turns out that the model in section \[subsec:Model\] includes this extended model. The secret-key capacity for this new model is identical to the secret-key capacity of a particular model in section \[subsec:Model\] defined by: ${\bar{\rvyr}}= (\rvyr, {\mathsf{s}_{r}})$ and ${\bar{\rvye}}= (\rvye, {\mathsf{s}_{e}})$ and the channel transition probability $$p(\bar{y}_r, \bar{y}_e|s_t,x) = \sum_{s_0} p(y_r, y_e|s_0,s_r,s_e,s_t,x)p(s_0,s_r,s_e|s_t).\label{eq:equiv}$$ The equivalence can be established by noting that the modified channel preserves the same knowledge of the side information sequences as the original problem, the rate and equivocation terms only depend on the joint distribution $p(\bar{y}_r^n, \bar{y}_e^n,x^n,s_t^n)$ and for any input distribution $p(x^n|s_t^n)$, the extended channel satisfies $$p(\bar{y}_r^n, \bar{y}_e^n| x^n,s_t^n) = \prod_{i=1}^n p(\bar{y}_{ri}, \bar{y}_{ei}|x_i, s_{ti}), 
\label{eq:jointPDF}$$ where each term on the right hand side of  obeys .

We omit a detailed proof in interest of space and point to the reader to [@sideInfo pp. 17—25][@ElGamalKim Chapter 7, pp. 7-54] for an analogous observation. Note that our model inherently uses the asymmetry in channel state knowledge between the eavesdropper and the legitimate receiver for secret key generation. While as discussed in this subsection, it can be easily extended to incorporate receiver side information, for simplicity in exposition we will suppress the availability of side information at the receivers.

Main 