On obfuscating point functions

We investigate the possibility of obfuscating point functions in the framework of Barak et al. from Crypto '01. A point function is a Boolean function that assumes the value 1 at exactly one point. Our main results are as follows:We provide a simple construction of efficient obfuscators for point functions for a slightly relaxed notion of obfuscation, for which obfuscating general circuits is nonetheless impossible. Our construction relies on the existence of a very strong one-way permutation, and yields the first non-trivial obfuscator under general assumptions in the standard model. We also obtain obfuscators for point functions with multi-bit output and for prefix matching.Our assumption is that there is a one-way permutation wherein any polynomial-sized circuit inverts the permutation on at most a polynomial number of inputs. We show that a similar assumption is in fact necessary, and that our assumption holds relative to a random permutation oracle.Finally, we establish two impossibility results which indicate that the limitations on our construction, namely simulating only adversaries with single-bit output and using nonuniform advice in our simulator, are in some sense inherent.Previous work gave negative results for the general class of circuits (Barak et al., Crypto '01) and positive results in the random oracle model (Lynn et al., Eurocrypt '04) or under non-standard number-theoretic assumptions (Canetti, Crypto '97). This work represents the first effort to bridge the gap between the two for a natural class of functionalities.