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Breaking the Size Barrier: Universal Circuits Meet Lookup Tables

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Advances in Cryptology – ASIACRYPT 2023 (ASIACRYPT 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14438))

Abstract

A Universal Circuit (UC) is a Boolean circuit of size \(\varTheta (n \log n)\) that can simulate any Boolean function up to a certain size n. Valiant (STOC’76) provided the first two UC constructions of asymptotic sizes \(\sim 5 n\log n\) and \(\sim 4.75 n\log n\), and today’s most efficient construction of Liu et al. (CRYPTO’21) has size \(\sim 3n\log n\). Evaluating a public UC with a secure Multi-Party Computation (MPC) protocol allows efficient Private Function Evaluation (PFE), where a private function is evaluated on private data.

Previously, most UC constructions have only been developed for circuits consisting of 2-input gates. In this work, we generalize UCs to simulate circuits consisting of (\(\rho \) \(\rightarrow \) \(\omega \))-Lookup Tables (LUTs) that map \(\rho \) input bits to \(\omega \) output bits. Our LUT-based UC (LUC) construction has an asymptotic size of \(1.5\rho \omega n \log \omega n\) and improves the size of the UC over the best previous UC construction of Liu et al. (CRYPTO’21) by factors 1.12\(\times \)–\(2.18\times \) for common functions. Our results show that the greatest size improvement is achieved for \(\rho =3\) inputs, and it decreases for \(\rho >3\).

Furthermore, we introduce Varying Universal Circuits (VUCs), which reduce circuit size at the expense of leaking the number of inputs \(\rho \) and outputs \(\omega \) of each LUT. Our benchmarks demonstrate that VUCs can improve over the size of the LUC construction by a factor of up to \(1.45\times \).

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Notes

  1. 1.

    UC-based PFE, unlike PFE based on Fully Homomorphic Encryption (FHE) [19, 32], relies primarily on symmetric encryption and involves far less computation.

  2. 2.

    Our code is published under the MIT license at: https://encrypto.de/code/LUC.

  3. 3.

    Note that a Universal Circuit can also compute circuits with less than the specified number of inputs, gates, and outputs by using dummy values with no functionality.

  4. 4.

    In Yao’s garbled circuit protocol [54], the UC’s universal gates can be implemented as garbled tables when the function holder takes over the garbling part.

  5. 5.

    We distinguish between EUGs and nested EUGs as the recursively constructed nested EUGs differ from its first EUG in Liu et al.’s construction [37].

  6. 6.

    An alternative would be to decompose the larger LUTs into multiple smaller ones using Shannon expansion [49].

  7. 7.

    \(\mathbbm {1}\) denotes the vector where each entry is 1.

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Acknowledgements

This project received funding from the ERC under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 850990 PSOTI). It was co-funded by the DFG within SFB 1119 CROSSING/236615297 and GRK 2050 Privacy & Trust/251805230.

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  1. Technical University of Darmstadt, Darmstadt, Germany

    Yann Disser, Daniel Günther, Thomas Schneider, Maximilian Stillger, Arthur Wigandt & Hossein Yalame

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Correspondence to Daniel Günther .

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  1. Nanyang Technological University, Singapore, Singapore

    Jian Guo

  2. Monash University, Melbourne, VIC, Australia

    Ron Steinfeld

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Disser, Y., Günther, D., Schneider, T., Stillger, M., Wigandt, A., Yalame, H. (2023). Breaking the Size Barrier: Universal Circuits Meet Lookup Tables. In: Guo, J., Steinfeld, R. (eds) Advances in Cryptology – ASIACRYPT 2023. ASIACRYPT 2023. Lecture Notes in Computer Science, vol 14438. Springer, Singapore. https://doi.org/10.1007/978-981-99-8721-4_1

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