Indistinguishability Obfuscation from Well-Founded Assumptions

TL;DR

This paper demonstrates how to construct indistinguishability obfuscation based on four well-founded assumptions, including LWE, LPN, SXDH, and a Boolean PRG, all under subexponential hardness, advancing cryptographic obfuscation theory.

Contribution

It introduces a new construction of indistinguishability obfuscation from four specific subexponentially hard assumptions, expanding the theoretical foundation of cryptographic obfuscation.

Findings

Indistinguishability obfuscation can be achieved from four well-founded assumptions.

The assumptions include SXDH, LWE, LPN, and a Boolean PRG in NC^0.

The construction is secure under subexponential hardness assumptions.

Abstract

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