Results {#sec:MainResults}
============

We summarize the main results of this paper in this section.

Capacity Bounds {#subsec:CapacityBounds}
---------------

We first provide an achievable rate (lower bound) on the secret-key capacity.

An achievable secret-key rate is $$\label{eq:lowerBoundNoDisc} R^- = \max_{p_\rvu, p_{\rvx|\rvs,\rvu}}
I(\rvu;\rvyr) - I(\rvu;\rvye),$$ where the maximization is over all auxiliary random variables $\rvu$ that satisfy the Markov condition $\rvu \rightarrow (\rvx, \rvs)
\rightarrow (\rvyr,\rvye)$ and furthermore satisfy the constraint that $$I(\rvu;\rvyr)-I(\rvu;\rvs) \ge 0. \label{eq:lowerBoundNoDiscCons}$$\[thm:lbNoDisc\]

The intuition behind the coding scheme is as follows. Upon observing $\rvs^n$, the sender communicates the best possible reproduction $\rvu^n$ of the state sequence to the receiver Now both the sender and the receiver observe a common sequence $\rvu^n$. The set of all codewords $\rvu^n$ is binned into $2^{nR^-}$ bins and the bin-index is declared to be the secret key. Note that the problem of communicating a state sequence with common knowledge to the receiver is studied in [@steinberg:08; @steinberg:09]. This setup requires that the reconstruction sequence satisfy a certain distortion measure with respect to the state sequence. In contrast the common reconstruction sequence in this problem is an intermediate step used to generate a common secret key.

While we do not have a matching upper 