Two typical fixed-length random number generation problems in information theory are considered for general sources. One is the source resolvability problem and the other is the intrinsic randomness problem. In each of these problems, the optimum achievable rate with respect to the given approximation measure is one of our main concerns and has been characterized using two different information quantities: the information spectrum and the smooth R\'enyi entropy. Recently, optimum achievable rates with respect to $f$-divergences have been characterized using the information spectrum quantity. The $f$-divergence is a general non-negative measure between two probability distributions on the basis of a convex function $f$. The class of f-divergences includes several important measures such as the variational distance, the KL divergence, the Hellinger distance and so on. Hence, it is meaningful to consider the random number generation problems with respect to $f$-divergences. However, optimum achievable rates with respect to $f$-divergences using the smooth R\'enyi entropy have not been clarified yet in both of two problems. In this paper we try to analyze the optimum achievable rates using the smooth R\'enyi entropy and to extend the class of $f$-divergence. To do so, we first derive general formulas of the first-order optimum achievable rates with respect to $f$-divergences in both problems under the same conditions as imposed by previous studies. Next, we relax the conditions on $f$-divergence and generalize the obtained general formulas. Then, we particularize our general formulas to several specified functions $f$. As a result, we reveal that it is easy to derive optimum achievable rates for several important measures from our general formulas. Furthermore, a kind of duality between the resolvability and the intrinsic randomness is revealed in terms of the smooth R\'enyi entropy.

翻译：本文针对一般信源，研究了信息论中两种典型的定长随机数生成问题：源可解性问题和内在随机性问题。在这两个问题中，给定近似度量下的最优可达速率是主要关注点之一，并已通过两种不同的信息量——信息谱和光滑Rényi熵——进行了刻画。近年来，基于$f$-散度的最优可达速率已通过信息谱量进行了表征。$f$-散度是一种基于凸函数$f$的、两个概率分布之间的一般性非负测度，其类别包含若干重要度量，如变分距离、KL散度、Hellinger距离等。因此，研究基于$f$-散度的随机数生成问题具有重要意义。然而，在这两个问题中，利用光滑Rényi熵的$f$-散度下的最优可达速率尚未明确。本文尝试利用光滑Rényi熵分析最优可达速率，并扩展$f$-散度的类别。为此，我们首先在与先前研究相同的条件下，推导了这两个问题中基于$f$-散度的一阶最优可达速率的通用公式。接着，我们放宽了$f$-散度的条件，并对所得通用公式进行推广。然后，将通用公式具体应用于若干特定函数$f$。结果表明，从通用公式中可简便导出若干重要度量的最优可达速率。此外，从光滑Rényi熵的角度揭示了可解性与内在随机性之间的一种对偶关系。