Rényi Resolvability and Its Applications to the Wiretap Channel

The conventional channel resolvability problem refers to the determination of the minimum rate required for an input process so that the output distribution approximates a target distribution in either the total variation distance or the relative entropy. In contrast to previous works, in this paper, we use the (normalized or unnormalized) R\'enyi divergence (with the R\'enyi parameter in $[0,2]\cup{\infty}$) to measure the level of approximation. We also provide asymptotic expressions for normalized R\'enyi divergence when the R\'enyi parameter is larger than or equal to $1$ as well as (lower and upper) bounds for the case when the same parameter is smaller than $1$. We characterize the R\'enyi resolvability, which is defined as the minimum rate required to ensure that the R\'enyi divergence vanishes asymptotically. The R\'enyi resolvabilities are the same for both the normalized and unnormalized divergence cases. In addition, when the R\'enyi parameter smaller than~$1$, consistent with the traditional case where the R\'enyi parameter is equal to~$1$, the R\'enyi resolvability equals the minimum mutual information over all input distributions that induce the target output distribution. When the R\'enyi parameter is larger than $1$ the R\'enyi resolvability is, in general, larger than the mutual information. The optimal R\'enyi divergence is proven to vanish at least exponentially fast for both of these two cases, as long as the code rate is larger than the R\'enyi resolvability. The optimal exponential rate of decay for i.i.d.\ random codes is also characterized exactly. We apply these results to the wiretap channel, and completely characterize the optimal tradeoff between the rates of the secret and non-secret messages when the leakage measure is given by the (unnormalized) R\'enyi divergence.