Private Capacity of Broadcast Channels

Abstract

The broadcast channel was introduced by T. M. Cover in 1972 [7]. In its simplified version, it has one sender (or encoder) E and two users (or decoders) $D_l, \ l=1, \ 2$. The sender E is required to send the messages m ₁ and m ₂ uniformly chosen from the message sets ${\cal M}_1$ and ${\cal M}_2$ respectively to D ₁ and D ₂ correctly with probability close to one. That is, the sender encodes the message (m ₁,m ₂) to an input sequence x ⁿ over a finite input alphabet ${\cal X}$ and sends it to the two users via two noisy channels W ⁿ and V ⁿ, respectively. The first (second) user D ₁ (D ₂) decodes the output sequence y ⁿ over the finite output alphabet ${\cal Y}$ of the channel W ⁿ (the output sequence z ⁿ over the finite output alphabet ${\cal Z}$ of the channel V ⁿ) to the first message $\hat{m}_1$ (the second message $\hat{m}_2$ ). In general, the capacity regions for this kind of channels are still unknown. Their determination is probably one of the hardest open problems in Multi-user Shannon Theory.