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This paper initiates a study of Fine Grained Secure Computation: i.e. the construction of secure computation primitives against “moderately complex” adversaries. We present definitions and constructions for compact Fully Homomorphic Encryption and Verifiable Computation secure against (non-uniform) adversaries. Our results do not require the existence of one-way functions and hold under a widely believed separation assumption, namely . We also present two application scenarios for our model: (i) hardware chips that prove their own correctness, and (ii) protocols against rational adversaries potentially relevant to the Verifier’s Dilemma in smart-contracts transactions such as Ethereum.