Adaptively Secure Garbled Circuits from One-Way Functions

A garbling scheme is used to garble a circuit C and an input x in a way that reveals the output C(x) but hides everything else. In many settings, the circuit can be garbled off-line without strict efficiency constraints, but the input must be garbled very efficiently on-line, with much lower complexity than evaluating the circuit. Yao's garbling scheme [31] has essentially optimal on-line complexity, but only achieves selective security, where the adversary must choose the input x prior to seeing the garbled circuit. It has remained an open problem to achieve adaptive security, where the adversary can choose x after seeing the garbled circuit, while preserving on-line efficiency. In this work, we modify Yao's scheme in a way that allows us to prove adaptive security under one-way functions. In our main instantiation we achieve on-line complexity only proportional to the width w of the circuit. Alternatively we can also get an instantiation with on-line complexity only proportional to the depth d (and the output size) of the circuit, albeit incurring in a 2(O(d)) security loss in our reduction. More broadly, we relate the on-line complexity of adaptively secure garbling schemes in our framework to a certain type of pebble complexity of the circuit. As our main tool, of independent interest, we develop a new notion of somewhere equivocal encryption, which allows us to efficiently equivocate on a small subset of the message bits.