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Invertible Polynomial Representation for Private Set Operations

  • Jung Hee Cheon
  • Hyunsook Hong
  • Hyung Tae Lee
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8565)

Abstract

In many private set operations, a set is represented by a polynomial over a ring \(\mathbb {Z}_\sigma \) for a composite integer \(\sigma \), where \(\mathbb {Z}_\sigma \) is the message space of some additive homomorphic encryption. While it is useful for implementing set operations with polynomial additions and multiplications, it has a limitation that it is hard to recover a set from a polynomial due to the hardness of polynomial factorization over \(\mathbb {Z}_\sigma \).

We propose a new representation of a set by a polynomial over \(\mathbb {Z}_\sigma \), in which \(\sigma \) is a composite integer with known factorization but a corresponding set can be efficiently recovered from a polynomial except negligible probability. Since \(\mathbb {Z}_\sigma [x]\) is not a unique factorization domain, a polynomial may be written as a product of linear factors in several ways. To exclude irrelevant linear factors, we introduce a special encoding function which supports early abort strategy. Our representation can be efficiently inverted by computing all the linear factors of a polynomial in \(\mathbb {Z}_\sigma [x]\) whose roots locate in the image of the encoding function.

As an application of our representation, we obtain a constant-round private set union protocol. Our construction improves the complexity than the previous without honest majority.

Keywords

Polynomial representation Polynomial factorization Root finding Privacy-preserving set union 

Notes

Acknowledgements

We thank Jae Hong Seo for helpful comments on our preliminary works and anonymous reviewers for their valuable comments. This work was supported by the IT R&D program of MSIP/KEIT. [No. 10047212, Development of homomorphic encryption supporting arithmetics on ciphertexts of size less than 1kB and its applications].

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Jung Hee Cheon
    • 1
  • Hyunsook Hong
    • 1
  • Hyung Tae Lee
    • 1
  1. 1.CHRI and Department of Mathematical SciencesSeoul National UniversitySeoulKorea

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