Wiretap channel

The wiretap channel is an information-theoretic model for communication in the presence of an eavesdropper. Suppose Adam wishes to communicate a message $M$ to Betty. However, Eve can eavesdrop on the channel between Adam and Betty, and Adam wishes to make sure that Eve cannot decode (or partly decode) the message $M$. A discrete-memoryless wiretap channel (X, p(y, z |x), Y×Z) is a broadcast channel with sender $X$ (Adam) , legitimate receiver $Y$ (Betty) and eavesdropper $Z$ (Eve). Adam wishes to communicate at a rate $R$ to Betty while ensuring a given level equivocation $H(M | Z)$ for Eve.

Capacity
The capacity of the wiretap channel is given by a rate-equivocation region which expresses the pairs $(R, E)$ such that Adam can encode a message to Betty at rate $R$ while maintaining a level of equivocation $E$ at the eavesdropper. A sub-case of this is the secrecy capacity, which refers to the maximum rate or reliable communication $R$ subject to $E = R$.

Theorem. The secrecy capacity of the wiretap channel $p(y, z |x)$ is given by \begin{align} C= max_{p(u,x)} (I (U ; Y ) − I (U ; Z )) \end{align} for $|U | \le |X |$

History
The wiretap channel was introduced by Aaron D. Wyner.