Differing-inputs obfuscation (diO) ⇒ iO

Fully-black-box reduction · standard model · folklore

Statement

A differing-inputs obfuscator for a circuit class is an indistinguishability obfuscator for the same class, with the same advantage — folklore.

Sketch

Functionally equivalent circuits $C_0,C_1$ have no differing input, so the sampler that always outputs $(C_0,C_1)$ is differing-inputs-hard, and the diO guarantee for it is the iO guarantee for the pair.

Notes

• The converse is known only in restricted form: iO for a class implies diO for pairs of circuits in the class that differ on at most polynomially many inputs — BCP14.
• If a special-purpose obfuscator exists for a specific circuit family, general-purpose diO with arbitrary auxiliary input does not exist — GGHW14.