On the Source Model Key Agreement Problem

We consider the source model key agreement problem involving two legitimate parties and an eavesdropper who observe n i.i.d. samples of X and Y and Z respectively. In this paper, we focus on one of the simplest instances where the key capacity remains open, specifically when X and Y are binary random variables and Z is a function of the pair ($X, Y$). The best-known upper bound on the key capacity is characterized by an inf-max optimization problem that generally lacks a closedform solution. In this paper, we solve the optimization for some class of sources, thereby providing simple expressions for the upper bound. We provide general conditions under which the upper bound reduces to $I(X; Y)$. As an example, we consider the XOR setting in which X and Y are binary, and Z is the XOR of X and Y. The upper bound reduces to $I(X; Y)$ for this source. Next, we conjecture that the rate $I(X; Y)$ is not achievable for the XOR source, and provide some ideas that might be useful for developing a new upper bound. ${ }^{1}$