Degraded broadcast channel with secrecy outside a bounded range

The $K$-receiver degraded broadcast channel with secrecy outside a bounded range is studied, in which a transmitter sends $K$ messages to $K$ receivers, and the channel quality gradually degrades from receiver $K$ to receiver 1. Each receiver $k$ is required to decode message $W {1},\ldots,W {k}$ , for $1\leq k\leq K$ , and to be kept ignorant of $W {k+2},\ldots,W {K}$ , for $k=1,\ldots, K-2$. Thus, each message $W {k}$ is kept secure from receivers with at least two-level worse channel quality, i.e., receivers 1, $\ldots $ , $k-2$. The secrecy capacity region is fully characterized. The achievable scheme designates one superposition layer to each message with binning employed for each layer. Joint embedded coding and binning are employed to protect all upper-layer messages from lower-layer receivers. Furthermore, the scheme allows adjacent layers to share rates so that part of the rate of each message can be shared with its immediate upper-layer message to enlarge the rate region. More importantly, an induction approach is developed to perform Fourier-Motzkin elimination of $2K$ variables from the order of $K^{2}$ bounds to obtain a close-form achievable rate region. An outer bound is developed that matches the achievable rate region, whose proof involves recursive construction of the rate bounds and exploits the intuition gained from the achievable scheme.