K-user degraded broadcast channel with secrecy outside a bounded range

A K-receiver degraded broadcast channel with secrecy outside a bounded range is studied, in which a transmitter sends K messages respectively to K receivers, and the channel quality gradually degrades from receiver K to receiver 1. Each receiver k is required to decode messages W₁, W_k, for 1 ≤ k ≤ K. Furthermore, each message W_k should be kept secure from receivers with two-level worse channel quality, i.e., receivers 1,., k-2. The secrecy capacity region is fully characterized. The achievable scheme designates one superposition layer to each message with random binning employed for each layer for protecting 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 potentially 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 over 2K variables among Θ(K²) bounds to obtain a close-form achievable rate region. A converse proof is developed that matches the achievable rate region, which involves recursive construction of the rate bounds.