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 divergence <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</i><sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</sub>. 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 of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</i><sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub>, we extend the OSRB framework via Rényi’s divergence with order infinity, denoted <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i><sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub>. 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,∞).