Distributed Randomness from Approximate Agreement
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Randomisation is a critical tool in designing distributed systems. The common coin primitive, enabling the system members to agree on an unpredictable random number, has proven to be particularly useful. We observe, however, that it is impossible to implement a truly random common coin protocol in a fault-prone asynchronous system. To circumvent this impossibility, we introduce two relaxations of the perfect common coin: (1) approximate common coin generating random numbers that are close to each other; and (2) Monte Carlo common coin generating a common random number with an arbitrarily small, but non-zero, probability of failure. Building atop the approximate agreement primitive, we obtain efficient asynchronous implementations of the two abstractions, tolerating up to one third of Byzantine processes. Our protocols do not assume trusted setup or public key infrastructure and converge to the perfect coin exponentially fast in the protocol running time. By plugging one of our protocols for Monte Carlo common coin in a well-known consensus algorithm, we manage to get a binary Byzantine agreement protocol with O(n^3 log n) communication complexity, resilient against an adaptive adversary, and tolerating the optimal number f<n/3 of failures without trusted setup or PKI. To the best of our knowledge, the best communication complexity for binary Byzantine agreement achieved so far in this setting is O(n^4). We also show how the approximate common coin, combined with a variant of Gray code, can be used to solve an interesting problem of Intersecting Random Subsets, which we introduce in this paper.
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