Secure Multi-Party Computation

The foundational property of SMPC: no party learns anything about another party's private input beyond what can be logically inferred from the output of the agreed-upon function. This is achieved through secret sharing schemes where private data is split into mathematically random shares distributed among participants. Even if an adversary corrupts a subset of parties up to a defined threshold, the underlying plaintext remains information-theoretically or computationally hidden. In genomic consortia, this means a hospital's variant allele frequencies remain opaque to partner institutions during joint computation.

The protocol ensures that the computed output is mathematically identical to what would have been produced if a trusted third party had performed the computation on the plaintext inputs. This is enforced through verifiable secret sharing and message authentication codes embedded in the circuit evaluation. Any active adversary attempting to deviate from the protocol by sending malformed messages will be detected, and the computation will abort or the honest majority will reconstruct the correct result. This guarantees that a federated GWAS p-value computed via SMPC is cryptographically equivalent to a pooled-data analysis.

SMPC protocols are classified by their adversarial tolerance:

• Honest Majority: Assumes more than half of parties follow the protocol correctly. Enables highly efficient protocols using information-theoretic secret sharing (e.g., Shamir's scheme) without expensive public-key cryptography. Suitable for cross-silo genomic federations with contractual trust.
• Dishonest Majority: Tolerates an arbitrary number of corrupt parties. Requires computationally intensive cryptographic primitives like oblivious transfer and zero-knowledge proofs. Essential for truly adversarial settings but imposes significant latency on complex genomic functions.

SMPC compiles a function into a circuit representation. The choice of circuit type dictates performance:

• Arithmetic Circuits: Operate over large finite fields, ideal for linear algebra, statistical computations, and GWAS regression models. Addition is free; multiplication requires communication.
• Boolean Circuits: Represent computation as AND/XOR gates on individual bits. Necessary for non-linear operations, comparison, and branching logic common in clinical decision support algorithms. Modern genomic SMPC frameworks employ mixed-protocol execution, dynamically switching between arithmetic and Boolean shares to optimize each computational sub-step.

The dominant bottleneck in SMPC is not local CPU cycles but the volume and rounds of network communication between parties. Every multiplication gate in an arithmetic circuit requires one round of interaction. For a genomic PCA computation involving thousands of variants, this translates to substantial inter-institutional data transfer. Optimizations include:

• Circuit depth minimization through algebraic restructuring
• Preprocessing model where input-independent correlated randomness (Beaver triples) is generated offline
• Batched multiplication to amortize round trips In practice, a federated logistic regression across three hospitals may require gigabytes of exchanged ciphertext.

SMPC protocols differ in their termination guarantees when faults occur:

• Guaranteed Output Delivery: All honest parties receive the result regardless of adversarial behavior. Requires honest majority and is computationally expensive.
• Fairness: If any honest party learns the output, all honest parties learn it. Prevents an adversary from aborting after receiving the result while others are denied.
• Security with Abort: The most practical model. If a cheating party is detected, the protocol halts and honest parties receive no output. Acceptable in genomic research where retrying with the honest subset is feasible. The choice directly impacts protocol selection for production federated learning pipelines.

A cryptographic technique enabling computation directly on encrypted ciphertext, producing an encrypted result that decrypts to the correct output. In genomic analysis, fully homomorphic encryption (FHE) allows a cloud server to execute variant calling algorithms on encrypted BAM files without ever seeing the raw reads.

• Partially Homomorphic Encryption (PHE): Supports only addition or multiplication, used for secure allele frequency counting
• Somewhat Homomorphic Encryption (SHE): Supports limited operations, sufficient for logistic regression on encrypted genotypes
• FHE schemes like CKKS and BFV are used for secure neural network inference on encrypted genomic embeddings

A foundational SMPC building block where a private value is split into multiple shares distributed among parties, such that no single share reveals any information about the original secret. In genomic consortia, additive secret sharing enables multiple hospitals to jointly compute allele frequencies without revealing individual patient genotypes.

• Shamir's Secret Sharing: Uses polynomial interpolation; requires t+1 of n shares to reconstruct
• Additive Secret Sharing: Splits a value x into random shares x₁ + x₂ = x modulo a prime field
• Replicated Secret Sharing: Optimized for three-party computation with extremely fast online phase

A two-party SMPC protocol where one party garbles a Boolean circuit representing the function to be computed, and the other party evaluates it obliviously. In genomic settings, garbled circuits enable privacy-preserving edit distance computation for sequence alignment without revealing the query or reference sequences.

• The garbler encrypts each gate's truth table using symmetric keys
• The evaluator uses oblivious transfer to obtain input keys without revealing their inputs
• Optimizations like half-gates and Free-XOR dramatically reduce communication overhead

A fundamental cryptographic primitive where a sender holds multiple messages and a receiver selects one without learning the others, while the sender remains oblivious to which message was chosen. OT extension protocols amplify a small number of expensive base OTs into millions of efficient OTs using symmetric-key operations.

• 1-out-of-2 OT: Receiver chooses one of two sender messages
• k-out-of-n OT: Generalization for selecting from larger sets
• Essential for garbled circuit input encoding and private set intersection in genomic cohort discovery

A lightweight SMPC protocol specifically designed for federated learning that allows a central server to compute the sum of model updates from multiple clients without inspecting any individual contribution. Google's SecAgg protocol uses pairwise masking and secret sharing to tolerate client dropouts.

• Each client masks their update with pairwise random seeds shared with other clients
• The server reconstructs the aggregate only after receiving sufficient unmasked shares
• Critical for federated genomic model training where gradient leakage could reveal patient-specific variants

A specialized SMPC protocol allowing two parties to compute the intersection of their private sets without revealing any elements outside the intersection. In genomic research, PSI enables hospitals to identify overlapping patient cohorts for multi-site studies without disclosing their full patient registries.

• Circuit-based PSI: Uses garbled circuits for exact matching
• Diffie-Hellman PSI: Leverages the hardness of the discrete logarithm problem
• Fuzzy PSI: Extends matching to approximate genomic sequence similarity using locality-sensitive hashing