Similarity Based Interactive Private Information Retrieval

Sashank Dara; V. N. Muralidhara

doi:10.1007/978-3-319-71501-8_9

International Conference on Security, Privacy, and Applied Cryptography Engineering

SPACE 2017: Security, Privacy, and Applied Cryptography Engineering pp 151-169 | Cite as

Similarity Based Interactive Private Information Retrieval

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Authors and affiliations

Sashank Dara
V. N. Muralidhara

Conference paper

First Online: 22 November 2017

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Part of the Lecture Notes in Computer Science book series (LNCS, volume 10662)

Abstract

Private Information Retrieval (PIR) schemes address users’ privacy concerns while querying public databases. Two major advancements that are needed for designing practical privacy preserving applications are: (i) constant communication complexity and (ii) private retrieval of matching documents. In this paper, we propose a new family of interactive schemes namely SIMPIR, that allow participating servers to interact with each other. Our methods are similarity based (i.e. the results could contain false positives but do not contain any false negatives). Importantly our approach has constant communication complexity agnostic of the size of database which is major improvement from known schemes. We achieve these results by slightly relaxing the traditional requirements of PIR schemes.

Keywords

Private information retrieval Encryption switching protocols Homomorphic encryption

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Notes

Acknowledgments

We would like to thank Cisco Systems for supporting this work.

A Computational - SIMPIR

We present the complete version of the advanced C-SIMPIR protocol here.

1.

Setup: The User configures creates the keypairs and Shares them among the auxiliary server (AS) and database server (DB) for encryption switching protocols to function.
2.

DB_Preparation, Query_Gen are same as basic protocol.
3.

Response_Gen: The database server generates $ \delta \textit{-psim}^a $ for each document $D_i$ in the database and initiates ciphertext encryption switching protocol (ESP) to switch the value to $ \delta \textit{-psim}^m $. Subsequently $ \prod _{j=1}^{j=n} \delta \textit{-psim}^m $ is computed w.r.t all the documents.
4.

Process_Res: User retrieves the response $\mathcal {RES}$ and decrypts it. If it is 0 then the input document $D_u$ matches with Server’s document $D_s$ as defined by the similarity factor (SF).
$$ Match = {\left\{ \begin{array}{ll} \textit{if } \mathcal {DEC}_{S_k}\left( \prod _{j=1}^{j=n} \delta \textit{-psim}^m \right) = 0 \textit{ then } \mathcal {SIM}(D_u,D_s) = \mathcal {SF}\\ else \quad \mathcal {SIM}(D_u,D_s) \ne \mathcal {SF} \end{array}\right. } $$

B Information Theoretic - SIMPIR

We present the complete version of the advanced IT-SIMPIR protocol here.

1.

DB_Preparation: The tf-idf vectors for all the documents are computed and replicated in k database servers $\mathcal {DB}_l$. Lets say for each document $D_s $ its vector is $ [b_1,b_2, \cdots , b_k]$ and its magnitude $||D_s|| = \sqrt{\sum _{j=1}^{j=k} b^2_{j}}$ is calculated.
2.

Query_Gen: Let user’s document $D_u = [a_{1},a_{2}, \dots , a_{k}] $ for which the private similarity check need to be performed. An Secret Sharing Scheme is used to share the terms of the document. Query vector $\mathcal {Q}^i_u = ([ q_1, q_2, \dots , q_k]$ for $\mathcal {DB}_i$ is defined as below. User also computes respective shares of $ (\mathcal {SF} \times ||D_u||))$ for each $\mathcal {DB}_i$.
$$\begin{aligned} \mathcal {Q}^i_u&=([Share_i(a_{1}), Share_i(a_{2}), .., Share_i(a_{k})] \quad \forall i \in (1,l)\; servers \end{aligned}$$
3.

Response_Gen: Each database server ($\mathcal {DB}_i$) receives its respective query vector, $\mathcal {Q}^i_u$, and computes the dot product for each document in the database. This is done blindfold using additive homomorphic properties of the secret sharing scheme. Subsequently the response per each server’s document $D_s$ is calculated, this would be a constant $\mathcal {RES}$ is:
$$\begin{aligned} |Q_u. D_{s}|&= {\sum _{j=1}^{j=k} q_j.b_{j}} = q_1 . b_1 + q_2 . b_2+ \cdots + q_k . b_k \\&= Share_i(a_{1}) . b_1 +Share_i(a_{2}) . b_2 +\cdots + Share_i(a_{k}) . b_k \\&= Share_i({\sum _{j=1}^{j=k} a_j.b_{j}}) \\ \mathcal {RES}&= Share_i({\sum _{j=1}^{j=k} a_j.b_{j}}) - Share_i(\mathcal {SF} \times ||D_u||) \times ||D_s|| \\&= Share_i({\sum _{j=1}^{j=k} a_j.b_{j}}) - Share_i (\mathcal {SF} \times ||D_u|| \times ||D_s|| )\\&= Share_i(\delta \textit{-sim}) \end{aligned}$$
Subsequently MPC protocol is initiated among all the database servers ( $\mathcal {DB}_i$ ) in order to compute their respective share of $Share_i(\prod _{j=1}^{j=n} (\delta \textit{-sim}_j))$.
4.

Process_Res: User retrieves the all the respective shares of the $\mathcal {RES}$ from the database servers $\mathcal {DB}_i$ and reconstructs $\delta \textit{-sim}$. If it is 0 then the input document $D_u$ matches with Server’s document $D_s$ as defined by the similarity factor (SF).
$$\begin{aligned} \delta \textit{-sim}&= reconstruct(Share_i(\delta \textit{-sim}))\forall i \in (1,l) \\ Match&= {\left\{ \begin{array}{ll} \textit{if } (\delta \textit{-psim} ) = 0 \textit{ then } \mathcal {SIM}(D_u,D_s) = \mathcal {SF}\\ else \quad \mathcal {SIM}(D_u,D_s) \ne \mathcal {SF} \end{array}\right. } \end{aligned}$$

The deviation from original assumptions of IT-PIR schemes is that in our protocols the servers communicate with each other in a non-colluding way to perform MPC protocols.

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Authors and Affiliations

Sashank Dara
- 1
- 2
Email author
V. N. Muralidhara
- 2

1.Cisco Systems Inc.BangaloreIndia
2.International Institute of Information Technology, BangaloreBangaloreIndia