We revisit previous formulations of zero knowledge in the random oracle model due to Bellare and Rogaway (CCS '93) and Pass (Crypto '03), and present a hierarchy for zero knowledge that includes both of these formulations. The hierarchy relates to the programmability of the random oracle, previously studied by Nielsen (Crypto '02). 1. We establish a subtle separation between the Bellare-Rogaway formulation and a weaker formulation, which yields a finer distinction than the separation in Nielsen's work. 2. We show that zero-knowledge according to each of these formulations is not preserved under sequential composition. We introduce stronger definitions wherein the adversary may receive auxiliary input that depends on the random oracle (as in Unruh (Crypto '07)) and establish closure under sequential composition for these definitions. We also present round-optimal protocols for NP satisfying the stronger requirements. 3. Motivated by our study of zero knowledge, we introduce a new definition of proof of knowledge in the random oracle model that accounts for oracle-dependent auxiliary input. We show that two rounds of interaction are necessary and sufficient to achieve zero-knowledge proofs of knowledge according to this new definition, whereas one round of interaction is sufficient in previous definitions. 4. Extending our work on zero knowledge, we present a hierarchy for circuit obfuscation in the random oracle model, the weakest being that achieved in the work of Lynn, Prabhakaran and Sahai (Eurocrypt '04). We show that the stronger notions capture precisely the class of circuits that is efficiently and exactly learnable under membership queries.
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