The notion of differing-inputs obfuscation (diO) was introduced by Barak et al. (CRYPTO 2001). It guarantees that for any two circuits $C_0, C_1$, for which it is difficult to use their description and come up with an input $x$ on which they differ $C_0(x) \\neq C_1(x)$, it should also be difficult to distinguish the obfuscation of $C_0$ from that of $C_1$. This is a strengthening of indistinguishability obfuscation, where the above is only guaranteed for circuits that agree on all inputs: $C_0(x) = C_1(x)$ for all $x$. Two recent works of Ananth et al. (ePrint 2013) and Boyle et al. (TCC 2014) study the notion of diO in the setting where the attacker is also given some auxiliary information related to the circuits, showing that this notion leads to many interesting applications. One of these applications is extractable witness encryption, defined by Goldwasswer et al. (CRYPTO \'13), which in turn leads to constructions of functional encryption and reusable garbling schemes that work directly on Turing Machines rather than circuits.
In this work, we show that the existence of general-purpose diO with general auxiliary input leads to a surprising ``implausible\'\' consequence: in particular, it would show the impossibility of obfuscating a specific circuit $C^*$ with specific auxiliary input $\\aux^*$ in a way that hides some specific information. In particular, we put forth the assumption that such special-purpose obfuscation exists, and under this assumption, we show that general-purpose diO does not exist. This special-purpose obfuscation assumption isn\'t implied by diO itself and hence we do not get an unconditional impossibility result. However, the special-purpose obfuscation assumption is a falsifiable assumption which we do not know how to break for candidate obfuscation schemes. Showing the existence of general-purpose diO with general auxiliary input would necessitate showing how to break this assumption. We also give a similar result showing the impossibility of extractable witness encryption under the same special-purpose obfuscation assumption.