In this work, we study what minimal sets of assumptions suffice for constructing indistinguishability obfuscation ($imathcal{O}$). We prove:

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{bf Theorem}(Informal): Assume sub-exponential security of the following assumptions:

– the Learning Parity with Noise ($mathsf{LPN}$) assumption over general prime fields $mathbb{F}_p$ with
polynomially many $mathsf{LPN}$ samples and error rate $1/k^delta$, where $k$ is the dimension of the $mathsf{LPN}$ secret, and $delta>0$ is any constant;

– the existence of a Boolean Pseudo-Random Generator ($mathsf{PRG}$) in $mathsf{NC}^0$ with stretch $n^{1+tau}$, where $n$ is the length of the $mathsf{PRG}$ seed, and $tau>0$ is any constant;

– the Decision Linear ($mathsf{DLIN}$) assumption on symmetric bilinear groups of prime order.

Then, (subexponentially secure) indistinguishability obfuscation for all polynomial-size circuits exists. Further, assuming only polynomial security of the aforementioned assumptions, there exists collusion resistant public-key functional encryption for all polynomial-size circuits.}

This removes the reliance on the Learning With Errors (LWE) assumption from the recent work of [Jain, Lin, Sahai STOC’21]. As a consequence, we obtain the first fully homomorphic encryption scheme that does not rely on any lattice-based hardness assumption.

Our techniques feature a new notion of randomized encoding called Preprocessing Randomized Encoding (PRE) that, essentially, can be computed in the exponent of pairing groups. When combined with other new techniques, PRE gives a much more streamlined construction of $iO$ while still maintaining reliance only on well-studied assumptions.

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