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Improved Secure Integer Comparison via Homomorphic Encryption

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Topics in Cryptology – CT-RSA 2020 (CT-RSA 2020)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 12006))

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Abstract

Secure integer comparison has been one of the first problems introduced in cryptography, both for its simplicity to describe and for its applications. The first formulation of the problem was to enable two parties to compare their inputs without revealing the exact value of those inputs, also called the Millionaires’ problem [45]. The recent rise of fully homomorphic encryption has given a new formulation to this problem. In this new setting, one party blindly computes an encryption of the boolean \((a<b)\) given only ciphertexts encrypting a and b.

In this paper, we present new solutions for the problem of secure integer comparison in both of these settings. The underlying idea for both schemes is to avoid decomposing the integers in binary in order to improve the performances. On the one hand, our fully homomorphic based solution is inspired by [9], and makes use of the fast bootstrapping techniques developed by [12, 14, 23] to obtain scalability for large integers while preserving high efficiency. On the other hand, our solution to the original Millionaires’ problem is inspired by the protocol of [10], based on partially homomorphic encryption. We tweak their protocol in order to minimize the number of interactions required, while preserving the advantage of comparing non-binary integers.

Both our techniques provide efficient solutions to the problem of secure integer comparison for large (even a-priori unbounded in our first scenario) integers with minimum interactions.

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Notes

  1. 1.

    This is not exactly the original LWE definition since we here consider binary coefficients for the secret key, as in [9, 12, 13]. Nevertheless, we will still refer to it as LWE for sake of simplicity.

  2. 2.

    The binary sign can be obtained by applying the techniques of [9] instead of our ternary sign in the last step.

  3. 3.

    We will provide more details on the parameters in Sect. 4.1.

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Acknowledgements

This work is supported by the European Union PROMETHEUS project (Horizon 2020 Research and Innovation Program, grant 780701) and PAPAYA project (Horizon 2020 Innovation Program, grant 786767). The authors are also grateful for the support of the ANR through project ANR-16-CE39-0014 PERSOCLOUD.

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Authors and Affiliations

  1. Orange Labs, Applied Crypto Group, Cesson-Sévigné, France

    Florian Bourse & Olivier Sanders

  2. Orange Labs, Applied Crypto Group, Caen, France

    Jacques Traoré

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  1. Florian Bourse
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  2. Olivier Sanders
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Correspondence to Olivier Sanders .

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  1. University of California, Irvine, CA, USA

    Stanislaw Jarecki

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Bourse, F., Sanders, O., Traoré, J. (2020). Improved Secure Integer Comparison via Homomorphic Encryption. In: Jarecki, S. (eds) Topics in Cryptology – CT-RSA 2020. CT-RSA 2020. Lecture Notes in Computer Science(), vol 12006. Springer, Cham. https://doi.org/10.1007/978-3-030-40186-3_17

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