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.

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与有根据的假设的不可区分性混淆

在这项工作中，我们展示了如何从四个有充分根据的假设的次指数硬度构建不可区分性混淆。我们证明：令 $\tau \in (0,\infty), \delta \in (0,1), \epsilon \in (0,1)$ 为任意常数。假设以下假设的次指数安全性，其中 $\lambda$ 是安全参数，下面的参数 $\ell,k,n$ 是 $\lambda$ 中足够大的多项式： - 非对称双线性群上的 SXDH 假设质数阶 $p = O(2^\lambda)$, - 对 $\mathbb{Z}_{p}$ 的 LWE 假设，具有次指数模噪比 $2^{k^\epsilon}$，其中 $k$ 是 LWE 秘密的维度， - 对 $\mathbb{Z}_p$ 的 LPN 假设具有多项式多个 LPN 样本和错误率 $1/\ell^\delta$，其中 $\ell$ 是维度LPN 的秘密，