channel with noncausal transmiter channel state information (CSI) is $$R =\max_{p_{\rvu},p_{\rvx|\rvu,\rvs}}I(\rvu;\rvyr) -
\max\left(I(\rvu;\rvs), I(\rvu;\rvye)\right).\label{eq:secMsgRate}$$

We note that the secret-key rate  is in general strictly better than the secret-message rate .

Secret Keys from Dirty Paper Coding {#subsec:Gaussian}
-----------------------------------

We study the Gaussian case under an average power constraint. The channel to the legitimate receiver and the eavesdropper is expressed as: $$\begin{aligned}
\rvyr &= \rvx + \rvs + \rvzr\\
\rvye &= \rvx + \rvs + \rvze,
\end{aligned}\label{eq:GaussianModel}$$ where $\rvzr \sim \cN(0,1)$ and $\rvze \sim \cN(0,1+\Delta)$ denote the additive white Gaussian nose and are assumed to be sampled independently. The state parameter $\rvs \sim \cN(0,Q)$ is also sampled i.i.d. at each time instance and is independent of both $\rvzr$ and $\rvze$. Furthermore, the channel input satisfies an average power constraint $E[\rvx^2] \le P$. We assume $\rvs^n$ to be noncausally known to the sender but not to any other terminals.

Thus the parameter $P$ denotes the signal-to-noise ratio, the parameter $Q$ denotes the interference-to-noise-ratio, whereas $\Delta$ denotes the degradation level of the eavesdropper. We now provide lower and upper bounds on the secret-key capacity[^4]. We limit our analysis to the case when $P \ge 1$.

![image](INRFIG1.eps){width="10cm"}

![image](INRFIG2.eps){width="10cm"}

Assuming that $P \ge 1$, a 