Wiretap Secret Key Capacity of Tree-PIN

Secret key agreement (SKA) is an essential primitive in cryptography and information security. In a multiterminal key agreement problem, there are a set of terminals each having access to a component of a vector random variable, and the goal of the terminals is to establish a shared key among a designated subset of terminals. This problem has been studied under different assumptions about the adversary. In the most general model, the adversary has access to a random variable $Z$, that is correlated with all terminals' variables. The single-letter characterization of the secret key capacity of this model, known as the wiretap secret key capacity, is not known for an arbitrary $Z$. In this paper, we calculate the wiretap secret key capacity of a Tree-PIN, when $Z$ consists of noisy version of terminals' variables. We also derive an upper bound and a lower bound for the wiretap secret key capacity of a PIN, and prove their tightness for some special cases.