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Actively Secure Setup for SPDZ

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Abstract

We present the first actively secure, practical protocol to generate the distributed secret keys needed in the SPDZ offline protocol. As an added bonus our protocol results in the resulting distribution of the public and secret keys are such that the associated SHE ‘noise’ analysis is the same as if the distributed keys were generated by a trusted setup. We implemented the presented protocol for distributed BGV key generation within the SCALE-MAMBA  framework. Our method makes use of a new method for creating doubly (or even more) authenticated bits in different MPC engines, which has applications in other areas of MPC-based secure computation. We were able to generate keys for two parties and a plaintext size of 64 bits in around 5 min, and a little more than 18 min for a 128-bit prime.

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Notes

  1. We use SCALE-MAMBA as a reference work throughout this paper as it gives a fixed target (including key sizes) for the final distributed keys we are trying to produce.

  2. Carsten Baum has pointed out that we can remove this reduction to the subset-sum by increasing, in some (important) cases, the number of bits we throw away. This however results in a less efficient protocol; thus, we rely on the Multiple Subset-Sum Problem to obtain an efficient protocol. As our focus is primarily on trying to obtain as efficient a protocol as possible we prefer to keep the reliance on the MSSP problem.

  3. If the underlying MPC system is SPDZ based, then a more efficient way to perform the method is using the FHE pre-processing instead of directly within the Offline phase as implied by the given protocol. But this assumes the pre-processing is FHE based, which it will not be in our application.

  4. In our security proof we show that this specific information can be perfectly simulated by the simulator and leaks no information about the actual shared value.

  5. The density of a standard subset sum problem is given by \(d = \frac{\nu }{\max _i \log a_i}\).

  6. See the proof of the theorem below.

  7. Note that \(\gamma =1\) since \(p_0, p_1\) are both big.

  8. Of course in practice we generate the secure bits in batches and hence this is just the minimal number of rounds required.

  9. Our implementations are now included in the SCALE-MAMBA code-base.

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Acknowledgements

The authors would like to thank Carsten Baum and Emmanuela Orsini for suggestions in relation to the work in this paper and Claudio Orlandi in elaborating on the early history of the BDOZ and Tiny-OT work. The work in this paper was carried out, while Dragos Rotaru and Tim Wood were PhD students at the University of Bristol and were employed by KU Leuven. This work has been supported in part by ERC Advanced Grant ERC-2015-AdG-IMPaCT, by the Defense Advanced Research Projects Agency (DARPA) and Space and Naval Warfare Systems Center, Pacific (SSC Pacific) under contract No. N66001-15-C-4070 and FA8750-19-C-0502, by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA) via Contract No. 2019-1902070006, by the FWO under an Odysseus project GOH9718N and by CyberSecurity Research Flanders with reference number VR20192203. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the ERC, ODNI, US Air Force, IARPA, DARPA, the US Government or FWO. The US Government is authorised to reproduce and distribute reprints for governmental purposes notwithstanding any copyright annotation therein.

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Authors and Affiliations

  1. Cape Privacy, New York, NY, USA

    Dragos Rotaru

  2. imec-COSIC, KU Leuven, Leuven, Belgium

    Dragos Rotaru, Nigel P. Smart, Titouan Tanguy, Frederik Vercauteren & Tim Wood

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  1. Dragos Rotaru
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Correspondence to Nigel P. Smart.

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Communicated by Jonathan Katz.

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Rotaru, D., Smart, N.P., Tanguy, T. et al. Actively Secure Setup for SPDZ. J Cryptol 35, 5 (2022). https://doi.org/10.1007/s00145-021-09416-w

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