Achieving the Secrecy Capacity of Wiretap Channels Using Polar Codes

Abstract

Suppose that Alice wishes to send messages to Bob through a communication channel$C_{1}$, but her transmissions also reach an eavesdropper Eve through another channel$C_{2}$. This is the wiretap channel model introduced by Wyner in 1975. The goal is to design a coding scheme that makes it possible for Alice to communicate both reliably and securely. Reliability is measured in terms of Bob's probability of error in recovering the message, while security is measured in terms of the mutual information between the message and Eve's observations. Wyner showed that the situation is characterized by a single constant${\cal C}_{s}$, called the secrecy capacity, which has the following meaning: for all$\varepsilon \!\! > \!\! 0$, there exist coding schemes of rate$R\! \geqslant {\cal C}_{s} \! \! - \! \varepsilon$that asymptotically achieve the reliability and security objectives. However, his proof of this result is based upon a random-coding argument. To date, despite considerable research effort, the only case where we know how to construct coding schemes that achieve secrecy capacity is when Eve's channel$C_{2}$is an erasure channel, or a combinatorial variation thereof.