How Many Queries Will Resolve Common Randomness?

TL;DR

This paper establishes bounds on the number of queries needed to resolve common randomness among multiple terminals, linking it to secret key capacity and providing a new strong converse result.

Contribution

It introduces a universal upper bound on query numbers for common randomness resolution and connects it to secret key capacity with a tight, single-letter formula.

Findings

Upper bound on query number applicable to all common randomness scenarios

Tight bound for i.i.d. signals leading to a single-letter formula

New strong converse for secret key capacity

Abstract

A set of m terminals, observing correlated signals, communicate interactively to generate common randomness for a given subset of them. Knowing only the communication, how many direct queries of the value of the common randomness will resolve it? A general upper bound, valid for arbitrary signal alphabets, is developed for the number of such queries by using a query strategy that applies to all common randomness and associated communication. When the underlying signals are independent and identically distributed repetitions of m correlated random variables, the number of queries can be exponential in signal length. For this case, the mentioned upper bound is tight and leads to a single-letter formula for the largest query exponent, which coincides with the secret key capacity of a corresponding multiterminal source model. In fact, the upper bound constitutes a strong converse for the optimum query exponent, and implies also a new strong converse for secret key capacity. A key tool, estimating the size of a large probability set in terms of Renyi entropy, is interpreted separately, too, as a lossless block coding result for general sources. As a particularization, it yields the classic result for a discrete memoryless source.