On Treewidth, Separators and Yao’s Garbling

Abstract

We show that Yao’s garbling scheme is adaptively indistinguishable for the class of Boolean circuits of size \(S\) and treewidth \(w\) with only a \({S^{O({w})}}\) loss in security. For instance, circuits with constant treewidth are as a result adaptively indistinguishable with only a polynomial loss. This (partially) complements a negative result of Applebaum et al. (Crypto 2013), which showed (assuming one-way functions) that Yao’s garbling scheme cannot be adaptively simulatable. As main technical contributions, we introduce a new pebble game that abstracts out our security reduction and then present a pebbling strategy for this game where the number of pebbles used is roughly \({O{(\delta w\log (S))}}\), \(\delta \) being the fan-out of the circuit. The design of the strategy relies on separators, a graph-theoretic notion with connections to circuit complexity.