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 W <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> , …, W <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</inf> , for 1 ≤ k ≤ K. Furthermore, each message W <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</inf> 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(K <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) 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.