Deterministic Circuit Range Avoidance is (Likely) Intractable

Abstract

Circuit Range Avoidance (denoted Avoid) is a computational problem where, given a Boolean circuit with more output bits than input bits, one must output a string outside of the range of the circuit. A simple counting argument implies that such a string must always exist and also guarantees that outputting a uniformly random string is correct with good probability. A natural question is whether this can be derandomized: does there exist an efficient deterministic algorithm for Avoid? We give the first evidence that deterministically solving Avoid is intractable. We show that there is no polynomial-time algorithm for Avoid under plausible assumptions in complexity theory and cryptography. Specifically, our assumptions are that NP ≠ coNP and that subexponentially-secure indistinguishability obfuscation exists.