Abstract
In this work we consider information-theoretically secure MPC against an mixed adversary who can corrupt \(t_p\) parties passively, \(t_a\) parties actively, and can make \(t_f\) parties fail-stop. With perfect security, it is known that every function can be computed securely if and only if \(3t_a + 2t_p + t_f < n\), and for statistical security the bound is \(2t_a + 2t_p + t_f < n\).
These results say that for each given set of parameters \((t_a, t_p, t_f)\) respecting the inequality, there exists a protocol secure against this particular choice of corruption thresholds. In this work we consider a dynamic adversary. Here, the goal is a single protocol that is secure, no matter which set of corruption thresholds \((t_a, t_p, t_f)\) from a certain class is chosen by the adversary. A dynamic adversary can choose a corruption strategy after seeing the protocol and so is much stronger than a standard adversary.
Dynamically secure protocols have been considered before for computational security. Also the information theoretic case has been studied, but only considering non-threshold general adversaries, leading to inefficient protocols.
We consider threshold dynamic adversaries and information theoretic security. For statistical security we show that efficient dynamic secure function evaluation (SFE) is possible if and only if \(2t_a + 2t_p + t_f < n\), but any dynamically secure protocol must use \(\varOmega (n)\) rounds, even if only fairness is required. Further, general reactive MPC is possible if we assume in addition that \(2t_a+2t_f \le n\), but fair reactive MPC only requires \(2t_a + 2t_p + t_f < n\).
For perfect security we show that both dynamic SFE and verifiable secret sharing (VSS) are impossible if we only assume \(3t_a + 2t_p + t_f < n\) and remain impossible even if we also assume \(t_f=0\). On the other hand, perfect dynamic SFE with guaranteed output delivery (G.O.D.) is possible when either \(t_p = 0\) or \(t_a = 0\) i.e. if instead we assume \(3t_a+t_f < n\) or \(2t_p +t_f < n\). Further, perfect dynamic VSS with G.O.D. is possible under the additional conditions \(3t_a + 3/2t_f \le n\) or \(2t_p + 2t_f \le n\). These conditions are also sufficient for dynamic perfect reactive MPC.
Work done while Daniel Escudero was at Aarhus University.
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Notes
- 1.
In a bit more detail, our construction needs as subprotocol a general non-dynamic SFE protocol \(\pi \), and the complexity we obtain is n times that of \(\pi \). Efficient non-constant round protocol \(\pi \) exists for all functions, so our construction is always efficient if we do not insist on asymptotically tight (but still polynomial) round complexity. However, if \(\pi \) is constant round we obtain O(n) rounds. Such a protocol \(\pi \) exists for all functions but is not always computationally efficient. Of course, it would be nice if our O(n) result could be shown with computational efficiency for all functions, but this would be extremely surprising: if the number of players is constant, it would imply constant-round, information theoretically secure and computationally efficient protocol for all functions. Doing this, even for a constant number of players, has been open for decades and is probably a very hard problem. On the other hand, if the function in question has an efficient non-dynamic constant-round protocol, as many functions do, then we can use that one as subprotocol and get an efficient dynamic O(n)-round protocol.
- 2.
In the case of statistical security, this includes the message that those parties were about to send on the broadcast channel, even if no one is actively or passively corrupted.
- 3.
Observe that there may be false-positives, that is, parties who did not fail to send a message in the actual round, but failed to send the signal bit in the heartbeat round. However, this is acceptable in the protocols we consider in this work.
- 4.
Here, it is implicitly assumed that the function output depends on honest parties’ inputs i.e. it could not have been computed locally by \(\mathcal {A}^\mathsf {stat}\) using corrupt parties’ inputs. Thereby, the argument for fairness can be invoked.
- 5.
This restriction is easily removed by modifying the sharing mechanism to include multiple key-tag pairs.
- 6.
This is a loose bound chosen for simplicity as it suffices for our purpose.
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Acknowledgments
Divya Ravi was funded by the European Research Council (ERC) under the European Unions’s Horizon 2020 research and innovation programme under grant agreement No 803096 (SPEC). During his time in Aarhus University, Daniel Escudero was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 669255 (MPCPRO).
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Damgård, I., Escudero, D., Ravi, D. (2021). Information-Theoretically Secure MPC Against Mixed Dynamic Adversaries. In: Nissim, K., Waters, B. (eds) Theory of Cryptography. TCC 2021. Lecture Notes in Computer Science(), vol 13042. Springer, Cham. https://doi.org/10.1007/978-3-030-90459-3_20
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