**Invertible Polynomial Representation for Private Set Operations**

*Jung Hee Cheon and Hyunsook Hong and Hyung Tae Lee*

**Abstract: **In many private set operations, a set is represented by a polynomial over a ring $\Z_{\sigma}$ for a composite integer $\sigma$, where $\Z_\sigma$ is the message space of some additive homomorphic encryption. While it is useful for implementing set operations with polynomial additions and multiplications, a polynomial representation has a limitation due to the hardness of polynomial factorization over $\Z_\sigma$. That is, it is hard to recover a corresponding set from a resulting polynomial over $\Z_\sigma$ if $\sigma$ is not a prime.

In this paper, we propose a new representation of a set by a polynomial over $\Z_\sigma$, in which $\sigma$ is a composite integer with {\em known factorization} but a corresponding set can be efficiently recovered from a polynomial except negligible probability in the security parameter. Note that $\Z_\sigma[x]$ is not a unique factorization domain, so 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. As a result, our representation can be efficiently inverted by computing all the linear factors of a polynomial in $\Z_{\sigma}[x]$ whose roots locate in the image of the encoding function.

When we consider group decryption as in most private set operation protocols, inverting polynomial representations should be done without a single party possessing the secret of the utilized additive homomorphic encryption. This is very hard for Paillier's encryption whose message space is $\Z_N$ with unknown factorization of $N$. Instead, we detour this problem by using Naccache-Stern encryption with message space $\Z_\sigma$ where $\sigma$ is a smooth integer with public factorization.

As an application of our representation, we obtain a constant round privacy-preserving set union protocol. Our construction improves the complexity than the previous without an honest majority. It can be also used for a constant round multi-set union protocol and a private set intersection protocol even when decryptors do not possess a superset of the resulting set.

**Category / Keywords: **cryptographic protocols /

**Date: **received 6 Sep 2012, last revised 8 Jul 2013

**Contact author: **htsm1138 at snu ac kr

**Available format(s): **PDF | BibTeX Citation

**Version: **20130708:165011 (All versions of this report)

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