This document analyzes the potential for implementing privacy-preserving computation using HDC primitives. We explore the parallels with homomorphic encryption, the possibility of cloud-based reasoning on encrypted knowledge, information leakage risks, and applications in federated learning.
The fundamental question: Can we perform reasoning on knowledge without the reasoner knowing what the knowledge represents?
In homomorphic encryption, we can compute f(E(x)) = E(f(x)) - compute on encrypted data and get encrypted results that decrypt to the correct answer.
In HDC, can we achieve: reason(Encode(knowledge)) → Encode(conclusions)?
HDC has properties that suggest privacy-preserving potential:
The key insight: If the atomic concept vectors are secret keys, can the composite structures be safely processed by untrusted parties?
The cloud sees only high-dimensional vectors. Without knowing the atom vectors:
Unfortunately, HDC representations leak information in several ways. This section analyzes what an adversary can learn.
The bundled KB vector doesn't hide how many facts were bundled. An adversary can estimate the number of facts from the vector's statistical properties.
If the same concept appears in multiple facts, the KB will have higher similarity to that concept's vector. By probing with random vectors, an adversary can detect "hot spots" - frequently occurring patterns.
If the client sends multiple queries, the adversary can correlate them. Queries about related concepts will have detectable similarity patterns.
The fundamental problem: similarity is observable.
This is analogous to the problem of order-preserving encryption leaking ordering information.
If the adversary knows the domain vocabulary (e.g., all possible relation names):
| Information Type | Hidden? | Notes |
|---|---|---|
| Exact concept names | Yes (if seed secret) | For hash/PRNG strategies, names can be hashed and not stored; EXACT uses a session-local dictionary for atom IDs |
| Number of facts | Partially | Estimable from vector statistics |
| Structural patterns | No | Similarity reveals shared structure |
| Query patterns | No | Repeated/similar queries are detectable |
| Reasoning complexity | No | Proof depth visible from computation |
| Knowledge graph topology | Partially | Connectivity patterns may leak |
True FHE provides:
HDC provides none of these in a cryptographic sense.
| Property | FHE | HDC "Privacy" |
|---|---|---|
| Security model | Cryptographic (provable) | Obfuscation (heuristic) |
| Information leakage | None (semantic security) | Structural patterns leak |
| Computation speed | Very slow (10000x+ overhead) | Near real-time |
| Supported operations | Any (with circuit conversion) | HDC primitives only |
| Key management | Complex (bootstrapping) | Simple (single seed) |
| Practical today? | Limited applications | Yes, if leakage acceptable |
It provides practical obfuscation with known leakage, not cryptographic security with provable guarantees. Use cases must accept this limitation.
While not fully homomorphic, HDC does exhibit some homomorphic-like properties:
HDC's properties make it interesting for federated scenarios where multiple parties contribute knowledge without revealing it.
HDC's bundling operation is amenable to differential privacy:
class PrivateHDCClient {
constructor(masterSeed) {
this.seed = masterSeed; // Keep secret!
this.atomCache = new Map();
}
// Generate deterministic but secret atom vector
getAtom(conceptName) {
if (!this.atomCache.has(conceptName)) {
const hash = SHA256(this.seed + conceptName);
const vec = PRNG_Vector(hash, this.geometry);
this.atomCache.set(conceptName, vec);
}
return this.atomCache.get(conceptName);
}
// Encode a fact using secret atoms
encodeFact(relation, ...args) {
let vec = this.getAtom(relation);
for (let i = 0; i < args.length; i++) {
const posVec = this.getAtom(`Pos${i+1}`);
const argVec = this.getAtom(args[i]);
vec = bind(vec, bind(posVec, argVec));
}
return vec; // Safe to send to cloud
}
// Decode result by matching against known atoms
decodeResult(resultVec, candidates) {
return candidates
.map(c => ({ name: c, sim: similarity(resultVec, this.getAtom(c)) }))
.sort((a, b) => b.sim - a.sim);
}
}
class CloudHDCService {
constructor() {
this.kb = null; // Bundled vector, semantics unknown
}
// Receive encoded KB from client
loadKB(encodedKB) {
this.kb = encodedKB; // Just a vector, no meaning
}
// Process query - pure HDC operations
query(queryVec) {
// Unbind query from KB
const result = bind(this.kb, queryVec);
return result; // Client will decode
}
// Prove goal - returns proof structure, not semantics
prove(goalVec, ruleVecs) {
// Standard backward chaining on vectors
// Cloud doesn't know what's being proven
return this.backwardChain(goalVec, ruleVecs);
}
// Federated aggregation
aggregate(kbVectors) {
return bundle(kbVectors);
}
}
Combine HDC's efficiency with FHE's security for critical operations:
Apply techniques from secure multi-party computation:
Develop protocols where similarity can be computed without revealing either vector:
Formal analysis of information leakage:
| Claim | Reality |
|---|---|
| "HDC is homomorphic encryption" | False. No cryptographic security guarantees. |
| "HDC provides perfect privacy" | False. Structural patterns leak. |
| "HDC offers practical obfuscation" | True. Semantic content hidden if atoms secret. |
| "HDC enables efficient federated computation" | True. With acceptable leakage trade-offs. |
| "HDC is better than nothing" | True. Defense in depth, data minimization. |