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Communicationless Evaluation of Quadratic Functions over Secret Shared Dynamic Database

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Abstract

One of the most active fields of research in cryptography is finding efficient homomorphic encryption schemes, particularly information-theoretically secure schemes which are not based on unproven computational hardness assumptions. We suggest here an information-theoretically secure secret sharing scheme based on Shamir’s secret sharing scheme. While Shamir’s scheme supports no homomorphic multiplications of secrets, our scheme efficiently supports one homomorphic multiplication of secrets in addition to homomorphic additions of, practically, any number of such multiplied secrets. We focus on the single-client–multi-server setting. Therefore, our scheme enables a single user to share a database of m records (secrets) among N semi-honest servers with \(O(m^2)\) ciphertext, using a novel variant of Shamir’s secret sharing scheme and polynomials of degree \(N-1\). Then, our scheme enables homomorphic evaluation of quadratic functions and 2-CNF circuits over the database with no communication between the servers. Our scheme is perfectly secure against attacks of a single server and information-theoretically statistically secure against attacks of coalitions of less than \(N-1\) servers. One of the main advantages of our scheme over known schemes is enabling the evaluation of quadratic functions and 2-CNF secrets over a dynamic database of secrets. A dynamic database of secrets is a database of secrets that can grow in the future with no need for storing and re-sharing existing secrets by the user. To the best of our knowledge, the challenging support for the dynamic property was not obtained in this setting elsewhere before.

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Notes

  1. Each of the \({\overline{0}}\)s refers to the zero vector of the vector space it belongs to. We include these zero vectors in \({\mathcal {V}}_p\) for technical reasons explained below.

  2. Clearly, one can use the proof of Proposition 1 to implement stage 1 in time O(n).

  3. In fact, given the \(y_j\)s, g(0) can be computed without finding g. That procedure is not of our main interests.

  4. We omit the index j and write \(\alpha ,y,y'\).

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Acknowledgements

With pleasure, we thank Amos Beimel and Niv Gilboa for useful inputs.

Funding

Research partially supported by the Lynne and William Frankel Center for Computer Science, the Rita Altura Trust Chair in Computer Science and also supported by a grant from the Ministry of Science, Technology and Space, Infrastructure Research in the Field of Advanced Computing and Cyber Security, the Israel & the Japan Science and Technology Agency (JST), and the German Research Funding Organization (DFG, Grant#8767581199).

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Authors and Affiliations

  1. Department of Mathematics, Ben-Gurion University of the Negev, Beer Sheva, Israel

    Daniel Berend & Dor Bitan

  2. Department of Computer Science, Ben-Gurion University of the Negev, Beer Sheva, Israel

    Daniel Berend & Shlomi Dolev

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  1. Daniel Berend
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  2. Dor Bitan
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  3. Shlomi Dolev
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Correspondence to Dor Bitan.

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Berend, D., Bitan, D. & Dolev, S. Communicationless Evaluation of Quadratic Functions over Secret Shared Dynamic Database. SN COMPUT. SCI. 3, 174 (2022). https://doi.org/10.1007/s42979-022-01073-5

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