FE and iO for Turing Machines from Minimal Assumptions.

We construct Indistinguishability Obfuscation (\(\mathsf {iO}\)) and Functional Encryption (\(\mathsf {FE}\)) schemes in the Turing machine model from the minimal assumption of compact \(\mathsf {FE}\) for circuits (\(\mathsf {CktFE}\)). Our constructions overcome the barrier of sub-exponential loss incurred by all prior work. Our contributions are: 1. We construct \(\mathsf {iO}\) in the Turing machine model from the same assumptions as required in the circuit model, namely, sub-exponentially secure \(\mathsf {FE}\) for circuits. The previous best constructions [6, 41] require sub-exponentially secure \(\mathsf {iO}\) for circuits, which in turn requires sub-exponentially secure \(\mathsf {FE}\) for circuits [5, 15]. 2. We provide a new construction of single input \(\mathsf {FE}\) for Turing machines with unbounded length inputs and optimal parameters from polynomially secure, compact \(\mathsf {FE}\) for circuits. The previously best known construction by Ananth and Sahai [7] relies on \(\mathsf {iO}\) for circuits, or equivalently, sub-exponentially secure \(\mathsf {FE}\) for circuits. 3. We provide a new construction of multi-input \(\mathsf {FE}\) for Turing machines. Our construction supports a fixed number of encryptors (say k), who may each encrypt a string \(\mathbf {x}_i\) of unbounded length. We rely on sub-exponentially secure \(\mathsf {FE}\) for circuits, while the only previous construction [10] relies on a strong knowledge type assumption, namely, public coin differing inputs obfuscation.