How to Garble Arithmetic Circuits

Yao\u27s garbled circuit construction transforms a boolean circuit $C:\{0,1\}^n\to\{0,1\}^m$ into a ``garbled circuit\u27\u27 $\hat{C}$ along with $n$ pairs of $k$-bit keys, one for each input bit, such that $\hat{C}$ together with the $n$ keys corresponding to an input $x$ reveal $C(x)$ and no additional information about $x$. The garbled circuit construction is a central tool for constant-round secure computation and has several other applications. Motivated by these applications, we suggest an efficient arithmetic variant of Yao\u27s original construction. Our construction transforms an arithmetic circuit $C : \mathbb{Z}^n\to\mathbb{Z}^m$ over integers from a bounded (but possibly exponential) range into a garbled circuit $\hat{C}$ along with $n$ affine functions $L_i : \mathbb{Z}\to \mathbb{Z}^k$ such that $\hat{C}$ together with the $n$ integer vectors $L_i(x_i)$ reveal $C(x)$ and no additional information about $x$. The security of our construction relies on the intractability of the learning with errors (LWE) problem