Jamming-based security

Jamming-based secure communication protocols exploit the superposition property of the wireless medium, by intentionally creating interference to confuse an illegitimate eavesdropper.

Secret Key Agreement by Public Discussion from Common Information
The result by Csisz$\overset{´}{a}$r and K$\overset{¨}{o}$rner for the Broadcast channel with confidential messages, suggests that positive secrecy rates are only achievable when legitimate receivers enjoy more favorable channel conditions than eavesdroppers. In an attempt to disprove this suggestion in a different setting, Maurer considered a modified version of the secrecy problem introduced by Shannon where both the main and eavesdropper's channels are noiseless. The modification lies in the assumption that the legitimate receiver is active and can transmit bits that are received by the message source and the eavesdropper through binary symmetric channels with crossover probabilities $\epsilon$ and $\delta$, respectively. A random bit $X$ is broadcast by the legitimate receiver, where $P(X=1)=0.5$. The received signals at both the message source and the eavesdropper are given by $Y = X \oplus E$ and $Z=X \oplus D$, respectively, where $P(E=1)=\epsilon$ and $P(D=1)=\delta$. Now, let the message bit be $W$ and $V= W \oplus Y$ be transmitted over the public discussion channel. The legitimate receiver can compute $V \oplus X = W \oplus E$, thus having an equivalent binary symmetric channel with the message source that has a crossover probability $\epsilon$. Assuming the eavesdropper compute $V \oplus Z= W \oplus E \oplus D$, it will have an equivalent binary symmetric channel with the message source with crossover probability $\epsilon(1-\delta)+\delta(1-\epsilon)$, which can not be better than the equivalent channel between the two communicating nodes. In fact, this turns out to be the best strategy for the eavesdropper as it was shown that the secrecy capacity for this channel is $H\left(\epsilon\left(1-\delta\right)+\delta\left(1-\epsilon\right)\right) - H(\epsilon)$. This shows that strictly positive secrecy rates are achievable in this setting unless $\epsilon=0.5$ or $\delta=0$ or $\delta=1$. i.e. unless the main channel capacity is zero or the eavesdropper have an access to a noiseless version to the receiver's broadcast.

The Wire-Tap Channel with Feedback: Encryption over the Channel
Maurer's work suggested the possibility of achieving strictly positive secrecy rates in the case where the reliable communication capacity of the main channel is lower than that of the eavesdropper's. In, a more practical setting is studied where, unlike the work of Maurer, a public discussion channel is absent and the receiver uses the same channel that is used for communicating the source message, to send its feedback. More specifically, the receiver inserts artificial noise in the channel to create a secrecy advantage. Consider the above example where both the main and eavesdropper's channels are binary symmetric, and assume that the legitimate receiver can transmit and receive symbols simultaneously (i.e. full-duplex), then the receiver can broadcast a random bit $X$ where $P(X=1)=0.5$, and at the same time the source will transmit the message bit $W$. The received signals at the legitimate receiver and the eavesdropper are now given by $Y=W \oplus E$ and $Z= W \oplus X \oplus D$, respectively, where $E$ and $D$ are binary random variables denoting the states of the main and eavesdropper's channels, respectively. Now, the eavesdropper's channel has zero capacity due to the presence of the random bit inserted by the receiver, which is equally likely to be $1$ or $0$. It follows that the secrecy capacity of the wiretap channel with a full-duplex receiver is the same as its main channel capacity (where no secrecy constraint is imposed).

The case where the legitimate receiver can only transmit or receive at any given time (half-duplex) was also studied. In this case, in each symbol time, it chooses to either transmit the random bit $X$ or receive the message bit $W$. Now, assuming noiseless channels, the artificial noise will turn the main channel into a binary erasure channel and the eavesdropper's channel into a binary symmetric erroneous channel, where the probability of erasure in the main channel is the same as the probability of error in the eavesdropper's channel, and is defined by the fraction of time during which, the legitimate receiver broadcasts random noise. It is obvious that in this case, positive secrecy rates are achievable by choosing a probability of transmission of artificial noise $p$ such that $H(p)\geq p$. This suggests the achievability of perfectly secure communication for the more practical setting when legitimate nodes are half-duplex, without requiring an inherent secrecy advantage in the main channel over the eavesdropper's.

Note: to be completed: half-duplex case with noisy channels, and extension to Gaussian channels