Indistinguishability Obfuscation from Well-Founded Assumptions

Abstract

In this work, we show how to construct indistinguishability obfuscation from subexponential hardness of four well-founded assumptions. We prove: Let $τ\in (0,\infty), δ\in (0,1), ε\in (0,1)$ be arbitrary constants. Assume sub-exponential security of the following assumptions, where $λ$ is a security parameter, and the parameters $\ell,k,n$ below are large enough polynomials in $λ$: - The SXDH assumption on asymmetric bilinear groups of a prime order $p = O(2^λ)$, - The LWE assumption over $\mathbb{Z}_{p}$ with subexponential modulus-to-noise ratio $2^{k^ε}$, where $k$ is the dimension of the LWE secret, - The LPN assumption over $\mathbb{Z}_p$ with polynomially many LPN samples and error rate $1/\ell^δ$, where $\ell$ is the dimension of the LPN secret, - The existence of a Boolean PRG in $\mathsf{NC}^0$ with stretch $n^{1+τ}$, Then, (subexponentially secure) indistinguishability obfuscation for all polynomial-size circuits exists.