Indistinguishability Obfuscation with Non-trivial Efficiency

It is well known that inefficient indistinguishability obfuscators $$\mathbf{iO } $$ with running time $${{\mathrm{poly}}}|C|,\lambda \cdot 2^n$$ , where C is the circuit to be obfuscated, $$\lambda $$ is the security parameter, and n is the input length of C, exists unconditionally: simply output the function table of C i.e., the output of C on all possible inputs. Such inefficient obfuscators, however, are not useful for applications. We here consider $$\mathbf{iO } $$ with a slightly "non-trivial" notion of efficiency: the running-time of the obfuscator may still be "trivial" namely, $${{\mathrm{poly}}}|C|,\lambda \cdot 2^{n}$$ , but we now require that the obfuscated code is just slightly smaller than the truth table of C namely $${{\mathrm{poly}}}|C|,\lambda \cdot 2^{n1-\epsilon }$$ , where $$\epsilon >0$$ ; we refer to this notion as iOwith exponential efficiency, or simply exponentially-efficientiOXio. We show that, perhaps surprisingly, under the subexponential LWE assumption, subexponentially-secure XiO for polynomial-size circuits implies polynomial-time computable iO for all polynomial-size circuits.