Wyner wiretap model for wireless

Wyner’s wiretap channel model consists of a transmitter, legitimate receiver, and an eavesdropper. The channel between the transmitter and legitimate receiver is called the main channel, and the channel between the transmitter and the eavesdropper is called the eavesdropper’s channel. The eavesdropper overhears the transmission over the main channel through a wiretap, hence its received signal is a noisy version of the signal received at the legitimate receiver.

Communication Theory of Secrecy Systems
The wiretap model came as a follow up to Claude Shannon’s work on perfect secrecy. The secrecy problem is that of communicating a message through the main channel without conveying information about the message through the eavesdropper’s channel. Shannon assumed both the main and eavesdropper’s channels to be noiseless, and the availability of a secret key at both the transmitter and legitimate receiver. One possible reason for the latter assumption is that at the time Shannon published his paper on this problem, most existing solutions assumed a shared secret key between the two legitimate communicating parties. In an attempt to understand the fundamental limits of this problem without regard to the detection strategy employed by the eavesdropper, Shannon called the communication over the main channel perfectly secure if and only if the entropy of the message $M$ given the eavesdropper’s observation $Z$ to be equivalent to the entropy of the message $M$. i.e. $Z$ does not contain any information about $M$. More precisely, perfectly secure communication takes place when $H(M|Z)=H(M)$. Shannon then proved that perfectly secure communication can take place if and only if the entropy of the shared secret key $K$ is at least equal to that of the message. i.e. $H(K) \geq H(M)$. For the case when both the key and message are binary encoded, perfect secrecy can be achieved when $H(K) \geq H(M)$ through the one-time-pad scheme introduced by Vernam.

The Wire-Tap Channel
Shannon’s result discouraged further research in information theoretic secrecy, since the availability of a scheme that distributes a secret key at both communicating nodes with entropy exceeding that of the message, means that the message can be shared securely between the two nodes using that scheme. Wyner revisited this problem with two relaxed assumptions, namely,
 * The noiseless communication assumption of Shannon was relaxed by assuming a possibly noisy main channel and an eavesdropper channel that is a noisy version of the signal received at the legitimate receiver. More specifically, let $X$, $Y$ and $Z$ be the transmitted signal and the received signals at both the legitimate receiver and the eavesdropper, respectively, then $X - Y - Z$ is a Markov Chain.


 * The perfect secrecy condition was relaxed to a weaker condition that only requires the leakage of information at the eavesdropper to go to zero, when normalized by the block length. i.e. $\frac{1}{n} I(M;{\bf Z}) \rightarrow 0$ as $n \rightarrow \infty$, where ${\bf Z}$ is the length $n$ observation vector at the eavesdropper, and $I(A;B)$ is the mutual information between two random variables $A$ and $B$.

Wyner showed that the capacity of the wire-tap channel is given by, $C = \max_{p_X} I(X;Y) - I(X;Z)$ where $p_X$ is the probability distribution function of the transmitted signal. In particular, positive secure rates of communication are achievable without sharing a secret key a priori between the communicating nodes.

Applications to Wireless Channels
The relaxation in the channel model introduced by Wyner, is of practical significance when perceived in the context of wireless communications, because of the natural impairments (e.g. attenuation in signal strength) that make the received signal different from the one originally transmitted. Moreover, the achievability of secure communication without the need to share a secret key, or what is now called as the key-less security approach suggested a new paradigm of secure communication protocols. That is, exploiting properties of the wireless medium that to satisfy the secrecy constraints. Taking advantage of the inherent noise in the channel is one example. Another makes use of the superposition property of the medium (interference) to hide a message from the eavesdropper. This last approach is known as jamming-based security.

Gaussian Wire-Tap Channel
A commonly considered simple model for wireless channels is that of the additive white-Gaussian noise channel. A natural extension of Wyner’s problem to Gaussian channel models was provided by Cheong and Hellman, where the noise processes over the main and wire-tap channels are i.i.d. Gaussian over different channel uses, with zero mean and variances $\sigma_1^2$ and $\sigma_2^2$, respectively. Under an average power constraint of $P$ over the transmitted symbols, the maximum achievable secure rate of communication with a weak secrecy constraint (secrecy capacity) of the Gaussian wire-tap channel was shown to be $C_M – C_{MW}$, where $C_M$ is the capacity of the main channel and $C_{MW}$ is the capacity of the eavesdropper’s channel, which is a concatenation of the main and wire-tap channels. More specifically, the secrecy capacity $C_S$ is given by,
 * $$\ C_S = \frac{1}{2} \log\left(1+\frac{P}{\sigma_1^2}\right) - \frac{1}{2} \log \left(1+\frac{P}{\sigma_1^2+\sigma_2^2}\right) $$

Broadcast Channel with Confidential Messages
Csisz$\overset{´}{a}$r and K$\overset{¨}{o}$rner extended Wyner’s result to a model that is more faithful to the wireless channel. They considered a setting where the wire-tap assumption on the eavesdropper’s channel is removed. More precisely, the transmitter broadcasts its transmission to both the legitimate receiver and the eavesdropper. In this case, the secrecy capacity is given by,
 * $$\ C_S = \max_{P_V} I(V;Y) – I(V;Z) $$

Where $P_V$ is the probability distribution function of an auxiliary random variable $V$ such that $V – X – Y$ is a Markov chain. The auxiliary variable acts as a prefix channel inserted by the source in hope of increasing the secrecy rate. In particular, this result shows that $\max_{P_X} I(X;Y) – I(X;Z)$ is achievable by setting $V=X$. Assuming both the main and eavesdropper’s channel to be AWGN, then this result implies that a strictly positive secrecy rate is achievable whenever the signal to noise ratio at the legitimate receive is higher than that at the eavesdropper.