Arbitrarily varying wiretap channels with type constrained states

Determining a single-letter secrecy-capacity formula for the arbitrarily varying wiretap channel (AVWTC) is an open problem largely because of two main challenges. Not only does it capture the difficulty of the compound wiretap channel (another open problem), it also requires that secrecy is ensured with respect to exponentially many possible channel state sequences. By extending the strong soft-covering lemma, recently derived by the authors, to the heterogeneous scenario, this paper accounts for the exponential number of secrecy constraints while only imposing single-letter constraints on the communication rate. Through this approach, we derive a single-letter characterization of the correlated-random (CR)-assisted semantic-security (SS) capacity of an AVWTC with a type constraint on the allowed state sequences. The allowed state sequences are the ones in a typical set around a single constraining type. The stringent SS requirement is established by showing that the mutual information between the message and the eavesdropper's observations is negligible even when maximized over all message distributions, choices of state sequences, and realizations of the CR-code. Both the achievability and the converse proofs of the type-constrained coding theorem rely on stronger claims than actually required. The direct part establishes a novel single-letter lower bound on the CR-assisted SS-capacity of an AVWTC with state sequences constrained by any convex and closed set of state probability mass functions. This bound achieves the best known single-letter secrecy rates for a corresponding compound wiretap channel over the same constraint set. In contrast to other single-letter results in the AVWTC literature, this paper does not assume the existence of a best channel to the eavesdropper. Instead, SS follows by leveraging the heterogeneous version of the strong soft-covering lemma and a CR-code reduction argument. Optimality is a consequence of a max-inf upper bound on the CR-assisted SS-capacity of an AVWTC with state sequences constrained to any collection of type-classes. When adjusted to the aforementioned compound WTC, the upper bound simplifies to a max-min structure, thus strengthening the previously best known single-letter upper bound by Liang et al. that has a min-max form. The proof of the upper bound uses a novel distribution coupling argument. The capacity formula shows that the legitimate users effectively see an averaged main channel, while security must be ensured versus an eavesdropper with perfect channel state information. An example visualizes our single-letter results, and their relation to the past multi-letter secrecy-capacity characterization of the AVWTC is highlighted.