A stochastic model of attack process for the evaluation of security metrics

To trust a computer system that is supposed to be secure, it is necessary to predict the degree to which the system’s security level can be achieved when operating in a specific environment under cyber attacks. In this paper, we propose a state-based stochastic model for obtaining quantitative security metrics representing the level of a system’s security. The main focus of the study is on how to model the progression of an attack process over time. The basic assumption of our model is that the time parameter plays the essential role in capturing the nature of an attack process. In practice, the attack process will terminate successfully, possibly after a number of unsuccessful attempts. What is important is, indeed, the estimation of how long it takes to be conducted. The proposed stochastic model is parameterized based on a suitable definition of time distributions describing attacker’s actions and system’s reactions over time. For this purpose, probability distribution functions are defined and assigned to transitions of the model for characterizing the temporal aspects of the attacker and system behavior. With the definition of the distributions, the stochastic model will be recognized to be a semi-Markov chain. This mathematical model will be analytically solved to calculate the desirable quantitative security metrics, such as mean time to security failure and steady-state security. The proposed method shows a systematic development of the stochastic modeling techniques and concepts, used frequently in the area of dependability evaluation, for attack process modeling. Like any other modeling method, the proposed model is also constructed based on some underlying assumptions, which are specific to the context of security analysis.