Generalized and Unified Equivalences between Hardness and Pseudoentropy

Abstract

Pseudoentropy characterizations give quantitatively precise formulations of the relationship between computational hardness and computational randomness. We prove a unified pseudoentropy characterization that generalizes and strengthens previous results in both uniform and nonuniform models of computation. Our characterization applies to a general family of entropy notions, including Shannon entropy and min-entropy as special cases. Moreover, the characterizations for these different entropy notions can be witnessed simultaneously by a single universal function, which captures both computational hardness and computational randomness. A key technical insight is that weight-restricted calibration, from the recent literature on algorithmic fairness, together with standard computational indistinguishability (known as multiaccuracy in the fairness literature), suffices for proving pseudoentropy characterizations for general entropy notions. To obtain this combination of properties, we prove an enhanced version of the Leakage Simulation Lemma (Jetchev and Pietrzak, 2014), which in turn extends the Complexity Theoretic-Regularity Lemma (Trevisan, Tulsiani, and Vadhan, 2009) from boolean functions to ones over a larger alphabet. Our Enhanced Regularity/Leakage-Simulation Lemma enables us to obtain an exponential improvement in the dependence on the alphabet size compared with the pseudoentropy characterizations of Casacuberta, Dwork, and Vadhan (2024), which are based on the stronger notion of multicalibration. We also show that this exponential dependence on the alphabet size is inevitable for multicalibration and even for the weaker notion of calibrated multiaccuracy.