AN CRYPTO CLOUD IN SOCIAL NETWORKS BY COOPERATIVE GAME THEORY

. 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 Com- puting, 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.


1.
Introduction. In the last decays, social networks have been a very important structure for users who are interconnecting through a variety of relations. Some popular social networking platforms are Facebook, Twitter, YouTube, etc. Social networks allow users to share information and form connections between one another, helping to improve the internet usability by storing content in cloud storage. In recent years, these kinds of interactions have been constructed towards the direction of how cryptographic tools can be employed to address a game-theoretical problem in the field of social networks (see [38]). Researchers think that Crypto Cloud Computing system can be designed so as to satisfy the needs of many users of the cloud by using game theory [1,19,38]. Cloud providers such as Google App Engine, Amazon EC2/S3, Microsoft Azure, Eucalyptus and Nimbus offer access into scalable virtualized resources [19].
On social network websites and cloud services, one of the magnificent concerns is the security and privacy of personal data. To control these information, being shared with other users and social applications is too important to be denied [25]. Cloud computing security corresponds to the technology which is used for protecting data and applications of the cloud from threats like disruption of services, unauthorized access, modification, etc. [1].
In the sequel, proposing the discrete logarithm problem by the help of the group of points on an elliptic curve defined over a finite field was proposed by Miller [45] and Koblitz [39] independently, elliptic curve cryptosystems (ECC) have attracted so much interest from the community of researchers. In fact, the attractability of ECC is given because of the fact that no subexponential algorithm employed for solving discrete logarithm problem on the chosen elliptic curve and the underlying field properly. For this reason, one can benefit from an elliptic curve group which is smaller in size than in other systems such as RSA and DSA, while having the same level of security. Involving smaller key sizes result in storage space and bandwidth savings, and faster implementations. This makes ECC appropriate for constrained devices like smart cards and cellular phones.
From the advent of elliptic curve cryptosystems, many methods are proposed to accelerate the arithmetic on elliptic curves. The implication of different coordinate systems for representation of group elements and the usage of alternate forms of elliptic curves are two of them. Different coordinate systems such as projective, jacobian, inverted, etc., have been deeply studied in [11,30]. Alternate forms of elliptic curves to the well-known Weierstrass curve can be classified in Edwards curves [9,12,13,23,31], Jacobi intersections and Jacobi quartics [14,27,31,32,42], Hessian curves [10,26,29,34,52], Huff curves [20,22,33,35,48,55], and their variants. The group structure of these curves has been already studied in [11] because of having some nice properties such as resistance to the side-channel attacks. Applying the unified addition formula, meaning that point addition and point doubling have the same formula, provides a countermeasure to these attacks. In our work, we propose a novel public key scheme by using elliptic curves over finite fields which fulfills the property of semantic security. Furthermore, the cost of the proposed scheme varies depending on the models of elliptic curves and the type of coordinate systems.
Forming a coalition is very important and necessary for providers to prevent from low security which induces a risk for its customers [43]. This leads us to Cooperative Game Theory, where the players can possibly evoke extra gains or save costs by working together, and to share them in a fair way. One way is to study general properties of games arising from a particular type of an Operational Research (OR) problem and to apply it to a suitable game-theoretical solution. Another way is to consider a suitable allocation rule [16].
In our model, we follow an algorithm to create a minimum cost spanning tree (mcst), which is an OR situation related with a graph. After constructing an mcst, an allocation problem has to be found for minimizing total costs. This allocation problem is proposed by Claus and Kleitman [21]. Moreover, Bird [15] took into account the problem by using game theory and proposed an allocation rule, named the Bird rule. Furthermore, we consider the Shapley value [50], which is used in most of the models in cooperative game theory [2].
On the other hand, in many real-life situations, uncertainty exists and influences the values of the coalitions. Hence, cooperative game theory has been extended to different models providing decision making in situations which are characterized by including uncertainty implied. In these models, the characteristic functions are not crisp like in the classical case. The outcome of cooperation includes uncertainty in different forms such as stochastic uncertainty, fuzzy uncertainty, interval uncertainty, ellipsoidal uncertainty, grey uncertainty, etc. [5,28,44,53]. Cooperative interval games and related interval solution concepts are suitable models which give an aid to decision making in collaborative situations under uncertainty [4,18]. The model of cooperative interval games supposes that for each coalition a lower and an upper bound of the outcome of cooperation can be forecasted, without any probabilistic assumptions [17].
In a majority of the real-life situations, players who are considering cooperation sign a contract without knowing the payoffs of the coalitions. But, with certainty they know their lower and upper bounds. These kinds of contracts are made to specify how interval uncertainty regarding the coalition values is incorporated in the allocation of the worth of the grand coalition before its uncertainty is resolved, and how the realization of payoff for the grand coalition is eventually allocated among the players [3]. An important issue which the players have to agree upon in order to construct cooperation in the grand coalition is how to transform an interval allocation into a crisp payoff when the uncertainty regarding the grand coalition's value is removed. A technique to transform an interval allocation into a payoff vector establishes a basic tool of contracts which players have to sign when they cannot assess with certainty the coalition payoffs [17].
In [38], the theory of Crypto-Cloud Computing with an efficient encryption algorithm under XTR by bringing together main topics of Cloud Computing, Cooperative Game Theory and Cryptology is introduced. The most interesting property of this work is the synergy achieved between cryptographic solutions and the cooperative game theory world in financial problems of Cloud Computing application areas. Uncertainty is a daily basis of real life. In many cases, we can not know the crisp values of the coalitions' values. Hence, we construct a model with interval costs.
In this paper, inspired by [38], we implement social networks to Crypto-Cloud Computing by constructing a cooperative game model. Here, a main novelty is to associate Crypto Cloud Computing with Cooperative Game Theory in the presence of uncertainty. In the sequel, we build the connection by retrieving data from the database of Amazon Web Service. Furthermore, we construct a cooperative interval game model and apply this model to Social Networks with this information. Moreover, 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 rest of the paper continues as follows. First, we give some preliminaries from cooperative interval games, graph theory and related solution concepts in Section 2. Section 3 introduces the elliptic curves over finite fields and proposes a novel encryption scheme that the security depends on the elliptic curve discrete logarithm problem and elliptic curve Diffie-Hellman problem. Information about Amazon Web Service and a cooperative interval game application on Social Networks with some interval solutions are stated in Section 4. Section 5 concludes with some final remarks and recommendations about future research.  [18].
A minimum interval cost spanning tree situation (micst) is a situation, where N = {1, 2, . . . , n} is the set of players willing to be connected as cheap as possible to a source denoted by 0, based on an interval-valued cost function [46]. In an micst situation, for each player i ∈ N the cost of the first edge on the unique path from player i to the player source constructs the Bird's tree allocation denoted by β R (T ) [15,16].
Consider a tuple given by ) as a set of arcs, and where 0 is the root. Furthermore, c : A → I (R + ) is a nonnegative interval function defined on the set of arcs, and b (k) of k ∈ N is the possible best connection (for details see [24,54]).
An interval solution concept on IG N is a map assigning to each interval game c ∈ IG N a set of n-dimensional vectors whose components belong to I(R).
In this investigation, we use the interval Bird rule as a solution concept. The interval Bird allocation (cf. [4]) is .., n. A game < N, c > is named as size monotonic if < N, |c| > is monotonic. Here, SM IG N stands for the class of size monotonic interval games with player set N . Moreover, Π(N ) is the set of permutations σ : N → N . Let be given some c ∈ SM IG N . The interval marginal operator corresponding to σ and the interval marginal vector of c with respect to σ are notated through m σ and m σ (c), respectively. If we denote the set of predecessors of i in σ by P σ In this study, we also involve the interval Shapley value (cf. [18]) as a solution concept. The interval Shapley value Φ : SM IG N → I(R) N is defined as the combination Now, we deal with the cooperation under interval uncertainty inside of the set of N players. The players use an interval solution concept named as Ψ, related with the associated cooperative interval game < N, c >. Here, an interval allocation Ψ(c) = (J 1 , . . . , J n ) ∈ I(R) N guarantees for each player i ∈ N a payoff eventually within the interval J i = [J i , J i ] if the value of the grand coalition c(N ) is known. Obviously, c(N ) = i∈N J i and c(N ) = i∈N J i . For each i ∈ N the interval [J i , J i ] can be seen as the interval claim of i on the realization R ∈ c(N ) of the payoff for the grand coalition N (c(N ) ≤ R ≤ c(N )). We determine the payoffs as The amount R to be divided between the players is smaller than i∈N J i implying the bankruptcy rules. These rules are appropriate candidates for transforming an interval allocation (J 1 , . . . , J n ) into a payoff vector (x 1 , . . . , x n ) ∈ R N [17].
A bankruptcy situation with a set of players N means a pair (E, d), where E ≥ 0 is the allocation to be divided and d ∈ R N + is the vector of claims such that i∈N d i ≥ E. In this paper, we use a bankruptcy rule, namely the proportional rule (P ROP ). The rule P ROP is defined by 3. Elliptic Curves. Cryptography is the most important tool to enhance security of cloud computing, which is possible to make by using symmetric key or public key algorithms. Here, we suggest a novel public key scheme by incorporating the more common Weierstrass curve and the alternate models of elliptic curves. This scheme essentially utilizes an ephemeral-static Elliptic Curve Diffie-Hellman key exchange algorithm. The security of the proposed scheme, which also fulfills the property of semantic security, depends on the difficulty of solving Elliptic Curve Discrete Logarithm Problem (ECDLP) and Elliptic Curve Diffie-Hellman Problem (ECDHP).
3.1. Introduction to elliptic curves over finite fields. Let F q be a finite field with q = p n . Then the algebraic closure of F q is given by F q = i≥1 F q i . An elliptic curve over F q with characteristic p > 3 is the set of solutions in F q × F q of a Weierstrass curve given by with regard to the coefficients a, b ∈ F q and ∆ = −16(4a 3 + 27b 2 ) = 0 in F q . The solution set of E W over F q defines an additive group, extended by the point at infinity (identity element of E W ) denoted by ∞ [40]. The explicit formula in affine coordinates for the addition of two points in the curve E W defined over F q of characteristic p > 3 is given by applying the chord-and-tangent rule. Let P = (x 1 , y 1 ) and Q = (x 2 , y 2 ) be the points on E W with P, Q = ∞ and Q = −P . Then, we briefly give the addition and doubling formula below: • Addition: In an analogous way, Weierstrass form of the curves over F q with characteristic p = 2, 3 can be simplified by a similar formula as above (see [51, Appendix A]).
It is possible to convert the aforementioned point multiplication formula to the different coordinate systems: projective, Jacobian, mixed, etc. [30]. These coordinate systems do not involve field inversions; thus, employing them supplies efficiency. For instance, by applying the method of homogenization with x = X/Z and y = Y /Z for Z = 0 relation to (1), we have the subsequent homogeneous equation in projective coordinates: where a, b ∈ F q . The curve E W,h has a unique point with coordinate Z equal to 0, namely (0 : 1 : 0), which has been called previously as the point at infinity ∞. On the other side, for a more efficient computation on elliptic curves we can use the alternate forms of elliptic curves, such as Edwards curves, Jacobi intersections and Jacobi quartics, Hessian curves, Huff curves and their variants. The group structure of these curves has already been surveyed in [8,11]. They also permit the unified addition formula, meaning that the point addition formula can be employed for the doublings; this allows for a resisting against the side-channel attacks.
3.2. Proposed scheme. Now, we propose a novel Elliptic Curve ElGamal based encryption scheme, inspired by the works in [7,37,38].
Scheme 1. Let E be any models of elliptic curve in affine coordinates over F q with characteristic p > 3 and P be an agreed-upon and publicly known point of prime order n on the curve E. Let A and B be two parties, which correspond to Alice and Bob, respectively. B randomly selects a static private key k B in the interval [1, n − 1] and he computes his static public key Q = k B P .
Public Parameters: P , Q. Private Parameters: k A , k B . Encryption: A encrypts a message m ∈ F q as follows: i) A randomly selects an ephemeral private key k A in the interval [1, n − 1]. Then, she computes her ephemeral public key k A P . ii) A computes k A k B P = (x 1 , y 1 ) using the static public key Q = k B P of B. iii) A calculates c = m + (x 1 + y 1 ) ∈ F q . Then, she sends the ciphertext c along with her ephemeral public key k A P .
Decryption: B recovers the message m ∈ F q as follows: 3.3. Computational cost of scheme 1. The novel scheme does not require any multiplication in F q , because of the encryption and decryption process. The computational costs of novel scheme just depend on the point multiplication of elliptic curves. In total, we have computations of 3 point multiplication; 2 point multiplication for the encryption part, 1 point multiplication for the decryption part. In Table 1, the detailed computational costs of the point multiplication on alternate forms of elliptic curves are stated; they are more deeply analyzed in [11]. Therefore, the costs of proposed scheme vary depending on the forms of elliptic curves and the coordinate systems. Here, M, S and D enumerate the cost of multiplication, squaring and multiplication by a constant in F q , respectively. 3.4. Security analysis of scheme 1. First, we raise the subsequent well-known setting and problems about the elliptic curves. It is evident that the proposed scheme is tractable if one can solve both ECDLP and ECDHP. On the other hand, the security of the scheme depends on identifying the pairwise point k A k B P = (x 1 , y 1 ) from x 1 + y 1 ∈ F q . In [47], it is proved that there exist q−1 2 ways to split x 1 + y 1 ∈ F q . However, as far as we survey, there is no methodology known yet to identify the pairwise point k A k B P = (x 1 , y 1 ) among them. Proof. Suppose that m 1 ∈ F q and m 2 ∈ F q are two known messages from the enemy E, and E sends these two messages to A for encryption. A encrypts the message m 1 or m 2 using k A k B P = (x 1 , y 1 ) such that c = m + (x 1 + y 1 ), and she sends the ciphertext c to the enemy E. Hence, by using c, m 1 , m 2 and public parameters, E can receive c − m 1 = d 1 + (x 1 + y 1 ) (resp. c − m 2 = d 2 + (x 1 + y 1 )) by substracting m 1 (resp. m 2 ) from c, where d 1 = m − m 1 (resp. d 2 = m − m 2 ). This fact implies that computing d 1 (resp. d 2 ) is equivalent with identifying the pairwise point k A k B P = (x 1 , y 1 ) from x 1 + y 1 , which is not possible. Consequently, E is not able to find whether the ciphertext c is the encryption of m 1 or m 2 with a probability non-negligibly larger than 1/2.
4. An application: The cooperative interval game model in social networks.

4.1.
Amazon web services. Amazon Web Services (AWS) are a scable cloudcomputing platform constructed for high availability and dependability that provides tools enabling us to run a wide range of applications. In this study, we utilize the subsequently named web services as follows: Amazon Elastic Compute Control (Amazon EC2) is a web service providing a resizable computational capacity within the cloud. It is designed to make web-scale cloud computing easier for developers. Amazon S3 is storage for the Internet and it is constructed to make web-scale computing easier for developers. Amazon Route 53 is a highly available and scable DNS (Domain Name Server) service, which is constructed to give developers a cost effective way to route users for Internet applications. Amazon CloudFront is a web service for content delivery, which delivers content using a global network of edge locations and works seamlessly with Amazon S3, that permanently stores the original versions of files. Amazon RDS is a web service which facilitates it to set up, operate, and scale a relational database in the cloud. Amazon DynamoDB is a high performance non-relational database service, which is easy to set up, operate, and scale. It is created to handle basic problems of database management, performance, scalability, and reliability. It also provides a predictable high performance and a low latency at scale [6].

4.2.
The model. In our study, it is presumed that we have 3 new social network companies launching social web applications. These websites are three-tier web applications, leveraging open-source content management and publishing soft wares, store and serve large amounts of static media content through content delivery networks, and use relational databases to deliver a personalized user experience to their visitors. These three companies pursue common objectives in terms of work safety. The data of website users are encrypted in a cryptology system; then the companies keep their own user's data in each others' data storage. In this way, the companies aim to demonstrate a reliable network configuration. However, the companies do not have any historical data or experience in launching such an application. This "enterprise" has the potential to bring in a lot of advertising revenue, but they have no idea whether the websites be useful.
To support the websites, each company has got 1 Load Balancer, 2 Web Servers, 2 Application Servers, and 6 High Availability Database Servers. In the twocoalitions, however, the companies use 1 Web Server, 1 Application Server and 4 High Availability Database Servers. In the grand coalition, they use 6 High Availability Database Servers. The type of these servers in all the coalitions are Linux on m4.4xlarge. The properties of Social Network Companies (SNC) and some additional storage for cloud computing services are stated in Table 2.
An illustration of our model's Amazon Cloud Services can be seen in Figure 1.
Here, cost accounting and pricing mechanisms for social network firms are received. The cryptology system as a source locates in the private cloud for safety reasons. The social network companies are placed in a public cloud; herewith, the model runs on a hybrid cloud. The cryptology part of the model refers to Platform as a Service (PaaS) (for details see [38]).
The companies strive to move data from an unencrypted volume to an encrypted volume. They create a snapshot of the unencrypted volume, then they create an unencrypted copy of that snapshot and, finally, restore the encrypted snapshot to  a new volume (the other companies' data store), which will also be encrypted. The model with cryptology system works for this target and creates cryptographic costs to the companies.
There is uncertainty about the realization of a snapshot creation, about the process in one day, and the cryptographic cost is taken as a degenerate internal. We assume that there exists a standard cost in cryptography (see Section 3). In order to work on the ambiguity levels of 0% and 100%, respectively, we use the cooperative game under interval uncertainty. The total costs of Amazon Web Services for each company and each coalitions can be found in Table 3. In the model, ψ = ψ, ψ , being the cost of the required proposed encryption algorithm (Scheme 1) is added to the costs constructed from the social network companies and the cryptology system. Then, the following costs are respectively obtained: 13063.02 + ψ, 35506.80 + ψ , 64401.07 + ψ, 91333.57 + ψ , 116776.67 + ψ, 188596.67 + ψ . Hence, the total costs are calculated from storing the encrypted information of other social network companies' data stores. Figure 2 illustrates the model with closer details. 4.3. The interval solutions. In our model, social network cloud services are constructed in a cooperative manner. In the sequel, the costs are allocated by using the interval Bird rule and the interval Shapley value. We note that ψ = [ψ, ψ] is our encryption cost. First, we use the Bird rule (see Figure 2). Then, the optimal solution is 196360.98 + ψ, 447731.16 + ψ . This yields us the Bird allocation β R (Γ) as follows:  Table 4 illustrates the interval costs of the coalitions.

4.4.
The PROP rule. In this section, we suggest a one-point solution by the help of our interval solutions. Here, we apply the proportional rule (PROP) to get a one-point solution from an interval solution (for details see [17]).
First, we employ the interval Bird rule and assume that the realizations of c(N ) are R 1 = 200000 + ψ, R 2 = 250000 + ψ, R 3 = 450000 + ψ. Now, we calculate the individual crisp allocations. Then, we distribute the amount E i , i = 1, 2, 3, among three social network companies as follows: The claims d i , i = 1, 2, 3, of each company on the realizations R 1 , R 2 , R 3 are as follows: We note that the total claim is 251370.18. Table 5 illustrates the one-point PROP solution by using interval Bird rule.
Second, we apply the interval Shapley rule. The claims d i , i = 1, 2, 3, of each company on the realizations R 1 , R 2 , R 3 are as follows: We note that the total claim is 251370.18. Table 6 illustrates the one-point PROP solution by using interval Shapley rule.

Conclusion.
Recently, Crypto Cloud Computing has become an interesting research area with many technical, security, commercial and financial aspects, goals and consequences. Cloud computing comes along with its share of challenges, in terms of security, data privacy, compliance, availability, lack of standards, etc. These challenges are highlighted more in regulated and security-sensitive environments, such as Social Networks. Considering the cooperative functionality of Crypto Cloud Computing, the use of game theory in that area has been understood to become very beneficial [1,19,38]. Uncertainty is present in almost every real-world situation, it is influencing and questioning our decisions. What in the past is regarded as a matter left alone to the soft human and social sciences, now enters core areas of hard research, computation and calibration. This has been transforming the view on uncertainty, supported by approaches such as uncertainty quantization, grey numbers, robust counterparts of optimization and of stochastic optimal control, e.g. related to stochastic hybrid systems with jumps [36,41,49,53].
In this study, we construct a model by using cooperative game theory under uncertainty, which associates to Crypto Cloud Computing. In the sequel, we propose a novel encryption algorithm by using elliptic curves over finite fields having the property of semantic security. Hence, we retrieve data from the database of Amazon Web Service. The most interesting and important property of our work is combining the cryptography and cooperative game theory in social networks used in cloud-computing applications. By implementing the cryptographic solution to the cooperative interval games, we both must be behind from cooperative game and cryptography sides. Before closing, we note that our study deals with cooperative interval games without any probability included. However, in future research and application, our model and related solution concepts can be applied to the different forms such as stochastic, fuzzy or grey uncertainty [44,49,53].