Cryptology ePrint Archive: Report 2021/1334

Indistinguishability Obfuscation from LPN over F_p, DLIN, and PRGs in NC^0

Aayush Jain and Huijia Lin and Amit Sahai

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

{\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.

Category / Keywords: foundations / Indistinguishability Obfuscation, Homomorphic Encryption,

Date: received 2 Oct 2021, last revised 5 Oct 2021

Contact author: aayushjain1728 at gmail com, rachel at cs washington edu, sahai at cs ucla edu

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Version: 20211005:173535 (All versions of this report)

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