Homomorphic Secret Sharing¶
A secret-sharing primitive in which servers locally evaluate an allowed function on input shares so their output shares reconstruct the function value without revealing the inputs.
Core Idea¶
Homomorphic secret sharing (HSS) lets a dealer encode an input \(x\) into shares \((x_1,\ldots,x_m)\) so each server locally computes an output share from its own input share and a public function \(f\). Combining the output shares reconstructs \(f(x)\), while any forbidden coalition's input shares hide \(x\). Interaction is absent or tightly limited during evaluation.
Unlike ordinary secret sharing, the shares support a declared function class beyond reconstruction. Unlike homomorphic encryption, HSS produces distributed shares rather than one ciphertext and can obtain information-theoretic or computational privacy under different constructions.
Scope of Application¶
HSS supports private database analytics, distributed point and interval queries, secure aggregation, low-interaction multiparty computation, private inference, and computation on data replicated across noncolluding servers. Schemes vary sharply by function class—linear functions, low-degree polynomials, branching programs, or richer circuits—and by number of parties and corruption threshold.
Claims must include leakage from output, setup, metadata, aborts, and malicious behavior; input privacy alone is not a deployed-system security proof.
Clarity¶
Specify algorithms, parties, function family, correctness error, coalition threshold, semi-honest or malicious adversary, setup, computational assumption, share size, output size, and interaction count. Distinguish input shares, function keys, ciphertexts, and output shares. Do not use “homomorphic” without naming the supported operations.
Manages Complexity¶
HSS separates a global private computation into parallel local maps plus a small recombination step. The abstraction makes a three-way tradeoff explicit: expressiveness, communication/share size, and security assumptions. It also isolates trust: privacy can survive compromise of fewer than the threshold servers without requiring them to communicate about the secret.
Abstract Reasoning¶
- Fix the function class and security parameter.
- Define input sharing and coalition privacy.
- Define local evaluation using only one share and public data.
- Define output reconstruction.
- Prove correctness for every allowed input and function.
- Prove simulation or indistinguishability for the adversary view.
- Account for preprocessing, randomness, errors, and leakage.
- Measure share length, local work, download, and interaction.
- Compare with FHE, FSS, and MPC under aligned assumptions.
Knowledge Transfer¶
The portable pattern is decompose protected state so independent local transformations recombine into the transformation of the whole. It transfers to distributed aggregates, coded computation, federated analytics, and privacy-preserving outsourcing. The proposed immediate parent is Decomposition.
Relationships to Other Abstractions¶
Current abstraction Homomorphic Secret Sharing Domain-specific
Parents (1) — more general patterns this builds on
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Homomorphic Secret Sharing is a kind of Decomposition Prime
Decomposition is the proposed immediate parent.
Hierarchy path (1) — routes to 1 parentless root
- Homomorphic Secret Sharing → Decomposition
Neighborhood in Abstraction Space¶
Homomorphic Secret Sharing sits in a sparse region of the domain-specific corpus (92nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (1565 abstractions)
Nearest neighbors
- Dirichlet convolution — 0.78
- Monotonic Function — 0.78
- Real-valued function — 0.78
- Graph of a Function — 0.77
- Pseudo-polynomial transformation — 0.77
Computed from structural-signature embeddings · 2026-09-08