**Indistinguishability Obfuscation Without Multilinear Maps: iO from LWE, Bilinear Maps, and Weak Pseudorandomness**

*Prabhanjan Ananth and Aayush Jain and Amit Sahai*

**Abstract: **The existence of secure indistinguishability obfuscators (iO) has far-reaching implications, significantly expanding the scope of problems amenable to cryptographic study. All known approaches to
constructing iO rely on $d$-linear maps which allow the encoding of elements from a large domain, evaluating degree $d$ polynomials on them, and testing if the output is zero. While secure bilinear maps are well established in cryptographic literature, the security of candidates for $d>2$ is poorly understood.

We propose a new approach to constructing iO for general circuits. Unlike all previously known realizations of iO, we avoid the use of $d$-linear maps of degree $d \ge 3$.

At the heart of our approach is the assumption that a new weak pseudorandom object exists, that we call a perturbation resilient generator ($\Delta\mathsf{RG}$). Informally, a $\Delta\mathsf{RG}$ maps $n$ integers to $m$ integers, and has the property that for any sufficiently short vector $a \in \mathbb{Z}^m$, all efficient adversaries must fail to distinguish the distributions $\Delta\mathsf{RG}(s)$ and ($\Delta\mathsf{RG}(s)+a$), with at least some probability that is inverse polynomial in the security parameter. $\Delta\mathsf{RG}$s have further implementability requirements; most notably they must be computable by a family of degree-3 polynomials over $\mathbb{Z}$. We use techniques building upon the Dense Model Theorem to deal with adversaries that have nontrivial but non-overwhelming distinguishing advantage. In particular, we obtain a new security amplification theorem for functional encryption.

As a result, we obtain iO for general circuits assuming:

\begin{itemize}

\item Subexponentially secure LWE

\item Bilinear Maps

\item $\poly(\lambda)$-secure 3-block-local PRGs

\item $(1-1/\poly(\lambda))$-secure $\Delta\mathsf{RG}$s

\end{itemize}

**Category / Keywords: **Indistinguishability Obfuscation

**Date: **received 17 Jun 2018, last revised 24 Dec 2018

**Contact author: **prabhanjan va at gmail com, aayushjainiitd at gmail com, sahai at cs ucla edu

**Available format(s): **PDF | BibTeX Citation

**Version: **20181225:014335 (All versions of this report)

**Short URL: **ia.cr/2018/615

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