Encoding Individual Sequences for the Wiretap Channel

We consider the problem of encoding an individual source sequence for the degraded wiretap channel using encoders and decoders that can be implemented as finite-state machines. Our first main result is a converse bound for reliable and secure transmission in terms of the given source sequence, the bandwidth expansion factor, the secrecy capacity, and the numbers of states of the encoder and the decoder. The bound is asymptotically achievable by Lempel-Ziv compression followed by good channel coding for the wiretap channel. Given that the lower bound is saturated, we also derive a lower bound on the minimum necessary rate of purely random bits needed for local randomness at the encoder in order to meet the security constraint. This bound too is achieved by the same achievability scheme. Finally, we extend the main results to the case where the legitimate decoder has access to a side information sequence, which is another individual sequence, and a noisy version of the side information sequence leaks to the wiretapper.