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Efficient Two-Party Exponentiation from Quotient Transfer

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Applied Cryptography and Network Security (ACNS 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13269))

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Abstract

Secure multi-party computation (MPC) allows participating parties to jointly compute a function over their inputs while keeping them private. In particular, MPC based on additive secret sharing has been widely studied as a tool to obtain efficient protocols secure against a dishonest majority, including the important two-party case. In this paper, we propose a two-party protocol for an exponentiation functionality based on an additive secret sharing scheme. Our proposed protocol aims to securely compute a public base exponentiation \(a^x \mathrm {mod} \,~p\) for some prime p, where the exponent \(x \in \mathbb {Z}_p\) is a (shared) secret and the base \(a \in \mathbb {Z}_p\) is public. Our protocol is based on a new simple but efficient approach involving quotient transfer that allows the parties to perform the most expensive part of the computation locally, and requires 3 rounds and 4 invocations of multiplication. As an intermediate primitive for our efficient two-party exponentiation protocol, we propose an efficient modulus conversion protocol. This protocol might be of independent interest.

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Notes

  1. 1.

    Since the multiplication MPC protocol is dominant in the communication, the communication complexity of MPC is usually measured by the number of invocation of multiplication.

  2. 2.

    To construct an exponentiation protocol over additive secret sharing, we could consider utilizing share conversion between Shamir and Additive secret sharing. However, [AAN18] additionally assumes the base and the exponent are shared by different moduli, which implies an additional modulus conversion is needed. These aspects make this approach more expensive.

  3. 3.

    QT was implicitly defined by [KIM+18]. In addition, in [OWIO19], a part of their protocol can be seen as a QT protocol based on bit-decomposition, even though they did not directly highlight this as a QT protocol.

  4. 4.

    In our actual exponentiation protocol given in Algorithm 1, each party locally sets the shares \([y_i]^i_p = y_i\) and \([y_i]^{1-i}_p = 0\) (and does not send their shares to each other) in order to optimize the round complexity.

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Acknowledgements

A part of this work was supported by JST SPRING JPMJSP2106, JST OPERA JPMJOP1612, JST CREST JPMJCR2113, JSPS KAKENHI JP19H01109, JP21H04879, and MIC JPJ000254.

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Authors and Affiliations

  1. Tokyo Institute of Technology, Tokyo, Japan

    Yi Lu & Keisuke Tanaka

  2. National Institute of Advanced Industrial Science and Technology (AIST), Tokyo, Japan

    Yi Lu, Keisuke Hara, Kazuma Ohara & Jacob Schuldt

  3. Yokohama National University, Yokohama, Japan

    Keisuke Hara

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  1. Yi Lu
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Correspondence to Yi Lu .

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  1. Stevens Institute of Technology, Hoboken, NJ, USA

    Giuseppe Ateniese

  2. Sapienza University of Rome, Rome, Italy

    Daniele Venturi

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Lu, Y., Hara, K., Ohara, K., Schuldt, J., Tanaka, K. (2022). Efficient Two-Party Exponentiation from Quotient Transfer. In: Ateniese, G., Venturi, D. (eds) Applied Cryptography and Network Security. ACNS 2022. Lecture Notes in Computer Science, vol 13269. Springer, Cham. https://doi.org/10.1007/978-3-031-09234-3_32

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  • DOI: https://doi.org/10.1007/978-3-031-09234-3_32

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