-   At time $i=1,2,\ldots, n$ the encoder transmits symbol $\rvx_i$ generated by sampling the distribution $p_{\rvx|\rvu,\rvs}(\cdot| u_i, s_i)$.

### Secret-key generation

-   The decoder upon observing $\rvyr^n$ finds a sequence $\rvu^n$ jointly typical with $\rvyr^n$.

-   Both encoder and the decoder declare the bin-index of $\rvu^n$ to be the secret-key.

### Error Probability Analysis

An error occurs only if one of the following events occur: $$\begin{aligned}
\cE_1 &= \{(\rvu^n(l), \rvs^n) \notin \cT_\eps^n(\rvu,\rvs) \text{ for all } 1 \le l \le T \}\\
\cE_2 &= \{(\rvu^n(L), \rvyr^n) \notin \cT_\eps^n(\rvu,\rvyr)  \}\\
\cE_3 &= \{(\rvu^n(l), \rvyr^n) \in \cT_\eps^n(\rvu,\rvyr) \text{ for some }  l \neq L \}\end{aligned}$$ Since the number of sequences $T > 2^{nI(\rvu;\rvs)}$ it follows from the Covering Lemma [@ElGamalKim Chapter 3] that $\Pr(\cE_1) \rightarrow 0$ as $n\rightarrow \infty$. Furthermore let $\cE_1^c = \{(\rvu^n,\rvs^n,\rvx^n) \in \cT_{\eps'}^n(\rvu,\rvs,\rvx)\}$ and $\Pr(\cE_1^c) \rightarrow 1$ as $n \rightarrow \infty$ for any $\eps' < \eps$. Since $p(\rvyr^n|\rvu^n(L), \rvx^n,\rvs^n) = \prod_{i=1}^n p(\rvy_{ri}|\rvu_i,\rvx_i,\rvs_i)$ it follows from the conditional typicality Lemma [@ElGamalKim Chapter 2] that $\Pr(\cE_2 \cap \cE_1^c) \rightarrow 0$ as $n\rightarrow\infty$. Finally since every $\rvu^n(l)$ is generated i.i.d. $p_{\rvu}(u_i)$ and is independent of $\rvyr^n$ for $l \neq L$ it follows from the Packing Lemma [@ElGamalKim Chapter 3] that $\Pr(\cE_3) \rightarrow 