# American Institute of Mathematical Sciences

August  2017, 11(3): 481-502. doi: 10.3934/amc.2017040

## Private set intersection: New generic constructions and feasibility results

 1 Dipartimento di Informatica, University of Salerno, 84084 Fisciano (SA), Italy 2 MACIMTE, Area de Matemática Aplicada, U. Rey Juan Carlos, c/ Tulipán, s/n, 28933, Móstoles, Madrid, Spain 3 Telefónica Research, Barcelona, Spain 4 Florida Atlantic University (FAU), 777 Glades Rd, Boca Raton, FL 33431, USA

* Corresponding author

Received  July 2015 Revised  December 2015 Published  August 2017

Fund Project: The first three and the last author were partially supported by the Spanish Ministerio de Economía y Competitividad through the project grant MTM-2010-15167. This research is also partially supported by the Italian PRIN project GenData 2020

In this paper we focus on protocols for private set intersection (PSI), through which two parties, each holding a set of inputs drawn from a ground set, jointly compute the intersection of their sets. Ideally, no further information than which elements are actually shared is compromised to the other party, yet the input set sizes are often considered as admissible leakage.

In the unconditional setting we evidence that PSI is impossible to realize and that unconditionally secure size-hiding PSI is possible assuming a set-up authority is present in an set up phase. In the computational setting we give a generic construction using smooth projective hash functions for languages derived from perfectly-binding commitments. Further, we give two size-hiding constructions: the first one is theoretical and evidences the equivalence between PSI, oblivious transfer and the secure computation of the AND function. The second one is a twist on the oblivious polynomial evaluation construction of Freedman et al. from EUROCRYPT 2004. We further sketch a generalization of the latter using algebraic-geometric techniques. Finally, assuming again there is a set-up authority (yet not necessarily trusted) we present very simple and efficient constructions that only hide the size of the client's set.

Citation: 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
##### References:

show all references

##### References:
An unconditionally secure size-hiding set intersection protocol
A generic PSI protocol from smooth projective hashing
A computationally secure size-hiding set intersection protocol
OT protocol based on trapdoor permutations
Polynomial-based construction for $|\mathcal{C}|,|\mathcal{S}| \le M$
Algebraic PSI construction for $|\mathcal{C}|,|\mathcal{S}| \le M$
The PRF-PSI-protocol
Setup phase of the OPRF-PSI-protocol
Performance comparison of SPH-based implementations vs prior public key implementations for PSI
 Protocol Comm. Overhead Server Exp. Client Exp. [20] $\mathcal{O}(v + w)$ $\mathcal{O}(w(\log\log v))$ $\mathcal{O}(w+v)$ [31] $\mathcal{O}(w +v)$ $\mathcal{O}(vw)$ $\mathcal{O}(v+w)$ SPH-DDH $\mathcal{O}(vw)$ $\mathcal{O}(vw)$ $\mathcal{O}(v)$
 Protocol Comm. Overhead Server Exp. Client Exp. [20] $\mathcal{O}(v + w)$ $\mathcal{O}(w(\log\log v))$ $\mathcal{O}(w+v)$ [31] $\mathcal{O}(w +v)$ $\mathcal{O}(vw)$ $\mathcal{O}(v+w)$ SPH-DDH $\mathcal{O}(vw)$ $\mathcal{O}(vw)$ $\mathcal{O}(v)$
 [1] 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 [2] Xueyan Wu. An algorithm for reversible information hiding of encrypted medical images in homomorphic encrypted domain. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1441-1455. doi: 10.3934/dcdss.2019099 [3] Alberto Bressan, Anders Nordli. The Riemann solver for traffic flow at an intersection with buffer of vanishing size. Networks & Heterogeneous Media, 2017, 12 (2) : 173-189. doi: 10.3934/nhm.2017007 [4] Yang Lu, Jiguo Li. Forward-secure identity-based encryption with direct chosen-ciphertext security in the standard model. Advances in Mathematics of Communications, 2017, 11 (1) : 161-177. doi: 10.3934/amc.2017010 [5] Maxim Arnold, Walter Craig. On the size of the Navier - Stokes singular set. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1165-1178. doi: 10.3934/dcds.2010.28.1165 [6] Marcela Mejía, J. Urías. An asymptotically perfect pseudorandom generator. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 115-126. doi: 10.3934/dcds.2001.7.115 [7] Pablo Sánchez, Jaume Sempere. Conflict, private and communal property. Journal of Dynamics & Games, 2016, 3 (4) : 355-369. doi: 10.3934/jdg.2016019 [8] Angsuman Das, Avishek Adhikari, Kouichi Sakurai. Plaintext checkable encryption with designated checker. Advances in Mathematics of Communications, 2015, 9 (1) : 37-53. doi: 10.3934/amc.2015.9.37 [9] Serafin Bautista, Carlos A. Morales. On the intersection of sectional-hyperbolic sets. Journal of Modern Dynamics, 2015, 9: 203-218. doi: 10.3934/jmd.2015.9.203 [10] 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 [11] 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 [12] S. Bautista, C. Morales, M. J. Pacifico. On the intersection of homoclinic classes on singular-hyperbolic sets. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 761-775. doi: 10.3934/dcds.2007.19.761 [13] Feimin Zhong, Wei Zeng, Zhongbao Zhou. Mechanism design in a supply chain with ambiguity in private information. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-27. doi: 10.3934/jimo.2018151 [14] Alina Ostafe, Igor E. Shparlinski, Arne Winterhof. On the generalized joint linear complexity profile of a class of nonlinear pseudorandom multisequences. Advances in Mathematics of Communications, 2010, 4 (3) : 369-379. doi: 10.3934/amc.2010.4.369 [15] 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 [16] 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 [17] 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 [18] Lisa C. Jeffrey and Frances C. Kirwan. Intersection pairings in moduli spaces of holomorphic bundles on a Riemann surface. Electronic Research Announcements, 1995, 1: 57-71. [19] Marta Faias, Emma Moreno-García, Myrna Wooders. A strategic market game approach for the private provision of public goods. Journal of Dynamics & Games, 2014, 1 (2) : 283-298. doi: 10.3934/jdg.2014.1.283 [20] Fei Gao. Data encryption algorithm for e-commerce platform based on blockchain technology. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1457-1470. doi: 10.3934/dcdss.2019100

2018 Impact Factor: 0.879