On the Private Key Capacity of the <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math> </inline-formula>-Relay Pairwise Independent Network

Abstract

We study the problem of private key generation in a cooperative pairwise independent network (PIN), with $M+2$ terminals (Alice, Bob, and $M$ relays), $M\geq 2$ . In the PIN, the correlated source observed by every pair of terminals is independent of the sources observed by any other pairs of terminals. Moreover, all terminals can communicate with each other over a public channel, which is also observed by Eve, noiselessly. The objective is to generate a private key between Alice and Bob with the help of the $M$ relays; such a private key needs to be protected not only from Eve but also from all relays. A single-letter expression for the private key capacity of this PIN model is obtained, where the achievability part is established by proposing a random binning (RB)-based key generation algorithm, and the converse part is established by deriving upper bounds of $M$ enhanced source models. Next, we consider a cooperative wireless network and use the estimates of fading channels to generate private keys. It has been shown that the proposed RB key generation algorithm can achieve a multiplexing gain of $M-1$ , which is an improvement compared with the existing XOR algorithm, whose achievable multiplexing gain is $\lfloor M /2\rfloor $ .