Statistics of Random Binning Based on Tsallis Divergence

Random binning is a widely utilized tool in information theory, particularly for proving achievability bounds. In this paper, we investigate the output statistics of random binning (OSRB) for two cases: where the binned sequence is i.i.d. generated, and randomly chosen from an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\epsilon$</tex>-typical set using the Tsallis divergence <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$T_{\alpha}$</tex> measure for all values of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha\in(0, \infty)$</tex>. For <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha=\infty$</tex>, due to the unbounded nature of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$T_{\infty}$</tex>, we analyze the OSRB framework using Rényi's divergence criterion with the order of infinity, denoted as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$D_{\infty}$</tex>. While exploring OSRB, we encounter a specific form of Renyi's conditional entropy and delve into its properties. Additionally, we demonstrate the effectiveness of this framework in establishing achievability results for wiretap channels, where Tsallis divergence serves as a security measure. The secure rate we obtain is equal to the capacity for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha\in(0.2]$</tex>.