Precise Cryptography

Precise zero knowledge guarantees that the view of any verifier V can be simulated in timeclosely related to the actual (as opposed to worst-case) time spent by V in the generated view.We generalize this notion in two ways:1. We provide definitions and constructions of precise encryption, precise proofs of knowledgeand precise secure computation.2. We introduce relaxed notions of precise zero knowledge—which relate the expectation(or higher moments) of the running time of the simulation with the expectation (higher-moments) of the running-time of the verifier—and provide conditions under which thesenotions are implied by the traditional (non-precise) notion of zero knowledge.