# American Institute of Mathematical Sciences

February  2018, 12(1): 17-47. doi: 10.3934/amc.2018002

## Private set-intersection with common set-up

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

* Corresponding author: Sanjit Chatterjee

Current affilation: IST Austria

Received  September 2015 Revised  June 2016 Published  March 2018

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.

Citation: Sanjit Chatterjee, Chethan Kamath, Vikas Kumar. Private set-intersection with common set-up. Advances in Mathematics of Communications, 2018, 12 (1) : 17-47. doi: 10.3934/amc.2018002
##### References:
 [1] M. Abe, G. Fuchsbauer, J. Groth, K. Haralambiev and M. Ohkubo, Structure-preserving signatures and commitments to group elements, in Advances in Cryptology -CRYPTO 2010, Springer, 2010,209-236. [2] G. Ateniese, E. De Cristofaro and G. Tsudik, (If) Size matters: Size-hiding private set intersection, in Public Key Cryptography -PKC 2011, Springer, 2011,156-173. [3] P. Baldi, R. Baronio, E. De Cristofaro, P. Gasti and G. Tsudik, Countering gattaca: Efficient and secure testing of fully-sequenced human genomes, in Proc. 18th ACM Conf. Comp. Commun. Secur. -CCS'11, ACM, New York, 2011,691-702. [4] M. Bellare, R. Canetti and H. Krawczyk, A modular approach to the design and analysis of authentication and key exchange protocols (extended abstract), in Proc. 30th Ann. ACM Symp. Theory Comp. -STOC'98, ACM, New York, 1998,419-428. [5] M. Bellare, C. Namprempre, D. Pointcheval and M. Semanko, The one-more-RSA-inversion problems and the security of Chaum's blind signature scheme, J. Cryptology, 16 (2003), 185-215. [6] M. Bellare and P. Rogaway, Random oracles are practical: a paradigm for designing efficient protocols, in Proc. 1st ACM Conf. Comp. Commun. Secur. -CCS '93, ACM, New York, 1993, 62-73. [7] M. Bellare and P. Rogaway, The exact security of digital signatures -how to sign with RSA and Rabin, in Advances in Cryptology -EUROCRYPT 1996, Springer, 1996,399-416. [8] A. Boldyreva, Threshold signatures, multisignatures and blind signatures based on the gapDiffie-Hellman group signature scheme, in Public Key Cryptography -PKC 2003, Springer, 2002, 31-46. [9] E. Bursztein, M. Hamburg, J. Lagarenne and D. Boneh, Openconflict: Preventing real time map hacks in online games, in Proc. 32nd IEEE Symp. Secur. Privacy, IEEE Comp. Soc., Berkeley, 2011,506-520. [10] J. Camenisch and G. M. Zaverucha, Private intersection of certified sets, in Financial Cryptography and Data Security, Springer, 2009,108-127. [11] R. Canetti, Security and composition of multiparty cryptographic protocols, J. Cryptology, 13 (2000), 143-202. [12] R. Canetti and H. Krawczyk, Analysis of key-exchange protocols and their use for building secure channels, in Advances in Cryptology -EUROCRYPT 2001, Springer, 2001,453-474. [13] R. Canetti and H. Krawczyk, Universally composable notions of key exchange and secure channels, in Advances in Cryptology -EUROCRYPT 2002, Springer, 2002, 337-351. [14] R. Canetti and T. Rabin, Universal composition with joint state, in Advances in Cryptology -CRYPTO 2003, Springer, 2003,265-281. [15] D. Chaum, Blind signatures for untraceable payments, in Advances in Cryptology -Proc. CRYPTO '82, Plenum Press, New York, 1982,199-203. [16] J.-H. Cheon, S. Jarecki and J.-H. Seo, Multi-party privacy-preserving set intersection with quasi-linear complexity, IEICE Trans., 95-A (2012), 1366-1378. [17] J. -S. Coron, On the exact security of full domain hash, in Advances in Cryptology -CRYPTO 2000, Springer, 2000,229-235. [18] D. Dachman-Soled, T. Malkin, M. Raykova and M. Yung, Efficient robust private set intersection, Int. J. Appl. Crypt., 2 (2012), 289-303. [19] E. De Cristofaro, J. Kim and G. Tsudik, Linear-complexity private set intersection protocols secure in malicious model, in Advances in Cryptology -ASIACRYPT 2010, Springer, 2010,213-231. [20] E. De Cristofaro and G. Tsudik, Practical private set intersection protocols with linear complexity, in Financial Cryptography and Data Security, Springer, 2010,143-159. [21] E. De Cristofaro and G. Tsudik, Experimenting with fast private set intersection, in Trust and Trustworthy Computing, Springer, 2012, 55-73. [22] C. Dong, L. Chen and Z. Wen, When private set intersection meets big data: an efficient and scalable protocol, in Proc. 2013 ACM SIGSAC Conf. Comp. Commun. Secur. -CCS '13, ACM, New York, 2013,789-800. [23] M. Freedman, K. Nissim and B. Pinkas, Efficient private matching and set intersection, in Advances in Cryptology -EUROCRYPT 2004, Springer, 2004, 1-19. [24] O. Goldreich, The Foundations of Cryptography -Volume 2, Basic Applications, Cambridge Univ. Press, 2004. [25] S. Goldwasser and S. Micali, Probabilistic encryption & how to play mental poker keeping secret all partial information, in Proc. 14th Ann. ACM Symp. Theory Comp. STOC '82, ACM, New York, 1982,365-377. [26] C. Hazey, Oblivious polynomial evaluation and secure set-intersection from algebraic PRFs, In Proc. 12th Theory Crypt. Conf. -TCC 2015, Springer, 90-120. [27] C. Hazay and Y. Lindell, Efficient protocols for set intersection and pattern matching with security against malicious and covert adversaries, in Theory of Cryptography, Springer, 2008,155-175. [28] C. Hazay and Y. Lindell, Efficient protocols for set intersection and pattern matching with security against malicious and covert adversaries, J. Cryptology, 23 (2010), 422-456. [29] C. Hazay and K. Nissim, Efficient set operations in the presence of malicious adversaries, in Public Key Cryptography -PKC 2010, Springer, 2010, 312-331. [30] Y. Huang, D. Evans and J. Katz, Private set intersection: Are garbled circuits better than custom protocols?, in 19th Ann. Network Distrib. System Secur. Symp. 2012, San Diego, California, 2012. [31] S. Jarecki and X. Liu, Efficient oblivious pseudorandom function with applications to adaptive OT and secure computation of set intersection, in Theory of Cryptography, Springer, 2009,577-594. [32] S. Jarecki and X. Liu, Fast secure computation of set intersection, in Security and Cryptography for Networks, Springer, 2010,418-435. [33] A. Juels, M. Luby and R. Ostrovsky, Security of blind digital signatures, in Advances in Cryptology -CRYPTO '97, Springer, 1997,150-164. [34] L. Kissner and D. Song, Privacy-preserving set operations, in Advances in Cryptology -CRYPTO 2005 (ed. V. Shoup), Springer, 2005,241-257. [35] Y. Lindell and B. Pinkas, Privacy preserving data mining, J. Cryptology, 15 (2002), 177-206. [36] A. Menezes, P. van Oorschot and S. Vanstone, Handbook of Applied Cryptography, CRC, 1996. [37] B. Pinkas, T. Schneider and M. Zohner, Faster private set intersection based on ot extension, in Proc. 23rd USENIX Secur. Symp., 797-812, 2014. [38] E. Stefanov, E. Shi and D. Song, Policy-enhanced private set intersection: Sharing information while enforcing privacy policies, in Public Key Cryptography -PKC 2012 (M. Fischlin, J. Buchmann and M. Manulis), Springer, 2012,413-430.

show all references

##### References:
 [1] M. Abe, G. Fuchsbauer, J. Groth, K. Haralambiev and M. Ohkubo, Structure-preserving signatures and commitments to group elements, in Advances in Cryptology -CRYPTO 2010, Springer, 2010,209-236. [2] G. Ateniese, E. De Cristofaro and G. Tsudik, (If) Size matters: Size-hiding private set intersection, in Public Key Cryptography -PKC 2011, Springer, 2011,156-173. [3] P. Baldi, R. Baronio, E. De Cristofaro, P. Gasti and G. Tsudik, Countering gattaca: Efficient and secure testing of fully-sequenced human genomes, in Proc. 18th ACM Conf. Comp. Commun. Secur. -CCS'11, ACM, New York, 2011,691-702. [4] M. Bellare, R. Canetti and H. Krawczyk, A modular approach to the design and analysis of authentication and key exchange protocols (extended abstract), in Proc. 30th Ann. ACM Symp. Theory Comp. -STOC'98, ACM, New York, 1998,419-428. [5] M. Bellare, C. Namprempre, D. Pointcheval and M. Semanko, The one-more-RSA-inversion problems and the security of Chaum's blind signature scheme, J. Cryptology, 16 (2003), 185-215. [6] M. Bellare and P. Rogaway, Random oracles are practical: a paradigm for designing efficient protocols, in Proc. 1st ACM Conf. Comp. Commun. Secur. -CCS '93, ACM, New York, 1993, 62-73. [7] M. Bellare and P. Rogaway, The exact security of digital signatures -how to sign with RSA and Rabin, in Advances in Cryptology -EUROCRYPT 1996, Springer, 1996,399-416. [8] A. Boldyreva, Threshold signatures, multisignatures and blind signatures based on the gapDiffie-Hellman group signature scheme, in Public Key Cryptography -PKC 2003, Springer, 2002, 31-46. [9] E. Bursztein, M. Hamburg, J. Lagarenne and D. Boneh, Openconflict: Preventing real time map hacks in online games, in Proc. 32nd IEEE Symp. Secur. Privacy, IEEE Comp. Soc., Berkeley, 2011,506-520. [10] J. Camenisch and G. M. Zaverucha, Private intersection of certified sets, in Financial Cryptography and Data Security, Springer, 2009,108-127. [11] R. Canetti, Security and composition of multiparty cryptographic protocols, J. Cryptology, 13 (2000), 143-202. [12] R. Canetti and H. Krawczyk, Analysis of key-exchange protocols and their use for building secure channels, in Advances in Cryptology -EUROCRYPT 2001, Springer, 2001,453-474. [13] R. Canetti and H. Krawczyk, Universally composable notions of key exchange and secure channels, in Advances in Cryptology -EUROCRYPT 2002, Springer, 2002, 337-351. [14] R. Canetti and T. Rabin, Universal composition with joint state, in Advances in Cryptology -CRYPTO 2003, Springer, 2003,265-281. [15] D. Chaum, Blind signatures for untraceable payments, in Advances in Cryptology -Proc. CRYPTO '82, Plenum Press, New York, 1982,199-203. [16] J.-H. Cheon, S. Jarecki and J.-H. Seo, Multi-party privacy-preserving set intersection with quasi-linear complexity, IEICE Trans., 95-A (2012), 1366-1378. [17] J. -S. Coron, On the exact security of full domain hash, in Advances in Cryptology -CRYPTO 2000, Springer, 2000,229-235. [18] D. Dachman-Soled, T. Malkin, M. Raykova and M. Yung, Efficient robust private set intersection, Int. J. Appl. Crypt., 2 (2012), 289-303. [19] E. De Cristofaro, J. Kim and G. Tsudik, Linear-complexity private set intersection protocols secure in malicious model, in Advances in Cryptology -ASIACRYPT 2010, Springer, 2010,213-231. [20] E. De Cristofaro and G. Tsudik, Practical private set intersection protocols with linear complexity, in Financial Cryptography and Data Security, Springer, 2010,143-159. [21] E. De Cristofaro and G. Tsudik, Experimenting with fast private set intersection, in Trust and Trustworthy Computing, Springer, 2012, 55-73. [22] C. Dong, L. Chen and Z. Wen, When private set intersection meets big data: an efficient and scalable protocol, in Proc. 2013 ACM SIGSAC Conf. Comp. Commun. Secur. -CCS '13, ACM, New York, 2013,789-800. [23] M. Freedman, K. Nissim and B. Pinkas, Efficient private matching and set intersection, in Advances in Cryptology -EUROCRYPT 2004, Springer, 2004, 1-19. [24] O. Goldreich, The Foundations of Cryptography -Volume 2, Basic Applications, Cambridge Univ. Press, 2004. [25] S. Goldwasser and S. Micali, Probabilistic encryption & how to play mental poker keeping secret all partial information, in Proc. 14th Ann. ACM Symp. Theory Comp. STOC '82, ACM, New York, 1982,365-377. [26] C. Hazey, Oblivious polynomial evaluation and secure set-intersection from algebraic PRFs, In Proc. 12th Theory Crypt. Conf. -TCC 2015, Springer, 90-120. [27] C. Hazay and Y. Lindell, Efficient protocols for set intersection and pattern matching with security against malicious and covert adversaries, in Theory of Cryptography, Springer, 2008,155-175. [28] C. Hazay and Y. Lindell, Efficient protocols for set intersection and pattern matching with security against malicious and covert adversaries, J. Cryptology, 23 (2010), 422-456. [29] C. Hazay and K. Nissim, Efficient set operations in the presence of malicious adversaries, in Public Key Cryptography -PKC 2010, Springer, 2010, 312-331. [30] Y. Huang, D. Evans and J. Katz, Private set intersection: Are garbled circuits better than custom protocols?, in 19th Ann. Network Distrib. System Secur. Symp. 2012, San Diego, California, 2012. [31] S. Jarecki and X. Liu, Efficient oblivious pseudorandom function with applications to adaptive OT and secure computation of set intersection, in Theory of Cryptography, Springer, 2009,577-594. [32] S. Jarecki and X. Liu, Fast secure computation of set intersection, in Security and Cryptography for Networks, Springer, 2010,418-435. [33] A. Juels, M. Luby and R. Ostrovsky, Security of blind digital signatures, in Advances in Cryptology -CRYPTO '97, Springer, 1997,150-164. [34] L. Kissner and D. Song, Privacy-preserving set operations, in Advances in Cryptology -CRYPTO 2005 (ed. V. Shoup), Springer, 2005,241-257. [35] Y. Lindell and B. Pinkas, Privacy preserving data mining, J. Cryptology, 15 (2002), 177-206. [36] A. Menezes, P. van Oorschot and S. Vanstone, Handbook of Applied Cryptography, CRC, 1996. [37] B. Pinkas, T. Schneider and M. Zohner, Faster private set intersection based on ot extension, in Proc. 23rd USENIX Secur. Symp., 797-812, 2014. [38] E. Stefanov, E. Shi and D. Song, Policy-enhanced private set intersection: Sharing information while enforcing privacy policies, in Public Key Cryptography -PKC 2012 (M. Fischlin, J. Buchmann and M. Manulis), Springer, 2012,413-430.
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.
Relationship between the security definitions for client privacy
Protocol Σ
Protocol Π
Protocol Ψ
Protocol F4
F3-protocol in a general cyclic-group setting
Protocol Σ: reduction for server unlinkability
Protocol Π: security argument for server privacy
Protocol Ψ: security argument for server privacy
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 Client Server 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 $2v+2$ $v+w+1$ $2(v+1)|p|+w\tau$ $\textsf{GDH}$
 Protocol MI-secure Computation (Exp.)(bits) Communication Assumption Client Server 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 $2v+2$ $v+w+1$ $2(v+1)|p|+w\tau$ $\textsf{GDH}$
 [1] Neal Koblitz, Alfred Menezes. Critical perspectives on provable security: Fifteen years of "another look" papers. Advances in Mathematics of Communications, 2019, 13 (4) : 517-558. doi: 10.3934/amc.2019034 [2] Sumit Kumar Debnath, Pantelimon Stǎnicǎ, Nibedita Kundu, Tanmay Choudhury. Secure and efficient multiparty private set intersection cardinality. Advances in Mathematics of Communications, 2021, 15 (2) : 365-386. doi: 10.3934/amc.2020071 [3] Paolo D'Arco, María Isabel González Vasco, Angel L. Pérez del Pozo, Claudio Soriente, Rainer Steinwandt. Private set intersection: New generic constructions and feasibility results. Advances in Mathematics of Communications, 2017, 11 (3) : 481-502. doi: 10.3934/amc.2017040 [4] Ramprasad Sarkar, Mriganka Mandal, Sourav Mukhopadhyay. Quantum-safe identity-based broadcast encryption with provable security from multivariate cryptography. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022026 [5] Neal Koblitz, Alfred Menezes. Another look at security definitions. Advances in Mathematics of Communications, 2013, 7 (1) : 1-38. doi: 10.3934/amc.2013.7.1 [6] Isabelle Déchène. On the security of generalized Jacobian cryptosystems. Advances in Mathematics of Communications, 2007, 1 (4) : 413-426. doi: 10.3934/amc.2007.1.413 [7] Palash Sarkar, Subhadip Singha. Verifying solutions to LWE with implications for concrete security. Advances in Mathematics of Communications, 2021, 15 (2) : 257-266. doi: 10.3934/amc.2020057 [8] Roberto Civino, Riccardo Longo. Formal security proof for a scheme on a topological network. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021009 [9] Riccardo Aragona, Alessio Meneghetti. Type-preserving matrices and security of block ciphers. Advances in Mathematics of Communications, 2019, 13 (2) : 235-251. doi: 10.3934/amc.2019016 [10] Archana Prashanth Joshi, Meng Han, Yan Wang. A survey on security and privacy issues of blockchain technology. Mathematical Foundations of Computing, 2018, 1 (2) : 121-147. doi: 10.3934/mfc.2018007 [11] Philip Lafrance, Alfred Menezes. On the security of the WOTS-PRF signature scheme. Advances in Mathematics of Communications, 2019, 13 (1) : 185-193. doi: 10.3934/amc.2019012 [12] Meenakshi Kansal, Ratna Dutta, Sourav Mukhopadhyay. Group signature from lattices preserving forward security in dynamic setting. Advances in Mathematics of Communications, 2020, 14 (4) : 535-553. doi: 10.3934/amc.2020027 [13] Jian Mao, Qixiao Lin, Jingdong Bian. Application of learning algorithms in smart home IoT system security. Mathematical Foundations of Computing, 2018, 1 (1) : 63-76. doi: 10.3934/mfc.2018004 [14] Liqun Qi, Zheng yan, Hongxia Yin. Semismooth reformulation and Newton's method for the security region problem of power systems. Journal of Industrial and Management Optimization, 2008, 4 (1) : 143-153. doi: 10.3934/jimo.2008.4.143 [15] Palash Sarkar, Subhadip Singha. Classical reduction of gap SVP to LWE: A concrete security analysis. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021004 [16] Yu-Chi Chen. Security analysis of public key encryption with filtered equality test. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021053 [17] Stamatios Katsikas, Vassilli Kolokoltsov. Evolutionary, mean-field and pressure-resistance game modelling of networks security. Journal of Dynamics and Games, 2019, 6 (4) : 315-335. doi: 10.3934/jdg.2019021 [18] Zongmin Li, Jiuping Xu, Wenjing Shen, Benjamin Lev, Xiao Lei. Bilevel multi-objective construction site security planning with twofold random phenomenon. Journal of Industrial and Management Optimization, 2015, 11 (2) : 595-617. doi: 10.3934/jimo.2015.11.595 [19] Jose-Luis Roca-Gonzalez. Designing dynamical systems for security and defence network knowledge management. A case of study: Airport bird control falconers organizations. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1311-1329. doi: 10.3934/dcdss.2015.8.1311 [20] Shuai Ren, Tao Zhang, Fangxia Shi, Zongzong Lou. The application of improved-DAA for the vehicle network node security in single- and multi-trusted domain. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1301-1309. doi: 10.3934/dcdss.2015.8.1301

2021 Impact Factor: 1.015