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.
翻译:我们考虑的问题是,如何用一个编码器和解码器将已退化的电线频道的确定源序列编码(a.k.a.a./ 单项序列),而该编码器和解码器都可以同时作为限定状态的机器使用。我们的第一个主要结果是可靠和安全地传输特定源序列、带宽扩展系数、保密能力、编码器状态和解码器状态数目的必要条件。同样,这一必要条件也可以作为最小可实现的带宽扩展系数的对应约束(即较低约束)来提出。该约束由Lempel-Ziv压缩完成,然后是精密的频道编码。鉴于下限饱和,我们还根据在编码器本地随机性所需的最起码的纯随机比特率,以满足安全限制。这一约束也可以由同样的可实现性计划来完成。最后,我们将主要结果扩展到一个案例,在这个案例中,合法的解码-Ziv压缩机的压缩机被精选,然后是电路序列与另一个序列相联通。鉴于较低约束的顺序,我们从一个合法的解码序列进入另一个序列,而该序列与该序列相联。