What a node sees, and what happens if nodes team up.
Compare a worker’s view with the real data, then see how many colluding nodes it would take to piece anything together.
What a node sees
This is one real task, sampled from the running system. On the left is the matrix the server actually wants multiplied. On the right is the share one node received. The share is the matrix plus a fresh random matrix scaled by a public constant, so every entry is uniformly random: it carries no information about the layer, the sample, or the numbers underneath. Before sharing, the server also pads, shuffles and mixes the matrices with a secret random transform, so even a leaked copy is a dense, scrambled one. Only by combining enough shares from enough different nodes can the server recover the real product.
What if nodes team up?
Sharing is a threshold scheme. With T-private sharing, any T nodes comparing notes learn nothing at all, however much computing power they have. T + 1 nodes holding shares of the same product could rebuild it. That is not a bug, it is what the threshold means, and no scheme that lets strangers do the arithmetic on real numbers can avoid it. So the system stacks four defences, and the numbers below show what each buys.
- A higher threshold. Each step up in T needs two more shares per product, so an attacker must control far more of the network. The cost is that nodes do 2T + 1 times the arithmetic.
- Random placement. The server picks nodes for every share at random, so an attacker with a fraction of the nodes cannot choose to receive several shares of the same product.
- A hiding layer. Before any sharing, both matrices are padded to a standard size, shuffled, and mixed by a secret random transform that spreads every entry across the whole matrix. A coalition that does cross the threshold gets a dense scrambled copy: values, order, zero positions and exact layer sizes are gone. It is not a proof of secrecy, but it leaves nothing readable.
- Admission rules. The operator can require that no two shares of a product go to the same network address, and cap how many nodes one address can run.
Chance that an attacker sees enough shares of one product
| Setting | Shares | Work | Chance per product | Per 1,000 products |
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