Fine-Grained Cryptography

Fine-grained cryptographic primitives are ones that are secure against adversaries with an apriori 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 (H̊astad, 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. NC-cryptography: Under the assumption that NC 6= ⊕L/poly, 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 NC and secure against all NC circuits. Our results rely heavily on the notion of randomized encodings pioneered by Applebaum, Ishai and Kushilevitz, and crucially, make non-blackbox use of randomized encodings for logspace classes. 2. AC-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 AC and secure against all AC circuits. Previously, one-way permutations and pseudo-random generators (with linear stretch) computable in AC and secure against AC circuits were known from the works of H̊astad and Braverman. ∗MIT. E-mail: {akshayd,vinodv,prashvas}@mit.edu. Research supported in part by NSF Grants CNS-1350619 and CNS-1414119, Alfred P. Sloan Research Fellowship, Microsoft Faculty Fellowship, the NEC Corporation, a Steven and Renee Finn Career Development Chair from MIT. This work was also sponsored in part by the Defense Advanced Research Projects Agency (DARPA) and the U.S. Army Research Office under contracts W911NF-15-C-0226.