American Institute of Mathematical Sciences

Advanced
February  2018, 1(1): 77-100. doi: 10.3934/mfc.2018005

CP_ABSC: An attribute-based signcryption scheme to secure multicast communications in smart grids

Chunqiang Hu 1,2,, , Jiguo Yu 3, , Xiuzhen Cheng 4, , Zhi Tian 5, , Kemal Akkaya 6, and  and Limin Sun 7,

1. 

Key Laboratory of Dependable Service Computing in Cyber Physical Society, Ministry of Education, Chongqing University, China
2. 

School of Software Engineering, Chongqing University, China
3. 

School of Information Science and Engineering, Qufu Normal University, China
4. 

Department of Computer Science, The George Washington University, USA
5. 

Electrical & Computer Engineering Department, George Mason University, USA
6. 

Electrical & Computer Engineering Department, Florida International University, USA
7. 

Beijing Key Laboratory of IOT Information Security Technology, Institute of Information Engineering, Chinese Academy of Sciences (CAS), China

*Corresponding author: Chunqiang Hu

The preliminary version of this paper appears in [16]

Received  September 2017 Revised  November 2017 Published  February 2018

Fund Project: This research was partially supported by the National Natural Science Foundation of China under grants 61702062,61373027 and 61472418, and the National Science Foundation of the US under grants CCF-1442642, IIS-1343976, CNS-1318872, and CNS-1550313

In this paper, we present a signcryption scheme called CP_ABSC based on Ciphertext-Policy Attribute Based Encryption (CP_ABE) [7] to secure the multicast communications in smart grids that require access control, data encryption, and authentication to ensure message integrity and confidentiality. CP_ABSC provides algorithms for key management, signcryption, and designcryption. It can be used to signcrypt a message based on the access rights specified by the message itself. A user can designcrypt a ciphertext if and only if it possesses the attributes required by the access structure of the data. Thus CP_ABSC effectively defines a multicast group based on the access rights of the data specified by the data itself, which differs significantly from the traditional Internet based multicast where the destination group is predetermined and must be known by the data source. CP_ABSC provides collusion attack resistance, message authentication, forgery prevention, and confidentiality. It can be easily applied in smart grids to secure the instructions/commands broadcast from a utility company to multiple smart meters (push-based multicast) and the data retrieved from a smart meter to multiple destinations (pull-based multicast). Compared to CP_ABE, CP_ABSC combines encryption with signature at a lower computational cost for signcryption and a slightly higher cost in designcryption for signature verification. We also consider the adoption of attribute-based signature (ABS), and conclude that CP_ABSC has a much lower computational cost than ABS.

Keywords: Smart meters, smart grids, secure communication mechanisms, access control, bilinear maps, signcryption.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Chunqiang Hu, Jiguo Yu, Xiuzhen Cheng, Zhi Tian, Kemal Akkaya, and Limin Sun. CP_ABSC: An attribute-based signcryption scheme to secure multicast communications in smart grids. Mathematical Foundations of Computing, 2018, 1 (1) : 77-100. doi: 10.3934/mfc.2018005
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J. A. Akinyele, C. Garman, I. Miers, M. W. Pagano, M. Rushanan, M. Green and A. D. Rubin, Charm: A framework for rapidly prototyping cryptosystems, Journal of Cryptographic Engineering, 3 (2013), 111-128. doi: 10.1007/s13389-013-0057-3. Google Scholar

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A. Alrawais, A. Alhothaily, J. Yu, C. Hu and X. Cheng, Secureguard: A certificate validation system in public key infrastructure, to appear in IEEE Transactions on Vehicular Technology.Google Scholar

[4]

Z. A. Baig and A.-R. Amoudi, An analysis of smart grid attacks and countermeasures, Journal of Communications, 8 (2013), 473-479. doi: 10.12720/jcm.8.8.473-479. Google Scholar

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G. Baker and A. Berg, Supervisory control and data acquisition (scada) systems, The Critical Infrastructure Protection Report, 1 (2002), 5-6. Google Scholar

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P. S. Barreto and M. Naehrig, Pairing-friendly elliptic curves of prime order, in Selected areas in cryptography, Springer, 2006, 319-331. Google Scholar

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J. Bethencourt, A. Sahai and B. Waters, Ciphertext-policy attribute-based encryption, in Security and Privacy, 2007. SP'07. IEEE Symposium on, IEEE, 2007, 321-334. doi: 10.1109/SP.2007.11. Google Scholar

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Z. Cai, Z.-Z. Chen and G. Lin, A 3.4713-approximation algorithm for the capacitated multicast tree routing problem, Theoretical Computer Science, 410 (2009), 5415-5424. doi: 10.1016/j.tcs.2009.05.013. Google Scholar

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M. Gagné, S. Narayan and R. Safavi-Naini, Threshold attribute-based signcryption, in Security and Cryptography for Networks, Springer, 2010, 154-171.Google Scholar

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V. Goyal, O. Pandey, A. Sahai and B. Waters, Attribute-based encryption for fine-grained access control of encrypted data, in Proceedings of the 13th ACM conference on Computer and communications security, ACM, 2006, 89-98. doi: 10.1145/1180405.1180418. Google Scholar

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C. Hu, X. Cheng, Z. Tian, J. Yu, K. Akkaya and L. Sun, An attribute-based signcryption scheme to secure attribute-defined multicast communications, in International Conference on Security and Privacy in Communication Systems, Springer, 2015, 418-437. doi: 10.1007/978-3-319-28865-9_23. Google Scholar

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A. Lewko and B. Waters, Decentralizing attribute-based encryption, Advances in Cryptology-EUROCRYPT 2011, 6632 (2011), 568-588. Google Scholar

[22]

D. Li, Z. Aung, S. Sampalli, J. Williams and A. Sanchez, Privacy preservation scheme for multicast communications in smart buildings of the smart grid, Smart Grid and Renewable Energy, 4 (2013), Article ID: 33928, 12 pages. doi: 10.4236/sgre.2013.44038. Google Scholar

[23]

Q. Li and G. Cao, Multicast authentication in the smart grid with one-time signature, Smart Grid, IEEE Transactions on, 2 (2011), 686-696. doi: 10.1109/TSG.2011.2138172. Google Scholar

[24]

J. Liu, Y. Xiao, S. Li, W. Liang and C. Chen, Cyber security and privacy issues in smart grids, Communications Surveys & Tutorials, IEEE, 14 (2012), 981-997. doi: 10.1109/SURV.2011.122111.00145. Google Scholar

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Y. Liu, P. Ning and M. Reiter, False data injection attacks against state estimation in electric power grids, Proceeding: CCS '09 Proceedings of the 16th ACM conference on Computer and communications security, (2009), 21-32. doi: 10.1145/1653662.1653666. Google Scholar

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H. K. Maji, M. Prabhakaran and M. Rosulek, Attribute-based signatures, in Topics in Cryptology-CT-RSA 2011, Springer, 2011, 376-392. Google Scholar

[29]

A. Metke and R. Ekl, Security technology for smart grid networks, Smart Grid, IEEE Transactions on, 1 (2010), 99-107. doi: 10.1109/TSG.2010.2046347. Google Scholar

[30]

A. Molina-Markham, P. Shenoy, K. Fu, E. Cecchet and D. Irwin, Private memoirs of a smart meter, in Proceedings of the 2nd ACM Workshop on Embedded Sensing Systems for EnergyEfficiency in Building, ACM, 2010, 61-66. doi: 10.1145/1878431.1878446. Google Scholar

[31]

W. Neumann, Horse: An extension of an r-time signature scheme with fast signing and verification, in Information Technology: Coding and Computing, 2004. Proceedings. ITCC 2004. International Conference on, IEEE, 1 (2004), 129-134. doi: 10.1109/ITCC.2004.1286438. Google Scholar

[32]

H. Nicanfar, P. Jokar and V. C. Leung, Smart grid authentication and key management for unicast and multicast communications, in Innovative Smart Grid Technologies Asia (ISGT), 2011 IEEE PES, IEEE, 2011, 1-8.Google Scholar

[33]

A. Perrig, The biba one-time signature and broadcast authentication protocol, in Proceedings of the 8th ACM conference on Computer and Communications Security, ACM, 2001, 28-37. doi: 10.1145/501983.501988. Google Scholar

[34]

A. Perrig, R. Canetti, J. Tygar and D. Song, The tesla broadcast authentication protocol, CryptoBytes, 5 (2002), 2-13. Google Scholar

[35]

M. Pirretti, P. Traynor, P. McDaniel and B. Waters, Secure attribute-based systems, in Proceedings of the 13th ACM conference on Computer and communications security, ACM, 2006, 99-112. doi: 10.1145/1180405.1180419. Google Scholar

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[37]

S. Ruj, A. Nayak and I. Stojmenovic, A security architecture for data aggregation and access control in smart grids, arXiv: 1111.2619.Google Scholar

[38]

A. Sahai and B. Waters, Fuzzy identity-based encryption, Advances in Cryptology-EUROCRYPT 2005, 3494 (2005), 457-473. Google Scholar

[39]

N. Saputro, K. Akkaya and S. Uludag, A survey of routing protocols for smart grid communications, Computer Networks, 56 (2012), 2742-2771. doi: 10.1016/j.comnet.2012.03.027. Google Scholar

[40]

A. Shamir, How to share a secret, Communications of the ACM, 22 (1979), 612-613. doi: 10.1145/359168.359176. Google Scholar

[41]

A. Shamir, Identity-based cryptosystems and signature schemes, in Advances in cryptology, Springer, 196 (1985), 47-53. Google Scholar

[42]

H. So, S. Kwok, E. Lam and K. Lui, Zero-configuration identity-based signcryption scheme for smart grid, in IEEE International Conference on Smart Grid Communications, IEEE, 2010, 321-326. doi: 10.1109/SMARTGRID.2010.5622061. Google Scholar

[43]

C. Valli, A. Woodward, C. Carpene, P. Hannay and M. Brand, Eavesdropping on the smart grid, in Australian Digital Forensics Conference, 2012, 54-60.Google Scholar

[44]

Q. Wang, H. Khurana, Y. Huang and K. Nahrstedt, Time valid one-time signature for timecritical multicast data authentication, in INFOCOM 2009, IEEE, IEEE, 2009, 1233-1241. doi: 10.1109/INFCOM.2009.5062037. Google Scholar

[45]

W. WANG and Z. LU, Cyber security in the smart grid: Survey and challenges, Computer networks, 57 (2013), 1344-1371. doi: 10.1016/j.comnet.2012.12.017. Google Scholar

[46]

C. K. Wong, M. Gouda and S. S. Lam, Secure group communications using key graphs, SIGCOMM '98 Proceedings of the ACM SIGCOMM '98 conference on Applications, Technologies, Architectures, and Protocols for Computer Communication, (1998), 68-79. doi: 10.1145/285237.285260. Google Scholar

[47]

K. Xing, C. Hu, J. Yu, X. Cheng and F. Zhang, Mutual privacy preserving k-means clustering in social participatory sensing, IEEE Transactions on Industrial Informatics, 13 (2017), 2066-2076. doi: 10.1109/TII.2017.2695487. Google Scholar

[48]

L. Zhang, Z. Cai and X. Wang, Fakemask: a novel privacy preserving approach for smartphones, IEEE Transactions on Network and Service Management, 13 (2016), 335-348. doi: 10.1109/TNSM.2016.2559448. Google Scholar

[49]

S. Zhongwei, H. Sitian, M. Yaning and S. Fengjie, Security mechanism for smart distribution grid using ethernet passive optical network, 2010 2nd International Conference on Advanced Computer Control, 3 (2010), 246-250. doi: 10.1109/ICACC.2010.5486625. Google Scholar

[50]

Z. Zhou and D. Huang, On efficient ciphertext-policy attribute based encryption and broadcast encryption, in Proceedings of the 17th ACM conference on Computer and communications security, ACM, 2010, 753-755. doi: 10.1145/1866307.1866420. Google Scholar

[51]

C. Zimmer and F. Mueller, Fault tolerant network routing through software overlays for intelligent power grids, in Parallel and Distributed Systems (ICPADS), 2010 IEEE 16th International Conference on, IEEE, 2010, 542-549. doi: 10.1109/ICPADS.2010.47. Google Scholar

show all references

References:
[1]

Guidelines for smart grid cyber security (vol. 1 to 3), http://csrc.nist.gov/publications/PubsNISTIRs.html, 2010.Google Scholar

[2]

J. A. Akinyele, C. Garman, I. Miers, M. W. Pagano, M. Rushanan, M. Green and A. D. Rubin, Charm: A framework for rapidly prototyping cryptosystems, Journal of Cryptographic Engineering, 3 (2013), 111-128. doi: 10.1007/s13389-013-0057-3. Google Scholar

[3]

A. Alrawais, A. Alhothaily, J. Yu, C. Hu and X. Cheng, Secureguard: A certificate validation system in public key infrastructure, to appear in IEEE Transactions on Vehicular Technology.Google Scholar

[4]

Z. A. Baig and A.-R. Amoudi, An analysis of smart grid attacks and countermeasures, Journal of Communications, 8 (2013), 473-479. doi: 10.12720/jcm.8.8.473-479. Google Scholar

[5]

G. Baker and A. Berg, Supervisory control and data acquisition (scada) systems, The Critical Infrastructure Protection Report, 1 (2002), 5-6. Google Scholar

[6]

P. S. Barreto and M. Naehrig, Pairing-friendly elliptic curves of prime order, in Selected areas in cryptography, Springer, 2006, 319-331. Google Scholar

[7]

J. Bethencourt, A. Sahai and B. Waters, Ciphertext-policy attribute-based encryption, in Security and Privacy, 2007. SP'07. IEEE Symposium on, IEEE, 2007, 321-334. doi: 10.1109/SP.2007.11. Google Scholar

[8]

Z. Cai, Z.-Z. Chen and G. Lin, A 3.4713-approximation algorithm for the capacitated multicast tree routing problem, Theoretical Computer Science, 410 (2009), 5415-5424. doi: 10.1016/j.tcs.2009.05.013. Google Scholar

[9]

Z. Cai, R. Goebel and G. Lin, Size-constrained tree partitioning: Approximating the multicast k-tree routing problem, Theoretical Computer Science, 412 (2011), 240-245. doi: 10.1016/j.tcs.2009.05.031. Google Scholar

[10]

Z. Cai, Z. He, X. Guan and Y. Li, Collective data-sanitization for preventing sensitive information inference attacks in social networks, IEEE Transactions on Dependable and Secure Computing, PP (2016), 1-1. doi: 10.1109/TDSC.2016.2613521. Google Scholar

[11]

Z. Cai, G. Lin and G. Xue, Improved approximation algorithms for the capacitated multicast routing problem, Computing and combinatorics, Lecture Notes in Comput. Sci., Springer, Berlin, 3595 (2005), 136-145. Google Scholar

[12]

Z. Erkin, J. R. Troncoso-Pastoriza, R. L. Lagendijk and F. Perez-Gonzalez, Privacy-preserving data aggregation in smart metering systems: An overview, Signal Processing Magazine, IEEE, 30 (2013), 75-86. doi: 10.1109/MSP.2012.2228343. Google Scholar

[13]

Z. Fadlullah, N. Kato, R. Lu, X. Shen and Y. Nozaki, Toward secure targeted broadcast in smart grid, Communications Magazine, IEEE, 50 (2012), 150-156. doi: 10.1109/MCOM.2012.6194396. Google Scholar

[14]

M. Gagné, S. Narayan and R. Safavi-Naini, Threshold attribute-based signcryption, in Security and Cryptography for Networks, Springer, 2010, 154-171.Google Scholar

[15]

V. Goyal, O. Pandey, A. Sahai and B. Waters, Attribute-based encryption for fine-grained access control of encrypted data, in Proceedings of the 13th ACM conference on Computer and communications security, ACM, 2006, 89-98. doi: 10.1145/1180405.1180418. Google Scholar

[16]

C. Hu, X. Cheng, Z. Tian, J. Yu, K. Akkaya and L. Sun, An attribute-based signcryption scheme to secure attribute-defined multicast communications, in International Conference on Security and Privacy in Communication Systems, Springer, 2015, 418-437. doi: 10.1007/978-3-319-28865-9_23. Google Scholar

[17]

C. Hu, W. Li, X. Cheng, J. Yu, S. Wang and R. Bie, A secure and verifiable access control scheme for big data storage in clouds, IEEE Transactions on Big Data.Google Scholar

[18]

C. Hu, X. Liao and X. Cheng, Verifiable multi-secret sharing based on lrsr sequences, Theoretical Computer Science, 445 (2012), 52-62. doi: 10.1016/j.tcs.2012.05.006. Google Scholar

[19]

M. Kgwadi and T. Kunz, Securing RDS broadcast messages for smart grid applications, Proceeding: IWCMC '10 Proceedings of the 6th International Wireless Communications and Mobile Computing Conference, (2011), 1177-1181. doi: 10.1145/1815396.1815666. Google Scholar

[20]

Y. Kim, A. Perrig and G. Tsudik, Simple and fault-tolerant key agreement for dynamic collaborative groups, in Proceedings of the 7th ACM conference on Computer and communications security, ACM, 2000, 235-244. doi: 10.1145/352600.352638. Google Scholar

[21]

A. Lewko and B. Waters, Decentralizing attribute-based encryption, Advances in Cryptology-EUROCRYPT 2011, 6632 (2011), 568-588. Google Scholar

[22]

D. Li, Z. Aung, S. Sampalli, J. Williams and A. Sanchez, Privacy preservation scheme for multicast communications in smart buildings of the smart grid, Smart Grid and Renewable Energy, 4 (2013), Article ID: 33928, 12 pages. doi: 10.4236/sgre.2013.44038. Google Scholar

[23]

Q. Li and G. Cao, Multicast authentication in the smart grid with one-time signature, Smart Grid, IEEE Transactions on, 2 (2011), 686-696. doi: 10.1109/TSG.2011.2138172. Google Scholar

[24]

J. Liu, Y. Xiao, S. Li, W. Liang and C. Chen, Cyber security and privacy issues in smart grids, Communications Surveys & Tutorials, IEEE, 14 (2012), 981-997. doi: 10.1109/SURV.2011.122111.00145. Google Scholar

[25]

Y. Liu, P. Ning and M. Reiter, False data injection attacks against state estimation in electric power grids, Proceeding: CCS '09 Proceedings of the 16th ACM conference on Computer and communications security, (2009), 21-32. doi: 10.1145/1653662.1653666. Google Scholar

[26]

R. Lu, X. Liang, X. Li, X. Lin, X. Shen et al., Eppa: An efficient and privacy-preserving aggregation scheme for secure smart grid communications, IEEE Trans. on Parallel and Distributed Systems.Google Scholar

[27]

B. Lynn, On the Implementation of Pairing-Based Cryptosystems, PhD thesis, Stanford University, 2007.Google Scholar

[28]

H. K. Maji, M. Prabhakaran and M. Rosulek, Attribute-based signatures, in Topics in Cryptology-CT-RSA 2011, Springer, 2011, 376-392. Google Scholar

[29]

A. Metke and R. Ekl, Security technology for smart grid networks, Smart Grid, IEEE Transactions on, 1 (2010), 99-107. doi: 10.1109/TSG.2010.2046347. Google Scholar

[30]

A. Molina-Markham, P. Shenoy, K. Fu, E. Cecchet and D. Irwin, Private memoirs of a smart meter, in Proceedings of the 2nd ACM Workshop on Embedded Sensing Systems for EnergyEfficiency in Building, ACM, 2010, 61-66. doi: 10.1145/1878431.1878446. Google Scholar

[31]

W. Neumann, Horse: An extension of an r-time signature scheme with fast signing and verification, in Information Technology: Coding and Computing, 2004. Proceedings. ITCC 2004. International Conference on, IEEE, 1 (2004), 129-134. doi: 10.1109/ITCC.2004.1286438. Google Scholar

[32]

H. Nicanfar, P. Jokar and V. C. Leung, Smart grid authentication and key management for unicast and multicast communications, in Innovative Smart Grid Technologies Asia (ISGT), 2011 IEEE PES, IEEE, 2011, 1-8.Google Scholar

[33]

A. Perrig, The biba one-time signature and broadcast authentication protocol, in Proceedings of the 8th ACM conference on Computer and Communications Security, ACM, 2001, 28-37. doi: 10.1145/501983.501988. Google Scholar

[34]

A. Perrig, R. Canetti, J. Tygar and D. Song, The tesla broadcast authentication protocol, CryptoBytes, 5 (2002), 2-13. Google Scholar

[35]

M. Pirretti, P. Traynor, P. McDaniel and B. Waters, Secure attribute-based systems, in Proceedings of the 13th ACM conference on Computer and communications security, ACM, 2006, 99-112. doi: 10.1145/1180405.1180419. Google Scholar

[36]

L. Reyzin and N. Reyzin, Better than biba: Short one-time signatures with fast signing and verifying, in Information Security and Privacy, Springer, 2002, 144-153. doi: 10.1007/3-540-45450-0_11. Google Scholar

[37]

S. Ruj, A. Nayak and I. Stojmenovic, A security architecture for data aggregation and access control in smart grids, arXiv: 1111.2619.Google Scholar

[38]

A. Sahai and B. Waters, Fuzzy identity-based encryption, Advances in Cryptology-EUROCRYPT 2005, 3494 (2005), 457-473. Google Scholar

[39]

N. Saputro, K. Akkaya and S. Uludag, A survey of routing protocols for smart grid communications, Computer Networks, 56 (2012), 2742-2771. doi: 10.1016/j.comnet.2012.03.027. Google Scholar

[40]

A. Shamir, How to share a secret, Communications of the ACM, 22 (1979), 612-613. doi: 10.1145/359168.359176. Google Scholar

[41]

A. Shamir, Identity-based cryptosystems and signature schemes, in Advances in cryptology, Springer, 196 (1985), 47-53. Google Scholar

[42]

H. So, S. Kwok, E. Lam and K. Lui, Zero-configuration identity-based signcryption scheme for smart grid, in IEEE International Conference on Smart Grid Communications, IEEE, 2010, 321-326. doi: 10.1109/SMARTGRID.2010.5622061. Google Scholar

[43]

C. Valli, A. Woodward, C. Carpene, P. Hannay and M. Brand, Eavesdropping on the smart grid, in Australian Digital Forensics Conference, 2012, 54-60.Google Scholar

[44]

Q. Wang, H. Khurana, Y. Huang and K. Nahrstedt, Time valid one-time signature for timecritical multicast data authentication, in INFOCOM 2009, IEEE, IEEE, 2009, 1233-1241. doi: 10.1109/INFCOM.2009.5062037. Google Scholar

[45]

W. WANG and Z. LU, Cyber security in the smart grid: Survey and challenges, Computer networks, 57 (2013), 1344-1371. doi: 10.1016/j.comnet.2012.12.017. Google Scholar

[46]

C. K. Wong, M. Gouda and S. S. Lam, Secure group communications using key graphs, SIGCOMM '98 Proceedings of the ACM SIGCOMM '98 conference on Applications, Technologies, Architectures, and Protocols for Computer Communication, (1998), 68-79. doi: 10.1145/285237.285260. Google Scholar

[47]

K. Xing, C. Hu, J. Yu, X. Cheng and F. Zhang, Mutual privacy preserving k-means clustering in social participatory sensing, IEEE Transactions on Industrial Informatics, 13 (2017), 2066-2076. doi: 10.1109/TII.2017.2695487. Google Scholar

[48]

L. Zhang, Z. Cai and X. Wang, Fakemask: a novel privacy preserving approach for smartphones, IEEE Transactions on Network and Service Management, 13 (2016), 335-348. doi: 10.1109/TNSM.2016.2559448. Google Scholar

[49]

S. Zhongwei, H. Sitian, M. Yaning and S. Fengjie, Security mechanism for smart distribution grid using ethernet passive optical network, 2010 2nd International Conference on Advanced Computer Control, 3 (2010), 246-250. doi: 10.1109/ICACC.2010.5486625. Google Scholar

[50]

Z. Zhou and D. Huang, On efficient ciphertext-policy attribute based encryption and broadcast encryption, in Proceedings of the 17th ACM conference on Computer and communications security, ACM, 2010, 753-755. doi: 10.1145/1866307.1866420. Google Scholar

[51]

C. Zimmer and F. Mueller, Fault tolerant network routing through software overlays for intelligent power grids, in Parallel and Distributed Systems (ICPADS), 2010 IEEE 16th International Conference on, IEEE, 2010, 542-549. doi: 10.1109/ICPADS.2010.47. Google Scholar

Figure 1.  A communication architecture in smart grid systems
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Figure 2.  An access control tree structure
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Figure 3.  An example access control structure in Smart Grid
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Figure 4.  Key generation time
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Figure 5.  Encryption time
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Figure 6.  Decryption time
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Figure 7.  ABS signature running-time
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Figure 8.  ABS verification running-time
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Table 1.  The Computational Cost of Different Functions and Operations between CP_ABE and our scheme
CP_ABE [7] CP_ABSC
Key Generation $n{{\mathbb{G}}_{1}} + (n+2){{\mathbb{G}}_{2}} + nH_{{{\mathbb{G}}_{2}}}$ $(2n+5){{\mathbb{G}}_{2}}$
Encryption $(k+1){{\mathbb{G}}_{1}} + k{{\mathbb{G}}_{2}} + 1{{\mathbb{G}}_{3}} + kH_{{{\mathbb{G}}_{2}}}$ $2((k+1){{\mathbb{G}}_{1}} +{{\mathbb{G}}_{2}}+{{\mathbb{G}}_{3}})+2$ (pairings)
Decryption $(2k^\prime + 1)$ (pairings) $1{{\mathbb{G}}_{3}} + (2k^\prime+3)$ (pairings)
Notes: ${{\mathbb{G}}_{1}}$ in the table means an exponentiation operation in ${{\mathbb{G}}_{1}}$ group; ${{\mathbb{G}}_{2}}$ and ${{\mathbb{G}}_{3}}$ are defined similarly. $H_{{{\mathbb{G}}_{1}}}$ means hashing an attribute string or a message into an element in ${{\mathbb{G}}_{1}}$; $H_{{{\mathbb{G}}_{2}}}$ is defined similarly.
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CP_ABE [7] CP_ABSC
Key Generation $n{{\mathbb{G}}_{1}} + (n+2){{\mathbb{G}}_{2}} + nH_{{{\mathbb{G}}_{2}}}$ $(2n+5){{\mathbb{G}}_{2}}$
Encryption $(k+1){{\mathbb{G}}_{1}} + k{{\mathbb{G}}_{2}} + 1{{\mathbb{G}}_{3}} + kH_{{{\mathbb{G}}_{2}}}$ $2((k+1){{\mathbb{G}}_{1}} +{{\mathbb{G}}_{2}}+{{\mathbb{G}}_{3}})+2$ (pairings)
Decryption $(2k^\prime + 1)$ (pairings) $1{{\mathbb{G}}_{3}} + (2k^\prime+3)$ (pairings)
Notes: ${{\mathbb{G}}_{1}}$ in the table means an exponentiation operation in ${{\mathbb{G}}_{1}}$ group; ${{\mathbb{G}}_{2}}$ and ${{\mathbb{G}}_{3}}$ are defined similarly. $H_{{{\mathbb{G}}_{1}}}$ means hashing an attribute string or a message into an element in ${{\mathbb{G}}_{1}}$; $H_{{{\mathbb{G}}_{2}}}$ is defined similarly.
Table 2.  The Computational Cost of Different Operations in Charm Library
Group ${{\mathbb{G}}_{1}}$ ${{\mathbb{G}}_{2}}$ ${{\mathbb{G}}_{3}}$ (pairings) $H_{{{\mathbb{G}}_{1}}}$ $H_{{{\mathbb{G}}_{2}}}$
SS512 3.73 3.70 0.48 3.92 8.34 8.39
MNT159 1.12 9.84 2.62 8.42 0.10 34.82
Notes: Time is in ms. The result in this table is the average of 1000 runs.
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Group ${{\mathbb{G}}_{1}}$ ${{\mathbb{G}}_{2}}$ ${{\mathbb{G}}_{3}}$ (pairings) $H_{{{\mathbb{G}}_{1}}}$ $H_{{{\mathbb{G}}_{2}}}$
SS512 3.73 3.70 0.48 3.92 8.34 8.39
MNT159 1.12 9.84 2.62 8.42 0.10 34.82
Notes: Time is in ms. The result in this table is the average of 1000 runs.
Table 3.  Comparison between CP_ABE and CP_ABSC
The scheme System Initial. KeyGeneration Encryption Decryption
CP_ABE [7] symmetric groups private key encryption decryption
CP_ABSC asymmetric groups (sign+verify) key signcrypt. decrypt.&verify.
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The scheme System Initial. KeyGeneration Encryption Decryption
CP_ABE [7] symmetric groups private key encryption decryption
CP_ABSC asymmetric groups (sign+verify) key signcrypt. decrypt.&verify.
Table 4.  Number of operations in the Maji's ABS scheme
TSetup() 1${{\mathbb{G}}_{1}}$ /user
AttrGen() 1${{\mathbb{G}}_{1}}$ / attribute
Sign() 2${{\mathbb{G}}_{1}}$+3($\ell_r$)${{\mathbb{G}}_{1}}$+2($\ell - \ell_r$)${{\mathbb{G}}_{1}}$ + 2($\ell \cdot t$)${{\mathbb{G}}_{2}}$
Verify() 1${{\mathbb{G}}_{1}}$+2($\ell \cdot t + t$)${{\mathbb{G}}_{2}}$+($\ell+4$)(pairings)
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TSetup() 1${{\mathbb{G}}_{1}}$ /user
AttrGen() 1${{\mathbb{G}}_{1}}$ / attribute
Sign() 2${{\mathbb{G}}_{1}}$+3($\ell_r$)${{\mathbb{G}}_{1}}$+2($\ell - \ell_r$)${{\mathbb{G}}_{1}}$ + 2($\ell \cdot t$)${{\mathbb{G}}_{2}}$
Verify() 1${{\mathbb{G}}_{1}}$+2($\ell \cdot t + t$)${{\mathbb{G}}_{2}}$+($\ell+4$)(pairings)
Table 5.  Key generation per attribute of the Maji's ABS scheme
SS512 MNT159 MNT159.S BN.S
3.67 ms 9.72 ms 1.13 ms 2.30 ms
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SS512 MNT159 MNT159.S BN.S
3.67 ms 9.72 ms 1.13 ms 2.30 ms
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