Abstract
We prove a lower bound on the communication complexity of unconditionally secure multiparty computation, both in the standard model with \(n=2t+1\) parties of which t are corrupted, and in the preprocessing model with \(n=t+1\). In both cases, we show that for any \(g \in \mathbb {N}\) there exists a Boolean circuit C with g gates, where any secure protocol implementing C must communicate \(\varOmega (n g)\) bits, even if only passive and statistical security is required. The results easily extends to constructing similar circuits over any fixed finite field. This shows that for all sizes of circuits, the O(n) overhead of all known protocols when t is maximal is inherent. It also shows that security comes at a price: the circuit we consider could namely be computed among n parties with communication only O(g) bits if no security was required. Our results extend to the case where the threshold t is suboptimal. For the honest majority case, this shows that the known optimizations via packed secret-sharing can only be obtained if one accepts that the threshold is \(t= (1/2 - c)n\) for a constant c. For the honest majority case, we also show an upper bound that matches the lower bound up to a constant factor (existing upper bounds are a factor \(\lg n\) off for Boolean circuits).
I. Damgård—Supported by the ERC Advanced Grant MPCPRO.
K. G. Larsen—Supported by a Villum Young Investigator grant and an AUFF starting grant. Part of this work was done while KGL was a long term visitor at the Simons Institute for Theory of Computing.
J. B. Nielsen—Supported by the Independent Research Fund Denmark project BETHE.
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Notes
- 1.
This is a much harder question of a completely different nature: for instance, if you are given a circuit to evaluate securely, there might exist a much smaller circuit computing the same function, so proving something on the overhead over the circuit size in general seems out of the question unless we are “magically” given the smallest circuit for the function in question.
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Damgård, I., Larsen, K.G., Nielsen, J.B. (2019). Communication Lower Bounds for Statistically Secure MPC, With or Without Preprocessing. In: Boldyreva, A., Micciancio, D. (eds) Advances in Cryptology – CRYPTO 2019. CRYPTO 2019. Lecture Notes in Computer Science(), vol 11693. Springer, Cham. https://doi.org/10.1007/978-3-030-26951-7_3
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