Abstract
Topology-Hiding Computation (THC) enables parties to securely compute a function on an incomplete network without revealing the network topology. It is known that secure computation on a complete network can be based on oblivious transfer (OT), even if a majority of the participating parties are corrupt. In contrast, THC in the dishonest majority setting is only known from assumptions that imply (additively) homomorphic encryption, such as Quadratic Residuosity, Decisional Diffie-Hellman, or Learning With Errors.
In this work we move towards closing the gap between MPC and THC by presenting a protocol for THC on general graphs secure against all-but-one semi-honest corruptions from constant-round constant-overhead secure two-party computation. Our protocol is therefore the first to achieve THC on arbitrary networks without relying on assumptions with rich algebraic structure. As a technical tool, we introduce the notion of locally simulatable MPC, which we believe to be of independent interest.
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Notes
- 1.
In contrast, this is trivial to achieve (in the semi-honest setting) if hiding network topology is not a concern: simply forward the message through the network.
- 2.
MPC with constant computational overhead means that a circuit of size s(n) can be securely evaluated in time \(O(s(n))+{\text {poly}}(\lambda )\), where the latter term is a fixed polynomial of the security parameter.
- 3.
[HMTZ16] gave an early construction of a more efficient protocol for such graphs from the decisional Diffie-Hellman assumption.
- 4.
On the other hand, it is known that oblivious transfer is necessary to simply communicate in a topology-hiding manner in the presence of a dishonest majority. In particular, OT is implied by topology-hiding broadcast with a dishonest majority for graphs with just 4 nodes [BBMM18]. Again, because the broadcast functionality does not hide its inputs it is trivial to realize without hiding the topology. [BBC+20] showed that OT is necessary for topology-hiding anonymous broadcast on even simpler graphs.
- 5.
In fact, the protocol we describe can be seen as a conceptually simpler alternative to Ball et al.’s [BBC+19, Theorem 4.1] 1-secure, semi-honest, information-theoretic topology-hiding anonymous broadcast on the class of all cycles with a given vertex set.
- 6.
If the box was clonable, a party could make a local copy then learn the partial OR of the inputs of all parties who previously handled the box by simply plugging 0s until they receive an output.
- 7.
The stationary distribution is not uniform, but nevertheless each party has non-negligible probability of being the last party.
- 8.
While this is beyond the scope of this exposition, we could quantify the leakage in terms of the electric conductance of the graph. What this means additionally is that the protocol is even insecure against a single corruption as a party can learn information from just counting the number of rounds between two consecutive visits of the box.
- 9.
However our final protocol will turn out to be both topology-hiding and locally simulatable.
- 10.
Note that the distinction into \(\mu \) different simulators instead of \(\mu \) copies of the same simulator is solely for the sake of clarity.
- 11.
A walk in a graph is an alternating sequence of adjacent vertices and edges; both vertices and edges may be repeated. A covering walk contains each vertex at least once.
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Acknowledgments
We thank Elette Boyle, Ran Cohen, and Tal Moran for helpful discussions. Marshall Ball is supported in part by the Simons Foundation. Lisa Kohl is funded by NWO Gravitation project QSC. Pierre Meyer was supported by ERC Project HSS (852952) and by AFOSR Award FA9550-21-1-0046. We thank the anonymous reviewers of TCC for helpful feedback regarding the presentation of our results.
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Ball, M., Bienstock, A., Kohl, L., Meyer, P. (2023). Towards Topology-Hiding Computation from Oblivious Transfer. In: Rothblum, G., Wee, H. (eds) Theory of Cryptography. TCC 2023. Lecture Notes in Computer Science, vol 14369. Springer, Cham. https://doi.org/10.1007/978-3-031-48615-9_13
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