Indistinguishability Obfuscation for Turing Machines: Constant Overhead and Amortization

We study the asymptotic efficiency of indistinguishability obfuscation (\(\mathsf {i}\mathcal {O}\)) on two fronts: Obfuscation size: Present constructions of indistinguishability obfuscation (\(\mathsf {i}\mathcal {O}\)) create obfuscated programs where the size of the obfuscated program is at least a multiplicative factor of security parameter larger than the size of the original program. In this work, we construct the first \(\mathsf {i}\mathcal {O}\) scheme for (bounded-input) Turing machines that achieves only a constant multiplicative overhead in size. The constant in our scheme is, in fact, 2. Amortization: Suppose we want to obfuscate an arbitrary polynomial number of (bounded-input) Turing machines \(M_1,\ldots ,M_n\). We ask whether it is possible to obfuscate \(M_1,\ldots ,M_n\) using a single application of an \(\mathsf {i}\mathcal {O}\) scheme for a circuit family where the size of any circuit is independent of n as well the size of any Turing machine \(M_i\). In this work, we resolve this question in the affirmative, obtaining a new bootstrapping theorem for obfuscating arbitrarily many Turing machines.