Capacity results for classes of wiretap channels

We consider two communication systems which are time-discrete and memoryless, both depend on a state, in terms of information-theoretic secure data transmission. The compound channel consists of a finite or infinite set of channels which is known to both the sender and the receiver, but unfortunately it is not known which channel is in use for any codeword transmission. In contrast the state of an arbitrarily varying channel may change in an arbitrary but also unknown manner from letter to letter in the transmission of any single codeword. For both scenarios we require secrecy from eavesdropping by a possibly existing third party. The associated model of secure data transmission (or transmission of private messages) is described by a wiretap channel. For a more realistic investigation of practical wireless communication systems we consider the wiretap channel under channel uncertainty. We call the resulting models the compound wiretap channel and the arbitrarily varying wiretap channel. For both systems we derive results for the secrecy capacity, which is defined in terms of the average error probability and the strong secrecy criterion. We derive a lower bound on the secrecy capacity of the compound wiretap channel with channel state information at the transmitter which matches the general upper bound on the secrecy capacity of general compound wiretap channels and thus establishing a full coding theorem in this case. We achieve this for the strong secrecy criterion and with a decoder that is robust against the effect of randomisation in the encoding. This relieves us from the need of decoding the randomisation parameter which is in general not possible within this model. Moreover we prove a lower bound on the secrecy capacity of the compound wiretap channel without channel state information and derive a multi-letter expression for the capacity in this communication scenario. For the arbitrarily varying wiretap channels AVWC we derive a lower bound on the random code secrecy capacity in the case of a“best”channel to the eavesdropper. We show that, provided that the channel to the legitimate receiver is not symmetrisable, the deterministic code secrecy capacity of the AVWC equals the random code