An application of crypto cloud computing in social networks by cooperative game theory

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doi: 10.3934/jimo.2019036

An application of crypto cloud computing in social networks by cooperative game theory

Serap Ergün ^1, , Bariş Bülent Kırlar ^2, , Sırma Zeynep Alparslan Gök ^2,, and Gerhard-Wilhelm Weber ^3,

1.	Department of Electrical and Electronic Engineering, Isparta University of Applied Sciences, Isparta, Turkey
2.	Department of Mathematics, Süleyman Demirel University, Isparta, Turkey, Institute of Applied Mathematics, METU, Ankara, Turkey
3.	Faculty of Engineering Management, Chair of Marketing and Economic Engineering, Poznan University of Technology, Poznan, Poland, Institute of Applied Mathematics, METU, Ankara, Turkey

^* Corresponding author: Sırma Zeynep Alparslan Gök

Received June 2018 Revised December 2018 Published May 2019

In this paper, we mathematically associate Crypto Cloud Computing, that has become an emerging research area, with Cooperative Game Theory in the presence of uncertainty. In the sequel, we retrieve data from the database of Amazon Web Service. The joint view upon Crypto Cloud Computing, Cooperative Game Theory and Uncertainty management is a novel approach. For this purpose, we construct a cooperative interval game model and apply this model to Social Networks. Then, we suggest some interval solutions related with the model by proposing a novel elliptic curve public key encryption scheme over finite fields having the property of semantic security. The paper ends with concluding words and an outlook to future studies.

Keywords: Cooperative game theory, cloud computing, elliptic curves, social networks, uncertainty, interval solutions.

Mathematics Subject Classification: Primary: 91A12, 94A60; Secondary: 68P25.

Citation: Serap Ergün, Bariş Bülent Kırlar, Sırma Zeynep Alparslan Gök, Gerhard-Wilhelm Weber. An application of crypto cloud computing in social networks by cooperative game theory. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019036

References:

[1]	S. P. Ahuja and B. Moore, A Survey of Cloud Computing and Social Networks, Network and Communication Technologies, 2 (2013), 11-16. doi: 10.5539/nct.v2n2p11. Google Scholar
[2]	S. Z. Alparslan Gök, R. Branzei and S. Tijs, The interval Shapley value: an axiomatization, Central European Journal of Operations Research, 18 (2010), 131-140. doi: 10.1007/s10100-009-0096-0. Google Scholar
[3]	S. Z. Alparslan Gök, R. Branzei and S. Tijs, Convex interval games, Journal of Applied Mathematics and Decision Sciences, 2009 (2009), Article ID 342089, 14 pages. doi: 10.1155/2009/342089. Google Scholar
[4]	S. Z. Alparslan Gök, O. Palancıand and M. O. Olgun, Cooperative interval games: Mountain situations with interval data, Journal of Computational and Applied Mathematics, 259 (2014), 622-632. doi: 10.1016/j.cam.2013.01.021. Google Scholar
[5]	S. Z. Alparslan Gök and G.-W. Weber, On dominance core and stable sets for cooperative ellipsoidal games, Optimization, 62 (2013), 1297-1308. doi: 10.1080/02331934.2013.793327. Google Scholar
[6]	Amazon Web Services, Available from: http://calculator.s3.amazonaws.com/index.html.Google Scholar
[7]	M. Ashraf and B. B. Kırlar, Message transmission for GH- public key cryptosystem, Journal of Computational and Applied Mathematics, 259 (2014), 578-585. doi: 10.1016/j.cam.2013.10.005. Google Scholar
[8]	M. Ashraf and B. B. Kırlar, On the Alternate Models of Elliptic Curves, International Journal of Information Security Science, 1 (2012), 49-66. Google Scholar
[9]	D. Bernstein, P. Birkner, M. Joye, T. Lange and C. Peters, Twisted Edwards curves, Progress in Cryptology - Africacrypt 2008, Lecture Notes in Computer Science, 5023 (2008), Springer, 389–405. doi: 10.1007/978-3-540-68164-9_26. Google Scholar
[10]	D. Bernstein, C. Chuengsatiansup, D. Kohel and T. Lange, Twisted Hessian curves, Progress in Cryptology—LATINCRYPT 2015, 269–294, Lecture Notes in Comput. Sci., 9230, Springer, Cham, 2015. Available from https://eprint.iacr.org/2015/781.pdf. doi: 10.1007/978-3-319-22174-8_15. Google Scholar
[11]	D. Bernstein and T. Lange, Explicit Formulas Database, Available from http://www.hyperelliptic.org/EFD.Google Scholar
[12]	D. Bernstein and T. Lange, Faster addition and doubling on elliptic curves, Progress in Cryptology - Asiacrypt 2007, Lecture Notes in Computer Science, 4833 (2007), Springer, 29–50. doi: 10.1007/978-3-540-76900-2_3. Google Scholar
[13]	D. Bernstein, T. Lange and R. R. Farashahi, Binary Edwards Curves, Cryptographic Hardware and Embedded Systems - CHES 2008, Lecture Notes in Computer Science, 5154 (2008), Springer, 244–265. doi: 10.1007/978-3-540-85053-3_16. Google Scholar
[14]	O. Billet and M. Joye, The Jacobi model of an elliptic curve and side-channel analysis, AAECC 2003, Lecture Notes in Computer Science, 2643 (2003), Springer-Verlag, 34–42. doi: 10.1007/3-540-44828-4_5. Google Scholar
[15]	C. G. Bird, On cost allocation for a spanning tree: A game theoretic approach, Networks, 6 (1976), 335-350. doi: 10.1002/net.3230060404. Google Scholar
[16]	P. Borm, H. Hamers and R. Hendrickx, Operations research games: A survey, TOP, 9 (2001), 139-216. doi: 10.1007/BF02579075. Google Scholar
[17]	R. Branzei, S. Tijs and S. Z. Alparslan Gök, How to handle interval solutions for cooperative interval games, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 18 (2010), 123-132. doi: 10.1142/S0218488510006441. Google Scholar
[18]	R. Branzei, S. Z. Alparslan Gök and O. Branzei, Cooperative games under interval uncertainty: on the convexity of the interval undominated cores, Central European Journal of Operations Research, 19 (2011), 523-532. doi: 10.1007/s10100-010-0141-z. Google Scholar
[19]	K. Chard, S. Caton, O. Rana and K. Bubendorfer, Social cloud: Cloud computing in social networks, IEEE 3rd International Conference on Cloud Computing, (2010), 99–106. doi: 10.1109/CLOUD.2010.28. Google Scholar
[20]	A. A. Ciss and D. Sow, On a New Generalization of Huff Curves, 2011. Available from http://eprint.iacr.org/2011/580.pdf.Google Scholar
[21]	A. Claus and D. J. Kleitman, Cost allocation for a spanning tree, Networks, 3 (1973), 289-304. doi: 10.1002/net.3230030402. Google Scholar
[22]	J. Devigne and M. Joye, Binary Huff Curves, Topics in Cryptology - CT-RSA 2011, Lecture Notes in Computer Science, 6558 (2011), Springer, 340–355. doi: 10.1007/978-3-642-19074-2_22. Google Scholar
[23]	H. Edwards, A normal form for elliptic curves, Bulletin of the American Mathematical Society, 44 (2007), 393-422. doi: 10.1090/S0273-0979-07-01153-6. Google Scholar
[24]	J. R. Evans and E. Minieka, Optimization Algorithms for Networks and Graphs, CRC Press, 1992. Google Scholar
[25]	K. A. Falahi, Y. Atif and S. Elnaffar, Social networks: Challenges and new opportunities, In Proceedings of the 2010 IEEE/ACM Int'l Conference on Green Computing and Communications & Int'l Conference on Cyber, Physical and Social Computing, (2010), 804–808. doi: 10.1109/GreenCom-CPSCom.2010.14. Google Scholar
[26]	R. R. Farashahi and M. Joye, Efficient Arithmetic on Hessian Curves, Public Key Cryptography - PKC 2010, Lecture Notes in Computer Science, 6056 (2010), Springer, 243–260. doi: 10.1007/978-3-642-13013-7_15. Google Scholar
[27]	R. Feng, M. Nie and H. Wu, Twisted jacobi intersections curves, Theory and Applications of Models of Computation, 2010,199–210, Available from http://eprint.iacr.org/2009/597.pdf. doi: 10.1007/978-3-642-13562-0_19. Google Scholar
[28]	D. Granot, Cooperative games in stochastic characteristic function form, Management Science, 23 (1977), 621-630. doi: 10.1287/mnsc.23.6.621. Google Scholar
[29]	T. S. Gustavsen and K. Ranestad, A simple point counting algorithm for hessian elliptic curves in characteristic three, Appl. Algebra Eng. Commun. Comput., 17 (2006), 141-150. doi: 10.1007/s00200-006-0013-x. Google Scholar
[30]	D. Hankerson, A. Menezes and S. Vanstone, Guide to Elliptic Curve Cryptography, Springer, 2004. doi: 10.1016/s0012-365x(04)00102-5. Google Scholar
[31]	H. Hisil, K. Koon-Ho Wong, G. Carter and E. Dawson, Twisted Edwards Curves Revisited, Advances in Cryptology - Asiacrypt 2008, Lecture Notes in Computer Science, 5350 (2008), Springer-Verlag, 326–343. doi: 10.1007/978-3-540-89255-7_20. Google Scholar
[32]	H. Hisil, K. Koon-Ho Wong, G. Carter and E. Dawson, Jacobi quartic curves revisited, ACISP, 2009,452–468. doi: 10.1007/978-3-642-02620-1_31. Google Scholar
[33]	G. Huff, Diophantine problems in geometry and elliptic ternary forms, Duke Math. J., 15 (1948), 443-453. doi: 10.1215/S0012-7094-48-01543-9. Google Scholar
[34]	M. Joye and J. Quisquater, Hessian elliptic curves and sidechannel attacks, Cryptographic Hardware and Embedded Systems - CHES 2001, Lecture Notes in Computer Science, 2162 (2001), Springer, 402–410. doi: 10.1007/3-540-44709-1_33. Google Scholar
[35]	M. Joye, M. Tibbouchi and D. Vergnaud, Huff's Model for Elliptic Curves, Algorithmic Number Theory - ANTS-IX, Lecture Notes in Computer Science, 6197 (2010), Springer, 234–250. doi: 10.1007/978-3-642-14518-6_20. Google Scholar
[36]	E. Kilic, A. Karimov and G.-W. Weber, Applications of stochastic hybrid systems in portfolio optimization, In: Thomaidis N, DashGHJr, editors. Recent Advances in Computational Finance. (NY): Nova Science.Google Scholar
[37]	B. B. Kırlar and M. Çil, On the k-th order LFSR sequence with public key cryptosystems, Mathematica Slovaca, 67 (2017), 601-610. doi: 10.1515/ms-2016-0294. Google Scholar
[38]	B. B. Kırlar, S. Ergün, S. Z. Alparslan Gök and G.-W. Weber, A game-theoretical and cryptographical approach to crypto-cloud computing and its economical and financial aspects, Annals of Operations Research, 260 (2018), 217-231. doi: 10.1007/s10479-016-2139-y. Google Scholar
[39]	N. Koblitz, Elliptic curve cryptosystems, Mathematics of Computation, 48 (1987), 203-209. doi: 10.1090/S0025-5718-1987-0866109-5. Google Scholar
[40]	N. Koblitz, A. Menezes and S. Vanstone, The State of Elliptic Curve Cryptography, Designs, Codes and Cryptography, 19 (2000), 173-193. doi: 10.1023/A:1008354106356. Google Scholar
[41]	E. Kropat, G.-W. Weber and J.-J. Rückmann., Regression analysis for clusters in gene environment networks based on ellipsoidal calculus and optimization., Dyn. Cont. Dis. Impulsive Syst. Ser. B., 17 (2010), 639-657. Google Scholar
[42]	P. Liardet and N. Smart, Preventing SPA/DPA in ECC systems using the Jacobi form, Cryptographic Hardware and Embedded Systems - CHES 2001, Lecture Notes in Computer Science, 2162 (2001), Springer-Verlag, 391–401. doi: 10.1007/3-540-44709-1_32. Google Scholar
[43]	P. Maillé, P. Reichl and B. Tuffin, Of threats and costs: A game-theoretic approach to security risk management, In: Performance Models and Risk Management in Communications Systems, 46 (2011), Springer, New York, 33–53. doi: 10.1007/978-1-4419-0534-5_2. Google Scholar
[44]	M. Mares, Fuzzy Cooperative Games: Cooperation with Vague Expectations, Physica Verlag, Heidelberg, 2001. doi: 10.1007/978-3-7908-1820-8. Google Scholar
[45]	V. Miller, Use of elliptic curves in cryptography, Advances in Cryptology – CRYPTO –85, Lecture Notes in Computer Science, 218 (1986), 417-426. doi: 10.1007/3-540-39799-X_31. Google Scholar
[46]	S. Moretti, S. Z. Alparslan Gök, R. Branzei and S. Tijs, Connection situations under uncertainty and cost monotonic solutions, Computers & Operations Research, 38 (2011), 1638-1645. doi: 10.1016/j.cor.2011.02.004. Google Scholar
[47]	A. Muratovic-Ribic and Q. Wang, Partitions and Compositions over Finite Fields, The Electronic Journal of Combinatorics, 20 (2013), Paper 34, 14 pp. Google Scholar
[48]	N. G. Orhon and H. Hisil, Speeding up Huff Form of Elliptic Curves, Designs, Codes and Cryptography, 86 (2018), 2807-2823. doi: 10.1007/s10623-018-0475-4. Google Scholar
[49]	O. Palancı, S. Z. Alparslan Gök, S. Ergün and G.-W. Weber, Cooperative grey games and the grey Shapley value, Optimization, 64 (2015), 1657-1668. doi: 10.1080/02331934.2014.956743. Google Scholar
[50]	L. S. Shapley, A value for n-person games, Annals of Mathematics Studies, 28 (1953), 307-317. Google Scholar
[51]	J. Silverman, The Arithmetic of Elliptic Curves, Springer, Berlin, 1986. doi: 10.1007/978-1-4757-1920-8. Google Scholar
[52]	N. Smart and E. J. Westwood, Point multiplication on ordinary elliptic curves over fields of characteristic three, Appl. Algebra Eng. Commun. Comput., 13 (2003), 485-497. doi: 10.1007/s00200-002-0114-0. Google Scholar
[53]	J. Suijs, P. Borm, A. De Waegenaere and S. Tijs, Cooperative games with stochastic payoffs, European Journal of Operational Research, 113 (1999), 193-205. doi: 10.1016/S0377-2217(97)00421-9. Google Scholar
[54]	D. B. West, Introduction to Graph Theory, Prentice Hall, Inc., Upper Saddle River, NJ, 1996. Google Scholar
[55]	H. Wu and R. Feng, Elliptic curves in Huff's model, Wuhan University Journal of Natural Sciences, 17 (2012), 473–480. Available from http://eprint.iacr.org/2010/390.pdf. doi: 10.1007/s11859-012-0873-9. Google Scholar

show all references

References:

[1]	S. P. Ahuja and B. Moore, A Survey of Cloud Computing and Social Networks, Network and Communication Technologies, 2 (2013), 11-16. doi: 10.5539/nct.v2n2p11. Google Scholar
[2]	S. Z. Alparslan Gök, R. Branzei and S. Tijs, The interval Shapley value: an axiomatization, Central European Journal of Operations Research, 18 (2010), 131-140. doi: 10.1007/s10100-009-0096-0. Google Scholar
[3]	S. Z. Alparslan Gök, R. Branzei and S. Tijs, Convex interval games, Journal of Applied Mathematics and Decision Sciences, 2009 (2009), Article ID 342089, 14 pages. doi: 10.1155/2009/342089. Google Scholar
[4]	S. Z. Alparslan Gök, O. Palancıand and M. O. Olgun, Cooperative interval games: Mountain situations with interval data, Journal of Computational and Applied Mathematics, 259 (2014), 622-632. doi: 10.1016/j.cam.2013.01.021. Google Scholar
[5]	S. Z. Alparslan Gök and G.-W. Weber, On dominance core and stable sets for cooperative ellipsoidal games, Optimization, 62 (2013), 1297-1308. doi: 10.1080/02331934.2013.793327. Google Scholar
[6]	Amazon Web Services, Available from: http://calculator.s3.amazonaws.com/index.html.Google Scholar
[7]	M. Ashraf and B. B. Kırlar, Message transmission for GH- public key cryptosystem, Journal of Computational and Applied Mathematics, 259 (2014), 578-585. doi: 10.1016/j.cam.2013.10.005. Google Scholar
[8]	M. Ashraf and B. B. Kırlar, On the Alternate Models of Elliptic Curves, International Journal of Information Security Science, 1 (2012), 49-66. Google Scholar
[9]	D. Bernstein, P. Birkner, M. Joye, T. Lange and C. Peters, Twisted Edwards curves, Progress in Cryptology - Africacrypt 2008, Lecture Notes in Computer Science, 5023 (2008), Springer, 389–405. doi: 10.1007/978-3-540-68164-9_26. Google Scholar
[10]	D. Bernstein, C. Chuengsatiansup, D. Kohel and T. Lange, Twisted Hessian curves, Progress in Cryptology—LATINCRYPT 2015, 269–294, Lecture Notes in Comput. Sci., 9230, Springer, Cham, 2015. Available from https://eprint.iacr.org/2015/781.pdf. doi: 10.1007/978-3-319-22174-8_15. Google Scholar
[11]	D. Bernstein and T. Lange, Explicit Formulas Database, Available from http://www.hyperelliptic.org/EFD.Google Scholar
[12]	D. Bernstein and T. Lange, Faster addition and doubling on elliptic curves, Progress in Cryptology - Asiacrypt 2007, Lecture Notes in Computer Science, 4833 (2007), Springer, 29–50. doi: 10.1007/978-3-540-76900-2_3. Google Scholar
[13]	D. Bernstein, T. Lange and R. R. Farashahi, Binary Edwards Curves, Cryptographic Hardware and Embedded Systems - CHES 2008, Lecture Notes in Computer Science, 5154 (2008), Springer, 244–265. doi: 10.1007/978-3-540-85053-3_16. Google Scholar
[14]	O. Billet and M. Joye, The Jacobi model of an elliptic curve and side-channel analysis, AAECC 2003, Lecture Notes in Computer Science, 2643 (2003), Springer-Verlag, 34–42. doi: 10.1007/3-540-44828-4_5. Google Scholar
[15]	C. G. Bird, On cost allocation for a spanning tree: A game theoretic approach, Networks, 6 (1976), 335-350. doi: 10.1002/net.3230060404. Google Scholar
[16]	P. Borm, H. Hamers and R. Hendrickx, Operations research games: A survey, TOP, 9 (2001), 139-216. doi: 10.1007/BF02579075. Google Scholar
[17]	R. Branzei, S. Tijs and S. Z. Alparslan Gök, How to handle interval solutions for cooperative interval games, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 18 (2010), 123-132. doi: 10.1142/S0218488510006441. Google Scholar
[18]	R. Branzei, S. Z. Alparslan Gök and O. Branzei, Cooperative games under interval uncertainty: on the convexity of the interval undominated cores, Central European Journal of Operations Research, 19 (2011), 523-532. doi: 10.1007/s10100-010-0141-z. Google Scholar
[19]	K. Chard, S. Caton, O. Rana and K. Bubendorfer, Social cloud: Cloud computing in social networks, IEEE 3rd International Conference on Cloud Computing, (2010), 99–106. doi: 10.1109/CLOUD.2010.28. Google Scholar
[20]	A. A. Ciss and D. Sow, On a New Generalization of Huff Curves, 2011. Available from http://eprint.iacr.org/2011/580.pdf.Google Scholar
[21]	A. Claus and D. J. Kleitman, Cost allocation for a spanning tree, Networks, 3 (1973), 289-304. doi: 10.1002/net.3230030402. Google Scholar
[22]	J. Devigne and M. Joye, Binary Huff Curves, Topics in Cryptology - CT-RSA 2011, Lecture Notes in Computer Science, 6558 (2011), Springer, 340–355. doi: 10.1007/978-3-642-19074-2_22. Google Scholar
[23]	H. Edwards, A normal form for elliptic curves, Bulletin of the American Mathematical Society, 44 (2007), 393-422. doi: 10.1090/S0273-0979-07-01153-6. Google Scholar
[24]	J. R. Evans and E. Minieka, Optimization Algorithms for Networks and Graphs, CRC Press, 1992. Google Scholar
[25]	K. A. Falahi, Y. Atif and S. Elnaffar, Social networks: Challenges and new opportunities, In Proceedings of the 2010 IEEE/ACM Int'l Conference on Green Computing and Communications & Int'l Conference on Cyber, Physical and Social Computing, (2010), 804–808. doi: 10.1109/GreenCom-CPSCom.2010.14. Google Scholar
[26]	R. R. Farashahi and M. Joye, Efficient Arithmetic on Hessian Curves, Public Key Cryptography - PKC 2010, Lecture Notes in Computer Science, 6056 (2010), Springer, 243–260. doi: 10.1007/978-3-642-13013-7_15. Google Scholar
[27]	R. Feng, M. Nie and H. Wu, Twisted jacobi intersections curves, Theory and Applications of Models of Computation, 2010,199–210, Available from http://eprint.iacr.org/2009/597.pdf. doi: 10.1007/978-3-642-13562-0_19. Google Scholar
[28]	D. Granot, Cooperative games in stochastic characteristic function form, Management Science, 23 (1977), 621-630. doi: 10.1287/mnsc.23.6.621. Google Scholar
[29]	T. S. Gustavsen and K. Ranestad, A simple point counting algorithm for hessian elliptic curves in characteristic three, Appl. Algebra Eng. Commun. Comput., 17 (2006), 141-150. doi: 10.1007/s00200-006-0013-x. Google Scholar
[30]	D. Hankerson, A. Menezes and S. Vanstone, Guide to Elliptic Curve Cryptography, Springer, 2004. doi: 10.1016/s0012-365x(04)00102-5. Google Scholar
[31]	H. Hisil, K. Koon-Ho Wong, G. Carter and E. Dawson, Twisted Edwards Curves Revisited, Advances in Cryptology - Asiacrypt 2008, Lecture Notes in Computer Science, 5350 (2008), Springer-Verlag, 326–343. doi: 10.1007/978-3-540-89255-7_20. Google Scholar
[32]	H. Hisil, K. Koon-Ho Wong, G. Carter and E. Dawson, Jacobi quartic curves revisited, ACISP, 2009,452–468. doi: 10.1007/978-3-642-02620-1_31. Google Scholar
[33]	G. Huff, Diophantine problems in geometry and elliptic ternary forms, Duke Math. J., 15 (1948), 443-453. doi: 10.1215/S0012-7094-48-01543-9. Google Scholar
[34]	M. Joye and J. Quisquater, Hessian elliptic curves and sidechannel attacks, Cryptographic Hardware and Embedded Systems - CHES 2001, Lecture Notes in Computer Science, 2162 (2001), Springer, 402–410. doi: 10.1007/3-540-44709-1_33. Google Scholar
[35]	M. Joye, M. Tibbouchi and D. Vergnaud, Huff's Model for Elliptic Curves, Algorithmic Number Theory - ANTS-IX, Lecture Notes in Computer Science, 6197 (2010), Springer, 234–250. doi: 10.1007/978-3-642-14518-6_20. Google Scholar
[36]	E. Kilic, A. Karimov and G.-W. Weber, Applications of stochastic hybrid systems in portfolio optimization, In: Thomaidis N, DashGHJr, editors. Recent Advances in Computational Finance. (NY): Nova Science.Google Scholar
[37]	B. B. Kırlar and M. Çil, On the k-th order LFSR sequence with public key cryptosystems, Mathematica Slovaca, 67 (2017), 601-610. doi: 10.1515/ms-2016-0294. Google Scholar
[38]	B. B. Kırlar, S. Ergün, S. Z. Alparslan Gök and G.-W. Weber, A game-theoretical and cryptographical approach to crypto-cloud computing and its economical and financial aspects, Annals of Operations Research, 260 (2018), 217-231. doi: 10.1007/s10479-016-2139-y. Google Scholar
[39]	N. Koblitz, Elliptic curve cryptosystems, Mathematics of Computation, 48 (1987), 203-209. doi: 10.1090/S0025-5718-1987-0866109-5. Google Scholar
[40]	N. Koblitz, A. Menezes and S. Vanstone, The State of Elliptic Curve Cryptography, Designs, Codes and Cryptography, 19 (2000), 173-193. doi: 10.1023/A:1008354106356. Google Scholar
[41]	E. Kropat, G.-W. Weber and J.-J. Rückmann., Regression analysis for clusters in gene environment networks based on ellipsoidal calculus and optimization., Dyn. Cont. Dis. Impulsive Syst. Ser. B., 17 (2010), 639-657. Google Scholar
[42]	P. Liardet and N. Smart, Preventing SPA/DPA in ECC systems using the Jacobi form, Cryptographic Hardware and Embedded Systems - CHES 2001, Lecture Notes in Computer Science, 2162 (2001), Springer-Verlag, 391–401. doi: 10.1007/3-540-44709-1_32. Google Scholar
[43]	P. Maillé, P. Reichl and B. Tuffin, Of threats and costs: A game-theoretic approach to security risk management, In: Performance Models and Risk Management in Communications Systems, 46 (2011), Springer, New York, 33–53. doi: 10.1007/978-1-4419-0534-5_2. Google Scholar
[44]	M. Mares, Fuzzy Cooperative Games: Cooperation with Vague Expectations, Physica Verlag, Heidelberg, 2001. doi: 10.1007/978-3-7908-1820-8. Google Scholar
[45]	V. Miller, Use of elliptic curves in cryptography, Advances in Cryptology – CRYPTO –85, Lecture Notes in Computer Science, 218 (1986), 417-426. doi: 10.1007/3-540-39799-X_31. Google Scholar
[46]	S. Moretti, S. Z. Alparslan Gök, R. Branzei and S. Tijs, Connection situations under uncertainty and cost monotonic solutions, Computers & Operations Research, 38 (2011), 1638-1645. doi: 10.1016/j.cor.2011.02.004. Google Scholar
[47]	A. Muratovic-Ribic and Q. Wang, Partitions and Compositions over Finite Fields, The Electronic Journal of Combinatorics, 20 (2013), Paper 34, 14 pp. Google Scholar
[48]	N. G. Orhon and H. Hisil, Speeding up Huff Form of Elliptic Curves, Designs, Codes and Cryptography, 86 (2018), 2807-2823. doi: 10.1007/s10623-018-0475-4. Google Scholar
[49]	O. Palancı, S. Z. Alparslan Gök, S. Ergün and G.-W. Weber, Cooperative grey games and the grey Shapley value, Optimization, 64 (2015), 1657-1668. doi: 10.1080/02331934.2014.956743. Google Scholar
[50]	L. S. Shapley, A value for n-person games, Annals of Mathematics Studies, 28 (1953), 307-317. Google Scholar
[51]	J. Silverman, The Arithmetic of Elliptic Curves, Springer, Berlin, 1986. doi: 10.1007/978-1-4757-1920-8. Google Scholar
[52]	N. Smart and E. J. Westwood, Point multiplication on ordinary elliptic curves over fields of characteristic three, Appl. Algebra Eng. Commun. Comput., 13 (2003), 485-497. doi: 10.1007/s00200-002-0114-0. Google Scholar
[53]	J. Suijs, P. Borm, A. De Waegenaere and S. Tijs, Cooperative games with stochastic payoffs, European Journal of Operational Research, 113 (1999), 193-205. doi: 10.1016/S0377-2217(97)00421-9. Google Scholar
[54]	D. B. West, Introduction to Graph Theory, Prentice Hall, Inc., Upper Saddle River, NJ, 1996. Google Scholar
[55]	H. Wu and R. Feng, Elliptic curves in Huff's model, Wuhan University Journal of Natural Sciences, 17 (2012), 473–480. Available from http://eprint.iacr.org/2010/390.pdf. doi: 10.1007/s11859-012-0873-9. Google Scholar

Figure 1. The Amazon Cloud Service properties of one social network company

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Figure 2. The crypto-computing model of the study

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Table 1. Cost of arithmetic on alternate forms of elliptic curves

Form of elliptic curves	Coordinates	Unified addition
Weierstrass	Projective	11M+5S+1D
Edwards [23]	Projective	10M+1S+1D
	Projective	10M+1S+2D
Twisted Edwards [9,31]	Inverted	9M+1S+2D
	Extended	9M+2D
Jacobi Intersections [14]	Projective	13M+2S+1D
Twisted Jacobi Intersections [27]	Projective	13M+2S+5D
Extended Jacobi Quartics [32]	Jacobian	10M+3S+1D
	Extended Projective	8M+3S+2D
Hessian Curves [34]	Projective	12M
Generalized Hessian Curves [26]	Projective	12M+1D
Twisted Hessian Curves [10]	Projective	11M
Huff Curves [35]	Projective	11M
Generalized Huff Curves [55]	Projective	11M+3D
New Generalized Huff Curves [20]	Projective	12M+4D
Extended Huff Curves [48]	Projective	10M

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Form of elliptic curves	Coordinates	Unified addition
Weierstrass	Projective	11M+5S+1D
Edwards [23]	Projective	10M+1S+1D
	Projective	10M+1S+2D
Twisted Edwards [9,31]	Inverted	9M+1S+2D
	Extended	9M+2D
Jacobi Intersections [14]	Projective	13M+2S+1D
Twisted Jacobi Intersections [27]	Projective	13M+2S+5D
Extended Jacobi Quartics [32]	Jacobian	10M+3S+1D
	Extended Projective	8M+3S+2D
Hessian Curves [34]	Projective	12M
Generalized Hessian Curves [26]	Projective	12M+1D
Twisted Hessian Curves [10]	Projective	11M
Huff Curves [35]	Projective	11M
Generalized Huff Curves [55]	Projective	11M+3D
New Generalized Huff Curves [20]	Projective	12M+4D
Extended Huff Curves [48]	Projective	10M

Table 2. The parameters of companies

PARAMETERS	SNC1	SNC2	SNC3	SNC1-SNC2	SNC1-SNC3	SNC2-SNC3	SNC1-SNC2-SNC3
Load Balancer	500	500	3000	1000	3500	3500	4000
(GB/Month) for EC2
Web Server	1/2	1/2	1/2	1/4	1/4	1/4	1/6
(Year/Piece) for EC2
App Server	1/2	1/2	1/2	1/4	1/4	1/4	1/6
(Year/Piece) for EC2
Storage: EBS Volume	6/2500	6/3000	6/8000	12/5500	12/10500	12/11000	18/13500
(Volume/GB) for EC2
Storage	10	100	200	110	210	300	310
(TB) for S3
Data Transfer Out	200	900	6400	1100	6600	7300	7700
(GB/Month) for EC2
Data Transfer In (GB/Month)	1000	500	10000	1500	11000	10500	11500
for EC2
Data Transfer Out	1000	3000	10000	4000	11000	13000	11000
(GB/Month) for CloudFront
Data Storage	30	200	350	230	380	550	380
(TB) for Dynoma
Data Transfer Out	200	250	1500	450	17000	1750	1700
(GB/Month) for Dynoma

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PARAMETERS	SNC1	SNC2	SNC3	SNC1-SNC2	SNC1-SNC3	SNC2-SNC3	SNC1-SNC2-SNC3
Load Balancer	500	500	3000	1000	3500	3500	4000
(GB/Month) for EC2
Web Server	1/2	1/2	1/2	1/4	1/4	1/4	1/6
(Year/Piece) for EC2
App Server	1/2	1/2	1/2	1/4	1/4	1/4	1/6
(Year/Piece) for EC2
Storage: EBS Volume	6/2500	6/3000	6/8000	12/5500	12/10500	12/11000	18/13500
(Volume/GB) for EC2
Storage	10	100	200	110	210	300	310
(TB) for S3
Data Transfer Out	200	900	6400	1100	6600	7300	7700
(GB/Month) for EC2
Data Transfer In (GB/Month)	1000	500	10000	1500	11000	10500	11500
for EC2
Data Transfer Out	1000	3000	10000	4000	11000	13000	11000
(GB/Month) for CloudFront
Data Storage	30	200	350	230	380	550	380
(TB) for Dynoma
Data Transfer Out	200	250	1500	450	17000	1750	1700
(GB/Month) for Dynoma

Table 3. The total costs

Amazon Web Services	Total Cost of Company (＄) $ \left( \left[ 0\%,100\%\right] \right) $
SNC1	$ \left[ 13063.02,35506.80\right] $
SNC2	$ \left[ 64401.07,91333.57\right] $
SNC3	$ \left[ 116776.67,188596.67\right] $
SNC1-SNC2	$ \left[ 41587.70,81986.54\right] $
SNC1-SNC3	$ \left[ 141710.26,330237.82\right] $
SNC2-SNC3	$ \left[ 193574.13,391079.13\right] $
SNC1-SNC2-SNC3	$ \left[ 168389.68,531978.52\right] $

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Amazon Web Services	Total Cost of Company (＄) $ \left( \left[ 0\%,100\%\right] \right) $
SNC1	$ \left[ 13063.02,35506.80\right] $
SNC2	$ \left[ 64401.07,91333.57\right] $
SNC3	$ \left[ 116776.67,188596.67\right] $
SNC1-SNC2	$ \left[ 41587.70,81986.54\right] $
SNC1-SNC3	$ \left[ 141710.26,330237.82\right] $
SNC2-SNC3	$ \left[ 193574.13,391079.13\right] $
SNC1-SNC2-SNC3	$ \left[ 168389.68,531978.52\right] $

Table 4. The interval costs of the coalitions

$ c\left( \left\{ \emptyset \right\} \right) =\left[ 0,0\right] $
$ c\left( \left\{ 1\right\} \right) =\left[ 13063.02+\underline{\psi },35506.80+\underline{\psi }\right] $
$ c\left( \left\{ 2\right\} \right) =\left[ 64401.07+\underline{\psi },91333.57+\underline{\psi }\right] $
$ c\left( \left\{ 3\right\} \right) =\left[ 116776.67+\underline{\psi },188596.67+\underline{\psi }\right] $
$ c\left( \left\{ 1,2\right\} \right) =\left[ 54650.72+\underline{\psi },117493.34+\underline{\psi }\right] $
$ c\left( \left\{ 1,3\right\} \right) =\left[ 154773.28+\underline{\psi },365744.62+\underline{\psi }\right] $
$ c\left( \left\{ 2,3\right\} \right) =\left[ 257975.2+\underline{\psi },482412.7+\underline{\psi }\right] $
$ c\left( \left\{ 1,2,3\right\} \right) =\left[ 196360.98+\underline{\psi },447731.16+\underline{\psi }\right] $

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$ c\left( \left\{ \emptyset \right\} \right) =\left[ 0,0\right] $
$ c\left( \left\{ 1\right\} \right) =\left[ 13063.02+\underline{\psi },35506.80+\underline{\psi }\right] $
$ c\left( \left\{ 2\right\} \right) =\left[ 64401.07+\underline{\psi },91333.57+\underline{\psi }\right] $
$ c\left( \left\{ 3\right\} \right) =\left[ 116776.67+\underline{\psi },188596.67+\underline{\psi }\right] $
$ c\left( \left\{ 1,2\right\} \right) =\left[ 54650.72+\underline{\psi },117493.34+\underline{\psi }\right] $
$ c\left( \left\{ 1,3\right\} \right) =\left[ 154773.28+\underline{\psi },365744.62+\underline{\psi }\right] $
$ c\left( \left\{ 2,3\right\} \right) =\left[ 257975.2+\underline{\psi },482412.7+\underline{\psi }\right] $
$ c\left( \left\{ 1,2,3\right\} \right) =\left[ 196360.98+\underline{\psi },447731.16+\underline{\psi }\right] $

Table 5. The one-point solutions by using PROP for the interval Bird rule

$ \ \ \ \ \ \ \ \ \ f $	$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ d $
$ PROP(E_{1},d) $	$ \left( 324.91,584.83,2729.26\right) $
$ PROP(E_{2},d) $	$ \left( 4789.20,8620.57,40229.25\right) $
$ PROP(E_{3},d) $	$ \left( 22646.36,40763.47,190229.19\right) $

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$ \ \ \ \ \ \ \ \ \ f $	$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ d $
$ PROP(E_{1},d) $	$ \left( 324.91,584.83,2729.26\right) $
$ PROP(E_{2},d) $	$ \left( 4789.20,8620.57,40229.25\right) $
$ PROP(E_{3},d) $	$ \left( 22646.36,40763.47,190229.19\right) $

Table 6. The one-point solutions by using PROP for the interval Shapley rule

$ \ \ \ \ \ \ \ f $	$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ d $
$ PROP(E_{1},d) $	$ \left( 660.66,790.62,2187.74\right) $
$ PROP(E_{2},d) $	$ \left( 9738.05,11653.72,32247.25\right) $
$ PROP(E_{3},d) $	$ \left( 46047.64,55106.10,152485.28\right) $

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$ \ \ \ \ \ \ \ f $	$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ d $
$ PROP(E_{1},d) $	$ \left( 660.66,790.62,2187.74\right) $
$ PROP(E_{2},d) $	$ \left( 9738.05,11653.72,32247.25\right) $
$ PROP(E_{3},d) $	$ \left( 46047.64,55106.10,152485.28\right) $

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American Institute of Mathematical Sciences