Towards a Theory of Extractable Functions

Extractable functions are functions where any adversary that outputs a point in the range of the function is guaranteed to "know" a corresponding preimage. Here, knowledge is captured by the existence of an efficient extractor that recovers the preimage from the internal state of the adversary . Extractability of functions was defined by the authors (ICALP'08) in the context of perfectly one-way functions. It can be regarded as an abstraction from specific knowledge assumptions, such as the Knowledge of Exponent assumption (Hada and Tanaka, Crypto 1998). We initiate a more general study of extractable functions. We explore two different approaches. The first approach is aimed at understanding the concept of extractability in of itself; in particular we demonstrate that a weak notion of extraction implies a strong one, and make rigorous the intuition that extraction and obfuscation are complementary notions. In the second approach, we study the possibility of constructing cryptographic primitives from simpler or weaker ones while maintaining extractability. Results are generally positive. Specifically, we show that several cryptographic reductions are either "knowledge-preserving" or can be modified to be so. Examples include reductions from extractable weak one-way functions to extractable strong ones, from extractable pseudorandom generators to extractable pseudorandom functions, and from extractable one-way functions to extractable commitments. Other questions, such as constructing extractable pseudorandom generators from extractable one way functions, remain open.