Communication-Rounds Tradeoffs for Common Randomness and Secret Key Generation

TL;DR

This paper investigates how the number of communication rounds affects the ability to generate common randomness and secret keys, demonstrating new separations between round complexity and communication cost.

Contribution

It constructs distributions showing exponential separation in communication complexity between different numbers of interaction rounds for CRG and SKG.

Findings

Feasible with r+1 rounds and low communication

Infeasible with fewer than r/2 - 3 rounds without high communication

Establishes the first known separations for r ≥ 2 rounds

Abstract

We study the role of interaction in the Common Randomness Generation (CRG) and Secret Key Generation (SKG) problems. In the CRG problem, two players, Alice and Bob, respectively get samples $X_1,X_2,\dots$ and $Y_1,Y_2,\dots$ with the pairs $(X_1,Y_1)$, $(X_2, Y_2)$, $\dots$ being drawn independently from some known probability distribution $\mu$. They wish to communicate so as to agree on $L$ bits of randomness. The SKG problem is the restriction of the CRG problem to the case where the key is required to be close to random even to an eavesdropper who can listen to their communication (but does not have access to the inputs of Alice and Bob). In this work, we study the relationship between the amount of communication and the number of rounds of interaction in both the CRG and the SKG problems. Specifically, we construct a family of distributions $\mu = \mu_{r, n,L}$, parametrized by integers $r$, $n$ and $L$, such that for every $r$ there exists a constant $b = b(r)$ for which CRG (respectively SKG) is feasible when $(X_i,Y_i) \sim \mu_{r,n,L}$ with $r+1$ rounds of communication, each consisting of $O(\log n)$ bits, but when restricted to $r/2 - 3$ rounds of interaction, the total communication must exceed $\Omega(n/\log^{b}(n))$ bits. Prior to our work no separations were known for $r \geq 2$.