Encoding Individual Source Sequences for the Wiretap Channel

TL;DR

This paper establishes fundamental limits and achievable schemes for encoding individual source sequences securely over a degraded wiretap channel using finite-state machines, incorporating source compression, channel coding, and randomness.

Contribution

It provides a necessary condition and a converse bound on bandwidth expansion for secure transmission of individual sequences with finite-state encoders and decoders.

Findings

Lower bound on bandwidth expansion factor for secure transmission.

Achievability of the bound via Lempel-Ziv compression and channel coding.

Bound on the rate of local randomness needed for security.

Abstract

We consider the problem of encoding a deterministic source sequence (a.k.a.\ individual sequence) for the degraded wiretap channel by means of an encoder and decoder that can both be implemented as finite--state machines. Our first main result is a necessary condition for both reliable and secure transmission in terms of the given source sequence, the bandwidth expansion factor, the secrecy capacity, the number of states of the encoder and the number of states of the decoder. Equivalently, this necessary condition can be presented as a converse bound (i.e., a lower bound) on the smallest achievable bandwidth expansion factor. 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 that may be related to the source sequence, and a noisy version of the side information sequence leaks to the wiretapper.