Characterizing the Distinguishability of Product Distributions through Multicalibration

TL;DR

The work presents a complexity-theoretic framework to compare computational indistinguishability of product distributions with their information-theoretic counterparts via multicalibration. It constructs intermediate distributions $\widetilde{X}_0,\widetilde{X}_1$ (and variants like $\widehat{X}_0$) through multicalibrated partitions, converting computational questions into statistical ones and yielding an instance-optimal characterization $k = Θ\left(d_H^{-2}(\widetilde{X}_0,\widetilde{X}_1)\right)$ for constant advantage. The framework recovers and clarifies prior results (e.g., Halevi-Rabin and Geier) while introducing a pseudo-Hellinger distance to quantify computational distinguishability in terms of information-theoretic distance. By linking efficient distinguishers to the partition-based representations and showing how to implement optimal distinguishers with small circuits, the paper provides a unified toolkit for analyzing distinguishing problems across general product distributions. The approach highlights a powerful translation from computational hardness to statistical indistinguishability, with potential implications for hardness amplification and cryptographic constructions.

Abstract

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 \[ k = Θ\left(d_H^{-2}\left(\widetilde{X}_0, \widetilde{X}_1\right)\right), \] 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.

Theorems & Definitions (65)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Definition 1.4
• Theorem 1.5
• Definition 1.6
• Theorem 1.7
• Definition 1.8
• Definition 1.9
• Definition 1.10