Random Selection with an Adversarial Majority

Abstract

Abstract. We consider the problem of random selection, where p players follow a protocol to jointly select a random element of a universe of size n. However, some of the players may be adversarial and collude to force the output to lie in a small subset of the universe. We describe essentially the first protocols that solve this problem in the presence of a dishonest majority in the full-information model (where the adversary is computationally unbounded and all communication is via nonsimultaneous broadcast). Our protocols are nearly optimal in several parameters, including the round complexity (as a function of n), the randomness complexity, the communication complexity, and the tradeoffs between the fraction of honest players, the probability that the output lies in a small subset of the universe, and the density of this subset.

High Entropy Random Selection Protocols

“…Our results are not completely comparable with those of [4]; the protocols of [4] only achieve entropy n − O(log n) whereas the entropy n − O(1) of our protocol implies only (µ, O(1/ log(1/µ)))-resilience for all µ > 0. Their (1/n 2 , O(1/n))-resilient protocol, non-explicit matches our non-explicit protocol from Section 4.1 in terms of entropy but our protocol can be extended to also achieve high (n/2 − O(log n)) min-entropy at the cost of additional 3 rounds.…”

“…We show the existence, nonexplicitly, of a protocol that has 6 rounds, 2n + 8 log n bits of communication and yields entropy n − O(log n) and min-entropy n/2 − O(log n). Our protocol achieves the same entropy bound as the recent, also non-explicit, protocol of Gradwohl et al [4], however achieves much higher min-entropy: n/2 − O(log n) versus O(log n). Finally we exhibit very simple explicit protocols.…”

“…Finally our explicit geometric protocol only obtains 3n/4 entropy and thus performs worse than the explicit protocol from [4], that achieves for µ = 1/ log n entropy n − o(n). Our explicit protocols though still have min-entropy n/2 − O(log n) outperforming [4], that only gets min-entropy O(log n).…”

“…Our explicit protocols though still have min-entropy n/2 − O(log n) outperforming [4], that only gets min-entropy O(log n).…”

“…Recently, Gradwohl et al [4], who also considered protocols with more than 2 players, constructed for each µ a O(log * n)-round protocol that is (µ, O( √ µ))-resilient and that uses linear communication. Our results are not completely comparable with those of [4]; the protocols of [4] only achieve entropy n − O(log n) whereas the entropy n − O(1) of our protocol implies only (µ, O(1/ log(1/µ)))-resilience for all µ > 0.…”