Paper 2018/973

How to leverage hardness of constant-degree expanding polynomials over $\mathbb{R}$ to build iO

Aayush Jain and Amit Sahai


In this work, we introduce and construct $D$-restricted Functional Encryption (FE) for any constant $D \ge 3$, based only on the SXDH assumption over bilinear groups. This generalizes the notion of $3$-restricted FE recently introduced and constructed by Ananth et al. (ePrint 2018) in the generic bilinear group model. A $D=(d+2)$-restricted FE scheme is a secret key FE scheme that allows an encryptor to efficiently encrypt a message of the form $M=(\vec{x},\vec{y},\vec{z})$. Here, $\vec{x}\in \F_{\prm}^{d\times n}$ and $\vec{y},\vec{z}\in \F_{\prm}^n$. Function keys can be issued for a function $f=\Sigma_{\vec{I}= (i_1,..,i_d,j,k)}\ c_{\vec{I}}\cdot \vec{x}[1,i_1] \cdots \vec{x}[d,i_d] \cdot \vec{y}[j]\cdot \vec{z}[k]$ where the coefficients $c_{\vec{I}}\in \F_{\prm}$. Knowing the function key and the ciphertext, one can learn $f(\vec{x},\vec{y},\vec{z})$, if this value is bounded in absolute value by some polynomial in the security parameter and $n$. The security requirement is that the ciphertext hides $\vec{y}$ and $\vec{z}$, although it is not required to hide $\vec{x}$. Thus $\vec{x}$ can be seen as a public attribute. $D$-restricted FE allows for useful evaluation of constant-degree polynomials, while only requiring the SXDH assumption over bilinear groups. As such, it is a powerful tool for leveraging hardness that exists in constant-degree expanding families of polynomials over $\mathbb{R}$. In particular, we build upon the work of Ananth et al. to show how to build indistinguishability obfuscation (iO) assuming only SXDH over bilinear groups, LWE, and assumptions relating to weak pseudorandom properties of constant-degree expanding polynomials over $\mathbb{R}$.

Available format(s)
Publication info
Preprint. MINOR revision.
Indistinguishability Obfuscation
Contact author(s)
aayushjain @ cs ucla edu
sahai @ cs ucla edu
2018-10-15: received
Short URL
Creative Commons Attribution


      author = {Aayush Jain and Amit Sahai},
      title = {How to leverage hardness of constant-degree expanding polynomials over $\mathbb{R}$ to build iO},
      howpublished = {Cryptology ePrint Archive, Paper 2018/973},
      year = {2018},
      note = {\url{}},
      url = {}
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