International Journal of Civil Engineering and Technology (IJCIET) Volume 10, Issue 04, April 2019, pp. 280-291, Article ID: IJCIET_10_04_030 Available online at http://www.iaeme.com/ijciet/issues.asp?JType=IJCIET&VType=10&IType=04 ISSN Print: 0976-6308 and ISSN Online: 0976-6316 © IAEME Publication Scopus Indexed OPTIMIZATION OF DECISION-MAKING ON FINANCING OF MEANS OF CYBER SECURITY IN THE CONDITIONS OF THE FISSILE COUNTERACTION TO THE ATTACKING PARTY B. Akhmetov Associate professor Yessenov University, Kazakhstan, Aktau V. Lakhno Professor Yessenov University, Kazakhstan, Aktau L. Kydyralina Doctoral Candidate, Kazakh National Pedagogical University named after Abay, Almaty, Kazakhstan V. Malyukov Professor Department of Computer systems and networks, National University of Life and Environmental Sciences of Ukraine, Kyiv, Ukraine T. Kartbayev Head of the Department of IT-engineering Almaty University of Power Engineering and Telecommunications, Kazakhstan B.Tussupova Associate professor of the Department of IT-engineering Almaty University of Power Engineering and Telecommunications, Kazakhstan A.Doszhanova Associate professor of the Department of IT-engineering, Almaty University of Power Engineering and Telecommunications ABSTRACT The article is devoted to the current problem of acceptance of an optimal solution on financing of means of cyber security in the conditions of the fissile counteraction to the burglars of informatization objects. The model is developed for the decision support system of financing process in the means of cyber security for informatization object. http://www.iaeme.com/IJCIET/index.asp 280 editor@iaeme.com B. Akhmetov, V. Lakhno, L. Kydyralina, V. Malyukov, T. Kartbayev, B. Tussupova and A. Doszhanova The model is based on use of tools of a game theory. The received decision gives the chance to estimate efficiently the risks in processes of financing of means of cyber security of informatization objects. The model differs from the existing approaches by the decision of a bilinear multistep quality game with several terminal surfaces. There was found a solution of a bilinear multistep quality game with the dependent movements. On the basis of the decision of a single-step game received by application of a method of the dominance developed for the infinite antagonistic games there were made conclusion about the risks for players. Results of a computing experiment within the program realization of decision support system in the sphere of financing of means of cyber security of an informatization object are described. At the same time any ratios of the parameters describing financing process are considered, despite the attacking party (hackers) financial actions. Key words: multistep quality game, cyber security, optimal strategy of investment, risks of financing, decision support system Cite this Article: B. Akhmetov, V. Lakhno, L. Kydyralina, V. Malyukov, T. Kartbayev, B. Tussupova and A. Doszhanova, Optimization of Decision-Making on Financing of Means of Cyber Security in the Conditions of the Fissile Counteraction to the Attacking Party, International Journal of Civil Engineering and Technology, 10(04), 2019,pp. 280-291 http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=10&IType=04 1. INTRODUCTION In the conditions of growth of quantity and complexity of the destructive impacts from intruders (hackers) on various computerized systems (for example, information systems – IS) of different objects of informatization [1] is one of the major tasks facing services of operation the problem of ensuring their cyber protection. It demands the corresponding financial investment. In turn the decision making on financing of systems and resources of cyber security (RCS) of objects of informatization (OBI) has to be based on the procedures allowing to carry out financing taking into account all factors inherent to the providing the problems of information security. It is possible if the decision support system (DSS) allowing to make rational decisions on an investment of financial means on development of tools of protection of OBI or IS are developed and introduced. Core of the modern DSS in problems of cyber security [2, 3] are various mathematical models and algorithms giving the chance to experts to intellectualize decisions support. The model for DSS according to the discrete procedure of financing of CS of OBI is considered. The model is based on the decision of a bilinear multistep quality game with two terminal surfaces. 2. FORMULATION OF THE PROBLEM There are two players – the protector of OBI (for example, IS) and the intruder (hacker). Respectively players 1 and 2. The simulated players operate dynamic system which is set by the system of the bilinear discrete equations with the dependent movements. Sets of strategy of players U, V are respectively defined. Two M 0 , N 0 terminal surfaces are set. The aim of 1 player (hereinafter – protector) to move a dynamic system by means of the control strategy on a terminal surface M 0 , despite how financially the player 2 (then – hacker) acted. The purpose of the hacker is to move a dynamic system by means of the control strategy on a terminal http://www.iaeme.com/IJCIET/index.asp 281 editor@iaeme.com Optimization of Decision-Making on Financing of Means of Cyber Security in the Conditions of the Fissile Counteraction to the Attacking Party surface N 0 , despite how financially the protector acted. The decision consists in finding of a set of initial states of objects and their strategy which allow objects to move system on that, or other surface [4]. By consideration of restriction for interaction time with one step we will receive the decision of a single-step game in a class of the mixed strategy. The solution is found by means of dominance methods for the infinite multistep games [5]. As a result of comparison of decisions of two games – multistep and single-step is come into coincidence of sets of initial states of financial resources with the following property. Property: the set of preference of the player proceeding from which he achieves the objectives for steps coincides with a set of reference states of financial resources from which he achieves the objectives for one step at application of optimal mixed strategy at optimal counteraction to it by other player in a class of the mixed strategy with probability 1 . It means that time restriction "is compensated" by T expansion of a class of the used strategy, namely, instead of pure strategy it is necessary to use the mixed strategy. Let's note that sets of reference states from which the player achieves the objectives for one step with probability 1 are sets of preference for the player for T of steps. T That is the probability 1 T means that from such states the player can achieve the objectives with probability 1 for T of steps. Besides, sets of reference states from which the player achieves the objectives for one step with probability 1 are sets of risk for players. The T probability 1 T means risk of achievement of the goal by one player and, on the other hand, for other player – risk not to achieve the goal by other player. On the language of "finance" it is interpreted as risk of loss of financial resources by players 1 and 2 (the protector and the hacker. 3. REVIEW OF LITERATURE Assessment of effectiveness of financing of RCS of OBI is one of the most important in the sphere of digitalization of economy. Rather large number of researches is devoted to this subject [6, 7]. A lack of many works is the lack of actual recommendations about development of strategy of financing of RCS of OBI. The works devoted to application various expert [8, 9] and the decision support systems [10-12] making the choice of strategy of financing of RCS became the self-contained direction of researches. This circumstance causes need of development of new models for DSS which would give the chance, in particular, to estimate risk of loss of financial resources at financing RCS. Such option is possible if the party of protection inaccurately chose incompatible or unefficient RCS. The solution of similar tasks is possible, in particular, for the account application of methods of the theory of differential and multistep quality games with several terminal surfaces [15, 16]. For such differential and multistep games the approaches explained in works [4, 5] are not used as within the scheme of positional differential and multistep games which the player opponent can apply not any managements, for example, no measurable functions, and, at least, measurable functions. As the analysis of the last researches in this area showed, a relevant problem is the further development of models for DSS in problems of financing of various RCS, and first of all, objects of critical informational infrastructure. 4. MODELS AND METHODS Both players need financial resources for the decision of the task. For example, hacker may buy some special software for hacking or bribe stuff. We suppose, that for the given period of http://www.iaeme.com/IJCIET/index.asp 282 editor@iaeme.com B. Akhmetov, V. Lakhno, L. Kydyralina, V. Malyukov, T. Kartbayev, B. Tussupova and A. Doszhanova time 1,...,T (Т–natural number) the 1 player have x 0 financial resources and the 2player have – y 0 . These resources define expected, in an instant, value of financial resources which are possessed by players on achievement of the purposes. At initial time point t a protector multiplies value x 0 by the coefficient (rate of change, rising) t and sets value u t u t 0,1, which define protectors’ resource percentage t xt , allocated by 1 player, at time point t . Similarly, at time point t , 2 player multiplies value y t by the coefficient (rate of change, rising) t and sets value vt vt 0,1, which define hacker’s resource percentage t yt , allocated by him to hacking OBI at time point t . r1 is efficiency of investments of financial resources to RCS. I.e. this is a coefficient, which shows, how many financial resources are needed to the hacker, to hack OBI, the secure of which spent a unit of protectors’ financial resource. r2 is efficiency of investments of financial resources to the software which hacks OBI. I.e. this is a coefficient, which shows, how many financial resources needed defender to secure OBI, to hacking of which spent a unit of hacker’s financial resource. Then the dynamics of changes in financial resources of the first and second players is defined by the following systems of discrete equations: xt 1 t xt ut t xt r2 vt t yt ; (1) yt 1 t yt vt t yt r1 ut t xt . (2) Then at time point it is possible implementation of one of four conditions: 1) xt 0, y t 0; 2) xt 0, y t 0; 3) xt 0, y t 0; 4) xt 0, y t 0. If the first condition is satisfied, then we will say that the procedure of financing of RCS is complete and the attacker of OBI didn't have enough financial resources to hack security. If the second condition is satisfied, then we will say that the procedure of financing of RCS is complete and the defender of OBI didn't have enough financial resources for his protection. If the third condition is satisfied, then we will say that the procedure of financing of RCS is complete both at the defender of OBI and the attacker didn't have enough financial resources for achievement of the purposes. If the fourth condition is satisfied, then the procedure of financing of RCS continues further. Values xT , yT show result of financing of RCS OBI on a planned interval 0, T . The given process of financing of RCS will be considered within the scheme of a multistep game with full information [4, 5].Within this scheme, this process creates two tasks. The first task is from the point of view of the first player-ally. The second task is from the point of view of the second player-ally [4, 5]. Because of symmetry, we confine ourselves to the task from the point of view of the first ally player. The second problem is solved similarly. We denote by T * multiplicity 0,1,...,T . Definition. Strategy of the first player ally is function u : T 0,1 0,1 0,1 , which puts the state of information t , x, y value u t , x, y : 0 u t , x, y 1. * Thus, the strategy of the first ally player is a function (rule), which puts the state of information at the time t value u t , x, y . This value determines the share of the financial resource of the defense party, which she planned to spend on RCS at the time point t . http://www.iaeme.com/IJCIET/index.asp 283 editor@iaeme.com Optimization of Decision-Making on Financing of Means of Cyber Security in the Conditions of the Fissile Counteraction to the Attacking Party Concerning notice of the player opponent (within the scheme of a positional multistage game) no assumptions become. It is equivalent to assumption that the player opponent chooses the operating influence on the basis of any information.Having defined strategy in a task 1, we will define a set of "preference" of the first player. Also, W1 – is a set of such reference states x0, y0 financial resources of the party of protection and the hacker which have below the formulated property. Property: for reference states W1 there is a strategy of the defender OBI, which, for any realization of strategy of the hacker, "brings", into one of instants t , condition of system x0, y0 in it at which the condition will be satisfied (1). At the same time, the second player has no strategy which can "lead" to realization of conditions (2) or (3), in one of the previous instants. We will call the strategy (a financial component) of the defender having the specified property optimum. The solution of a task 1 consists in finding of a set of preference of the first player and his optimum strategy. The task from the point of view of the second player ally is similarly set. Owing to symmetry of statement of tasks, it is enough to be limited to the solution of a task 1 as the solution of a task 2 is found in the same way. The solution of 1 task is found by tools of the theory of multistage plays of quality with the complete information [4, 5, 16, 17]. These tools allow to find the solution at any ratios of parameters of a game. The decision, i.e. sets of "preference" W1 and optimum strategy u* .,. is provided in article at all ratios of parameters of a game. Case а) . x0, y0: k i 1 y0 i W , i 1,.. 1 r x0 k i 2 y0 1 2 u* u* 0, x, y ,...,u* i 1, x, y , u* t , x, y 1 r2 y / x , at x, y R , x r2 y, not defined – otherwise; t 0,1,..., i 1. There k i 1 r1 r2 r1 / k i 1; k 1 0, k 0 1 r1 r2 ; W1 W1i . i 1 0.5 1 r r 2 / 2 y0 Beam r1 2 x0 1 r1 r2 1 2 4 r r / 1 2 will be a barrier [4-5]. Barrier – case when from states x0, y0 : r1 x0 0.5 1 r r 2 1 2 1 r1 r2 / 2 y 0 4 r1 r2 the first player cannot achieve the objectives in some instant. Case b) , r1 r2 1. http://www.iaeme.com/IJCIET/index.asp 284 editor@iaeme.com B. Akhmetov, V. Lakhno, L. Kydyralina, V. Malyukov, T. Kartbayev, B. Tussupova and A. Doszhanova In this case set of preference of the first player W1 will be join of a finite number of sets W1i , exactly N 2 sets, where N : k i r1 r2 , i 0,..., N 1; k N r1 r2 , i x0 , y 0 : k i 1 y 0 W1 , i 1,..., N 1 r1 x0 k i 2 y0 x0, y0 : r1 r2 y0 N 2 W1 . r1 x0 k N y0 Optimal strategy u* u* 0, x, y ,..., u* N 1, x, y is defined as: 2 u* 0, x, y 0, at x, y R , x r2 y , not defined – otherwise}, u* t , x, y 1 r2 y / x , at x, y R , x r2 y, not defined – otherwise; t 1,..., N 1}. 2 Case c) , r1 r2 1. In this case set of preference of the first player W1 also will be join of a finite number of sets W1i . Exactly N i* 2 sets, where N : k i , i 0,..., N 1; k N ; i the minimum integral nonnegative number * determined by inequality k N / * r1 r2 . i 1 Then x0, y0 : k i 1 y0 i W1 , i 1,..., N 1 r1 x0 k i 2 y0 If i* 0, then x0, y0 : r1 r2 y0 i 1,..., N 1; W1N 2 . r1 x0 k N y0 Record of optimum strategy in this case just the same, as well as in a case b). If i* 0, then i x0 , y 0 : k i 1 y 0 W1 , r1 x0 k i 2 y0 N 1 j W1 j x0 , y 0 : k N y 0 , j 1 r1 x0 k N y 0 i 1,..., i* ; http://www.iaeme.com/IJCIET/index.asp 285 editor@iaeme.com Optimization of Decision-Making on Financing of Means of Cyber Security in the Conditions of the Fissile Counteraction to the Attacking Party N 1 i* W1 x0 , y0 : r1 r2 y0 i* . r1 x 0 k N y 0 Optimal strategy u* u* 0, x, y ,..., u* N 1 i* , x, y in this case is defined as follows: u* i, x, y 0, at x, y R , x r2 y, 2 not defined – otherwise; i 0,..., i* , u* i , x, y 1 r2 y / x , at x, y R2 , x r2 y, i i* 1 not defined – otherwise; t 1,..., N 1. In the same way, the task 2 from the point of view of the second player ally is solved. It allows to present a positive orthant to the planes x0, y0 in the form of three sets (cones with top in a point 0,0 ). One set (cone) adjoining an axis 0 X , is a set preferable to the defender. The second set (cone) is a set preferable to the hacker. The third set (cone) is a set neutral, from the point of view of both players. Actually this set characterizes property of balance for the players occupied with financing of protection and breaking. That is players, for the states belonging to this set have strategy allowing players to continue somehow long financings RCS and hacking OBI. That is conditions will be satisfied xt 0, yt 0 for any instant t . Let's note that the beams which are borders of cones are set by means of coefficients, representing a combination of the parameters setting dynamics of the budgetary process on RCS and hacking. Therefore, if initial sizes are set x0, y0 financial resources of the parties of protection OBI and hacking, that can be varied, for example, these parameters. In particular, to demand that the parameters setting dynamics of change of financial resources were that that, a point x0, y0 was in area of balance. Or, on a balance beam if the cone dividing two sets of preference is a beam. If, some parameters defining dynamics of change of financial resources are recorded, then it is possible to demand that values x0, y0 and a part of unstable parameters were that that a point x0, y0 got to the area of balance. It, in turn, can influence both process of financing, and recommendations at the choice of strategy of financing RCS OBI. If it is impossible to change anything, then the above-stated decision of a game in a task 1, or the decision of a game in a task 2, will point out possible result of carrying out financing RCS and hacking, within assumptions at which tasks 1 and 2 were considered. In case of restriction for time of "interaction" of the defender and the hacker, for example one step, a multistage game is transformed to the infinite antagonistic game on a simple square with a payoff function K : 1, at xt 0, yt 0; K x, y 1, at xt 0, y t 0; 0, in other cases. The solution of such game is found by means of the methods of dominance developed in [4]. In a class of clear strategy such game has no decision. The decision exists in a class of the mixed strategy. In [4] the algorithm of finding of optimum mixed strategy is developed. http://www.iaeme.com/IJCIET/index.asp 286 editor@iaeme.com B. Akhmetov, V. Lakhno, L. Kydyralina, V. Malyukov, T. Kartbayev, B. Tussupova and A. Doszhanova Optimum mixed strategy represent probability measures, the concentrated in a finite number of points, and probabilities of realization of each such point identical. For example, for conditions of the players who are in a set in which the probability of achievement of the goal is equal 1 / N , the optimum strategy of the player represents a probability measure. According to this measure, are available points from a simple segment, the probability of realization of each of which is equal 1 / N . At the decision of such antagonistic game there are sets having property: for the financial resources belonging to such sets there are optimum mixed strategy of players at which application achieves the objectives. For example, the first player with probability 1/N, at application of optimum mixed strategy by it. It turns out that such sets coincide with sets of financial resources from which in a multistage game, for example the first player, achieves the objectives exactly for N steps. Thus, one may say, use of tools of a game theory gives the chance to efficiently solve problems of estimation of risk in processes of financing of means of cyber security of objects of informatization. 5. RESULTS OF COMPUTING EXPERIMENTS The computing experiment was made in the environment of Mathcad. The model was also realized in the program module for the system of support of a decision making [3, 16]. Three test computing experiments, see fig. 1-3 are executed. During the experiment situations when two players operate dynamic system were considered. The purpose of an experiment to define sets of strategy of players – the defender and the hacker and, respectively, to simulate scratches of loss of financial resources of players. Cases when the strategy of players bring them to the corresponding terminal surfaces are considered M 0 , N 0 . During the experiment there are sets of reference states of objects and their strategy which allow objects to give system on that, or other terminal surface. On the plane an axis X – financial resources of the defender. Axis Y – financial resources of the hacker. Area under a beam – area of "preference" of the defender. Area over a beam – area of "preference" of the hacker. The beam of balance is shown by a solid line with round markers. Values of points are received during the experiment. The trajectory of movements of players is shown by a dashed line with triangular markers. Trajectories are in area of preference of players. The received results show effectiveness of the offered approach. During testing of model the correctness of the received results is established. Approbation «SSDMI» it is executed also for actual investment projects in the sphere of cyber security of various objects of informatization of Ukraine and Kazakhstan [3, 16, 17]. Figure 1. Results of a computing experiment. A path of motion of the first player (defender of an information system) http://www.iaeme.com/IJCIET/index.asp 287 editor@iaeme.com Optimization of Decision-Making on Financing of Means of Cyber Security in the Conditions of the Fissile Counteraction to the Attacking Party Figure 2. Results of a computing experiment 2. A path of motion of the second player (the hacker attacking object of informatization) Figure 3. Results of a computing experiment 3. ("stability" of system) The figure 1 illustrates a situation when the first player has advantage in the ratio of initial financial resources, i.e. they are in a set of preference of the first player. In this case the first player, applying the optimum strategy, will achieve the objectives, namely reduction of a condition of system on "the" terminal surface. The positive orthant on the plane undertakes. Further, in this orthant the set of beams, coming from points is considered (0,0). These beams are set by a ratio: y 1.5 1 / n x. These beams set sets of preference of the first player for n of steps. For example, set W1n this set: x0, y0 : x0, y0 R2 , 1.5 1 /n 1 x0 y0 1.5 1 / n x0. For example, at n=1 will be http://www.iaeme.com/IJCIET/index.asp 288 editor@iaeme.com B. Akhmetov, V. Lakhno, L. Kydyralina, V. Malyukov, T. Kartbayev, B. Tussupova and A. Doszhanova n 2 W1 x0 , y 0 : x0 , y 0 R , 0 y 0 0.5 x0 . Beam: y 0 1.5 x0 will be a balance beam. Set W1n are sets of conditions of the players having property that if a game begins from them, then the first player for one step will achieve the objectives for one step with probability of 1 / n at application by players of the optimum mixed strategy. The figure 2 shows a situation in which the second player (hacker), using non-optimum behavior of the defender in an initial instant, tries to obtain that the condition of system on "the" terminal surface "brings". The positive orthant on the plane is accepted. In this orthant the set of beams, coming from points is considered (0,0). These beams are set by a ratio: y 2 1 / n x. These beams set sets of preference of the second player (hacker) for n of steps. x0, y0 : x0, y0 R2 , n . For example, set W1 is set: 2 1 / n 1 x 0 2 1 / n x 0 2 1 x0 , y 0 : x0 , y 0 R , . At n=1 we have: W1 0 y 0 3 x 0 Beam y0 2 x0 will be a balance beam. Set W1n are sets of conditions of the players having property that if a game begins from them, then the first player for one step will achieve the objectives for one step with probability of 1 / n at application by players of the optimum mixed strategy. The figure 3 corresponds to a case when the reference state of system is on a balance beam. And players, applying the optimum strategy "move" on this beam. It "satisfies" at the same time both players. In [3, 16] acceptable accuracy of operation of the SPR program module in the ratio with results of computing experiments in Mathcad is confirmed. The divergence did not exceed 6–7%. Let's notice that the offered model describes process of prediction of results of investment into RCS for OBI. The revealed lack of model, the fact that the obtained investments into RCS, given projection at the choice of strategy, not always coincide with actual data is. During the computing experiments and data of practical approbation [3, 16], it is established that the offered model within the scheme of a bilinear differential play of quality for SSDR during management of financing in RCS, allows to describe adequately the dependent movements by means of bilinear functions. It gives efficient tools for players of investment process in means of CS. In comparison with the available models, the proposed solution improves efficiency factors and predictability for the investor on average for 11–15% [2, 3, 6, 18, 19]. The further prospects of development of this research is transferring of the accumulated experience to actual investment projects on perfecting of systems of cyber security of various objects of informatization, in particular information systems of transport in Ukraine and the Republic of Kazakhstan. Work is performed within the competition on grant financing on scientific and scientific and technical projects for 2018-2020 of the Republic of Kazakhstan the registration number AP05132723 "Development of Adaptive Expert Systems in the field of Cyber Security of Crucial Objects of Informatization". http://www.iaeme.com/IJCIET/index.asp 289 editor@iaeme.com Optimization of Decision-Making on Financing of Means of Cyber Security in the Conditions of the Fissile Counteraction to the Attacking Party 6. CONCLUSION In article the following results are received: the model is developed for the system of support of a decision making in the course of financing in means of cyber security for object of informatization. The model is based on use of tools of a game theory and gives the chance to efficiently estimate risks in processes of financing of means of cyber security of objects of informatization. The novelty motels is that it differs from the existing approaches in the decision of a bilinear multistage play of quality with several terminal surfaces. The solution of a bilinear multistage play of quality with the dependent movements is found. On the basis of the decision of a single-step game received by application of a method of the dominance developed for the infinite antagonistic games it is concluded scratches for players; The results of a computing experiment are given. During the experiment various ratios of the parameters describing financing process were considered, kind of the attacking party (hackers) financially did not work. The class of games considered in work, allows to describe adequately process and to find the optimum investment strategy of cyber defense by the party (any object of informatization) in means of information protection. 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