Obfuscation of Probabilistic Circuits and Applications

This paper studies the question of how to define, construct, and use obfuscators for

probabilistic programs

. Such obfuscators compile a possibly randomized program into a

deterministic

one, which achieves computationally indistinguishable behavior from the original program as long as it is run on each input at most once. For obfuscation, we propose a notion that extends

indistinguishability obfuscation

to probabilistic circuits: It should be hard to distinguish between the obfuscations of any two circuits whose output distributions at each input are computationally indistinguishable, possibly in presence of some auxiliary input. We call the resulting notion

probabilistic indistinguishability obfuscation (

pIO

).

We define several variants of

pIO

, and study relations among them. Moreover, we give a construction of one of our variants, called

X

-

pIO

, from sub-exponentially hard indistinguishability obfuscation (for deterministic circuits) and one-way functions.

We then move on to show a number of applications of

pIO

. In particular, we first give a general and natural methodology to achieve fully homomorphic encryption (FHE) from variants of

pIO

and of semantically secure encryption schemes. In particular, one instantiation leads to FHE from any

X

-

pIO

obfuscator and any re-randomizable encryption scheme that’s slightly super-polynomially secure.

We note that this constitutes the first construction of full-fledged FHE that does not rely on encryption with circular security.

Moreover, assuming sub-exponentially secure puncturable PRFs computable in

NC

1

, sub-exponentially-secure indistinguishability obfuscation for (deterministic)

NC

1

circuits can be bootstrapped to obtain indistinguishability obfuscation for arbitrary (deterministic) poly-size circuits (previously such bootstrapping was known only assuming FHE with

NC

1

decryption algorithm).