Supersimulators
Cynthia Dwork, Pranay Tankala
TL;DR
The paper addresses the challenge of fooling highly powerful distinguishers with simulators whose size does not necessarily scale with the target function. It introduces supersimulators—circuits whose size can depend on the target function yet remain bounded—constructed via iterative regularity techniques adapted from graph theory, and leverages calibrated multiaccuracy to refine the computational indistinguishability of product distributions. Two main results are established: (i) a CMA-based refinement that yields proxied distributions with controllable statistical and computational distances, and (ii) a supersimulator-based refinement that eliminates the remaining complexity gap by allowing the distinguisher size bound to depend on the actual distribution pair. Collectively, these contributions strengthen the complexity-theoretic regularity lemma, extend its applicability to product distributions, and potentially impact cryptography, learning theory, and related areas by enabling robust indistinguishability guarantees even against substantially more powerful distinguishers.
Abstract
We prove that every randomized Boolean function admits a supersimulator: a randomized polynomial-size circuit whose output on random inputs cannot be efficiently distinguished from reality with constant advantage, even by polynomially larger distinguishers. Our result builds on the landmark complexity-theoretic regularity lemma of Trevisan, Tulsiani and Vadhan (2009), which, in contrast, provides a simulator that fools smaller distinguishers. We circumvent lower bounds for the simulator size by letting the distinguisher size bound vary with the target function, while remaining below an absolute upper bound independent of the target function. This dependence on the target function arises naturally from our use of an iteration technique originating in the graph regularity literature. The simulators provided by the regularity lemma and recent refinements thereof, known as multiaccurate and multicalibrated predictors, respectively, as per Hebert-Johnson et al. (2018), have previously been shown to have myriad applications in complexity theory, cryptography, learning theory, and beyond. We first show that a recent multicalibration-based characterization of the computational indistinguishability of product distributions actually requires only (calibrated) multiaccuracy. We then show that supersimulators yield an even tighter result in this application domain, closing a complexity gap present in prior versions of the characterization.