Output Statistics of Random Binning: Tsallis Divergence and Its Applications

Random binning is a widely used technique in information theory with diverse applications. In this paper, we focus on the output statistics of random binning (OSRB) using the Tsallis divergenceTα. We analyze all values of α ∈ (0,∞)∪{∞} and consider three scenarios: (i) the binned sequence is generated i.i.d., (ii) the sequence is randomly chosen from an ϵ-typical set, and (iii) the sequence originates from an ϵ-typical set and is passed through a non-memoryless virtual channel. Our proofs cover both achievability and converse results. To address the unbounded nature ofT∞, we extend the OSRB framework via Rényi’s divergence with order infinity, denotedD∞. As part of our exploration, we analyze a specific form of Rényi’s conditional entropy and its properties. Additionally, we demonstrate the application of this framework in deriving achievability results for the wiretap channel, where Tsallis divergence serves as a security measure. The secure rate we obtain through the OSRB analysis matches the secure capacity for α ∈ (0, 2] ∪ {∞} and serves as a potential candidate for the secure capacity when α ∈ (2,∞).