Rényi Resolvability and Its Applications to the Wiretap Channel

The conventional channel resolvability problem refers to the determination of the minimum rate needed for an input process to approximate the output distribution of a channel 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 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 minimum rate needed to ensure that the R\'enyi resolvability vanishes asymptotically. The optimal rates are the same for both the normalized and unnormalized cases. In addition, the minimum rate when the R\'enyi parameter no larger than $1$ equals the minimum mutual information over all input distributions that induce the target output distribution similarly to the traditional case. When the R\'enyi parameter is larger than $1$ the minimum rate is, in general, larger than the mutual information. The optimal R\'enyi resolvability is proven to vanish at least exponentially fast for both of these two cases, as long as the code rate is larger than the minimum admissible one. 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. This tradeoff differs from the conventional setting when the leakage is measured by the mutual information.