Random coding bound for $E$-capacity region of the broadcast channel with confidential messages

Abstract

We study the discrete memoryless broadcast channel with confidential messages (BC-C). It involves two discrete memoryless channels with two sources, one encoder and two receivers. A common message must be transmitted at rate R 0 to both receivers and a private message to the intended receiver at rate R 1 while keeping the other receiver ignorant of it with equivocation rate Re. We consider error probability exponents (reliabilities) E 1 , E 2 , E 3 , of exponentially decrease of error probability, respectively, of the first decoder, the second decoder and of the decoder trying to find the confidential message. For E = (E 1 , E 2 , E 3 ) the E-capacity region is the set of all achievable rate triples R 0 , R 1 , Re of codes with given reliabilities E 1 , E 2 , E 3 . We construct a random coding bound for E-capacity region of the BCC. When error probability exponents are going to zero, the limit of this bound coincides with the capacity region of the BCC obtained by Csiszár and Körner. Meanwhile the attainable error probability exponents as a function of given rate triple proposed by Hayashi and Matsumoto are positive in the region which can be smaller than the capacity region of the BCC.