State space approach to security quantification

In this paper, we describe three different state space models for analyzing the security of a software system. In the first part of this paper, we utilize a semi-Markov process (SMP) to model the transitions between the security states of an abstract software system. The SMP model can be solved to obtain the probability of reaching security failed states along with the meantime to security failure (MTTSF). In the second part of the paper, we use a discrete event dynamic system model of security dynamics. We show how to derive events and transitions from existing security taxonomies. We then apply theory of discrete event control to define safety properties of the computer system in terms of the basic concepts of controllability used in discrete event control for two special sublanguages K_s and K_v. These languages correspond to maximally robust controllable sub-languages. In the third approach, we show that by associating cost with the state transitions, the security quantification problem can be casted as Markov decision problem (MDP). This MOP can be solved to obtain an optimal controllable language K_s⊆K_v the gives the minimal cost safe security policy.