Saurabh Sanghvi and Salil Vadhan
We study the round complexity of two-party protocols for generating a random
n-bit string such that the output is guaranteed to have bounded bias (according
to some measure) even if one of the two parties deviates from the protocol
(even using unlimited computational resources). Specifically, we require that
the output's statistical difference from the uniform distribution on {0,1}n
is bounded by a constant less than 1.
We present a protocol for the above problem that has 2log^*n+O(1) rounds,
improving a 2n-round protocol that follows from the work of Goldreich,
Goldwasser, and Linial (FOCS `91). Like the GGL protocol, our protocol
actually provides a stronger guarantee, ensuring that the output lands in any
set T of density mu with probability at most O(mu+delta)^{1/2}, where delta is
an arbitarily small constant.
We then prove a matching lower bound, showing that any protocol guaranteeing
bounded statistical difference requires at least log^*n-log^*log^*n-O(1)
rounds. As far as we know, this is the first nontrivial lower bound on the
round complexity of random selection protocols (of any type) that does not
impose additional constraints (e.g. on communication or
"simulatability").
We also prove several results for the case when the output's bias is measured by the maximum multiplicative factor by which a party can increase the probability of a set T.