# American Institute of Mathematical Sciences

November  2019, 13(4): 733-757. doi: 10.3934/amc.2019043

## The secure link prediction problem

 1 Cryptology and Security Research Unit, Indian Statistical Institute, Kolkata, India 2 R C Bose Centre for Cryptology and Security, Indian Statistical Institute, Kolkata, India

* Corresponding author

Received  October 2018 Revised  January 2019 Published  June 2019

Link Prediction is an important and well-studied problem for social networks. Given a snapshot of a graph, the link prediction problem predicts which new interactions between members are most likely to occur in the near future. As networks grow in size, data owners are forced to store the data in remote cloud servers which reveals sensitive information about the network. The graphs are therefore stored in encrypted form.

We study the link prediction problem on encrypted graphs. To the best of our knowledge, this secure link prediction problem has not been studied before. We use the number of common neighbors for prediction. We present three algorithms for the secure link prediction problem. We design prototypes of the schemes and formally prove their security. We execute our algorithms in real-life datasets.

Citation: Laltu Sardar, Sushmita Ruj. The secure link prediction problem. Advances in Mathematics of Communications, 2019, 13 (4) : 733-757. doi: 10.3934/amc.2019043
##### References:
 [1] [2] G. Asharov, Y. Lindell, T. Schneider and M. Zohner, More efficient oblivious transfer and extensions for faster secure computation, in 2013 ACM SIGSAC Conference on Computer and Communications Security, CCS'13, Berlin, Germany, November 4-8, 2013, 2013, 535–548. doi: 10.1145/2508859.2516738. [3] L. Backstrom, C. Dwork and J. M. Kleinberg, Wherefore art thou r3579x: anonymized social networks, hidden patterns, and structural steganography, WWW '07 Proceedings of the 16th international conference on World Wide Web, (2007), 181–190. doi: 10.1145/1242572.1242598. [4] D. Boneh, E. Goh and K. Nissim, Evaluating 2-dnf formulas on ciphertexts, in Theory of Cryptography, Second Theory of Cryptography Conference, TCC 2005, Cambridge, MA, USA, February 10-12, 2005, Proceedings, 3378 (2005), 325–341. doi: 10.1007/978-3-540-30576-7_18. [5] C. Bösch, A. Peter, B. Leenders, H. W. Lim, Q. Tang, H. Wang, P. H. Hartel and W. Jonker, Distributed searchable symmetric encryption, in 2014 Twelfth Annual International Conference on Privacy, Security and Trust, Toronto, ON, Canada, July 23-24, 2014, 2014, 330–337. [6] M. Chase and S. Kamara, Structured encryption and controlled disclosure, in Advances in Cryptology - ASIACRYPT 2010 - 16th International Conference on the Theory and Application of Cryptology and Information Security, Singapore, December 5-9, 2010. Proceedings, 6417 (2010), 577–594. doi: 10.1007/978-3-642-17373-8_33. [7] V. Kolesnikov, A. Sadeghi and T. Schneider, Improved garbled circuit building blocks and applications to auctions and computing minima, in Cryptology and Network Security, 8th International Conference, CANS 2009, Kanazawa, Japan, December 12-14, 2009. Proceedings, 2009, 1–20. [8] V. Kolesnikov and T. Schneider, Improved garbled circuit: Free XOR gates and applications, in Automata, Languages and Programming, 35th International Colloquium, ICALP 2008, Reykjavik, Iceland, July 7-11, 2008, Proceedings, Part II - Track B: Logic, Semantics, and Theory of Programming & Track C: Security and Cryptography Foundations, 5126 (2008), 486–498. doi: 10.1007/978-3-540-70583-3_40. [9] J. Leskovec and A. Krevl, SNAP Datasets: Stanford large network dataset collection, http://snap.stanford.edu/data, 2014. [10] D. Liben-Nowell and J. M. Kleinberg, The link prediction problem for social networks, in Proceedings of the 2003 ACM CIKM International Conference on Information and Knowledge Management, New Orleans, Louisiana, USA, November 2-8, 2003, 2003, 556–559. doi: 10.1145/956863.956972. [11] Y. Lindell and B. Pinkas, A proof of security of yao's protocol for two-party computation, J. Cryptology, 22 (2009), 161-188.  doi: 10.1007/s00145-008-9036-8. [12] C. Liu, L. Zhu and J. Chen, Graph encryption for top-k nearest keyword search queries on cloud, T-SUSC, 2 (2017), 371-381.  doi: 10.1109/TSUSC.2017.2704163. [13] A. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of Applied Cryptography, CRC Press, Boca Raton, FL, 1997. [14] X. Meng, S. Kamara, K. Nissim and G. Kollios, GRECS: graph encryption for approximate shortest distance queries, in Proceedings of the 22nd ACM SIGSAC Conference on Computer and Communications Security, Denver, CO, USA, October 12-6, 2015, 2015, 504–517. doi: 10.1145/2810103.2813672. [15] M. Naor and B. Pinkas, Efficient oblivious transfer protocols, in Proceedings of the Twelfth Annual Symposium on Discrete Algorithms, January 7-9, 2001, Washington, DC, USA., 2001, 448–457. [16] K. Nayak, X. S. Wang, S. Ioannidis, U. Weinsberg, N. Taft and E. Shi, Graphsc: Parallel secure computation made easy, in 2015 IEEE Symposium on Security and Privacy, SP 2015, San Jose, CA, USA, May 17-21, 2015, 2015, 377–394. doi: 10.1109/SP.2015.30. [17] PBC Library, The Pairing-based Cryptography Library, https://crypto.stanford.edu/pbc/. [18] L. Sardar and S. Ruj, Prototypes of secure link prediction schemes, GitHub repository, https://github.com/sardarlaltu/SecureLinkPrediction. [19] M. Shen, B. Ma, L. Zhu, R. Mijumbi, X. Du and J. Hu, Cloud-based approximate constrained shortest distance queries over encrypted graphs with privacy protection, IEEE Trans. Information Forensics and Security, 13 (2018), 940–953. doi: 10.1109/TIFS.2017.2774451. [20] D. X. Song, D. A. Wagner and A. Perrig, Practical techniques for searches on encrypted data, in 2000 IEEE Symposium on Security and Privacy, Berkeley, California, USA, May 14-17, 2000, 2000, 44–55. [21] E. Stefanov, M. van Dijk, E. Shi, C. W. Fletcher, L. Ren, X. Yu and S. Devadas, Path ORAM: An extremely simple oblivious RAM protocol, J. ACM, 65 (2018), Art. 18, 26 pp. doi: 10.1145/3177872. [22] Q. Wang, K. Ren, M. Du, Q. Li and A. Mohaisen, Secgdb: Graph encryption for exact shortest distance queries with efficient updates, in Financial Cryptography and Data Security - FC 2017, Sliema, Malta, April 3-7, 2017, Revised Selected Papers, 10322 (2017), 79–97. [23] A. C. Yao, Protocols for secure computations (extended abstract), in 23rd Annual Symposium on Foundations of Computer Science, Chicago, Illinois, USA, 3-5 November 1982, 1982, 160–164. [24] Y. Zheng, B. Wang, W. Lou and Y. T. Hou, Privacy-preserving link prediction in decentralized online social networks, in Computer Security - ESORICS 2015 - Vienna, Austria, September 21-25, 2015, Proceedings, Part II, 9327 (2015), 61–80. doi: 10.1007/978-3-319-24177-7_4.

show all references

##### References:
 [1] [2] G. Asharov, Y. Lindell, T. Schneider and M. Zohner, More efficient oblivious transfer and extensions for faster secure computation, in 2013 ACM SIGSAC Conference on Computer and Communications Security, CCS'13, Berlin, Germany, November 4-8, 2013, 2013, 535–548. doi: 10.1145/2508859.2516738. [3] L. Backstrom, C. Dwork and J. M. Kleinberg, Wherefore art thou r3579x: anonymized social networks, hidden patterns, and structural steganography, WWW '07 Proceedings of the 16th international conference on World Wide Web, (2007), 181–190. doi: 10.1145/1242572.1242598. [4] D. Boneh, E. Goh and K. Nissim, Evaluating 2-dnf formulas on ciphertexts, in Theory of Cryptography, Second Theory of Cryptography Conference, TCC 2005, Cambridge, MA, USA, February 10-12, 2005, Proceedings, 3378 (2005), 325–341. doi: 10.1007/978-3-540-30576-7_18. [5] C. Bösch, A. Peter, B. Leenders, H. W. Lim, Q. Tang, H. Wang, P. H. Hartel and W. Jonker, Distributed searchable symmetric encryption, in 2014 Twelfth Annual International Conference on Privacy, Security and Trust, Toronto, ON, Canada, July 23-24, 2014, 2014, 330–337. [6] M. Chase and S. Kamara, Structured encryption and controlled disclosure, in Advances in Cryptology - ASIACRYPT 2010 - 16th International Conference on the Theory and Application of Cryptology and Information Security, Singapore, December 5-9, 2010. Proceedings, 6417 (2010), 577–594. doi: 10.1007/978-3-642-17373-8_33. [7] V. Kolesnikov, A. Sadeghi and T. Schneider, Improved garbled circuit building blocks and applications to auctions and computing minima, in Cryptology and Network Security, 8th International Conference, CANS 2009, Kanazawa, Japan, December 12-14, 2009. Proceedings, 2009, 1–20. [8] V. Kolesnikov and T. Schneider, Improved garbled circuit: Free XOR gates and applications, in Automata, Languages and Programming, 35th International Colloquium, ICALP 2008, Reykjavik, Iceland, July 7-11, 2008, Proceedings, Part II - Track B: Logic, Semantics, and Theory of Programming & Track C: Security and Cryptography Foundations, 5126 (2008), 486–498. doi: 10.1007/978-3-540-70583-3_40. [9] J. Leskovec and A. Krevl, SNAP Datasets: Stanford large network dataset collection, http://snap.stanford.edu/data, 2014. [10] D. Liben-Nowell and J. M. Kleinberg, The link prediction problem for social networks, in Proceedings of the 2003 ACM CIKM International Conference on Information and Knowledge Management, New Orleans, Louisiana, USA, November 2-8, 2003, 2003, 556–559. doi: 10.1145/956863.956972. [11] Y. Lindell and B. Pinkas, A proof of security of yao's protocol for two-party computation, J. Cryptology, 22 (2009), 161-188.  doi: 10.1007/s00145-008-9036-8. [12] C. Liu, L. Zhu and J. Chen, Graph encryption for top-k nearest keyword search queries on cloud, T-SUSC, 2 (2017), 371-381.  doi: 10.1109/TSUSC.2017.2704163. [13] A. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of Applied Cryptography, CRC Press, Boca Raton, FL, 1997. [14] X. Meng, S. Kamara, K. Nissim and G. Kollios, GRECS: graph encryption for approximate shortest distance queries, in Proceedings of the 22nd ACM SIGSAC Conference on Computer and Communications Security, Denver, CO, USA, October 12-6, 2015, 2015, 504–517. doi: 10.1145/2810103.2813672. [15] M. Naor and B. Pinkas, Efficient oblivious transfer protocols, in Proceedings of the Twelfth Annual Symposium on Discrete Algorithms, January 7-9, 2001, Washington, DC, USA., 2001, 448–457. [16] K. Nayak, X. S. Wang, S. Ioannidis, U. Weinsberg, N. Taft and E. Shi, Graphsc: Parallel secure computation made easy, in 2015 IEEE Symposium on Security and Privacy, SP 2015, San Jose, CA, USA, May 17-21, 2015, 2015, 377–394. doi: 10.1109/SP.2015.30. [17] PBC Library, The Pairing-based Cryptography Library, https://crypto.stanford.edu/pbc/. [18] L. Sardar and S. Ruj, Prototypes of secure link prediction schemes, GitHub repository, https://github.com/sardarlaltu/SecureLinkPrediction. [19] M. Shen, B. Ma, L. Zhu, R. Mijumbi, X. Du and J. Hu, Cloud-based approximate constrained shortest distance queries over encrypted graphs with privacy protection, IEEE Trans. Information Forensics and Security, 13 (2018), 940–953. doi: 10.1109/TIFS.2017.2774451. [20] D. X. Song, D. A. Wagner and A. Perrig, Practical techniques for searches on encrypted data, in 2000 IEEE Symposium on Security and Privacy, Berkeley, California, USA, May 14-17, 2000, 2000, 44–55. [21] E. Stefanov, M. van Dijk, E. Shi, C. W. Fletcher, L. Ren, X. Yu and S. Devadas, Path ORAM: An extremely simple oblivious RAM protocol, J. ACM, 65 (2018), Art. 18, 26 pp. doi: 10.1145/3177872. [22] Q. Wang, K. Ren, M. Du, Q. Li and A. Mohaisen, Secgdb: Graph encryption for exact shortest distance queries with efficient updates, in Financial Cryptography and Data Security - FC 2017, Sliema, Malta, April 3-7, 2017, Revised Selected Papers, 10322 (2017), 79–97. [23] A. C. Yao, Protocols for secure computations (extended abstract), in 23rd Annual Symposium on Foundations of Computer Science, Chicago, Illinois, USA, 3-5 November 1982, 1982, 160–164. [24] Y. Zheng, B. Wang, W. Lou and Y. T. Hou, Privacy-preserving link prediction in decentralized online social networks, in Computer Security - ESORICS 2015 - Vienna, Austria, September 21-25, 2015, Proceedings, Part II, 9327 (2015), 61–80. doi: 10.1007/978-3-319-24177-7_4.
System model
Example of a Maximum circuit with $N = 7$
Different max blocks used in $\mathtt{MAXIMUM}$ circuit
Few circuit blocks
Number of vertices and edges of the subgraphs
comparison between $\mathtt{SLP}$-$\mathtt{I}$ and $\mathtt{SLP}$-$\mathtt{II}$ w.r.t. computation time when the primes are of 128 bits each
Time taken by the proxy in $\mathtt{SLP}$-$\mathtt{II}$ for different datasets considering 128-bit primes
Computational time in $\mathtt{SLP}$-$\mathtt{I}$ with 128,256 and 512-bit primes
Computational time in $\mathtt{SLP}$-$\mathtt{II}$ with 128,256 and 512-bit primes
Complexity Comparison Table
 Param Entity $\mathtt{SLP}$-$\mathtt{I}$ $\mathtt{SLP}$-$\mathtt{II}$ $\mathtt{SLP}$-$\mathtt{III}$ Leakage CS $|V|$, $\tau_{v_1},\tau_{v_2},\ldots$ $|V|$, $\tau_{v_1},\tau_{v_2},\ldots$ $|V|$, $\tau_{v_1},\tau_{v_2},\ldots$ PS $S_{v},i_{res}$ $S'_{v},i_{res}$ $i_{res}$ client $\lambda$ bits $\lambda$ bits $\lambda$ bits Storage CS $|V|^2\rho$ bits $2|V|^2\rho$ bits $|V|^2\rho$ bits PS $\rho$ bits $\rho$ bits $\rho$ bits client $|V|^2(\mathsf{M}+\mathsf{A})$ $|V|^2(\mathsf{M}+\mathsf{A}+\mathsf{M_1}+\mathsf{A_1})$ $|V|^2(\mathsf{M}+\mathsf{A})$ Compu- CS $|V|^2$ $\mathsf{P}$ + $|V|$ $\mathsf{E}$ $|V|^2$ $\mathsf{P}$ + $|V|^2$ $\mathsf{P}$ + $4|V|$ $\mathsf{E}$ tation + ($|V|^2+ |V|$) $\mathsf{M}$ ($|V|^2+ 2|V|$) $\mathsf{M}$ + ($|V|^2+ 3|V|$) $\mathsf{M}$ + $MGC_{const}{(\log |V|,|V|)}$ PS $|V|log|V| (\mathsf{M+C}+\mathsf{M_1+C_1})$ $|V| (\mathsf{M_1+C_1})$ + $|V| (\mathsf{M+C}+\mathsf{M_1+C_1})$+ +$|V|log|V| \mathsf{C}$ +$|V| log|V| \mathsf{C}$ $MGC_{eval}{(\log |V|,|V|)}$ client$\rightarrow$CS $|V|^2 \rho$ bits $2 |V|^2 \rho$ bits $|V|^2\rho$ bits Commu- CS$\rightarrow$PS $2|V|\rho$ bits $|V|\rho$ bits $2|V|\rho$ bits + $|V|OT ^{(\log |V| +1)}_{snd}$+ nication $MGC_{size}{(\log |V|,|V|)}$ bits PS$\rightarrow$CS - - $|V|OT ^{(\log |V| +1)}_{rcv}$ PS$\rightarrow$client $\log |V|$ bits $2|V| \log |V|$ bits $\log |V|$ bits $S_{v}$ - Set of scores of $v$ with all other vertices, $S'_{v}$- a subset of $S_{v}$, $\rho$- length of elements in $\mathbb{G}$ or $\mathbb{G}_1$, $\mathsf{C}$- comparison in $\mathbb{G}$, $\mathsf{C_1}$- comparison in $\mathbb{G}_1$, $\mathsf{M}$- multiplication in $\mathbb{G}$, $\mathsf{M_1}$- multiplication in $\mathbb{G}_1$, $\mathsf{E}$- exponentiation in $\mathbb{G}$, $\mathsf{E_1}$- exponentiation in $\mathbb{G}_1$, $\mathsf{P}$- pairing/ bilinear map computation, $MGC_{size}{(\log |V|,|V|)}$- size of $MGC$ with $|V|$ $\log |V|$-bit inputs, $MGC_{const}{(\log |V|,|V|)}$- $MGC$ contraction with $|V|$ $\log |V|$-bit inputs, $MGC_{eval}{(\log |V|,|V|)}$- $MGC$ evaluation with $|V|$ $\log |V|$-bit inputs, $OT ^{(\log |V| +1)}_{snd}$- information to send for $(\log |V|+1)$-bit $OT$, $OT ^{(\log |V| +1)}_{rcv}$- information to receive for $\log |V|$-bit $OT$.
 Param Entity $\mathtt{SLP}$-$\mathtt{I}$ $\mathtt{SLP}$-$\mathtt{II}$ $\mathtt{SLP}$-$\mathtt{III}$ Leakage CS $|V|$, $\tau_{v_1},\tau_{v_2},\ldots$ $|V|$, $\tau_{v_1},\tau_{v_2},\ldots$ $|V|$, $\tau_{v_1},\tau_{v_2},\ldots$ PS $S_{v},i_{res}$ $S'_{v},i_{res}$ $i_{res}$ client $\lambda$ bits $\lambda$ bits $\lambda$ bits Storage CS $|V|^2\rho$ bits $2|V|^2\rho$ bits $|V|^2\rho$ bits PS $\rho$ bits $\rho$ bits $\rho$ bits client $|V|^2(\mathsf{M}+\mathsf{A})$ $|V|^2(\mathsf{M}+\mathsf{A}+\mathsf{M_1}+\mathsf{A_1})$ $|V|^2(\mathsf{M}+\mathsf{A})$ Compu- CS $|V|^2$ $\mathsf{P}$ + $|V|$ $\mathsf{E}$ $|V|^2$ $\mathsf{P}$ + $|V|^2$ $\mathsf{P}$ + $4|V|$ $\mathsf{E}$ tation + ($|V|^2+ |V|$) $\mathsf{M}$ ($|V|^2+ 2|V|$) $\mathsf{M}$ + ($|V|^2+ 3|V|$) $\mathsf{M}$ + $MGC_{const}{(\log |V|,|V|)}$ PS $|V|log|V| (\mathsf{M+C}+\mathsf{M_1+C_1})$ $|V| (\mathsf{M_1+C_1})$ + $|V| (\mathsf{M+C}+\mathsf{M_1+C_1})$+ +$|V|log|V| \mathsf{C}$ +$|V| log|V| \mathsf{C}$ $MGC_{eval}{(\log |V|,|V|)}$ client$\rightarrow$CS $|V|^2 \rho$ bits $2 |V|^2 \rho$ bits $|V|^2\rho$ bits Commu- CS$\rightarrow$PS $2|V|\rho$ bits $|V|\rho$ bits $2|V|\rho$ bits + $|V|OT ^{(\log |V| +1)}_{snd}$+ nication $MGC_{size}{(\log |V|,|V|)}$ bits PS$\rightarrow$CS - - $|V|OT ^{(\log |V| +1)}_{rcv}$ PS$\rightarrow$client $\log |V|$ bits $2|V| \log |V|$ bits $\log |V|$ bits $S_{v}$ - Set of scores of $v$ with all other vertices, $S'_{v}$- a subset of $S_{v}$, $\rho$- length of elements in $\mathbb{G}$ or $\mathbb{G}_1$, $\mathsf{C}$- comparison in $\mathbb{G}$, $\mathsf{C_1}$- comparison in $\mathbb{G}_1$, $\mathsf{M}$- multiplication in $\mathbb{G}$, $\mathsf{M_1}$- multiplication in $\mathbb{G}_1$, $\mathsf{E}$- exponentiation in $\mathbb{G}$, $\mathsf{E_1}$- exponentiation in $\mathbb{G}_1$, $\mathsf{P}$- pairing/ bilinear map computation, $MGC_{size}{(\log |V|,|V|)}$- size of $MGC$ with $|V|$ $\log |V|$-bit inputs, $MGC_{const}{(\log |V|,|V|)}$- $MGC$ contraction with $|V|$ $\log |V|$-bit inputs, $MGC_{eval}{(\log |V|,|V|)}$- $MGC$ evaluation with $|V|$ $\log |V|$-bit inputs, $OT ^{(\log |V| +1)}_{snd}$- information to send for $(\log |V|+1)$-bit $OT$, $OT ^{(\log |V| +1)}_{rcv}$- information to receive for $\log |V|$-bit $OT$.
Detail of the graph datasets
 Dataset Name #Nodes #Edges bitcoin-alpha 3,783 24,186 ego-facebook 4,039 88,234 email-Enron 36,692 183,831 email-Eu-core 1,005 25,571 Wiki-Vote 7,115 103,689
 Dataset Name #Nodes #Edges bitcoin-alpha 3,783 24,186 ego-facebook 4,039 88,234 email-Enron 36,692 183,831 email-Eu-core 1,005 25,571 Wiki-Vote 7,115 103,689
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