Fine-Grained Cryptography

Fine-grained cryptographic primitives are ones that are secure against adversaries with an a-priori bounded polynomial amount of resources time, space or parallel-time, where the honest algorithms use less resources than the adversaries they are designed to fool. Such primitives were previously studied in the context of time-bounded adversaries Merkle, CACM 1978, space-bounded adversaries Cachin and Maurer, CRYPTO 1997 and parallel-time-bounded adversaries Hastad, IPL 1987. Our goal is come up with fine-grained primitives in the setting of parallel-time-bounded adversaries and to show unconditional security of these constructions when possible, or base security on widely believed separation of worst-case complexity classes. We show:1. $${\textsf {NC}^{1}}$$ -cryptography: Under the assumption that [InlineEquation not available: see fulltext.], we construct one-way functions, pseudo-random generators with sub-linear stretch, collision-resistant hash functions and most importantly, public-key encryption schemes, all computable in $${\textsf {NC}^{1}}$$ and secure against all $${\textsf {NC}^{1}}$$ circuits. Our results rely heavily on the notion of randomized encodings pioneered by Applebaum, Ishai and Kushilevitz, and crucially, make non-black-box use of randomized encodings for logspace classes.2. $${\textsf {AC}^{0}}$$ -cryptography: We construct unconditionally secure pseudo-random generators with arbitrary polynomial stretch, weak pseudo-random functions, secret-key encryption and perhaps most interestingly, collision-resistant hash functions, computable in $${\textsf {AC}^{0}}$$ and secure against all $${\textsf {AC}^{0}}$$ circuits. Previously, one-way permutations and pseudo-random generators with linear stretch computable in $${\textsf {AC}^{0}}$$ and secure against $${\textsf {AC}^{0}}$$ circuits were known from the works of Hastad and Braverman.