Secure multiparty machine learning for privacy-preserving model training
  • Example: Collaborative training of a neural network across multiple hospitals
• Secure auction systems protecting bid privacy while determining the winner
  • Example: Second-price sealed-bid auctions without a trusted auctioneer
• Privacy-preserving voting schemes preserving voter privacy and ensuring correct tallies
  • Example: Homomorphic encryption-based e-voting systems
• Secure multiparty linear regression for collaborative data analysis
  • Example: Analyzing financial trends across multiple banks without sharing raw data

Security and Efficiency of Secure Multi-Party Computation Protocols

Security Analysis and Models

• Semi-honest model assumes parties follow protocol but may attempt to learn additional information
• Malicious model considers adversaries who may arbitrarily deviate from the protocol
• Covert model addresses adversaries who may cheat but fear detection
• Information-theoretic security provides unconditional protection against unbounded adversaries
• Computational security relies on hardness assumptions of mathematical problems
  • Example: Difficulty of factoring large numbers in RSA encryption
• Protocol composability ensures security when combining multiple MPC primitives
• Side-channel attack resistance protects against implementation-specific vulnerabilities
  • Example: Timing attacks on cryptographic operations

Efficiency Considerations and Trade-offs

• Communication complexity measures round complexity and bandwidth requirements
  • Example: Number of messages exchanged in a distributed key generation protocol
• Computational complexity assesses scalability with respect to parties and input size
  • Example: Time complexity of secure multiparty sorting algorithms
• Trade-offs between security guarantees and efficiency in practical implementations
  • Example: Using garbled circuits vs. homomorphic encryption for different computations
• Scalability challenges in large-scale MPC deployments
  • Example: Performance degradation in protocols with many participants
• Optimization techniques for improving MPC efficiency
  • Example: Preprocessing phases to reduce online computation time