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doi: 10.3934/jimo.2019036

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

 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.

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
The Amazon Cloud Service properties of one social network company
The crypto-computing model of the study
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
 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
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
 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
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 $\left[ 168389.68,531978.52\right]$
 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 $\left[ 168389.68,531978.52\right]$
The interval costs of the coalitions
 $c\left( \left\{ \emptyset \right\} \right) =\left[ 0,0\right]$ $c\left( \left\{ 1\right\} \right) =\left[ 13063.02+\underline{\psi },35506.80+\underline{\psi }\right]$ $c\left( \left\{ 2\right\} \right) =\left[ 64401.07+\underline{\psi },91333.57+\underline{\psi }\right]$ $c\left( \left\{ 3\right\} \right) =\left[ 116776.67+\underline{\psi },188596.67+\underline{\psi }\right]$ $c\left( \left\{ 1,2\right\} \right) =\left[ 54650.72+\underline{\psi },117493.34+\underline{\psi }\right]$ $c\left( \left\{ 1,3\right\} \right) =\left[ 154773.28+\underline{\psi },365744.62+\underline{\psi }\right]$ $c\left( \left\{ 2,3\right\} \right) =\left[ 257975.2+\underline{\psi },482412.7+\underline{\psi }\right]$ $c\left( \left\{ 1,2,3\right\} \right) =\left[ 196360.98+\underline{\psi },447731.16+\underline{\psi }\right]$
 $c\left( \left\{ \emptyset \right\} \right) =\left[ 0,0\right]$ $c\left( \left\{ 1\right\} \right) =\left[ 13063.02+\underline{\psi },35506.80+\underline{\psi }\right]$ $c\left( \left\{ 2\right\} \right) =\left[ 64401.07+\underline{\psi },91333.57+\underline{\psi }\right]$ $c\left( \left\{ 3\right\} \right) =\left[ 116776.67+\underline{\psi },188596.67+\underline{\psi }\right]$ $c\left( \left\{ 1,2\right\} \right) =\left[ 54650.72+\underline{\psi },117493.34+\underline{\psi }\right]$ $c\left( \left\{ 1,3\right\} \right) =\left[ 154773.28+\underline{\psi },365744.62+\underline{\psi }\right]$ $c\left( \left\{ 2,3\right\} \right) =\left[ 257975.2+\underline{\psi },482412.7+\underline{\psi }\right]$ $c\left( \left\{ 1,2,3\right\} \right) =\left[ 196360.98+\underline{\psi },447731.16+\underline{\psi }\right]$
The one-point solutions by using PROP for the interval Bird rule
 $\ \ \ \ \ \ \ \ \ f$ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ d$ $PROP(E_{1},d)$ $\left( 324.91,584.83,2729.26\right)$ $PROP(E_{2},d)$ $\left( 4789.20,8620.57,40229.25\right)$ $PROP(E_{3},d)$ $\left( 22646.36,40763.47,190229.19\right)$
 $\ \ \ \ \ \ \ \ \ f$ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ d$ $PROP(E_{1},d)$ $\left( 324.91,584.83,2729.26\right)$ $PROP(E_{2},d)$ $\left( 4789.20,8620.57,40229.25\right)$ $PROP(E_{3},d)$ $\left( 22646.36,40763.47,190229.19\right)$
The one-point solutions by using PROP for the interval Shapley rule
 $\ \ \ \ \ \ \ f$ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ d$ $PROP(E_{1},d)$ $\left( 660.66,790.62,2187.74\right)$ $PROP(E_{2},d)$ $\left( 9738.05,11653.72,32247.25\right)$ $PROP(E_{3},d)$ $\left( 46047.64,55106.10,152485.28\right)$
 $\ \ \ \ \ \ \ f$ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ d$ $PROP(E_{1},d)$ $\left( 660.66,790.62,2187.74\right)$ $PROP(E_{2},d)$ $\left( 9738.05,11653.72,32247.25\right)$ $PROP(E_{3},d)$ $\left( 46047.64,55106.10,152485.28\right)$
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