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Invertible Polynomial Representation for Private Set Operations

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Information Security and Cryptology -- ICISC 2013 (ICISC 2013)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 8565))

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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.

This work includes some part of the third author’s PhD thesis [14].

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Notes

  1. 1.

    Note that one has to solve \(\bar{\ell }\) DLPs over a group of order \(q_j\) for one decryption in the NS encryption scheme. In Step 3 (b), one has to solve \(2d=2nk\) DLPs over a group of order \(q_j\) for each \(q_j\). It requires \(O(\sqrt{dq_j})\) multiplications to solve \(d\) DLPs over a group of order \(q_j\) [13] and hence total complexity of this step is \(O(\bar{\ell }\sqrt{dq_j})\) multiplications.

  2. 2.

    Due to the space limitation, the detailed computation of Eqs. (5) and (6) are given in the full version of this paper [3].

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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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Correspondence to Jung Hee Cheon .

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A Proof of Theorem 1

A Proof of Theorem 1

Let \({\mathsf E}_j\) be the expected number of linkable pairs of \(j\)-tuple in \(\mathbb {Z}_{q_1}\times \cdots \times \mathbb {Z}_{q_j}\) for \(j\ge 2\). For \(1\le j\le j'\le \bar{\ell }\), let \({\mathsf S}_{j'-j+1}(i_{j}, \ldots , i_{j'})\) be the event that \((s_{j}^{(i_j)}, \ldots , s_{j'}^{(i_{j'})})\) is a linkable pair. Then,

$$\begin{aligned} {\mathsf E}_2&= \sum _{i_1, i_2\in \{1, \ldots , d\}}1\cdot \Pr [{\mathsf S}_2(i_1, i_2)]\\&= \sum _{i_1,i_2\in \{1, \ldots , d\}} \Pr [{\mathsf S}_2(i_1, i_2)\wedge (i_1=i_2)] + \sum _{i_1,i_2\in \{1,\ldots , d\}} \Pr [{\mathsf S}_2(i_1, i_2)\wedge (i_1\ne i_2)]\\&= d+ d(d-1)\frac{1}{2^{2\tau }}=d\left( 1+\frac{d-1}{2^{2\tau }}\right) \end{aligned}$$

since \(\Pr [{\mathsf S}_2(i_1,i_1)]=1\) for \(i_1\in \{1,\ldots , d\}\) and \(\Pr [{\mathsf S}_2(i_1, i_2)]=\frac{1}{2^{2\tau }}\) for distinct \(i_1, i_2\in \{1,\ldots , d\}\) from the Eq. (1).

Now, we consider the relation between \({\mathsf E}_j\) and \({\mathsf E}_{j+1}\). When \((s_{1}^{(i_1)}, \ldots , s_{j}^{(i_j)})\) is a linkable pair, consider the case that \((s_{1}^{(i_1)}, \ldots , s_{j}^{(i_j)}, s_{j+1}^{(i_{j+1})})\) is a linkable pair. One can classify this case into the following three cases:

  1. 1.

    \(i_{j+1}=i_{j}\),

  2. 2.

    \(\left( i_{j+1}\ne i_{j}\right) ~\wedge ~\left( i_{j+1}= i_{j-1}\right) \),

  3. 3.

    \(\left( i_{j+1}\ne i_{j}\right) ~\wedge ~\left( i_{j+1}\ne i_{j-1}\right) \).

At the first case, if \(i_{j+1}=i_{j}\) and \((s_{1}^{(i_1)}, \ldots , s_{j}^{(i_j)})\) is a linkable pair, then \((s_{1}^{(i_1)}, \ldots , s_{j}^{(i_j)}, s_{j+1}^{(i_{j+1})})\) is always a linkable pair. Hence,

$$\begin{aligned} {\mathsf E}_{j+1}^{(1)}&:= \sum _{i_1,\ldots , i_{j+1}} \Pr \left[ {\mathsf S}_{j+1}(i_1,\ldots ,i_{j}, i_{j+1}) \wedge (i_{j+1}=i_j)\right] \\&= \sum _{i_1,\ldots , i_j} \Pr \left[ {\mathsf S}_{j}(i_1,\ldots ,i_{j})\right] ={\mathsf E}_{j}. \end{aligned}$$

At the second case, if \(i_{j+1}=i_{j-1}\ne i_j\) and \((s_{1}^{(i_1)}, \ldots , s_{j}^{(i_j)})\) is a linkable pair, then the relation \(s_{i_{j-1}, j+1}=s_{i_{j}, j+1}=s_{i_{j+1}, j+1}\) is satisfied from the encoding rule of \(\iota \). Hence,Footnote 2

$$\begin{aligned} {\mathsf E}_{j+1}^{(2)}&:= \sum _{i_1,\ldots , i_{j+1}\in \{1,\ldots , d\}} \Pr [{\mathsf S}_{j+1}(i_1,\ldots ,i_{j}, i_{j+1})\wedge (i_{j+1}=i_{j-1}\ne i_{j})]\nonumber \\&\le \frac{1}{2^{\tau }}\sum _{i_1,\ldots , i_{j}\in \{1,\ldots , d\}} \Pr \left[ {\mathsf S}_{j}(i_1,\ldots ,i_{j})\right] =\frac{1}{2^{\tau }}{\mathsf E}_{j}. \end{aligned}$$
(5)

At the last case, we can obtain the following result:

$$\begin{aligned} {\mathsf E}_{j+1}^{(3)}&:= \sum _{i_1,\ldots , i_{j+1}\in \{1,\ldots , d\}} \Pr [{\mathsf S}_{j+1}(i_1,\ldots ,i_{j}, i_{j+1})\wedge \left( (i_{j+1}\ne i_{j}) ~\wedge ~(i_{j+1}\ne i_{j-1})\right) ]\nonumber \\&\le \frac{d-1}{2^{2\tau }}\sum _{i_1,\ldots , i_{j}\in \{1,\ldots , d\}} \Pr \left[ {\mathsf S}_{j}(i_1,\ldots ,i_{j})\right] =\frac{d-1}{2^{2\tau }}{\mathsf E}_{j}. \end{aligned}$$
(6)

From the above results, we obtain the recurrence formula of \({\mathsf E}_j\) as follows:

$$\begin{aligned} {\mathsf E}_{j+1} ={\mathsf E}_{j+1}^{(1)}+{\mathsf E}_{j+1}^{(2)}+{\mathsf E}_{j+1}^{(3)} \le \left( 1+\frac{1}{2^{\tau }}+\frac{d-1}{2^{2\tau }}\right) {\mathsf E}_j \end{aligned}$$

for \(j\ge 2\) and hence \( {\mathsf E}_{\bar{\ell }}\le d\left( 1+\frac{1}{2^{\tau }}+\frac{d-1}{2^{2\tau }}\right) ^{\bar{\ell }-1} \) since \({\mathsf E}_2=d\left( 1+\frac{d-1}{2^{2\tau }}\right) \le d\left( 1+\frac{1}{2^{\tau }}+\frac{d-1}{2^{2\tau }}\right) \).

Now, we show that \(\bar{\ell }\le \frac{2^{2\tau }}{2^{\tau }+d}\). From the parameter setting, it is satisfied that \(\bar{\ell }\le \min \{d, \frac{\lfloor \log {N}\rfloor -2}{3\tau }\}\). When \(d_0\ge 8d\), it holds

$$\begin{aligned} \min \left\{ d, \frac{\lfloor \log {N}\rfloor -2}{3\tau }\right\} \le d \le \frac{d_0^{1/3}d^{2/3}}{2}. \end{aligned}$$

Consider the case that \(d_0<8d\). Then, it also holds

$$\begin{aligned} \min \left\{ d, \frac{\lfloor \log {N}\rfloor -2}{3\tau }\right\} \le \frac{\lfloor \log {N}\rfloor -2}{3\tau }\le \frac{d_0}{3\tau }\le \frac{d_0^{1/3}d^{2/3}}{2} \end{aligned}$$

since \(\tau \ge 3\). Hence

$$\begin{aligned} \bar{\ell }\le \min \left\{ d, \frac{\lfloor \log {N}\rfloor -2}{3\tau }\right\} \le \frac{d_0^{1/3}d^{2/3}}{2} \le \frac{\left( {d_0^{2}d}\right) ^{2/3}}{2d_0}\le \frac{2^{2\tau }}{2^{\tau }+d} \end{aligned}$$

since \(2d_0>2^{\tau }+d\). Therefore we obtain the following result:

$$\begin{aligned} {\mathsf E}_{\bar{\ell }}\le d\left( 1+\frac{1}{2^{\tau }}+\frac{d-1}{2^{2\tau }}\right) ^{\bar{\ell }-1}<ed<3d, \end{aligned}$$

where \(e\approx 2.718\) is the base of the natural logarithm. In other words, the upper bound of the expected number of linkable pairs of \(\bar{\ell }\)-tuple is \(3d\). \(\Box \)

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Cheon, J.H., Hong, H., Lee, H.T. (2014). Invertible Polynomial Representation for Private Set Operations. In: Lee, HS., Han, DG. (eds) Information Security and Cryptology -- ICISC 2013. ICISC 2013. Lecture Notes in Computer Science(), vol 8565. Springer, Cham. https://doi.org/10.1007/978-3-319-12160-4_17

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