Paper 2023/501

New Ways to Garble Arithmetic Circuits

Marshall Ball, New York University
Hanjun Li, University of Washington
Huijia Lin, University of Washington
Tianren Liu, Peking University

The beautiful work of Applebaum, Ishai, and Kushilevitz [FOCS'11] initiated the study of arithmetic variants of Yao's garbled circuits. An arithmetic garbling scheme is an efficient transformation that converts an arithmetic circuit $C: \mathcal{R}^n \rightarrow \mathcal{R}^m$ over a ring $\mathcal{R}$ into a garbled circuit $\widehat C$ and $n$ affine functions $L_i$ for $i \in [n]$, such that $\widehat C$ and $L_i(x_i)$ reveals only the output $C(x)$ and no other information of $x$. AIK presented the first arithmetic garbling scheme supporting computation over integers from a bounded (possibly exponentially large) range, based on Learning With Errors (LWE). In contrast, converting $C$ into a Boolean circuit and applying Yao's garbled circuit treats the inputs as bit strings instead of ring elements, and hence is not "arithmetic". In this work, we present new ways to garble arithmetic circuits, which improve the state-of-the-art on efficiency, modularity, and functionality. To measure efficiency, we define the rate of a garbling scheme as the maximal ratio between the bit-length of the garbled circuit $|\widehat C|$ and that of the computation tableau $|C|\ell$ in the clear, where $\ell$ is the bit length of wire values (e.g., Yao's garbled circuit has rate $O(\lambda)$). $\bullet$ We present the first constant-rate arithmetic garbled circuit for computation over large integers based on the Decisional Composite Residuosity (DCR) assumption, significantly improving the efficiency of the schemes of Applebaum, Ishai, and Kushilevitz. $\bullet$ We construct an arithmetic garbling scheme for modular computation over $\mathcal{R} = \mathbb{Z}_p$ for any integer modulus $p$, based on either DCR or LWE. The DCR-based instantiation achieves rate $O(\lambda)$ for large $p$. Furthermore, our construction is modular and makes black-box use of the underlying ring and a simple key extension gadget. $\bullet$ We describe a variant of the first scheme supporting arithmetic circuits over bounded integers that are augmented with Boolean computation (e.g., truncation of an integer value, and comparison between two values), while keeping the constant rate when garbling the arithmetic part. To the best of our knowledge, constant-rate (Boolean or arithmetic) garbling was only achieved before using the powerful primitive of indistinguishability obfuscation, or for restricted circuits with small depth.

Available format(s)
Publication info
A major revision of an IACR publication in EUROCRYPT 2023
garbled circuitssecure computationarithmetic circuits
Contact author(s)
marshall @ cs nyu edu
hanjul @ cs washington edu
rachel @ cs washington edu
trl @ pku edu cn
2023-04-07: approved
2023-04-06: received
See all versions
Short URL
Creative Commons Attribution


      author = {Marshall Ball and Hanjun Li and Huijia Lin and Tianren Liu},
      title = {New Ways to Garble Arithmetic Circuits},
      howpublished = {Cryptology ePrint Archive, Paper 2023/501},
      year = {2023},
      note = {\url{}},
      url = {}
Note: In order to protect the privacy of readers, does not use cookies or embedded third party content.