Minimization and Reliability Analyses of Attack Graphs

Abstract : An attack graph is a succinct representation of all paths through a system that end in a state where an intruder has successfully achieved his goal. Today Red Teams determine the vulnerability of networked systems by drawing gigantic attack graphs by hand. Constructing attack graphs by hand is tedious, error-prone, and impractical for large systems. By viewing an attack as a violation of a safety property, we can use model checking to produce attack graphs automatically: a successful path from the intruder's viewpoint is a counterexample produced by the model checker. In this paper we present an algorithm for generating attack graphs using model checking. Security analysts use attack graphs for detection, defense, and forensics. In this paper we present a minimization technique that allows analysts to decide which minimal set of security measures would guarantee the safety of the system. We provide a formal characterization of this problem: we prove that it is polynomially equivalent to the minimum hitting set problem and we present a greedy algorithm with provable bounds. We also present a reliability technique that allows analysts to perform a simple cost-benefit analysis depending on the likelihoods of attacks. By interpreting attack graphs as Markov Decision Processes we can use a standard MDP value iteration algorithm to compute the probabilities of intruder success for each attack the graph. We illustrate our work in the context of a small example that includes models of a firewall and an intrusion detection system.