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.

[2]

S. Z. Alparslan GökR. 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.

[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.

[4]

S. Z. Alparslan GökO. 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.

[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.

[6]

Amazon Web Services, Available from: http://calculator.s3.amazonaws.com/index.html.

[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.

[8]

M. Ashraf and B. B. Kırlar, On the Alternate Models of Elliptic Curves, International Journal of Information Security Science, 1 (2012), 49-66.

[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.

[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.

[11]

D. Bernstein and T. Lange, Explicit Formulas Database, Available from http://www.hyperelliptic.org/EFD.

[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.

[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.

[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.

[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.

[16]

P. BormH. Hamers and R. Hendrickx, Operations research games: A survey, TOP, 9 (2001), 139-216. doi: 10.1007/BF02579075.

[17]

R. BranzeiS. 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.

[18]

R. BranzeiS. 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.

[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.

[20]

A. A. Ciss and D. Sow, On a New Generalization of Huff Curves, 2011. Available from http://eprint.iacr.org/2011/580.pdf.

[21]

A. Claus and D. J. Kleitman, Cost allocation for a spanning tree, Networks, 3 (1973), 289-304. doi: 10.1002/net.3230030402.

[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.

[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.

[24]

J. R. Evans and E. Minieka, Optimization Algorithms for Networks and Graphs, CRC Press, 1992.

[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.

[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.

[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.

[28]

D. Granot, Cooperative games in stochastic characteristic function form, Management Science, 23 (1977), 621-630. doi: 10.1287/mnsc.23.6.621.

[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.

[30]

D. Hankerson, A. Menezes and S. Vanstone, Guide to Elliptic Curve Cryptography, Springer, 2004. doi: 10.1016/s0012-365x(04)00102-5.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[38]

B. B. KırlarS. ErgünS. 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.

[39]

N. Koblitz, Elliptic curve cryptosystems, Mathematics of Computation, 48 (1987), 203-209. doi: 10.1090/S0025-5718-1987-0866109-5.

[40]

N. KoblitzA. Menezes and S. Vanstone, The State of Elliptic Curve Cryptography, Designs, Codes and Cryptography, 19 (2000), 173-193. doi: 10.1023/A:1008354106356.

[41]

E. KropatG.-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.

[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.

[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.

[44]

M. Mares, Fuzzy Cooperative Games: Cooperation with Vague Expectations, Physica Verlag, Heidelberg, 2001. doi: 10.1007/978-3-7908-1820-8.

[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.

[46]

S. MorettiS. Z. Alparslan GökR. 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.

[47]

A. Muratovic-Ribic and Q. Wang, Partitions and Compositions over Finite Fields, The Electronic Journal of Combinatorics, 20 (2013), Paper 34, 14 pp.

[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.

[49]

O. PalancıS. Z. Alparslan GökS. Ergün and G.-W. Weber, Cooperative grey games and the grey Shapley value, Optimization, 64 (2015), 1657-1668. doi: 10.1080/02331934.2014.956743.

[50]

L. S. Shapley, A value for n-person games, Annals of Mathematics Studies, 28 (1953), 307-317.

[51]

J. Silverman, The Arithmetic of Elliptic Curves, Springer, Berlin, 1986. doi: 10.1007/978-1-4757-1920-8.

[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.

[53]

J. SuijsP. BormA. 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.

[54]

D. B. West, Introduction to Graph Theory, Prentice Hall, Inc., Upper Saddle River, NJ, 1996.

[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.

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.

[2]

S. Z. Alparslan GökR. 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.

[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.

[4]

S. Z. Alparslan GökO. 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.

[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.

[6]

Amazon Web Services, Available from: http://calculator.s3.amazonaws.com/index.html.

[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.

[8]

M. Ashraf and B. B. Kırlar, On the Alternate Models of Elliptic Curves, International Journal of Information Security Science, 1 (2012), 49-66.

[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.

[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.

[11]

D. Bernstein and T. Lange, Explicit Formulas Database, Available from http://www.hyperelliptic.org/EFD.

[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.

[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.

[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.

[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.

[16]

P. BormH. Hamers and R. Hendrickx, Operations research games: A survey, TOP, 9 (2001), 139-216. doi: 10.1007/BF02579075.

[17]

R. BranzeiS. 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.

[18]

R. BranzeiS. 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.

[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.

[20]

A. A. Ciss and D. Sow, On a New Generalization of Huff Curves, 2011. Available from http://eprint.iacr.org/2011/580.pdf.

[21]

A. Claus and D. J. Kleitman, Cost allocation for a spanning tree, Networks, 3 (1973), 289-304. doi: 10.1002/net.3230030402.

[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.

[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.

[24]

J. R. Evans and E. Minieka, Optimization Algorithms for Networks and Graphs, CRC Press, 1992.

[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.

[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.

[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.

[28]

D. Granot, Cooperative games in stochastic characteristic function form, Management Science, 23 (1977), 621-630. doi: 10.1287/mnsc.23.6.621.

[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.

[30]

D. Hankerson, A. Menezes and S. Vanstone, Guide to Elliptic Curve Cryptography, Springer, 2004. doi: 10.1016/s0012-365x(04)00102-5.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[38]

B. B. KırlarS. ErgünS. 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.

[39]

N. Koblitz, Elliptic curve cryptosystems, Mathematics of Computation, 48 (1987), 203-209. doi: 10.1090/S0025-5718-1987-0866109-5.

[40]

N. KoblitzA. Menezes and S. Vanstone, The State of Elliptic Curve Cryptography, Designs, Codes and Cryptography, 19 (2000), 173-193. doi: 10.1023/A:1008354106356.

[41]

E. KropatG.-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.

[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.

[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.

[44]

M. Mares, Fuzzy Cooperative Games: Cooperation with Vague Expectations, Physica Verlag, Heidelberg, 2001. doi: 10.1007/978-3-7908-1820-8.

[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.

[46]

S. MorettiS. Z. Alparslan GökR. 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.

[47]

A. Muratovic-Ribic and Q. Wang, Partitions and Compositions over Finite Fields, The Electronic Journal of Combinatorics, 20 (2013), Paper 34, 14 pp.

[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.

[49]

O. PalancıS. Z. Alparslan GökS. Ergün and G.-W. Weber, Cooperative grey games and the grey Shapley value, Optimization, 64 (2015), 1657-1668. doi: 10.1080/02331934.2014.956743.

[50]

L. S. Shapley, A value for n-person games, Annals of Mathematics Studies, 28 (1953), 307-317.

[51]

J. Silverman, The Arithmetic of Elliptic Curves, Springer, Berlin, 1986. doi: 10.1007/978-1-4757-1920-8.

[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.

[53]

J. SuijsP. BormA. 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.

[54]

D. B. West, Introduction to Graph Theory, Prentice Hall, Inc., Upper Saddle River, NJ, 1996.

[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.

Figure 1.  The Amazon Cloud Service properties of one social network company
Figure 2.  The crypto-computing model of the study
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
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
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] $
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] $
$ 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) $
$ \ \ \ \ \ \ \ \ \ 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) $
$ \ \ \ \ \ \ \ 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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