Absrfucr-Given two discrete memoryless channels @MC's) with a common input, it is desired to transmit private messagea to receiver 1 at r&R, andcommon meswgea to both receivers at rate R,, while keeping receiver 2 as ignorant of tbe private messages as possible. Measurhg ignorance by equivocation, a single-letter characterization is given of the. achievable trfplea (RI&R,-,) where 4 is the equivocation rate. Based on this channel ding result, the related source-channel matdng problem is also settled. l%ese results generahe those of Wyner on the wiretap channel and of Kiirner-Marton on tke broahxst channel. I. INTR~DU~I~N W E CONSIDER a broadcast channel with two receivers, i.e., a pair of discrete memoryless channels (DMC's) with common input alphabet % and output alphabets 9 and %. In his celebrated paper [2], Cover raised the problem of determining the possible rates R,, R,,R, such that one can send separate messages to receivers located at the two outputs and respective rates R, and R,, and a common message to both at rate R,. This problem is still open in the general case (i.e., no single-letter characterization of the set of achievable rates is known). If, however, we assume that no separate message is sent to receiver 2, i.e., R, =O, a single-letter characterization of the capacity region has been given in [5]. Our model, which we call a broadcast channel with confidential messages (BCC), has the additional feature that the separate message sent to receiver 1 is confidential, i.e., receiver 2 should be kept as ignorant of it as possible. This point of view has been introduced into channel coding by Wyner [8] in his study of wiretap channels. Following Wyner [8], we shall measure confidentiality by equivocation. Our main result is a single-letter characterization of the set of triples (R,, R,, R,) such that, in addition to a common message at rate R,, a private message can be sent reliably at rate R, to receiver 1 with equivocation at least R, per channel use at receiver 2. This constitutes a generalization of the results of [8], where the above problem is solved if the channel to receiver 2 is a degraded version of that to receiver 1 and no common message is sent (cf. Corollary 4 below). It also constitutes a generalization of the results of [5], where no confidentiality condition is imposed (cf. Corollary 5 below), although we do not prove a strong converse. On the other hand, our converse proof is simpler. Notation: We designate sets by capital letters and random variables (RV) ranging over these sets by the same italic capitals. All RV's will have finite ranges. The number of elements of a set Ex will be denoted by 11% 11. The
The result of the wiretap channel was generalized in REF for the BC, where one of the messages needs to be kept secret from the unintended receiver.
Broadcast channels with confidential messages
IEEE Trans. Inf. Theory
IEEE Trans. Inf. Theory
Computer Science Mathematics