chain $\rvu \rightarrow (\rvx,\rvs) \rightarrow (\rvyr,\rvye)$. Additionally it suffices to limit the cardinality of the auxiliary variable to $|\cS|(1 + |\cX|)$ in . \[thm:symCSI\]

The achievability in  follows from  by augmenting ${\bar{\rvyr}}= (\rvyr, \rvs)$. Observe that  is redundant as $I(\rvu;\rvyr,\rvs)-I(\rvu;\rvs) \ge 0$ holds. Furthermore the expression in  can be simplified as follows $$\begin{aligned}
R^- &= \max_{p_\rvu, p_{\rvx|\rvs,\rvu}}
I(\rvu;\rvyr,\rvs) - I(\rvu;\rvye)  \notag\\
&= \max_{p_\rvu, p_{\rvx|\rvs,\rvu}}
I(\rvu;\rvyr |\rvs) - I(\rvu;\rvye | \rvs)  + I(\rvs ; \rvu |\rvye) \label{eq:ft1}\\
&= \max_{p_\rvu, p_{\rvx|\rvs,\rvu}} 
I(\rvu;\rvyr |\rvs) - I(\rvu;\rvye | \rvs)  + H(\rvs |\rvye) \label{eq:ft2}\end{aligned}$$ where the last relation follows by noting that if $\rvu$ is an optimal choice in  then by selecting $\rvu^\star = (\rvu,\rvs)$ will leave the difference in the two mutual information terms unchanged but increase the second term $H(\rvs|\rvye)$ as specified in . Notice that  is identical to . The converse follows by an application of Csiszar’s Lemma and is provided in section \[subsec:converse\]

We provide another achievability scheme for Theorem \[thm:symCSI\] that only requires causal knowledge of $\rvs^n$ at the encoder. The scheme is based on the following interpretation of . The term $I(\rvu;\rvyr|\rvs) - I(\rvu;\rvye|\rvs)$ is the rate of a multiplexed wiretap codebook constructed assuming that all the three terminals 