Private set-intersection with common set-up

Sanjit Chatterjee; Chethan Kamath; Vikas Kumar

doi:10.3934/amc.2018002

Article Contents

2018, Volume 12, Issue 1: 17-47. Doi: 10.3934/amc.2018002

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Private set-intersection with common set-up

Department of Computer Science and Automation, Indian Institute of Science, Bangalore 560012, India

* Corresponding author: Sanjit Chatterjee
* Corresponding author: Sanjit Chatterjee

^† Current affilation: IST Austria

Received: September 2015

Revised: June 2016

Published: February 2018

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Abstract

The problem of private set-intersection (PSI) has been traditionally treated as an instance of the more general problem of multi-party computation (MPC). Consequently, in order to argue security, or compose these protocols one has to rely on the general theory that was developed for the purpose of MPC. The pursuit of efficient protocols, however, has resulted in designs that exploit properties pertaining to PSI. In almost all practical applications where a PSI protocol is deployed, it is expected to be executed multiple times, possibly on related inputs. In this work we initiate a dedicated study of PSI in the multi-interaction (MI) setting. In this model a server sets up the common system parameters and executes set-intersection multiple times with potentially different clients. We discuss a few attacks that arise when protocols are naïvely composed in this manner and, accordingly, craft security definitions for the MI setting and study their inter-relation. Finally, we suggest a set of protocols that are MI-secure, at the same time almost as efficient as their parent, stand-alone, protocols.

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Mathematics Subject Classification: 94A60, 11T71, 68P20.

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Figure 2. Relationship between the security definitions for server privacy. $\textsf{A}\rightarrow\textsf{B}$ implies that if a protocol is secure according to definition $\textsf{A}$, then it is also secure according to definition $\textsf{B}$. $\textsf{A} \nrightarrow \textsf{B}$ indicates a separation.

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Figure 3. Relationship between the security definitions for client privacy

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Figure 1. Protocol Σ

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Figure 4. Protocol Π

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Figure 5. Protocol Ψ

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Figure 7. Protocol F4

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Figure 6. F3-protocol in a general cyclic-group setting

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Figure 8. Protocol Σ: reduction for server unlinkability

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Figure 9. Protocol Π: security argument for server privacy

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Figure 10. Protocol Ψ: security argument for server privacy

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Table 1. Comparison of protocols; cardinality of client (resp. server) set is $v$ (resp. $w$). In protocols F4, $\Sigma$ and $\Pi$ the server takes $v+w$ exponentiations where both the exponent and modulus are of size $|N|$ bits. Since the server knows the factorization of $N$ ($p$ and $q$), by using the Chinese remainder theorem, the computation cost for the server can be reduced to $2(v+w)$ exponentiations, where both the exponent and modulus are of size $|N|/2$ bits (refer to [36,Fact 14.75] and [21]). Note that we give an improved security analysis of protocol F3 (the original reduction is based on one-more GDH assumption). See §5 for further details

Protocol	MI-secure	Computation (Exp.) (bits)		Communication	Assumption
Protocol	MI-secure	Client	Server	Communication	Assumption
F4 [20]	No	$v$	$2(v+w)$	$2v\|N\|+w\tau$	$\textsf{OMRSA}$
$\Sigma$	No	$v$	$2(v+w)$	$2v\|N\|+w\tau +l$	$\textsf{RSA}$
$\Pi$	Yes	$v$	$2(v+w)$	$2v\|N\|+w\tau +l$	$\textsf{RSA}$
$\Psi$	Yes	$2v$	$v+w$	$2v\|p\| + w\tau$	$\textsf{GDH}$
F3 [20]	Yes	$2v+2$	$v+w+1$	$2(v+1)\|p\|+w\tau$	$\textsf{GDH}$

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