Given a sequence of samples $x_1, \dots , x_k$ promised to be drawn from one of two distributions $X_0, X_1$, a well-studied problem in statistics is to decide $\textit{which}$ distribution the samples are from. Information theoretically, the maximum advantage in distinguishing the two distributions given $k$ samples is captured by the total variation distance between $X_0^{\otimes k}$ and $X_1^{\otimes k}$. However, when we restrict our attention to $\textit{efficient distinguishers}$ (i.e., small circuits) of these two distributions, exactly characterizing the ability to distinguish $X_0^{\otimes k}$ and $X_1^{\otimes k}$ is more involved and less understood. In this work, we give a general way to reduce bounds on the computational indistinguishability of $X_0$ and $X_1$ to bounds on the $\textit{information-theoretic}$ indistinguishability of some specific, related variables $\widetilde{X}_0$ and $\widetilde{X}_1$. As a consequence, we prove a new, tight characterization of the number of samples $k$ needed to efficiently distinguish $X_0^{\otimes k}$ and $X_1^{\otimes k}$ with constant advantage as \[ k = \Theta\left(d_H^{-2}\left(\widetilde{X}_0, \widetilde{X}_1\right)\right), \] which is the inverse of the squared Hellinger distance $d_H$ between two distributions $\widetilde{X}_0$ and $\widetilde{X}_1$ that are computationally indistinguishable from $X_0$ and $X_1$. Likewise, our framework can be used to re-derive a result of Halevi and Rabin (TCC 2008) and Geier (TCC 2022), proving nearly-tight bounds on how computational indistinguishability scales with the number of samples for arbitrary product distributions.

翻译：给定一个样本序列 $x_1, \\dots , x_k$，已知其来自两个分布 $X_0$ 或 $X_1$ 之一，统计学中的一个经典问题是判断样本究竟来自哪个分布。从信息论角度看，给定 $k$ 个样本时区分这两个分布的最大优势由 $X_0^{\\otimes k}$ 与 $X_1^{\\otimes k}$ 之间的总变差距离所刻画。然而，当我们将注意力限制在这两个分布的高效区分器（即小型电路）时，精确刻画区分 $X_0^{\\otimes k}$ 和 $X_1^{\\otimes k}$ 的能力则更为复杂且研究较少。本文提出一种通用方法，将 $X_0$ 和 $X_1$ 的计算不可区分性界约化为某些特定相关变量 $\\widetilde{X}_0$ 和 $\\widetilde{X}_1$ 的信息论不可区分性界。由此，我们证明了以恒定优势高效区分 $X_0^{\\otimes k}$ 和 $X_1^{\\otimes k}$ 所需样本数 $k$ 的新紧致特征：\\[ k = \\Theta\\left(d_H^{-2}\\left(\\widetilde{X}_0, \\widetilde{X}_1\\right)\\right), \\] 其中 $d_H$ 表示与 $X_0$ 和 $X_1$ 计算不可区分的两个分布 $\\widetilde{X}_0$ 和 $\\widetilde{X}_1$ 之间的 Hellinger 距离平方的倒数。同样，我们的框架可用于重新推导 Halevi 和 Rabin（TCC 2008）以及 Geier（TCC 2022）的结果，证明任意乘积分布的计算不可区分性随样本数量变化的近紧致界。